chapter 1 photonic crystals: properties and applications

Chapter 1

Photonic crystals: properties and

applications

This chapter is intended as a brief overview of the history, concepts, characteristics

and applications of photonic crystals (PhCs). A single chapter is insufficient to re-

view a field that continues to grow almost exponentially in the literature almost 20

years after it began, so attention is given to aspects most relevant to the work pre-

sented in the remainder of this thesis. Sections 1.1–1.4 introduce the concepts of one-,

two-, and three-dimensional photonic crystals and bandgaps and the motivation that

led to their development. The theory of band structures and Bloch modes of uniform

two-dimensional photonic crystals and photonic crystal slabs is then considered in Sec-

tions 1.5 and 1.6. Section 1.7 is concerned with the properties and applications of defects

in PhCs, and specifically those designed for resonant behaviour and waveguiding. Inte-

gration of these two functions into photonic devices is discussed in the context of two

basic operations: coupling and add/drop filtering. Finally, Section 1.8 focuses on a dif-

ferent class of applications, namely those that exploit the unusual dispersion properties

of the Bloch modes that exist in uniform photonic crystals.

1.1 Introduction

The optical properties of periodic structures can be observed throughout the natural

world, from the changing colours of an opal held up to the light to the patterns on a

butterfly’s wings. Nature has been exploiting photonic crystals for millions of years [1],

but humans have only recently started to realise their potential. One-dimensional peri-

odic structures in the form of thin film stacks have been studied for many years [2], but

three-dimensional photonic crystal were first proposed by Yablonovitch [3] and John [4]

in 1987. Yablonovitch proposed that three-dimensional periodic dielectric structures

could exhibit an electromagnetic bandgap - a range of frequencies at which light cannot

propagate through the structure in any direction. He also predicted that unwanted

spontaneous emission within a semiconductor can be prevented by structuring the ma-

5

6 CHAPTER 1. PHOTONIC CRYSTALS: PROPERTIES AND APPLICATIONS

terial so that the frequencies of these emissions fall within a photonic bandgap; since

no propagating states exist at that frequency, emission is effectively forbidden. John [4]

showed that many of the properties of PhCs survive even when the periodic lattice

becomes disordered. In such structures, if the index contrast is sufficiently large, strong

light localisation can still occur, in analogy to the electronic bandgaps of amorphous

semiconductors. Perhaps more relevant to much of the PhC research that has followed,

and to the topic of this thesis, was Yablonovitch’s [3] interpretation of the cavity modes

that can be introduced into a periodic structure by creating a defect or “phase slip”.

While resonant cavities in distributed feedback lasers had had already been demon-

strated using this approach [5], Yablonovitch showed that modes could be localised in

three-dimensions and explained the effect in terms of defect states in the photonic band-

gap. From this observation, and the initial proposals for limiting spontaneous emission,

the concept of controlling light with periodic structures has developed rapidly into a

topic of worldwide research.

Bandgaps in periodic materials were already well understood from solid-state physics,

where the presence of electronic bandgaps in semiconductors has revolutionised electron-

ics. Many of the concepts from solid-state research have been carried over to photonic

crystals including the notation and nomenclature, and perhaps this is what has allowed

the field to make such rapid progress in less than twenty years.

1.2 One-dimensional photonic crystals

Although the term photonic crystal (PhC) is relatively recent, simple one-dimensional

(1D) PhCs in the form of periodic dielectric stacks have been used for considerably

longer [2]. Their wavelength-selective reflection properties see them used in a wide

range of applications including high-efficiency mirrors, Fabry-Perot cavities, optical

filters and distributed feedback lasers. As illustrated in Fig. 1.1, the simplest PhC is

an alternating stack of two different dielectric materials. When light is incident on

such a stack, each interface reflects some of the field. If the thickness of each layer is

chosen appropriately, the reflected fields can combine in phase, resulting in constructive

interference and strong reflectance, also known as Bragg reflection. In contrast to

two- and three-dimensional PhCs, 1D Bragg reflection occurs regardless of the index

contrast, although a large number of periods is required to achieve a high reflectance

if the contrast is small. Since the absorption in dielectric optical materials is very

low, mirrors made from dielectric stacks are extremely efficient, and can be designed

to reflect almost 100% of the incident light within a small range of frequencies. The

main limitation of these dielectric mirrors is that they only operate for a limited range

of angles close to normal incidence.

Another, more recent application of 1D PhCs is the fibre Bragg grating (FBG), in

which the refractive index of the fibre core is varied periodically along its axis, typically

approximating a sinusoidal profile. This case is somewhat more complex because the

1.3. TWO-DIMENSIONAL PHOTONIC CRYSTALS 7

x

y

zd

Figure 1.1: Schematic of a one-dimensional photonic crystal consisting of a periodic

stack of dielectric layers with period d.

refractive index varies continuously, rather than discretely, as in the previous example,

but the properties are essentially the same. The main difference is that the refractive

index contrast in the FBG is so small (∆n ≤ 0.5%) that the operational bandwidth

is very narrow and thousands of periods are typically required to obtain the desired

reflectance properties. FBGs are now an integral part of fibre optic systems, being used

in dispersion compensation, filters, and a wide range of other applications [6].

1.3 Two-dimensional photonic crystals

Both two-dimensional (2D) and three-dimensional (3D) PhCs can be thought of as

generalisations to the 1D case where a full 2D or 3D bandgap appears only if the 1D

Bragg reflection condition is satisfied simultaneously for all propagation directions in

which the structure is periodic. For most 2D periodic lattices this occurs providing the

index contrast is sufficiently large, but for 3D structures only certain lattice geometries

display the necessary properties, and only then for large enough index contrasts (Ref. [7],

Chapter 6). The 3D case is discussed in more detail in Section 1.4.

Instead of a stack of uniform dielectric layers, 2D PhCs typically consist of an array

of dielectric cylinders in a homogeneous dielectric background material, as illustrated

in Fig. 1.2, although there are many other possible geometries. If the refractive index

contrast between the cylinders and the background is sufficiently large, 2D bandgaps can

occur for propagation in the plane of periodicity — perpendicular to the rods. Light at

a frequency within the bandgap experiences Bragg reflection in all directions due to the

periodic array of cylinders. However, as in the 1D case where light could still propagate

in two-dimensions, in a 2D PhC propagation can still occur in the non-periodic direction,

parallel to the cylinders. Thus, an alternative means of confinement is required in the

third dimension to avoid excessive losses due to diffraction and scattering. This issue

is discussed further in Section 1.6.

As in semiconductor devices, much of the interest in photonic crystals arises not


(e)

(c) (d)

(a) (b)

Figure 1.2: (a) Schematic of a 2D hole-type PhC slab consisting of low refractive

index cylinders in a high-index slab. (b) Scanning electron microscope (SEM) image

of a fabricated hole-type PhC in a silicon slab (from Ref. [8]). (c) Schematic of a 2D

rod-type PhC consisting of high refractive index cylinders in a low-index background.

(d) SEM image of a fabricated rod-type PhC formed from GaAs rods on a low-index

aluminium oxide layer (from Ref. [9]).

from the presence of a bandgap alone, but rather from the ability to create localised

defect states within the bandgap by introducing a structural defect into an otherwise

regular lattice. For example, the removal of a single cylinder from a 2D PhC creates

a point-like defect or resonant cavity, and the removal of a line of cylinders can create

a waveguide that supports propagating modes. Many potential applications based on

this concept have been proposed and demonstrated, a number of which are discussed

in more detail in Section 1.7.

A second class of 2D PhC applications exploits the unique properties of the propa-

gating modes that exist outside the bandgaps in defect-free PhCs. The discrete transla-

tional symmetry of PhCs imposes strict phase conditions on the field distributions that

they support. As a result, only a discrete number of modes are supported for any given

frequency and light propagating in these Bloch modes can have very different proper-

ties to light in a homogeneous medium. We review some of the potential applications

related to this in Section 1.8.

1.4. THREE-DIMENSIONAL PHOTONIC CRYSTALS 9

1.4 Three-dimensional photonic crystals

Three-dimensional PhCs have proved to be the most challenging PhC structures to

fabricate. Whereas 2D PhC research has gained significant benefit from well-established

1D PhC thin-film and semiconductor processing technology such as plasma deposition

and electron-beam lithography, fabrication of 3D PhCs has required the development

of entirely new techniques. For this reason, it was more than three years after the

initial proposal for 3D band gap materials [3, 4] before a structure was calculated to

exhibit a bandgap for all directions and all polarizations [10]. The design consisted

of dielectric spheres positioned at the vertices of a diamond lattice. This followed

experimental reports the previous year in which a partial bandgap in a face-centred-

cubic (FCC) lattice of spheres was mistakenly identified as a complete bandgap [11].

This latter result highlighted the requirement for rigorous theoretical and computational

tools capable of dealing with high-index contrast dielectrics.

Since these early studies, a wide range of 3D PhC geometries exhibiting complete

bandgaps have been demonstrated both in theory and experiment. As an example, a

3D “woodpile” PhC is shown in Fig. 1.3. Due to the challenges involved in fabricating

high-quality structures with features on the scale of optical wavelengths, early photonic

crystal experiments were performed at microwave and mid-infrared frequencies [12–

14]. With the improvement of fabrication and materials processing methods, smaller

structures have become feasible, and in 1999 the first 3D PhC with a bandgap at

telecommunications frequencies was reported [15]. Since then, various lattice geometries

have been reported for operation at similar frequencies [16–18].

Waveguiding and the introduction of intentional defects in 3D PhCs has not pro-

gressed as rapidly as in 2D PhCs, largely due to the fabrication difficulties and the more

complex geometry required to achieve 3D bandgaps. Theoretical studies have demon-

strated the potential for novel photonic circuit designs [19, 20], but to date only a few

experimental results have been reported [21, 22]. Although much of the recent interest

in PhCs has focused on telecommunication related applications, the original concept

of controlling spontaneous emission has not been forgotten. Recent experiments have

demonstrated both inhibition and enhancement of spontaneous emission from quantum

dots embedded in both in 2D [23] and 3D [21, 24] PhCs. The presence of a photonic

bandgap at black-body radiation frequencies has also been shown to modify the thermal

emission properties of heated tungsten 3D PhCs [25,26].

3D PhCs formed in low-index contrast materials such as silica or polymer are also

potentially useful for applications such as those described in Section 1.8 where a com-

plete bandgap is not required. Superprism effects have been calculated in 3D polymer

PhCs [27], and tunable bandgap effects have also been demonstrated using both non-

linear [28] and liquid crystal tuning [29,30].


(a) (b)

Figure 1.3: An example of a 3D PhC woodpile structure known to exhibit a full photonic

bandgap. (a) Schematic of an ideal woodpile PhC. (b) SEM image of a real 3D woodpile

structure fabricated in silicon (from Ref. [14]).

1.5 Band structure and Bloch modes of 2D pho-

tonic crystals

The remainder of this thesis is concerned with 2D PhCs and their application to optical

processing and photonic integrated circuits. In this section we review the properties of

the 2D band structure and associated Bloch modes of uniform PhC lattices. A truly

2D PhC is invariant in the direction parallel to the cylinder axis, and thus has cylin-

ders of infinite length. For a rigorous analysis of such a structure, the out-of-plane

wavevector component must be included in a full 3D calculation, but this can be im-

practical for large structures given the computational demands of 3D calculations. In

a 2D calculation only in-plane propagation is considered, but there are some simple

modifications that can be made to correct partially for out-of-plane wavevector com-

ponents. Although real PhC structures have finite-length cylinders and usually rely

on a slab waveguide geometry to prevent out-of-plane losses, the underlying physics is

the same, and significant insight can be obtained by considering the ideal case. Unless

otherwise stated, the structures studied throughout this thesis are treated as 2D and

the light is taken to be propagating in the plane of the PhC. The implications of this

approximation are discussed in general terms in Section 1.6.4, and more specifically in

the chapters relating to each of the devices.

1.5.1 Band structure calculation

Figures 1.4(a) and (d) show two typical 2D PhC geometries consisting of a square

and a triangular array of cylinders of radius r and refractive index ncyl embedded in a

background dielectric of refractive index nb. The choice of index contrast (ncyl > nb or

ncyl < nb) has a number of important consequences, not only from the point-of-view of

fabrication, but also in terms of fundamental properties. To distinguish between these

1.5. BAND STRUCTURE AND BLOCH MODES OF 2D PHOTONIC CRYSTALS11

two geometries, we will refer to the former as rod-type and the latter as hole-type PhCs.

Examples of both are shown in Fig. 1.2.

Many of the techniques for solving quantum mechanical problems in solid state

physics can be used for electromagnetic fields by casting Maxwell’s equations in the

form of an eigenvalue equation in terms of either the electric (E) or magnetic (H) field

∇2E = −(

ωc

)2εE, (1.1)

∇×(

1ε∇×

)H =

(ωc

)2H , (1.2)

where ε = ε(x, y, z) is the dielectric constant, which may depend also on frequency, and

c is the speed of light. A time dependence of exp(−iωt) has been included implicitly

and we assume that ε is everywhere real and positive, and that µ = µ0 corresponding to

a lossless, non-magnetic material. Whereas plane wave based numerical methods (see

Section 1.9) typically solve Eq. (1.2), the Bloch mode matrix method described in Chap-

ter 2 involves the solution of Eq. (1.1). We discuss here the band structure calculations

in terms of Eq. (1.1), although the conclusions are equally valid for Eq. (1.2). In Eq. (1.1)

the eigenvalue term (ω/c)2ε is analogous to the energy eigenvalue in Schrodinger’s equa-

tion describing the propagation of electrons in a potential.

For a 2D PhC such as those illustrated in Fig. 1.2, the periodic modulation of the

dielectric constant can be written expressed as ε(r) = ε(r+lpq). Here, lpq = pe1+qe2 is

a general lattice vector defined in terms of the basis vectors e1 and e2, and r = r(x, y)

is a vector in the plane perpendicular to the cylinders. The propagation of a wave in

such a periodic medium is governed by the Floquet-Bloch theorem which states that the

solutions to Eq. (1.1) correspond to plane wave fields modulated by a periodic function.

These Bloch modes have the form

Emk(r) = eik0.rumk(r), (1.3)

where umk(r) has the periodicity of the PhC, i.e. umk(r) = umk(r + lpq). Thus, for a

given Bloch vector k0 = (kx, ky), the eigenvalue equation (1.1) can be solved to yield

a collection of Bloch mode eigenfunctions Emk and corresponding eigenvalues ωm(k0).

When plotted over reciprocal space, the ωm(k0) form discrete bands, where each band

is a continuous function of k0, indexed by m = 1, 2, 3 . . . with increasing frequency

as seen in Fig. 1.5. A 2D bandgap occurs at frequencies where no solutions exist

for all possible (real) k0-vectors. Note that an equivalent calculation for the modes

of a uniform dielectric of refractive index n yields a conical band surface defined by

(ω/c)2 = (k0/n)2, where k0 is the free-space wavevector.

The eigensolutions are also periodic functions of k0 on the reciprocal lattice defined

by the basis vectors b1 and b2, where ei.bj = 2πδij, as shown in Figs. 1.4(b) and

(e). Thus, to characterise the band structure completely, it is necessary only to find

solutions for k0 vectors within the first Brillouin zone, defined as the region of reciprocal

space centred on k0 = 0 in which any two wavevectors are separated by less than a

reciprocal lattice vector. The first Brillouin zones of a square and a triangular lattice

are illustrated in Figs. 1.4(c) and (f) respectively.


For lattices with rotational and/or mirror symmetry in addition to translational

symmetry, the complete band structure can be obtained by applying these symmetry

operations to the solutions in an even smaller region of reciprocal space - the irreducible

Brillouin zone, indicated by the white triangles in Figs. 1.4(c) and (f). The origin of

the Brillouin zone is labelled as the Γ-point in both cases, while the other two vertices

are labelled M and X for the square lattice and M and K for the triangular lattice.

For many applications it is sufficient to calculate only the modes corresponding to

k0 vectors lying on the edge of the irreducible Brillouin zone, since the local minima

and maxima of the band structure tend to lie on the symmetry axes and at the high-

symmetry points of the Brillouin zone. Thus, a typical band diagram for a 2D PhC

shows only the band structure along the edges of the irreducible Brillouin zone between

each pair of labelled symmetry points, as can be seen in Figs. 1.5(c) and (d). From

such a diagram, the position and width of all bandgaps can be determined, along with

many other characteristic features of the band structure. In Section 1.5.3 we consider

the two most common lattice types used for 2D PhCs – square and triangular – and

give examples of the band structure for these lattices.

1.5.2 TE and TM modes

In the analysis described in Section 1.5.1, no assumptions are made about the polarisa-

tion of the electromagnetic fields. However, in the case of a 2D PhC, where propagation

is restricted to a plane perpendicular to the cylinders, the solutions decouple into two

distinct polarisation states - transverse electric (TE) and transverse magnetic (TM).

For TE (TM) modes, the electric (magnetic) field lies in the xy-plane, while the mag-

netic (electric) field is aligned with the z-axis, along the cylinders. In general the band

structures of the TE and TM states are quite different due to the boundary conditions

imposed at the dielectric interfaces. Therefore most 2D PhCs are designed for operation

in a single polarisation state. The choice depends largely on the geometry of the PhC

and the desired properties; in rod-type geometries, the largest bandgaps tend to occur

for TM polarisation, so for bandgap applications this is usually the best choice; the

reverse is the case in hole-type PhCs where TE polarised states tend to have the largest

bandgaps (Ref. [7], Chapter 5).

1.5.3 Band structures of square and triangular lattices

The first 2D PhC structures found to exhibit photonic bandgaps were square and trian-

gular lattices of cylinders [31,32]. These geometries are still the most common, and the

ones considered in this thesis. Systematic studies of both rod- and hole-type geometries

have been undertaken to identify the optimal choice of cylinder radius, index contrast

and lattice type in order to maximise the frequency range of the bandgap for either

one or both polarisations [33, 34]. In all cases, there is a minimum refractive index

contrast below which gaps do not exist for either polarisation, and not all geometries

1.5. BAND STRUCTURE AND BLOCH MODES OF 2D PHOTONIC CRYSTALS13

e1

e2

d

d e1

e2

G

M

X

G

KM

b2

b1

b1

2

d

p

b2

Real lattice Reciprocal lattice Brillouin zone

x

y

kx

ky

Square

Triangular

(f)(e)(d)

(b) (c)(a)

Figure 1.4: (a) and (d): Diagrams of square and triangular PhC lattices in real space

generated by the basis vectors e2 and e2. (b) and (e): The corresponding reciprocal

lattices and basis vectors b1 and b2. The boundaries of first Brillouin zones are indicated

by the dotted lines. (c) and (f): Representation of the first Brillouin zones and the

irreducible region (white triangles) of the two lattices.

have overlapping TE and TM bandgaps, regardless of the index contrast.

Figure 1.4 shows a square and a triangular lattice represented in both real and

reciprocal space along with the basis vectors used to generate any lattice vector. The

first Brillouin zone of each lattice type is shown in parts (b), (c), (e) and (f) of the figure.

In (c) and (f), the irreducible Brillouin zones are defined by the lines of symmetry joining

the high symmetry points, which are labelled according to convention.

As an example, we consider here the band structures of two typical PhC geometries:

the first is a rod-type PhC consisting of a square lattice of silicon cylinders of index

ncyl = 3.4 and radius r = 0.25 d embedded in silica with index nb = 1.46, and the

second is a hole-type PhC formed by a triangular lattice of air holes (ncyl = 1) of

radius r = 0.32 d in silicon (nb = 3.4). When calculating the properties of PhCs, it

is standard practice to express all spatial dimensions relative to the lattice period d

as this allows structures to be scaled for operation at any desired frequency, although

material dispersion must be taken into account for large changes of scale. Thus, the

cylinder radius is typically written as r/d and the wavelength as λ/d. The dimensionless

frequency d/λ is equivalent to the parameter ωd/2πc used in some texts.


Figure 1.5(a) shows the first four TM polarised bands (see Section 1.5.2) of the

square lattice PhC plotted as a function of frequency over first Brillouin zone. Since

the eigenvalues ωm(k0) are continuous functions of k0 within each band, they form

band surfaces. Observe that there is a bandgap between the first and second bands.

Figure 1.5(b) is a contour plot of the second band showing the equifrequency, or isofre-

quency contours over the Brillouin zone. Visualising the bands in this manner can be

informative for some applications, but as mentioned in Section 1.5.1, it is often sufficient

to calculate the bands along the perimeter of the irreducible Brillouin zone, indicated

by the dashed line. Figure 1.5(c) shows such a band diagram for the same PhC, for

both the TE and TM polarised modes. The TM bandgap is clearly seen over the nor-

malised frequency range of 0.249 < d/λ < 0.298, but there is no equivalent gap for TE

polarisation. Figure 1.5(d) shows the band diagram for the triangular lattice PhC with

the parameters given earlier in this section. In this case, there is a large bandgap for

TE polarisation between 0.216 < d/λ < 0.298 but none for TM.

��

��

� ππ� π π ��

��

Γ

�

�

"!�# "!�#%$ "!�#'& "!�#)( "!�#)* "!+& "!+&,$

-/.λ

01 λ

243

��5�� 6��

�� 7��8��

9;:=< 9;>?<

24@ 24324@

Γ A B Γ Γ ΓB C

��

01 λ

01 λ

Figure 1.5: (a) TM band surfaces for a 2D square rod-type photonic crystal with the

parameters defined in the text. (b) Equifrequency contour plot for the second band in

(a). (c) Band diagram for the same structure showing TE and TM modes on the edge

of the irreducible Brillouin zone. (d) Band diagram for the 2D triangular hole-type

photonic crystal with the parameters given in the text.

For bandgap applications, the position and size of the gap are usually the critical

1.6. PHOTONIC CRYSTAL SLABS 15

parameters, while the details of the band structure above and below the gap are much

less important. Recently, however, the unusual propagation properties of the Bloch

modes have been attracting renewed attention and a number of novel applications have

been proposed. These applications require the band surfaces to be engineered to have

very specific properties, so a full calculation of the bands over the Brillouin zone is

required. We discuss some of these in-band applications in Section 1.8.

1.6 Photonic crystal slabs

We have so far discussed only the properties of purely 2D PhC geometries which are

invariant along the z-axis, and thus have cylinders of infinite length. This approximation

is valid if the cylinders are many wavelengths long, but for practical purposes this is

not always desirable or possible to achieve. Even in geometries that can be considered

as purely 2D, diffraction losses are high since light is free to propagate out of the xy-

plane even when there is a bandgap for in-plane propagation. An exception to this

are photonic bandgap fibres in which light propagates almost parallel to the cylinders

and is confined to a central core by the periodic cladding [35]. But for most other

applications, a method is required to confine the light in the out-of-plane direction

within a 2D PhC of finite height. One way to achieve this is to cap the slab above

and below with another PhC structure that has a bandgap for propagation in the z-

direction. Suggestions have been made for 1D [36, 37], 2D [38] and 3D [39] capping

layers, but to date these approaches have not been demonstrated experimentally.

A more common technique is to confine the light in the slab using total internal

reflection (TIR) at the slab/cladding interface as in a planar dielectric waveguide. This

approach was proposed in 1994 [40], and demonstrated experimentally in 1996 [41], but

was not studied rigorously until 1999 [42] for uniform PhC slabs and later for PhC

waveguides [38, 43, 44]. To achieve TIR, the refractive index of the cladding must be

lower than that of the slab, or in the case of rod-type PhCs, the high-index rods. A

bulk dielectric such as air or silica is typically used for the low index cladding material,

as illustrated in the examples of Fig. 1.2. A more detailed discussion of slab design in

both rod- and hole-type PhCs is provided in Sections 1.6.2 and 1.6.3.

1.6.1 The lightline

As with any index-guiding structure, for light to be confined within the slab, it must

satisfy a total internal reflection condition to ensure that light in the core does not

couple to the radiation modes of the cladding. The dispersion relationship for a mode

in a homogeneous dielectric cladding of refractive index ncl is given by(nclω

c

)2

= k20 + k2

z , (1.4)


where k0 = (kx, ky) is the wavevector component in the plane of the slab. Thus,

if k20 > (nclω/c)

2, then kz is imaginary and the field in the cladding is evanescent,

corresponding to TIR. If k20 < (nclω/c)

2 however, kz is real and light can propagate

in the cladding. When plotted in k − ω-space, the relation k0 = (nclω/c)2 defines the

light cone, as illustrated by the white cone in Fig. 1.6. In a conventional band diagram

this appears as a lightline, such as those in Fig. 1.7. Above the lightline a continuum

of propagating modes exists in the cladding, whereas no propagation is allowed below.

Therefore, modes of the PhC that lie inside the cone (above the lightline) can couple to

the radiation modes and leak into the cladding but modes that lie below the lightline

decay exponentially in the cladding and are guided within the slab.

G

M K

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

d/l

kx

ky

43p

-43p

00

-23p

23p

Figure 1.6: Band surface plot showing the first four bands of the hole-type PhC con-

sidered in Section 1.5.3, but with a finite slab of height 0.4 d surrounded by air, as in

Fig. 1.5(d). The white surface centred at Γ represents the light cone. Modes outside the

cone are guided in the slab, while those inside the cone can radiate into the cladding.

The band properties of a PhC slab are in many ways similar to those of the 2D PhCs

discussed in Section 1.5.3, but a number of additional effects must also be considered

including the slab thickness, refractive index contrast and symmetry properties of the

slab and the cladding. One of the most important differences is the effect of out-of-plane

propagation on the mode structure. In the 2D analysis, the TE and TM polarised modes

are decoupled and can be treated separately, but this is not the case for the modes of

a PhC slab. However, if the slab and cladding are symmetric about the z = 0 plane,


these modes can be classified according to the symmetry of the Hz field component

with respect to the reflection plane. In this case, the TE and TM modes of a purely

2D structure map onto even and odd slab modes respectively and bandgaps can exist

for one or other of these symmetries. These are not true bandgaps because light can

always propagate above the lightline in cladding modes. However they correspond to

gaps in the spectrum of guided modes, and thus share many of the properties of 2D

photonic bandgaps.

1.6.2 Hole-type photonic crystal slabs

The fabrication and experimental demonstration of PhC-based devices has progressed

much more rapidly for hole-type PhCs than for rod-type structures, despite both ge-

ometries receiving almost equal attention in theoretical studies. This difference is in

part due to the fabrication challenges involved in forming circular rods with smooth,

vertical sides, but is also related to the issue of vertical confinement. In a hole-type

PhC, the high-index material surrounding the holes can also serve as a high-index slab

for TIR guidance, but in rod-type PhCs, where the high-index regions are separated

from one another, alternative methods are required to achieve TIR while maintaining

structural integrity. This issue is discussed further in Section 1.6.3.

There are two cladding designs that are used most for hole-type PhC slabs. The first

of these consists of a high refractive index PhC slab on top of a lower refractive index

substrate with an air cladding above. Such structures are relatively easy to fabricate

using existing thin film deposition techniques to lay down the multiple layers [41, 45],

but the resulting asymmetric slabs do not allow for the simple splitting of the odd and

even modes. More recently, methods have been developed to fabricate thin membranes

of PhC material suspended in air, producing symmetric air-clad PhC slabs, also known

as membrane-type or air-bridge PhCs. High-quality low-loss PhCs have been fabricated

in both Si and GaAs slabs using this method [8,46].

Figure 1.7(a) shows the band diagram calculated for the same photonic crystal as

in Fig. 1.5(d), but now for a finite slab of height 0.4 d with air above and below. The

corresponding band surfaces are shown in Fig. 1.6. Only the modes below the lightline

are plotted, while the region above the lightline is shaded light grey and contains a

continuum of radiation modes. The even mode bandgap in the range 0.292 ≤ d/λ ≤0.388 is shaded dark grey and corresponds to the TE bandgap in the 2D calculation,

although it has shifted to higher frequencies.

1.6.3 Rod-type photonic crystal slabs

Rod-type PhC slab geometries are inherently different from hole-type structures because

the regions of high refractive index are not connected, and thus the rods must be

attached to a substrate material. Index guidance at the bottom of the rods is achieved

by choosing a substrate of lower refractive index, or even etching the rods into the


Γ M K Γ

odd

even

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.0

d/l

Γ X M Γ

odd

even

0.5

0.4

0.3

0.2

0.1

0.0

d/l

(a) (b)

Figure 1.7: (a) Band diagram of a hole-type PhC slab with the same parameters as in

Fig. 1.5(d) and a slab height of 0.4 d. The cladding above and below the slab is air. (b)

Band diagram of a rod-type PhC with the same parameters as in Figs. 1.5(a)–(c), but

for cylinders of height 1.5 d surrounded above and below by a silica cladding. Only the

guided modes below the cladding lightline are plotted. The regions shaded light grey

lie above the lightline and contain a continuum of cladding modes.

substrate so that the index of the rods is lower at the bottom than at the top. If

the rods are left surrounded by air, then index confinement at the top of the rods

is provided by the rod-air interface [47]. The main drawback of this geometry is the

strong asymmetry due to the different cladding materials above and below the rods.

One solution is to fill the region between and above the rods with a low-index material

that matches the substrate index, thus creating an up-down symmetric array of finite

rods analogous to the hole-type PhC membrane structure [48, 49]. A similar design

was proposed by Martinez et al., [50] in which two different materials were used – one

to fill the spaces between the rods, and another with a slightly lower index for the

cladding above and below. However it is not clear whether such a small index contrast

contributes significantly to the guidance, given the already strong contrast between the

rods and the cladding. In these last two approaches, it is important that a sufficiently

large index contrast is maintained between the rods and the surrounding material to

ensure that the in-plane bandgap is not destroyed.

Figure 1.7(b) shows the band diagram calculated for the same photonic crystal

as in Figs. 1.5(a)–(c), but now for finite rods of height 1.5 d with the SiO2 back-

ground extending above and below. The bandgap for odd-symmetry modes in the

range 0.293 ≤ d/λ ≤ 0.314 corresponds to the TM bandgap in the 2D calculation, again

shifted to higher frequencies.


1.6.4 2D vs 3D modelling of PhC slabs

Rigorous theoretical analysis of PhC slabs requires full 3D modelling tools, but these

involve intensive numerical calculations that can take days or weeks to run on a desk-

top computer, or may be impossible due to memory requirements. Although parallel

computing methods and access to supercomputers may be an option, significant results

and insight can be obtained using 2D methods.

When operating below the lightline, the main difference between 2D and 3D calcula-

tions is a frequency shift in the band structure. This occurs because the band structure

is largely determined by the wavevector component k0 that lies in the plane of the PhC,

rather than the total k. Since k0 = k − kz, out-of-plane propagation has the effect of

reducing k0 for a given frequency. Thus, band features associated with a particular

k0 are shifted to higher frequencies, as observed in Sections 1.6.2 and 1.6.3. In many

cases this effect can be corrected to some extent by replacing the refractive index of

the background material in the 2D calculation with an effective refractive index. For

hole-type PhCs, an appropriate effective index is found by first calculating the guided

modes of the slab without any holes. The effective indices of the slab modes are pro-

portional to their in-plane wavevector. Since most PhC slabs are designed to support

only a single slab mode, calculating the 2D band structure with the effective index of

the fundamental slab mode approximates the effect of out-of-plane propagation [51].

For relatively low slab-cladding index contrasts in both symmetric and asymmetric

hole-type PhC slabs, the effective index method generally agrees well with the full 3D

band structure and waveguide mode calculations [51]. Experimental results have also

shown good agreement with 2D calculations [52, 53]. The approximation deteriorates

for high index contrasts, such as in membrane-type slabs, as the tighter slab mode

confinement introduces significant waveguide dispersion and other effects not easily

accounted for in 2D calculations. The method is also unsuitable for studying strongly

resonant behaviour in cavities as the operation of these structures can be severely

degraded by out-of-plane losses. In these cases it is essential to understand and control

the radiation properties [54] and hence 3D calculations are generally required. Another

case where 3D methods are needed is the interaction region between modes of different

symmetry [55].

Applying the effective index approximation to rod-type PhCs is somewhat less

straightforward since the material between the rods is often the same as the cladding.

In this case, or for hole-type PhCs with large holes, it may be appropriate to calculate

first an average refractive index of the PhC, taking into account the refractive indices

and the relative fill-fractions of the cylinders and background material (Ref. [56], Chap-

ter 10). The average index value could then be used as the slab index when calculating

the guided slab modes. Although applying an average index to a PhC is strictly valid

only when the wavelength is significantly larger than the lattice period, this approach

may at least allow the effective index method to be applied to rod-type PhCs as an

intermediate approximation between 2D and full 3D calculations. However there are


no reports in the literature of this method being tested.

In Chapters 3 and 4 we present several novel PhC waveguide-based devices, all of

which are studied using the purely 2D techniques described in Chapter 2. Although the

devices have not been explicitly designed for implementation in a PhC slab geometry,

their operation relies on generic PhC properties such as waveguiding, mode coupling

and omnidirectional reflection. The exception to this is Chapter 5, where 3D numerical

calculations are used to verify the efficient coupling characteristics of rod-type PhCs.

Rather than presenting devices optimised for fabrication, this thesis is intended to

highlight some of the unique properties of 2D photonic crystals and demonstrate novel

ways to utilise these properties.

1.7 Defect states in PhCs: cavities and waveguides

Guiding and trapping light using waveguides and resonant cavities are two fundamental

optical functions that enable a range of all-optical devices to be created. Waveguides

not only perform the tasks of their electrical analogues, wires, by transferring light from

one part of a circuit to another, but are used in many other devices such as couplers,

junctions and interferometers. Resonant cavities have many potential applications that

make use of the sharp spectral response and very strong field intensities that occur at

resonance. There are various methods for achieving efficient waveguiding and many

others for producing high quality optical cavities, but very few single technologies allow

both to be engineered in a single integrated structure. Two dimensional photonic

crystals can provide just such a combination due to their versatile geometry that allows

simultaneous fine tuning of a number of parameters. 3D PhCs could potentially provide

even greater control over light, but they are considerably more challenging to fabricate.

In addition, the complex lattice structures required to achieve full 3D bandgaps can

make it very difficult to create high quality defects.

The possibility of producing cavities and waveguides in PhCs by introducing defects

was recognised not long after the first 2D bandgap structures were proposed. In 1994,

Meade et al. suggested that both cavities and waveguides could be created in 2D

PhCs by simply changing or removing a single cylinder or a whole line of cylinders,

respectively [40]. A defect created in this way is surrounded by the uniform PhC that

acts as an omnidirectional mirror for light at frequencies in the bandgap, thus trapping

light in the defect region. In a cavity defect, this can result in sharp resonant behaviour,

whereas a linear defect allows light to propagate along the defect while being confined

in the transverse direction by perfectly reflecting PhC walls. In a PhC slab, this is only

true for light confined vertically by TIR, but the behaviour is essentially the same.

1.7. DEFECT STATES IN PHCS: CAVITIES AND WAVEGUIDES 21

1.7.1 Cavities

Resonant cavities provide both sharp spectral responses and large field enhancement

within the cavity when the resonance condition is satisfied. The former can be used for

narrow bandwidth filters and wavelength selective couplers, both of which are required

in wavelength-division-multiplexing (WDM) optical systems for operating on individual

frequency channels. High field intensities due to light being trapped in a small cavity

can enhance light-matter interaction, making them ideal for photonics applications such

as lasers and nonlinear optics. They are also useful in sensing applications and for more

fundamental research into cavity quantum electrodynamics and control of spontaneous

emission. A review of many of these applications is given in Ref. [57].

An ideal optical cavity would trap light indefinitely and have a single resonant

frequency, but real cavities are limited by losses, both due to radiation and absorption.

A measure of how close a cavity approximates an ideal resonator is given by the quality

factor Q, which is proportional to the lifetime of light in the cavity,

Q = ω0stored energy

power loss per cycle= −ω0

U

dU/dt, (1.5)

where ω0 is the resonant frequency, and U is the energy stored in the cavity (Ref. [58],

chapter 8). Therefore, an ideal cavity would have an infinite Q. A more practical

definition for the purposes of measuring Q in experiment or simulation is given by the

relative width of the resonance,

Q =ω0

∆ω, (1.6)

where ∆ω is the full-width at half-maximum (FWHM) of the resonant power spectrum.

Field intensities are also determined by the volume of the cavity V , so to achieve the

optimum field enhancement, the ratio Q/V should be maximised. Silica microspheres

hold the record for the highest Q values, with experimental demonstrations of Q > 109

[59], but the geometry is not easily integrated with other photonic devices and the

modal volumes are relatively large. Toroid microcavities are perhaps more promising

as they have been demonstrated with Q > 108 in an on-chip silicon-on-insulator (SOI)

geometry [60]. However, despite being fabricated on-chip, a tapered optical fibre was

still required for coupling to the toroid. While this method allows fine-tuning, it may

not be suitable for dense integration as the condition for coupling between the fibre and

the resonator is likely to be delicate and hence sensitive to perturbations. In contrast,

PhC cavities can be coupled into directly from nearby waveguides in the same structure,

requiring no additional alignment or coupling control.

Whereas a cavity in a 3D PhC could theoretically provide perfect confinement,

cavities in PhC slabs rely on TIR for confinement in the vertical direction, and so

their performance is largely dependent on out-of-plane losses. For this reason, it is

informative to separate the contributions due to in-plane effects Q‖ and out-of-plane


effects Q⊥. These are related to the total Q of the cavity by

1

Q=

1

Q‖+

1

Q⊥. (1.7)

From this equation it can be seen that the cavity Q is limited by the smaller of Q‖ and

Q⊥. Q‖ depends on a number of factors including the number of PhC layers surrounding

the cavity and the scattering losses due to imperfections in the lattice. Q⊥ depends on

the out-of-plane radiation, most of which occurs when light is reflected at the interfaces

surrounding the cavity [61].

Recall from Section 1.5.1 that Bloch modes of a 2D PhC extend throughout the

crystal lattice and can thus be characterised by a single frequency eigenvalue ωm(k0)

for every in-plane wavevector k0. When light is confined to a small cavity, it is no

longer represented by a single wavevector, but the modal field instead contains a super-

position of spatial frequencies in the same way that a finite beam can be expressed as a

superposition of plane waves. Any components with k20 < (nclω/c)

2 lie inside the light

cone, and can couple to radiation modes leading to losses that reduce Q⊥ and hence

result in significant reductions in the total Q of the cavity.

The first PhC cavities to be fabricated exhibited Q values of only a few hundred

[62,63]. These low values were due not only to out-of-plane losses inherent in the cavity

design, but also to scattering losses introduced by structural variations and surface

roughness. In 2001, Vuckovic et al. reported a systematic theoretical study of various

cavity designs to establish methods for reducing radiation losses [64]. Cavities with

Q > 104 were calculated by varying the radius and shape of the cylinders in the area

surrounding the defect. Structures based on some of these designs were also fabricated

and measured to have Q = 2800 [65].

Many of these early PhC cavities were designed for lasing and included quantum

wells in the slab structure that produce photoluminescence when optically pumped

from outside the slab but the first experimental report of direct coupling between a

PhC cavity and a waveguide was in 2001 [66], where a cavity was created in a section

of PhC between the ends of two waveguides. When light was coupled into the end of

one waveguide, a sharp resonant transmission of Q = 816 was observed

Although radiation losses were considered in the various cavity designs discussed

above, the direct relationship between Q and the wavevector components of the mode

was not explored. The wavevector distribution of a mode at the resonant frequency is

given by the 2D spatial Fourier transform of the field. In this k-space representation,

the light cone forms a circle of radius (nclω/c). Any component of the Fourier transform

that lies within the circle contributes to radiation losses and hence reduces Q⊥ and the

total Q of the cavity. Theoretical studies have shown that by careful adjustment of the

size and shape of the cavity, it is possible to redistribute almost all of the mode energy

outside the light cone [67, 68]. The first experimental demonstration of this design

approach produced Q values of 1.3 × 104 [69]. An alternative approach to reducing

radiation losses was proposed by Johnson et al. whereby the cavity is modified to


achieve cancellation of specific radiated field components in the far-field [70].

There are many possible ways to optimise the geometry of a cavity, and it is not

immediately obvious which one should be taken. In Ref [69], the defect was formed by

reducing the size of many holes in a large defect region, with the holes near the centre

of the region being changed more than the ones closer to the outside. A number of

other ‘graded lattice’ designs have been proposed with predicted Q values of more than

105 [71, 72]. The feature of all these designs is that there are no abrupt changes in the

lattice structure at the edges of the defect. Using a simple 1D model, Akahane et al.

[54] showed that this feature is the key to high-Q cavity design; the modal field should

vary smoothly across the edges of the defect. By shifting the position of two holes, a

Q of 4.5× 104 was measured in a cavity formed by the removal of only three holes in a

triangular lattice. The small cavity size resulted inQ/V = 1.2×105/λ3, at least an order

of magnitude higher than any other optical resonator at the time ∗. Further refinements

in the cavity design have more than doubled these values [74]. The same technique

was also used to design a different cavity geometry with an experimentally measured

Q = 6 × 105 [75]. In the last three examples, light was coupled into the cavity from

a PhC waveguide separated by a number of lattice periods - a simple, yet convincing

demonstration of the possibilities for integrated optical circuitry. Bistable switching for

powers as low as 40µW was demonstrated recently using a PhC cavity coupled between

two waveguides that exhibited a linear transmission resonance of Q = 3.3× 104 [76].

Although the study of high-Q cavities is not a major theme of this thesis, resonance

effects are considered in detail and some of the design rules used to reduce radiation

losses in cavities could be applied to the resonant coupling devices presented here.

1.7.2 Waveguides

The concept of creating a linear defect in a PhC slab for waveguiding was already

introduced at the start of Section 1.7, but we now consider the properties in more

detail. In many ways the design of efficient waveguides is considerably easier than the

design of resonant cavities because the confinement of light by the PhC is required

only in one direction within the slab. There are two conditions for propagating modes

to exist in any waveguide, whether metal, dielectric or PhC: first, the walls must be

reflective, and second a phase condition must be satisfied. The latter requirement

results in a discrete spectrum of guided modes. Hollow metal waveguides are used for

guiding microwaves as metals have relatively low absorption at those frequencies, and

they reflect light for all incident angles. Dielectric waveguides, on the other hand, rely

on total internal reflection, so they only guide modes that satisfy the TIR condition

on all the walls. PhC slab waveguides use the in-plane photonic bandgap for lateral

mode confinement and TIR for vertical confinement. Since the PhC walls reflect light

for all incident angles, PhC waveguides have many similarities to metal waveguides and

∗Higher values have since been reported in toroid microcavities [73].


some of the concepts of microwave circuit design such as impedance matching have been

applied to PhCs [77,78].

There is justifiably some debate as to whether photonic crystal waveguides provide

any benefit over high-index contrast dielectric waveguides, such as those based on the

well-established SOI technology [79]. For simply transferring light from one part of an

optical circuit to another, SOI waveguides not only tend to have lower losses, but can

also be positioned anywhere, rather than having to conform to a PhC lattice. PhC

waveguides, on the other hand, can be directly integrated with other PhC structures

such as cavities to produce functional devices. In addition, light guided by a photonic

bandgap can exhibit very different behaviour to TIR guided light, including very strong

dispersion and strong reflections that we exploit in Chapters 3 and 4.

1.7.3 Linear waveguide modes

A linear defect is typically created in a PhC by removing or changing nearest-neighbour

cylinders along one of the symmetry directions of the lattice. In a square lattice this

usually corresponds to the Γ−X direction, while in a triangular lattice the waveguide

lies along Γ−K, as illustrated in Fig. 1.8(b) and (c). Since the waveguide only breaks

the PhC symmetry in one direction, the structure remains periodic along the waveguide

axis. Thus the modes are characterised by a wavevector k directed along the guide and

a corresponding frequency. As for any 1D periodic lattice, the waveguide dispersion

curves can be represented on a band-diagram showing the mode frequency as a function

of the Bloch vector k0 over the 1D Brillouin zone defined by −π/d ≤ k0 ≤ π/d, where

d is the periodicity in the waveguide direction. For the common waveguide directions

described above for the square and triangular lattices, the period along the waveguide

is just the lattice period d = d, but for waveguides in other directions, generally d 6= d.

The bands of the PhC surrounding the waveguide can be represented on the same

plot by projecting the 2D band diagram onto the k−axis of the waveguide. In a square

lattice for example, the Γ−X band structure is plotted as before, but the modes lying

in the rest of the Brillouin zone are plotted as a function of their kx component only.

Such a combined waveguide dispersion and projected PhC band diagram clearly shows

where the waveguide modes lie in relation to the bandgap and the Bloch modes of the

PhC. For 3D calculations, the lightline can be projected onto the waveguide axis in the

same way. More details on the construction of these diagrams are given by Johnson et

al. [38]. As an example, Fig. 1.8 shows the dispersion curves and modal field profiles

for a waveguide formed by leaving out a single line of holes in the Γ − K direction

of the hole-type PhC considered in Section 1.5.3. For TE polarisation there are two

modes: one with even symmetry, extending down from the top of the bandgap to the

cutoff frequency d/λ = 0.221, and one with odd symmetry that lies in the frequency

range 0.245 ≤ d/λ ≤ 0.256. The properties of these modes are discussed in more detail

in Section 1.7.4.

Waveguide modes lying below the lightline are lossless in an ideal PhC slab. In prac-


0.2

0.22

0.24

0.26

0.28

0.3

0.32

0.180 0.50.25

k d( )2 /p

d/l

PhC bands

PhC bands(a)

(b)

(c)

Figure 1.8: (a) Dispersion curves for a waveguide formed in the triangular PhC lattice of

Section 1.5.3 by removing a line of cylinders along the Γ−K direction. Two waveguide

modes exist in the bandgap: an even symmetry mode (solid curve) and an odd symmetry

mode (dashed curve). Hz field distributions of the odd and even modes are shown in

(b) and (c) respectively for k = π/d, clearly illustrating the difference in symmetry.

tice however, propagation losses are significant. The first experimental demonstration

of PhC waveguides was performed using millimetre waves in a lattice of alumina rods

with a period of 1.27 mm, in which propagation losses of 0.3 dB/cm were measured [80].

The lattice in this case was essentially 2D, since the rods were considerably longer than

the wavelength, and there was no guidance in the vertical direction. Waveguiding at

telecommunications wavelengths in PhC slabs was first observed in 1999 but transmis-

sion spectra and loss measurements were not reported until 2001 [81, 82]. Much effort

has gone into reducing loss since then with recent reports of 15 dB/cm propagation loss

in SiO2 clad structures [83] and < 5 dB/cm in air-clad Si slabs [84,85]. These losses are

within an order of magnitude of high-index-contrast SOI waveguides, for which losses

of 0.8 dB/cm have been reported [86].

Apart from the initial millimetre-wave experiments, the great majority of wave-

guiding demonstrations in PhC slabs have involved hole-type PhCs. Although a few

experimental demonstrations of waveguiding in rod-type PhCs have been reported, the

propagation losses are at least an order of magnitude greater the lowest-loss waveguides

in hole-type structures [47,48].

Below the lightline, the main contribution to waveguide loss is thought to be struc-

tural disorder. The length scales of this disorder can range from several tens of nm,

in the case of hole position and radius, down to only a few nm, in the case of surface

roughness. Positional and size disorder has been shown to reduce the size of the band-

gap in bulk PhC, and simultaneously decrease the transmission bandwidth and total

transmission of PhC waveguides [87, 88]. Calculations have also shown that the first


row of cylinders on the waveguide edges is the most important for minimizing in-plane

scattering losses; as long as these cylinders are in the correct places, disorder in the

rest of the lattice has a minimal effect on the waveguide transmission properties [89].

Surface roughness has only recently been included rigorously in scattering loss calcu-

lations for PhC waveguides using a Green function theory. Very good agreement with

experimental loss measurements was demonstrated, suggesting that the dominant loss

mechanisms were identified [84].

The prospect of low-loss bends in PhC waveguides has perhaps attracted more

attention than any other aspect of PhC applications [90]. High-efficiency bends have

been demonstrated in many different photonic crystal geometries [48, 80, 82, 91]. Since

the PhC waveguide walls reflect light for all incident angles, sharp bends do not suffer

from the same bend losses that occur in dielectric waveguides [92], but this also means

that without careful design, they can reflect a considerable amount of light as well. In

contrast, if a dielectric waveguide is bent too sharply, the TIR condition is violated

and light radiates into the cladding. Although this is major drawback of low-index-

contrast dielectric waveguides originally proposed for integrated optics applications,

the stronger index confinement in SOI waveguides allows low-loss curved waveguide

bends of less than 5µm in radius [93]. Alternative methods have also been proposed to

achieve even sharper bends [94].

1.7.4 Dispersion

One aspect where PhC waveguides differ significantly from TIR-based waveguides is

in their dispersion properties. Large and variable dispersive effects are a characteris-

tic feature of light propagation in all periodic media, originating from the wavelength

dependence of coherent scattering in these materials. Dispersion control is particularly

important when transmitting short pulses through waveguides because of the corre-

spondingly broad frequency spectrum. If the dispersion properties change significantly

over the spectral range of the pulse it can lead to distortion or, in the worst case de-

struction of the signal. This effect can also be used to great advantage, however if the

properties can be controlled. By tuning the dispersion appropriately, dispersive effects

of other components in the system can be compensated for, or alternatively pulses can

be slowed down, compressed or stretched. Nonlinear interaction lengths can also be

significantly reduced by slowing the light down.

The dispersion properties of a waveguide mode are characterised in part by the

group velocity, vg = dω/dk0 and the group velocity dispersion GVD = dvg/dk0; the

slope and the curvature of the mode dispersion curve, respectively. Figure 1.8(a) shows

a typical pair of dispersion curves for a waveguide formed in a triangular hole-type

lattice. Consider first the even mode represented by the solid curve. The curve is

almost straight over most of its frequency range, indicating a constant group velocity

and a near-zero GVD. As the mode approaches cutoff however, it flattens out and

vg → 0 at the edge of the Brillouin zone, as expected for the modes of most periodic


structures. The sharp change in the slope just before cutoff also results in a large

GVD at these frequencies, a property first measured experimentally with an indirect

interferometric technique [95]. In contrast, the odd mode, represented by the dashed

curve in Fig. 1.8(a) has zero vg in the middle of the Brillouin zone, as well as at the

edges. Group velocities as low as 0.02c, where c is the speed of light in a vacuum, have

been demonstrated experimentally via group delay measurements [83]. Even smaller

values of vg < 0.001c have been inferred from direct time-resolved observations of pulse

propagation in PhC waveguides, although these have not been confirmed rigorously

using other measurement techniques [96].

One of the advantages of the geometrical complexity of PhC waveguides is that

there are a vast number of possibilities for tailoring the dispersion properties to suit the

desired application. Some of these include changing the waveguide width [97], changing

the radius of some of the cylinders [98, 99], or even placing additional cylinders along

the waveguide to create a coupled cavity waveguide (CCW) [100, 101]. Other, more

complicated coupled resonator designs have been proposed for stopping light altogether,

raising the possibility of optical data storage or buffering [102].

1.7.5 Integrated waveguide and cavity devices

Waveguides and cavities form the building blocks for a vast range of optical components

and devices. In this section we consider two classes of device that have been successfully

demonstrated in photonic crystals by integrating waveguides and resonant cavities.

Waveguide couplers

Directional couplers perform various functions in conventional optical circuits, includ-

ing power splitting and combining and wavelength-selective coupling. Switching and

wavelength tuning can also be achieved by incorporating nonlinear or other tunable

elements into the structure (See, for example, Ref. [103], Chapters 17, 18).

In its most basic form a coupler consists of two parallel identical waveguides running

together over some distance L. When placed closely together, the modes in each guide

interact to produce supermodes — modes of the coupled waveguide system. If each

waveguide in isolation supports only a single mode, then the coupled system generally

splits these into two supermodes, one at a lower frequency and one at a higher frequency

than the original. For coupled dielectric waveguides, the fundamental, lowest frequency

mode is always symmetric (even) with respect to the symmetry axis of the waveguide

pair, while the second mode is antisymmetric (odd), with a nodal line running along

the axis. Two identical coupled PhC waveguides can also support a symmetric and an

antisymmetric mode, but the lowest frequency mode is not always the symmetric one.

This issue is studied and explained in more detail in Chapter 3. Although this property

may introduce some additional design considerations for PhC waveguide couplers, the

coupling mechanism is unchanged as it depends only on the magnitude of the mode


splitting, and not on the order of the two modes. The mode splitting is characterised by

the frequency-dependent parameter ∆β = |β+−β−|/2, where the β± are the propagation

constants (or wavevectors) of the even and odd supermodes respectively.

If monochromatic light of unit power is injected into one of the waveguides and

allowed to propagate, it couples into the other waveguide. After a propagation distance

L, the power remaining in the input guide is given by cos2(∆βL), while the power

transferred to the second guide is sin2(∆βL). The coupling length is the distance over

which a π phase difference accumulates between the two modes

Lc =π

2∆β=

π

|β+ − β−|, (1.8)

corresponding to the propagation distance required for all of the light in one guide to

be transferred into the other. Couplers of length Lc/2 transfer 50% of the light into the

other guide and are widely used as 3 dB power splitters and combiners. The shortest

coupling lengths achievable in dielectric waveguides are several tens of microns, and

even this requires novel coupler designs [93, 104]. Coupled PhC waveguides however,

exhibit very strong mode splitting, and coupling lengths of only a few wavelengths have

been predicted [105,106]. Various PhC directional coupler designs have been proposed

and demonstrated experimentally [107, 108]. Modified coupler designs incorporating a

short waveguide cavity between the two guides have also been suggested to reduce the

coupling length further and improve coupling efficiency [109].

The sensitivity of PhC waveguide dispersion properties to small variations in ge-

ometry or refractive index makes them ideal for coupler-based switching applications.

Optical switches based on both electro-optic and nonlinear refractive index modula-

tion have been proposed as methods of tuning the coupling strength to allow or block

coupling between the guides [110, 111]. A basic add/drop filter can also be formed by

arranging multiple couplers along a single waveguide, each designed to couple out a

single wavelength channel of a WDM signal [112]. One limitation of using a standard

directional coupler response for wavelength switching or filtering is that the transmis-

sion spectrum is almost sinusoidal. To achieve a transmission band narrow enough to

filter a single WDM or DWDM channel, the coupler must be many coupling lengths

long, and even then there are transmission peaks close by on either side of the desired

frequency. Although it has been suggested that this property could be used for a chan-

nel interleaving device, it is by no means ideal for add/drop functions. Some alternative

designs for improved add/drop performance will now be considered.

Channel add/drop filters

The ability to add or remove one or more wavelength channels from a multiplexed signal

is an essential function in optical networks, and as such, all-optical add/drop filters are

one of the components required for future all-optical processing. Almost all proposed

PhC add/drop filters utilise a resonant cavity coupled to one or more waveguides to

improve the wavelength selectivity of the device. Light propagating along a waveguide


will couple to a nearby cavity if the frequency matches, or is close to the resonant

frequency of the cavity. Once in the cavity, the light can be dropped into a nearby

waveguide, or even radiated out of the plane for monitoring purposes or coupling into

an optical fibre. Schematics of these approaches are shown in Fig. 1.9.

forwarddrop port

backwarddrop port

transmissioninput port

l1

l1, 2 3l l, l2,l3

resonantcavityresonantcavity

drop port

l1, 2 3l l,

l1

l2,l3

resonantcavity



l1 radiativedrop port(out-of-plane)

l2,l3

resonantcavity

(a)

(b)

(c)

l1, 2 3l l,

Figure 1.9: Schematics showing three different add/drop coupler geometries. (a) and

(b) Two alternative methods for in-plane cavity coupled waveguides. (c) Out-of-plane

channel drop filter.

A geometry for in-plane channel dropping from one waveguide to a parallel wave-

guide as shown in Fig. 1.9(a) was first analysed by Fan et al., [113] who showed that two

resonant states with different symmetries are required for complete power transfer, and

furthermore, that these states must be degenerate. These conditions are necessary to

ensure light is not coupled into the backward drop port or reflected back into the input

waveguide. Various PhC based designs displaying these properties have been proposed

and demonstrated experimentally either using two distinct cavities or a single cavity

supporting two modes [83, 114, 115]. The use of multiple coupled cavities can also be

used to improve the spectral shape of the coupling response. A single cavity resonance

typically exhibits a Lorentzian response, whereas multiple coupled cavities can be de-


signed to have a flat-topped response with steeper sides, thereby reducing overlap with

nearby channels while improving the transmission within the desired channel [74,116].

The filter design shown in Fig. 1.9(b) works in a similar fashion to that in (a), although

it is not clear from the numerical results whether mode symmetry plays the same role

in determining the efficiency [83,117]. One clear advantage, however is the ability to in-

clude additional drop ports along the same input waveguide, thereby allowing multiple

channels to be dropped.

An alternative to coupling light into another waveguide is to radiate it out of the

plane, as illustrated in Fig. 1.9(c). The main drawback of this technique is that the

maximum theoretical coupling efficiency is only 50%, with the remainder coupling back

into the input waveguide in both the forward and backward directions [118, 119]. Un-

like the cavities required for designs (a) and (b), where radiation losses have to be

minimised, the cavities in this case must be intentionally designed to radiate, while still

exhibiting a sharp resonance. Experimental demonstration of a multiple wavelength

filter incorporating seven different cavities was recently reported with drop efficiencies

of up to 40% [120]. Thermo-optic tuning of a single cavity with a blue laser has also

been used to achieve dynamic control of the coupling frequency, with shifts of nine

times the resonance width demonstrated [121].

1.8 In-band applications

In this section we consider the possible applications of uniform PhCs operating at

frequencies where propagating states exist. As the band diagrams in Fig. 1.5 illustrate,

the Bloch modes can have complicated dispersion profiles, and these lead to a range

of unusual propagation effects, some of which are discussed in this section. Here the

behaviour is illustrated by considering the transmission of light from a homogeneous

dielectric into a PhC, or vice-versa, but interfaces between two PhCs or between a

waveguide and a PhC have qualitatively similar properties.

1.8.1 Coupling to Bloch modes

The coupling of plane waves from a dielectric to the Bloch modes of a PhC can be

understood in terms of momentum conservation or as a generalised version of Snell’s

law. Figure 1.10(b) illustrates a graphical construction of Snell’s law for an interface

between two dielectrics labelled 1 and 2, as shown in part (a) of the figure. A plane wave

of frequency ω propagating in medium 1 is characterised in k-space by a wavevector of

magnitude k1 = n1ω/c pointing in the direction of propagation. The set of all possible

wavevectors at this frequency can thus be represented by a circle of radius k1, indicated

by the solid green circle in the diagram. Similarly, the set of plane waves in medium 2

lie on a circle of radius k2 = n2ω/c, indicated by the dashed circle. These circles are

equifrequency contours (EFCs) of the conical band surfaces of media 1 and 2. If one

1.8. IN-BAND APPLICATIONS 31

or both of the media is anisotropic, the EFCs become elliptical, rather than circular,

since n depends on the propagation direction [122].

A refraction problem can be constructed for a given incident angle θi by first drawing

the incident wavevector into the previous diagram, indicated by the solid green arrow.

Momentum conservation requires the wavevector component parallel to the interface,

k‖ = k1 sin(θi) to be the same in both media, as represented by the dashed-dotted

line perpendicular to the interface. The wavevector of the refracted wave is given by

the intersection of this line with the EFC of medium 2, indicated by the dashed blue

arrow. Since the light is being transmitted through the interface, it is necessary to

choose the intersection corresponding to energy flowing away from the interface. The

energy flow is given by the group velocity vg = ∇k0ω(k0), which is a vector normal to

the EFC in the direction of increasing frequency. For an isotropic dielectric, the group

and phase velocities are parallel, and hence vg ‖ kpc. Snell’s law can be found using

simple geometry to show n1 sin(θi) = n2 sin(θ2).

A similar method is followed for an interface between a dielectric and PhC, but there

are three significant differences. First, it is clear from Fig. 1.5(b) that the EFCs of PhC

Bloch modes are not circular, so the refraction properties are in general anisotropic.

Second, the phase and group velocities are not necessarily in the same direction. Third,

the momentum conservation rule must be generalised to take account of the periodicity

in the direction along the interface, dint. Hence, the construction lines perpendicular

to the interface representing conservation of k‖ must be repeated for k‖ + m(2π/dint),

where 2π/dint is a reciprocal lattice vector parallel to the interface andm = 0,±1,±2, . . .

[123,124]. In the example shown in Fig. 1.10(d), only the first Brillouin zone is drawn,

but in general the surrounding zones must also be considered.

For a dielectric, the circular EFCs have a radius proportional to the frequency, so kpc

and vg are both directed radially outwards in the same direction. In a PhC, however, the

situation can be very different, as illustrated by the example in Fig. 1.10(d) which shows

an EFC for a band with a maximum at the Γ-point. In this case, the EFCs are almost

hexagonal and their size decreases with increasing frequency so that vg is directed

inwards from the contour. To ensure that energy flows away from the interface the

upper intersection point is chosen, resulting in the kpc and vg indicated. θpc is given by

the direction of vg. Coupling to PhCs is not always as simple as the example given here.

Depending on the shape of the band surfaces, it is sometimes possible for light to couple

into more than one part of a band, or into more than one band simultaneously, although

these situations are not desirable for most practical applications. Similarly, coupling

into higher-order Brillouin zones can sometimes occur for non-normal incidence, or

when the interface is not aligned to a symmetry axis of the crystal [125]

Although this construction method provides a simple and intuitive way of under-

standing where the light can go, like Snell’s law, it does not provide information about

transmission and reflection amplitudes at the interface. For dielectric interfaces, these

are characterised by the Fresnel coefficients [122]. In Chapter 2 we introduce a gen-

eralised form of Fresnel coefficient that characterises transmission between dielectric


k1

k2

k//

qi

q2

kx

ky

n2

n1

qi

q2

x

y

qi

n1

qpc

kpc

k1

vg

kx

qpc

qi

(b)(a)

(c) (d)

Figure 1.10: Diagrams of refraction at an interface between (a) two homogeneous

isotropic dielectrics and (c) a dielectric and a PhC. (a), (c) Ray diagram of incident,

reflected and refracted light for incidence from above. (b), (d): Construction diagrams

for refraction in k-space showing the incident wavevectors k1, refracted wavevectors k2

and kpc, and the group velocity vectors of the refracted light vg. Solid green circles in

(b) and (d) are the equifrequency contours of the medium 1 at the incident frequency.

Dashed blue curves are the EFCs of the dielectric medium 2 and the PhC.

and PhC or between two different PhCs. The issue of improving transmission from a

dielectric to a uniform PhC is considered in detail in Chapter 5.

1.8.2 p and q parameters

From Section 1.8.1 it is clear that θpc has a strong dependence on the local shape of

the EFC where it intersects the k‖ conservation line in the construction diagram. Thus,

changing the incident angle or the frequency can result in large variations of θpc. These

two effects are characterised by the parameters

p ≡ ∂θpc

∂θi

∣∣∣∣ω

, q ≡ ∂θpc

∂ω

∣∣∣∣θi

, (1.9)


which were first introduced by Baba et al. in the context of PhC superprisms [126].

In Sections 1.8.4 and 1.8.5 we consider two examples of unusual propagation behaviour

in uniform PhCs that require careful control of p and q. First, however, we consider

a more general property exhibited when light is coupled into a PhC, namely negative

refraction.

1.8.3 Negative refraction

Theoretical studies predict that materials with both a negative permittivity ε and per-

meability µ should exhibit a range of unusual properties including a negative refractive

index and anti-parallel group and phase velocities [127]. Furthermore, a material with

n = −1 has been shown to act as a perfect lens, capable of sub-wavelength resolu-

tion [128]. In contrast, the resolution limit of conventional optics in the far-field is

equal to the wavelength, regardless of the size and precision of the lens. Although

no naturally-occurring materials are known to exhibit negative refractive indices, com-

posite “meta-materials” with these properties have been proposed and demonstrated

experimentally for microwave frequencies [129].

Photonic crystals have been known for some time to exhibit unusual refraction

properties, including ultra-refraction, where a beam incident from air is refracted away

from the normal, implying a refractive index of n < 1 [130, 131]. Negative refraction

in PhCs, where the refracted and incident beams lie on the same side of the normal

was first observed experimentally in a 3D PhC [132] and since then has been studied in

detail in both theory and experiment [123,133–136]. Negative refraction is exhibited in

PhCs wherever the local curvature of the EFC is such that vg is directed away from the

normal on the same side as the incident beam. Moreover, if the EFCs are approximately

circular, and vg is directed radially inwards, then the PhC displays negative refraction

for all incident angles. For this situation, the concept of an effective refractive index

based on the radius of the EFC has been proposed [123,133].

Although PhCs are not true negative refractive index materials, they can still exhibit

some similar novel properties due to their negative refractive behaviour. A number

of theoretical studies have predicted that a finite slab of PhC that exhibits negative

refraction for all incident angles can focus an image to sub-wavelength resolution [137,

138]. This effect has been demonstrated experimentally with microwaves [139,140] but

has not yet been reported at telecommunications wavelengths.

1.8.4 Auto-collimation

Auto-collimation, or self-guiding, in photonic crystals allows a narrow beam to prop-

agate without any significant broadening or change in the beam profile. This effect

was first observed experimentally in a 3D PhC [141], where it was shown to occur at

inflection points on EFCs. A subsequent theoretical analysis provided a framework for

the design of auto-collimating 2D PhCs, and also demonstrated that PhCs designed for


collimation properties do not need to exhibit bandgaps [142]. This is an advantage of

many in-band applications: since they rely on the shape of the bands, rather than a

bandgap, many of the properties occur for lower refractive index contrasts, and hence a

larger range of materials become available for fabrication. Auto-collimation of an inci-

dent beam was first demonstrated experimentally in a PhC slab for light incident from

a waveguide. Relatively low-loss collimated beams were observed propagating through

the crystal for a range of angles [143].

Collimation occurs when an incident beam is coupled to a Bloch mode in a region

where the EFC is locally straight. Quantitatively, this can be expressed in terms of the

p-parameter (1.9), which indicates the sensitivity of θpc to variations in θi at a fixed

frequency. An incident Gaussian beam is not represented by a single wavevector, but

can be considered as a superposition of plane waves with a small angular spread centred

on the beam direction. Thus, an incident Gaussian beam in the construction diagram

of Fig 1.10 becomes a narrow cone of wavevectors that couple into a region of the EFC,

rather than a single point. Figure 1.11(a) illustrates the result of coupling to a region of

the EFC with strong curvature (large p); the plane wave components are refracted over

a range of angles ∆θpc and hence the beam broadens. In extreme cases the beam can

even be split into multiple beams [125]. Therefore, collimation requires a locally flat

region of the EFC where p ≈ 0, as shown in Fig. 1.11(b) [126]. A recent analytic study

of the p and q parameters has shown that every PhC band has regions where p = 0, but

in some cases this occurs over only a small range of incident angles [144]. If an EFC

has long straight sides, rather than just local points of inflection, then collimation is

possible for a range of incident angles. Square lattice PhCs exhibiting collimation for

incident angles in the range −54◦ ≤ θi ≤ 54◦ have been calculated [142].

Since the collimated beams do not require a waveguide or even a bandgap for lateral

confinement auto-collimation has been proposed as a novel method to achieve compact,

reconfigurable photonic circuits. Multiple beams can, in principle, pass through each

other without any resulting cross-talk [145, 146] and various circuit components such

as beam splitters [147, 148] and mirrors have been demonstrated experimentally [146].

Circuits based on auto-collimation would also be more tolerant to lateral coupling mis-

alignment than PhC waveguides which are typically only one or two lattice periods

wide. In Chapter 5 we consider the issue of coupling through dielectric–PhC inter-

faces, and present examples of high-efficiency coupling into a PhC designed to exhibit

auto-collimation for a range of incident angles.

1.8.5 Superprism effects

Superprism effects, where the angle of a transmitted beam is depends strongly on the

incident angle or frequency, have been proposed as a method of achieving ultra-compact

WDM multiplexing and demultiplexing in photonic circuits. Superprism behaviour

was first reported in 3D PhCs in a series of papers by Kosaka et al., where large beam

deflections were demonstrated for changes in both the incident angle and frequency [132,


k1

vg

kx

qi

Dqi

Dqpc

vg

kx

qi

k1

Dqi

Dqpc~0

ky

(a) (b)

ky

Figure 1.11: Refraction construction diagrams similar to Fig. 1.10(d), but for an incident

beam of finite width, and hence a spread of incident angles. The incident wavevector

k1 and group velocity vg of the refracted beam are shown. (a) Example when |p| � 1

showing the spreading of the incident beam by the sharply curved EFC. (b) Example

when the EFC is straight, and hence |p| ≈ 0, resulting in collimation of the refracted

beam.

149]. Wavelength separations of 1 nm were shown to produce a 3◦ deflection - around

500 times that obtained in conventional prisms. Further, the authors suggested that

a PhC only 190µm long would be sufficient to separate a WDM signal into individual

channels [149].

Although studies of PhC prisms were reported earlier than the work described above,

the parameters were chosen so that the EFCs were round. The prisms therefore dis-

played refraction properties similar to an isotropic medium, albeit with larger dispersion

and a corresponding moderate improvement in the wavelength resolution [150]. The

difference between this early study and those that followed was the use of non-circular

EFCs to enhance the sensitivity significantly for a small range of incident angles and

frequencies. The large deflections observed in Ref [149] were attributed to a sharply

curved section of the EFC close to the region being accessed by the incident beam.

Small changes in the input conditions moved the outgoing wavevector along the curve,

resulting in rapid variation in the group velocity direction, and hence the angle of the

transmitted beam. Similar results have been demonstrated in 2D PhC slabs, with

angular resolutions of up to 1◦/nm at telecommunications frequencies [151,152].

In order for a superprism to resolve two closely spaced wavelengths, the angular

separation must be converted into a spatial separation sufficient to resolve the two

beams. The required separation therefore depends on the beam width; wider beams


require greater separation. Unfortunately the sharp curves in the EFC that produce

strong angular resolution also cause spreading of narrow beams, as was discussed in

Section 1.8.4, so a balance is required. To quantify these two competing effects, Baba

et al. [126] introduced the collimation and angular resolution parameters, p and q re-

spectively (1.9) and proposed that a realistic measure of resolution should be the ratio

r = q/p, not simply the value of q as suggested in earlier superprism discussions [149].

This alternative measure led to the conclusion that PhCs of more than 1 cm2 would be

required to resolve the channels of a DWDM network, a similar size to conventional

silica-based arrayed waveguide grating filters.

One reason for such long PhCs is the requirement that beams be separated spatially

before leaving the PhC. Such a condition is only necessary if the input and output

interfaces are parallel since in this case the separated wavelengths all leave the PhC in

the same direction. This can be seen by considering again the diagram in Fig 1.10(d).

The construction lines are unchanged for coupling out of the crystal, and thus the

transmitted beams are composed of the same wavevectors as the incident beam. A

more compact superprism is possible if the transmitted beams maintain their angular

separation after leaving the PhC. One way to achieve this is to angle the second interface

relative to the first, thereby changing the k‖ conservation condition for the output

coupling. This approach has been studied in two different geometries in Refs [153]

and [154].

1.9 Computational methods

In Section 1.5 the basic theory of band structures was discussed, and the eigenvalue

equations (1.1) and (1.2) that determine the Bloch modes were introduced. Although

the solutions and their significance were considered, we have not yet described how the

equation should be solved. Furthermore, aside from identifying the bandgaps, the band

structure of an infinite periodic lattice provides little information about transmission

and reflection or many other aspects of PhC operation. No single formulation provides

the computational tools necessary for an efficient and rigorous study of all photonic

crystal devices. Hence, many different theoretical methods have been developed, some

of which are discussed in this section.

The most appropriate numerical method for a given problem depends largely on the

structure being studied and the data that is required. Some methods provide versatility

at the expense of efficiency and/or accuracy, while others are highly efficient but can

only deal with a limited range of geometries or specific problems. Another consider-

ation is whether a method solves the problem in the time or frequency domain. In

time domain methods, the solution is calculated as a function of time, and can thus be

used to track the propagation of a pulse or observe other transient behaviour. This is

especially important when studying many nonlinear effects for which time dependent

material properties must be included explicitly. A frequency spectrum can be calcu-

1.9. COMPUTATIONAL METHODS 37

lated by launching a short pulse and taking a Fourier transform of the time response,

however long simulations are required to obtain the steady-state response of high-Qstructures. Frequency domain methods on the other hand, calculate the solution as a

function of frequency, which makes them more appropriate for calculating steady-state

frequency responses and band structures, and for studying systems with strongly dis-

persive materials. There is also considerable overlap between the application of these

two methods; while the frequency spectra can be obtained by taking a Fourier trans-

form of a time response, it is also possible to calculate the time response of a device if

the full frequency response is known.

Finite-Difference-Time-Domain (FDTD) methods are the most general and widely-

used tool for PhC calculations as they are flexible and can handle almost any geom-

etry [155]. The method involves the solution of Maxwell’s equations via numerical

integration on a discretised grid in both time and space. Since the grid must be fine

enough to resolve the smallest feature in the problem, dealing with extended PhC struc-

tures can be very demanding of computational time and memory. FDTD can be used

to simulate finite or periodic structures through the appropriate choice of absorbing or

periodic boundary conditions.

Another numerical propagation technique, this time in the frequency domain, is the

beam propagation method (BPM) which is more commonly used for studying optical

fibres and dielectric waveguides [156,157], but has also been applied to PhCs [158]. This

approach involves the launch of an initial field distribution u(x, y, z = 0) which is then

propagated numerically in the z direction. The method was initially implemented for

forward propagation only and is most efficient for structures that vary slowly along the

propagation direction. Adaptations of the original method have allowed bi-directional

propagation and improved the versatility, but BPM can not deal with the same scope

of problems as FDTD methods.

Also in the frequency domain, plane wave methods are the most appropriate for

band structure calculations and other problems that can be modelled as periodic sys-

tems [159]. In order to solve the eigenvalue form of Maxwell’s equation (1.2), the

refractive index profile and the electric and magnetic fields are first decomposed into

their respective Fourier components. The expansions are truncated to a finite number of

plane wave orders, and substituted into Eq. (1.2), resulting in a matrix eigenvalue equa-

tion that is solved numerically. Plane wave methods are not limited to band structure

calculations of simple lattices. For instance, waveguides and cavities can be simulated

by creating a periodic lattice of supercells, each one containing an array of cylinders

with the desired defect introduced. If the supercell is sufficiently large that defects in

nearby cells do not interact significantly, then the solutions to the eigenvalue problem

will approximate those for a single isolated defect. Waveguide dispersion curves are

commonly calculated in this way. Supercells are also used when calculating the 3D

band structure of a 2D PhC slab; the slab and surrounding cladding are replicated

periodically in the vertical direction with enough cladding between each slab to ensure

negligible coupling. Although this method is useful for calculating modes bound within


the slab, cladding modes can couple to nearby supercells, and are therefore not an

accurate approximation of the modes of an isolated slab. As with all supercell treat-

ments, the plane wave method is not able to calculate radiation losses, since every cell

is surrounded by others, and any light leaking out of one supercell passes into the ones

nearby and vice-versa.

A range of alternative frequency domain methods have also been developed, with

each one using a different set of orthogonal basis functions to represent the fields and

the index profile of the PhC. Although the plane wave method is versatile and can be

used to represent many different structures, it is not efficient for calculating complex or

tightly bound defect modes as this requires many thousands of plane waves. One way

to improve this is to expand the fields using functions that more closely represent the

modal profiles, and therefore require fewer terms in the expansion. One approach that

is most suitable for studying localised modes is based on Wannier functions which are

derived from the Bloch modes of the uniform PhC and therefore exhibit many of the

symmetry properties of the underlying lattice. The calculation of Wannier functions

from the Bloch modes is non-trivial, but when optimised, they provide a highly-efficient

basis set for PhC waveguide and defect calculations [160].

For PhCs consisting of circular cylinders, one of the most efficient modal techniques

is the multipole method [161–163]. In this case, the basis functions are cylindrical

harmonics centred on each of the cylinders, thereby exploiting the circular geometry

and allowing the boundary conditions to be expressed analytically. For this reason,

the multipole method is accurate and efficient when studying structures with very high

index contrasts or absorbing materials. Multipole methods can be applied to either

finite clusters of cylinders or periodic systems, with the former allowing radiation losses

and edge effects to be investigated.

Transfer matrix methods refer to a large class of techniques in which the structure

to be modelled is considered as a stack of individual layers, where transmission and

reflection from each layer can be characterised by scattering matrices. Transmission

through the whole stack is calculated by multiplying the scattering matrices together in

a recursive manner. Thus, if there are multiple identical layers, the scattering matrices

need only be calculated once, and can then be reused every time the layer appears

in the stack. Transfer matrix methods are a common technique for modelling thin-

film stacks, where the layer-by-layer approach is an obvious choice [164], but they can

also be applied to PhCs, which can often be treated as a periodic stack of diffraction

gratings [165, 166]. Calculating the band structure of a uniform PhC thus requires

only the knowledge of the scattering properties of a single layer. Various numerical

methods can be used to calculate the scattering matrices, including differential Fourier

methods [167,168] and finite-element methods [165].

The computational method chosen for most of the numerical calculations in this the-

sis is a recently developed 2D Bloch mode scattering matrix method [169, 170], which

is highly efficient for studying 2D PhC structures consisting of distinct segments. The

formulation is based on transfer matrix techniques where the fields are expanded in

1.10. DISCUSSION AND THESIS OUTLINE 39

terms of the natural Bloch mode basis of the PhCs. As we discuss in Chapter 2, the

significant advantage and novelty of this approach is the derivation of Bloch mode scat-

tering matrices which play the role of generalised Fresnel coefficients to describe the

reflection and scattering of light at a PhC interface. This allows many complicated

PhC structures to be treated using methods developed for thin-film optics, and also

provides a systematic approach for deriving semianalytic models of many structures.

The numerical implementation of the Bloch mode method used here also takes advan-

tage of the multipole method for grating calculations, and as a result gains some of

the additional benefits of this highly efficient and accurate technique for dealing with

circular cylinders.

1.10 Discussion and thesis outline

We have reviewed in this chapter some of the unique properties of photonic crystals and

the opportunities they present for the design of efficient, compact devices for a range of

photonics applications. As the most recent experimental results demonstrate, photonic

crystal research is entering a new phase. Fabrication techniques can now provide the

quality and control of structural detail necessary for optimisation and fine-tuning of

device designs, and as a consequence, modelling and simulation of new designs will

be more relevant than ever. Given the inherently complex nature of photonic crystal

structures, there is little doubt that they will remain more difficult to fabricate than

alternative integrated photonics technologies such as high-index SOI based components.

Therefore, to compete seriously with more conventional technologies, PhC devices must

either provide functions that are not easy to achieve using other methods, or perform

the same functions in a cheaper, more efficient and compact fashion that outweighs the

additional fabrication effort. Therefore, the ultimate goal is to demonstrate compact,

integrated photonic devices that satisfy both of these criteria. In the following chapters

we present results and techniques that may contribute towards the future realisation of

this goal.

Chapter 2 provides an overview of the Bloch mode scattering matrix method which

is the main computational tool that we use in later chapters. This method is not only

an efficient and accurate 2D numerical tool, but can also be used to derive semianalytic

treatments for many structures. We describe first the general formulation of the method,

then discuss several issues regarding the application of the method to specific structures

considered in this thesis.

In Chapter 3 we introduce a new class of device in which directional coupling and

resonant reflection effects are combined. The characteristics of these devices vary con-

siderably with the geometry, which allows us to demonstrate a high-Q resonant filter,

a wide bandwidth, high-transmission Y-junction and a coupled cavity delay line. The

devices are analysed using accurate 2D numerical results obtained from the Bloch mode

method, and semianalytic models that provide insight into the underlying physics.


The concepts of mode coupling and resonances are extended in Chapter 4, where

we study the effects of mode recirculation in Mach-Zehnder interferometers (MZIs)

and show that a perfectly reflecting cladding such as a photonic crystal can result in

manifestly different transmission characteristics compared with conventional, dielectric

waveguide-based MZIs. We derive a semianalytic model for this class of device and

demonstrate that rather than degrading the performance, recirculation can provide an

enhanced response. Numerical results are presented for both rod- and hole-type PhC

structures to verify these effects.

Chapter 5 concerns an entirely different type of resonance and mode coupling be-

haviour that we have identified in uniform rod-type PhCs. We show that scattering

resonances in the high-index rods provide a mechanism for light to be coupled effi-

ciently into the Bloch modes of the PhC for a wide range of angles. These results

may lead to more efficient designs for in-band applications such as superprisms and

auto-collimation devices. We demonstrate one such example by optimising a structure

to exhibit both auto collimation and very low reflectance for a wide range of incident

angles, and confirm the 2D results with rigorous 3D simulations.

The thesis concludes with a brief outlook on the future of photonic crystal devices

and a discussion of out-of-plane losses and other issues that must be considered in order

for the devices we have proposed to be demonstrated experimentally.

Chapter 2

Bloch mode scattering matrix

method

This chapter provides an overview of the Bloch mode scattering matrix method – the

main numerical tool used in this thesis. It is not the intention to provide a full derivation

of the formulation, but rather to present the details of the method that relate directly

to the results that follow in Chapters 3–5. The majority of this chapter is based on the

work of Botten et al. in Refs. [169] and [171], with the exception of Sections 2.4 and 2.5

in which we discuss a number of issues relating to the application of the Bloch mode

method to the specific structures of relevance to this thesis.

2.1 Introduction

While the calculation of 2D PhC band structures has become routine and can be per-

formed with a minimum of computational expense, studying propagation in more com-

plicated PhC structures remains a challenge. The main computational method used

throughout this thesis is the recently developed Bloch mode scattering matrix method

(BMM), which is not only an efficient 2D numerical tool but also provides a systematic

and intuitive approach for deriving semianalytic models of many structures. We use

both of these features here to gain insight into the underlying physics of resonances and

mode coupling behaviour in photonic crystal devices.

The computational efficiency of the method is a result of two key techniques that

are combined in the BMM. The first of these is the transfer matrix method, which has

been used for many years in the field of thin-film optics [2], and more recently has been

applied to PhC problems [165,166,172]. The second technique is the use of the natural,

Bloch mode basis to expand the fields.

To apply transfer matrix methods to a complex structure, it is necessary to break

the structure down into basic units – layers that can be characterised relatively easily

in terms of their transmission and reflection properties. In a conventional thin-film

stack the basic layer is typically a homogeneous dielectric material. The interfaces

41

42 CHAPTER 2. BLOCH MODE SCATTERING MATRIX METHOD

between layers are characterised by Fresnel transmission and reflection coefficients while

propagation within a layer is described by plane waves. When applied to a periodic

geometry such as a slab of PhC consisting of periodic gratings as shown in Fig. 2.1,

the fields are expressed in terms of the Bloch modes of the slab, and we can derive

generalised Fresnel coefficients (in this case matrices) that describe transmission and

reflection of these modes at the interfaces of the slab. The derivation of these Fresnel

matrices is the main conceptual advance of the method we describe here, as it allows

many relatively complicated PhC structures to be treated using the framework of thin-

film optics.

In order to represent accurately the fields in a photonic crystal device using a Bloch

mode expansion, it is necessary to include all of the modes, both propagating and

evanescent. Recall from Section 1.5.1 that the standard technique for solving Eq. (1.1)

is to calculate the eigenvalues (ω/c)2ε as a function of the real Bloch vector k0, which

leads to the band structure. The approach described here involves an alternative eigen-

value equation which we solve at a fixed frequency to yield the Bloch vectors. If the

frequency lies in a bandgap, then all of the Bloch wavevector solutions are complex,

corresponding to evanescent Bloch states that can be excited at an interface but which

decay exponentially within the PhC. For frequencies where propagating modes exist,

there are a finite number of solutions for which k0 is real, corresponding to these modes,

and an infinite number of solutions for which k0 is complex, again corresponding to

evanescent states.

Bloch modes can be expanded as a superposition of plane wave orders, where both

propagating and evanescent plane waves are required to represent the modes accurately.

Hence, propagating Bloch modes may have evanescent plane wave components and

evanescent Bloch modes may have propagating plane wave components. While the

evanescent Bloch states carry no energy through an infinite structure, they can couple

energy through thin PhC barriers via evanescent coupling, much like frustrated total

internal reflection in conventional optics. The evanescent states are important when

matching fields at an interface between two different PhCs or between a homogeneous

medium and a PhC. In practice, the field expansions are truncated to include only a

finite number of modes where the truncation order depends on the specific geometry and

the required accuracy. For many structures, the essential physics is described accurately

by the propagating modes alone, in which case the evanescent Bloch modes can be

dropped from the treatment, resulting in simple expressions involving only a few modes.

We provide numerical examples of this asymptotic approximation in Section 3.3.2.

A typical structure that we consider is illustrated in Fig. 2.1, consisting of multiple

segments, M1, M2, . . . , MN , each of which is formed by a stack of identical layers with

a transverse period of Dx. Media M1 and MN are semi-infinite in the vertical direction

and form the input and output domains of the problem. While the example shows all

segments containing photonic crystal material, they may also contain a homogeneous

medium such as a dielectric or free-space. Each individual layer is treated as a diffraction

grating with a period Dx corresponding to the size of the supercell. A uniform 2D PhC

2.1. INTRODUCTION 43

M1

M3

M2

M4

layer

stack of L2 layers

(n = 0)

(n = L2)

Dx

x

yy = y1

y = y2

Figure 2.1: Schematic of a photonic crystal structure formed by joining segments M1−MN , each consisting of identical grating layers (here N = 4). All layers must have the

same supercell period Dx, but the positions and properties of the cylinders within a

layer can all be different. Segments M2 −MN−1 contain Lm layers, segments M1 and

MN are semi-infinite. Layer interfaces within each segment are numbered n = 0 to

n = Lm.

structure is formed if the grating consists of a periodic array of cylinders with period d,

and Dx is an integer multiple of d. Functional elements such as waveguides are created

in the supercell by removing or modifying cylinders from the grating as shown. When

operated in the bandgap of the bulk photonic crystal, the supercell must be sufficiently

large to effectively isolate the elements in one cell from those in neighbouring cells. For

many structures we find that a separation of at least 10d is sufficient for coupling effects

to be neglected, but more may be required when modelling strongly resonant structures

such as those studied in Chapter 3. We discuss convergence with supercell size in more

detail in Section 2.4.

In Section 2.2 we describe the steps required to obtain the Bloch modes for a typical

segment in Fig. 2.1 from the scattering properties of a single grating layer. We then

proceed to couple segments together in Section 2.3, considering initially a single interface

between two semi-infinite segments which allows the derivation of generalised Bloch

mode Fresnel matrices. Calculation of the complete multi-segment structure follows

in direct analogy to the modelling of multi-layer thin-film stacks via recursion. In

Sections 2.4 and 2.5 we discuss several practical issues relating to the application of the

Bloch mode method to modelling PhC devices.


2.2 From a grating to a PhC

In this section we outline the derivation of the Bloch modes of a typical PhC segment

in Fig. 2.1 formed from a stack of identical periodic grating layers. In the most general

case, each layer is offset from the previous one by δx in the horizontal direction and δyin vertical direction, as shown in Fig. 2.2. The choice of these offsets determines the

symmetry properties of the lattice as we discuss in Section 2.2.2.

2.2.1 Grating characterisation

Since each layer in a segment is a periodic grating with period Dx, both the layers and

the segments that they form are coupled together by plane wave diffraction orders s,

the directions θs of which are governed by the grating equation

k sin θs = αs = α0 +2πs

Dx

, s = 0,±1,±2, . . . (2.1)

where α0 = k0x is the component of the Bloch vector along the direction of periodicity

of the gratings, and k = 2π/λ is the wavenumber in the background medium. The

integers s in Eq. (2.1) run over a finite set of propagating orders, and an infinite set

of evanescent orders. Between each grating layer the background medium is taken

to be homogeneous, and so the fields can be represented in the plane wave basis by

an expansion over all plane wave orders. The response of the grating to an incident

field is characterised by the plane wave reflection and transmission scattering matrices

R, T and R′, T

′, with the two pairs corresponding to incidence from above and below,

respectively. For example, the element Rpq specifies the reflected amplitude in plane-

wave order p due to unit amplitude incidence from above in plane-wave order q. The

tilde notation is used here to differentiate between the plane-wave scattering matrices

for a single grating and those of a PhC segment, which we introduce in Section 2.3.

The grating scattering matrices can be calculated using a variety of techniques,

including integral equation methods, differential Fourier methods and finite-element

methods [165, 167, 168, 173–175]. The BMM implementation described here uses the

multipole method [176], which is one of the most efficient numerical techniques for

dealing with circular cylinders. The main disadvantage of the multipole method is that

individual grating layers cannot interpenetrate. Although this limits somewhat the

geometries that can be studied, in situations where it is valid, the multipole method

provides many significant advantages over other techniques. For example, the energy

conservation and reciprocity relations are embedded in the multipole formulation and

these features are inherited by the scattering matrices and the Bloch modes. This

issue has some important consequences regarding the verification of the method and

convergence of solutions, as we discuss further in Section 2.4.

2.2. FROM A GRATING TO A PHC 45

e2f1

+

f2

-

f1

-

f2

+

y

x

dx

dy

Figure 2.2: Geometry of a single grating layer in a periodic stack defined by the lattice

vector e2 = (δx, δy). The plane wave field coefficients above the grating are given by f−1and f+

1 , corresponding to upward and downward propagating components respectively.

Similarly, the fields below the grating are given by f−2 and f+2 .

2.2.2 Basic PhC lattices

Here we consider briefly the steps required to form a segment of square or triangular

PhC from a stack of identical grating layers with a supercell period Dx, containing

multiple cylinders. The cylinders in each supercell are typically arranged with a local

period d, and Dx is chosen to be an integral multiple of d so that dislocations are not

formed at the edges of the supercell.

The diagrams in Fig. 2.3 illustrate how a PhC segment is formed from a stack of

gratings where each successive grating is shifted by δx along the x-axis and δy along

the y-axis. Stacking gratings of local period d with δx = 0 and δy = d for example,

results in a square lattice oriented with the interface parallel to the Γ − X direction

as shown in Fig. 2.3(a). A straight waveguide is created simply by removing one or

more cylinders from each layer. As mentioned in Section 2.2.1, in order to use the

multipole method for calculating the grating scattering matrices, the layers must not

interpenetrate. Therefore we are restricted to cylinder radii r < δy/2. This is not a

problem for the square array in Fig. 2.3(a) since the gratings do not begin to penetrate

until the cylinders themselves touch at r = d/2, but as we see in the next examples,

the interpenetration can limit the study of other lattice types and orientations. We

note again that this is a limitation of the multipole grating formulation, and not a

fundamental limitation of the Bloch mode method.

The square lattice in (a) can also be oriented with the interface parallel to the Γ−Mdirection by starting with a grating layer of period d′ =

√2d, and stacking them with

δx = d′/2 =√

2d/2 and δy = d′/2 =√

2d/2 as shown in Fig. 2.3(b). In this case,

interpenetration of layers restricts the cylinder radius to r <√

2d/4. Observe that

the waveguide formed by removing a single cylinder from each layer is identical to the

waveguide in Fig. 2.3(a).

Figures. 2.3(c) and (d) show triangular lattices of period d oriented respectively with

interfaces parallel to the Γ−K and the Γ−M directions. The example in (c) is formed


G

M

X

dG M

Xd

d’

G

KM

d

d

(a) (b)

(c) (d)

G

K

M

d’

d

dx

dy

Figure 2.3: Diagram showing four PhC lattices with period d formed by stacking in-

dividual grating layers of period d′ with a horizontal offset δx and vertical offset δy as

given in the text. The waveguides are formed by removing one cylinder from each layer.

(a) Square lattice oriented with the interface parallel to the Γ−X direction. (b) Square

lattice with interface parallel to Γ−M . (c) Triangular lattice with interface parallel to

Γ−K. (d) Triangular lattice with interface parallel to Γ−M .

from gratings of period d with δx = d/2 and δy =√

3d/2 and the example in (d) has

a grating period of d′ =√

3d, stacked with δx = d′/2 =√

3d/2 and δy =√

3d′/6 = d/2.

Hence the maximum cylinder radii to avoid interpenetration in these two cases are

respectively r =√

3d/4 and r = d/4.

The horizontal offset δx is included in the formulation by applying the translation

operator Q = diag[exp(iαsδx/2)] to the original grating scattering matrices [177]. This

translation operator adjusts the phase of each scattered plane wave order to account for

the relative shift between the upper and lower interfaces. For example, the plane-wave

transmission matrix T becomes QTQ, and the reflection matrix R becomes QRQ−1.

A similar translation operator can also be defined for the vertical offset between each

layer. In the following treatment, it is assumed that the plane-wave scattering matrices

have already been transformed in this way.

2.2.3 Bloch mode calculation

On the interface between two grating layers in a given segment, the plane wave field ex-

pansions are represented by vectors of plane wave amplitudes f− and f+, corresponding

respectively to the downward and upward propagating plane wave orders of Eq. (2.1).

Both propagating and evanescent orders must be included to represent the fields accu-

2.2. FROM A GRATING TO A PHC 47

rately. The fields f1 and f2 on either side of the grating in Fig. 2.2 are related via a

transfer matrix T which can be derived from the plane wave scattering matrices such

that

f2 = T f1, where fj =

[f−jf+j

], (2.2)

and

T =

[T− R

′T′−1

R R′T′−1

−T′−1

R T′−1

]. (2.3)

The periodic stack of gratings must also satisfy a 1D Bloch condition, f2 = µf1, where

µ = exp(−ik0.e2), k0 is the Bloch vector and e2 is the basis vector of the lattice

defined in Fig. 2.2. Combining the Bloch condition with Eq. (2.2) leads to the eigenvalue

equation

T f = µf . (2.4)

In practice, the form of the transfer matrix given in Eq. (2.3) results in a numerically

unstable eigenvalue equation, so Eq. (2.4) must be expressed in an alternative, more

robust form before the eigensolutions can be calculated [178, 179]. We do not discuss

the details here.

2.2.4 Bloch mode partitioning

The solutions to Eq. (2.4) yield the Bloch modes, expressed as a superposition of plane

waves, and their associated eigenvalues, µ. In practice, the plane wave field expansions

must be truncated to a finite number of orders, running from s = −Npw to s = Npw,

resulting in transfer matrices of even order with dimension 2(2Npw + 1) × 2(2Npw +

1). This allows the Bloch modes to be partitioned into equal numbers of downward

and upward modes. The details of the partitioning are given in Ref. [169], but for

the purposes of this discussion it is sufficient to note that the non-propagating states

with |µ| 6= 1 are classified according to the direction in which they decay, while the

directionality of the propagating modes (|µ| = 1) is inferred from the energy flux normal

to the grating. Hence the non-propagating modes with |µ| < 1 are classified as forward

(downward) states, while those with |µ| > 1 decay in the opposite direction, and are

thus classified as backward (upward) states. With the modes partitioned in this way,

the transfer matrix can be written in its diagonalised form

T = FLF−1, (2.5)

which includes the whole family of eigenvalue equations T F = FL from Eq. (2.4).

Here, F is a matrix whose columns are the eigenvectors of T , and L is a diagonal ma-

trix containing the corresponding eigenvalues µ. When partitioned using the approach

described above, F and L have the forms

F =

[F− F ′

−F+ F ′

+

], L =

[Λ 0

0 Λ′

]. (2.6)


The columns of the matrices F∓, F ′∓ are formed from the downward and upward plane-

wave components of the eigenvectors for the downward and upward Bloch modes respec-

tively, and Λ and Λ′ are diagonal matrices containing the corresponding eigenvalues µj.

The columns of F are scaled so that physical properties such as reciprocity and energy

conservation, and modal orthogonality are represented in a physically normalised form.

The matrices F∓ and F ′∓ provide a direct mapping between the Bloch mode basis

and the plane wave basis. We can thus treat propagation within a segment in terms of

the Bloch modes of that segment, but transform the fields into the common plane-wave

basis to match the fields at interfaces between different segments. This is the topic of

the next section.

2.3 From a PhC segment to a PhC device

Propagation of light through a complex PhC structure such as that illustrated in Fig. 2.1

can be characterised by two basic processes. The first is propagation within a segment

of identical gratings and the second is reflection and transmission at a single interface

between two different segments. In the BMM, the first of these is described in terms of

the Bloch modes of that segment, while the second is characterised in terms of Bloch

mode transmission and reflection matrices that play the role of generalised Fresnel

coefficients. We note that the interface could instead be characterised by plane-wave

scattering matrices, but as we demonstrate in the remainder of this chapter, the Bloch

mode scattering matrices allow photonic crystal structures to be treated using the

formalism developed for thin-film optics.

2.3.1 Propagation in a single PhC medium

Consider first the propagation of light in a single finite segment of PhC consisting of

L grating layers as illustrated in Fig. 2.4. Light incident on the segment from above,

whether from free-space or another PhC, excites downward propagating Bloch modes

at the interface with amplitudes given by the vector c−. Similarly, light incident from

below excites upward propagating Bloch modes, c+. Hence, at interface n inside the

segment, the downward and upward Bloch mode amplitudes are given by c−(n) = Λnc−and c+(n) = Λ′ −(L−n)c+, where the Λ matrices introduced in Section 2.2.4 describe

propagation through a single layer, and hence Λn describes propagation through n

layers. Similarly, Λ′ is describes propagation in the backward direction. The plane-wave

expansion coefficients f(n) for fields at interface n are obtained via a simple change of

basis using F∓ and F ′∓ which gives

f(n) =

[f−(n)

f+(n)

]=

[F−F+

]c−(n) +

[F ′−

F ′+

]c+(n). (2.7)

We use this plane wave expansion in Sections 2.3.2 and 2.3.3 in order to match the

fields between two different segments with different Bloch mode bases.

2.3. FROM A PHC SEGMENT TO A PHC DEVICE 49

n = 0

n = L

n

L - n

c-c-(n) = Λnc-

c+(n) = ΛL-nc+c+

Figure 2.4: Downward and upward Bloch mode amplitudes c−(n) and c+(n) defined

at interface n in a segment of length L in terms of Bloch modes sourced respectively

from the interfaces above (c−) and below (c+) the segment. Each grating layer in the

segment is identical.

Although the full complement of propagating and evanescent Bloch modes may be

required to match the fields at input and output interfaces, the evanescent modes excited

at the top interface decay inside the segment as µn, since |µ| < 1 (see Section 2.2.4), and

similarly for evanescent fields excited from below. Thus, in a long segment (typically

L > 5) that supports both propagating and evanescent modes, the propagating modes

dominate and in many cases the evanescent states can be dropped from the calculation.

Examples of this approximation are given in Section 3.3.2 and [180], and a rigorous

asymptotic analysis is described in Ref. [169]. Note that while this simplification reduces

the number of Bloch modes in the analysis, a sufficiently large set of both propagating

and evanescent plane wave orders is still required to represent each of the propagating

Bloch modes accurately.

2.3.2 Interfacing two semi-infinite segments

In this section we consider the reflection and transmission of Bloch modes at an interface

between two semi-infinite media, M1 and M2, as shown in Fig. 2.5, and derive the PhC

analogues of Fresnel reflection and transmission coefficients. These generalised Fresnel

coefficients are Bloch mode scattering matrices with the domain and range defined by

the Bloch modes of the input and output media.

The interface shown in Fig. 2.5 is a fictitious line between the bottom grating layer

of M1 and the top grating layer of M2. In M1, the downward and upward mode Bloch

mode amplitudes at the interface are given by the vectors c−1 and c+1 respectively, and

similarly in M2. Since the interface lies in the homogeneous medium between layers


c2

+c2

-

c1

+c1

-

M1

M2

Figure 2.5: Diagram of a single interface between two semi-infinite media, M1 and M2.

The Bloch mode amplitudes at the interface are given by the vectors c∓1 in M1 and c∓2in M2.

there is no change in material properties and thus the fields and their upward and

downward components must be continuous across it. As the Bloch mode expansions

in two different media cannot be directly compared, the fields on either side of the

interface are first transformed into the common plane-wave basis. The field continuity

relationship therefore becomes[F−

1

F +1

]c−1 +

[F−′

1

F +′

1

]c+

1 =

[F−

2

F +2

]c−2 +

[F−′

2

F +′

2

]c+

2 , (2.8)

where the F∓j are the partitioned eigenvector matrices introduced in Section 2.2.4,

defined in medium j.

The generalised Fresnel (Bloch mode) transmission and reflection matrices are de-

fined by the relations

c+1

def= R12c

−1 + T21c

+2 (2.9)

c−2def= T12c

−1 + R21c

+2 , (2.10)

which express the Bloch modes propagating away from the interface in each region in

terms of the incident Bloch modes. The Bloch mode scattering matrices are set in sans

serif typeface to distinguish them from the plane-wave scattering matrices, which are

presented in standard Roman typeface throughout. Solving Eq. (2.8) and expressing the

outgoing fields in terms of the incoming fields then leads to the following expressions

for the Bloch mode reflection and transmission matrices:

R12 = (F +′

1 )−1(I −R∞

2 R∞′

1

)−1

(R∞2 −R∞

1 ) F−1 , (2.11a)

T12 = (F−′

2 )−1(I −R∞′

1 R∞2

)−1 (I −R∞′

1 R∞1

)F−

1 , (2.11b)

where

R∞j = F +

j (F−j )−1, R∞′

j = F−′

j (F +′

j )−1. (2.11c)


Similar expressions may be obtained for R21 and T21. In the most general treatment, ad-

ditional Bloch mode matrices are required for each propagation direction. For instance

T12 describes forward (downward) propagation from M1 to M2 in while T′12 describes

backward (upward) propagation from M1 to M2. To simplify the notation, we assume

throughout this thesis that the segments are numbered in order from top to bottom as

in Fig. 2.1, and thus the propagation direction is implicit in the Bloch mode matrix

subscripts.

In Eq. (2.11c), R∞1 denotes the plane-wave reflection matrix that characterises re-

flection of a plane-wave field incident from above on a semi-infinite slab of M1. Cor-

respondingly, R∞′

1 is the reflection matrix for plane-wave incidence from below. An

arbitrary lateral shift can be introduced at the interface by applying a translation oper-

ator Q of the same form as in Section 2.2.1 to the plane-wave scattering matrices of one

or both segments. This operation is used in Section 3.3.1 to study transmission through

a waveguide dislocation. In Chapter 5 we study the coupling of light from free-space

into the Bloch modes of a uniform PhC. For this application, R∞ proves particularly

useful as it allows the coupling properties to be separated completely from the effects

of finite-thickness PhCs. The Bloch modes are normalised so that the reflectance of

a propagating plane wave order p into reflected order q is simply given by the square

modulus of the corresponding element of R∞, |R∞qp|2.

2.3.3 Propagation through three segments

We now combine the results of the previous two sections to consider propagation from

M1 to M3 in the three segment structure illustrated in Fig. 2.6, where segments M1 and

M3 are semi-infinite and segment M2 consists of L grating layers. This analysis reveals

clearly the similarity of the Bloch mode method to the standard approach in thin-film

optics.

In the most general case, each interface in Fig. 2.6 is described by four Fresnel

matrices, Rij, Tij, Rji, and Tji, which can be calculated from Eqs. (2.11). Here, the

subscript ij corresponds to propagation from segment Mi to Mj through a single in-

terface. Transmission from M1 to M3 is characterised by an incident field composed

of downward propagating Bloch modes with amplitudes c−1 , a reflected field of up-

ward propagating modes c+1 = R13c

−1 , and a transmitted field of downward propagating

modes, c+3 = T13c

−1 in M3.

In segment M2, the fields have both downward and upward Bloch mode components,

c−2 and c+2 , originating from the top and bottom interfaces respectively, as shown in

Fig. 2.6. The field matching equations at the two interfaces are then

c+1 = R12c

−1 + T12Λ

Lc+2 , c−2 = T12c

−1 + R21Λ

Lc+2 , (2.12a)

c+2 = R23Λ

Lc−2 , c−3 = T23ΛLc−2 , (2.12b)

where Eqs. (2.12a) and (2.12b) correspond to field matching at the top and bottom

interfaces respectively. Note that Λ−1 has been substituted here for Λ′, a result due


c2

+

c2

-

c1

+c1

-c1

+c1

-M1

c3

+c3

-

M2

M3

L

LL +c2

LL -c2

Figure 2.6: Diagram of a three-segment structure consisting of semi-infinite segments

M1 and M3 above and below segment M2, which is formed from L grating layers.

to the forward/backward pairing of the Bloch mode eigenvalues that holds for all PhC

lattices, as shown in Ref. [169]. Expressions for T13 and R13 are obtained by solving

these four equations, which gives

R13 = T21ΛLR23Λ

L(I − R21Λ

LR23ΛL)−1

T12 + R12, (2.13a)

T13 = T23ΛL

(I − R21Λ

LR23ΛL)−1

T12. (2.13b)

Given that the BMM is cast in a form that resembles thin-film optics, it is not

surprising that Eqs. (2.13) are generalisations of the Airy formulae for a Fabry-Perot

interferometer (see for example Ref. [122], Chapter 7). The term (I − R21ΛLR23Λ

L)−1

in both expressions encapsulates the multiple reflections within segment M2, while the

remaining terms correspond to reflection and transmission at the interfaces when the

light enters and leaves M2.

2.3.4 Propagation through N segments

With the results of Sections 2.3.2 and 2.3.3, we are now in a position to analyse a

generic N -segment structure of the type shown in Fig. 2.7, consisting of semi-infinite

media M1 and MN , and finite segments M2–MN−1 of lengths L2–LN−1. This is achieved

via a straightforward process of recursion, starting from the interface between segments

MN and MN−1 and working upwards to the complete structure which is characterised

by the Bloch mode scattering matrices R1,N and T1,N , defined according to

c+1 = R1,Nc

−1 , c−N = T1,Nc−1 . (2.14)


Mn

MN-1

M2

M1 (semi-infinite)

MN (semi-infinite)

��

��

��

��

� � ��

� � ��

Mn-1

Mn+1

Figure 2.7: Recursive coupling of grating stacks. An incident field from above, c−1 , in

semi-infinite stack M1, is reflected back into M1 (c+1 ) and transmitted through the N

stacks to semi-infinite stack MN (c−N).

We begin by considering the fields at an interface between segments Mn and Mn+1

as shown in Fig. 2.7. Following the notation used in Eq. (2.15a), we define Bloch mode

scattering matrices that describe the reflection and transmission properties of the stack

comprising segments Mn+1, Mn+2 . . .MN , such that

c+n = Rn,Nc−n , c−N = Tn,Nc−n . (2.15a)

Turning our attention now to the interface at the top of Mn, we consider propagation

from Mn−1 through Mn and eventually into MN . Using Eq. (2.15a), we can treat the

stack of segments below Mn as a single interface separating Mn and MN described

by Fresnel matrices Rn,N and Tn,N which reduces the problem to the three segment

structure considered in Section 2.3.3, where the three segments are Mn+1, Mn and MN .

Substituting the Bloch mode scattering matrices for the two interfaces into Eqs. (2.13),

we derive the recurrence relations

Rn−1,N = Rn−1,n + Tn,n−1ΛLnn Rn,NΛLn

n

·(I− Rn,n−1Λ

Lnn Rn,NΛLn

n

)−1Tn−1,N

(2.15b)

Tn−1,N = Tn,NΛLnn

(I− Rn,n−1Λ

Lnn Rn,NΛLn

n

)−1Tn−1,N . (2.15c)

Thus, R1,N and T1,N can be derived for any number of segments by repeated application

of Eqs. (2.15b) and (2.15c), starting from the last interface and working upwards in

Fig. 2.7.


2.4 Numerical aspects

For the numerical implementation of the BMM we use a combination of Mathematica

and FORTRAN code. Mathematica is used as a front-end and for all the matrix ma-

nipulation and solution of eigenvalue problems, while the more numerically intensive

grating scattering calculations are implemented in FORTRAN, with the two linked by

MathLink. While the code has been undergoing development during the course of this

thesis, and is by no means optimised, the majority of the results presented here were

calculated on a laptop computer with a 1.8 GHz Pentium 4 processor and 512 MB of

memory.

The numerical accuracy of the Bloch mode method depends on four main param-

eters: the truncation orders of the multipole, plane wave and Bloch mode expansions

and the size of the supercell. Since energy conservation and reciprocity are both pre-

served analytically in the formulation as a result of using the multipole method (see

Section 2.2.1), these physical properties cannot be used to verify the accuracy of a

calculation. Hence, the main test for accuracy is numerical convergence with respect

to the truncation orders and the supercell dimension. In the current numerical im-

plementation, an equal number of plane wave and Bloch mode orders are retained in

the full calculation, so it is necessary to check convergence of only three parameters,

rather than four. In practice it is often convenient to set “safe” values that guarantee

convergence for a wide range of structures. Although this means sacrificing some nu-

merical efficiency, it reduces the need for convergence tests every time a new structure

is calculated. It is still important, however, to identify how the geometry of a struc-

ture influences the accuracy of the calculation, and the optimal truncation orders and

supercell size.

As a general rule, the multipole truncation order Nm, required for a given accu-

racy depends most strongly on the relative cylinder separation within the grating; more

multipole orders are needed as the relative cylinder radius, r/d increases. The BMM

calculations presented throughout thesis have all been performed with a multipole trun-

cation order of Nm = 7 ie. 2Nm + 1 = 15 multipole terms are used in the field

expansions centred on each cylinder within the supercell. This is sufficient to ensure

convergence for the largest cylinders considered, which have a radius of r = 0.35d.

The number of plane-waves required for convergence depends on the size of the

supercell since the relative spacing of orders is given by the grating equation (2.1). We

have already noted that Dx must be sufficiently large to ensure that there is negligible

coupling between functional elements such as waveguides or cavities in neighbouring

supercells. Table 2.1 gives an example of the convergence behaviour as a function of Dx

and the plane-wave truncation order, Npw. The value listed in the table is the modal

propagation constant β = arg(µ) (units of d) calculated for a single waveguide in a

square lattice, rod-type PhC with cylinders of radius r = 0.3d and refractive index

n = 3 in air (n = 1). The data is calculated for a normalised wavelength of λ = 3.15d.

Each column corresponds to a different sized supercell, and the β values are listed as a

2.5. MODELLING COMPLEX STRUCTURES 55

Dx/d

Npw 11 13 15 17 19 21 23

26 1.115906 1.115406 1.115293 1.115246 1.115148 1.115033 1.115011

28 1.115907 1.115406 1.115297 1.115261 1.115240 1.115104 1.115024

30 1.115907 1.115407 1.115297 1.115271 1.115245 1.115213 1.115074

32 1.115907 1.115407 1.115297 1.115273 1.115260 1.115240 1.115169

34 1.115907 1.115407 1.115297 1.115273 1.115266 1.115250 1.115238

36 1.115907 1.115407 1.115298 1.115273 1.115268 1.115261 1.115242

38 1.115907 1.115407 1.115298 1.115273 1.115268 1.115265 1.115254

40 1.115907 1.115407 1.115298 1.115274 1.115268 1.115267 1.115262

42 1.115907 1.115407 1.115298 1.115274 1.115268 1.115267 1.115266

44 1.115907 1.115407 1.115298 1.115274 1.115268 1.115267 1.115266

46 1.115907 1.115407 1.115298 1.115274 1.115268 1.115267 1.115267

Table 2.1: Convergence of the modal propagation constant β = arg(µ) (units of d),

as a function of the plane-wave truncation order Npw and supercell period Dx for the

waveguide described in the text.

function of Npw. Note that the plane-wave expansion is over all orders s in the range

−Npw ≤ s ≤ Npw, so the total number of plane waves is 2Npw + 1.

Observe that within each column of Table 2.1 β converges to 7 figures accuracy, and

these converged values approach a stable solution as Dx increases. All of the BMM

results given in Chapters 3 and 4 are calculated with Dx = 21d and Npw = 44. For

these parameters, β is stable to 6 figures in Table 2.1. While most calculations do

not require such a high level of accuracy, it can be necessary when studying strongly

resonant structures with extremely narrow spectral features. We note, finally, that we

have confirmed the accuracy of the BMM by comparison with a recently developed

Wannier function method [160] and found agreement of results to better than 1 part in

1000.

2.5 Modelling complex structures

In this section we consider two examples that highlight several important aspects relat-

ing to the application of the Bloch mode matrix method to PhC structures consisting of

multiple segments. While the Bloch mode scattering matrices allow simple expressions

such as Eqs. (2.13) to be written down, the calculation of the grating scattering matri-

ces is still a significant numerical task. Efficient modelling approaches can reduce the

computational demand significantly. Any structure that is modelled using the BMM

must be treated as a stack of individual segments formed from gratings. As we demon-

strate in the first example there may be more than one way to do this so it is important

to consider the geometry of the structure carefully before starting a calculation. For


M1

M2

(a)

M1

M2

M3

(b)

Figure 2.8: Diagram showing two alternative ways to model the waveguide dislocation

shown in (a) using the Bloch mode method. (a) A single interface formed from two

identical waveguide segments displaced laterally. (b) The same dislocation as in (a)

rotated by 45◦ to form a three-segment structure.

complicated devices it is often beneficial to break the structure down into basic func-

tional components which can be characterised individually before being combined into

the complete structure. We consider such an example in Section 2.5.2.

2.5.1 Lattice orientation

The first example we consider is taken from Section 3.3.1, where we study transmission

through a dislocation in an otherwise straight waveguide extending infinitely above and

below the dislocation. The structure illustrated in Fig. 2.8(a) is an example of such a

dislocation formed in a square lattice of period d by shifting the lower section of the

PhC by 2d′ = 2√

2d along the Γ − M direction, where d′ is the local period of the

grating. If the structure is rotated anti-clockwise by 45◦, we obtain the structure shown

in Fig. 2.8(b), which is identical to the folded directional coupler geometry considered in

Section 3.4. The shading of the cylinders indicates the two different ways to model the

dislocation using the BMM. The structure in (a) is treated as a single interface between

two identical semi-infinite sections of PhC with a lateral displacement, whereas the

structure in (b) is treated as a three-segment problem consisting of identical top and

bottom segments M1 and M3 and a short segment of double waveguide M2.

Consider first the structure in Fig. 2.8(a). Since there is only a single interface, the

transmission and reflection characteristics are given by the Fresnel matrices defined in

Eqs. (2.11) which are functions of the plane-wave scattering matrices R∞1 and R∞

2 , of

the semi-infinite segments. Since M1 and M2 are identical, R∞1 and R∞

2 are related by a

simple translation operation as described in Section 2.3.2 (see also Section 3.3.1). Thus,

to calculate the transmission properties of the dislocation at one wavelength requires

the scattering matrices for a single grating, after which the remaining computation

involves only matrix manipulation.

When the dislocation is oriented as in Fig. 2.8(b), it becomes a three-segment struc-

ture of the type treated in Section 2.3.3, where the Bloch mode transmission and re-

2.5. MODELLING COMPLEX STRUCTURES 57

M1

M2

M3

M4

M5

M6

M7

L2

L3

L4

L5

L6

Figure 2.9: Diagram of a PhC Mach-Zehnder interferometer of the type studied in

Chapter 4. The dashed lines indicated the six interfaces that must be considered in the

BMM analysis of the device. The cylinders that form each segment are indicated by

the shading.

flection properties are given by Eqs. (2.13). Modelling this structure therefore requires

the characterisation of two interfaces, plus an additional grating calculation since the

gratings that form segment M2 are different to those in M1 and M3. As the calculation

of the grating scattering matrices is the most numerically intensive part of method, it

is clearly preferable to treat the dislocation as a single interface as in Fig. 2.8(a). We

have performed both calculations and confirmed that they are numerically consistent.

There are, of course other advantages to the single-interface treatment, not least being

the option of non-integer displacements. Quantitatively, the three-segment calculation

takes approximately 70% longer than the single interface. While this is a matter of

only a few seconds for the simple structure considered here, it illustrates the impor-

tance of lattice orientation in the BMM, and may be important in larger, more complex

structures.

2.5.2 Composite devices

One of the many advantages of the Bloch mode method is that it provides a systematic

approach for combining basic functional components into more complex PhC devices.

We have already seen how segments consisting of identical gratings can be coupled to-

gether using Bloch mode scattering matrices and simple recursion equations (2.15), but

this approach can be extended to the coupling of basic components such as junctions,

PhC mirrors and waveguides. Here we consider one such example: the Mach-Zehnder

interferometer that we study in detail in Chapter 4.

The diagram in Fig. 2.9 shows a symmetric Mach-Zehnder interferometer formed in

a triangular lattice PhC. At a basic level, a symmetric Mach-Zehnder consists of two

beam splitters or junctions joined by two identical arms as shown. The details of the


design shown here are discussed in Chapter 4 but are unnecessary for the purposes of

this section. As indicated by the dashed lines, the device can be modelled as a seven-

segment structure, requiring a minimum of four distinct types of grating; gratings in

M1 and M7 are missing one cylinder; gratings in M2 and M6 are missing two cylinders;

gratings in M3 and M5 are missing three cylinders; and the gratings forming M4 have

two missing cylinders separated by three cylinders. Observe that segments M6 and M2

are included to ensure the structure remains symmetric with respect to the input and

output waveguides. This is necessary as result of the lateral shift introduced between

each grating to create the triangular lattice. An equivalent device formed in a square

lattice can be modelled using only five segments and three different types of grating, as

can be seen in Fig. 4.6(a).

Propagation between M1 and M7 can be calculated using the recurrence relations

(2.15), which start by characterising propagation from M5 to M7 then work backwards

to the M1–M2 intersection. When designing a Mach-Zehnder interferometer, however,

it may be necessary to characterise the properties of the junctions independently of the

complete structure. Consider first the input junction consisting of segments M1–M4.

The transmission and reflection properties of the junction are given by the Bloch mode

scattering matrices R14 and T14, where T14 describes the transmission amplitudes of the

Bloch modes from the single input waveguide into the two arms of the interferometer

in M4. These matrices, and their counterparts corresponding to propagation from M4

into M1, are calculated using Eqs. (2.15) with N = 4. Similarly, the output junction is

characterised by the matrices R47 and T47.

Once the complete set of scattering matrices has been calculated for the junctions, it

is straightforward to calculate the properties of the complete structure using Eq. (2.3.3),

where the scattering matrices of the junctions replace the scattering matrices of the

interfaces, such that

R17 = T41ΛL44 R47Λ

L44

(I− R41Λ

L44 R47Λ

L44

)−1T14 + R14, (2.16a)

T17 = T47ΛL44

(I− R41Λ

L44 R47Λ

L44

)−1T14. (2.16b)

This approach is simply a re-ordering of the recursion process described in Section 2.3.4,

and therefore does not provide any computational advantage for a single calculation.

However, the advantages of this component-based approach are obtained when the

Mach-Zehnder interferometer is modified but the junctions are left unchanged. This

occurs in Chapter 4, for instance, when varying the length of the arms or introducing

a phase delay in one arm. In this case the junction scattering matrices need only be

calculated once for each wavelength, and can then be recalled whenever the junction is

used, saving considerable computational expense. Even further efficiencies are gained

if the device is described accurately by only the propagating Bloch modes, as then the

complete junction is characterised by just nine numbers – the elements of the scattering

matrices corresponding to the single propagating mode in M1 and the two modes in

M4. Another example where the component-based approach is particularly useful is

2.6. DISCUSSION 59

in the analysis of the Fabry-Perot interferometer considered in Section 3.3.2 where we

show that a PhC partial mirror is described accurately by a single pair of (complex)

scalar Fresnel coefficients.

As we have demonstrated here, the modelling of even relatively simple devices can

be simplified further by considering first, the functional components that make up

the device. Ultimately, this approach could lead to the development of a component

“library” containing the Bloch mode scattering matrices for each structure. This would

provide a rapid and efficient tool for developing more complex PhC circuits with many

coupled components.

2.6 Discussion

In this chapter we have described the theoretical framework of the Bloch mode scatter-

ing matrix method, and discussed a number of issues relating to the practical application

of the method to studying photonic crystal devices. We have shown that the structure

of the formulation closely resembles that of thin-film optics, with familiar scalar quan-

tities such as Fresnel coefficients being generalised to appropriate matrix forms. While

the method is a highly efficient numerical tool for studying many PhC devices, there

are geometrical limitations imposed by the transfer matrix approach, since all struc-

tures must be described in terms of discrete PhC segments. As we demonstrate in the

remainder of this thesis, however, a wide range of PhC-based devices can be analysed

in this way.

A geometry that we have not discussed in detail is coupling between a homogeneous

dielectric medium such as air, and a PhC. Here, the same supercell must be used in

the dielectric and in the PhC so the Bloch modes outside the PhC are the diffracted

plane wave orders given by the grating equation (2.2). In Chapter 5 we consider the

transmission of an incident plane wave into a uniform PhC, but the method is not

limited to a single plane wave order. If the supercell period Dx is large, and hence the

plane wave orders are closely spaced, it is possible to simulate an incident beam of finite

width by taking an appropriate superposition of plane wave orders.

Perhaps the most important advantage of the Bloch mode method compared to

many other numerical techniques is the physical insight that it provides as a result of

using the natural Bloch mode basis. This allows many seemingly complicated scattering

problems to be expressed in terms of familiar, well understood structures that can be

analysed from first-principles. While we have not demonstrated this aspect here, the

following chapters provide numerous examples.

Chapter 3

Coupled waveguide devices

In this chapter we introduce a new class of photonic crystal device in which directional

coupling and resonant reflection effects are combined. We demonstrate, using both 2D

numerical calculations and simple semianalytic models, that these devices exhibit not

only high-Q reflection resonances but also high transmission over wide-bandwidths. We

present three novel PhC devices that exploit these features – a notch-rejection filter, a

Y-junction and a coupled waveguide delay line.

Much of the work presented in this chapter has been published or is due to appear.

In particular, Sections 3.3.1, 3.3.2, and 3.4–3.6 are based on Refs. [170, 171, 181, 182],

with additional discussions and analysis provided where necessary.

3.1 Introduction

Waveguide coupling and resonant cavities were discussed separately in Sections 1.7.1

and 1.7.5, and in Section 1.7.5 we reviewed several results where a single waveguide

was used to couple light to and from a resonant cavity. In contrast, here we combine

waveguide coupling and resonance effects in a single structure to obtain a range of

unusual transmission characteristics. We present rigorous 2D numerical calculations

using the Bloch mode method, but also show that these structures can be modelled

accurately using simple modal methods and semianalytic techniques. These approaches

provide physical insight into the behaviour of the devices allow rapid and efficient

parameter optimization. The physical interpretation involves two different forms of

waveguide coupling: coupling between parallel waveguides and end-coupling between

two waveguide segments. These two effects are discussed separately in Sections 3.2 and

3.3.

In Section 3.2 we justify numerically the application of conventional tight-binding

approximations [164] to PhC waveguides and present a detailed discussion of coupled

PhC waveguide dispersion properties. Two examples of end-coupling are considered in

Section 3.3: waveguide dislocations and a waveguide-based Fabry-Perot interferometer.

These examples not only provide insight into the coupling of Bloch modes through

61

62 CHAPTER 3. COUPLED WAVEGUIDE DEVICES

waveguide discontinuities, but also demonstrate the efficiency of the Bloch mode method

when studying such structures.

The second half of the chapter consists of Sections 3.4–3.6 where we present three

closely related structures that derive their properties from a combination of waveguide

coupling and resonant reflection at waveguide interfaces. The first of these devices is the

“folded directional coupler” (FDC) - a simple, high-Q notch-rejection filter, the second

is a wide-bandwidth symmetrical Y-junction and the third is a delay line formed by

coupling together multiple FDCs. The properties of all three structures are described

accurately by the semianalytic model derived in Section 3.4.1.

3.1.1 Photonic crystal and numerical parameters

Unless otherwise stated, the numerical examples provided throughout this chapter are

based on two generic photonic crystal lattices. PC1 is a rod-type PhC consisting of a

square lattice of rods of radius r = 0.3d and index ncyl = 3.0 in a background of air (nb =

1). This structure has a TM bandgap in the wavelength range 2.986 < λ/d < 3.774,

corresponding to 0.265 < d/λ < 0.337. The results presented for PC1 are therefore

for TM polarised light. PC2 is a hole-type PhC consisting of a triangular lattice of air

holes (ncyl = 1) in a dielectric background of index nb = 3.37, corresponding to GaAs.

The hole radius is r = 0.3d, which gives a TE bandgap in the range 3.335 < λ/d <

4.629 (0.216 < d/λ < 0.299). Hence the results for PC2 are for TE polarised light.

We consider in this chapter a range of PhC structures and devices, and provide

numerical examples of each to demonstrate the typical characteristics. Although ex-

amples are provided for either PC1 or PC2 in each case, the results are general to

both rod- and hole-type PhCs. The semianalytic analysis of these structures devel-

ops physical insight into device behaviour, thereby allowing simple design rules to be

identified. Within the assumptions of these models, the design rules are not specific

to the underlying PhC lattice and can thus be applied to a range of PhC geometries.

However, this does not imply that all PhCs are equally suitable for a particular device;

the dispersion and modal properties can vary widely, and it is these parameters that

have a significant effect on device characteristics.

The full Bloch mode calculations presented in this chapter were performed with a

supercell period of Dx = 21d to ensure that coupling between supercells is negligible.

Eighty-nine plane wave orders ([−44,+44]) were used to represent the fields between

each layer of cylinders, and the scattering matrices were calculated with 17 multipole

orders ([−8,+8]). For many of the calculations presented here, a supercell of 11d and

45 plane wave orders provides satisfactory accuracy. However, the larger supercell is

required for studying the strong resonant behaviour in Section 3.4, since a very high

spectral resolution is required to resolve the resonant features.

3.2. COUPLED WAVEGUIDE MODES 63

3.2 Coupled waveguide modes

The derivations of the semianalytic methods in Sections 3.4 and Chapter 4 rely on

conventional weak-coupling approximations to describe the modes of parallel coupled

waveguides. Although this is a standard technique for studying conventional dielectric

waveguides, its application to PhC waveguides, which can exhibit strong coupling, has

not been studied systematically. In this section we consider PhC waveguide coupling

in detail and demonstrate numerically that weak-coupling theories predict accurately

the modal fields of coupled PhC waveguides, even when the mode interaction is sig-

nificant. The results also highlight important differences between conventional coupled

waveguides and PhC waveguides, such as mode ordering and symmetry properties.

Recall from Section 1.7 that the supermodes of a two-waveguide system are classified

as either even (+) or odd (−) according to their symmetry with respect to the mirror

plane between the guides. When infinitely far apart, the waveguides are isolated from

each other and so the supermodes are degenerate, being characterised by the propaga-

tion constant β of the single waveguides. As the guides are brought together, the modes

in each waveguide begin to interact and the supermodes split into two non-degenerate

modes |ψ±〉 with propagation constants β±. This splitting is analogous to the splitting

of atomic energy levels when two atoms approach each other.

Coupling between more than two identical waveguides is essentially the same as

the two-waveguide case, however the modes of the individual guides are split into N

supermodes, whereN is the number of waveguides. Approximate modal solutions can be

derived for weakly coupled conventional waveguides using perturbation methods and a

tight-binding approximation that considers only coupling between adjacent waveguides.

We briefly outline here the results derived in Chapter 11 of Ref. [164], using bra-ket

notation to represent the modes. The supermodes in this case are written as linear

combinations of the individual waveguide modes,

|Ψs〉 = 1AN

∑Nn=1Cn|ψn〉,

where Cn = sin(

nsπN+1

), and AN =

(∑Nn=1 |Cn|2

)1/2

, (3.1)

for s = 1, 2, 3, . . . , N and |ψn〉 is the unperturbed mode of the nth waveguide. Here

AN is a normalization factor to ensure 〈Ψ|Ψ〉 = 1, and hence each supermode carries

unit power. In this approximation the propagation constants of the supermodes can be

determined by overlap integrals of the modes in neighbouring waveguides, although we

do not use this result in our analysis.

The application of this simple coupled-mode-theory (CMT) to photonic crystal

waveguides has not been studied rigorously, although a similar approach reported by

Kuchinsky et al. showed good agreement with 2D numerical results in the case of two

coupled waveguides [183]. In this chapter and in Chapter 4 we apply CMT to several

PhC waveguide-based devices and derive semianalytic models that display excellent

agreement with full 2D numerical calculations. The devices consist of both two and


three coupled waveguide systems, the properties of which we consider in more detail in

Sections 3.2.1 and 3.2.2.

3.2.1 Coupling of two waveguides

We consider first the simplest coupled PhC waveguide example: two identical wave-

guides in PC1, each formed by removing a single line of cylinders in the Γ−X direction,

as shown in Figs. 3.1(e) and (f). The electric field of a single waveguide mode, calcu-

lated using the Bloch mode method, is plotted in Figs. 3.1(a) and (d) for a normalised

frequency of d/λ = 0.32. The dark regions in (d) correspond to negative phase and the

white regions to positive phase. The profile in (a) is calculated along the white line

shown between the cylinders in (d). Note that the field extends several periods into the

PhC walls of the waveguide.

In the tight-binding approximation the odd and even supermodes are given by

Eq. (3.1) which yields

|ψ±〉 = (|ψL〉 ± |ψR〉)/√

2 . (3.2)

Here |ψL〉 and |ψR〉 are the unperturbed modes of the left and right waveguides, respec-

tively, which are simply obtained by applying a lateral translation to the waveguide

mode in Fig. 3.1(a). Figures. 3.1(b) and (c) show the electric fields of |ψ±〉 calculated

using Eq. (3.2) (dashed curves) and the full Bloch mode method (solid curves) for two

PhC waveguides separated by a single line of cylinders as shown in Figs. 3.1(e) and (f).

While the tight-binding approximation is most accurate between the layers of cylin-

ders (red curves), reasonable agreement is also obtained for the fields in the cylinders

(blue curves). Such good agreement is surprising given that the waveguides are only

separated by a single line of cylinders and thus interact strongly with each other. The

tight-binding treatment also provides a reasonable approximation for coupled hole-type

PhC waveguides, although we do not present numerical results here. We note however,

that the approximation was applied successfully to design the hole-type PhC Y-junction

used in the Mach-Zehnder interferometer in Section 4.4.

Figure 3.2(a) shows the projected band structure and dispersion curve for the single

waveguide shown in the inset. The PhC bands are shaded dark grey and the bandgap

is shaded light grey. In contrast to the waveguide modes of the hole-type PhC shown

in Fig. 1.8, there is at most one guided mode for all frequencies in the bandgap. This is

a property common to most single line defect waveguides in rod-type PhCs, and one of

their advantages over hole-type PhCs. Observe also that the dispersion slope of the rod-

type PhC waveguide is positive, while the even mode of the hole-type PhC waveguide

has a negative dispersion slope. Figures 3.2(b)–(f) show the supermode dispersion

curves for coupled waveguides with separations of Nc = 5 to Nc = 1 respectively,

where Nc is the number of cylinder layers between the waveguides, as shown in the

inset diagrams. The even and odd modes are indicated by dotted and dashed curves

respectively. For comparison, the single waveguide dispersion curve from (a) is also

plotted in each of (b)–(f), clearly showing that the mode splitting is almost symmetric


(a) (c)(b)

(d) (e) (f)

Figure 3.1: Electric fields of the modes in a single waveguide (a,d) and two waveguides

separated by a row of cylinders (b,c,e,f). The solid curves in (a)–(c) are calculated along

the corresponding red and blue lines in (d)–(f) using the full Bloch mode method. The

dashed curves in (b) and (c) are the approximate result given by Eq. (3.2). The light

and dark regions in (d)–(f) correspond respectively to positive and negative phases.

about the single waveguide mode, and hence the average of the two modes β = (β+ +

β−)/2 ≈ β. This feature is important for the semianalytic analysis in Section 3.4. The

strong mode coupling for Nc = 1 and Nc = 2 results in large mode splitting ∆β = |β+−β−|/2, a feature that has attracted much attention recently. Recall from Section 1.7.5

that large mode splitting results in short coupling lengths (1.8) and therefore provides

the possibility of compact coupler-based devices [105–107,110–112].

The dispersion curves in Fig. 3.2 also illustrate an unusual mode ordering property

of coupled PhC waveguides which occurs when the dominant Bloch factor of the bulk

crystal, µ, is negative [184]. When Nc = 1, 3, 5, the lowest frequency (fundamental)

mode has odd symmetry, whereas for Nc = 2, 4, the fundamental mode is even. In

contrast, the fundamental mode of a symmetric coupler formed in conventional wave-

guides is always even and the second mode is odd. An interesting consequence of this

is the ability to design waveguide structures that support only an odd mode. Consider,

for instance, the dispersion curves for the Nc = 1 case in Fig. 3.2(f). The even mode,

indicated by the dotted curve, is cut off at a frequency of d/λ = 0.305, below which


0 0.1 0.2 0.3 0.4 0.50.22

0.24

0.26

0.28

0.3

0.32

0.34

0.36

0 0.1 0.2 0.3 0.4 0.50.22

0.24

0.26

0.28

0.3

0.32

0.34

0.36

0 0.1 0.2 0.3 0.4 0.50.22

0.24

0.26

0.28

0.3

0.32

0.34

0.36

0 0.1 0.2 0.3 0.4 0.50.22

0.24

0.26

0.28

0.3

0.32

0.34

0.36

0 0.1 0.2 0.3 0.4 0.50.22

0.24

0.26

0.28

0.3

0.32

0.34

0.36

0 0.1 0.2 0.3 0.4 0.50.22

0.24

0.26

0.28

0.3

0.32

0.34

0.36

b pd/2

b pd/2

b pd/2

b pd/2

b pd/2

b pd/2

d/l

d/l

d/l d/l

d/l

d/l

N =5c

N =2c

N =4c N =3c

N =1c

(a) (b)

(c) (d)

(e) (f)

Figure 3.2: Projected band structures and dispersion curves for the various waveguide

structures shown in the insets. Dark grey regions represent PhC bands, while the light

grey region shows the bandgap. (a) Single waveguide (b)-(f): Two coupled waveguides

separated by Nc = 5 − 1 rows of cylinders respectively. The dotted curve corresponds

to the even supermode, and the dashed curve corresponds to the odd supermode. The

solid curve is the single waveguide dispersion plotted in (a).

only the odd supermode can propagate. In an ideal symmetric structure, an even mode

incident from another waveguide would be perfectly reflected by the waveguide pair.

Similar concepts of mode symmetry filtering play an important role in the recirculating

Mach-Zehnder interferometer studied in Chapter 4.

3.2.2 Coupling of three waveguides

We now consider the effect of adding a third waveguide to the system in Section 3.2.1

to form a waveguide structure that supports three supermodes, |ψ1,2,3〉 respectively. In

this case the correct linear superpositions of waveguide modes is not as obvious as in

the two waveguide example. These are calculated by substituting N = 3 into Eq. (3.1)


(a) (b) (c)

(d) (e) (f)

Figure 3.3: Contour and line plots of the supermode electric fields in a three waveguide

system with Nc = 1(a–c) and Nc = 2 (d–f). The line plots above each figure show the

electric field profile along the white lines in the contour plots. The solid curves show

the rigorous numerical field calculation and the dashed curves are the coupled mode

approximation (3.1).

which gives

|ψ1〉 = (|ψL〉+√

2 |ψC〉+ |ψR〉)/2,|ψ2〉 = (|ψL〉 − |ψR〉)/

√2, (3.3)

|ψ3〉 = −(|ψL〉 −√

2 |ψC〉+ |ψR〉)/2,

where |ψL〉, |ψC〉 and |ψR〉 are the unperturbed modes of the left, centre and right

waveguides respectively. Observe that |ψ1〉 and |ψ3〉 are even modes, whereas |ψ2〉 is

odd. For conventional waveguides, |ψ1〉 is the fundamental mode, followed by |ψ2〉 and

|ψ3〉 but this is not necessarily the case for PhC waveguides. The contour plots in

Fig. 3.3 show the mode fields for Nc = 1 (a–c) and Nc = 2 (d–f) at d/λ = 0.32. The

modes are listed in order of decreasing β, where β = 1.473, 1.171, 0.377 for modes (a)–

(c) and β = 1.320, 1.164, 0.907 for (d)–(f). As for double waveguides, the mode order


0 0.1 0.2 0.3 0.4 0.50.22

0.24

0.26

0.28

0.3

0.32

0.34

0.36

b pd/2

d/l

(a)

0 0.1 0.2 0.3 0.4 0.50.22

0.24

0.26

0.28

0.3

0.32

0.34

0.36

b pd/2

d/l

(b)

(b)(a)

Figure 3.4: Dispersion curves for three coupled PhC waveguides with (a) Nc = 1 and

(b) Nc = 2. The dotted and dashed-dotted represent the two even modes |ψ1〉 and |ψ3〉respectively, and the dashed curve corresponds to the odd mode |ψ2〉. The solid line in

both plots is the dispersion of a single waveguide mode.

changes depending on whether Nc is even or odd, but here it is the two even modes,

|ψ1〉 and |ψ3〉, that swap positions. This is clearly seen in the line plots above each of

the figures which show the electric field along the white lines in each of the contour

plots (solid curves) and the approximate result given by Eqs. (3.3) (dashed lines). Once

again the agreement between the rigorous numerical calculation and the coupled mode

approximation is excellent, with the two curves being indistinguishable for Nc = 2.

The supermode dispersion curves are shown in Fig. 3.4 for Nc = 1 and Nc = 2,

where the dotted and dashed-dotted curves correspond to the two even modes |ψ1〉 and

|ψ3〉 respectively, and the dashed curves corresponds to the odd mode |ψ2〉. The single

waveguide dispersion is also indicated by the solid curve. Since the three supermodes

are split almost symmetrically relative to the single waveguide mode, the dispersion

curve of |ψ2〉 is very close to that of the unperturbed waveguide. Note that in contrast

to double waveguides, it is not possible to design a triple waveguide structure that only

supports an odd mode since |ψ2〉 always lies between the two even modes.

3.3 Waveguide coupling at interfaces

The second type of waveguide coupling that plays an important role in the context

of this thesis is mode coupling through an interface between the ends of two PhC

waveguides. In this case, the coupling properties are often a combination of direct

transmission and resonant coupling effects. The Bloch mode scattering matrix method

is an ideal numerical tool for studying this form of coupling as it allows interface effects

to be treated separately from propagation along sections of straight waveguide. In

particular, the generalised Fresnel matrices introduced in Section 2.3.2 describe the

coupling from one Bloch mode basis to another. Here we present two examples that

provide physical insight into Bloch mode coupling behaviour, and at the same time

demonstrate the advantages of the Bloch mode method for studying such problems. In

3.3. WAVEGUIDE COUPLING AT INTERFACES 69

Section 3.3.1 we consider a lateral dislocation in a waveguide that can be treated as a

single interface between two identical PhC stacks and therefore requires the calculation

of only one set of scattering matrices. Coupling through multiple interfaces in the form

of a Fabry-Perot cavity is considered in Section 3.3.2 where we show that the cavity

resonance is described accurately by considering only the propagating Bloch modes.

3.3.1 Waveguide dislocations

Recall from Section 1.7.2 that an ideal, straight PhC waveguide can guide light without

loss or reflection. In practical situations, however, non-uniformities in the crystal or

waveguide structure introduce losses due to reflection and scattering. In this section

we consider the effect of a linear dislocation through the crystal and waveguide. Two

structures are considered: a dislocation perpendicular to the guide direction (along the

Γ − X axis of the crystal), as in Fig. 3.5(a), and a dislocation at a 45◦ angle to the

waveguide (along the Γ−M crystal axis) (Fig. 3.5(b)). The bulk crystal in Fig. 3.5(b) is

the same square lattice of period d as in (a), but rotated by 45◦. Thus, when calculating

the scattering matrices for the rotated crystal, the lattice period within each grating

layer is d′ =√

2d.

The dislocation is modelled as an interface between two identical semi-infinite crys-

tals as discussed in Section 2.5.1. To introduce a lateral shift of sx parallel to the

grating layers as shown in Fig. 3.5, we apply a translation operator Qs = diag[eiαpsx/2

]to the appropriate R∞ matrices and define R∞

sx = Q−1s R∞Qs and R∞′

sx = QsR∞′

Q−1s as

described in Section 2.3.2. Here, the switching of Qs and Q−1s simply changes the sign

of sx depending on whether the dislocation is viewed from above or below. The Fresnel

matrices are then calculated using Eqs. (2.11) with R∞sx replacing R∞. Figure 3.6 shows

the transmission through the dislocated waveguides in Fig. 3.5 as a function of displace-

ment sx. The wavelength is fixed at λ = 3.25d. Since the period of the supercell used

in these calculations is Dx = 21d, the transmission curves have the same periodicity.

Most of the features in the spectra of Fig. 3.6 are easily understood. Both cases

exhibit 100% transmission for sx = 0, since this corresponds to an infinite guide with no

dislocation. However, the two curves show very different behaviour when a dislocation is

introduced. In the case of the perpendicular dislocation (Fig. 3.5(a)), the transmission

curve is symmetric about sx = 0, as expected since the guide is symmetric about the

axis transverse to the interface. The transmission peaks correspond to local maxima

in the overlap of the fields in the two waveguide sections. These peaks occur close to

integer values of sx/d since at these points the cylinders are aligned at the interface

between the two sections so the only dislocation occurs at the waveguide ends.

For the 45◦ dislocation, the transmission curve is strongly asymmetric in sx, which

can be understood from the diagram in Fig. 3.5(b). When sx > 0 (as in the diagram),

the structure resembles two overlapping waveguides at 45◦ separated by an arrangement

of cylinders. In this case, the overlap region allows strong coupling between the two

waveguides, and hence strong resonant transmission peaks. The first two of these


D =21dx

D =21d’x

d

d’= 2 d

sx

s ’x

(a)

(b)

Figure 3.5: (a) Straight waveguide with dislocation along the Γ−X crystal axis. The

circles represent the cross-section of the dielectric cylinders forming the 2D photonic

crystal. The cylinder shading indicates the two distinct waveguide sections considered

in the calculations. (b) Identical waveguide, but with dislocation along the Γ − M

crystal axis. Both diagrams show the supercell used in the calculations, and the dash-

dot line indicates the line of dislocation. The waveguides extend infinitely in the vertical

direction.

peaks occur at sx ≈ 1d′ and sx ≈ 2d′, for which the transmission exceeds 99.5%. For

integer values of sx/d′, the structure resembles the folded directional coupler discussed

in Section 3.4. If the guide is displaced in the sx < 0 direction, however, the two guides

are effectively being pulled apart in both directions, and so the transmission is seen

to decrease rapidly below 1% and stay low as |sx| is increased. Such an asymmetry

in transmission with respect to displacement could potentially be used in a sensitive

directional motion detector [185].

3.3.2 Fabry-Perot cavity

Resonant devices are an essential component in modern optics, being used in filters,

switches, couplers and many other other applications. We reviewed in Section 1.7 some

of the properties and applications of resonant PhC cavities, most of which were formed

by removing or modifying one or several cylinders in an otherwise uniform PhC. Here

we consider a more conventional resonant cavity design – a Fabry-Perot cavity – and

show that it can be modelled accurately using semianalytic methods.

The Fabry-Perot (FP) resonator is one of the most basic resonant devices, consist-


10 5 0 5 10Displacement, sx (units of d)

0.2

0.4

0.6

0.8

1

T

Figure 3.6: Transmission of dislocated straight guides, for dislocations parallel to the

Γ−X (solid curve) and Γ−M (dashed curve) crystal axes at λ/d = 3.25.

ing of two high-reflectivity mirrors on either end of a cavity. Away from resonance,

the transmission of the Fabry-Perot device is typically very small, depending on the

reflectivity of the individual mirrors. On resonance, however, the small transmitted

amplitude through the first mirror interferes constructively inside the cavity, and high

transmission is possible. If the mirrors are identical (balanced), the resonant transmis-

sion can reach 100% but this is reduced if the reflectivities are unequal. The finesse,

F , of a balanced FP cavity is defined as the ratio of the fringe separation (free spectral

range) to the full width at half maximum (FWHM) of the transmission peak. It is a

function of the mirror reflectivity only, given by

F = π√R/(1−R), (3.4)

where R = |r|2 is the reflectance of each mirror (Ref. [122], Chapter 7). In a physical

sense, the finesse is a measure of how many times light circulates through the cavity be-

fore it has all leaked away. Thus, for high finesse cavities, the mirrors must be strongly

reflecting, requiring R > 0.9985 for F > 2000. The resonance width is characterised by

the “quality factor” Q, defined in Section 1.7.1 as the ratio of the resonant frequency

(ω0) to the FWHM (∆ω) of the transmission intensity peak (1.6). In practical PhC

devices, however, radiation losses can reduce Q significantly. As we discussed in Sec-

tion 1.7.1, cavity losses can be minimised with careful cavity design, for instance by

modifying the cavity walls to reduce coupling to radiation modes [54]. Since the issue of

losses is relevant to all of the structures considered in this thesis, we provide a general

discussion of the matter in Chapter 6.


L

d

mirror depth

air holes

Figure 3.7: Diagram of the Fabry-Perot resonator formed in a single-mode waveguide

in PC2. The mirrors are formed by “plugs” of cylinders. For the example here, all

cylinders are identical.

Mirror properties

Fabry-Perot devices in photonic crystals have been studied previously in various con-

texts [112,186–188]. Here we apply the Bloch mode method to the simple Fabry-Perot

cavity shown in Fig. 3.7, formed by placing two “plugs” of Nplug cylinders in a single-

mode waveguide formed in the hole-type PhC, PC2. Although their general behaviour

is similar to that of conventional Fabry-Perot resonators, PhC-based Fabry-Perot de-

vices have one significant difference that arises because the mirrors in a PhC FP are

distributed reflectors, in which the light penetrates several layers into the crystal struc-

ture. This introduces additional spectral features such as resonances in the mirror

reflectivity due to reflections within the mirror itself. The reflectivity of the mirrors can

be increased by adding more cylinders, or by changing the properties of the cylinders

forming the mirror, such as the radius [186]. Figure 3.8(a) shows the reflectivity of a

single mirror as Nplug is increased from 1 to 6 cylinders. As expected, the reflectivity

increases with the number of cylinders, however the effect is not monotonic at all wave-

lengths due to resonant reflections from either end of the plug, as seen in the curves

corresponding to Nplug = 3− 6.

The properties of an individual mirror can be calculated with the Bloch mode

method as a 3-section waveguide structure, where the two end sections are identical


10-7

10-5

10-3

10-1

123

456

Mirror thickness (cylinders)

T

3.4 3.5 3.6 3.7 3.8

d/l

10-11

10-8

10-5

10-2

10-14

T

(a)

(b)

Figure 3.8: (a) Transmittance (log scale) as a function of normalised wavelength,

through a single mirror (plug) formed in a single-mode waveguide in PC2 for mir-

rors with Nplug = 1 − 6. (b) Transmittance of a Fabry-Perot cavity of length L = 5d

for mirrors with Nplug = 1− 6.

single waveguides, and the central section is composed of complete layers of cylinders

(possibly with changes made to the central one). The transmission and reflection of the

mirror are therefore described by Eqs. (2.13). The scattering matrices are calculated

for each of the interfaces, with any symmetries being taken into account to save com-

putation time. In this case the mirror is symmetric with identical waveguides on both

sides, so T21 = T23 and R21 = R23, which reduces significantly the required scattering

matrix calculations. Many calculations can benefit from an understanding of the sym-

metry properties of the structure as demonstrated by the examples in this chapter and

Chapter 4.

Unless the central blocking cylinder is made very small, the plug section does not

support any propagating modes, and hence the reflection of the propagating modes at

the 1−2 interface is 100%. However, the mirror is of finite thickness so energy is coupled

into the cavity by evanescent modes within the plug, giving a non-zero transmission of


the propagating Bloch mode from section 1 to section 3.

Fabry-Perot properties

As shown in Fig. 3.7, the FP cavity is formed by placing two such mirrors in a single-

mode waveguide, to form a cavity of length L. The transmission through the complete

structure can be calculated in two ways. The first approach is to apply Eq. (2.15) with

N = 5, thereby treating it as a 5 layer structure consisting of the input and output

waveguides, the two mirrors and the cavity. The second, and more efficient, method

is to treat each mirror as a single interface characterised by the scattering matrices

Rplug = R and Tplug = T which are calculated from Eqs. (2.13) as described above. This

allows the whole FP structure to again be treated as a 3-layer problem using Eqs. (2.13),

where the matrices T12 = T21 = T23 = Tplug describe the transmission of Bloch modes

through the mirror, and R12 = R21 = R23 = Rplug describe reflection from the mirror.

Thus, once the mirror properties have been calculated, the length of the cavity can be

varied without the need to repeat any scattering matrix calculations.

Figure 3.8(b) shows the transmission spectra for a cavity of length L = 5d with

increasing mirror thickness. As expected from Eq. (3.4), the peak width decreases

(Q increases) as the reflectivity of the mirrors is increased. At frequencies far from

the reflection resonances shown in Fig. 3.8(a), the reflectivity of the mirrors increases

exponentially with Nplug so Q also increases rapidly with the size of the mirror. The

transmission spectra become more complicated in the vicinity of a mirror reflection

resonance, since the reflectance changes significantly across the width of the cavity

resonance. As a consequence of this, Eq. (3.4), which is calculated for a fixed R loses

some accuracy.

Table 3.1 shows the quality factor of the Fabry-Perot resonance near d/λ = 0.369

as a function of the number of cylinders in the mirror for Nplug = 1 − 4. Since these

calculations are purely 2D, out-of-plane losses would have to be controlled to achieve

such large Q values in realistic slab geometries. We discuss this issue in Chapter 6.

Experimental measurements ofQ = 7300 have been reported for a similar FP cavity [83]

and Q = 1.2 × 104 was measured for a structure with multi-mode rather than single-

mode waveguides [188]. Although these values are several orders of magnitude lower

than the highest-Q PhC cavities discussed in Section 1.7.1 [75], the in-line geometry of

the FP structure provides a simple design for an integrated notch-pass filter.

Propagating mode analysis

The results presented so far include both propagating and evanescent Bloch mode or-

ders. It is not always necessary, however, to include the evanescent orders and in these

cases, many of the complicated matrix expressions can be replaced with semianalytic

scalar forms [180]. For the FP cavity described above, the mirror sections have no prop-

agating modes and therefore rely solely on evanescent modes to carry energy through

the reflecting layers into the cavity. Between the mirrors, however, most of the energy


Nplug λ0(d) (full) λ0(d) (prop) Q (full) Q (prop) F [= π√R/(1−R)]

1 3.68692 3.68692 78 78 6.8

2 3.69090 3.69091 1200 1200 97

3 3.690642 3.690656 6.6× 104 6.4× 104 5.4× 103

4 3.6905974 3.6906108 1.3× 106 1.1× 106 1.1× 105

Table 3.1: Resonant wavelength (units of d), Q and F for a Fabry-Perot cavity of

length L = 5d in PC2 for mirrors of thickness Nplug. Resonant wavelength and Qvalues are given for both the full numerical calculation (full) and the propagating mode

approximation (prop).

is carried by the single propagating Bloch mode, suggesting that a propagating mode

analysis might be appropriate. We describe here the application of this approach to the

FP structure, although we note that a rigorous asymptotic analysis for a general class

of three layer structures was reported by White et al. [170].

The most significant advantage in using Bloch mode methods derives from their

capacity to model efficiently, long propagation spans in a fixed medium. Since this

is characterised by the matrix ΛL in Eqs. (2.13), it follows that for sufficiently long

spans (typically L > 5), the evanescent modes can be disregarded since they decay

exponentially with L and therefore the calculations are dominated by the propagating

states. To apply this to the Fabry-Perot resonator in Fig. 3.7, we first calculate the

transmission and reflection matrices through a single mirror using the full Bloch mode

calculation. Since these describe coupling between single-mode waveguides on either

side of the mirror, they reduce to complex scalar transmission and reflection coeffi-

cients when only the single propagating mode is considered in each segment. Similarly,

the matrix Λ in Eqs. (2.13) reduces to the eigenvalue µ corresponding to the single

propagating mode of the waveguide. Substituting these scalar values into Eq. (2.13b)

results in a simple scalar equation that gives the transmission through the complete FP

structure as a function of the cavity length, L. Since the properties of the mirror are

wavelength-dependent, the scattering matrices must be calculated for each wavelength

of interest. In practice, once the matrices have been calculated in full, only the elements

corresponding to the propagating mode(s) need to be saved.

The effect of neglecting the evanescent modes within the cavity can be seen in

Table 3.1, where the resonant wavelength and Q for the FP cavity of length 5d are

compared for the full evanescent field calculation and the reduced, propagating mode

calculation. For all but the highest Q cavities, the agreement is excellent. If the cavity

length is reduced, it might be expected that the evanescent fields would have a more

significant effect on the results. However, even for cavity lengths of 2 lattice periods, the

difference between the propagating mode result and the full result is relatively small.

For mirrors that are 3 layers thick, the difference in resonant frequency calculated with

the two methods is less than 0.05%, and the Q values agree to 2 figures. Table 3.1 also

shows the finesse calculated using the standard FP equation (3.4) with the reflectance


values R = 1− T obtained from the transmission data plotted in Fig. 3.8(a).

The two examples in this section demonstrate clearly the advantages of the Bloch

mode matrix method for studying PhC waveguide structures consisting of a few different

waveguide sections. We have shown that the calculational demands can be reduced

by taking into account the symmetry properties of the structures being considered,

and even greater reductions can be made by considering only the propagating modes.

When these two simplifications are combined with the coupled mode theory results

of Section 3.2, they provide a powerful set of tools for developing simple semianalytic

models to describe relatively complex photonic crystal structures. In the remainder of

this chapter and in Chapter 4, we study several different PhC devices that exhibit the

two types of coupling discussed in Sections 3.2 and 3.3 in addition to Fabry-Perot-like

resonances. Elegant semianalytic models are derived for each device, and the results

are compared to the rigorous 2D numerical results obtained with the full Bloch mode

method.

3.4 Folded directional coupler

In this section we discuss the folded directional coupler (FDC): a novel notch-rejection

filter that exploits the mode coupling properties of a directional coupler and the sharp

resonances of a Fabry-Perot resonator. This same concept is extended in Sections 3.5

and 3.6 to design an efficient, wide-bandwidth junction and a novel coupled resonator

waveguide that could be used as a delay line.

Figure 3.9(a) illustrates the basic geometry of the FDC which consists of a single-

mode input waveguide (region M1), a single-mode output waveguide (M3), and an

overlap region (M2), of length L, where the modes of the two guides couple. The

similarity to a directional coupler is obvious, but in addition to coupling behaviour, re-

flections from the closed waveguide ends result in resonant transmission characteristics.

We demonstrate that an ideal FDC structure that is less than 3µm long can be tuned

to have a notch-rejection width of less than 10 GHz, corresponding to Q > 104, and

with minor structural modifications, this can be increased to Q > 6× 105.

The concept of combining Fabry-Perot effects in a directional coupler geometry was

studied previously by Safaai-Jazi et al. [189] as a way to improve mode discrimination

and longitudinal mode spacing in resonant cavities. Although the FDC shares similari-

ties with the earlier device, there are also distinct differences. In Ref. [189] the resonant

cavity is formed by placing two partially reflecting mirrors across a pair of coupled

waveguides to form a four-port device. In contrast, the FDC is a two-port device since

one of the waveguides at each end of the cavity is blocked by a fully reflective photonic

crystal mirror, while the other waveguide is left open. This difference in the reflection

mechanism allows the FDC to exhibit complementary transmission behaviour to FP

devices. FP structures are normally highly reflective, only transmitting when a res-

onance condition is satisfied (notch-pass filter), whereas the FDC can be designed to

3.4. FOLDED DIRECTIONAL COUPLER 77

L

(a) (b)

M2

M3

M1

��

��

��

��

Figure 3.9: (a) Geometry of the FDC consisting of three sections M1, M2 and M3,

of identical cylinders. The shading indicates the three waveguide different sections.

(b) Schematics showing all of the propagating modes in the structure. The input and

output guides each support a single mode, while region M2 supports an even and an

odd supermode, |ψ±〉, indicated by the dashed and dotted curves respectively.

transmit at most frequencies and only reflect on resonance (notch-rejection filter).

In general, FDC structures exhibit a complicated and varied range of transmission

spectra that depend strongly on the structural parameters. Several examples are shown

in Fig. 3.10, where the transmission is plotted as a function of frequency for different

cavity lengths (a) and guide separations (b). These curves were calculated using the

full numerical formulation of the Bloch Mode method. Observe that there are several

common features in the spectra in Fig. 3.10 such as resonances where the transmission

vanishes, and regions where the transmission is flat and close to 100%. The solid

curves in both parts of Fig. 3.10 correspond to the same FDC with a cavity length of

L = 5d and guide separation of Nc = 1. This structure exhibits a sharp resonance at

d/λ = 0.331 where the transmittance drops from almost 100% to < 0.1% for a very

narrow range of frequencies. In Section 3.4.3 we use the semianalytic model derived

in Section 3.4.1 to identify the origin of this resonance, and find an expression for the

resonance width in terms of the waveguide dispersion parameters.

In Sections 3.4.1 and 3.4.2 we describe two alternative but equivalent derivations of

a simple semianalytic model for the FDC structure illustrated in Fig. 3.9(a), both of

which are based on mode coupling and symmetry arguments. In the first method, we

begin from the coupled mode approximation for two identical coupled waveguides (3.2)

and follow the propagation of an input mode through the FDC. The second derivation

is based on the Bloch mode scattering matrix formulation, which is first reduced to

include only the propagating modes, as in Section 3.3.2, then simplified further by

replacing the numerically calculated scattering matrices by idealised expressions. The


0.29 0.3 0.31 0.32 0.33

d/l

0

0.2

0.4

0.6

0.8

1

T

0.29 0.3 0.31 0.32 0.33

d/l

0

0.2

0.4

0.6

0.8

1

T

(a) (b)

Figure 3.10: FDC transmission spectra for (a) FDCs with Nc = 1 and cavity lengths

L = 4d (dashed), L = 5d (solid) and L = 6d (dashed-dotted) and (b) FDCs of length

L = 5d and guide separations of Nc = 1 (solid), Nc = 2 (dashed) and Nc = 3 (dashed-

dotted).

resulting model requires numerical input of only the waveguide dispersion properties

and the structure length to describe accurately the transmission properties of the FDC.

It also provides physical insight into many of the spectral features observed in Fig. 3.10.

We compare the results to full numerical calculations using the Bloch mode method.

3.4.1 Semianalytic model: coupling analysis

Consider first the modes in each section of the FDC shown in Fig. 3.9(b). The input

and output waveguides each support a single mode, |ψin〉 and |ψout〉, respectively, while

the coupling region supports even and odd supermodes |ψ±〉 with propagation constants

β±. As in Sections 1.7.5 and 3.2.1, we define ∆β = |β+ − β−|/2 and β = (β+ + β−)/2,

so that the coupling length Lc = π/(2∆β).

Recall from Section 3.2.1 that the supermodes of two identical coupled waveguides

can be approximated by linear superpositions of the individual waveguide modes using

Eq. (3.2). The comparison with rigorous numerical results in Fig. 3.1 clearly demon-

strates that this approximation remains accurate even for PhC waveguides separated

by a single line of cylinders and can therefore be applied to the modes in region M2

of the FDC. Since the input and output waveguides continue into region M2, the cou-

pled mode result provides a simple relationship between the modes in each of the three

regions. From Eq. (3.2) we can write

|ψin/out〉 = (|ψ+〉 ± |ψ−〉)/√

2, |ψ±〉 = (|ψin〉 ± |ψout〉)/√

2 . (3.5)

We now follow the propagation of a mode as it passes through the FDC. Light

entering from the input waveguide in mode |ψin〉 is coupled into the supermodes of

region M2 in the ratio given by Eq. (3.5), where we ignore back reflection at this first


interface since the input waveguide is continuous. The even and odd supermodes then

propagate a distance L through region M2, advancing in phase by factors exp(iβ+L)

and exp(iβ−L) respectively so that the field at the output interface is given by

Ψ(L)(1) =(eiβ+L|ψ+〉+ eiβ−L|ψ−〉

)/√

2 ,

=(eiβ+L + eiβ−L

)|ψin〉/2 +

(eiβ+L − eiβ−L

)|ψout〉/2 ,

= eiβL (cos(∆βL)|ψin〉+ i sin(∆βL)|ψout〉) , (3.6)

where we have used the definitions of β and ∆β to write β± = β±∆β. The superscript

(1) on the left hand side of equation (3.6) indicates that this is the first pass of the

light through M2. Notice that the modes of the two waveguides have a relative phase

difference of π/2 due to the mode coupling, while they share a common phase term

exp(iβL) which depends on the average propagation constant.

At the M2–M3 interface, the right-hand waveguide continues into M3 without in-

terruption, whereas the left-hand waveguide ends. We therefore assume that the field

component in mode |ψout〉 is transmitted with amplitude τ1 = i exp(iβL) sin(∆βL),

while the remaining field in mode |ψin〉 is reflected by the photonic crystal back into

the left-hand waveguide. For the purposes of this model, we neglect any phase change

that may occur when |ψin〉 is reflected, and so the reflected mode has an amplitude

of exp(iβL) cos(∆βL). Note that if the FDC was formed from conventional dielec-

tric waveguides, most of the light in the closed waveguide would be radiated into the

cladding and lost. The reflected field propagates back through region M2 to the first

interface, where it is again expressed in terms of the individual waveguide modes as

Ψ(0)(1) = e2iβL(cos2(∆βL)|ψin〉+ i cos(∆βL) sin(∆βL)|ψout〉

). (3.7)

Here, as at the output interface, the field in the open guide is transmitted into the input

waveguide with amplitude ρ1 = exp(2iβL) cos2(∆βL) while the field in the right-hand

waveguide is reflected back into region M2. We again see that both terms in Eq. (3.7)

have gained an additional factor of exp(iβL) and the individual waveguide modes are

again π/2 out of phase.

The total transmission and reflection amplitudes of the FDC, denoted τ and ρ

respectively, are found by repeating this calculation for each pass through M2 and

adding coherently the transmitted and reflected components, τn and ρn respectively,


which yields

τ = ieiβL sin(∆βL) + ie3iβL cos2(∆βL) sin(∆βL)

− ie5iβL cos2(∆βL) sin3(∆βL) + ie7iβL cos2(∆βL) sin5(∆βL)− . . .

= i exp(iβL) sin(∆βL)1 + exp(2iβL)

1 + sin2(∆βL) exp(2iβL)

(3.8a)

ρ = e2iβL cos2(∆βL)− e4iβL cos2(∆βL) sin2(∆βL)

+ ie6iβL cos2(∆βL) sin4(∆βL)− e8iβL cos2(∆βL) sin6(∆βL)− . . .

=cos2(∆βL) exp(2iβL)

1 + sin2(∆βL) exp(2iβ)L.

(3.8b)

Hence the reflectance is

R = |ρ|2 =cos4(∆βL)

cos4(∆βL) + 4 sin2(∆βL) cos2(βL), (3.9)

and the transmittance is simply T = 1−R. Note that the summation for τ in Eq. (3.8a)

is a geometric series from the second term onwards, whereas the summation for ρ in

Eq. (3.8b) is geometric series from the first term, as for a standard FP interferometer

[122]. The relative phase of each term is determined by the value of exp(iβL), which is

associated with a single pass through M2. We return to this property when considering

reflection resonances in Section 3.4.3.

Here we have derived a semianalytic model to describe the transmission properties

of the FDC that requires numerical input of only the modal dispersion properties in

region M2. These are calculated easily using the Bloch mode method or another PhC

calculation tool, and substituted into Eq. (3.9) to provide a very efficient formulation

to study FDC behaviour. The properties predicted by the model are discussed in

Section 3.4.3, but first we outline an alternative derivation of Eqs. (3.8) based on Bloch

mode scattering matrices.

3.4.2 Semianalytic model: Bloch mode matrix method

The FDC shown in Fig. 3.9(a) can be analysed using the Bloch mode method as a

three layer structure, and is therefore described by Eqs. (2.13) in terms of the Fresnel

matrices at each interface. Propagation from M1 to M3 is characterised by six Fresnel

matrices, R12, R21, R23, T12, T21 and T23. We showed in Section 3.3.2 that it is not

always necessary to retain the evanescent Bloch modes in the calculation if the interfaces

in the structure are sufficiently far apart. In this limit, accurate results can often be

obtained by including only the propagating modes in each section. This approach can

be applied to the FDC, however in contrast to the Fabry-Perot example, there are two

propagating modes in the central section, rather than one.


The truncated matrix dimensions are determined by the number of propagating

modes on either side of the relevant interface. For instance, R12 describes modes incident

and reflected in section M1, which only supports a single mode, hence R12 is a 1 ×1 matrix, and for the same reason, R21 and R23 are 2 × 2 matrices. T12 describes

transmission of the single mode in M1 into the two modes of M2, and hence it is a

column vector with two rows (2× 1). Similarly, T21 and T23 are 1× 2 row vectors. The

matrix Λ, which represents propagation through region M2, reduces to a 2×2 diagonal

matrix of eigenvalues µi = exp(iβjd), where j = 1, 2. When these reduced matrices are

substituted into Eq. (2.13b), the resulting transmission coefficient is a complex scalar

quantity.

While the transmission properties of the FDC are described accurately by such a

propagating mode treatment, the computational requirements are only modestly re-

duced since the full plane wave basis is still required to represent the Bloch modes.

We can however, determine approximate analytic expressions for the Fresnel matrices

using the same assumptions as in Section 3.4.1. At the first interface, |ψin〉 is perfectly

transmitted into |ψ±〉 with amplitudes given by Eq. (3.5) which gives

R12 = 0 , T12 =

[1/√

2

1/√

2

]. (3.10)

At the M2–M3 interface, the field in the right-hand guide is transmitted freely into the

output guide, so we can again use Eq. (3.5) to write

T23 = [ 1/√

2 −1/√

2 ]. (3.11)

The reflection matrix at the second interface is a 2 × 2 matrix, where element (i, j)

is the reflection coefficient for the reflection of mode j into mode i (travelling in the

opposite direction), and modes 1 and 2 are the even and odd supermodes, respectively.

Once again, assuming that a mode in the right guide is completely transmitted while a

mode in the left guide is completely reflected back into the left guide, we can write

R23 .

[1/√

2 1/√

2

1/√

2 −1/√

2

]=

[1/√

2 0

1/√

2 0

]⇒ R23 =

[1/2 1/2

1/2 1/2

]. (3.12)

Similar arguments are used to find the final two scattering matrices,

R21 =

[1/2 −1/2

−1/2 1/2

]and T21 = [ 1/

√2 1/

√2 ] . (3.13)

Equations (3.8) are derived by substituting these ideal Fresnel matrices into Eqs. (2.13)

and making the appropriate substitutions for β and ∆β. Although the result is iden-

tical to that derived using the coupled mode analysis in Section 3.4.1, the technique

described in this section provides a systematic approach that is more easily applied to

complicated structures. In addition, it is possible to improve the accuracy of the model

by replacing the approximate matrix entries derived above by more accurate estimates.


For instance, it is straightforward to include a small reflection at the first interface by

setting R12 6= 0 and making the appropriate modifications to T12 to maintain energy

conservation.

3.4.3 Transmission properties

In this section we investigate the transmission characteristics of the FDC and compare

the semianalytic model derived in Section 3.4.1 to full numerical calculations using the

Bloch mode method. The model is used to identify parameters for which the FDC

operates as a high-Q notch-rejection filter with a resonance width determined largely

by the mode splitting parameter ∆β. We also discuss methods of tuning the FDC

geometry to increase Q and reduced the device length.

The transmission spectra in Fig. 3.10 exhibit a number of high reflection and high

transmission features that we now identify using the model derived in Section 3.4.1.

Each of these features is driven by one of two physical effects; the features associated

with particular values of ∆βL result from coupling between the waveguides inM2, while

those features associated with particular βL values are caused by mode recirculation

in M2. We consider first the high transmission features. From Eq. (3.9) it can be seen

that all incident light is transmitted when cos(∆βL) = 0 which corresponds to L = Lc.

The physical explanation for this is clear from Fig. 3.9: light incident in mode |ψin〉is completely coupled into |ψout〉 at the end of M2 where it is transmitted into the

output waveguide with minimal reflection. This single-pass behaviour can be seen in

the field intensity plot of Fig. 3.11(a), calculated for a FDC with L = 4d and Nc = 1

at d/λ = 0.319. For these parameters, T > 99.9%. Provided that cos(βLc) 6= 0, the

reflection zero of Eq. (3.9) has a quartic dependence on cos(∆βL), resulting in the flat-

topped, high-transmission parts of the spectra in Fig. 3.10(a). The Nc = 2 and Nc = 3

curves in Fig. 3.10(b) do not exhibit this behaviour since the coupling is much weaker

and hence L < Lc.

If the cavity length is doubled to L = 2Lc, then all of the light is coupled back

into mode |ψin〉 by the end of M2, where it is reflected back through M2 and out

of the input guide. From Eq. (3.9) it is seen that L = 2Lc gives sin(∆βL) = 0, and

hence R = 1. Figure 3.11(b) shows the field intensity in a FDC with Nc = 1 and

L = 8d, calculated at d/λ = 0.317, for which T < 1%. The small frequency difference

is attributed to an additional phase shift introduced when the mode is reflected – recall

that we ignored any phase change on reflection by setting the reflection coefficient equal

to 1 in Section 3.4.1.

For the purpose of designing a notch-rejection filter, we are interested in the second

condition for which R = 1 in Eq. (3.9), namely when cos(βL) = 0, or exp(iβL) = ± i.Unlike the transmission and reflection features that we have just discussed, this reflec-

tion resonance is a result of mode recirculation through the double guides in M2. To

see this, consider again the series for ρ and τ in Eqs. (3.8a) and (3.8b) which corre-

spond to the transmitted and reflected field amplitudes on each pass through section


0 0.2 0.4 0.6 0.8 1

Field intensity (a.u.)

(a) (b)

Figure 3.11: (a) Field intensity in a FDC of length L = 4d at d/λ = 0.319, for which

the reflected power is less than 0.1%. (b) Field intensity for a FDC of length L = 8d at

d/λ = 0.317d, for which R > 99.9%. The geometry of the two devices is identical aside

from the length.

M2. When the resonance condition is satisfied the field transmitted on the first pass,

τ1 is 180◦ out-of-phase relative to transmitted fields on each subsequent pass. On the

other hand, the reflection terms ρn are all in-phase and hence they combine construc-

tively to produce the resonant reflection. This resonance is the cause of all of the local

transmission minima seen in Fig. 3.10 since L < 2Lc in each example and hence the

first R = 1 condition identified above is not satisfied. Observe that the resonance width

varies considerably for the different structures, but appears to be narrowest when the

transmission on either side of the resonance is close to 1. This is not a coincidence, as

we now show.

Suppose that the resonance occurs at a frequency ω = ω0, and denote the values of

∆β and β at resonance by ∆β0 and β0. For frequencies ω = ω0 + δω close to ω0, ∆β

and β are expanded to first order as

∆β = ∆β0 + δω∆β1 , β = β0 + δωβ1 , (3.14)

where

∆β1 =

(d∆β

dω

)ω=ω0

and β1 =

(dβ

dω

)ω=ω0

.

Since we are interested in the resonance related to the zeros of cos(βL), we set β0 = nπ/(2L)

for n = 1, 3, 5, . . . and substitute Eqs. (3.14) into Eq. (3.8b) to obtain

ρ =−1

1− 2iδω/∆ω, R =

1

1 + (2δω/∆ω)2, (3.15)


where R = 1/2 at ω = ω0 ±∆ω/2 and

∆ω =cos2(∆β0L)

β1L. (3.16)

Note that Eqs. (3.15) describe a classic Lorentzian reflection resonance, as expected for

a simple resonant cavity [113,190].

Our earlier observation that the resonance is narrowest when the surrounding trans-

mission is high is verified by Eq. (3.16), since bothR→ 0 and ∆ω → 0 as cos(∆β0L) → 0.

This is a desirable feature for notch-rejection filters which ideally transmit all frequen-

cies ω 6= ω0. We note also that the β1 term in Eq. (3.16) is a function of the group

velocities of |ψ+〉 and |ψ−〉. Since β is well approximated by the dispersion of a single

waveguide for frequencies well above cutoff (see Section 3.2.1), we have β1 ≈ 1/vg. Thus

Q is approximately inversely proportional to the group velocity of the waveguides at the

resonant frequency. A similar dependence is displayed by other coupled-cavity designs,

and this has been used to improve their resonant characteristics [83, 191]. Equations

(3.9),(3.15) and (3.16) also show that the resonant frequency for a FDC of length L is

determined largely by β, whereas the resonance width depends strongly on ∆β. Since

∆β is more sensitive than β to changes in the coupling conditions, it is therefore possible

to tune the resonance width by modifying the structure of region M2, for instance by

changing the waveguide separation or modifying the cylinders between the two guides.

An example of the latter method is given in Section 3.4.4, where we demonstrate tuning

of Q by almost four orders of magnitude.

The results of Eqs. (3.9) and (3.15) (dashed curves) are compared to the full nu-

merical calculations (solid curves) in Figs. 3.12(a) and (b) for a FDC of length L = 5

and Nc = 1. The dashed curve in (a) is given by Eq. (3.9) with L = 6.067d, which was

chosen to give the best agreement with the numerical result. Although L was defined

in Fig. 3.9 as the number of PhC layers forming region M2, this is not necessarily

the most accurate value to use in Eq. (3.9) since the waveguide ends act as distributed

reflectors and hence the phase change on reflection can be relatively large. This effect

was neglected in Section 3.4.1, but can be partly corrected for by adjusting L. In many

cases a fixed L value gives reasonable agreement for a small range of frequencies, as can

be seen in Figs. 3.12(a) and (b), but this does not take into account the wavelength

dependence of the phase change. Attempts to include a wavelength-dependent length

correction by considering the phase of the reflected mode have proved no more accurate

than the simple result, suggesting that there are additional physical effects that are not

accounted for in our model. In most cases, the length adjustment necessary for the best

agreement with numerical results is approximately one lattice period or less.

The transmission spectrum in Fig. 3.12(b) is a high-resolution plot of the sharp

reflection resonance observed at d/λ = 0.33107 in (a) where the resonant reflection

exceeds 99.95%. For operation at λ = 1.55µm, corresponding to d = 513 nm, the

length of M2 is only L = 5d = 2.6µm or 1.7 wavelengths, and the resonance width is

approximately 0.11 nm. From the resonance we calculateQ = 1.3×104 for the numerical


0.31 0.315 0.32 0.325 0.330

0.2

0.4

0.6

0.8

1

d/l

T

0.2

0.4

0.6

0.8

1

0

T

d/l

(a)

(b)

(c)

105

100

102

103

Field intensity (a.u)

104

101

0.33102 0.33106 0.33110 0.33114

Figure 3.12: (a) Transmission spectrum as a function of normalised frequency for an

un-optimised FDC of length L = 5d and Nc = 1. The solid curve in (a) and (b) is

the full numerical result and the dashed curve is given by Eq. (3.9) with L = 6.066.

(b) High resolution plot of the resonance in (a) centred at d/λ = 0.33107. (c) Field

intensity in the FDC on resonance. Note that the colour scaling is logarithmic and

spans five orders of magnitude.

result (solid curve) and Q = 2.4 × 104 for the approximate result (dashed curve). As

expected, there is a strong field enhancement in regionM2 when the resonance condition

is satisfied, as shown in Fig. 3.12(c), where the field intensity on resonance is plotted

on a log scale. The peak intensity is more than 4 orders of magnitude greater than the

input intensity, which is consistent with the Q of the resonance. Despite the factor of

two difference in the peak width, the simple model predicts the broader characteristics

of the FDC with reasonable accuracy, and as we show in Section 3.4.4, it can be used

as an efficient aid to device optimisation.

3.4.4 High-Q tuning

In this section we demonstrate fine-tuning of the FDC response by changing the radius

of the cylinders that separate the two guides in region M2. The resonance width is

varied by almost four orders of magnitude to obtain an optimised FDC design of length

L = 4d that exhibitsQ > 6×105. The change inQ is predicted accurately by Eq. (3.16).

We begin with the FDC illustrated in Fig. 3.11 (a), which has a length of L = 4d.


0.312 0.314 0.316 0.318 0.32

0.2

0.4

0.6

0.8

1

00.31

d/l

T

0.26 0.28 0.3 0.32 0.3410

103

104

105

106

102

Q

rc

Bloch mode method (L=4d)Model (L = 4.83d)Model (L = 4.90d)

r =0.25dc

r =0.26d

r =0.27d

r =0.28d

r =0.29d

r =0.30d

r =0.31d

r =0.32d

r =0.33d

r =0.34d

r =0.35d

c

c

c

c

c

c

c

c

c

c(a)

(b)

Figure 3.13: (a) Transmission spectra for a FDC of length L = 4d as the cylinder radius

between the two waveguides (rc) is changed between 0.25d and 0.35d. The black curve

corresponds to the original structure with rc = 0.34d. (b) Variation of Q with rc. The

black points correspond to the curves in (a), while the red and blue points are given by

Eq. (3.16) for two different value of L.

The transmission spectrum for the unmodified structure is indicated by the dashed line

in Fig. 3.10(a) and exhibits a resonant reflection close to d/λ = 0.315 with Q = 1300.

The same curve is shown in Fig. 3.13(a) for a narrower range of frequencies, where

the black curve corresponds to the original FDC with a cylinder radius of r = 0.3d.

The remaining curves in Fig. 3.13(a) show the transmission spectrum as the radius of

the four cylinders between the guides is varied between rc = 0.25d and rc = 0.35d.

Observe that the resonant frequency decreases with increasing radius, while the peak

width initially narrows to a minimum at rc = 0.27d, then broadens again.

Figure 3.13(b) is a plot of Q as a function of rc, where the black circles are calculated

directly from the transmission spectra in (a). Observe that Q varies by almost four

orders of magnitude, and reaches a maximum of Q = 6× 105 when rc = 0.27d. In this


105

100

102

103

Field intensity (a.u)10

410

610

1

Figure 3.14: Resonant field intensity in the optimised FDC with L = 4d and Nc = 1.

The four central cylinders have a radius of rc = 0.27d: 0.03d smaller than the other

cylinders. Note that the colour scaling is logarithmic and spans six orders of magnitude.

purely 2D geometry, Q can be made arbitrarily large in principle by adjusting rc to

match the ∆ω = 0 condition given by Eq. (3.16). The red and blue points plotted in

Fig. 3.13 are given by Eq. (3.16) for L = 4.83d and L = 4.90d respectively, where ∆β0

and β1 have been calculated for each value of rc. The choice of L = 4.83d gives the best

agreement between Eq. (3.9) and the rc = 0.3d curve in Fig. 3.13, whereas L = 4.9d

provides the best agreement with the numerically calculated Q values.

The resonant field intensity inside the optimised FDC is shown in Fig. 3.14. Observe

that the peak intensity inside region M2 is more than five orders of magnitude larger

than the incident field, clearly illustrating the high-Q properties of the FDC.

The results in this section provide a convincing demonstration of how the semian-

alytic model can be used to great advantage when optimizing design parameters. In

this example, the results are summarised by the plot in Fig. 3.13(b) which indicates the

variation of Q as a function of rc. Each black point in (b) corresponds to a single curve

in (a), each of which must be calculated with enough points to resolve the resonance.

The curves in (a) each consist of approximately 250 points, and many more would be

required if the frequency grid was uniform. One data point takes approximately 17 s to


calculate using the Bloch mode method on a Pentium 4 (2.8GHz) desktop computer,

which gives a total computation time of 13 hours to generate the 11 curves in Fig. 3.13.

In contrast, the blue and red data points in Fig. 3.13(b) are given by Eq. (3.16), for

which we only need to calculate the coupled waveguide dispersion curves for each value

of rc. Since β and ∆β are slowly varying functions of frequency only 11 data points

were used to calculate the dispersion curves and then a polynomial function was fitted

to allow interpolation and differentiation. Each point takes approximately 7 s to cal-

culate, so the total calculation time for the 11 different rc values was approximately

810 s – fifty-five times faster than the full calculation. Once the dispersion data was

calculated, the red and blue data points in Fig. 3.13(b) were generated in a few seconds

from Eq. (3.16).

Although the accuracy of the model depends on the choice of L, this can be esti-

mated using a number of methods. For instance, the red points in Fig. 3.13(b) were

calculated using the L value that gave the best match between Eq. (3.9) and the original

spectrum. Even if this meant one full transmission spectrum was required, the reduc-

tion in computation time is still significant. Approximate methods do not replace full

numerical calculations, which would still be required for the final optimization process,

however they can be used to restrict the computational demand to a minimum.

3.4.5 Comparison with side-coupled cavity

Although it is not immediately obvious from the geometry, the FDC is closely related

to other waveguide-coupled cavities such as those in Refs. [115, 120, 121, 192], in which

a short line-defect is coupled by evanescent fields to a straight PhC waveguide as il-

lustrated in Fig. 3.15(a). A number of semianalytic theories have been developed for

describing the coupling in this type of structure based on tight-binding approximations

and scattering-theory [113,190,193,194]. Here we show that the simple mode-coupling

analysis developed in Section 3.4.1 also provides an accurate approximation for simple

side-coupled cavity geometries. A feature of these designs is that the in-plane Q, in-

creases exponentially with the distance between the cavity and the waveguide [192]. In

contrast, it is clear from Fig. 3.10 that the FDC resonances start to disappear as the

waveguide separation increases. To understand this difference we consider the simple

side-coupled cavity (SCC) design illustrated in Fig. 3.15(a), which resembles the FDC

in that it consists of single input and output waveguides coupled into a section of double

waveguide. This structure can be modelled using the same semianalytic method applied

to the FDC in Section 3.4.1. The only difference in the analysis is that the light in the

right-hand waveguide is reflected at the second interface of the SCC, whereas in the

FDC the light in the left-hand waveguide was reflected. The equation for the reflected

power of the SCC is thus found to be

R =sin4(∆βL)

sin4(∆βL) + 4 cos2(∆βL) sin2(βL), (3.17)


2 3 4 5 6 710

2

103

104

105

106

Nc

Q

L

Nc

(a) (b)

Figure 3.15: (a) Diagram of the side-coupled cavity geometry considered in Sec-

tion 3.4.5. The example shown here is for Nc = 6. (b) Plot of Q as a function of

Nc for a cavity with L = 2, as in (a). Black points are calculated using the Bloch mode

method and the red points are given by Eq. (3.18) for L = 2.882.

which is identical in form to Eq. (3.9) but with the cos(∆βL) and sin(∆βL) terms

swapped and sin(βL) in place of cos(βL). In analogy to the FDC, the resonance of

interest occurs in the SCC for sin(βL) = 0 and has a width approximated by

∆ω =sin2(∆β0L)

β1L, (3.18)

which is the equivalent of Eq. (3.16).

Let us now compare Eqs. (3.16) and (3.18) to identify the differences between the

FDC and the SCC. For both geometries, the position of the resonance is determined by

βL whereas the resonance width depends on ∆βL and β1L. Recall from Section 3.4.3,

that the maximum Q of the FDC occurs when cos(∆βL) ≈ 0, and thus we were able

to tune the Q in Section 3.4.4 by modifying the coupling conditions, and hence the

value of ∆β in the cavity region. Similarly, to maximise the Q of the SCC, we want

sin(∆βL) = 0 in Eq. (3.18). This can be achieved in one of two ways. The first is to

design the cavity such that L = 2mLc, where m is an integer, and hence ∆βL = mπ,

but this requires a cavity twice the length of the equivalent FDC. The second option is

to let ∆β → 0, which can be done by increasing the waveguide separation. This latter

approach for tuning Q is the most common technique used in the literature (see for

example Ref. [192]).

While we have only considered a basic SCC geometry, it is interesting to compare

the relative sizes of a FDC and a SCC with the same Q. In Section 3.4.4 we designed a

FDC with L = 4d and Q = 6× 105. This resonance corresponded to βL = 3π/2 – the

second zero of cos(βL). A shorter FDC would not have been possible in this geometry,

since we require L = Lc in order to maximise Q, and hence the shortest length is

determined by the maximum ∆β that can be achieved. The SCC does not have the

same length limitation since Q can always be increased by decreasing ∆β. Hence, the


first practical resonance occurs when βL = π. This condition is satisfied for d/λ ≈ 0.316

for a cavity of L = 2d as illustrated in Fig. 3.15(a). Figure 3.15(b) is a plot of Q as

a function of Nc for Nc = 1 − 7 where the black points were calculated using the full

Bloch mode method and the red points are given by Eq. (3.18) with L = 2.882d. This

value of L was chosen so that the resonant frequency predicted by Eq. (3.17) matched

the numerical results for Nc = 4. As in the FDC example, the semianalytic model

provides an accurate estimate of the resonant properties of the cavity.

Despite the exponential increase of Q with Nc, the SCC still requires Nc > 6 for

Q > 105, which corresponds to a distance of more than 7d between the cavity and

the waveguide. This is approximately twice the size of the FDC design considered

in Section 3.4.4. Although a rigorous comparison of relative sizes would include the

area of PhC required around each design to confine the light in the plane, the simple

comparison here suggests that the FDC compares favourably with alternative high-Qfilter designs.

3.4.6 Discussion

In this section we have demonstrated a novel filter design that exhibits high Q reso-

nances in a compact geometry. Since it is essentially an in-line device it can be easily

coupled to, and the resonance properties can be optimised by modifying the geometry

of the coupling region. As with any resonant cavity, however, the control of out-of-

plane losses is critical if such high-Q values are to be achieved in practice. Although we

have not considered losses in our analysis, it is expected that low-loss designs could be

optimised in much the same way as the cavities discussed in Section 1.7.1. We return

to this issue in Chapter6.

The comparison between the FDC and a simple side-coupled cavity filter in Sec-

tion 3.4.5 shows that the two devices have similar resonance characteristics when oper-

ated as narrow-band filters, but we have not compared other functions such as add-drop

behaviour or nonlinear switching characteristics. A more detailed study of these prop-

erties would be required in order to establish the relative merits of the two geometries.

Although we do not consider the structure here, we note that the modification pro-

posed by Fan [195] to achieve asymmetric line shapes in side-coupled cavities can also

be applied in the case of the FDC.

In addition to the high-Q resonances studied in this section, the transmission spectra

in Fig. 3.10(a) also exhibit regions of flat, almost perfect transmission. This character-

istic behaviour is exploited in Section 3.5 to design an efficient wide-bandwidth sym-

metric Y-junction that is incorporated into a PhC-based Mach-Zehnder interferometer

in Chapter 4. We also study the transmission properties and band structure of a chain

of identical FDCs in Section 3.6.

3.5. COUPLED Y-JUNCTION 91

3.5 Coupled Y-junction

Efficient, wide-bandwidth Y-junctions or beam splitters are required for compact inte-

gration of multiple optical devices on photonic chips. As a basic component of many

integrated optical devices, these junctions must be designed with a minimum back

reflection and maximum transmission over the operating bandwidth. A number of

studies have been made into the optimal method of splitting a single guided mode into

two modes, and the approaches taken can be broadly classified into two groups. The

first design approach has been to optimise simple Y-junction designs, while the sec-

ond approach has been to design novel alternative structures exploiting the properties

of photonic crystals. Good transmission has been demonstrated in Y-junctions both

theoretically [77,196–198], and experimentally [198,199] using optimisation techniques

such as placing one, or several, “tuning” cylinders near the junction of three wave-

guides. High transmission junctions, with calculated transmissions up to 99% have

been designed, however in most cases, the transmission bandwidth decreases rapidly

as the maximum transmission approaches 100% [77]. Typical bandwidths reported in

the literature for hole-type PhC junctions are on the order of 10 − 40 nm for 95%

transmission.

An alternative to the Y-junction is essentially a directional coupler with L = Lc/2

so that light incident in one of the guides is split evenly between the two guides. Such

a structure was proposed by Martinez et al. [200] as a means of splitting an input

beam for a Mach-Zehnder interferometer application. The optimal coupling length is

wavelength dependent, but may be robust enough to give a satisfactory bandwidth.

One of the big advantages of a coupler-based beam splitter is that the back reflection

can be negligible. If the coupler is not perfectly matched to the wavelength, the splitting

between the output guides changes, but the total transmission of the splitter remains

high. This is an important issue in compact devices where back reflections can cause

unwanted interference effects. A modified version of the directional coupler junction

was demonstrated by Sugimoto et al. [109] in which a cavity was placed between the two

waveguides to create a similar geometry to the coupled cavity add-drop filter illustrated

in Fig. 1.9(a). Couplers of this type have been successfully integrated into PhC Mach-

Zehnder interferometers [201].

Here, we present a novel coupled Y-junction design which operates similarly to

a directional coupler junction, but with the symmetry properties of a Y-junction that

ensure the transmission into each output arm is identical even when the coupling length

is not perfectly matched to the wavelength. In addition, recirculation inside the junction

results in high transmission over a wide bandwidth. We show that the junction is

closely related to the folded directional coupler, and a modal analysis similar to that in

Section 3.4 yields identical expressions for the transmission and reflection. While similar

structures have been studied in dielectric waveguides [202, 203], the PhC-based device

gains a significant advantage from the perfectly reflecting cladding which provides mode

recirculation that increases the transmission bandwidth.


L

(a) (b)

��

��

��

��

Figure 3.16: (a) Geometry of the coupled Y-junction. (b) Schematics of the symmetric

propagating modes in the three waveguide sections.

The general geometry of the coupled Y-junction is illustrated in Fig. 3.16(a). A

single input guide enters a coupling region of length L consisting of three identical

waveguides. At the output interface, the central waveguide ends and the two remaining

waveguides continue as outputs. Light entering the triple guide region in the central

waveguide is coupled into the two guides on either side. If the length of the triple guide

region is such that all of the light has been transferred into the outer guides at the end

of the triple guide region, then almost all light propagates into the two output guides.

If the output guides are sufficiently far apart, the coupling between them is negligible.

In the junction shown in Fig. 3.16 the output guides are separated by five layers of

cylinders so the coupling length is several hundred lattice periods. Hence, for compact

devices the guides can be treated as isolated from each other.

A semianalytic model for the coupled Y-junction is derived in the same way as

the FDC model in Sec. 3.4.1, and we show here the two structures are described by

an identical set of equations. We must first consider the propagating modes within

each of the three regions. The input guide supports a single even mode, the output

guides support an odd and an even mode, and the central triple guide section supports

three modes - two even and one odd - as discussed in Section 3.2.2. However, since

the structure is symmetric and the input waveguide only supports a single even mode,

light can only couple to the even modes in each section. Hence, the single input mode

couples into two modes in the central region, which in turn couple into a single output

mode, in much the same way as in the FDC. Schematics of the even modes in each

region of the coupled Y-junction are shown in Fig. 3.16(b).

Recall from Section 3.2.2 that the supermodes of the triple guide region can be

approximated by linear superpositions of the individual waveguide modes. Hence we

3.5. COUPLED Y-JUNCTION 93

can use Eq. (3.3) to write

|ψ1〉 = (|ψL〉+√

2 |ψin〉+ |ψR〉)/2 = (|ψin〉+ |ψout〉)/√

2,

|ψ3〉 = −(|ψL〉 −√

2 |ψin〉+ |ψR〉)/2 = (|ψin〉 − |ψout〉)/√

2,

(3.19)

where |ψL〉, |ψin〉 and |ψR〉 are the modes of the left, centre (input) and right waveguides

respectively, and |ψout〉 = (|ψL〉+ |ψR〉)/√

2 is the even mode of the output waveguides.

As in the case of the FDC, the propagation constants of |ψ1〉 and |ψ3〉 are expressed as

β1 = β + ∆β and β3 = β −∆β, where β = (β1 + β3)/2 and ∆β = |β1 − β3|/2.

Equations (3.19) reveal the direct mapping between the coupled Y-junction and

the FDC; |ψ1〉 and |ψ3〉 are equivalent to |ψ+〉 and |ψ−〉 in Eq. (3.5) while the input

and output modes of the two structures are also equivalent. Hence, the transmission

properties of the coupled Y-junction are described by Eqs. (3.8) and (3.9) with β and

∆β calculated for the even modes of the triple guide section.

Although the coupled Y-junction also exhibits the sharp resonances of the FDC, it

is the flat, high-transmission regions of the spectrum that we wish to exploit for the

purposes of this device. In Section 3.4.3 we identified this feature with the condition

cos(∆βL) = 0, which corresponds to L = Lc. When this condition is satisfied, all of the

incident light is coupled from the central input waveguide into the output waveguides

on the first pass through the triple guide section, and so the transmission is high.

Since the reflectance given by Eq. (3.9) is quartic in cos(∆βL), the high transmission

is typically flat-topped and broad. In order to maximise the transmission bandwidth,

it is desirable to have cos(βLc) ≈ 1 so that the reflection resonances are far from the

maximum transmission frequency, although reasonable bandwidths are possible without

this condition, as demonstrated in Section 4.4.1.

Figures 3.17(a) and (b) show the transmission through two different coupled Y-

junctions, with lengths L = 5d and L = 7d respectively. The junction is formed in

the PC1 lattice, with the output guides separated by 5 rows of cylinders, and each of

the guides in the central region separated by 2 rows of cylinders. The lattice period

is d = 491 nm in both examples, corresponding to a cylinder radius of 173 nm. For

lengths of L = 5d and L = 7d, the Y-junction has 95% transmission bandwidths of

approximately 92 nm and 83 nm, respectively at λ = 1.55µm, and 99% transmission

bandwidths of about 80 nm, and 76 nm. The dashed curves in Figs. 3.17(a) and (b) are

the semianalytic result (3.9) calculated with L = 5.75d and L = 7.80d respectively for

the two devices. The solid curves show the transmission calculated using the full Bloch

mode method. As in the examples of Section 3.4, the difference between the length

defined in Fig. 3.16 and the length parameter in the model results from phase changes

upon reflection, and the ambiguity in defining a reflection plane at either end of the triple

guide region. The semianalytic model was used to design the two junctions, and it can

be seen that the transmission bandwidth obtained from the full Bloch mode calculation

exceeds the bandwidth predicted by the model. In Section 4.4.1 we present two other

coupled Y-junction designs - one in a rod-type PhC and the other in a triangular


0.2

0.4

0.6

0.8

1

T

1.48 1.5 1.52 1.54 1.56 1.58 1.6

0.2

0.4

0.6

0.8

1

T

92nm

83nm

λ (µm)

(a)

(b)

L = 5d

L = 7d0

Figure 3.17: Transmission curves for the coupled Y-junction with (a) L = 5d and (b)

L = 7d. Wavelengths have been scaled so that the high transmission bands occur near

λ = 1.55µm. The 95% transmission bandwidths are 92 nm and 83 nm respectively. The

solid curves are calculated using the full Bloch mode method, and the dashed curves

are the semianalytic result given by Eq. (3.9) for L = 5.75d and L = 7.80d.

lattice hole-type PhC. The hole-type PhC junction has a 95% transmission bandwidth

of 39 nm, which compares favourably with alternative junction designs reported in the

literature [77,197,198]. It is expected that the bandwidths could be further increased by

tuning the structure of the triple guide region as demonstrated in Section 3.4.4 for the

FDC. We note that since this work was reported in Ref. [204], similar junction designs

have been proposed by other authors [205, 206], however the structures we present

here exhibit superior bandwidth and transmission properties. There have also been

several experimental demonstrations of efficient junction designs obtained via numerical

optimisation of the structure [199,207].

When designing the coupled Y-junction for maximum bandwidth, the splitting of

the modes in the triple guide region must be considered. Since the successful operation

of the device requires two propagating even modes to exist in this region, we must

operate in the frequency range above the cut-off of the second even mode, and below

3.6. SERPENTINE WAVEGUIDE 95

the high-frequency edge of the bandgap. If the three guides are too close together,

the mode splitting becomes very large, and the frequency range for operation becomes

limited. This is clearly seen in the triple waveguide dispersion curves for Nc = 1

in Fig. 3.4. On the other hand, if the guides are too far apart, the coupling length

becomes prohibitively large. Thus, the choice of guide separation and its effect on

mode splitting must be considered. In the examples chosen above, the second even mode

cutoff occurs at λ ≈ 3.32d, giving a maximum bandwidth of ∆λ = 0.33d corresponding

to approximately 160 nm at λ = 1.55µm. The transmission bandwidth can also be

increased by operating at frequencies where the dispersion curves of all three modes

are approximately parallel, such that ∆β and hence Lc are very weak functions of

wavelength [205].

The coupled Y-junction design provides a novel solution to the problem of achieving

high-transmission, wide-bandwidth beamsplitting, by combining the efficient coupling

properties of a directional coupler with the symmetrical splitting of more conventional

Y-junction designs. Such properties make this junction ideal for use in PhC-based

Mach-Zehnder interferometers where high performance junctions are required. We

demonstrate this application in Chapter 4. While the 2D results presented here are

promising, further work is required to give an accurate comparison with alternative

Y-junction designs, not only in terms of transmission and bandwidth, but also regard-

ing fabrication tolerance. One issue of particular interest is the sensitivity to structural

variations that break the symmetry properties of the junction. We have only considered

the even modes in our analysis, however any breaking of the symmetry allows light to

also couple into the odd modes. Although this is likely to result in unequal splitting

between the two output waveguides, it is not clear whether the back reflection would

increase significantly.

3.6 Serpentine waveguide

In this section we consider the serpentine waveguide – a period coupled-cavity waveguide

formed by cascading multiple FDC structures as shown in Fig. 3.18. Periodic coupled

resonators such as coupled-resonator optical waveguides (CROWs) and coupled ring

resonators have been studied previously for a range of applications, including add-drop

filtering, slow-light propagation, and nonlinear switching [101, 102, 208–213]. The ser-

pentine waveguide is of particular interest as a slow-light structure due to the existence

of very low group velocity bands as we demonstrate here. In Section 3.6.1 we derive a

dispersion relation for the serpentine waveguide in terms of the transmission and reflec-

tion properties of a single FDC. The dispersion properties of the periodic structure can

thus be evaluated using either the full Bloch mode method or the semianalytic expres-

sions in Eq. (3.8). We calculate the band structure and transmission properties of two

different serpentine waveguides in Section 3.6.2 and discuss their general properties.


L1

L2

L1

L /22

Dy

y3

y2

y1

L /22

(a) (b)

Figure 3.18: Two equivalent serpentine waveguide geometries (assuming no tunnelling

through the guide ends). Both are characterised by the period Dy and the double and

single guide lengths L1 and L2.

3.6.1 Derivation of the dispersion relation

The unit cell of the serpentine waveguide is a pair of coupled FDC devices joined input-

to-output by a single waveguide of length L2 to create a periodic structure with a period

Dy = 2(L2 + L1) as shown in Fig. 3.18(a). Each of the individual components of this

structure is essentially the same as those of the isolated FDC device. In the sections of

length L1 there are two parallel waveguides that support a pair of propagating modes

which allow light to couple from one guide to the other. The single guide sections of

length L2 act as the input/output guides to the next/previous coupling region. If L2

is sufficiently long that evanescent tunnelling between the blocked waveguide ends is

negligible, the only connection between the coupling regions is via the one propagating

mode of the single guide. In this limit, the structures shown in Figs. 3.18(a) and (b) are

equivalent and hence a single FDC, rather than a complete unit cell, is the minimum

segment of the serpentine required to determine the band structure. As we show in

Section 3.6.3, this property holds for the structure in Fig. 3.18(a) even when the single

mode approximation is invalid.

We now proceed to analyse the structure using a single mode approximation which is


valid when the L2 is sufficiently long to ensure coupling between the double waveguide

segments in Fig. 3.18 occurs only via the propagating mode in the single waveguide

section. Consider first the transmission and reflection characteristics of a single period

of the serpentine waveguide, bounded by the interfaces y1 and y3 shown in Fig. 3.18.

Using the notation of Chapter 2 we can express the fields at interface yj in terms of

their upward and downward components b+j and b−j respectively. In the propagating

mode analysis here, the b±j are complex amplitudes of the single propagating waveguide

mode, which are related via a transfer matrix according to[b−3b+3

]= Ts

[b−1b+1

]. (3.20)

Denoting the complex transmission and reflection amplitudes of a single period by

τs and ρs respectively, we can write down the transfer matrix

Ts =

[τs − ρ2

s/τs ρs/τs−ρs/τs 1/τs

]. (3.21)

Since the basic unit of the serpentine is the FDC bounded by y1 and y2 in Fig. 3.18,

we can rewrite Eq. (3.20) as

Ts = T 2f , where Tf =

[τf − ρ2

f/τf ρf/τf−ρf/τf 1/τf

](3.22)

is the transfer matrix for a FDC with input and output guides of length L2/2. Here,

τf and ρf are the transmission and reflection coefficients of a FDC with an additional

phase term exp(iβL2) to include propagation between the FDC and interfaces y1 and

y2. β is the propagation constant of the single input and output mode. Note that a

more general relationship between Ts and Tf , valid for all L2, is derived in Section 3.6.3.

The dispersion relation for the periodic serpentine structure is obtained by solving

the eigenvalue problem Tsb = µb, where the eigenvalues are µ = exp(iqDy) and q is

the Bloch factor in the longitudinal direction of the serpentine. Since Ts = T 2f , the

eigenvalue equation can be written as Tfb = ±µ1/2b, which has solutions that satisfy

µ+ 1

∓µ1/2=τ 2f − ρ2

f + 1

τf. (3.23)

This expression can be further simplified using the energy conservation properties

|ρf |2 + |τf |2 = 1 and arg(τf )− arg(ρf ) = ±π/2, to yield the dispersion relation

∓ cos(qDy/2) = <(1/τf ). (3.24)

The right hand side of Eq. (3.24) can be calculated using either the full Bloch mode

method or the semianalytic expression in Eq. (3.8), with τf = τ exp(iβL2) to include the

additional phase due to propagation through the single waveguide sections. The latter


method results in a semianalytic form of the right hand side, which can provide a better

understanding of the serpentine waveguide properties. Substituting the semianalytic

expression for τf into Eq. (3.24) yields

cos(qDy/2) = sin(∆βL) sin(βL1 + βL2) +cos(∆βL1) sin(2βL1 + βL2)

2 sin(∆βL1) cos(βL1), (3.25)

which can be evaluated after numerical input of the waveguide dispersion properties

and the lengths L1 and L2. In the next section we consider the properties predicted by

this dispersion relation and compare the results to full numerical calculations.

3.6.2 Serpentine waveguide properties

We consider first the different solutions to Eq. (3.25), and identify band features that

are directly related to the properties of a single FDC, and others that result from the

periodicity. The serpentine waveguide exhibits transmission bands for frequencies where

the right hand side of Eq. (3.25), <(1/τf ), has magnitude less than unity. Correspond-

ingly, band gaps, which lie between the transmission bands, occur when |<(1/τf )| > 1.

There are three distinct types of band gap, each originating through a distinct physical

effect. Following from Eq. (3.25), or from the analysis of a single FDC in Section 3.4.3,

there are two conditions for which τf = 0. The first occurs when sin(∆βL1) = 0, which

corresponds to L1 being an integer multiple of the coupling length Lc. The second

condition occurs when cos(βL) = 0, corresponding to the reflection resonance identified

in Section 3.4.3 as being suitable for narrow-band filtering. When either of these con-

ditions is satisfied, the right hand side of Eq. (3.25) diverges, and a band gap appears

in the serpentine band diagram. Similar resonator band gaps are displayed by coupled

ring-resonator structures [211] and other periodic coupled resonator systems [193]. We

label the two types of resonator band gaps identified here as RG1 and RG2 respectively.

The third class of band gap that appears in the serpentine band diagram is a Bragg

gap [211] that occurs as a result of the overall periodicity of the serpentine structure,

rather than a single feature of the FDC.

Figure 3.19(a) shows the band diagram of a serpentine waveguide with L1 = L2 = 5d,

formed in the square rod-type PhC, PC1. The FDCs making up this device are identical

to the L = 5d structure shown in Fig. 3.12(c), which has the transmission properties

plotted in Fig. 3.12(a) and (b). The very narrow RG2-type band gap near d/λ = 0.33

corresponds to the sharp reflection resonance of the FDC. In Fig. 3.19(b), the intensity

transmission is shown for the single FDC (half a period), a single whole period and two

whole periods of the serpentine waveguide. The band diagram was calculated using

τf obtained from the full Bloch mode method and the transmission spectra were also

calculated using the full numerical approach. Counting from the top of Fig. 3.19(a),

we see that top two resonator gaps are both of type RG2, and hence they correspond

to strong features in all three of the transmission curves. In contrast, the Bragg gaps

do not correspond to any obvious features in the FDC transmission spectrum, and


0.29

0.3

0.31

0.32

0.33d/

λ

��

1

Transmission

Bragg gap

Resonator gap type RG2


Bragg gap

Bragg gap

ππ/2π/4 3π/40� � ��

(a) (b)

Figure 3.19: (a) Band diagram for a serpentine waveguide with L1 = L2 = 5d, where q is

the Bloch coefficient along the waveguide and Dy = L1 +L2. (b) Transmission through

a FDC with the same parameters (dashed), one period of the serpentine waveguide

(dotted) and two periods (solid). Note that for frequencies below d/λ, the double guide

cavity only supports a single mode (see Fig. 3.2(f)), and thus the analytic result of

Eq. (3.25) does not apply.

only begin to appear in the multiple FDC serpentine structure. Given the resonance

conditions for RG1 and RG2, and since β > ∆β, it is apparent that resonator gaps of

type RG1 occur less frequently than RG2-type gaps.

If the structure is made slightly longer, a RG1-type gap appears, as shown in

Fig. 3.20(a), which shows the band structure for a serpentine waveguide with L1 =

L2 = 7d. The solid curves in (a) are the full numerical result given by Eq. (3.24) cal-

culated using the Bloch mode method, and the dashed curves are calculated using the

semianalytic expression (3.25), with L1 = 7.5d and L2 = 6.7d, which were chosen to

give the best agreement to the numerical result. Figure 3.20(b) shows the transmis-

sion spectrum through a single FDC with L = 7d (dashed curve), and two and three

periods of the serpentine waveguide, indicated by the dotted and solid curves respec-

tively. Observe that for these parameters, the resonator gaps are relatively strong even

for such a small number of periods, whereas the Bragg gaps only appear as very weak

perturbations in the transmission spectrum for two periods.

Another feature associated with both types of resonator band gap is the positioning

of the bands above and below the gaps. In a one-dimensional Kronig-Penney model, the

right hand side (RHS) of the dispersion relation is continuous, and hence only direct

gaps exist since the RHS cannot change sign without passing through zero (see, for

example Ref. [164], chapter 6). In the case of the resonator gaps however, the RHS

of the dispersion relation (3.25) diverges when either of the resonance conditions is

satisfied. When this happens, the RHS can change sign without passing back through

zero, resulting in an indirect band gap. Indirect band gaps can be seen in Figs. 3.19(a)

and Fig. 3.20(a) at each of the resonance gaps. Although indirect band gaps occur


0.31

0.315

0.32

0.325

0.33d/

λ


Bragg gap


Bragg gap

Transmission0 0.2 0.4 0.6 0.8 1�� ππ/2π/4 3π/40

(a) (b)

Figure 3.20: (a) Band diagram for a serpentine waveguide with L1 = L2 = 7d. The

solid curve is calculated with the full numerical simulation while the dashed curve is

calculated using Eq. (3.25). (b) Transmission through a FDC with L = 7d (dashed), 2

periods of the serpentine guide (dotted) and 3 periods of the serpentine guide.

commonly in two-dimensional photonic crystals, such features have only recently been

observed in other coupled resonator waveguides consisting of a linear periodic array of

resonant elements [211].

The serpentine waveguide also exhibits a number of other interesting band structures

that can be created with appropriate choices of the lengths L1 and L2 and waveguide

separation, Nc. For example, the sharp resonance at d/λ = 0.331 in Fig. 3.19(a) creates

very flat bands which correspond to low group velocities, and hence large group delays.

We showed in Section 3.4.4 that the width of a RG2-type resonance is very sensitive to

structural changes in the double waveguide section, so a similar tuning method could be

applied to adjust the group velocity response. An alternative mechanism for achieving

low group velocity bands is discussed below.

Another unusual band feature occurs when two or more consecutive band gaps

are resonator gaps, as in Fig. 3.20 where the band gaps at centre frequencies d/λ =

0.3135 and d/λ = 0.3183 are resonator gaps of type RG1 and RG2 respectively. Since

each of these gaps is indirect, the three bands above, below and between the gaps are

almost parallel to one other and all have a positive slope. With optimisation of the

geometry it may be possible to design a band structure with several equally-spaced

bands, each exhibiting a similar group-velocity profile. The width of the central band

is also very sensitive to the relative positions of the two resonances. Since the RG1 gap

depends on ∆β and the RG2 gap depends on β, it is possible to change their relative

positions by changing L1 or tuning the geometry of the double guide region. If the

two gaps are brought close together, the band between them becomes flat, and as the

two resonances pass through each other, the slope of the band, and hence the group

velocity changes sign. The group velocity could thus be controlled by tuning the relative

resonance positions. Similar properties have been identified in other coupled-resonator


y1

y2

y3

y1

y2

� R1�

~

T1�

~

FDC1

FDC2� R

2�

~

T2�

~

y2

y3

y1

y2

�

�

y2

y3

R2’ �

~

T1’ �

~

T2’ �

~

R1’ �

~

Forward Backward

(a) (b)

L1

L /22

L2

L1

L /22

Figure 3.21: (a) Schematic of the serpentine waveguide structure shown in Fig. 3.18(a).

(b) The FDC structures that are combined to form the structure in (a). Each FDC is

characterised by plane wave scattering matrices for a field δ incident from above (T, R)

and below (T′, R

′).

waveguides, and have been proposed as a basis for all-optical trapping of light [102,214].

Further work is required to determine whether a serpentine structure can be designed

to exhibit the same properties.

3.6.3 Symmetry properties of the serpentine waveguide

In Section 3.6.1 we derived the dispersion properties of the serpentine waveguide in

the limit where the single waveguide regions in Figs. 3.18(a) and (b) were sufficiently

long to neglect evanescent coupling between the ends of the blocked waveguides. In

this limit, the Bloch mode transfer matrix of the full serpentine period is equal to the

square of the transfer matrix of a single FDC (3.22). Here we use symmetry relations

to derive a more general form of Eq. (3.22) for the structure in Fig. 3.18(a) which is

valid for all L2.

Consider first the serpentine structure shown in Fig. 3.21 which consists of two

FDCs (FDC1 and FDC2) joined by a single waveguide. Since FDC2 is identical to

FDC1 after reflection about the dashed-dotted line in Fig. 3.21(a), the plane wave


scattering matrices of the two structures are related according to

R1 = SR2S T1 = ST2S

R′1 = SR

′2S T

′1 = ST

′2S, (3.26a)

where S is the reversing permutation matrix

S =

0 · · · 1

· · · 1 ·· · · · ·· 1 · · ·1 · · · 0

.

Here, the plane wave scattering matrices T1, R1, T2, R2 correspond to transmission

and reflection of a field δ incident in the forward (downward) direction and T′1, R

′1, T

′2, R

′2

correspond to incidence in the backward (upward) direction as shown in Fig. 3.21(b).

Furthermore, it can be seen from Fig. 3.21(a) that forward propagation from y1 to y2

is equivalent to backward propagation from y3 to y2 such that

T1 = T′2 T2 = T

′1

R1 = R′2 R2 = R

′1. (3.26b)

Following the notation of Chapter 2, the plane wave fields at either end of FDC1

and FDC2 are related by the transfer matrices

T pwf1 =

T1 − R′1 T

′ −1

1 R1 R′1 T

′ −1

1

−T′ −1

1 R1 T′ −1

1

, (3.27a)

T pwf2 =

T2 − R′2 T

′ −1

2 R2 R′2 T

′ −1

2

−T′ −1

2 R2 T′ −1

2

. (3.27b)

Hence, given the relations (3.26a) and (3.26b), it follows that

T pwf2 = S T pw

f1 S, where S =

[S

S

]. (3.28)

We now convert the above transfer matrix formulation from the plane wave basis

to the waveguide mode basis (Bloch mode basis). At interface yi in Fig. 3.22, the field

can be represented as an expansion of plane waves with amplitudes f±i or Bloch modes

with amplitudes c±i . At y1 and y2 these expansions are related according to[f−1f+1

]= F1

[c−1c+

1

] [f−2f+2

]= F2

[c−2c+

2

], (3.29a)


f1

±

f2

±

f3

±

c1

±

c2

±

c3

±

y1

y2

y3

�f�

BM

�f�

BM

�f�

pw

�f�

pw

�s

BM

Figure 3.22: Schematic of the unit cell of the serpentine waveguide formed from two

FDC structures. The forward and backward propagating fields at interfaces y1 − y3

are represented in the basis of plane waves f±i or Bloch modes c±i . The fields at each

interface are related by plane wave and Bloch mode transfer matrices, T pw and T BM

respectively.

where the F i provide the change of basis from plane waves to Bloch modes as defined

Section 2.2.4.

By definition, the plane wave transfer matrix T pwf1 relates[

f−2f+2

]= T pw

f1

[f−1f+1

], (3.29b)

which can be converted to Bloch mode form using Eqs. (3.29a) such that[c−2c+

2

]= F−1

2 T pwf1 F1

[c−1c+

1

]= T BM

f1

[c−1c+

1

], (3.29c)

where

T BMf1 = F−1

2 T pwf1 F1. (3.29d)

Similarly, the equivalent Bloch mode transfer matrix for FDC2 is

T BMf2 = F−1

1 T pwf2 F2, (3.29e)

which can be expressed in terms of T BMf1 using Eqs. (3.28) and (3.29d) as

T BMf2 = F−1

1 ST pwf1 SF2 = F−1

1 SF2T BMf1 F−1

1 SF2. (3.29f)

Therefore the Bloch mode transfer matrix for the full period of the serpentine waveguide

in Fig. 3.22 is

T BMs = T BM

f2 T BMf1 = QT BM

f1 QT BMf1 =

(QT BM

f1

)2, (3.30)


where

Q = F−11 SF2. (3.31)

Recall from Section 3.6.1 that the dispersion relation for the serpentine waveguide

is obtained by solving for the eigenvalues of T BMs . In the monomodal approximation

we observed that T BMs = T BM

f1

2and hence the eigenvalues µ were found by solving the

eigenvalue equation T BMf1 b = ±µ1/2b. Equation (3.30) shows that the general solution

can also be found in terms of a single FDC, where the relevant eigenvalue equation is(QT BM

f1

)B = µ1/2B. (3.32)

Here B is a matrix of eigenvectors expanded in the full Bloch mode basis containing

both propagating and evanescent modes. Although we do not derive the result here, Qis a diagonal matrix of elements ±1, and so (Q)2 = I. In the limit where L2 is large,

T BMf1 is dominated by the single propagating Bloch mode in the waveguide, and hence

Q has little effect on the solutions of Eq. (3.32), which then reduces to the single mode

result derived in Section 3.6.1.

3.7 Conclusion

We have covered in this chapter a range of topics associated with directional coupling

and resonant reflection in photonic crystals, and have presented several devices that

exhibit unusual transmission characteristics as a result of combining the two effects.

These devices are described accurately by semianalytic models which allow rapid and

efficient optimization and provide physical insight into the properties observed in full

numerical simulations.

The semianalytic formulations and the physical interpretation of the numerical re-

sults are based on the two coupling effects considered in Sections 3.2 and 3.2. Coupling

between parallel waveguides was considered in Section 3.2 where we demonstrated that

weak-coupling approximations typically used for dielectric waveguides can also be ap-

plied to photonic crystal waveguides. The second type of coupling occurs when two

different waveguide sections are interfaced end-to-end, as in the two examples presented

in Section 3.3. In this case, the coupling properties are influenced not only by direct

propagation from one waveguide to the other, but also by resonant coupling through

the photonic crystal barrier, should one exist. The devices presented in Sections 3.4 —

3.6 derive their properties from a combination of these two types of coupling.

Chapter 4

Recirculating Mach-Zehnder

interferometer

In this chapter we use many of the techniques and results from Chapter 3 to investigate

the characteristics of photonic crystal Mach-Zehnder interferometers (MZIs). We show

that the MZIs formed from waveguides in a perfectly reflecting cladding can display

manifestly different transmission characteristics from conventional MZIs due to mode

recirculation and resonant reflection. In devices of this type, recirculation effects can

result in a significantly sharper switching response than in conventional interferometers.

A simple and accurate analytic model is derived to describe this behaviour and we

propose specific PhC structures in both rod-type and hole-type geometries that display

these properties.

This chapter is based largely on the work published in Ref. [204], with some addi-

tional details included relating to model in Section 4.2 and the design and simulation of

the photonic crystal based Mach-Zehnder interferometer in Section 4.4. The discussion

in Section 4.6 is based on work to be published in Ref. [182].

4.1 Introduction

Mach-Zehnder interferometers (MZIs) are used extensively in optical systems as filtering

and switching devices and in phase measurement and detection applications. Their

many uses and functions have led to the fabrication of MZI devices in bulk-optics [103],

fibres [215], planar dielectric waveguides [216], rib waveguides [217], and more recently

2D PhCs [200,201,218–220]. Although the morphology of all MZIs is the same regardless

of the type of waveguide in which they are formed, the operational characteristics can

be strongly influenced by the waveguide and cladding properties.

Three generic MZI designs are illustrated in Figure 4.1. The example in (a) is a

bulk-optics MZI, while those in (b) and (c) are waveguide-based designs that can be

formed in either dielectric or PhC waveguides. In each case, an incident beam is split

into two beams at the first junction, with each one taking a different path through

105

106 CHAPTER 4. RECIRCULATING MACH-ZEHNDER INTERFEROMETER

(a) (c)

in

out 1

out 2

in out

(b)

in 1

in 2

out 1

out 2

Figure 4.1: Examples of three generic symmetric MZI designs with different types of

junction. (a) Bulk-optics MZI using 50/50 beam-splitters and mirrors. (b) Directional

coupler-based MZI with two possible input and output waveguides. (c) Y-junction MZI

with single input and output waveguides.

the interferometer before being recombined at the second junction. In a symmetric,

or balanced, MZI, the two paths are identical as in the diagram, whereas the paths

of an unbalanced MZI are different so as to introduce a relative phase delay. The

devices illustrated in Fig. 4.1 (a) and (b) each have two output ports where the power

coupled into each depends on the relative phase delay between the two arms of the

interferometer. For an ideal device, the combined output power is equal to the input

power, and the light only passes through the interferometer once. In contrast, the MZI

in Fig. 4.1(c) only has a single output waveguide. In this case, any light that is not

coupled into the output waveguide must either be radiated out of the device or reflected

back into the interferometer. If the junction is formed in a dielectric waveguide relying

on TIR for guidance, most of the non-transmitted light is radiated into the cladding, as

happened with the terminated waveguides discussed in Chapter 3. For a PhC waveguide

junction however, the cladding is perfectly reflecting so radiation cannot occur and thus

any light that is not transmitted must be reflected. The resulting recirculating MZI

exhibits strong resonance effects that are not usually observed in conventional dielectric

MZIs or coupler-based MZIs of the type illustrated in Figs. 4.1(a) and (b). The clear

observation of these effects relies critically on the use of efficient, symmetric beam

splitters that exhibit very low reflectance (< 2%).

In Section 4.2 we present a simple modal analysis to explain the transmission char-

acteristics of recirculating MZIs and derive an elegant semianalytic expression that

accurately describes their behaviour. The properties of these devices are strikingly

different from conventional MZIs, and our analysis shows that they are typical of a gen-

eral class of MZI with perfectly reflecting cladding and single-mode input and output

waveguides. One feature of these devices is an almost square transmission response

with phase and thus a significantly sharper switching response than conventional MZIs.

Such behaviour is a highly desirable characteristic for both linear and nonlinear switch-

ing applications. As in Chapter 3, the analysis and numerical results are purely 2D,

but the results hold generally if radiation losses are low. This issue is discussed in more

4.2. MODAL ANALYSIS 107

detail in Chapter 6.

Two specific examples of PhC waveguide-based recirculating MZIs are proposed in

Section 4.4. Both examples use the coupled Y-junction design presented in Section 3.5

which provides the necessary high-transmission and mode rejection properties to observe

the recirculation effects. In this case, the perfectly reflecting cladding is provided by

the photonic bandgap of the PhC waveguide walls. While it has been proposed that the

strong reflections and associated interference observed in many PhC based devices may

limit their practical use in optical systems [221], we show here that these effects can

also be used to great advantage. Similar recirculation characteristics can be obtained

in a bulk-optics geometry by including two additional mirrors in the system illustrated

in Fig. 4.1(a). We analyse such a system in Section 4.6 and show that the transmission

properties are analogous to those of the PhC MZIs.

4.2 Modal analysis

Consider first the MZI illustrated in Fig. 4.2(a), consisting of single input and output

waveguides, and two junctions Y1 and Y2 joined by waveguide arms A1 and A2 of length

L. In this analysis we assume that the Y-junctions are ideal and split light equally

between A1 and A2. If an element is placed in A2 to introduce a phase difference ϕ

between the two arms, the intensity of light transmitted into the output depends on

ϕ. Active devices such as switches and filters can be made by varying ϕ to modulate

the output [218, 220, 222], or in passive devices, the measured output can be used to

determine ϕ [215]. In a symmetric, or balanced, MZI ϕ = 0, so light entering from

the input guide recombines in-phase at Y2 and is transmitted into the output guide - a

balanced MZI transmits equally well at all wavelengths.

ϕ

input outputwaveguide junctionsY1 Y2

A1

A2

(a) (b)

Figure 4.2: (a) Schematic of a simple MZI with single input and output waveguides

joined by arms A1 and A2 via Y-junctions Y1 and Y2. (b) Schematic of the propagating

modes in each of the waveguide sections in (a). The green and red curves show the

even and odd superpositions of the individual waveguide modes.

The transmission properties of the MZI in Fig. 4.2(a) can be derived by considering

the propagating modes in each of the waveguide sections, in a procedure similar to that


used to analyse the coupled waveguide devices in Chapter 3. An alternative derivation

based on ideal Bloch mode scattering matrices is outlined in Section 4.5. All of the

waveguides in Fig. 4.2 are taken to be identical, with each supporting a single mode

of even symmetry with respect to the waveguide axis. Thus, the input and output

guides support one mode each, denoted by |ψin〉 and |ψout〉 respectively, and the two-

guide region supports one mode in each of A1 and A2, denoted |ψ1〉 and |ψ2〉, as shown

in Fig. 4.2(b). Alternatively, the modes in A1 and A2 are well approximated as a

combination of two supermodes of even (+) and odd (−) symmetry with respect to the

axis of the interferometer, indicated by the green and red curves in Fig. 4.2(b). The

supermodes are approximately related to the modes in the individual guides by

|ψ±〉 =1√2

(|ψ1〉 ± |ψ2〉) , |ψ1,2〉 =1√2

(|ψ+〉 ± |ψ−〉) . (4.1)

In an ideal structure, A1 and A2 are separated sufficiently that coupling between the

two guides is negligible, in which case the propagation constants of |ψ±〉 are equal to

the propagation constant of an isolated waveguide, β+ = β− = β respectively. This ap-

proximation is not always accurate however, in which case an alternative approximation

can be made to the semianalytic model, as discussed in Section 4.4.3.

Since mode |ψin〉 is symmetric with respect to A1 and A2, it can only couple to the

supermode with the same symmetry, |ψ+〉, so for an ideal Y1 junction, mode |ψin〉 is

fully transmitted into |ψ+〉. When this mode propagates through A1 and A2, the effect

of ϕ is to introduce an asymmetric field component, equivalent to transferring some of

the energy into |ψ−〉. As at Y1, the |ψ+〉 component of the field entering Y2 can couple

into the output guide, but the |ψ−〉 component cannot and must be either radiated or

reflected.

To this point, the modal description holds for MZIs in general. The essential differ-

ence lies in what happens to |ψ−〉 at Y2. In dielectric waveguides, most of the light in

|ψ−〉 is radiated into the cladding and lost, and only a very small amount is reflected

back into the arms of the interferometer [223]. However, if light is unable to radiate into

the cladding, any light in |ψ−〉 must be reflected at Y2, back into the interferometer.

This odd mode reflection results in the unique resonant behaviour of recirculating MZIs

with single-mode input and output waveguides.

We now follow the propagation of a mode Ψ with unit energy 〈Ψ|Ψ〉 = 1 entering an

unbalanced MZI from the input guide and consider the effect of odd mode reflection at

Y2. After passing through Y1, the resulting symmetric field distribution can be expressed

in terms of |ψ1〉 and |ψ2〉 using Eq. (4.1) to give

|Ψ(0)〉(1) = |ψ+〉 =1√2

(|ψ1〉+ |ψ2〉) .

Here the superscript (1) indicates that we are considering the fields on the first pass

through the interferometer. The modes propagate through each arm essentially inde-

pendently of each other, advancing in phase as they propagate. At Y2, the combined

4.2. MODAL ANALYSIS 109

field in A1 and A2 is

|Ψ(L)〉(1) =(eiβL|ψ1〉+ ei(βL+ϕ)|ψ2〉

)/√

2 = eiχ (cos(ϕ/2)|ψ+〉 − i sin(ϕ/2)|ψ−〉) ,(4.2)

where χ = βL + ϕ/2 is the average phase of the two arms and β is the propagation

constant of the single propagating mode of an isolated waveguide. At Y2, the even field

component is transmitted into the output guide, while the odd mode is reflected back

into A1 and A2. Note that in a conventional MZI the odd mode is radiated into the

cladding and the transmission is simply given by the even mode term in Eq. (4.2),

T = |eiχ cos(ϕ/2)|2 = cos2(ϕ/2) =1

1 + tan2(ϕ/2). (4.3)

In the PhC structure, however, the reflected odd field component, −i exp(iχ) sin(ϕ/2)|ψ−〉,propagates back through the two arms, accumulating a further phase ϕ so that the field

arriving back at Y1 is

|Ψ(0)〉(2) = −ieiχ sin(ϕ/2)(eiβL|ψ1〉 − ei(βL+ϕ)|ψ2〉

)/√

2

= −ie2iχ sin(ϕ/2) (cos(ϕ/2)|ψ−〉 − i sin(ϕ/2)|ψ+〉) . (4.4)

Once again the odd field component is reflected back into the interferometer, while the

even component is transmitted through the junction, in this case back into the input

waveguide.

The total transmission is given by the superposition of the fields transmitted through

Y2 each time the odd mode is reflected. Denoting the contribution after n passes through

the interferometer by tn, the transmitted field is

t =∞∑

n=1

tn (4.5)

= eiχ cos(ϕ/2)− e3iχ sin2(ϕ/2) cos(ϕ/2)− e5iχ sin2(ϕ/2) cos3(ϕ/2)− . . . (4.6)

= eiχ cos(ϕ/2)− e3iχ sin2(ϕ/2) cos(ϕ/2)

1− e2iχ cos2(ϕ/2), (4.7)

where the second term in (4.7) is the sum of the geometric series∑∞

n=2 tn. Hence, the

transmitted intensity is given by

T = |t|2 =4 sin2(χ) cos2(ϕ/2)/ sin4(ϕ/2)

1 + 4 sin2(χ) cos2(ϕ/2)/ sin4(ϕ/2). (4.8)

Note that the form of Eq. (4.7) closely resembles the Airy-formula [122] for reflection

from a Fabry-Perot cavity – a feature we return to in Section 4.3. The reflected in-

tensity (R = 1 − T ) can be derived in the same way by summing the field amplitudes

reflected back into the input waveguide. Observe that Eq. (4.8) is a function of both

ϕ and βL = β(λ)L, in contrast to the conventional MZI transmission (4.3), which is a


function of ϕ only. Thus, the waveguide dispersion properties are especially important

in recirculating MZIs. Apart from the different functional forms of Eqs. (4.3) and (4.8),

the additional dependence on wavelength and structure length leads to significantly dif-

ferent transmission properties. The explicit dependence on structure length is a direct

result of the mode recirculation inside the device.

4.3 Transmission characteristics

In this section we study the transmission characteristics of recirculating MZIs described

by Eqs. (4.7) and (4.8). For generality the normalised units βL are used, although we

note that when modelling a realistic device, the waveguide dispersion β = β(λ) must be

taken into account. Hence for a specific MZI design, the dispersion properties provide

a direct mapping between βL and wavelength. The effects of waveguide dispersion are

discussed in Section 4.4, where the transmission properties of two PhC-based MZIs are

presented in terms of wavelength.

Figures 4.3(a) and (b) are contour plots of transmittance as a function of βL and ϕ

for a conventional and a recirculating MZI respectively, where the red areas correspond

to high transmittance and the blue areas correspond to low transmittance. The trans-

mittance of the conventional MZI is given by Eq. (4.3) so there is no variation with βL,

but only with ϕ. For the recirculating MZI however, there is a strong dependence on

both parameters but it can be seen that T = 1 when ϕ = 0 and T = 0 when ϕ = π,

as for the conventional MZI. In (b) the slightly tilted, low transmittance ‘valleys’ cor-

respond to strong resonances due to the multiple reflections of the odd mode. At the

centre of the valleys, the transmittance drops to zero and the valleys become rapidly

narrower as ϕ approaches 0 or 2π. The slight tilt of the valleys from vertical is due to

the dependence of χ on both βL and ϕ in Eq. (4.8).

We consider first the properties of Eq. (4.8) as ϕ is varied at a fixed βL, correspond-

ing to traversing a vertical line across Fig. 4.3(b). Figures 4.4(a)–(e) show the trans-

mittance of a recirculating MZI as a function of ϕ for βL = 0, π/8, π/4, 3π/8 and π/2

respectively, indicated by the solid curves. The corresponding transmittance of a con-

ventional MZI (4.3) is indicated in each plot by the dashed curves. Observe from

Fig. 4.3(b) that for most values of βL, a vertical line of varying ϕ intersects one end

of a low transmission valley, resulting in a response with a strong resonant reflection

as seen in Fig. 4.4(b). The width of the resonance decreases to zero as ϕ → 0 or

ϕ→ 2π, at which point T = 1. Of particular interest is the response of the MZI when

βL = π/2 , 3π/2 , 5π/2 . . ., where the diagonal valleys in Fig. 4.3(b) intersect the low

transmittance band at ϕ = π. The solid curve in Fig. 4.4(e) shows the response as

a function of ϕ when βL = π/2. Comparing this with the dashed curve of the con-

ventional MZI, it is seen that the recirculating MZI response exhibits a significantly

sharper transition between ‘on’ and ‘off’ states. Substituting the condition identified

4.3. TRANSMISSION CHARACTERISTICS 111

0

p

2p

p

2

32p

j

0 2p 4p 6p

bL

0 0.2 0.4 0.6 0.8 1

0

p

2p

p

2

32p

j

0 2p 4p 6p

bL

(a) (b)Transmission

Figure 4.3: Contour plots of transmittance as a function of the dimensionless parameter

βL and the phase difference ϕ for (a) a conventional MZI described by Eq. (4.3) and

(b) a recirculating MZI described by Eq. (4.8)

above for βL into Eq. (4.8), we find the transmittance to be

T =4 cot4(ϕ/2)

1 + 4 cot4(ϕ/2)=

1

1 + tan4(ϕ/2)/4. (4.9)

Comparing Eqs. (4.3) and (4.9), we see that the recirculating MZI response is quar-

tic in tan(ϕ/2), compared with the quadratic response of a conventional MZI. The

change in ϕ required to switch the transmittance from 0.9 to 0.1 in the recirculating

MZI is approximately 0.32π, whereas a conventional MZI would require a change of

0.59π. Although cos4(ϕ/2) responses can be achieved with a pair of conventional MZIs,

and higher order behaviour occurs for a chain of such devices [224], we find that at

least six identical conventional MZIs would be required to obtain a superior switching

performance to that of a single recirculating MZI. Note also, that the transmittance

given by Eq. (4.9) has a fourth-order variation near both T = 0 and T = 1, whereas a

chain of MZIs always exhibits a quadratic variation in the region around T = 1. The

steep, almost square response of the recirculating MZI is highly desirable for switching

applications, and could be used to improve performance in both linear and nonlinear

devices.

If βL is varied while ϕ is fixed, as in Figs. 4.4(f)–(j), the transmission properties are

very different from those shown in Figs. 4.4(a)–(e). For a given ϕ, the transmittance

T = T (βL) given by Eq. (4.8) resembles the expression for the reflectance of a Fabry-

Perot etalon [122] with finesse

F =π| cos(ϕ/2)|

1− | cos(ϕ/2)|. (4.10)


bL=0

bL=p/2

bL=3p/8

bL=p/4

bL=p/8

j=0

j=p/4

j=p/2

j=3p/4

j=p

0

1

0.2

0.4

0.6

0.8

T

0

1

0.2

0.4

0.6

0.8

T

0

1

0.2

0.4

0.6

0.8

T

0

1

0.2

0.4

0.6

0.8

T

0

1

0.2

0.4

0.6

0.8

T

0 p 2pp

23p

2

0 p 2p 3p 6p5p4p

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

j bL

Figure 4.4: Solid curves: (a)–(e): Power transmitted by an ideal recirculating MZI as a

function of ϕ for βL = 0, π/8, π/4, 3π/8, π/2 respectively. (f)–(j): Power transmitted by

an ideal recirculating MZI as a function of βL for ϕ = 0, π/4, π/2, 3π/4, π respectively.

Dashed curves: Power transmitted by a conventional MZI for the same parameters.

Thus, the transmission behaviour of the recirculating MZI is essentially the reverse of

a Fabry-Perot etalon, exhibiting resonant reflection, rather than resonant transmission.

This is seen by considering horizontal lines of constant ϕ in Fig. 4.3(b), and the examples

in Figs. 4.4 (f)–(j). As ϕ approaches 0 or 2π, the resonances become narrower, and the

finesse increases according to Eq. (4.10). As ϕ approaches π, both the fringe visibility

and the finesse decrease. These reversed Fabry-Perot transmission characteristics could

be used in the design of a tunable notch rejection filter where the finesse and resonance

position could be adjusted by varying β and ϕ.

In practice both β and ϕ are functions of wavelength, and so it is difficult to design a

device to exhibit transmission spectra identical to those shown in Figs. 4.3(f)–(j). Any

realistic measurement of transmission spectra is likely to correspond to the transmission

along a curved line in Fig. 4.3(b), with the exact shape of such a curve being determined

4.4. PHOTONIC CRYSTAL RECIRCULATING MZI STRUCTURES 113

by the method used to induce ϕ. This aspect is discussed further in Section 4.4.3.

4.4 Photonic crystal recirculating MZI structures

In Section 4.2 we derived a simple model for the transmission of an ideal MZI with

a perfectly reflecting cladding and single-mode input and output guides, and showed

that recirculation and resonant properties occur due to the reflection of modes with

odd symmetry at the waveguide junctions. In this section we present both rod- and

hole-type PhC MZI designs that display these characteristics and compare full numer-

ical calculations with the semianalytic results of Eq. (4.8). For these comparisons, the

waveguide dispersion properties are required as numerical input into Eq. (4.8), however

this is a relatively simple calculation that does not require additional knowledge of the

MZI structure.

The rod-type PhC considered throughout this section consists of a square lattice

of rods with index ncyl = 3.4 and radius r = 0.18d in air. This structure has a TM

bandgap in the normalised frequency range 0.302 ≤ d/λ ≤ 0.445. We present results

for a structure with lattice period d = 570 nm, chosen so that the MZI operates close

to λ = 1.55µm, within the bandgap spanning 1.28µm ≤ λ ≤ 1.88µm. A waveguide

formed by removing a single line of cylinders supports a single propagating mode of

even symmetry in the wavelength range 1.28µm ≤ λ ≤ 1.83µm.

For the hole-type PhC we use the same parameters as in Section 1.5.3, namely a

triangular lattice of air holes with radius r = 0.32d in a silicon background of nb = 3.4.

A lattice period of 350 nm is chosen so that the TE bandgap in the range 0.216 <

d/λ < 0.298 corresponds to a wavelength range 1.17µm ≤ λ ≤ 1.62µm. Since a W1

waveguide can support two modes in the bandgap (see Fig. 1.8), we operate only in the

low-frequency region of the bandgap where a single mode is supported for 1.435µm ≤λ ≤ 1.580µm. This mode has a large group velocity dispersion close to cutoff at the

low frequency end of this range which distorts the transmission profile of the MZI

considerably, as seen in Section 4.4.3.

4.4.1 Design

The most important components in a PhC-based recirculating MZI are the Y-junctions

or beam splitters. Recall from Section 4.2 that the Y-junctions in the model were

taken to be ideal, transmitting modes of even symmetry and reflecting modes of odd

symmetry. In practice, the even mode is not perfectly transmitted, and the additional

reflected field components can result in extra Fabry-Perot effects [218] that make the

odd mode resonances difficult to identify. From our numerical studies, we find that

the odd mode effects are most clearly observed when the even mode reflectance is less

than 2%. Thus, to observe the properties predicted by Eq. (4.8), the junction design is

critical. In Section 3.5 we proposed a PhC coupled Y-junction design that exhibits very

high transmission over a wide bandwidth due to a combination of directional coupling


and mode recirculation behaviour. This type of junction is ideal for use in recirculating

PhC MZI designs as demonstrated by the examples in this section. We note that Wang

et al. proposed a similar coupled junction design for a surface plasmon MZI based

on metal gap waveguides, although results were only reported for a balanced MZI so

recirculation effects were not observed [225].

1.5 1.55 1.60.95

0.96

0.97

0.98

0.99

1

T

l m( m)

1.65

88nm 67nm

(a) (b)

(c) (d)

36nm

1.54 1.55 1.56 1.57

l m( m)

0.95

0.96

0.97

0.98

0.99

1

T

L =5dy L =11dy

Figure 4.5: (a) and (b): Y-junction designs used for the rod- and hole-type PhC MZIs

illustrated in Figs. 4.6(a) and (b) respectively. (c) Solid curve: transmission spectrum of

the Y-junction shown in (a). Dashed curve: transmission of the PhC MZI in Fig. 4.6(a)

with ϕ = 0 and LMZ = 15d. (d) Solid curve: transmission spectrum of the Y-junction

shown in (b). Dashed curve: Transmission of the PhC MZI in Fig. 4.6(b) with ϕ = 0

and LMZ = 32d. Note that the transmission axis starts at 95% in both plots.

Figures 4.5 (a) and (b) show two coupled Y-junctions designed for wide bandwidth

transmission in the rod- and hole- type PhCs defined at the start of Section 4.4. The

transmission spectra for these junctions are plotted in (c) and (d) of the same figure

(solid curves). The spectrum in (c) has two transmission bands of widths 88 nm and

67 nm respectively where T > 98%, but between these two regions is a sharp resonant

reflection, the cause of which is discussed in Section 3.5. Transmission spectra are cal-

culated over the combined wavelength range of the two bands and thus the resonant

reflection feature can be seen in some of the results for the rod-type PhC MZI. Since

the resonance is sharp, however, the MZI properties are only affected in a very nar-

row wavelength region. The hole-type PhC junction shown in Fig. 4.5(b) has a 98%

transmission bandwidth of 36 nm, as shown by the solid curve in Fig. 4.5(d). When


the junctions are operated in reverse as beam combiners, the even mode is transmitted

into the single output waveguide with the same efficiencies shown in Figs. 4.5(c) and

(d), while the symmetrical design of the junctions provides the necessary odd mode

reflection properties.

The PhC junctions illustrated in Figs. 4.5(a) and (b) are formed into MZIs by joining

two identical junctions together with PhC waveguides as shown in Fig. 4.6. The total

length of the device, including the junctions of length LY is given by LMZ as shown. In

Section 4.4.2 we demonstrate that a phase difference can be introduced between the two

arms by modifying the radius of several cylinders on the edge of one waveguide, thereby

changing the optical path length relative to the other arm. The number of cylinders

modified on each edge is given by the parameter Lϕ, defined in Fig. 4.6, where the

modified cylinders are shaded a different colour for illustration purposes. Recall that

we discussed in Section 2.5.2 some of the issues involved with modelling a balanced

PhC MZI using the Bloch mode method. To introduce the modified cylinders in one

arm we must include a new segment in the structure, and thus an additional grating

calculation is required. To ensure that the modified segment does not interfere with

the junction operation, at least two layers of symmetric double waveguide are retained

where the arms enter the junctions. Thus, the MZI in Fig. 4.6(a) must be calculated as

a seven segment structure, while the one in Fig. 4.6(b) is calculated as a nine segment

structure by adding an extra segment in the middle of M4 in Fig. 2.9. The Bloch mode

method results presented in this chapter we calculated with a grating supercell period

of Dx = 21d and a plane-wave truncation order of Npw = 44, where Dx and Npw are

defined in Chapter 2.

Recall from Section 4.2 that when ϕ = 0 an MZI is symmetric (balanced) and

therefore transmits equally well at all wavelengths. In a practical PhC based device,

this is only expected for wavelengths within the transmission band of the junction.

The dashed curves in Figs. 4.5(c) and (d) show the transmission spectra for balanced

MZIs of length LMZ = 15d (rod-type PhC) and LMZ = 32d (hole-type PhC) respectively.

Note that although the transmission bandwidth is reduced slightly relative to the single

junction in both cases, the transmission remains high across the band with only a few

shallow ripples due to weak Fabry-Perot resonances of the even mode.

4.4.2 Tuning ϕ

Most MZI applications require ϕ to be tuned for the purposes of modulating or switching

the output signal. This has been demonstrated in PhC based MZIs using thermo-

optic [218] and nonlinear [220] effects to induce a small refractive index change in one

arm. Liquid crystal tuning of the refractive index has also been proposed [222]. For

the purposes of demonstrating the response of recirculating MZIs, here we generate ϕ

by changing the radii of Lϕ/d pairs of cylinders on either side of one waveguide from

r to r as indicated in Fig. 4.6. This approach is chosen for numerical convenience as

it is easier to modify the properties of individual cylinders rather than the background


Lj

LY

LMZ

Lj

LY

LMZ

(a)

(b)

Figure 4.6: Diagrams showing the basic geometry of the PhC MZIs studied in this

chapter. (a) Rod-type PhC MZI formed from the junction shown in Fig. 4.5(a). (b)

Hole-type PhC MZI formed from the junction shown in Fig. 4.5(b). The cylinders

highlighted in (a) and (b) are modified to generate ϕ as described in Section 4.4.2.

refractive index in the Bloch-mode scattering matrix method.

The modal propagation constant in the modified waveguide section differs from

the original waveguide by ∆β � β, so a mode propagating through the modified

waveguide arm accumulates a total phase difference of approximately ϕ = ∆βLϕ relative

to the other arm. Thus, the maximum ϕ that can be introduced depends on Lϕ and

the change in radius ∆r = r − r. However ∆r must be sufficiently small to avoid

significant reflection caused by the change in waveguide properties and hence many

cylinders typically need to be modified to achieve even a relatively small phase change.

This is shown in Fig. 4.7(a) where ϕ is plotted as a function of ∆r for Lϕ = 5d

and Lϕ = 10d for the rod-type PhC waveguides used in Fig. 4.1(a) at λ = 1.553µm.

Hole-type PhC waveguides exhibit similar behaviour when tuned in the same way.

Observe that ϕ is doubled when Lϕ is increased from 5d to 10d as expected. The total

reflected power from the modified waveguide section is also plotted for the two lengths

in Fig. 4.7(b). The reflectance does not increase linearly with Lϕ because the reflections

originate only at the two ends of the modified section. Since any additional reflection


effects could interfere with the recirculation properties of the MZI, we limit ∆r such

that the R . 1%.

In Section 4.4.3 we use the Bloch mode method to calculate the transmission spectra

of the PhC MZIs shown in Fig. 4.6 for different values of Lϕ. Since ∆β and hence ϕ are

both functions of wavelength, this calculation is not directly equivalent to the results

plotted in Fig. 4.4(f)–(j), where ϕ was held constant. In order to compare Eq. (4.8) with

the full numerical results, the wavelength dependence variation of ϕ must be included.

The dashed and dash-dot curves plotted in Fig. 4.8 show ϕ as a function of wavelength

for the different Lϕ used in the numerical calculations of Section 4.4.3.

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0-0.02 0.02 0.04

R0

0-0.02 0.02 0.04

j

p

2

p

2 (a) (b)

L dj=5

L dj=10

L dj=5

L dj=10

Dr/d Dr/d

Figure 4.7: (a) Phase difference induced by changing the radii of Lϕ pairs of cylinders

by ∆r in one arm of the rod-type PhC MZI in Fig. 4.6(a). (b) Reflectance of the

modified waveguide section for the same parameters. Note that both ϕ = 0 and R = 0

when r = r = 0.18d (∆r = 0). The calculations are for λ = 1.553µm, corresponding to

λ/d = 2.689.

4.4.3 Numerical results

Before proceeding to full numerical calculations, we consider first the transmission prop-

erties predicted by Eq. (4.8) for the two MZIs illustrated in Fig. 4.6. Figures 4.8(a) and

(b) are contour plots of transmittance as a function of wavelength and ϕ calculated

using Eq. (4.8) for rod- and hole-type PhC MZIs of length L = 36d and L = 40d respec-

tively. The waveguide dispersion β = β(λ) was calculated using the Bloch mode method

and used as numerical input. These plots are essentially the same as Fig. 4.3(b), where

the horizontal axis has been rescaled to take into account the length and dispersion

parameters of the devices in Fig. 4.6.

There are some clear differences between the contour plots in Figs. 4.8(a) and (b).

First, the low transmittance valleys run in the opposite diagonal direction because of

the different dispersion slopes of the waveguide modes in the two PhC structures. Sec-

ond, in (b) the tilt and the separation of the resonances changes significantly at longer

wavelengths, whereas in (a) the valleys remain almost parallel and equally spaced. This

is also a result of the different dispersion slopes. The waveguide modes of the MZI in


1.5 1.55 1.6

2p

p

0

32p

p

2

l m( m)

j

1.52 1.54 1.56 1.58

2p

p

0

32p

p

2

l m( m)

j

0 0.2 0.4 0.6 0.8 1

Transmission

Figure 4.8: (a) Contour plot of transmittance as a function of λ and ϕ calculated using

Eq. (4.8) for the PhC waveguides in Fig. 4.6(a) with LMZ = 36d. The dashed and

dashed-dotted curves show the phase change induced by ∆r = 0.02d for Lϕ = 8d and

Lϕ = 16d respectively as described in Section 4.4.3. (b) As in (a) but for the waveguides

in Fig. 4.6(b) with LMZ = 40d. The dashed and dashed-dotted curves correspond to

the phase change for ∆r = −0.02d and Lϕ = 4d and Lϕ = 8d respectively.

Fig. 4.6(a) are well away from cutoff, so the dispersion slope is relatively constant,

resulting in the regular features of Fig. 4.8(a). In contrast, the waveguide modes of

the MZI in Fig. 4.6(b) are closer to cutoff and the resulting strong dispersion distorts

the transmission profile. The spacing of the resonances decreases with increasing wave-

length, while the resonances become narrower, maintaining a constant finesse. Despite

this distortion, the square response predicted by Eq. (4.9) still occurs when βL satisfies

the conditions identified in Section 4.3.

Rod-type PhC MZI

Figures 4.9(a) and (b) show the transmission spectra of two MZIs of the type shown

in Fig. 4.6(a) with Lϕ = 8d and Lϕ = 16d respectively for r = 0.2d (∆r = 0.02d) and

LMZ = 36d. The solid curves are calculated using the full Bloch mode numerical calcu-

lation, and the dashed and dashed-dotted curves are the result of Eq. (4.8), calculated

along the corresponding curves in Fig. 4.8(a) to account for the wavelength depen-

dence of ϕ. Both spectra exhibit the reversed Fabry-Perot-like behaviour predicted by

Eq. (4.8) with the finesse of the reflection resonances decreasing with increasing phase

according to Eq. (4.10). Note also that the sharp reflection feature exhibited by the

Y-junction in Fig. 4.5(a) can just be seen in Fig. 4.9(a) at λ ≈ 1.57µm, very close to

a resonance of the MZI. The two resonances coincide exactly in Fig. 4.9(b) and cannot


0.2

0.4

0.6

0.8

1

T

0.2

0.4

0.6

0.8

1

T

l m( m)

1.54 1.55 1.56 1.57

0.2

0.4

0.6

0.8

1

0

1.54 1.55 1.56 1.57

0.2

0.4

0.6

0.8

1

0

1.5 1.54 1.58 1.6200

TT

l m( m)

l m( m)l m( m)

1.5 1.54 1.58 1.62

(a) (b)

(c) (d)

Figure 4.9: (a) and (b): Transmission spectra of the rod-type PhC MZI (Fig. 4.6(a))

calculated using the Bloch mode matrix method with LMZ = 36d and ∆r = 0.02d for

(a) Lϕ = 8d and (b) Lϕ = 16d (solid curve). Dashed and dashed-dotted curves are

given by Eq. (4.8) calculated along the corresponding curves in Fig. 4.8(a). (c) and

(d): Transmission spectra of the hole-type PhC MZI (Fig. 4.6(b)) calculated using the

Bloch mode matrix method with LMZ = 40d and ∆r = −0.02d for (a) Lϕ = 4d and

(b) Lϕ = 8d (solid curve). Dashed and dashed-dotted curves are given by Eq. (4.8)

calculated along the corresponding curves in Fig. 4.8(b).

be distinguished.

The parameters L and Lϕ used in Eq. (4.8) are not necessarily identical to those

defining the structure in Fig. 4.3(d). Since PhCs behave as distributed reflectors, defin-

ing a structure length is somewhat ambiguous. We find that the best agreement in

resonance position occurs when we take L in Eq. (4.8) to be the total length of the

structure including the junctions, LMZ, as shown in Fig. 4.6, with a slight correction

for the phase change introduced by reflection from the PhC. For the comparisons in

Figs. 4.9(a) and (b) we found the best agreement for L = 36.75d. Under these condi-

tions, the agreement between the rigorous numerical result and the semianalytic model

is excellent. In particular, the model predicts the resonance widths to a high accu-

racy, and with the corrected length parameter the resonance positions are also closely

matched.


In Section 4.3 we showed that recirculating MZIs exhibit a considerably sharper

response to ϕ than conventional, single-pass MZIs, with that response described by

Eq. (4.9) when βL = π/2 , 3π/2 , 5π/2 . . . To demonstrate this property in a full nu-

merical calculation ϕ must be varied while keeping the wavelength fixed. Recall from

Section 4.4.2 that the maximum ϕ that can be induced by varying the cylinder radii is

limited by Lϕ and the maximum ∆r that can be used without introducing additional

reflections into the system. From Fig. 4.7(b) it can be seen that the reflectance remains

below approximately 1% in the range −0.025 ≤ ∆r/d ≤ 0.035. This results in a phase

variation of −0.36π . ϕ . 0.36π for an MZI with Lϕ = 10d, as shown by the dashed

curve in Fig. 4.7(a).

0

0.2

0.4

0.6

0.8

1

T

p

p/2

- /2p

-p

0 j

Dr-0.02 -0.01 0 0.01 0.02 0.03

Figure 4.10: Solid (blue) curve: Transmission as a function of ∆r for a rod-type PhC

MZI (Fig. 4.6(a)) with LMZ = 36d and Lϕ = 22d at λ = 1.553µm. Dashed-dotted

curve: phase difference induced by the radius modification ∆r. Dashed (red) curve:

Quartic response given by Eq. (4.9).

The range of ϕ is maximised in the rod-type PhC MZI of length LMZ = 36d by

choosing Lϕ = 22d so that there are only two lattice periods of undisturbed waveguide

separating the modified waveguide section and junctions at either end. The dashed-

dotted curve in Fig. 4.10 shows ϕ as a function of ∆r for λ = 1.553µm and the solid

curve shows the MZI transmittance calculated using the Bloch mode method for the

same wavelength. The dashed curve is calculated using Eq. (4.9) with ϕ = ϕ(∆r) given

by the dashed-dotted curve. Again there is very good agreement between the rigorous

numerical calculation and the semianalytic model derived in Section 4.2. Given the

restrictions placed on ϕ by the choice of tuning method, a longer structure would be

required to obtain a full 2π phase change, but shorter devices may be possible with

alternative tuning methods.


Hole-type PhC MZI

Figures 4.9(c) and (d) show the transmittance as a function of wavelength for two

MZIs of the type shown in Fig. 4.6(b) with Lϕ = 4d and Lϕ = 8d respectively for

r = 0.30d (∆r = −0.02d) and LMZ = 40d. The solid curves are the full numerical result

and the dashed and dashed-dotted curves are given by Eq. (4.8), calculated along the

dashed (Lϕ = 4d) and dashed-dotted (Lϕ = 8d) curves in Fig. 4.8(b), corresponding to

the wavelength variation of ϕ = ∆βLϕ. As in the rod-type PhC example, the parameter

L used in Eq. (4.8) does not match LMZ exactly. For the comparisons in Figs. 4.9(c)

and (d) the best agreement was found for L = 39.26d.

Once again we see that the semianalytic model predicts the finesse of the resonances

with good accuracy. The resonance positions line up closely at the shorter wavelength

end of the transmission region, however the agreement is poorer at longer wavelengths,

as a result of the strong modal dispersion close to cutoff. We have found that the agree-

ment is improved at longer wavelengths if β is replaced by the propagation constant of

the odd supermode β− in Eq. (4.8). This modification is reasonable since the resonances

inside the interferometer are driven by multiple reflections of the odd mode. For the

rod-type PhC examples in Figs. 4.9(a) and (b), the modes are well away from cutoff

and β and β− are almost equal, so the comparison is not improved significantly by this

modification. For consistency however, the semianalytic curves plotted in Figs. 4.9(a)

and (b) are also calculated using β− in place of β.

5 10 15 20 25 30

0.2

0.4

0.6

0.8

1

00

-p

- p2

0

T

L /dj

j

-p/2

- p3 /2

Figure 4.11: Blue data points: transmission as a function of Lϕ for a hole-type PhC

MZI (Fig. 4.6(b)) with LMZ = 72d and r = 0.266 d (∆r = −0.054) at λ = 1.5392µm.

The dotted line between the points is an aid to the eye. Dashed-dotted curve: phase

difference induced by ∆r. Dashed (red) curve: quartic response given by Eq. (4.9).

In order to demonstrate the response of the hole-type MZI as ϕ is varied, we now

consider a similar hole-type PhC MZI to the previous example, but with a length of


LMZ = 72 d. This longer structure allows us to change ϕ through a complete 2π period.

For this example we fix the radius of the modified cylinders to r = 0.266 d, corresponding

to ∆r = −0.054 d, then vary ϕ by changing the number of modified cylinders, Lϕ.

Figure 4.11 shows the transmitted power as a function of Lϕ for 0d ≤ Lϕ/2 ≤ 32d. The

blue data points were calculated using the full Bloch mode method, and the dashed red

curve is the quartic response given by Eq. (4.9), where the dependence of ϕ on Lϕ is given

by the dashed-dotted curve and the right-hand axis. Observe that there is excellent

agreement between the approximate and the numerical calculation for Lϕ < 15d, but

for Lϕ > 15d the numerical results appear to be shifted relative to the model. We

attribute this to the change in ∆β that results from the radius change in the MZI arm.

In the derivation of the semianalytic model in Section 4.2, we assume that β changes

by ϕ/2, but that ∆β remains small and approximately zero. This is valid when ϕ is

small, but becomes less accurate as ϕ → π. Although this may make it difficult to

obtain the predicted quartic response over a full 2π change in ϕ, the steep switching

behaviour between T = 0 and T = 1 is still displayed in the example shown in Fig. 4.11.

Alternative tuning methods may also make it possible to reduce this effect.

4.5 Semianalytic Bloch mode matrix derivation

In Section 4.2, the transmission equations (4.7) and (4.8) were derived from a simple

modal analysis and compared in Section 4.4.3 with full numerical results calculated

using the Bloch mode method. Identical expressions can be obtained via the Bloch

mode matrix method by considering only the propagating modes in each region, and

filling the Bloch mode scattering matrices with idealised values corresponding to the

assumptions of Section 4.2. This alternative approach is outlined here.

The MZI of Fig. 4.2 can be modelled as a three layer structure as shown in Fig. 4.12.

Two semi-infinite regions form the input and output waveguides on either end of the

central section which consists of the two arms of the interferometer. Transmission

through the three-layer structure is given by the matrix equation

t = T23P (I− R21PR23P)−1 T21, (4.11)

where the operator P describes Bloch mode propagation between interfaces 1 and 2.

We can write this as

P =

[1/√

2 1/√

2

1/√

2 −1/√

2

]−1 [ei(χ−ϕ/2) 0

0 ei(χ+ϕ/2)

] [1/√

2 1/√

2

1/√

2 −1/√

2

](4.12a)

=

[eiχ cos(ϕ/2) −ieiχ sin(ϕ/2)

−ieiχ sin(ϕ/2) eiχ cos(ϕ/2)

], (4.12b)

where the diagonal matrix in Eq. (4.12a) describes propagation through the two arms of

the interferometer, while the other two matrices provide a change of basis between the

odd/even supermodes and the modes of the individual arms. The diagonal elements

4.5. SEMIANALYTIC BLOCH MODE MATRIX DERIVATION 123

T12

R12

T23

R23

T21

R21

P

Interface 1 Interface 2

1 2 3

Figure 4.12: Diagram showing the three layer structure considered in Section 4.5 to

derive a semianalytic Bloch mode scattering matrix model of a recirculating MZI. In-

terface 1 lies between the input waveguide and the two arms and is characterised by

the scattering matrices R12, T12, R21 and T21. Interface 2 lies between the two arms

and the output waveguide and is characterised by the scattering matrices R23 and T23.

of the matrix in Eq. (4.12b) can also be obtained directly from Eqs. (4.2) and (4.4) by

extracting the odd and even mode amplitudes before and after a single pass through

the interferometer.

Analytic approximations to the scattering matrices in Eq. (4.11) are obtained by

considering the transmission and reflection of modes at the interfaces between the input

and output waveguides and interferometer arms. In Section 4.2 we assumed that a mode

in the input guide is perfectly transmitted into the even mode inside the interferometer.

In the Bloch mode matrix notation, this corresponds to setting R12 = 0 and

T12 =

[1

0

],

where the first and second elements of T12 represent transmission into the even and

odd supermodes respectively. Light incident on either interface from inside the inter-

ferometer is filtered by the junction according to the odd and even mode components;

the even mode is perfectly transmitted, while the odd mode is reflected. Hence, the

remaining Bloch mode scattering matrices in Eq. (4.11) can be written as

R21 = R23 =

[0 0

0 1

]T21 = T23 = [1 0] .

Equations (4.7) and (4.8) can be derived by substituting the analytic matrix expressions

above into Eq. (4.11).

Although this method of deriving the semianalytic result may seem identical to the

modal method in Section 4.2, it provides an intermediate step between an ideal model


and the a full numerical treatment. For instance, if the full Bloch mode matrices are

truncated to include only the propagating modes, the complex reflection and trans-

mission coefficients are retained but the matrices have the dimensions of the idealised

version above and can thus be manipulated very efficiently. Alternatively, the effects of

non-ideal transmission and reflection can be included in an intuitive way by modifying

the analytic matrix expressions appropriately.

4.6 Recirculation in coupler-based MZIs

Recall from Section 4.2 that mode recirculation occurred because the input and output

waveguides of the MZI only supported a single mode of even symmetry, and thus any

odd field component created inside the interferometer is reflected by both junctions.

In Section 4.3 we showed that rather than degrading the performance of the device,

recirculation enhances the response for some wavelengths, and so it may be a desirable

feature to generate in other MZI geometries. While mode recirculation is likely to occur

in PhC-based MZIs of the type Fig. 4.1(c), MZIs of the type illustrated in Figs. 4.1(a)

and (b) do not exhibit the same behaviour because the beam splitter and coupler based

junctions can transmit all incident fields. Mode recirculation can be achieved in a

such MZIs, however, by blocking one each of the input and output ports and reflecting

the light back into the interferometer. The resulting transmission characteristics are

identical to those derived in Section 4.2. Here we use a conventional MZI analysis with

the addition of reflections from mirrorsm1 andm2 to model the bulk-optics recirculating

MZI of the type illustrated in Fig. 4.1(a). A similar approach could also be applied to

the coupler-based MZI in (b).

in

out 1

50/50 beamsplitters

m1

in

out

50/50 beamsplitters

h

h

out 2(a) (b)

m2

L1

L2

L1

L2

B1

B2

B1

B2

Figure 4.13: (a) Conventional bulk-optics MZI using beam splitters. (b) Modified bulk-

optics MZI designed to exhibit recirculation effects. The two additional mirrors m1 and

m2 reflect light back into the interferometer.

Figures 4.13(a) and (b) respectively show the conventional bulk optics MZI from

4.7. DISCUSSION 125

Fig. 4.1(a) and a modified version designed to exhibit mode recirculation. Consider first

the design illustrated in (a). Light entering via the input port is split at the first beam

splitter B1 and the two beams travel different paths before being recombined at the

second beam splitter B2. Suppose that the two arms of the interferometer have lengths

L1 and L2 = L1 + δL respectively, as shown in Fig. 4.13. A beam propagating along

path 1 therefore advances in phase by Φ1 = k0L1, while a beam propagating along path

2 advances by Φ2 = k0(L1 + δL) = Φ1 + ϕ, where k0 = 2π/λ. The light transmitted

into output 1 is a superposition of two beams; the first is transmitted through B1 then

propagates a distance L1 and is reflected at B2; the second beam is reflected at B1 then

propagates a distance L2 and is transmitted through B2. The superposition of these

two beams and their accumulated phases is given by

t1 = τeiΦ1ρ+ ρeiΦ2τ = 2ρτeiΦ1 cos(ϕ/2), (4.13)

where we have assumed that ρ = 1/√

2 exp(iφr) and τ = 1/√

2 exp(iφt) are the re-

flection and transmission coefficients of the beam splitters B1 and B2. Therefore, the

transmitted intensity for the single pass MZI is T1 = cos2(ϕ/2), identical to Eq. (4.3).

Similarly, the field transmitted into output 2 is given by

t2 = τeiΦ1τ + ρeiΦ2ρ = ie2iφreΦ1 sin(ϕ/2), (4.14)

where we have taken the fields reflected by the beam splitters to be π/2 out of phase

relative to the transmitted fields i.e. φt = φr + π/2. Comparing Eqs. (4.13) and (4.14)

with Eq. (4.2) it can be seen that the field transmitted at output 1 is equivalent to the

even mode component in Eq. (4.2), while the field transmitted at output 2 is equivalent

to the odd mode component.

Now consider the effect of placing two additional mirrors m1 and m2 in the positions

indicated in Fig. 4.13(b). Mirror m1 reflects the field t2 back into the interferometer

with an additional phase of exp(2ik0h) due to propagation between B2 and m1. The

reflected beam splits and propagates through the interferometer back to B1 where some

is transmitted into the input port, while the rest is reflected at m2. This selective re-

flection at either end of the interferometer is analogous to the odd-mode recirculation

between the junctions of the PhC MZIs in Section 4.4, and results in identical recir-

culation properties. By tracking the recirculating beam back and forth through the

interferometer, the total transmitted power is found to be given by Eq. (4.8), where

χ = βL+ ϕ/2 is replaced by χ = k0(L1 + 2h) + φr + ϕ/2.

4.7 Discussion

In this chapter we have shown that MZIs formed from waveguides with a perfectly

reflecting cladding display significantly different transmission characteristics to conven-

tional MZIs. In particular, reflections due to modal symmetry mismatches result in

mode recirculation and resonant transmission behaviour. Photonic crystals provide one


means of obtaining a reflecting cladding with the required properties and we have pro-

posed two PhC MZI designs that exhibit the predicted mode recirculation behaviour.

The semianalytic model derived in Section 4.2 gives insight into the physics of MZIs

and provides a powerful design tool for predicting the properties of relatively com-

plicated devices knowing only the dispersion properties of a single waveguide. When

combined with rigorous numerical techniques such as the Bloch mode matrix method,

new designs can be optimised rapidly over a broad parameter space. An example of this

was given in Sections 4.3 and 4.4.3 where we first identified using Eq. (4.8), the condi-

tions under which a recirculating MZI exhibits a quartic phase dependence (4.9), then

demonstrated the predicted behaviour in a PhC MZI (Fig. 4.10). The model provided

an accurate estimate of the optimum wavelength and structural parameters, thereby

saving a considerable amount of computational time compared to an all-numerical pa-

rameter search.

Although several PhC MZI designs have been fabricated, the recirculation properties

that we predict here have not been identified experimentally for several reasons. First,

the devices studied in Refs. [200, 201, 220] all use directional coupler-based junctions

that transmit both odd and even symmetry modes, and are thus inherently single-pass

MZIs. Second, although the designs in Refs [218] and [219] both have single input

and output waveguides, and would thus be expected to exhibit recirculation properties,

the junctions do not have the necessary high transmission properties identified in Sec-

tion 4.4.1 to observe clearly the odd-mode resonance effects. The coupled Y-junction

proposed in Chapter 3 exhibits high transmission over a wide bandwidth and is thus

an ideal junction for designing efficient MZIs.

A theoretical and experimental investigation of an MZI with a realistic tuning mech-

anism, rather than the model adopted for convenience in Section 4.4.2 would be of

interest. Among other things, this would provide a better estimate of the length re-

quirements for tuning the phase. Three-dimensional modelling in a slab design would

also be valuable.

Chapter 5

Efficient wide-angle transmission

into rod-type photonic crystals

In this chapter we demonstrate highly efficient coupling into uniform rod-type photonic

crystals with unmodified interfaces. Reflected powers of less than 1% are calculated for

two-dimensional structures at incident angles up to 80◦. These properties are closely

related to forward scattering resonances of the high index cylinders, and as a con-

sequence, are not displayed by rod-type photonic crystals. We use these results to

optimise a realistic PhC slab structure to exhibit both high efficiency coupling and

auto-collimation and present numerical results for two- and three-dimensional finite

difference time domain calculations. The results could also be applied to the design of

efficient superprisms and other in-band dispersion based PhC devices.

This chapter contains a substantial amount of unpublished work in Sections 5.1–5.4

which is currently being prepared for publication. The results presented in Section 5.5

are based on Ref. [226] which is due to appear in Applied Physics Letters in September

2005.

5.1 Introduction

We now turn our attention away from photonic bandgap devices and consider what is

perhaps a more fundamental photonic crystal problem, that of coupling light from a

homogeneous dielectric material such as free-space into the propagating Bloch modes of

a uniform PhC. As we saw in Section 1.8, the unusual dispersion properties of PhC bands

lead to a range of interesting effects such as superprism behaviour and auto-collimation.

To use these effects in practical devices it is essential that light can be coupled efficiently

into and out of the PhC structure with minimal back-reflection. This is necessary not

only to keep the insertion losses low, but also to prevent light from being scattered

into other parts of a device where it may cause interference and cross-talk. The latter

issue will especially important in compact integrated optical circuits consisting of many

components on a single chip.

127

128 CHAPTER 5. EFFICIENT WIDE-ANGLE TRANSMISSION

While one of the main attractions of PhC devices is their compact size, this also

presents many practical challenges when interfacing these PhC components to con-

ventional optical systems. Several efficient methods have been developed for coupling

to waveguide-based devices [8, 227–230], but the challenge remains to find a practical

solution for coupling light into the extended Bloch modes of a uniform PhC lattice for in-

band applications. The main approach reported in the literature for air-hole type PhCs

has been to modify one or several of the interface layers to produce an anti-reflection

coating on the front and back faces of the structure. One such method is analogous to

the application of thin-film anti-reflecting coatings to optical elements, where resonant

effects are used to enhance the transmission properties over a narrow frequency band-

width, although the bandwidth can be improved with multi-layered gratings [231]. A

second approach, proposed by Baba et al. [232] is to change the size and/or shape of the

air holes to gradually modify the field profile as it approaches the PhC slab, effectively

creating an apodised interface. A third possibility suggested by Witzens et al. [233] is

to place a waveguide above a PhC slab and use evanescent coupling to transfer light

from the waveguide into counter-propagating Bloch modes of the PhC.

Here we study the transmission properties of rod-type, rather than hole-type, PhCs

and demonstrate that it is possible to couple more than 99% of incident light into such

structures for a wide range of incident angles without making any modifications to the

interface. We start in Section 5.2 by considering the basic 2D geometry illustrated in

Fig. 5.1, in which a plane wave is incident on a semi-infinite PhC from a homogeneous,

semi-infinite region above. The incident medium is taken to continue into the PhC

as the background material. Both square and triangular rod-type PhCs oriented as

in Figs. 5.1(a) and (b) respectively, are found to exhibit almost perfect wide-angle

transmission features for a wide range of structural parameters.

In Sections 5.3 and 5.4 we show that this behaviour is closely related to the scattering

properties of the individual cylinders which in turn influence the transmission properties

of the gratings that form the PhC lattice. We identify the conditions under which

high efficiency coupling can occur and show in Section 5.4 that these provide a simple

explanation as to why efficient coupling is achieved more easily in rod-type PhCs than

in hole-type PhCs. In Section 5.5 we optimise a purely 2D rod-type PhC to exhibit

simultaneously very low reflectance and auto-collimation. The successful combination

of these two properties is demonstrated for an incident Gaussian beam using Finite

Difference Time Domain (FDTD) simulations, first for a purely 2D geometry, and finally

in a realistic rod-type PhC slab consisting of finite length silicon rods in a background

of silica.

5.2 Plane wave coupling to semi-infinite PhCs

In this section we consider the two geometries shown in Figs. 5.1(a) and (b), where a

plane wave is incident at an angle θi onto a semi-infinite 2D PhC consisting of dielectric

5.2. PLANE WAVE COUPLING TO SEMI-INFINITE PHCS 129

0 20 40 60 800.1

0.2

0.3

0.4

0.5

0.6

0.7

d/l

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 20 40 60 80

d/l

Transmission

0 0.2 0.4 0.6 0.8 1

G X

Mqi

(a)

KM

G

qi

(b)

(c) (d)

qi (degrees) qi (degrees)

Figure 5.1: Geometry and transmittance contour plots for a plane wave incident on a

semi-infinite PhC of rods (ncyl = 3) in air (n = 1). (a) Square lattice oriented with

Γ − X direction normal to the interface. (b) Triangular lattice with Γ −M direction

normal to the interface. (c) and (d) Transmittance contour plots for varying θi and d/λ

for the PhCs in (a) and (b) respectively. Below the dashed white line there is only a

single propagating plane wave grating order.

rods of refractive index ncyl, and radius r embedded in a dielectric background nb. To

calculate the transmission into the PhC, we use the Bloch mode method to calculate the

matrix R∞, defined in Section 2.3.2 as the plane wave reflection scattering matrix for a

plane wave incident from free-space onto a semi-infinite PhC. Since we are interested in

the reflected power, and no energy is carried by the evanescent states, R∞ is truncated

to contain only those elements corresponding to reflection into propagating plane wave

orders. Thus, when only a single reflected order is supported, R∞ can be treated as a

scalar and the total reflected power is R = |R∞|2. When multiple reflected orders are

present, the total reflectance is calculated as a sum over the reflectance into each order.

For the purely 2D structures that we consider first, the transmittance is then simply

calculated as T = 1−R.

The Bloch mode method is ideal for these calculations, as it is very efficient and can


20

40

60

80

00.25 0.3 0.35 0.4

qm

ax(d

egre

es)

r/d

T>99%T>95%T>90%

20 40 60 80

0.992

0.994

0.996

0.998

1

00.99

qi (degrees)

T

(a) (b)

Figure 5.2: (a) Transmittance as a function of incident angle for the square (solid

curve) and triangular (dashed curve) lattices in Fig. 5.1. Note the vertical axis starts

at T = 99%. (b) Maximum acceptance angle θmax plotted as a function of normalised

radius for a square lattice of rods with ncyl = 3.0 in air. The three curves correspond

to T > 90%, 95%, and 99% in the range 0 ≤ θi ≤ θmax.

deal with semi-infinite structures without the need for absorbing boundary conditions

or other techniques required by many other simulation methods to avoid reflection from

the rear surface of finite structures. In addition, the method implicitly calculates the

scattering properties of a single grating, from which the PhC lattices are formed. As we

show in Section 5.4, studying the properties of a single grating provides an intermediate

case between a single cylinder and the properties of a full 2D lattice.

The contour plots in Figs. 5.1(c) and (d) show the transmittance as a function

of incident angle and normalised frequency for a TM-polarised plane wave incident

on the semi-infinite PhCs shown in (a) and (b) respectively. The PhCs both have

period d, and consist of rods with ncyl = 3.0 and r = 0.34d embedded in free-space

(nb = 1). Recall from Section 1.8.1 that when coupling into a PhC, the incident

wavevector component parallel to the interface (kx here) must be conserved and so the

incident field can only couple to specific regions of each PhC band. We can therefore

understand the relatively complex structure of the transmission plots in Figs. 5.1(c)

and (d) as a projection of the band diagram onto the kx component of the incident

wavevector, where kx = 2π/λ sin(θi). Hence, the dark blue regions of zero transmission

correspond to full and partial bandgaps in the projected band structure. Note that the

transmission into the lowest frequency band does not exceed 90%, but almost perfect

transmission is possible into some of the higher order bands.

One of the most striking features in Fig. 5.1 is the dark red, high transmission

‘finger’ extending almost horizontally across both transmission plots at approximately

d/λ = 0.34, corresponding to transmission into the second band in both cases. At this

frequency, indicated by the dotted green lines in Figs. 5.1(c) and (d), the transmittance

into the square lattice PhC exceeds 99% for incident angles 0◦ ≤ θi ≤ 80◦. This can

5.2. PLANE WAVE COUPLING TO SEMI-INFINITE PHCS 131

be seen in Fig. 5.2(a), which shows the transmittance as a function of incident angle at

d/λ = 0.34 for the two lattices. The triangular lattice exhibits T > 99% for all angles

in the range 0◦ ≤ θi ≤ 37◦. Observe that a second region of almost perfect transmission

occurs at frequencies close to d/λ = 0.6, for which the transmission exceeds 99% for

θi ≤ 25◦ in both cases. Such efficient coupling over a wide range of angles is far

superior to the published results for air hole PhCs. In the remainder of this section and

Sections 5.3–5.4 we identify the physical origin of this behaviour and use the results to

explain why hole-type PhCs do not have similar properties.

The high transmission features exhibited by the PhCs in Fig. 5.1 are not unique

to the structural parameters chosen for the examples. In fact, highly efficient, wide-

angle coupling appears to be a common feature of rod-type PhCs, occurring for incident

fields of both TE and TM polarisations and for hexagonal and square lattices with a

range of refractive indices and radii. As an example of the robust nature of the high

transmission, Fig. 5.2(b) shows the maximum acceptance angle θmax for transmission

into the second PhC band, plotted as a function of cylinder radius for a square lattice

of rods with ncyl = 3.0 in air. The three curves correspond to T > 90%, 95%, and 99%

for all incident angles 0 ≤ θi ≤ θmax.

2.5 3.0 3.5 4.0

0.24

0.22

0.2

0.18

0.16

0.14

0.12

0.1

0.08

r/l

normalised lattice period (d/r)

T

0

0.2

0.4

0.6

0.8

1

Figure 5.3: Plane wave transmission for normal incidence into a square lattice of rods

with fixed radius r and ncyl = 3.0 plotted as a function of normalised lattice period d/r

and normalised frequency r/λ. The dashed black curves are the T = 98% contours. The

green dotted lines at r/λ = 0.115 and r/λ = 0.204 indicate the maximum transmission

frequencies for r = 0.34d (d/r = 2.94) as shown in Figs. 5.1(c) and (d).

Another unusual property of the high transmission features is weak dependence on

the lattice geometry. While the frequency at which they occur is a strong function of

the cylinder radius and refractive index, it is a much weaker function of the cylinder


position. Evidence of this is seen in Figs. 5.1(c) and (d) where the high transmission

bands occur at d/λ = 0.34 and d/λ = 0.6 for both the square and the triangular

lattice. Perhaps more surprising is the observation that the maximum transmission

frequencies change very little when the cylinder separation (lattice period) is varied but

the cylinders remain unchanged. This is demonstrated in Fig. 5.3, which is a contour

plot of the plane wave transmittance for normal incidence into a square lattice similar

to that of Fig. 5.1(a), but with a fixed cylinder radius and varying lattice period d. The

horizontal axis is expressed as the normalised lattice period d/r and the vertical axis is

the frequency r/λ, also normalised to the cylinder radius. In these units, the structure

calculated in Fig. 5.1(c) with r = 0.34d corresponds to d/r = 2.94. Observe that the

two high transmission bands occur at approximately r/λ = 0.115 and r/λ = 0.205 and

do not change significantly as the lattice period is varied by more than 60%. Other

features in the same transmission plot are clearly influenced more strongly by a change

in lattice period. This very weak dependence on structure suggests that the efficient

coupling in rod-type PhCs is driven by a property of the individual rods, rather than

by a property of the periodic lattice. We investigate this further in Sections 5.3 and

5.4.

5.3 Single cylinder scattering

In this section we show that the high transmission features of rod-type PhCs are closely

related to the scattering characteristics of the individual rods. We identify two scat-

tering parameters, namely the scattering cross-section and the asymmetry parameter,

that provide an intuitive physical interpretation of the results in Section 5.2.

Although photonic crystals rely intrinsically on coherent scattering for many of their

unique properties, the individual scattering elements contribute to the detailed charac-

teristics of the structure. In Refs. [234,235] it was shown that there is a correspondence

between the position of bandgaps in three-dimensional (3D) PhCs formed from dielec-

tric spheres, and the Mie resonances of the spheres. Similar results have been shown for

2D PhCs consisting of dielectric cylinders [236] but to our knowledge, the relationship

between scattering resonances and PhC transmittance has not been directly studied.

Here we show that there is not only a correspondence between cylinder resonances and

band position, but also a direct relationship between forward scattering and the trans-

mittance of plane waves incident on a PhC. We note that although the single cylinder

treatment presented in this section is strictly 2D, the results are also applicable to

realistic PhC slab structures, as demonstrated in Section 5.5.

The scattering characteristics of an infinite dielectric cylinder are well understood,

and can be calculated for non-conical incidence following the treatment of van de

Hulst [237]. Results for TM polarization are presented here, but we note that sim-

ilar scattering behaviour also occurs for TE polarization. Consider first a TM polarised

wave in a background medium of index (nb = 1), incident on a dielectric cylinder of

5.3. SINGLE CYLINDER SCATTERING 133

r

q

r

Figure 5.4: Single cylinder scattering geometry. An incident plane wave from above is

scattered in directions θ, where θ = 0 corresponds to forward scattering.

radius r and index ncyl in the plane perpendicular to cylinder axis as shown in Fig. 5.4.

The total field outside the cylinder is expressed as the sum of the incident and scattered

components, which allows weak scattering effects to be separated from the dominant

incident field. At large distances ρ from the cylinder, the scattered field can be written

as [237]

Ez ∝√

2

πkρexp [−ikρ+ iωt− i3π/4]T (θ), (5.1)

where k = 2π/λ,

T (θ) = b0 + 2∞∑

n=1

bn cos(nθ), (5.2)

and1

bn= 1 + i

ncylJ′n(ncylkr)Yn(kr)− Jn(ncylkr)Y

′n(kr)

ncylJ ′n(ncylkr)Jn(kr)− Jn(ncylkr)J ′n(kr).(5.3)

Note that these equations can be applied to an arbitrary index contrast after appropriate

scaling of ncyl and k.

Two scattering parameters relevant to this discussion are the scattering cross-section

per unit length of the cylinder,

cscat =2

πk

∫ 2π

0

|T (θ)|2dθ =4

k

∞∑n=−∞

|bn|2, (5.4)

and the asymmetry parameter, g = 〈cos(θ)〉, the average cosine of the scattering angle.

The scattering cross-section is a measure of the total energy scattered by the cylinder

in all directions, while the asymmetry parameter gives an indication of whether the

scattered energy is in the forward or backward direction. Here, |g| < 1 and g > 0

(g < 0) corresponds to dominant forward (backward) scattering. In the long wavelength


limit, as k → 0, the bn coefficients (5.3) scale approximately as k2(n+1), and hence the

field expansion (5.1) is dominated by the b0 term, and cscat → 0.

0.1 0.2 0.3 0.4 0.5 0.6

0.2

0.4

0.6

0.8

1

0 0.70

d/l

T

0.05 0.1 0.15 0.2

(c) (g)(f)(e)(d)

r/l

csca

t,g

(a.u

)

0

(a)

(b)

Square

Triangular

cscat

g

0

Figure 5.5: (a) Normal incidence transmission spectrum into the square (solid curve)

and the triangular (dashed curve) PhCs in Fig. 5.2.(b) Single-cylinder scattering cross-

section (dashed curve) and asymmetry parameter g (dashed-dotted curve) plotted as a

function of r/λ for the same frequency range as (a). (c)–(g) Polar plots the scattering

pattern |T (θ)|2/k for r/λ = 0.034, 0.085, 0.116, 0.163 and 0.204. The light is incident

from above.

Figure 5.5(a) shows the normal incidence transmittance spectrum for the square

and triangular lattices considered in Fig. 5.1. Note that the transmission peaks at

d/λ = 0.34 and d/λ = 0.6 almost coincide for the two structures, even though the band

gap frequencies shift significantly. The curves plotted in Fig. 5.5(b) show the scattering

cross-section and asymmetry parameter as a function of r/λ for light scattered from

a single cylinder of index ncyl = 3.0 in air. Figures 5.5 (c)-(g) are polar plots of the

scattering profile given by |T (θ)|2/k, where T (θ) is defined in Eq. (5.2). The profiles

are plotted for frequencies in the first band, the first gap, the second band, the second

gap and the third band respectively. Observe from Eq. (5.4) that the area enclosed by

these scattering diagrams is proportional to the scattering cross-section.

Figure 5.5 shows a clear correspondence between the position of the bands (indicated

by non-zero transmittance) and peaks in cscat, as has been observed by others [235,236].

5.4. GRATING PROPERTIES 135

Observe also that the transmittance into the second and third bands is better than

99%, whereas transmittance into the first band does not exceed 90% – a feature we

also identified in Section 5.2. The asymmetry parameter curve and the polar plots

provide an explanation for this difference. In the second and third bands, where the

transmittance is highest, the peak in cscat coincides approximately with a peak in g,

indicating a forward scattering resonance. In the first band g remains positive but

approaches zero, despite there being a small, broad peak in cscat. Thus, although

there is still significant scattering from each cylinder at low frequencies, the scattering

direction is almost isotropic, with Eqs. (5.1), (5.2), and (5.4) being dominated by the

b0 term as described above. This is clearly seen in the scattering patterns plotted

in Figs. 5.5(c) and (d). As the frequency is increased, higher-order scattering terms

become significant, thus introducing dipole-like and higher order scattering behaviour

which can result in strong directional scattering. For example, the cscat peaks near

d/λ = 0.36 and d/λ = 0.60 correspond to maxima in b1 and b2 respectively, with both

resonances producing strong forward scattering, indicated by the g curve of Fig. 5.5(b)

and the scattering patterns in Figs. 5.5(e) and (g).

The observation that forward scattering resonances enhance the transmission into a

PhC is a physically intuitive explanation for the results presented in Sec. 5.2, however

we have not discussed the possibility of coupling between closely spaced cylinders. In

Section 5.4 we consider the properties of a grating formed from a single line of cylinders,

which provides an intermediate case between the single cylinder results of Section 5.3

and the full 2D PhC results in Section 5.2.

5.4 Grating properties

Recall from Chapter 2 that we consider a PhC to be a stack of identical cylinder grating

layers that are coupled together by diffracted plane wave orders given by the grating

equation (2.2). In this chapter we consider only gratings with period d consisting of

identical cylinders and so we would expect the scattering properties of a grating to

approach those of the individual cylinders as d/λ → ∞. For finite lattice periods we

would still expect the forward scattering resonances of the single cylinders identified in

Section 5.3 to have strong influence on the scattering properties of a grating formed

from those cylinders.

We consider first the transmission properties of the grating that is used to form the

PhCs shown in Fig. 5.1. The square and triangular lattices are both formed from the

same grating by stacking each layer as described in Section 2.2.2. Figure 5.6(a) shows

the transmittance as a function of θi and d/λ for the grating of period d formed from rods

with ncyl = 3.0 and r = 0.34d. The results are calculated directly from the plane-wave

scattering matrix T, where we sum the total transmittance over all propagating grating

orders. Observe that the two high-transmission features of the grating at d/λ = 0.34

and d/λ = 0.6 resemble closely those in Figs. 5.1 (c) and (d). The agreement is even


more apparent in Fig. 5.6(b) where we compare the normal incidence reflectance of the

grating and the square PhC with the same parameters. The strong correlation between

low grating reflectance and efficient coupling into the PhC further reinforces the single

cylinder scattering explanation. It also explains why the transmission peaks occur at

the same frequencies in the square and triangular lattices.

0 20 40 60 80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

qi

d/l

0 0.2 0.4 0.60.001

0.01

0.1

1

d/l

R

(a) (b)

T

0

0.2

0.4

0.6

0.8

1

Figure 5.6: (a) Plane wave transmittance through a grating of period d consisting of rods

with ncyl = 3.0 and r = 0.34d, plotted as a function of incident angle and normalised

frequency. (b) Plane wave reflectance spectra (log scale) for normal incidence on the

grating (dashed red curve) and a semi-infinite square PhC lattice formed from the

grating (solid blue curve).

The challenge of establishing a rigorous proof of the relationship between the forward

scattering resonances of a cylinder and high grating transmission is a topic of ongoing

research, and is not solved in this thesis. The multipole method may be particularly

helpful in this regard as it includes implicitly the scattering properties of the individual

cylinders in the grating scattering formulation [176] thereby providing a mathematical

connection between the two structures. An alternative approach may be to consider

first the interaction of the scattered fields from two neighbouring cylinders, then use a

tight-binding approximation similar to that of Lidorikis et al. [236] to characterise the

scattering from a grating. We show here, however that the single-cylinder interpreta-

tion provides a simple physical explanation as to why rod-type PhCs exhibit superior

coupling properties compared to hole-type PhCs in agreement with our earlier results.

This argument relies on key observations: the first is that strong forward scattering is

required from each cylinder, and the second is that there must only be a single propagat-

ing diffracted grating order. The second condition can be understood in the following

way.

When a plane-wave is incident on a grating with period d, the number of propagating

grating orders depends on both the ratio of the incident wavelength to the period and

the incident angle. The condition for only a single reflected and single transmitted

grating order to exist can be derived from the grating equation (2.1). If the background

5.5. EFFICIENT COUPLING FOR AUTO-COLLIMATION 137

material has a refractive index nb, then the single order condition can be expressed as

d

λ<

1

nb(1 + sin(θi)), (5.5)

where λ is the free-space wavelength. Consider now the properties of the grating at a

forward scattering resonance of the cylinders. If Eq. (5.5) is satisfied, the light scattered

forward by each cylinder can only couple into the zeroth (specular) grating order, and

thus there is very little overlap between the scattered fields resulting in high transmission

through the grating. If multiple grating orders are present, they provide a means for

the fields scattered by each cylinder to interfere with each other via coupling to higher

diffraction orders. When this occurs, phase differences between the zeroth and higher

orders become an additional complicating factor. Moreover, it is considerably easier to

satisfy the conditions for zero reflectance into a single grating order than into multiple

reflected orders simultaneously. The relationship between single cylinder scattering and

grating transmission thus becomes much more complex and the simple arguments that

we applied in Section 5.3 are no longer valid. This interpretation is consistent with our

observations that the zero reflectance features in Fig. 5.6(b) disappear at frequencies

where multiple grating orders exist.

The results and observations of this section and Section 5.3 provide a simple expla-

nation as to why rod-type PhCs typically exhibit highly efficient coupling over a wide

range of incident angles, whereas air-hole PhCs may require interface modifications in

order to achieve satisfactory results [231–233]. In Section 5.3 we showed that the reso-

nance condition for strong forward scattering is not satisfied at low frequencies where

light is scattered almost isotropically. The same applies to scattering from cylinders

with a low refractive index contrast relative to the background. Efficient coupling is

therefore more likely to occur in the second PhC band or higher, while additional grating

orders appear if the frequency is too high. For rod-type PhCs, where the background

refractive index is lower than that of the rods, these two conditions are relatively easy to

satisfy, as shown in Figs. 5.1(c) and (d) where the dashed white lines indicate indicate

the upper frequency limit given by Eq. (5.5). In air hole PC structures it is much more

difficult to satisfy both conditions simultaneously since non-isotropic scattering from

air holes only occurs at relatively high frequencies but at the same time the high index

background material reduces the frequency at which multiple plane wave orders appear.

For example, if the background material is silicon (nb = 3.4) the first non-zero grating

order appears for large angle of incidence at frequencies above d/λ = 0.15, which is well

below the second band for most air-hole PhCs.

5.5 Efficient coupling for auto-collimation

So far in this chapter we have demonstrated very low reflection coupling to rod-type

PhCs over a wide range of incident angles, however we have not considered how the light


propagates once inside the structure. In order to exploit the efficient coupling for in-

band applications such as superprism effects and auto-collimation, we must optimise the

PhC structure so that the light is coupled into bands which have the desired dispersion

properties for these effects. In this section we present a realistic rod-type PhC slab

design that exhibits auto-collimation and less than 1.5% reflection loss at each interface

for angles of incidence up to θi = 25◦. We first describe in Section 5.5.1 the optimisation

process using 2D calculations, then in Sections 5.5.2 and 5.5.3 we present both 2D and

3D results that demonstrate the successful combination of the two effects. We also show

that this simple PhC structure can be used as a collimating beam combiner.

n = 1.460slab

n = 1.458clad

n = 1.458clad

Si rodsn = 3.4cyl

h =3d

x

zy

qi

Figure 5.7: Geometry of the rod-type PhC slab consisting of a triangular lattice of

silicon rods (ncyl = 3.4) in a slab of silica (nslab = 1.460) of height h = 3d where d is

the lattice period. The structure is clad above and below by a cladding with refractive

index nclad = 1.458.

Figure 5.7 shows the PhC slab geometry considered here, consisting of a triangular

lattice of silicon rods (ncyl = 3.4) embedded in a slab of silica with index nslab = 1.46.

The lattice is oriented such that the interface is parallel to the Γ −M direction. The

slab and the cylinders have a finite height of h = 3d, and they are capped above

and below with a cladding of index nclad = 1.458. As we demonstrate in this section

a cylinder radius of r = 0.35d results in both low reflectance and auto-collimation

properties. This geometry resembles that proposed by Martinez et al. [50], although

the slab height is increased to improve the mode overlap with the incident Gaussian

beam in the 3D simulations. The design and modelling of this structure is performed

using a combination of 2D and 3D Finite Difference Time Domain (FDTD) simulations,

in addition to Bloch mode method calculations and the use of commercial plane-wave

band structure software.


5.5.1 Design

We describe here the process of optimising the PhC in Fig. 5.7 for auto-collimation

and transmittance at incident angles of θi = 22.5◦, as this allows the PhC to be used

as a simple beam combiner for two incident beams with an angular separation of 45◦.

Although designed for a specific incident angle, the properties of the optimised structure

are relatively robust. The reflectance remains below 1.5% for all incident angles θi <

25◦, and auto-collimation is observed throughout this range.

We consider first the transmission of a plane wave incident at θi = 22.5◦ on the PhC

in Fig. 5.7, which we approximate as a 2D PhC lattice of rods with index ncyl = 3.4 in

a background of index nb = 1.458. As we demonstrate in Sections 5.5.2 and 5.5.3, the

2D results provide a good approximation to the 3D slab structure, thus providing an

efficient tool for optimizing parameters before launching computationally intensive 3D

simulations. The contour plot in Fig. 5.8(a) shows the transmittance into the second

band of the PhC as a function of the cylinder radius, where the contours correspond

to transmittances of T = 90%, 95%, 98% and 99%. These results were obtained using

the Bloch mode method as described in Section 5.2. Observe that for all cylinder radii

in the range 0.31 ≤ r ≤ 0.4 there are frequencies for which the transmittance exceeds

99%. This region defines the ideal part of parameter space for obtaining efficient auto-

collimation.

r/d

d/l

0.3 0.32 0.34 0.36 0.38 0.4

0.24

0.26

0.28

0.3

0.32

0.34

kx

ky

p/2

p

- /2p

-p

p/2 p- /2p-p

0

0

T>99%

(a) (b)

Figure 5.8: (a) Transmittance contour plot for a plane wave incident at θi = 22.5◦ onto

a 2D triangular PhC lattice of silicon rods embedded in silica. The contour levels are

at T = 90%, 95%, 98% and 99%. The green points represent the optimum frequency

for collimation at θi = 22.5◦. (b) Background: contour plot of 1/p calculated over the

Brillouin zone for the second band of the PhC in (a) with r = 0.35d. White regions

correspond to maximum 1/p (strong collimation). Blue curves: equi-incident angle

contours for θi = ±22.5◦. Red curve: d/λ = 0.29 equifrequency contour.


Recall from Section 1.8.4 that auto-collimation occurs when an incident beam cou-

ples into a Bloch mode where there is a point of inflection in the equifrequency curve

[141]. At this point, the small range of incident k-vectors present in the beam all

propagate in approximately the same direction within the crystal, and thus the beam is

collimated. If the equifrequency curve is relatively flat, then two beams can be launched

into the crystal at different angles, but once inside, they are collimated and propagate

through the crystal in approximately the same direction. Quantitatively, collimation is

measured by the parameter p defined in Eq. (1.9) [126]. For good collimation, we re-

quire small p values so that the beam direction inside the crystal θc, varies slowly with

θi. The green circles plotted on top of the transmission contours in Fig. 5.8 (a) show

the optimum collimation frequency as a function of the cylinder radius, for θi = 22.5◦.

The diagram in Fig. 5.8(b) illustrates how the optimum collimation frequency is

calculated for each cylinder radius using a similar approach to Baba et al. [238]. We

first calculate the 2D band surface for the second band using the commercial software

BandSOLVE from which we evaluate the p parameter over the Brillouin zone as shown

in Fig. 5.8(b), where the grey contours in the background represent the value of 1/p,

calculated for a cylinder radius of r = 0.35d. The narrow white regions of the 1/p plot

correspond to large 1/p values, and hence indicate the regions where collimation occurs

(the horizontal white line on the ky = 0 axis indicates the interface). The blue curves

in Fig. 5.8(b) are the equi-incident angle contours [126] that indicate the regions of the

Brillouin zone that can be accessed by varying the frequency while θi = 22.5◦. The

green circle indicates the region we wish to couple into with the incident beam, defined

by the intersection of the blue equi-incident angle contour and the white collimation

region. Thus, the optimum frequency for collimation at θi = 22.5◦ is given by the

frequency of the second band at this intersection point. In the example shown here, the

optimum collimation frequency for r = 0.35d is approximately d/λ = 0.290, indicated

by the red equifrequency curve in Fig. 5.8(b). To plot the green line in Fig. 5.8(a)

we repeat the process described above for a range of cylinder radii. The results show

that while high transmittance and auto-collimation are independent properties, they

are not incompatible. Figure 5.8(a) shows that both properties occur simultaneously

for cylinder radii in the range 0.335d ≤ r ≤ 0.382. For the results presented in the

remainder of this chapter we consider rods with r = 0.35d, for which the optimum

collimation occurs at approximately d/λ = 0.29, as shown in Fig. 5.8(b).

5.5.2 Results: 2D vs 3D coupling properties

Before presenting the collimation results, we demonstrate first that the efficient coupling

conditions identified with the 2D calculations in Section 5.5.1 provide a good approxi-

mation to the slab structure in Fig. 5.7, which requires full 3D numerical simulations.

The commercial FDTD software FullWave is used for the 3D calculations with perfectly

matched layer boundary conditions on all sides of the computational domain to simulate

a semi-infinite cladding above and below the slab. We consider an incident Gaussian


beam with an elliptical cross-section with full widths at half maximum (FWHM) of

wxy = 5d and wz = 2.5d, launched with TM polarization. While there is some loss at

the interface of the PhC due to a mismatch in the beam profile, the reflected power

remains very low, which is essential to avoid interference effects in compact devices.

Figure 5.9 shows the 3D reflectance spectra for Gaussian beams with θi = 0◦ (solid

curve) and θi = 22.5◦ (long dashed curve) incident on 16 layers of the PhC in Fig. 5.7.

Also shown are the equivalent reflectance spectra for plane waves incident on a 2D

semi-infinite PhC, calculated using the Bloch mode matrix method. On all curves the

minimum reflected power occurs for scaled frequencies between 0.29 < d/λ < 0.295,

with bandwidths of more than 15% of the central frequency in which the reflectance is

below 10%. The 3D results show a minimum reflectance of 0.2% and 1.4% for θi = 0◦

and θi = 22.5◦ respectively, compared to < 0.1% in the 2D plane-wave results, a

variation attributed to the angular spread of the Gaussian beam in the 3D calculations.

Note also that the Fabry-Perot fringes in the FDTD results are much stronger for

the normally incident beam since the finite beam width weakens multiple reflection

interference for oblique incidence.

0.24 0.26 0.28 0.3 0.32 0.34 0.36 0.38 0.40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

d/l

Re

fle

cte

d p

ow

er

2D R

2D R∞

∞

i

i

, = 0

, = 22.5

3D FDTD, 16 layers, = 0

3D FDTD, 16 layers, = 22.5

q

q

q

q

i

i

o

o

o

o

Figure 5.9: 2D plane wave reflectance spectra (R∞: Bloch mode method) for a semi-

infinite PC and 3D FDTD reflectance spectra for a Gaussian beam of width wxy = 5d

and height wz = 2.5d incident on 16 layers of cylinders. Results are shown for θi = 0◦

and θi = 22.5◦.

The agreement between the 2D and 3D calculations in Fig. 5.9 is surprisingly good

given that the 2D calculation is for plane waves incident on a semi-infinite 2D PhC,


while the 3D results are calculated for a relatively narrow Gaussian beam incident on a

PhC slab consisting of only 16 grating layers. This means that the 2D calculations can

provide an efficient tool for initial optimization calculations over a broad parameter

space as demonstrated in Sec. 5.5.1. The results in Section 5.5 show that the 2D

optimisation of the collimation parameter p is also a good approximation to the 3D

results.

5.5.3 Results: collimation

We consider now the second optimised feature of the PhC – auto-collimation. Ob-

serve that the approximately hexagonal equifrequency curve in Fig. 5.8(b) has slightly

concave, rather than flat sides. This is a consequence of optimising the structure for col-

limation at non-normal incidence. We have found however, that reasonable collimation

is achieved for all incident angles θi ≤ 25◦.

(a) (b)

PC

Figure 5.10: (a) Electric field from a 2D FDTD simulation for an incident Gaussian

beam launched from below at a frequency of d/λ at θi = 22.5◦ incident on 30 layers of

PhC. The dashed lines indicate the upper and lower interfaces of the lattice. The cylin-

ders are omitted so that the field inside the PhC can be seen more clearly. (b) Electric

field for the same Gaussian beam propagating in the uniform background material with

no PhC present.

Figure 5.10(a) shows the electric field for an incident Gaussian beam at θi = 22.5◦

launched from below with a FWHM of 5.7d onto a PhC consisting of 30 grating layers.

The cylinders have not been drawn in the figure so that the fields inside the PhC can

be more easily seen, but the dashed lines indicate the upper and lower boundaries of

the structure. The results were calculated using the commercial 2D FDTD software

FullWAVE for a frequency of d/λ = 0.29. For comparison, Fig. 5.10(b) shows the fields

in the background medium when the PhC is removed. The collimation effects of the

PhC can be clearly seen; the beam is deflected when it enters the PhC, and propagates

through the structure at an angle θc ≈ 0. At the rear interface the beam leaves the


PhC and continues to propagate in the direction of the original beam. The beam on

either side of the PhC does not broaden significantly, and remains narrower than the

beam propagating in the dielectric background. As expected from the 2D calculations

in Fig. 5.9, the reflectance is less than 1%, and no reflected fields can be observed

Fig. 5.10(a).

45o

45o

(a) (b)

Figure 5.11: (a) Electric field calculated using the 2D FDTD method for a PhC beam

combiner. Two Gaussian beams of width 5.7d are incident on the front face of a 40-layer

PhC with θi = ±22.5◦. (b) Electric field for the same parameters as (a) but without

the PhC.

Since the beam direction inside the crystal is approximately normal to the interface

of the PhC, two beams incident at θi = ±22.5◦ will propagate in approximately the

same direction through the crystal. We use this property to demonstrate a simple PhC

beam combiner exhibiting low reflection, wide angle coupling and collimation that could

be used to interact multiple beams for studying nonlinear effects in PCs. In Fig. 5.11(a)

we present 2D results for the case where two Gaussian beams with the same parameters

as in Fig. 5.10 are focused onto the front face of a 40-layer PC. Interference of the beams

results in the three-lobed pattern in front of the interface in Fig. 5.11(a). Inside the PC,

the beams are collimated and propagate in approximately the same direction, emerging


through the rear interface where they continue to propagate in their original directions.

For comparison, Fig. 5.11(b) shows the propagation of the beams in the background

material without the PhC. As we demonstrate in the 3D examples in Fig. 5.12, the field

profile at the input interface is effectively transferred through the crystal to the rear

interface with only minor distortion. This behaviour is maintained regardless of the

relative phase difference between the beams. PhC designs of this type could potentially

be used for transferring more complex field modulations such as images from one side of

the crystal to the other. In contrast, if the two beams were coupled into a single finite

waveguide, the conversion between waveguide and free space modes would destroy the

original field profile without careful waveguide design.

In this final example we demonstrate auto-collimation behaviour using full 3D nu-

merical simulations for the PhC slab illustrated in Fig. 5.7 with the parameters opti-

mised for the 2D PhC in Section 5.5.1. Figure 5.9 shows that the minimum reflectance

for the 3D structure occurs for a frequency of approximately d/λ = 0.295, slightly

higher than for the purely 2D lattice. Hence the results that follow are calculated at

this frequency, for which the reflectance is approximately 1.4%. Figure 5.12(a) shows

the fields inside a PhC of 16 layers calculated using 3D FDTD software for two incident

Gaussian beams as in Fig. 5.11. The beams have a width of wxy = 5d and height of

wz = 2.5d and are launched in phase with other.

These 3D simulations are computationally intensive and are limited by the memory

requirements for large structures. We are thus limited to relatively small computational

domains that do not allow us to consider long propagation regions before and after the

PhC. Instead, the overlapping Gaussian beams are launched just before the front face

of the PhC to reproduce the input field of the two converging beams in Fig. 5.11. With

this modification, we have been able to simulate propagation through 16 layers of PhC,

but this is close to the limit of current computational resources.

As in the 2D results in Fig. 5.11(a), the fields at the front and rear interfaces of

the PhC in Fig. 5.12(a) have a very similar profile. This can be seen more clearly

in Fig. 5.12(b) and (c) where the field intensities at the front and rear interfaces are

plotted for different phase differences between the two input beams. The dashed curve

in Fig. 5.12(b) is a Gaussian profile with a width of 5d cos(22.5◦), corresponding to the

envelope of the two input beams, while the dashed curve in Fig. 5.12(c) is a Gaussian

of width 5.8d cos(22.5◦), indicating that the beams are not perfectly collimated. Apart

from this slight broadening, the field profiles at the back interface closely resemble those

at the front interface for each of the relative phase differences. We have confirmed that

this occurs regardless of the lateral alignment of the beams with the cylinders of the

interface.

While we did not explicitly consider the out-of-plane losses in the optimisation, the

operating frequency of d/λ = 0.29 lies close to the light line, and so the light is relatively

well confined once it enters the PhC region where the high index rods provide a large

index contrast in the vertical direction. The beam profiles in Figs. 5.12 however indicate

a decrease in the peak intensity. This is due in part to the spreading of the beam as it


propagates, but also to a mismatch between the incident Gaussian beam profile and the

vertical width of the guided mode, which results in additional scattering losses. The

good agreement between the 2D and 3D reflectance spectra in Fig. 5.9 suggests that

improved mode matching in the vertical plane would increase the transmission into the

PhC without a significant increase in the reflected power.

-6 -4 -2 0 2 4 6x (d)

1

0

0.2

0.4

0.6

0.8

Inte

nsity (

a.u

.)

j

j p

j p

j p

= 0

= /2

=

= 3 /2

(b)

-6 -4 -2 0 2 4 6

1

0

0.2

0.4

0.6

0.8

Inte

nsity (

a.u

.)

x (d)

(c)

(a)

Figure 5.12: (a) 3D FDTD simulation showing the |Ey| field component in the PC

for two in-phase Gaussian beams launched with θi = ±22.5◦. (b) Field intensity (nor-

malised) at the input interface for beams with phase differences ϕ = 0, π/2, π, and 3π/2.

The dashed curve is a Gaussian profile of width wxy = 5d cos(22.5◦). (c) Intensity at

the output interface for the phase differences in (b). The dashed curve is a Gaussian

profile of width 5.8d cos(22.5◦).


5.6 Discussion

In this chapter we have studied the coupling of light from a homogeneous dielectric into

the Bloch modes of uniform rod-type PhCs and have shown that very efficient coupling

is possible over a wide range of angles without modifying the interface of the crystal.

Numerical studies reveal that this behaviour is a generic property of many rod-type

PhCs, and thus they present an attractive class of structure for studying in-band PhC

applications such as auto-collimation and superprism effects. In contrast, coupling to

air-hole type PhCs is typically much less efficient and requires interface modifications or

novel coupling techniques such as those proposed in Refs. [231–233]. While a rigorous

proof of the single cylinder scattering interpretation is still being developed, there is

sufficient evidence to connect the efficient coupling properties of the PhCs with forward

scattering resonances of the cylinders. We have shown that this interpretation provides

a simple and intuitive physical explanation for the coupling properties of rod-type PhCs

and the difference between these and hole-type PhCs.

The recognition that forward scattering resonances play a large rope in determining

the transmission properties of a PhC could also lead to improved designs for hole-type

PhCs in which the hole shape is optimised to produce strong forward scattering. If

this was to work in practice, the modifications would require careful design so as not

to interfere with other desirable properties. While the additional fabrication challenges

may be comparable to the interface modifications that have been proposed, a theoretical

study of this approach may be worthwhile.

Analogies can be made between the transmission properties observed here and loss

mechanism in a new class of microstructured optical fibre consisting of high refrac-

tive index cylindrical inclusions in a low index background [239]. These fibres rely on

“anti-resonant reflection” and exhibit high loss features in the transmission spectrum

that correspond to forward scattering resonances of the individual cylinders [240]. We

note also that there are similarities between our results and recent studies of perfect

transmission through periodically structured metallic films [241–243]. In particular,

Garcıa et al. identified Mie resonances as the cause of enhanced transmission through

periodic arrays of silica spheres embedded in a metallic film [242]. In another recent

paper, Mandatori et al. demonstrated theoretically, wide-angle transmission into 1D

multilayer structures consisting of birefringent materials [244]. High transmissions were

obtained for incident angles up to 80◦ by choosing the birefringent properties of each

layer to balance the angular transmission dependence due to increased path length and

variation of Fresnel coefficients. While it is not immediately clear whether the wide-

angle PhC transmission investigated in this chapter can be interpreted in a similar way,

a future comparison of the two structures is warranted.

Since rod-type PhCs exhibit efficient coupling over a wide parameter space, this

provides significant flexibility to optimise a structure for additional characteristics as

demonstrated in Section 5.5 for auto-collimation. These results have been used to

design an efficient PC self-collimating beam combiner that could be used for studying

5.6. DISCUSSION 147

nonlinear interactions in PhCs. Many other in-band applications could benefit from

these design techniques and results. For example PhC superprisms require light to be

coupled efficiently through both interfaces while maintaining a good beam shape [126]

and there are also proposals for optical chips based on self collimated beams [146,245],

for which this work would clearly be of great benefit.

The ability to combine auto-collimation and efficient coupling without complex in-

terface modification is surprising if considered from the viewpoint of conventional optics.

At a dielectric interface, strong beam deflections require a large index contrast but this

is accompanied by high reflectance. In contrast, here we have demonstrated deflections

of more than 20◦ with a reflectance of less than 1%. This is only possible as a result of

the unique dispersion properties of photonic crystals.

In conclusion, we have found that coupling into regular rod-type PhCs can be signif-

icantly more efficient over a wide range of incident angles than for PCs with air holes,

and this can be achieved without modifications to the interface. We have demonstrated

that 2D simulations of this behaviour can be used to optimise the parameters for real-

istic PC slab geometries, and that these properties are retained by the 3D structures.

Chapter 6

Discussion and conclusions

In this thesis we have considered a number of photonic crystal structures and devices

and identified the basic physical effects that determine their behaviour. While we have

focussed on two aspects of these unique structure – resonances and mode coupling

properties – the theoretical approach taken here is much more widely applicable. In

particular, the combined use of simple physical models and accurate numerical methods

provides a powerful and efficient array of tools for studying many structures. As we

demonstrated, this is aided greatly by the Bloch mode scattering matrix method, which

is not only an efficient numerical method but also provides framework for deriving

semi-analytic models. Perhaps one of the greatest advantages to be gained from this

approach is the physical insight, and as a consequence, intuition which can lead to

simple design rules. These will become ever more important as photonic crystal devices

enter commercial applications.

In Chapters 3 and 4 we presented several PhC waveguide based devices designed for

operation in a photonic bandgap. The three main structures we considered are the FDC,

the coupled Y-junction and the recirculating Mach-Zehnder interferometer. While these

devices have been demonstrated only in a purely 2D geometry, the underlying concepts

are general and rely predominantly on modal dispersion properties and characteristics

common to most PhCs rather than specific details of the waveguides. The geometries

that we used for the numerical demonstration of these concepts were designed without

considering the out-of-plane losses that are present in realistic PhC slab structures.

As such, the specific design parameters are intended as examples only, and do not

necessarily correspond to practical devices. The logical next step, therefore, is to use

the insights gained from this research to develop complete designs for experimental

demonstration.

As we have mentioned several times throughout this thesis, one of the important

challenges in designing efficient resonant devices is controlling out-of-plane losses. Al-

though the Y-junction and the Mach-Zehnder interferometer presented here are not

high-Q resonant structures like the FDC, the unique properties of all three structures

are a result of multiple reflections between the ends of blocked waveguides, and so the

149

150 CHAPTER 6. DISCUSSION AND CONCLUSIONS

design considerations are common. We discuss here three distinct issues that must be

addressed in order to achieve realistic device designs. They are:

1. Control of waveguide dispersion properties for modes guided below the lightline.

2. Out-of-plane scattering losses at the reflective interfaces.

3. Tolerance to symmetry breaking and other fabrication variations.

These issues are common to many PhC slab-based devices, and thus there are many

examples in the literature where they have been successfully addressed.

Let us consider first the waveguide dispersion properties. As the semi-analytic mod-

els derived in Sections 3.4.1 and 4.2 show, the performance of these devices depends

strongly on the dispersion. While dispersion tuning is relatively straightforward in the

2D analysis where radiation losses are ignored, the available parameter space is reduced

considerably in slab structures by the requirement to operate below the lightline. How-

ever, techniques such as changing the waveguide width [97], or tuning the radius of the

cylinders close to waveguide [98] provide significant control over the dispersion. We

expect that tuning mechanisms such as these can be used do optimise the waveguide

design to permit our devices to operate below the lightline.

The second issue – reducing scattering losses at the reflective interfaces – is some-

what more of a challenge. While waveguide modes that lie below the cladding lightline

are theoretically lossless in a straight waveguide, considerable losses may still occur

when additional features such as junctions, bends or mirrors are introduced. Since the

devices we have proposed all require light to be reflected by the PhC at the end of a

blocked waveguide, the control of losses at these mirrors is essential. While there has

been considerable effort to reduce the radiation losses of high-Q PhC cavities via vari-

ous structural tuning methods as discussed in Section 1.7.1, [54,61,68,70], the radiation

losses that occur at a single reflective interface have received less attention. The physi-

cal mechanisms of this loss have been studied in some detail in Refs. [61,246,247], and

a number of approaches have been proposed to minimise the effects. Although the re-

sults were reported for waveguides with a 1D Bragg mirror, the physical interpretation

applies equally to 2D PhC mirrors in a slab geometry.

In Ref. [247], Bragg reflection from a PhC mirror is described as a three-step process;

first the forward propagating mode in the waveguide must couple to the fundamental

forward (non-propagating) Bloch mode of the PhC; second, light is coupled from the

forward travelling Bloch mode into its counterpropagating pair; and finally the back-

ward travelling Bloch mode is coupled into the backward propagating waveguide mode.

While the second process is lossless provided the Bloch modes are guided by the slab,

radiation losses may occur during the first and last stages of the process due to possible

mode mismatches between the modes of the PhC and the modes of the waveguide. Sev-

eral adiabatic tapering techniques have been proposed to optimise the mode-matching

in 1D periodic structures [246, 247] for high reflectance and low radiation losses. A

151

similar approach could be applied to the blocked waveguides in the FDC and the other

devices to minimise the radiation loss at these interfaces.

An alternative highly efficient waveguide mirror was demonstrated recently by Song

et al. [248], formed by coupling two slightly different waveguides. Almost perfect re-

flection occurs at frequencies where the input waveguide supports a propagating mode

but the output waveguide supports no propagating modes (ie: in the frequency interval

between the waveguide mode cutoff and the bottom of the bandgap). These results

are consistent with a mode-matching interpretation since the very similar geometry of

the two waveguides is likely to allow efficient coupling between the propagating and

nonpropagating modes. While this design is suitable for a single waveguide mirror,

some modification would be required to apply to the FDC or the Y-junction, where

waveguides on both sides of the interface are required to support propagating modes.

Reducing the scattering losses upon reflection is likely to result in efficient Y-junction

and Mach-Zehnder designs, but in order to achieve the predicted high-Q resonances in

the FDC, it may be necessary to consider the complete cavity design. While reducing

the out-of-plane scattering losses at each reflective interface is critical to achieve a high-

Q response, a second mechanism due to “radiation-loss recycling” has also been shown

to play a significant role in these structures [61]. Since this effect depends on the details

of the cavity as well as the interfaces, more complicated optimisation approaches may

be required. In Section 3.4.5 we show that the FDC is closely related to other common

PhC cavity designs, so the same optimisation techniques developed for those structures

could be applied [54,67,68,70,74].

The third and final consideration for developing practical PhC device designs is

their tolerance to fabrication errors, although recent demonstrations of ultrahigh-Qcavities [74] and low-loss waveguides [83–85] indicate that these errors can be tightly

controlled. While we cannot predict the sensitivity of the devices proposed here without

a numerical study, we can make some general observations. One aspect common to

all of our structures, is their dependence on modal symmetry, a property we used to

develop the semi-analytic models in Sections 3.4.1 and 4.2. Although it is unlikely

that small structural variations which break the symmetry will destroy the properties

of these devices, they may be more sensitive to this type of fabrication error. The

Y-junction in particular requires further study in this regard, since an asymmetric

structure would allow light to couple into the odd supermode of the triple waveguide

section – a possibility forbidden in the symmetric geometry considered here.

To summarise the results of this discussion, the waveguide-based devices that we pro-

pose in this thesis are expected to present the same challenges as many other PhC based

structures that have been successfully demonstrated in experiment. By addressing the

three issues listed above, we hope to develop realistic designs for future fabrication.

The PhC structures that we study in Chapter 5 face similar loss-related issues

which we have already discussed briefly in Section 5.6. Provided the Bloch modes

being coupled into lie below the cladding lightline of the slab, the dominant cause of

loss is the vertical mode-mismatch at the interface. The width of the incident Gaussian

152 CHAPTER 6. DISCUSSION AND CONCLUSIONS

beam in the 3D simulations was chosen to match approximately the observed width

of the guided mode in the slab, but no further attempt was made to optimise the

coupling. To minimise the radiation loss at the interface one could either tune the

incident beam parameters, or alternatively couple light into the crystal via a dielectric

waveguide [143,249], which may provide better mode matching conditions.

In conclusion, we have studied in this thesis a number of photonic crystal structures

and devices that derive their characteristic properties from resonance and mode coupling

effects that are common to most PhC geometries. The combination of these two features

in a single device results in a range of different behaviours that we have exploited to

design several novel and promising devices. In Chapter 3 we presented three such

structures: a high-Q resonant filter, a wide-bandwidth symmetric Y-junction and a

periodic coupled waveguide that can be tuned to exhibit very low group velocities.

These same design concepts and analysis methods were extended in Chapter 4 where we

showed that mode recirculation in PhC Mach-Zehnder interferometers can be harnessed

to enhance the response of these devices.

In Chapter 5 we considered an entirely different aspect of resonance and mode

coupling: the transmission of light from free-space into the propagating Bloch modes of

a uniform PhC. The demonstration of highly efficient wide-angle coupling in rod-type

geometries, and the combination of this property with auto-collimation effects could

motivate further research efforts to fabricate rod-type PhCs for in-band applications.

Finally, it should be noted that the structures we have studied are intended to

exploit the unique properties of photonic crystals, rather than to replicate conventional

devices on a smaller scale. As we have demonstrated, this design philosophy, combined

with physical insight can lead to simple but efficient devices with highly attractive

properties. The combination of powerful general methods and semi-analytic solutions

for particular structures is one which will help the rapid progress in photonic crystal

optics to continue.

Bibliography

[1] A. Parker, In the blink of an eye: the most dramatic event in the history of life

(Free press, London, 2003).

[2] H. A. Macleod, Thin film optical filters, 3rd ed. (Institute of Physics, London,

2001).

[3] E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and elec-

tronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).

[4] S. John, “Strong localization of photons in certain disordered dielectric superlat-

tices,” Phys. Rev. Lett. 58, 2486–2489 (1987).

[5] H. A. Hauss and C. V. Shank, “Antisymmetric taper of distributed feedback

lasers,” IEEE J. Quantum Electron. QE-12, 532–539 (1976).

[6] R. Kashyap, Fiber Bragg Gratings (Academic Press, San Diego, 1999).

[7] J. Joannopoulos, R. Meade, and J. Winn, Photonic Crystals (Princeton Univer-

sity, Princeton, N.J., 1995).

[8] S. J. McNab, N. Moll, and Y. A. Vlasov, “Ultra-low loss photonic integrated

circuit with membrane-type photonic crystal waveguides,” Opt. Express 11, 2927–

2939 (2003).

[9] P. Bienstman, S. Assefa, S. G. Johnson, J. D. Joannopoulos, G. Petrich, and

L. A. Kolodziejski, “Taper structures for coupling into photonic crystal slab wave-

guides,” J. Opt. Soc. Am. B 20, 1817–1821 (2003).

[10] K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in

periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990).

[11] E. Yablonovitch and T. J. Gmitter, “Photonic band structure: the face-centered-

cubic case,” Phys. Rev. Lett. 63, 1950–1953 (1989).

[12] E. Yablonovitch, T. J. Gmitter, and K. M. Leung, “Photonic band structure: the

face-centered-cubic case employing non-spherical atoms,” Phys. Rev. Lett. 67,

2295–2298 (1991).

153

154 BIBLIOGRAPHY

[13] A. Chelnokov, S. Rowson, J. M. Lourtioz, L. Duvillaret, and J. L. Coutaz, “Tera-

hertz characterisation of mechanically machined 3D photonic crystal,” Electron.

Lett. 33, 1981–1983 (1997).

[14] S. Y. Linn, J. G. Fleming, D. L. Hetherington, B. K. Smith, R. Biswas, K. M.

Ho, M. M. Sigalas, W. Zubrzycki, S. R. Kurtz, and J. Bur, “A three-dimensional

photonic crystal operating at infrared wavelengths,” Nature 394, 251–253 (1998).

[15] J. G. Fleming and S. Y. Lin, “Three-dimensional photonic crystal with a stop

band from 1.35 to 1.95µm,” Opt. Lett. 24, 49–51 (1999).

[16] S. Noda, K. Tomoda, N. Yamamoto, and A. Chutinan, “Full three-dimensional

photonic bandgap crystals at near-infrared wavelengths,” Science 289, 604–606

(2000).

[17] A. Blanco, E. Chomski, S. Grabtchak, M. Ibisate, S. John, S. W. Leonard,

C. Lopez, F. Meseguer, H. Miguez, J. P. Mondia, G. A. Ozin, O. Toader, and

H. M. van Driel, “Large-scale synthesis of a silicon photonic crystal with a com-

plete three-dimensional bandgap near 1.5 micrometres,” Nature 405, 437–440

(2000).

[18] M. O. Jensen and M. J. Brett, “Square spiral 3D photonic bandgap crystals at

telecommunications frequencies,” Opt. Express 13, 3348–3354 (2005).

[19] E. Lidorikis, M. L. Povinelli, S. G. Johnson, and J. D. Joannopoulos,

“Polarization-independent linear waveguides in 3D photonic crystals,” Phys. Rev.

Lett. 91, 023902 (2003).

[20] R. Wang and S. John, “Engineering the electromagnetic vacuum for controlling

light with light in a photonic-band-gap microchip,” Phys. Rev. A. 70, 043805

(2004).

[21] S. Ogawa, M. Imada, Y. Susumu, M. Okana, and S. Noda, “Control of light

emission by 3D photonic crystals,” Science 305, 227–229 (2004).

[22] C. Sell, C. Christensen, G. Tuttle, Z. Y. Li, and K. M. Ho, “Propagation and loss

in three-dimensional photonic crystal waveguides with imperfect confinement,”

Phys. Rev. B. 68, 113106 (2003).

[23] M. Fujita, S. Takahashi, Y. Tanaka, T. Asano, and S. Noda, “Simultaneous in-

hibition and redistribution of spontaneuos light emission in photonic crystals,”

Science 308, 1296–1298 (2005).

[24] P. Lodahl, A. F. van Driel, I. S. Nikolaev, A. Irman, K. Overgaag, D. Vanmaekel-

bergh, and W. L. Vos, “Controlling the dynamics of spontaneous emission from

quantum dots by photonic crystals,” Nature 430, 654–657 (2004).

BIBLIOGRAPHY 155

[25] S. Y. Lin, J. Moreno, and J. G. Fleming, “Three-dimensional photonic-crystal

emitter for thermal photovoltaic power generation,” Appl. Phys. Lett. 83, 380–

382 (2003).

[26] S. Y. Lin, J. G. Fleming, and I. El-Kady, “Three-dimensional photonic-crystal

emission through thermal emission,” Opt. Lett. 28, 1909–1911 (2003).

[27] T. Prasad, V. Colvin, and D. Mittleman, “Superprism phenomenon in three-

dimensional macroporous polymer photonic crystals,” Phys. Rev. B 67, 165103

(2003).

[28] X. Hu, Q. Zhiang, Y. Liu, B. Cheng, and D. Zhang, “Ultrafast three-dimensional

tunable photonic crystal,” Appl. Phys. Lett. 83, 2518–2520 (2003).

[29] S. Kubo, Z. Z. Gu, K. Takahashi, A. Fujishima, H. Segawa, and O. Sato, “Tun-

able photonic band gap crystals based on a liquid crystal-infiltrated inverse opal

structure,” J. Am. Chem. Soc. 126, 8314–8319 (2004).

[30] D. McPhail, M. Straub, and M. Gu, “Electrical tuning of three-dimensional pho-

tonic crystals using polymer dispersed liquid crystals,” Appl. Phys. Lett. 86,

051103 (2005).

[31] M. Plihal, A. Shambrook, and A. A. Maradudin, “Two-dimensional photonic

bandgap structures,” Opt. Comm. 80, 199–204 (1991).

[32] M. Plihal and A. A. Maradudin, “Photonic band structure of two-dimensional

systems: the triangular lattice,” Phys. Rev. B. 44, 8565–8571 (1991).

[33] P. R. Villeneuve and M. Piche, “Photonic band gaps in two-dimensional square

and hexagonal lattices,” Phys. Rev. B. 46, 4969–4972 (1992).

[34] R. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Existence of

a photonic band gap in two dimensions,” Appl. Phys. Lett. 61, 495–497 (1992).

[35] J. C. Knight, J. Broeng, T. A. Birks, and P. St. J. Russel, “Photonic bandgap

guidance in optical fibres,” Science 282, 1476–1478 (1998).

[36] C. C. Chen, C. Y. Chen, W. K. Wang, F. H. Huang, C. K. Lin, W. Y. Chiu, and

Y. J. Chan, “Photonic crystal directional couplers formed by InAlGaAs nano-

rods,” Opt. Express 13, 38–43 (2005).

[37] A. Chutinan and S. John, “Diffractionless flow of light in two- and three-

dimensional photonic band gap heterostructures: Theory, design rules, and sim-

ulations,” Phys. Rev. E. 71, 026605 (2005).

[38] S. G. Johnson, P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Linear wave-

guides in photonic-crystal slabs,” Phys. Rev. B 62, 8212–8222 (2000).

156 BIBLIOGRAPHY

[39] A. Chutinan, S. John, and O. Toader, “Diffractionless flow of light in all-optical

microchips,” Phys. Rev. Lett. 90, 123901 (2003).

[40] R. D. Meade, A. Devenyi, J. D. Joannopoulos, O. L. Alerhand, D. A. Smith, and

K. Kash, “Novel applications of photonic bandgap materials: Low-loss bands and

high Q cavities,” J. Appl. Phys. 75, 4753–4755 (1994).

[41] T. F. Krauss, R. M. De La Rue, and S. Brand, “Two-dimensional photonic-

bandgap structures operating at near infrared wavelengths,” Nature 383, 699–702

(1996).

[42] S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A.

Kolodziejski, “Guided modes in photonic crystal slabs,” Phys. Rev. B. 60, 5751–

5758 (1999).

[43] S. Kuchinsky, D. C. Allan, N. F. Borrelli, and J. C. Cotteverte, “3D localization

in a channel waveguide in a photonic crystal with 2D periodicity,” Opt. Commun.

175, 147–152 (2000).

[44] M. Loncar, T. Doll, J. Vuckovic, and A. Scherer, “Design and fabrication of silicon

photonic crystal optical waveguides,” J. Lightwave Technol. 18, 1402–1411 (2000).

[45] E. Chow, S. Y. Lin, S. G. Johnson, P. R. Villeneuve, J. D. Joannopoulos,

J. R. Wendt, G. A. Vawter, W. Zubrzycki, H. Hou, and A. Alleman, “Three-

dimensional control of light in a two-dimensional photonic crystal slab,” Nature

407, 983–986 (2000).

[46] Y. Sugimoto, Y. Tanaka, N. Ikeda, Y. Nakamura, and K. Asakawa, “Low propa-

gation loss of 0.76 dB/mm in GaAs-based single-line-defect two-dimensional pho-

tonic crystal slab waveguides up to 1 cm in length,” Opt. Express 12, 1090–1096

(2004).

[47] A. Assefa, P. T. Rakich, P. Bienstman, S. G. Johnson, G. S. Petrich, J. D.

Joannopoulos, L. A. Kolodziejski, E. P. Ippen, and H. I. Smith, “Guiding 1.5µm

light in photonic crystals based on dielectric rods,” Appl. Phys. Lett. 85, 6110–

6112 (2004).

[48] M. Tokushima, H. Yamada, and Y. Arakawa, “1.5µm-wavelength light guiding in

waveguides in square-lattice-of-rod-photonic crystal slab,” Appl. Phys. Lett. 84,

4298–4300 (2004).

[49] M. Tokushima, “Experimental demonstration of waveguides in arrayed-rod pho-

tonic crystals for integrated optical buffers,” in Optical Fiber Communication

Conference and Cxposition and the National Fiber Optic Engineers Conference

on CD-ROM (Optical Socity of America, Washington, DC, USA, 2005). Paper

OWD2.

BIBLIOGRAPHY 157

[50] A. Martinez, J. Garcia, G. Sanchez, and J. Marti, “Planar photonic crystal struc-

ture with inherently single-mode waveguides,” J. Opt. Soc. Am. A 20, 2131–2136

(2003).

[51] M. Qiu, “Effective index method for heterostructure-slab-waveguide-based two-

dimensional photonic crystals,” Appl. Phys. Lett. 81, 1163–1165 (2002).

[52] H. Benisty, C. Weisbuch, D. Labilloy, M. Rattier, C. J. M. Smith, T. F. Krauss,

R. M. De La Rue, R. Houdre, U. Oesterle, C. Jouanin, and D. Cassagne, “Optical

and confinement properties of two-dimensional photonic crystals,” J. Lightwave

Technol. 17, 2063–2077 (1999).

[53] S. Olivier, H. Benisty, M. Rattier, C. Weisbuch, M. Qiu, A. Karlsson, C. J. M.

Smith, R. Houdre, and U. Oesterle, “Resonant and nonresonant transmission

through waveguide bends in a planar photonic crystal,” Appl. Phys. Lett. 79,

2514–2516 (2001).

[54] Y. Akahane, T. Asano, B. S. Song, and S. Noda, “High-Q photonic nanocavity

in a two-dimensional photonic crystal,” Nature 425, 944–947 (2003).

[55] R. Zoli, “2D Photonic crystals: theory and applications,” Ph.D. thesis, Univerita

degli Studi di Bologna, Bologna, Italy (2004).

[56] G. W. Milton, The theory of composites, 1st ed. (Cambridge University Press,

Cambridge, 2002).

[57] K. J. Vahala, “Optical microcavities,” Nature 424, 839–846 (2003).

[58] J. D. Jackson, Classical electrodynamics, 2nd ed. (John Wiley and Sons, New

York, 1975).

[59] D. W. Vernooy, V. S. Ilchenko, H. Mabuchi, E. W. Streed, and H. J. Kimble,

“High-Q measurements of fused-silica microspheres in the near infrared,” Opt.

Lett. 23, 247–249 (1998).

[60] D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q

toroid microcavity on a chip,” Nature 421, 925–928 (2003).

[61] P. Lalanne, S. Milas, and J. P. Hugonin, “Two physical mechanisms for boosting

the quality factor to cavity volume ratio of photonic crystal microcavities,” Opt.

Express 12, 458–467 (2004).

[62] O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and

I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science 284,

1819–1821 (1999).

158 BIBLIOGRAPHY

[63] O. J. Painter, A. Husain, A. Scherer, J. D. O’Brien, I. Kim, and P. D. Dapkus,

“Room temperature photonic crystal defect lasers at near-infrared wavelengths

in InGaAsP,” J. Lightwave Technol. 17, 2082–2088 (1999).

[64] J. Vuckovic, M. Loncar, H. Mabuchi, and A. Scherer, “Design of photonic crystal

microcavities for cavity QED,” Phys. Rev. E 65, 016608 (2001).

[65] T. Yoshie, J. Vuckovic, A. Scherer, H. Chen, and D. Deppe, “High quality two-

dimensional photonic crystal slab cavities,” Appl. Phys. Lett. 79, 4289–4291

(2001).

[66] S. Y. Lin, E. Chow, S. G. Johnson, and J. D. Joannopoulos, “Direct measurement

of the quality factor in a two-dimensional photonic-crystal microcavity,” Opt.

Lett. 26, 1903–1905 (2001).

[67] J. Vuckovic, M. Loncar, H. Mabuchi, and A. Scherer, “Optimization of the Q

factor in photonic crystal microcavities,” IEEE J. Quantum Electron. 38, 850–

856 (2002).

[68] K. Srinivasan and O. Painter, “Momentum space design of high-Q photonic crystal

optical cavities,” Opt. Express 10, 670–684 (2002).

[69] K. Srinivasan, P. E. Barklay, O. Painter, J. Chen, A. Y. Cho, and C. Gmachl, “Ex-

perimental demonstration of a high quality factor photonic crystal microcavity,”

Appl. Phys. Lett. 83, 1915–1917 (2003).

[70] S. G. Johnson, S. Fan, A. Mekis, and J. D. Joannopoulos, “Multipole-cancellation

mechanism for high-Q cavities in the absence of a complete photonic band gap,”

Appl. Phys. Lett. 78, 3388–3390 (2001).

[71] K. Srinivasan and O. Painter, “Fourier space design of high-Q cavities in standard

and compressed hexagonal lattice photonic crystals,” Opt. Express 11, 579–593

(2003).

[72] K. Srinivasan, P. E. Barclay, and O. Painter, “Fabrication-tolerant high quality

factor photonic crystal microcavities,” Opt. Express 12, 1458–1463 (2004).

[73] T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Demonstration of ultra-

high-Q small mode volume toroid microcavities on a chip,” Appl. Phys. Lett. 85,

6113–6115 (2004).

[74] Y. Akahane, T. Asano, B. S. Song, and S. Noda, “Fine-tuned high-Q photonic-

crystal nanocavity,” Opt. Express 13, 1203–1214 (2005).

[75] B. S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double-

heterostructure nanocavity,” Nature Materials 4, 207–210 (2005).

BIBLIOGRAPHY 159

[76] M. Notomi, A. Shinya, S. Mitsugi, G. Kira, E. Kuramochi, and T. Tanabe, “Op-

tical bistable switching action of Si high-Q photonic-crystal nanocavities,” Opt.

Express 13, 2678–2687 (2005).

[77] S. Boscolo, M. Midrio, and T. F. Krauss, “Y junctions in photonic crystal channel

waveguides: high transmission and impedance matching,” Opt. Lett. 27 (2002).

[78] R. Biswas, Z. Y. Li, and K. M. Ho, “Impedance of photonic crystals and photonic

crystal waveguides,” Appl. Phys. Lett. 84, 1254–1256 (2004).

[79] W. Bogaerts, R. Baets, P. Dumon, V. Wiaux, S. Beckx, D. Taillaert, B. Luys-

saert, J. van Campenhout, P. Bienstman, and D. Van Thourhout, “Nanophotonic

waveguides in silicon-on-insulator fabricated with CMOS technology,” J. Light-

wave Technol. 23, 401–412 (2005).

[80] S. Y. Lin, E. Chow, V. Hietala, P. R. Villeneuve, and J. D. Joannopoulos, “Ex-

perimental demonstration of guiding and bending of electromagnetic waves in a

photonic crystal,” Science 282, 274–276 (1998).

[81] S. Y. Lin, E. Chow, S. G. Johnson, and J. G. Joannopoulos, “Demonstration of

highly efficient waveguiding in photonic crystal slab at the 1.5µm wavelength,”

Opt. Lett. 25, 1297–1299 (2000).

[82] M. Loncar, D. Nedeljkovic, T. Doll, and J. Vuckovic, “Waveguiding in planar

photonic crystals,” Appl. Phys. Lett. 77, 1937–1939 (2000).

[83] M. Notomi, A. Shinya, S. Mitsugi, E. Kuramochi, and H. Y. Ryu, “Waveguides,

resonators and their coupled elements in photonic crystal slabs,” Opt. Express

12, 1551–1561 (2004).

[84] S. Hughes, L. Ramunno, J. F. Young, and J. E. Sipe, “Extrinsic optical scattering

loss in photonic crystal waveguides: role of fabrication disorder and photon group

velocity,” Phys. Rev. Lett. 94, 033903 (2005).

[85] E. Dulkeith, S. J. McNab, and Y. A. Vlasov, “Mapping the optical properties

of SOI-type photonic crystal waveguides,” in Conference on Lasers and Electro-

Optics/Quantum Electronics and Laser Science Conference (CLEO/QELS) 2005

(Optical Society of America, Wachington, DC, USA, 2005). Paper QWD3.

[86] K. K. Lee, D. R. Lim, and L. C. Kimerling, “Fabrication of ultralow-loss Si/SiO2

waveguides by roughness reduction,” Opt. Lett. 26, 1888–1890 (2001).

[87] M. M. Sigalas, C. M. Soukoulis, C. T. Chan, and D. Turner, “Localization of

electromagnetic waves in two-dimensional disordered systems,” Phys. Rev. B 53,

8340–8348 (1996).

160 BIBLIOGRAPHY

[88] M. Bayindir, E. Cubukcu, I. Bulu, T. Tut, and E. Ozbay, “Photonic band gaps,

defect characteristics, and waveguiding in two-dimensional disordered dielectrics

and metallic photonic crystals,” Phys. Rev. B 64, 195113 (2001).

[89] K. C. Kwan, X. Zhang, Z. Q. Zhang, and C. T. Chan, “Effects of disorder on

photonic crystal-based waveguides,” Appl. Phys. Lett. 82, 4414–4416 (2003).

[90] A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. Joannopoulos,

“High Transmission through sharp bends in photonic crystal waveguides,” Phys.

Rev. Lett. 77, 3787–3790 (1996).

[91] E. Chow, S. Y. Lin, J. R. Wendt, S. G. Johnson, and J. D. Joannopoulos, “Quan-

titative analysis of bending efficiency in photonic-crystal waveguide bends at

λ = 1.55µm wavelengths,” Opt. Lett. 26, 286–288 (2001).

[92] E. A. J. Marcatilli, “Bends in optical dielectric waveguides,” Bell Syst. Tech. J.

48, 2103–2132 (1969).

[93] T. Tsuchizawa, K. Yamada, H. Fukuda, T. Watanabe, J. Takahashi, M. Taka-

hashi, T. Shoji, E. Tamechika, S. Itabashi, and H. Morita, “Microphotonics de-

vices based on silicon microfabrication technology,” IEEE Sel. Top. Quantum

Electron. 11, 232–240 (2005).

[94] R. U. Ahmad, F. Pizzuto, G. S. Camarda, R. L. Espinola, H. Rao, and R. M. Os-

good, Jr., “Ultracompact corner-mirrors and T-branches in silicon-on-insulator,”

IEEE Photon. Technol. Lett. 14, 65–67 (2002).

[95] M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama,

“Extremely large group-velocity dispersion of line-defect waveguides in photonic

crystal slabs,” Phys. Rev. Lett. 87, 253902 (2001).

[96] H. Gersen, T. J. Karke, R. J. P. Engelen, W. Bogaerts, J. P. Korterik, N. F. van

Hulst, T. F. Krauss, and L. Kuipers, “Real-space observation of ultraslow light

in photonic crystal waveguides,” Phys. Rev. Lett. 94, 073903 (2005).

[97] M. Notomi, A. Shinya, K. Yamada, J. Takahashi, C. Takahashi, and I. Yoko-

hama, “Structural tuning of guiding modes of line-defect waveguides of silicon-

on-insulator photonic crystal slabs,” IEEE J. Quantum Electron. 38, 736–742

(2002).

[98] A. Y. Petrov and M. Eich, “Zero dispersion at small group velocities in photonic

crystal waveguides,” Appl. Phys. Lett. 85, 4866–4868 (2004).

[99] T. Baba, “Fast spontaneous emission and slow light in photonic crystals,” in

Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Sci-

ence Conference (CLEO/QELS) technical digest, Paper QWD1 (Baltimore, USA,

2005).

BIBLIOGRAPHY 161

[100] E. Ozbay, M. Bayindir, I. Bulu, and E. Cubukcu, “Investigation of localized

coupled-cavity modes in two-dimensional photonic band gap structures,” IEEE

J. Quantum Electron. 38, 837–843 (2002).

[101] T. J. Karle, Y. J. Chai, C. N. Morgan, I. H. White, and T. F. Krauss, “Observation

of pulse compression in photonic crystal coupled cavity waveguides,” J. Lightwave

Technol. 22, 514–519 (2004).

[102] M. F. Yanik and S. Fan, “Stopping light all optically,” Phys. Rev. Lett. 92, 083901

(2004).

[103] B. E. A. Saleh and M. C. Teich, Fundamentals of photonics (John Wiley and

Sons, New York, 1991).

[104] S. C. Cho, B. G. Kim, Y. Moon, and A. Shakouri, “Ultra short two-section vertical

directional coupler switches with high extinction ratios,” Jpn. J. Appl. Phys. 40,

4045–4050 (2001).

[105] S. Boscolo, M. Midrio, and C. G. Someda, “Coupling and decoupling of electro-

magnetic waves in parallel 2-D photonic crystal waveguides,” IEEE J. Quantum

Electron. 38, 47–53 (2002).

[106] A. Martinez, F. Cuesta, and J. Marti, “Ultrashort 2-D photonic crystal directional

couplers,” IEEE Photon. Technol. Lett. 15, 694–696 (2003).

[107] M. Tokushima and H. Yamada, “Photonic crystal line defect waveguide,” Elec-

tron. Lett. 37, 1454–1455 (2001).

[108] Y. Tanaka, H. Nakamura, Y. Sugimoto, N. Ikeda, K. Asakawa, and K. Inoue,

“Coupling properties in a 2-D photonic crystal slab directional coupler with a

triangular lattice of airholes,” IEEE J. Quantum Electron. 41, 76–84 (2005).

[109] Y. Sugimoto, Y. Tanaka, N. Ikeda, T. Yang, H. Nakamura, K. Asakawa, K. Inoue,

T. Maruyama, K. Miyashita, K. Ishida, and Y. Watanabe, “Design, fabrication,

and characterization of coupling-strength-controlled direcional coupler based on

two-dimensional photonic-crystal slab waveguides,” Appl. Phys. Lett. 83, 3236–

3238 (2003).

[110] A. Sharkawy, S. Shi, D. Prather, and R. Soref, “Electro-optical switching using

coupled photonic crystal waveguides,” Opt. Express 10, 1048–1059 (2002).

[111] F. Cuesta, A. Martinez, J. Garcıa, F. Ramos, P. Sanchis, J. Blasco, and J. Martı,

“All-optical switching structure based on a photonic crystal directional coupler,”

Opt. Express 12, 161–167 (2004).

[112] M. Koshiba, “Wavelength division multiplexing and demultiplexing with photonic

crystal waveguide couplers,” J. Lightwave Technol. 19, 1970–1975 (2001).

162 BIBLIOGRAPHY

[113] S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “Channel Drop Tunneling

through Localized States,” Phys. Rev. Lett. 80, 960–963 (1998).

[114] S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and H. A. Haus, “Channel drop

filters in photonic crystals,” Opt. Express 3, 4–11 (1998).

[115] H. Takano, Y. Akahane, T. Asano, and S. Noda, “In-plane-type channel drop

filter in a two-dimensional photonic crystal slab,” Appl. Phys. Lett. 84, 2226–

2228 (2004).

[116] S. Fan, P. R. Villeneuve, J. D. Joannopoulos, M. J. Khan, C. Manolatou, and

H. A. Haus, “Theoretical analysis of channel drop tunneling process,” Phys. Rev.

B 59, 882–892 (1999).

[117] A. Sharkawy, S. Shi, and D. W. Prather, “Multichannel wavelength division mul-

tiplexing with photonic crystals,” Appl. Opt. 40, 2247–2252 (2001).

[118] S. Noda, A. Chutinan, and M. Imada, “Trapping and emission of photons by a

single defect in a photonic bandgap structure,” Nature 407, 608–610 (2000).

[119] A. Chutinan, M. Mochizuki, M. Imada, and S. Noda, “Surface-emitting channel

drop filters using single defects in two-dimensional photonic crystal slabs,” Appl.

Phys. Lett. 79, 2690–2692 (2001).

[120] B. S. Song, T. Asano, Y. Akahane, Y. Tanaka, and S. Noda, “Multichannel

add/drop filter based on in-plane hetero photonic crystals,” J. Lightwave Technol.

23, 1449–1455 (2005).

[121] T. Asano, M. Nakamura, B. S. Song, and S. Noda, “Dynamic wavelength tuning

of channel-drop device in two-dimensional photonic crystal slab,” Electron. Lett.

41, 37–38 (2005).

[122] M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975).

[123] M. Notomi, “Theory of light propagation in strongly modulated photonic crystals:

refractionlike behavior in the vicinity of the photonic band gap,” Phys. Rev. B

62, 10696–10705 (2000).

[124] S. Foteinopoulou and C. M. Soukoulis, “EM wave propagation in two-dimensional

photonic crystals: a study of anomalous refractive effects,” arXiv:cond-

mat/0403542 (2003).

[125] T. Baba and M. Nakamura, “Photonic crystal light deflect devices using the

superprism effect,” IEEE J. Quantum Electron. 38, 909–914 (2002).

[126] T. Baba and T. Matsumoto, “Resolution of photonic crystal superprism,” Appl.

Phys. Lett. 81, 2325–2327 (2002).

BIBLIOGRAPHY 163

[127] V. G. Veselago, “The electrodynamics of substances with simultaneously negative

values of ε and µ,” Sov. Phys. Usp. 10, 509–514 (1968).

[128] J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85,

3966–3969 (2000).

[129] D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz,

“Composite medium with simultaneously negative permeability and permittiv-

ity,” Phys. Rev. Lett. 84, 4184–4187 (2000).

[130] J. P. Dowling, “Anomalous index of refraction in photonic bandgap materials,”

J. Mod. Opt. 41, 345–351 (1994).

[131] S. Enoch, G. Tayeb, and D. Maystre, “Numerical evidence of ultrarefractive optics

in photonic crystals,” Opt. Comm. 161, 171–176 (1994).

[132] H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and

S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58,

R10096–R10099 (1998).

[133] B. Gralak, S. Enoch, and G. Tayeb, “Anomalous refractive properties of photonic

crystals,” J. Opt. Soc. Am. A 17, 1012–1020 (2000).

[134] S. Foteinopoulou, E. N. Economou, and C. M. Soukoulis, “Refraction in media

with a negative refractive index,” Phys. Rev. Lett. 90, 107402 (2003).

[135] E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, and C. M. Soukoulis, “Neg-

ative refraction by photonic crystals,” Nature 423, 604–605 (2003).

[136] A. Berrier, M. Mulot, M. Swillo, M. Qiu, L. Thylen, A. Talneau, and S. Anand,

“Negative refraction at infrared wavelengths in two-dimensional photonic crystal,”

Phys. Rev. Lett. 93, 073902 (2004).

[137] C. Luo, S. Johnson, J. D. Joannopoulos, and J. B. Pendry, “Subwavelength imag-

ing in photonic crystals,” Phys. Rev. B 68, 045115 (2003).

[138] Z. Y. Li and L. L. Lin, “Evaluation of lensing in photonic crystal slabs exhibiting

negative refraction,” Phys. Rev. B 68, 245110 (2003).

[139] E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, and S. M. Soukoulis, “Sub-

wavelength resolution in a two-dimensional photonic-crystal-based superlens,”

Phys. Rev. Lett. 91, 207401 (2003).

[140] P. V. Parimi, W. T. Lu, P. Vodo, and S. Sridar, “Imaging by a flat lens using

negative refraction,” Nature 426, 404 (2003).

164 BIBLIOGRAPHY


S. Kawakami, “Self-collimating phenomena in photonic crystals,” Appl. Phys.

Lett. 74, 1212–1214 (1999).

[142] J. Witzens, M. Loncar, and A. Scherer, “Self-Collimation in planar photonic

crystals,” IEEE J. Sel. Top. Quantum Electron. 8, 1246–1257 (2002).

[143] L. Wu, M. Mazilu, and T. Krauss, “Beam steering in planar-photonic crystals:

from superprism to supercollimator,” J. Lightwave Technol. 21, 561–566 (2003).

[144] M. J. Steel, R. Zoli, C. Grillet, R. C. McPhedran, C. M. de Sterke, A. Norton,

P. Bassi, and B. J. Eggleton, “Analytic properties of photonic crystal superprism

paramters,” Phys. Rev. E 71, 056608 (2005).

[145] D. N. Chigrin, S. Enoch, C. M. Sotomayer Torres, and G. Tayeb, “Self-guiding in

two-dimensional photonic crystals,” Opt. Express 11, 1203–1211 (2003).

[146] D. W. Prather, S. Shi, D. M. Pustai, C. Chen, S. Venkataraman, A. Sharkawy,

G. J. Schneider, and J. Murakowski, “Dispersion-based optical routing in photonic

crystals,” Opt. Lett. 29, 50–52 (2004).

[147] X. Yu and S. Fan, “Bends and splitters for self-collimated beams in photonic

crystals,” Appl. Phys. Lett. 83, 3251–3253 (2003).

[148] S. Shi, A. Sharkawy, C. Chen, D. M. Pustai, and D. W. Prather, “Dispersion-

based beam splitter in photonic crystals,” Opt. Lett. 29, 617–619 (2004).


S. Kawakami, “Photonic crystals for micro lightwave circuits using wavelength

dependent angular beam steering,” Appl. Phys. Lett. 74, 1370–1372 (1999).

[150] S. Y. Lin, V. M. Hietala, L. Wang, and E. D. Jones, “Highly dispersive photonic

band-gap prism,” Opt. Lett. 21, 1771–1773 (1996).

[151] L. Wu, M. Mazilu, T. Karle, and T. F. Krauss, “Superprism phenomena in planar

photonic crystals,” IEEE J. Quantum Electron. 38, 915–918 (2002).

[152] J. J. Baumberg, N. M. B. Perney, M. C. Netti, M. D. C. Charlton, M. Zoorob,

and G. J. Parker, “Visible-wavelength super-refraction in photonic crystal super-

prisms,” Appl. Phys. Lett. 85, 354–356 (2004).

[153] T. Matsumoto and T. Baba, “Photonic crystal k-vector superprism,” J. Lightwave

Technol. 22, 917–922 (2004).

[154] C. Luo, M. Soljacic, and J. D. Joannopoulos, “Superprism effect based on phase

velocities,” Opt. Lett. 29, 745–747 (2004).

BIBLIOGRAPHY 165

[155] C. T. Chan, Q. L. Yu, and K. M. Ho, “Order-N spectral methods for electromag-

netic waves,” Phys. Rev. B 51, 16635–16642 (1995).

[156] M. D. Feit and J. A. Fleck, Jr., “Computation of mode properties in optical fiber

waveguides by a propagating beam method,” Appl. Opt. 19, 1154–1164 (1980).

[157] B. J. Eggleton, P. S. Westbrook, C. A. White, C. Kerbage, R. S. Windeler,

and G. L. Burge, “Cladding-mode-resonances in air-silica microstructured optical

fibers,” J. Lightwave Technol. 18, 1084–1100 (2000).

[158] M. Koshiba, Y. Tsuji, and M. Hikari, “Time-domain beam propagation method

and its application to photonic crystal circuits,” J. Lightwave Technol. 18, 102–

110 (2000).

[159] K. M. Leung and Y. F. Liu, “Full vector wave calculation of photonic band struc-

tures in face-centered-cubic dielectric Media,” Phys. Rev. Lett. 65, 2646–2649

(1990).

[160] K. Busch, S. F. Mingaleev, A. Garcia-Martin, M. Schillinger, and D. Hermann,

“The Wannier function approach to photonic crystal circuits,” J. Phys.:Condens.

Matter 15, R1233–R1256 (2003).

[161] N. A. Nicorovici, R. C. McPhedran, and L. C. Botten, “Photonic band gaps for

arrays of perfectly conducting cylinders,” Phys. Rev. E 52, 1135–1145 (1995).

[162] D. Felbacq, G. Tayeb, and D. Maystre, “Scattering by a random set of parallel

cylinders,” J. Opt. Soc. Am. A 11, 2526–2538 (1994).

[163] L. M. Li and Z. Q. Zhang, “Multiple-scattering approach to finite-sized photonic

band-gap materials,” Phys. Rev. B 58, 9587–9590 (1998).

[164] P. Yeh, Optical waves in layered media (Wiley, New York, 1988).

[165] J. B. Pendry and A. MacKinnon, “Calculation of photon dispersion relations,”

Phys. Rev. Lett. 69, 2772–2775 (1992).

[166] M. Sigalas, C. M. Soukoulis, E. N. Economou, C. T. Chan, and K. M. Ho, “Pho-

tonic band gaps and defects in two dimensions: Studies of the transmission coef-

ficient,” Phys. Rev. B 48, 14121–14126 (1993).

[167] L. Li, “Formulation and comparison of two recursive matrix algorithms for mod-

eling layered diffraction gratings,” J. Opt. Soc. Am. A 13, 1024–1035 (1996).

[168] E. Silberstein, P. Lalanne, and J.-P. Hugonin, “Use of grating theories in inte-

grated optics,” J. Opt. Soc. Am. A 18, 2865–2875 (2001).

166 BIBLIOGRAPHY

[169] L. C. Botten, T. P. White, A. A. Asatryan, T. N. Langtry, C. M. de Sterke, and

R. C. McPhedran, “Bloch mode scattering matrix methods for modeling extended

photonic crystal structures. Part I: Theory,” Phys. Rev. E 70, 056606 (2004).

[170] T. P. White, L. C. Botten, C. M. de Sterke, R. C. McPhedran, A. A. Asatryan,

and T. N. Langtry, “Bloch mode scattering matrix methods for modeling ex-

tended photonic crystal structures. Part II: Applications,” Phys. Rev. E 70,

056607 (2004).

[171] L. C. Botten, T. P. White, C. M. de Sterke, R. C. McPhedran, A. A. Asatryan,

and T. N. Langtry, “Photonic crystal devices modelled as grating stacks: matrix

generalizations of thin film optics,” Opt. Express 12, 1592–1604 (2004).

[172] P. Bienstman and R. Baets, “Optical modelling of photonic crystals and VC-

SELs using eigenmode expansion and perfectly matched layers,” Opt. Quantum

Electron. 33, 327–341 (2001).

[173] M. Neviere and E. Popov, Light propagation in periodic media (Marcel Dekker,

New York, 2003).

[174] R. Petit, ed., Electromagnetic theory of gratings (Springer-verlag, Heidelberg,

1980).

[175] S. F. Mingaleev and K. Busch, “Scattering matrix approach to large-scale pho-

tonic crystal circuits,” Opt. Lett. 28, 619–621 (2003).

[176] L. C. Botten, N. A. Nicorovici, A. A. Asatryan, R. C. McPhedran, C. M. de Sterke,

and P. A. Robinson, “Formulation for Electromagnetic Scattering and Propaga-

tion through Grating Stacks of Metallic and Dielectric Cylinders for Photonic

Crystal Calculations. Part 1: Method,” J. Opt. Soc. Am. A 17, 2165–2176 (2000).

[177] L. C. Botten, N. A. Nicorovici, R. C. McPhedran, C. M. de Sterke, and A. A.

Asatryan, “Photonic band structure calculations using scattering matrices,” Phys.

Rev. E 64, 046603 (2001).

[178] G. H. Smith, L. C. Botten, R. C. McPhedran, and N. A. Nicorovici, “Cylinder

graitings in conical incidence with applications to woodpile structures,” Phys.

Rev. E 67, 056620 (2003).

[179] B. Gralak, S. Enoch, and G. Tayeb, “From scattering or impedance matrices to

Bloch modes of photonic crystals,” J. Opt. Soc. Am. A 19, 1547–1554 (2002).

[180] L. C. Botten, A. A. Asatryan, T. N. Langtry, T. P. White, C. M. de Sterke, and

R. C. McPhedran, “Semianalytic treatment for propagation in finite photonic

crystal waveguides,” Opt. Lett. 28, 854–856 (2003).

BIBLIOGRAPHY 167

[181] T. P. White, L. C. Botten, R. C. McPhedran, and C. M. de Sterke, “Ultracompact

resonant filters in photonic crystals,” Opt. Lett. 28, 2452–2454 (2003).

[182] L. C. Botten, R. C. McPhedran, C. M. de Sterke, N. A. Nicorovici, A. A. Asatryan,

G. H. Smith, T. N. Langtry, T. P. White, D. P. Fussell, and B. T. Kuhlmey, “From

multipole methods to photonic crystal device modeling,” in Electromagnetic the-

ory and applications for photonic crystals, K. Yasomoto, ed. (CRC Taylor and

Francis, Boca Raton, 2006).

[183] S. Kuchinsky, V. Y. Golyatin, A. Y. Kutinov, T. P. Pearsall, and D. Nedeljkovic,

“Coupling between photonic crystal waveguides,” IEEE J. Quantum Electron.

38, 1349–1352 (2002).

[184] C. M. de Sterke, L. C. Botten, A. A. Asatryan, T. P. White, and R. C. McPhedran,

“Modes of coupled photonic crystal waveguides,” Opt. Lett. 29, 1384–1386 (2004).

[185] O. Levy, B. Z. Steinberg, M. Nathan, and A. Boag, “Ultrasensitive displacement

sensor using photonic crystal waveguides,” Appl. Phys. Lett. 86, 104102 (2005).

[186] R. Costa, A. Melloni, and M. Martinelli, “Bandpass resonant filters in photonic-

crystal waveguides,” IEEE Photon. Technol. Lett. 15, 401–403 (2003).

[187] A. S. Jugessur, P. Pottier, and R. M. De La Rue, “Engineering the filter response

of photonic crystal microcavity filters,” Opt. Express 12, 1304–1312 (2004).

[188] J. Zimmermann, B. K. Saravanan, R. Marz, M. Kamp, A. Forchel, and S. Anand,

“Large dispersion in photonic crystal waveguide resonator,” Electron. Lett. 41,

414–415 (2005).

[189] A. Safaai-Jazi and C. C. Chang, “Spectral characteristics of coupled-waveguide

Fabry-Perot resonators and filters,” IEEE J. Quantum Electron. 32, 1063–1069

(1996).

[190] C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D.

Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,”

IEEE J. Quantum Electron. 35, 1322–1331 (1999).

[191] E. Drouard, H. T. Hattori, C. Grillet, A. Kazmierczak, X. Letarte, P. Rojo-

Romero, and P. Viktorovitch, “Directional channel-drop filter based on a slow

Bloch mode photonic crystal waveguide section,” Opt. Express 13, 3037–3048

(2005).

[192] Y. Akahane, M. Mochizuki, T. Asano, Y. Tanaka, and S. Noda, “Design of a

channel drop filter by using a donor-type cavity with high-quality factor in a

two-dimensional photonic crystal slab,” Appl. Phys. Lett. 82, 1341–1343 (2003).

168 BIBLIOGRAPHY

[193] Y. Xu, Y. Li, R. K. Lee, and A. Yariv, “Scattering-theory analysis of waveguide-

resonator coupling,” Phys. Rev. E 62, 7389–7404 (2000).

[194] E. Waks and J. Vuckovic, “Coupled mode theory for photonic crystal cavity-

waveguide interaction,” Opt. Express 13, 5064–5073 (2005).

[195] S. Fan, “Sharp asymmetric line shapes in side-coupled waveguide-cavity systems,”

Appl. Phys. Lett. 80, 908–910 (2002).

[196] S. Fan, S. G. Johnson, and J. D. Joannopoulos, “Waveguide branches in photonic

crystals,” J. Opt. Soc. Am. B 18, 162–165 (2001).

[197] W. J. Kim and J. D. O’Brien, “Optimization of a two-dimensional photonic-

crystal waveguide branch by simulated annealing and the finite-element method,”

J. Opt. Soc. Am. B 21, 289–295 (2004).

[198] R. Wilson, T. J. Karle, I. Moerman, and T. F. Krauss, “Efficient photonic crystal

Y-junctions,” J. Opt. A 5, S76–S80 (2003).

[199] M. Ayre, T. J. Karle, L. Wu, T. Davies, and T. F. Krauss, “Experimental verifi-

cation of numerically optimised photonic crystal injector, Y-splitter, and bend,”

IEEE J. Sel. Area. Comm. 23, 1390–1395 (2005).

[200] A. Martinez, A. Griol, P. Sanchis, and J. Marti, “Mach-Zehnder interferometer

employing coupled-resonator optical waveguides,” Opt. Lett. 28, 405–407 (2003).

[201] Y. Sugimoto, H. Nakamura, Y. Tanaka, N. Ikeda, and K. Asakawa, “High-

precision optical interference in Mach-Zehnder-type photonic crystal photonic

crystal waveguide,” Opt. Express 13, 96–105 (2005).

[202] C. Elachi and C. Yeh, “Distribution networks and electrically controllable couplers

for integrated optics,” Appl. Opt. 13, 1372–1375 (1974).

[203] H. A. Haus and L. Molter-Orr, “Coupled multiple waveguide systems,” IEEE J.

Quantum Electron. QE-19, 840–844 (1983).

[204] T. P. White, C. M. de Sterke, R. C. McPhedran, T. Huang, and L. C. Botten,

“Recirculation-enhanced switching in photonic crystal Mach-Zehnder interferom-

eters,” Opt. Express 12, 3035–3045 (2004).

[205] I. Park, H. S. Lee, H. J. Kim, K. M. Moon, S. G. Lee, B. H. O, S. G. Park, and

E. H. Lee, “Photonic crystal power-splitter based on directional coupler,” Opt.

Express 12, 3599–3604 (2004).

[206] T. Liu, A. R. Zakharian, M. Fallahi, J. V. Maloney, and M. Mansuripur, “Multi-

mode interference-based photonic crystal waveguide power splitter,” J. Lightwave

Technol. 22, 2842–2846 (2004).

BIBLIOGRAPHY 169

[207] P. I. Boral, L. H. Frandsen, A. Harpøth, M. Kristensen, J. S. Jensen, and O. Sig-

mund, “Topology optimised broadband photonic crystal Y-splitter,” Electron.

Lett. 41, 69–71 (2005).

[208] B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J. P. Laine, “Microring resonator

channel dropping filters,” J. Lightwave Technol. 15, 998–1005 (1997).

[209] A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled-resonator optical waveguide:

a proposal and analysis,” Opt. Lett. 24, 711–713 (1999).

[210] M. Bayindir, B. Temelkuran, and E. Ozbay, “Tight-binding description of the

coupled defect modes in three-dimensional photonic crystals,” Phys. Rev. Lett.

84, 2140–2143 (2000).

[211] S. Pereira, P. Chak, and J. Sipe, “Gap-soliton switching in short microresonator

structures,” J. Opt. Soc. Am. B 19, 2191–2202 (2002).

[212] S. Pereira, P. Chak, and J. E. Sipe, “All-optical AND gate by use of a Kerr

nonlinear microresonator structure,” Opt. Lett 28, 444–446 (2003).

[213] B. E. Little, S. T. Chu, J. V. Hryniewicz, and P. P. Absil, “Filter synthesis for

periodically coupled microring resonators,” Opt. Lett. 25, 344–346 (2000).

[214] M. F. Yanik and S. Fan, “Stopping and storing light coherently,” Phys. Rev. A

71, 013803 (2005).

[215] A. N. Chester, S. Martellucci, and A. M. V. Scheggi, eds., Optical fiber sensors

(Martinus Nijhoff, Dordrecht, 1987).

[216] G. V. Treyz, “Silicon Mach-Zehnder waveguide interferometers operating at

1.3µm,” Electron. Lett. 27, 118–120 (1991).

[217] C. Rolland, R. S. Moore, F. Shepherd, and G. Hillier, “10Gbit/s, 1.56µm Mul-

tiquantum well InP/InGaAsP Mach-Zehnder optical modulator,” Electron. Lett.

29, 471–472 (1993).

[218] E. A. Camargo, H. M. H. Chong, and R. M. De La Rue, “Photonic crystal Mach-

Zehnder structures for thermo-optic switching,” Opt. Express 12, 588–592 (2004).

[219] M. H. Shih, W. J. Kim, W. Kuang, J. R. Cao, H. Yukawa, S. J. Choi, J. D.

O’Brien, and P. D. Dapkus, “Two-dimensional photonic crystal Mach-Zehnder

interferometers,” Appl. Phys. Lett. 84, 460–462 (2004).

[220] H. Nakamura, Y. Sugimoto, K. Kanamoto, N. Ikeda, Y. Tanaka, Y. Nakamura,

S. Ohkouchi, Y. Watanabe, K. Inoue, H. Ishikawa, and K. Asakawa, “Ultra-fast

photonic crystal/quantum dot all-optical switch for future photonic networks,”

Opt. Express 12, 6606–6614 (2004).

170 BIBLIOGRAPHY

[221] A. Lan, K. Kanamoto, T. Yang, A. Nishikawa, Y. Sugimoto, N. Ikeda, H. Naka-

mura, K. Asakawa, and H. Ishikawa, “Similar role of waveguide bends in photonic

crystal circuits and disordered defects in coupled cavity waveguides: An intrinsic

problem in realizing photonic crystal circuits,” Phys. Rev. B 67, 115208 (2003).

[222] C. Y. Liu and L. W. Chen, “Tunable photonic-crystal waveguide Mach-Zehnder

interferometer achieved by nematic liquid-crystal phase modulation,” Opt. Ex-

press 12, 2616–2624 (2004).

[223] T. Ranganath and S. Wang, “Ti-diffused LiNbO3 branched-waveguide modula-

tors: performance and design,” IEEE J. Quantum Electron. QE-13, 290–295

(1977).

[224] G. P. Agrawal, Fiber-optic communication systems, 3rd ed. (John Wiley and Sons,

New York, 2002).

[225] B. Wang and G. P. Wang, “Surface plasmon polariton propagation in nanoscale

metal gap waveguides,” Opt. Lett. 29, 1992–1994 (2004).

[226] T. P. White, C. M. de Sterke, R. C. McPhedran, and L. C. Botten, “Highly

efficient wide-angle coupling into uniform rod-type photonic crystals,” Appl. Phys.

Lett. 87, 111107 (2005).

[227] A. Mekis and J. D. Joannopoulos, “Tapered couplers for efficient interfacing be-

tween dielectric and photonic crystal waveguides,” J. Lightwave Technol. 19, 861–

865 (2001).

[228] W. Kuang, C. Kim, A. Stapleton, and J. D. O’Brien, “Grating-assisted coupling of

optical fibers and photonic crystal waveguides,” Opt. Lett. 27, 1604–1606 (2002).

[229] A. Talneau, P. Lalanne, M. Agio, and C. M. Soukoulis, “Low-reflection photonic-

crystal taper for efficient coupling between guide sections of arbitrary widths,”

Opt. Lett. 27, 1522–1524 (2002).

[230] P. E. Barclay, K. Srinivasan, M. Borselli, and O. Painter, “Probing the dispersive

and spatial properties of photonic crystal waveguides via highly efficient coupling

from fiber tapers,” Appl. Phys. Lett. 85, 4–6 (2004).

[231] J. Witzens, M. Hochberg, T. Baehr-Jones, and A. Scherer, “Mode matching in-

terface for efficient coupling of light into planar photonic crystals,” Phys. Rev. E.

69, 046609 (2004).

[232] T. Baba and D. Ohsaki, “Interfaces of photonic crystals for high efficiency light

transmission,” Jpn. J. Appl. Phys. 40, 5920–5924 (2001).

BIBLIOGRAPHY 171

[233] J. Witzens and A. Scherer, “Efficient excitation of self-collimated beams and

single Bloch modes in planar photonic crystals,” J. Opt. Soc. Am. A 20, 935–940

(2003).

[234] E. N. Economou and A. Zdetsis, “Classical wave propagation in periodic struc-

tures,” Phys. Rev. B 40, 1334–1337 (1989).

[235] C. M. Soukoulis, ed., Photonic bandgaps and localization (Plenum Press, New

York, 1993).

[236] E. Lidorikis, M. M. Sigalas, E. N. Economou, and C. M. Soukoulis, “Tight-binding

parametrization for photonic band gap materials,” Phys. Rev. Lett. 81, 1405–1408

(1998).

[237] H. C. van de Hulst, Light scattering by small particles (John Wiley and Sons,

New York, 1957).

[238] T. Baba, T. Matsumoto, and M. Echizen, “Finite difference time domain study of

high efficiency photonic crystal superprisms,” Opt. Express 12, 4608–4613 (2004).

[239] R. T. Bise, R. S. Windeler, K. S. Kranz, C. Kerbage, B. J. Eggleton, and D. J.

Trevor, “Tunable photonic bandgap fiber,” in Optical Fiber Communication Con-

ference, Postconference Edition, vol. 70 of OSA Trends in Optics and Photonics,

pp. 466–468 (Optical Society of America, Washington, DC, USA, 2002).

[240] T. P. White, R. C. McPhedran, C. M. de Sterke, N. M. Litchinitser, and B. J.

Eggleton, “Resonance and scattering in microstructured optical fibers,” Opt. Lett.

27, 1977–1979 (2002).

[241] T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraor-

dinary optical transmission through sub-wavelength hole arrays,” Nature 391,

667–669 (1998).

[242] F. J. Garcıa de Abajo, G. Gomez-Santos, L. A. Borislav, and S. V. Shabanov,

“Tunneling mechanism of light transmission through metallic films,” Phys. Rev.

Lett. 95, 067403 (2005).

[243] F. J. Garcıa de Abajo, R. Gomez-Medina, and J. J. Saenz, “Full transmission

through perfect-conductor subwavelength hole arrays,” Phys. Rev. E 72, 016608

(2005).

[244] A. Mandatori, C. Sibilia, M. Bertolotti, S. Zhukovsky, and S. V. Gaponenko, “An-

alytic demonstration of omnidirectional transmission enhancement in dispersive

birefringent photonic-bandgap structures,” J. Opt. Soc. Am. B 22, 1785–1792

(2005).

172 BIBLIOGRAPHY

[245] T. Yamashita and C. J. Summers, “Evaluation of self-collimated beams in pho-

tonic crystals for optical interconnect,” IEEE J. Sel. Area. Comm. 23, 1341–1347

(2005).

[246] M. Palamaru and P. Lalanne, “Photonic crystal waveguides: out-of-plane losses

and adiabatic modal conversion,” Appl. Phys. Lett. 78, 1466–1468 (2001).

[247] P. Lalanne and J. P. Hugonin, “Bloch-wave engineering for high-Q, small-V mi-

crocavities,” IEEE J. Quantum Electron. 39, 1430–1438 (2003).

[248] B. S. Song, T. Asano, Y. Akahane, Y. Tanaka, and S. Noda, “Transmission and

reflection characteristics of in-plane hetero-photonic crystals,” Appl. Phys. Lett.

85, 4591–4593 (2004).

[249] J. Witzens, T. Baehr-Jones, and A. Scherer, “Hybrid superprism with low inser-

tion losses and suppressed cross-talk,” Phys. Rev. E 71, 026604 (2005).

Appendix A

List of author’s publications

Journal papers

[1] M. J. Steel, T. P. White, C. M. de Sterke, and R. C. McPhedran, “Symmetry and

degeneracy in microstructured optical fibers,” Opt. Lett. 26, 488-490 (2001).

[2] T. P. White, R. C. McPhedran, C. M. de Sterke, L. C. Botten, and M. J. Steel,

“Confinement losses in microstructured optical fibers,” Opt. Lett. 26, 1660-1662

(2001).

[3] T. P. White, R. C. McPhedran, L. C. Botten, G. H. Smith, and C. M. de Sterke,

“Calculations of air-guided modes in photonic crystal fibers using the multipole

method,” Opt. Express 9, 721-732 (2001).

[4] T. P. White, B. T. Kuhlmey, R. C. McPhedran, D. Maystre, G. Renversez, C. M.

de Sterke, and L. C. Botten, “Multipole method for microstructured optical fibers.

I. Formulation,” J. Opt. Soc. Am. B 19, 2322-2330 (2002).

[5] B. T. Kuhlmey, T. P. White, G. Renversez, D. Maystre, L. C. Botten, C. M.

de Sterke, and R. C. McPhedran, “Multipole method for microstructured optical

fibers. II. Implementation and results,” J. Opt. Soc. Am. B 19, 2331-2340 (2002).

[6] T. P. White, R. C. McPhedran, C. M. de Sterke, N. M. Litchinitser and B. J.

Eggleton, “Resonance and scattering in microstructured optical fibers,” Opt. Lett.

27, 1977-1979 (2002).

[7] M. Sumetsky, Y. Dulashko, T. P. White, T. Olsen, P. S. Westbrook, and B. J.

Eggleton, “Interferometric characterization of phase masks,” Appl. Opt. 42, 2336-

2340 (2003).

[8] L. C. Botten, A. A. Asatryan, T. N. Langtry, T. P. White, C. M. de Sterke, and

R. C. McPhedran, “Semianalytic treatment for propagation in finite photonic crystal

waveguides,” Opt. Lett. 28, 854–856 (2003).

173

174 APPENDIX A. LIST OF AUTHOR’S PUBLICATIONS

[9] N. M. Litchinitser, S. C. Dunn, B. Usner, B. J. Eggleton, T. P. White, R. C. McPhe-

dran, and C. M. de Sterke, “Resonances in microstructured optical waveguides,”

Opt. Express 10, 1243-1251 (2003).

[10] T. P. White, L. C. Botten, R. C. McPhedran, and C. M. de Sterke, “Ultracompact

resonant filters in photonic crystals,” Opt. Lett. 28, 2452–2454 (2003).

[11] N. M. Litchinitser, S. C. Dunn, P. E. Steinvurzel, B. J. Eggleton, T. P. White, R. C.

McPhedran and C. M. de Sterke, “Application of an ARROW model for designing

tunable photonic devices,” Opt. Express 12, 1540-1550 (2004).

[12] L. C. Botten, T. P. White, C. M. de Sterke, R. C. McPhedran, A. A. Asatryan,

and T. N. Langtry, “Photonic crystal devices modelled as grating stacks: matrix

generalizations of thin film optics,” Opt. Express 12, 1592–1604 (2004).

[13] C. M. de Sterke, L. C. Botten, A. A. Asatryan, T. P. White, and R. C. McPhedran,

“Modes of coupled photonic crystal waveguides,” Opt. Lett. 29, 1384–1386 (2004).

[14] T. P. White, C. M. de Sterke, R. C. McPhedran, T. Huang, and L. C. Botten,

“Recirculation-enhanced switching in photonic crystal Mach-Zehnder interferome-

ters,” Opt. Express 12, 3035–3045 (2004).

[15] M. Sumetsky, T. P. White, H. M. Dyson, P. S. Westbrook, and B. J. Eggleton,

“Interferometric characterization of nonflat transmission diffraction gratings,” IEEE

Photon. Technol. Lett. 16, 2314-2316 (2004).

[16] L. C. Botten, T. P. White, A. A. Asatryan, T. N. Langtry, C. M. de Sterke, and

R. C. McPhedran, “Bloch mode scattering matrix methods for modelling extended

photonic crystal structures. Part I: Theory,” Phys. Rev. E 70, 056606 (2004).

[17] T. P. White, L. C. Botten, C. M. de Sterke, R. C. McPhedran, A. A. Asatryan,

and T. N. Langtry, “Bloch mode scattering matrix methods for modelling extended

photonic crystal structures. Part II: Applications,” Phys. Rev. E 70, 056607 (2004).

[18] P. Steinvurzel, B. T. Kuhlmey, T. P. White, M. J. Steel, C. M. de Sterke, and

B. J. Eggleton, “Long wavelength anti-resonant guidance in high index inclusion

microstructured fibers,” Opt. Express 12, 5424-5433 (2004).

[19] T. P. White, C. M. de Sterke, R. C. McPhedran, and L. C. Botten, “Highly efficient

wide-angle coupling into uniform rod-type photonic crystals,” Appl. Phys. Lett.

87, 111107 (2005).

175

Book chapter

[20] L. C. Botten, R. C. McPhedran, C. M. de Sterke, N. A. Nicorovici, A. A. Asatryan,

G. H. Smith, T. N. Langtry, T. P. White, D. P. Fussell, and B. T. Kuhlmey, “From

multipole methods to photonic crystal device modeling,” in Electromagnetic theory

and applications for photonic crystals, K. Yasomoto, ed. (CRC Taylor and Francis,

Boca Raton 2006).

chapter 1 photonic crystals: properties and applications

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