study of a ferrite circulator for pbg waveguides in...

i

JOSÉ LUIS ARRIETA CONCHA

STUDY OF A FERRITE CIRCULATOR FOR PBG

WAVEGUIDES IN THE MICROWAVE BAND

CAMPINAS

2014

ESTUDO DE UM CIRCULADOR DE FERRITA PARA GUIAS PBG

NA FAIXA DE MICRO-ONDAS

iii

UNIVERSIDADE ESTADUAL DE CAMPINAS

Faculdade de Engenharia Elétrica e de Computação

CAMPINAS

2014

Thesis presented to the School of Electrical and

Computer Engineering of the University of Campinas,

in partial fulfillment of the requirements for the degree

of Master of Science in Electrical Engineering, in the

field of Telecommunications and Telematics.

Tese apresentada à Faculdade de Engenharia Elétrica e

de Computação da Universidade Estadual de Campinas

para obtenção do título de Mestre em Engenharia

Elétrica, na área de Telecomunicações e Telemática.

JOSÉ LUIS ARRIETA CONCHA

STUDY OF A FERRITE CIRCULATOR FOR PBG WAVEGUIDES IN THE

MICROWAVE BAND

ESTUDO DE UM CIRCULADOR DE FERRITA PARA GUIAS PBG NA FAIXA DE

MICRO-ONDAS

Orientador: Prof. Dr. HUGO ENRIQUE HERNÁNDEZ FIGUEROA

ESTE EXEMPLAR CORRESPONDE À VERSÃO FINAL DA TESE

DEFENDIDA PELO ALUNO JOSE LUIS ARRIETA CONCHA

E ORIENTADO PELO PROF. DR. HUGO ENRIQUE HERNÁNDEZ FIGUEROA

Assinatura do Orientador

_______________________________________________________________________

vii

ABSTRACT

Optical communication networks are part of today’s information based society, in which optical

fibers play a very important role, since they are widely used in such communication networks and

other state of the art communication systems. However, the use of electronic devices in various

electronic-optical networks represent a major bottleneck for the optimal utilization of the large

bandwidth and high speed data rates allowed by optical fibers. In order to avoid such congestion

of network traffic, devices based on photonic technologies are one of the proposed solutions,

because they provide the capability to maintain transmission speeds with both high data rates and

low losses. That is why such photonic devices are continuously being researched in order to

improve their effectiveness, obtain greater confinement and guiding of the optical signal, as well

as to reduce the insertion loss when it is integrated with other devices.

To achieve a greater understanding of photonic devices, the work presented in this paper allow for

a pedagogical and yet thorough understanding of photonic crystals and magnetic effects. This

pedagogical approach permits a hands-on experience for students and researchers on a complex

subject, a kind of experimental implementation that usually demands sophisticated photonic

fabrication resources. The microwave approach allows one to construct conceptually equivalent

but hand-manipulable prototypes.

Here, we present the theoretical study, numerical analysis, computer modeling and electromagnetic

simulations for a 3-port ferrite circulator in a photonic grid structure built for the 2.3 GHz to 2.9

GHz microwave range. For a photonic device operating in the optical communication range, we

may expect a qualitative analogous behavior.

Keywords: Waveguide circulators, Photonic crystals, Ferrite (Magnetic Materials),

Microwave devices.

ix

RESUMO

Redes de comunicação ópticas são parte da sociedade de informação de hoje, em que as fibras

ópticas têm um papel muito importante, uma vez que são amplamente utilizados em redes de

comunicação e outros sistemas de comunicação atuais. No entanto, o uso de dispositivos

eletrônicos em várias redes eletro-ópticas representa um grande engarrafamento para a utilização

ideal da grande largura de banda e as taxas de dados de alta velocidade permitida por fibras ópticas.

A fim de evitar tal congestionamento de tráfego da rede, dispositivos baseados em tecnologias

fotônicas são uma das soluções propostas, porque eles fornecem a capacidade de manter altas

velocidades de transmissão com taxas elevadas de dados e baixas perdas. Por isso, tais dispositivos

fotônicos são continuamente pesquisados, a fim de melhorar a sua eficácia, a obtenção do maior

confinamento e encaminhamento do sinal óptico, e melhorar a perda de inserção quando ele é

integrado com outros dispositivos.

Para alcançar uma maior compreensão dos dispositivos fotônicos, o trabalho apresentado aqui

permite uma compreensão pedagógica e ainda completa de cristais fotônicos e efeitos magnéticos.

Esta abordagem pedagógica permite uma experiência em mãos para os estudantes e pesquisadores

sobre um tema complexo, um tipo de implementação experimental que normalmente exige

sofisticados recursos de fabricação fotônicos. A abordagem em micro-ondas permite construir

protótipos conceitualmente equivalentes e manipuláveis.

Aqui, apresentamos um estudo teórico, análise numérica, modelagem de computador e simulações

eletromagnéticas para um circulador de ferrita de 3 portas em uma estrutura de grade fotônico

construído para a faixa de micro-ondas de 2,3GHz - 2,9GHz.

Palavras chave: Circuladores de guia de ondas, Cristais fotônicos, Ferrita (Materiais magnéticos),

Dispositivos de micro-ondas.

xi

TABLE OF CONTENTS

1. INTRODUCTION ........................................................................................................ 1

2. PHOTONIC CRYSTALS ............................................................................................ 3

2.1 BASIC DEFINITIONS ..................................................................................................... 3

2.2 MAXWELL’S EQUATIONS ............................................................................................. 9

2.3 CRYSTALLINE STRUCTURE ........................................................................................ 12

2.4 PHOTONIC DEVICES BASED ON TWO-DIMENSIONAL PHOTONIC CRYSTALS.............. 17

2.4.1 Waveguide ......................................................................................................... 17

2.4.2 Resonant Cavity ................................................................................................ 20

2.4.3 Waveguide Splitter ............................................................................................ 21

3. FERRITES AND THE CIRCULATOR .................................................................... 22

3.1 THEORY OF FERRIMAGNETIC COMPONENTS ............................................................. 22

3.1.1 The Permeability Tensor .................................................................................. 23

3.2 CIRCULATORS ............................................................................................................ 24

3.3 FERRITE Y-JUNCTION CIRCULATORS........................................................................ 26

3.3.1 Description of the Ferrite Circulator Operation ............................................. 26

3.3.2 Design of a Ferrite Junction Circulator ........................................................... 28

4. NUMERICAL ANALYSIS AND PROJECT DESIGN ............................................ 30

4.1 DESIGN AND MODELING OF THE PHOTONIC GRID ..................................................... 30

4.2 PBG WAVEGUIDE DESIGN ......................................................................................... 34

4.3 DESIGN OF THE FERRITE CYLINDER AND CIRCULATOR ............................................ 37

5. RESULTS ................................................................................................................... 38

5.1 COMPUTER SIMULATIONS ......................................................................................... 38

5.1.1 Straight Photonic Waveguide ........................................................................... 38

5.1.2 Photonic Waveguide Y-Splitter ........................................................................ 41

5.1.3 3-Port Photonic Junction Circulator ................................................................ 43

6. FINAL CONLUSIONS AND CONSIDERATIONS ................................................. 55

7. REFERENCES ........................................................................................................... 57

xiii

DEDICATÓRIA

Dedico este trabalho a minha esposa Nella, que tem sido o meu apoio constante ao longo deste

desafio e como a musa que me inspira a cada dia. Aos meus pais, Delia Rosa e Francisco; e minha

irmã Cynthia, que têm me guiado e acompanhado durante toda a minha vida, sempre me dando o

seu apoio constante para alcançar meus sonhos. Aos professores que tive durante toda a minha vida

acadêmica e me orientaram para alcançar meus objetivos. Enfim, a todos os meus amigos e colegas,

muitos dos quais também andam este caminho e de algum modo são parte desta conquista.

............................................................................................................................................................

Dedico este trabajo a mi esposa Nella, quien ha sido mi apoyo constante a lo largo de este reto y

por ser la musa que me inspira día a día. A mis padres, Delia Rosa y Francisco; y mi hermana

Cynthia, quienes me han guiado y acompañado a lo largo de mi vida, brindándome siempre su

apoyo constante para alcanzar mis sueños. A todos los profesores que tuve a lo largo de mi vida

académica y que supieron orientarme para alcanzar mis metas. Finalmente, a todos mis amigos y

compañeros de estudio, muchos de los cuales también recorren este camino y de alguna manera

participan de este logro.

xv

AGRADECIMENTOS

Agradeço, primeiramente, a Deus, por nunca me desamparar, à minha esposa e a minha família,

pelo apoio incondicional, e aos meus amigos, que sempre torceram por mim.

Agradeço ao Prof. Dr. Hugo Enrique Hernández Figueroa pela orientação deste trabalho, ao CNPq

pelo suporte financeiro e aos colegas do Laboratório de Eletromagnetismo Aplicado e

Computacional (LEMAC) pelas dúvidas solucionadas e pelas conversas.

Agradeço, também, ao Prof. Sombra, que confeccionou as cerâmicas utilizadas nesse projeto e ao

Sr. Antunes do Talher de Mecânica do CCS pela ajuda com a fabricação do protótipo utilizado

neste projeto.

Enfim, agradeço a todos que, de alguma forma, colaboraram para que esse trabalho fosse possível.

xvii

“Everything must be made

as simple as possible.

But not simpler.”

Albert Einstein

“The only way of discovering the limits of the possible

is to venture a little way past them into the impossible”

Arthur C. Clarke

xix

LIST OF FIGURES

Figure 1: Scattering regimes - (a) Homogeneous Medium (b) Incoherent scattering, as in

geometrical optics (c) Coherent scattering as in a photonic crystal.3 ............................................ 5

Figure 2: Examples of photonic crystals a) one-dimensional (1D), b) two-dimensional (2D) and

three-dimensional (3D). The different shades represent different materials with different

dielectric constants 4. ....................................................................................................................... 6

Figure 3: A two-dimensional photonic crystal with square lattice arrangement of dielectric

columns. The material is homogeneous along the z-axis and periodic along the x and y-axis. ...... 7

Figure 4: a) square and b) triangular lattice photonic structures cross sections. (εa and εb are the

relative permittivity of the background material, respectively, a: lattice constant, L: cavity

length). 5 ........................................................................................................................................ 12

Figure 5: A schematic view of a) a square lattice photonic crystal with primitive vectors; and b)

its representation in reciprocal space with reciprocal primitive vectors. 5 .................................. 13

Figure 6: A schematic view of a) a triangular lattice photonic crystal with primitive vectors; and

b) its representation in reciprocal space with reciprocal primitive vectors. 5 .............................. 13

Figure 7: The square lattice. On the left is the network of lattice points in real space. In the

middle is the corresponding reciprocal lattice. On the right is the construction of the first

Brillouin zone 2. ............................................................................................................................. 16

Figure 8: The triangular lattice. On the left is the network of lattice points in real space. In the

middle is the corresponding reciprocal lattice, which in this case is a rotated version of the

original. On the right is the Brillouin zone construction. In this case, the first Brillouin zone is a

hexagon centered on the origin 2. .................................................................................................. 16

Figure 9: Line-defect waveguide formed by a missing row in a square lattice of dielectric rods in

air. 2 ............................................................................................................................................... 18

Figure 10: 𝐸𝑧 field in a right-angle bend waveguide, showing 100% transmission at the PBG

range. 2 .......................................................................................................................................... 19

Figure 11: Point-defect cavity formed by a single missing rod in a square lattice. The cavity

supports a single mode inside the TM band gap, whose electric field 𝐸𝑧 is shown. 2 .................. 20

xx

Figure 12: Left: 𝐸𝑧 field in a “T” splitter for a waveguide, showing essentially 100%

transmission (50% in each branch). Right: Abstract model, treating junction as weak resonance,

predicting 100% transmission when 1𝜏1 = 1𝜏2 + 1𝜏3. 2 ............................................................ 21

Figure 13: Ferrite circulator, principle of operation. ................................................................. 25

Figure 14: Stripline Y-junction circulator .................................................................................... 27

Figure 15: a) Dipolar mode of a dielectric disk (𝐻𝑖𝑛𝑡 = 0 𝑜𝑟 ∞). b) Analogous pattern for a

magnetized disk (𝐻𝑖𝑛𝑡 𝑓𝑜𝑟 𝑐𝑖𝑟𝑐𝑢𝑙𝑎𝑡𝑖𝑜𝑛). The pattern for the magnetized disk has been rotated to

isolate port 3. ................................................................................................................................. 28

Figure 16: Unit-cell array configurations for different 𝑟𝑎 ratios, as viewed by the MPB tool ... 32

Figure 17: (a) Unit cell (highlighted by the parallelogram) to a hexagonal arrangement and (b)

respective Brillouin zone. The irreducible zone is a triangular wedge of vertices Γ, K and M. ... 32

Figure 18: Band Gap diagram calculated for a hexagonal array of dielectric columns

surrounded by air, with geometric ratio 𝑟𝑎 = 0.2167, for the TM polarization parameters. .... 33

Figure 19: From left to right: Super-cell configurations for five, four, three, two and one

dielectric elements (on each side) respectively. Each super-cell has a line-defect along the middle

to allow for the guided mode propagation. ................................................................................... 35

Figure 20: TM guided mode for the super-cell configuration of three elements per side. ........... 36

Figure 21: Straight Photonic Waveguide with two coaxial ports ................................................. 38

Figure 22: S-parameters for the straight photonic waveguide. Input signal sweeps from 1.5GHz

to 3.5GHz. ...................................................................................................................................... 39

Figure 23: Zoom in on the S21 parameter (transmission coefficient). In this case it shows good

transmission (under -1dB in losses) from 2.35 𝐺𝐻𝑧 < 𝑓 < 2.90 𝐺𝐻𝑧........................................ 40

Figure 24: Three-Port Photonic Waveguide Y-Splitter ................................................................ 41

Figure 25: S-parameters for the photonic waveguide splitter. S21 and S31 are overlapping each

other. They both show equal signal output with about -0.78dB less than ideally expected, caused

by losses in the waveguide as well as some impedance mismatch (reflections). .......................... 42

Figure 26: 3-Port Photonic Junction Circulator .......................................................................... 43

Figure 27: S-Parameters for the 3-port Junction Circulator with a ferrite-biasing field of 1500

Gauss. Signal input is done only at Port-1 since all ports are symmetric .................................... 44

xxi






Gauss. Signal input is done at all three ports, sequentially .......................................................... 47

Figure 31: S-parameters for the three-port photonic junction circulator with biasing field 2125

Gauss and input signal at Port 1. .................................................................................................. 48

Figure 32: Representation of the E-fields for the circulator operation with input at Port 1. We

observe very good transmission from Port 1 to Port 2 and good isolation at Port 3. .................. 49









xxiii

LIST OF TABLES

Table 1: Analysis of band gap size for different 𝑟𝑎 ratios of grid arrays. .................................... 31

Table 2: Analysis for the Guided Mode Gap as a function of the number of dielectric elements on

each side of the line-defect. ........................................................................................................... 35

Table 3: Comparison between a top of the line circulator (military grade) and our proposed

design for a photonic circulator .................................................................................................... 54

1

1. INTRODUCTION

Communication networks are part of today’s knowledge society, in which optical fibers

play an important role since they are widely used in communication networks and other the state

of the art communication systems. However, the use of electronic devices in various electronic-

optical networks still represents a major bottleneck for the full utilization of the large bandwidth

and high-speed data rates provided by optical fibers. In order to avoid such congestion of network

traffic, devices based on optical technologies are needed because they provide higher data

transmission rates with fewer losses. Therefore, there is ongoing research on these devices based

on photonic technologies, so we are able to improve their effectiveness, greater confinement and

guiding of the optical signal, as well as reduce the insertion loss when it’s integrated with other

devices. For all of the above, this is why it is very important for us to achieve a thorough

understanding of photonic crystals and waveguide structures built with these technologies.

In Chapter 2, we present the Photonic Crystals, which in simple terms are periodic

structures with high contrast ratios in their refraction indexes and obeying a certain symmetry,

being the rectangular and triangular ones the most common. Lord Rayleigh has studied these

crystals properties since the late nineteenth century, with the analysis of multilayers with different

indices of refraction; however, their study has continued until nowadays because of the increasing

need for high transmission rates. The main feature of these structures is to prohibit the propagation

of electromagnetic waves of determined frequencies within the crystal structure, acting similarly

to a microwave conductor (SIBILIA et al, 2008).

There are two major problems in transmissions of optical signals: (1) the miniaturization of

the required circuitry and (2) the difficulty of obtaining guidance with low losses, since metallic

materials used for guiding micrometric electromagnetic waves (microwaves), have high losses for

optical frequencies. However, these problems can be solved by using waveguides made from

photonic crystals. The dimensions of the elements in a photonic crystal are of the order of

magnitude of the wavelength of the incident signal, that is, for optical signals they are of some

micrometers, which is very useful for miniaturization of the devices. Also by confining a signal

between two photonic crystal structures or removing some elements (forming a path) and then

2

focusing a signal in this region, one obtains a wave-guide with low loss. It is also possible to guide

the desired signal in a path, with twists and turns, and building resonant cavities, as it is currently

being done for conventional microwave signals.

In Chapter 3, we present the theory of ferrimagnetic components and some background on

one of its most common applications: the ferrite circulator. Nowadays, microwave integrated

systems include some of these nonreciprocal structures (circulators, isolators, phase shifters)

operating over a broad frequency band (NOVA MICROWAVE, 2014). However, the potential

drawbacks of these structures are high insertion losses associated with a ferrite slab, which it is

often several wavelengths long. Recent research of ferrite-coupled devices focus on improvement

of these parameters such as insertion losses and isolation. These devices are realized in finline,

slotline or microstrip technology (MARYNOWSKI; MAZUR, 2009).

In Chapter 4, we present the numerical analysis we conducted for the design of the photonic

grid, the photonic waveguides, the photonic splitter and the ferrite disk. These numerical analyses

were conducted using software already developed for these applications (COMSOL

MULTIPHYSICS, 2008), or in the case of the ferrite, using already known calculations presented

in previous works.

In Chapter 5, we present the computer modeling of these photonic structures and conducted

electromagnetic simulations on them (ANDONEGUI; GARCIA-ADEVA, 2013). The results we

obtained match very well with the theory behind their design.

Finally, it is the intention of the present work, to achieve a greater understanding of photonic

devices, photonic band gap techniques and magnetic effects, in order to construct an experimental

device as means for a pedagogical approach and thorough understanding of these technologies.

This pedagogical approach permits a hands-on experience for students and researchers on a

complex subject, a kind of experimental implementation that usually demands sophisticated

photonic fabrication resources. The microwave approach (YAO, 2012a; 2012b) allows one to

construct conceptually equivalent but hand-manipulable prototypes.

3

2. PHOTONIC CRYSTALS

This chapter presents the analogy between the electrons in a semiconductor crystalline

lattice and the photons in a photonic crystal. Then we present the Maxwell equations, which are

used to analyze the structures of photonic crystals. Finally, some devices based on two-dimensional

photonic crystals, which can be used in communication systems, are presented at the end of the

chapter.

2.1 BASIC DEFINITIONS

In simple terms, a crystal is a periodic arrangement of atoms or molecules, and the pattern

in which the atoms or molecules are repeated in space is the crystal lattice. Therefore the crystal

presents a periodic potential to an electron propagating through it, and both the constituents of the

crystal and the geometry of the lattice dictate the conduction properties of the crystal

(JOANNOPOULOS et al, 2011).

The theory of quantum mechanics in a periodic potential explains that electrons propagate

as waves, and waves that meet certain criteria can travel through a periodic potential without

scattering (although they will be scattered by defects and impurities). Importantly, however, the

lattice can also prohibit the propagation of certain waves, meaning that there may be gaps in the

energy band structure of the crystal in which the electrons are forbidden to propagate with certain

energies in certain directions. If the lattice potential is strong enough, the gap can extend to cover

all possible propagation directions, resulting in a complete band gap. For example, a semiconductor

has a complete band gap between the valence and conduction energy bands.

The optical analogue is the photonic crystal, in which the atoms or molecules are replaced

by macroscopic media with differing dielectric constants, and the periodic potential is replaced by

a periodic dielectric function (or, equivalently, a periodic index of refraction). If the dielectric

constants of the materials in the crystal are sufficiently different, and if the absorption of light by

the materials is minimal, then the refractions and reflections of light from all of the various

4

interfaces can produce many of the same phenomena for photons (light modes) that the atomic

potential produces for electrons. One solution to the problem of optical control and manipulation

is thus a photonic crystal, a low-loss periodic dielectric medium. In particular, we can design and

construct photonic crystals with photonic band gaps, preventing light from propagating in certain

directions with specified frequencies (i.e., a certain range of wavelengths, or “colors,” of light). We

will also see that a photonic crystal can allow propagation in anomalous and useful ways.

Although the photonic crystal properties are well known since the end of the 19th century,

in recent decades they have been widely studied because of their ability for presenting these band

gaps for the propagation of electromagnetic signals in certain wavelengths, thus allowing the

guidance and control of these signals. This is very important because the reflective materials

commonly used for microwave frequencies (conducting metals), have very large losses in the

optical frequencies. With photonic crystals, designs with steep curves, intersections and junctions

can be built in the waveguides and still present very low losses. This added to the fact that they can

be also integrated into microelectronics, since they can be fabricated on silicon wafers, which is

the base material of such technology.

To develop this concept further, consider how metallic waveguides and cavities relate to

photonic crystals. Metallic waveguides and cavities are widely used to control microwave

propagation. The walls of a metallic cavity prohibit the propagation of electromagnetic waves with

frequencies below a certain threshold frequency, and a metallic waveguide allows propagation only

along its axis. It would be extremely useful to have these same capabilities for electromagnetic

waves with frequencies outside the microwave regime, such as visible light. However, visible light

energy is quickly dissipated within metallic components, which makes this method of optical

control impossible to generalize. Photonic crystals allow the useful properties of cavities and

waveguides to be generalized and scaled to encompass a wider range of frequencies. We may

construct a photonic crystal of a given geometry with millimeter dimensions for microwave control,

or with micron dimensions for infrared control.

5

The operating regime of photonic crystals is between two regimes of scattering. They have

comparable dimensions to the wavelength of the propagating signal. Under this regime, when the

dimensions are much smaller than the wavelength (Figure 1a), the medium can be considered

homogeneous. Above this limit, when the dimensions are much larger than the wavelength (Figure

1b), there is an incoherent scattering, entering the regime of geometrical optics. However in the

intermediate limit, when the dimensions are comparable to the propagating wavelength (Figure 1c),

the regime becomes one of coherent scattering. In the intermediate situation, a periodic

arrangement of elements can perform an addition of scattered fields, which leads to the emergence

of the “band gap”.

Figure 1: Scattering regimes - (a) Homogeneous Medium (b) Incoherent scattering, as in

geometrical optics (c) Coherent scattering as in a photonic crystal.(PASETTO, 2011)

6

Photonic crystals can be classified by the number of dimensions for which they exhibit

symmetry in the network. As can be seen in Figure 2, there are crystals with symmetry

in one dimension (1D - multilayers), with two dimensions of symmetry (2D) and three

dimensions of symmetry (3D – woodpile). This following work will focus only on 2D photonic

crystals.

Figure 2: Examples of photonic crystals a) one-dimensional (1D), b) two-dimensional (2D) and

three-dimensional (3D). The different shades represent different materials with different

dielectric constants (ROBINSON; NAKKEERAN, 2013).

A two-dimensional photonic crystal is periodic along two axes of symmetry (x-axis and y-

axis as in Figure 3) and smooth on the other axis (z-axis in Figure 3). A typical arrangement consists

of a set of cylinders of different dielectric constant than the background material (primarily high

dielectric constant immersed in air or air holes in a material of high dielectric constant). Depending

on the dielectric constant, the dimensions of the periodic arrangement (radius of the cylinders and

spacing between their centers) and their distribution, the crystal would behave as having a rejection

to certain frequencies (band gap) of the signals propagating in the x-y plane (JOANNOPOULOS

et al, 2011).

7

Figure 3: A two-dimensional photonic crystal with square lattice arrangement of dielectric

columns. The material is homogeneous along the z-axis and periodic along the x and y-axis.

An example of a widely used optical device is a multilayer dielectric mirror, such as a

quarter-wave stack, consisting of alternating layers of material with different dielectric constants.

Light of the proper wavelength, when incident on such a layered material, is completely reflected.

The reason is that the light wave is partially reflected at each layer interface and, if the spacing is

periodic, the multiple reflections of the incident wave interfere destructively to eliminate the

forward-propagating wave. This well-known phenomenon, first explained by Lord Rayleigh in

1887, is the basis of many devices, including dielectric mirrors, dielectric Fabry-Perot filters, and

distributed feedback lasers. All contain low-loss dielectrics that are periodic in one dimension, and

by our definition, they are one-dimensional photonic crystals. Even these simplest of photonic

crystals can have surprising properties. We will see that layered media can be designed to reflect

light that is incident from any angle, with any polarization (an omnidirectional reflector) despite

the common intuition that reflection can be arranged only for near-normal incidence.

8

If, for some frequency range, a photonic crystal prohibits the propagation of

electromagnetic waves of any polarization traveling in any direction from any source, we say that

the crystal has a complete photonic band gap. A crystal with a complete band gap will obviously

be an omnidirectional reflector, but the opposite is not necessarily true. Usually, in order to create

a complete photonic band gap, one must arrange for the dielectric lattice to be periodic along three

axes, constituting a three-dimensional photonic crystal, but there are exceptions. A small amount

of disorder in an otherwise periodic medium will not destroy a band gap, and even a highly

disordered medium can prevent propagation in a useful way (SOUZA, 2012).

9

2.2 MAXWELL’S EQUATIONS

The following Maxwell equations were used in the analysis of the periodic structures of the

Photonic Crystals:

∇ ∙ �⃗⃗� = 𝜌, (2.1)

∇ × �⃗� +𝜕�⃗�

𝜕𝑡= 0, (2.2)

∇ ∙ �⃗� = 0, (2.3)

∇ × �⃗⃗� −𝜕�⃗⃗�

𝜕𝑡= 𝐽 , (2.4)

Where �⃗� is the electric field, �⃗⃗� is the magnetic field, �⃗⃗� the electric flux density, �⃗� the

magnetic flux density, 𝜌 is the density of free charges and 𝐽 is the electric current density. The

following constitutive relations are also needed:

�⃗� = 𝜇�⃗⃗� (2.5)

�⃗⃗� = 𝜀�⃗� (2.6)

Considering a medium free of charges and currents, it is well known that 𝐽 = 0 and 𝜌 = 0.

For periodic structures, we have that 𝜀(𝑟 ) = 𝜀0𝜀𝑟(𝑟 ) and since the materials used are not magnetic

then 𝜇(𝑟 ) = 𝜇0. Substituting the above expressions and constitutive relations in Maxwell’s

equations we obtain:

∇ ∙ [𝜀𝑟(𝑟 )�⃗� (𝑟 , 𝑡)] = 0 (2.7)

∇ × �⃗� (𝑟 , 𝑡) + 𝜇0𝜕�⃗⃗� (𝑟 ,𝑡)

𝜕𝑡= 0 (2.8)

∇ ∙ �⃗⃗� (𝑟 , 𝑡) = 0 (2.9)

∇ × �⃗⃗� (𝑟 , 𝑡) − 𝜀0𝜀𝑟𝜕�⃗� (𝑟 ,𝑡)

𝜕𝑡= 0 (2.10)

10

The �⃗� and �⃗⃗� fields are functions of time and space, however, since Maxwell's equations

are linear, they can be written as the product of their spatial dependence and temporal dependence

of the form of a complex exponential, like this:

�⃗� (𝑟 , 𝑡) = �⃗� (𝑟 )𝑒−𝑖𝜔𝑡 (2.11)

�⃗⃗� (𝑟 , 𝑡) = �⃗⃗� (𝑟 )𝑒−𝑖𝜔𝑡 (2.12)

To find the equation describing the modes at a given frequency, simply enter these

equations in (2.9) and (2.10) as (2.7) and (2.8) only ensures the free space and without load.

∇ × �⃗� (𝑟 ) − 𝑖𝜔𝜇0�⃗⃗� (𝑟 ) = 0 (2.13)

∇ × �⃗⃗� (𝑟 ) + 𝑖𝜔𝜀0𝜀𝑟(𝑟 )�⃗� (𝑟 ) = 0 (2.14)

From equations (2.13) and (2.14), the following equation can be obtained:

∇ × (1

𝜀𝑟(𝑟 )∇ × �⃗⃗� (𝑟 )) = (

𝜔

𝑐)2

�⃗⃗� (𝑟 ) (2.15)

Since it is known that 𝜀𝑟(𝑟 ) is a periodic function in space, we conclude that �⃗⃗� (𝑟 ) is too.

From equation (2.15), (2.7) and (2.8), we calculate the modes of �⃗⃗� (𝑟 ) and their corresponding

frequencies. From which we can determine the frequencies where propagation is not possible,

which is called the Electromagnetic Band Gap (EBG).

Note that this equation is invariant to scale. Equation (2.15) will apply the same both for

microwave frequencies as for optic frequencies. For optical or photonic frequencies, the band gap

region is called the Photonic Band Gap (PBG).

Considering now a mode for �⃗⃗� (𝑟 ), with a frequency 𝜔 and in a periodic dielectric grid

given by 𝜀𝑟(𝑟 ), we use equation (2.15), to study what happens with the mode when you scale the

grid by a factor of 𝑠. In this case the following changes will be made: 𝜀𝑟′(𝑟 ) = 𝜀𝑟(𝑟 𝑠⁄ ), 𝑟′ = 𝑠𝑟

11

and ∇′= ∇ 𝑠⁄ . Which after some mathematic manipulation and substitutions (JOANNOPOULOS

et al, 2011) we arrive at the following equation:

∇′ × (1

𝜀𝑟′ (𝑟 ′)

∇′ × �⃗⃗� ′(𝑟 ′)) = (𝜔′

𝑐)2

�⃗⃗� ′(𝑟 ′) (2.16)

Thus, it can be concluded that by resizing the dielectric grid, there will be new modes of a

frequency inversely proportional to the scale resizing, or in other words, for any variation in the

scale of the grid, we also use the same scale for the wavelength of the modes. By using this property

we can conclude that any result that we will obtain in optical frequencies (photonic) can also be

verified in microwave frequencies, as long as we scale the grid accordingly to the escalation in

wavelength, or vice versa.

12

2.3 CRYSTALLINE STRUCTURE

An ideal crystal can be defined as a regular repetition of a block that is filling all space. The

smallest structure to be periodically repeated to build the crystal is usually called a unit cell. An

ideal crystal can have one, two or three dimensions of crystalline symmetry. In three dimensions,

a crystal is determined by three basic vectors of translation: 𝑎 1, 𝑎 2 𝑎𝑛𝑑 𝑎 3. Thus, for a point

represented by 𝑟 , is equivalent to another point represented by 𝑟 ′:

𝑟 ′ = 𝑟 + 𝑙1𝑎 1 + 𝑙2𝑎 2 + 𝑙3𝑎 3 = 𝑟 + �⃗� (2.17)

Where 𝑙1, 𝑙2 𝑎𝑛𝑑 𝑙3 are integers and �⃗� , called the lattice vector, is any vector translating a

point in one cell to an equivalent point in the crystal lattice. In Figure 4a and in Figure 4b, can be

seen two examples of two-dimensional crystal structures: the square lattice and the triangular

lattice, respectively. In both cases, the elements are separated from each other by a distance 𝑎 from

the center of each other.

Figure 4: a) square and b) triangular lattice photonic structures cross sections. (εa and εb are the

relative permittivity of the background material, respectively, a: lattice constant, L: cavity

length). (KOBA, 2013)

13

Schemes in Figure 5a and Figure 6a show a view of photonic crystal cross sections in a x-

y plane, with cylinders arranged in square or triangular lattice with period 𝑎, and with depicted

primitive vectors 𝑎1 𝑎𝑛𝑑 𝑎2.

Figure 5: A schematic view of a) a square lattice photonic crystal with primitive vectors; and b)

its representation in reciprocal space with reciprocal primitive vectors. (KOBA, 2013)

Figure 6: A schematic view of a) a triangular lattice photonic crystal with primitive vectors; and

b) its representation in reciprocal space with reciprocal primitive vectors. (KOBA, 2013)

14

Figure 5b and Figure 6b show the reciprocal lattices corresponding, respectively, to the real

square and triangular lattices. In the described case, the nodes of a two-dimensional structure can

be expressed by:

𝑥∥ = 𝑙1𝑎 1 + 𝑙2𝑎 2 (2.18)

Where 𝑎 1 and 𝑎 2 are primitive vectors, 𝑙1 and 𝑙2 are arbitrary integers, 𝑥∥ specifies the

placement on the plane 𝑥∥ = �̂�𝑥 + �̂�𝑦, where �̂� and �̂� are unit vectors along x and y axis,

respectively. Primitive vectors for square lattice are described by the expressions: 𝑎 1 = (𝑎, 0),

𝑎 2 = (0, 𝑎) and for the triangular lattice: 𝑎 1 = (𝑎√3 2⁄ , 𝑎 2⁄ ), 𝑎 2 = (0, 𝑎).

In general, the reciprocal vectors can be written in the following form:

𝐺(ℎ) = ℎ1�⃗� 1 + ℎ2�⃗� 2 (2.19)

Where ℎ1 and ℎ2 are arbitrary integers, �⃗� 1 and �⃗� 2 are the primitive vectors of the two-

dimensional reciprocal space, which are expressed by the following equations (KOBA, 2013):

�⃗� 1 =2𝜋

𝑎𝑐(𝑎𝑦

(2), −𝑎𝑥

(2)) (2.20)

�⃗� 2 =2𝜋

𝑎𝑐(−𝑎𝑦

(1), 𝑎𝑥

(1)) (2.21)

Where 𝑎𝑗(𝑖)

is the j-th Cartesian component (x or y) of the 𝑎 𝑖 vector (𝑖 = 1 𝑜𝑟 2). The areas

of primitive cells are 𝑎𝑐 = |𝑎 1 × 𝑎 2| = 𝑎2 in case of the square lattices and 𝑎𝑐 = |𝑎 1 × 𝑎 2| =

√3𝑎2 2⁄ in case of the triangular lattices.

From the above equations, and using the expressions for square and triangular lattice

primitive vectors and primitive cell areas, the reciprocal primitive vectors are described by the

following formulas:

15

�⃗� 1 = (2𝜋 𝑎⁄ , 0), �⃗� 2 = (0, 2𝜋 𝑎⁄ ) --- Square lattice (2.22)

�⃗� 1 = (4𝜋 √3𝑎⁄ , 0), �⃗� 2 = (−2𝜋 √3𝑎⁄ , 2𝜋 𝑎⁄ ) --- Triangular lattice (2.23)

Finally, we present the concept of the Brillouin zone, which is a uniquely defined primitive

cell in reciprocal space. The boundaries of this cell are given by planes related to points on the

reciprocal lattice. The importance of the Brillouin zone stems from the Bloch wave description of

waves in a periodic medium, in which it is found that the solutions can be completely characterized

by their behavior in a single Brillouin zone.

Taking surfaces at the same distance from one element of the lattice and its neighbors, the

volume included is the first Brillouin zone. Another definition is as the set of points in k-space that

can be reached from the origin without crossing any Bragg plane.

There are also second, third, etc., Brillouin zones, corresponding to a sequence of disjoint

regions (all with the same volume) at increasing distances from the origin, but these are used less

frequently. As a result, the first Brillouin zone is often called simply the Brillouin zone. (In general,

the n-th Brillouin zone consists of the set of points that can be reached from the origin by crossing

exactly n − 1 distinct Bragg planes.)

A related concept is that of the irreducible Brillouin zone, which is the first Brillouin zone

reduced by all of the symmetries in the point group of the lattice.

16

Figure 7: The square lattice. On the left is the network of lattice points in real space. In the

middle is the corresponding reciprocal lattice. On the right is the construction of the first

Brillouin zone (JOANNOPOULOS et al, 2011).

Figure 8: The triangular lattice. On the left is the network of lattice points in real space. In the

middle is the corresponding reciprocal lattice, which in this case is a rotated version of the

original. On the right is the Brillouin zone construction. In this case, the first Brillouin zone is a

hexagon centered on the origin (JOANNOPOULOS et al, 2011).

17

2.4 PHOTONIC DEVICES BASED ON TWO-DIMENSIONAL PHOTONIC CRYSTALS

The ability to control the flow of electromagnetic waves, in particular for the microwave

range, has been developed for some time. One of the most common devices used in this process is

the metallic waveguide, whose main function is to guide electromagnetic waves from one point to

another by following a particular direction of propagation.

However, these materials are not suitable for controlling the flow of electromagnetic waves

with frequency higher than the microwave, like visible light for example, because the losses in this

frequency range are large enough to cripple the process.

In order to overcome this difficulty, photonic crystals can be used, since the dielectric

materials commonly used in the synthesis of these crystals do not suffer as much with the

dissipative losses as metallic materials, at frequencies above the microwave range.

The following are some devices based on two-dimensional photonic crystals, which can be

used in a communication system.

2.4.1 Waveguide

A waveguide has the function of transporting electromagnetic waves from one point to

another in a system. A dielectric waveguide, based on two-dimensional photonic crystals, uses the

property of such crystals to be able to reflect electromagnetic waves often located in a specific

band, known as Photonic Band Gap (PBG).

Considering a 2D photonic crystal, a perfectly periodic one, like the ones shown in Figure

5 or Figure 6; you can create a simple waveguide by removing a row of dielectric cylinders

(creating a linear defect), as we can see in Figure 9.

18

Figure 9: Line-defect waveguide formed by a missing row in a square lattice of dielectric rods in

air. (JOANNOPOULOS et al, 2011)

Since the structure with a linear defect is excited with electromagnetic waves in the

frequency range of the PBG, then it can be said that these waves are confined outside the crystal

lattice, since the walls surrounding the defect are reflective to those frequencies. Also in Figure 9,

it is presented the profile of the electromagnetic field along a dielectric waveguide, considering

propagation in the range of the PBG.

It is also interesting to point out that a PBG waveguide may present sharp curves, which is

not possible in optical fibers (due to the bending losses). For example in Figure 10, we present a

waveguide with a 90° bend.

19

Figure 10: 𝐸𝑧 field in a right-angle bend waveguide, showing 100% transmission at the PBG

range. (JOANNOPOULOS et al, 2011)

20

2.4.2 Resonant Cavity

A resonant cavity can be accomplished through the creation of a point defect in a two-

dimensional photonic crystal. This defect can be created from the change of parameters of a single

dielectric cylinder, the simple removal of it or the alteration of the radius.

Considering the light with frequency located in the PBG range and located within the

defect, it can be said that it is prevented from leaving the defect, given that in this situation the

walls surrounding the defect reflect light. These cavities are fundamental in laser systems and can

be used in the manufacture of filters.

In Figure 11, we have the profile of the electromagnetic field inside a cavity, which was

designed using PBG properties.

Figure 11: Point-defect cavity formed by a single missing rod in a square lattice. The cavity

supports a single mode inside the TM band gap, whose electric field 𝐸𝑧 is shown.

(JOANNOPOULOS et al, 2011)

21

2.4.3 Waveguide Splitter

Another very interesting device based on the existence of the PBG in two-dimensional

photonic crystal is the waveguide splitter. The basic function of this device is to

divide the input power between two waveguides at the output. In Figure 2.6, we have an accurate

representation of this device operating in the PBG range.

The design of the photonic circulator will be based mostly on the design of a waveguide

splitter considering 120º angles in between the waveguides and a weak resonant cavity in the

middle.

Figure 12: Left: 𝐸𝑧 field in a “T” splitter for a waveguide, showing essentially 100% transmission

(50% in each branch). Right: Abstract model, treating junction as weak resonance, predicting

100% transmission when 1 𝜏1⁄ = 1 𝜏2⁄ + 1 𝜏3⁄ . (JOANNOPOULOS et al, 2011)

22

3. FERRITES AND THE CIRCULATOR

3.1 THEORY OF FERRIMAGNETIC COMPONENTS

Usually most materials studied in electromagnetism are isotropic (meaning that they behave

in the same way independent of the direction of the signal polarization), however in order to

implement devices that present directional properties when a signal has different polarization (such

as the circulator) we must use anisotropic materials.

The most practical anisotropic materials for microwave applications are ferrimagnetic

compounds such as YIG (yttrium iron garnet)(M POZAR, 2005), or ferrites which are ceramic

compounds consisting of a mixture between iron oxide and one or more other metals, such as

aluminum, cobalt, manganese, and nickel (BRITUN; DANYLOV; ZAGORODNY, 2002) which

as a result end up forming a cubic crystal structure. In contrast to ferromagnetic materials (e.g.,

iron, steel), ferrimagnetic compounds have high resistivity and a significant amount of anisotropy

at microwave frequencies, which makes them suitable in high-frequency electrical components

such as antennas.

The magnetic anisotropy of a ferrimagnetic material is actually induced by applying a DC

magnetic bias field. This field aligns the magnetic dipoles in the ferrite material to produce a net

(nonzero) magnetic dipole moment, and causes the magnetic dipoles to precess at a frequency

controlled by the strength of the bias field. A microwave signal circularly polarized in the same

direction as this precession will interact strongly with the dipole moments, while an oppositely

polarized field will interact less strongly. Since, for a given direction of rotation, the sense of

polarization changes with the direction of propagation, a microwave signal will propagate through

a ferrite differently in different directions. This effect can be utilized to fabricate directional devices

such as isolators, circulators, and gyrators. Another useful characteristic of ferrimagnetic materials

is that the interaction with an applied microwave signal can be controlled by adjusting the strength

of the bias field. This effect leads to a variety of control devices such as phase shifters, switches,

and tunable resonators and filters.

23

We will begin by considering the microscopic behavior of a ferrimagnetic material and its

interaction with a microwave signal to derive the permeability tensor. This macroscopic description

of the material can then be used with Maxwell's equations to analyze wave propagation in an

infinite ferrite medium, and in a ferrite-loaded waveguide.

3.1.1 The Permeability Tensor

The magnetic properties of a material are due to the existence of magnetic dipole moments,

which arise primarily from electron spin. In most solids, electron spins occur in pairs with opposite

signs so the overall magnetic moment is negligible. However, in a magnetic material, a large

fraction of the electron spins are unpaired (more left-hand spins than right-hand spins, or vice

versa), but are generally oriented in random directions so that the net magnetic moment is still

small. An external magnetic field, however, can cause the dipole moments to align in the same

direction to produce a large overall magnetic moment. The existence of exchange forces can keep

adjacent electron spins aligned after the external field is removed; the material is then said to be

permanently magnetized.

For this reason when a ferrite is subjected to a static magnetic bias field 𝐻0, it exhibits

anisotropic properties, which can be described by means of the magnetic permeability tensor. The

anisotropy of the magnetic permeability is responsible for the non-reciprocal behavior found in

ferrite circulators. Losses can be neglected, however, whereas the applied bias magnetic field is

strong enough so that the ferrite is in magnetic saturation. Under this condition, the permeability

tensor of a magnetized ferrite the positive direction of z-axis is given by (M POZAR, 2005):

[𝜇] = [

𝜇 𝑗𝜅 0−𝑗𝜅 𝜇 00 0 𝜇0

] �̂� (3.1)

24

Where the elements of the permeability tensor are defined by:

𝜇 = 𝜇0 (1 +𝜔0𝜔𝑚

𝜔02−𝜔2) (3.2)

𝜅 = 𝜇0 (𝜔 𝜔𝑚

𝜔02−𝜔2) (3.3)

Where 𝜇0 is the permeability of free space, 𝜔 is the angular frequency, 𝜔0 is the precession

frequency or Larmor frequency for the electron spin in the presence of an applied magnetic field

𝐻0 (𝜔0 = 𝜇0𝛾𝐻0), 𝜔𝑚 is the Larmor frequency for the electron when the ferrite is under

magnetization saturation 𝑀𝑠 (𝜔𝑚 = 𝜇0𝛾𝑀𝑠), and 𝛾 is the gyromagnetic radius of the electron.

3.2 CIRCULATORS

Circulators are devices that use the non-reciprocal properties of ferrimagnetic materials,

which means that in the presence of a magnetic field, allow for the control and transmission of

microwave energy.

This interaction between microwave signals and the electrons within the ferrimagnetic

material is usually for a specific resonant frequency. This resonant frequency, which is determined

by the atomic composition of a ferrite component, allows for non-reciprocal behavior in the ferrite

that makes microwave energy travel in the transmission path in only one direction with a small

loss, but also absorbs the energy when applied from the opposite direction (ZAHWE et al, 2009).

In this sense, a ferrite circulator can be a passive non-reciprocal three-port or four-port

device, in which a microwave or radio frequency signal entering any port is transmitted to the next

port in rotation (only one way). A port in this context is a point where an external waveguide or

transmission line (such as a microstrip line, coaxial cable or PBG waveguide) connects to the

device. For a three-port circulator, as shown in Figure 13, a signal applied to port 1 only comes out

of port 2; a signal applied to port 2 only comes out of port 3; a signal applied to port 3 only comes

out of port 1 (PHILIPS SEMICONDUCTORS, 1998).

25

Figure 13: Ferrite circulator, principle of operation.

Circulators fall into two main classes: four-port waveguide circulators based on Faraday

rotation of waves propagating in a magnetized material; and three-port "Y-junction" circulators

based on cancellation of waves propagating over two different paths near a magnetized material.

Waveguide circulators may be of either type, while smaller devices based on striplines are

of the three-port type. Sometimes two or more Y-junctions are combined in a single component to

give four or more ports, but these differ in behavior from a true 4-port circulator.

Circulators exist for many frequency bands, ranging from VHF up to optical frequencies,

the latter being used in optical fiber networks. For example, radio frequency circulators are

composed of magnetized ferrite materials in which a permanent magnet produces the magnetic flux

through the waveguide, however for optical frequencies, ferrimagnetic crystals are used to make

optical circulators. At frequencies much below VHF, ferrite circulators become impractically large.

It is however possible to simulate circulator behavior all the way down to DC using op-amp circuits.

Unlike ferrite circulators, these active circulators are not lossless passive devices but require a

supply of power to run. Also the power handling capability and linearity and signal to noise ratio

of transistor-based circulators is not as high as those made from ferrites.

26

3.3 FERRITE Y-JUNCTION CIRCULATORS

The ferrite junction circulator is a versatile microwave device, whose main characteristic is

that in addition to its use as a circulator, it also can be used as an isolator or as a switch. The three-

port version of the ferrite junction circulator, usually called the Y-junction circulator, it is the most

commonly used, and it can be constructed to join striplines, metallic waveguides or, in our case

study, PBG waveguides.

3.3.1 Description of the Ferrite Circulator Operation

The easiest case study for a ferrite circulator is the stripline Y-junction ferrite circulator,

which consists of two ferrite cylinders filling the space between a metallic conducting center disk

and two conducting ground plates (TEOH; DAVIS, 2001). Thus, presenting the simplest

geometrical arrangement and, therefore, is the easiest to treat analytically. This circulator in its

basic form is illustrated in Figure 14. The connections to the center disk are in the form of three

strip-line center conductors attached to the disk at points 120° apart around its circumference. A

magnetizing field is applied parallel to the axis of the ferrite cylinders.

From several experimental results (FAY; COMSTOCK, 1965), it was found that the Y-

circulator had some of the properties of a low-loss transmission cavity. At its resonant frequency,

it was well matched but slightly under coupled, and a standing wave existed in the structure. The

maximum isolation occurred at the frequency at which the insertion loss was a minimum. The

isolation at the third port and the return loss from the input port correspond quite well as the

frequency is changed. The above experimental evidence suggests a resonance of the center disk

structure as being an essential feature of the operation of the circulator. The lowest frequency

resonance of the circular disk structure of Figure 14 is the dipolar mode in which the electric field

vectors are perpendicular to the plane of the disk and the RF magnetic field vectors lie parallel to

the plane of the disk. This mode, as excited at port 1, is illustrated in the unmagnetized case by the

standing-wave pattern of Figure 15a.

27

Figure 14: Stripline Y-junction circulator

With ferrite cylinders on each side of the disk, the bottom cylinder behaves as a mirror

image of the top one, and the analysis of the device need only be concerned with one cylinder.

Ports 2 and 3, if open-circuited, will see voltages that are 180° out of phase with the input voltage,

and about half of the value of the input voltage (LIN; AFSAR, 2006). If the standing-wave pattern

is rotated, as in Figure 15b then port 3 is situated at the voltage null of the disk and the voltages at

ports 1 and 2 are equal. The device is equivalent to a transmission cavity between ports 1 and 2,

and port 3 is isolated.

28

Figure 15: a) Dipolar mode of a dielectric disk (𝐻𝑖𝑛𝑡 = 0 𝑜𝑟 ∞). b) Analogous pattern for a

magnetized disk (𝐻𝑖𝑛𝑡 𝑓𝑜𝑟 𝑐𝑖𝑟𝑐𝑢𝑙𝑎𝑡𝑖𝑜𝑛). The pattern for the magnetized disk has been rotated to

isolate port 3.

3.3.2 Design of a Ferrite Junction Circulator

There are several sources (FAY; COMSTOCK, 1965), (M POZAR, 2005) that deals with

circulator design by the use of electromagnetic theory. We start by solving the Maxwell equations

(2.1), (2.2), (2.3) and (2.4) and applying the permeability tensor (3.1) and the boundary conditions.

Since the geometry of the ferrite is disk shaped, the permeability tensor should be applied in

cylindrical coordinates.

29

The resonant modes can be found by enforcing the electromagnetic boundary conditions. If

the ferrite is not magnetized, resonance occurs when:

𝐽0(𝑘𝑅) −𝐽1(𝑘𝑅)

𝑘𝑅= 𝐽1(𝑘𝑅) = 0 (3.4)

Where 𝐽𝑛(𝑘𝑅) is the n-order Bessel function of the first kind. 𝑅 is the radius of the ferrite

disk, 𝑘 is the wavenumber. The first root of (3.4) is at 𝑘𝑅 = 1.841. For a more detailed

mathematical deduction of this equation, this can be found in (FAY; COMSTOCK, 1965).

In order to find the value of 𝑘, we need to take into account the material properties of the

ferrite disk, in this case the value of 𝑘 is determined by:

𝑘2 = 𝜔2𝜀 𝜀0 𝜇𝑒𝑓𝑓 𝜇0 = 𝜔2𝜀 𝜀0 𝜇0𝜇2−𝜅2

𝜇 (3.5)

Finally, from (3.4) and (3.5), we find the radius 𝑅 of the ferrite disk. However, we need to

adjust another important property: the magnetic biasing field. This outer magnetic field, which is

perpendicular to the axis of the ferrite disk, has to be large enough to rotate the standing wave

pattern in the gyromagnetic resonator by 30 degrees. This condition can be satisfied by different

magnetic fields, either below resonance or above resonance operation; however, the most important

condition that should be met, is that the ferrite should be magnetically saturated, since the formulas

obtained in the analysis of the permeability tensor are only valid when magnetic saturation was

considered.

Choosing the proper magnetic field can be facilitated by the use of the hysteresis loop of the

applied ferrite material that gives advice for the smallest field needed for saturation.

30

4. NUMERICAL ANALYSIS AND PROJECT DESIGN

The development of this project consisted of two stages. On the first stage, we design and

model the dielectric elements that will be used in the Photonic Array, for this we use the Photonic

Band Gap analysis and with this information determine the size (radius), spacing and configuration

of the array. Also as part of this first stage, we will design the ferrite circulator and thus choose the

radius that will match the frequencies chosen for our project.

On a second stage, we will test our design on an Electromagnetic Analysis Software. For

this, we chose one of the most reliable software’s in the market, which is the CST Microwave Studio

2013®. The results from this stage will assure us that the theoretical design was well projected,

and thus allow us to test these results on an experimental basis, which will be the objective of a

future paper.

4.1 DESIGN AND MODELING OF THE PHOTONIC GRID

For the design of two-dimensional photonic crystal array, first, we use a mathematical tool

to calculate the band diagrams. Through the band diagrams it is possible to determine if the

structure we are proposing presents a photonic "gap" (region of forbidden frequencies propagating

in all directions of the plane), and to establish the amount and ranges of possible forbidden

frequencies.

For the current work, the mathematical tool we will use is the MPB(JOHNSON, [s.d.]) or

MIT Photonic-Bands package, which is a free program for computing the band structures

(dispersion relations) and electromagnetic modes of periodic dielectric structures. With this tool,

we are able to analyze the electromagnetic band gap of different materials (in our case dielectric

rods) as well as the different configurations of them.

The geometry used to generate the 2D photonic crystal array consists of dielectric rods

(𝜀𝑟 = 10) surrounded by air (𝜀𝑟 = 1), forming an hexagonal arrangement (triangular lattice), we

http://www.gnu.org/philosophy/free-sw.html

31

choose this configuration since for our final design (three-port PBG circulator) it has the adequate

geometrical setup so we can create line defects (for the waveguides) with 120 degrees of separation

in between them. These dielectric rods were developed by the research group of Prof. Dr. Sombra

(ALMEIDA et al, 2008), which he provided as collaboration for the experimental tests based on

this design.

For the MBP tool analysis, we are required to input the lattice configuration (triangular

lattice in this case), the dielectric rods relative permittivity to the surrounding material (𝜀𝑟 = 10),

and finally the fill factor ratio 𝑟 𝑎⁄ . Since the dielectric rods we are using for this project already

have a fixed radius 𝑟 = 9.75 𝑚𝑚, then the remaining parameter 𝑎 (the lattice constant) is the one

we will use in order to determine the best setup for the design. We then proceeded to make several

analysis for different values of 𝑎 (as shown in Table 1), which shows us that a value of 𝑎 = 45𝑚𝑚

or 𝑟 𝑎⁄ = 0.2167, will be the one which will keep the grid as small as possible (dimension wise)

as well as provide the largest band gap necessary for the guided mode.

Radius “r”

(mm)

Periodicity

“a” (mm)

r/a

ratio

Start

of Gap

End

of Gap

Band

Gap

9.75 20 0.4875 None None None

9.75 25 0.3900 0.2232 0.2592 14.93%

9.75 30 0.3250 0.2373 0.3119 27.17%

9.75 35 0.2786 0.2530 0.3604 35.02%

9.75 40 0.2438 0.2698 0.4044 39.94%

9.75 45 0.2167 0.2870 0.4439 42.94%

9.75 50 0.1950 0.3040 0.4772 42.34%

9.75 55 0.1773 0.3210 0.5039 41.23%

9.75 60 0.1625 0.3373 0.5233 39.98%

Table 1: Analysis of band gap size for different 𝑟 𝑎⁄ ratios of grid arrays.

32

Figure 16: Unit-cell array configurations for different 𝑟 𝑎⁄ ratios, as viewed by the MPB tool

In Figure 16, we show the different unit-cell configurations for the different 𝑟 𝑎⁄ ratios as

shown in Table 1. Figure 17 shows the unit cell and the corresponding Brillouin zone for the

hexagonal array.

Figure 17: (a) Unit cell (highlighted by the parallelogram) to a hexagonal arrangement and (b)

respective Brillouin zone. The irreducible zone is a triangular wedge of vertices Γ, K and M.

r/a = 0.4875 r/a = 0.3900 r/a = 0.3250 r/a = 0.2786

r/a = 0.2438 r/a = 0.2167 r/a = 0.1950 r/a = 0.1773

33

As can be seen in Figure 18, a photonic band gap exists between the first and second

band in all directions of the photonic crystal, which corresponds to the normalized frequency band

between 0.28697 < 𝜔𝑎 2𝜋𝑐⁄ < 0.44391. In this range, there are no propagation modes.

Figure 18: Band Gap diagram calculated for a hexagonal array of dielectric columns surrounded

by air, with geometric ratio 𝑟 𝑎⁄ = 0.2167, for the TM polarization parameters.

34

4.2 PBG WAVEGUIDE DESIGN

The main property of photonic crystals is the existence of forbidden frequency bands that

impede the propagation of electromagnetic waves in those bands. However, for these crystals to be

used on the construction of devices (both active and passive), it is necessary the insertion of defects.

Within these defects, the electromagnetic wave, with a wavelength in the range of the PBG, is

confined, so it cannot propagate into other regions of the photonic crystal. This way, small devices

with very low losses can be fabricated, as well as waveguides (with very narrow bandwidths and

curved sections).

In our proposed design, waveguides in photonic crystals are generated by a linear defect

obtained by removing rows or columns of the dielectric material along the direction of propagation,

as shown in Figure 9. The dispersion relation and the electric field distribution are also calculated,

considering the hexagonal symmetry for TM polarization. The TM polarization corresponds to

modes whose electric field vector is parallel to the z-axis.

The analysis of the waveguides and the guided modes are also conducted by using the MPB

tool. For this part of the analysis, we run the MPB for different super-cells configurations as shown

in Figure 19. From there we obtain the results shown in Table 1 and thus obtain the band gaps in

normalized frequency.

35

Figure 19: From left to right: Super-cell configurations for five, four, three, two and one

dielectric elements (on each side) respectively. Each super-cell has a line-defect along the middle

to allow for the guided mode propagation.

# of elements

(each side)

Start

of Gap

End

of Gap

Guided

Mode Gap

11 0.28452 0.34313 0.10078

9 0.28359 0.34307 0.10084

7 0.28259 0.34269 0.10122

5 0.28105 0.34022 0.10369

4 0.27639 0.34998 0.09393

3 0.27496 0.32230 0.12161

2 0.26679 0.39143 0.05245

1 0.50000 0.76376 No Gap

Table 2: Analysis for the Guided Mode Gap as a function of the number of dielectric elements on

each side of the line-defect.

36

From the results in Table 2, we observe that the introduction of the line-defect has reduced

the band gap (End of Gap) from the previous value of 0.4439 to about 0.3223, this for the super

cell configuration of three elements, which means that a guided mode has appeared in the gap

difference between the two band-gap ranges (the one without and the one with the line defect). In

this case, we also observe that super-cell configurations of three or more elements have the guided

mode well confined, then we choose to implement the waveguide using the lowest possible super-

cell size so that we require fewer elements when we design the PBG circulator.

In Figure 20, we plot the guided mode corresponding to the super-cell of three elements in

a normalized frequency scale. As we can observe, this mode is guided inside the PBG waveguide

between 0.3222 < 𝜔𝑎 2𝜋𝑐⁄ < 0.4439. This way, if we replace the value of our lattice period of

𝑎 = 45𝑚𝑚, the photonic waveguide we designed will have a guided mode that is confined in the

frequency range of 2.33 GHz < f < 2.96 GHz. Frequencies above this range will be spreading

waves inside the crystal lattice, while frequencies below this range will be completely reflected.

Figure 20: TM guided mode for the super-cell configuration of three elements per side.

37

4.3 DESIGN OF THE FERRITE CYLINDER AND CIRCULATOR

From the results of section 4.2, we know that the proposed waveguide will have a guided

mode between 2.33 GHz < f < 2.96 GHz. If we consider the central frequency of this range as the

value for which we want the best transmission/isolation in our three-port circulator, then we choose

our frequency for the rest of the calculations as 𝑓𝑐 = 2.6 𝐺𝐻𝑧.

Below are the material properties for the ferrite component, which was also given to us by

Prof. Dr. Sombra as collaboration for this thesis work:

Relative permittivity = 14.45

Saturation magnetization (Ms) = 1786.5 Gauss

Dielectric loss (tan ) = 2.73 x 10-5

Resonance linewidth (H) = 60 Gauss

Height of the Disk = 12mm

Applying the calculated frequency 𝑓𝑐 and the above data for the ferrite material to equations

(3.4) and (3.5), we obtain that the radius of the ferrite disk needs to be 𝑅 = 8,9 𝑚𝑚. This value

will allow for the best transmission/isolation at ports 2/3, when we input a signal around the 𝑓𝑐 =

2.6 𝐺𝐻𝑧 value.

38

5. RESULTS

5.1 COMPUTER SIMULATIONS

5.1.1 Straight Photonic Waveguide

For the electromagnetic simulations, we use the aid of the computer software CST

Microwave Studio 2013® (COMPUTER SIMULATION TECHNOLOGY, 2013). We start our

analysis by first modeling a straight photonic waveguide (no bends or corners), which was designed

using the line-defect technique. This means the removal of one row of dielectric rods in a photonic

grid based on triangular lattice. In Figure 21, we show such waveguide with two coaxial ports,

which we are going to excite with frequencies between 2GHz to 3GHz.

Figure 21: Straight Photonic Waveguide with two coaxial ports

39

The first result displayed in Figure 22 and Figure 23, shows the S-parameters for a signal

injected at Port 1 with a frequency range from 1.5GHz to 3.5GHz. In this case, the S21 parameter

(transmission coefficient) indicates very good transmission from Port 1 to Port 2 (meaning less

than -1dB in transmission losses) in the frequency range from 2.35GHz to 2.90GHz. This result

agrees very well with the numerical result we obtained in Section 4.2 where we predicted that the

guided mode in this photonic waveguide would be between 2.33 GHz < f < 2.96 GHz.

Figure 22: S-parameters for the straight photonic waveguide. Input signal sweeps from 1.5GHz to

3.5GHz.

40

Figure 23: Zoom in on the S21 parameter (transmission coefficient). In this case it shows good

transmission (under -1dB in losses) from 2.35 GHz < f < 2.90 GHz.

41

5.1.2 Photonic Waveguide Y-Splitter

Next, in our computer-aided design using CST Microwave Studio 2013®, we model a three-

port photonic waveguide Y-Splitter, with waveguides at 120º from each other, as shown in Figure

24.

In this case, all three ports can be inputs for the test signal, which will also make the other

two ports as outputs. This way each output port should receive 50% percent of the input signal

(theoretically). However, since it is our objective to compare the obtained results with experimental

ones to be done in later work, we are simulating this model considering possible losses in the

materials (metal and dielectric components). In addition, we are modeling the standard SMA

coaxial inputs in order to feed the test signal, which means there will be reflection losses for

possible impedance mismatches that we will also try to minimize by designing impedance match

connectors to be used in the experimental work too.

Figure 24: Three-Port Photonic Waveguide Y-Splitter

42

The results displayed in Figure 25, show the S-parameters for a signal injected at Port 1,

with frequencies from 2.2GHz to 3GHz. In this case, the S21 and S31 parameters (transmission

coefficients) are equal and overlapped, and show a -3.78dB loss in each output port. The ideal Y-

splitter (with no losses) will output -3dB on each output port (50% of the input signal power),

which means that in this case, the remaining -0.78dB are losses due to energy absorption in the

materials, scattering and reflections due to impedance mismatch. From it, we can infer that our

Photonic Y-Splitter is working as designed since it is splitting the energy of the input signal by half

and delivering it to the output ports with minimal losses (-0.78 dB loss at each port).

Figure 25: S-parameters for the photonic waveguide splitter. S21 and S31 are overlapping each

other. They both show equal signal output with about -0.78dB less than ideally expected, caused

by losses in the waveguide as well as some impedance mismatch (reflections).

43

5.1.3 3-Port Photonic Junction Circulator

Next in Figure 26, we present the schematic for the 3-Port Photonic Junction Circulator. As

seen in the previous section, this circulator is modeled by adding a Ferrite Circulator in the junction

that exists in the waveguide Y-splitter. This way, and after adjusting the right magnetic biasing

field, we can control the transmission/isolation from an input signal in any port.

Figure 26: 3-Port Photonic Junction Circulator

From the Junction Circulator theory presented in Section 3.3.2, we know that in order to

achieve circulation on the ferrite component we must operate with a magnetically saturated ferrite.

From the ferrite material properties shown in section 4.3, we know that the minimum value for

Saturation magnetization (4Ms) is 1786.5 Gauss. In Figure 27, we verify that with a value of 1500

Gauss of biasing field we still not see a transmission/isolation behavior, meaning we need more

biasing field to achieve circulator operation.

44


Gauss. Signal input is done only at Port-1 since all ports are symmetric

In Figure 28, we show the S-parameters for input at Port 1 and outputs at Port 2

(transmission) and Port 3 (isolation). In here, the value of the biasing field is of 1700 Gauss, which

is just below the ferrite magnetization saturation. In addition, we start seeing the circulation

behavior we expected, although not at the frequency we designed the circulator. This means we

must increase the biasing field in order to adjust the dipole rotation for our desired frequency.

45



46

In Figure 29, we adjust the biasing field of the ferrite for 2300 Gauss. The results below

show that our operation frequency has shifted to the right and it is now at 2.63GHz. Since we want

the operation to be as close as possible to 2.60GHz then we adjust the biasing field one more time

and in Figure 30, we present our final results (for all 3 ports) in which the biasing field was at

adjusted for 2125 Gauss and the circulator operating frequency stands at 2.60GHz.



47


Gauss. Signal input is done at all three ports, sequentially

48

In Figure 31 and Figure 32, we present the results for the circulator operation with input

signal at Port 1. We verify that we have a very good transmission coefficient S21 (Port 1 Port

2) with only -0.212dB of losses, which is a very good value compared to commercial circulators

available for the same frequency range. Signal isolation at Port 3 is below -20dB, same with the

reflection losses on Port 1, which means we have good impedance match.


Gauss and input signal at Port 1.

49


observe very good transmission from Port 1 to Port 2 and good isolation at Port 3.

50

In Figure 33 and Figure 34, we have similar results for circulator operation with input signal

at Port 2. We verify that we have a very good transmission coefficient S32 (Port 2 Port 3) as well,

with only -0.211dB of losses. Signal isolation at Port 1 is below -20dB, same with the reflection

losses on Port 2, which means we have good impedance match on this port as well.



51



52

Again, in Figure 35 and Figure 36, we have similar results for circulator operation with

input signal at Port 3. We verify that we have a very good transmission coefficient S13 (port 3

port 1) with only -0.214dB of losses. Isolation is yet again below -20dB on port 2, and the reflection

losses on Port 3 are about the same, which means we have again good impedance match.



53



54

Finally, we performed measurements of the bandwidth of our designed circulator. For

comparison with these measurements, we obtained some parameters from data sheets available

from a top of the line circulator (military specifications) (RENAISSANCE ELECTRONICS,

2006), which also operates in our microwave band (S-Band). From these measurements, we obtain

the following Table 3.

Parameters Commercial Circulator

Bandwidth

Photonic Circulator

Bandwidth

Insertion Loss (-0.25dB Max) 200 MHz 103 MHz

Isolation (-20dB Min) 200 MHz 48 MHz

Reflection “VSWR” (-23 dB Min) 200 MHz 43 MHz

Table 3: Comparison between a top of the line circulator (military grade) and our proposed

design for a photonic circulator

55

6. FINAL CONLUSIONS AND CONSIDERATIONS

The work presented on this thesis aimed for a pedagogical, but yet thorough, approach at

understanding photonic crystals, photonic devices and magnetic effects. This pedagogical approach

at the theoretical concepts, computer aided modeling, electromagnetic simulations and subsequent

experimental implementation; allows for an easier introduction on a complex subject both for

students and researchers alike. In that sense, the microwave approach into photonics, allows for the

fabrication of a conceptually equivalent but hand-manipulable prototype, which would be very

difficult to fabricate and manipulate if we were to use optic frequencies, for that approach usually

demands sophisticated photonic fabrication resources and techniques.

The grid distribution or photonic lattice for the dielectric elements was chosen as a

triangular one, since that grid arrangement was the one that provided the best band gap and was

easily configurable when designing the photonic waveguides (separated by 120º) for the design of

the three-port splitter and later three-port circulator. The size and periodicity of the grid elements

were chosen by calculating the PBG diagrams with the aid of the MPB® mathematical tool created

by MIT, which we used for computing the band structures (dispersion relations) and

electromagnetic modes of our periodic dielectric structure. Finally, we proceeded to create a

photonic crystal waveguide by inserting a linear defect in the crystal structure.

The ferrite dimensions were designed based on the frequency of operation for our structure,

as well as the material properties for the ferrite component. The electromagnetic relations for that

design were already summarized in Chapter 3.

On a next step, we proceeded to model the entire structure using CST Microwave Studio

2013®. In the modeling process we based the design on the dimensions and grid periodicity

obtained from the numerical analysis, these dimensions were final and were not changed or fine-

tuned later to provide better results, on the contrary, the electromagnetic simulation was performed,

keeping in mind the need to fabricate a prototype and later replicate these results experimentally

for subsequent work.

56

After the computer assisted modeling was completed, we proceeded to perform

electromagnetic simulations to verify if our numerical analysis and design were correct. The

simulation results shown in Chapter 5 conclude that the numerical analysis we performed was well

thought of, and in every case was consistent with the theory we used to design and project the PBG

Circulator.

Finally, we compared the performance of our designed PBG circulator, obtained from the

analysis of the electromagnetic simulation results (reflection, insertion loss and isolation), with the

performance of a commercially available circulator (military specs). From this comparison, we

learned that our design still needs a few improvements in order to achieve the same performance

as this top of the line microwave device, but still demonstrates the feasibility of constructing

efficient microwave devices using photonic band gap techniques.

57

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