electromagnetic properties of double-periodic...

SLOVAK UNIVERSITY OF TECHNOLOGY

Faculty of Electrical Engineeringand Information Technology

Electromagnetic propertiesof double-periodic structures

Dissertation thesis

2014 Gergely Kajtár

SLOVAK UNIVERSITY OF TECHNOLOGYIN BRATISLAVA

Faculty of Electrical Engineeringand Information Technology

Reg. No.: FEI-10832-62224

Electromagnetic propertiesof double-periodic structures

Dissertation thesis

Study programme: Physical EngineeringStudy field number: 3940Study field: 5.2.48 Physical EngineeringTraining workplace: Institute of Nuclear and Physical EngineeringSupervisor: prof. RNDr. Peter Markoš, DrSc.

Bratislava 2014 Mgr. Gergely Kajtár

Slovenská technická univerzita v BratislaveÚstav jadrového a fyzikálneho inžinierstva

Fakulta elektrotechniky a informatikyAkademický rok: 2014/2015Evidenčné číslo: FEI-10832-62224

ZADANIE DIZERTAČNEJ PRÁCE

Študent: Mgr. Gergely KajtárID študenta: 62224Študijný program: Fyzikálne inžinierstvoŠtudijný odbor: 5.2.48 fyzikálne inžinierstvoVedúci práce: doc. RNDr. Peter Markoš, DrSc.

Miesto vypracovania: ÚJFI

Názov práce: Elektromagnetické vlastnosti dvojrozmerne periodických štruktúr

Špecifikácia zadania:

Cieľom práce je kvantitatívna analýza elektromagnetických vlastností periodických dvojrozmerných štruktúr:difrakčných mriežok, tenkých fotonických vrstiev, kovových štruktúrovaných vrstiev a multivrstiev,s aplikáciou na štúdium elektromagnetických vlastností metamateriálov. Súčasťou práce bude tvorba vlastnýchnumerických programov vo frekvenčnej oblasti.

Výstupom práce budú vypočítané hodnoty odrazu, absorpcie a prechodu elektromagnetických vĺn cezkonkrétne študované štruktúry, analýza závislosti elektromagnetických vlastností od fyzikálnycha priestorových parametrov štruktúr, fyzikálna interpretácia procesov vyvolaných prechodomelektromagnetickej vlny cez štruktúru.

Riešenie zadania práce od: 05. 09. 2011

Dátum odovzdania práce: 05. 09. 2014

L. S.

Mgr. Gergely Kajtárriešiteľ

prof. Ing. Vladimír Slugeň, DrSc.vedúci pracoviska

prof. Ing. Julius Cirák, CSc.garant študijného programu

Abstract

There are numerous fields in optics, where diffractive properties of optical elements (beamsplitters, multilayered mirrors, grating couplers, polarizers, filters, modulators, photonicfibers, deflectors, etc.) are studied, therefore it is inevitable to have a theoretical model,which helps to design these structures. RCWA (rigorous coupled wave analysis) is a rela-tively new method, applied at nanophotonics and optics.

The goal of the presented thesis is to introduce and implement RCWA in order to studywave propagation through various periodic structures (such as lamellar gratings, thin pho-tonic layers, metallic gratings, structures made of left handed material, photonic crystals –1D, 2D and 3D). Part of the thesis is developing a numerical program which calculates trans-mission, reflection and absorption of these structures in the frequency domain. These physicalquantities are dependent on the optical properties and geometry (dielectric constant, spatialperiod, thickness of layers, filling factor) of the structure of interest. We provide quantitativeanalysis how the diffraction efficiency depends on those parameters.

The structure of the thesis is organized as follows. Introduction briefly summarizes his-torical aspects of RCWA. Qualitative descriptions of various numerical methods in opticsare presented and compared. The next three chapters thoroughly introduce RCWA – fordouble-periodic structures, single-periodic structures with conical and classical mounting.The analysis is derived as generally as possible. Special case of anisotropic lamellar gratingwith randomly distributed permittivity tensors is discussed in Chapter 4. The last chapteris dedicated to our numerical program – RawDog – developed for diffraction efficiency cal-culation of lamellar gratings. This chapter also includes results and applications achievedby RawDog for certain physical problems: mode guiding of inside periodic dielectric slab,plasmonic resonance on a metallic slab, study of anisotropic photonic crystal.

Keywords: diffraction efficiency, RCWA, numerical analysis, grating

Abstrakt

V oblasti optiky existuje početné využitie difrakčných prvkov (optické mriežky, hranoly,zrkadlá, polarizátory, atď.). Výroba a použitie týchto prvkov vyžaduje, aby šírenie elek-tromagnetickej vlny cez takéto štruktúry bolo adekvátne modelované a opísané. Preto jedôležité mať teoretický model, pomocou ktorého je možné navrhovať optické prvky. Analýzazviazaných vĺn (RCWA – rigorous coupled wave analysis) je relatívne nová metóda v oblastiteoretickej optiky, ktorá má početné uplatnenia aj v nanofotonike.

Cieľom dizertačnej práce je podrobný popis metódy RCWA, ktorá je použitá na analýzušírenia elektromagnetickej vlny cez rôzne periodické štruktúry (obyčajné optické mriežky,1D, 2D, a 3D fotonické kryštály, tenké vrstvy, kovové periodické štruktúry, metamateriálovémriežky) a na výpočet ich difrakčnej účinnosti. Súčasťou dizertačnej práce je vývoj a aplikácianumerického programu, ktorý je schopný numericky rátať vo frekvenčnej oblasti odrazivosť(reflektancia), priepustnosť (transmitancia) a absorpčné vlastnosti vyššie uvedených štruk-túr. Tieto fyzikálne veličiny závisia od dopadajúcej vlny (polarizácia, vlnová dĺžka, uholdopadu), geometrie a optických vlastností štruktúr (index lomu, hrúbka, perióda). Kvanti-tatívne vyšetríme, ako difrakčná účinnosť závisí od týchto faktorov.

Štruktúra dizertačnej práce je nasledovná: Úvodná časť stručne rekapituluje dejiny, sú-časný stav a aspekty metódy. Okrem toho obsahuje aj kvalitatívne porovnanie metódy s in-ými známymi numerickými metódami, ktoré sa bežne používajú v praxi. Ďalsie tri kapitolyvenujeme opisu RCWA pre rôzne periodické štruktúry, napríklad pre dvojnásobne periodickúštruktúru s ľubovoľným dopadom elektromagnetickej vlny alebo jednonásobne periodickúštruktúru s ľubovoľným a klasickým dopadom vlny. Špeciálny prípad anizotrópnej optickejmriežky s náhodne rozloženou permitivitou je diskutovaný v Kapitole 4. Posledná kapitolapráce je venovaná nášmu numerickému programu (RawDog) a aplikácii programu na rôznefyzikálne problémy: výpočet zakázaných pásov fotonického kryštálu, hľadanie vlastných re-zonancií štruktúry, šírenie povrchových plasmónov v kovových nanoštruktúrach, elektromag-netické vlny v anizotrópnom fotonickom kryštáli a pod.

Kľúčové slová: difrakčná účinnosť, RCWA, numerická analýza, mriežka

Statement of originality

The work contained in this thesis has not been previously submitted at any other highereducation institution. To the best of my knowledge and belief, the thesis contains no materialpreviously published or written by another person except where due references are made.

Bratislava, 18. 06. 2014

Gergely Kajtár

Contents

Introduction 8History of the method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

RCWA for double-periodic structures 15Maxwell’s equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15Inverse rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16Grating region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24Superstrate region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25Substrate region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27Transfer matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30Renormalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31Multilayered structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32Diffraction efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34Fourier decomposition of double-periodic structures . . . . . . . . . . . . . . . . . . 36

RCWA for single-periodic structures with conical mounting 39Differences between 2D and 1D models . . . . . . . . . . . . . . . . . . . . . . . . . 39Solving the eigenproblems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41Fourier decomposition of single-periodic structures . . . . . . . . . . . . . . . . . . 45

RCWA for single-periodic structures with classical mounting 47Differences between conical and classical mounting . . . . . . . . . . . . . . . . . . 47TE polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49TM polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

RCWA for anisotropic and disordered grating 54Fourier decomposition of disordered structures . . . . . . . . . . . . . . . . . . . . 57

Applications of RCWA 59Testing the program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60Photonic crystal with linear defect . . . . . . . . . . . . . . . . . . . . . . . . . . . 64Plasmonic resonance analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67Anisotropic photonic crystal with randomly distributed permittivity . . . . . . . . 78Guided mode resonance of a dielectric slab . . . . . . . . . . . . . . . . . . . . . . . 81

Summary 86

Resumé 87

Bibliography 98

Introduction

Introduction

There is a thirst of humanity, called understanding the nature of things. Understandingthe nature of things means revealing the connections between them, disclosing causalities,categorizing, observing properties, quantifying them and last but not least constructing socalled models. Coming up with models and theories is a standard procedure of science toexplain the reality – the experiments we make –, since we can solely rely on phenomena weobserve. No theory is absolutely correct, but correct for the time being, until an observationemerges which can not be explained by that theory. On the other hand, every working modelis inevitable correct within its domain of validity, since models are usually tailored to explaina particular physical phenomenon.

In optics, various models were invented from time to time to keep the pace with emergingexperiments. For instance, geometrical optics (also known as ray optics) was capable to modelrefraction of light (Snell’s law), but when came to diffraction on slits (Young’s experiment),geometrical optics failed [1]. There was a need to explain phenomena of diffraction andinterference, thus Huygens introduced the scalar wave theory of light. Being a scalar theory,it is not capable to model diffraction on subwavelength features. A more precise theorywas introduced by Maxwell in 1861. His vector theory of electromagnetism, unlike scalartheories, deals with subwavelength apertures, evanescent waves (near-field) and boundaryconditions, therefore it is more rigorous than a scalar theory.

Based on Maxwell’s theory, various wave propagation models have been evolved. Eachof them has its own advantages and drawbacks, each of them is used to model a particularproblem. They are usually divided into two main categories: time-domain and frequency-domain methods. Time-domain methods model how the wave propagates in time and space,therefore it discretizes time and space. The most well-known time-domain method is probablyFDTD – finite difference time domain method [2]. Frequency-domain methods work withtime independent complex amplitudes of the electric and magnetic field. Therefore they aresuitable to simulate problems for a single frequency of the incident wave. The most widelyused analytic methods are: C-method – coordinate transformation method [3], FEM - finiteelement method [4], PWE – plane wave expansion method [5] and RCWA [6]. Every methodhas its own strengths and weaknesses. Table 1 contains the comparison of three methods:FDTD, FEM and RCWA.

Since our aim is to analyze problems involving periodic structures, our choice of numericalmethod is RCWA. For the sake of convenience, advantages and drawbacks of RCWA is brieflysummarized in Table 2. Finally, we give a very short description of RCWA.

8

Introduction

Table 1: Comparison of the three most used numerical methods for wave propagation.

Features FDTD FEM RCWA

Geometry Any Any Periodic structures

Discretization Time and space Space None

Output Field distribution Field distributionField distributionand diffraction

efficiency

Dispersionmodeling No Yes Yes

Multiplewavelengthsimulation

Yes No No

Computingspeed Slow

Slow meshing, fastcomputing

Fast for single-,medium for

double-periodic

Inaccuracyfactor

Discretization ofgeometry and rounding

error

Discretization ofgeometry androunding error

Truncation ofFourier expansions

Convergenceproblems

Metallic, subwavelengthand dispersive

materialsNone

Negativepermittivity orpermeability

RCWA in a nutshell

The main principle of RCWA consists of the Fourier decomposition of allthe space dependent physical quantities, such as electric and magneticfield amplitudes, spatial relative permeability and permittivity, using theFloquet-Bloch condition. Resolving Maxwell’s equations in each regionsurrounded by interfaces – superstrate, grating region and substrate –leads to a system of ordinary differential equations. This system is solvedin each region, then they are matched at the interfaces by the boundaryconditions, which lead to a linear algebraic problem. Finally, diffractionefficiencies are calculated for the given structure using field amplitudes.

9

Introduction

Table 2: Advantages and drawbacks of RCWA.

Advantages Drawbacks

Relatively fast computations Only periodic structures

Easy to implement Approximatesstructure’s shape

Solution for asingle frequency

Convergence problem fornegative permittivity

or permeability

Deals with step-functions Complicated eigenproblemsolving process

History of the method

For the very first time, Rayleigh tried to explain Wood’s anomaly by analytically cal-culating the amplitude of different diffraction orders in 1907 [7]. In his model, he used amodal expansion method to define the fields. However, he limited his analysis to the TEpolarization case (electric field vector is parallel to the grating grooves) only.

Six decades later, Burckhardt attempted to rigorously analyze diffraction on a thickhologram with a periodic grating structure [8]. To simplify the problem, he assumed thatthe grating had a sinusoidal profile. He used a Fourier series expansion to represent theelectromagnetic field inside the grating slab, solved for the TE and TM polarization caseseparately, and matched the field at the interfaces to obtain the electromagnetic distributionin each layer. Similar derivation was made by Kogelnik in 1969, however, he worked withmore approximations [9]. In 1973, Kaspar modeled first time a lamellar (with rectangularshape) grating and complex refractive index [10]. He used Fourier series expansion andFloquet’s theorem to model the periodic refractive index and to decompose the field withrespect to the grating vector, respectively. A few years later, Peng solved different types ofgrating profiles (sinusoidal, rectangular, curvy) for multilayered grating and discussed severaltypes of matrix propagation methods for matching the fields at the interfaces [11].

Magnusson and Gaylord extended the coupled wave method introduced by Kogelnikto more general conditions for describing diffraction properties of thick gratings and provedthat the coupled wave method is equivalent to the modal method when including all diffrac-tion orders and retaining the second derivatives of the electromagnetic field [12, 13]. ThenGaylord together with Moharam presented a more rigorous coupled wave analysis, wherethere introduced the role of higher order modes and the second partial derivatives of theelectromagnetic field, yielding a better accuracy for the calculated diffraction efficiency [14].Since then, we speak of RCWA. Later on, they also extended the method to various pro-file shapes, such as trapezoidal gratings, and for arbitrary complex permittivity, such as forreal metals [15, 16]. Until this point, models included single-periodic gratings with classical

10

Introduction

mounting (incident plane of wave is perpendicular to grating vector). The general descriptionof a more general description of the problem – conical mounting – was given by Moharamand Gaylord in 1983 [17]. Double-periodic structure implementation was introduced byMoharam in 1988 [18].

Later, Li compared the RCWA method and the modal method [19]. The main differencebetween these two methods is that RCWA approximates the discontinuous grating profile asa continuous function of Fourier components, then matches the fields on interfaces betweenthe grating and the superstrate/substrate (Fig. 1). On the other hand, modal method triesto find the exact solution of the field inside the grating, therefore it uses boundary conditionsfor interfaces between grating walls, too. He pointed out that the convergence problem ofthe TM polarized case (vector of magnetic field is parallel to the grating grooves) was causedby the incorrect way of using Fourier expansion for the permittivity function.

Fig. 1: Comparison of RCWA with the modal method for lamellar grating (a). Thediscontinuous function of index of refraction is Fourier decomposed as a continuous functionfor RCWA (b). Then fields are matched at the boundaries of the superstrate and substrate(c). For the modal method, fields are matched at the boundaries inside the grating, too (d).Figure taken from [20].

This remark was taken as a stimulus by the people who implemented the RCWA method,and was followed by a series of publications that showed the improvements of the method. Forinstance, Chateau and Hugonin introduced a more numerically efficient matrix approachin the RCWA method that is applicable for a multilayered grating profile [21]. Moharamand his co-workers also took a step by formulating the RCWA method in a more stable andefficient way for both binary and arbitrary profile gratings [22].

In 1996, Lalanne and Morris reformulated the Fourier representation of the permit-tivity for the TM polarization, yielding a faster convergence [23]. Later that year, Li demon-strated in mathematical terms the reason why the proposed numerical implementations inthe previously mentioned paper improves the convergence problem [24]. He also made an ex-tension of the method by modeling anisotropic and double-periodic grating with symmetricmounting [25, 26]. Since then, RCWA is also known as Fourier modal method – FMM. Otherimprovements of the method not mentioned here can be found in Ref. [20].

For modeling a multilayered structure a matrix approach is needed to deal with matching

11

Introduction

the the fields at interfaces, after obtaining the solutions in each grating layer. There are twomain propagation matrix methods commonly used: transfer and scattering matrix method– see Fig. 2. Transfer matrix method is easier to implement, however, it has tendency toproduce numerical instabilities since it creates increasing exponential terms of evanescentwaves.

Fig. 2: Transfer matrix relates waves on the left side of the interface (r, o) with waves onthe right side (t, 0). Scattering matrix relates waves propagating to the right side (o, t) withwaves propagating to the left (r, 0).

Li compared the existing matrix methods and described the applicability of each [27].According to his paper, one alternative to overcome the numerical problems of transfer matrixis using scattering matrix, which is inherently stable due to the matrix inversion for allincreasing exponential terms. Another alternative for a more stable matrix propagationalgorithm is the R-matrix, which links the variables and its derivatives at the interface byusing trigonometric relations. The third alternative is to use the transfer matrix with arenormalization method, which eliminates the inversion of exponentially growing terms [28].

Fig. 3: Around the discontinuity jump (left, point X) the original coordinate system S isstretched (middle) resulting a new system S′. Red dashed curve denotes a transformation,which remaps the space around the discontinuity with a higher resolution in the new systemS′. Red continuous line denotes identity transformation S′ = S. From the original system Sthe new coordinate system S′ (blue web) looks deformed (right).

A few extensions of RCWA were made by means of coordinate transformation along thegrating direction. Granet introduced an adaptive spatial resolution such that more pointsare taken close to the discontinuity jump of the permittivity function (as seen in Fig. 3) [29].

12

Introduction

This technique improves the convergence rate and gives higher resolution in presence ofresonance peaks. The adaptive coordinate transformation was further improved by Valliusand Honkanen in 2002 [30].

Another extension was applied such that RCWA can be used to simulate non-periodicstructures, for instance waveguide and photonic band gap problems. This extension was fol-lowing the introduced technique of the perfectly matched layer (PML) boundary conditioninstead of periodic boundary condition [31]. Hugonin and Lalanne have applied a con-tinuous complex coordinate transformation to achieve absorbing boundaries and effectivelysimulate a non-periodic structure (see Fig. 4) [32].

/centre for analysis, scientific computing and applications

Motivation Model equations and standard RCWA RCWA for non-periodic structures Computations

RCWA for isolated structures

P. Lalanne, et al. (2000), RCWA for isolated structuresFig. 4: Non-periodic structure (top) can be simulated as a periodic structure with absorbingboundary condition (bottom). PML is the shaded area around the unit cell.

Schuster with his co-workers introduced a normal vector method for crossed grat-ings [33]. This method is based on the transformation of the components of the electro-magnetic field in a lateral plane (or slice) from global coordinates into local normal andtangential coordinates relative to the boundaries between two different materials – Fig. 5.This method is based on the work of Neviere and Popov. Later they extended the normalvector method for arbitrarily shaped gratings.

Fig. 5: If the structure is not lamellar, it has to be approximated by a staircase shape(left). Introducing local normal and tangential components staircasing is no more required,speeding up the computation and improving convergence.

Recently, RCWA got combined with other popular numerical methods to exploit theadvantages of them. One of them is the hybrid C-RCWA method, which combines theChandezon-method (C-method or coordinate transformation method [3]) with RCWA and

13

Introduction

was introduced by Bischoff in 2009 [34]. Unlike RCWA, which creates a multilayer systemin the same coordinate system, the original C-method introduces a new coordinate system,in which the interface is flat (Fig. 6). The hybrid method is therefore suitable, where thegrating surface is given by a continuous function, eliminating the slicing to multilayers.

Fig. 6: Comparison of RCWA and C-method. RCWA removes z dependency by introducinga multilayer structure. C-method transforms the coordinate system, where the interface isflat.

14

RCWA for double-periodic structures

Chapter 1


Maxwell’s equations

Since RCWA is a rigorous theory, it requires electromagnetic field to be treated as avector quantity. The whole analysis is based on Maxwell’s equations. The most general formof Maxwell’s equations is the macroscopic Maxwell’s equations (also known as Maxwell’sequations in matter):

∇ · ε̄Ê = ρ (1.1)

∇ · µ̄Ĥ = 0 (1.2)

∇× Ê = −∂µ̄Ĥ∂t

(1.3)

∇× Ĥ = ∂ε̄Ê∂t

+ J (1.4)

Here, ρ and J are the free charge density and the free current, respectively. Unlike the micro-scopic Maxwell’s equations, where total charge (= free charges + bound charges) and totalcurrent (= free current + bound current) is considered, macroscopic equations express electricand magnetic field at a larger scale than atomic, therefore bound charges and bound currenton atomic levels are included in material properties ε̄ and µ̄. All material characteristics (e.g.resistance, magnetic response) are included in permittivity tensor ε̄ and permeability tensorµ̄.

Since RCWA operates in the frequency domain, calculations are made for plane waveswith a single frequency

ω =k

√ε0µ0

(1.5)

being k = 2π/λ the wavenumber in vacuum, ε0 and µ0 the permittivity and permeability ofvacuum. Electric and magnetic field (Ê, Ĥ) are expressed as monochromatic time-harmonicfields

Ê(x, y, z, t) = E(x, y, z) cosωt = <{E(x, y, z)e−iωt

}(1.6)

Ĥ(x, y, z, t) = H(x, y, z) cosωt = <{H(x, y, z)e−iωt

}(1.7)

15


where E(x, y, z) and H(x, y, z) are so called complex amplitudes of electric and magnetic field.For the sake of simplicity, we will use the complex form of field vectors instead of its real part.In this analysis we only consider what is called linear optics, where the time-harmonic settingis relevant and there are no time-frequency conversions, so that the different wavelengths maybe treated independently of each other.

Henceforward, if not stated differently, ρ = 0 (no free charges in material) and J = 0 (nofree currents in material). Also, permittivity and permeability are scalars and are given inrespect of the permittivity and permeability of vacuum (ε̄ = εε0, µ̄ = µµ0). Grating made ofanisotropic material is discussed in a separate chapter. Plugging Eqs. 1.6–1.7 to Eq. 1.3–1.4we obtain

εε0E =i

ω∇×H (1.8)

µµ0H = −i

ω∇×E (1.9)

In the following we suppose that relative permittivity ε and relative permeability µ areperiodic space dependent variables. In this chapter they are periodic in the x and y direction,however, they are independent of z. This is an essential condition of RCWA. This implies thatx, y and z components of E and H have to be periodic in the x and y direction, in addition,this periodicity is conserved in the substrate and superstrate. If the periodic structure is notuniform in the z direction, it is sliced up and treated as a sandwich of lamellar gratings (seeFig. 11). Part of the analysis is Fourier decomposition of functions ε, µ, E and H also ofinverse functions 1/ε = ε̃ and 1/µ = µ̃.

Inverse rule

In general, coefficients from Fourier decomposition of an arbitrary periodic (with spatialperiod p) function f(x) are not the inverted coefficients from Fourier decomposition of itsinverse function 1/f(x). Mathematically, if

f(x) =∑n

Fnei 2πpnx (1.10)

and

1

f(x)= f̃(x) =

∑n

F̃nei 2πpnx (1.11)

then Fn 6= F̃−1n . Therefore it has to be carefully examined whether the correct form of Eq. 1.8is written as it is or in the form

ε0E =1

ε

i

ω∇×H (1.12)

because the two (Eq. 1.8 and Eq. 1.12) are not equivalent. The same problem concerns toEq. 1.9. This issue was thoroughly discussed in [35] and mathematically explained in [24]. Ifthe equations are not treated correctly, RCWA refuses to converge to an exact value whichis obtained when keeping a finite number of Fourier modes.

16


Let us assume that functions f(x) and g(x) are discontinuous with pairwise complemen-tary jump discontinuities. Fourier decomposition of continuous function h(x) = f(x)g(x)is not simply the product of Fourier decompositions of functions f(x) and g(x). Fouriercoefficients of function h(x) are derived as

h(x) =N∑

n=−NFne

i 2πpnx

N∑m=−N

Gmei 2πpmx (1.13)

h(x) =N∑

n=−N

N∑m=−N

FnGmei 2πp(n+m)x (1.14)

Substituting n+m→ u and n→ u−m we have

h(x) =

N∑u=−N

Huei 2πpux

=

N∑u=−N

N∑m=−N

Fu−mGmei 2πpux (1.15)

This equation holds for any x. Further manipulation leads to Hu expressed in vector form

Hu = Fu−mGm (1.16)

where Hu and Gm are vectors of Fourier coefficients of functions h(x) and g(x), respectively.Summation is assumed over dummy index m, index u is the free index.

Hu =

H−N

H−N+1...

HN

Gm =

G−N

G−N+1...

GN

(1.17)

Toeplitz matrix Fu−m is defined as

Fu−m =

F0 F−1 · · · F−2N+1 F−2N

F1 F0 · · · F−2N+2 F−2N+1...

.... . .

......

F2N−1 F2N−2 · · · F0 F−1

F2N F2N−1 · · · F1 F0

(1.18)

Eq. 1.16 is called the Laurent’s rule, which is valid only if N →∞. If the equation is truncated(keeping a finite number of orders), then this rule is not longer valid (see Fig. 7). Instead ofusing Laurent’s rule, reformulating equation h(x) = f(x)g(x) in form

h(x)1

f(x)= h(x)f̃(x) = g(x) (1.19)

we have discontinuous functions on both sides of the equation. Then deriving Fourier coeffi-cients Hu we obtain

Hu =(F̃)−1u−m

Gm (1.20)

17


Equation 1.20 is called the inverse rule, which is correct for any N . The correct form of h(x)is therefore

h(x) =(f̃(x)

)−1g(x) (1.21)

This way discontinuity is removed in function h(x) and its Fourier decomposition preservesuniform convergence for any x as it is shown in Fig. 7.

−0.5 0 0.50

0.5

1

1.5

2

2.5

3

3.5

one period

f(x)

inverted f(x)

−0.5 0 0.50.99

0.995

1

1.005

1.01

1.015

1.02

1.025

one period

Laurent rule

Inverse rule

Fig. 7: Fourier expansion is plotted for step function f(x) between values 3 and 2 and itsinverse function f−1 (left). Their product ff−1 is obtained using both Laurent rule andinverse rule (right) for number of modes N = 50. Inverse rule gives the correct value of 1.

The same problem arises in Eq. 1.8 and Eq. 1.9, where on the left side of the equation isa product of two discontinuous functions (εE, µH). To have a continuous terms on the rightsides of the equations the correct forms are expressed as

iωε0ε̃−1Ex =

∂Hy∂z− ∂Hz

∂y(1.22)

iωε0ε̃−1Ey =

∂Hz∂x− ∂Hx

∂z(1.23)

iωε0εEz =∂Hx∂y− ∂Hy

∂x(1.24)

iωµ0µ̃−1Hx =

∂Ez∂y− ∂Ey

∂z(1.25)

iωµ0µ̃−1Hy =

∂Ex∂z− ∂Ez

∂x(1.26)

iωµ0µHz =∂Ey∂x− ∂Ex

∂y(1.27)

In Eqs. 1.22, 1.23, 1.25 and 1.26 the inverse rule is applied, because the x and the y com-ponents of electric and magnetic field are perpendicular to the interface inside the grating,thus they are discontinuous. For these equations the right side of the equation is always

18


continuous (Fig. 8 left). As for Eqs. 1.24 and 1.27, the z component of E and H has no jumpdiscontinuity because it is parallel to the grating wall inside the layer. Therefore it has nodiscontinuity (Fig. 8 right) and functions ε and µ are not inverted here. The right side ofthese two equations is always discontinuous.

Fig. 8: When the x component of the electric (magnetic) field is discontinuous then the ycomponent of the magnetic (electric) field is continuous and vice versa. The z component offields is always continuous because it is parallel to the grating wall.

o r

t

I G II

z

xy

εεrr(x, y) = ε(x, y) = εrr(x+p(x+pxx, y+p, y+pyy) ) μμrr(x, y) = μ(x, y) = μrr(x+p(x+pxx, y+p, y+pyy))

εε22 μ μ22

εε11 μ μ11

Fig. 9: Incident wave (o) is reflected (r) in the superstrate (I) and transmitted (t) into thesubstrate (II) by the periodic structure (G). Material properties (ε and µ) are periodic in thex and y direction with periods px and py, respectively.

As mentioned before, the basic principle of RCWA is the Fourier decomposition of periodicfunctions and separation of space variables x, y and z. Functions ε, ε̃, µ, µ̃, Ex, Ey, Ez, Hx,Hy and Hz are periodic with spatial period px in the x direction and spatial period py in they direction, respectively (Fig. 9). These quantities are Fourier decomposed as follows:

Ex (x, y, z) =∑s,t

Es,tx (z) eiksxxeik

tyy (1.28)

Ey (x, y, z) =∑s,t

Es,ty (z) eiksxxeik

tyy (1.29)

Ez (x, y, z) =∑s,t

Es,tz (z) eiksxxeik

tyy (1.30)

19


Hx (x, y, z) =∑s,t

Hs,tx (z) eiksxxeik

tyy (1.31)

Hy (x, y, z) =∑s,t

Hs,ty (z) eiksxxeik

tyy (1.32)

Hz (x, y, z) =∑s,t

Hs,tz (z) eiksxxeik

tyy (1.33)

Indices s and t run from −N → −∞ to N → ∞. The total number of diffracted orders is2(N + 1)2, at this point it is infinite. Number N is called number of modes. Eqs. 1.28–1.32are valid in the grating region (G), superstrate (I) and substrate region (II) with differentE(z) and H(z) (Fig. 9). Here, kx and ky are components of the wavevector k = (kx, ky, kz)corresponding to each diffracted order given by the Floquet-Bloch theorem

ksx = k√ε1µ1 sinϕ cos θ +

2π

pxs (1.34)

kty = k√ε1µ1 sinϕ sin θ +

2π

pyt (1.35)

As we see, the grating periodicity modulates the initial wavevector components in the x andy direction. Tangential components kx and ky are conserved and are the same in each region.Angle ϕ is the angle of incidence, denotes the angle between incident k and the z-axis (seeFig. 10). Angle θ is the angle of plane of incidence (also known as conical angle or mountangle) between the plane of incidence and the x-axis. If ϕ = 0 then θ = 0 by definition.

E

kφ

ψ

θ

z

x

y

zz

Fig. 10: Vector k and the z axis define plane of incidence (green). Angle ψ is the polarizationangle. For TE polarized wave ψ = 0◦. For TM polarized wave E lies in plane of incidenceand ψ = 90◦.

As for the permittivity and permeability functions, they are decomposed in the gratingregion (G) as

ε =∑m,n

εm,nei 2πpxxm

ei 2πpyyn (1.36)

µ =∑m,n

µm,nei 2πpxxm

ei 2πpyyn (1.37)

20


ε̃ =∑m,n

ε̃m,nei 2πpxxm

ei 2πpyyn (1.38)

µ̃ =∑m,n

µ̃m,nei 2πpxxm

ei 2πpyyn (1.39)

Indices m and n run from −N → −∞ to N →∞. The superstrate (I) and substrate (II) arehomogeneous, non-periodic, thus ε = ε̃ = ε1 and µ = µ̃ = µ1 for the superstrate, ε = ε̃ = ε2and µ = µ̃ = µ2 for the substrate.

Plugging Eqs. 1.28–1.39 into Eqs. 1.22–1.27 we aim to derive a system of second orderdifferential equations for x and y components of fields E and H in form

∂2

∂z2

Ex

Ey

Hx

Hy

= −C

Ex

Ey

Hx

Hy

(1.40)

Here, matrixC couples all the diffracted orders (including evanescent and propagating orders)of fieldsE andH. This is the reason why this analysis is called “coupled”. MatrixC is differentfor each region (I, II and G), Eq. 1.40 is solved separately for each region. To obtain the formof Eq. 1.40, field amplitudes Es,tx , Es,ty , Hs,tx and Hs,ty are rearranged and expressed in vectorforms Ex, Ey, Hx and Hy as follows:

Es,tx = Ex =

E−N,−Nx

E−N,−N+1x...

E−N,N−1x

E−N,Nx

E−N+1,−Nx...

E−N+1,Nx...

EN,−Nx...

EN,Nx

(1.41)

Here, limit N →∞ is considered. Vectors Ey, Hx and Hy are similarly defined, they consistof (2N + 1)2 elements, therefore size of matrix C is 4(2N + 1)2 × 4(2N + 1)2.

21


Reformulation of Eqs. 1.22–1.27 to their vector forms is not straightforward. In thefollowing we will derive it for Eq. 1.22. For the rest of the equations (1.23–1.27) the derivationis analogous. Plugging Eqs. 1.32, 1.28 and 1.38 into 1.22 we obtain

iωε0∑s,t

Es,tx eiksxxeik

tyy =

=∑m,n

ε̃m,nei 2πpxxm

ei 2πpyyn

(∂

∂z

∑s,t

Hs,ty eiksxxeik

tyy − ∂

∂y

∑s,t

Hs,tz eiksxxeik

tyy

)(1.42)

Further manipulation leads to

iωε0∑s,t

Es,tx eiksxxeik

tyy =

=∑m,n,s,t

ε̃m,n∂

∂zHs,ty e

iks+mx xeikt+ny y −

∑m,n,s,t

ε̃m,nHs,tz k

tye

iks+mx xeikt+ny y (1.43)

here we used that

ksx +2π

pxm = ks+mx (1.44)

kty +2π

pyn = kt+nx (1.45)

which follows from Eqs. 1.34–1.35. To obtain Eq. 1.43 in vector form, we substitute on theright side s+m→ u, so m→ u− s. Similarly, t+ n→ v, so n→ v − t. On the left side wesubstitute s→ u and t→ v, respectively. Then we express Eq. 1.43 as

iωε0∑u,v

Eu,vx eikuxxeik

vyy =

∑u,v,s,t

ε̃u−sv−t

∂

∂zHs,ty e

ikuxxeikvyy −

∑u,v,s,t

ε̃u−sv−t

Hs,tz ktye

ikuxxeikvyy (1.46)

Here, indices u, v, s, t runs independently from −N → −∞ to N → ∞. Now the equationcan be simplified, since it is a sum over the same indices u and v, coefficients of eikuxxeik

vyy are

equal. Terms of sum Eq. 1.46 now can be formulated as elements of vectors using Einstein’ssummation notation as

iωε0Eu,vx = ε̃u−s

v−t

∂Hs,ty∂z

− ε̃u−sv−t

ktyHs,tz (1.47)

Here, vector Eu,vx is the same as defined in Eq. 1.41, vectors Hs,ty = Hy and Hs,tz = Hz are

defined similarly. Free indices are u and v, dummy indices are s and t. Matrix ε̃u−sv−t

= ε̃ is a

Toeplitz matrix with size (2N + 1)2 × (2N + 1)2 containing rearranged terms εi,j from sumEq. 1.38 and it is defined as

ε̃ = ε̃u−sv−t

=

ε̃i=0 ε̃i=−1 · · · ε̃i=−2N+1 ε̃i=−2N

ε̃i=1 ε̃i=0 · · · ε̃i=−2N+2 ε̃i=−2N+1...

.... . .

......

ε̃i=2N−1 ε̃i=2N−2 · · · ε̃i=0 ε̃i=−1

ε̃i=2N ε̃i=2N−1 · · · ε̃i=1 ε̃i=0

(1.48)

22


where

ε̃i =

ε̃i,0 ε̃i,−1 · · · ε̃i,−2N+1 ε̃i,−2N

ε̃i,1 ε̃i,0 · · · ε̃i,−2N+2 ε̃i,−2N+1...

.... . .

......

ε̃i,2N−1 ε̃i,2N−2 · · · ε̃i,0 ε̃i,−1

ε̃i,2N ε̃i,2N−1 · · · ε̃i,1 ε̃i,0

(1.49)

Matrix ky is a diagonal square matrix with elements

kty = k√ε1µ1 sinϕ sin θ +

2π

pyt (1.50)

on its diagonal. Sequence t is given by the second index of Es,tx in Eq. 1.41 as

t = (−N,−N + 1 . . . , N − 1, N,︸︷︷︸2N+1 elements

. . . ,−N,−N + 1 . . . , N − 1, N︸︷︷︸2N+1 elements︸︷︷︸

2N+1 elements

) (1.51)

Now, Eq. 1.22 is finally rewritten in vector form

∂

∂zHy = ikyHz + iωε0ε̃

−1Ex (1.52)

Equations 1.23–1.27 are similarly derived and expressed as

∂

∂zHx = ikxHz − iωε0ε̃−1Ey (1.53)

Ez =1

ωε0ε−1 (kyHx − kxHy) (1.54)

∂

∂zEy = ikyEz − iωµ0µ̃−1Hx (1.55)

∂

∂zEx = ikxEz + iωµ0µ̃

−1Hy (1.56)

Hz =1

ωµ0µ−1 (kxEy − kyEx) (1.57)

Vectors Ey, Ez, Hx, Hy and Hz are defined analogously as vector Ex in Eq. 1.41. Matricesε, µ and µ̃ are defined analogously as matrix ε̃ in Eqs. 1.48–1.49. Matrix kx is given asdiagonal square matrix with elements


2π

pxs (1.58)

on its diagonal, where sequence s is defined by the first index of Es,tx in Eq. 1.41:

s = (−N, . . . ,−N,︸︷︷︸2N+1 elements

−N + 1, . . . ,−N + 1,︸︷︷︸2N+1 elements

. . . , N, . . . , N︸︷︷︸2N+1 elements︸︷︷︸

2N+1 elements

) (1.59)

23


Both kx and ky have (2N + 1)2 elements on its diagonal.Using Eqs. 1.52–1.57 a system of second order differential equations can be constructed

as expressed in Eq. 1.40. In the following sections this system is solved as an eigenproblemindependently in each region: superstrate (I), grating region (G) and substrate (II).

Grating region

For the grating region, the system of differential equations defined by Eq. 1.40 can beseparated for the electric and the magnetic field (E and H) since matrix

C =

Ce 00 Ch

(1.60)does not couple them. The eigenproblem is expressed for the electric field as

∂2

∂z2

EGxEGy

= −Ce EGx

EGy

(1.61)where

Ce = k2µ̃−1ε̃−1 − µ̃−1kyµ−1ky − kxε−1kxε̃−1 µ̃−1kyµ−1kx − kxε−1kyε̃−1µ̃−1kxµ

−1ky − kyε−1kxε̃−1 k2µ̃−1ε̃−1 − µ̃−1kxµ−1kx − kyε−1kyε̃−1

(1.62)

The same eigenproblem is derived for magnetic field

∂2

∂z2

HGxHGy

= −Ch HGx

HGy

(1.63)where

Ch = k2ε̃−1µ̃−1 − ε̃−1kyε−1ky − kxµ−1kxµ̃−1 ε̃−1kyε−1kx − kxµ−1kyµ̃−1ε̃−1kxε

−1ky − kyµ−1kxµ̃−1 k2ε̃−1µ̃−1 − ε̃−1kxε−1kx − kyµ−1kyµ̃−1

(1.64)

There are two conjugate solutions of Eqs. 1.61 and 1.63, which are

EG(z) =

EGxEGy

= Qee−i√Lezve1 + Qeei√Lezve2 (1.65)

24


HG(z) =

HGxHGy

= Qhe−i√Lhzvh1 + Qhei√Lhzvh2 (1.66)Le and Lh are diagonal matrices containing the eigenvalues1 of Ce and Ch on the diagonal.Qe and Qh are matrices whose columns are the corresponding eigenvectors of Ce and Ch,respectively. It holds that QL = CQ. The size of Ce and Ch is 2(2N + 1)2 × 2(2N + 1)2.Vectors ve1, ve2, vh1 and vh2 are unknown integrating constants. The term with v1 stands forthe wave which propagates in the opposite direction of z (up), the term with v2 stands for thewave which propagates in the z direction (down) inside the grating. Equations 1.65 and 1.66are solutions for the z dependent amplitudes of total fields of Eqs. 1.28–1.29 and 1.31–1.32in the grating region.

Superstrate region

In the superstrate (above the grating) the permittivity and permeability are constant,therefore ε̃ = ε = ε1 and µ̃ = µ = µ1. This simplifies the coupling matrix C and theeigenproblem is obtained as

∂2

∂z2

EIxEIy

= ε1µ1k2 − k2y − k2x 0

0 ε1µ1k2 − k2y − k2x

EIx

EIy

(1.67)for the electric field and

∂2

∂z2

HIxHIy

= ε1µ1k2 − k2y − k2x 0

0 ε1µ1k2 − k2y − k2x

HIx

HIy

(1.68)for the magnetic field. Matrix k2 is a diagonal matrix with elements k2 on its diagonal. Thereis only one solution with physical meaning for Eqs. 1.67 and 1.68, since there are no incomingwave orders except the incident wave in the zeroth order. Therefore the second (conjugate)solution is excluded. The only solution is expressed as

EI(z) =

EIxEIy

= e−ikIzz rex

rey

(1.69)and

HI(z) =

HIxHIy

= e−ikIzz rhx

rhy

(1.70)

1Elements of Le and Lh are the same but they might be in different order. Important is that they mustcorrespond to eigenvectors Qe and Qh.

25


where diagonal matrix kIz is a diagonal matrix containing the square rooted eigenvalues ofthe coupling matrix

kIz =

√ε1µ1k2 − k2x − k2y 0

0√ε1µ1k2 − k2x − k2y

(1.71)The total field in the superstrate consists not only of the reflected wave but also of theincident wave as well in the zeroth order. Therefore the z dependent electric and magneticfield amplitudes of Eqs. 1.28–1.29 and 1.31–1.32 are given as

EI(z) = e−ikIzz

rexrey

+ eikIzz oex

oey

(1.72)and

HI(z) = e−ikIzz

rhxrhy

+ eikIzz ohx

ohy

(1.73)Unknown coefficients re and rh are yet to be determined. The incident planar wave propagatesin opposite direction as reflected wave. The amplitude of electric field of the incident waveequals to unity by convention (|(oex,oey,oez)| = 1). For the x and the y components of theincident electric amplitude we obtain

oex =

...

0

sinψ cos θ cosϕ− cosψ sin θ

0

...

(1.74)

oey =

...

0

sinψ sin θ cosϕ+ cosψ cos θ

0

...

(1.75)

Since the incident wave propagates in the zeroth order, it has zero amplitudes apart fromthe zeroth order terms (in the middle of the vectors). Angle ψ denotes the polarizationangle between the incident electric field vector and the xy plane (or between the incidentmagnetic field vector and the plane of incidence). For normal incidence ψ denotes anglebetween incident electric field vector and the y-axis. For ψ = 90◦ wave is TM polarized, forψ = 0◦ wave is TE polarized (Fig. 10).

26


Substrate region

In the substrate (below the grating) permittivity and permeability is constant, thereforeε̃ = ε = ε2 and µ̃ = µ = µ2. This leads to the coupling matrix C and the eigenproblem isexpressed for the electric field as

∂2

∂z2

EIIxEIIy

= ε2µ2k2 − k2y − k2x 0

0 ε2µ2k2 − k2y − k2x

EIIx

EIIy

(1.76)and for the magnetic field as

∂2

∂z2

HIIxHIIy

= ε2µ2k2 − k2y − k2x 0

0 ε2µ2k2 − k2y − k2x

HIIx

HIIy

(1.77)Matrix k2 is a diagonal matrix with elements k2 on its diagonal. Solutions for Eqs. 1.76 and1.77 are

EII(z) =

EIIxEIIy

= eikIIz z tex

tey

(1.78)and

HII(z) =

HIIxHIIy

= eikIIz z thx

thy

(1.79)where diagonal matrix kIIz is given as

kIIz =

√ε2µ2k2 − k2x − k2y 0

0√ε2µ2k2 − k2x − k2y

(1.80)Since in the substrate transmitted wave propagates away from the grating and there is noincoming wave there, only one solution has physical meaning. Therefore the second (con-jugate) solution is omitted. Vectors te and th are unknown integrating constants yet to bedetermined.

Boundary conditions

Our goal is to find field amplitudes of diffracted orders in the superstrate (reflected waves)and in the substrate (transmitted waves). Using boundary conditions – tangential compo-nents (here x and y components) of E and H are continuous across the interfaces superstrate-grating and grating-substrate – a system of linear equations is constructed, then solved forelectric field amplitudes

re =

rexrey

(1.81)27


te =

textey

(1.82)Let us summarize the z dependent amplitudes of fields in each region (I, G and II):

EI (z) = eikIzzoe + e−ik

Izzre (1.83)

HI (z) = eikIzzoh + e−ik

Izzrh (1.84)

EG (z) = Qee−i√Lezve1 + Q

eei√Lezve2 (1.85)

HG (z) = Qhe−i√Lhzvh1 + Q

hei√Lhzvh2 (1.86)

EII (z) = eikIIz zte (1.87)

HII (z) = eikIIz zth (1.88)

We used substitutions above (besides Eqs. 1.81–1.82)

rh =

rhxrhy

(1.89)

th =

thxthy

(1.90)

oh =

ohxohy

(1.91)

oe =

oexoey

(1.92)Boundary conditions hold for any x and y on interface between the superstrate and the

grating region at z = 0, also between the grating region and the substrate at z = d, whered is the thickness of the grating. Having a look at Eqs. 1.28–1.33 it can be concluded, thatequations are reducible by exponential terms eikuxxeik

vyy, then boundary conditions

EI(z = 0) = EG(z = 0) (1.93)

HI(z = 0) = HG(z = 0) (1.94)

EG(z = d) = EII(z = d) (1.95)

HG(z = d) = HII(z = d) (1.96)

28


are applied to z dependent amplitudes (Eqs. 1.83–1.88) leading to four algebraic vector equa-tions

oe + re = Qe(ve1 + ve2) (1.97)

oh + rh = Qh(vh1 + vh2 ) (1.98)

Qe(

e−i√Ledve1 + e

i√Ledve2

)= eik

IIz dte (1.99)

Qh(

e−i√Lhdvh1 + e

i√Lhdvh2

)= eik

IIz dth (1.100)

Although the above four equations have eight unknown amplitudes (re, rh, ve1, vh1 , ve2, vh2 ,te and th), magnetic field variables (indexed h) will be expressed in respect of electric fieldvariables (indexed e)

e−ikIzzrh = Rre

−ikIzzre (1.101)

eikIzzoh = Roe

ikIzzoe (1.102)

e−i√Lhzvh1 = Rv1e

−i√Lezve1 (1.103)

ei√Lhzvh2 = Rv2e

i√Lezve2 (1.104)

eikIIz zth = Rte

ikIIz zte (1.105)

Using Eqs. 1.52, 1.53 and 1.57 we express the x and the y components of magnetic field givenby x and y components of electric field:

∂

∂zHx =

i

ωµ0kxµ

−1 (kxEy − kyEx)− iωε0ε̃−1Ey (1.106)

∂

∂zHy =

i

ωµ0kyµ

−1 (kxEy − kyEx) + iωε0ε̃−1Ex (1.107)

Substituting Eqs. 1.83–1.88 to these two equations, block diagonal matrices R with size2(2N + 1)2 × 2(2N + 1)2 are derived:

Rr =ε0ω

µ1k2(kIz)−1 kxky ε1µ1k2 − k2x

k2y − ε1µ1k2 −kxky

(1.108)

Ro = −ε0ω

µ1k2(kIz)−1 kxky ε1µ1k2 − k2x


(1.109)

Rv1 =ε0ω

k2

(Qh√Lh)−1 k2ε̃−1Qey − kxµ−1kxQey + kyµ−1kxQex

kyµ−1kyQ

ex − kxµ−1kyQey − k2ε̃−1Qex

(1.110)Rv2 = −Rv1 (1.111)

29


Rt = −ε0ω

µ2k2(kIIz)−1 kxky ε2µ2k2 − k2x


(1.112)Here Qx and Qy are taken from matrix Q

Qe =

QexQey

(1.113)

Qh =

QhxQhy

(1.114)Size of Qx and Qy is (2N + 1)2 × 2(2N + 1)2.

Transfer matrix

Substituting Eqs. 1.108–1.112 to boundary equations 1.97–1.100, we have four equationswith four unknown vectors re, ve1, ve2 and te. Our goal is to have solutions for re and te.There are different ways to achieve this. We use transfer matrix formalism to solve thisalgebraic problem [36]. Transfer matrix relates the waves on one side of the interface withthe waves on the other side of the interface (Fig. 2). The main advantage of this approachis the easy application to multilayered systems (explained later in this chapter). A transfermatrix can be constructed for each interface, firstly for the interface between the superstrateand the grating layer using Eqs. 1.97 and 1.98 1 1

Rr Ro

re

oe

= Qe Qe

QhRv1 QhRv2

ve1

ve2

(1.115)then for the interface between grating layer and substrate using Eqs. 1.99 and 1.100 we get Qee−i

√Led Qeei

√Led

QhRv1e−i√Led QhRv2e

i√Led

ve1

ve2

= 0 eikIIz d

0 RteikIIz d

0

te

(1.116)Combining these two equations and excluding variables ve1 and ve2, variables re and te arerelated (including incident wave amplitude oe) by the transfer matrix M re

oe

= M 0

te

= M11 M12

M21 M22

0

te

(1.117)The transfer matrix is derived considering Eqs. 1.115 and 1.116 as

M =

1 1Rr Ro

−1

P

0 eikIIz d0 Rte

ikIIz d

(1.118)30


where matrix

P =

Qe QeQhRv1 Q

hRv2

ei

√Led 0

0 e−i√Led

Qe Qe

QhRv1 QhRv2

−1

(1.119)

characterizes the grating region. Finally, unknown variables are expressed from Eq. 1.117:

re = M12M−122 o

e (1.120)

te = M−122 oe (1.121)

Renormalization

In Eqs. 1.120 and 1.121 there is an inversion of matrixM22. Since square root of eigenvaluevector (

√Le) may contain imaginary numbers in case of evanescent waves, exponential terms

e−i√Led and ei

√Led in Eq. 1.119 may reach very large values. Inversion of a matrix with

such a large values are numerically problematic. Therefore a renormalization method is usedaccording Ref. [28] to deal with the inversion of M22. First of all, exponential terms in matrix ei

√Led 0

0 e−i√Led

are rearranged such way, that terms with large values are swappedto e−i

√Led, terms with small values are swapped to exponential term ei

√Led. It is always

possible since e−i√Led and ei

√Led form a conjugate pair, their product is e−i

√Ledei

√Led =

1. Therefore one of the terms (its absolute value) is greater than 1, the second one issmaller. If |e−i

√Led| = |ei

√Led| = 1, then the swap is not necessary. When swapping elements

between exponential matrices e−i√Led and ei

√Led, swapping the corresponding column in

matrix

Qe QeQhRv1 Q

hRv2

−1

is also required. Now M22 is expressed as

M22 =

(A1 A2

) e−i√Led 0

0 ei√Led

B1

B2

(1.122)where

(A1 A2

)=

(0 1

) 1 1Rr Ro

−1 Qe Qe

QhRv1 QhRv2

(1.123)and B1

B2

= Qe Qe

QhRv1 QhRv2

−1 0 eikIIz d

0 RteikIIz d

0

1

(1.124)

31


The simple trick of the renormalization is that exponential term e−i√Led with the large values

is extracted in Eq. 1.122

M22 =

(A1 A2

) 1ei√LedB2B

−11 e

i√Led

e−i√LedB1 = Xe−i√LdB1 (1.125)where

X = A1 + A2ei√LedB2B

−11 e

i√Led (1.126)

Finally, the inversion of M22 is simply obtained as

M−122 = B−11 e

i√LedX−1 (1.127)

Matrices B1 and X do not contain large values. In such a way no inversion is made withmatrix containing large values. Matrix M12 can be derived similarly, the only difference willbe that

(A1 A2

)=

(1 0

) 1 1Rr Ro

−1 Qe Qe

QhRv1 QhRv2

(1.128)

Multilayered structure

The main advantage of the transfer matrix formalism manifests itself when applied toa multilayered structure. Structures with z dependent permittivity/permeability are alsotreated as multilayered structure with z independent sublayers (Fig. 11).

Fig. 11: If the structure is not uniform in the z direction, it is sliced up in order toget a sandwich of z independent lamellar gratings. More sublayers mean more accurateapproximation.

Consider a multilayered structure (Fig. 12) with number of layers w. There are w + 1interfaces. Boundary conditions are applied for each interface which lead to a system of w+1equations: 1 1

Rr Ro

re

oe

= Qe Qe

QhRv1 QhRv2

1

ve1ve2

1

(1.129)

32


... Qee−i√Led Qeei

√Led


i√Led

j

ve1ve2

j

=

=

Qee−i√Led Qeei

√Led


i√Led

j+1

ve1ve2

j+1

(1.130)

... Qee−i√Led Qeei

√Led


i√Led

w

ve1ve2

w

=

0 eikIIz d0 Rte

ikIIz d

0

te

(1.131)Index j represents the j-th layer. Using Eqs. 1.129–1.131 transfer matrix can be constructedsimilarly to Eq. 1.118. The only difference will be that matrix P (see Eq. 1.119) is given bythe product of matrices Pj characterizing each sublayer

P =

w∏j=1

Pj (1.132)

where

Pj =

Qe QeQhRv1 Q

hRv2

j

ei√Led 0

0 e−i√Led

j

Qe QeQhRv1 Q

hRv2

−1

j

(1.133)

Please note that here d is the thickness of the j-th sublayer, also Q, L and R might bedifferent for each sublayer. Renormalization introduced in the previous section can be appliedfor multilayered structures, too.

Interface no. 1 2 3 w w + 1

Layer no. I G1 G2

...Gw II

rto ...

GjGj-1

jd1 d2

dj – 1

zZ = 0

dwdj

Fig. 12: Multilayered system with many interfaces.

33


Diffraction efficiency

At this point we have expressed re and te (Eqs. 1.120 and 1.121). Using these parametersdiffraction efficiency can be derived, which is generally defined as

η =I cosϕ′

I0 cosϕ(1.134)

where I is the intensity of the diffracted wave, ϕ′ is the angle under which the diffracted waveis propagating, I0 is the intensity of the incident wave, ϕ is the angle of incidence. Diffractionefficiency is not defined for evanescent waves, only for non-evanescent (propagating) waves.

The intensity of electromagnetic field is given by the magnitude of time-averaged Poyntingvector

I = |〈Ê× Ĥ〉| (1.135)

According to Ref. [37] if a calculation involves the product of field vectors, like Poyntingvector, one must use the real form of field vectors instead of the complex representation.Therefore we substitute the real electric and magnetic field vectors

Ê = E0 cos (k · r + ωt) (1.136)

Ĥ = H0 cos (k · r + ωt) (1.137)

into Eq. 1.135, then we calculate the temporal average over the time T = 2π/ω. Here k · ris the phase of the complex amplitude for both the electric and magnetic field vector, sincethey are always in phase speaking of a plane wave.

I =1

T

∫ T0|E0 ×H0| cos2 (k · r + ωt) dt (1.138)

Amplitudes E0 and H0 are perpendicular to each other, so we obtain for the intensity

I =1

2|E0||H0| (1.139)

Since we have solution for re and te, which denote amplitudes of electric field, it is necessaryto express intensity through electric field amplitude only (E0). We learned that complexamplitude of electric and magnetic field in homogeneous medium yields to solutions

E = E0eik·r (1.140)

H = H0eik·r (1.141)

being E0 = (E0x, E0y, E0z) and H0 = (H0x, H0y, H0z) the magnitudes of electric and mag-netic field, k = (kx, ky, kz) the wavevector, its modulus is |k| =

√εµk and r = (x, y, z).

Inserting these solutions to the 3rd or 4th Maxwell equation, one obtains the relationshipbetween |E0| and |H0|

|H0| = |E0|√εε0µµ0

(1.142)

34


At this point we can express intensity in general form

I =1

2

√εε0µµ0|E0|2 (1.143)

Diffraction angle ϕ′ is expressed as

cosϕ′ =kz√εµk

(1.144)

Recalling that in the superstrate E0 = (rex, rey, rez) and in the substrate E0 = (tex, tey, tez),respectively, inserting Eqs. 1.143, 1.144 to 1.134 one obtains the diffraction efficiency forreflected orders (also known as reflectance) in the superstrate as

ηr =kIz√

ε1µ1k cosϕ

(|rex|2 + |rey|2 + |rez|2

)(1.145)

and diffraction efficiency for transmitted orders (also known as transmittance) in the substrateas

ηt =

√µ1k

IIz√

ε1µ2k cosϕ

(|tex|2 + |tey|2 + |tez|2

)(1.146)

Here, kIz =√ε1µ1k2 − k2x − k2y and kIIz =

√ε2µ2k2 − k2x − k2y, respectively. For the incident

wave amplitude we substituted |E0| = 1. Only real kz components are considered. The xand y components of re and te are expressed in Eqs. 1.120 and 1.121. The z components arederived from the condition that k is perpendicular to E0, therefore

(kx,ky,kz) ·(rex, r

ey, r

ez

)= 0 (1.147)

(kx,ky,kz) ·(tex, t

ey, t

ez

)= 0 (1.148)

yielding to

rz = −kxkIz

rex −kykIz

rey (1.149)

and

tz = −kxkIIz

tex −kykIIz

tey (1.150)

AbsorptanceDiffraction efficiency describes the proportion of the incident energy propagating in each

diffracted order. If the grating is not absorbing (relative permittivity and permeability arereal), the incident energy equals to reflected plus transmitted energy. This is the law ofenergy conversation. By definition, incident energy equals to unity. Therefore∑

s

(ηrs + η

ts

)= 1 (1.151)

35


It is important to note that evanescent waves do not transfer any energy. If the material isabsorbing (relative permittivity and/or permeability has imaginary part), the absorptance(or absorptivity) is expressed by equation

A = 1−∑s

(ηrs + η

ts

)(1.152)

TruncationNow we have diffraction efficiencies derived for diffracted orders. The last step of the

analysis is the truncation of equations. Thus far, our equations contained infinite numberof modes N . Of course, for numerical computations, it is necessary to have a finite numberof equations. Therefore we choose N to be a finite number. The choice of N depends onthe actual physical problem. It has to be always carefully examined what is the appropriatenumber of modes. The higher this number is, the results more converge to the exact value, forthe cost of computation time and resource. Truncation is the only approximation of RCWA,besides this the method is rigorous.

Fourier decomposition of double-periodic structures

Until this point we derived RCWA in general form for any double-periodic structure.Fourier coefficients stand for any periodic structure in general given by Eqs. 1.36–1.39. Inthis section we will introduce analytic Fourier coefficients for the two most common structurepatterns, square pattern and hexagonal pattern.

Square patternConsider a double-periodic structure with a square pattern as pictured in Fig. 13. The

structure is periodic along the x direction with spatial period px, along the y direction withspatial period py. The size of the square pattern is qxpx × qypy, where q is the filling factorin the given direction. Reasonably, q is a number between 0 and 1. Material properties forthe square element are �2, µ2. Around the square the embedding material is characterizedby �1, µ1.

x

y

px

py

qxpx

qypyϵ2 μ2

ϵ1 μ1

Fig. 13: Double-periodic structure with a square pattern. Elementary cell is pictured onthe right.

36


The coefficients of the Fourier series

ε =∑m,n

εm,nei 2πpxxm

ei 2πpyyn (1.153)

are analitically expressed as

εm,n = (�2 − �1) qxqy sincmqx sincnqy + �1 sincm sincn (1.154)

Fourier coefficients for the inverse function

ε̃ =∑m,n

ε̃m,nei 2πpxxm

ei 2πpyyn (1.155)

are expressed as

ε̃m,n =

(1

�2− 1�1

)qxqy sincmqx sincnqy +

1

�1sincm sincn (1.156)

where function sincx is defined as

sincx ≡

sinπxπx if x 6= 0

1 if x = 0(1.157)

Relative permittivity coefficients µm,n and µ̃m,n are expressed analogously.

Hexagonal patternHexagonal pattern consists of two square patterns (see Fig. 14). Each of the square

patterns has its own filling factors, they are made of the same material. In addition we canchoose their mutual position dx and dy as seen in Fig. 14.

x

y

px

pyqxpxqypy

ϵ2 μ2

ϵ1 μ1

ϵ2 μ2

qypy

qxpx

dypy

dxpx1

1

2

2

Fig. 14: Double-periodic structure with a square pattern. Elementary cell is pictured onthe right.

Fourier coefficients of ε, ε̃ are

εm,n = (�2 − �1)(q1xq

1y sincmq

1x sincnq

1y + q

2xq

2y sincmq

2x sincnq

2ye

i2πdxmei2πdyn)

+

+ �1 sincm sincn (1.158)

37


ε̃m,n =

(1

�2− 1�1

)(q1xq

1y sincmq

1x sincnq

1y + q

2xq

2y sincmq

2x sincnq

2ye

i2πdxmei2πdyn)

+

+1

�1sincm sincn (1.159)

Truncation of orders causes approximation of the actual structure. The higher the numberof Fourier modes, the structure more resembles of the actual structure. Fig. 15 shows a Fourierdecomposed chessboard structure for N = 7 and N = 14 modes.

1

2

3

4

5

6

7

8

Fig. 15: Comparison of Fourier decomposed chessboard pattern for 7 (left) and 14 (right)modes. The actual structure has parameters �2 = 7 and �1 = 1.

Homogeneous structureIt is often required to model wave propagation through a homogeneous slab. For example,

in the case of a photonic crystal (Fig. 16) wave propagates through a sandwich of homogeneouslayers and lamellar gratings. RCWA is of course capable to simulate such a multilayeredstructure. For a homogeneous slab with � and µ the Fourier coefficients are zeros except thezeroth coefficient ε0,0 = �, µ0,0 = µ. This leads to a simple diagonal coupling matrix for thehomogeneous slab

Ce = Ch =

�µk2 − k2y − k2x 00 �µk2 − k2y − k2x

(1.160)Then the calculation is realized with transfer matrix formalism for multilayered structuresdescribed previously.

Fig. 16: From left to right: 1D, 2D and 3D photonic crystal with finite thickness d. Wavepropagates in the z direction, photonic crystals are infinitely extended in the xy plane.

38

RCWA for single-periodic structures with conical mounting

Chapter 2

RCWA for single-periodic structureswith conical mounting

In this chapter RCWA for single-periodic structures is introduced for arbitrary polar-ization and wavevector orientation (conical mounting). One may argue that RCWA fordouble-periodic structures also includes the analysis of single-periodic structures since asingle-periodic structure is a special case of a double-periodic structure. Although this istrue, the analysis for single-periodic structures can be greatly simplified by tailoring it specif-ically to structures with a single period. In the previous chapter we introduced matrices withsize 2(2N+1)2×2(2N+1)2. If we consider single-periodic structures, physical quantities areFourier decomposed in respect of only one spatial period, reducing the number of free indicesto one. This also reduces the number of Fourier coefficients to 2N + 1 instead of (2N + 1)2

leading to matrices with size 2(2N+1)×2(2N+1). This makes numerical calculations muchmore faster. The situation is even more simplified if the plane of incident wave is perpendic-ular to the grating grooves (classical mounting – the incident wave and the diffracted orderspropagate in the same plane) because TE and TM polarization can be separated. These twocases (conical and classical mounting) are discussed in separate chapters.

However, the analysis hereinafter is confined only to the differences between the doubleand single-periodic case.

Differences between 2D and 1D models

The first difference for the single-periodic case is in Maxwell’s equations (Eqs. 1.22–1.27).The structure is no longer periodic in the y direction (Fig. 17).

Therefore the inverse rule is not applied for Ey and Hy because they are not discontinuousquantities anymore:

iωε0ε̃−1Ex =

∂Hy∂z− ∂Hz

∂y(2.1)

iωε0εEy =∂Hz∂x− ∂Hx

∂z(2.2)

iωε0εEz =∂Hx∂y− ∂Hy

∂x(2.3)

39


Fig. 17: Single-periodic structure is periodic in the x direction only.

iωµ0µ̃−1Hx =

∂Ez∂y− ∂Ey

∂z(2.4)

iωµ0µHy =∂Ex∂z− ∂Ez

∂x(2.5)

iωµ0µHz =∂Ey∂x− ∂Ex

∂y(2.6)

The second difference is that physical quantities are Fourier decomposed only in onedirection: x. This leads to equations:

Ex (x, y, z) =∑s

Esx (z) eikyyeik

sxx (2.7)

Ey (x, y, z) =∑s

Esy (z) eikyyeik

sxx (2.8)

Ez (x, y, z) =∑s

Esz (z) eikyyeik

sxx (2.9)

Hx (x, y, z) =∑s

Hsx (z) eikyyeik

sxx (2.10)

Hy (x, y, z) =∑s

Hsy (z) eikyyeik

sxx (2.11)

Hz (x, y, z) =∑s

Hsz (z) eikyyeik

sxx (2.12)

The total number of diffracted orders is 2(N + 1) for each component.The third difference is, wavevector k is no longer modulated in the y direction. Under

this condition the Floquet-Bloch expressed as


2π

pxs (2.13)

ky = k√ε1µ1 sinϕ sin θ (2.14)

Here, ky is a scalar and corresponds to the y component of the incident wavevector k.

40


The last difference is the Fourier expansion of relative permittivity and permeabilityfunctions in the grating region, which are not anymore dependent on y. As seen in Fig. 17,the grating is uniformly extended in the y direction.

ε =∑m

εmei 2πpxxm (2.15)

µ =∑m

µmei 2πpxxm (2.16)

ε̃ =∑m

ε̃mei 2πpxxm (2.17)

µ̃ =∑m

µ̃mei 2πpxxm (2.18)

Solving the eigenproblems

Using the same approach we used in the previous chapter, Eqs. 2.7–2.18 are plugged intoEqs. 2.1–2.6, then a system of second order differential equations is constructed for x and ycomponents of fields E and H in form

∂2

∂z2

Ex

Ey

Hx

Hy

= −C

Ex

Ey

Hx

Hy

(2.19)

Here, vectors Ex, Ey, Hx and Hy contain rearranged terms of Esx, Esy, Hsx and Hsy as itselements. Example is given for Ex, others are defined alike.

Esx = Ex =

E−Nx

E−N+1x...

EN−1x

ENx

(2.20)

Ex has 2N + 1 components, size of matrix C is 4(2N + 1)× 4(2N + 1).Maxwell’s equations Eqs. 2.1–2.6 are now reformulated to their vector form. Equations

are expressed as

∂

∂zHy = ikyHz + iωε0ε̃

−1Ex (2.21)

∂

∂zHx = ikxHz − iωε0εEy (2.22)

41


Ez =1

ωε0ε−1 (kyHx − kxHy) (2.23)

∂

∂zEy = ikyEz − iωµ0µ̃−1Hx (2.24)

∂

∂zEx = ikxEz + iωµ0µHy (2.25)

Hz =1

ωµ0µ−1 (kxEy − kyEx) (2.26)

Toeplitz matrices ε, ε̃, µ and µ̃ are given by equation

ε =

ε0 ε−1 · · · ε−2N+1 ε−2N

ε1 ε0 · · · ε−2N+2 ε−2N+1...

.... . .

......

ε2N−1 ε2N−2 · · · ε0 ε−1

ε2N ε2N−1 · · · ε1 ε0

(2.27)

Its size is (2N + 1) × (2N + 1), its elements εi are taken from sum Eq. 2.15. Matrix kx isgiven as diagonal square matrix with elements


2π

pxs (2.28)

on its diagonal, where sequence s is defined as the index of Esx in Eq. 2.20

s = (−N,−N + 1,−N + 2, . . . ,−2,−1, 0, 1, 2, . . . , N − 2, N − 1, N︸︷︷︸2N+1 elements

) (2.29)

Using Eqs. 2.21–2.26 system of differential equations can be constructed as expressed inEq. 2.19. Inside the grating it obtains form

∂2

∂z2

EGxEGy

= −Ce EGx

EGy

(2.30)where

Ce =

k2µε̃−1 − k2y − kxε−1kxε̃−1 0kyµ̃

−1kxµ−1 − kyε−1kxε̃−1 k2µ̃−1ε− µ̃−1kxµ−1kx − k2y

(2.31)The same eigenproblem is derived for magnetic field

∂2

∂z2

HGxHGy

= −Ch HGx

HGy

(2.32)where

Ch =

k2εµ̃−1 − k2y − kxµ−1kxµ̃−1 0kyε̃−1kxε

−1 − kyµ−1kxµ̃−1 k2ε̃−1µ− ε̃−1kxε−1kx − k2y

(2.33)42


We used identities εε̃ = µµ̃ = 1. Matrix ky is a diagonal matrix with elementsky = k

√ε1µ1 sinϕ sin θ on its diagonal. Solutions of Eqs. 2.30 and 2.32 are

EG(z) =

EGxEGy

= Qee−i√Lezve1 + Qeei√Lezve2 (2.34)

HG(z) =

HGxHGy

= Qhe−i√Lhzvh1 + Qhei√Lhzvh2 (2.35)The size of Ce and Ch is 2(2N + 1) × 2(2N + 1). Vectors ve1, ve2, vh1 and vh2 are unknownintegrating constants.

In the superstrate region above the grating, where ε̃ = ε = ε1 and µ̃ = µ = µ1, eigen-problem is formulated as

∂2

∂z2

EIxEIy

= ε1µ1k2 − k2y − k2x 0

0 ε1µ1k2 − k2y − k2x

EIx

EIy

(2.36)and

∂2

∂z2

HIxHIy

= ε1µ1k2 − k2y − k2x 0

0 ε1µ1k2 − k2y − k2x

HIx

HIy

(2.37)The solutions for these problems are

EI(z) =

EIxEIy

= e−ikIzz rex

rey

(2.38)and

HI(z) =

HIxHIy

= e−ikIzz rhx

rhy

(2.39)where diagonal matrix kIz is a diagonal matrix containing the square rooted eigenvalues ofthe coupling matrix

kIz =

√ε1µ1k2 − k2x − k2y 0

0√ε1µ1k2 − k2x − k2y

(2.40)Including the incident wave, the total z dependent field amplitudes are given by equations

EI(z) = e−ikIzz

rexrey

+ eikIzz oex

oey

(2.41)

43


and

HI(z) = e−ikIzz

rhxrhy

+ eikIzz ohx

ohy

(2.42)Integrating constants are re and rh. Vector oe is already defined in the previous chapter.

Regarding the substrate, derivation is similar to the superstrate above. The only differenceis in kIIz , which is written as

kIIz =

√ε2µ2k2 − k2x − k2y 0

0√ε2µ2k2 − k2x − k2y

(2.43)This leads to z dependent field solutions

EII(z) = eikIIz z

textey

(2.44)and

HII(z) = eikIIz z

thxthy

(2.45)From this point the derivation is exactly the same as in the previous chapter until

Eqs. 1.106–1.107, which are modified to

∂

∂zHx =

i

ωµ0kxµ

−1 (kxEy − kyEx)− iωε0εEy (2.46)

∂

∂zHy =

i

ωµ0kyµ

−1 (kxEy − kyEx) + iωε0ε̃−1Ex (2.47)

which lead to relation matrices

Rr =ε0ω

µ1k2(kIz)−1 kykx ε1µ1k2 − k2x

k2y − ε1µ1k2 −kykx

(2.48)

Ro = −ε0ω

µ1k2(kIz)−1 kykx ε1µ1k2 − k2x


(2.49)

Rv1 =ε0ω

k2

(Qh√Lh)−1 k2εQey − kxµ−1kxQey + kyµ−1kxQex

k2yµ−1Qex − kykxµ−1Qey − k2ε̃−1Qex

(2.50)Rv2 = −Rv1 (2.51)

Rt = −ε0ω

µ2k2(kIIz)−1 kykx ε2µ2k2 − k2x


(2.52)44


The rest of the derivation is identical to the double-periodic structure analysis as far asEq. 1.151.

Fourier decomposition of single-periodic structures

Lamellar gratingLamellar (or binary) grating is the simplest single-periodic structure (Fig. 17). It consists

of uniformly extended rods in the y direction, periodically placed in the x direction withspatial period px. It has thickness d in the z direction. The grating material has fillingfactor q, relative permittivity and permeability �2 and µ2, respectively. Around the gratingthe surrounding material is characterized by �1 and µ1 (Fig. 18). Fourier coefficients ofEqs. 2.15–2.18 are given as

εm = (�2 − �1) q sincmq + �1 sincm (2.53)

and

ε̃m =

(1

�2− 1�1

)q sincmq +

1

�1sincm (2.54)

Fourier coefficients for relative permeability are expressed similarly.

−0.5 0 0.51.8

2

2.2

2.4

2.6

2.8

3

3.2

One period

Rel. p

erm

ittivity

−0.5 0 0.51.8

2

2.2

2.4

2.6

2.8

3

3.2

One period

Rel. p

erm

ittivity

Fig. 18: Function of relative permittivity of a lamellar grating is a discontinuous stepfunction (left). Its Fourier expansion (right) is a continuous function. Parameters are �2 = 3,�1 = 2, q = 0.5 and N = 50.

Surface relief gratingSurface relief grating is a grating with z dependent profile. For example in Fig. 11 there

is a grating with sinusoidal profile. RCWA is capable to simulate such a structure. Thisstructure is sliced up in the z direction and treated as a multilayered system of lamellargratings each having a different filling factor q. Of course, the more sublayers we have, themore accurate we mimic the original structure.

45


Gradient gratingWe speak of a gradient grating when the function of permittivity (or permeability) is

not given by step a function but an arbitrary continuous function (Fig. 19). In this case,the function is Fourier decomposed as well. Gradient grating with a harmonic (sinusoidal)function of permittivity is the easiest type of gradient grating, since its Fourier coefficientsare zeros except the 0th, 1st and −1st order.

−10 −5 0 5 101

1.5

2

2.5

3

3.5

4

x

Rel. p

erm

ittivity

Fig. 19: Harmonic gradient grating (left) has a sinusoidal relative permittivity function ofx (right).

46

RCWA for single-periodic structures with classical mounting

Chapter 3

RCWA for single-periodic structureswith classical mounting

In the previous chapter we presented RCWA applied for single-periodic structure witharbitrary incident wave orientation. This model can be further simplified by introducingclassical (also known as orthogonal) mounting. If the mounting is classical – wave propagatesin the xz plane (see Fig. 20) – TE and TM polarization can be separated and studiedindependently. Thereby the size of coupling matrix C is halved, making a single calculationup to eight times faster. On the whole, arbitrarily polarized problem is solved up to fourtimes faster compared to the previous analysis.

G

kφ

z

x

q*ppϵ2

ε1

ε2

dϵ1

o r

t

I

II μ2

μ1

μ2μ1

Fig. 20: Cross section of a lamellar grating in the xz plane.

Differences between conical and classical mounting

For classical mounting conical angle becomes zero (θ = 0). Consequently, ky = 0 meaningno physical quantity is dependent on y but x only. Fourier expansions of quantities areexpressed in form

Ex (x, z) =∑s

Esx (z) eiksxx (3.1)

Ey (x, z) =∑s

Esy (z) eiksxx (3.2)

Ez (x, z) =∑s

Esz (z) eiksxx (3.3)

47


Hx (x, z) =∑s

Hsx (z) eiksxx (3.4)

Hy (x, z) =∑s

Hsy (z) eiksxx (3.5)

Hz (x, z) =∑s

Hsz (z) eiksxx (3.6)

where kx is given by Floquet-Bloch relation

ksx = k√ε1µ1 sinϕ+

2π

pxs (3.7)

Maxwell’s equations from the previous chapter (Eqs. 2.21–2.26) are further simplified to:

∂

∂zHy = iωε0ε̃

−1Ex (3.8)

∂

∂zHx = ikxHz − iωε0εEy (3.9)

Ez = −1

ωε0ε−1kxHy (3.10)

∂

∂zEy = −iωµ0µ̃−1Hx (3.11)

∂

∂zEx = ikxEz + iωµ0µHy (3.12)

Hz =1

ωµ0µ−1kxEy (3.13)

Manipulating these equations and deriving eigenproblems one can immediately see that notonly the electric and magnetic field components are separable but the individual x and ycomponents are no longer related. This leads to four eigenproblems

∂2

∂z2Ex = −

(k2µε̃−1 − kxε−1kxε̃−1

)Ex (3.14)

∂2

∂z2Ey = −

(k2µ̃−1ε− µ̃−1kxµ−1kx

)Ey (3.15)

∂2

∂z2Hx = −

(k2εµ̃−1 − kxµ−1kxµ̃−1

)Hx (3.16)

∂2

∂z2Hy = −

(k2ε̃−1µ− ε̃−1kxε−1kx

)Hy (3.17)

Obviously, independence of x and y components means the conservation of polarization states.For example, if the incident wave is TE polarized, all the reflected and transmitted diffractionorders remain TE polarized. Of course the same applies to TM polarized incident wave.Therefore an arbitrarily polarized wave can be decomposed into TE and TM polarized wave(Fig. 21) and solved individually. The following paragraphs deal with them one by one.

48


Fig. 21: Arbitrarily polarized incident wave can be decomposed as a sum of TE and TMpolarized incident waves.

TE polarization

For classical mounting TE polarization occurs, when vector E = (Ex, Ey, Ez) is parallelto the y-axis (ψ = 0), therefore Ex = 0 and Hy = 0. Also, Ez = 0. Only Eq. 3.15 is solvedin each region. There is no need to solve eigenproblem for Hx because the fact that Ez = 0allows us to directly express Hx from Eq. 3.11 as

Hx =i

ωµ0µ̃∂

∂zEy (3.18)

In the grating region Eq. 3.15 has solution

EGy (z) = Qee−i

√Lezve1 + Q

eei√Lezve2 (3.19)

Where Le is the vector of eigenvalues, Qe is the matrix of corresponding eigenvectors ofcoupling matrix

Ce = k2µ̃−1ε− µ̃−1kxµ−1kx (3.20)

Matrix Ce is sized (2N + 1)× (2N + 1). In the superstrate Eq. 3.15 obtains form

∂2

∂z2EIy = −

(ε1µ1k

2 − k2x)EIy (3.21)

The solution for the eigenproblem above including the incident wave is

EIy(z) = e−ikIzzrey + e

ikIzzoey (3.22)

Where

kIz =√ε1µ1k2 − k2x (3.23)

and

oey =

...

0

1

0

...

(3.24)

49


because of TE polarization oex = oez = 0. In the substrate region the eigenproblem is expressedby equation

∂2

∂z2EIIy = −

(ε2µ2k

2 − k2x)EIIy (3.25)

Its solution is

EIIy (z) = eikIIz ztey (3.26)

where

kIIz =√ε2µ2k2 − k2x (3.27)

Magnetic field vectorHx is derived from Eq. 3.18 for each region by substituting Eqs. 3.19,3.22 and 3.26 to it yielding to

HGx (z) =1

ωµ0µ̃Qe√Le(

e−i√Lezve1 − ei

√Lezve2

)(3.28)

HIx(z) =1

ωµ0µ1kIz

(eik

Izzrey − eik

Izzoey

)(3.29)

HIIx (z) = −1

ωµ0µ1kIIz e

ikIIz ztey (3.30)

Using boundary conditions

EIy(z = 0) = EGy (z = 0) (3.31)

HIx(z = 0) = HGx (z = 0) (3.32)

EGy (z = d) = EIIy (z = d) (3.33)

HGx (z = d) = HIIx (z = d) (3.34)

with Eqs. 3.19, 3.22, 3.26, 3.28–3.30 a transfer matrix can be constructed for rey and tey: reyoey

= M 0

tey

= M11 M12

M21 M22

0

tey

(3.35)where transfer matrix M is expressed as

M =

1 11µ1kIz − 1µ1k

Iz

−1

P

0 eikIIz d0 − 1µ2k

IIz e

ikIIz d

(3.36)where matrix P characterizes the grating region

P = Qe Qeµ̃Qe√Le −µ̃Qe

√Le

ei

√Led 0

0 e−i√Led

Qe Qeµ̃Qe√Le −µ̃Qe

√Le

−1

(3.37)

50


Finally, unknown variables are expressed in form:

rey = M12M−122 o

ey (3.38)

tey = M−122 o

ey (3.39)

Renormalization and multilayer expansion is applied the same way explained in previouschapters.

Diffraction efficiency is given by Eq. 1.145–1.146, considering rex = rez = tex = tez = 0 theyare written in form

ηrTE =kIz√

ε1µ1k cosϕ|rey|2 (3.40)

ηtTE =

√µ1k

IIz√

ε1µ2k cosϕ|tey|2 (3.41)

TM polarization

By definition, wave is TM polarized, if vector H = (Hx, Hy, Hz) is parallel to the y-axis(ψ = 90◦), therefore Hx = 0 and Ey = 0. Also, Hz = 0. Eigenproblem given by Eq. 3.17is solved in each region. There is no need to solve eigenproblem for Ex, because of Hz = 0vector Ex from Eq. 3.8 is expressed as

Ex = −i

ωε0ε̃∂

∂zHy (3.42)

In the grating region Eq. 3.17 has solution

HGy (z) = Qhe−i

√Lhzvh1 + Q

hei√Lhzvh2 (3.43)

Where Lh is the vector of eigenvalues, Qh the matrix of corresponding eigenvectors of couplingmatrix

Ch = k2ε̃−1µ− ε̃−1kxε−1kx (3.44)

Matrix Ch is sized (2N + 1)× (2N + 1). In the superstrate Eq. 3.17 obtains form

∂2

∂z2HIy = −

(ε1µ1k

2 − k2x)HIy (3.45)

The solution for the eigenproblem above including the incident wave is

HIy(z) = e−ikIzzrhy + e

ikIzzohy (3.46)

Where

kIz =√ε1µ1k2 − k2x (3.47)

51


and

ohy =

...

0

1

0

...

(3.48)

In the substrate region the eigenproblem is expressed by equation

∂2

∂z2HIIy = −

(ε2µ2k

2 − k2x)HIIy (3.49)

Its solution is

HIIy (z) = eikIIz zthy (3.50)

where

kIIz =√ε2µ2k2 − k2x (3.51)

Electric field vector Ex is derived from Eq. 3.42 for each region by substituting Eqs. 3.43,3.46 and 3.50.

EGx (z) =1

ωε0ε̃Qh√Lh(

e−i√Lhzvh1 − ei

√Lhzvh2

)(3.52)

EIx(z) =1

ωε0ε1kIz

(eik

Izzrhy − eik

Izzohy

)(3.53)

EIIx (z) = −1

ωε0ε1kIIz e

ikIIz zthy (3.54)

Using boundary conditions

EIy(z = 0) = EGy (z = 0) (3.55)

HIx(z = 0) = HGx (z = 0) (3.56)

EGy (z = d) = EIIy (z = d) (3.57)

HGx (z = d) = HIIx (z = d) (3.58)

with Eqs. 2.35, 3.46, 3.50, 3.52–3.54 a transfer matrix can be constructed for rhy and thy : rhyohy

= M 0

thy

= M11 M12

M21 M22

0

thy

(3.59)where transfer matrix M is expressed as

M =

1 11ε1kIz − 1ε1k

Iz

−1

P

0 eikIIz d0 − 1ε2k

IIz e

ikIIz d

(3.60)52


where matrix P characterizes the grating region

P = Qh Qhε̃Qh√Lh −ε̃Qh

√Lh

ei

√Lhd 0

0 e−i√Lhd

Qh Qhε̃Qh√Lh −ε̃Qh

√Lh

−1

(3.61)

Finally, unknown variables are expressed in form:

rhy = M12M−122 o

hy (3.62)

thy = M−122 o

hy (3.63)

For TM polarization, diffraction efficiency is derived through magnetic field vector inten-sity. From Eq. 1.142 we e

electromagnetic properties of double-periodic...

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