modal method based on subsectional gegenbauer polynomial expansion for lamellar gratings

Modal method based on subsectional Gegenbauerpolynomial expansion for lamellar gratings

Kofi Edee1,2

1Clermont Université, Université Blaise Pascal, LASMEA, BP 10448, F-63000 Clermont-Ferrand, France2CNRS, UMR 6602, LASMEA, F-63177 Aubiere, France

*(kofi.edee@univ‐bpclermont.fr)

Received April 15, 2011; revised July 25, 2011; accepted July 29, 2011;posted July 29, 2011 (Doc. ID 146026); published September 8, 2011

A first approach of a modal method by Gegenbauer polynomial expansion (MMGE1) is presented for a plane wavediffraction by a lamellar grating. Modal methods are among the most popular methods that are used to solve theproblem of lamellar gratings. They consist in describing the electromagnetic field in terms of eigenfunctions andeigenvalues of an operator. In the particular case of the Fourier modal method (FMM), the eigenfunctions areapproximated by a finite Fourier sum, and this approximation can lead to a poor convergence of the FMM.The Wilbraham–Gibbs phenomenon may be one of the reasons for this poor convergence. Thus, it is interestingto investigate other basis functions that may represent the fields more accurately. The approach proposed in thispaper consists in subdividing the pattern in homogeneous layers, according to the periodicity axis. The field isexpanded, in each layer, on the basis of Gegenbauer’s polynomials. Boundary conditions are rigorously writtenbetween adjacent layers; thus, an eigenvalue equation is obtained. The approach presented in this paper proves todescribe the fields accurately. Finally, it is demonstrated that the results obtained with the MMGE1 are moreaccurate than several existing modal methods, such as the classical and the parametric FMM. © 2011 OpticalSociety of America

OCIS codes: 050.1950, 050.1755.

1. INTRODUCTIONThe calculation of the electromagnetic field diffracted by alamellar grating through a modal method consists in describ-ing the electromagnetic field in terms of the eigenfunctionsand eigenvalues of an operator, which represents the propa-gation through the structure. This operator will be called theoperator of diffraction. The classical modal method [1,2] is arigorous method of calculation of these eigenfunctions.Indeed, this allows one to obtain an analytical expression ofthe eigenfunctions that depend on the eigenvalues of the op-erator of diffraction. The eigenvalues are required as the zerosof a transcendental equation. That is not easy to solve in gen-eral. The Fourier modal method (FMM), introduced by Knop[3] in the case of a dielectric grating, consists in approximatingthe eigenfunctions of the operator of diffraction by a partialFourier sum and solving an eigenvalue equation. The case ofmetallic gratings challenged for a long time the community ofthe electromagneticians, especially the case of TM polariza-tion. The authors of [4–6], suggested new calculation rules al-lowing fast convergence of the series of the partial Fouriersum. Despite this improvement, the problem of the FMM re-mains the representation of nonsmooth fields through finiteFourier sums that lead, inevitably, to undesired oscillations.This phenomenon, known as traditional Gibbs phenomenon,was probably discovered for the first time by Euler in 1755 byreconstituting the function f ðxÞ ¼ x, defined on a finite sup-port, from its Fourier series. Many eminent physicists andmathematicians from Wilbraham [7] to Michelson, Stratton(1898), and Gibbs have learned on the analysis, the interpreta-tion and the sense of the representation of an irregular func-tion by a linear combination of continuous functions. It isGibbs that came to the conclusion with this matter in a paper

published in Nature [8,9], by suggesting a correct analysis ofthis phenomenon, which was later named Gibbs phenomenon.

Several techniques were developed in order to reduce con-siderably the Gibbs phenomenon when a function is reconsti-tuted from a partial sum of its Fourier coefficients [10–13].Some of these techniques are based on a polynomial recon-struction [14,15]. Let us consider a function g defined onan interval Ω, in these techniques, Ω is subdivided in subinter-vals Ωs so that the restriction gΩs of g on Ωs is continuous. Morfin [16] suggested an efficient modal method for the resolutionof the diffraction problem by a lamellar grating. He suggestedsubdividing the pattern in homogeneous regions ðΩsÞs and ap-proximating the eigenfunctions of the operator of diffractionon each subinterval by Legendre or Chebyshev polynomials.

In this paper, we present a more general method based onthe use of the ultraspherical polynomials or Gegenbauer’spolynomials [17] for computation of the eigenvalues andeigenfunctions of the diffraction operator. The choice ofGegenbauer’s polynomials was inspired by the work ofGottlieb and Shu [14]. The rest of the paper is organized asfollows: Section 2 briefly reconsiders the framework of theFMM. In Section 3, the calculation of the eigenfunctionsand eigenvalues of the operator of diffraction with theGegenbauer’s polynomials is presented. Finally, Section 4 isdevoted to the presentation of the numerical results.

2. STATEMENT OF THE PROBLEMThe configuration under study is depicted in Fig. 1, where az-invariant lamellar grating is illuminated by a 1D monochro-matic plane wave with angular frequency ω and angle ofincidence θ. The incident field is also assumed to be invariantalong Oz axis. Consequently, the scattering field is also

2006 J. Opt. Soc. Am. A / Vol. 28, No. 10 / October 2011 K. Edee

1084-7529/11/102006-08$15.00/0 © 2011 Optical Society of America

invariant along Oz axis and can be either TE (the only nonnullcomponents of the field are Ez, Hx, and Hy) or TM (the onlynonnull components of the field areHz, Ex, and Ey) polarized.It is well known that in these cases of polarization, all the fieldcomponents can be expressed in terms of the Ez componentin the TE case or in terms of theHz component in the TM case.The time dependence of the fields will be held by the termexpðiωtÞ. The space is subdivided into three regions alongthe Oy axis. The incident medium and the substrate are homo-geneous media with refractive indices ν1 and ν3. The twomedia are separated by an inhomogeneous medium, whoserefractive index ν2ðxÞ is described as follows:

ν2ðxÞ ¼�ν21; x ∈ ½0; f d�;ν22; otherwise:

ð1Þ

Here, f denotes the fill factor of the grating. It can be shownthat the z component of the electromagnetic field, which willbe denoted by Fðx; yÞ, satisfies a second-order differentialequation:

LFðx; yÞ ¼ −1k2

∂2yFðx; yÞ; ð2Þwhere

TE:L ¼ 1k2

∂2x þ ν2ðxÞ; ð3aÞ

TM:L ¼ 1

k2ν2ðxÞ∂x

1

ν2ðxÞ ∂x þ ν2ðxÞ: ð3bÞ

Here, k ¼ 2π=λ denotes the wave number and λ is the wave-length. The separation of variables method is used to computethe solutions of Eq. (2):

Fðx; yÞ ¼ XðxÞYðyÞ: ð4Þ

One can easily prove that YðyÞ ¼ e�ikβy, consequently X is theeigenfunction of the operator of diffraction L:

LXðxÞ ¼ β2XðxÞ; ð5Þ

that satisfies the pseudoperiodicity condition, i.e., Xðxþ dÞ ¼e−ikν1 sin θdXðxÞ. Hence, the spectrum of the L operator is dis-crete. The eigenfunctions XpðxÞ form a complete set of vec-tors [1,2] and this property allows the representation of anypseudoperiodic function. In practice, Fðx; yÞ (i.e., Ezðx; yÞor Hzðx; yÞ) is approximated by a finite sum of eigenfunctionsXpðxÞ:

Fðx; yÞ ¼X2Nþ1

p¼1

ðApeikβpy þ Bpe−ikβpyÞXpðxÞ: ð6Þ

The square root βp of the eigenvalue β2p is chosen in such away that

ImðβpÞ < 0 or βp > 0 if βp is real: ð7Þ

According to Eqs. (6) and (7), ApeikβpyXpðxÞ is associated withdownward waves and Bpe−ikβpyXpðxÞ with upward waves.

The FMM consists in expanding any eigenfunction Xp in afinite partial Fourier sum:

XpðxÞ ¼XNn¼−N

XpnenðxÞ; ð8Þ

where enðxÞ ¼ exp½−ixðkν1 sin θ þ 2πnd Þ�. The main disadvan-

tage of the FMM approach lies in the fact that it is difficult,and even sometimes impossible, to describe in a satisfactoryway a strong discontinuity of the electromagnetic field from apartial Fourier sum. It is often the case for materials withstrong contrast of refractive index but also with strong spacecontrast, i.e., f ≪ 1 or f ≈ 1. One solution to this problem con-sists in writing in a rigorous way the boundary conditions inthe vicinities of the physical discontinuities of the material.

3. POLYNOMIAL APPROACHLet us consider an interval Ω on which Xp presents strong dis-continuities. A partition fΩi; 1 ≤ i ≤ NΩg of Ω is defined andeach subinterval Ωi can be divided in layers Ωij , 1 ≤ j ≤ Ni.For example, in Fig. 1, Ω is subdivided into two homogeneoussubintervals Ω1 and Ω2 and each subinterval Ωi, ði ¼ 1; 2Þ con-tainsNi layers. The eigenfunctions XpðxÞ of theL operator aredescribed in each homogeneous layer Ωij as follows:

jXi;jp i ¼

XNn¼1

ai;jnpjbi;jn i; ð9Þ

where ðjbi;jn iÞn are basis functions defined on the interval Ωij .

The vector jXi;jp i is the restriction of the eigenfunction jXpi

to the homogeneous layer Ωij , characterized by its refractive

index νi. jXi;jp i satisfies the Helmholtz equation identical for

both TE and TM polarizations:

Li;jjXi;jp i ¼ β2pjXi;j

p i; ð10Þwith

Li;j ¼ 1

k2d2

dx2þ ν2i : ð11Þ

From Eq. (9), it follows that Eq. (10) may be written as

Li;jXNn¼1

ai;jnpjbi;jn i ¼ β2pXNn¼1

ai;jnpjbi;jn i; ð12Þ

or

XNn¼1

ai;jnpLi;jjbi;jn i ¼ β2pXNn¼1

ai;jnpjbi;jn i: ð13Þ

Fig. 1. The grating configuration: one period is depicted.

K. Edee Vol. 28, No. 10 / October 2011 / J. Opt. Soc. Am. A 2007

By projecting Eq. (13) on the set of N − 2 basis functionsðjbi;jm iÞm∈½1:N−2�, we obtain

XNn¼1

ai;jnphbi;jm jLi;jjbi;jn i ¼ β2pXNn¼1

ai;jnphbi;jm jbi;jn i; ∀ m ∈ ½1:N − 2�;

ð14Þ

with

hbi;jm jLi;jjbi;jn i ¼ 1k2

�bi;jm

��d2bi;jndx2

�þ ν2i hbi;jm jbi;jn i: ð15Þ

Let us set

hbi;jm jbi;jn i ¼ Gi;jmn; ð16aÞ

�bi;jm

��d2bi;jndx2

�¼ Di;j

mn: ð16bÞ

Equation (14) is then written as follows:

�1k2

Di;j þ ν2iGi;j

�ai;j½1;N � ¼ β2pGi;jai;j½1;N �; ð17Þ

where Gi;jðDi;jÞ is a matrix, which elements are Gi;jmn (Di;j

mn,respectively). The vector ai;j½1;N � is a column matrix formedby the coefficients ai;jnp, n ∈ ½1; N �:

ai;j½1;N � ¼ ½ai;j1p;…; ai;jNp�t: ð18Þ

Since the matrices Gi;j and Di;j are ðN − 2Þ × N dimension, weobtain, for each layer Ωij, a system of N − 2 equations with Nunknowns. This system of equations is completed with:

• For each layer Ωij of the same subinterval Ωi,ði; jÞ ∈ ½1:NΩ − 1� × ½1:Ni�, the boundary equations obtainedby writing the continuity of the tangential components ofthe electromagnetic field at the interfaces ðxjÞj∈½1:Ni−1� separ-ating two adjacent layers, Ωij and Ωi;jþ1:

Xi;jp ðxjÞ ¼ Xi;jþ1

p ðxjÞ; ð19aÞ�dXi;j

p

dx

�x¼xj

¼�dXi;jþ1

p

dx

�x¼xj

: ð19bÞ

• At the interfaces separating two adjacent subintervalsΩi and Ωiþ1 i ∈ ½1:NΩ − 1�,

Xi;Nip ðxNi

Þ ¼ Xiþ1;1p ðxNi

Þ; ð20aÞ

1ηi�dXi;Ni

p

dx

�x¼xNi

¼ 1ηiþ1

�dXiþ1;1

p

dx

�x¼xNi

; ð20bÞ

where ηi ¼ 1 in TE polarization and ηi ¼ ν2i in TM polarization.• For the subinterval ΩNΩ , Eq. (19) is written for the first

layer ðΩijÞij , i ¼ NΩ, 1 ≤ j ≤ NNΩ − 1 and the pseudoperiodiccondition for j ¼ NNΩ :

XNΩ;NNΩp ðdÞ ¼ eikν1 sin θdX1;1

p ð0Þ; ð21aÞ

1ηNΩ

"dX

NΩ;NNΩp

dx

#x¼d

¼ eikν1 sin θd

η1

"dX1;1

p

dx

#x¼0

: ð21bÞ

Equations (19)–(21) allow to express the vector

aΩ½N−1;N � ¼ha1½N−1;N �; a

2½N−1;N �;…; aNΩ

½N−1;N �it; ð22Þ

as follows:

aΩ½N−1;N � ¼ TaΩ½1;N−2�; ð23Þ

where

aΩ½1;N−2� ¼ha1½1;N−2�; a

2½1;N−2�;…; aNΩ

½1;N−2�it; ð24Þ

with ai½N−1;N � ¼ ½aiN−1;p; aiNp�t, ai½1;N−2� ¼ ½ai1p;…; aiN−2;p�t and

ai½r;s� ¼ ½ai;1½r;s�; ai;2½r;s�…ai;Ni½r;s� �t. We are led to the computation of

the eigenvectors ai½1;N−2� associated with the eigenvalues

β2p of a matrix with dimension Nmax2 ¼ ðNΩðN − 2ÞÞ×ðNΩðN − 2ÞÞ. We choose as basis functions jbi;jn i an ultrasphe-rical polynomials, also called Gegenbauer polynomials jCα

mi,defined for all ξ ∈ ½−1; 1� as follows:

CαmðξÞ ¼

1ΓðαÞ

X½m�

q¼0

ð−1Þq Γðαþm − qÞðqþ 1Þ!ð1þm − 2qÞ! ð2ξÞ

m−2q: ð25Þ

The Gegenbauer polynomials Cαm are m degree orthogonal

polynomials on the interval ½−1; 1� that satisfy

hCαm; Cα

ni ¼Z

1

−1ð1 − ξ2Þα−:5Cα

mðξÞCαnðξÞdξ ¼ δnmhαn; ð26Þ

where δnm denotes the Kronecker’s symbol and where

hαn ¼ π12Cα

nð1ÞΓðαþ 1

2ÞΓðαÞðnþ αÞ ; ð27Þ

with

Cαnð1Þ ¼

Γðnþ 2αÞΓð2αÞΓðnþ 1Þ : ð28Þ

In numerical implementations presented in the currentpaper, the inner product defined by Eq. (26) is computed byconvolving and integrating the polynomials; the correspond-ing weight function ð1 − ξ2Þα−:5 is not taken into account:

hCαm; Cα

ni ¼Z

1

−1Cα

mðξÞCαnðξÞdξ; ð29Þ

�Cα

m;dCα

n

dξ

�¼

Z1

−1Cα

mðξÞddξC

αnðξÞdξ: ð30Þ

Doing so, we lose the advantage offered by the orthogon-ality properties of the polynomials in the computing of the


matrix hbi;jm ; bi;jn i. Nevertheless, the numerical computation of

the terms hbi;jm ; d2bi;jndx2

i becomes simple. For α ¼ 0:5, the weightfunction of the inner product disappears and the Gegenbauerpolynomials are just the Legendre ones. For a problem ofdiffraction grating, the wave equation is solved in the threemedia: incident medium, substrate, and the inhomogeneousmedium. Then the field coefficients are obtained with the helpof boundary conditions, i.e., the continuity of the componentsðEz;HxÞ in TE polarization and ðEx;HzÞ in TM polarization,at the interfaces separating these three media. Finally, theresulting set of algebraic equations is processed throughthe S-matrix algorithm.

4. NUMERICAL RESULTSA. Comparison with Literature Results and MethodPerformanceIn this section, a comparison of our results with some pub-lished data is performed. For that purpose, we consider threeexamples:

1. a metallic grating,2. an ideal lossless metallic grating,3. a dielectric grating.

The cases of highly conductive metallic grating and of a grat-ing presenting a great contrast of the refractive index butalso a great spatial contrast will be studied in Subsection 4.B.The first example concerns a metallic grating in a resonanceconfiguration, with the following parameters: d ¼ λ ¼ 1,h ¼ λ ¼ 1, f ¼ 0:5, θ ¼ 30°, ν21 ¼ 1, ν22 ¼ ν3 ¼ 0:22 − 6:71i.This example was studied by L. Li [18] with FMM. Lalanne[5] and Granet [4] used FMM combined with Li-factorizationrules. This example was also studied by Granet [19] with theparametric Fourier modal method (PFMM) and later otherauthors, in particular Lalanne [20], Armeanu [21], and Granet[22]. Lalanne used a finite difference method with nonuniformsampling (NSFDM), while B-spline expansions were used in[21,22]. The minus first-order efficiency R−1 in TE polarizationand the zeroth-order efficiency R0 in TM polarization, for dif-ferent precisions and different values of Nmax (size of diffrac-tion matrix), are presented in Table 1. These results areobtained with the FMM, the PFMM [19], and the NSFDM[20]. From all these works and for this example, it turnsout that the most efficient method remains the PFMM. Table 2shows results obtained with the polynomial method describedin this paper (MMGE1). The grating is subdivided into twosubintervals in the x direction (NΩ ¼ 2). Each subinterval cor-responding to a homogeneous region is not subdivided intolayers (N1 ¼ N2 ¼ 1). This table presents values of R−1 (TEpolarization) and R0 (TM polarization) for different values ofthe parameter α, the number N of polynomial on eachinterval, and Nmax (the size of matrices appearing in theeigenvalue problem). Generally, the convergence of themethod strongly depends on the parameter α and the case ofpolarization. Indeed, in the case of TE polarization, α hardlyinfluences the results contrarily to TM polarization case. For aprecision of 10−3 and 10−4, the best result is obtained for α ¼ 1:R−1 ¼ 848e − 3 for Nmax ¼ 12 and R−1 ¼ 8485e − 4 forNmax ¼ 16. Nevertheless, a higher precision is obtained forα ¼ 0:05: R−1 ¼ 84848e − 5 and this for Nmax ¼ 24. The TMcase is a typical case of field presenting a strong discontinuity.

Indeed, the Ex component is discontinuous at x ¼ f d. Conse-quently, a description of this component with a partial sum ofFourier coefficients is not a judicious choice. Figure 2 showsthe normalized modulus, (L1 norm) of Ex associated to theeigenvalue β ¼ 1:0507 − 0:0018i. In this figure, we compareExðxÞ obtained by the FMM and MMGE1. In the case of theFMM, despite the great number of Fourier coefficientNmax ¼ 101, the Gibbs phenomenon remains visible closeto the discontinuity. However, the polynomial piecewise ap-proach, MMGE1, with very few Gegenbauer’s polynomials(N ¼ 13, Nmax ¼ 22) allows to eliminate these oscillations.In the second example, we consider the case of a grating withpure metallic material in TM polarization [20,23]. This exam-ple is identical to the previous one except that the real partof the grating material refractive index vanishes: ν22 ¼ν3 ¼ −6:71i. In Ref. [23], the authors also approximate the

Table 1. Diffraction Efficiencies of the Minus

First-Order (TE) and the Zeroth-Order (TM)

Reflected of a Metallic Gratinga

TE Polarization

Nmax FMM NSFDM PFMM

21 7:6227e − 1 6:5601e − 1 7:3422e − 125 7:5181e − 1 6:8540e − 1 7:3427e − 129 7:3857e − 1 7:0082e − 1 7:3428e − 1

TM Polarization

Nmax FMM NSFDM PFMM

21 8:4211e − 1 8:4592e − 1 8:4831e − 125 8:4717e − 1 8:3960e − 1 8:4862e − 129 8:4241e − 1 8:4774e − 1 8:4850e − 1

aResults obtained with FMM, NSFDM, and PFMM. Numerical parameters:θ ¼ 30°, ν3 ¼ ν21 ¼ :22 − 6:71i, ν1 ¼ ν22 ¼ 1, d ¼ λ, f ¼ 0:5, h ¼ λ.



Reflected of a Metallic Gratinga

TE Polarization

Nmax N α ¼ 0:05 α ¼ 0:5 α ¼ 0:8 α ¼ 1

4 4 9:4531e − 1 7:8428e − 1 6:7880e − 1 6:2186e − 112 8 7:2255e − 1 7:2234e − 1 7:2213e − 1 7:2197e − 116 10 7:3386e − 1 7:3383e − 1 7:3379e − 1 7:3376e − 120 12 7:3428e − 1 7:3427e − 1 7:3427e − 1 7:3426e − 124 14 7:3428e − 1 7:3428e − 1 7:3428e − 1 7:3428e − 128 16 7:3428e − 1 7:3428e − 1 7:3428e − 1 7:3428e − 136 20 7:3428e − 1 7:3428e − 1 7:3428e − 1 7:3428e − 144 24 7:3428e − 1 7:3428e − 1 7:3428e − 1 7:3428e − 1

TM Polarization

Nmax N α ¼ 0:05 α ¼ 0:5 α ¼ 0:8 α ¼ 1

4 4 9:1248e − 1 8:8936e − 1 8:7856e − 1 8:7270e − 112 8 8:4749e − 1 8:4777e − 1 8:4809e − 1 8:4834e − 116 10 8:4826e − 1 8:4831e − 1 8:4840e − 1 8:4847e − 120 12 8:4847e − 1 8:4849e − 1 8:4851e − 1 8:4853e − 124 14 8:4848e − 1 8:4849e − 1 8:4850e − 1 8:4852e − 128 16 8:4848e − 1 8:4849e − 1 8:4850e − 1 8:4851e − 136 20 8:4848e − 1 8:4849e − 1 8:4849e − 1 8:4849e − 144 24 8:4848e − 1 8:4848e − 1 8:4849e − 1 8:4849e − 1aResults obtained with MMGE1. Numerical parameters: θ ¼ 30°, N1 ¼

N2 ¼ 1, ν3 ¼ ν21 ¼ :22 − 6:71i, ν1 ¼ ν22 ¼ 1, d ¼ λ, f ¼ 0:5, h ¼ λ.


eigenfunctions by a piecewise polynomials by using a Fourier-matching pseudospectral modal method (PSMMðf Þ). Table 3presents the zeroth-order (TM) and minus first-order (TE) re-flected efficiencies obtained by MMGE1. Exact values are0.84899 in TE polarization and 0.89298 in TM polarization.In the case of TE polarization, for the PSMMðf Þ, this resultis obtained by using 73 eigenfunctions (each eigenfunction isdescribed with 142 coefficients), while the FMM needs 301eigenfunctions. In the case of TM polarization, PSMMðf Þneeds 133 eigenfunctions (each eigenfunction is describedwith 262 coefficients), while FMM needs at least 401 eigen-functions. These results are obtained for Nmax ¼ 28 in bothTE and TM polarization with the MMGE1 approach proposedin this paper.

In the third example, a dielectric grating is considered withthe following numerical parameters: θ ¼ 30°, N1 ¼ 2, N2 ¼ 3,λ ¼ 1, ν1 ¼ 1, ν21 ¼ 2:3, ν22 ¼ 1, ν3 ¼ 1:5, d ¼ λ, f ¼ 0:234,h ¼ λ. The choice of the numbers of layers N1 and N2 willbe briefly discussed in Subsection 4.B. Figure 3 shows thefirst-order efficiency in TM polarization; GPMM1 is more effi-cient than FMM and PSMMðf Þ.

B. Algorithm StabilityNevertheless, the algorithm may quickly become unstable.That is due to the fact that Gegenbauer’s polynomials co-efficients are expressed through the Gamma function (seeFig. 4). The coefficients of Gegenbauer’s polynomials quicklyincrease with respect to the degree of polynomials but alsowith respect to the parameter α [Eq. (25)]. This fast growthcreates numerical instabilities when one wishes to observethe convergence of the method at infinity. An alternative tothis problem consists in increasing the number of homoge-neous layers according to the Ox direction. For illustration,we consider

0 0.2 0.4 0.6 0.8 1−1

−0.5

0

0.5

1

1.5

2

2.5x 10

−3

x/λ

|Ex(x

)|

Gegenbauer piecewise expansion (Nmax=22)Fourier expansion (Nmax=101)

Fig. 2. Modulus of Ex associated with the eigenvalue β ¼1:0507 − 0:0018i. Despite the great number of Fourier coefficientNmax ¼ 101, the Gibbs phenomenon remains visible close to thediscontinuity by using FMM. However, a very few Gegenbauer’s poly-nomials (N ¼ 13, Nmax ¼ 22) allows to eliminate these oscillations byusing MMGE1.



Reflected of an Ideal Lossless Metala

TE Polarization

Nmax N α ¼ 0:05 α ¼ 0:5 α ¼ 0:8 α ¼ 1

4 4 9:9506e − 1 8:3486e − 1 7:2481e − 1 6:6479e − 112 8 8:3110e − 1 8:3091e − 1 8:3076e − 1 8:3064e − 120 12 8:4896e − 1 8:4896e − 1 8:4895e − 1 8:4895e − 128 16 8:4899e − 1 8:4899e − 1 8:4899e − 1 8:4899e − 136 20 8:4898e − 1 8:4899e − 1 8:4899e − 1 8:4899e − 144 24 8:4898e − 1 8:4899e − 1 8:4899e − 1 8:4899e − 1

TM Polarization

Nmax N α ¼ 0:05 α ¼ 0:5 α ¼ 0:8 α ¼ 1

4 4 9:5448e − 1 9:3691e − 1 9:2844e − 1 9:2381e − 112 8 8:9305e − 1 8:9326e − 1 8:9354e − 1 8:9376e − 120 12 8:9303e − 1 8:9300e − 1 8:9299e − 1 8:9301e − 128 16 8:9298e − 1 8:9298e − 1 8:9298e − 1 8:9299e − 136 20 8:9298e − 1 8:9298e − 1 8:9298e − 1 8:9298e − 144 24 8:9298e − 1 8:9298e − 1 8:9298e − 1 8:9298e − 1aNumerical parameters: θ ¼ 30°, N1 ¼ N2 ¼ 1, ν3 ¼ ν21 ¼ −6:71i, ν1 ¼

ν22 ¼ 1, d ¼ λ, f ¼ 0:5, h ¼ λ.

50 60 70 80 90 100 110 120 130 1400.5102

0.5104

0.5106

0.5108

0.511

0.5112

0.5114

Nmax

α=0.05α=0.5α=0.8α=1FMM

Fig. 3. First-order transmitted efficiency of a dielectric grating in TMpolarization. Numerical parameters: θ ¼ 30°, N1 ¼ 2, N2 ¼ 3, λ ¼ 1,ν1 ¼ 1, ν21 ¼ 2:3, ν22 ¼ 1, ν3 ¼ 1:5, d ¼ λ, f ¼ 0:234, h ¼ λ.

0 5 10 15 20 250

2

4

6

8

10

12

14x 10

16

ζ

Γ(ζ)

Fig. 4. Representation of the gamma function.


1. a dielectric grating presenting a great contrast of therefractive index but also a great spatial contrast,

2. a highly conductive metallic grating.

The numerical parameters of the first example are: d=λ ¼ 10,f ¼ 0:1ðf d=λ ¼ 1=100Þ, ν3 ¼ ν21 ¼ 5, ν1 ¼ ν22 ¼ 1, θ ¼ 10°,h=λ ¼ 1. Table 4 presents the zeroth-order efficiency R0 forboth polarizations, with respect to the number N of polyno-mials and for different values of α ∈ f0:05; 0:5; 0:8; 1g. Thegrating is divided into two homogeneous subintervals Ω1

and Ω2, and the number N of the polynomials is identicalfor both intervals. Ω1 corresponds to a homogeneous mediumwith width f d ¼ 0:1d and refractive index ν21 ¼ 5; Ω2 is ahomogeneous medium with width ð1 − f Þd ¼ 0:9d and refrac-tive index ν22 ¼ 1. We notice a poor convergence of results forall values of α. The results given by the FMM are also not veryprecise: R0 ¼ 392e − 3 for TE polarization with Nmax ¼ 201and 394e − 3 for TM polarization with Nmax ¼ 401. Let us high-light that the PFMM fails in this example. This probably comesfrom the poor convergence of the Fourier series of the para-metric function. We foresee the necessity of a great number Nof polynomials to represent the field. In order to observe theaccuracy of the method, we have to make a partition of thesubintervals Ω1 (resp Ω2) inN1 (respN2) homogeneous layers.Table 5 is a first illustration. In this case, N1 ¼ N2 ¼ 2.

In TE polarization, the accuracy of 10−3, that is to sayR0 ¼ 392e − 3, given by the FMM for Nmax ¼ 201, could bereached for N equal to 18 and regardless of the α valueschosen. This matches with Nmax ¼ 64. In TM polarization,an accuracy of 10−2 is reached for this value of N . Whenincreasing the number of layers Ni, the number of polyno-mials N necessary to the field description goes down. Table 6illustrates this result. In this table, values of the zeroth re-flected order efficiency R0 are presented for values ofN1 ¼ N2 ¼ 6, for both polarizations. In the case of TE polar-ization, the N number of polynomials necessary to get an

accuracy at least of 10−3 on the R0 value is N ¼ 8(Nmax ¼ 72) for all values of α. In the case of TM polarization,an accuracy of 10−2 is reached for N ¼ 8 ðNmax ¼ 72Þ. Mainly,the algorithm based on subdividing in homogeneous layersoffers a better numerical stability when the field descrip-tion would need a great number of polynomials. The major

Table 4. Zeroth Reflected Order Efficiency

of Dielectric Grating with Great Spatial

Contrast with N1 � 1 and N2 � 1a

TE Polarization

Nmax N α ¼ 0:05 α ¼ 0:5 α ¼ 0:8 α ¼ 1

4 4 0.7228 0.4424 0.3509 0.315612 8 0.4170 0.3114 0.3886 0.416028 16 0.4076 0.3999 0.3958 0.394560 32 0.3843 0.3861 0.3985 0.388368 36 0.3910 0.3892 0.3961 0.388276 40 0.3938 0.3846 0.3839 0.390284 44 0.3874 0.3841 0.3851 0.3973

TM Polarization

Nmax N α ¼ 0:05 α ¼ 0:5 α ¼ 0:8 α ¼ 1

4 4 0.5645 0.4386 0.4109 0.402212 8 0.3718 0.3509 0.3319 0.321928 16 0.4124 0.4135 0.4140 0.414260 32 0.3706 0.3837 0.3923 0.395368 36 1.1087 0.3966 0.3932 0.384176 40 9.6987 1.5102 0.7771 0.377084 44 1.2643 0.4694 1.0546 0.9098aNumerical parameters: θ ¼ 10°, λ ¼ 1, ν3 ¼ ν21 ¼ 5, ν1 ¼ ν22 ¼ 1, d ¼ 10λ,

f ¼ 0:1, h ¼ λ.




TE Polarization

Nmax N α ¼ 0:05 α ¼ 0:5 α ¼ 0:8 α ¼ 1

8 4 0.6503 0.4054 0.2691 0.229524 8 0.4055 0.3921 0.3847 0.380840 12 0.3873 0.3884 0.3891 0.389456 16 0.3932 0.3933 0.3933 0.393464 18 0.3923 0.3924 0.3925 0.392672 20 0.3923 0.3924 0.3924 0.392580 22 0.3926 0.3925 0.3926 0.392688 24 0.3925 0.3925 0.3925 0.3924

TM Polarization

Nmax N α ¼ 0:05 α ¼ 0:5 α ¼ 0:8 α ¼ 1

8 4 0.4512 0.4264 0.3862 0.359624 8 0.4014 0.4035 0.4041 0.404540 12 0.3945 0.3944 0.3947 0.394956 16 0.3941 0.3935 0.3933 0.393264 18 0.3959 0.3958 0.3956 0.395472 20 0.3963 0.3961 0.3959 0.395780 22 0.3956 0.3955 0.3954 0.395288 24 0.3949 0.3948 0.3947 0.3943aNumerical parameters: θ ¼ 10°, λ ¼ 1, ν3 ¼ ν21 ¼ 5, ν1 ¼ ν22 ¼ 1, d ¼ 10λ,

f ¼ 0:1, h ¼ λ.




TE Polarization

Nmax N α ¼ 0:05 α ¼ 0:5 α ¼ 0:8 α ¼ 1

24 4 0.3462 0.3689 0.3761 0.379072 8 0.3924 0.3926 0.3927 0.3927120 12 0.3924 0.3924 0.3924 0.3925168 16 0.3924 0.3924 0.3924 0.3924192 18 0.3924 0.3924 0.3924 0.3924216 20 0.3925 0.3925 0.3925 0.3925240 22 0.3924 0.3924 0.3924 0.3924264 24 0.3924 0.3924 0.3924 0.3924

TM Polarization

Nmax N α ¼ 0:05 α ¼ 0:5 α ¼ 0:8 α ¼ 1


f ¼ 0:1, h ¼ λ.


inconvenience of this approach is that the matrix size in-creases sensibly with respect to the partitions number. Itwould be suitable to perform an intelligent partitioning in or-der to keep both stability offered by this algorithm and the fastconvergence. We suppose that the number of layers of a sub-interval Ωi must be proportional to its width. In Table 7, Ω1 of0:1dwidth is subdivided in two subintervals (N1 ¼ 2) while Ω2

of 0:9d width is a pileup of N2 ¼ 10 layers. The global size ofthe matrix is identical to the case of N1 ¼ N2 ¼ 6 (Table 6);however, the results obtained are the best in terms of fast con-vergence and stability. Figure 5 shows the convergence of R0

with respect to 1=Nmax in TM polarization. In this figure,results obtained from FMM and MMGE1 are presented. Wecan see that the results obtained with MMGE1 convergeremarkably fast and remain stable.

The case of the highly conductive grating (θ ¼arcsinðλ=2dÞ, λ ¼ 1 ν3 ¼ ν21 ¼ 1 − 40i, ν22 ¼ 1, ν1 ¼ ν22 ¼ 1,d ¼ 1:2361λ, f ¼ 0:57, h ¼ 0:4λ) is presented in Fig. 6 forTE and Fig. 7 for TM polarization. Results are compared tothose obtained with FMM; it can be seen that MMGE1 is alsomore efficient than the classical FMM for this example.Lalanne in Ref. [20] also deals with this example by using aNSFDM. Here, MMGE1 is likely to be more efficient thanthe NSFDM.

5. CONCLUSIONIn this paper, we presented an efficient method to calculatethe electromagnetic field diffracted by a metallic lamellargrating. This method, which is inspired by modal methods,allows us to accelerate the convergence of the numerical re-sults thanks to a piecewise interpolation technique of modesdescribing the electromagnetic field. The FMM, which provedto be efficient for the study of lamellar gratings in the past, islimited when studying gratings with great contrast of refrac-tive index or gratings with great spatial contrast, i.e., a fill fac-tor close to 0 or 1. The issue of the slow convergence causedin those cases probably comes from the Gibbs phenomenon,

0 0.005 0.01 0.015 0.02 0.0250.36

0.37

0.38

0.39

0.4

0.41

0.42

0.43

0.44

0.45

1/Nmax

zero

th r

efle

cted

ord

er

α=0.5FMM

Fig. 5. Zeroth-order reflected efficiency of dielectric grating withgreat spatial contrast in TM polarization. Numerical parameters:θ ¼ 10°, N1 ¼ 2, N2 ¼ 10, λ ¼ 1, ν3 ¼ ν21 ¼ 5, ν1 ¼ ν22 ¼ 1, d ¼ 10λ,f ¼ 0:1, h ¼ λ.

0.006 0.008 0.01 0.012 0.0140.6

0.602

0.604

0.606

0.608

0.61

1/Nmax

min

us−

first

ref

lect

ed o

rder

α=0.05α=0.5α=0.8α=1FMM

Fig. 6. Minus first-order reflected efficiency of highly conductivemetal in TE polarization. Numerical parameters: θ ¼ arcsinðλ=2dÞ,N1 ¼ 4, N2 ¼ 4, λ ¼ 1, ν3 ¼ ν21 ¼ 1 − 40i, ν1 ¼ ν22 ¼ 1, d ¼ 1:2361λ,f ¼ 0:57, h ¼ 0:4λ.

0.006 0.008 0.01 0.012 0.0140.55

0.6

0.65

0.7

0.75

0.8

1/Nmax

min

us−

first

ref

lect

ed o

rder

α=0.05α=0.5α=0.8α=1FMM

Fig. 7. Minus first-order reflected efficiency of highly conductivemetal in TM polarization. Numerical parameters: θ ¼ arcsinðλ=2dÞ,N1 ¼ 4, N2 ¼ 4, λ ¼ 1, ν3 ¼ ν21 ¼ 1 − 40i, ν1 ¼ ν22 ¼ 1, d ¼ 1:2361λ,f ¼ 0:57, h ¼ 0:4λ.




TE Polarization

Nmax N α ¼ 0:05 α ¼ 0:5 α ¼ 0:8 α ¼ 1

24 4 0.3938 0.3944 0.3950 0.395272 8 0.3830 0.3825 0.3821 0.3820120 12 0.3922 0.3922 0.3922 0.3922168 16 0.3924 0.3924 0.3924 0.3924192 18 0.3924 0.3924 0.3924 0.3924216 20 0.3924 0.3924 0.3924 0.3924240 22 0.3924 0.3924 0.3924 0.3924264 24 0.3924 0.3924 0.3924 0.3924

TM Polarization

Nmax N α ¼ 0:05 α ¼ 0:5 α ¼ 0:8 α ¼ 1


f ¼ 0:1, h ¼ λ.


which appears when partial sums of Fourier series are trun-cated. From our point of view, the polynomial approach withGegenbauer polynomials brings a satisfactory solution to thisproblem. Questions remain regarding the choice of the param-eter α, but also regarding the optimal cutting scheme. Tryingto make a discussion about the optimal choice on the numberof Gegenbauer polynomials N is similar to finding the optimaltruncation orderNmax in FMM. Regarding the parameter α, wecannot make a complete discussion without introducing andclarifying the role of the weight function and concepts such astruncation error, regularization error, and Gibbs complemen-tary basis. A simple subsection of the current paper cannotdeal with all these questions in a satisfactory way. Neverthe-less, these questions will be topics of a future paper. The algo-rithm may turn out to be unstable when the polynomialnumber necessary to describe the field is important. In ouropinion, this is caused by—among other things—the fact thatpolynomials coefficients defined by Gamma functions in-crease excessively with respect to the degree of polynomialsand the parameter α. A subdivision of the grating in variouslayers turns out to be necessary for a stable algorithm.

ACKNOWLEDGMENTSThe author wishes to thank Brahim Guizal, Gerard Granet,and Jean-Pierre Plumey for fruitful discussions.

REFERENCES1. L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and

J. R. Andrewartha, “The dielectric lamellar diffraction grating,”Opt. Acta 28, 413–428 (1981).

2. L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, andJ. R. Andrewartha, “The finitely conducting lamellar diffractiongrating,” Opt. Acta 28, 1087–1102 (1981).

3. K. Knop, “Rigorous diffraction theory for transmission phasegratings with deep rectangular grooves,” J. Opt. Soc. Am. 68,1206–1210 (1978).

4. G. Granet and B. Guizal, “Efficient implementation of thecoupled-wave method for metallic lamellar gratings in TM polar-ization,” J. Opt. Soc. Am. A 13, 1019–1023 (1996).

5. Ph. Lalanne and G. M. Morris, “Highly improved convergenceof the coupled-wave method for TM polarization,” J. Opt.Soc. Am. A 13, 779–784 (1996).

6. L. Li, “Use of Fourier series in the analysis of discontinuousperiodic structures,” J. Opt. Soc. Am. A 13, 1870–1876(1996).

7. H. Wilbraham, “On a certain periodic function,” Cambridge andDublin Math. J. 3, 198–201 (1848).

8. J. W. Gibbs, “Fourier’s series,” Nature 59, 200 (1898).9. J. W. Gibbs, “Fourier’s series,” Nature 59, 606 (1899).10. W. Cai, D. Gottlieb, and C. W. Shu, “Essentially non oscillatory

spectral Fourier methods for shock wave calculations,” Math.Comput. 52, 389–410 (1989).

11. K. S. Eckhoff, “Accurate and efficient reconstruction of discon-tinuous functions from truncated series expansions,” Math.Comput. 61, 745–763 (1993).

12. K. S. Eckhoff, “Reconstructions of functions of finite regularityfrom truncated Fourier series expansions,” Math. Comput. 64,671–690 (1995).

13. D. Gottlieb and C. W. Shu, “General theory for the resolutionof the Gibbs phenomenon,” Accademia Nazionale Del Lincey,ATTI Del Convegni Lincey 147, 39–48 (1998).

14. D. Gottlieb and C. W. Shu, “On the Gibbs phenomenon IV: reco-vering exponential accuracy in a subinterval from a Gegenbauerpartial sum of a piecewise analytic function,”Math. Comput. 64,1081–1095 (1995).

15. M. Min, T. Lee, P. F. Fisher, and S. K. Gray, “Fourier spectralsimulation and Gegenbauer reconstructions for electromagneticwave in the presence of a metal nanoparticule,” J. Comput.Phys. 213, 730–747 (2006).

16. R. H. Morf, “Exponentially convergent and numerically efficientsolution of Maxwell’s equations for lamellar gratings,” J. Opt.Soc. Am. A 12, 1043–1056 (1995).

17. U. W. Hochstrasser, “Orthogonal polynomials,” in Handbook ofMathematical Functions, M. Abramowitz and I. A. Stegun (eds.)(Dover, 1965), pp. 771–802.

18. L. Li and C. W. Haggans, “Convergence of the coupled-wavemethod for metallic lamellar diffraction gratings,” J. Opt. Soc.Am. A 10, 1184–1189 (1993).

19. G. Granet, “Reformulation of the lamellar grating problemthrough the concept of adaptative spatial resolution,” J. Opt.Soc. Am. A 16, 2510–2516 (1999).

20. P. Lalanne, “Numerical performance of finite-differencemodal methods for the electromagnetic analysis of one-dimensional lamellar gratings,” J. Opt. Soc. Am. A 17,1033–1042 (2000).

21. A. M. Armeanu, K. Edee, G. Granet, and P. Schiavone, “Modalmethod based on spline expansion for the electromagneticanalysis of the lamellar grating,” Prog. Electromagn. Res. 106,243–261 (2010).

22. G. Granet, A. L. Bakonirina, R. Karyl, A. M. Armeanu, and K.Edee, “Modal analysis of lamellar gratings using the momentmethod with subsectional basis and adaptive spatial resolution,”J. Opt. Soc. Am. A 27, 1303–1310 (2010).

23. D. Song, L. Yuan, and Y. Y. Lu, “Fourier-matching pseudospec-tral modal method for diffraction gratings,” J. Opt. Soc. Am. A28, 613–220 (2011).


modal method based on subsectional gegenbauer polynomial expansion for lamellar gratings

Documents