Field singularities at lossless metal-dielectric right-angleedges and their ramifications to the numerical
modeling of gratings
Lifeng Li1,* and Gérard Granet2,3
1State Key Laboratory of Precision Measurement Technology and Instruments, Department of Precision Instruments,Tsinghua University, Beijing 100084, China
2Clermont Universités, Université Blaise Pascal, LASMEA, BP 10448, F-63000 Clermont-Ferrand, France3CNRS, UMR 6602, F-63177 Aubière, France
1. INTRODUCTIONFor modeling lamellar diffraction gratings, i.e., gratings ofrectangular groove profiles, the family of methods that canbe categorically called the modal methods is undoubtedlythe most efficient. It includes the analytic modal method(AMM) [1,2], the Fourier modal method (FMM) [3] (also re-ferred to as the rigorous coupled-wave approach), and othervariants based on finite difference [4], B spline [5,6], and otherbasis functions [7]. Many people, including us, thought thatthe modal methods (in particular, the popular Fourier modalmethod after the improvements [8–11] since 1996) can handleany kind of lamellar gratings. This is why the two recent pa-pers of Gundu and Mafi [12,13] came as a big surprise. Theauthors found some cases for which both the AMM and theFMM fail to converge. A typical set of grating parameters thatthey considered, referring to Fig. 1, are εd ¼ εs ¼ 1:96,εc ¼ 2:25, εm ¼ −2:5, grating period d ¼ 0:2 μm, groove depthh ¼ 0:28 μm, groove width w ¼ d=2, wavelength λ ¼ 0:62 μm,normal incidence, and TM polarization (the magnetic fieldvector parallel to the grating grooves). This specific parametercombination is referred to as the Gundu case in this paper.Using the AMM as a reference method, they convincingly ruleout inaccuracy of eigen solutions, which inevitably occur withthe other types of modal methods, as a possible cause for thenonconvergence and attributed the convergence problem tousing square matrix truncation in matching the electromag-netic boundary conditions between fields in the grating regionand the two exterior regions. They further proposed to userectangular matrix truncation and a least-squares mini-mization procedure to solve the resulting system of linearequations. In [12], the least-squares minimization is uncon-
strained, leading to violation of energy conservation; thedrawback was quickly removed in [13] by using energy con-servation as a constraint in the minimization. Both papersshow seemingly convergent results for the grating problemdescribed above, but the final reflectivity numbers are notice-ably different.
One may suspect that the nonconvergence reported in[12,13] is due to insufficiently large matrix orders used.As a supplement to Figs. 5(a) and 5(b) in [12], we provideFigs. 2(a) and 2(b); in both, the truncation order N is the orderof the square matrices and equal to the number 2M þ 1 in[12,13]. Evidently, convergence is still unreachable at suchhigh truncation orders. Although these additional data cannotquench the suspicion, they cast doubt on the possibility of get-ting convergence with higher truncation orders as long assquare matrix truncation is used.
The high significance of the work of Gundu and Mafi is thatthey presented some cases for which the modal methods failto converge. Although they correctly focused attention on theboundary matching and proposed a solution method, the rootcause for the convergence problem was not found. In our in-itial investigation, we made numerous numerical tests, whichshowed that the nonconvergence depends solely on the rela-tive magnitudes of the permittivities εc, εs, εd, and εm, indepen-dent of all other optical and geometrical parameters, e.g.,wavelength-to-period ratio, groove depth, groove duty cycleðw=dÞ, and angle of incidence, and led us to the present in-depth study. In this paper, we mathematically prove and nu-merically demonstrate that the source of the nonconvergenceis the electromagnetic field singularity at the edges of the rec-tangular grating profile. Field singularity is, of course, always
738 J. Opt. Soc. Am. A / Vol. 28, No. 5 / May 2011 L. Li and G. Granet
present in all lamellar gratings under TM illumination, butit becomes fundamentally more important when a losslessmetallic wedge is surrounded by dielectric media. We rigor-ously derive a set of precise criteria with which, given a la-mellar grating, one can predict whether the conventionalimplementation (square matrix truncation) of a modal methodof any kind will converge without doing actual numerical testcomputations.
2. CLASSIFICATION OF EDGESINGULARITIESSharp edges are frequently encountered in electromagneticand optical structures. It is well known that when a dielectric,metallic, or perfectly conducting wedge is illuminated by anelectromagnetic wave, the transverse components of the elec-tric and magnetic fields may be singular at the mathematicallysharp edge of the wedge, while the longitudinal componentsare not singular (longitudinal means parallel to the edge direc-
tion and transverse means perpendicular to it). The knowl-edge of field singular behavior is important in two ways:taking it into account in structural design can avoid electricalbreakdown and properly incorporating it into electromagneticmodeling can improve numerical convergence. Extensive re-search has been done in this area [14], but a comprehensiveliterature survey is beyond the scope of this paper. We arecontent to make the following observation: while field singu-larity at edges has been extensively studied in electromagneticwave and microwave waveguide communities, it is seldomdiscussed in grating theory. There are few exceptions. Mays-tre [15] and Rathsfeld et al. [16] incorporated edge singularityinto the integral method of gratings. Lalanne and Jurek [17]simulated field singularities in dielectric and metallic gratingsin TM polarization using the FMM.
The English word “wedge”means a piece of material (invar-iant in one direction) having one thick end and tapering to athin edge. In this paper, “wedge assembly” means a collectionof wedges whose edges meet along a common line and whoseangles add up to 2π and “lossless metal-dielectric edge”meansthe edge of a wedge assembly consisting of pure dielectricmedia and a lossless metal (whose permittivity is real andnegative).
The electromagnetic field components near the edge of awedge assembly have complex spatial structures. In the polarcoordinate system ðρ;φÞ, with its origin located at the edge,they are given by the generalized Meixner series, which aredouble power series of the radial ordinate and the logarithmof the radial ordinate [18–20]. In the vicinity of the edge, thefield behavior is dominated by the leading term of the serieswhose radial dependence obeys a power law. In TM polariza-tion, the transverse electric field has the following asymptoticform:
EðjÞρ ¼ ρτ−1AðjÞ
ρ ðφÞ; EðjÞφ ¼ ρτ−1AðjÞ
φ ðφÞ; ð1Þ
where the subscripts ρ and φ denote the radial and azimuthcomponents, the superscript j in parentheses distinguishesdifferent wedge regions, and τ is the singularity exponent. Gi-ven a wedge assembly, the most important task in studying thefield singularity is to find the τ that has the least nonnegativereal part. Because jρτ−1j ¼ ρRe½τ�−1, the transverse field compo-nents are not singular when Re½τ� ≥ 1. When 0 < Re½τ� < 1,they are singular but the edge condition is still satisfied,i.e., any finite volume including the edge contains a finiteamount of electromagnetic energy.
A. General ConsiderationsAn explicit expression for the singularity exponent of theright-angle four-wedge assembly shown schematically inFig. 3(a) was first given by Bobrovnikov and Zamaraeva [21]:
A similar expression was later derived by Bressan andGamba [22]:
Fig. 1. Geometric parameters and permittivities of a lamellar grating.
Fig. 2. Nonconvergence of reflectivity of a grating case studied in[12] computed with the (a) analytical modal method and the(b) FMM, both with square matrix truncation (see Fig. 5 in [12]).The legends are explained in Section 2.
L. Li and G. Granet Vol. 28, No. 5 / May 2011 / J. Opt. Soc. Am. A 739
τ ¼ 2π arcsin
ffiffiffiffiΔ
p; Δ ¼ 1 −Δ0: ð4Þ
For convenience, we work with Eq. (4) in this paper.We are mainly concerned with the two special cases of
Fig. 3(a), as shown in Figs. 3(b) and 3(c); however, we firstconsider some general properties of Eq. (4). As a func-tion of four variables, Δ has these easily verifiable symmetryproperties:
where κ is any nonzero complex number. Additional symme-try relations can be deduced from Eq. (5). Let us study theright-hand side of Eq. (4) as a complex variableΔ ¼ z ¼ xþ i y, where x and y are real numbers andi ¼ ffiffiffiffiffiffi
−1p
. The arcsine function in Eq. (4) having its real partin the interval ½0; π=2� is the principal value of the multivaluedfunction whose infinite number of branches have the follow-ing form:
Arc sinffiffiffiz
p ¼ nπ � arcsinffiffiffiz
p; ð6Þ
where n is any integer. Clearly, only the principal value hasthe least nonnegative real part. In Figs. 4(a), 4(b), and 5,we plot the real and imaginary parts of τ over the complexplane z and along the real axis, respectively. As shown inFig. 4, τðzÞ has three branch points, z ¼ 0, 1, and ∞, andthe two branch cuts are chosen as the semi-infinite real linesð−∞; 0� and ½1;∞Þ. It can be rigorously shown that 0 ≤ Re½τ� ≤ 1for all z and the first and second equalities hold only when z ison the left and right branch cut lines, respectively; however,jIm½τ�j is unbounded, increasing like ln jzj as jzj → ∞. On thereal line, as shown in Fig. 5, Re½τ� ¼ 0 for x < 0, 0 ≤ Re½τ� ≤ 1,Im½τ� ¼ 0 for 0 ≤ x ≤ 1, and Re½τ� ¼ 1 for x ≥ 1. The sign of Im½τ�on the two branch cut lines depends on whether z is infinite-simally above or below the real line.
Based on the above analysis, we conclude that, as far as thewedge assemblies shown in Fig. 3 are concerned, there can bethree types of wedge assemblies. In type I,Δ is away from thetwo branch cuts, so the transverse field components are sin-gular. In type II, Δ falls on the left branch cut, so the trans-verse field components are also singular, but the singularityis stronger than that in type I. In type III, Δ falls on the rightbranch cut, so no field components are singular. For conve-nience, we say the field singularity in type I is regular and thatin type II is irregular.
We mention in passing that if all four permittivities inFig. 3(a) are positive, then 0 < Δ ≤ 1 and the equality holds
if, and only if, ε1ε3 ¼ ε2ε4. Thus, the transverse field compo-nents at an all-dielectric right-angle edge are always regularlysingular or nonsingular. If Ohm loss is present in any memberwedge of a wedge assembly, Δ is complex, excluding rare ac-cidental cases, and the field singularity is regular. In addition,all wedge assemblies obtained by symmetry manipulations orscaling of the four permittivities as shown in Eq. (5) have thesame singularity exponent.
The irregular edge field singularity appears to have notbeen studied previously. This is probably because the typeII situation cannot occur for an all-dielectric wedge assemblyor a wedge assembly containing a lossy metallic body. In Sub-section 2.B, we show that it takes a lossless metal-dielectricwedge assembly meeting certain conditions to render a type IIsituation.
B. Wedge Assembly of Fig. 3(b)Setting ε1 ¼ ε2 ¼ εc, ε3 ¼ εd, and ε4 ¼ εm in Eq. (2), we obtainthe Δ function for the wedge assembly in Fig. 3(b):
Δ ¼ 1 −εc ðεd − εmÞ2
2 ðεc þ εdÞ ðεc þ εmÞ ðεd þ εmÞ: ð7Þ
Introducing new symbols rc ¼ −εc=εm and rd ¼ −εd=εm trans-forms Eq. (7) into
Fig. 4. (Color online) (a) Real and (b) imaginary parts of functionτðzÞ ¼ ð2=πÞ arcsinðz1=2Þ in the complex z. The two branch cutsare chosen as the semi-infinite real lines ð−∞; 0� and ½1;∞Þ.
Fig. 3. Three configurations of right-angle wedge assemblies.
740 J. Opt. Soc. Am. A / Vol. 28, No. 5 / May 2011 L. Li and G. Granet
Δ ¼ 1 −rc ð1þ rdÞ2
2 ð1 − rcÞ ð1 − rdÞ ðrc þ rdÞ: ð8Þ
Note that, by definition, rc > 0 and rd > 0, so Δ is a real num-ber. Therefore, the task of singularity classification for thewedge assembly in Fig. 3(b) becomes that of identifyingthe different regions where Δ < 0, 0 < Δ < 1, and Δ > 1 inthe two-dimensional space ðrc; rdÞ. Such identification is givenin Fig. 6. In regions I1 and I2, 0 < Δ < 1; in regions II1 and II2,Δ < 0; in regions III1 and III2, Δ > 1. As the point ðrc; rdÞ ap-proaches the straight line rc ¼ 1 or rd ¼ 1 from the side ofregion II1 or II2,Δ tends to −∞. Approaching from the oppositeside gives þ∞. On the boundaries between regions I and II,Δ ¼ 0. The two dashed–dotted lines at rc ¼ 2 and rd ¼ 1markthe asymptotes of the hyperbolic boundary between regions I2and II2. It follows from Eq. (5) that regions R1 and R2 areimages of each other with respect to coordinate changeðrc; rdÞ → ð1=rc; 1=rdÞ, where R ¼ I, II, and III, and, at eachpair of image points, Δ has the same value. This map makesthe identification of edge field singularity very easy.
C. Wedge Assembly of Fig. 3(c)Setting εd ¼ εc in Eq. (7) and rd ¼ rc ¼ r in Eq. (8), we obtainthe Δ function for the wedge assembly in Fig. 3(c):
Δ ¼ 1 −14
� εc − εmεc þ εm
�2¼ 1 −
14
�1þ r1 − r
�2: ð9Þ
It is unnecessary to analyze this Δ in detail; the solution isalready given by the dashed diagonal line in Fig. 6. We onlyneed to note that the diagonal line intercepts the boundariesbetween regions I and II at points ð1=3; 1=3Þ and ð3; 3Þ.
D. Crossed GratingsThe above analysis obviously also applies to crossed gratingsof rectangular pillars or holes. From Eq. (9), it can be seen thatwhether the periodic pattern is formed by pillars or holes (i.e.,whether the wedge is salient or reentrant), the singularity ex-ponents are the same. It is interesting to note that, for a loss-less metal-dielectric checkerboard grating (ε1 ¼ ε3 ¼ εd > 0and ε2 ¼ ε4 ¼ εm < 0), Δ ¼ 4εdεm=ðεd þ εmÞ2 < 0. Thus, thetransverse fields at the edges of such crossed gratings are al-ways irregularly singular, independent of the relative magni-tudes of the two permittivities.
3. RAMIFICATION OF EDGE SINGULARITYTO GRATING MODELINGA. Edge Singularity and Fourier ExpansionRamification of edge singularity to grating modeling manifestsin the matching of boundary conditions. In the modal meth-ods, the total fields in the grating region are given by modalfield expansions, and those in the upper and lower homoge-neous semi-infinite regions are given by the Rayleigh expan-sions. For TM polarization, the X component (see Fig. 1) ofthe electric field at the two interfaces between the three re-gions must be continuous. However, with respect to the localcylindrical coordinate systems centered at the periodicallyspaced edges along an interface, EX is just the radial compo-nent that may be singular at the edges. Although the individualmodal fields and plane waves are finite, a superposition of aninfinite number of them in principle may be capable of build-ing up the singular field. How well and how efficiently thesuperposition can represent the singular field, which directlydetermines the quality of boundary condition matchingstrongly depends on the singularity exponent.
From amathematical point of view, a Rayleigh expansion ata constant Y ordinate is simply a Fourier series. Since theFourier basis functions are infinitely differentiable, Fourierseries are not at all adept to represent discontinuous func-tions, not to mention singular functions. The modal expansionis expected to perform better because its basis functions (forEX) have appropriate discontinuities built in. For this reason,we focus our attention in this paper on the Fourier expansionof EX .
Suppose the field singularity at the edges along the upper orlower boundary of the grating region is regular. Then, its nthFourier coefficient has the following asymptotic estimate for alarge jnj:
EXn ¼ Oðjnj−Re½τ�Þ; ð10Þ
where Re½τ� > 0. Thus, if the Fourier expansion of EX con-verges, its convergence rate depends on Re½τ�. The greaterRe½τ� is, the faster it converges. If Re½τ� is very small, althoughpositive, it can be expected that EX converges slowly. Equa-tion (10) can be established by using the theory of a complex
Fig. 5. (Color online) Real (solid curve) and imaginary parts (dashedcurve and dashed–dotted curve) of function τðzÞ ¼ ð2=πÞ arcsinðz1=2Þon the real axis x. In the legend, “above”means z ¼ zþ iδ and “below”
means z ¼ z − iδ as 0 < δ → 0.
Fig. 6. (Color online) Map of Δðrc; rdÞ. In regions I1 andI2, 0 < Δ < 1; in regions II1 and II2, Δ < 0; in regions III1 and III2,Δ > 1.
L. Li and G. Granet Vol. 28, No. 5 / May 2011 / J. Opt. Soc. Am. A 741
variable, but, for the sake of saving space, we do not give theproof here.
If the field singularity at the edges along the upper or lowerboundary of the grating region is irregular, as can be extrapo-lated from Eq. (10), the Fourier coefficients do not tend tozero and the Fourier expansion of EX does not converge.By examining the form of the field near an edge, one canget a feel about why the Fourier expansion fails to converge.For a complex τ, we have
ρτ−1 ¼ ρRe½τ�−1 expðiIm½τ� ln ρÞ: ð11Þ
When Re½τ� ¼ 0 and Im½τ� ≠ 0, the right-hand side is an oscil-latory function of ρ with infinitely increasing both amplitudeand frequency as ρ → 0. Having seen this, the reader may be-gin to understand why the numerical efforts represented byFig. 2 failed.
B. Convergence CriteriaWe now can state the convergence criteria for the modalmethods as follows. If Δ ≤ 0, the modal methods do not con-verge. If Δ > 0 or complex, the modal methods in principlealways converge, but the rate of convergence depends onthe location of the Δ value in the complex Δ plane. Thefarther away from the negative real line, the faster the conver-gence. Specifically for the lossless metal-dielectric right-anglewedge assemblies of Figs. 3(b) and 3(c), the convergencecriteria are summarized in Table 1.
C. Numerical VerificationIn the Gundu case, the wedge assemblies in the upper gratingboundary are that of Fig. 3(b), for which rc ¼ 0:9, rd ¼ 0:676,Δ ¼ −23:7548, and τ ¼ i1:4562. In the lower grating boundary,the wedge assemblies are that of Fig. 3(c), for whichr ¼ 0:676, Δ ¼ −5:68957, and τ ¼ i1:02099. Thus, it is not sur-prising that the modal methods do not converge. Another loss-less metallic lamellar grating case often cited in the literatureis that of Popov et al. [23]. In this case, r ¼ 0:01,Δ ¼ 0:739797,and τ ¼ 0:659216, which explains why the AMM convergeswithout any problem.
Besides the above two published lossless metallic lamellargrating cases, we present Fig. 7 computed with the AMM.Figures 7(a)–7(f) share the following parameters: normal in-cidence and TM polarization, wavelength λ ¼ 0:6 μm, groovedepth h ¼ 0:5 μm, groove width w ¼ d=2, and εm ¼ −6. Thevalues of rc, rd, Δ, and τ are given in each figure, from whichthe values of εc and εd can be deduced. For simplicity, we haveset εs ¼ εc, so only one singularity exponent is involved in agrating. The grating period d ¼ 0:25 μm in all but Fig. 7(e),where d ¼ 0:85 μm. The reason for the exception is that,for the particular permittivity combination in Fig. 7(e) withd ¼ 0:25 μm, the periodic region does not support any propa-
gating mode, so the reflectivity is 100%, disqualifying the pa-rameter combination as a valid convergence test case. Locat-ing the ðrc; rdÞ values in Fig. 6, we see that the predictions ofTable 1 are all verified. We have conducted numerous similarnumerical tests and have not found a counterexample.
The applicability of our field singularity analysis is by nomeans limited to the modal methods for lamellar gratings.In Fig. 8, we show some results of the C method [24,25] forright-angle triangular gratings. All subfigures share the follow-ing parameters: TM polarization, wavelength λ ¼ 0:9 μm, angleof incidence 30°, left base angle 30°, grating period d ¼ 1:0 μm,and εm ¼ −6 [see the inset in Fig. 8(a)]. The values of εc can bededuced from the r values given in the legends. Note that, ascommented before for crossed gratings, the field singularityexponents at the two edges per grating period are the same.
In the C method, the grating contour is one of the coordi-nate surfaces, and, in TM polarization, the tangential electricfield component on the grating contour is given by a Fourierseries in terms of a spatial variable that, as the variable X inthe lamellar grating case, periodically runs through the edgesin the locally radial directions. Consequently, although theconvergence or divergence behaviors are different from thoseof the AMM results for lamellar gratings, the fundamental con-clusion is the same: all predictions in Table 1 are still valid.
4. DISCUSSIONWe have focused on the idealized situations: infinitely sharpedges and lossless metals; however, overall, the study is notacademic, except for some strange permittivity combinationsin Figs. 7 and 8, which are chosen for the completeness of thestudy. The edges of a real lamellar grating are not infinitelysharp, but the transverse fields are much stronger near thanfar away from the edges. In certain spectral regions, some me-tals have very small losses. Studying the idealized situationsallows us to understand better the limitations of the existingnumerical methods and, hopefully, also find more effectivemethods to deal with the difficult, but realistic, grating pro-blems. In [12], the authors show that changing εm from−2:5 to −2:5þ i can effectively improve the convergence ofthe FMM for the Gundu case. Based on the theory in Section 3,we know that this is because the change increases the realparts of the singularity exponents at the upper and loweredges from zero to 0.593713 and 0.474794, respectively.
The wedge assemblies of lamellar gratings in most practicalapplications are of type I defined in Subsection 2.A. Theauthors of [12,13] may be the only researchers who have ap-plied the modal methods to lamellar gratings having type IIwedge assemblies. It has been generally believed that the con-vergence rate of the AMM and the FMM in TM polarizationstrongly depends on the contrast of the medium permittivitiesin the grating layer. The higher the contrast, the slowerthese two methods converge. The present study has shownconvincingly that the complex Δ value, not the contrast, de-termines the convergence rate.
To overcome the nonconvergence problem associated withthe irregular edge singularity, the authors of [12,13] suggestusing the constrained least-squares minimization procedureto match the boundary conditions. We are unsure of this re-medy’s reliability. Figure 9 shows our result for the Gunducase using the FMM under the same conditions as those ofFig. 3 in [13], but reaching a much greater truncation order.
Table 1. Convergence Criteria for the
Modal Methods and the C Method when
Applied to Gratings Containing Lossless
Metal-Dielectric Right-Angle Wedge
Assemblies (see Fig. 6)
I1, I2 II1, II2 III1, III2
Slow Divergent Fast
742 J. Opt. Soc. Am. A / Vol. 28, No. 5 / May 2011 L. Li and G. Granet
(As for Fig. 2, our N equals 2M þ 1 in [13].) Our mathematicalformulation behind the code specifically written for this studyexactly follows that of [13] (however, the mirror-reflectionsymmetry is used to reduce the computation time andmemoryconsumption). The two pairs of horizontal and vertical axesare used to display the same set of data. The data trend ismore visible with the logarithmic horizontal axis. The datado not seem to have well converged.
We also question the validity of the least-squares minimiza-tion formulations in [12,13]. Both formulations allow, besidesthe only incident plane wave, additional plane waves to comefrom infinity onto the grating, which violates the radiationcondition. In the constrained least-squares approach, energy
conservation is achieved as a result of adjusting the energyfluxes among the true incident wave, the diffracted waves,and the additional incoming waves, but the third part shouldnot be present at all.
In the literature on edge field singularity, it has always beenassumed that the real part of the singularity exponent ispositive, i.e., 0 < Re½τ� < 1, as required by the edge condition.In this paper, we have stepped into the unexplored area ofRe½τ� ¼ 0; therefore, it is absolutely necessary that we checkthe edge condition is still satisfied. Otherwise, all solutionswith Re½τ� ¼ 0 would be nonphysical. Since the transverseelectric field is singular in TM polarization, only the elec-tric part of the edge condition needs to be verified. For a
Fig. 7. Numerical convergence tests with the analytical modal method. The ðrc; rdÞ values in (a)–(f) are located in regions I1–III2 on the map ofFig. 6, respectively.
L. Li and G. Granet Vol. 28, No. 5 / May 2011 / J. Opt. Soc. Am. A 743
cylindrical volume of a unit length along the edge directionand a radius 0 < δ ≪ λ centered at an edge, from Eq. (1)we have
Z δ
0
Z2π
0εðjEρj2 þ jEφj2Þρdρdφ ¼
Z δ
0ρ2Re½τ�−1dρ
Z2π
0εðjAρj2
þ jAφj2Þdφ; ð12Þ
where the superscripts labeling different wedges have beenomitted. If Re½τ� ¼ 0, the radial integral diverges, so the onlychance for the edge condition still being satisfied is that theazimuth integral vanishes. In Appendix A, we show that it in-deed vanishes when Re½τ� ¼ 0 and Im½τ� ≠ 0.
When the edge singularity at the upper or lower boundaryof a lamellar grating is irregular, we have seen that the Fourierexpansion of EX on the boundary does not converge (at leastin the sense of function). However, the Rayleigh expansionevaluated on the boundary is nothing but a Fourier series. Si-milarly, in the C method, the electric field component tangentto the grating profile and perpendicular to the groove direc-tion is also expressed as a Fourier series in terms of the spatialvariable that runs along the tangent dimension. Therefore, weconclude that the failure of modal methods to converge for theGundu case and the like has nothing to do with the modal nat-ure of these methods. Any method that uses a Fourier expan-sion to express the transverse electric field component alonga line passing through sharp edges of type II wedge assembliesought to fail to converge.
Thus far, we have assumed that, when a wedge assemblymeets one of the criteria given in Sections 2 and 3, the trans-verse electric field components at the edge are singular. Ingeneral, the existence of singular fields depends not only onthe wedge assembly, but also on the excitation condition[14,26] (having zero field amplitudes for the terms associatedwith the singularity exponent in the generalized Meixnerseries is a valid solution of Maxwell’s equations). In the cur-rent context, the angle of incidence may determine whethersingular fields are actually excited. However, in our limitednumber of numerical experiments of varying angle ofincidence, we have not found a case where the AMM,FMM, and the C method converge for a grating having type IIwedge assemblies.
Fig. 8. Numerical convergence tests with the C method. The r values in (a), (b), (c), and (d) are located along the diagonal line in regions I1, I2, II1,and II2 on the map of Fig. 6, respectively.
Fig. 9. Reflectivity of a grating case studied in [13] computed withthe FMMwith rectangular matrix truncation and the constrained least-squares minimization procedure. See Fig. 3 in [13].
744 J. Opt. Soc. Am. A / Vol. 28, No. 5 / May 2011 L. Li and G. Granet
5. CONCLUSIONWe have mathematically proven and numerically demon-strated that the failure to converge of the AMM and theFMM for modeling some lossless metal-dielectric lamellargratings in TM polarization recently reported by Gundu andMafi is due to the existence of irregular singularities at theright-angle edges of the gratings. We further have given aset of precise and simple criteria to discern if the transverseelectric field components at the edge of a given wedge assem-bly are not singular, regularly singular, or irregularly singular.We have shown that Fourier series are incapable of represent-ing the transverse electric field components in the vicinity ofan edge of irregular singularity. This is the root cause of theconvergence problem; consequently, any method, not neces-sarily of modal type, using Fourier series in this way isdoomed to fail.
In this work, we have focused on finding and understandingthe cause of the problem. The results obtained will point thesearch for a method of solving the problem in the right direc-tion. Our work in this direction is currently underway.
APPENDIX AWithin each homogeneous wedge, the electric field vectorsatisfies this vectorial wave equation:
∇2EðjÞ þ kðjÞ2EðjÞ ¼ 0; ðA1Þ
where the superscript j in parentheses distinguishes differentwedge regions and kðjÞ is the wavenumber of the wedgemedium. In the cylindrical coordinate system ðρ;φ; zÞ, thecomponent form of Eq. (A1) is
ρ−1∂ρðρ∂ρEðjÞρ Þ þ ρ−2∂2φφEðjÞ
ρ − ρ−2EðjÞρ − 2ρ−2∂φEðjÞ
φ
þ kðjÞ2EðjÞρ ¼ 0; ðA2aÞ
ρ−1∂ρðρ∂ρEðjÞφ Þ þ ρ−2∂2φφEðjÞ
φ − ρ−2EðjÞφ þ 2ρ−2∂φEðjÞ
ρ
þ kðjÞ2EðjÞφ ¼ 0: ðA2bÞ
Since we are dealing with TM polarization and the wedge is zinvariant, there are no ∂2zz terms and Ez ¼ 0. Substituting theasymptotic form of the transverse electric field componentsgiven by Eq. (1) in the main text into Eqs. (A2a) and (A2b)and eliminating the common ρ dependence, we obtain
∂2φφAðjÞρ þ ðτ2 − 2τÞAðjÞ
ρ − 2∂φAðjÞφ ¼ 0; ðA3aÞ
∂2φφAðjÞφ þ ðτ2 − 2τÞAðjÞ
φ þ 2∂φAðjÞρ ¼ 0; ðA3bÞ
where the terms proportional to kðjÞ2 have been dropped be-cause ρ ≪ λ in the vicinity of the edge. Equations (A3a) and(A3b) can be rewritten in a matrix form
�∂2φφ −2∂φ2∂φ ∂2φφ
��AðjÞρ
AðjÞφ
�¼ ð2τ − τ2Þ
�AðjÞρ
AðjÞφ
�: ðA4Þ
It should be solved together with the boundary conditionsbetween wedges,
AðjÞρ ðφjÞ ¼ Aðjþ1Þ
ρ ðφjÞ; εðjÞAðjÞφ ðφjÞ ¼ εðjþ1ÞAðjþ1Þ
φ ðφjÞ;AðNÞρ ðφNÞ ¼ Að1Þ
ρ ðφ0Þ; εðNÞAðNÞφ ðφNÞ ¼ εð1ÞAð1Þ
φ ðφ0Þ;ðA5Þ
where φj and φj−1 are the boundaries of the jth wedge,j ¼ 1; 2;…; N − 1, φN ¼ φ0 þ 2π, and N is the total numberof the wedges in the assembly.
Equations (A4) and (A5) can be used to derive some of theresults given in [22], but we use them differently here. We firstdefine the inner product with weight function εðφÞ betweentwo vector-valued functions FðφÞ ¼ ðFρ; FφÞT and GðφÞ ¼ðGρ; GφÞT as
hF;Gi ¼Z
2π
0εF†Gdφ; ðA6Þ
where the superscripts T and † stand for matrix transpose andconjugate, respectively. Let L be the 2 × 2 matrix differentialoperator, A be the 2 × 1 eigenvector, and χ ¼ 2τ − τ2 be theeigenvalue in Eq. (A4). By using integration by parts in eachwedge range, we obtain
ðχ − �χÞhA;Ai ¼ hA; LAi − hLA;Ai ¼ 0; ðA7Þ
where Eq. (A5), ε being a piecewise constant real function ofφ, and the relations
∂φAρ ¼ τAφ; ∂φAφ ¼ −τAρ ðA8Þ
have been used. Equation (A8) follows from Maxwell’s equa-tions and the static limit assumption. Since χ − �χ ¼4iIm½τ�ð1 − Re½τ�Þ, we assert that, when Re½τ� ≠ 1 and Im½τ� ≠ 0,
hA;Ai ¼Z
2π
0εðjAρj2 þ jAφj2Þdφ ¼ 0: ðA9Þ
In this appendix, nowhere is the wedge assembly assumed tobe of the right-angle type, so the proof is valid for any losslessmetal-dielectric N -wedge assemblies.
ACKNOWLEDGMENTSThis work was initiated and partially completed during arecent one-month visit by L. Li to Université Blaise Pascal be-ginning in mid-November 2010. We gratefully acknowledgethe financial support provided by the university that madethe visit and this collaboration possible.
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, and J. R.Andrewartha, “The finitely conducting lamellar diffraction grat-ing,” Opt. Acta 28, 1087–1102 (1981).
3. L. Li, “Mathematical reflections on the Fourier modal method ingrating theory,” in Mathematical Modeling in Optical Science,G. Bao, L. Cowsar, and W. Masters, eds., SIAM Frontiers inApplied Mathematics (SIAM, 2001), pp. 111–139.
4. P. Lalanne and J.-P. Hugonin, “Numerical performance of finite-difference modal method for the electromagnetic analysis ofone-dimensional lamellar gratings,” J. Opt. Soc. Am. A 17,1033–1042 (2000).
L. Li and G. Granet Vol. 28, No. 5 / May 2011 / J. Opt. Soc. Am. A 745
5. K. Edee, P. Schiavone, and G. Granet, “Analysis of defect inextreme UV lithography mask using a modal method basedon nodal B-spline expansion,” Jpn. J. Appl. Phys. 44,6458–6462 (2005).
6. P. Bouchon, F. Pardo, R. Haïdar, and J.-L. Pelouard, “Fast modalmethod for subwavelength gratings based on B-spline formula-tion,” J. Opt. Soc. Am. A 27, 696–702 (2010).
7. G. Granet, L. B. Andriamanampisoa, K. Raniriharinosy, A. M.Armeanu, and K. Edee, “Modal analysis of lamellar gratingsusing the moment method with subsectional basis and adap-tive spatial resolution,” J. Opt. Soc. Am. A 27, 1303–1310(2010).
8. P. Lalanne and G. M. Morris, “Highly improved convergence ofthe coupled-wave method for TM polarization,” J. Opt. Soc. Am.A 13, 779–784 (1996).
9. 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).
10. L. Li, “Use of Fourier series in the analysis of discontinuousperiodic structures,” J. Opt. Soc. Am. A 13, 1870–1876(1996).
11. N. M. Lyndin, O. Parriaux, and A. V. Tishchenko, “Modal analysisand suppression of the Fourier modal method instabilities inhighly conductive gratings,” J. Opt. Soc. Am. A 24, 3781–3788(2007).
12. K. M. Gundu and A. Mafi, “Reliable computation of scatteringfrom metallic binary gratings using Fourier-based modal meth-ods,” J. Opt. Soc. Am. A 27, 1694–1700 (2010).
13. K. M. Gundu and A. Mafi, “Constrained least squares Fouriermodal method for computing scattering from metallic binarygratings,” J. Opt. Soc. Am. A 27, 2375–2380 (2010).
14. J. van Bladel, Singular Electromagnetic Fields and Sources(Clarendon, 1991), pp. 116–162.
15. D. Maystre, “Sur la diffraction et l’absorption par les réseauxutilisés dans l’infrarouge, le visible et l’ultraviolet,” Ph.D. disser-tation (Université d'Aix-Marseille III, 1974).
16. A. Rathsfeld, G. Schmidt, and B. H. Kleemann, “On a fast integralequation method for diffraction gratings,” Commun. Comput.Phys. 1, 984–1009 (2006).
17. P. Lalanne and M. P. Jurek, “Computation of the near-fieldpattern with the coupled-wave method for transverse magneticpolarization,” J. Mod. Opt. 45, 1357–1374 (1998).
18. J. Meixner, “The behavior of electromagnetic fields at edges,”IEEE Trans. Antennas Propag. 20, 442–446 (1972).
20. G. I. Makarov and A. V. Osipov, “Structure of Meixner’s series,”Radiophys. Quantum Electron. 29, 544–549 (1986).
21. M. S. Bobrovnikov and V. P. Zamaraeva, “On the singularity ofthe field near a dielectric wedge,” Sov. Phys. J. 16, 1230–1232(1973).
22. M. Bressan and P. Gamba, “Analytical expressions of field sin-gularities at the edge of four right wedges,” IEEE Microw.Guided Wave Lett. 4, 3–5 (1994).
23. E. Popov, B. Chernov, M. Nevière, and N. Bonod, “Differentialtheory: application to highly conducting gratings,” J. Opt. Soc.Am. A 21, 199–206 (2004).
24. J. Chandezon, D. Maystre, and G. Raoult, “A new theoreticalmethod for diffraction gratings and its numerical application,”J. Opt. 11, 235–241 (1980).
25. L. Li and J. Chandezon, “Improvement of the coordinate trans-formation method for surface-relief gratings with sharp edges,”J. Opt. Soc. Am. A 13, 2247–2255 (1996).
26. P. Y. Ufimtsev, B. Khayatian, and Y. Rahmat-Samii,“Singular edge behavior: to impose or not impose—that is thequestion,” Microw. Opt. Technol. Lett. 24, 218–223 (2000).
746 J. Opt. Soc. Am. A / Vol. 28, No. 5 / May 2011 L. Li and G. Granet
