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A. Khavasi and K. Mehrany Vol. 26, No. 6 /June 2009/J. Opt. Soc. Am. A 1467

Artifact-free analysis of highly conducting binarygratings by using the Legendre polynomial

expansion method

Amin Khavasi and Khashayar Mehrany*

Integrated Photonics Laboratory, Department of Electrical Engineering, Sharif University of Technology,P.O. Box 11555-4363, Tehran, Iran

*Corresponding author: mehrany@sharif.edu

Received February 17, 2009; revised April 27, 2009; accepted April 28, 2009;posted April 29, 2009 (Doc. ID 107590); published May 28, 2009

Analysis of highly conducting binary gratings in TM polarization has been problematic as the Fourier factor-ization fails and thus unwanted numerical artifacts appear. The Legendre polynomial expansion method(LPEM) is employed here, and the erroneous harsh variations attributed to the violation of the inverse rulevalidity in applying the Fourier factorization are filtered out. In this fashion, stable and artifact-free numericalresults are obtained. The observed phenomenon is clearly demonstrated via several numerical examples and isexplained by inspecting the transverse electromagnetic field profile. © 2009 Optical Society of America

OCIS codes: 050.0050, 050.1950, 050.1960, 050.2770.

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. INTRODUCTIONespite the great success of Fourier-based methods and

he Li factorization rule [1], unwanted numerical artifactsmerge in the analysis of highly-conducting gratings thatre illuminated by TM polarized waves [2–5]. This prob-em has been attributed to the presence of both positivend negative permittivity values within the grating re-ion, and consequently, to the violation of the inverse rulealidity in the Fourier expansion of the permittivity pro-le [5]. A number of solutions have been so far suggestedo overcome this particular difficulty and to eliminate un-anted numerical artifacts. Two different approachesave been proposed by Popov et al. in [2]. First, the blockatrices appearing in the governing set of differential

quations are truncated in two steps, and an additionalruncation is performed to eliminate the more erroneouslements lying at the extremities of the matrices. This ap-roach cannot however guarantee the elimination of allumerical artifacts [2]. Second, a lossier material is intro-uced within the bulk of the highly conducting regions.nasmuch as the incident wave does not penetrate deeplynto the highly conducting bulk and is strongly reflectedrom the grating surface, the introduced perturbationoes not significantly alter the diffractive properties, yett does improve the numerical behavior. Despite beinguite general, the second method fails to calculate the dif-raction efficiency of lossless metallic gratings precisely.his latter technique was later improved by Watanabe in

3], where an extrapolation technique is applied to furtherdjust the sought-after diffraction efficiency.More recently, Lyndin et al. employed a modal ap-

roach to delve into the source of the problem and suc-essfully traced it to the presence of high-order spurious

1084-7529/09/061467-5/$15.00 © 2

odes engendered by the unwanted yet inevitable trun-ation of the Fourier series [4]. They therefore proposedn astute algorithm to filter out the troublesome spuriousodes whose interference with each other had caused the

bserved artifacts. It was also shown that the high-orderature of spurious modes, i.e., their being at the end ofhe modal spectrum, makes them less significant. Thisoint was independently demonstrated in [5], where therror of using the inverse rule in Fourier factorizationas studied to show that unwanted numerical artifacts

an be subdued by increasing the total number of retainedpace harmonics.

The described difficulty, however, remains a critical is-ue and seems to be an unwanted yet inherent flaw of allhe Fourier- based methods that use Fourier expansion ofermittivity profile and electromagnetic field. However,pplication of the nonmodal Fourier-based approach, e.g.,he Legendre polynomial expansion method (LPEM) [6,7],o a lossless metallic binary grating has been found to re-ult in stable solutions and, surprisingly, filtering out ofhe unwanted numerical artifacts . This unexpected phe-omenon is here numerically demonstrated and is thenxplained by studying the electromagnetic field profile in-ide the grating region. It is also shown that this methodan become time-consuming and memory-hungry whenpplied to thick lossless metallic gratings. It does notowever require extra management of spurious modes ly-

ng behind the numerical artifacts.This paper is organized as follows. In Section 2, several

umerical examples are provided to demonstrate that ap-lying the LPEM can in fact eliminate numerical arti-acts. The observed results are then verified by inspectinglectromagnetic field profiles. The time and memory con-

009 Optical Society of America

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1468 J. Opt. Soc. Am. A/Vol. 26, No. 6 /June 2009 A. Khavasi and K. Mehrany

traints in applying the LPEM for thick highly conductingratings are also explained. Finally, conclusions are maden Section 3.

. ELIMINATION OF NUMERICALRTIFACTS

he structure studied here is a typical surface relief bi-ary grating. The grating parameters, as depicted in Fig., are set in accordance with the example already exam-ned in the literature: d=�G=500 nm, nc=1, and ns−10j. This structure is then illuminated by a TM polar-

zed plane wave (the H field is parallel to the y axis)hose vacuum wavelength is �=632.8 nm, and which is

ncident at the angle �=30°. In Fig. 2(a), the minus-firsteflected order is plotted versus groove width �g�. Thelotted results are obtained by applying the rigorousoupled wave analysis (RCWA) [1,8] with N=35, where Ns the truncation order of the Fourier series, and conse-uently 2N+1 is the total number of the retained spacearmonics. The unwanted numerical artifacts are clearlybservable in this figure. In Fig. 2(b), on the other hand,he same example is reexamined and artifact-free nu-erical results are obtained by applying the LPEM with=35 and M=7. Here, M stands for the number of re-

ained Legendre polynomial basis functions in terms ofhich each space harmonic is expanded [6]. Although thePEM solves the same set of coupled differential equa-ions as does the RCWA, and although the inverse rule inpplying the LPEM has become invalid under the consid-red circumstances, the otherwise present numerical ar-ifacts are obviously filtered out when the latter approachs followed. Interestingly, the obtained results in Fig. 2(b)

Fig. 1. Geometry of a typical binary grating.

ig. 2. Minus-first reflected order versus the groove width g=35 and M=7 (solid curve) and the extrapolation technique baig. 1 illuminated by a TM polarized plane wave at �=30° and wi=� =500 nm, n =1, n =−10j.

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irtually overlap with those obtained by using the ex-rapolation technique with N=60, shown by the dashedurve [3]. The observed agreement verifies that applyinghe LPEM with N=35 and M=7 has been as accurate ashe extrapolation technique with N=60.

To explain the observed phenomenon, one can arguehat harsh and harmonically rich variations of electro-agnetic fields arising from the interference of spuriousodes as suggested in [4] are automatically filtered outhen each space harmonic is projected in the linear space

panned by keeping M Legendre basis functions. In fact,f M is large enough to accurately approximate the physi-al variation of electromagnetic fields, and if it is smallnough to filter out the very fast changes of Fourier spacearmonics that are caused by the interference of high-patial-frequency spurious modes [4], then obtaining ac-urate and stable results is guaranteed. This argument isurther expounded in Fig. 3, where the groove width isxed at g=250 nm and the minus-first order reflected dif-raction efficiency calculated by using the LPEM with M6 (dotted curve), M=7 (dashed curve), and M=8 (solidurve) is plotted versus the space-harmonic truncation or-er N. The result obtained by using the extrapolationechnique with 2N+1=301 space harmonics is also givenn this figure as a reference.

As is clearly shown, increasing M results in a more ac-urate value, but it needs a large enough number of spacearmonics N. This is expected, as calculation of electro-agnetic fields by using the LPEM with larger M is more

iable to follow the harsh variations of space harmonicsroduced by the inevitable error of the truncated Fouriereries. Fortunately, increasing the space-harmonic trun-ation order N is helpful and stabilizes the obtained re-ults by further sharpening the numerical artifacts of thelectromagnetic field profile in such a manner that it cano longer be approximated by keeping M basis functions.f M tends to infinity and the space harmonics are de-cribed accurately by Legendre polynomials, i.e., whenPEM is mathematically equivalent to the RCWA, thepace-harmonic truncation order N must also be infiniteo produce stable results [6]. On the other hand, if M isoo small then the correct and physical variation of elec-romagnetic field cannot be accurately approximated byhe retained Legendre polynomials, the truncation error

d by using (a) the RCWA with N=35 and (b) the LPEM withthe RCWA with N=60 (dashed curve) for the grating shown in

vacuum wavelength �=632.8 nm. Other parameters in Fig. 1 are

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A. Khavasi and K. Mehrany Vol. 26, No. 6 /June 2009/J. Opt. Soc. Am. A 1469

f the Legendre polynomial expansion prevails, and thebtained results, though artifact-free, will be erroneous.he compromise is therefore that the number of retainedegendre basis functions should neither be unnecessarily

arge nor erroneously small.To further justify the statements we made in the previ-

us paragraph, the transverse magnetic field profile of theighly conducting binary grating is more closely studiedt g=360.4 nm where a large numerical artifact was con-picuous in Fig. 2(a). In Fig. 4(a), the RCWA with 2N+171 space harmonics is employed and the magnitude of

he transverse magnetic field is plotted. This figurelearly shows that the violation of the inverse rule valid-ty leads to harsh and erroneous variations within thelectromagnetic profile. A similar figure is also providedn [4], where the deteriorating interference effect of spu-ious modes is demonstrated (see Fig. 6 in [4]). In Fig.(b), on the other hand, the LPEM with N=35 and M=7s employed and the magnitude of the transverse mag-etic field is once again plotted. This time, the harsh be-avior of the magnetic field is filtered out and the physi-al field profile is accurately approximated in the linearpace spanned by keeping M=7 Legendre basis functions.

ig. 3. Minus-first reflected order versus the truncation orderN� obtained by LPEM with M=8 (solid curve), M=7 (dashedurve), and M=6 (dotted curve). The groove width of the gratings fixed at g=250 nm. The reference value is obtained by usinghe extrapolation technique based on the RCWA with N=150 ap-lied to a slightly lossier structure.

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he accuracy of this approximation is confirmed by com-aring the obtained field profile in Fig. 4(b) to the fieldrofile in Fig. 4(c) obtained by using the RCWA with=35 applied to the slightly lossier structure with ns0.05−10j. It should be noted that N=35 in the analysisf this lossy structure has been large enough to guaranteen error level below 0.5%. It is thus demonstrated that=7 in the analysis of a highly conducting binary gratingith a thickness of about d�=0.8 has been large enough

o approximate the field profile accurately and smallnough to eliminate the unwanted harsh variations aris-ng from the truncated Fourier series.

For highly conducting binary gratings with largerhickness, however, the number of retained Legendreolynomials M and therefore the truncation order ofpace harmonics N should both be increased. As an ex-mple, a very similar highly conducting binary grating,his time with the grating thickness of d=1 �m and theroove width of g=252.4 nm, is considered. In Fig. 5, theinus-first order reflected diffraction efficiency calculated

y using the LPEM with M=8 (dotted curve), M=9dashed curve), and M=10 (solid curve) is plotted versushe space-harmonic truncation order N. The result ob-ained by using the extrapolation technique with 2N+1301 space harmonic is also given as a reference. This fig-re clearly demonstrates that more Legendre basis func-ions are required to accurately approximate the physicallectromagnetic field profile. Unfortunately, more Leg-ndre basis functions call for more space harmonics to en-ure the numerical stability. This is confirmed in Figs.–8 , where the magnitude of the transverse magneticeld profile is plotted by using different approaches. Inigs. 6 and 7, the LPEM with N=29 and M=10 and thePEM with N=40 and M=10 are applied to the same

ossless structure with ns=−10j, respectively. In Fig. 8, onhe other hand, the RCWA with N=35 is applied to theame structure made of a slightly lossier material withs=0.05−10j. Although keeping M=10 Legendre basis

unction is good enough to approximate the space har-onics accurately, the truncation order of space harmon-

cs N=29 results in a transverse electromagnetic fieldrofile with noisy fluctuations and thus should be in-reased to N=40, which removes nonphysical field varia-ions. These figures thus show that increasing N is indeedelpful to eliminate numerical artifacts.

ig. 4. Spatial distribution of �Hy� calculated by (a) RCWA with N=35 (b), LPEM with N=35 and M=7, and (c) by RCWA with N=35pplied to a slightly lossier structure with n =0.05−10j. The groove width of the grating is fixed at g=360.4 nm.

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1470 J. Opt. Soc. Am. A/Vol. 26, No. 6 /June 2009 A. Khavasi and K. Mehrany

Regrettably, accurate analysis of thick gratings by ap-lying the LPEM requires more Legendre polynomials6]. Therefore, as is demonstrated in previous figures, re-aining a large number of space harmonics is necessary toecure the stability of analyzing highly conducting thickinary gratings with the LPEM. Therefore, the wholetrategy becomes time-consuming and memory-hungry.lthough it might seem that this problem can be easilyvercome by decomposing the thick structure into suffi-iently thin slices and by using the appropriate R-matrixlgorithm [7], the presence of numerical artifacts is inten-ified once a matrix propagation algorithm is applied.herefore, the required truncation order of space harmon-

cs N should be very large to ensure numerical stability.or example, if the diffraction efficiency of the latter ex-mple was to be stably extracted by applying the-matrix algorithm and by decomposing the grating to

wo layers each with thickness of d=500 nm, then M=6egendre polynomial terms and 2N+1=101 space har-onics would have been necessary in the calculation.ven though M=10 would have been reduced to M=6, theumber of required space harmonics would have undesir-bly increased from 81 to 101.

. CONCLUSIONe have shown that the numerical difficulties arising

rom the violation of the inverse rule validity in Fourier-ased analysis of lossless metallic gratings can be elimi-ated by applying the LPEM. The unexpected stability ofhe LPEM as a Fourier- based method relying on the vio-ated inverse rule is attributed to the fact that the trun-ated linear space spanned by the Legendre basis func-ions can automatically filter out unwanted numericalrtifacts. The LPEM is shown to be quite fast in artifact-ree analysis of thin highly conducting binary gratingsut time-consuming and memory-hungry when applied tohick highly conducting binary gratings.

ig. 8. Spatial distribution of �Hy� calculated by RCWA with=35 applied to a slightly lossier structure with ns=0.05−10j.he groove width of the grating is once again fixed at=252.4 nm and the grating thickness is d=1 �m.

ig. 5. Minus-first reflected order versus the truncation orderN� obtained by LPEM with M=10 (solid curve), M=9 (dashedurve), and M=8 (dotted curve). The groove width of the gratings g=252 nm and its thickness is d=1 �m. The reference value isbtained by using an extrapolation technique based on RCWA

ig. 6. Spatial distribution of �Hy� calculated by LPEM with=29 and M=10 applied to the lossless metallic grating with

s=−10j. The groove width of the grating is fixed at g

ig. 7. Spatial distribution of �Hy� calculated by LPEM with=40 and M=10 applied to the lossless metallic grating with

s=−10j. The groove width of the grating is fixed at=252.4 nm and its thickness is d=1 �m.

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A. Khavasi and K. Mehrany Vol. 26, No. 6 /June 2009/J. Opt. Soc. Am. A 1471

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