Evgeny Popov, Michel Neviere, Boris Gralak, and Gerard Tayeb
Institut Fresnel, Unite Mixte de Recherche du Centre National de la Recherche Scientifique 6133,Faculte des Sciences et Techniques de St-Jerome, 13397 Marseille Cedex 20, France
Received March 23, 2001; revised manuscript received May 31, 2001; accepted May 31, 2001
1. INTRODUCTIONWhen almost 25 years ago Moharam and Gaylord pub-lished their first work1 on a method for modeling diffrac-tion gratings, a method later called the ‘‘rigorous coupled-wave (RCW) method,’’ they could probably hardly imaginehow long would be the history of this method. A yearlater2 they proposed to use the method for surface reliefgratings. Although the method was initially designatedfor lamellar gratings, the idea that each grating profilecould be represented as a staircase approximation of sev-eral or more layers of lamellar gratings naturally came tomind to extend the universality of the method. However,as happened sooner or later with all the methods, prob-lems arose in dealing with deeper gratings. The first rea-son was that the growing exponential terms3,4 eliminatedusing an approach derived from chemistry5,6 known asthe R-matrix propagation algorithm, which was soon re-placed by the much simpler S-matrix propagationalgorithm.7,8 This, however, did not help when highlyconducting gratings were used in TM polarization, withthe magnetic-field vector parallel to the grooves. Re-cently Lalanne et al.9 found an empirical solution for thisproblem, which was put on a strong mathematical basisby Li.10 The rules pointed out by Li were used by Popovand Neviere11,12 to propose a fast converging formulationof the differential theory for arbitrary-shaped gratings,which gave spectacular results in TM polarization.
The current situation is that the RCW theory is verysuitable for lamellar metallic and dielectric gratings be-cause it gives rapidly converging results. The questionthat remains is whether the method can have the samesuccess when applied to arbitrary profiles. Despite the
long history of its development, there are only a few nu-merical results published in the literature that addressthe question. Moreover, these studies concern either di-electric gratings or metallic gratings in TE polarizationonly. The few published studies on deep metallic grat-ings with profile different from the lamellar one13 do notpresent sufficient accuracy, because the staircase approxi-mation only contains only a few ‘‘levels’’ (or ‘‘slices’’).14
The aim of this paper is to present a comparative studyof three different methods based on the differentialtheory. These are the classical differential method, re-cently improved by using the fast Fourier factorization(FFF) technique,11,12 the RCW method, and the modalmethod.15–18 The choice is linked to the fact that thethree methods are similar: The solution of Maxwell dif-ferential equations is sought by projecting the solution onsome basis of functions of x (see Fig. 1), which are periodicalong the grating surface. Then the field dependence inthe vertical ( y) direction along the groove height is foundby using different techniques:
1. The classical differential method uses Fourier basisin x with numerical integration of a finite set of ordinarydifferential equations in y.
2. The RCW method also uses Fourier basis in x, butthe grating is lamellar. Thus the coefficients of the dif-ferential system are constant along y, and the methodavoids numerical integration by using an eigenvalue/eigenvector technique to find the solution of the set of dif-ferential equations.
3. The modal method projects the field components on abasis of functions of x, which are rigorous solutions of theMaxwell equations, and boundary conditions along x
2002 Optical Society of America
34 J. Opt. Soc. Am. A/Vol. 19, No. 1 /January 2002 Popov et al.
(modes), assuming infinite height of the grooves along y.The solution of the diffraction problem is found in this ba-sis by matching a superposition of the modes (with un-known coefficients) with the Rayleigh expansion in thesubstrate and the cladding.
It is clear that the RCW and the modal method are welladapted to lamellar gratings, while the classical differen-tial method can be used for arbitrary profiles but pays aprice by using numerical integration. In general, themodal method requires fewer numbers of basis functionsthan the methods based on the Fourier representation.On the other hand, finding the modes is more difficult formetallic gratings, but the problem is already solved.15–18
It is important to note that, when applied to a givengrating, consisting of a smooth profile represented with afixed number of slices in the staircase approximation, thedifferential and the RCW methods give identical results,even when the number of the Fourier components is notsufficient to correctly represent the field. This fact showsthat if a problem exists, it is not due to the techniquesused to integrate the set of equations, i.e., it is not an al-gorithmic artifact. However, the results obtained for thediscretized (staircaselike) and nondiscretized profile maydiffer significantly. The aim of this paper is to study thisproblem.
2. SINUSOIDAL ALUMINUM GRATING INTE POLARIZATIONWe begin by studying the staircase approximation in TEpolarization (electric field vector parallel to the grooves).The grating under investigation is made of aluminum andhas a sinusoidal profile with period d 5 0.5 mm andgroove depth h 5 0.2 mm. This is a typical widely usedgrating in lasers and spectroscopy, because it is character-ized by almost perfect blazing (very-high diffraction effi-ciency) in TM polarization in a wide spectral range in thevisible.
The grating is illuminated at an angle u i 5 40° withlight of a wavelength l 5 0.6328 mm; thus two diffractedorders propagate in air. The aluminum complex refrac-tive index is nAl 5 1.3 1 i7.6. In what follows we are in-terested mainly in the convergence rate of the differentmethods as a function of the so-called truncation param-eter N, which characterizes the number of Fourier compo-nents (or modes) used in the field expansion along x, this
Fig. 1. Schematic representation of a sinusoidal profile and afive-stair approximation of the same profile, together with somenotations used in the text.
number being equal to 2N 1 1 (varying from 2N to N inthe case of Fourier basis). However, to start with, it isinteresting to know the number M of slices that are nec-essary to correctly represent the sinusoidal profile (Fig.1). As a rule of thumb, one can expect that the charac-teristic dimensions of the stairs must be less than l/50.And, indeed, Fig. 2 points out that a sufficient accuracy(.1%) is already obtained with approximately 15 slices.The figure represents the mean error, equal to the sum ofthe relative errors in all the propagating orders, dividedby their number. In the calculations, the truncation pa-rameter N is fixed equal to 90, a value sufficient to pro-vide convergent results for both the differential and theRCW method when they are applied to the same dis-cretized profile. The error decreases monotonically andbecomes close to 1024 for M . 200. The reference valuesto calculate the error are calculated by two independentmethods, namely, the integral method19 and the methodof fictitious sources.20
The convergence with respect to the truncation param-eter N for the three methods is presented in Fig. 3. Thedifferential method is applied to the smooth sinusoidal
Fig. 2. Mean error in the reflected diffraction orders as a func-tion of the number of slices in the staircase approximation (Fig.1) of an aluminum grating with a sinusoidal groove profile, pe-riod d 5 0.5 mm, groove depth h 5 0.2 mm, aluminum refractiveindex nAl 5 1.3 1 i7.6, illuminated at 40° incidence with TE po-larized light with wavelength l 5 0.6328 mm.
Fig. 3. Convergence of the mean error for the grating with pa-rameters given in Fig. 2 as a function of the truncation param-eter N, the number of the Fourier components or modes beingequal to 2N 1 1. The RCW method (heavy solid curve) and themodal method (thin curve) are used with M 5 200 slices, the dif-ferential method (curve ‘‘diff.’’) is applied to the smooth sinu-soidal profile. TE polarization.
Popov et al. Vol. 19, No. 1 /January 2002 /J. Opt. Soc. Am. A 35
profile, and the modal and the RCW methods are appliedto the staircase profile with M 5 200 slices. The resultsof the modal method are not presented for large N, be-cause the corresponding computer code deals with themore general case of crossed gratings in conical diffrac-tion, which needs more computational resources for thesame values of N.
For this polarization we can conclude that the threemethods lead to the same error for a given truncation andthat a relative error of less than 0.1% is obtained withN 5 10, which is a good indication for short computationtimes.
3. TM CONVERGENCEIn TM polarization, the zeroth-order efficiency is veryclose to zero, so that the relative error is an unreliablecharacteristic, and thus in what follows we present theconvergence rates of the 21st order efficiency. Figure 4presents the calculated efficiency as a function of N forthe different methods and the different numbers of slicesM. Again, the differential method is applied to the realsmooth sinusoidal profile. Its numerical implementationis described in Refs. 11 and 12; i.e., a set of first-order dif-ferential equations is integrated taking as unknown func-tions the Fourier components of both electric and mag-netic fields; the FFF is used to ameliorate theconvergence. Again we find that when applied to thestaircase profile, the differential method gives the sameresults as the RCW method. This proves that the resultsdo not depend on the algorithm used for the integration:when implementing the differential method, we indepen-dently applied the Runge–Kutta and Adams–Moultontechniques; in addition, two codes developed indepen-dently by two authors have been used to confirm theresults.
For the smooth profile (curve ‘‘diff.’’ in Fig. 4), theasymptotic value approaches 1% as soon as N 5 11 (23Fourier components). Using thirteen layers in theS-matrix propagation algorithm and four steps of integra-tion per layer, the calculation time is 1 s on a PC PentiumIII (800 MHz).
When M 5 1, the sinusoidal grating is replaced by alamellar one; Fig. 4(a) shows that the RCW and the modalmethod give results having the same convergence rate asa function of N. The convergence rate for the discretizedprofile is the same as for the smooth sinusoidal profile(curve ‘‘diff.’’), although the asymptotic efficiency differssignificantly for the two profiles, which is not surprising.This shows that the single-step approximation is ratherpoor and that the number of slices M must be increased.When M is increased to ;20 [Fig. 4(b)], the asymptoticvalue of the efficiency approaches to within 1% of the ef-ficiency of the smooth profile (almost in the same way asin the TE polarization, where M 5 15 was sufficient).However, the number of Fourier coefficients (or modes, forthe modal method) required to approach the asymptoticvalue increases for both the RCW and the modal methods.To obtain the same accuracy (error ,1%), it is necessaryto increase N up to 20 for the modal method and up to50 for the RCW method (i.e., 101 Fourier spatial harmon-ics), requiring 30 s calculation time, which has to be com-
pared with 1 s necessary for the differential method whenmodeling the smooth sinusoidal profile with the sameaccuracy.
The situation becomes worse for M 5 200, Fig. 4(c).Even N 5 100 is not sufficient for the RCW method to
Fig. 4. Convergence of the 21st order efficiency in TM polariza-tion of the RCW and modal methods (indicated in the figure) (a)for a single-step (M 5 1) lamellar grating, (b) for a staircasegrating with M 5 20, (c) M 5 200, compared with the conver-gence of the differential method for a smooth sinusoidal profile(curve ‘‘diff.’’). The grating parameters are given in Fig. 2.
36 J. Opt. Soc. Am. A/Vol. 19, No. 1 /January 2002 Popov et al.
converge (we stress again that the same results are ob-tained when the differential method is applied to thestaircase profile). To explain this surprising deteriora-tion of the convergence rate, it is necessary to study thefield map inside the grooves.
4. NEAR-FIELD MAPS:TM POLARIZATIONIn order to understand the poor convergence of all themethods for staircase profiles in TM polarization, we ana-lyzed the near-field maps for two discretized profiles, withfive and twenty slices, as well as the field map of thesmooth sinusoidal profile. In TM polarization themagnetic-field vector is parallel to the grooves (i.e., to thez axis) and is a continuous function even across the grat-ing surface. The electric field has x and y components, Exand Ey , respectively. For five slices, Fig. 5 shows thesurface of uExu2 as a function of x and y inside the groove,characterized by a smoothly varying background withsharp peaks close to the profile ridges. The field maps fora staircase profile were obtained with the RCW method,while the differential method was used for the sinusoidalprofile (see Fig. 8 below). For a better analysis, Figs. 6(a)and 6(b) represent gray-scale maps of the two componentsuExu2 and uEyu2, respectively. Strong maxima near thestep ridges are observed, which could be expected for ametallic grating having sharp ridges, owing to the chargeaccumulation at the ridges. However, the maxima ofuExu2 and uEyu2 do not occur at the same location. This isdue to the fact that the two components of the electric-field vector are discontinuous across the different seg-ments of the profile: uExu2 is continuous across the hori-zontal part (parallel to the x axis). Its jump across thevertical segments (which are due to the large jump of therefractive index) allows for the existence of the sharpspeaks on the vertical segments, while the continuity
Fig. 5. Three-dimensional view of the distribution of the x com-ponent of the field squared inside the groove for a five-step stair-case approximation in TM polarization.
along the y axis and the fact that the electric field is weakinside the metal limits uExu2 in the vicinity of the horizon-tal segments. The opposite is true for uEyu2, which ex-plains why its maxima occur on the horizontal segments.In any case, the maxima of the two components occurclose to the ridge, as already discussed.
It must be stressed that the existence of these sharppeaks is not due to Gibbs phenomena arising from therepresentation of the discontinuous functions Ex and Eyby truncated Fourier series. The convergence of the fieldmaps was checked and no visible difference was foundwhen going from 161 to 321 Fourier components.
A greater number of stairs introduces a larger numberof edges and smaller features of the profile segments. Athree-dimensional view of uExu2 and uEyu2 as a function ofx and y for M 5 20 is shown in Fig. 7 with a zoom insidethe groove region where the field is stronger. Well-pronounced peaks of uExu2 and uEyu2 are observed close tothe edges in the same manner as for M 5 5, and againthe peaks of uExu2 and uEyu2 are spatially separated. Theshorter length of the profile segments leads to narrowerpeaks.
The fact that close to the ridges of the staircase profileone observes regions with field enhancement explains thedeterioration of the convergence rate of the methods withincreasing number of slices: The greater the number ofstairs, the closer and thinner the maxima and the greatertheir number and thus the greater the number of Fouriercomponents required to correctly represent the field inthe form of truncated series. In the modal method, thisleads to the increase in the number of modes inside thegrooves, although not to as great an extent as for theRCW method, probably because the modes are not equi-distant spectrally, as the Fourier harmonics are.
The field enhancement close to the ridges occurs whenany of the three methods are used, and it is a real physi-cal effect because the smooth profile is replaced with astaircaselike profile having ridges. To observe the differ-ence, it is sufficient to compare the previous figures withthe field map of the sinusoidal (nondiscretized) grating,presented in Fig. 8(a), obtained by use of the differentialtheory. A closer view [Fig. 8(b)] of the same region aswas presented in Fig. 7(a) shows no peaks close to the pro-file, and the field inside the groove [on the left part of Fig.8(b)] is the same as the background field (outside the peakregion) inside the grooves in Fig. 7(a) (approximatelyequal to 0.30). The weak variations of the field in Fig. 8come from the Gibbs phenomena, as the calculations weremade using 201 Fourier components. The absence ofsharp and narrow peaks of the field for the smooth profileexplains the fast convergence of the differential methodwith respect to the number of Fourier components. Asdiscussed in Section 3, 23 Fourier components aresufficient.
5. NEAR-FIELD MAP: TE POLARIZATIONIt is now necessary to explain the fast convergence of thethree methods in TE polarization that are independent ofthe number of slices used in the staircase approximation.For that reason, it is worth studying the field maps.Only the region of the groove where the field is the stron-gest will be shown.
Popov et al. Vol. 19, No. 1 /January 2002 /J. Opt. Soc. Am. A 37
Fig. 6. Field map inside the groove of a five-step staircase grating in TM polarization. (a) uExu2, (b) uEyu2.
38 J. Opt. Soc. Am. A/Vol. 19, No. 1 /January 2002 Popov et al.
In TE polarization, both electric and magnetic fieldsare continuous across the grating surface, because weconsider nonmagnetic media. For M 5 5, Figs. 9(a) and9(b) show, respectively, the field maps of uHxu2 and uHyu2
for the same grating, studied in the previous sections, butin TE polarization. Peaks are found in the maps, butthey rise to a lesser height from the smooth backgroundthan in the TM case. This is because the two componentsof the magnetic field (as well as the only component,the transverse one, of the electric field) are continuousfunctions when crossing the profile. As can be observed,the field ‘‘enters’’ the metal inside the stairs, and its varia-tions are not so rapid as in the TM case. This fact
Fig. 7. Spatial field distribution in the vicinity of several stepsinside the groove of a 20-step staircase profile, used to approxi-mate the sinusoidal grating under study. TM polarization, (a)uExu2, (b) uEyu2.
permits averaging the peaks when the number of slices isincreased so that the step dimensions decrease. The ef-fect could be observed for the 20-stairs approximation, asshown in Fig. 10, where the height of the peaks is furtherreduced, compared with five slices. Thus when M is in-creased in the TE case, it is not necessary to increase thenumber of Fourier harmonics in the field representation,which explains the better convergence rate, already ob-served in Fig. 3.
Fig. 8. Spatial distribution of uExu2 calculated for a nondis-cretized sinusoidal profile in TM polarization. (a) The entiregroove region as in Fig. 5, (b) the same region as presented inFig. 7.
Popov et al. Vol. 19, No. 1 /January 2002 /J. Opt. Soc. Am. A 39
6. LIMIT FOR AN INFINITE NUMBER OFSTAIRSThe effect of field ‘‘homogenization,’’ i.e., the averaging ofthe peaks when the number of slices is increased in theTE case (as shown in Figs. 9 and 10), raises the questionof the possibility of such a homogenization in TM polar-ization, when the number of slices becomes so large thatthe adjacent maxima on the consecutive ridges start tooverlap. Unfortunately, numerical requirements put a
Fig. 9. Spatial field distribution inside the groove region of afive-step staircase grating, presented in Fig. 1, in TE polariza-tion. (a) uHxu2, (b) uHyu2.
limit on the numerical investigations, so that no analysisof the near-field map for, say, M 5 200 or 1000 is possible,because this would require at least ten times more Fou-rier components to distinguish between Gibbs phenomenaand the real-field fluctuations close to the ridges.
In any case, Fig. 4(c) already gives an answer about theeffect when going to M 5 200; the convergence is poor.Some unreported results have been obtained for M5 500; the convergence rate is not better. This meansthat no homogenization appears in the TM case, whateverthe number of slices may be. In order to model this phe-nomenon, instead of increasing M (and thus N), we maychoose an alternative approach that consists of keeping Mand N unchanged but increasing the wavelength. Infact, increasing the wavelength while preserving thenumber of slices (and thus their dimensions) is equivalentin the homogenization study to reducing the slice thick-ness and keeping the wavelength unchanged. The fieldmaps inside the same groove region as shown in Fig. 7 arerepresented in Fig. 11 when the wavelength l is increased20 times, so that the dimensions of each step in the stair-case profile with M 5 20 are of the order of l/1300. Asobserved, no homogenization occurs, and sharp peaks arestill present close to the step ridges. It is important tonote that the three numerical methods use the sameequations [see Eqs. (5)–(6) below] and the same boundaryconditions along the vertical and the horizontal segmentsof the steps. Thus the physically strange result that nohomogenization of the near field occurs may be linked tothat fact; i.e., it may be a numerical artifact. The nextpart of this section deals with this problem.
In order to clarify whether these sharp peaks exist inthe limit when M → ` and why they continue to play animportant role in the far-field (efficiency) properties, al-though their width decreases with increasing the numberof slices M, it is necessary to consider the limits of theMaxwell equations and of the boundary conditions for theelectromagnetic field.
Fig. 10. Distribution of uHxu2 inside the groove for a 20-stepstaircase approximation of the sinusoidal grating in TE polariza-tion. Same groove region as in Fig. 7(a).
40 J. Opt. Soc. Am. A/Vol. 19, No. 1 /January 2002 Popov et al.
Recent studies,10–12 already mentioned in the introduc-tion, have shown that correctly writing the Maxwell equa-tions in the infinite Fourier basis does not guarantee thatcorrect and stable numerical results will be obtained aftertruncation. The properties of truncated Fourier series ofdiscontinuous functions require that the projection of theMaxwell equations on a truncated Fourier basis must de-pend on the direction of the vector normal to the gratingsurface, where the field and permittivity are discontinu-ous. Thus different forms of the equations have to be in-tegrated for the smooth sinusoidal profile and for thestaircaselike profile, whatever the number M of the stairsmay be, including the case M → `.
Without going into details, which can be foundelsewhere,21 the equations in the truncated Fourier spacein TM polarization are
Fig. 11. Same as in Fig. 7, except for the wavelength l5 13 mm.
d@Hz#
dy5 2ivH S ieiiNy
2i 1 I 1
eI21
iNx2i D @Ex#
2 S iei 2 I 1
eI21D iNxNyi@Ey#J , (1)
d@Ex#
dy5 2ivm0@Hz# 1 ia@Ey#, (2)
in which @Ey# is derived from
@Ey# 5 H ieiiNx2i 1 I 1
eI21
iNyi2J 21H a
v@Hz#
1 S iei 2 I 1
eI21D iNxNyi@Ex#J . (3)
Nx and Ny are the components of the unit vector locallynormal to the profile, square brackets denote a columnvector symmetrically filled with Fourier components,double-straight-line brackets denote the truncatedToeplitz matrix composed of the Fourier components ofthe corresponding quantity:
i fimn 5 @ f#m2n . (4)
e is the permittivity, which depends on x and y, m0 is thevacuum magnetic permeability, v is the circular fre-quency, and a is a diagonal matrix with elements equal toamn 5 (2p/l)dmn@sin ui 1 m(l/d)#. Although the compo-nents of the normal vector are defined on the profile only,it is necessary to continue them everywhere in space in anappropriate way,11,12 in order to obtain their Fourier coef-ficients.
The lamellar profile (or any of the vertical segments ofthe staircase profile) is characterized by Nx 5 1 and Ny5 0. Then Eqs. (1)–(3) are reduced to
d@Hz#
dy5 2ivI 1
eI21
@Ex#, (5)
d@Ex#
dy5 2ivm0@Hz# 1 iaiei21
a
v@Hz#. (6)
The classical differential method without the FFF im-provement uses a similar set of equations derived fromEqs. (1)–(3) with Nx 5 0 and Ny 5 1:
d@Hz#
dy5 2iviei@Ex#, (7)
d@Ex#
dy5 2ivm0@Hz# 1 iaI 1
eI a
v@Hz#. (8)
Without truncation, i.e., when an infinite number ofFourier components is considered, iei 5 i1/ei21 and thethree formulations are equivalent. However, this is notpossible numerically, and they give quite different resultsfor a finite N.
It is important to note that whatever the value of M, inthe case of the staircase profile both the differential andthe RCW method use Eqs. (5) and (6), whereas for thesmooth profile one has to use the system of Eqs. (1)–(3).This means that in the limit of M → `, the sets of equa-
Popov et al. Vol. 19, No. 1 /January 2002 /J. Opt. Soc. Am. A 41
tions to integrate are different if N is finite. It is thusinappropriate to expect identical results for the smoothand the staircase profiles. Moreover, if we now considerthe different field components on the grating profile,whatever the value of M is, Ex is continuous across thehorizontal segments (where Ey is discontinuous) and dis-continuous along the vertical parts (where Ey is continu-ous). None of the two situations is valid for a continuoussinusoidal profile, for which the continuous (tangential)and discontinuous (normal) components are neither hori-zontal nor vertical (except at the bottom and top of thegrooves). Thus neither the equations nor the field com-ponents of the staircase approximation tend toward theequations, or the field components for a smooth (sinusoid-al) profile, however finite the truncation N is.
One of the possibilities for improving the convergenceof the staircase approximation when the number of slicesis large, in order to optionally avoid the numerical inte-gration, is to use the set of Eqs. (1)–(3) with the vectornormal to the sinusoidal profile (or the other nonstaircaseprofile to which the staircase approximation is applied)instead of Eqs. (5) and (6). This will not be correct for asmaller number of slices, because the set of Eqs. (1)–(3)must be used only for the true profile, whereas Eqs. (5)and (6) are valid for the multilamellar profile. But com-putations have shown that when M → `, Eqs. (1)–(3) are
Fig. 12. Convergence rates of the RCW method compared whenusing the three different sets of equations as marked on the fig-ure [Eqs. (1)–(3), (5) and (6), and (7) and (8)], compared with thedifferential method used for the sinusoidal profile. (a) l5 0.6328 mm, M 5 20, and M 5 200; (b) l 5 13 mm and M5 20. The convergence of the differential method for a sinu-soidal profile is shown by a solid curve marked ‘‘diff.’’
better suited to describe the fact that Ex or Ey is, in gen-eral, not continuous on the surface of an arbitrary shapedgrating.
To that end, Fig. 12 presents the convergence rates ofthe different sets of Equations (1)–(3), (5) and (6), and (7)and (8) for a 20-step staircase profile. The numericalprocedure is the following: Along the vertical y coordi-nate inside each slice (step), the coefficients in the differ-ential set of equations to be integrated are taken constant(to permit use of the RCW method); however, these coef-ficients are calculated in three different manners [Eqs.(1)–(3), (5) and (6), and (7) and (8)]. As can be observed[Fig. 12(a)], the results are not satisfactory: While Eqs.(5) and (6) converge slowly but monotonically, Eqs. (7) and(8) converge more slowly; applying Eqs. (1)–(3) to thestaircase profile gives better results for smaller N, but theimprovement is not preserved for larger N. This showsthat the number of slices M 5 20 is not sufficient to rep-resent the ‘‘smooth’’ profile with the ‘‘smooth’’ set of Eqs.(1)–(3). However, if the wavelength is again increased to13 mm [Fig. 12(b)], as is already done in Fig. 11, the con-vergence rate of Eqs. (1)–(3) is quite rapid and the resultsare more stable (but still less stable than those obtainedwith the differential method). Note the zoom of the ordi-nate to reveal the differences between the different meth-ods. The same conclusion is valid when the number ofslices is increased to 200 slices with l 5 0.6328 mm; theresults obtained with Eqs. (1)–(3) and M 5 200 nearly co-incide with the results of the differential method [alwaysEqs. (1)–(3)] applied to the sinusoidal profile [curve ‘‘diff.’’in Fig. 4(a)]. Moreover, as already observed in Fig. 8, inthat case the field is homogenized and no sharp peaks oc-cur.
The conclusion is that if one is interested in the studyof a true staircase profile having a comparatively lownumber of slices, so that the slice height is comparablewith the wavelength, it is correct to use Eqs. (5) and (6),because they correctly describe the field discontinuitiesand the edge effects (sharp field peaks close to the ridges).When the methods are applied to the staircase approxi-mation of a smooth arbitrary profile with a very largenumber of slices (theoretically, in the limit when M → `),it is much better to use the set of Eqs. (1)–(3), which bet-ter describe the effect of homogenization. However, thisapproach is equivalent to using the differential method(improved by the FFF technique) with the worse algo-rithm of integrating the differential equations: the rect-angular rule. And indeed, when the number of slices islarge and the coefficients of the system to be integratedare taken to be constant within each slice, whatever themethod of integration may be, this is equivalent to inte-grating the system by using the rectangular rule withequidistant and fixed points of integration. It is wellknown that standard integration algorithms (e.g.,Runge–Kutta or Adams–Moulton techniques) do muchbetter.
Our conclusion, that when the slice height tends to-ward zero it is better to use Eqs. (1)–(3) than Eqs. (5) and(6), explains the observation made by Lalanne.22 He con-sidered a lamellar (M 5 1) grating in the low-modulationlimit (h/d → 0). He found that the classical Eqs. (7) and(8) worked better than Eqs. (5) and (6), contrary to what
42 J. Opt. Soc. Am. A/Vol. 19, No. 1 /January 2002 Popov et al.
happened for a finite-depth lamellar grating, for whichEqs. (7) and (8) failed to converge rapidly. As concludedhere, when the slice height (equal to h/d for a single-stepprofile) tends toward zero, it is better to use Eqs. (1)–(3).However, in the case of a single-step lamella with h/d→ 0, Eqs. (1)–(3) tend toward Eqs. (7) and (8), becauseNx → 0 and Ny → 1 everywhere along the profile. Thusit is better to use Eqs. (7) and (8) instead of Eqs. (5) and(6) when h/d → 0. To repeat, the three approacheswould become equivalent only without truncation.
7. CONCLUSIONThe validity of the staircase approximation used to de-scribe arbitrary-shaped gratings in the RCW and themodal methods is studied numerically. Although the ap-proximation leads to reliable results in TE polarization,we show that for metallic gratings used in TM polariza-tion, the profile ridges introduced by the staircase ap-proximation cause sharp maxima in the local field.These maxima are independent of the method of modelingand are due to the replacement of the smooth profile by astaircaselike one. The maxima do not exist for thesmooth profile. The number of maxima increase with thenumber of steps, which requires a greater number of basisfunctions (Fourier components or modes) to represent thefield. No homogenization is observed when the numberof the slices and/or the wavelength are increased, evenwhen the number of steps exceeds 500 (with the step di-mensions as small as wavelength/1300). This resultsfrom the choice of the way in which Maxwell equationsare projected onto a truncated Fourier basis. When thecorrect projection is used by applying the rules for Fourierfactorization of the products of discontinuous functions,as is done with the differential method, the convergence ismuch more rapid.
The conclusion is that although the RCW and themodal method are well suited for lamellar profiles or forgratings consisting of a stack of rectangular rods, becausethey lead to a short computation time they cannot com-pete with the differential method in the study ofarbitrary-shaped metallic gratings in TM polarization.
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