Vol. 6, No. 4/April 1989/J. Opt. Soc. Am. A 543

Analysis of diffraction from echelette gratings, using astrip-current model

Amir Boag, Yehuda Leviatan, and Alona Boag

Department of Electrical Engineering, Technion-Israel Institute of Technology, Haifa 32000, Israel

Received May 19, 1988; accepted December 20, 1988

A method is presented for analyzing electromagnetic scattering from an echelette grating separating two contrastinghomogeneous media and illuminated by a plane wave. The reduction of the general problem to a consideration ofthe fields over a suitably selected period, referred to as the unit cell, is facilitated by the Floquet theorem. Thesolution, in the p-polarization case (electric field parallel to the grooves), uses sets of spatially periodic and properlymodulated fictitious electric-current strips to simulate the field scattered by the grating boundary surface and thefield penetrated through the surface. In the s-polarization case (magnetic field parallel to the grooves), which is notexamined in this paper, sets of fictitious magnetic-current strips, instead of electric ones, should be used. The fieldsradiated by the current strips are expressed in terms of Floquet modes and are adjusted to fit the continuityconditions for the tangential components of the electric and magnetic fields at a finite number of points on thegrating surface within the unit cell. Special attention is given to the behavior of the fields at the corners. Theprocedure is simple to perform and is applicable to gratings of arbitrary cross section. Perfectly conductinggratings are treated as reduced cases of the general procedure. Results are given and compared with existing data.The efficiency of the suggested method is demonstrated.

1. INTRODUCTION

Two-dimensional electromagnetic scattering of a plane wavefrom echelette gratings has been the subject of many studies.These studies are of great interest because diffraction grat-ings with relatively deep grooves and small grating periods,which are now easy to fabricate, have numerous applicationsin optics. Some of the previous theoretical and experimen-tal treatments of this subject can be found in Refs. 1-6.

In this paper we extend the results of a previous paper7 inwhich we showed how plane-wave scattering from nonplanarperiodic surfaces of arbitrary smooth shape can be analyzedin a simple and efficient manner by using a strip-currentmodel. The basic idea in the approach suggested in Ref. 7 isas follows: Instead of using surface integral equations tosolve for conventional electric and magnetic surface cur-rents, we solve for fictitious strip currents that lie a distanceaway from the surface by adjusting the amplitudes of thesesources to fit the continuity conditions at a finite number ofpoints on the boundary. More specifically, we set up simu-lated equivalences for the regions on both sides of the peri-odic interface by using two sets of spatially periodic andproperly modulated strips of electric current. The fieldscattered by the surface is simulated by the field of a set ofelectric-current strips placed a distance beneath the inter-face, and the field refracted into the region beneath theinterface is simulated by the field of a set of electric-currentstrips placed a distance above the interface. The strips arecharacterized by a common Fourier-transformable electric-current-density profile, which for each periodic source ismultiplied by a yet-to-be-determined constant complex am-plitude. The strips are of infinite extent in the axial direc-tion, parallel to the direction of the periodicity, and areassumed to radiate in an unbounded homogeneous spaceoccupied with the same medium as that of the correspondingregion whose fields are simulated. Since the strips are par-

allel to the direction of the periodicity, the field that eachperiodic source is radiating can be represented analyticallyby means of a discrete spectrum of outgoing and decayingplane waves known as Floquet modes. Furthermore, asthese current strips lie some distance away from the bound-ary surface, their fields constitute a set of smooth periodicfield functions on the surface that may potentially span theactual smooth field on the boundary. The above character-istic, namely, that these currents not only yield smoothfields on the surface but also enable us to determine thesefields anywhere analytically, is the key advantage of theproposed approach. This quality is highly appealing, as onecan avoid laborious surface-current integrations when com-puting the fields at the various stages of the solution. Inaddition, since we are actually using a basis of smooth fieldfunctions for representing fields on the boundary, a simplepoint-matching procedure can be selected conveniently fortesting.

In this paper we develop further the technique describedin Ref. 7 by extending it to handle echelette gratings. Theanalysis proceeds along parallel routes, but special care mustnow be given to the edges, where the field can become singu-lar. The modification is effected by locating sources close tothe edges in addition to those that can well represent thesmooth fields on the boundary. These extra sources renderthe spanning of the fields near the edges, at least to somedesired accuracy, feasible.

2. PROBLEM SPECIFICATION

Consider an echelette grating formed by a boundary surfaceseparating two contrasting media. The grating is periodicalong the x axis, with a period p, and uniform along the y axisof a rectangular coordinate system. A cross section of thegeometry together with a relevant coordinate system isshown in Fig. 1. It should be noted that, according to our

0740-3232/89/040543-07$02.00 © 1989 Optical Society of America

Boag et al.

544 J. Opt. Soc. Am. A/Vol. 6, No. 4/April 1989

-ZFig. 1. General problem of scattering from an echelette grating.

convention, the z axis is oriented downward. For futureconvenience, we refer to the upper region in Fig. 1 as region I,characterized by a permeability ,I and a permittivity eI.Similarly, the lower region in Fig. 1 is referred to as region II,characterized by a permeability a,, and a permittivity ell.Both media can be dissipative. Thus !,, EI, Atn, and ElI areallowed to be complex.

In the case of p polarization (electric field parallel to thegrooves), a plane wave given by

Einc = exp[-j(k inx + k incz)] (1)

with harmonic (eiwt) time dependence assumed and sup-pressed, is incident upon the grating from region I. Here uydenotes the unit vector in the y direction and kxinc and kzincdenote, respectively, the x and z components of the wavevector of the incident field. An explicit expression for theincident magnetic field HinC is not given here, as it is readilyderivable from Einc of Eq. (1). Because Einc is y directed andindependent of the spatial y direction and since the gratingis also uniform along the y direction, the electric fields inboth regions are also y directed and independent of thespatial y direction. The problem is thus a two-dimensionalone and can be worked out entirely in some y = constantplane. To this is adjoined the Floquet theorem, whichstates that the field distribution over a periodic structureilluminated by a plane wave remains unchanged under atranslation of the observation point through a period p in thedirection of the periodicity, and its amplitude is multipliedby a complex constant exp(-jkxincp), which corresponds tothe variation of the incident field with this direction. Byinvoking the Floquet theorem, the problem can be reducedreadily to a consideration of the fields over a single period ofwidth p, referred to also as the unit cell. For future conve-nience C will denote the portion of the boundary surface thatis included in a unit cell. Our objective, in general, is todetermine the excitation of the various spectral orders re-flected into region I and refracted into region II.

3. FORMULATION

We now describe how the simulated equivalences for theoriginal situations in regions I and II are set up. Thesesimulated equivalences are shown in Figs. 2 and 3. Forconciseness we use the character r to indicate parameterspertinent to region I (r = I) or to region II (r = II). In thesimulated equivalence for region r, the electromagnetic field(Er, Hr) diffracted from the surface is simulated by the fieldof a set of N fictitious periodic strips of electric current JrJ',i = 1, 2, ... , Nr, each centered at a source point (r, Zr)outside what was originally region r at some distance away

X from C. These sources are assumed to radiate in an un-bounded homogeneous space characterized by (ur, Er). Theelectric field Er at an arbitrary observation point (x, z) inregion r is given by

where Eyr is the electric field that is due to the ith periodicsource described by the current-density distribution

jir = Ir6(z - zir)exp[jkxin(x - Xr)] f(x - Xr - np),

X ~~~~~~~~~~n=-(3)

in which Ii is a yet-to-be-determined constant complex am-plitude. Again, an explicit expression for the magnetic fieldH' is not given, as it is readily derivable from E' of Eq. (2).The function f(x) in Eq. (3) is a real-valued window functionof width s (s << p) characterized by a continuous profilethat is zero for all x outside the interval -s12 < x S s/2 andhas a piecewise continuous derivative on that interval. Un-der these conditions the periodic extension of f (x) appearing

unbounded homogeneous space

(LI ,I)

(EI +EincHI-Hinc)

Einc

Hinc

mathematicalboundary

unit cell

K- P -H

]\~~~~~~~~~~~~~~~~~~~~~~~~~~

\An

periodic current strips

Einc

Hinc

gratingboundary

region I

HLICI)

n

N

E' = ay E Eiyr,i=1

(2)

Fig. 2. Simulated equivalence for region I.

Boag et al.

Vol. 6, No. 4/April 1989/J. Opt. Soc. Am. A 545

unbounded homogeneous space

('aI )

unit cell

I- p -4II

I II II I n

In Fig. 2, (E', HI) is the electromagnetic field that is due tothe fictitious currents, and (Einc, Hiuc) is the field of theincident wave as given by Eq. (1). The total field (E' + Einc,HI + Hinc) in region I in Fig. 2 is an approximation of theactual field in region I in the original situation. The field(E"l, HI) in region II in Fig. 3 is an approximation of theactual field in region II in the original situation. It is impor-tant to point out that the location of the sources in the twosimulated equivalences has not been specified yet. As far asthe formulation is concerned, the location of the sources canbe arbitrary. The question of selecting source locationssuitable for a numerical solution is an important one. Wegive this issue further attention in Section 5.

The relationship between the electromagnetic fields (E' +EinC, HI + Hinc) and (E"l, HII) in the simulated equivalentsituations shown in Figs. 2 and 3, respectively, is dictated bythe field-continuity conditions at the boundary surface Cwithin the unit cell. This leads to the operator equations

zFig. 3. Simulated equivalence for region II.

in Eq. (3) can be represented by a Fourier series, and thefield Eyr originating from Jr can be expressed in terms ofFloquet modes. Thus we have

EY=-'2 i a exp-j[k(x - xir) + kzrlz- Zrl]I,2~rk~ri -, exJxk n=-m z

(4)

where -lr and kr are, respectively, the intrinsic impedanceand the wave number in region r, and

(5)

(6)

kxn = kxin + 27rn,

k, r = (k 2 k 2)1/2

h X (EI-EII) =-h X Einc on C,

h X (HI-HII) = -h X Hnc on C,

where h is a unit vector normal to C.To obtain an approximate solution, the two continuity

conditions [Eqs. (9) and (10)] are imposed at NC = (NI +NII)/2 selected points on C. This leads to a matrix equation

[Z]i= Vwhere

[Ze'] [Ze"] 1

L [Zh'] [Zh"] j

I V 1

P=V7hj

subject to the requirements that Re(kznr) > 0 and Im(kznr) <0, which stem directly from the radiation condition at IzI a. Here, kxn is the x component of the wave vector of thenth Floquet mode in region r. Also, kr and -kznr are,respectively, the z components of the wave vectors of the z-and -z-traveling nth Floquet modes in region r. The coeffi-cients an in Eq. (4) are the Fourier-series coefficients for theperiodic extension of f(x). We have

1 C'2 / 2r~n

an - f(x)expi -x)dx. (7)

A specific choice for f(x) that has been used in our numericalsolution is

f(x) = 0.42 + 0.5 cost2)rx + 0.08 4rx) (8)

It should be noted that the convergence rate of the Fourierseries to the periodic extension of f(x) in Eq. (8) is affectedby the choice of the strip-width parameter s. Clearly, whens is larger, the spectral spread of f(x) is narrower, and conse-quently the corresponding Fourier series converges faster.However, this would unavoidably occur at the expense of adegradation in the ability of the sources to simulate rapidfield variations.

(14)

In Eq. (12), [Zer] and [Zhr] are NC X N' matrices whose (1, i)elements are, respectively, the electric field Er and thetangential component of the magnetic field for r = I or thenegative of the electric field Er and the negative of thetangential component of the magnetic field for r = II bothdue to a periodic strip current Jr of unit amplitude (Ir = 1)evaluated at an observation point (xl, z) on C. It should bepointed out that, whereas the generalized impedance matrix[Z] usually depends on the scatterer but not on the excita-tion, in the present case [Z] clearly depends on the angle ofincidence. In Eq. (13), Jr is an Nr-element column vectorwhose ith element is the unknown amplitude Ir. Finally, inEq. (14), V and Vh are NC-element column vectors whoseIth elements are, respectively, the negative of Einc and thenegative of the tangential component of the incident mag-netic field at an observation point (xl, z) on C.

Having formulated the matrix Eq. (11), we can solve theunknown current vector in a straightforward manner. If[Z] is invertible, the solution to Eq. (11) is

= [Z] -,V, (15)

periodic currentstrips

(9)

(10)

(11)

(12)

(13)

Boag et al.

546 J. Opt. Soc. Am. A/Vol. 6, No. 4/April 1989

where [Z]-l is the inverse [Z]. Once the unknown currentvector is derived, from Eq. (15), one can readily evaluate theapproximate field in each region and other related quanti-ties of interest by exploiting the analytically known fields ofthe strip-current sources.

It should be pointed out that the case in which region II isa perfect conductor can be treated as a special case of theabove procedure. Formulating the simulated equivalencefor region I alone by using NI periodic current strips andsubsequently imposing the boundary condition on the tan-gential component of the electric field at NC = NI points onC leads to the matrix equation

[Ze'fi' = Ve' (16)

where [Zel], 71, and Pe are the same constituents that appearin Eqs. (12), (13), and (14), respectively, and are definedthereafter. Once Eq. (16) is formulated, the solution can beobtained in a way similar to that used in the general case.

We stated earlier that, if the boundary conditions [Eqs. (9)and (10)] were satisfied strictly by the Maxwellian andsource-free (in their respective regions) periodic fields (E',HI) and (EII, HII), then these fields would be the true fields.In our solution, however, we enforce these conditions only ata finite number of selected points within a unit cell on theboundary between regions I and II. Naturally, one canquestion the behavior of the fields on the boundary betweenthe matching points, as they can in general be quite differentfrom what is required by the boundary condition, therebyrendering the results for the field values in the two regionsinaccurate. Toward this end, we study the convergence ofthe boundary-condition errors AEbc and AHbc, defined by

AEbc = Ih X (E' + Einc - E") on C (20)

Ih X (HI + Hinc - HIY on Cbc- ---- H I... (21)

4. REPRESENTATION OF THE SCATTEREDAND THE REFRACTED FIELDS IN TERMS OFFLOQUET MODES

An alternative representation for the field El that is valid forobservation points (x, z) in region r above the highest (for r =I) or below the lowest (for r = II) point of C can be obtainedby substituting Eq. (4) into Eq. (2), using the fact that, forobservation points in the region r = I,

z < z for all i

and, for observation points in the region r = II,

z > zir for all i,

(17a)

(17b)

and subsequently interchanging the order of the two sum-mations involved. One obtains

n=-Er= a E Fnrexp[-j(kxnx =F kznrz r)], (18)

n=-

where

F~~~r=~~2kr = _ rGrnE rexp~j(k n~r F k2 nrzir)I. (19)

The upper and lower signs in the arguments of the exponen-tial functions in Eqs. (18) and (19) apply, respectively, toquantities pertinent to regions r = I and r = II. Equation(18), with Fnr given by Eq. (19), constitutes a field represen-tation for Er at observation points in region r above thehighest (for r = I) or below the lowest (for r = II) point of C interms of an infinite discrete set of Floquet modes.

The coefficients Fnr in Eq. (18), given by Eq. (19), arereferred to as the space-harmonic expansion coefficients forthe scattered and the refracted fields. They are parametersof interest in problems involving diffraction gratings.

5. NUMERICAL RESULTS

A computer program has been developed, using the formula-tion of the preceding sections. Some representative compu-tations obtained with this program are given in this section.

These boundary-condition checks certainly do not totallyvalidate the solution, but they can immediately indicatefaulty results.

An additional criterion that must be satisfied in the loss-less case is the conservation of the power flow along the zaxis. Hence it is important to determine whether the sum ofthe scattered and refracted powers equals the incident pow-er. To that end we examine the power-conservation errorAP, defined by

|PiC E Ypn'- _Epn II|n n

A'p = li. (22)

where the primed sums indicate that the summations aretaken over only the values of n for which knl and kll arereal. In Eq. (22),

in k h (23)

is the power flow per unit area in the z direction of theincident field, and

p r = k IFrI2 (24)

is the power flow per unit area in the negative z direction forr = I and in the positive z direction for r = II of the propagat-ing Floquet modes of order n. The power-conservationcheck also does not totally validate the solution. However,we surmise that a solution of a relatively large number ofunknowns, which satisfies not only the boundary conditionbut also the power-conservation law within some acceptablylow error, is not likely to be defective.

We consider a homogeneous plane wave that is traveling infree space, in which case k = ho, and is incident upon anechelette grating at some angle inc in the x-z plane mea-sured from the z axis (Fig. 1). The incident field is given byEq. (1), with

kin, = kosin 0inc (25)

Boag et al.

Vol. 6, No. 4/April 1989/J. Opt. Soc. Am. A 547

and

kinc = kocos 6inc- (26)

The wavelength in free space of the incident field is denotedby X. In the following examples, we consider the case inwhich X = 0.546 ,um and test the method on a symmetricgrating of unit slope and period p = 1.25 ,m whose surface isdescribed by

{-X= x, -0 < x < p/2f-,-p/2 < x<0O (27)

Results of our previous study7 of scattering by smoothperiodic nonplanar surfaces showed that it is advantageousto place the sources on a surface that is similar in shape tothe boundary. For echelette gratings, sources must also belocated near the edges in order to negotiate the field behav-ior there. Along this line of thinking, in the simulatedequivalence for region r, r = I, II, Nar = Na sources arespaced evenly on a surface

Zar(X) = dr + z(X), (28)

which is displaced a distance dar (daI > 0 and daII < 0) fromthe boundary, and 2Nbr = 2Nb additional sources are con-centrated around both protruding and reentrant edges at adistance dbr from the boundary. The schematic locations ofthe sources for r = I are shown in Fig. 4. We use a total of Na+ 2Nb = Nr = N strip currents. Similarly, Na match pointsare distributed evenly over z(x), and an additional Nbpoints are concentrated around each corner (the locations ofthe match points also are indicated in Fig. 4).

First, we consider a metallic grating. This case at normalincidence inc = 0 was investigated by Petit' and Kalhor andNeureuther.2 Normalized space-harmonic coefficientsIF,'I 2 calculated by the method described in this paper arecompared with those obtained by Petit and by Kalhor andNeureuther in Table 1. The maximum and average bound-ary-condition errors and the energy-conservation error asso-ciated with our results are also tabulated. Two sets of re-sults pertinent to our proposed method are presented.These results were computed, respectively, with the follow-ing distinct parameter selections: (a) N = 50 at d =0.00625 m and NbI = 0, and (b) N = 44 at daI = 0.01 ,um andNb = 1 at db = 0.002 ,um. In both cases current strips ofwidth s = 0.0009 m are used. It should be pointed out thatthe current strips used here are significantly narrower than

-p/2 p/2

I~~ ~~R Nb current strips

I -

I ,

I - N0,

d I ,f

°L 'A Z

a motchpoints

r X/

r si

current strips

i 1-Id

-I

Fig. 4. Schematic location of the fictitious sources in the simulatedequivalence for region I.

Table 1. Comparison of Space-Harmonic ExpansionCoefficients Obtained by Three Different Methodsa

Value Obtained by Method Described inThis Paper

Na 50 N 44Parameter n Ref. 1 Ref. 2 Nb = 0 Nb 1

FAI|2 0 0.457 0.472 0.4732 0.4859-1, 1 0.114 0.110 0.1113 0.1079-2, 2 0.343 0.334 0.3356 0.3280

AP (%) 3.4 0.5 0.005 0.060max(AEb,) (%) 36.6 7.9av(AEbk) (%) 3.2 2.1

a For the case of normal incidence upon a symmetric metallic grating of unitslope and period p = 1.25 ,m at X = 0.546 um.

those used in the solution of smooth nonplanar periodicsurfaces. 7 Here, those sources close to the edges must benarrow so as to make the simulation of the rapid field varia-tions attainable there. For the sake of simplicity, all thecurrent strips are of the same width in our program, in spiteof the fact that the strips located farther away could havebeen wider. In case (a), setting Nb equal to zero implies thatno refinement is made toward dealing with the actual edges,and in effect the edges are smoothed. Nevertheless thischoice results in space-harmonic coefficients relatively closeto those obtained by Kalhor and Neureuther. However, theconsiderably high value of AEb, occurring near the edges isan indication that the solution might not be sufficientlyaccurate for some purposes. In case (b), a single strip sourcein the proximity of each edge and an additional matching ofthe boundary conditions right at the edges appear to be morethan enough to reduce the boundary-condition error signifi-cantly. Thus the latter choice should be preferred. Figure5 shows plots of grating efficiencies PnI/Pinc for the variousspectral orders as a function of °ic. Also shown is the totalnormalized scattered power. The error in the power conser-vation for all angles of incidence is less than 0.2%.

Finally, the method is applied to the case of an echelettegrating separating two contrasting dielectric half-spacescharacterized by e = o, rn = o, eII = 3, and au = o. Theinterface between the media is described again by Eq. (27).We take an equal number of sources Na = Na = NaI and Nb= NbI = NbII. We let Na = 44, Nb = 6, daI =-daII = 0.08 ,m,and dbI = -dbII = 0.015 ,m. The width of the current stripsis s = 0.006 ,um. Compared with the current strips used inthe metallic case, the current strips here are wider and arenot placed so close to the boundary. This relaxation can beeffected because the fields near the dielectric edges are lesssingular. Note that Na + 2Nb represents in this case half thenumber of unknowns. Table 2 presents the efficiencies PI/Pinc and PnII/Pinc in the various spectral orders for variousangles of incidence. Also shown are the power-conservationerror AP, the maximum boundary-condition errors max(AEb,)and max(AHbC), and the average boundary-condition errorsav(AEbc) and av(AHbc). Plots of the variation of AEbc andAHbc with x across a half-period for various values of theparameters N and Nb at normal incidence are presented inFigs. 6(a) and 6(b), respectively. The boundary-conditionerrors, which are zero at the matching points, increase

I , .

i -

Boag et al.

548 J. Opt. Soc. Am. A/Vol. 6, No. 4/April 1989

Total Normalized Scattered Power1.0

0.6 ~ ~ ~ .

0.4

0.62 \ /

0.2 ,.~~~~ \SS ,.i'="

0 30 60 90

Oino (degrees)

Fig. 5. Grating efficiencies P/inc versus Oinc for the case of asymmetric metallic grating of unit slope and period p = 1.25,um at X= 0.546 Am.

It should be emphasized that for any calculated quantityof interest, one should examine the numerical convergenceby comparing the results for various numbers of sources andmatch points. If the computed results are sufficiently close,it can be assumed that a satisfactory accurracy has beenachieved. To validate the solution, one should check at thesame time the behavior of the error in the boundary condi-tion between the matching points and examine the decreasein the power-conservation error. These checks are executedeasily by means of summations of analytic terms, which donot seriously tax the computing system. As explainedabove, strictly speaking neither of these checks totally vali-dates the result. However, they can undoubtedly serve asuseful tools for determining whether the result is trustwor-thy. One can also, without appreciable difficulty, consideradditional source locations, each providing a check againstthe others. It may be added that, even for choices less thanoptimal in source location, the solution converges. The rateof convergence might be sensitive to the location of thesources, though usually not heavily.

Table 2. Grating Efficiencies Pn'/Pinc and Pn"/Pincand Errors for Various Angles of Incidencea

Value for the Following OincParameter

(units) n 00 15° 450 750

Pn'/Pinc -4 0.0070-3 0.0498 0.0320-2 0.0093 0.0053 0.0067 0.0143-1 0.0027 0.0219 0.0013 0.0120

0 0.0071 0.0009 0.0026 0.15381 0.0027 0.00092 0.0093

Pn"/Pinc -6 0.0001-5 0.0010 0.0006-4 0.0243 0.0021 0.0023-3 0.0296 0.0087 0.0141 0.0370-2 0.0797 0.1588 0.0805 0.1718-1 0.3104 0.1307 0.0688 0.1819

0 0.0062 0.0025 0.0560 0.11441 0.3104 0.5022 0.6454 0.27162 0.1412 0.0948 0.07213 0.0296 0.0486

AP (%) 0.02 0.02 0.05 0.12max(AEbk) (%) 0.73 1.33 0.75 1.92max(AHb) (%) 4.57 8.83 10.7 11.3av(AEb,) (%) 0.10 0.22 0.16 0.32av(AHbc) (%) 0.94 1.81 0.97 1.97

a For the case of a symmetric penetrable grating of unit slope and period p =1.25 gm separating dielectric half-spaces of gnl = jui = go and fit = 3el = 3O at X- 0.546 gim.

smoothly and reach their maxima between the points.Here, the effect of proper edge treatment is most pro-nounced. On one hand, setting Nb = 0 results, as expected,in high errors around the edges. On the other hand, using anexcessive number of sources near the edges, for instance,setting Nb = 10 for the sourcts near the edges and only N. =40 for the distant ones, leads to undesired fluctuations of thefields on the smooth portions of the surface.

AEb (o)

91

0

0.00

AHbc ()

807 ..

60: :i. .

40 . ji.,:

20

Na=40, Nb=6

--- Na=50, Nb=O

Na=40, Nb=10

0.13 0.25 0.38 0.50 0.63

X (m)

(a)

___ Na=40, NbO6

-il Na=50, Nb=O

- Na=40, Nb=10

ot...... II....'I.... '-''.........0.00 0.13 0.25 0.38 0.50 0.63

x (Aum)

(b)

Fig. 6. Plots of boundary-condition errors (a) AE6, and (b) AHb,versus x along a half-period for various numbers of current strips Naand Nb at normal incidence for the cage of a symmetric penotrablegrating of unit slope and period p = 1.25 gum separating dielectrichalf-spaces of iII = tq = gio and eii = 3&I = 3co at X = 0.546 gim.

Boag et al.

Vol. 6, No. 4/April 1989/J. Opt. Soc. Am. A 549

6. DISCUSSION

The technique presented in this paper provides, by a simpleand efficient moment procedure, a complete analysis of scat-tering by echelette gratings separating homogeneous dielec-tric half-spaces illuminated by a p-polarized plane wave.Metallic gratings can be treated readily as reduced cases ofthe general procedure. It has been demonstrated that thenumerical solution converges appropriately to available so-lutions. The technique can also be extended to treat the s-polarization case. In this case, one should simulate thefields in the two regions by using fictitious periodic strips ofmagnetic, rather than electric, currents in a way analogous tothat followed in Ref. 8, in which transverse electric (relativeto the cylinder axis) scattering from homogeneous dielectriccylinders was treated by using a multifilament magneticcurrent model (as opposed to the electric one used in the caseof transverse magnetic scattering9 ).

REFERENCES

1. R. Petit, "Diffraction of a plane wave by metal gratings," Rev.Opt. Theor. Instrum. 45, 249-276 (1966) (in French).

2. H. A. Kalhor and A. R. Neureuther, "Numerical method foranalysis if diffraction gratings," J. Opt. Soc. Am. 61, 43-48(1971).

3. S. Jovicevic and S. Sesnic, "Diffraction of parallel- and perpen-dicular-polarized wave from an echelette grating," Appl. Opt. 1,421-429 (1962).

4. D. Marcuse, "Exact theory of TE-wave scattering from blazeddielectric gratings," Bell Syst. Tech. J. 55, 1295-1317 (1976).

5. M. G. Moharam and T. K. Gaylord, "Diffraction analysis ofdielectric surface-relief gratings," J. Opt. Soc. Am. 72, 1385-1392(1982).

6. J. Strong, Concepts of Classical Optics (Freeman, San Francisco,Calif., 1958).

7. A. Boag, Y. Leviatan, and A. Boag, "Analysis of two-dimensionalelectromagnetic scattering from non-planar periodic surfaces us-ing a strip current mode," IEEE Trans. Antennas Propag. (to bepublished).

8. Y. Leviatan, A. Boag, and A. Boag, "Analysis of TE scatteringfrom dielectric cylinders using a multifilament magnetic currentmode," IEEE Trans. Antennas Propag. AP-36, 1026-1031(1988).

9. Y. Leviatan and A. Boag, "Analysis of electromagnetic scatteringfrom dielectric cylinders using a multifilament current mode,"IEEE Trans. Antennas Propag. AP-35, 1119-1127 (1987):

Boag et al.