1554 J. Opt. Soc. Am. A/Vol. 14, No. 7 /July 1997 B. Layet and M. R. Taghizadeh

Electromagnetic analysis of fan-out gratings anddiffractive cylindrical

lens arrays by field stitching

Ben Layet and Mohammad R. Taghizadeh

Department of Physics, Heriot-Watt University, Edinburgh EH14 4AS, UK

Received September 24, 1996; accepted January 7, 1997; revised manuscript received January 22, 1997

The field-stitching method introduced previously [Opt. Lett. 21, 1508 (1996)] allows large-period gratings con-taining small-scale local structure to be analyzed without requiring excessive computing power. Multiplescattering and the polarization of the electromagnetic field are taken into account by standard rigorous analy-sis. A variety of calculations demonstrating the accuracy and the speed of the method are made. Also, wereport on the optimization by simulated annealing of fan-out gratings with periods of up to 240l and of dif-fractive cylindrical lenses with a period of 1200l, which has been made possible by field-stitching analysis.© 1997 Optical Society of America [S0740-3232(97)00707-2]

1. INTRODUCTIONThe analysis of diffraction gratings with features of a sizesimilar to the wavelength of the incident radiation re-quires the use of a rigorous diffraction theory, that is, afull electromagnetic treatment of the problem that takesinto account polarization and multiple scattering. Manysuch theories exist (e.g., Refs. 1–3). In the grating re-gion, they allow the coupling of each diffracted order toevery other diffracted order. And, consequently, thecomputational cost increases rapidly as the period of thegrating grows. In most cases, however, it is unlikely thatthere will be significant coupling of the field scatteredfrom widely separated grating features. The field-stitching (FS) method4 takes advantage of the weaknessof the long-range coupling to provide a means of rapidlycalculating the diffracted field of a large-period grating.This field closely approximates that found by a fullanalysis.The FS process can be summarized as follows. The

grating is considered to be divided into overlapping sec-tions, and a rigorous grating method is used to solve thediffraction problem for each section as if it were the wholeperiod of a smaller-period grating. If the section size, orlocal period, is large enough, then the calculated fieldnear the middle of the section is very similar to the fieldthat would be found there by a full rigorous calculation ofthe whole grating. Therefore we can use the central fieldfrom each grating section to get a piecewise representa-tion of the field across the whole grating. At the surfaceof the grating region this is matched to a Rayleigh expan-sion containing all the nonevanescent orders, to yield thepropagating field. We term the method ‘‘FS’’ owing to thepiecewise evaluation of the field. A summary of themathematical details is given in Section 2, and a compu-tational assessment is given in Section 3, wherein we ex-amine the effect of the local-period size of the calculationon the accuracy of the results. Also, we compare thespeed and memory requirements with those of a typical

0740-3232/97/0701554-08$10.00 ©

rigorous method (RM) to demonstrate the enormous com-putational savings that can be obtained.Array generators, or fan-out gratings, are in common

use, particularly in optical computing applications.5 Inthe majority of cases scalar analysis is perfectly adequatefor understanding their behavior. But satisfaction of thedesire for more compact optical systems leads to a reduc-tion of the grating period into a regime in which scalartheories are inaccurate and RM’s computationally inten-sive. There have been a number of studies of the perfor-mance of fan-outs of small period6,7 and of rigorousoptimization.8–10 Using FS, we examine the effect of pe-riod size on the efficiency and the uniformity and attemptto improve the performance by optimization, using simu-lated annealing. We demonstrate the significance of thesize of the grid on which surface-relief transitions are con-strained to lie and indicate that the performance at smallperiods can be maintained without necessarily requiringa reduction in the minimum feature size (compared withthe scalar design at that period). We have also studieddiffractive cylindrical lens arrays with small f-numbersand, consequently, small feature sizes. Previous relatedresearch has been based on the optimization of the localgrating structure of the lenses.11 We look at lenses withtwo surface-relief levels, which are not well suited totreatment by this method.

2. THEORYThe mathematical details of the FS method were pub-lished in Ref. 4. However, a brief summary that allowsus to set up some notation is appropriate. The formula-tion of the method does not depend on the polarization ofthe incident radiation or on the nature of grating refrac-tive indices. These are taken care of by the local rigorouscalculation, which can be treated as a black box. As ex-plained above, the local diffraction calculations for thegrating sections are made at regular intervals along the

1997 Optical Society of America

B. Layet and M. R. Taghizadeh Vol. 14, No. 7 /July 1997 /J. Opt. Soc. Am. A 1555

x axis. The sections are numbered from 1 to N, and theparameters associated with the nth section are shown inFig. 1, where dl is the local period, w is the size of the cen-tral segment used from each local calculation, and xs isthe distance from the start of each local period to the cen-tral utilized segment.To find the transmitted diffracted-order amplitudes,

Tp , it is simply a case of matching the real Rayleigh ex-pansion with the local ones, which describe the field onthe grating exit surface in a piecewise manner. (A simi-lar process holds for the reflected diffracted orders.) El-ementary Fourier mathematics leads to the expression forthe diffracted-order amplitudes:

Tp 51d (

n51

N

(m5m~2!

m~1!

Tm~n !dn exp$igm8 @xs 1 ~1 2 n !w#%

3 E~n21 !w

nw

exp~igm8 x !exp~2igpx !dx, (1)

where Tm(n) are the diffracted-order amplitudes calculated

from the nth grating section; gp and gm8 are x-directionpropagation constants from the real and the local Ray-leigh expansions, respectively; and dn is a simple constantthat arises from the variation in the phase of the incidentplane wave across the grating. The integral can easily beevaluated analytically to give an expression for Tp that isa simple double sum. In our previous paper we discussedthe necessary range over which the inner sum should becalculated to ensure accurate results. We concluded thatthis depends on the largest value of interest of the order,p. At the least, it would normally include all the propa-gating orders of the local calculations.We now discuss an implication of the FS method out-

lined above. Consider that one can study aperiodic struc-tures of finite extent by embedding them in large feature-less regions and treating the resulting structure asperiodic.12,13 The near field can be found very simply be-cause, as long as the period is large enough, the diffractedfield near the structure is unaffected by the periodic na-ture of the calculation. Finding the far field requires atransformation to an appropriate integral representation.The usefulness of this approach is limited if one is re-stricted to a standard RM, but it becomes much more at-tractive when combined with FS, which is fast and canhandle extremely large periods. It should also be notedthat nonuniform plane-wave illumination can be treated.

Fig. 1. Schematic representation of a grating and associated pa-rameters.

There are two possible implementations. If the intensityvariation is small over each local period, then the inten-sity profile is achieved by appropriate scaling of the localcalculation incident plane-wave intensities. Alterna-tively, many RM’s allow the existence of numerous inci-dent plane waves, possessing the same propagation con-stants as the reflected diffracted orders, withoutincreasing the complexity of the problem. An arbitraryincident field is represented by a sum of plane waves.This implementation allows rapidly varying incidentfields.

3. COMPUTATIONAL TESTSBefore we can trust the FS method, the accuracy and theconvergence of its results must be examined. In a previ-ous paper4 the accuracy of the FS method was demon-strated for a 1 3 81 fan-out grating analyzed under TE-polarized illumination with two period sizes, namely,100l and 10,000l. The smaller-period FS result wascompared with a full rigorous calculation made by theBurckhardt–Kaspar–Knop (BKK) method14 (which is theRM used throughout this study), and the larger one wascompared with a scalar diffraction analysis. The accu-racy of the match in the individual diffracted orders wasfound to depend on their intensity. In both cases almostall the propagating diffracted orders containing greaterthan 1025 of the total incident energy agreed to within10%. We found that the large-magnitude orders weretypically much more accurate. For example, the 81 cen-tral orders were all matched to better than 1% (except forthe zeroth order of the 10,000l period grating). These re-sults are good evidence for the validity of the FS ap-proach.We expect the FS method to produce more-accurate re-

sults when the distance xs has large values, because lesslong-range electromagnetic interactions are ignored. Toinvestigate this possibility we examine the behavior ofseveral parameters as the local period of the FS analysisis increased. One of these parameters is a convergencemeasure that we call the total order error (TOE). The or-der error is defined as the absolute difference of two in-tensity values, from different calculations, of a particulardiffracted order. The TOE is the sum over every (trans-mitted) propagating diffracted order. It is equal to twicethe total energy that must be redistributed among the dif-fracted orders so that the two calculations match. In theexamples that we present here 40l local-period resultswere used to obtain a TOE for smaller local periods. Weshould also note that the segment size, w, remained con-stant at 10l and that the number of Rayleigh orders in-cluded in the local calculation was enough to ensure con-vergence of the underlying RM.We consider first a two-level 1 3 13 fan-out grating

with a period of 200l and a substrate refractive index of1.5. The transmission medium has an index of 1. Theserefractive indices are common to all the designs in thispaper and imply a surface-relief depth of 1l. We also as-sume TE-polarized incident illumination throughout. InFig. 2 are plotted the calculated efficiency and uniformity(both defined in Section 4), and the TOE. Unsurpris-ingly, their variation with the local period is not smooth,because the local structures that are being analyzed

1556 J. Opt. Soc. Am. A/Vol. 14, No. 7 /July 1997 B. Layet and M. R. Taghizadeh

change suddenly as further grating features are encom-passed by the widening local period. However, the mag-nitude of the oscillation of the efficiency and the unifor-mity is small for all the regions shown, and it decreases asthe local period is increased. The TOE indicates thatonly approximately 0.05% of the energy needs to be redis-tributed among the orders to obtain convergence. It isinteresting to compare the fan-out performance measureswith the results of an independent analysis. The gratingwas rigorously analyzed in Ref. 10, wherein the efficiencywas found to be 0.7524 and the uniformity 0.0725. Thesecompare well with the average values toward which ourFS results tend (efficiency, 0.7525; uniformity, 0.075).The rigorously obtained uniformity is not within thebounds of the oscillation of the FS uniformity plot, butthere is a difference of only a few percent. This may bebecause uniformity is largely dependent on the intensityof a single order and is typically a highly sensitive param-eter.Our second example is a two-level f/2 diffractive lens

array with a period of 300l. In Fig. 3 are plotted the fo-cal spot intensity and shape measures a and b (both de-fined in Section 4), and the TOE. Again there is smallbut rapid variation of the performance measures as thelocal period changes. The TOE falls to a value of ap-proximately 0.002. There are no independent resultsavailable for the lens array. However, the similaritywith the fan-out behavior gives us confidence in the FSanalysis.

Fig. 2. Efficiency, uniformity, and TOE of a 200l period 13 13 fan-out grating as a function of the local period of the FScalculation.

Finally, in Fig. 4 the TOE’s of two gratings typical ofthose optimized in this paper are shown. The 1 3 33fan-out grating has a period of 240l. It is noticeable thatthe TOE grows larger as the local period increases to ap-proximately 24l and does not fall as low as the TOE of the1 3 13 as the local period approaches 40l. We believethat this is due to the greater number of features that thegrating contains and to the interaction among the local-period size, the segment size, and the average featuresize. Even at its worst, the TOE is below 0.006. So thepossibility of such an interaction is not a real concern.The f/3 diffractive lens array has a period of 1200l. Westill obtain a low TOE value of 0.0007. This suggeststhat no significant problems arise in the FS method whenthe total period becomes large. In fact, one can see fromthe expression for Tp in Eq. (1) that the algorithm is asaccurate with many sections as with few. It is the aver-age error in the total field per unit distance along thegrating that affects the values of the diffracted orders.Of course, the accuracy will depend on the total period be-cause, when that increases, the length of grating thateach local calculation ignores becomes greater. However,after a certain point the region of grating being ignored isso far from the region of the local calculation that the ef-fect of the former will be negligible anyway. Throughoutour investigation we have optimized gratings by using alocal period of 30l, because the degree of uncertainty inthe figures of merit is acceptable. There is no need for

Fig. 3. Focal spot intensity measure a, shape measure b, andthe TOE of a 300l period f/2 cylindrical diffractive lens array asa function of the local period of the FS calculation.

B. Layet and M. R. Taghizadeh Vol. 14, No. 7 /July 1997 /J. Opt. Soc. Am. A 1557

analysis accurate to many decimal places when one isconcerned with the optimization of practical diffractiveoptical elements because fabrication introduces muchgreater errors.We have seen that in most cases of practical interest

FS is accurate enough to use in place of a single rigorouscalculation. It is appropriate to ask how computationallyintensive the two approaches are. A fair comparison canbe made if one ensures that they have both converged tothe same accuracy. To do this, we accurately obtained

Fig. 4. TOE’s of a 1 3 33 fan-out grating and an f/3 lens as afunction of the local period of the FS calculation.

Fig. 5. Calculation time and number of matrix elements ofequally accurate BKK and FS calculations as a function of thegrating period.

the diffracted fields of gratings with periods of up to 100l,using a well-converged BKK calculation. And we ana-lyzed the same gratings by using FS, with a local-periodsize of 30l and a segment size of 10l. The speed andmemory requirements are compared with those of an in-completely converged BKK analysis that approaches theaccurate answer about as closely as does the FS analysis.The time taken on our Digital Equipment Corporation Al-pha computer and the number of elements in the main ar-rays are illustrated in Fig. 5. The advantages of the FSmethod are obvious. There are no memory constraintson the size of the period of the grating under analysis, ex-cept if the number of diffracted orders is enormous (theyneed to be stored, too). And the time taken increases lin-early with the period size.

4. OPTIMIZATIONThere are great advantages to be had from using fieldstitching as the analytic tool in an optimization. Section3 showed the speed increase that is possible comparedwith a typical RM. In fact, because of the piecewise na-ture of the calculation, even greater gains are possibleduring the repeated analysis of similar structures. Thisoften occurs in an optimization when a minor trial changeis made. In this case, only grating sections in which thestructure has changed since the last iteration need be re-calculated (provided that the local diffracted orders of theother sections are stored). Furthermore, the effects ofeach grating section on the diffracted orders, Tp , are in-dependent. It is possible to add and subtract the contri-butions of specific sections without reevaluating the en-tire sum in Eq. (1) from the beginning.A final shortcut is available for structures with reflec-

tion symmetry, provided that the incident illuminationpossesses the same symmetry. One example is a nor-mally illuminated lens. The local diffracted orders, Tm ,of mirror-image grating sections are simply related,Tm(original) 5 T2m(mirror), and a rigorous analysis ofonly one of the sections needs to be done. Ensuring thesymmetry prevents an optimization from searching a so-lution space that includes asymmetric structures, provid-ing a further speed gain.We used the simulated annealing algorithm to perform

the optimizations reported in this paper. The grating isdivided into many equally sized regions, or teeth, inwhich the surface relief may be either high or low. (Allthe gratings have two-level surface-relief structures.)The optimization proceeds by trial changes in the heightof several adjacent teeth. This arrangement allows theuse of the local calculation storage technique describedabove. The number of teeth and the period size imply agrid on which the transition points lie. (We use the termtransition points to describe the positions at which thesurface-relief height changes.) We choose the number ofteeth so that the grid size is 0.1l. This also makes theminimum permitted feature size 0.1l.In our optimization of diffractive cylindrical lenses

(Section 6) we wish to control the intensity and the shapeof the focal spot. To define these terms mathematically,we use the following discretized cross correlations and au-tocorrelations:

1558 J. Opt. Soc. Am. A/Vol. 14, No. 7 /July 1997 B. Layet and M. R. Taghizadeh

Ctt 5 (m51

N

tm2,

Caa 5 (m51

N

am2,

Cta 5 (m51

N

tmam ,

where N is the number of samples of the focal spot inten-sity, m is the sample position indexing variable, tm is thetarget intensity at position m, and am is the actual inten-sity at position m. The intensity parameter, a, and theshape parameter, b, are then defined as

a 5 Cta /Ctt ,

b2 5 1 2 Cta2/CttCaa .

Only the relative values of the target intensities, i.e., thetarget shape, affects the b parameter, which varies be-tween 0 and 1. A lower value indicates a better fit. Thea parameter measures the ratio of the energy in the ac-tual profile compared with the target profile and, obvi-ously, has no set bounds. The merit function, which is tobe maximized, is a weighted combination of the two pa-rameters:

merit 5 waa 2 wbb.

We choose the values of the weights wa and wb accordingto the typical values of a and b and their relative impor-tance in the optimization.Fan-out gratings are normally designed to generate a

number of diffracted orders, called the signal orders,which are of equal intensity. Their performance is usu-ally described in terms of the efficiency, h, which is thefraction of the incident energy present in the signal, andthe reconstruction error or uniformity, Dr, which mea-sures the maximum deviation of one signal order effi-ciency from the average. The uniformity is defined as fol-lows:

Dr 5 maxm→1...N

u1 2 Nam /hu,

where N is the number of signal orders and am is the in-tensity of themth signal order. To optimize fan-out grat-ings the following merit function is commonly used:

merit 5 (m51

N

~am 2 hg /N !2,

where hg is a predetermined goal efficiency. In the finalstages of some optimizations we used the uniformity asthe merit function. This can be done without significantalteration of the efficiency.

5. ONE-DIMENSIONAL FAN-OUTGRATINGSIn this section we study two-level fan-out gratings thatgenerate 7 and 33 beams, respectively, in a symmetricalarrangement about the zeroth order. The predictions ofFourier optics for the scalar designs that we analyze are

as follows: an efficiency of 0.786 and a uniformity of0.004 for the 1 3 7, and an efficiency of 0.802 and a uni-formity of 0.001 for the 1 3 33. When comparing thesevalues with FS results, one should remember that the ex-istence of reflected orders and of Fresnel reflection is ig-nored in the scalar analysis. The performance of thesedesigns when the period becomes small has been calcu-lated with FS and is shown in Figs. 6 and 7. Also shownare FS-optimized fan-outs with periods of 60l, 120l,180l, and 240l. The minimum and average feature sizesare critical in determining how well the scalar designsperform for small periods, because the effects of the smallfeatures are not well predicted by scalar diffractiontheory. This explains the poor performance of the 13 33 compared with the 1 3 7. Table 1 shows the fea-ture sizes of the scalar and the FS-optimized designs forthe specific periods mentioned. Some of the optimalstructures contain a small feature (of the order of thewavelength, or less, in size), and this is the apparent dif-ference from a typical structure designed by scalartheory. The benefit of such features has already beennoted in Ref. 10. The features were deliberately added inthat case; it is interesting to see them appear naturally inthe optimization. However, at least for the cases that wehave examined, they do not appear to be essential. Theminimum feature sizes for the smaller-period 1 3 7’s areactually slightly larger than those in the scalar designs.This means that the benefits of a proper electromagneticanalysis can be brought to fan-out design without requir-ing more advanced fabrication techniques. Of course, ifthe period is small, then all the features are small, andfabrication becomes difficult again. Another consider-

Fig. 6. Efficiency and uniformity of scalar designed (line) andFS optimized (bars) 1 3 7 fan-out gratings as a function of thegrating period.

B. Layet and M. R. Taghizadeh Vol. 14, No. 7 /July 1997 /J. Opt. Soc. Am. A 1559

eration is that the grid on which the structure transitionpoints lie seems to be critical in determining the successof the optimization. For example, we optimized with agrid size of 0.01l and found a 60l period 1 3 33 with a0.72 efficiency and a 0.056 uniformity. So, even if the de-signed structure has relatively large features, it may per-form poorly in practice because the resolution of the fab-rication process limits the accuracy of positioning of thetransition points. Recall that for an optimal two-levelscalar design fabricated by photolithography and reactiveion etching one typically observes an experimental unifor-mity of several percent above the predicted value.15 Fi-nally, it should be remembered that in many applicationshigh-angle off-axis beams are undesirable. So

Fig. 7. Efficiency and uniformity of scalar designed (line) andFS optimized (bars) 1 3 33 fan-out gratings as a function of thegrating period.

Table 1. (a) Minimum Feature Sizes and (b)Average Feature Sizes of Scalar and FS-Designed

1 3 7 and 1 3 33 Fan-out Gratings

Period/l 60 120 180 240

(a)1 3 7 (scalar) 6.1 12.1 18.2 24.21 3 7 (FS optimized) 6.3 12.3 0.2 0.11 3 33 (scalar) 1.1 2.1 3.2 4.31 3 33 (FS optimized) 0.3 1 1.5 1

(b)1 3 7 (scalar) 15 30 45 601 3 7 (FS optimized) 15 30 22.5 301 3 33 (scalar) 3.3 6.6 10 13.31 3 33 (FS optimized) 3 5.5 8.2 10

small-period designs are not always useful, even if theycan be manufactured without difficulty.

6. DIFFRACTIVE CYLINDRICAL LENSESA fast diffractive lens contains small features at the edgesof its aperture, which makes a full electromagnetic analy-sis of its behavior appropriate. We consider two-level cy-lindrical lens arrays under uniform, normal illumination.The aperture size of the lenses is 1200l. In Fig. 8 is plot-ted the variation in efficiency with the f-number of stan-dard designs based on simple geometrical theory.16

These are compared with the efficiencies of three FS-optimized lenses. In Fig. 9 the focal spot profiles areplotted. The performance of the standard designs variesin a complicated fashion over a range of several percent.An examination of the focal spot shapes shows that theyare quite sensitive to the f-number, which produces theobserved efficiency curve. This result can perhaps be ex-plained by the variation in the efficiency and phase of thefirst diffracted order of the local grating structure, whichoccurs when the features become small and resonance do-main effects are significant. However, the smallest fea-ture size of the f/6 lens is approximately 6l. One mightexpect less variation at this end of the scale (largerf-numbers), which is not observed.The optimized lenses all contain at least 5% more of the

incident energy than do the standard lenses with thesame f-number. From the three examples given it seemsthat resonance domain effects are better utilized by thesmaller-f-number lenses, which permits higher efficien-cies to be achieved. All the optimized designs possessfeatures of the minimum size allowed by the transitionpoint grid size of 0.1l. In particular, the edges of the cen-tral Fresnel zones often have high-spatial-frequency fea-tures. They can be viewed as regions of artificial indexmedia that permit a smoother variation of phase acrossthe lens aperture and consequently a more efficient focus.Unfortunately, these features are impractical to fabricatewith current technology, since the maximum surface-

Fig. 8. Efficiency of standard (line) and FS optimized (bars)1200l period f/3 cylindrical diffractive lens arrays as a functionof the f-number.

1560 J. Opt. Soc. Am. A/Vol. 14, No. 7 /July 1997 B. Layet and M. R. Taghizadeh

relief aspect ratio is 10. In the future, the optimizationshould be designed to prevent them from appearing.

7. CONCLUSIONSWe have presented an examination of the field-stitching(FS) method, which is a general method for the analysis ofdiffraction from gratings of arbitrarily large period, evenif they contain wavelength-scale features. Independentverification of some results and the convergence proper-ties lead us to believe that the method is accurate and re-liable. Its primary advantages when implemented on acomputer are speed and limited memory requirements.For these reasons, it is ideally suited for use in the analy-sis stage of an iterative optimization procedure. The ap-proximation introduced by the discarding of long-rangeresonance effects is unlikely to be important comparedwith the fabrication error that is introduced in the manu-facturing of a real component. Finally, we note that thepiecewise calculation of the field across the grating sur-face permits the storage of local fields in lookup tables,which further facilitates optimization.We have demonstrated that the performance of fan-out

gratings can be maintained when the period is small andwhen scalar theory is inaccurate, without necessarily re-quiring more features or smaller minimum feature sizes.The uniformity error, which becomes particularly bad for

Fig. 9. Focal intensity (axial component of the Poynting vector)of standard (solid curve) and FS optimized (dashed curve) 1200lperiod cylindrical diffractive lens arrays of various f-numbers asa function of the position.

designs that include the zeroth order in the signal, can bereduced to the desired level if the grid on which transitionpoints lie is fine enough. The sensitivity of these designsto transition point inaccuracy arising from fabrication er-ror should be examined. This will indicate the range ofusefulness of small-period fan-outs that exploit the be-havior of the electromagnetic field in the resonance re-gime. We note that in the future we will use a separatechoice of grid size and minimum feature size during theoptimization of practical components because the fabrica-tion constraints are different for each and should betreated accordingly.Our investigation of two-level surface-relief cylindrical

diffractive lenses has shown that the efficiency can varyconsiderably depending on the exact f-number. Thismay be important in the choice of lens for a particular ap-plication. It is also clear that optimization by means ofFS should allow the focal spot profile to be tailored forspecific needs, for example, the reduction of sidelobes or acarefully controlled shape, when scalar analysis is inaccu-rate. Further research is needed to design lenses withreasonable minimum feature sizes. Also, single lenses aswell as lens arrays should be studied.We have looked at two-level gratings with a surface-

relief depth of 1l. It is intuitively reasonable that thedepth of the grating will largely determine the range ofthe resonance effects and, hence, the necessary size of thelocal period. Thus multilevel analysis will require theuse of larger local periods if one is to obtain the same ac-curacy. However, as long as the fundamental principle ofthe insignificance of long-range coupling of the field isvalid, FS will remain reliable and useful.

REFERENCES1. M. G. Moharam and T. K. Gaylord, ‘‘Rigorous coupled-wave

analysis of planar-grating diffraction,’’ J. Opt. Soc. Am. 71,811–818 (1981).

2. I. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams,and J. R. Andrewartha, ‘‘The dielectric lamellar diffractiongrating,’’ Opt. Acta 28, 413–428 (1981).

3. R. Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1980).

4. B. Layet and M. R. Taghizadeh, ‘‘Analysis of gratings withlarge periods and small feature sizes by stitching of theelectromagnetic field,’’ Opt. Lett. 21, 1508–1510 (1996).

5. C. P. Barrett, P. Blair, G. S. Buller, D. T. Neilson, B. Rob-ertson, E. C. Smith, M. R. Taghizadeh, and A. C. Walker,‘‘Components for implementation of free-space opticalcrossbars,’’ Appl. Opt. 35, 6934–6944 (1996).

6. A. Vasara, E. Noponen, J. Turunen, J. M. Miller, and M. R.Taghizadeh, ‘‘Rigorous diffraction analysis of Dammanngratings,’’ Opt. Commun. 81, 337–342 (1991).

7. J. M. Miller, M. R. Taghizadeh, J. Turunen, N. Ross, E.Noponen, and A. Vasara, ‘‘Kinoform array illuminators infused silica,’’ J. Mod. Opt. 40, 723–732 (1993).

8. E. Sidick, A. Knoesen, and J. Mait, ‘‘Design and rigorousanalysis of high-efficiency array generators,’’ Appl. Opt. 32,2599–2605 (1993).

9. E. Noponen, J. Turunen, and F. Wyrowski, ‘‘Synthesis ofparaxial-domain diffractive elements by rigorous electro-magnetic theory,’’ J. Opt. Soc. Am. A 12, 1128–1133 (1995).

10. R. Brauer and O. Bryngdahl, ‘‘Design strategy of diffractiveelements with prescribed diffraction angles in nonparaxialregion,’’ Opt. Commun. 115, 411–416 (1995).

11. E. Noponen, J. Turunen, and A. Vasara, ‘‘Electromagnetictheory and design of diffractive-lens arrays,’’ J. Opt. Soc.Am. A 10, 434–443 (1993).

B. Layet and M. R. Taghizadeh Vol. 14, No. 7 /July 1997 /J. Opt. Soc. Am. A 1561

12. J. T. Sheridan and C. J. R. Sheppard, ‘‘Coherent imaging ofperiodic thick fine isolated structures,’’ J. Opt. Soc. Am. A 4,614–632 (1993).

13. F. Montiel and M. Neviere, ‘‘Electromagnetic theory ofBragg–Fresnel linear zone plates,’’ J. Opt. Soc. Am. A 12,2672–2678 (1995).

14. K. Knop, ‘‘Rigorous diffraction theory for transmissionphase gratings with deep rectangular grooves,’’ J. Opt. Soc.Am. 68, 1206–1210 (1978).

15. A. Vasara, M. R. Taghizadeh, J. Turunen, J. Westerholm,E. Noponen, H. Ichikawa, J. M. Miller, T. Jaakkola, and S.Kuisma, ‘‘Binary surface-relief gratings for array illumina-tion in digital optics,’’ Appl. Opt. 31, 3320–3336 (1992).

16. H. Nishihara and T. Suhara, ‘‘Micro Fresnel lenses,’’ inProgress in Optics XXIV, E. Wolf, ed. (North-Holland, Am-sterdam, 1987), pp. 1–40.