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858 J. Opt. Soc. Am. A/Vol. 23, No. 4 /April 2006 Harvey et al.

Nonparaxial scalar treatment of sinusoidal phasegratings

James E. Harvey, Andrey Krywonos, and Dijana Bogunovic

Center for Research and Education in Optics and Lasers, (CREOL), P.O. Box 162700,4000 Central Florida Boulevard, University of Central Florida, Orlando, Florida 32816

Received May 2, 2005; revised August 19, 2005; accepted August 24, 2005; posted October 14, 2005 (Doc. ID 61867)

Scalar diffraction theory is frequently considered inadequate for predicting diffraction efficiencies for gratingapplications where � /d�0.1. It has also been stated that scalar theory imposes energy upon the evanescentdiffracted orders. These notions, as well as several other common misconceptions, are driven more by an un-necessary paraxial approximation in the traditional Fourier treatment of scalar diffraction theory than by thescalar limitation. By scaling the spatial variables by the wavelength, we have previously shown that diffractedradiance is shift invariant in direction cosine space. Thus simple Fourier techniques can now be used to predicta variety of wide-angle (nonparaxial) diffraction grating effects. These include (1) the redistribution of energyfrom the evanescent orders to the propagating ones, (2) the angular broadening (and apparent shifting) ofwide-angle diffracted orders, and (3) nonparaxial diffraction efficiencies predicted with an accuracy usuallythought to require rigorous electromagnetic theory. © 2006 Optical Society of America

OCIS codes: 050.0050, 050.1950, 050.1940, 050.2770, 260.1960.

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. INTRODUCTIONince the late 1960s, holographic gratings, fabricated byhe exposure of photoresist by a stationary sinusoidal in-erference fringe field, have become commonplace. Thehotoresist substrate is chemically developed after expo-ure to produce a master holographic grating with sinu-oidal groove profiles. These master holographic gratingsre routinely coated and replicated, as are master ruledratings.

Prior to the widespread use of holographic gratings, theiffraction characteristics of sinusoidal phase gratingsere of interest primarily because other groove profiles

lamellar and blazed gratings) can be Fourier analyzednto a superposition of sinusoidal profiles.1 Likewise, ar-itrary scattering surfaces are routinely modeled as a su-erposition of sinusoidal surfaces of different amplitudes,eriods, and orientations.2–4

Scalar diffraction theory is frequently considered inad-quate for predicting diffraction efficiencies for gratingpplications where � /d�0.1.5–7 It has also been statedhat scalar theory imposes energy upon the evanescentiffracted orders.5 These notions, as well as several otherommon misconceptions, are actually driven more by annnecessary paraxial approximation in the traditionalourier treatment of scalar diffraction theory than thecalar limitation. We have found only one recent paper inhe literature that seems to support our claim that scalariffraction theory may indeed be valid for feature sizesuch smaller than popular opinion seems to dictate.8

To minimize any chance for confusion concerning ourotivation or claims in this paper, we want to emphasize

ere that we are not trying to replace rigorous electro-agnetic (vector) theory with scalar theory. And we are

ot claiming that our new nonparaxial scalar theoryomehow provides us with the sometimes dramatic dis-inctions in the behavior of orthogonal polarizations of

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ight that can be provided by electromagnetic theory.owever, we have developed a more convenient Fourier

reatment of scalar theory (not restricted to paraxial ap-lications). Furthermore, the nonparaxial diffraction be-avior predicted by this scalar theory agrees well with theehavior of TE-polarized light, not TM or unpolarizedight.

. HISTORICAL BACKGROUNDn a review article on diffraction gratings in 1984,aystre9 discussed a variety of rigorous vector theories

ncluding the Rayleigh method, the Waterman method,is own integral vector method, and other differential andodal methods. He presented a comparison (duplicatedere as Fig. 1) of the diffraction efficiency (TE polariza-ion) of the first diffracted order of a perfectly conductinginusoidal grating �h /d=0.20� in the Littrow condition asalculated by the classical Beckmann–Kirchhoff theorynd his own rigorous integral vector theory.9 For this spe-ial case (�m=−�i in Maystre’s sign convention), theeckmann geometric factor2 reduces to

F = sec �m

1 + cos��i + �m�

cos �i + cos �m=

1

cos2 �m, �1�

nd the classical Beckmann–Kirchhoff theory predicts aiffraction efficiency of

�1 = J12�k

h

2cos �m�� cos4 �m, �2�

here h is the peak-to-peak amplitude of the sinusoidalrating surface profile.

We have added to Fig. 1 the paraxial scalar predictionf diffraction efficiency for sinusoidal phase gratings pro-ided by Goodman10 and others11–13:

006 Optical Society of America

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Harvey et al. Vol. 23, No. 4 /April 2006 /J. Opt. Soc. Am. A 859

�m = Jm2 �a/2�, �3�

here a=2kh cos �i.The format of Fig. 1 is commonly used to display dif-

raction efficiency data because the Littrow condition�m−�i, in our standard sign convention) allows one toeave the detector and the source fixed and to merely ro-ate the grating between measurements. An angular de-iation of 2–8 deg between the fixed source and detector isften used for convenience (thus not strictly satisfying theittrow condition).14 Every data point in the above dif-

raction efficiency curve therefore requires a different in-ident angle. For small � /d, there are many diffracted or-ers, but they all have small diffraction angles; hence theeft edge of the curve is the paraxial regime. As the grat-ng is rotated to increase � /d, both the angle of incidencend the diffraction angles increase, and the higher dif-racted orders start going evanescent. For the right twohirds of the curves in Fig. 1 there can be at most only tworopagating orders, the zero order and the +1 order, theatter being maintained in the Littrow condition. All otherrders are evanescent.

Note that all three curves agree well for � /d�0.4, i.e.,n the paraxial regime. The Beckmann–Kirchhoff theorys better than the paraxial theory for � /d�0.8 but blowsp for values of � /d�1.5. The paraxial scalar theorygrees well not only in the paraxial regime but also in theong wavelength region, i.e., the smooth surface regime.lthough many authors state categorically that scalar

heory can be applied only in the paraxial regime �� /d0.1�, they would probably concede that in the smooth

urface regime the higher orders of a sinusoidal phaserating contain a negligible fraction of the energy and theredicted efficiency is not significantly affected whenhose orders go evanescent. It is in the midportion of thebove diffraction efficiency curve that the traditionalaraxial scalar diffraction theory is totally inadequate.owever, we will now show that it is not the scalar limi-

ation, but instead the unnecessary paraxial approxima-ion implicit in the traditional Fourier treatment of scalariffraction theory that has prevented its useful applica-ion to diffraction grating problems.

ig. 1. Diffraction grating efficiency of the first order of a sinu-oidal phase grating �h /d=0.20� in the Littrow condition as pre-icted by the nonrigorous Beckmann–Kirchhoff theory, thearaxial scalar theory, and the rigorous integral vector theory.M, electromagnetic.

. LINEAR SYSTEM’S FORMULATION OFONPARAXIAL SCALAR DIFFRACTIONHEORYhe fundamental diffraction problem consists of twoarts: (i) determining the effects of introducing the dif-racting screen (or grating) upon the field immediately be-ind the screen and (ii) determining how it affects theeld downstream from the diffracting screen (i.e., what ishe field immediately behind the grating and how does itropagate). Harvey et al. have generalized Goodman’sourier treatment of scalar diffraction theory to includeew insight into the phenomenon of diffraction through-ut the whole space in which it occurs.15–17 By using acaled coordinate system in which all of the spatial vari-bles are normalized by the wavelength of the light,

x = x/�, y = y/�, z = z/�, etc . , �4�

he reciprocal variables in Fourier-transform space be-ome the direction cosines of the propagation vectors ofhe plane-wave components

� = x/r, � = y/r, � = z/r �5�

n the angular spectrum of plane waves discussed byatcliff,18 Goodman,10 and Gaskill.19 Recall that this an-ular spectrum approach leads to a transfer function ofree space whose Fourier transform (the impulse responsef the diffraction process) is a precise mathematical de-cription of a Huygens wavelet, complete with a cosinebliquity factor and a /2 phase delay.10,15 The convolu-ion of that impulse response with the optical disturbancemerging from a diffracting aperture is precisely the fa-iliar Rayleigh–Sommerfeld diffraction integral for near-eld �z�� diffraction:

U�x2, y2; z� = − i�−�

� �−�

�

U0�x1, y1;0�z

�

exp�i2��

�dx1dy1,

�6�

here � is the distance between an arbitrary point in theiffracting aperture (grating) to an arbitrary point in thebservation plane.

The Rayleigh–Sommerfeld diffraction integral is rathernwieldy to solve explicitly for most problems of practical

nterest. The Fresnel and Fraunhofer diffraction formulasre obtained by retaining only the first two terms in theinomial expansion for the quantity � in the exponent ofhe Rayleigh–Sommerfeld diffraction integral. These ex-licit approximations impose severe restrictions uponoth the size of the aperture (relative to the observationistance) and the diffraction angle. In fact, Goodmantates that the Fresnel approximation is equivalent to aaraxial approximation.10 So that we do not impose theseestrictions, all terms from the binomial expansion muste retained. This can be accomplished by rewriting Eq. (6)s a Fourier-transform integral of a generalized pupilunction that includes phase variations that resembleonventional aberrations.20,21 Any departures of the ac-ual diffracted wave field from that predicted by the Fou-ier transform of the aperture function are shown to havehe same functional form as the conventional wavefront

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berrations of imaging systems. These aberrations, whichre inherent to the diffraction process, are precisely theffects ignored when making the usual Fresnel andraunhofer approximations. Significant insight is ob-ained by recognizing that near-field diffraction patternsre merely aberrated Fraunhofer diffraction patterns. Itollows that Fresnel diffraction patterns are merely defo-used Fraunhofer diffraction patterns.20–22

When a spherical wave is incident upon a diffractingperture and the observation space is a hemisphere cen-ered upon the aperture, the phase variations mentionedbove are frequently negligible.15 For normal incidence,he diffracted wave field on the hemisphere is then givenirectly by the Fourier transform of the aperture function

U��,�; r� = ��exp�i2r�/�ir��FU0�x, y;0�, �7�

here F is the Fourier transform operator, E0�U0�x , y ;0��2 is the irradiance in the plane of the diffract-

ng aperture, and P0=E0As is the total radiant powerassing through an aperture of area As. Furthermore,his Fourier-transform relationship is valid not merelyver a small region about the optical axis, but over the en-ire hemisphere (with certain restrictions depending uponhe residual phase variations).20

Now consider the situation in which the incident radia-ion strikes the diffracting aperture at an angle �i as il-ustrated in Fig. 2. This is equivalent to introducing a lin-ar phase variation across the aperture and attenuatinghe irradiance in the plane of the aperture by the factori=cos��i�. By applying the shift theorem of Fourier-ransform theory to Eq. (7), we find that the complex am-litude distribution in direction cosine space is a functionf �−�0,

U��,� − �0; r� = ��exp�i2r�/�ir��FU0��x, y;0�exp�i2�0y�

�8�

here

ig. 2. Geometric configuration when the incident beam strikeshe diffracting aperture at an arbitrary angle.

U0��x, y;0� = ��iU0�x, y;0�. �9�

ere � is the direction cosine of the position vector of thebservation point, and �0 is the direction cosine of the po-ition vector of the undiffracted beam. Note that the di-ection cosines are obtained by merely projecting the re-pective points on the hemisphere back onto the plane ofhe aperture and normalizing to a unit radius. The com-lex amplitude distribution at an arbitrary point on theemisphere can now be said to be a function of the dis-ance of the observation point from the undiffracted beamn direction cosine space. Furthermore, �=cos � is just aosine obliquity factor.

Hence nonparaxial diffraction phenomena have beenhown to be linear, shift-invariant phenomena with re-pect to incident angle if we formulate the problem inerms of the direction cosines of the propagation vectors.urthermore, for a uniformly illuminated diffracting ap-rture of area As, it is the diffracted radiance (not irradi-nce or intensity) that is shift invariant in direction co-ine space16:

L��,� − �0� =�2

As�FU0��x, y;0�exp�i2�0y��2. �10�

Rayleigh’s (Parseval’s) theorem from Fourier-transformheory states that the integral over all space of thequared modulus of any function is equal to the integralver all space of the squared modulus of its Fourierransform.10,16 Hence we can write

�−�

� �−�

�

�U0��x, y;0�exp�i2�0y��2dx dy

=�−�

� �−�

�

�FU0��x, y;0�exp�i2�0y��2d� d�. �11�

ubstituting Eq. (10) into Eq. (11),

�−�

� �−�

�


=As

�2�−�

� �−�

�

L��,� − �0�d� d�. �12�

ecall that only those plane-wave components that lie in-ide the unit circle in direction cosine space ��2+�2�1�re real and propagate. Those that lie outside of the unitircle are imaginary and are referred to as evanescentaves (and thus do not propagate).10,15,16 All (real) space

s therefore represented by a unit circle in the two-imensional direction cosine space. Hence all of the radi-nt power emanating from the diffracting aperture is con-ained in that portion of the diffracted radianceistribution function lying inside a unit circle in directionosine space (the direction cosines of a vector must satisfyhe equation �2+�2+�2=1). Therefore Eq. (12) can also beritten as

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�−�

� �−�

�


=As

�2�−1

1 �−�1−�2

�1−�2

L��,� − �0�d� d�, �13�

here we have used L�� ,�−�0� to indicate the real dif-racted radiance distribution that lies inside of the unitircle. Note that the left side of Eq. (13) is merely the in-egral of the radiant exitance over the (scaled) aperture.t is therefore proportional to the total radiant powerransmitted through the diffracting aperture, which di-inishes with the cosine of the incident angle.

�−�

� �−�

�

�U0��x, y;0�exp�i2�0y��2dx dy =PT��i�

�2 =E0As�i

�2 .

�14�

ubstituting Eqs. (14) into Eqs. (12) and (13), it is nowvident that, for the case of a uniformly illuminated dif-racting aperture, the total radiant power diffracted fromhe aperture is just the area of the aperture times the in-egral of the diffracted radiance:

PT = As�−�

� �−�

�

L��,� − �0�d� d�

= As�−1

1 �−�1−�2

�1−�2

L��,� − �0�d� d�, �15�

here L�� ,�−�0� is given by Eq. (10) and

L��,� − �0� = K L��,� − �0�. �16�

he real radiance distribution function is thus merely aenormalized version of the original radiance distributionunction L�� ,�−�0�.16 The renormalization constant K isiven by the ratio of the integral of L�� ,�−�0� over infi-ite limits to the integral of L�� ,�−�0� over the unitircle in direction cosine space16:

K =

��=−�

� ��=−�

�

L��,� − �0�d� d�

��=−1

1 ��=−�1−�2

�1−�2

L��,� − �0�d� d�

. �17�

For those cases where the diffracted radiance distribu-ion function given by Eq. (10) extends beyond the unitircle in direction cosine space, the real radiance distribu-ion is given by

L��,� − �0� = K�2

As�FU0��x, y;0�exp�i2�0y��2

for �2 + �2 � 1

L��,� − �0� = 0 for �2 + �2 � 1. �18�

The renormalization constant differs from unity only ifhe radiance distribution function extends beyond thenit circle in direction cosine space (i.e., only if evanes-ent waves are produced). The well-known Wood’s anoma-

ies that occur in diffraction grating efficiency measure-ents are entirely consistent with this predicted

enormalization in the presence of evanescent waves.22

hese equations have also been applied to more generalcattering surfaces and have successfully explained sometherwise nonintuitive surface scatter effects.16 Thisenormalization process is also consistent with the law ofonservation of energy. However, it is significant that thisinear systems formulation of nonparaxial scalar diffrac-ion theory has been derived by the application of Parse-al’s theorem10 and not by merely heuristically imposinghe law of conservation of energy. The physics has nothanged from the Rayleigh–Sommerfeld theory, but weave reformulated it into a Fourier treatment that can beasily solved for a wide variety of nonparaxial applica-ions.

. NONPARAXIAL (WIDE-ANGLE)EHAVIOR OF SINUSOIDAL PHASERATINGS

here are substantial advantages of modeling diffractionrating behavior (especially for conical diffraction con-gurations with large obliquely incident beams) in termsf the direction consines of the propagation vectors of theiffracted orders and the incident beam.23 In particular,se of direction cosine diagrams can be shown to simplifyhe analysis of the number and angular location of thearious propagating orders for different grating periodsnd orientations. In addition, several qualitative aspectsf the intensity distributions diffracted from gratings be-ome apparent. These include the broadening and appar-nt shifting of diffracted orders at large diffractionngles.16,24

However, in this paper we will use the nonparaxial sca-ar diffraction theory summarized in Section 3 to makeuantitative predictions of diffraction efficiency for per-ectly conducting sinusoidal phase gratings. Hopefully,hese results will dispel several common misconceptionshat continue to persist due to the paraxial scalar predic-ion of Eq. (3). Two related misconceptions are (i) the no-ion that scalar theory imposes energy on the evanescentiffracted orders5 and (ii) the prediction, from Eq. (3) andllustrated in Fig. 1, that it is impossible to obtain a dif-raction efficiency greater than 0.3386 for the first dif-racted order of a sinusoidal phase grating.

We could model either a uniformly illuminated gratingf finite size or a small beam of some specific size andhape underfilling a large grating. In the first case the to-al radiant power being transmitted through the apertures reduced by the cos �i, whereas in the second case theotal transmitted radiant power remains the same but theeam footprint in the plane of the grating is increased byos �i and the exitance emerging from the plane of therating is reduced by cos �i. Both cases should yield theame results for diffraction efficiency, which is defined ashe ratio of the radiant power in a given diffracted ordero the radiant power in the incident beam. We have cho-en to analyze the first case to be consistent with Good-an’s paraxial treatment.10

Using a symbolic notation for special functions popular-zed by Goodman10 and Gaskill,19 a square sinusoidal

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hase grating (of finite size, b b) can be defined by theomplex amplitude transmittance function

t�x, y� = rect� x

b,y

b�exp i

a

2sin�2y/d�� . �19�

lluminating this grating with a uniform amplitude planeave with an angle of incidence �i and using the followingathematical identity

exp ia

2sin�2y/d�� = �

m=−�

�

Jm�a

2�exp�i2my/d�,

�20�

e can now rewrite Eq. (10) as

L��,� − �0� = �iE0

�2

As� �

m=−�

�

F�rect� x

b,y

b�

exp�i2�iy�Jm�a

2�exp�i2my/d��2

.

�21�

his is an expression for the diffracted radiance emanat-ng from a sinusoidal phase grating with its grating linesgrooves) aligned parallel to the x axis when illuminatedith a uniform amplitude plane wave whose propagationector lies in the y–z plane. We thus have planar diffrac-ion (all of the diffracted orders lie in the y–z plane) andhe grating equation can be written as

�m + �i = m/d, �22�

here �m=sin �m and �i=sin �i.Applying the convolution theorem of Fourier-transform

heory to Eq. (21), we obtain

L��,� − �0� = �iE0

�2

As� �

m=−�

�

F�rect� x

b,y

b�exp�i2�iy��

� � F�Jm�a

2�exp�i2my/d��2

, �23�

here �� denotes a two-dimensional convolutionperation.19 Applying the shift theorem and the similarityheorem of Fourier-transform theory, and noting that theuantity exp�i2my / d� merely Fourier transforms into ahifted delta function, �� ,�−m / d�, we have

L��,� − �0� = �iE0

�2

As� �

m=−�

� 1

1/b2sinc� �

1/b,� + �i

1/b�

� � Jm�a

2��,� − m/d��2

. �24�

ince any function convolved with a delta function merelyeplicates that function at the location of the delta func-ion,

L��,� − �0� = �iE0

�2

As� �

m=−�

�

Jm�a

2� 1

1/b2

sinc� �

1/b,� + �i − m/d

1/b��2

. �25�

rom the grating equation expressed in Eq. (22), the ar-ument of the sinc function becomes

��,� − �0�

= �iE0

�2

As� �

m=−�

�

Jm�a

2� 1

1/b2sinc� �

1/b,� − �m

1/b��2

�26�

ince ��=1/ d1/ b, there is negligible overlap betweenhe individual sinc functions; hence there are no crosserms in the squared modulus of the above summation.actoring b2 outside of the summation sign and notinghat As=b2, we can write

L��,� − �0� = �iE0 �m=−�

�

Jm2 �a

2� 1

1/b2sinc2� �

1/b,� − �m

1/b�� .

�27�

ubstituting Eq. (27) into Eq. (16) and then into Eq. (15)nd interchanging the order of the summation and the in-egral operation, we have

PT = �iE0KAs �m=min

max

Jm2 �a

2��−1

1 �−�1−�2

�1−�2

1

1/b2sinc2� �

1/b,� − �m

1/b��d� d�, �28�

here the summation is taken only over the diffracted or-ers lying inside the unit circle in direction cosine spacei.e., the propagating diffracted orders). The quantity inquare brackets is merely a unit volume sinc2 function.19

ince b1, the sinc2 function is very narrow and we willssume that, even for large diffracted angles, it lies com-letely inside the unit circle; hence the above integral isqual to unity and

PT = �iE0KAs �m=min

max

Jm2 �a

2� . �29�

he radiant power in a given diffracted order is thuslearly equal to

Pm = �iE0KAsJm2 �a

2� , �30�

nd the diffraction efficiency of the mth diffracted order isiven by

�m =Pm

PT=

Jm2 � a

2�

�m=min

max

Jm2 � a

2�. �31�

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Note that the nonparaxial Eq. (31) for diffraction effi-iency reduces to the paraxial Eq. (3) when the denomi-ator is a summation from minus infinity to plus infinity.ecall that the quantity a in the argument of the aboveessel functions is the peak-to-peak phase variation in-

roduced by the grating. Figure 3 illustrates that a variesot only with the angle of incidence, but with the dif-racted order as well:

a = �2/��h1 + h2� = 2h�cos �i + cos �m�. �32�

Equations (31) and (32) can now be used to calculatehe diffraction efficiency of a perfectly conducting sinu-oidal reflection grating. Figure 4 shows these predictionsor the first order in the Littrow condition and compareshem with the predictions shown in Fig. 1.

The above nonparaxial scalar diffraction theory pro-ides remarkably good agreement with rigorous integrallectromagnetic theory, not merely in the paraxial regimend the smooth surface (shallow grating) regime, but overhe entire range of � /d.

Similar predictions for sinusoidal reflection gratings at/d values of 0.05, 0.15, and 0.30 are shown in Figs. 5–7,espectively. As expected, Fig. 5 shows that all of the theo-ies agree well with rigorous calculations for low values of/d. The rigorous data in Figs. 5–7 are taken from Ref. 1,. 185.Figure 6 not only shows good agreement between our

onparaxial scalar theory and rigorous calculations, itlso demonstrates that our nonparaxial scalar diffraction

ig. 3. Illustration of the peak-to-peak phase variation intro-uced into a given diffracted order by reflection from a sinusoidalurface.

ig. 4. Diffraction grating efficiency of the first order of a per-ectly conducting sinusoidal phase grating �h /d=0.20� in the Lit-row condition as predicted by the Beckmann–Kerchhoff theory,he paraxial scalar theory, the nonparaxial (NP) scalar diffrac-ion theory presented in this paper, and a rigorous integral vectorheory. EM, electromagnetic.

ig. 5. Diffraction grating efficiency of the first order of a sinu-oidal phase grating �h /d=0.05� in the Littrow condition as pre-icted by the Beckmann–Kerchhoff theory, the paraxial scalarheory, our nonparaxial (NP) scalar diffraction theory, and a rig-rous integral vector theory. EM, electromagnetic.

ig. 6. Diffraction grating efficiency of the first order of a per-ectly conducting sinusoidal phase grating �h /d=0.15� in the Lit-row condition as predicted by the Beckmann–Kerchhoff theory,he paraxial scalar theory, our nonparaxial (NP) scalar diffrac-ion theory, and a rigorous integral vector theory. EM,

ig. 7. Diffraction grating efficiency of the first order of a per-ectly conducting sinusoidal phase grating �h /d=0.30� in the Lit-row condition as predicted by the Beckmann–Kerchhoff theory,he paraxial scalar theory, our nonparaxial (NP) scalar diffrac-ion theory, and a rigorous integral vector theory. EM,lectromagnetic.

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heory indeed predicts the Rayleigh anomalies1,7,25 thatccur when a propagating order goes evanescent. Note thebrupt increase in diffraction efficiency at � /d=0.67. Thiss precisely the value of � /d at which the −1 and +2 dif-racted orders go evanescent.

Likewise, Fig. 7 shows that our nonparaxial scalarheory continues to predict the major features of the dif-raction efficiency curve even for h /d values of 0.30. Theiminishing agreement for � /d�0.6 is probably due tohe fact that our theory does not include shadowing andultiple-scattering effects. However, it does still predict a

econd Rayleigh anomaly at � /d=0.40 where the −2 and3 diffracted orders go evanescent.Note that there is a substantial difference in the peak

alues of the oscillatory behavior in the paraxial regimes predicted by the paraxial and the nonparaxial scalarheories in Figs. 4, 6, and 7. The diffraction efficiency pre-icted for the nonparaxial scalar theory is given by Eq.31). As � /d becomes very small (paraxial regime), theumber of diffracted orders becomes very large and theenominator approaches unity resulting in the well-nown result given by �m=Jm

2 �a /2�. Clearly for a finiteumber of propagating orders the denominator is lesshan unity. This results in a higher value for the non-araxial prediction. Our nonparaxial scalar theory as-umes that the energy previously contained in the eva-escent orders is distributed uniformly over theemaining propagating orders. This may or may not be anccurate assumption. Additional rigorous data are neededo quantitatively evaluate the deviation between the non-araxial scalar theory and the rigorous theory in thearaxial regime.

. SUMMARY AND CONCLUSIONSpparently much of the grating community believes thatcalar diffraction theory is valid only for � /d�0.1.5–7 Weelieve that this limitation is due to an unnecessaryaraxial approximation in the traditional Fourier treat-ent of scalar diffraction theory, not a limitation of scalar

heory itself. We have therefore summarized the develop-ent of a linear system’s formulation of nonparaxial sca-

ar diffraction theory that greatly expands the range ofarameters over which accurate predictions can be madeith simple Fourier techniques. This theory was then ap-lied to the prediction of diffraction efficiency from per-ectly conducting sinusoidal phase (holographic) gratings.hese diffraction efficiency predictions were comparedith those of the classical Beckmann–Kirchhoff theory, a

imple paraxial theory, and a rigorous integral vectorheory. The results for the diffraction efficiency of the +1rder in the Littrow condition were plotted in the usualanner as a function of � /d for values from 0.1 to 2.0. Ex-

ellent agreement with rigorous electromagnetic theoryas shown for h /d values up to 0.3. This is quite remark-ble since this entire range �0.1�� /d�2.0� lies outside ofhat is usually thought of as the scalar regime �� /d0.1�. Furthermore, our nonparaxial scalar theory even

redicts the Rayleigh anomalies1,7,22 that are associatedith the redistribution of energy from the evanescent or-ers to the remaining propagating orders. Hopefully,hese results completely dispel the notion that scalar

heory imposes energy upon the evanescent orders.5 Were fully aware that there are two distinct types of effi-iency anomalies: Rayleigh anomalies and resonancenomalies. The resonance anomalies are an electromag-etic effect that depends upon the optical constants of apecific material, and thus do require a rigorous vectorheory (they should be nonexistent in the perfectly con-ucting gratings modeled in this paper). Also, we havenly compared our scalar predictions with rigorous calcu-ations for TE polarization. We will report at a later daten our attempt to quasi-vectorize our nonparaxial scalarheory to see if we can predict some of the anomalous TMehavior.Corresponding author James E. Harvey’s e-mail ad-

ress is [email protected].

EFERENCES1. R. Petit, Electromagnetic Theory of Gratings (Springer-

Verlag, 1980), p. 98.2. P. Beckman and A. Spizzichino, The Scattering of

Electromagnetic Waves from Rough Surfaces (Pergamon,1963).

3. J. M. Bennett and L. Mattsson, Introduction to SurfaceRoughness and Scattering, 2nd ed. (Optical Society ofAmerica, 1999).

4. J. C. Stover, Optical Scattering, Measurement andAnalysis, 2nd ed. (SPIE, 1995).

5. D. A. Gremaux and N. C. Gallager, “Limits of scalardiffraction theory for conducting gratings,” Appl. Opt. 32,1048–1953 (1993).

6. D. A. Pommet, M. G. Moharam, and E. B. Grann, “Limits ofscalar diffraction theory for diffractive phase elements,” J.Opt. Soc. Am. A 11, 1827–1834 (1994).

7. E. G. Loewen and E. Popov, Diffraction Gratings andApplications (Marcel Dekker, 1997).

8. S. D. Mellin and G. P. Nordin, “Limits of scalar diffractiontheory and an iterative angular spectrum algorithm forfinite aperture diffractive optical element design,” Opt.Express 8, 705–722 (2001).

9. D. Maystre, “Rigorous vector theories of diffractiongratings,” in Progress in Optics XXI, E. Wolf, ed. (ElsevierScience, 1984).

0. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

1. M. Born and E. Wolf, Principles of Optics (Pergamon,1980), p. 598.

2. T. K. Gaylord and M. G. Moharam, “Analysis andapplications of optical diffraction by gratings,” Proc. IEEE73, 894–937 (1985).

3. C. V. Raman and N. S. N. Nath, “The diffraction of light byhigh frequency sound waves,” Proc. Ind. Acad. Sci. A 2,406–413 (1935).

4. C. Palmer, Diffraction Grating Handbook, 4th ed.(Richardson Grating Laboratory, 2000), p. 15.

5. J. E. Harvey, “Fourier treatment of near-field scalardiffraction theory,” Am. J. Phys. 47, 974–980 (1979).

6. J. E. Harvey, C. L. Vernold, A. Krywonos, and P. L.Thompson, “Diffracted radiance: a fundamental quantity ina nonparaxial scalar diffraction theory,” Appl. Opt. 38,6469–6481 (1999).

7. J. E. Harvey, C. L. Vernold, A. Krywonos, and P. L.Thompson, “Diffracted radiance: a fundamental quantity ina nonparaxial scalar diffraction theory: errata,” Appl. Opt.39, 6374–6375 (2000).

8. J. A. Ratcliff, “Some aspects of diffraction theory and theirapplication to the ionosphere,” in Reports on Progress inPhysics, A. C. Strickland, ed. (Physical Society, 1956), Vol.XIX.

9. J. D. Gaskill, Linear Systems, Fourier Transforms, andOptics (Wiley, 1978).

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0. J. E. Harvey and R. V. Shack, “Aberrations of diffractedwave fields,” Appl. Opt. 17, 3003–3009 (1978).

1. J. E. Harvey, A. Krywonos, and D. Bogunovic, “Toleranceon defocus precisely locates the far field (exactly where isthat far field anyway?)” Appl. Opt. 41, 2586–2588 (2002).

2. R. W. Wood, “On a remarkable case of uneven distributionof light in a diffraction grating spectrum,” Philos. Mag. 4,396–410 (1902).

3. J. E. Harvey and C. L. Vernold, “Description of diffraction

grating behavior in direction cosine space,” Appl. Opt. 37,8158–8160 (1998).

4. J. E. Harvey and E. A. Nevis, “Angular grating anomalies:effects of finite beam size upon wide-angle diffractionphenomena,” Appl. Opt. 31, 6783–6788 (1992).

5. A. Hessel and A. A. Oliner, “A new theory of Wood’sanomalies on optical gratings,” Appl. Opt. 4, 1275–1297(1965).

nonparaxial scalar treatment of sinusoidal phase gratings

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