1. INTRODUCTIONIn recent years refinements in fabrication methods suchas microphotolithography, laser-beam writing, andelectron-beam writing have made it possible to producediffractive optical elements (DOE’s) with small featuresizes. These diffractive elements have applications insuch areas as laser-beam focusing, coupling, feedback,spectral filtering, correlation filtering, wavelength-division multiplexing, signal processing, optical diskreadout, beam array generation, and others.1 In parallelwith advances in fabrication, analysis methods are con-tinually being developed and refined.
In classical diffraction theory, various scalar integralmethods have been developed that calculate the diffractedfields by asserting scalar boundary conditions that relatethe total fields on the boundary of the diffractive struc-ture and/or their normal derivatives to the incident fieldsand/or their normal derivatives.2–6 The boundary condi-tions are known as Kirchhoff boundary conditions in thecase in which both the fields and their normal derivativesare specified, Dirichlet boundary conditions if only thefields are specified, and Neumann boundary conditions ifonly the normal derivatives of the fields are specified.These assumed boundary conditions can be combinedwith Huygen’s principle and the appropriate Green’sfunctions to obtain the Kirchhoff diffraction integral andthe two Rayleigh–Sommerfeld diffraction integrals.2
Further approximations can be made to these theories for
observation points that are a relatively large distancefrom the boundary of the DOE by approximating theGreen’s functions. Examples of such formulations arethe Fresnel and Fraunhofer diffraction integrals, in whichthe diffracted-wave phase fronts are approximated asquadratic and linear functions, respectively. Anotherbranch of the scalar theory exists that also asserts thescalar boundary conditions but uses the angular spectrumof plane waves to propagate the fields from the boundaryto the observation point.3,7
Some early investigations of the accuracy of the variousscalar diffraction theories involved comparisons of thefields predicted by various methods for diffraction by ap-ertures in three-dimensional black screens.8–13 How-ever, because no exact analytic methods existed for gen-eral geometries, the scalar methods could be comparedonly with other approximations or with experimentaldata. Consistent with the growth of interest in diffrac-tive phase elements, more recent investigations havebeen made by Totzeck and co-workers14–16 into the valid-ity of various scalar methods for diffraction by dielectricobjects. In these studies the structures were either veryweak rectangular phase objects with cross-sectional area, l0
2 or extremely thin, elongated rectangular phase ob-jects. In addition, only near fields were calculated, andthe scalar calculations were compared with either 3-cmmicrowave measurements or moment-method calcula-tions. Pommet et al.17 studied the range of validity of
1998 Optical Society of America
Bendickson et al. Vol. 15, No. 7 /July 1998 /J. Opt. Soc. Am. A 1823
scalar theory for diffractive phase elements by using therigorous coupled-wave analysis to determine the exactfields needed for comparison. However, this techniqueapplies only to periodic or quasi-periodic structures suchas gratings and should not be used to analyze aperiodicstructures such as small-diameter diffractive lenses withonly a few Fresnel zones.
In the formulation of the scalar methods, two impor-tant assumptions are made. First, it is required that thefeature sizes of the diffractive object be large in compari-son with the wavelength of illumination, and second, theobservation point must be located a large distance fromthe diffractive object relative to its dimensions. How-ever, with the trend of reduced feature sizes of DOE’s andfaster-focusing diffractive lenses, such conditions nolonger hold for many modern DOE’s. This developmenthas motivated research into rigorous diffraction methodsthat attempt to solve Maxwell’s equations on and aroundthe DOE. Significant progress has been made in bothdifferential and integral methods. Recently, a number ofresearchers have investigated the analysis and the designof relatively large diffractive lenses or arrays of smallerlenses by using a variety of differential methods.18–20
Because of the relatively large number of Fresnel zonesand long-range periodicity of these structures, they can beviewed as a grating with a continuously varying period.This is particularly convenient as a result of the extensivedevelopment of rigorous diffraction analysis methods ap-plied to gratings. For smaller-diameter diffractive lenseswith fewer Fresnel zones and other aperiodic diffractiveelements, solutions to integral forms of Maxwell’s equa-tions have been found to be particularly appropriate.21–28
These integral methods can be rigorous and thus providesolutions that are exact to within the limits of numericalprecision. One such integral method is an open-regionformulation of the boundary element method (BEM)21
that provides a rigorous means for calculating diffractedfields from quite general two-region DOE geometries.
The hierarchical relationships among the various rigor-ous and scalar integral methods are not well understood.Furthermore, there is a need for a unified representationof the integral methods that systematically identifies theapproximations inherent to each particular formulation.Finally, the accuracy of the various scalar methods hasnot yet been evaluated quantitatively and in a practicalsetting such as that of focusing diffractive lenses. Togain a true measure of the accuracy of these scalar meth-ods, it is necessary to have a reference rigorous diffractionmethod that can be used for comparison.
In the present work, the above needs are fulfilled. Thevarious rigorous and scalar integral methods are unifiedinto a single framework for the first time. Such a frame-work shows the interrelations among the methods andthe approximations that are asserted in each formulation.Each scalar integral method is then computationallyimplemented and applied to cylindrical diffractive focus-ing lenses with f-numbers f/2, f/1, and f/0.5. For the firsttime, the accuracy of the various scalar methods is quan-titatively analyzed for the practical application of focusingby diffractive two-dimensional (2-D) cylindrical lenses.Performance metrics are calculated by means of scalarmethods and the rigorous open-region BEM,21 including
focal spot size, diffraction efficiency, reflected and trans-mitted powers, and focal-plane sidelobe power. As ex-pected, the errors in the approximate scalar methods be-come more significant for faster-focusing lenses.However, this analysis quantifies these errors and illus-trates how different performance metrics may be more orless sensitive to violations of the assumptions on whichthe various scalar theories are based.
In Section 2 the fundamental integral equations arepresented along with the BEM and the scalar integral dif-fraction methods; also, the equations needed to calculatetransmitted and reflected powers and diffraction effi-ciency are included. A brief overview of diffractive lensdesign is given in Section 3. In Section 4 results are pre-sented for transmitted near fields, focal-plane fields, fo-cusing diffraction efficiency and focal-plane sidelobepower, and transmitted and reflected powers for f/2, f/1,and f/0.5 lenses with 2, 4, 8, and 16 levels. Section 5 fol-lows with conclusions.
2. INTEGRAL EQUATION METHODSA. Integral EquationsThe scattering problem treated here consists of a 2-Dspace filled by two open regions of dielectric material asshown in Fig. 1. The plane of incidence is the (x, y)plane, and the boundary G divides all space into two semi-infinite regions S1 and S2 . In the scalar case, thisboundary reduces to the lines Gr and G t , and the surface-relief profile of the DOE is incorporated into an appropri-ate phase delay factor for the reflected and transmittedfields, respectively. Region Si has index ni (i 5 1, 2),and n̂ is the unit normal vector directed into region S1 .By applying Green’s theorem to Maxwell’s equations andincorporating the 2-D radiation condition, the integralequations for the total fields in regions S1 and S2 are21–23
where f 5 Ez and pi 5 1 for TE polarization and f5 Hz and pi 5 ni
2 for TM polarization (i 5 1, 2). Thequantities ft and f inc represent the total and incidentfields, respectively, and subscripts 1, 2, and G refer toquantities in region S1 , in region S2 , and on the bound-ary G, respectively. Gi is the Green’s function for regionSi , and H 0
(2) is the zero-order Hankel function of the sec-ond kind. The vectors r1 , r2 , and rG8 are the position vec-
1824 J. Opt. Soc. Am. A/Vol. 15, No. 7 /July 1998 Bendickson et al.
Fig. 1. Geometry associated with the open-region integral equation formulation used in this paper. The boundary G divides all spaceinto two open, semi-infinite regions S1 and S2 with refractive indices n1 and n2 , respectively. A wave is incident from region S1 , andn̂ represents the normal to the boundary G. The surface-relief profile of the lens is given by h(x), which is a stepwise function for themultilevel lenses considered. The linear boundaries Gr and G t are the boundaries used in the scalar methods to determine the reflectedand the transmitted fields, respectively.
tors in region S1 , in region S2 , and on the boundary G,and ki 5 nik0 , where k0 5 2p/l0 and l0 is the free-space wavelength. In addition, the electromagneticboundary conditions that must hold on G are (continuityof tangential electric- and magnetic-field components)
f1t ~rG! 5 f2
t ~rG! [ fG~rG!, (4)
1p1
]f1t
]n~rG! 5
1p2
]f2t
]n~rG! [ cG~rG!. (5)
The total as well as the diffracted fields can be obtainedfrom Eqs. (1) and (2) when the boundary fields fG andtheir normal derivatives cG are known. The BEM (Ref.23) represents a rigorous solution of the boundary fieldsand correspondingly of the diffracted fields. The scalarmethods (Rayleigh–Sommerfeld, Kirchhoff, plane-wavespectrum, Fresnel, and Fraunhofer) can all be describedwith the help of Eqs. (1)–(3) with approximations of theboundary fields (and the normal derivatives) as well aspossible approximations of the Green’s functions. TheBEM and the approximations that can be made to theboundary fields and to the Green’s functions are summa-rized next.
B. Boundary Element MethodWhen r1 or r2 approaches a point rG on G, Eqs. (1) and (2)become21–23
S uG
2p2 1 DfG~rG! 1 E
G– FfG~rG8 !
]G1
]n~rG , rG8 !
2 p1G1~rG , rG8 !cG~rG8 !Gdl8 5 2f inc~rG!, (6)
S uG
2p DfG~rG! 1 EG
– FfG~rG8 !]G2
]n~rG , rG8 !
2 p2G2~rG , rG8 !cG~rG8 !Gdl8 5 0, (7)
where uG is the internal angle of G at rG and *– denotesCauchy’s principal value of integration.23 The two equa-tions above may be cast into the form of a set of linearequations by expanding fG and cG over quadratic ele-ments as fG 5 $N%T$fG%e and cG 5 $N%T$cG%e , where thecomponents of $fG%e and $cG%e are the values of fG and cG
at three nodes of an element, respectively, and $N% is theshape function vector of the element. The superscript Tindicates the transpose operation. After one solves thesystem of equations, the boundary fields fG and their nor-mal derivatives cG are specified, allowing the total field atany point in regions S1 and S2 to be determined fromEqs. (1) and (2), respectively.
C. Approximations to Green’s FunctionsTo determine the diffracted fields at a distance from thelens, several approximations can be made that simplifythe above formulation and reduce the computationaltime. While the fields on the boundary have been deter-mined rigorously by the BEM, the Green’s functions andtheir normal derivatives may be represented exactly orapproximated as
Exact:
Gi 52j4
H0~2 !~kiuri 2 rG8 u!, (8a)
Bendickson et al. Vol. 15, No. 7 /July 1998 /J. Opt. Soc. Am. A 1825
]Gi
]n5
2jki
4cos~g!H1
~2 !~kiuri 2 rG8 u!; (8b)
LA-Hankel:
Gi '2j4
exp~ jp/4!A 2pkiuri 2 rG8 u
3 exp~2jkiuri 2 rG8 u!, (9a)
]Gi
]n'
ki
4cos~g!exp~ jp/4!A 2
pkiuri 2 rG8 u
3 exp~2jkiuri 2 rG8 u!; (9b)
Fresnel:
Gi '2j4
exp~ jp/4!A 2pkiuri 2 rG8 u
exp~2jkiuyi 2 yG8 u!
3 expF2jki
~xi 2 xG8 !2
2uyi 2 yG8 u G , (10a)
]Gi
]n'
ki
4cos~g!exp~ jp/4 !A 2
pkiuri 2 rG8 u
3 exp~2jkiuyi 2 yG8 u!
3 expF2jki
~xi 2 xG8 !2
2uyi 2 yG8 u G ; (10b)
Fraunhofer:
Gi '2j4
exp~ jp/4!A 2pkiuri 2 rG8 u
exp~2jkiuyi 2 yG8 u!
3 expS 2jkixi
2
2uyi 2 yG8 u D expS jki
xixG8
uyi 2 yG8 u D ,
(11a)
]Gi
]n'
ki
4cos~g!exp~ jp/4!A 2
pkiuri 2 rG8 u
3 exp~2jkiuyi 2 yG8 u!
3 expS 2jkixi
2
2uyi 2 yG8 u D expS jki
xixG8
uyi 2 yG8 u D ,
(11b)
where (xi , yi) and (xG8 , yG8 ) are the coordinates of the ob-servation and integration points, respectively. Also, grepresents the angle between the vectors n̂ and ri 2 rG8such that cos(g) 5 n̂ • (ri 2 rG8 )/uri 2 rG8 u. For the scalarformulations where the boundary lies along the x axis,this expression reduces to cos(g) 5 ( yi 2 yG8)/uri 2 rG8 u.Equations (8) are the exact Green’s function and its nor-mal derivative for propagating the fields by using Eqs. (1)and (2). For distances of more than a few wavelengthsfrom the diffractive structure, the Hankel functions maybe replaced by their large-argument (LA-Hankel)approximations29 as in Eqs. (9), resulting in what we willrefer to as the asymptotic Green’s-function method. Theevaluation of the Hankel functions is computationally in-tensive, especially if the diffracted fields need to be deter-mined for a large number of locations. The large-
argument approximations reduce the computational time,and for sufficiently large values of kiuri 2 rG8 u they pro-duce negligible errors. If the distance from the diffrac-tive structure to the observation point is large with re-spect to the dimensions of the diffractive structure andyi @ xi , then the phase term involving uri 2 rG8 u can beexpanded as a binomial series. If only the quadratic andlower-order terms of the resulting series are retained, theGreen’s function and its normal derivative reduce to Eqs.(10). This form of the Green’s function, in which thephase function has a quadratic dependence on xi 2 xG8 , isoften referred to as Fresnel diffraction theory. Finally,at very large distances from the diffractive structure (inthe far field), the phase fronts can be approximated byplane waves, and only linear phase terms need to be re-tained as in relations (11); this approximation is com-monly referred to as Fraunhofer diffraction.
D. Scalar ApproximationsAlthough rigorous numerical methods such as the BEMallow the determination of the exact fields that satisfy thecomplete electromagnetic boundary conditions at an in-terface, it is often desirable to develop a simpler, if some-what less exact, approach. To begin with, the boundaryof the lens G is deformed into the linear boundaries Gr andG t as illustrated in Fig. 1. Although Gr and G t are spa-tially identical, they represent distinct boundaries onwhich boundary conditions will be defined for determin-ing reflected and transmitted diffracted fields, respec-tively. In the remainder of the paper, Gs will be used todenote either of the scalar boundaries Gr or G t . As willbe explained in further detail below, the effect of the lensis approximated by a reflection/transmission coefficientand an appropriate phase delay factor. In general, atleast three different sets of approximate boundary condi-tions may be used, which greatly reduces the complexityof the diffraction problem. In each case the approximateboundary conditions specify the complex fields (Dirichletboundary conditions) and/or their normal derivatives(Neumann boundary conditions) in terms of the incidentfields and/or their normal derivatives. Perhaps the best-known case is the Kirchhoff or physical optics approxima-tion, in which both the fields on the boundary and theirnormal derivatives are assumed to be equal to the inci-dent fields and their normal derivatives. Under these as-sumptions the diffracted fields in regions S1 and S2 areapproximated as2–6
f1K~r1! 5 f1
inc~r1! 1 EGr
FfGr
inc~rGr8 !
]G1
]n~r1 , rGr
8 !
2 p1G1~r1 , rGr8 !cGr
inc~rGr8 !Gdl8, r1 P S1 ,
(12)
f2K~r2! 5 2E
GFfGt
inc~rGt8 !
]G2
]n~r2 , rGt
8 !
2 p2G2~r2 , rGt8 !cGt
inc~rGt8 !Gdl8, r2 P S2 .
(13)
1826 J. Opt. Soc. Am. A/Vol. 15, No. 7 /July 1998 Bendickson et al.
Notice that these equations are essentially identical toEqs. (1) and (2), with the exception that fG and cG havebeen replaced by f Gs
inc and c Gs
inc . It is important to recog-nize that the incident fields and their normal derivativesare different for the cases of reflection into region S1 andtransmission into region S2 . Assuming that a quasi-plane wave is normally incident from region S1 , the sca-lar approximation of the incident fields and their normalderivatives on the boundary are given by
fGr
inc~rGr! 5 Rf0w~x !exp@2jd ~x !#
]fGr
inc
]n~rGr
! 5 2jk1Rf0w~x !exp@2jd ~x !#J , r P Gr ,
(14)
fGt
inc~rGt! 5 Tf0w~x !exp@2jD~x !#
]fGt
inc
]n~rGt
! 5 jk2Tf0w~x !exp@2jD~x !#J , r P G t ,
(15)
where the phase functions that are due to the lens profileare given by
d ~x ! 5 22k0n1h~x !, (16)
D~x ! 5 k0~n2 2 n1!h~x !. (17)
Here f0 is the amplitude of the incident wave, R and Trepresent the Fresnel reflection and transmission coeffi-cients, h(x) is the lens thickness as shown in Fig. 1, andw(x) is a window function for the incident beam profilegiven as
abrupt rectangular profile. More details about w(x) canbe found in Subsections 4.D and 5.A and Appendix A ofRef. 21.
Although the boundary fields differ from those deter-mined by the rigorous BEM, the fields are propagatedwith the same Green’s functions as those in Subsection2.C. Thus the Green’s function and its normal derivativein Eqs. (12) and (13) can be replaced by either the exactexpressions in Eqs. (8) or the approximate expressions inrelations (9)–(11). In practice, the Green’s functions andtheir normal derivatives are often further approximatedwhen used with scalar theory. One simplification is thatyG8 5 0 over the entire boundary Gs . In addition, the ap-proximation uri 2 rGs
8 u ' uriu 5 ri is often made in theamplitude terms of the Fresnel and Fraunhofer cases,since the observation point is assumed to be a large dis-tance from the diffractive structure. Another commonlymade approximation in the Fresnel and Fraunhofer casesis that yi /ri ' 1 if it is assumed that the relation yi@ xi holds.
The formulation of Eqs. (12) and (13) is somewhat prob-lematic from a strictly mathematical standpoint, becausethe boundary conditions overspecify the problem.2,3 Aunique solution would require knowledge of either fGs
orcGs
on the boundary, but in the Kirchhoff diffraction inte-gral, both are assumed to be known. To remedy thisproblem, two alternative formulations have beendeveloped2 in which only one of the two possible boundaryconditions is needed. To eliminate the need for both thefield and its normal derivative on the boundary, appropri-
GiRS1 5 0,
]GiRS1
]n5 2
]Gi
]n
GiRS2 5 2Gi ,
]GiRS2
]n5 0 6 for r18 5 r28 5 rGs
8 ,
(21)
(22)
w~x ! 5 51, 0 < uxu < D/2 2 l
cos2S uxu 2 D/2 1 l4l
p D , D/2 2 l < uxu < D/2 1 l,
0, D/2 1 l < uxu , `
(18)
where D is the lens diameter and l is a measure of thedegree of apodization of the incident wave. The windowfunction is used to represent the finite extent of the inci-dent fields, and its edge shape minimizes the numericalerror of the BEM formulation. If an abrupt rectangularwindow function is used instead of the smoother edge pro-file, there is extensive self-diffraction of the incident beamand a greater percentage of power is lost into regions out-side the width of the calculational region. This self-diffraction is directly manifested in a poorer conservationof power than can be obtained by apodizing the windowfunction, as is done here. Another interpretation of thisidea, which can be treated in some mathematical detail, isthat the window function with the smoother edge profilesatisfies the Helmholtz equation more closely than an
ate choices of alternative Green’s functions are made.Suppose that r18 and r28 are defined to be mirror images ofeach other with respect to the linear boundary Gs , suchthat (x28 , y28) 5 (x18 , 2y18) as in Fig. 1. The alternativeGreen’s functions may be defined in terms of the originalGreen’s functions as
where (i, j) 5 (1, 2) for Gr and (i, j) 5 (2, 1) for G t .Since the integration is to be performed over the bound-ary, the values of the alternative Green’s functions at theboundary are needed. One finds that
Bendickson et al. Vol. 15, No. 7 /July 1998 /J. Opt. Soc. Am. A 1827
where the Gi and the ]Gi /]n represent the originalGreen’s function and its normal derivative as defined inEqs. (8) and relations (9)–(11). If the alternative Green’sfunctions are substituted into Eqs. (12) and (13), then thetwo Rayleigh–Sommerfeld formulations
f1RS1~r1! 5 f1
inc~r1! 1 EGr
FfGr
inc~rGr8 !
]G1RS1
]n~r1 , rGr
8 !Gdl8,
r1 P S1 , (23)
f2RS1~r2! 5 2E
Gt
FfGt
inc~rGt8 !
]G2RS1
]n~r2 , rGt
8 !Gdl8,
r2 P S2 , (24)
f1RS2~r1! 5 f1
inc~r1! 2 EGr
@ p1G1RS2~r1 , rGr
8 !cGr
inc~rGr8 !#dl8,
r1 P S1 , (25)
f2RS2~r2! 5 E
Gt
@ p2G2RS2~r2 , rGt
8 !cGt
inc~rGt8 !#dl8,
r2 P S2, (26)
are obtained. The quantities f iRS1 and f i
RS2 representthe fields in region Si corresponding to Dirichlet and Neu-mann boundary conditions, respectively. Because of thevarious boundary conditions that are used, all three sca-lar methods discussed thus far (f i
K , f iRS1, and f i
RS2) willproduce different approximations to the exact fields as de-termined by the rigorous BEM.
Because the first alternative Green’s function GiRS1 and
the normal derivative of the second alternative Green’sfunction ]Gi
RS2/]n vanish on the boundary, the corre-sponding diffraction integrals depend on only one of thetwo field quantities. This eliminates the mathematicalinconsistency of the Kirchhoff formulation and reducesthe amount of information needed by a factor of 2. It isinteresting to note that the Kirchhoff formulation pro-duces diffracted fields that are exactly the arithmetic av-erage of the fields determined by the two Rayleigh–Sommerfeld methods. That is, f i
K(ri) 5 @f iRS1(ri)
1 f iRS2(ri)]/2.3 As mentioned above, once the assumed
boundary conditions have been chosen, the Green’s func-tion and its normal derivative may take any of the formsdescribed by Eqs. (8) and relations (9)–(11). However,even if the exact forms for the Green’s function and itsnormal derivative are used in the scalar diffraction inte-grals, the resulting diffracted fields will still be approxi-mate, because the boundary fields themselves are ap-proximations of their exact counterparts.
All of the methods described to this point are con-structed as spatial integrals over the product of theboundary fields and an appropriate Green’s function.This type of representation follows from Huygen’s prin-ciple, in which diffraction is explained by reradiation ofcylindrical waves (for 2-D diffraction problems) from allpoints on the diffracting object. In terms of linear sys-tems theory, the Green’s function is the impulse response,and the diffraction integral is a spatial convolution inte-gral used to determine the diffracted field. As is oftendone in linear systems theory, one may also use transferfunctions and Fourier transform theory to determine theoutput of a system. For diffraction problems this can be
done by Fourier-transforming the known fields along theboundary Gs , producing what is called an angular spec-trum of plane waves.3,7 The distribution of plane waveshas a certain complex amplitude associated with all pos-sible directions of propagation. Once the angular spec-trum is determined, all the plane waves are propagatedfrom the diffractive object to the observation point bymultiplying by the appropriate complex exponential. Atthe observation point, the modified angular spectrum isinverse Fourier transformed to obtain the field at that lo-cation. Mathematically, the angular spectrum of planewaves may be expressed as
F~kx! 5 E2`
`
fGs
inc~rGs!exp~ jkxx !dx, (27)
where kx is the spatial frequency in the x direction. Thediffracted fields may now be written as
f1PW1~r1! 5 f1
inc~r1! 11
2p E2`
`
F~kx!
3 exp@2j~kxx 1 kyy !#dkx , (28)
f2PW1~r2! 5
12p E
2`
`
F~kx!exp@2j~kxx 1 kyy !#dkx , (29)
where
ky 5 H 6Aki2 2 kx
2 for kx2 < ki
2
7jAkx2 2 ki
2 for kx2 . ki
2, (30)
where the upper sign applies for region S1 and the lowersign applies for region S2 . The quantity ky is the spatialfrequency in the y direction and can be either real, corre-sponding to propagating waves, or imaginary, correspond-ing to evanescent waves. Although the integral extendsfrom 2` to 1` in Eqs. (28) and (29), the limits can bechanged to 2ki and 1ki if the observation point is a suf-ficient distance from the diffracting object. This is due tothe fact that the contribution of the evanescent waves tothe diffracted field becomes negligible for distances ofmore than a few wavelengths from the diffractive object.
If Neumann boundary conditions are to be used, a simi-lar plane-wave representation can be used to determinethe fields. By taking the partial derivative of both sidesof Eqs. (28) and (29) with respect to the normal, settingy 5 0, and Fourier-transforming both sides of the equa-tions, one can show that
F8~kx! 5 2jkyF~kx!, (31)
where
F8~kx! 5 E2`
` ]f inc
]n~rGs
!exp~ jkxx !dx. (32)
Equation (31) can then be substituted back into Eqs. (28)and (29) to obtain
f1PW2~r1! 5 f1
inc~r1! 11
2p E2`
` F2F8~kx!
jkyG
3 exp@2j~kxx 1 kyy !#dkx , (33)
1828 J. Opt. Soc. Am. A/Vol. 15, No. 7 /July 1998 Bendickson et al.
f2PW2~r2! 5
12p E
2`
` F2F8~kx!
jkyG
3 exp@2j~kxx 1 kyy !#dkx . (34)
Although the integral expressions for f iPW1 and f i
PW2 ap-pear quite different from those for f i
RS1 and f iRS2 (i
5 1, 2), both sets of expressions provide exact solutionsto the wave equation for the appropriate boundary condi-tions. For this reason it must be true that f i
PW1 5 f iRS1
and f iPW2 5 f i
RS2. However, one must remember thatthese fields will not be the same as the actual fields cal-culated by the rigorous BEM because the assumed bound-ary conditions are approximations to the complete electro-magnetic boundary conditions. In principle, one couldalso develop an angular plane-wave spectrum expressionfor f i(ri) that incorporates the Kirchhoff boundary condi-
tions. This expression would simply be the arithmeticaverage of f2
PW1 and f2PW2 and would produce the same
fields as Eqs. (12) and (13).Despite the widespread application of numerous rigor-
ous and scalar integral diffraction formulations, the uni-fication of the various methods into a single logical frame-work is lacking in the current literature. One of theprimary goals of this paper is to construct such a unifieddescription and illustrate the interrelationships amongthe various integral methods that have been discussed.In Fig. 2 the methods are shown schematically, in whicheach box represents either one particular formulation or acollection of similar formulations. Only those methodslying above the dashed line are rigorous, and everythinglying below the dashed line is approximate in some sense.The approximations that have been made to produce aparticular formulation are described next to the lines that
Fig. 2. Hierarchical diagram of the various rigorous and scalar integral diffraction methods.
Bendickson et al. Vol. 15, No. 7 /July 1998 /J. Opt. Soc. Am. A 1829
connect the various methods. In general, the relative ac-curacy of the formulations decreases as one moves downthe diagram. Determining just how accurate a particu-lar approximation is in a practical setting requires mak-ing quantitative comparisons between the approximationand a rigorous theory such as the BEM in the form of anumber of performance metrics. The methods by whichthese metrics can be calculated are discussed next.
E. Power and Diffraction EfficiencyTo examine the relative performance of the various meth-ods that have been discussed, it is useful to have severalperformance metrics that may be calculated and com-pared. A few of the most important metrics are the re-flected and transmitted powers and the diffraction effi-ciency. To calculate these quantities for TE polarization,the angular spectrum of the backward- and forward-scattered field is determined (by using a fast Fouriertransform) as
Ai~rn! 51
M (m52M/2
M/221
Ezis ~mDx, yi!exp~ jrnmDx !,
(35)
where Ez1s 5 f1
t 2 f1inc and Ez2
s 5 f2t . The quantity M
is the number of sampling points used in the fast Fouriertransform, and 2L/2 < x < L/2. Here rn 5 2np/L,where L is the size of the sampled region and n can havevalues ranging from 2M/2 to M/2 2 1. Additionally,Dx 5 L/M, and yi 5 y1 or y2 is the y coordinate of the ob-servation point. Once the Ai(rn) have been determined,the reflected and the transmitted power may be calcu-lated as21,22
Ps 5 ReF L
2h i(
m52M/2
M/221b im*
kiuAi~rm!u2G , (36)
where Re( ) denotes the real part, h i 5 Am0 /ni2e0, and
b im 5 Aki2 2 rm
2. If i 5 1, then Ps represents the re-flected power, and if i 5 2, then Ps represents the trans-mitted power. If the diffractive structure is a lens withfocal length f, the amount of power focused within a de-tection slit of width d in region S2 is
Pf 5 ReH d
2h2(
m52M/2
M/221
(n52M/2
M/221b2m*
k2@A2~rm!#*
3 A2~rn!sinc@~rm 2 rn!d/2#J , (37)
where sinc(x) 5 sin(x)/x. The diffraction efficiency is de-fined as DE 5 Pf /P inc , where P inc is the incident power.The expressions that apply for TM polarization may beobtained from Eqs. (36) and (37) by simply replacing h i by1/h i , b im by b im* , and Ai(r) by @Ai(r)#* .
3. LENS DESIGNThe diffractive lenses analyzed in this paper were de-signed to convert a normally incident plane wave (inci-dent from region 1) into a focusing cylindrical wave in re-gion 2. A detailed discussion of lens design can be found
in Ref. 30, but the fundamental equations are also pre-sented here for completeness. The focal point is assumedto be located at the coordinates (0, 2f ), where f is the fo-cal length of the lens. It is also assumed that n1 . n2 ,where n1 and n2 are the refractive indices in regions S1and S2 , respectively. For constructive interference atthe focal point, the phase delay associated with a particu-lar x location should obey the relationship
u ~x ! 5 k0n2~f 2 Af 2 1 x2!. (38)
For a lens designed for normal incidence, the etch depthh(x) needed to produce the desired phase delay for a con-tinuous (except at zone boundaries) Fresnel surface-reliefprofile is
h~x ! 5n2
n1 2 n2~Af 2 1 x2 2 f 2 ml2!,
xm < uxu < min~xm11 , D/2!, (39)
where l2 5 l0 /n2 , m 5 0, 1, ..., and D is the lens diam-eter. The values of xm represent the zone boundariesand are given by xm 5 @2mf l2 1 (ml2)2#1/2. For dis-cretized, multilevel diffractive lenses, the phase transi-tion points (xi and xi11 in Fig. 1) occur at solutions of
h~xi! 2 ihmax
N5 0, xm < uxu < min~xm11 , D/2!
~i 5 1, 2, ..., N 2 1 !, (40)
where N is the number of levels (equal to 2K if K is thenumber of phase masks used to fabricate the lens) andhmax 5 l0 /(n1 2 n2).
The preceding equations were used to design threefamilies of lenses with f-numbers f/2, f/1, and f/0.5. Foreach of these f-numbers, lenses with 2, 4, 8, and 16 levelswere designed, all having a 50-mm diameter. For thepurposes of this analysis, the f-number is defined as fn5 f/D. With this definition one can see that the focallengths of the lenses are 100.0 mm, 50.0 mm, and 25.0 mm.The free-space wavelength was assumed to be 1.0 mm,and the indices of refraction were n1 5 1.5 and n2 5 1.0for all cases considered.
4. DIFFRACTIVE LENS RESULTSA. Transmitted Near FieldsAs mentioned in Subsection 2.D, the scalar methods cal-culate the diffracted fields by using various approxima-tions of the boundary fields and/or their normal deriva-tives. Because of these approximations, one wouldexpect the near fields calculated by the rigorous BEM,Rayleigh–Sommerfeld 1, Rayleigh–Sommerfeld 2, andKirchhoff approaches to differ substantially from one an-other even when the exact Green’s functions are used topropagate the fields. In Figs. 3 and 4, the normalizedfield intensity and the phase are shown 0.01 wavelengthon the transmitted side of an f/2 lens with two and eightlevels, respectively. The lens profile has also been plot-ted beneath the intensity and the phase. Since the re-sults are symmetric about the centerline of the lens, onlyone-half the lens is shown in the figures and the y dimen-sion has been exaggerated to show the lens transition
1830 J. Opt. Soc. Am. A/Vol. 15, No. 7 /July 1998 Bendickson et al.
points more clearly. The plots clearly illustrate that allfour theories predict approximately the same phase shiftsat appropriate transition points along the lens. How-ever, the intensity profiles calculated by the variousmethods show little resemblance to one another. TheRayleigh–Sommerfeld 1 intensity profile has far fewer os-cillations than the other methods; in fact, there are only afew appreciable changes in the Rayleigh–Sommerfeld 1intensity profile, and these occur at discontinuities in thelens. The Kirchhoff method produces diffracted fieldsthat are an average of those produced by the twoRayleigh–Sommerfeld methods; this averaging can beseen in the intensity plots, wherein the Kirchhoff inten-sity profiles lie somewhere between the smooth intensityprofiles of the Rayleigh–Sommerfeld 1 method and thehighly oscillatory profiles of the Rayleigh–Sommerfeld 2method. Perhaps the most important conclusion thatone should draw from the figures is that none of the scalarmethods produces satisfactory approximations of the true
intensity profile as determined by the rigorous BEM.This occurs since the assumptions made in formulatingthe scalar theories are strongly violated for the near-fieldcalculations.
B. Focal-Plane FieldsThe intensity profile in the focal plane is perhaps themost important consideration for focusing applications.To illustrate the effects of using scalar approximations indetermining the focal-plane fields, the focal-plane inten-sity profiles have been plotted in Fig. 5 for the rigorousBEM and the three scalar methods for two eight-levellenses. In all of these cases, exact Green’s functions areused to propagate the fields. The focal length is 25 mmand 50 mm for the f/0.5 and f/1 lenses, respectively. Al-though there is a significant difference in the relative sizeof the focal-plane intensity profiles of all four methods inthe case of the f/0.5 lens, the general shapes of all the
Fig. 3. Near-field normalized intensity and phase profiles for a two-level f/2 lens at 0.01 wavelength from the transmitted side of thelens. One-half of the lens profile is shown, with the dotted–dashed line being its axis of symmetry. The y dimension is exaggerated forclarity. The plots compare the rigorous BEM (for TE incidence) and three scalar methods, all using exact Green’s functions.
Bendickson et al. Vol. 15, No. 7 /July 1998 /J. Opt. Soc. Am. A 1831
Fig. 4. Same as Fig. 3 but for an eight-level f/2 lens.
curves are very similar, which was not the case with thenear fields. This similarity between the fields calculatedby the scalar methods and the exact fields calculated bythe BEM is quite noteworthy upon considering that theassumptions of the scalar theories are not at all satisfiedin the focal plane of this lens. Specifically, the observa-tion point is not at a large distance from the diffractivelens with respect to the size of the diameter of the lens.For the f/1 lens, the differences between the various sca-lar method intensity profiles are greatly reduced, alongwith the differences between intensity profiles of the sca-lar methods and the rigorous BEM. For f/2 and slowerlenses, the intensity profiles of the three scalar methodscannot be distinguished from one another on plots similarto those in Fig. 5, and they can barely be distinguishedfrom the BEM profile. As one expects, the general trendis that for slower-focusing lenses, there is less differencebetween the focal-plane intensity profiles of the variousscalar methods, and all the scalar methods more accu-rately approximate the BEM. However, the fact that the
scalar methods perform as well as they do even in thecase in which the distance to the focal plane is only one-half the size of the lens diameter is somewhat remark-able.
An additional observation can be made about the size ofthe focal-plane intensity profiles of the three scalar meth-ods relative to one another. One can observe that the in-tensity profiles in the focal plane calculated by theRayleigh–Sommerfeld 2 method are always larger thanthose calculated by the Rayleigh–Sommerfeld 1 method.As expected, the intensity profiles of the Kirchhoffmethod lie somewhere between the intensity profiles ofthe two Rayleigh–Sommerfeld methods. The origin ofthe difference between the two Rayleigh–Sommerfeldmethods may easily be deduced by comparing the two for-mulations, assuming that a large-argument approxima-tion of the Hankel functions is valid (which it is in thiscase, since k2ur2 2 rG8 u > 157 @ 1) and that the wave vec-tor of the incident wave is normal to the boundary. Oncomparison of the formulations, the only difference is that
1832 J. Opt. Soc. Am. A/Vol. 15, No. 7 /July 1998 Bendickson et al.
Table 1. Focal Spot Size for Scalar and Rigorous Methods for Lenses with Various f-Numbersand Numbers of Levels
the integrand of the Rayleigh–Sommerfeld 1 formulationcontains an extra weighting term of the form y2 /ur22 rG8 u. Since this term is always less than unity, it fol-lows that the Rayleigh–Sommerfeld 1 method will alwaysproduce smaller diffracted fields and intensity profilesthan those from the Rayleigh–Sommerfeld 2 method. Toget a first-order estimate for the scaling difference be-tween the two Rayleigh–Sommerfeld methods, one cansimply determine the average value of the weightingfunction by integrating over the lens aperture. Perform-ing this calculation for the observation point being in thefocal plane and along the axis of the lens at (0, 2f ), oneobtains an average value given by the expression
Fig. 5. Focal-plane normalized intensity profiles for eight-levelf/1 and f/0.5 lenses. The same methods as those in Figs. 3 and4 are compared, and again exact Green’s functions are used.The x axis has been scaled by the diffraction-limited spot size d0 .
Average Value 5 2 fn lnF 1
2 fn1 AS 1
2 fnD 2
1 1G ,
(41)
where fn represents the f-number of the lens. To first or-der, this value represents a ratio of the magnitude of thefield at the center of the focal plane as computed by theRayleigh–Sommerfeld 1 method to that of the Rayleigh–Sommerfeld 2 method. Indeed, by squaring this valueand comparing it with the actual ratio of the computed in-tensity profiles, the agreement is better than 4% for all ofthe cases that were tested. It is simple to show that asthe f-number increases, this term tends toward unity(which it must, since the weighting factor approachesunity for all points over which the integral is taken).Therefore it is quite reasonable that, for slower-focusinglenses, the three scalar methods will produce nearly iden-tical results in the focal plane for points close to the lensaxis.
Another consideration of interest is the focal spot sizepredicted by the scalar and rigorous methods and howthis spot size compares with the diffraction-limited spotsize. For simplicity we define the spot size as theminimum-to-minimum width of the main lobe of the focal-plane intensity profile. The spot sizes found from the in-tensity profiles are displayed in Table 1, along with thediffraction-limited spot size defined as d0 5 8f/k2D. Allthree scalar methods and the rigorous BEM for both TEand TM polarizations have been included, and again, ex-act Green’s functions have been used. In general, thescalar methods always result in spot sizes slightly largerthan those corresponding to the rigorous TE or TM spotsizes. Because the diffraction-limited spot size is basedon the 1/e2 points of the main lobe in the focal plane, it isexpected that the spot sizes from the various integralmethods will be somewhat larger than the diffraction-limited spot size.
Another type of approximation that can be made in-volves the way in which the fields are propagated to thefocal plane. The asymptotic Green’s-function method isvalid when the argument of the Hankel function of the ex-
Bendickson et al. Vol. 15, No. 7 /July 1998 /J. Opt. Soc. Am. A 1833
act Green’s function is much greater than unity, which, inthis case, reduces to k2ur2 2 rG8 u @ 1. For the focal-planefields, the smallest that the argument can become (for fo-cal length > 25 mm) is approximately k2ur2 2 rG8 u 5 157,which is more than adequate for the approximation to bevalid. Thus, for focal-plane fields and intensity profiles,it is impossible to differentiate between calculations madeby using exact Green’s functions and ones made by usingthe large-argument approximation.
The Fresnel method is another technique for propagat-ing the boundary fields and involves further approximat-ing the Green’s functions with a quadratic phase term.Whether such an approximation is valid in the case of fo-cusing diffractive lenses depends on the f-number of thelens under consideration. For f/2 and slower-focusinglenses, the Fresnel method produces fields and intensitiesthat are nearly identical to those produced by using exactGreen’s functions. However, for the faster f/1 and f/0.5lenses, using the Fresnel method results in significant er-ror. In Fig. 6 the focal-plane intensity has been plottedfor f/1 and f/0.5 eight-level lenses for various methods ofpropagation. All the intensity profiles were calculated byusing scalar methods with Dirichlet-type boundary condi-tions. As shown in the figure, there is no significant dif-ference in the focal-plane intensity profiles produced byusing exact Green’s functions, asymptotic Green’s func-tions, or the plane-wave spectrum approach. On theother hand, the Fresnel method results in significantbroadening and reduction of the focal spot for the f/1 andf/0.5 lenses. Therefore it appears that use of the Fresnelmethod should be restricted to lenses that are f/2 orslower for the parameters used in this paper.
The Fraunhofer methods involve further approxima-tions of the Green’s functions in which the phase is mod-eled as a linear function. This level of approximation re-quires that the observation point be in the far field of thediffracting object, a condition that is not even nearly sat-isfied at the focal plane of the lenses. For this reason theFraunhofer results have been omitted from the plots inFig. 6. From Fig. 6 it is apparent that the propagationwith use of the plane-wave spectrum produces results
nearly identical to those obtained with the exact Green’sfunctions. This is as expected, since the plane-wavespectrum formulation merely makes use of Fourier trans-form theory and plane-wave propagation but involves noapproximations to the exact propagation of the boundaryfields (except for the possible omission of the evanescentwaves).
C. Diffraction Efficiency and Sidelobe PowerTwo quantitative measures of the performance of the fo-cusing diffractive lenses are the diffraction efficiency and
Fig. 6. Focal-plane normalized intensity profiles for eight-levelf/1 and f/0.5 lenses. Dirichlet boundary conditions are used inall methods, and various propagation methods are compared.The solid curves represent propagation with exact Green’s func-tions, asymptotic Green’s functions, and the plane-wave spec-trum, and the dashed curves correspond to Fresnel propagation.
Table 2. Diffraction Efficiency for Scalar and Rigorous Methods for Lenses with Various f-Numbersand Numbers of Levels
1834 J. Opt. Soc. Am. A/Vol. 15, No. 7 /July 1998 Bendickson et al.
Table 3. Diffraction Efficiency under Dirichlet Boundary Conditions with Various Propagation Methodsfor Lenses with Various f-Numbers and Numbers of Levels
the sidelobe power. The diffraction efficiency is definedas the percentage of incident power that is focused withinthe full width of the main lobe (minimum to minimum) inthe focal plane. The sidelobe power is defined as the frac-tion of the incident power contained within the full widthof one of the first sidelobes in the focal plane. Both ofthese quantities were calculated by using the equationspresented in Subsection 2.E. To ensure numerical con-vergence, 4096 sample points were used in the focal planeat evenly spaced intervals from 2100 mm to 100 mm. InTable 2 the diffraction efficiency is shown as calculated bythe rigorous BEM (both TE and TM polarization) and forthe three scalar methods. In this table the exact Green’sfunctions are used in all cases, so that any differences indiffraction efficiency can be attributed to the scalar ap-proximation of the boundary fields or the various exactboundary conditions corresponding to TE or TM inci-dence. In Table 3 the diffraction efficiency is shown formethods using Dirichlet boundary conditions but various
methods of propagation. Similar results can be observedby using other types of boundary conditions (scalar or ex-act) and identical methods of propagating the fields.However, it is important to realize that the plane-wavespectrum approach can be used only with the scalarmethods, because the fields must be known along a line,whereas the boundary used in the rigorous BEM is notstraight. Finally, in Table 4, the sidelobe power is shownfor various lenses as calculated by the rigorous BEM(both TE and TM polarization) and the three scalar meth-ods; once again, exact Green’s functions are used topropagate the boundary fields.
From Table 2 it is clear that the scalar method diffrac-tion efficiencies become less and less accurate for faster-focusing lenses. This is a direct result of the closeness ofthe observation point to the lens boundary and the reduc-tion in the feature sizes of the DOE’s (more zones aresqueezed into the same lens diameter). In addition, thescalar method diffraction efficiencies are larger than both
Bendickson et al. Vol. 15, No. 7 /July 1998 /J. Opt. Soc. Am. A 1835
the rigorous TE and TM results for almost all the lensesconsidered. In Fig. 7 the diffraction efficiency error ofthe various scalar methods relative to the rigorous TEand TM results has been plotted for lenses with f-numberranging from f/2 to f/0.5 and 2 to 16 levels. It is evidentfrom the figure that the error tends to increase to largerpositive values as the number of levels increases and thatthe error increases sharply as the f-number decreases.Additionally, the Rayleigh–Sommerfeld 1 method almostalways produces the most accurate results, and all thescalar methods predict diffraction efficiencies that arecloser to the rigorous TM results than to the TE results.Finally, the fact that each group of three error curvesspreads apart as the f-number decreases indicates thatthe scalar methods differ more from one another forfaster-focusing lenses. All of these conclusions are con-sistent with the observations made of the focal-plane in-tensity profiles.
Table 3 shows that for all lenses considered, the diffrac-tion efficiency calculations are nearly identical when theboundary fields are propagated by using exact Green’sfunctions, the large-argument approximation of theGreen’s functions, or the plane-wave spectrum. It isworth noting that the plane-wave spectrum method takesfar less time to execute computationally as a result of theefficiency of the fast Fourier transform and inverse fastFourier transform algorithms. The Fresnel method ofpropagation produces accurate results for the f/2 lenses,results in significant error for the f/1 lenses, and fails en-tirely for the f/0.5 lenses. These results confirm the ideathat for the Fresnel method to yield accurate results, thedistance from the diffracting object must be at least largerthan the dimensions of the diffracting object. The Fraun-hofer method results are not shown in Table 3 becausethis method fails to produce any accurate results in thefocal plane of the lenses.
Calculating the sidelobe powers is one way of measur-ing the ability of the scalar methods to make accuratepredictions about the diffracted fields away from the axisof the diffractive lenses. As can be seen in Table 4, onlyin the cases of the f/2 lenses and the two-level f/1 and
f/0.5 lenses are the scalar method sidelobe power calcula-tions reasonably accurate relative to those of the rigorousmethod. For the f/1 and f/0.5 lenses with more than twolevels, the scalar methods predict sidelobe powers rang-ing from 2 to 12 times larger than those calculated by therigorous BEM. Even for the f/2 lens, the error in the sca-lar method sidelobe powers is much greater than the cor-responding error in diffraction efficiency. This observa-tion points out an important characteristic of the scalar
Fig. 7. Percent error in the diffraction efficiency calculated byscalar methods as compared with the rigorous BEM for TE or TMincidence. The f/2, f/1, and f/0.5 lenses with 2, 4, 8, and 16 lev-els have been considered. Exact Green’s functions were used inall cases to propagate the fields. The dotted lines indicate thezero-error point.
Table 5. Transmitted Power for Scalar and Rigorous Methods for Lenses with Various f-Numbersand Numbers of Levels
1836 J. Opt. Soc. Am. A/Vol. 15, No. 7 /July 1998 Bendickson et al.
integral methods. Namely, for a given DOE configura-tion, some performance metrics may be calculated with areasonable degree of accuracy by a particular scalar ap-proximation, whereas other performance metrics may beoff by as much as 1 order of magnitude for the same scalarapproximation and DOE configuration. These resultsare consistent with observations of the mainlobes and thesidelobes in the intensity profiles of Fig. 5.
D. Transmitted and Reflected PowerThe final metrics used in this paper to evaluate the per-formance of the scalar methods relative to the BEM arethe total transmitted and the total reflected power. Forthe rigorous BEM, these powers are computed very closeto the lens (less than 1 mm from either side) so as to mini-mize the region over which the fields needed to be calcu-lated. For the BEM calculations, convergence of thetransmitted and the reflected power was obtained by us-ing 4096 points spaced evenly from x 5 2100 mm to x5 1100 mm. The scalar method transmitted fields werecalculated by using 8192 evenly spaced points in the focalplane ranging from x 5 21600 mm to x 5 11600 mm.The reflected fields were calculated in the plane y5 116 mm by using 8192 points spaced evenly from x5 2400 mm to x 5 1400 mm. To ensure convergence,representative cases were tested with more samplingpoints and larger calculational regions until the resultschanged by only a negligible amount.
Table 5 shows the resulting transmitted powers for therigorous TE and TM methods and the three scalar meth-ods. Once again, the scalar methods produce the mostaccurate results for the f/2 lenses, whereas errors ofnearly 30% result for the f/0.5 lenses. As expected, theRayleigh–Sommerfeld 2 method always predicts the larg-est transmitted power of the three scalar methods for rea-sons discussed in Subsection 4.B. It is important to no-tice that the transmitted powers predicted by the scalarmethods are not, in general, larger than the rigorous re-sults, as is the case with the diffraction efficiencies. In-stead, the Rayleigh–Sommerfeld 1 method almost alwayspredicts transmitted powers less than the rigorous TE orTM transmitted powers, whereas the Rayleigh–Sommerfeld 2 transmitted powers are almost alwayslarger. In fact, for the f/0.5 16-level lens, the fraction oftransmitted power calculated by using the Rayleigh–Sommerfeld 2 method is greater than unity—clearly, anunphysical result. In general, total power is not con-served by the scalar methods. This should come as nosurprise, however, since the boundary conditions thatprovide the starting point of the scalar methods do notsatisfy the true electromagnetic boundary conditions forthe diffractive lenses. The reflected powers were alsocalculated, and in almost every case, the scalar methodreflected powers were significantly smaller than thoseproduced by the BEM. Once again, this fact emphasizesthat the accuracy of the scalar methods depends not onlyon the DOE configuration and dimensions but also on theperformance metric in which one is interested. At thispoint the reasons that the scalar methods yield far lessaccurate results for the reflected powers than for thetransmitted powers are not well understood. The differ-ence may result from the fact that only a small portion of
the incident power is reflected, as a result of the relativelysmall refractive-index contrast. Further investigationsusing germanium-type lenses (n 5 4.0) or metallic DOE’sdesigned to operate in reflection may assist in clarifyingthis issue.
5. CONCLUSIONSIn this paper the interrelationships that exist amongvarious integral diffraction methods have been presented.The rigorous and scalar method formulations have beenplaced into a unified framework that systematically illus-trates the hierarchical relationships among various meth-ods as well as the approximations incorporated into each.Furthermore, the methods have been applied to the prac-tical application of focusing cylindrical diffractive lenses,and several performance metrics have been calculated toexamine quantitatively the accuracy of the scalar ap-proaches. In general, it is found that it is imperative toapply rigorous methods (such as the BEM) when calculat-ing near-field quantities, on account of the boundary con-dition approximations inherent in the scalar methods. Itis also clear that although the scalar methods can provideaccurate approximations of the diffraction efficiency andthe focal-plane intensity profile, the approximations be-come progressively worse for faster-focusing lenses.Nevertheless, it has been shown that the scalar methodscan still perform relatively well, even when the assump-tions used to formulate the scalar methods are not satis-fied. Additionally, the accuracy of the scalar methodsshows a strong dependence on the performance metric ofinterest; specifically, the scalar methods appear to be farless capable of making accurate predictions of quantitiessuch as reflected power and sidelobe power for diffractivecylindrical lenses.
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