1. INTRODUCTIONDiffractive optical elements (DOE’s) are widely used in avariety of applications including optical interconnections,laser-beam focusing, coupling, feedback, spectral filtering,correlation filtering, wavelength-division multiplexing,signal processing, optical disk readout, beam array gen-eration, and others.1 DOE’s have a number of advan-tages over conventional optical components includingsmaller size (in terms of thickness) and weight, greaterversatility, and lower cost. In recent years advances inthe design and the fabrication of DOE’s have led to min-iaturized components with greater flexibility and im-proved performance. Among the most commonly used el-ements are focusing diffractive lenses and focusingdiffractive mirrors. Focusing diffractive elements havean extensive history, progressing from amplitude Fresnelzone plates to phase Fresnel mirrors and lenses with step-wise or continuous profiles. An overview of the varioustypes of zone plates and Fresnel focusing elements can befound in Refs. 2 and 3. Recent advances in microphoto-lithography have brought about renewed interest in usingintegrated diffractive focusing elements for a number ofoptical applications. However, the primary research em-phasis has been on the design, the analysis, and the fab-rication of transmissive focusing diffractive lenses,whereas there has been a relatively small amount of re-search into metallic or reflective focusing diffractive mir-rors. Metallic surface-relief focusing diffractive mirrorshave been proposed for applications such as laser-beamfocusing4 and for optical interconnections.5–13
Some early investigations were made by Swanson andVeldkamp14 into the design, the fabrication, and the per-formance of metallic focusing diffractive mirrors for wave-
lengths in the mid-infrared. They designed mirrors toconvert a plane wave at an arbitrary angle of incidenceinto a spherical wave converging to an arbitrary off-axislocation, fabricated the mirrors by using gold for the re-flective surface, and measured experimental diffractionefficiencies of more than 95%. Lenkova15 developed ex-pressions for the zone-boundary topology of a similar fo-cusing configuration, with the exception that the inci-dence and focusing angles were required to be equal.Dubik et al.16 presented similar expressions for the zoneboundaries in the case in which the incident wave radi-ates from a point source. More recently, Haidner et al.17
addressed the optimization of focusing diffractive cylin-drical mirrors with small f-numbers. In that work mir-rors were assumed to be perfectly conducting, and rigor-ous grating theory was used with a simulated annealingalgorithm to optimize the performance of the mirrors.Other researchers have also examined optimizing variousaspects of focusing diffractive mirrors.18,19 Another re-lated area of recent interest has been the use of subwave-length structures acting as artificial dielectrics to producefocusing diffractive mirrors.17,20
In parallel with advances in the design and the optimi-zation of focusing diffractive mirrors, progress has alsobeen made in the analysis of these structures. Althoughvarious scalar integral methods have traditionally beenused for diffraction analysis at optical wavelengths, thesemethods become increasingly inaccurate as the featuresize of the diffractive element is reduced.21 As improve-ments in fabrication technology have made possibleDOE’s with submicrometer feature sizes, it is clear thatmore rigorous methods of analysis are required. Foranalyzing aperiodic structures such as focusing diffrac-
1999 Optical Society of America
114 J. Opt. Soc. Am. A/Vol. 16, No. 1 /January 1999 Bendickson et al.
tive mirrors, solutions to integral forms of Maxwell’sequations have been found to be particularly appropriate.One method that provides a rigorous solution for the dif-fracted fields and is well suited for open-region geom-etries is the boundary element method (BEM).22,23 Vari-ous researchers24–30 have used the BEM to analyzescattering from both dielectric and perfectly conductingdiffractive structures with various aperiodic shapes.Finite-conductivity metallic gratings of arbitrary shapehave also been analyzed with the BEM by Nakata andKoshiba.31 Continuous and multilevel dielectric diffrac-tive cylindrical lenses were analyzed by Hirayama et al.32
for a range of incident beam profiles and angles of inci-dence through an open-region formulation of the BEM,and Bendickson et al.33 used the same BEM formulationto investigate the validity of various scalar methods foranalyzing focusing diffractive cylindrical lenses. Also,the BEM was recently used by Prather et al.34 to analyzevarious types of DOE, including both dielectric diffractivelenses and perfectly conducting diffractive mirrors for on-axis and off-axis focusing. In contrast to the integralmethods used by most researchers, Judkins andZiolkowski35 illustrated the use of the finite-differencetime-domain method for modeling finite-conductivity me-tallic gratings, including one example of a binary focusingdiffractive mirror.
Despite the extensive research that has been done, anaccurate and complete description is still lacking for thesurface-relief profile of a diffractive cylindrical mirror de-signed to convert a plane wave incident at an arbitraryangle to a cylindrical wave converging to an arbitrarypoint. Many of the existing designs are for three-dimensional geometries or for normal-incidence and on-axis focusing configurations. In addition, essentially allrigorous analyses of such mirrors have been for perfectconductors.17,19,24–26,34 For this reason there is a need forrigorous, quantitative analysis of finite-conductivity me-tallic focusing diffractive cylindrical mirrors.
In the present paper, the above needs are fulfilled.Initially, two design procedures are presented that deter-mine the zone-boundary locations and the surface-reliefprofile for continuous and multilevel focusing diffractivecylindrical mirrors. One design explicitly accounts forthe finite thickness of the mirrors whereas the other doesnot, and both are based on geometrical optics. Intu-itively, the finite-thickness design should produce supe-rior performance. However, since both design proce-dures are based on geometrical optics, they both need tobe evaluated by using a full electromagnetic analysis toestablish their region of validity. The mirrors are de-signed to focus a plane wave incident at an arbitraryangle to a point either on or off the axis of the mirror.Both zero-thickness and finite-thickness design proce-dures are used to design mirrors with various types ofcontinuous and multilevel profile, f-numbers, and on-axisand off-axis incidence and focusing configurations.These mirrors are then analyzed for both TE and TM in-cidence by a rigorous open-region formulation of theBEM.32 In contrast with previous work on aperiodic dif-fractive optics, the metal is not approximated as a perfectconductor but is modeled by using the complex refractiveindex for a real metal at a particular wavelength. A
number of performance metrics including diffraction effi-ciency, sidelobe power, total reflected power, and focalspot size are determined from the diffracted fields tostudy quantitatively the effects of various parameters onthe performance of the mirrors. The mirrors are alsoanalyzed by using various scalar integral methods, sothat the accuracy of the scalar methods may be evaluatedin comparison with the rigorous BEM results.
In Section 2 the zero-thickness and finite-thickness de-sign procedures for the diffractive mirrors are presented.The fundamental integral equations are summarized,along with the BEM and the scalar integral diffractionmethods, in Section 3. Also included are the equationsneeded to calculate reflected power and diffraction effi-ciency. In Section 4 results are presented for the diffrac-tive mirrors in terms of various performance metrics toreveal the effects of mirror profile type, f-number, on-axisor off-axis configuration, and zero-thickness versus finite-thickness design. Section 5 follows with conclusions.
2. MIRROR DESIGNThe diffractive mirrors analyzed in this paper are de-signed to focus a plane wave incident at an arbitrary in-plane angle to an arbitrary point either on or off the axisof the mirror. The two-dimensional (2-D) cross section ofthe infinite cylindrical structure is shown in Fig. 1, alongwith various parameters used in the design and theanalysis of these structures. The boundary G representsthe cross section through the DOE, and the structure isassumed to be infinite in the z dimension. The upper re-gion S1 is lossless with refractive index n1 , and the lowerregion S2 is composed of finite-conductivity metal charac-terized by its complex refractive index n2 5 n2 2 jk2 ,where n2 is the real part of the refractive index and k2 isthe extinction coefficient. Without loss of generality, theangle of incidence a may be restricted to lie in the range0° < a , 90°. The designs allow the focal point to be atany location in region S1 , either on or off the y axis. Thefocusing angle b is defined as the angle between the posi-tive y axis and the line connecting the origin to the focalpoint, as shown in Figs. 1 and 2(a). The focal length ofthe mirror is represented by f, so that the focal point islocated at coordinates (xf , f ), where xf 5 f tan b and blies in the range 290° , b , 90°. The mirror is cen-tered on the y axis and has width D such that its surface-relief profile extends over the range 2D/2 < x < D/2.Two mirror designs have been developed to accomplishthe desired focusing. The first design is relatively simplebut does not account for the finite thickness of the mirrorin determining its surface-relief profile. The second de-sign does account for the finite mirror thickness but isslightly more complicated.
A. Zone BoundariesThe first step in both mirror designs is to determine thelocation of the Fresnel zone boundaries. The zone bound-aries are identical for both the zero-thickness and finite-thickness mirror designs. Before determining all of thezone boundaries, one must first calculate x0 , the centerpoint of the central Fresnel zone, which is where the slopeof the surface-relief mirror profile becomes zero. The
Bendickson et al. Vol. 16, No. 1 /January 1999 /J. Opt. Soc. Am. A 115
construction in Fig. 2(a) provides a simple method of cal-culating the value of x0 and is based on the idea that a rayreflected from x0 that obeys the optical law of reflectionshould be reflected to the point (xf , f ) for optimal focus-ing. Simple trigonometry then leads to the expression
x0 5 f~tan b 2 tan a!. (1)
This expression can be verified once the surface-relief pro-file has been determined by differentiating h(x) andshowing that h8(x0) 5 0. The remainder of the zone-boundary locations are determined by requiring the pathlength accumulated by a ray incident at xm
6 , the innerboundary of the mth zone, to be ml1 greater than that ofa ray incident on the mirror at x0 , where l1 5 l0 /n1 .Mathematically, this is equivalent to [see Fig. 2(a)]
~xm6 2 x0!sin a 1 @~xm
6 2 xf!2 1 f 2#1/2
2 @~x0 2 xf!2 1 f 2#1/2 5 ml1 , m 5 1, 2,..., (2)
where m is the zone number, x02 5 x0
1 5 x0 , and xm2 and
xm1 are the zone boundaries to the left and to the right,
respectively, of the central zone. The first term on theleft-hand side is due to the difference in path length ac-cumulated before reflection from the mirror surface, andthe second and third terms represent the difference inpath length accumulated after reflection from the mirror.Solving this equation for the zone-boundary locations xm
6
yields
xm6 5 x0 1 @2ml1 sin a
6 ~m2l12 1 2ml1f cos a!1/2#sec2 a. (3)
Note that the only effect of changing the focusing angle bis to shift all the zone boundaries by the same amountthat x0 is shifted. Meanwhile, changing the incidentangle a causes both an overall shift of all the zone bound-aries and a shift of each of the zone boundaries relative toone another. For the special case where a 5 b 5 0°(normal incidence and on-axis focusing), the general formof the zone boundaries reduces to the familiar expression
xm1 ua5b50° 5 2xm
2 ua5b50° 5 ~m2l12 1 2ml1 f !1/2. (4)
Because of the symmetry of this geometry, the positiveand negative zone boundaries for each zone are mirrorimages of one another with respect to the y axis. Once allthe locations of the zone boundaries have been deter-mined, the zero-thickness and finite-thickness surface-relief mirror profiles may be calculated.
B. Zero-Thickness DesignFor the zero-thickness design, the diffractive mirror ismodeled as an infinitesimally thin phase-shifting ele-ment. This approximation of the geometry may be desir-able, as it produces a relatively simple, closed-form ex-pression for the surface-relief profile of the diffractivemirror. The procedure employed here differs from thatwhich is commonly used,36,37 in which the surface-reliefprofile is derived directly from the desired phase delay ofthe focusing element. Instead, a procedure based onequating optical path lengths is used, so that only minormodifications are needed to incorporate the effects of fi-nite mirror thickness. A similar approach has previouslybeen used for finite-thickness designs of normal-incidence, on-axis focusing lenses.3,38 For constructive
Fig. 1. Geometry for the open-region integral equation formulation and various parameters related to diffractive mirror designs. Theboundary G divides all space into the open, semi-infinite regions S1 with real refractive index n1 and S2 with complex refractive indexn2 . The linear boundary Gr is used in the scalar methods to determine the reflected fields. A wave is incident at an arbitrary angle afrom region S1 and is to be focused by the metallic diffractive mirror to the point (xf , f ), which makes an angle b with the positive y axis.The surface-relief profile of the metallic diffractive mirror is given by h(x), which can be a stepwise function as shown, or a continuousor piecewise-continuous function depending on the mirror type. The quantity x0 gives the location of the center of the central Fresnelzone, xm
2 and xm112 are Fresnel zone boundaries, and xm,i
2 is the location of a step transition within the mth zone of a multilevel mirror.
116 J. Opt. Soc. Am. A/Vol. 16, No. 1 /January 1999 Bendickson et al.
interference at the focal point, the phase difference accu-mulated by a ray incident at an arbitrary point on themirror and a ray incident at x0 should be an integer mul-tiple of 2p. In mathematical terms,
k1~x 2 x0!sin a 1 Fmir~x ! 1 k1@~x 2 xf!2 1 f 2#1/2
2 k1@~x0 2 xf!2 1 f 2#1/2 5 2pm, (5)
where Fmir(x) 5 22k1hzt(x) represents the phase shiftthat is due to the mirror and hzt(x) is the surface-reliefmirror profile for the zero-thickness design. The firstterm on the left-hand side of the equation is the phase dif-ference accumulated before the rays are incident on themirror, and the third and fourth terms give the phase dif-ference accumulated by the rays traveling to the focalpoint after reflection. By solving Eq. (5) for hzt(x), onecan write the surface-relief profile of the diffractive mir-ror as
Fig. 2. (a) Geometry used to determine the center point of thecentral Fresnel zone x0 of a diffractive mirror for arbitrary inci-dence and focusing angles. The ray that is incident at x0 isspecularly reflected to the focal point (xf , f ) according to the op-tical law of reflection. (b) Geometry used to determine thesurface-relief profile h ft(x) for the finite-thickness mirror design.The dashed lines indicate planes of constant phase. The dis-tance between them, d (x), indicates the path difference betweena ray incident at an arbitrary point within a particular Fresnelzone of the mirror and a ray incident at the inner zone boundaryof the same zone.
hzt~x ! 512 $x sin a 1 @~x 2 xf!
2 1 f 2#1/2
2 f~cos a 1 tan b sin a! 2 ml1%
for xm1 < x < min~xm11
1 , D/2! if x > x0 ,for max~xm11
2 , 2D/2! < x < xm2 if x < x0 . (6)
Within each zone the mirror height increases from zero atthe inner zone boundary to a maximum height given byhmax 5 l1/2 at the outer zone boundary. From the pre-ceding equation, the special case of a 5 b 5 0° (normalincidence and on-axis focusing) leads to the following ex-pression for the surface-relief profile of the mirror:
hzt~x !ua5b50° 512 ~Ax2 1 f 2 2 f 2 ml1!. (7)
The surface-relief profiles developed thus far apply to con-tinuous mirrors (with discontinuities only at zone bound-aries). For discretized, multilevel diffractive mirrors, thephase transition points occur at solutions of
hzt~xm,i6 ! 2 i
hmax
N5 0 for i 5 1, 2,..., N 2 1, (8)
where N is the number of levels (usually of the form N5 2K, where K is the number of phase masks used in thefabrication process), hmax is the maximum thickness ofthe continuous diffractive mirror, and xm,i
6 refers to the lo-cation of the transition to the ith step within the mthzone to the left or to the right of x0 .
C. Finite-Thickness DesignThe simple expression given in Eq. (6) for the surface-relief profile of the diffractive mirror is adequate for manyfocusing configurations. However, in some cases, ap-proximating the mirror as an infinitesimally thin phase-shifting element causes degradation of the focusing char-acteristics of the mirror. To remedy this problem, thefinite-thickness mirror design may be used, in which themirror thickness is explicitly accounted for in the deriva-tion of the surface-relief profile. For constructive phaseinterference of all rays reflected within the mth zone ofthe diffractive mirror,
k1d~x ! 1 k1$~x 2 xf!2 1 @f 2 h ft~x !#2%1/2
2 k1@~xm6 2 xf!
2 1 f 2#1/2 5 0 (9)
must be satisfied, where h ft(x) is the surface-relief profileof the finite-thickness design and d(x) is the path-lengthdifference accumulated prior to reflection for a ray inci-dent at an arbitrary point within the mth zone of the mir-ror and a ray incident at xm
6 as shown in Fig. 2(b). Usingbasic trigonometry, one finds that d(x) 5 (x 2 xm
6)sin a2 hf t(x)cos a. In this mirror design, the surface-reliefprofile of each zone is determined independently. For ex-ample, although the incident ray on the left in Fig. 2(b)would actually be reflected by the zeroth zone beforereaching the left zone boundary of the first zone, this ef-fect is ignored in determining the surface-relief profile ofthe first zone. The terms involving xm
6 can be eliminatedby using Eq. (2), so that the resulting expression involvesonly the focal length f, the incidence and focusing angles aand b, the wavelength l1 , the zone number m, and the
Bendickson et al. Vol. 16, No. 1 /January 1999 /J. Opt. Soc. Am. A 117
location along the x axis. Off-axis incidence (a Þ 0°)produces a quadratic equation for h f t(x), so that
h f t~x ! 52B 2 ~B2 2 4AC !1/2
2A, (10)
where
A 5 sin2 a,
B 5 22 f~1 1 cos2 a!
22 cos a~ f tan b sin a 2 x sin a 1 ml1!,
C 5 ~x 2 f tan b!2 1 f 2
2 ~ f cos a 1 f tan b sin a 2 x sin a 1 ml1!2.
(11)
The expression for h f t(x) in Eq. (10) is valid over the sameset of intervals as that in Eq. (6). For the special case ofnormal incidence (a 5 0°), the equation for h f t(x) be-comes linear, and the solution is simply given by h f t(x)5 2C/B, where B and C are as defined in Eqs. (11). Aswith the zero-thickness design, the mirror profile for thespecial case of normal incidence and on-axis focusing canbe obtained from the more general expression by substi-tuting a 5 b 5 0°, resulting in
h f t~x !ua5b50° 5x2 2 2 fml1 2 m2l1
2
4f 1 2ml1. (12)
It is important to note that the maximum thickness of themirror is different at each zone boundary for the finite-thickness design, as shown in Fig. 2(b). Specifically, themaximum thickness of the mth zone to the left and to theright of x0 is given by hm6
max 5 hf t(xm116 ). This issue be-
comes quite important when one considers discretizingthe finite-thickness mirror designs into N equal levels.One way of treating this problem is to discretize the mir-ror in the same manner as that in Eq. (8), with the excep-tion that hmax is replaced by hmax, which is defined to bethe average of the maximum thicknesses hm6
max for all thezones in the mirror.
3. ANALYSISA. Integral EquationsThe scattering problem treated here consists of a 2-Dspace divided into two semi-infinite regions S1 and S2 byan arbitrarily shaped boundary G as shown in Fig. 1. Asmentioned in Section 2, the boundary G represents thesurface-relief profile of the DOE, and the structure is as-sumed to be infinite in the z dimension. Also, recall thatthe upper region S1 is lossless with refractive index n1 ,and the lower region S2 is composed of finite-conductivitymetal characterized by its complex refractive index n25 n2 2 jk2 . The excitation is provided by a plane wave(actually a finite-width quasi-plane wave), which is inci-dent from region S1 . The plane of incidence is the (x, y)plane, and the incident wave vector makes an angle of awith respect to the positive y axis. For scalar methods ofanalysis, the boundary G reduces to the straight line Gr ,and the surface-relief profile of the DOE is incorporatedinto a phase delay factor appropriate for determining thereflected fields.33 Because of the 2-D geometry of the
DOE and the in-plane incidence of the plane wave, thescalar Helmholtz equation can be used to solve for TE andTM polarizations separately. If Green’s theorem is ap-plied to the 2-D Helmholtz equation and the Sommerfeldradiation condition is incorporated, the integral equationsfor the total fields in regions S1 and S2 are22,23,29,32
2f1t ~r1! 1 E
GFfG~rG8 !
]G1
]n8~r1 , rG8 !
2 p1G1~r1 , rG8 !cG~rG8 !Gdl8 5 2f inc~r1!,
r1 P S1 , (13)
f2t ~r2! 1 E
GFfG~rG8 !
]G2
]n8~r2 , rG8 !
2 p2G2~r2 , rG8 !cG~rG8 !Gdl8 5 0, r2 P S2 , (14)
with
Gi~ri , rG8 ! 52j4
H0~2 !~kiuri 2 rG8 u! ~i 5 1, 2 !, (15)
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 H0
(2) is the zero-order Hankel function of the sec-ond kind. In calculating diffracted fields that are ten ormore wavelengths from the boundary G, it is possible toapproximate the Hankel function and its normal deriva-tive by their large-argument values.33,39 These approxi-mations significantly reduce computational time, and forsufficiently large values of the argument kiuri 2 rG8 u theerrors introduced are negligible. The vectors r1 , r2 , andrG8 are the position vectors in region S1 , in region S2 , andon the boundary G, respectively, and ki 5 nik0 , wherek0 5 2p/l0 and l0 is the free-space wavelength. In ad-dition, the electromagnetic boundary conditions thatmust hold on G are the continuity of tangential electric-and magnetic-field components:
f1t ~rG! 5 f2
t ~rG! [ fG~rG!, (16)
1p1
]f1t
]n~rG! 5
1p2
]f2t
]n~rG! [ cG~rG!. (17)
The total as well as the diffracted fields can be obtainedfrom Eqs. (13) and (14) when the boundary fields fG andtheir normal derivatives cG are known. The BEM (Refs.22 and 23) represents a rigorous solution of the boundaryfields and correspondingly of the diffracted fields. Thescalar methods are based on similar integral equations,but the boundary fields and/or their normal derivativesare approximated by simple modifications of the incidentfields; hence the diffracted fields calculated by means ofscalar methods will be approximations of the exact dif-fracted fields.
B. Boundary Element MethodWhen r1 or r2 approaches a point rG on G, Eqs. (13) and(14) become22,23,29,32
118 J. Opt. Soc. Am. A/Vol. 16, No. 1 /January 1999 Bendickson et al.
S uG
2p2 1 DfG~rG! 1
«GFfG~rG8 !
]G1
]n8~rG , rG8 !
2 p1G1~rG , rG8 !cG~rG8 !Gdl8 5 2f inc~rG!, (18)
S uG
2p DfG~rG! 1«G
FfG~rG8 !]G2
]n8~rG , rG8 !
2 p2G2~rG , rG8 !cG~rG8 !Gdl8 5 0, (19)
where uG is the internal angle of G at rG and W 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. After the system of equations issolved, the boundary fields fG and their normal deriva-tives cG are specified, allowing the total field at any pointin either region S1 or S2 to be determined from Eqs. (13)and (14).
C. Scalar ApproximationsScalar methods rely on a simpler but approximate ap-proach to determining the boundary fields and their nor-mal derivatives. To begin with, the surface-relief profileof the boundary G is replaced by the straight boundary Gralong the line y 5 0. The boundary fields and their nor-mal derivatives are constructed by modifying the incidentfields to account for the surface-relief profile of the DOEas well as for changes in amplitude and phase that aredue to the reflection from the metallic surface. The inci-dent field can be written as
f inc~r1! 5 f0w~x !exp~2jk1x sin a!exp~ jk1 y cos a!,
r1 P S1 , (20)
where f0 is the amplitude of the incident wave and w(x)is a cosine-squared window function used to apodize theincident wave. The apodization serves to improve thenumerical convergence of the BEM and is used in the sca-lar methods as well for consistency. Further discussionof the window function can be found in Subsections 4.Dand 5.A and in Appendix A of Ref. 32. The scalar ap-proximation of the boundary field and its normal deriva-tive can now be represented by
tude for the quasi-plane wave reflecting from the corru-gated surface of the DOE. For off-axis incidence theFresnel reflection coefficient is polarization dependent.The phase function D(x) can be written as
D~x ! 5 22gk1h~x !, (22)
where h(x) is the DOE thickness as shown in Fig. 1 andg 5 1 or g 5 sec a. The parameter g is associated withhow the path length is determined in calculating thephase function. From a ray optics point of view, for off-axis incidence the incident wave travels diagonally bothinto and out of the grooves of the DOE. Because of thisdiagonal path (as opposed to light going normally into andout of the grooves), the path length is increased by a fac-tor of sec a. Thus g 5 sec a when this effect is accountedfor in the phase function, and g 5 1 when the correctionfor off-axis incidence is not made. As will be shown inSection 4, the absence or the presence of this correctionfactor in determining the scalar boundary fields and theirnormal derivatives greatly affects the diffracted fields forrelatively large angles of incidence.
Once the boundary fields and their normal derivativesare known, a variety of scalar integral formulations canbe used to determine the diffracted fields. The Kirchhoffdiffraction integral is obtained by substituting the ap-proximate scalar boundary fields and their normal deriva-tives directly into Eq. (13), giving21,33,40–42
f1K~r1! 5 f1
inc~r1! 1 EGr
FfGr~rGr
8 !]G1
]n8~r1 , rGr
8 !
2 G1~r1 , rGr8 !
]fGr
]n8~rGr
8 !Gdl8, r1 P S1 .
(23)
Replacing the true boundary fields and their normal de-rivatives by the approximate scalar values that are de-rived directly from the incident fields is an implementa-tion of what is known as the Kirchhoff or physical opticsboundary conditions. The Rayleigh–Sommerfeld scalardiffraction integrals are constructed in a similar manner,with the exception that only the boundary fields or theirnormal derivatives are required, but not both. This re-duction of the number of required boundary conditions ismade possible through the application of alternativeGreen’s functions. The Rayleigh–Sommerfeld 1 formula-tion uses only the boundary fields (Dirichlet boundary
fGr~rGr
! 5 Rf0w~x !exp~2jk1x sin a!exp@2jD~x !#
]fGr
]n~rGr
! 5 2jk1 cosa Rf0w~x !exp~2jk1x sin a!exp@2jD~x !#J r P Gr , (21)
where R is the Fresnel reflection coefficient and D(x) is aphase function that accounts for the surface-relief profileof the DOE. The Fresnel reflection coefficient representsthe change in phase and amplitude experienced by an in-finite plane wave upon reflection from a planar boundarybut is used here to model the change in phase and ampli-
conditions) and can be expressed as21,33,40–42
f1RS1~r1! 5 f1
inc~r1! 1 2EGr
FfGr~rGr
8 !]G1
]n8~r1 , rGr
8 !Gdl8,
r1 P S1 . (24)
Bendickson et al. Vol. 16, No. 1 /January 1999 /J. Opt. Soc. Am. A 119
Similarly, the Rayleigh–Sommerfeld 2 formulation re-quires only the normal derivatives of the boundary fields(Neumann boundary conditions) and takes theform21,33,40–42
f1RS2~r1! 5 f1
inc~r1! 2 2EGr
FG1~r1 , rGr8 !
]fGr
]n8~rGr
8 !Gdl8,
r1 P S1 . (25)
From inspection of Eqs. (23)–(25), it is clear that theKirchhoff diffraction integral is the average of the twoRayleigh–Sommerfeld diffraction integrals. All the fieldquantities on the left-hand sides of Eqs. (23)–(25) repre-sent the total fields in region S1 . To obtain the scatteredfield, the incident fields are simply subtracted from theseexpressions.
In practice, calculating the diffraction integrals abovecan become quite time consuming if there is a very largenumber of observation points. However, the computa-tional time can be dramatically reduced by using theplane-wave spectrum methods to determine the diffractedfields. These methods use Fourier transform theory andsimple plane-wave propagation to take advantage of thecomputational efficiency of the fast Fourier transform. Itis possible to develop three such formulations, one foreach set of boundary conditions (Kirchhoff, Dirichlet,Neumann), which calculate fields that are essentiallyequivalent to those calculated by the Kirchhoff andRayleigh–Sommerfeld diffraction integrals. As one ex-ample, consider the plane-wave spectrum method corre-sponding to Dirichlet boundary conditions. Mathemati-cally, the angular spectrum of plane waves may beexpressed as
F~kx! 5 E2`
`
fGr~rGr
!exp~ jkx x !dx, (26)
where kx is the spatial frequency in the x direction. Thediffracted fields in region S1 can be represented as33,40,43
f1PW1~r1! 5 f1
inc~r1! 11
2p E2`
`
F~kx!
3 exp@2j~kx x 1 ky y !#dkx , (27)
where
ky 5 H Ak12 2 kx
2 for kx2 < k1
2
2jAkx2 2 k1
2 for kx2 . k1
2. (28)
The quantity ky is the spatial frequency in the y directionand can be either real, corresponding to propagatingwaves, or imaginary, corresponding to exponentially de-caying or evanescent waves. The plane-wave spectrummethods have similar representations for Neumann andKirchhoff boundary conditions. More detail on the plane-wave spectrum methods can be found in Refs. 33, 40, and43.
D. Power and Diffraction EfficiencyTo obtain quantitative measures of the performance ofthe diffractive mirrors designed and analyzed in thiswork, it is useful to have several performance metrics
that may be calculated and compared. Two of the mostimportant metrics are the total reflected power Pr and thediffraction efficiency DE. To calculate these quantitiesfor TE polarization, the angular spectrum of the back-ward scattered field is determined (by using a fast Fouriertransform) as
A1~rn , y 5 y1! 51
M (m52M/2
M/221
Ez1s ~mDx, y1!
3 exp~ jrnmDx !, (29)
where Ez1s 5 f1
t 2 f1inc . The quantity M is the number
of sampling points used in the fast Fourier transform and2L/2 < x < L/2. Here rn 5 2np/L, where L is the sizeof the sampled region and n can have values ranging from2M/2 to M/2 2 1. Additionally, Dx 5 L/M, and y1 isthe y coordinate of the observation point. Once theA1(rn) have been determined, the reflected power may becalculated as29,32
Pr 5 ReF L
2h1(
m52M/2
M/221b1m*
k1uA1~rm , y 5 y1!u2G , (30)
where Re( • ) denotes the real part, h1 5 (m0 /n12e0)1/2,
b1m 5 (k12 2 rm
2)1/2, and a superscript * denotes thecomplex conjugate. If the diffractive structure is a focus-ing mirror with focal length f, the amount of power fo-cused within a detection slit extending over the rangea < x < b in region S1 is
Pf 5 ReH b 2 a
2h1(
m52M/2
M/221
(n52M/2
M/221
expF j~rm 2 rn!
3S b 1 a
2 D G b1m*
k1@A1~rm , y 5 f !#*
3 A1~rn , y 5 f !sincF ~rm 2 rn!S b 2 a
2 D G J ,
(31)
where sinc(x) [ (sin x)/x. The diffraction efficiency isdefined as DE 5 Pf /P0 , where P0 is the incident power.The expressions that apply for TM polarization may beobtained from Eqs. (30) and (31) by simply replacing h iwith 1 /h i , b im with b im* , rm 2 rn with rn 2 rm , andAi(r, y 5 y1) with @Ai(r, y 5 y1)#* .
4. FOCUSING DIFFRACTIVE MIRRORPERFORMANCEA variety of focusing diffractive mirrors were designed byusing the methods discussed in Section 2. However, allof the mirrors designed and analyzed in this paper haveseveral characteristics in common. In all cases the inci-dent region S1 is free space with refractive index n15 1, and the free-space wavelength of the monochro-matic incident radiation is l0 5 10.0 mm, correspondingto the far-IR region of the spectrum. The reflective ma-terial making up region S2 is gold with a refractive indexof n2 5 11.5 2 j67.5 at l0 5 10.0 mm (from Ref. 44).Also, all mirrors had a width of 500 mm (50l0). The
120 J. Opt. Soc. Am. A/Vol. 16, No. 1 /January 1999 Bendickson et al.
most time-consuming processes in the rigorous BEManalysis are filling and inverting the matrix used to rep-resent the system of equations. For diffractive mirrors ofthe size considered in this paper, and a discretization often elements per free-space wavelength, the BEM modelhas on the order of 2000 computational nodes and thus4000 unknowns. The 4000 3 4000 BEM matrix thentakes approximately 30 min to fill and 45 min to invert ona HP 9000 J280 machine.
A. Effect of Mirror ProfileFabrication of DOE’s is often performed by photolithogra-phy in which a series of K phase masks is used to producea DOE with 2K discrete levels. For this reason it is in-teresting to study the effects of such a discretization onthe performance of the focusing diffractive mirrors consid-ered in this paper. For this study six different mirrortypes were considered. The first type, referred to here asthe continuous Fresnel mirror, has a continuous profilewith the exception of discontinuities at the zone bound-aries. The next four mirror types are discrete multilevelapproximations of the continuous Fresnel mirror with 2,4, 8, and 16 levels, respectively. The final mirror type,referred to here as continuous nondiffractive, is identicalto the continuous Fresnel mirror type but without thezone-boundary discontinuities. As suggested by itsname, this mirror is not a diffractive design, and it will bemuch thicker than the five other types of mirror. All thesimulations for this particular study used mirrors de-signed for normal incidence and on-axis focusing and hav-ing focal length f 5 375 mm corresponding to anf-number 5 f/D 5 0.75. The six mirrors were analyzedby using the rigorous BEM for TE and TM incidence andthree scalar methods corresponding to Kirchhoff, Dirich-let, and Neumann boundary conditions.
A 2-D mapping of the scattered field intensity as deter-mined by the BEM for TE incidence is shown in Fig. 3 forfour of the six types of mirror (two-level, four-level, eight-level, and continuous nondiffractive). For all BEManalysis in this work, the boundary was discretized suchthat the maximum element length was 1/10 of a free-space wavelength. No significant changes were observedin the calculated fields when a finer discretization wasused in a few test cases, indicating that the method hadconverged to the solution. In the plots dark regions cor-respond to areas of large scattered field intensity, andlight regions correspond to areas of small scattered fieldintensity. At the bottom of each scattered field plot, themirror profile is shown as a solid black curve. It is clearfrom the plots that the focusing is greatly improved in go-ing from a two-level to a four-level diffractive mirror andthat significant additional improvement results in goingfrom a four-level to an eight-level diffractive mirror.Further increases in the smoothness of the diffractivemirrors do not significantly change the focusing behavior.In fact, the 2-D field intensity plots for the 16-level andcontinuous Fresnel mirrors look almost identical to thatof the eight-level mirror and are not shown for this rea-son. Significant differences are seen between the con-tinuous nondiffractive mirror and the five diffractive mir-rors. Not only is there far less scattering from thecontinuous nondiffractive mirror (on account of the ab-
sence of zone-boundary discontinuities), but the focalpoint has shifted approximately 30 mm in the positive ydirection from its designed location. This is a result ofusing the zero-thickness mirror design for a mirror withthickness greater than 35 mm (3.5l0).
To compare quantitatively the performance of the vari-ous types of mirror, a number of different metrics wereevaluated for each mirror type. The first step in theanalysis was to determine the focal length by finding thelocation of the maximum field intensity along the y axis.For both TE and TM incidence and the five diffractivemirror designs, the focal length determined by the BEMwas within 3% of the designed focal length of 375 mm.Additionally, the focal lengths determined by the threescalar methods agreed with the rigorous results to within1%. However, the focal length of the continuous nondif-fractive mirror was determined to be 406 mm by the BEMfor both TE and TM incidence—a difference of more than30 mm from the designed focal length. As mentionedabove, this difference results from the fact that the zero-thickness design was used. In contrast, the scalar meth-ods predicted focal lengths between 373 and 374 mm forthe continuous nondiffractive mirror. Because thesurface-relief profile of the DOE is deformed into the lin-ear boundary Gr for the scalar methods, they are unableto predict accurately the shift in the focal length of thismirror. The scalar methods do not produce significanterrors for the focal length of the diffractive mirrors, onlybecause the maximum thickness of these mirrors is lessthan or equal to 5 mm (0.5l0).
Another important metric in many applications is thefocal spot size d, which is defined here as the minimum-to-minimum width of the main lobe evaluated in the focalplane of the mirrors. For all six mirrors, both rigorousand scalar methods predicted d 5 17 6 0.5 mm. Thediffraction-limited spot size, d0 5 8f/k1D, is 9.55 mm forthese mirrors. The difference between the actual anddiffraction-limited spot sizes is due in part to the fact thatd0 is determined analytically from the 1/e2 points of themain lobe as opposed to the minima. However, it is alsoexpected that small f-number mirrors such as the ones be-ing analyzed here, with f-number 5 0.75, will not per-form well enough to achieve diffraction-limited focusingwhen designed by scalar methods.
The total reflected power Pr was also calculated by themethod discussed in Subsection 3.D for the various mirrortypes and analysis methods, and the normalized reflectedpower Pr /P0 is shown in Table 1. In the calculation ofthe reflected power, 4096 samples of the diffracted fieldwere used over a calculational window L 5 2000 mmwide in a plane less than one wavelength away from thesurface of the diffractive mirrors. The total reflectedpower has been normalized by the incident power so thatit always has values between zero and unity, with Pr /P05 1 corresponding to all of the incident power being re-flected. For the rigorous analysis, the normalized totalreflected power will never quite be equal to unity becausea small percentage of the power will be transmitted andabsorbed by the metal as a result of its finite conductivity.With use of the BEM, Pr /P0 was found to be between 0.97and 0.99 for both TE and TM incidence and all mirrortypes. However, the scalar method predictions of the re-
Bendickson et al. Vol. 16, No. 1 /January 1999 /J. Opt. Soc. Am. A 121
Fig. 3. Diffracted field intensity for two-level, four-level, and eight-level diffractive mirrors and a continuous nondiffractive mirror, alldesigned for normal incidence and on-axis focusing. All mirrors are designed with D 5 500 mm, f 5 375 mm, and n2 5 11.5 2 j67.5corresponding to gold for l0 5 10 mm. The diffracted fields are determined by using the BEM for TE incidence. Dark regions indicateareas of high field intensity.
Table 1. Normalized Total Reflected Power Obtained with the BEM and Scalar Methodsfor Various Types of Focusing Mirror Profile
flected power showed a much greater variation among thevarious mirror types. As shown in Table 1, the scalarmethod reflected powers are significantly smaller thanthe correct results for the two-level and four-level mir-rors; in fact, the diffracted fields corresponding to Dirich-let boundary conditions result in values of Pr /P0 onlyslightly larger than 0.80 for these mirrors. The fact that
the Kirchhoff diffracted fields are an average of the Di-richlet and Neumann fields is also clear from the table,since the Kirchhoff reflected power is approximately anaverage of the reflected powers of the two other scalarmethods. As the mirror profiles become smoother, thescalar method reflected powers increase significantly.For the 16-level, continuous Fresnel, and continuous non-
122 J. Opt. Soc. Am. A/Vol. 16, No. 1 /January 1999 Bendickson et al.
diffractive mirrors, the Kirchhoff reflected power is shownto be in very close agreement with the rigorous results forboth polarizations.
Typically, the most important performance metric for afocusing element is the diffraction efficiency DE, definedhere as the fraction of the incident power P0 that is fo-cused within the minimum-to-minimum width of themain lobe in the focal plane. The diffraction efficiencywas calculated with Eq. (31) by using the same calcula-tional window and number of sample points as those forthe reflected power, but the diffracted field was sampledin the focal plane instead of close to the mirror surface.The diffraction efficiencies for all the mirror types andmethods of analysis are given in Table 2. Consistentwith the observations of Fig. 3, the diffraction efficiencyapproximately doubles in going from a two-level to a four-level design and increases significantly in adding onemore phase mask to obtain an eight-level design. Be-yond this point, using even smoother diffractive mirrorsresults in no more than a 5% increase in diffraction effi-ciency. With the exception of the two-level diffractiveand continuous nondiffractive mirrors, the diffraction ef-ficiency as determined by the rigorous BEM is consis-tently higher for TM polarization than for TE polarizationby approximately 10%. This higher efficiency for the TMpolarization is consistent with the general experimentalresults from surface-relief metallic gratings presented inRef. 45. Despite the very low f-number of these mirrors,the scalar method diffraction efficiencies are in goodagreement with the rigorous results, as can be seen in thetable. However, it is impossible to say that any of thethree scalar methods is the ‘‘best’’ in a general sense, be-
cause the agreement between the various scalar methodsand the rigorous BEM depends strongly on both the po-larization and the type of mirror.
A final performance metric that is calculated is thesidelobe power Psl . The sidelobe power is computed ac-cording to the expression for focused power Pf in Eq. (31),but the slit extends over one of the first sidelobes, whoselimits are defined by the first and the second minimum tothe left or to the right of the main lobe in the focal-planeintensity profile. In Table 3 the normalized sidelobepowers Psl /P0 are shown for the same mirrors and analy-sis methods as those above. As with the diffraction effi-ciencies, the rigorous results indicate that the sidelobepowers are larger for TM polarization than for TE polar-ization for the diffractive mirrors, but to a much greaterextent in terms of relative size. In fact, for all but thetwo-level and continuous nondiffractive mirrors, the side-lobe powers for TM incidence are more than 50% largerthan those for TE incidence. It is also interesting to notethat among the five diffractive mirrors, and for both TEand TM incidence, the four-level mirrors exhibit the larg-est sidelobes by a substantial amount. This is in contrastto the diffraction efficiency, which increased monotoni-cally as the smoothness of the mirrors increased. Thescalar methods do not accurately predict this effect; in-stead, the sidelobe powers increase monotonically, follow-ing a trend similar to that of the scalar diffraction effi-ciencies. As the BEM results in Table 3 indicate, thecontinuous nondiffractive mirrors have much larger side-lobe powers than those of any of the diffractive mirrortypes. Rigorous results indicate that the sidelobes of thecontinuous nondiffractive mirrors are almost twice as
Table 2. Diffraction Efficiency Obtained with the BEM and Scalar Methodsfor Various Types of Focusing Mirror Profile
Bendickson et al. Vol. 16, No. 1 /January 1999 /J. Opt. Soc. Am. A 123
broad as those of the other mirrors; this is due to the sub-optimal performance of the zero-thickness mirror design.This increase in the sidelobe width is the primary causefor the large sidelobe powers shown in Table 3 for the con-tinuous nondiffractive mirror. Once again, the scalarmethods, although appropriate in some cases, do not ac-curately model the broadening of the sidelobes and thecorresponding increase in the sidelobe power that are ob-served in the rigorous results for the continuous nondif-fractive mirror.
B. Effect of f-NumberOne of the most important parameters of a focusing sys-tem, whether a lens or a mirror, is the f-number. For dif-fractive focusing elements, it is well known that as thef-number of a system decreases, the feature size of theDOE’s becomes small and scalar methods become less ac-curate. Quantitative results comparing scalar methodswith a rigorous BEM have recently been presented thatconfirm this for the case of focusing diffractive lenses in atransmission configuration.33 In addition, it is expectedthat the performance of diffractive focusing elementswhose design is based on scalar diffraction methods willbe degraded as the f-number is reduced. In this subsec-tion the performance of metallic focusing diffractive mir-rors is analyzed by both the rigorous BEM and scalarmethods for four different f-number mirrors: f/2, f/1,f/0.75, and f/0.5. As above, the width of all mirrors is 500mm, so that the focal lengths are 1000, 500, 375, and 250mm, respectively. In all cases the diffractive mirrors aredesigned with eight discrete levels, all simulations are fornormal incidence and on-axis focusing, and all mirrorsare zero-thickness designs.
In Fig. 4 various performance metrics are displayed, in-cluding diffraction efficiency DE, normalized sidelobepower Psl /P0 , normalized total reflected power Pr /P0 ,and normalized spot size d/d0 , versus the f-number of themirrors. From Fig. 4 it is clear that all four performancemetrics calculated by the three scalar methods differmuch less from one another and rapidly converge to therigorous results as the f-number of the mirrors is in-creased. Also, the rigorous BEM results for TE and TMincidence become more similar for all metrics except thetotal reflected power as the f-number is increased. Con-sistent with the results of Subsection 4.A, the diffractionefficiency and the sidelobe power are consistently largerfor TM incidence than for TE incidence. Another impor-tant distinction between the two polarizations is that thediffraction efficiency for TM incidence is reduced by only9% as the f-number is changed from f/2 to f/0.5, whereasfor TE incidence it is reduced by 19%, more than twice asmuch, for the same change in the f-number. As seen inFig. 4(c), rigorous calculations for the reflected power re-main fairly constant as the f-number varies, staying be-tween 0.97 and 0.99 for both polarizations. As for thescalar methods, the normalized Kirchhoff reflected powerPr /P0 shows little variation with f-number, staying be-tween 0.95 and 0.97, whereas the normalized Dirichletand Neumann reflected powers change significantly asthe f-number is reduced. In fact, for an f/0.5 mirror, thenormalized reflected power Pr /P0 drops to less than 0.84for the Dirichlet method, whereas it becomes greater than
1.08 for the Neumann method, clearly an unphysical re-sult. Finally, the spot size d approaches the diffraction-limited spot size d0 as the f-number increases, as illus-trated in Fig. 4(d). In addition, the scalar methods are ingood agreement with the rigorous results for the spot size,even when the f-number becomes as small as 0.5.
C. Effect of On-Axis and Off-Axis Incidence/FocusingIn many applications of metallic diffractive focusing mir-rors, the incoming wave may not be incident normal tothe surface of the mirror, and/or the focusing may not bealong the axis of the mirror. A variety of mirrors de-signed for such configurations are analyzed in this sub-section by rigorous and scalar methods to address several
Fig. 4. (a) Diffraction efficiency, (b) normalized sidelobe power,(c) normalized reflected power, and (d) normalized spot size ver-sus f-number for eight-level diffractive mirrors designed for nor-mal incidence and on-axis focusing. All mirrors are designedwith D 5 500 mm and n2 5 11.5 2 j67.5. The mirrors havebeen analyzed by the rigorous BEM for both TE and TM inci-dence and by the Dirichlet, Neumann, and Kirchhoff scalar inte-gral methods.
124 J. Opt. Soc. Am. A/Vol. 16, No. 1 /January 1999 Bendickson et al.
important issues. First, several performance metricshave been evaluated from rigorously determined dif-fracted fields to quantify the degradation in performancefor varying degrees of off-axis incidence and/or off-axis fo-cusing. Second, these same metrics have been evaluatedfrom scalar diffracted fields to determine the accuracy ofthe scalar methods for varying degrees of off-axis inci-dence or off-axis focusing. Finally, scalar method calcu-lations have been made both without (g 5 1) and with(g 5 sec a) the off-axis incidence correction factor. Theeffects of including this correction factor for scalar analy-sis can then be determined. For all of these studies, zero-thickness eight-level diffractive mirror designs have beenused with f-number 5 0.75, corresponding to a focallength of 375 mm. The following three focusing configu-rations were studied: (1) a 5 0° and b 5 5°, 25°, 45°;(2) a 5 5°, 25°, 45° and b 5 0°; and (3) a 5 b 5 5°, 25°,45°. As above, the analysis of the mirrors may be dividedinto five cases: the rigorous BEM for TE and TM inci-dence, and the Dirichlet, Neumann, and Kirchhoff scalarintegral methods.
Before a discussion of the results, some comment as to
how various metrics were calculated is appropriate. Ini-tially, a 2-D mapping of the diffracted field intensity wasconstructed to obtain a qualitative understanding of theoverall scattering behavior and also to determine the lo-cation of the focal plane. Then a slice of the diffractedfield intensity was taken through the focal plane so thatthe main lobe and the first sidelobes could be identifiedalong with their associated minima locations. From thisinformation the spot size was determined, and the diffrac-tion efficiency, the sidelobe power, and the total reflectedpower could be calculated. As in the cases of normal in-cidence and on-axis focusing, the diffraction efficiency, thesidelobe power, and the total reflected power correspondto measures of the time-averaged power flow in the direc-tion of the positive y axis. The diffraction efficiency DE,the normalized total reflected power Pr /P0 , and the nor-malized average sidelobe power (Psl
L 1 PslR)/2P0 for each
of the three focusing configurations and the five types ofanalysis are summarized in Tables 4–6. The quantitiesPsl
L and PslR denote the power in the left and right side-
lobes, respectively. The scalar method calculations inthese tables were made without including the off-axis in-
Table 4. Diffraction Efficiency Obtained with the BEM and Scalar Methods for Eight-Level f/0.75Diffractive Mirrors in Various Off-Axis Incidence and/or Off-Axis Focusing Configurations
Table 5. Normalized Total Reflected Power Obtained with the BEM and Scalar Methods for Eight-Levelf/0.75 Diffractive Mirrors in Various Off-Axis Incidence and/or Off-Axis Focusing Configurations
Bendickson et al. Vol. 16, No. 1 /January 1999 /J. Opt. Soc. Am. A 125
Table 6. Normalized Average Sidelobe Power Obtained with the BEM and Scalar Methods for Eight-Levelf/0.75 Diffractive Mirrors in Various Off-Axis Incidence and/or Off-Axis Focusing Configurations
cidence correction factor, so that g 5 1 in all cases shown.The differences that result from using g 5 sec a will beexamined in the discussion of the scalar methods. Theaverage sidelobe powers given in each case in Table 6 arethe averages of the left and right sidelobe powers. Foreach focusing configuration, there is a detailed discussionof the performance of the mirrors for TE and TM inci-dence as determined by the rigorous BEM, followed bysome brief comments concerning the scalar method re-sults.
a 5 0°, b Þ 0°. Consider the configuration of normalincidence (a 5 0°) and off-axis focusing (b Þ 0°). Fromthe BEM results in Table 4, it is evident that as the fo-cusing angle is increased, the diffraction efficiency is re-duced much more for TE incidence than for TM incidence.In fact, the diffraction efficiency for b 5 45° is reducedfrom its on-axis focusing value by nearly 30% for TE inci-dence, whereas it is reduced by less than 6% for TM inci-dence. In Fig. 5, 2-D mappings of the scattered field in-tensity have been shown for TE and TM incidence in thecase where a 5 0° and b 5 45°. From these plots onecan see that the drastic reduction in the diffraction effi-ciency for TE incidence results from a large portion of theincident light being scattered specularly back in the direc-tion of the positive y axis. Meanwhile, there appears tobe far less scattering in this direction for TM incidence.As seen in Table 5, the normalized total reflected powerremains very close to unity for all focusing angles andboth polarizations, always being slightly larger for theTM polarization. The normalized average sidelobe powershown in Table 6 also stays relatively constant for variousfocusing angles, but the TM sidelobe power is signifi-cantly larger than the TE sidelobe power by a factor rang-ing between 1.3 and 2. Calculations also indicate thatthe focal spot size more than doubles for both polariza-tions as the focusing angle increases from 0° to 45°. Theincrease in the spot size is partially due to the fact thatthe spot size is determined from the intensity profilealong a line parallel to the x axis, even though the overalldirection of the focused beam is not along the positive yaxis but at some angle (5°, 25°, 45°) with respect to thepositive y axis. Although one would expect this effect to
increase the spot size by a factor of sec b, the spot size ac-tually increases by more than this factor, indicating thatthe general focusing quality of the mirror is degraded asthe focusing angle is increased.
As seen in Table 5, the scalar method reflected powersbecome very different from one another as the focusingangle is increased, especially for b 5 45°. Whereas theKirchhoff reflected power remains relatively close to therigorous values for increasing b, the Neumann methodnormalized reflected power becomes much larger, and theDirichlet method normalized reflected power becomesmuch smaller. The diffraction efficiency and the sidelobepowers exhibit a similar trend.
a Þ 0°, b 5 0°. Next, consider the configuration withoff-axis incidence (a Þ 0°) and on-axis focusing (b5 0°). The BEM results for the diffraction efficiency inTable 4 are very similar to those of the previous case. Asthe angle of incidence increases to a 5 45°, the TE dif-fraction efficiency drops by more than 36% from itsnormal-incidence value, whereas the TM diffraction effi-ciency is reduced by less than 6%. In Fig. 6, 2-D map-pings of the diffracted field intensity are shown for bothpolarizations in the case in which a 5 45° and b 5 0°.As above, the difference between the TE and TM diffrac-tion efficiencies stems from the specular reflection (whichin this case is shown by the beam at 45° with respect toboth axes), observed to be quite strong for TE incidenceand relatively weak in the TM case. The fact that thespecularly reflected component is present in the TM case,though to a lesser extent, explains the slight decrease inthe TM diffraction efficiency. As with the previous con-figuration, the sidelobe power does not change apprecia-bly as the angle of incidence increases, and the sidelobepower is always significantly larger for TM polarizationthan for TE polarization, as shown in Table 6. However,in contrast to the first configuration, the focal spot sizedoes not change significantly as the angle of incidence isincreased.
In this configuration the absence or the presence of theoff-axis incidence correction factor has a significant effecton the scalar performance metrics for a 5 45° but only a
126 J. Opt. Soc. Am. A/Vol. 16, No. 1 /January 1999 Bendickson et al.
slight effect for a 5 5° or a 5 25°. As seen in Table 5,the reflected powers for g 5 1 follow a trend opposite thatof the previous configuration, with the Dirichlet reflectedpowers becoming large and the Neumann reflected pow-ers becoming small. The inclusion of the correction fac-tor causes a reduction of the reflected power of the variousscalar methods by approximately 20%, and similar effectsare seen for the diffraction efficiency and the sidelobepower.
a 5 b Þ 0°. The final focusing configuration consistsof off-axis incidence and off-axis focusing, with the inci-dence and focusing angles being equal to each other (a5 b Þ 0°). This means that a plane wave incident fromthe upper left at some angle with respect to the positive yaxis will be reflected and focused to a point to the upperright of the mirror that makes an equal angle but on theopposite side of the positive y axis. The rigorous BEM re-sults for this configuration show many differences withthose of the two previous configurations. In Figs. 7(a),7(b), and 7(c), the 2-D intensity plots are shown for thisconfiguration (TE incidence only) as the incidence and fo-cusing angles take on the values 5°, 25°, and 45°, respec-tively. Whereas the focusing seems to be quite good forthe 5° and 25° cases in Figs. 7(a) and 7(b), it appears tobecome much worse for the 45° case. For a 5 b 5 45°the diffraction efficiency for both TE and TM incidence
has been reduced by more than 20% from that of the a5 b 5 0° case. Also, the focal spot size increases by afactor of 3 for the same change in incidence and focusingangles, resulting in spot sizes of approximately 50 mm forTE and TM polarizations when a 5 b 5 45°. This blur-ring of the focal spot is clearly visible in Fig. 7(c). A largeincrease in sidelobe power is another important indica-tion of the degradation in the performance of these mir-rors with increasing incidence and focusing angles. Asseen in Table 6, the sidelobe power for a 5 b 5 45° hasincreased by 165% and 130% over the a 5 b 5 0° valuesfor TE and TM polarizations, respectively. Finally, fora 5 b 5 45°, the focal length is greater than 390 mm forboth TE and TM incidence, more than 15 mm longer thanthe designed focal length.
In comparison with the two previous configurations,the reflected powers as determined by the scalar methodsdo not increase or decrease as sharply with increasing in-cidence and focusing angle. Inclusion of the off-axis inci-dence correction factor affects the reflected power in amanner similar to that of the previous configuration.With g 5 1 the scalar methods do not accurately modeleither the large reduction in diffraction efficiency or thesharp increase in sidelobe power predicted by the BEM asa and b are increased to 45°. Including the correctionfactor (g 5 sec a) produces results that are much closer
Fig. 5. Diffracted field intensity as determined by the BEM for TE and TM incidence on an eight-level diffractive mirror designed fornormal incidence and 45° off-axis focusing. The mirror is designed with D 5 500 mm, f 5 375 mm, and n2 5 11.5 2 j67.5.
Bendickson et al. Vol. 16, No. 1 /January 1999 /J. Opt. Soc. Am. A 127
Fig. 6. Same as Fig. 5, but for 45° off-axis incidence and on-axis focusing.
to the rigorous diffraction efficiency and sidelobe power,especially for TE incidence. However, the scalar methodpredictions for other parameters, such as the focal length,are significantly worse when the correction factor is in-cluded, making it difficult to establish whether its pres-ence is beneficial.
The preceding paragraphs have presented several ex-amples in which the performance of the mirrors is de-graded as the incidence angle and/or the focusing angle isincreased. Most of the significant degradation has beenobserved when a and/or b is equal to 45°. One shouldnote that the mirrors perform quite well for smallerangles. In fact, even for angles as large as 25°, the per-formance of the mirrors in terms of diffraction efficiency,sidelobe power, and focal spot size is within a few percentof the performance of focusing mirrors in a normal-incidence, on-axis configuration.
D. Finite-Thickness DesignsAll of the diffractive mirror results discussed thus farhave dealt with zero-thickness mirror designs hzt(x). Inthis subsection finite-thickness mirror designs h f t(x) areconsidered, and the performance differences between thetwo types of mirror design are investigated. Because it isthe performance of the mirrors that is of interest in thisstudy, and not the accuracy of various methods of analy-sis, only the rigorous BEM results are presented, for both
TE and TM incidence. The performances of the zero-thickness and finite-thickness mirror designs were com-pared. First, the finite-thickness f/0.75 mirrors designedfor normal incidence and on-axis focusing (a 5 b 5 0°)were simulated. Two mirror types, eight-level diffractiveand continuous nondiffractive, were considered. Second,the eight-level f/0.75 focusing diffractive mirrors designedfor the various off-axis configurations discussed in Sub-section 4.C were simulated. The results of the first set ofsimulations indicate that using a finite-thickness designfor the eight-level diffractive mirror increases the TE dif-fraction efficiency by 1.78%, whereas the TM diffractionefficiency remains essentially unchanged. The slight im-provement in the TE diffraction efficiency is accompaniedby a 6.67% increase in the sidelobe power, which may beundesirable in some applications. Using the finite-thickness design produces only small performancechanges because the diffractive mirror is relatively thin.For this reason one would expect similar results to holdfor other multilevel and continuous Fresnel diffractivemirrors. However, the continuous nondiffractive mirrorhas a much greater thickness (more than 3.5 free-spacewavelengths), and for this reason the finite-thickness de-sign significantly improves the performance of this mir-ror. In this case the diffraction efficiency for both polar-izations increases by more than 8%, whereas the sidelobepower is reduced by more than 50%.
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The purpose of the second set of simulations was to de-termine if the finite-thickness designs provided any per-formance improvement for diffractive mirrors in off-axisincidence and/or off-axis focusing configurations. It wasfound that the effects of replacing a zero-thickness designwith a finite-thickness design depended strongly on theoff-axis configuration. For the case of normal incidenceand off-axis focusing, the finite-thickness diffractive mir-rors performed slightly worse than the zero-thickness de-
signs for both polarizations and focusing angles of 5°, 25°,and 45°. However, for the off-axis incidence and on-axisfocusing configuration, the finite-thickness mirrors per-formed significantly better for TE incidence. In fact, foran incident angle of 45°, the TE diffraction efficiency wasincreased by nearly 23% as a result of using the finite-thickness mirror design. In contrast, the TM diffractionefficiency remained essentially unchanged in this configu-ration for a 5 5° and 25° and decreased by more than 7%
Fig. 7. Diffracted field intensity as determined by the BEM for TE incidence on an eight-level diffractive mirror designed for (a) 5°off-axis incidence and 5° off-axis focusing, (b) 25° off-axis incidence and 25° off-axis focusing, and (c) 45° off-axis incidence and 45° off-axisfocusing. All mirrors are designed with D 5 500 mm, f 5 375 mm, and n2 5 11.5 2 j67.5.
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for a 5 45°. Finally, consider the off-axis incidence andoff-axis focusing configuration. For the case where a5 b 5 45°, the diffraction efficiency for both polariza-tions was increased significantly by using a finite-thickness mirror design in place of a zero-thickness de-sign. The TE diffraction efficiency increased by nearly25%, and the TM diffraction efficiency increased by morethan 10%. As further evidence of improved performance,the average sidelobe power for the finite-thickness designwas less than half that of the zero-thickness design forboth polarizations. These results confirm that, at leastin some off-axis configurations, the finite-thickness mir-ror designs significantly outperform the zero-thicknessdesigns.
5. CONCLUSIONSMetallic diffractive mirrors have been designed for vari-ous focusing configurations and have been analyzed byboth rigorous and scalar integral methods. Two designmethods, zero-thickness and finite-thickness, have beendeveloped that give the mirror profile and the zone-boundary locations for the general case of off-axis plane-wave incidence and off-axis focusing. To the authors’knowledge, this is the first detailed presentation of thesedesign methods for cylindrical diffractive mirrors. Therigorous analysis consisted of a 2-D open-region formula-tion of the BEM suitable for application to metallic dif-fractive mirrors with finite conductivity, and three scalarintegral methods were used based on Dirichlet, Neu-mann, and Kirchhoff boundary conditions. Upon deter-mination of the diffracted fields, a variety of metrics in-cluding diffraction efficiency, sidelobe power, totalreflected power, and focal spot size have been used toquantify the performance of the focusing mirrors. To theauthors’ knowledge, this is the first such detailed analysisby rigorous and scalar methods for finite-conductivity dif-fractive mirrors.
With the use of these analysis methods and mirror de-signs, the effects of a number of different mirror param-eters were studied including the type of mirror profile, thef-number, the on-axis or off-axis incidence and focusingconfiguration, and the finite-thickness versus zero-thickness mirror design. Although the focusing behaviorof the mirrors was found to be enhanced significantly byincreasing the number of discrete levels from 2 to 4, andto a lesser extent from 4 to 8, finer discretizations andcontinuous profiles provided only minor improvements.Continuous nondiffractive mirrors were also analyzedand found to produce far less scattering and somewhathigher efficiencies. However, to avoid shifts in the focallength and suboptimal performance in terms of diffractionefficiency and sidelobe power, finite-thickness designsmust be used for the continuous nondiffractive mirrors.In contrast, for the diffractive mirrors in normal-incidence, on-axis focusing configurations, none of theperformance metrics showed appreciable differences be-tween zero-thickness and finite-thickness mirror designs.It was also shown that the mirrors consistently producedhigher diffraction efficiency for TM incidence than for TEincidence, particularly for the four-level, eight-level, 16-level, and continuous Fresnel diffractive mirrors.
Mirrors with various f-numbers were also analyzed todetermine the effect of f-number on focusing performanceand on the accuracy of the scalar methods relative to thatof the BEM. As above, the diffraction efficiency washigher for TM incidence than for TE incidence, but thiseffect became less exaggerated as the f-number of the mir-rors increased. As expected, the performance of the mir-rors for both polarizations improved in terms of diffrac-tion efficiency and focal spot size as the f-numberincreased. Also as expected, the scalar method resultsbecame more similar to one another and approximatedthe exact results more closely as the f-number was in-creased. Generally, for f-numbers greater than f/2 or f/4,the scalar methods are quite accurate.
In all configurations the focusing behavior was onlyslightly affected for incidence and/or focusing angles of 5°and was, in general, still quite good for angles of 25°.However, increasing the incidence and/or focusing anglesto 45°, in some cases, brought about dramatic negative ef-fects in terms of diffraction efficiency, sidelobe power, andfocal spot size. One interesting result of the simulationswas that for two configurations, normal incidence withoff-axis focusing and off-axis incidence with on-axis focus-ing, the diffraction efficiency was reduced much more forTE incidence than for TM incidence because of a largespecular reflection component that remains in the TEcase. For the other configuration, off-axis incidence andoff-axis focusing, both polarizations underwent large re-ductions in diffraction efficiency for angles of 45°. Theperformance metrics as predicted by various scalar meth-ods were shown to become significantly less accurate asthe incidence and/or focusing angle was increased, withthe exception of the focal spot size, which was in mostcases quite accurate even for angles of 45°. Finally, theeffects of including the off-axis incidence correction factorwere investigated. However, since the scalar methodsare inherently not well suited to make accurate predic-tions of the diffracted fields when relatively large anglesof incidence or observation points located far off-axis areinvolved, the inclusion of this factor has a somewhat un-predictable effect. Including this factor seemed to affectadversely the accuracy of the total reflected power andthe focal length. Simultaneously, however, it seemed tohave a positive effect on other metrics such as diffractionefficiency, sidelobe power, and focal spot size. Finally,the performance of finite-thickness mirror designs wasstudied relative to that of zero-thickness designs. Theresults clearly indicated that replacing the zero-thicknessdesigns with finite-thickness ones significantly improvedthe performance of the mirrors for focusing configurationsinvolving off-axis incidence.
ACKNOWLEDGMENTSThis research was supported in part by grant DAAH-04096-1-0161 from the Joint Services Electronics Pro-gram and by grant ERC-94-02723 from the National Sci-ence Foundation. J. M. Bendickson was supported bythe Department of Defense National Science and Engi-neering Graduate Fellowship Program. E. N. Glytsiswas partially supported by Office of Naval Research grantN00014-96-1-0926.
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Address all correspondence to Jon M. Bendicksonat the address on the title page or e-mail,[email protected].
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