curved diffractive optical elements: design and...

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E. Wolf, Progress in Optics 48© 2005 Elsevier B.V.All rights reserved

Chapter 3

Curved diffractive optical elements:Design and applications

by

Nándor Bokor

Department of Physics, Budapest University of Technology and Economics,Budafoki u. 8, 1111 Budapest, Hungarye-mail: [email protected]

Nir Davidson

Department of Physics of Complex Systems, Weizmann Institute of Science,76100 Rehovot, Israel

ISSN: 0079-6638 DOI: 10.1016/S0079-6638(05)48003-8107

mailto:[email protected]

Contents

Page

§ 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

§ 2. Spherical/cylindrical CDOEs for imaging and Fourier transform . . 112

§ 3. Spherical/cylindrical CDOEs for concentration of diffuse light

on flat targets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

§ 4. Uniform collimation and ideal concentration for arbitrary source and

target shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

§ 5. CDOEs for controlling the geometrical apodization factor . . . . . . 136

§ 6. Curved gratings for spectroscopy . . . . . . . . . . . . . . . . . . . 141

§ 7. CDOEs for optical coordinate transformations . . . . . . . . . . . . 143

§ 8. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

108

§ 1. Introduction

Diffractive optical elements (DOEs) are used in many areas in optics such asimaging, Fourier transform, multifocusing lenses, spectroscopy, interferometry,optical signal processing and optical transformations. They have the advantage oflight weight, small thickness, and the flexibility to encode complicated asphericphase functions on a single element. The phase function, i.e. the shape of thegrating grooves, can be used as an optimization parameter for specific tasks. Inmany cases, for simplicity, DOEs are fabricated on flat substrates, but when man-ufactured on curved surfaces they offer many additional important advantages.Curved spectroscopic gratings were invented and first produced by Rowland al-ready in the 19th century (Rowland [1883]), thus they represent the first man-made curved diffractive optical elements (CDOEs). Rowland gratings combinethe diffractive properties of a spectroscopic grating with the focusing propertiesof a spherical mirror in a single element, thereby minimizing throughput losses.They are widely used in spectroscopy, especially in the far-ultraviolet regime.Welford [1965]gave a detailed aberration analysis of spectroscopic gratings ruledon spherical substrates, andSchmahl and Rudolph [1976]reviewed spectroscopicgratings recorded on spherical and toroidal substrates by holography.

DOEs serving for visual display applications, and fabricated on curved sur-faces, were first considered much later, byMcCauley, Simpson and Murbach[1973]. In their article, however, the substrate shape was not used as a free pa-rameter, but was rather motivated and dictated by the specific application area(windshields, swimming goggles, etc.).Fairchild and Fienup [1982]proposed anumerical raytracing technique for analyzing and designing DOEs holographi-cally recorded with aspheric wavefronts on both flat and curved substrate shapes.Hybrid refractive–diffractive lenses, in which a diffractive pattern is manufac-tured onto a curved lens surface, have been considered for combined CD–DVDreaders, or thermal imagers operating in wide temperature ranges (Wood [1992],Zhang and Wang [2004]). Curved diffraction gratings can even be found in na-ture: curved surface relief gratings are partially responsible for the metallic colorsof certain species of spiders (Parker and Hegedus [2003]).

109

110 Curved diffractive optical elements: Design and applications [3, § 1

In this chapter we show that when a DOE is fabricated on a curved substrate,the shape of the substrate can provide additional degrees of freedom in the design.Figure 1shows the schematic diagram of a CDOE, working in transmission. Boththe substrate shapez(x, y) and the phase functionφ[x, y, z(x, y)] are free para-meters which can be controlled independently to optimize the CDOE, resulting inmuch better performance than is possible with DOEs fabricated on flat substrates.In a simplified ray-optics picture, the phaseφ[x, y, z(x, y)] determines the di-rections into which the incoming rays are diffracted by the grating, whereas theshapez(x, y) determines the spatial density of rays propagating in the directionsimposed byφ[x, y, z(x, y)], thereby serving as an effective local apodization fac-tor.

In recent years, improved fabrication methods, such as sophisticated ruling ma-chines and interferometric recording with the help of computer-generated holog-raphy or computer-originated holography, have provided great flexibility in creat-ing the phase functionφ[x, y, z(x, y)]. Additionally, it is now feasible to produceand replicate arbitrary substrate shapesz(x, y) with high precision. These twopractical developments, along with the appearance of new application areas anddesign principles, account for the recent increase of interest in CDOEs.

For many applications either the CDOE shape or the phase function or bothcan have very simple analytical forms. For example, an aplanatic imaging CDOEsatisfying the sine condition, and hence eliminating coma, can be fabricated ona spherical (in 3D) or cylindrical (in 2D) substrate, with the interference of two

Fig. 1. Schematic diagram of a CDOE, working in transmission.

3, § 1] Introduction 111

simple spherical or cylindrical waves, as was first pointed out byMurty [1960]and Welford [1973]. First, in Section2, we will review the properties of suchspherical/cylindrical CDOEs, and their applications for aberration-free imagingand Fourier transformation.

In Section3 we will show that aplanatic spherical/cylindrical CDOEs satisfyingthe sine condition are also capable of concentrating quasi-monochromatic diffuselight onto a flat target at the theoretical limit imposed by brightness conservation(or equivalently, the second law of thermodynamics, seeBassett, Welford andWinston [1989]). By simply reversing the direction of the light rays, it can beshown that these spherical/cylindrical CDOEs can also uniformly collimate thelight emitted from a flat Lambertian source.

In Section4 we will extend the sine condition, which leads to coma-freeimaging and ideal diffuse light concentration and collimation, to arbitrary targetand source shapes. This generalization leads to CDOE shapes other than spheri-cal/cylindrical operating at the thermodynamic limit.

The smallest possible spot sizes, essential for applications such as lithography,data storage, microscopy, optical tweezers and optical traps, require high numeri-cal aperture focusing optics. An important factor for such systems is the so-calledgeometrical apodization factor, which determines the relative intensity of lightrays coming towards the focus from different directions. In Section5 we willshow that the substrate shape of CDOEs can control the geometrical apodizationfactor and hence improve the performance of high numerical aperture focusingsystems.

In Section6 we will discuss recent developments in the very old field of curvedspectroscopic gratings where either aspheric substrate shape or aspheric phasefunction are shown to suppress various types of aberration. As mentioned above,earlier works on spherical spectroscopic gratings, both ruled and holographicallyrecorded, were reviewed byWelford [1965]andSchmahl and Rudolph [1976],respectively, and are hence only briefly discussed here.

In Section7 we will discuss the application of CDOEs for optical transforma-tions and illustrate the principle with optical systems capable of coordinate trans-formation. Here the CDOE has an appropriately designed substrate shape, and therecording waves are simple plane waves. Finally, we will conclude in Section8.

Throughout the chapter, except in Section5, we resort only to geometrical op-tics, since the sizes of CDOEs and of all other elements in the systems consideredare much larger than diffraction-limited sizes. Moreover, except in Section6, welimit the discussion to quasi-monochromatic light to suppress the large chromaticaberration of DOEs. We do not put emphasis on fabrication methods. We brieflymention that holography is a very flexible tool for recording the phase function


on the CDOE, provided that a substrate of the given shape can be coated with therecording material. For surfaces with zero curvature, like a cylinder, this can bedone, e.g., by glueing a photopolymer film to the substrate (Bokor and Davidson[2001b]), whereas for surfaces with nonzero curvature, like a sphere, dip-coatingprocesses can be used. We note that for most applications the performance of theCDOE depends only weakly on the exact shape of the substrate, and hence thefabricated shape does not have to be accurate within optical precision, as long asit is identical during recording and readout. CDOEs with straight or nearly straightgrating grooves can be formed using mechanical ruling engines. Finally, a litho-graphic method for fabricating CDOEs having axial symmetry was reported byXie, Lu, Li, Zhao and Weng [2002].

§ 2. Spherical/cylindrical CDOEs for imaging and Fourier transform

Consider an imaging system I with an object distancea and an image distanceb,as shown infig. 2 (all distances are taken to be positive when they are to the leftof I and negative if they are to the right). For a thin imaging lens I is the principalplane, and for a thin CDOE it is the surface of the CDOE. Abbe’s sine condition(Born and Wolf [1993]) states that first-order aberrations (i.e. aberrations thatgrow linearly with the object point’s distance from the optical axis) are eliminatedif

(2.1)sinα

sinβ= const= −b

a= −M,

whereM is the magnification andα andβ are arbitrary angles of the input andthe output rays, respectively, that connect the on-axis points A and B. A sys-tem that satisfies the Abbe sine condition(2.1) is called aplanatic and is free of

Fig. 2. An imaging system with a principal surface I;a andb are the object and image distances,respectively.

3, § 2] Spherical/cylindrical CDOEs for imaging and Fourier transform 113

first-order coma aberration.Welford [1973]showed that the surface that satisfiescondition(2.1) is a sphere with radius

(2.2)R = ab

a + b.

The center of the sphere is at a distance ofc = R from the origin (the commonpoint of I and the optical axis). Hence, if a CDOE is located on such an Abbe sur-face, it is free of first-order aberrations for off-axis points at an object distancea

and an image distanceb. Let us consider some specific cases:1. Transmission geometry (ab < 0) with unit magnification (M = −1). Here

the Abbe surface is a plane that is situated symmetrically between A and B(in this caseα = β; hence sinα/ sinβ = tanα/ tanβ = 1).

2. Reflection geometry (ab > 0) with unit magnification (M = 1). In this caseα = −β. Moreover, the input and output rays for on-axis points coincidecompletely. Hence the Abbe surface can be any arbitrary surface.

3. Reflection geometry with arbitrary (not unit) magnification. Here it followsfrom eq.(2.2) that the radius of curvature of the Abbe sphere is half theradius of the equivalent spherical mirror, i.e. the one with the same objectand image distances.

4. Plane-wave illumination, where, ifa = ∞, thenR = c = b for both trans-mission and reflection: Here the Abbe sine condition can simply be statedash = R sinβ, whereh is the distance of the incoming ray from the opticalaxis.

2.1. Spherical CDOE for imaging

Bokor and Davidson [2001b]investigated numerically and experimentally anaplanatic CDOE recorded as a reflection hologram. The performance of theCDOE was compared with that of a flat DOE, a reflection hologram recordedon a flat substrate, but having the same object and image distances. The objectdistance in both cases was infinity, hence the image distance was equal to the fo-cal length of the CDOEs. Numerical raytracing results showed severe coma forthe flat DOE, and a significantly larger spot size than for the spherical CDOE, asseen infig. 3. These spot diagrams were obtained for NA= 0.7, a focal length of35 mm, and an angular separation of 2 between adjacent spots.

The method was experimentally tested in a configuration for 1D imaging (cylin-drical CDOE). The recording setup is shown infig. 4. To form the CDOE, theinterference of a plane wave and a counterpropagating, diverging spherical wavewas recorded on a thin (∼20 µm) Dupont photopolymer film that was coated on


(a) (b)

Fig. 3. Spot diagrams obtained from numerical raytracing for (a) flat DOE, and (b) a CDOE satisfyingthe sine condition (Bokor and Davidson [2001b]).

Fig. 4. Recording setup of a CDOE satisfying the sine condition.

the outside of a 1 mm thick glass surface with a half-cylinder shape with radiusR = 35 mm. The spherical wave was produced by a 60× microscope objectivewhose focal point coincided with the center of the cylinder, yielding a beam witha large NA (>0.86). To ensure essentially 1D operation of the imaging CDOE –despite the spherical recording wave – the height of the CDOE was limited to3 mm, as compared to the width of 60 mm. 1D configuration can also be obtainedif a cylindrical lens, instead of the microscope objective, is used in the recordingsetup. For comparison, a flat DOE was also recorded with the same parameters.The focal length of both DOEs wasF = 35 mm. The 488 nm line of an argonlaser was used for both recording and reconstruction. The experimental focal im-ages are shown infig. 5. As expected, when the DOEs are reconstructed with anon-axis plane wave, no appreciable aberrations are present. However, when thereconstructing plane waves have an off-axis angle ofθ = 5, the flat DOE has se-


Fig. 5. Experimental results ofBokor and Davidson [2001b], showing a large coma for the flat DOEwhen it is illuminated with an off-axis plane wave; for the aplanatic CDOE the coma is eliminated.

vere coma, whereas the CDOE still has no appreciable aberration. This outcome isall the more impressive because the two DOEs had the same size of 60 mm, yield-ing NA = 30/35 = 0.86 for the CDOE and NA= 30/(302 + 352)1/2 = 0.65for the flat DOE. Thus the CDOE, even when its NA is much higher, performssignificantly better than the flat DOE.

Welford [1975]investigated the transmission CDOE setup shown infig. 6. Herethe holographic layer is on the concave surface of a spherical meniscus substrate.Each reconstructing ray (shown with solid arrow infig. 6) must pass the substratebefore reaching the hologram, and the meniscus will in general introduce bothspherical aberration and coma. The rays used for the recording are represented bybroken lines infig. 6. If the CDOE is reconstructed with a point source located at(xC, yC, zC), the Gaussian image point will be located at (xI, yI, zI ). For paraxialrays (x, y z),

(2.3)1

zI= 1

zC± µ

(1

zO− 1

zR

),

(2.4)xI

zI= xC

zC± µ

(xO

zO− xR

zR

),

(2.5)yI

zI= yC

zC± µ

(yO

zO− yR

zR

),

where (xO, yO, zO) and (xR, yR, zR) are the object and reference point locations,µ is the ratio between the reconstructing and the recording wavelengths, and the± sign refers to the±1st diffraction orders. The location of the paraxial imagepoint is independent ofR, the curvature of the CDOE. Both spherical aberrationand coma can be eliminated in the following way (Welford [1975]): the radiusof the inner surface of the meniscus and the distance of the reconstructing objectpoint are considered given. The location of the reconstructed image point can bechosen such that eq.(2.2) is satisfied, where nowa = zC andb = zI , hence coma


Fig. 6. Aplanatic transmission CDOE recorded on the inner surface of a meniscus substrate (Welford[1975]).

is canceled. The locations of the recording point sources must satisfy the paraxialimaging condition(2.3) which gives a relationship betweenzO andzR, but doesnot determine their values. They can be chosen such that the spherical aberrationintroduced by the meniscus is compensated, while relation(2.3) is maintained.

The application of aplanatic transmission CDOEs was proposed for optical diskpick-ups byTatsuno [1997]. The setup considered byTatsuno [1997]is similar tothe one shown infig. 6, except that the CDOE is recorded on the outer surface, andthe outer and inner surfaces of the substrate have a common center of curvaturewhich coincides with the focal point of the CDOE. This geometry ensures that thefinite thickness of the substrate does not introduce appreciable aberrations in thesystem.

All the third-order aberration terms – spherical aberration (S), coma (C),astigmatism (A), field curvature (F ) and distortion (D) – were calculated byJagoszewski [1985]andJagoszewski and Klakocar-Ciepacz [1986]for sphericalCDOEs recorded and reconstructed with spherical waves. The aberrations havethe following form:

(2.6)S = 1

z3C

− 1

z3I

± µ

(1

z3O

− 1

z3R

)+ 2

R

[1

z2C

− 1

z2I

± µ

(1

z2O

− 1

z2R

)],

Cx = xC

z3C

− xI

z3I

± µ

(xO

z3O

− xR

z3R

)

(2.7)+ 1

R

[xC

z2C

− xI

z2I

± µ

(xO

z2O

− xR

z2R

)],

(2.8)Ax = x2C

z3C

− x2I

z3I

± µ

(x2

O

z3O

− x2R

z3R

),


Axy = xCyC

z3C

− xIyI

z3I

± µ

(xOyO

z3O

− xRyR

z3R

),

F = x2C + y2

C

z3C

− x2I + y2

I

z3I

± µ

(x2

O + y2O

z3O

− x2R + y2

R

z3R

)

(2.9)+ 1

R

[x2

C + y2C

z2C

− x2I + y2

I

z2I

± µ

(x2

O + y2O

z2O

− x2R + y2

R

z2R

)],

Dx = (x2C + y2

C)xC

z3C

− (x2I + y2

I )xI

z3I

(2.10)± µ

((x2

O + y2O)xO

z3O

− (x2R + y2

R)xR

z3R

).

As seen,S, C andF depend onR, whileA andD are independent of it. Note thatnot only is coma canceled when the aplanatic condition(2.1) is satisfied, but thespherical aberration is also zero, provided thatzC = zR (or zC = zO) andµ = 1.For example, for a CDOE recorded with the setup shown infig. 4, zero sphericalaberration can be obtained when the CDOE is reconstructed either with a planewave or with a point source that is situated in the front focal plane.

A similar analytic calculation for the aberration terms of spherical CDOEs wasperformed byPeng and Frankena [1986]. Verboven and Lagasse [1986]provideda programmable formula for the numerical calculation of aberration coefficientsup to arbitrary order, and for CDOEs with arbitrary shape. Third-order aberrationcalculations were also done for spherical CDOEs byNowak and Zajac [1988]andJagoszewski and Talatinian [1991b], but they also considered the effect ofshifting the entrance pupil to a distancet from the CDOE, as shown infig. 7. Theradius of curvature of the CDOE and the position of the entrance pupil can beused as additional free parameters for numerical optimization, e.g., to achieve thesmallest possible spot size. Several DOEs were compared numerically (flat DOEand CDOE, with and without a shifted entrance pupil). It was shown byNowakand Zajac [1988]that the position of the entrance pupil has a stronger effect onthe image quality than does the radius of curvature. The best results were obtainedwith a CDOE and a shifted entrance pupil.Bokor and Davidson [2001b]alsoshowed that in terms of minimum spot size the optimal CDOE radius is not equalto the aplanatic radius given by eq.(2.2). However, for high NA (NA> 0.8) thisdiscrepancy between optimal and aplanatic radii is negligible. At NA= 0.86 andθ = 5, for example, the radius giving the minimum spot size is less than 2%different from the aplanatic radius.


Fig. 7. Imaging CDOE with an entrance pupil at a distancet from the CDOE.

As discussed above, spherical aberration can be eliminated by choosingzC = zR or zC = zO. Baskar and Singh [1992]investigated a CDOE which isrecorded with a spherical object wave and a plane reference wave, and is recon-structed by a spherical wave for whichzC = zR andzC = zO. The radiusR forwhich spherical aberration is canceled was calculated from eq.(2.6). Note that forsuch a spherical aberration corrected CDOE the object field angle must be small,because coma – which grows linearly with object field angle – is not eliminated.In all of these investigations, the CDOE recording waves were considered to beperfect spherical or plane waves.

Fisher [1989]proposed to use CDOEs for head-up displays in airplanes. Herethe motivation to use CDOEs instead of flat DOEs is the need for increased field ofview and large binocular overlap in a limited space. A spherical reflection CDOEwas considered whose grating function, refractive index modulation and thicknesswere free parameters for iterative numerical optimization. The results of thesenumerical calculations showed that for best performance, one of the recordingwavefronts has to be astigmatic, and the other has to have a complicated asphericshape.

A special application of CDOEs was reported for planar optics byBelostotskyand Leonov [1993]. Because of the plane geometry, astigmatism does not exist,and the only low-order aberrations that should be corrected in an imaging sys-tem are spherical aberration, coma and field curvature. Spherical aberration andcoma can be simultaneously eliminated by a circular CDOE which satisfies theAbbe sine condition and for whichzC = zR, as discussed above.Belostotsky


and Leonov [1993]proposed to use an aplanatic planar Fresnel lens to act as theCDOE. Since an aplanatic Fresnel lens and an aplanatic refractive lens have fieldcurvatures of opposite sign, they can be combined to form a hybrid system thateliminates field curvature too, as shown infig. 8. Belostotsky and Leonov [1993]also gave a generalization of the Abbe sine condition(2.1) for the extraordinarywaveguide mode in an anisotropic planar medium, for the special case of a uniax-ial crystal with its crystal axis lying in the plane of propagation and perpendicularto the optical axis of the system:

(2.11)sinα

sinβ

√1 + (n2

o/n2e − 1) sin2 β√

1 + (n2o/n2

e − 1) sin2 α

= const= −M,

whereno andne are the ordinary and extraordinary indices of refraction, respec-tively. A curve satisfying condition(2.11) is a section of an ellipse, instead of acircle as it was in the isotropic case. The semi-axes of the ellipse areab/(a + b)

and(ne/no)ab/(a + b), along the optical axis and along the crystal axis, respec-tively (seefig. 2 and cf. eq.(2.2)).

Fig. 8. The planar hybrid system ofBelostotsky and Leonov [1993]that eliminates spherical aberra-tion, coma and field curvature.


2.2. Spherical CDOE for Fourier transform

An ideal Fourier-transform lens converts an incoming plane wave of arbitraryslant angle into a diffraction-limited spot at the back focal plane, also called thefrequency plane. Each incoming plane wave represents a component in the angu-lar spectrum of an input transparency which is placed in the front focal plane, andhence the back focal plane shows the 2D spectrum of the input transparency. Thelocation of each diffraction-limited spot in the frequency plane should therefore bedirectly proportional to the spatial frequency component of the input transparencywhich is responsible for the given tilted plane wave. The spatial frequency of agrating is proportional to the sine of the tilt angleα of the diffracted plane wave,according to the grating relation

(2.12)sinα = λ1

Λ,

where 1/Λ is the spatial frequency of the grating andλ is the illuminating wave-length. It follows that the location of spots at the frequency plane should be pro-portional to sinα. As was pointed out byJagoszewski and Talatinian [1991a]andshown infig. 9, this condition is exactly fulfilled for an aplanatic CDOE for prin-cipal rays. Therefore, aplanatic CDOEs, which are also free of aberrations thatgrow linearly with the angle, appear to be naturally suited for Fourier-transformapplications. However, for simple aplanatic CDOEs that are recorded with per-fect spherical waves, and whose phase function is therefore called “spherical”,the residual aberrations such as astigmatism and field curvature still impair theperformance. In order to achieve minimum spot sizes over a given input angularrange, the phase function of the CDOE needs to be optimized.

Fig. 9. Aplanatic CDOE used for Fourier transform (Jagoszewski and Talatinian [1991a]).

3, § 3] Spherical/cylindrical CDOEs for concentration of diffuse light on flat targets 121

For the optimization of Fourier-transform CDOEsTalatinian [1991, 1992]pro-posed and demonstrated the so-called analytic raytracing technique developedfirst for flat Fourier-transform DOEs byHasman and Friesem [1989]. This opti-mization technique yields a DOE with low overall aberrations, provided that thereis an entrance pupil in front of the DOE. In this case each incoming plane waveilluminates only part of the DOE (a “local hologram”), as can be seen infig. 9.Talatinian [1991, 1992]compared numerically three types of CDOEs: (1) hav-ing spherical phase function, (2) having quadratic phase function, and (3) havingoptimized phase function obtained from analytic ray tracing. In each case sev-eral CDOEs with different radii of curvature were considered. In terms of spotsize, CDOEs with quadratic phase function perform almost as well as the opti-mized CDOEs. Additionally, the radius of curvature satisfying the aplanatic con-dition (2.2)does not yield the smallest spot sizes for a given input angular range,hence not only the grating function has to be optimized, but the radius of theCDOE too. Naturally, a CDOE having a spherical phase function can very easilybe fabricated, since only spherical wave fronts are required for the holographicrecording. On the other hand, CDOEs having optimized phase functions requirecomplicated aspherical wavefronts for the recording. Such wavefronts can be pro-duced, e.g., with computer-generated holograms.

§ 3. Spherical/cylindrical CDOEs for concentration of diffuse lighton flat targets

Concentration of diffuse light has many applications, including solar energy con-centration, light collection for optical instruments and efficient light coupling intofiber optics (Winston and Welford [1989], Davidson and Bokor [2003]). In thissection we show that aplanatic spherical/cylindrical CDOEs that eliminate first-order coma are inherently capable of concentrating quasi-monochromatic diffuselight onto a flat target at the theoretical limit imposed by the second law of ther-modynamics (or, equivalently, the conservation of brightness). This thermody-namic limit on the amount of concentration is 1/ sinαx (in beam width) in 1D and1/(sinαx sinαy) (in beam area) in 2D, whereαx andαy are the half-divergenceangles of the diffuse input beam in the transverse directions (Davidson and Bokor[2003]). Note that in the common terminology of the literature (Bassett, Welfordand Winston [1989]) “1D concentration”, i.e. concentration along one spatial di-rection, is achieved by “2D concentrator shapes”z(x), whereas “2D concentra-tion” is achieved by “3D concentrator shapes”z(x, y).


For 1D coma-free imaging the concentration ratio is (Bokor, Shechter, Friesemand Davidson [2001])

(3.1)CRideal = Xin

Xout= NA

sinαx

,

whereXin and Xout are the input and output lateral beam sizes, respectively.Note that as NA approaches 1, eq.(3.1) approaches the 1D thermodynamiclimit CRmax = 1/ sinαx . The normalized concentration ratio is defined asCR/CRmax = (Xin/Xout) sinαx , so that at the thermodynamic limit it isequal to 1.

In general, high-NA imaging concentrators, in particular those that use sim-ple parabolic mirrors, suffer from large aberrations which reduce the concentra-tion ratio far below the theoretical limit (for example, by a factor of 2 for 1Dand a factor of 4 for 2D concentration by optimal parabolic mirrors). AplanaticCDOEs cancel those aberrations that depend linearly on the incoming diver-gence angle, and hence are good candidates to achieve ideal concentration forquasi-monochromatic sources and small incoming diffusive angles. We note thatnonimaging concentrators, that were also shown to achieve diffuse-light concen-tration close to the thermodynamic limit (Winston and Welford [1989]), are lim-ited in use for small incoming diffuse angles, since they are extremely elongatedin this case.

Bokor, Shechter, Friesem and Davidson [2001]investigated an aplanatic CDOEtheoretically and experimentally for diffuse-light concentration. The normalizedconcentration ratio was calculated analytically, using a paraxial expansion of thediffraction relation for small diffusive angles, and confirmed by numerical raytracing. The results are shown for 1D concentration infig. 10, together with resultsfrom two other concentrators: the flat DOE and the parabolic mirror (PM). Thenormalized concentration ratios are NA, NA· (1 − NA2)1/2 and NA· (1 − NA2),for the CDOE, the PM and the flat DOE, respectively. As expected, for smalldiffusive angles the concentration capabilities of the CDOE are in accordancewith eq. (3.1), and hence approach the thermodynamic limit at high NA. Themaximum achievable concentration ratio for the PM and the flat DOE is 50% (atNA = 0.71) and 38% (at NA= 0.58) of the thermodynamic limit, respectively.For 2D concentration, the best normalized concentration ratio for the PM and theflat DOE is even lower, namely 25% and 14%, respectively, of the thermodynamiclimit, whereas for the CDOE concentration at the thermodynamic limit is stillachieved.

The failure of the parabolic concentrator at high NA can be understood by not-ing that at each point an incident light cone with half-angleαx is reflected as an

3, § 3] Spherical/cylindrical CDOEs for concentration of diffuse light on flat targets 123

Fig. 10. Calculated normalized concentration ratios as a function of numerical aperture (NA) for1D concentration on a flat target with a CDOE, a parabolic mirror (PM) and a flat DOE (Bokor,

Shechter, Friesem and Davidson [2001]).

identical cone towards the focus. However, as the cone intersects the optical axisat an angleθ (where NA= sinθmax), the focal spot includes an inclination fac-tor 1/ cosθ , which diverges for large NA, and the spot size reaches an optimumat NA = 0.71. For the CDOE the diffracted light cones for highθ are actuallynarrower than the incident cones by exactly cosθ . This result is directly obtainedfrom the diffraction relations of the grating for smallαx and is an excellent ap-proximation even for largeαx . Hence the effect of the inclination factor 1/ cosθ isexactly canceled, and an identical, diffraction-limited spot size is obtained even asthe NA approaches 1 (and hence a higher concentration ratio). For the flat DOE,the reflected light cone from each point is 1/ cosθ times wider than the incidentcone, resulting in an even worse concentration.

Bokor, Shechter, Friesem and Davidson [2001]demonstrated diffuse light con-centration experimentally with a CDOE using a cylindrical aplanatic CDOE,recorded as a reflection hologram on Dupont photopolymer film. The CDOE wasrecorded with a spherical wave and a plane wave and had NA= 0.86. An ex-perimental normalized concentration ratio of 0.79 was obtained (which is 92% ofthe theoretical limit at this NA), for an input beam with a diffusive half-angleof αx = 1.

The aplanatic CDOE has thus the advantage of ideal concentration of diffuselight at the thermodynamic limit. Additionally, it is very simple to fabricate sincea cylindrical/spherical substrate, and only readily available cylindrical/sphericaland plane waves are needed for the recording. However, in order to reach highdiffraction efficiency, Bragg CDOEs, or blazed surface-relief CDOEs should beused for concentration. In this case the angular selectivity of the CDOE puts an


upper limit on the usable input diffusive angleαx . The angleαx is also limited byhigher-order aberrations, which the aplanatic CDOE does not eliminate.

Furthermore, by simply reversing the direction of the light rays, the spheri-cal/cylindrical CDOEs can also uniformly collimate the light emitted from a flatLambertian source. This result is obtained as a special case from the uniform col-limator design principle, presented in Section4 for general Lambertian sources.

In a somewhat related approach,Kritchman, Friesem and Yekutieli [1979a,1979b]suggested a curved refractive Fresnel lens for 1D concentration of sun-light. The system is shown schematically infig. 11. The basic design principleis that two collimated light beams from the edges of the acceptance field shouldideally be aimed by the curved Fresnel lens toward two stigmatic focus points, re-spectively:a/2 = f tanαx , wherea is the size of the output aperture andf is theheight of the center of the lens above the output plane. If this requirement is satis-fied then every light ray included in the acceptance field will reach the output aper-ture. This principle is analogous to the so-called edge-ray principle (Winston andWelford [1989]) widely used for the design of nonimaging concentrators. Fromthe design principle, after symmetry considerations and application of Snell’s law,the lens shapez(x) and the primary angle of the prismatic groovesΦ(x) can befound numerically for givenαx andn, wheren is the index of refraction. Ifn issmall then the obtainable concentration ratio is much worse than the theoreticallimit (e.g., by a factor of 2.6 forn = 1.1 and by a factor of 1.33 forn = 1.5,whenαx → 0). In the limiting case ofαx → 0 andn = ∞, the curved Fresnel

Fig. 11. The curved Fresnel lens ofKritchman, Friesem and Yekutieli [1979a]for 1D concentration ofsunlight.

3, § 4] Uniform collimation and ideal concentration for arbitrary source and target shapes 125

lens has no grooves and can be treated as a thin aplanatic CDOE whose shape iscylindrical, in accordance with relation(2.2).

§ 4. Uniform collimation and ideal concentration for arbitrary source andtarget shapes

4.1. Uniform 1D collimation and concentration of diffuse light at thethermodynamic limit

Uniform plane wave illumination is of great importance in many areas of optics.A point source can be simply collimated with a parabolic mirror (PM) that re-flects all rays originating from the focal point into the same direction without anyaberrations. However, even for isotropic point sources, the PM yields a collimatedbeam of nonuniform intensity. The crux of the problem here is that the shape ofthe mirror required for collimation is uniquely determined by Snell’s reflectionlaw to be a parabola, whereas uniform intensity of the reflected light requires adifferent shape. FollowingBokor and Davidson [2002], in this section we firstpresent the design principle of a CDOE reflector that can independently fulfillthe two requirements for quasi-monochromatic light: its shape ensures uniformillumination for arbitrarily shaped diffuse sources, and its phase function ensuresoptimal collimation. Extension of the method for any specific nonuniform illumi-nation is straightforward.

The geometric arrangements for recording and readout of a curved holographiccollimator operating in 1D are shown infig. 12. Figure 12(a) shows a record-ing arrangement of the interference of a plane wave and a counter-propagatingcylindrical wave on a holographic film with shapez(x). Figure 12(b) showsthe readout arrangement with the same diverging cylindrical wave. As seen,the first diffraction order yields a perfectly collimated plane wave, regardlessof the CDOEs shape. Note that this readout geometry also automatically fulfillsthe Bragg condition for the entire CDOE, hence permitting high diffraction effi-ciencies. The shapez(x) can now be designed to yield uniform intensity of thecollimated beam for a given illumination source.

Consider a small source located atx = 0, z = 0, and emitting radiation in thexz plane with angular (far-field) intensity distributionS(β), whereβ is the anglebetween the direction of the radiation and thez axis. For an isotropic point sourceS(β) = const, for a planar diffuse Lambertian sourceS(β) ∝ cosβ, and for gen-eral elongated Lambertian sourcesS(β) ∝ ∆(β), where∆(β) is the projecteddiameter of the source as seen from directionβ. For a cylindrical Lambertian


(a) (b)

Fig. 12. (a) Recording and (b) readout geometry of a curved holographic collimator of arbitraryshapez(x).

sourceS(β) = const, the same as for an isotropic point source. IfI (x) is the localintensity of the output beam at vertical coordinatex, then from energy conserva-tion we get

(4.1)S(β) dβ = I (x) dx.

Thus to ensure uniform intensity alongx, i.e. I (x) = const, light rays emittedfrom the source toward directionβ must intersect the CDOE atx(β), obeying therelation

(4.2)dx

dβ= F · S(β),

where the constantF is the paraxial focal length of the collimator, chosen to bemuch larger than the source size. Uniform diffraction efficiency was assumed ineq.(4.2). x(β) is obtained from eq.(4.2)by integration, andz(β) is simply givenas (seefig. 12)

(4.3)z(β) = x(β)

tanβ.

Hence the shapez(x) of the uniform collimator CDOE is obtained.As an example, consider the simplest case, of an isotropic “1D” light

source (e.g., an elongated line source or a diffuse cylindrical source), whereS(β) = const. The shape of the uniform collimator CDOE for this case, derivedfrom eqs.(4.2) and (4.3), is

(4.4)z(x) = x

tan(x/F ).


Equation(4.4) is defined for the entire angular distribution−180 β < 180,where−πF < x < πF and−∞ < z F , and therefore uniformly collimatesthe entire source energy. Part ofz(x) of eq.(4.4)is shown infig. 13(b). As shown,the rays originating from the point source at equal angular density also have uni-form spatial density alongx, after collimation. On the other hand, such rays arecollimated by a PM (whose shape isz(x) = F − x2/(4F)) with nonuniformspatial density alongx, as seen infig. 13(a). For the PM the intensity of the col-limated beam satisfiesI (x) ∝ 4/[(x/F )2 + 4)], having a maximum atβ = 0(x = 0) and dropping monotonically to zero forβ = ±180 (x = ±∞), whereasfor the CDOEI (x) = const for allβ.

Next, we consider the finite angular spread of the collimated beam resultingfrom a diffuse source with finite size. Here the goal is to design a collimator witha minimum angular spread. Alternatively, when the direction of the rays is simplyreversed, this goal is equivalent to designing a concentrator that will concentratea uniform incoming diffuse beam on a small target of the given shape. Althoughin many applications light is concentrated on flat targets (e.g., detectors, solarcells), for other applications, such as cylindrical water pipes and cylindrical laserrods, the target shape is curved. For curved targets the thermodynamic limit onconcentration (discussed in Section3) must be applied to the circumference lengthof the target. For example, a diffuse beam may be concentrated on a cylindricaltarget with diameterπ times smaller than that of a flat target.

It can be analytically proved that a CDOE that fulfills eqs.(4.2) and (4.3)tocollimate light uniformly for a given light source shape is inherently also an ideal

(a) (b)

Fig. 13. (a) The PM that gives nonuniform collimation, (b) the CDOE shape designed byBokor andDavidson [2002]that yields uniform collimation.


diffuse light concentrator [collimator] for the same shape as a target [source], op-erating at the thermodynamic limit. For example, the CDOE offig. 13(b) is anideal collimator for small cylindrical sources and an ideal diffuse light concen-trator onto cylindrical targets. In the concentrator application the ray directionsof fig. 13(b) are simply reversed, and the target is placed in the focus so that itscenter coincides with the focal point.Figure 14(a) illustrates why a PM fails asa diffuse-light concentrator for 1D concentration on a uniform cylindrical target(see also our discussion of the PM for flat targets in Section3). At each pointon the PM the incident light cone with half angleα is reflected as an identicalcone toward the focus. As|β| approaches 180, the distance from the reflectionpoint to the focal point approaches infinity, and hence the spread of the reflectedlight cone at the focus also approaches infinity, resulting in a dramatic decrease inconcentration performance. On the other hand, for the uniform collimator CDOE,as|β| approaches 180, the diffracted light cones are actually much narrower thanthe incident light cones (α′ α), as illustrated infig. 14(b), and compensate forthe increased distance. Expansion of the diffraction relations of the grating forsmallα and simple algebra yield for the CDOE that the extreme incident rays atangles±α from thez axis are directed exactly tangent to the target, and all inter-mediate rays hit the target for allβ. Just like for flat targets, discussed in Section3,these considerations are similar to the edge-ray principle (Winston and Welford[1989]) used for the design of nonimaging concentrators.Figure 15presents thecalculated normalized concentration ratio onto a cylindrical target, as a function

(a) (b)

Fig. 14. Diffuse light concentration onto a small cylindrical target by (a) the PM, and (b) the uniformcollimator CDOE; the CDOE reaches the thermodynamic limit of light concentration (Bokor and

Davidson [2002]).


Fig. 15. Calculated normalized concentration ratios onto a small cylindrical target as a functionof |βmax| for the uniform collimator CDOE and the PM (Bokor and Davidson [2002]).

of |βmax| for both the uniform-collimator CDOE and the PM. For the CDOE thenormalized concentration ratio is|βmax|/π, which yields concentration at the ther-modynamic limit for|βmax| = 180. For the PM the normalized concentrationratio is sin|βmax|/π, resulting in an optimal concentration at|βmax| = 90, themaximum concentration ratio beingπ times below the thermodynamic limit.

Bokor and Davidson [2002]verified the design procedure experimentally byholographically recording a reflection CDOE of the shape given by eq.(4.4)(fig. 13(b)). The focal length of the CDOE wasF = 20 mm. Several exposureswere used to record a hologram for the range−90 β 90 (correspondingto NA = 1 and a lateral size of 2xmax = 58 mm). Since a spherical (instead of acylindrical) wave was used for the recording, a 5 mm slit in they direction ensuredthat the setup worked essentially as a 1D concentrator. To test the performance ofthe uniform collimator CDOE as an ideal concentrator, the entire aperture of theCDOE was illuminated with a monochromatic plane wave that was inclined atvariable angleα with respect to thez axis. A cylindrical black target with radiusr = 1.5 mm at the focal point of the CDOE blocked all the concentrated light untilα reached the value of±r/F = ±4.3. The unblocked light powerP was mea-sured by integrating the imaged intensity on a white diffuse screen placed behindthe target. The measuredP(α) graph, shown infig. 16with diamonds, features adiscontinuous sharp transition atα = r/F , as expected for an ideal concentrator.The theoreticalP(α) curve for the CDOE concentrator (fig. 16, solid line) hasa somewhat sharper, but also finite slope, which is due to the finite size of thetarget, corresponding to a finite input diffuse angle. Finally, the theoreticalP(α)

concentrator curve for the PM concentrator (fig. 16, dashed line) has a much gen-


Fig. 16. Experimental data byBokor and Davidson [2002], characterizing the performance of theuniform collimator CDOE as a diffuse light concentrator onto a small cylindrical target: the measuredlight intensity not blocked by the cylindrical target as a function of the illumination angle (diamonds),compared with calculations for the uniform collimator CDOE (solid curve) and the PM (dashed curve).

tler slope, indicating concentration well below the thermodynamic limit (Bokorand Davidson [2002]).

The design procedure for uniform collimator CDOE described by eqs.(4.2)and (4.3)can be applied for general Lambertian source shapes. For example, theuniform collimator [concentrator] CDOE for a one-sided flat source [target], com-mon in many practical cases, is found to have the half-cylinder shape that satisfiesthe Abbe sine condition (see Sections2 and 3). Equations(4.2) and (4.3)alsoyield simple analytic solutions for the CDOE shapes for Lambertian sources hav-ing oval, square, triangular, and many other cross-sections (Bokor and Davidson[2002]). We stress that the actual shape of the fabricated CDOE does not haveto follow the analytical shape obtained from eqs.(4.2) and (4.3)within an opti-cal precision. The reason for this is that the quality of concentration depends onlyweakly on the exact shape of the CDOE (as long as it is identical during recordingand reconstruction), making this technique insensitive to small fabrication errors.

4.2. Extension to finite distances

The design principle described above was implemented and generalized byDavidson and Bokor [2004a]for finite distances, both for reflection and transmis-sion geometries in 1D. The problem to be solved can be stated as follows: the lightof an elongated Lambertian source of given size and cross-sectional geometry is tobe uniformly concentrated onto an elongated target of given cross-sectional shape


and the smallest cross-sectional size that is permitted by brightness conservation.This general problem has great practical significance, e.g., for side pumping ofsolid-state laser rods, where high geometrical efficiency and uniform pump den-sity profile are important. First consider a reflection CDOE.Figure 17(a) showsschematically the design geometry for a reflection CDOE for 1D diffuse-lightconcentration. A small source is centered at point A and a small target is locatedat a distanced, centered at point B. The source emits radiation with angular (far-field) intensity distributionS1(α), whereα is the angle between the direction ofthe radiation and the line connecting A and B. As discussed above, for general 2DLambertian sourcesS1(α) ∝ ∆1(α), where∆1(α) is the projected diameter ofthe source as seen from directionα. Similarly, S2(β) is the local intensity of theoutput beam in directionβ, measured at the target, as shown infig. 17(a). Fromenergy conservation we get

(4.5)S1(α) dα = S2(β) dβ.

To ensure uniform illumination on a Lambertian target,S2(β) must be propor-tional to∆2(β), where∆2(β) is the projected diameter of the target as seen fromdirectionβ. Hence for Lambertian source and target, eq.(4.5) takes the form

(4.6)∆1(α) dα = ∆2(β) dβ.

The CDOE shaper(α) for specific∆1(α) and∆2(β) geometries can be obtainedby solving differential eq.(4.6) for β, and then substituting it into the relation

(4.7)r(α) = d sinβ(α)

sin[α − β(α)] ,

as seen infig. 17(a). The meaning ofd as the scaling parameter is similar tothat of F in eq. (4.2). This design principle together with eqs.(4.5) and (4.6)is also valid for the geometry of a transmission CDOE concentrator, presented in

(a) (b)

Fig. 17. Design geometry of a CDOE working (a) in reflection, (b) in transmission, for finite distanceconcentration of diffuse light.


fig. 17(b). However, since hereα is measured from the opposite direction, eq.(4.7)is modified to

(4.8)r(α) = d sinβ(α)

sin[α + β(α)] .Similarly as for the uniform collimator CDOE described in Section4.1, again byexpanding the diffraction relations of the holographic grating for small diffuse an-gles, i.e. when the source and target sizes are much smaller thand, it can be shownanalytically that for the CDOE designs of eqs.(4.6)–(4.8)the rays emitted tangen-tially by the source are directed exactly tangent to the target, and all intermediaterays emitted by the source hit the target for allα. Moreover, the phase-space areais conserved, i.e.(phase-space area)target = (phase-space area)source. Therefore,if there is no loss in power, such a device concentrates diffuse light on the theo-retical limit of brightness conservation, i.e. on the smallest possible target size ofthe given geometry. The experimental realization of such a holographic elementis extremely simple, since the curved hologram can be recorded using two sim-ple cylindrical laser beams, originating from locations A and B (seefigs. 17(a)and 17(b)).

As the first example of a CDOE designed from eqs.(4.6) and (4.7), considera cylindrical source emitting rays with uniform angular distribution in the entireangular range−180 α < 180, and a target that receives rays with uniformangular distribution in the limited angular range(−180/M) β < (180/M),whereM is the linear magnification of the CDOE. (The role of source and targetcan, of course, be interchanged, and thenM becomes demagnification.) Here both∆1(α) and∆2(β) are constants, and eq.(4.6)yields dα = M dβ. The solution isβ = α/M, and from eq.(4.7)we getr(α) = d sin(α/M)/sin[(1 − 1/M)α]. TheCDOE shape corresponding toM = ∞ is the uniform collimator/concentratordiscussed above and shown infig. 13(b). The CDOE shape corresponding toM = 2 is a simple cylinder (r(α) = d = const), where A is at the center andB is on the circumference. AsM → 1, the CDOE cross-sectional shape ap-proaches the so-called cochleoid curve, for whichr(α) ∝ (sinα)/α. TheM < ∞cases of the CDOE shapes can be applied to direct the rays coming from a cylin-drical source onto a cylindrical target, where the diameter of the target isM timeslarger than the diameter of the source.

Figure 18(b) shows how, for example, two CDOEs withM = 2 can be used ina double-lamp configuration in the pumping of solid-state lasers. The two cylin-drical pump lamps are located in the centers of the two cylindrical CDOEs, andthe axis of the cylindrical laser rod coincides with the straight line where the twoCDOEs touch. To avoid shadowing effects, neither of the two CDOEs is com-pletely closed, as shown infig. 18(b). This leads to a small geometrical loss in


(a)

(b)

Fig. 18. Double-lamp pumping configuration using (a) two elliptical mirrors, (b) two reflective CDOEs(Davidson and Bokor [2004a]).

power, since some of the rays emitted by the sources do not reach the CDOE.However, if the source and the target sizes are made much smaller than the DOEcross-sectional size, this geometrical loss becomes negligible. Moreover, since themagnification of both CDOEs isM = 2, the combined phase-space area of thetwo sources is expected to be equal to the phase-space area of the target. To sum-marize, the double-CDOE configuration offig. 18(b) conserves brightness andprovides a uniform pump density profile. For comparison,fig. 18(a) shows a dou-ble elliptical pumping configuration, which is widely used for solid state lasers.In order to provide the same magnificationM = 2 (and hence phase-space areaconservation) as in the CDOE case, an eccentricity of 1/3 was chosen for bothellipses. As seen infig. 18(a), combining the two ellipses causes relatively largeportions of both mirrors to be missing, and hence a large percentage of the rays


emitted by the sources does not arrive at the proper elliptical mirror surface andis essentially lost. The geometrical loss in this case is∼30%, which correspondsto an equal amount of loss in brightness.

An additional drawback of the double elliptical cavity, and elliptical mirrors ingeneral, is its highly nonuniform pump density profile, as illustrated byfig. 19(Davidson and Bokor [2004a]). Figures 19(a) and 19(b)show phase-space spotdiagrams of rays hitting the cylindrical target, for the double elliptical reflectorand for the double CDOE, respectively, obtained from numerical ray tracing forthe paraxial case (i.e. when the source and target sizes are much smaller thanthe size of the cavity). The horizontal axisl represents the location of the raysalong the circumference of the target, and the vertical axis represents the sine ofthe angleεl between the rays and the surface normal. The density of dots in anyregion on the phase-space diagrams gives directly the local brightness of lightin that region. The rays are emitted isotropically from the sources, half from theleft source (solid circles infig. 19) and half from the right source (open circles).Figure 19(a) is found to contain only∼70% of the illuminating dots, in agree-ment with the geometrical loss calculated above. As also seen, the phase space ofthe double elliptical reflector is highly nonuniform and contains empty regions.On the other hand, the double-CDOE configuration yields uniform phase-spacedensity, and light concentration at the thermodynamic limit of brightness conser-vation. Since uniform pump density profile, besides high pumping efficiency, isoften desirable in a laser rod, especially when fundamental transverse modes are

(a) (b)

Fig. 19. Phase-space spot diagrams obtained from numerical raytracing byDavidson and Bokor[2004a], for (a) the double elliptical mirror cavity offig. 18(a), and (b) the double CDOE cavity

of fig. 18(b).


to be excited, CDOEs may give a practical alternative to elliptical reflector tubesas pump cavities.

For the special case of a flat Lambertian source and a flat Lambertian target,eq. (4.6) takes the form cosα dα = M cosβ dβ, which gives back exactly theAbbe sine condition sinα/ sinβ = M, and the CDOE shape is a cylinder, inagreement with the aplanatic CDOE discussed in Sections2 and 3. The phase-space area conservation can be proved for this case by noting that although thetarget size isM times larger than the source size, the sine of the angular rangeof rays hitting the target is exactlyM times smaller than the sine of the angularrange of rays emitted by the source. ForM = 1 the radius of the cylinder becomesinfinity (corresponding to a flat DOE, placed halfway between A and B), and forM → ∞ the circle is centered at point A. Equation(4.6) can hence be thoughtof as a generalization of the Abbe condition for arbitrary Lambertian target andsource shapes.

For sources and targets having an axial symmetry the technique can be extendedto 3D CDOE shapes and 2D concentration/collimation in a straightforward way(Davidson and Bokor [2004a]). Because of the solid angles involved, the energyconservation equation(4.5) is modified to

(4.9)S1(α) sinα dα = S2(β) sinβ dβ,

whereS1(α) andS2(β) are defined in the same way as for the 1D concentrationcase. Interestingly, for a flat Lambertian source and a flat Lambertian target, the3D-shape design(4.9), which takes the form cosα sinα dα = M cosβ sinβ dβ,leads to Abbe’s sine condition again (here its form is sinα/ sinβ = √

M ), justas eq.(4.6)did for the 2D CDOE shapes. For this geometry, the 3D CDOE shapeis spherical, and can be thought of as a simple rotated version of the 2D shape.Another simple analytic shape is given by a spherical source and a flat target. Inthis case, eq.(4.9)can be written as sinα dα = C cosβ sinβ dβ, and the solutionis β = α/2. This is also a sphere, with the spherical source at its center and theflat target at its surface.

For infinitesimal source sizes the uniform collimator design eq.(4.9)gives cor-rect results, however, it can be shown analytically that the “edge ray principle”discussed above for elongated Lambertian sources and targets does not apply au-tomatically for 3D CDOE shapes. The reason for this is the effect of skew rays, i.e.rays that are not in the same plane as the symmetry axis of the system (Winstonand Welford [1989]). The diffuse ray pencils hitting the CDOE in the skew direc-tion are unaffected by the grating and are directed towards the target as identicaldiffuse ray pencils, similarly to simple reflections. For the case of a flat sourceand a flat target, where Abbe’s sine condition is satisfied, uniform concentration


at the thermodynamic limit is achieved for the 3D CDOE shapes too, whereas inother cases, such as spherical sources and spherical targets, skew rays impair theperformance of the CDOE and result in concentration below the thermodynamiclimit (Davidson and Bokor [2004a]).

§ 5. CDOEs for controlling the geometrical apodization factor in vectorialdiffraction problems

In this section we demonstrate how a CDOE with an appropriate shape can con-trol the principal surface, and hence the geometrical apodization factor, of a highNA focusing device, and thereby improve its performance (Davidson and Bokor[2004b], Bokor and Davidson [2004]). In our discussion both the focusing systemand the illumination intensity are assumed to have axial symmetry.

By use of vectorial diffraction theory for a focusing element, the electric fieldvector near the focus at point P can be obtained from the generalized Debye inte-gral (Richards and Wolf [1959]), expressed as

(5.1)E(P) = − i

λ

∫ ∫Ω

A2(θ)a(θ, φ) exp[ik(sxx + syy + szz)

]dsx dsy

sz.

Here E is the electric field vector,k = 2π/λ with λ the wavelength of illumina-tion, (sx, sy, sz) is the propagation unit vector toward the focus,θ is the focusingangle (the angle between the optical axis and the propagation unit vector) so thatNA = sin(θmax), φ is the azimuth angle, anda(θ, φ) is a unit vector representingthe polarization direction of the given electric field component. The amplitudefactor A2(θ) can be expressed asA2(θ) = A1(θ) · Aap(θ), whereA1(θ) is theamplitude distribution of the collimated input beam at the entrance pupil, andAap(θ) is the so-called apodization factor, obtained from energy conservation andgeometric considerations. The integration is done over a solid angleΩ that coversthe entrance pupil of the system. The Debye approximation is valid for high-NAsystems where the focal length is much larger thanλ.

The physical meaning of the geometrical apodization factorAap(θ) can be un-derstood with the help offig. 20. The focusing system can be characterized withthe geometrical shape of its principal surface. The principal surface consists of thepoints of intersection between the incoming ray directions and the directions ofthe corresponding rays that are deflected toward the focus. For example, for a thinlens (or a flat DOE) the principal surface is flat, for a PM it is a paraboloid, and foran aplanatic system (which, as discussed before, eliminates first order transverseaberrations) it is a sphere centered on the focal point. The shape of the principal

3, § 5] CDOEs for controlling the geometrical apodization factor 137

Fig. 20. Schematic diagram of a high-NA focusing system with principal surfaceρ(θ); the principalsurface can be controlled by CDOE shape.

surface is given by the functiong(θ) for which

(5.2)ρ = f · g(θ),

where ρ is the distance of the input ray from the optical axis, andf is thefocal length. For example, for an aplanatic systemg(θ) = sinθ , for a flatDOE g(θ) = tanθ , for a PMg(θ) = 2 tan(θ/2), and for a Herschel-type sys-tem (which eliminates first-order longitudinal aberration,Born and Wolf [1993])g(θ) = 2 sin(θ/2). As can be understood fromfig. 20, the input amplitude distri-butionA1(θ), which depends on the density of rays in the collimated input beam,is different from the output amplitude distributionA2(θ), which depends on theangular density of the rays directed toward the focus. The geometrical intensitylaw yields the following relationship betweenA2(θ) andA1(θ):

(5.3)A2(θ) = A1(θ)

∣∣∣∣g(θ)g′(θ)

sinθ

∣∣∣∣1/2

,

whereg′(θ) = dg/dθ . As seen,A2(θ) is the product of two terms, the first ofwhich depends only on the incident illumination and is independent of the focus-ing system, while the second depends only on the geometrical properties of thefocusing system. Hence

(5.4)Aap(θ) =∣∣∣∣g(θ)g′(θ)

sinθ

∣∣∣∣1/2

.

For an aplanatic systemAap(θ) = cos1/2 θ , for a flat DOEAap(θ) = 1/ cos3/2 θ ,for a PMAap(θ) = 2/(1 + cosθ), and for a Herschel-type systemAap(θ) = 1.


The apodization factorAap(θ) can be used as a free parameter when a system isdesigned to optimize various properties of the focused field, such as the focus area,the focal depth, and the height of the sidelobes. Lens systems can be designed tohave a pre-described set of principal surfaces, and hence, apodization factors.

A simple alternative way of controllingAap(θ) is made possible by the fact thatfor a thin curved diffractive focusing lens the principal surface is simply the shapeof the lens substrate (Davidson and Bokor [2004b]), as seen infig. 20. Hence ahigh-NA CDOE having an appropriate substrate shape can directly determine theapodization factorAap(θ) through eq.(5.4), whereg(θ) now describes the shapeof the CDOE. As explained before, the fabrication of such a CDOE is extremelysimple, since a plane wave and a spherical wave (originating from the focal point)can be used to record it holographically. However, since the grating period atthe edge of a high-NA CDOE approachesλ (where incoming rays are deflectedby a large angle toward the focus), special care must be taken to include in thecalculation of eq.(5.1) the space-variant diffraction efficiencies and phase delaysimposed by each local grating (Turunen, Kuittinen and Wyrowski [2000]). Thesecan be calculated from vectorial rigorous coupled-wave analysis (Moharam andGaylord [1982]), but are not considered here.

The situation is further complicated by the fact that CDOEs can also drasti-cally change the polarization state of the incoming beam. An important exceptionto this is when the CDOE receives purely TE or purely TM radiation, i.e. whenthe electric field vector is everywhere parallel with or everywhere perpendicu-lar to the grating lines. In these cases, the polarization state of the input beamis unchanged upon diffraction. High-NA focusing CDOEs have axial symmetryand concentric grating grooves. For such CDOEs the purely TM case correspondsto radial polarization, and the purely TE case to azimuthal polarization. Thesetwo polarization states are relevant in many high-NA applications. For example,asDorn, Quabis and Leuchs [2003]demonstrated both theoretically and experi-mentally for high-NA aplanatic lenses, a radially polarized input beam yields asharper spot at the focus than a linearly polarized input beam, especially for thelongitudinal electric field component.

As an example for the application of CDOEs to controlAap(θ), Bokor andDavidson [2004]considered the 4π focusing system shown schematically infig. 21. The basic idea of the 4π geometry (Hell and Stelzer [1992]) is to enhancethe axial (z-)resolution in 3D scanning confocal microscopy by coherently illu-minating the sample with two counter-propagating waves. However, the improvedaxial resolution is usually accompanied by high-intensity sidelobes in the axial di-rection, especially when both focusing objectives are aplanatic lenses and the illu-minating beams have linear polarization.Bokor and Davidson [2004]investigated

3, § 5] CDOEs for controlling the geometrical apodization factor 139

Fig. 21. Schematic diagram of a 4π focusing system with principal surfacesρ(θ), illuminated withtwo counterpropagating radially polarized doughnut beams with a relativeπ phase shift (Bokor and

Davidson [2004]).

the applicability of radially polarized beams and properly designed principal sur-face shapes to achieve a sharp, nearly spherical focal spot with uniformly lowsidelobe intensities in the longitudinal and transverse directions. Infig. 21the in-stantaneous electric field vectors are represented by dotted arrows, and the layer ofthe specimen under investigation is represented by a dashed line. To have a stronglongitudinal electric field component in the focus, the two counter-propagating ra-dially polarized beams need to have opposite phases, as shown infig. 21. Thus thepolarization distribution of the focused field strongly resembles that of a dipole ra-diation pattern, propagating outward from the focal point. The principal surfacesof the two focusing objectives are characterized by theirρ(θ) shape, accordingto relation(5.2). Several focusing systems, each having different apodization fac-tors, were investigated. The focal spots having the best spherical symmetry anduniformly low sidelobes along the longitudinal and transverse directions were ob-tained for a focusing system that had two Herschel-type principal surfaces. Thecalculated logarithmic contour map of the electric field intensity near the focus ofsuch a system is shown infig. 22. HereR andZ are the radial and axial directionsat the focus. Other geometries such as the Lagrange system (which can be thoughtof as the rotated version of the 2D uniform collimator of eq.(4.4), and for whichAap(θ) = (θ/ sinθ)1/2) and the parabolic surfaces also performed significantlybetter than the widely used aplanatic objectives.Figure 23presents the calcu-lated radial and axial intensity profiles at the focus for the Herschel-type system,along with the corresponding intensity profiles for an aplanatic system, showing


Fig. 22. Calculated intensity contour map near the focus in the(R, Z) plane for a Herschel-type4π focusing system and radial illumination polarization, yielding a nearly spherical central spot and

low radial and axial sidelobes (Bokor and Davidson [2004]).

(a) (b)

(c) (d)

Fig. 23. Calculated normalized intensity profiles near the focus, along (a,c) theR direction, and(b,d) theZ direction; (a) and (b) are for a Herschel-type 4π focusing system, and (c) and (d) for

an aplanatic 4π focusing system (Bokor and Davidson [2004]).

3, § 6] Curved gratings for spectroscopy 141

the poorer spherical symmetry and higher axial sidelobes for the aplanatic system.These results demonstrate the importance of controlling the geometrical apodiza-tion factor for specific purposes.

§ 6. Curved gratings for spectroscopy

As already stated in Section1, curved spectroscopic gratings had been reviewedby Welford [1965] and Schmahl and Rudolph [1976], so we mention this ap-plication only briefly here. We focus on recent developments, where either theCDOE shape or the grating-line shapes or both are optimized to eliminate or sup-press certain aberrations.

The basic setup for a curved grating spectrograph is shown infig. 24. TheCDOE is a curved reflection grating and the dotted line represents the well-knownRowland circle which has a diameter equal to the radius of the CDOE.Figure 24shows the spectrograph application, as opposed to the monochromator, where A isthe entrance slit, and the spectrum is observed at the exit aperture B. The Rowlandgeometry has several important advantages. First, if the object A is placed on theRowland circle then sharp spectroscopic lines, free of meridional coma, will beformed at the image B, also on the Rowland circle. Hence, the Rowland geometryis, in this sense, aplanatic. Second, the spectrograph consists of a single opticalelement, without any need for additional lenses or mirrors. The latter property isespecially useful in far-ultraviolet applications, where high reflection and trans-mission losses are a key issue. Although the simple Rowland geometry eliminates

Fig. 24. The basic setup for a Rowland-type curved grating spectrograph.


meridional coma at all wavelengths, several other aberrations, such as astigma-tism, sagittal coma and spherical aberration are still present. These aberrations ingeneral lead to deteriorated spectrum quality. In several applications astigmatismis especially important to correct, because astigmatism-corrected spectrographshave an enhanced intensity, and hence an improved signal-to-noise ratio at theexit aperture. Note that in contrast with Sections2 and 3, where both the sub-strate shape and the phase function of the spherical CDOE were symmetric aboutthe optical axis, here the grating lines are straight or almost straight.

Harada and Kita [1980]demonstrated a mechanical ruling engine which is ca-pable of numerically controlling the spacing and curvature of the grating lines, inorder to reduce aberrations associated with straight, equidistant grooves. Holog-raphy, which is capable of recording high-density gratings, is also a flexible toolto control the shape of the grating lines, and was proposed by several authors forthe optimization of spectroscopic gratings.McKinney and Palmer [1987]consid-ered a CDOE recorded with two point sources, and then used the location of thetwo recording sources, along with the location of the entrance slit and the detec-tor (not on the Rowland circle), as the parameters for numerical optimization, forcases where the Rowland circle must be abandoned, e.g. when the detector is flat.Noda, Harada and Koike [1989]proposed a recording setup in which at least oneof the two recording waves is aspherical, to provide free parameters for the opti-mization. To simplify fabrication, the aspheric wavefront is generated by having aspherical wave reflected from an off-axis spherical mirror of appropriate locationand tilt angle.Grange [1992, 1993]demonstrated how a simple recording setup,involving only spherical waves (but one of which needs usually to be converging)can simultaneously reduce sagittal coma and astigmatism.Duban, Lemaître andMalina [1998]proposed a method where one of the recording wavefronts for theCDOE is aspherical, generated by reflecting a spherical wave from a deformableplane mirror. This technique gives many free parameters that can be used for op-timization, however, the calculated mirror deformations are extremely small andmust be reproduced with optical precision.Duban [1999]suggested a methodin which the aspherical recording wavefront is generated by an auxiliary spheri-cal holographic grating. This setup involves only spherical surfaces and sphericalwaves, and still gives enough controllable parameters for correcting aberrationsup to the 4th order.

So far, only spherical CDOE substrates were considered, and the grating lineswere chosen to deviate from straight grooves. On the other hand, astigmatism,which is one of the dominant aberrations in the original Rowland configuration,can simply be eliminated if the grating shape is ellipsoidal or toroidal, as alreadynoted byWelford [1965]. Although both the ellipsoidal and the toroidal shapes

3, § 7] CDOEs for optical coordinate transformations 143

are capable of eliminating astigmatism, their behavior is different in terms ofhigher-order aberrations, e.g., the ellipsoidal substrate gives a better correctionof spherical aberration, as noted byGrange and Laget [1991].

Cash [1984]considered an ellipsoidal CDOE with straight, mechanically ruledgrating grooves. Besides eliminating astigmatism, sagittal coma can also be can-celed by a slight 3rd-order modification of the ellipsoidal shape. Such modifiedellipsoidal substrates are, however, extremely difficult to fabricate. A variationof the same method was proposed byContent, Trout, Davila and Wilson [1991],where a 4th-order deformation was added to the ellipsoid, leading to much bet-ter performance, but at the expense of more problematic fabrication issues due tosmall deviations from the ellipsoidal shape.Grange and Laget [1991]combined aregular ellipsoidal substrate, to eliminate astigmatism, with the method ofNoda,Harada and Koike [1989], mentioned above, to optimize the grating lines holo-graphically and eliminate sagittal coma.Content and Namioka [1993]consideredthe case when the Rowland geometry cannot be used (e.g., in case of multiple grat-ings), and suggested that an asymmetrically deformed ellipsoidal substrate withstraight grating grooves be used to minimize aberrations, which in this case mustinclude compensating for meridional coma. They also demonstrated the useful-ness of a setup in which one recording spherical wave is convergent and the otheris divergent. This is equivalent to the setup suggested byGrange [1992, 1993]andmentioned above.Namioka, Koike and Content [1994]gave analytical formulasto express ray-traced spot diagrams for CDOEs having a 4th-order deformed ellip-soidal substrate and several types of ruled grating lines. The formulas are derivedfrom 3rd-order approximations and can be used in grating optimization tasks.

Astigmatism can be corrected with a CDOE on a toroidal substrate too. For ex-ample, such a setup was proposed and demonstrated experimentally both for holo-graphically recorded metallic gratings (Huber, Jannitti, Lemaître and Tondello[1981]) and mechanically ruled gratings (Huber, Timothy, Morgan, Lemaître,Tondello, Jannitti and Scarin [1988]). The grating was created by first fabricat-ing a grating on a spherical substrate and then slightly deforming it by applyingelastic forces at four points on its circumference. This flexible design makes itpossible to use the same fabricated grating for different wavelength ranges byvarying the elastic deformation.

§ 7. CDOEs for optical coordinate transformations

Davidson, Friesem, Hasman and Shariv [1991]proposed and demonstrated anapproach for obtaining optical coordinate transformations which is based on


CDOEs. Unlike flat DOEs which must have complicated aspheric phase func-tions to achieve optical transformations, here the curvature provides an additionaldegree of freedom that makes it possible for CDOEs to have simple phase func-tions that can be simply recorded directly by optical means. The design method forsuch CDOEs is the following: consider a coordinate transformation for a 1D func-tion f (x),

(7.1)f (x) → f[u(x)

],

whereu(x) is the new coordinate. Such a transformation is illustrated graphicallyin fig. 25, where the input functionf (x) is drawn along thex axis and the outputfunctionf [u(x)] along thez axis (for simplicityfig. 25shows a binary function).From each pointxi on thex axis a vertical line is drawn, and from its intersectionwith the curvez = u(x) a horizontal line is drawn that intersects thez axis at thepoint zi = u(xi).

The transformation offig. 25may be realized directly by optical means: a trans-parency with transmittance functionf (x) at the input plane is illuminated by acoherent plane wave. If it is assumed that this plane wave is not diffracted by theinput pattern (geometrical shadow approximation), the wavefront after the trans-parency can still be described by parallel rays, where the coordinate of each rayis x and the intensity of the ray isf (x). The geometrical shadow approximation isvalid when the distance between the input and the output planes is small comparedwith λ−1f −2

max, wherefmax is the maximum spatial frequency of the input. Theseparallel rays are diffracted by a CDOE along a curvez = u(x) by exactly 90and hence arrive at the output plane at the desired locationz = u(x). This isa much larger diffraction angle than most computer-generated flat DOEs are ca-pable of. Thus for a given input size the distance between the input and outputplanes will be much smaller for the CDOE than for the flat DOE. Besides makingthe total optical system more compact, this shorter distance considerably improvesthe optical performance of the coordinate transformation. Specifically, the space-bandwidth product capabilities of the transformation are inversely proportionalto the square root of this distance (Davidson, Friesem and Hasman [1992]). TheCDOE may be generated optically by simply recording the interference pattern oftwo perpendicular plane waves on a curved holographic film, as shown infig. 26.

The CDOE approach can be generalized to 2D. Quasi-1D coordinate trans-formations of the form[x, y] → [u(x, y), y] can be achieved with the sameconfiguration as for the 1D case, except that now the CDOE is located on asurfacez = u(x, y). However, a general 2D coordinate transformation of theform [x, y] → [u(x, y), v(x, y)] requires two cascaded CDOEs. The first gen-erates the transformation[x, y] → [u(x, y), y] as discussed above, while the

3, § 7] CDOEs for optical coordinate transformations 145

Fig. 25. Graphical illustration of a 1D coordinate transformation.

Fig. 26. Recording setup for a CDOE performing 1D optical transformation (Davidson, Friesem, Has-man and Shariv [1991]).

second generates the transformation[u(x, y), y] → [u(x, y),w(u, y)], wherew(u, y) = v(x, y).

This CDOE approach can also be applied for tasks that represent coordinatetransformations only in a broader sense. One of the examples investigated exper-imentally byDavidson, Friesem, Hasman and Shariv [1991]was to transform aGaussian beam into one that is uniform in one of the transverse dimensions. Sucha conversion may be obtained by the following coordinate transformation:

(7.2)[x, y] →[erf

(√2x

r0

), y

],


(a) (b)

Fig. 27. Experimental cross-sections for the Gaussian-to-uniform beam converter ofDavidson,Friesem, Hasman and Shariv [1991]: (a) input, (b) output.

where erf is the error function (the integral of the Gaussian function). Experimen-tal intensity cross-sections for the input and output of the Gaussian-to-uniformbeam converter are shown infigs. 27(a) and 27(b), respectively, qualitatively indi-cating the validity of the concept.Davidson, Friesem, Hasman and Shariv [1991]have also used CDOEs recorded in a similar method, but having a different shape,for logarithmic coordinate transformation which is used for scale-invariant opticalcorrelation. We note that a related approach of a beam converter for diffuse whitelight was reported byBokor and Davidson [2001a], where, instead of using dif-fraction, the beam shaping is achieved by a single reflection from an appropriatelyshaped mirror consisting of many displaced parallel planar reflecting facets.

§ 8. Summary

We have discussed the main application areas of diffractive optical elements fab-ricated on a curved substrate. The versatility of the CDOE comes from the factthat the substrate shape and the grating function can be controlled independently.CDOEs can be used to achieve aberration-free imaging, uniform collimation andideal concentration of diffuse light at the thermodynamic limit, for general sourceand target shapes. In high-NA focusing CDOEs can be used to control the effec-tive apodization factor of the focused light rays, and thereby yield uniform andtight focal spots with very low sidelobe intensities.

For many applications – like the ideal concentration of diffuse light discussed inSections3 and 4, the vectorial diffraction problems mentioned in Section5, andthe optical coordinate transformation described in Section7 – it is sufficient tooptimize the shape of the CDOE, and the grating function can be recorded either

3] References 147

holographically with simple spherical or plane waves, or by direct laser lithog-raphy. In other cases the grating function must have a more complicated form;here additional optics or computer-generated holograms may be required. SinceCDOEs are diffractive elements, they usually have large chromatic aberrationswhich limit their use to quasi-monochromatic light, except for the spectrographapplications discussed in Section6. The history of CDOEs began in the mid-19thcentury with Rowland’s curved spectroscopic grating. Thanks to their versatility,and to recent advances in applications, in new design methods, and in holographicrecording materials that can be placed on arbitrary shapes and enable high diffrac-tion efficiency, CDOEs should remain a lively area of research in the future too.

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