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Diffractive optical elements in hybrid lenses:modeling and design by zone decomposition

Herve Sauer, Pierre Chavel, and Gabor Erdei

We propose to model hybrid optical systems ~i.e., lenses with conventional and diffractive optical ele-ments! as multiaperture systems in which the images formed by each zone of the diffractive opticalelement should be summed up coherently. This new zone decomposition concept is explained andcompared with the standard diffraction-order expansion with the help of a hybrid triplet example.© 1999 Optical Society of America

OCIS codes: 050.1970, 220.3620, 110.3000, 110.4100.

1. Introduction

We discuss the computation of the point-spread func-tion ~PSF! and the modulation transfer function ~MTF!of hybrid lenses, i.e., optical imaging systems that com-bine conventional ~refractive or reflective! surfaces anddiffractive optical elements ~DOE’s!. Such hybridsystems offer new attractive solutions to some opticaldesign problems; thanks to progress in manufacturingtechniques, they have been used in IR and visible bandfor a few years.1–3 However, standard design toolssuffer from limitation in their optical-quality analysis.In particular, most commercial software packages foroptical design model and design the system with onlyone diffractive order of the DOE, the so-called nominalorder—in most cases the 61 order. This generatesrrors that are noticeable as soon as a significant frac-ion of the light is diffracted into the other orders.he effect is well known. One rough corrective actionas been customary, although it is not, as far as wenow, implemented as such in commercial software:onnominal orders are usually significantly defocusednd participate in the image formation mostly throughuniform background that can be accounted for as a

When this research was performed, the authors were with Labo-ratoire Charles Fabry, Institut d’Optique Theorique et Appliquee,

entre National de la Recherche Scientifique, Unite Mixte de Re-herche 8501-Centre Universitaire, Batiment 503, B. P. 147-91403,rsay Cedex France. G. Erdei is now with the Department oftomic Physics, Optical Research Laboratory, Technical Universityf Budapest, H-1111 Budapest, Budafoki ut 8., Hungary. H. Sau-r’s e-mail address is herve.sauer@iota.u-psud.fr.Received 26 April 1999; revised manuscript received 12 July

999.0003-6935y99y316482-05$15.00y0© 1999 Optical Society of America

6482 APPLIED OPTICS y Vol. 38, No. 31 y 1 November 1999

central peak in the MTF; in this approximation, it issufficient to multiply, at all nonzero spatial frequen-cies, the nominal-order normalized MTF by the ratio ofthe useful light power to the total light power ~in themonochromatic case, the factor is simply the nominal-order efficiency!.4 However, this does not accountwell for the not uncommon case of a small number ofzones, where the nonnominal low-order PSF’s are onlya few times wider than the nominal-order PSF and,therefore, participate in some extended range of spa-tial frequencies. Moreover, it is impossible with suchan approximation to include those orders in the opti-mization process.

Indeed, infinite summation of orders is a rigoroussolution as far as the scalar approximation of opticalsystems holds; such summation should be performedcoherently for all orders at each wavelength. Theconvergence of the diffraction-orders series expan-sion, however, is known to be often impractical owingto its slow 1yM convergence, with M being the num-ber of summed-up orders. One may note thatincoherent-order summation, sometimes used by op-tical designers, has the same slow 1yM convergencebehavior but leads to a mathematically incorrect an-swer.

In the following, we propose a simple alternativemethod based on zone decomposition rather than onorder summation. The principle is described in Sec-tion 2, and one illustrative example is presented inSection 3.

2. Modeling Principle

First, let us note that in aberration-correction appli-cations, the groove spacing of the diffractive opticalelements is usually large compared with the wave-length. Thus, the theoretical background to our

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computations is the scalar theory of diffraction andmainly the Kirchhoff diffraction integral.5,6 ThePSF is then the two-dimensional ~2D! Fourier trans-form of the complex amplitude ~relative to the givenreference sphere! in the exit pupil of the system. Inour proposed method, we model the hybrid system asa multiaperture system, much in the same way as inmultipupil telescopes in which the images owing tothe various pupils combine coherently into one finalimage. In the present case, however, the variousapertures correspond to the DOE zones: namely,optical thickness jumps partition the DOE surfaceinto nonoverlapping apertures, which, in the commoncase of rotationally symmetric systems, are concen-tric annular areas. In fact, the discontinuity be-tween zones is designed to create a phase jump for theoutgoing wave front, and the main idea in DOE’s is tohave 2p phase jumps between adjacent zones ~or aninteger multiple thereof ! at least on axis for somenominal wavelength. Within each zone, DOE opti-cal thickness varies continuously like in any standardcomponent in a lens, and then, conventional analysismethods, like ray tracing, can be used there. Notethat in the case of the so-called binary optics manu-facturing method7 for which multiple masking stepsare used to approximate, on the outgoing wave front,a continuous phase variation over 2p by a p-stepstaircase function, each step defines one subzonewith 2pyp phase discontinuities. In such a case, theelementary aperture that should be used is the sub-zone.

The analysis method of the whole hybrid lens isthen straightforward: each monochromatic image isthe coherent sum of the images given by the systemfor each zone or each subzone. Any tool, includingcommercial optical design software, can be used tomodel the system behavior as soon as diffraction PSFof systems with nonstandard aperture and coherent~complex! summation are available. Let us empha-size that the infinite summation of orders is mathe-matically equivalent to our, potentially simpler, finitesummation based on zone decomposition. More-over, the zone decomposition still holds when thephase discontinuities do not have the same value atall zone boundaries, even though the order decompo-

Fig. 1. Layout of the hybrid triplet. This system is an fy1.7,21.1-mm focal length lens intended to work in the visible and theNIR band ~0.48–0.78 mm!.

sition no longer exists from a mathematical point ofview.

From the monochromatic PSF of the whole system,obtained as explained above, standard methods applyto derive the monochromatic MTF. As usual, oneperforms polychromatic analysis by use of incoherentweighted summation over wavelength.

3. One Example: A Fast Achromatic Hybrid Triplet

One of the major motivations for using DOE’s insidehybrid systems is the dispersive behavior of diffrac-tion, which provides efficient means to deal withchromatic aberration. In the order decompositionmethod, nonnominal orders essentially create defo-cused images that, in the absence of any other aber-ration, result in more or less uniform plateaus in thePSF. Our approach directly leads to the result ofinterferences between the plateaus of all orders ateach wavelength. We already presented such a re-sult8 and intend to publish more detailed examples ata later stage. Here, we emphasize the advantage ofthe proposed method on just one characteristic exam-ple for which the DOE corrects for combined aberra-tions rather than for pure chromatism. Although it

Fig. 2. Characteristic function and etching profile of the hybridtriplet DOE. Note the unusual shape of the characteristic func-tion curve that is mainly intended to correct for spherical aberra-tion ~mostly like a Schmidt plate! rather than for pure axialchromatism. lNominal 5 594 nm.

Table 1. Detailed Layout of the Hybrid Triplet

SurfaceNumber Surface Type

Radius ofCurvature

~mm!Thickness

~mm! Material

0 ~Object! Flat Infinity Infinity —1 ~Stop! Sphere 15.94 5.80 SK2 ~Schott!2 Sphere 211.53 6.50 SF4 ~Schott!3 Sphere 229.34 0.40 —4 Flat 1 DOEa Infinity 1.50 BK7 ~Schott!5 Flat Infinity 14.70 —6 ~Image! Flat Infinity — —

aThe specifications for the DOE were as follows: lNominal 5 594m; the nominal diffraction order was 11; the characteristic func-ion was a radial polynomial; and the phase at lNominal waspylNominal @~c1h2! 1 ~c2h4! 1 ~c3h6! 1 ~c4h8!#, where c1 54.903 3 1024 mm21, c2 5 3.263 3 1025 mm23, c3 5 23.012 3

1027 mm25, c4 5 25.268 3 1029 mm27, and h is the radial dis-tance from the optical axis.

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may not correspond to a very practical case in view ofsome manufacturing constraints, this example is in-tended to illustrate our point of view about the coher-ent multiaperture image summation.

Figure 1 sketches the system layout: It is avisible–near-infrared triplet built from a cementeddoublet and a nine-zone DOE with continuous zoneprofiles on a plane-parallel plate. The characteristicfunction and the etching profile of the DOE are shownon Fig. 2. The precise description can be found inTable 1. The result is a fairly good 21.1-mm focallength, fy1.7 achromat for infinity–focal-plane conju-gation in the wavelength range 0.48–0.78 mm.9 Thepolychromatic analyses are done with 11 equispacedwavelengths with equal weight. Figure 3 shows thetriplet polychromatic PSF in the immediate vicinityof the axis. The departure between the standardcalculation with only one order ~with 100% efficiencyat all wavelengths, as assumed by most commercialoptical design software! is fairly clear from the Strehlatio. Figure 4 gives the same PSF in a log scale formuch wider radial extent, showing the jagged as-

Fig. 3. On-axis polychromatic PSF of the hybrid triplet ~centralegion!. Thin line: 11 diffraction-order computation ~100% ef-

ficiency assumed for all wavelengths!. Thick line: zone-decomposition computation.

Fig. 4. On-axis polychromatic PSF of the hybrid triplet ~wideregion!. Thin line: 11 diffraction-order computation ~100% ef-ficiency assumed for all wavelengths!. Thick line: zone-decomposition computation.

484 APPLIED OPTICS y Vol. 38, No. 31 y 1 November 1999

ect of the real PSF far from center, where all dif-raction orders interfere in a fairly irregular fashion,ompared with the smoother and faster decline of the1 order PSF. Figure 5 shows the MTF. One maygain note the significant departure between the 11rder modeling and our more rigorous model. Thebsence of a sharp MTF drop at very low spatialrequencies means that higher orders are far fromniform defocusing plateaus, as can be seen from Fig.in which no such structure can be distinguished on

he thick solid curve. This rather uncommon behav-or is clear from Fig. 2, where it can be seen that theharacteristic function of the DOE is mainly intendedo correct combined defocus and spherical aberra-ions rather than pure chromatism.

For comparison purposes, Fig. 6 shows the MTF ofn achromatic cemented doublet, with the same focal

Fig. 5. On-axis polychromatic MTF of the hybrid triplet.Dashed line: diffraction limit. Thin solid line: 11 diffraction-order computation ~100% efficiency assumed for all wavelengths!.Thick solid line: zone-decomposition computation. The inset isa magnified view of the low spatial frequency region indicated bythe dashed rectangle.

Fig. 6. On-axis polychromatic MTF of an optimized conventionalcemented doublet. Dashed line: diffraction limit. Solid line:MTF of a conventional cemented doublet of same focal length andaperture as the hybrid triplet and optimized for polychromaticoperation over the same wavelength range ~0.48–0.78 mm!. Theinset is a magnified view of the low spatial frequency region indi-cated by the dashed rectangle. Note that it has a different verti-cal scale than the one of Fig. 5.

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length and f number as those of the hybrid triplet,ptimized for operation in the same visible–near-nfrared wavelength range. The usefulness of theybrid design is obvious!Finally, Fig. 7 shows the polychromatic MTF of

the hybrid triplet for low spatial frequencies, ascomputed from different approximations of the lensbehavior. Curves ~2! and ~3!, which correspond tohe 11 order and to zone-decomposition computa-ions, respectively, have already been plotted with aifferent scale on Fig. 5. Curve ~4! corresponds tohe computation mentioned in the introduction,here the stray light due to nonnominal orders is

aken into account through a uniform background.urves ~5! and ~6! show the results of the incoherentummation of 5 and 19 orders, respectively, taken

Fig. 7. On-axis polychromatic MTF of the hybrid triplet for lowspatial frequencies. ~1!: diffraction limit. ~2!: 11 diffractionorder ~100% efficiency assumed for all wavelengths!. ~3!: zonedecomposition. ~4!: 11 diffraction order with stray light fromnonnominal orders taken into account as a uniform background.~5!: incoherent summation of 5 diffraction orders ~21 . . . 13!.~6!: incoherent summation of 19 diffraction orders ~28 . . . 110!.See Table 2 for precise definitions of the curves.

Table 2. Precise Definitions of the Polychrom

Components andCurve Numbers

MTF Modulus of the cwhere the OTand the OTF a

OTFm, l MonochromaticOTFZD,l Monochromaticwl Spectral weightshm,l Efficiency of diff

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ymmetrically around the 11 order. Table 2 giveshe precise definitions of each computation. Forhe sake of clarity, the curves for coherent summa-ion of orders are not included; for the present ex-mple, they are closed to the incoherent ones. Inll cases, the differences to the more rigorous zoneecomposition computation @curve ~3!# are conspic-ous.Regarding the practical implementation of the

one decomposition for the example curve compu-ations, we extracted the entire hybrid system exitupil complex amplitude by means of ray tracingith the help of a commercial optical design soft-are. This lead to a discontinuous wave front withdirect relation between step locations and DOE

one boundaries. We thus neglected the Fresneliffraction that occurred during propagation fromhe DOE to the exit pupil; this approximation islearly valid if the DOE is near the exit pupil andeems to be applicable in most practical cases, inhe same way as central obscuration of reflectingelescopes are correctly modeled by diffraction PSFr MTF computed by present optical design soft-are. Then, we computed the PSF’s and MTF’sutside any commercial optical design software, tak-ng advantage of the rotational symmetry of the prob-em to get the 2D Fourier transform to high precisiony means of the more efficient one-dimensional ~1D!ero-order Hankel transform6 on a piecewise analyti-

cal approximation of the discontinuous wave front.Let us emphasize that, with proper care of disconti-nuities, this zone-decomposition implementation al-lows for computation of the PSF or the MTF of anentire hybrid system with mainly the same computa-tional effort as the one required for obtaining its 11diffraction-order PSF or MTF, by avoidance of any ex-plicit computation and coherent summation of thePSF’s of each DOE zone. This clearly outperformed,

MTF Curves of Fig. 7 and Their Components

Definitions

lex optical transfer function ~OTF!, i.e., MTF 5 uOTFu,he Fourier transform of the real PSF.6 The MTFormalized to 1 at the spatial frequency of 0 cycleymm.of the diffraction order m at l.at l for the zone-decomposition computation.¥l wl 5 1.

n order m at the wavelength l. We haveall l. In the case of the assumed perfect continuouse ~see Fig. 2!, we have hm,l 5 sinc2~m 2 lNominalyl!.7

! 5 u¥l OTF1,lwlu.lwlu.ckground 5 @u¥l OTF1,lwlh1,luy¥l wlh1,l# @¥l wlh1,l#, atfrequencies ~1 at 0 cycleymm!. Here, because of theice of lNominal, ¥l wlh1,l reaches 0.927.

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on both precision and computer time, the incoherent orthe coherent summation of orders.

4. Conclusions

We have proposed to model hybrid systems with amultiaperture approach by coherent summation ofthe images formed by the various zones of the diffrac-tive component, and we have shown on one examplethat such rigorous modeling ~in the framework ofscalar optics! is useful and indeed even necessary forome hybrid systems. In fact, we expect our ap-roach to be especially relevant for hybrid systems inhich the DOE has a fairly small number of zones, aighly nonparabolic characteristic function, or non-qual phase discontinuities over the zone boundaries.or the first two cases, the resulting entire PSF sig-ificantly departs from the PSF of the nominal orderurrounded by weak, large uniform plateaus. Thisuins the validity of simple approximations andherefore dictates the use of more rigorous modeling;lthough both zone decomposition and coherent-rder summation are mathematically equivalent, theormer is computationally more efficient for the pre-ision usually required. For the last case, no orderecomposition is available at all, and the zone decom-osition is the only possible approach.We also have briefly presented the zone-

ecomposition implementation we used to compute theaper curves. This implementation, based on rayracing and 1D zero-order Hankel transform, suffers,owever, from two limitations: It cannot be entirelyerformed inside a commercial optical design software,nd it is restricted to on-axis images for a rotationallyymmetric hybrid system. We are therefore workingo implement the zone-decomposition method as add-ns to commercial software packages with the help of

486 APPLIED OPTICS y Vol. 38, No. 31 y 1 November 1999

he 2D fast Fourier transform in order to relax theotational symmetry constraint; however, the accuracyf such an implementation depends on the used sam-ling grid. This, together with possible application toystem design, including optimization, will be the sub-ect of future publications.

References and Notes1. T. Stone and N. George, “Hybrid diffractive-refractive lenses

and achromats,” Appl. Opt. 27, 2960–2971 ~1988!.2. M. J. Riedl and J. T. McCann, “Analysis and performance limits

of diamond-turned diffractive lenses for the 3-5 and 8-12 micro-meter regions,” in Infrared Optical Design and Fabrication, R.Hartmann, M. Marietta, and W. J. Smith, eds., Vol. CR38 ofSPIE Critical Review Series ~SPIE, Bellingham, Wash., 1991!,pp. 153–163.

3. M. D. Missig and G. M. Morris, “Diffractive optics applied toeyepiece design,” Appl. Opt. 34, 2452–2461 ~1995!.

4. D. A. Buralli and G. M. Morris, “Effects of diffraction efficiencyon the modulation transfer function of diffractive lenses,” Appl.Opt. 31, 4389–4396 ~1992!.

5. M. Born and E. Wolf, Principles of Optics, 6th ed. ~Pergamon,Oxford, UK, 1980!.

6. J. W. Goodman, Introduction to Fourier Optics, 2nd ed.~McGraw-Hill, New York, 1996!.

7. G. J. Swanson, “Binary optics technology: the theory and de-sign of multi-level diffractive optical elements,” MIT, LincolnLab. Technical Report 854 ~Massachusetts Institute of Technol-ogy, Lexington, Mass., 1989!.

8. H. Sauer, G. Narcy, and P. Chavel, “Kinoform modeling forhybrid optical system design,” Diffractive Optics 97, Vol. 12 ofTopical Meetings Digests Series ~European Optical Society, Or-say, France, 1997!, communication D19, pp. 174–175.

9. This is a pure academic exercise, and this layout has beendeveloped independently of the layout of the Melles GriotAPO014 Dapromat commercial product that happens to havefairly similar external characteristics.