Diffractive refractive behavior of kinoform lenses - Apollo Optical ...

where g n is a function of an integer argument. Themain reason for the generalization in Eqs. 5 willbecome clear in the following developments when therefractive role of each zone is made evident. In particular,we have the conventional, g n n, and thehigher-order, g n pn, cases for the usual diffractivelens designs.To elicit the diffractive–refractive relation, we developsome on-axis calculations based on the Fresnelapproximation to the diffraction integral 7 given by2 expikzz iz0r Nexpirexp iz r2rdr, (6)where we have made use of the circular symmetry ofthe elements under consideration and where r N is theboundary of the outermost zone or the semiapertureof the lens. Use of the phase function of Eqs. 5 inthe diffraction integral yields the following, somewhatformidable, expression for the on-axis scalarfield: D z 2 0fexpi2g n 0fz Nz n1 g n g n1 g n g n1 sinc 0fz g n g n1 , (7)where subscript D indicates a diffractive lens andirrelevant phase terms were ignored.By using Eq. 6 we can also determine the scalarfield of a purely refractive lens with maximum phaseheight 2 0 , given by R z 2 0fzexpi2 0 0fz 0 sinc 0f z 0 , (8)with index R indicating a refractive lens. We notethat Eqs. 7 and 8 can be combined because thesummation over zones in the diffractive scalar fieldcan be seen as a sum over the field caused by localrefractive lenses. The result isN D z n1 expi2 0fn1g n g n1 R zg n z, (9)where n R is the phase function of a refractive lens ofmaximum phase height such that 0 g n g n1 .Because term g n g n1 is directly related to eachsingle zone, the diffractive element can be seen as theinterference of N refractive lenses whose characteristicsare determined locally by each zone. Noticethat this case is different from a lens array, which ismore akin to a grating in which each period may beconstituted by a refractive lens.The on-axis field that is due to each associatedrefractive lens is modulated by the complex interferenceterm, n l z expi2 0fn1g n g n1 , (10)zgwhich couples the refractive field amplitude of eachzone. The expression of the scalar field as in Eq. 9explicitly gives the contribution of each zone to formthe final interference pattern of the generalized diffractivelens. In the case of a higher-order lens wecan see thatg n g n1 pn pn 1 p, (11)implying that the refractive field associated witheach zone is independent of n and can be taken out ofthe summation. If we write n R z p R zto referto the higher-order lens, we can write Eq. 9 asN D z p R zn1 p expi2 0fzpn 1 p R zz, (12)where z represents the summation of phase termsover all zones and can be calculated exactly to bez Np expi 0fzN 1p sinc 0fzNpsinc 0 fzp .(13)It is also interesting to investigate how the pointspreadfunction PSF of the diffractive element relatesto that of the refractive lens. For this purpose,we consider a higher-order lens with maximum phaseheight 2p and N zones. Infinite conjugate imagingis assumed. The comparison to be carried outhere is with a refractive lens of maximum phaseheight 2Np and the same clear aperture as thekinoform. After straightforward albeit somewhattedious calculations, which again are based on Eq.6, we find that the PSF of the diffractive lens, I D z,is written in terms of the PSF, I R z, of the refractivelens asI D z I T zI R z, (14)where term I T z relating the diffractive–refractivelimits can be written ascosI T z 2 2 i1 0 fzpi1 cos 2 2 i1 0 fz p , (15)assuming that N 2 for an integer and i1 i 1 2... . Notice that in Eq. 14, term I R z correspondsto the purely refractive limit illustrated inFig. 1. At 0 the transition coefficient Eq. 15reduces to unity and the diffractive element behavesessentially as a refractive lens. This is expected becauseboth elements are identical at 0 , except for amodulo 2p transformation rule. However, for awavelength detuning from the design, the transitionterm becomes important and gives rise to those phenomenathat afflict a diffractive element such as re-1 January 1997 Vol. 36, No. 1 APPLIED OPTICS 255

Fig. 2. PSF’s illustrating the dispersive nature of a diffractivelens in comparison to a refractive one. The on-axis intensity isshown for three wavelength values: 0.5876 m index ofrefraction n D 1.5168 for BK-7 glass, 0.4861 m n F 1.5143, and 0.6563 m n C 1.5228. The diffractive lenswas designed to operate at 0 0.5876 m with a focal length of50 mm. In this case p 1 and N 32. The solid curve shows thebehavior at the design wavelength, which is identical for bothdiffractive and refractive elements. Note the opposite sign of thedispersion and the appearance of multiple orders.Fig. 3. Evolution of the PSF of a diffractive lens of 32 zones andp 1 toward the refractive limit or N 1. Design parameters: 0 0.5876 m, F 50 mm, and 0.4861 m. For a meaningfulcomparison, product Np is set to a constant equal to 32.Note that as N decreases p increases the diffractive pattern tendsto approximate the spectral behavior of the refractive lens solidcurve. However, even for N 2 the spatial intensity pattern isconsiderably distinct from the refractive case.duced efficiency and the appearance of additionaldiffracted orders. These effects are illustrated inFig. 2 for what could be called a conventional diffractiveelement or p 1.From the PSF results derived above, we now investigatethe behavior of the diffractive lens as parameterp increases while Np remains fixed. Thiscorresponds to a sequence similar to the one depictedin Fig. 1. In performing the calculations previouslydescribed we have assumed an error-free diffractiveelement. Although this is not entirely realistic, inthe presence of small errors the results should still bevalid. Some discussion of fabrication errors with respectto the diffractive–refractive transition can befound elsewhere. 4In Fig. 3 the evolution of the point-spread functionas a function of the number of zones is presented.The behavior of the element can be analyzed from twoimportant aspects: spatial distribution and spectralbehavior. These characteristics of the diffractionpattern are certainly related, but as the number ofzones decreases and p increases, they need not evolvein a similar fashion. Indeed, we note in Fig. 3 thatas the diffractive lens tends to the refractive limit thediffractive point spread function tends to the correspondingrefractive pattern. For instance, when thenumber of zones decreases from 32 to 16, the totalenergy on the first order is reduced while the secondorderfocal point undergoes an increase in efficiencyfollowed by a dislocation toward the paraxial refractivefocal point. Such a process continues as N decreases,indicating that the diffractive element tendsto recover the refractive focal point for the wavelengthin question. Such clustering indicates that,spectrally, the tuning of parameter p in fact leads toa more refractivelike diffractive lens, in accordancewith the idea that a higher order improves polychromaticbehavior. 5,6 In contrast, the spatial characteristicsof the point-spread function are also relevantbecause they reflect the efficiency of each diffractedorder and consequently provide important informationon image contrast. As we can see in Fig. 3, eventhough increasing p tends to reduce the contributionof spurious orders, it seems that the diffracted lensFig. 4. Transition term I T z, Eq. 15, as the diffractive elementtends to the refractive limit. The same design parameters of Fig.3 are adopted. In the purely refractive case, the transition coefficientis identically equal to one. As seen here, the basic dependencewith the number of zones remains essentially unaltered,except for a scaling term, up to N 2.256 APPLIED OPTICS Vol. 36, No. 1 1 January 1997

exhibits intensity profiles that are very peculiar up tothe case in which only two zones are present. In thisextreme limit, we might say that although parameterp tends to correct chromatic properties, it cannot correctthe spatial distribution of the produced beam.As a result, from the spatial point of view the kinoformlens preserves its intrinsically diffractive naturewith any nonunity number of zones. Such a resulthas also been verified numerically in other designcases. 4 This fact can be made more evident in Fig. 4,where a plot of I T z, Eq. 15, is shown for distinctvalues of p. The basic dependence of the transitionterm, except for a scaling factor, is unaltered for allvalues of N.3. Summary and ConclusionsThe design of higher-order diffractive lenses poses theproblem of the diffractive–refractive behavior in certainkinoform elements. We have expressed the onaxisscalar field of the diffractive lens in terms of aninterference pattern of associated refractive lensesrelated to each individual zone and modulated bya zone-dependent complex-valued coefficient. Althoughfor transverse fields the same idea shouldapply, it is not possible to obtain closed-form solutionsas in the on-axis case. We have found that it isalso possible to express the point-spread function ofthe diffractive lens as the product of a transition termand the intensity pattern of a refractive lens. In thisform, the connection diffractive–refractive is immediatelyestablished.Even though the refractive behavior becomes moreevident as the maximum phase height increases, thediffractive lens preserves a unique spatial identity upto the point at which only two zones are present buttends continuously to the refractive spectral behavior.This result has also been observed for gratingsby means of an altogether different method. 3 Althoughrefractive elements without proper achromatizationmay be acceptable in some situations, fordiffractive lenses there is greater difficulty becausethe amount of chromatic aberrations is dramatic. Ifthe problem in question requires operation of a fewdiscrete wavelengths, higher-order diffractive lensesprovide an elegant solution. As the number of zonesdecreases and higher values of maximum phase areattained, the diffractive behavior approaches that of arefractive lens but never completely reaches it; infact, for N 2 the spatial intensity pattern of the lensdeparts even more significantly from the refractivebehavior. Consequently, although intuitive, theconcept of a transition between diffractive behaviorand refractive behavior does not evolve identically inthe spatial-spectral sense. In broadband applications,diffractive optics seem more promising for useas hybrid diffractive–refractive elements.This research was supported in part by the NewYork State Center for Electronic Imaging Systems.T. R. M. Sales is pleased to acknowledge financialsupport from the Brazilian agency Fundação Coordenaçãode Aperfeiçoamento de Pessoal de Nível Superior.References1. D. A. Buralli, G. M. Morris, and J. R. Rogers, “Optical performanceof holographic kinoforms,” Appl. Opt. 28, 976–9831989.2. See, for example, D. Faklis and G. M. Morris, “Diffractive lensesin broadband optical system design,” Photon. Spectra 25, 131–134 1991.3. S. Sinzinger and M. Testorf, “Transition between diffractive andrefractive micro-optical components,” Appl. Opt. 34, 5970–59761995.4. M. Rossi, R. E. Kunz, and H. P. Herzig, “Refractive and diffractiveproperties of planar micro-optical elements,” Appl. Opt. 34,5996–6007 1995.5. D. Faklis and G. M. Morris, “Spectral properties of multiorderdiffractive lenses,” Appl. Opt. 34, 2462–2468 1995.6. D. W. Sweeney and G. E. Sommargren, “Harmonic diffractivelens,” Appl. Opt. 34, 2469–2475 1995.7. J. W. Goodman, Introduction to Fourier Optics McGraw-Hill,New York, 1968.1 January 1997 Vol. 36, No. 1 APPLIED OPTICS 257