5910 APPLIED OPTICS / Vol. 31, No. 28 / 1 October 1992

Parametric optimization of multilevel diffractive optical elements by electromagnetic theory Eero Noponen, Jari Turunen, and Antti Vasara

E. Noponen and A. Vasara are with the Department of Technical Physics, Helsinki University of Technology, SF-02150 Espoo, Finland; J. Turunen is with the Depart­ment of Physics, Heriot-Watt University, Riccarton, Ed­inburgh, EH13,4AS, UK. Received 4 February 1992. 0003-6935/92/285910-03$05.00/0. © 1992 Optical Society of America.

To maximize the efficiency of dielectric diffractive optical elements, we optimized the local groove shape using the rigorous diffraction theory of multilevel surface-relief grat­ings.

Key words: Micro-optics; Holographic optical elements; Diffraction.

The wave-front transformations produced by micro Fresnel lenses1 and other diffractive optical elements that can be treated locally as gratings are determined primarily by the variations in the local grating period across the aperture. However, the local first-order diffraction efficiency depends critically on the grating groove shape. Within the frame-work of paraxial scalar diffraction theory (Fourier optics) this efficiency is related to the first Fourier coefficient of the complex-amplitude transmission function, i.e., it is indepen­dent of both the grating period and the state of polarization of the incident light. Such predictions fail in the reso­nance region where the grating period d and the wave-length λ are of the same order of magnitude. In these conditions the electromagnetic theory of gratings2 must be used to predict the local characteristics of the diffractive element.3-4 It turns out that for dielectric gratings there is a significant drop of efficiency in the resonance region, which not only reduces the overall efficiency on the diffrac-

tive element but may also lead to strong undesired apodiza-tion effects if the local grating period varies considerably across the aperture. For metallic gratings the effects are much less pronounced, and a small adjustment of the relief depth cures the problem.5 While reflection-type diffractive optical elements are required in many substrate-mode configurations,6 they are not suitable for nearly all applica­tions, including those based on stacked planar optics.7

We show in this Technical Communication that the reso­nance region characteristics of dielectric diffractive optical elements can be improved substantially by optimizing the local groove shape by using the electromagnetic theory of gratings.

We concentrate here on dielectric K-\evel surface-relief gratings with staircase permittivity profiles of the form ε(x,y) = n2 if y < h(x) and ε(x,y) = 1 otherwise. Here h(x) = hk/K when d(k - 1)/K ≤ x< dk/K with k = 1,. . . , K and h = λ/(n - 1). We can fabricate such gratings by using the standard microlithographic techniques that gen­erate K=2N levels with N masks.4,8 According to Fourier optics, the first-order efficiency η of this grating, when illuminated from the -y direction (from dielectric to air), is η = η0 sinc2(l/K), where η0 = 4n/(n + l)2. Figure 1 shows the true resonance region first-order efficiency curves η versus d/λ for both TE and TM polarization. We calculated them using the multilevel extension9 of Knop's rigorous theory of lamellar dielectric gratings,10 assuming that n = 1.5. The first-order efficiency falls well below the Fourier-optics prediction at periods d < 10λ if K ≥ 4. Rather dramatically the efficiency of the 16-level grating [that behaves almost as the sawtooth profile h(x) = hx] drops below 10% for TE polarization at d ≈ 2λ. This is less than one half of the corresponding value for the binary profile, which is much easier to fabricate.

We can perform optimization of the local groove shape using either inverse scattering methods11 or parametric optimization.9,12 The former methods give a continuous-relief profile; we therefore apply parametric optimization,

Table 1. Parameters and Performance of Some Optimized Four-Level Gratings

Fig. 1. First-order diffraction efficiency as a function of the grating period for dielectric (refractive index 1.5) staircase surface relief gratings with K depth levels (K = 3, 4, 8, 16): (a) TE polariza­tion, (b) TM polarization. The horizontal lines represent the asymptotic predictions of Fourier optics.

mum, and several random starting configurations have to be tried.

In Table 1 we show explicitly some typical optimization results. Here the period is fixed to d = 2.5λ, and the results for all options that are given above are listed. The first-order efficiency improves significantly in all cases, and for the TE mode it often exceeds the Fourier optics prediction η ≈ 0.778 for the four-level staircase profile. Inspection of the transition point locations shows that the optimized groove shapes differ strongly from this staircase profile. In fact the feature size between the second and third transition points becomes very small (< λ/10), which is virtu­ally impossible to fabricate even by direct-write electron-beam lithography, at least for visible and near-IR light. Fortunate­ly, however, the performance of the grating structures does not deteriorate dramatically if such small features are removed. Realizing, e.g., the grating optimized for TE

definingthe grating profile as h(x) = hk/K when xk-1 < x < xk and using the transition points xk, k = 1, . . . , K - 1 as free optimization parameters. As a result of translation invariance and periodicity, x0 = 0 and xκ = d are fixed. The relief depth h can be used also as a free parameter if the period of the element is essentially the same over the aperture (e.g., circular gratings for generating Bessel beams,13 but in general it must be fixed if the multiple-mask technique is used. In both cases we assume that the K levels are equally spaced and n = 1.5. To avoid undercut boundaries, we also apply a constraint x k + 1 ≥ xk. Optimi­zation for one state of polarization only may be appropriate if the diffractive element has approximately straight fringes (e.g., a cylindrical or a highly off-axis micro Fresnel lens), but in general both states of polarization must be consid­ered simultaneously, and the total local efficiency is then given (at normal incidence, see Ref. 14) by η = η T E sin2 δ + ηTM cos2 δ, where η T E and η T M are the TE- and TM-mode efficiencies and δ gives the local fringe orientation with respect to the polarization vector.

Apart from binary gratings that are too inefficient for many applications and cannot be significantly improved by optimization, four-level gratings (N = 2 masks) are the easiest to fabricate, and we concentrate on them. We reoptimize the three transition-point locations xk, k = 1, 2, 3 and in some cases also the total relief depth h, using a gradient algorithm to maximize the first-order efficiency η either for one state of polarization alone or for both TE and TM modes simultaneously. The initial configuration is xk = kd/4 + ∆k, where ∆k is a random perturbation. Typically the gradient method converges into a local mini-

Fig. 2. (a) TE-mode first-order efficiency of a four-level staircase grating reproduced from Fig. 1(a) compared with optimized profiles when the transition points Xj alone, and those together with the total relief depth h, are used as optimization parameters, (b) Optimization for both states of polarization simultaneously with a fixed relief depth.

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Table 2. Parameters and Performance of the Gratings in Fig. 2(b)

polarization and fixed h (second row, Table 1) as a three-level structure with xk/λ = {1.08, 2.16, 2.16} gives η T E = 0.794 and η T M = 0.572.

In the resonance region the use of more than four levels does not necessarily improve the efficiency by a significant amount. We optimized an eight-level grating for TE polar­ization and free h, obtaining η T E = 0.869, i.e., only ~ . 0 l above the value for the four-level grating. We also investi­gated the possibility of using a stepped backwall instead of the abrupt drop at x = d, but the improvement provided by these new degrees of freedom was either marginal or nonexistent, depending on d.

Figure 2 illustrates optimization results for the construc­tion of arbitrary interferometric-type diffractive optical elements. The first-order efficiency is optimized here throughout the resonance region. Figure 2(a) shows the TE-mode efficiency with h either fixed or free, while in Fig. 2(b) both polarizations are considered simultaneously with fixed h. Parametric optimization is applied at integer and half-integer values of d/λ (filled and open circles, respective­ly) . The sharp downfall of the local diffraction efficiency in the resonance region is avoided, and the optimized efficien-cycurves are reasonably flat. Especially if the relief depth h is free, the efficiency at small periods is rather consis­tently above the asymptotic value (large d) of 0.778 pre­dicted by Fourier optics. To demonstrate that optimiza­tion results at discrete values of d can be used to construct diffractive elements with a continuously varying local pe­riod, we have scaled the period of each optimized solution in Fig. 2 up to halfway between the adjacent circles (the lines connecting the filled and open circles). The efficiency fluctuates especially when d ≈ 2λ, but by decreasing the spacing between the optimized points these fluctuations can be reduced to an arbitrarily small level. The optimiza­tion results of Fig. 2(b) are given explicitly in Table 2. It is not possible to prove that the results of this type of parametric optimization represent global optimum, but we believe that these efficiencies could be improved only slightly.

In conclusion we have demonstrated that the perfor­mance of diffractive optical elements could be enhanced significantly by parametric optimization of the local grating

groove shape. Further details and examples of the con­struction process (where the phase of the first diffraction order can also be important) will be given elsewhere.

We acknowledge the Jenny and Antti Wihuri Foundation (Finland), the Academy of Finland, and the Science and Engineering Research Council (UK) for financing this study.

References 1. H. Nishihara and T. Suhara, "Micro Fresnel lenses," in

Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1987), Vol. 24, pp. 1-40.

2. R. Petit, ed., Electromagnetic Theory of Gratings, Vol. 22 of Topics in Current Physics (Springer-Verlag, Berlin, 1980).

3. N. C. Gallagher, Jr., and S. S. Naqvi, "Diffractive optics: scalar and non-scalar design analysis," in Holographic Optics: Optically and Computer Generated, I. N. Cindrich and S. H. Lee, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1052, 32-40 (1989).

4. J. A. Cox, T. Werner, J. Lee, S. Nelson, B. Fritz, and J. Bergstrom, "Diffraction efficiency of binary optical elements," in Computer and Optically Formed Holographic Optics, I. N. Cindrich, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 1211, 116-124(1990).

5. T. Shiono, M. Kitagawa, K. Setsune, and T. Mitsuyu, "Reflec­tion micro-Fresnel lenses and their use in an integrated focus sensor," Appl. Opt. 28, 3434-3442 (1989).

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8. J. Jahns and S. J. Walker, "Two-dimensional array of diffrac­tive microlenses fabricated by thin film deposition," Appl. Opt. 29, 931-936 (1990).

9. A. Vasara, E. Noponen, J. Turunen, J. M. Miller, M. R. Taghizadeh, and J. Tuovinen, "Rigorous diffraction theory of binary optical interconnects," in Holographic Optics III: Principles and Applications, G. M. Morris, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 1507, 224-238 (1991).

10. K. Knop, "Rigorous diffraction theory for transmission phase gratings with deep rectangular grooves," J. Opt. Soc. Am. 68, 1206-1210 (1978).

11. A. Roger and D. Maystre, "Inverse scattering method in electromagnetic optics: application to diffraction gratings," J. Opt. Soc. Am. 70, 1483-1495 (1980).

12. E. Noponen, A. Vasara, J. Turunen, J. M. Miller, and M. R. Taghizadeh, "Synthetic diffractive optics in the resonance domain," J. Opt. Soc. Am. A 9, 1206-1213 (1992).

13. A. Vasara, J. Turunen, and A. T. Friberg, "Realization of general nondiffracting beams with computer-generated holo­grams," J. Opt. Soc. Am. A 6, 1748-1754 (1989).

14. M. G. Moharam and T. K. Gaylord, "Three-dimensional vector coupled-wave analysis of planar-grating diffraction," J. Opt. Soc. Am. 73, 1105-1112 (1983).

5912 APPLIED OPTICS / Vol. 31, No. 28 / 1 October 1992