## Analytic Approach for Optimal Quantization of Diffractive Optical Elements

Applied Optics, Vol. 38, Issue 26, pp. 5527-5532 (1999)

http://dx.doi.org/10.1364/AO.38.005527

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### Abstract

One of the most important factors that limit the performance of diffractive optical elements (DOE’s) is the depth accuracy of the relief structure. A common procedure for fabricating DOE’s is the binary optics procedure, in which binary masks are used for the fabrication of a multilevel relief structure. Here an analytic procedure for calculating the optimal depth levels of DOE’s, the phase bias, and the decision levels is presented. This approach is based on the minimization of the mean-squared error caused by the quantization of the continuous profile. As a result of the minimization an optimal value for the etching depth of each photolithographic mask is determined. The obtained depth values are, in general, different from the depth values used by the conventional multilevel approach. Comprehensive mathematical analysis is given, followed by several computer simulations that demonstrate the advantages of the proposed procedure.

© 1999 Optical Society of America

**OCIS Codes**

(050.1380) Diffraction and gratings : Binary optics

(050.1970) Diffraction and gratings : Diffractive optics

**Citation**

Uriel Levy, Nadav Cohen, and David Mendlovic, "Analytic Approach for Optimal Quantization of Diffractive Optical Elements," Appl. Opt. **38**, 5527-5532 (1999)

http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-38-26-5527

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### References

- F. Wyrowski and O. Bryngdahl, “Iterative Fourier transform algorithm applied to computer holography,” J. Opt. Soc. Am A 7, 961–969 (1988).
- J. L. Horner and J. R. Leger, “Pattern recognition with binary phase-only filters,” Appl. Opt. 24, 609–611 (1985).
- S. J. Walker and J. Jahns, “Optical clock distribution using integrated free-space optics,” Opt. Commun. 90, 359–371 (1992).
- M. T. Gale, M. Rossi, J. Pederson, and H. Schutz, “Fabrication of continuous-relief micro-optical elements by direct laser writing in photoresist,” Opt. Eng. 33, 3556–3566 (1994).
- H. M. Phillips and R. A. Sauerbrey, “Eximer-laser-produced nanostructures in polymers,” Opt. Eng. 32, 2424–2436 (1993).
- N. A. Vainos, S. Mailis, S. Pissadakis, L. Boutsikaris, P. J. M. Parmiter, P. Dainty, and T. J. Hall, “Excimer laser use for microetching computer-generated holographic structures,” Appl. Opt. 35, 6304–6319 (1996).
- M. Ekberg, M. Larsson, S. Hard, and B. Nilsson, “Multilevel phase holograms manufactured by electron-beam lithography,” Opt. Lett. 15, 568–569 (1990).
- M. Larsson, M. Ekberg, F. Nikolajeff, and S. Hard, “Successive-development optimization of resist kinoforms manufactured with direct writing, electron-beam lithography,” Appl. Opt. 33, 1176–1179 (1994).
- H. P. Herzig, Micro-optics: Elements, Systems and Applications (Taylor & Francis, London, 1997).
- G. J. Swanson and W. B. Weldkamp, “High-efficiency, multilevel, diffractive optical elements,” US. patent 4,895,790 (23 January 1987).
- W. H. Welch, J. E. Morris, and M. R. Feldman, “Iterative discrete on-axis encoding of radially symmetric computer-generated holograms,” J. Opt. Soc. Am. A 10, 1729–1738 (1993).
- M. Kuittinen and H. P. Herzig, “Encoding of efficient diffractive microlenses,” Opt. Lett. 20, 2156–2158 (1995).
- J. Fan, D. Zaleta, K. S. Urquhart, and S. H. Lee, “Efficient encoding algorithms for computer-aided design of diffractive optical elements by the use of electron-beam fabrication,” Appl. Opt. 34, 2522–2533 (1995).
- C. Chen and A. A. Sawchuk, “Nonlinear least-squares and phase-shifting quantization methods for diffractive optical element design,” Appl. Opt. 36, 7297–7306 (1997).
- V. Arrizon and S. Sinzinger, “Modified quantization schems for Fourier-type array generator,” Opt. Commun. 140, 309–315 (1997).
- A. K. Jain, Fundamentals of Digital Image Processing (Prentice-Hall, Engelwood Cliffs, N.J., 1989).
- J. Max, “Quantizing for minimum distortion,” IEEE Trans. Inf. Theory IT-6, 7–12 (1960).
- N. C. Gallagher, “Optimum quantization in digital holography,” Appl. Opt. 17, 109–115 (1978).
- G. J. Swanson and W. B. Weldkamp, “Diffractive optical elements for use in infrared systems,” Opt. Eng. 28, 605–608 (1989).
- R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

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