## Generalized prolate spheroidal wave functions for optical finite fractional Fourier and linear canonical transforms

JOSA A, Vol. 22, Issue 3, pp. 460-474 (2005)

http://dx.doi.org/10.1364/JOSAA.22.000460

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### Abstract

Prolate spheroidal wave functions (PSWFs) are known to be useful for analyzing the properties of the finite-extension Fourier transform (fi-FT). We extend the theory of PSWFs for the finite-extension fractional Fourier transform, the finite-extension linear canonical transform, and the finite-extension offset linear canonical transform. These finite transforms are more flexible than the fi-FT and can model much more generalized optical systems. We also illustrate how to use the generalized prolate spheroidal functions we derive to analyze the energy-preservation ratio, the self-imaging phenomenon, and the resonance phenomenon of the finite-sized one-stage or multiple-stage optical systems.

© 2005 Optical Society of America

**OCIS Codes**

(070.2580) Fourier optics and signal processing : Paraxial wave optics

(070.2590) Fourier optics and signal processing : ABCD transforms

(350.6980) Other areas of optics : Transforms

**Citation**

Soo-Chang Pei and Jian-Jiun Ding, "Generalized prolate spheroidal wave functions for optical finite fractional Fourier and linear canonical transforms," J. Opt. Soc. Am. A **22**, 460-474 (2005)

http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-22-3-460

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### References

- D. Slepian and H. O. Pollak, "Prolate spheroidal wave functions, Fourier analysis and uncertainty - I," Bell Syst. Tech. J. 40, 43-63 (1961).
- H. J. Landau and H. O. Pollak, "Prolate spheroidal wave functions, Fourier analysis and uncertainty - II," Bell Syst. Tech. J. 40, 65-84 (1961).
- H. J. Landau and H. O. Pollak, "Prolate spheroidal wave functions, Fourier analysis and uncertainty - III," Bell Syst. Tech. J. 41, 1295-1336 (1962).
- D. Slepian, "Some asymptotic expansions for prolate spheroidal wave functions," J. Math. Phys. 44, 99-140 (1965).
- D. Slepian, "Prolate spheroidal wave functions, Fourier analysis, and uncertainty - V: the discrete case," Bell Syst. Tech. J. 57, 1371-1430 (1977).
- J. H. Wilkinson, The Algebraic Eigenvalue Problem (Oxford U. Press, London, 1988).
- B. R. Frieden, "Evaluation, design and extrapolation methods for optical signals, based on use of the prolate functions," in Progress in Optics, Vol. IX , E. Wolf, ed. (North-Holland, Amsterdam, 1971), pp. 311-407.
- A. Papoulis and M. S. Bertran, "Digital filtering and prolate functions," IEEE Trans. Circuit Theory 19, 674-681 (1972).
- H. M. Ozaktas, M. A. Kutay, and Z. Zalevsky, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, New York, 2000).
- V. Namias, "The fractional order Fourier transform and its application to quantum mechanics," J. Inst. Math. Appl. 25, 241-265 (1980).
- K. B. Wolf, Integral Transforms in Science and Engineering (Plenum, New York, 1979), Chap. 9, pp. 381-416.
- L. M. Bernardo, "ABCD matrix formalism of fractional Fourier optics," Opt. Eng. (Bellingham) 35, 732-740 (1996).
- M. J. Bastiaans, "Propagation laws for the secondorder moments of the Wigner distribution function in first-order optical systems," Optik (Stuttgart) 82, 173-181 (1989).
- H. M. Ozaktas and D. Mendlovic, "Fractional Fourier optics," J. Opt. Soc. Am. A 12, 743-751 (1995).
- P. Pellat-Finet and G. Bonnet, "Fractional order Fourier transform and Fourier optics," Opt. Commun. 111, 141-154 (1994).
- S. Abe and J. T. Sheridan, "Optical operations on wave functions as the Abelian subgroups of the special affine Fourier transformation," Opt. Lett. 19, 1801-1803 (1994).
- S. C. Pei and J. J. Ding, "Eigenfunctions of the offset Fourier, fractional Fourier, and linear canonical transforms," J. Opt. Soc. Am. A 20, 522-532 (2003).
- K. Khare and N. George, "Fractional finite Fourier transform," J. Opt. Soc. Am. A 21, 1179-1185 (2004).
- J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1996).

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