Calculation of eigenfunctions of a bounded fractional Fourier transform
M.S. Kirilenko, R.O. Zubtsov, S.N. Khonina


Image Processing Systems Institute, Russian Academy of Sciences,

Samara State Aerospace University

Full text of article: Russian language.


In this paper we consider the use of a one-dimensional fractional Fourier transform for gradient-index optical waveguides. We calculate eigenfunctions of the transform in view of a limited range in the spatial and spectral domain.

fractional Fourier transform, bounded paraxial operator, eigenfunctions, Hermite-Gaussian modes, spheroidal wave functions.

Kirilenko MS, Zubtsov RO, Khonina SN. Calculation of eigenfunctions of a bounded fractional Fourier transform. Computer Optics 2015; 39(3): 332-8. DOI: 10.18287/0134-2452-2015-39-3-332-338.


  1. Namias V. The fractional  Fourier  transform  and  its  application  in quantum mechanics. J Inst Math Appl 1980; 25: 241-65.
  2. Abet S, Sheridant JT. Generalization of the fractional Fourier transformation to an arbitrary linear lossless transformation: an operator approach. J Phys A: Math Gen 1994; 27: 4179-87.
  3. Alieva T, Bastiaans MJ, Calvo ML. Fractional transforms in optical information processing. EURASIP J Appl Signal Processing 2005; 10: 1-22.
  4. Dorsch RG, Lohmann AW. Fractional Fourier transform used for a lens-design problem. Appl Opt 1995; 34(20): 4111-2.
  5. Cai LZ, Wang YQ. Optical implementation of scale invariant fractional Fourier transform of continuously variable orders with a two-lens system. Opt Laser Technol 2002; 34: 249-52.
  6. Malutin ��. Use of fractional Fourier-transformation in π/2-converters of laser modes. Quantum Electronics 2004; 2: 165-71.
  7. Hahn J. Optical implementation of iterative  fractional  Fourier transform algorithm. Opt Expr 2006; 14(23): 11103-12.
  8. Khonina SN, Karpeev SV, Ustinov AV. Functional enhancement of mode astigmatic converters �n the basis of application of diffractive optical elements. News of the Samara Science Center of the RAS 2009; 11(5): 13-23.
  9. Abramochkin E, Volostnikov V. Beams transformations and nontransformed beams. Opt Commun 1991; 83: 123-35.
  10. Beijersbergen MW, Allen L, van der Veen H.E.L.O., Woerdman JP. Astigmatic laser mode converters and transfer of orbital angular momentum. Opt Commun 1993; 96: 123-32.
  11. Ozaktas HM, Mendlovic D. Fourier transforms of fractional order and their optical interpretation. Opt Commun 1993; 101: 163-9.
  12. Mendlovic D, Ozaktas HM. Fractional Fourier transforms and their optical implementation: I. J Opt Soc Am A 1993; 10(9): 1875-81.
  13. Goodman JW. Introduction to Fourier optics. McGraw-Hill, 1996.
  14. McMullin JN. The ABCD matrix in arbitrarily tapered quadratic-index waveguides. Appl Opt 1986; 25: 2184.
  15. Striletz AS, Khonina SN. Matching and investigation methods based on differential and integral operators of laser radiation propagation in a medium with small inhomogeneities. Computer Optics 2008; 32(1): 33-8.
  16. Yariv A. Quantum Electronics. 2nd ed. New York: Wiley, 1975.
  17. Lanczos C. Linear Diffrential Operators. London: Van Nostrand, 1961.
  18. Slepian D, Sonnenblick E. Eigenvalues associated with prolate spheroidal wave functions of zero order. The Bell System Technical Journal 1965; 44: 1745-63.
  19. Khonina SN. Approximation of spheroidal wave functions by finite series. Computer Optics 1999; 19: 65-70.
  20. Khonina SN, Volotovskii SG, Soifer VA. A method of eigenvalue calculation of the zero order prolate spheroidal functions. Reports of the Russian Academy of Sciences 2001; 376(1): 30-2.
  21. Volotovskii SG, Kazanskii NL, Khonina SN. Analysis and development of the methods for calculating eigenvalues of prolate spheroidal functions of zero order. Pattern Recognition and Image Analysis 2001; 11(3): 633-48.
  22. Khonina SN. Investigation of the matrix method of the zero order prolate spheroidal functions calculating. News of the Samara Science Center of the RAS 2001; 3(1): 111-7.
  23. Slepian D, Pollak HO. Prolate spheroidal wave functions, Fourier analysis and uncertainty – I. Bell Syst Technol J 1961, 40: 43-63.
  24. Brovarova MA, Khonina SN. Increasing resolution using  pro­late spheroidal wave functions. Computer Optics 2001; 21: 53-7.
  25. Khonina SN, Kotlyar VV. Generating light fields matched to the spheroidal wave-function basis. Optical Memory and Neural Networks 2001; 10(4): 267-76.
  26. Kirilenko MS, Khonina SN. Coding of an optical signal by a superposition of spheroidal functions for undistorted transmission of information in the lens system. Proc SPIE 2014; 9156: 91560J.

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