Calculation of eigenfunctions for imaging two-lens system with axial symmetry
M.S. Kirilenko, S.N. Khonina

Image Processing Systems Institute, Russian Academy of Sciences,
Samara State Aerospace University

Full text of article: Russian language.

Abstract:
Eigenfunctions of the optical operator describing limited imaging system with two lenses are considered under axial symmetry. Resulting functions are analogous to the generalized spheroidal functions which are eigenfunctions of the zero-order Hankel transformation. The expression for eigenfunctions' calculation is obtained using operator representation of the optical system.
The influence of the spectrum width on the number of significant eigenvalues is investigated. Decomposition of circular, annular, and Gaussian beam by eigenfunctions is calculated and deviation from the original signals is estimated.

Key words:
optical operator, eigenfunctions, axial symmetry, Hankel transform, spectrum, two-dimensional convolution, optical signal.

References:

  1. Miller, D.A.B. Communicating with waves between volumes: evaluating orthogonal spatial channels and limits on coupling strength / D.A.B. Miller // Applied Optics. – 2000. – V. 39(11). – P. 1681-1699.
  2. Martinsson, P. Communication modes in scalar diffracti­on / P. Martinsson, P. Ma, A. Burvall, A.T. Friberg // Optik. – 2008. – V. 199(3). – P. 103-111.
  3. Khonina, S.N. Effect of diffraction on images matched with prolate spheroidal wave functions / S.N. Khonina, V.V. Kotlyar // Pattern Recognition and Image Analysis: Advances in Mathematical Theory and Applications. – 2001. – V. 11(3). – P. 521-528.
  4. Khonina, S.N. A method of eigenvalue calculation of zero-order prolate spheroidal functions / S.N. Khonina, S.G. Volotovskii, V.A. Soifer // Proceedings of the Russian Academy of Sciences. – 2001. – V. 376(1). – P. 30-32. – (In Russian).
  5. Volotovskii, S.G. Analysis and development of the methods for calculating eigenvalues of prolate spheroidal functions of zero order / S.G. Volotovskii, N.L. Kazanskii, S.N. Khonina // Pattern Recognition and Image Analysis: Advances in Mathematical Theory and Applications. – 2001. – V. 11(2). – P. 473-475.
  6. Khonina, S.N. А finite series approximation of spheroidal wave functions / S.N. Khonina // Computer Optics. – 1999. – V. 19. – P. 65-70. – (In Russian).
  7. Frieden, B.R. Evaluation, design and extrapolation methods for optical signals / B.R. Frieden // Progress in Optics. – 1971. – V. IX. – P. 311-407.
  8. Slepian, D. Prolate spheroidal wave functions, Fourier analysis and uncertainty – I / D. Slepian, H.O. Pollak // Bell System Technical Journal. – 1961. – V. 40(1). – P. 43-63.
  9. Landau, H.J. Prolate spheroidal wave functions, Fourier analysis and uncertainty – II / H.J. Landau, H.O. Pollak / Bell System Technical Journal. – 1961. – V. 40(1). – P. 65-84.
  10. Komarov, I.V. Spheroidal and Coulomb spheroidal functions / I.V. Komarov, L.I. Ponomarev, S.Yu. Slavyznov; Ed. by V.S. Bul­dyrev. – Moscow: “Nauka” Publisher, 1976. – 320 p. – (In Russian).
  11. Kirilenko, M.S. Forming of an optical signal matched with spheroidal functions for undistorted transmission in the lens system / M.S. Kirilenko, S.N. Khonina // The News of Samara Science Center of RAS. – 2013. – V. 15(6). – P. 31-34. – (In Russian).
  12. Kirilenko, M.S. Coding of an optical signal by a superposition of spheroidal functions for undistorted transmission of information in the lens system / M.S. Kirilenko, S.N. Khonina // Proceedings of SPIE. – 2014. – V. 9156. – P. 91560J (8 pp.).
  13. Pich´e, K. Experimental realization of optical eigenmode super-resolution / K. Pich´e, J. Leach, A.S. Johnson, J.Z. Salvail, M.I. Kolobov, R.W. Boyd // Optics Express. – 2012. – V. 20(24). – P. 26424-26433.
  14. Voroncov, M.A. Principles of adaptive Optics / M.A. Voron­cov, V.I. Shmalgauzen. – Moscow: “Nauka” Publisher, 1985. – (In Russian).
  15. Tyson, R.K. Principles of Adaptive Optics / R.K. Tyson. – CRC Press, Taylor and Francis Group, 2011.
  16. Itoh, Y. Evaluation of Aberrations Using the Generalized Prolate Spheroidal Wavefunctions / Y. Itoh // Journal of the Optical Society of America. – 1970. – V. 60(1). – P. 10-14.
  17. Vulikh, B.Z. Introduction to Functional Analysis / B.Z. Vu­likh. – Moscow: “Nauka” Publisher, 1976. – 416 p. – (In Russian) .

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