OSA's Digital Library

Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • Vol. 17, Iss. 12 — Dec. 1, 2000
  • pp: 2301–2314

Covariant discretization of axis-symmetric linear optical systems

Natig M. Atakishiyev, Shakir M. Nagiyev, Luis Edgar Vicent, and Kurt Bernardo Wolf  »View Author Affiliations

JOSA A, Vol. 17, Issue 12, pp. 2301-2314 (2000)

View Full Text Article

Acrobat PDF (292 KB)

Browse Journals / Lookup Meetings

Browse by Journal and Year


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools



We propose a discretization strategy for systems with axial symmetry. This strategy replaces the continuous position coordinates by a discrete set of sensor points, on which the discrete wave fields transform covariantly with the group of 2×2 symplectic matrices. We examine polar arrays of sensors (i.e., numbered by radius and angle) and find the complete, orthonormal sets of discrete-waveguide Meixner functions; when the sensors come closer together, these tend to the Laguerre eigenmodes of the continuous waveguide. In particular, the fractional Hankel transforms are discretized in order to define the fractional Hankel–Meixner transforms and similarly for all axis-symmetric linear optical maps. Coherent states appear in the discrete cylindrical waveguide. Covariant discretization leads to the same Wigner phase-space function for both the discrete and the continuum cases. This reinforces a Lie-theoretical model for the phase space of discrete systems.

© 2000 Optical Society of America

OCIS Codes
(000.3860) General : Mathematical methods in physics
(070.4560) Fourier optics and signal processing : Data processing by optical means
(200.4560) Optics in computing : Optical data processing
(350.6980) Other areas of optics : Transforms

Natig M. Atakishiyev, Shakir M. Nagiyev, Luis Edgar Vicent, and Kurt Bernardo Wolf, "Covariant discretization of axis-symmetric linear optical systems," J. Opt. Soc. Am. A 17, 2301-2314 (2000)

Sort:  Author  |  Year  |  Journal  |  Reset


  1. M. Moshinsky, T. H. Seligman, and K. B. Wolf, “Canonical transformations and the radial oscillator and Coulomb problems,” J. Math. Phys. 13, 901–907 (1972).
  2. V. Bargmann, “Unitary irreducible representations of the Lorentz group,” Ann. Math. 48, 568–640 (1947).
  3. N. M. Atakishiyev and K. B. Wolf, “Fractional Fourier–Kravchuk transform,” J. Opt. Soc. Am. A 14, 1467–1477 (1997).
  4. M. Krawtchouk, “Sur une générelization des polinômes d’Hermite,” C. R. Acad. Sci. Paris 189, 620–622 (1929); A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions (McGraw-Hill, New York, 1953), Vol. 2.
  5. N. M. Atakishiyev and S. K. Suslov, “Difference analogs of the harmonic oscillator,” Theor. Math. Phys. 85, 1055–1062 (1991); N. M. Atakishiyev and K. B. Wolf, “Approximation on a finite set of points through Kravchuk functions,” Rev. Mex. Fis. 40, 366–377 (1994).
  6. S. M. Candel, “An algorithm for the Fourier–Bessel transform,” Comput. Phys. Commun. 23, 343–352 (1981); S. M. Candel, “Simultaneous calculation of Fourier–Bessel transforms up to order N,” Comput. Phys. Commun. 44, 243–250 (1981); J. D. Secada, “Numerical evaluation of the Hankel transform,” Comput. Phys. Commun. CPHCBZ 116, 278–294 (1999).
  7. N. M. Atakishiyev, E. I. Jafarov, Sh. M. Nagiyev, and K. B. Wolf, “Meixner oscillators,” Rev. Mex. Fis. 44, 135–244 (1998).
  8. W. Miller, Jr., Symmetry and Separation of Variables (Addison-Wesley, Reading, Mass., 1978); E. G. Kalnins, Separation of Variables for Riemannian Spaces of Constant Curvature (Wiley, New York, 1986).
  9. R. Simon and K. B. Wolf, “The structure of paraxial optical systems,” J. Opt. Soc. Am. A 17, 342–355 (2000).
  10. N. Wiener, The Fourier Integral and Certain of Its Applications (Cambridge U. Press, Cambridge, UK, 1933).
  11. K. B. Wolf, Integral Transforms in Science and Engineering (Plenum, New York, 1979).
  12. K. B. Wolf, “Canonical transforms. II. Complex radial transforms,” J. Math. Phys. 15, 2101–2111 (1974).
  13. D. Basu and K. B. Wolf, “The unitary irreducible representations of SL(2, R) in all subgroup reductions,” J. Math. Phys. 23, 189–205 (1982).
  14. S. Flügge, Practical Quantum Mechanics (Springer-Verlag, Berlin, 1994).
  15. K. B. Wolf, “Equally-spaced energy spectra: the harmonic oscillator with a centrifugal barrier or with a centripetal well,” Kinam 3, 323–346 (1981); “Canonical transformations to phase variables in quantum oscillator systems. A group-theoretic solution,” 4, 293–332 (1982).
  16. J. Ojeda-Castañeda and A. Noyola-Iglesias, “Nondiffracting wavefields in GRIN and free-space,” Microwave Opt. Technol. Lett. 3, 430–433 (1990).
  17. L. D. Landau and E. Lifshitz, Quantum Mechanics (Addison-Wesley, Reading, Mass., 1968).
  18. K. B. Wolf, “Canonical transforms. IV. Hyperbolic transforms: continuous series of SL(2, R) representations,” J. Math. Phys. 21, 680–688 (1980).
  19. J. Meixner, “Orthogonale Polynomsysteme mit einer besonderen Gestalt der erzeugenden Funktion,” J. London Math. Soc. 9, 6–13 (1934).
  20. V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” J. Inst. Math. Appl. 25, 241–265 (1980); V. Namias, “Fractionalization of Hankel transforms,” J. Inst. Math. Appl. 26, 187–197 (1980).
  21. J.-P. Aubin, Applied Functional Analysis (Wiley-Interscience, New York, 1979); K. E. Atkinson, An Introduction to Numerical Analysis (Wiley, New York, 1989).
  22. A. M. Perelomov, Generalized Coherent States and Their Applications (Springer-Verlag, Berlin, 1986).
  23. A. O. Barut and L. Girardello, “New ‘coherent’ states associated with non-compact groups,” Commun. Math. Phys. 21, 41–55 (1971).
  24. N. M. Atakishiyev, L. E. Vicent, and K. B. Wolf, “Continuous vs. discrete fractional Fourier transforms,” J. Comput. Appl. Math. 107, 73–95 (1999).
  25. W. K. Wootters, “A Wigner-function formulation of finite-state quantum mechanics,” Ann. Phys. 176, 1–21 (1987).
  26. T. Hakioğlu, “Finite-dimensional Schwinger basis, deformed symmetries, Wigner function, and an algebraic approach to quantum phase,” J. Phys. A 31, 6975–6994 (1998); “Linear canonical transformations and quantum phase: a unified canonical and algebraic approach,” 32, 4111–4130 (1999); T. Hakioğlu and E. Tepedelenlioğlu, “The action-angle Wigner function: a discrete and algebraic phase space formalism,” J. Phys. A (to be published).
  27. E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932); H.-W. Lee, “Theory and application of the quantum phase-space distribution functions,” Phys. Rep. 259, 147–211 (1995); H. O. Bartelt, K.-H. Brenner, and H. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. OPCOB8 32, 32–38 (1980).
  28. D. D. Holm and K. B. Wolf, “Lie–Poisson description of Hamiltonian ray optics,” Physica D 51, 189–199 (1991).
  29. N. M. Atakishiyev, S. M. Chumakov, and K. B. Wolf, “Wigner distribution function for finite systems,” J. Math. Phys. 39, 6247–6261 (1998).
  30. S. T. Ali, N. M. Atakishiyev, S. M. Chumakov, and K. B. Wolf, “The Wigner function for general Lie groups and the wavelet transform,” Ann. Inst. Henri Poincaré Phys. Theor. 1, 685–714 (2000).
  31. G. S. Agarwal, “Relation between atomic coherent-state representation, state multipoles, and generalized phase-space distributions,” Phys. Rev. A 24, 2889–2896 (1981); J. C. Várilly and J. M. Gracia-Bondía, “The Moyal representation for spin,” Ann. Phys. (N.Y.) 190, 107–148 (1989); J. P. Dowling, G. S. Agarwal, and W. P. Schleich, “Wigner distribution of a general angular-momentum state: applications to a collection of two-level atoms,” Phys. Rev. A PLRAAN 49, 4101–4109 (1994).
  32. S. M. Chumakov, A. B. Klimov, and K. B. Wolf, “Connection between two Wigner functions for spin systems,” Phys. Rev. A 61, 034101–1–034101–3 (2000).
  33. S. M. Chumakov, A. Frank, and K. B. Wolf, “Finite Kerr medium: Schrödinger cats and Wigner functions on the sphere,” Phys. Rev. A 60, 1817–1823 (1999).
  34. M. Toller, “Three-dimensional Lorentz group and harmonic analysis of the scattering amplitude,” Nuovo Cimento 37, 631–657 (1965); J. F. Boyce, “Relation of the O(2, 1) partial-wave expansion to the Regge representation,” J. Math. Phys. 8, 675–684 (1967); M. Toller, “An expansion of the scattering amplitude at vanishing four-momentum transfer using the representations of the Lorentz group,” Nuovo Cimento ZZZZZZ 53, 671–715 (1968).
  35. M. Abramowitz and I. Stegun, eds., Handbook of Mathematical Functions, Vol. 55 of Applied Mathematics Series (National Bureau of Standards, Washington, D.C., 1964).

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited