Abstract

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

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  1. M. Moshinsky, T. H. Seligman, K. B. Wolf, “Canonical transformations and the radial oscillator and Coulomb problems,” J. Math. Phys. 13, 901–907 (1972).
    [CrossRef]
  2. V. Bargmann, “Unitary irreducible representations of the Lorentz group,” Ann. Math. 48, 568–640 (1947).
    [CrossRef]
  3. N. M. Atakishiyev, K. B. Wolf, “Fractional Fourier–Kravchuk transform,” J. Opt. Soc. Am. A 14, 1467–1477 (1997).
    [CrossRef]
  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, F. G. Tricomi, Higher Transcendental Functions (McGraw-Hill, New York, 1953), Vol. 2.
  5. N. M. Atakishiyev, S. K. Suslov, “Difference analogs of the harmonic oscillator,” Theor. Math. Phys. 85, 1055–1062 (1991);N. M. Atakishiyev, K. B. Wolf, “Approximation on a finite set of points through Kravchuk functions,” Rev. Mex. Fis. 40, 366–377 (1994).
    [CrossRef]
  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. 116, 278–294 (1999).
    [CrossRef]
  7. N. M. Atakishiyev, E. I. Jafarov, Sh. M. Nagiyev, K. B. Wolf, “Meixner oscillators,” Rev. Mex. Fis. 44, 135–244 (1998).
  8. W. Miller, 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, K. B. Wolf, “The structure of paraxial optical systems,” J. Opt. Soc. Am. A 17, 342–355 (2000).
    [CrossRef]
  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, K. B. Wolf, “The unitary irreducible representations of SL(2, R) in all subgroup reductions,” J. Math. Phys. 23, 189–205 (1982).
    [CrossRef]
  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,” Kinam 4, 293–332 (1982).
  16. J. Ojeda-Castañeda, A. Noyola-Iglesias, “Nondiffracting wavefields in GRIN and free-space,” Microwave Opt. Technol. Lett. 3, 430–433 (1990).
    [CrossRef]
  17. L. D. Landau, 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).
    [CrossRef]
  19. J. Meixner, “Orthogonale Polynomsysteme mit einer besonderen Gestalt der erzeugenden Funktion,” J. London Math. Soc. 9, 6–13 (1934).
    [CrossRef]
  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).
    [CrossRef]
  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, L. Girardello, “New ‘coherent’ states associated with non-compact groups,” Commun. Math. Phys. 21, 41–55 (1971).
    [CrossRef]
  24. N. M. Atakishiyev, L. E. Vicent, K. B. Wolf, “Continuous vs. discrete fractional Fourier transforms,” J. Comput. Appl. Math. 107, 73–95 (1999).
    [CrossRef]
  25. W. K. Wootters, “A Wigner-function formulation of finite-state quantum mechanics,” Ann. Phys. 176, 1–21 (1987).
    [CrossRef]
  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,” J. Phys. A 32, 4111–4130 (1999); T. Hakioğlu, E. Tepedelenlioğlu, “The action-angle Wigner function: a discrete and algebraic phase space formalism,” J. Phys. A (to be published).
    [CrossRef]
  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, H. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32–38 (1980).
    [CrossRef]
  28. D. D. Holm, K. B. Wolf, “Lie–Poisson description of Hamiltonian ray optics,” Physica D 51, 189–199 (1991).
    [CrossRef]
  29. N. M. Atakishiyev, S. M. Chumakov, K. B. Wolf, “Wigner distribution function for finite systems,” J. Math. Phys. 39, 6247–6261 (1998).
    [CrossRef]
  30. S. T. Ali, N. M. Atakishiyev, S. M. Chumakov, K. B. Wolf, “The Wigner function for general Lie groups and the wavelet transform,” Ann. Inst. Henri Poincaré Phys. Theor. 1, 685–714 (2000).
    [CrossRef]
  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, J. M. Gracia-Bondía, “The Moyal representation for spin,” Ann. Phys. (N.Y.) 190, 107–148 (1989);J. P. Dowling, G. S. Agarwal, W. P. Schleich, “Wigner distribution of a general angular-momentum state: applications to a collection of two-level atoms,” Phys. Rev. A 49, 4101–4109 (1994).
    [CrossRef] [PubMed]
  32. S. M. Chumakov, A. B. Klimov, K. B. Wolf, “Connection between two Wigner functions for spin systems,” Phys. Rev. A 61, 034101-1–034101-3 (2000).
    [CrossRef]
  33. S. M. Chumakov, A. Frank, K. B. Wolf, “Finite Kerr medium: Schrödinger cats and Wigner functions on the sphere,” Phys. Rev. A 60, 1817–1823 (1999).
    [CrossRef]
  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 53, 671–715 (1968).
    [CrossRef]
  35. M. Abramowitz, I. Stegun, eds., Handbook of Mathematical Functions, Vol. 55 of Applied Mathematics Series (National Bureau of Standards, Washington, D.C., 1964).

2000 (3)

R. Simon, K. B. Wolf, “The structure of paraxial optical systems,” J. Opt. Soc. Am. A 17, 342–355 (2000).
[CrossRef]

S. T. Ali, N. M. Atakishiyev, S. M. Chumakov, K. B. Wolf, “The Wigner function for general Lie groups and the wavelet transform,” Ann. Inst. Henri Poincaré Phys. Theor. 1, 685–714 (2000).
[CrossRef]

S. M. Chumakov, A. B. Klimov, K. B. Wolf, “Connection between two Wigner functions for spin systems,” Phys. Rev. A 61, 034101-1–034101-3 (2000).
[CrossRef]

1999 (2)

S. M. Chumakov, A. Frank, K. B. Wolf, “Finite Kerr medium: Schrödinger cats and Wigner functions on the sphere,” Phys. Rev. A 60, 1817–1823 (1999).
[CrossRef]

N. M. Atakishiyev, L. E. Vicent, K. B. Wolf, “Continuous vs. discrete fractional Fourier transforms,” J. Comput. Appl. Math. 107, 73–95 (1999).
[CrossRef]

1998 (3)

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,” J. Phys. A 32, 4111–4130 (1999); T. Hakioğlu, E. Tepedelenlioğlu, “The action-angle Wigner function: a discrete and algebraic phase space formalism,” J. Phys. A (to be published).
[CrossRef]

N. M. Atakishiyev, E. I. Jafarov, Sh. M. Nagiyev, K. B. Wolf, “Meixner oscillators,” Rev. Mex. Fis. 44, 135–244 (1998).

N. M. Atakishiyev, S. M. Chumakov, K. B. Wolf, “Wigner distribution function for finite systems,” J. Math. Phys. 39, 6247–6261 (1998).
[CrossRef]

1997 (1)

1991 (2)

N. M. Atakishiyev, S. K. Suslov, “Difference analogs of the harmonic oscillator,” Theor. Math. Phys. 85, 1055–1062 (1991);N. M. Atakishiyev, K. B. Wolf, “Approximation on a finite set of points through Kravchuk functions,” Rev. Mex. Fis. 40, 366–377 (1994).
[CrossRef]

D. D. Holm, K. B. Wolf, “Lie–Poisson description of Hamiltonian ray optics,” Physica D 51, 189–199 (1991).
[CrossRef]

1990 (1)

J. Ojeda-Castañeda, A. Noyola-Iglesias, “Nondiffracting wavefields in GRIN and free-space,” Microwave Opt. Technol. Lett. 3, 430–433 (1990).
[CrossRef]

1987 (1)

W. K. Wootters, “A Wigner-function formulation of finite-state quantum mechanics,” Ann. Phys. 176, 1–21 (1987).
[CrossRef]

1982 (1)

D. Basu, K. B. Wolf, “The unitary irreducible representations of SL(2, R) in all subgroup reductions,” J. Math. Phys. 23, 189–205 (1982).
[CrossRef]

1981 (3)

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,” Kinam 4, 293–332 (1982).

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. 116, 278–294 (1999).
[CrossRef]

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, J. M. Gracia-Bondía, “The Moyal representation for spin,” Ann. Phys. (N.Y.) 190, 107–148 (1989);J. P. Dowling, G. S. Agarwal, W. P. Schleich, “Wigner distribution of a general angular-momentum state: applications to a collection of two-level atoms,” Phys. Rev. A 49, 4101–4109 (1994).
[CrossRef] [PubMed]

1980 (2)

K. B. Wolf, “Canonical transforms. IV. Hyperbolic transforms: continuous series of SL(2, R) representations,” J. Math. Phys. 21, 680–688 (1980).
[CrossRef]

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).
[CrossRef]

1974 (1)

K. B. Wolf, “Canonical transforms. II. Complex radial transforms,” J. Math. Phys. 15, 2101–2111 (1974).

1972 (1)

M. Moshinsky, T. H. Seligman, K. B. Wolf, “Canonical transformations and the radial oscillator and Coulomb problems,” J. Math. Phys. 13, 901–907 (1972).
[CrossRef]

1971 (1)

A. O. Barut, L. Girardello, “New ‘coherent’ states associated with non-compact groups,” Commun. Math. Phys. 21, 41–55 (1971).
[CrossRef]

1965 (1)

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 53, 671–715 (1968).
[CrossRef]

1947 (1)

V. Bargmann, “Unitary irreducible representations of the Lorentz group,” Ann. Math. 48, 568–640 (1947).
[CrossRef]

1934 (1)

J. Meixner, “Orthogonale Polynomsysteme mit einer besonderen Gestalt der erzeugenden Funktion,” J. London Math. Soc. 9, 6–13 (1934).
[CrossRef]

1932 (1)

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, H. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32–38 (1980).
[CrossRef]

1929 (1)

M. Krawtchouk, “Sur une générelization des polinômes d’Hermite,” C. R. Acad. Sci. Paris 189, 620–622 (1929); A. Erdélyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Higher Transcendental Functions (McGraw-Hill, New York, 1953), Vol. 2.

Agarwal, G. S.

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, J. M. Gracia-Bondía, “The Moyal representation for spin,” Ann. Phys. (N.Y.) 190, 107–148 (1989);J. P. Dowling, G. S. Agarwal, W. P. Schleich, “Wigner distribution of a general angular-momentum state: applications to a collection of two-level atoms,” Phys. Rev. A 49, 4101–4109 (1994).
[CrossRef] [PubMed]

Ali, S. T.

S. T. Ali, N. M. Atakishiyev, S. M. Chumakov, K. B. Wolf, “The Wigner function for general Lie groups and the wavelet transform,” Ann. Inst. Henri Poincaré Phys. Theor. 1, 685–714 (2000).
[CrossRef]

Atakishiyev, N. M.

S. T. Ali, N. M. Atakishiyev, S. M. Chumakov, K. B. Wolf, “The Wigner function for general Lie groups and the wavelet transform,” Ann. Inst. Henri Poincaré Phys. Theor. 1, 685–714 (2000).
[CrossRef]

N. M. Atakishiyev, L. E. Vicent, K. B. Wolf, “Continuous vs. discrete fractional Fourier transforms,” J. Comput. Appl. Math. 107, 73–95 (1999).
[CrossRef]

N. M. Atakishiyev, S. M. Chumakov, K. B. Wolf, “Wigner distribution function for finite systems,” J. Math. Phys. 39, 6247–6261 (1998).
[CrossRef]

N. M. Atakishiyev, E. I. Jafarov, Sh. M. Nagiyev, K. B. Wolf, “Meixner oscillators,” Rev. Mex. Fis. 44, 135–244 (1998).

N. M. Atakishiyev, K. B. Wolf, “Fractional Fourier–Kravchuk transform,” J. Opt. Soc. Am. A 14, 1467–1477 (1997).
[CrossRef]

N. M. Atakishiyev, S. K. Suslov, “Difference analogs of the harmonic oscillator,” Theor. Math. Phys. 85, 1055–1062 (1991);N. M. Atakishiyev, K. B. Wolf, “Approximation on a finite set of points through Kravchuk functions,” Rev. Mex. Fis. 40, 366–377 (1994).
[CrossRef]

Aubin, J.-P.

J.-P. Aubin, Applied Functional Analysis (Wiley-Interscience, New York, 1979); K. E. Atkinson, An Introduction to Numerical Analysis (Wiley, New York, 1989).

Bargmann, V.

V. Bargmann, “Unitary irreducible representations of the Lorentz group,” Ann. Math. 48, 568–640 (1947).
[CrossRef]

Barut, A. O.

A. O. Barut, L. Girardello, “New ‘coherent’ states associated with non-compact groups,” Commun. Math. Phys. 21, 41–55 (1971).
[CrossRef]

Basu, D.

D. Basu, K. B. Wolf, “The unitary irreducible representations of SL(2, R) in all subgroup reductions,” J. Math. Phys. 23, 189–205 (1982).
[CrossRef]

Candel, S. M.

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. 116, 278–294 (1999).
[CrossRef]

Chumakov, S. M.

S. T. Ali, N. M. Atakishiyev, S. M. Chumakov, K. B. Wolf, “The Wigner function for general Lie groups and the wavelet transform,” Ann. Inst. Henri Poincaré Phys. Theor. 1, 685–714 (2000).
[CrossRef]

S. M. Chumakov, A. B. Klimov, K. B. Wolf, “Connection between two Wigner functions for spin systems,” Phys. Rev. A 61, 034101-1–034101-3 (2000).
[CrossRef]

S. M. Chumakov, A. Frank, K. B. Wolf, “Finite Kerr medium: Schrödinger cats and Wigner functions on the sphere,” Phys. Rev. A 60, 1817–1823 (1999).
[CrossRef]

N. M. Atakishiyev, S. M. Chumakov, K. B. Wolf, “Wigner distribution function for finite systems,” J. Math. Phys. 39, 6247–6261 (1998).
[CrossRef]

Flügge, S.

S. Flügge, Practical Quantum Mechanics (Springer-Verlag, Berlin, 1994).

Frank, A.

S. M. Chumakov, A. Frank, K. B. Wolf, “Finite Kerr medium: Schrödinger cats and Wigner functions on the sphere,” Phys. Rev. A 60, 1817–1823 (1999).
[CrossRef]

Girardello, L.

A. O. Barut, L. Girardello, “New ‘coherent’ states associated with non-compact groups,” Commun. Math. Phys. 21, 41–55 (1971).
[CrossRef]

Hakioglu, T.

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,” J. Phys. A 32, 4111–4130 (1999); T. Hakioğlu, E. Tepedelenlioğlu, “The action-angle Wigner function: a discrete and algebraic phase space formalism,” J. Phys. A (to be published).
[CrossRef]

Holm, D. D.

D. D. Holm, K. B. Wolf, “Lie–Poisson description of Hamiltonian ray optics,” Physica D 51, 189–199 (1991).
[CrossRef]

Jafarov, E. I.

N. M. Atakishiyev, E. I. Jafarov, Sh. M. Nagiyev, K. B. Wolf, “Meixner oscillators,” Rev. Mex. Fis. 44, 135–244 (1998).

Klimov, A. B.

S. M. Chumakov, A. B. Klimov, K. B. Wolf, “Connection between two Wigner functions for spin systems,” Phys. Rev. A 61, 034101-1–034101-3 (2000).
[CrossRef]

Krawtchouk, M.

M. Krawtchouk, “Sur une générelization des polinômes d’Hermite,” C. R. Acad. Sci. Paris 189, 620–622 (1929); A. Erdélyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Higher Transcendental Functions (McGraw-Hill, New York, 1953), Vol. 2.

Landau, L. D.

L. D. Landau, E. Lifshitz, Quantum Mechanics (Addison-Wesley, Reading, Mass., 1968).

Lifshitz, E.

L. D. Landau, E. Lifshitz, Quantum Mechanics (Addison-Wesley, Reading, Mass., 1968).

Meixner, J.

J. Meixner, “Orthogonale Polynomsysteme mit einer besonderen Gestalt der erzeugenden Funktion,” J. London Math. Soc. 9, 6–13 (1934).
[CrossRef]

Miller, W.

W. Miller, 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).

Moshinsky, M.

M. Moshinsky, T. H. Seligman, K. B. Wolf, “Canonical transformations and the radial oscillator and Coulomb problems,” J. Math. Phys. 13, 901–907 (1972).
[CrossRef]

Nagiyev, Sh. M.

N. M. Atakishiyev, E. I. Jafarov, Sh. M. Nagiyev, K. B. Wolf, “Meixner oscillators,” Rev. Mex. Fis. 44, 135–244 (1998).

Namias, V.

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).
[CrossRef]

Noyola-Iglesias, A.

J. Ojeda-Castañeda, A. Noyola-Iglesias, “Nondiffracting wavefields in GRIN and free-space,” Microwave Opt. Technol. Lett. 3, 430–433 (1990).
[CrossRef]

Ojeda-Castañeda, J.

J. Ojeda-Castañeda, A. Noyola-Iglesias, “Nondiffracting wavefields in GRIN and free-space,” Microwave Opt. Technol. Lett. 3, 430–433 (1990).
[CrossRef]

Perelomov, A. M.

A. M. Perelomov, Generalized Coherent States and Their Applications (Springer-Verlag, Berlin, 1986).

Seligman, T. H.

M. Moshinsky, T. H. Seligman, K. B. Wolf, “Canonical transformations and the radial oscillator and Coulomb problems,” J. Math. Phys. 13, 901–907 (1972).
[CrossRef]

Simon, R.

Suslov, S. K.

N. M. Atakishiyev, S. K. Suslov, “Difference analogs of the harmonic oscillator,” Theor. Math. Phys. 85, 1055–1062 (1991);N. M. Atakishiyev, K. B. Wolf, “Approximation on a finite set of points through Kravchuk functions,” Rev. Mex. Fis. 40, 366–377 (1994).
[CrossRef]

Toller, M.

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 53, 671–715 (1968).
[CrossRef]

Vicent, L. E.

N. M. Atakishiyev, L. E. Vicent, K. B. Wolf, “Continuous vs. discrete fractional Fourier transforms,” J. Comput. Appl. Math. 107, 73–95 (1999).
[CrossRef]

Wiener, N.

N. Wiener, The Fourier Integral and Certain of Its Applications (Cambridge U. Press, Cambridge, UK, 1933).

Wigner, E.

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, H. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32–38 (1980).
[CrossRef]

Wolf, K. B.

S. M. Chumakov, A. B. Klimov, K. B. Wolf, “Connection between two Wigner functions for spin systems,” Phys. Rev. A 61, 034101-1–034101-3 (2000).
[CrossRef]

S. T. Ali, N. M. Atakishiyev, S. M. Chumakov, K. B. Wolf, “The Wigner function for general Lie groups and the wavelet transform,” Ann. Inst. Henri Poincaré Phys. Theor. 1, 685–714 (2000).
[CrossRef]

R. Simon, K. B. Wolf, “The structure of paraxial optical systems,” J. Opt. Soc. Am. A 17, 342–355 (2000).
[CrossRef]

N. M. Atakishiyev, L. E. Vicent, K. B. Wolf, “Continuous vs. discrete fractional Fourier transforms,” J. Comput. Appl. Math. 107, 73–95 (1999).
[CrossRef]

S. M. Chumakov, A. Frank, K. B. Wolf, “Finite Kerr medium: Schrödinger cats and Wigner functions on the sphere,” Phys. Rev. A 60, 1817–1823 (1999).
[CrossRef]

N. M. Atakishiyev, S. M. Chumakov, K. B. Wolf, “Wigner distribution function for finite systems,” J. Math. Phys. 39, 6247–6261 (1998).
[CrossRef]

N. M. Atakishiyev, E. I. Jafarov, Sh. M. Nagiyev, K. B. Wolf, “Meixner oscillators,” Rev. Mex. Fis. 44, 135–244 (1998).

N. M. Atakishiyev, K. B. Wolf, “Fractional Fourier–Kravchuk transform,” J. Opt. Soc. Am. A 14, 1467–1477 (1997).
[CrossRef]

D. D. Holm, K. B. Wolf, “Lie–Poisson description of Hamiltonian ray optics,” Physica D 51, 189–199 (1991).
[CrossRef]

D. Basu, K. B. Wolf, “The unitary irreducible representations of SL(2, R) in all subgroup reductions,” J. Math. Phys. 23, 189–205 (1982).
[CrossRef]

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,” Kinam 4, 293–332 (1982).

K. B. Wolf, “Canonical transforms. IV. Hyperbolic transforms: continuous series of SL(2, R) representations,” J. Math. Phys. 21, 680–688 (1980).
[CrossRef]

K. B. Wolf, “Canonical transforms. II. Complex radial transforms,” J. Math. Phys. 15, 2101–2111 (1974).

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Figures (4)

Fig. 1
Fig. 1

Two-dimensional arrays of sensors admitting separation in (a) Cartesian and (b), (c), polar coordinates, further characterized in the text. Arrays (a) and (c) exhibit a homogeneous density of sensors on the plane. The arrays considered in this paper are of type (c).

Fig. 2
Fig. 2

Stratum surfaces of the generators Jx of Sp(2, R). Operators that lie on the same surface can be mapped onto each other by Sp(2, R) transformations. The J(ζ) are obtained from J0 through boosts generated by J2; the heavy curve marked ζ is the manifold of Meixner transforms Mζ. The fractional Hankel–Meixner transforms Hα are rotations, generated by J0, around the vertical axis of the figure by πα.

Fig. 3
Fig. 3

As ζ grows, J(ζ) asymptotically approaches the direction of J-; a change of scale by 2e-ζ is needed to make the vectors K(ζ) coincide with the desired limit.

Fig. 4
Fig. 4

Successive refinements of the set of orthogonality points nƵ0+ (filled points on the dashed curves) of the Meixner functions ψn(k;0, ζ)(n), n=(1/4)eζrn2, approximate the radial waveguide functions Ψm,nH(r) (solid curve). From bottom to top, the radial modes n=0, 1, 2, 3, 4. In each plot there are successive refinements corresponding to ζ=1, 3, 5, . (a) m=0 [k=(1/2)], (b) m=±1 (k=1), (c) m=±2 [k=(3/2)].

Equations (92)

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Sp(4, R)  Sp(2, R)xSp(2, R)y,
Sp(4, R)  SO(2)θSp(2, R)r,
qx=r cos θ,qy=r sin θ,rR0+,θS1,
rψ(r, θ)=12πm=-ψmH(r)eimθ,
ψmH(r)=12πS1dθrψ(r, θ)e-imθ.
(ϕ, ψ)L2=R0+drϕ(r)*ψ(r)
(F: ψ)(q)=12πR2dq exp(-iq·q)ψ(q).
exp(iq·q)=exp[irr cos(θ-θ)]=mƵimJm(rr)exp[im(θ-θ)],
(F : ψ)(r, θ)=mƵeimθ2πr(-i)m(Hm : ψmH)(r),
(Hm : ϕ)(r)=imr2π1/2S1dθe-imθ(F : ϕ)(r, θ)=R0+drHm1(r, r)ϕ(r),
Hm1(r, r)=rrJm(rr),
[C(M) : ψ](q)=R2dqCM(q, q)ψ(q),
M=abcd,
CM(q, q)=12πibexp id2b|q|2-q·qb+a2b|q|2.
[C(M) : ψ](r, θ)=mƵeimθ2πr[C(M)|m : ψmH](r),
[C(M)|m : ϕ](r)=R0+drCM|m|(r, r)ϕ(r),
CM|m|(r, r)=1i|m|+1brrexpid2br2J|m|rrb×expia2br2.
Ca0ca-1m: ϕ(r)=(sign a)m|a|expic2ar2ϕr|a|.
R0+drCM1|m|(r, r)CM2|m|(r, r)=CM1M2|m|(r, r).
Hmα=exp[i12π(|m|+1)α]C(Hα)|m,
Hα=cos12παsin12πα-sin12παcos12πα,
Hmα(r, r)=1sin (π/2)αrr expi12(r2+r2)×cotπ2αJmrrsin(π/2)α.
Hmα=exp[i12π(|m|+1)α]exp(-i12παHˆm),
limα02iπα{[C(Hα)|m-1] : ϕ}(r)=Hˆmϕ(r)=12-2r2+m2-(1/4)r2+r2ϕ(r).
HˆmΨm,nH(r)=En|m|Ψm,nH(r),
En|m|=2n+|m|+1,nƵ0+={0, 1, 2,},
Ψm,nH(r)=2n!(|m|+n)!1/2e-r2/2r|m|+1/2Ln|m|(r2).
(Hmα : Ψm,nH)(r)=exp[i12π(|m|+1-En|m|)α]Ψm,nH(r)=exp(-iπnα)Ψm,nH(r).
exp[iθJˆ0(m)]=Ccos(θ/2)-sin(θ/2)sin(θ/2)cos(θ/2)m
  Jˆ0(m)=14-2r2+m2-(1/4)r2+r2,
exp[iηJˆ1(m)]=Ccosh(η/2)-sinh(η/2)-sinh(η/2)cosh(η/2)m
  Jˆ1(m)=14-2r2+m2-(1/4)r2-r2,
exp[iζJˆ2(m)]=Ce-ζ/200eζ/2m
  Jˆ2(m)=-i2rr+12,
exp[ibJˆ+(m)]=C1-b01m
  Jˆ+(m)=Jˆ0(m)+Jˆ1(m)
=12-2r2+m2-(1/4)r2,
exp[icJˆ-(m)]=C10c1m Jˆ-(m)=Jˆ0(m)-Jˆ1(m)=12r2.
[Jˆ0(m), Jˆ1(m)]=iJˆ2(m),[Jˆ1(m), Jˆ2(m)]=-iJˆ0(m),
[Jˆ2(m), Jˆ0(m)]=iJˆ1(m).
Jˆ(m)2=Jˆ1(m)2+Jˆ2(m)2-Jˆ0(m)2=12(1-m2)12.
|m|=2k-1,k=12(|m|+1){12, 1,32, 2,}.
Dn,nk(M)=(Ψm,nH, Cabcd : Ψm,nH)L2
=[(d-a)-i(b+c)]n[(a-d)-i(b+c)]n[(a+d)+i(b-c)]-2k-n-n[n!n!Γ(2k+n)Γ(2k+n)]1/222kΓ(2k+n+n)×F-n,-n1-2k-n-n;a2+b2+c2+d2+2a2+b2+c2+d2-2,
Dn,nkcos θsin θ-sin θcos θ=exp[-2i(k+n)θ]δn,n.
[C(M)|k : ψ](n)=nƵ0+Dn,nk(M)ψ(n),
nƵ0+Dn,nk(M1)Dn,nk(M2)=Dn,nk(M1M2).
(ϕH, ψH)l2=nƵ0+ϕnH*ψnH.
J¯0(k)ψ(n)=(k+n)ψ(n),
J¯1(k)ψ(n)=12μ(k, n)ψ(n+1)+12μ(k, n-1)ψ(n-1),
J¯2(k)ψ(n)=12μ(k, n)ψ(n+1)-12iμ(k, n-1)ψ(n-1).
μ(k, n)=[(n+1)(2k+n)]1/2.
(H2k-1α : ψ)(n)=eiπkα[C(Hα)|k : ψ](n)=eiπkαnƵ0+Dn,nkcos12παsin12πα-sin12παcos12πα×ψ(n)=e-iπnαψ(n).
Jx=GJxG-1,Jx=x0J0+x1J1+x2J2,
Mζ=exp(iζJ2)=Ce-ζ/200eζ/2=(M-ζ)-1,
J(ζ)=MζJ0M-ζ=cosh ζJ0-sinh ζJ1=12e-ζJ++12eζJ-.
ψn(k;ζ)=Mζ : ψn(k;0).
ψn(k;0, ζ)(n)=(ψn(k;ζ), ψn(k;0))l2=(ψn(k;0), M-ζ : ψn(k;0))l2=Dn,nkeζ/200e-ζ/2=(-1)nΓ(2k+n+n)[n!n!Γ(2k+n)Γ(2k+n)]1/2tanhn+n (1/2)ζcosh2k (1/2)ζ,F-n,-n1-2k-n-n; coth212ζ,
=ψnM(n; 2k, tanh212ζ)=dk+n,k+nk(ζ)=(-1)nψn(k;-ζ, 0)(n),
ψn(k;0, ζ)(ξ)=ψnM(ξ; β, γ)=γn+ξ(1-γ)β(β)n(β)ξn!Γ(ξ+1)1/2Mn(ξ; β, γ),
Mn(ξ; β, γ)=F-n,-ξ β; 1-1γ=(β+ξ)n(β)nF-n,-ξ1-β-n-ξ ;1γ
ξƵ0+ψnM(ξ)ψnM(ξ)=δn,n,
nƵ0+ψnM(ξ)ψnM(ξ)=δξ,ξ,
K(ζ)=2e-ζJ(ζ)=J-+e-2ζJ+J-.
2e-ζn=12rn2ρ=12r2.
rn=2e-ζ/2nr,n=14eζrn2.
limζ Mn(eζρ; β, 1-2e-ζ)=n!(β)nLnβ-1(2ρ),
limζ12eζ/2ψn(k;0, ζ)14eζr2=Ψm,nH(r).
Hn,nα;2 : κ-1,ζ=(ψn(k;ζ), Hα : ψn(k;ζ))l2=nƵ0+ψn(k;ζ)(n)e-iπαnψn(k;ζ)(n).
e-iπkαHn,nα;|m|,ζ=(ψn(k;ζ), exp(iπαJ0 : ψn(k;ζ)))l2=Ce-ζ/200eζ/2 : ψn(k;0), Ccos12πα-sin12παsin12παcos12παCe-ζ/200eζ/2 : ψn(k;0)l2=ψn(k;0), Ceζ/200e-ζ/2cos12πα-sin12παsin12παcos12παe-ζ/200eζ/2 : ψn(k;0)l2=Dn,nkcos12πα-eζ sin12παe-ζ sin12παcos12πα.
(Hα|k : f)(n)(ζ)=nƵ0+Hn,nα;|m|,ζf(n).
nƵ0+Hn,nα1;|m|,ζHn,nα2;|m|,ζ=Hn,nα1+α2;|m|,ζ.
limζ Hn,nα;|m|,ζ=Hmα(r, r)=eiπkαCHα(m)(r, r),
n14eζr2,n14eζr2.
ψ(n, θ)=(F¯ : ψH)(n, θ)=mƵeimθ2π(-i)mnƵ0+Hn,n1;|m|,ζψmH(n).
θn{θn,0, θn,1,, θn,N-1},θn,l=θn,0+2πl/N,
exp(-imθn,0)N(n)lƵNexp(-2πiml/N)ψ(n, l)
=(-i)m2πnƵ0+Hn,n1;|m|,ζψmH(n).
ψ(n, l)=(F¯¯ : ψH)(n, θl)=12πmƵN(n)(-i)m exp(imθn, l)×nƵ0+Hn,n1;|m|,ζψmH(n),
ψmH(n)=(F¯¯-1 : ψ)m(n),
=2πimnƵ0+1N(n)(Hn, n1;|m|, ζ)*×lƵN(n)exp(-imθn, l)ψ(n, l),
ψn(0, 0)=12πnƵ0+Hn,n1;0,ζψ0H(n),
ψ0H(n)=2πnƵ0+1N(n)(Hn, n1;0, ζ)*lƵN(n)ψ(n, l).
(C(M)|k: ψ)(n, l)(ζ)=1N(n)mƵN(n) exp(imθn, l)×nƵ0+Cn,n(k;ζ)(M)ψ(n, m),
Cn,n(k;ζ)abcd=ψn(k;ζ), Cabcd : ψn(k;ζ)=ψn(k;0), Ceζ/200e-ζ/2abcd×e-ζ/200eζ/2 : ψn(k;0)=Dn,nkaeζbe-ζcd.
W(ϕ, ψ; x)=ϕ|W(x)|ψ,
W(ϕ, ψ; x)=nƵ0+nƵ0+[(ζ)n|ϕ]*× (ζ)n|W(x)|n(ζ)(ζ)n, ψ,
=R0+drR0+dr(r|ϕ)*r|W(x)|rrψ.
z=a2+b2+c2+d2+2a2+b2+c2+d2-2=[(a+d)+i(b-c)][(a+d)-i(b-c)][(a-d)+i(b+c)][(a-d)-i(b+c)].
F-n,-n1-2k-n-n; z=(-n)n(-z)n(1-2k-n-n)n×F-n,2k+n1-n+n;1z+{nn},
Dn,nkabcd=22k(n-n)!n!n!Γ(2k+n)Γ(2k+n)1/2×[(a+d)-i(b-c)]n[(a-d)-i(b+c)]n-n[(a+d)+i(b-c)]2k+n×F-n,2k+n1-n+n;[(a-d)+i(b+c)][(a-d)-i(b+c)][(a+d)+i(b-c)][(a+d)-i(b-c)],
Dn,nkcos θ-eζ sin θe-ζ sin θcos θin-n(n-n)!n!n!Γ(2k+n)Γ(2k+n)1/2=(cos θ+i sin θ cosh ζ)n(sin θ sinh ζ)n-n(cos θ-i sin θ cosh ζ)2k+n×F-n,2k+n1-n+n;sin2 θ sinh2 ζcos2 θ+sin2 θ cosh2 ζ,

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