Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 7 (2011), 053, 18 pages      arXiv:1106.0093
Contribution to the Special Issue “Symmetry, Separation, Super-integrability and Special Functions (S4)”

The Fourier U(2) Group and Separation of Discrete Variables

Kurt Bernardo Wolf a and Luis Edgar Vicent b
a) Instituto de Ciencias Físicas, Universidad Nacional Autónoma de México, Av. Universidad s/n, Cuernavaca, Mor. 62210, México
b) Deceased

Received February 19, 2011, in final form May 26, 2011; Published online June 01, 2011

The linear canonical transformations of geometric optics on two-dimensional screens form the group Sp(4,R), whose maximal compact subgroup is the Fourier group U(2)F; this includes isotropic and anisotropic Fourier transforms, screen rotations and gyrations in the phase space of ray positions and optical momenta. Deforming classical optics into a Hamiltonian system whose positions and momenta range over a finite set of values, leads us to the finite oscillator model, which is ruled by the Lie algebra so(4). Two distinct subalgebra chains are used to model arrays of N2 points placed along Cartesian or polar (radius and angle) coordinates, thus realizing one case of separation in two discrete coordinates. The N2-vectors in this space are digital (pixellated) images on either of these two grids, related by a unitary transformation. Here we examine the unitary action of the analogue Fourier group on such images, whose rotations are particularly visible.

Key words: discrete coordinates; Fourier U(2) group; Cartesian pixellation; polar pixellation.

pdf (1792 kb)   tex (1720 kb)


  1. Wolf K.B., Geometric optics on phase space, Texts and Monographs in Physics, Springer-Verlag, Berlin, 2004.
  2. Simon R., Wolf K.B., Structure of the set of paraxial optical systems, J. Opt. Soc. Amer. A 17 (2000), 342-355.
  3. Simon R., Wolf K.B., Fractional Fourier transforms in two dimensions, J. Opt. Soc. Amer. A 17 (2000), 2368-2381.
  4. Moshinsky M., Quesne C., Linear canonical transformations and their unitary representation, J. Math. Phys. 12 (1971), 1772-1779.
  5. Moshinsky M., Quesne C., Canonical transformations and matrix elements, J. Math. Phys. 12 (1971), 1780-1786.
  6. Wolf K.B., Integral transforms in science and engineering, Mathematical Concepts and Methods in Science and Engineering, Vol. 11, Plenum Press, New York - London, 1979.
  7. Collins S.A. Jr., Lens-system diffraction integral written in terms of matrix optics, J. Opt. Soc. Amer. A 60 (1970), 1168-1177.
  8. Ding J.-J., Research of fractional Fourier transform and linear canonical transform, Ph.D. Thesis, National Taiwan University, 2001.
  9. Hennelly B.M., Sheridan J.T., Fast numerical algorithm for the linear canonical transform, J. Opt. Soc. Amer. A 22 (2005), 928-937.
  10. Koç A., Ozaktas H.M., Candan C., Kutay M.A., Digital computation of linear canonical transforms, IEEE Trans. Signal Process. 56 (2008), 2382-2394.
  11. Healy J.J., Sheridan J.T., Sampling and discretization of the linear canonical transform, Signal Process. 89 (2009), 641-648.
  12. Gilmore R., Lie groups, Lie algebras, and some of their applications, John Wiley, New York, 1974.
  13. Atakishiyev N.M., Wolf K.B., Fractional Fourier-Kravchuk transform, J. Opt. Soc. Amer. A 14 (1997), 1467-1477.
  14. Atakishiyev N.M., Vicent L.E., Wolf K.B., Continuous vs. discrete fractional Fourier transforms, J. Comput. Appl. Math. 107 (1999), 73-95.
  15. Atakishiyev N.M., Pogosyan G.S., Vicent L.E., Wolf K.B., Finite two-dimensional oscillator. I. The Cartesian model, J. Phys. A: Math. Gen. 34 (2001), 9381-9398.
  16. Atakishiyev N.M., Pogosyan G.S., Wolf K.B., Finite models of the oscillator, Phys. Part. Nuclei 36 (2005), 247-265.
  17. Wolf K.B., Discrete systems and signals on phase space, Appl. Math. Inf. Sci. 4 (2010), 141-181.
  18. Atakishiyev N.M., Pogosyan G.S., Vicent L.E., Wolf K.B., Finite two-dimensional oscillator. II. The radial model, J. Phys. A: Math. Gen. 34 (2001), 9399-9415.
  19. Vicent L.E., Wolf K.B., Unitary transformation between Cartesian- and polar-pixellated screens, J. Opt. Soc. Amer. A 25 (2008), 1875-1884.
  20. Vicent L.E., Unitary rotation of square-pixellated images, Appl. Math. Comput. 221 (2009), 111-117.
  21. Wolf K.B., Alieva T., Rotation and gyration of finite two-dimensional modes, J. Opt. Soc. Amer. A 25 (2008), 365-370.
  22. Biedenharn L.C., Louck J.D., Angular momentum in quantum mechanics, in Encyclopedia of Mathematics and its Applications, Editor G.-C. Rota, Addison-Wesley, 1981, Section 3.6.
  23. Atakishiyev N.M., Suslov S.K., Difference analogs of the harmonic oscillator, Theoret. and Math. Phys. 85 (1991), 1055-1062.
  24. Krawtchouk M., Sur une généralization des polinômes d'Hermite, Compt. Rend. Acad. Sci. Paris 189 (1929), 620-622.
  25. Gel'fand I.M., Tsetlin M.L., Finite-dimensional representations of the group of unimodular matrices, Dokl. Akad. Nauk SSSR 71 (1950), 825-828 (English transl.: I.M. Gel'fand, Collected papers, Vol. II, Springer-Verlag, Berlin, 1987, 653-656).
  26. Miller W. Jr., Symmetry and separation of variables, Encyclopedia of Mathematics and Its Applications, Vol. 4, Editor by G.-C. Rota, Addison-Wesley Publ. Co., Reading, Mass., 1981.
  27. Atakishiyev N.M., Jafarov E.I., Nagiyev Sh.M., Wolf K.B., Meixner oscillators, Rev. Mexicana Fís. 44 (1998), 235-244.
  28. Wolf K.B., Mode analysis and signal restoration with Kravchuk functions, J. Opt. Soc. Amer. A 26 (2009), 509-516.
  29. Vicent L.E., Wolf K.B., Analysis of digital images into energy-angular momentum modes, J. Opt. Soc. Amer. A 28 (2011), 808-814.
  30. Grosche C., Karayan Kh.G., Pogosyan G.S., Sissakian A.N., Free motion on the three-dimensional sphere: the ellipso-cylindrical bases, J. Phys. A: Math. Gen. 30 (1997), 1629-1657.
  31. Bandres M.A., Gutirrez-Vega J.C., Elliptical beams, Opt. Express 16 (2008), 21087-21092.
  32. Atakishiev N.M., Pogosyan G.S., Wolf K.B., Work in progress.

Previous article   Next article   Contents of Volume 7 (2011)