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


SIGMA 7 (2011), 072, 34 pages      arXiv:1107.3625      https://doi.org/10.3842/SIGMA.2011.072
Contribution to the Special Issue “Symmetry, Separation, Super-integrability and Special Functions (S4)”

Appell Transformation and Canonical Transforms

Amalia Torre
ENEA UTAPRAD-MAT Laboratorio di Modellistica Matematica, via E. Fermi 45, 00044 Frascati (Rome), Italy

Received January 31, 2011, in final form July 11, 2011; Published online July 19, 2011

Abstract
The interpretation of the optical Appell transformation, as previously elaborated in relation to the free-space paraxial propagation under both a rectangular and a circular cylindrical symmetry, is reviewed. Then, the caloric Appell transformation, well known in the theory of heat equation, is shown to be amenable for a similar interpretation involving the Laplace transform rather than the Fourier transform, when dealing with the 1D heat equation. Accordingly, when considering the radial heat equation, suitably defined Hankel-type transforms come to be involved in the inherent Appell transformation. The analysis is aimed at outlining the link between the Appell transformation and the canonical transforms.

Key words: heat equation; paraxial wave equation; Appell transformation.

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References

  1. Appell M.P., Sur l'équation ∂2z/∂x2−∂z/∂y=0 et la théorie de la chaleur, J. Math. Pure Appl. 8 (1892), 187-216.
  2. Widder D.V., The heat equation, Pure and Applied Mathematics, Vol. 67, Academic Press, New York - London, 1975.
  3. Rosenbloom P.C., Widder D.V., Expansions in terms of heat polynomials and associated functions, Trans. Amer. Math. Soc. 92 (1959), 220-266.
  4. Widder D.V., Analytic solutions of the heat equation, Duke Math. J. 29 (1962), 497-503.
  5. Widder D.V., Expansions in series of homogeneous temperature functions of the first and second kinds, Duke Math. J. 36 (1969), 495-509.
  6. Leutwiler H., On the Appell transformation, in Potential Theory (Prague, 1987), Editors J. Kràl et. al., Plenum Press, New York, 1988, 215-222.
  7. Shimomura K., The determination of caloric morphisms on Euclidean domains, Nagoya Math. J. 158 (2000), 133-166.
  8. Brzezina M., Appell type transformation for the Kolmogorov operator, Math. Nachr. 169 (1994), 59-67.
  9. Torre A., The Appell transformation for the paraxial wave equation, J. Opt. 13 (2011), 015701, 14 pages.
  10. Kalnins E.G., Miller W. Jr., Lie theory and separation of variables. V. The equation iUt+Uxx=0 and iUt+Uxx−(c/x2)U=0, J. Math. Phys. 15 (1974), 1728-1737.
  11. Boyer C.P., Kalnins E.G., Miller W. Jr., Lie theory and separation of variables. VI. The equation iUt2U=0, J. Math. Phys. 16 (1975), 499-511.
  12. Miller W. Jr., Symmetry and separation of variables, Encyclopedia of Mathematics and its Applications, Vol. 4, Addison-Wesley Publishing Co., Reading, Mass. - London - Amsterdam, 1977.
  13. 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.
  14. Olver P.J., Applications of Lie groups to differential equations, 2nd ed., Graduate Texts in Mathematics, Vol. 107, Springer-Verlag, New York, 1993.
  15. Torre A., A note on the general solution of the paraxial wave equation: a Lie algebra view, J. Opt. A: Pure Appl. Opt. 10 (2008), 055006, 14 pages.
  16. Torre A., Separable-variable solutions of the wave equation from a general type of solutions of the paraxial wave equation, in Proceedings of the International Conference "Days on Diffraction" (May 26-29, 2009, St. Petersburg), 178-183.
  17. Torre A., Linear and quadratic exponential modulation of the solutions of the paraxial wave equation, J. Opt. 12 (2010), 035701, 11 pages.
  18. Torre A., Appell transformation and symmetry transformations for the paraxial wave equation, J. Opt. 13 (2011), 075710, 12 pages.
  19. Kato T., Perturbation theory for linear operators, 2nd ed., Grundlehren der Mathematischen Wissenschaften, Band 132, Springer-Verlag, Berlin - New York, 1976.
  20. Collins S.A. Jr., Lens-system diffraction integral written in terms of matrix optics, J. Opt. Soc. Amer. A 60 (1970), 1168-1177.
  21. Siegman A.E., Lasers, University Science Books, Mill Valley, CA, 1986.
  22. Ballentine L.E., Quantum mechanics, Prentice Hall, Englewood Cliffs, New Jersey, 1990.
  23. Bandres M.A., Gutiérrez-Vega J.C., Cartesian beams, Opt. Lett. 32 (2007), 3459-3461.
  24. Bandres M.A., Gutiérrez-Vega J.C., Circular beams, Opt. Lett. 33 (2008), 177-179.
  25. Bandres M.A., Gutiérrez-Vega J.C., Elliptical beams, Opt. Expr. 16 (2008), 21087-21092.
  26. Sudarshan E.C.G., Mukunda N., Simon R., Realization of first order optical systems using thin lenses, Opt. Acta 32 (1985), 855-872.
  27. Bandres M.A., Guizar-Sicairos M., Paraxial group, Opt. Lett. 34 (2009), 13-15.
  28. Wei J., Norman E., Lie algebraic solution of linear differential equations, J. Math. Phys. 4 (1963), 575-581.
    Dattoli G., Gallardo J.C., Torre A., An algebraic view to the operatorial ordering and its applications to optics, Riv. Nuovo Cimento (3) 11 (1988), no. 11, 1-79.
    Ban M., Decomposition formulas for su(1,1) and su(2) Lie algebras and their applications to quantum optics, J. Opt. Soc. Amer. B 10 (1993), 1347-1359.
  29. Magnus W., Oberhettinger F., Soni R.P., Formulas and theorems for the special functions of mathematical physics, 3rd ed., Die Grundlehren der mathematischen Wissenschaften, Band 52, Springer-Verlag, New York, 1966.
  30. Torre A., A note on the Airy beam in the light of the symmetry algebra based approach, J. Opt. A: Pure Appl. Opt. 11 (2009), 125701, 11 pages.
  31. Berry M.V., Balazs N.L., Nonspreading wave packets, Amer. J. Phys. 47 (1979), 264-267.
  32. Besieris I.M., Shaarawi A.M., Ziolkowski R.W., Nondispersive accelerating wave packets, Amer. J. Phys. 62 (1994), 519-521.
  33. Bandres M.A., Accelerating beams, Opt. Lett. 34 (2009), 3791-3793.
  34. Siviloglou G.A., Christodoulides D.N., Accelerating finite energy Airy beams, Opt. Lett. 32 (2007), 979-981.
    Siviloglou G.A., Broky J., Dogariu A., Christodoulides D.N., Observation of accelerating Airy beams, Phys. Rev. Lett. 99 (2007), 2139011, 4 pages.
    Siviloglou G.A., Broky J., Dogariu A., Christodoulides D.N., Ballistic dynamics of Airy beams, Opt. Lett. 33 (2008), 207-209.
  35. Besieris I.M., Shaarawi A.M., A note on an accelerating finite energy Airy beam, Opt. Lett. 32 (2007), 2447-2449.
  36. Broky J., Siviloglou G.A., Dogariu A., Christodoulides D.N., Self-healing properties of optical Airy beams, Opt. Expr. 16 (2008), 12880-12891.
  37. Morris J.E., Mazilu M., Baumgartl J., Cizmar T., Dholakia K., Propagation characteristics of Airy beams: dependence upon spatial coherence and wavelength, Opt. Expr. 17 (2009), 13236-13245.
  38. Dai H.T., Sun X.W., Luo D., Liu Y.J., Airy beams generated by binary phase element made of polymer-dispersed liquid crystals, Opt. Expr. 17 (2009), 19365-19370.
  39. Baumgartl J., Mazilu M., Dholakia K., Optically mediated particle clearing using Airy wavepackets, Nature Photonics 2 (2008), 675-678.
  40. Ellenbogen T., Voloch-Bloch N., Ganany-Padowicz A., Arie A., Nonlinear generation and manipulation of Airy beams, Nature Photonics 3 (2009), 395-398.
  41. Salandrino A., Christodoulides D.N., Airy plasmon: a nondiffracting surface wave, Opt. Lett. 35 (2010), 2082-2084.
  42. Mendlovic D., Ozaktas H.M., Fractional Fourier transforms and their optical implementation. I, J. Opt. Soc. Amer. A 10 (1993), 1875-1881.
    Mendlovic D., Ozaktas H.M., Fractional Fourier transforms and their optical implementation. II, J. Opt. Soc. Amer. A 10 (1993), 2522-2531.
    Ozaktas H.M., Kutay M.A., Mendlovic D., Introduction to the fractional Fourier transform and its applications, in Advances in Imaging and Electron Physics, Vol. 106, Editor P.W. Hawkes, Academic Press, San Diego, 1999, 239-291.
  43. Ozatkas H.M., Zalevsky Z., Kutay M.A., The fractional Fourier transform with applications in optics and signal processing, Wiley, New York, 2001.
  44. Lohmann A.W., Image rotation, Wigner rotation, and the fractional order Fourier transform, J. Opt. Soc. Amer. A 10 (1993), 2181-2186.
    Lohmann A.W., Mendlovic D., Zalevsky Z., Fractional transformations in optics, in Progress in Optics, Vol. 38, Editor E. Wolf, Elsevier, Amsterdam, 1997, 263-342.
  45. Torre A., The fractional Fourier transform and some of its applications to optics, in Progress in Optics, Vol. 43, Editor E. Wolf, Elsevier, Amsterdam, 2002, 531-596.
  46. Durnin J., Exact solutions for nondiffracting beams. I. The scalar theory, J. Opt. Soc. Amer. A (1987), 651-654.
    Durnin J., Miceli J.J., Eberly J.H., Diffraction-free beams, Phys. Rev. Lett. 58 (1987), 1499-1501.
  47. Sheppard C.J.R., Wilson T., Gaussian beams theory of lenses with annular aperture, IEE J. Microwaves Opt. Acoust. 2 (1978), 105-112,
    Gori F., Guattari G., Padovani C., Bessel-Gauss beams, Opt. Comm. 64 (1987), 491-495.
  48. Lohmann A.W., Ein neues Dualitatsprinzip in der Optik, Optik 11 (1954), 478-488.
    Lohmann A.W., Duality in optics, Optik 89 (1992), 93-97.
  49. Sheppard C.J.R., Beam duality, with application to generalized Bessel-Gaussian, and Hermite- and Laguerre-Gaussian beams, Opt. Expr. 15 (2009), 3690-3697.
  50. Heaviside O., Electrical papers, The Macmillan Co., New York and London, 1892.
    Heaviside O., Electromagnetic theory, The Electrician Printing & Publishing Co., London, Vol. 1, 1894; Vol. 2, 1899; Vol. 3, 1912.
  51. Nahin P.J., Oliver Heaviside: the life, work, and times of an electrical genius of the victorian age, The Johns Hopkins University Press, Baltimore, 2002.
  52. Tranter C.J., Integral transforms in mathematical physics, Methuen & Co., Ltd., London; John Wiley & Sons, Inc., New York, 1951.
  53. Davies B., Integral transforms and their applications, Applied Mathematical Sciences, Vol. 25, Springer-Verlag, New York - Heidelberg, 1978.
  54. Bargmann V., On a Hilbert space of analytic functions and an associated integral transform, Comm. Pure Appl. Math. 14 (1961), 187-214.
    Bargmann V., On a Hilbert space of analytic functions and an associated integral transform. II. A family of related function spaces. Application to distribution theory, Comm. Pure Appl. Math. 20 (1967), 1-101.
  55. Moshinsky M., Quesne C., Linear canonical transformations and their unitary representations, J. Math. Phys. 12 (1971), 1772-1780.
  56. Quesne C., Moshinsky M., Canonical transformations and matrix elements, J. Math. Phys. 12 (1971), 1780-1783.
  57. Wolf K.B., Canonical transforms. I. Complex linear transforms, J. Math. Phys. 15 (1974), 1295-1301.
  58. Wolf K.B., Canonical transforms. II. Complex radial transforms, J. Math. Phys. 15 (1974), 2102-2111.
  59. Kramer P., Moshinsky M., Seligman T.H., Complex extensions of canonical transformations and quantum mechanics, in Group Theory and Its Applications, Vol. III, Editor E.M. Loebl, Academic Press, New York, 1975, 249-332.
  60. Wolf K.B., Canonical transforms, separation of variables and similarity solutions for a class of parabolic differential equations, J. Math. Phys. 17 (1976), 601-613.
  61. Wolf K.B., On self-reciprocal functions under a class of integral transforms, J. Math. Phys. 18 (1977), 1046-1051.
  62. Namias V., The fractional order Fourier transform and its application to quantum mechanics, J. Inst. Appl. Math. 25 (1980), 241-265.
  63. Namias V., Fractionalization of Hankel transforms, J. Inst. Math. Appl. 26 (1980), 187-197.
  64. McBride A.C., Kerr F.H., On Namias's fractional Fourier transforms, IMA J. Appl. Math. 39 (1987), 159-175.
  65. Pei S.-C., Ding J.-J., Eigenfunctions of linear canonical transform, IEEE Trans. Signal Process. 50 (2002), 11-26.
  66. Torre A., Linear and radial transforms of fractional order, J. Comp. Appl. Math. 153 (2003), 477-486.
  67. Alieva T., Bastiaans M.J., Properties of the linear canonical integral transformation, J. Opt. Soc. Amer. A 24 (2007), 3658-3665.
  68. Stern A., Uncertainty principles in linear canonical transform domains and some of their implications in optics, J. Opt. Soc. Amer. A 25 (2008), 647-652.
  69. Deng B., Tao R., Wang Y., Convolution theorems for the linear canonical transform and their applications, Sci. China Ser. F 49 (2006), 592-603.
  70. Koç A., Ozaktas H.M., Hesselink L., Fast and accurate algorithm for the computation of complex linear canonical transforms, J. Opt. Soc. Amer. A 27 (2010), 1288-1302.
  71. Sharma K.K., Fractional Laplace transform, Signal Image Video Process 4 (2010), 377-379.
  72. Louck J.D., Moshinsky M., Wolf K.B., Canonical transformations and accidental degeneracy. I. The anisotropic oscillator, J. Math. Phys. 14 (1973), 692-695,
    Louck J.D., Moshinsky M., Wolf K.B., Canonical transformations and accidental degeneracy. II. The isotropic oscillator in a sector, J. Math. Phys. 14 (1973), 696-700.
  73. Barut A.O., Girardello L., New "coherent" states associated with non-compact groups, Comm. Math. Phys. 21 (1971), 41-55.
  74. Linares Linares M., Méndez Pérez J.M.R., A Hankel type integral transformation on certain space of distributions, Bull. Calcutta Math. Soc. 83 (1991), 447-546.
    Linares Linares M., Méndez Pérez J.M.R., Hankel complementary integral transformations of arbitrary order, Internat. J. Math. Math. Sci. 15 (1992), 323-332.
  75. Malgonde S.P., Debnath L., On Hankel type integral transformations of generalized functions, Integral Transforms Spec. Funct. 15 (2004), 421-430.
  76. Malgonde S.P., Bandewar S.R., Debnath L., Mixed Parseval equation and generalized Hankel-type integral transformation of distributions, Integral Transforms Spec. Funct. 15 (2004), 431-443.
  77. Torre A., Hankel-type integral transforms and their fractionalization: a note, Integral Transforms Spec. Funct. 19 (2008), 277-292.
  78. Bragg L.R., The radial heat polynomials and related functions, Trans. Amer. Math. Soc. 119 (1965), 270-290.
    Bragg L.R., The radial heat equation and Laplace transforms, SIAM J. Appl. Math. 14 (1966), 986-993.


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