Canonical transforms for paraxial wave optics

  • Octavio Castaños
  • Enrique López-Moreno
  • Kurt Bernardo Wolf
Conference paper
Part of the Lecture Notes in Physics book series (LNP, volume 250)


Paraxial geometric optics in N dimensions is well known to be described by the inhomogeneous symplectic group I2NSp(2N, ℜ). This applies to wave optics when we choose a particular (ray) representation of this group, corresponding to a true representation of its central extension and twofold cover \(\tilde \Gamma _N = W_N ^ \wedge Mp(2N,\Re )\). for wave optics, the representation distinguished by Nature is the oscillator one. There applies the theory of canonical integral transforms built in quantum mechanics. We translate the treatament of coherent states and other wave packets to lens and pupil systems. Some remarks are added on various topics, including a fundamental euclidean algebra and group for metaxial optics.


Coherent State Gaussian Beam Integral Kernel Wave Optic Free Propagation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. [1]
    H. Weyl, Quantenmechanik und Gruppentheorie, Z. Phys. 46, 1–46 (1928).Google Scholar
  2. [2]
    M. Moshinsky and C. Quesne, Oscillator systems. In Proceedings of the XV Solvay Conference in Physics (Brussels, 1970). E. Prigogine Ed., (Gordon and Breach, New York, 1975).Google Scholar
  3. [3]
    M. Moshinsky, Canonical transformations in quantum mechanics, SIAM J. Appl. Math. 25, 193–212 (1973).Google Scholar
  4. [4]
    K.B. Wolf, The Heisenberg-Weyl ring in quantum mechanics. In Group Theory and its Applications, Vol. 3, Ed. by E.M. Loebl (Academic Press, New York, 1975).Google Scholar
  5. [5]
    K.B. Wolf, Integral Transforms in Science and Engineering, (Plenum Publ. Corp., New York, 1979); Part IV.Google Scholar
  6. [6]
    M. Nazarathy and J. Shamir, Fourier optics described by operator algebra, J. Opt. Soc. Am. 70, 150–158 (1980); ib. First-order optics —a canonical operator representation. I. Lossless systems. J. Opt. Soc. Am. 72, 356–364 (1982).Google Scholar
  7. [7]
    O.N. Stavroudis, The Optics of Rays, Wavefronts, and Caustics, (Academic Press, New York, 1972).Google Scholar
  8. [8]
    A.J. Dragt, Lie algebraic theory of geometrical optics and optical aberrations, J. Opt. Soc. Am. 72, 373–379 (1983).Google Scholar
  9. [9]
    A.J. Dragt, Lectures on Nonlinear Orbit Dynamics, AIP Conference Proceedings N° 87 (American Institute of Physics, New York, 1982).Google Scholar
  10. [10]
    P.A.M. Dirac, The Principles of Quantum Mechanics, (Oxford University Press, 4th Ed., 1958).Google Scholar
  11. [11]
    G. Lions and M. Vergne, The Weil Representations, Maslov Index, and Theta Series, (Birkhäuser, Basel, 1980).Google Scholar
  12. [12]
    K.B. Wolf, Canonical transforms. I. Complex linear transforms, J. Math. Phys. 15, 1295–1301 (1974).Google Scholar
  13. [13]
    V. Bargmann, Irreducible unitary representations of the Lorentz group. Ann. Math. 48, 568–640 (1947).Google Scholar
  14. [14]
    V. Bargmann, Group representations in Hilbert spaces of analytic functions. In: Analytical Methods in Mathematical Physics, P. Gilbert and R. G. Newton Eds. (Gordon and Breach, New York, 1970); pp. 27–63.Google Scholar
  15. [15]
    M. Navarro-Saad and K.B. Wolf, Factorization of the phase-space transformation produced by an arbitrary refracting surface. Preprint CINVESTAV (March 1984); to appear in J. Opt. Soc. Am. Google Scholar
  16. [16]
    O. Castaños, E. López-Moreno, and K.B. Wolf, The Lie-theoretical description of geometric and wave gaussian optics (manuscript in preparation).Google Scholar
  17. [17]
    J.R. Klauder and E.C.G. Sudarshan, Fundamentals of Quantum Optics, (Benjamin, Reading, Mass., 1968).Google Scholar
  18. [18]
    V.V. Dodonov, E.V. Kurmyshev, and V.I. Man'ko, Generalized uncertainty relation in correlated coherent states, Phys. Lett. 79A, 150–152 (1980).Google Scholar
  19. [19]
    K.B. Wolf, On time-dependent quadratic quantum Hamiltonians, SIAM J. Appl. Math. 40, 419–431 (1981).Google Scholar
  20. [20]
    J.W. Goodman, Introduction to Fourier Optics (Mc Graw-Hill, New York, 1968).Google Scholar
  21. [21]
    M. Moshinsky, T.H. Seligman, and K.B. Wolf, Canonical transformations and the radial oscillator and Coulomb problems, J. Math. Phys. 13, 1634–1638 (1972).Google Scholar
  22. [22]
    K.B. Wolf, Canonical transforms. II. Complex radial transforms. J. Math. Phys. 15, 2102–2111 (1974).Google Scholar
  23. [23]
    A. Weil, Sur certaines groups d'operateurs unitairs, Acta Math. 11, 143–211 (1963).Google Scholar
  24. [24]
    K.B. Wolf, Recursive method for the computation of SO n, SO n,1, and ISO n representation matrix elements, J. Math. Phys. 12, 197–206 (1971).Google Scholar
  25. [25]
    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).Google Scholar
  26. [26]
    K.B. Wolf, Canonical transforms. IV. Hyperbolic transforms: continuous series of SL(2,R) representations, J. Math. Phys. 21, 680–688 (1980).Google Scholar
  27. [27]
    D. Basu and K.B. Wolf, The Clebsch-Gordan coefficients of the three-dimensional Lorentz algebra in the parabolic basis, J. Math. Phys. 24, 478–500 (1983).Google Scholar
  28. [28]
    A. Frank and K.B. Wolf, Lie algebras for systems with mixed spectra. The scattering Pöschl-Teller potential. J. Math. Phys. 26, 973–983 (1985).Google Scholar
  29. [29]
    H. Kogelnik, On the propagation of gaussian beams of light through lenslike media including those with a loss or gain variation, Appl. Opt. 4, 1562–1569 (1965).Google Scholar
  30. [30]
    K.B. Wolf, On self-reproducing functions under a class of integral transforms, J. Math. Phys. 18, 1046–1051 (1977).Google Scholar
  31. [31]
    V.I. Man'ko and K.B. Wolf, The influence of aberrations in the optics of gaussian beam propagation. Reporte de Investigación, Vol. 3, # 2 (1985), Departamento de Matemáticas, Universidad Autónoma Metropolitana14 Google Scholar
  32. [32]
    W. Schempp, Radar reception and nilpotent harmonic analysis. I–VI. C. R. Math. Rep. Acad. Sci. Canada 4, 43–48, 139–144, 219–224 (1982); ibid. 5, 121–126 (1983); 6, 179–182 (1984).Google Scholar
  33. [33]
    W. Schempp, On the Wigner quasi-probability distribution function. I–III. C. R. Math. Rep. Acad. Sci. Canada 4, 353–358 (1982); ibid. 5, 3–8, 35–40 (1983).Google Scholar
  34. [34]
    W. Schempp, Radar ambiguity function, nilpotent harmonic analysis, and holomorphic theta series. In Special Functions: Group Theoretical Aspects and Applications. Ed. by R.A. Askey, T.H. Koornwinder, and W. Schempp (Reidel, Dordrecht, 1984).Google Scholar
  35. [35]
    P.M. Woodward, Probability and Information Theory, with Applications to Radar, (Artech House, Dedham, Mass., 1980).Google Scholar
  36. [36]
    M.J. Bastiaans, Wigner distribution function and its applications to first-order optics. J. Opt. Soc. Am. 69, 1710–1716 (1979).Google Scholar
  37. [37]
    G. García-Calderón and M. Moshinsky, Wigner distribution functions and the representation of canonical transformations in quantum mechanics, J. Phys. A 13, L185–L188 (1980).Google Scholar
  38. [38]
    M. García-BuIlé, W. Lassner, and K.B. Wolf, The metaplectic group within the Heisenberg-Weyl ring. Reporte de Investigación, Vol. 2 # 20 (1985), Departamento de Matemáticas, Universidad Autónoma Metropolitana. To appear in J. Math. Phys. Google Scholar
  39. [39]
    A.J. Dragt and J.M. Finn, Lie series and invariant functions for analytic symplectic maps. J. Math. Phys. 17, 2215–2227 (1976).Google Scholar
  40. [40]
    W. Lassner, Symbol representations of noncommutative algebras. (submitted for publication, 1985).Google Scholar
  41. [41]
    M. Born and E. Wolf, Principles of Optics, (Pergamon Press, 6th Ed., 1980).Google Scholar
  42. [42]
    J. Ojeda-Castañeda and A. Boivin, The influence of wave aberrations: an operator approach (preprint, August 1984). To appear in Canadian J. Phys.; J. Ojeda-Castañeda, Focus-error operator and related special functions, J. Opt. Soc. Am. 73, 1042–1047 (1983); A.W. Lohman, J. Ojeda-Castañeda, and N. Streibl, The influence of wave aberrations on the Wigner distribution (preprint, 1984).Google Scholar
  43. [43]
    K.B. Wolf, A euclidean algebra of hamiltonian observables in Lie optics, Kinam 6, 141–156 (1985).Google Scholar
  44. [44]
    G.W. Mackey, A theorem of Stone and von Neumann, Duke Math. J. 16, 313–326 (1949).Google Scholar
  45. [45]
    M. Abramowitz and I. E. Stegun, Eds., Handbook of Mathematical Functions, Applied Mathematics Series, Vol. 55 (National Bureau of Standards, Washington D.C., 1st Ed., 1964).Google Scholar
  46. [46]
    C.P. Boyer and K.B. Wolf, The algebra and group deformations I m[SO(n)⊗SO(m)] ⇒ SO(n, m), I m[U(n)⊗U(m)] ⇒ U(n, m), and I m[Sp(n)⊗Sp(m)] ⇒ Sp(n,m), for 1mn. J. Math. Phys. 15, 2096–2100 (1974).Google Scholar
  47. [47]
    C.P. Boyer and K.B. Wolf, Canonical transforms. III. Configuration and phase descriptions of quantum systems possessing an sl(2,R) dynamical algebra, J. Math. Phys. 16, 1493–1502 (1975).Google Scholar
  48. [48]
    E.C.G. Sudarshan, R. Simon, and N. Mukunda, Paraxial-wave optics and relativistic front description. I. The scalar theory, Phys. Rev. A28, 2921–2932 (1983); ibid. The vector theory, Phys. Rev. A28, 2933–2942 (1983).Google Scholar
  49. [49]
    R. Simon, E.C.G. Sudarshan, and N. Mukunda, Generalized rays in first-order optics: transformation properties of gaussian Schell-model fields, Phys. Rev. A29, 3273–3279 (1984).Google Scholar
  50. [50]
    N. Mukunda, R. Simon, and E.C.G. Sudarshan, Fourier optics for the Maxwell field: formalism and applications, J. Opt. Soc. Am. A2, 416–426 (1985).Google Scholar

Copyright information

© Springer-Verlag 1986

Authors and Affiliations

  • Octavio Castaños
  • Enrique López-Moreno
  • Kurt Bernardo Wolf

There are no affiliations available

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