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Canonical transforms for paraxial wave optics

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Lie Methods in Optics

Part of the book series: Lecture Notes in Physics ((LNP,volume 250))

Abstract

Paraxial geometric optics in N dimensions is well known to be described by the inhomogeneous symplectic group I 2 N ∧ Sp(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.

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References

  1. H. Weyl, Quantenmechanik und Gruppentheorie, Z. Phys. 46, 1–46 (1928).

    Google Scholar 

  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. M. Moshinsky, Canonical transformations in quantum mechanics, SIAM J. Appl. Math. 25, 193–212 (1973).

    Google Scholar 

  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. K.B. Wolf, Integral Transforms in Science and Engineering, (Plenum Publ. Corp., New York, 1979); Part IV.

    Google Scholar 

  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. O.N. Stavroudis, The Optics of Rays, Wavefronts, and Caustics, (Academic Press, New York, 1972).

    Google Scholar 

  8. A.J. Dragt, Lie algebraic theory of geometrical optics and optical aberrations, J. Opt. Soc. Am. 72, 373–379 (1983).

    Google Scholar 

  9. A.J. Dragt, Lectures on Nonlinear Orbit Dynamics, AIP Conference Proceedings N° 87 (American Institute of Physics, New York, 1982).

    Google Scholar 

  10. P.A.M. Dirac, The Principles of Quantum Mechanics, (Oxford University Press, 4th Ed., 1958).

    Google Scholar 

  11. G. Lions and M. Vergne, The Weil Representations, Maslov Index, and Theta Series, (Birkhäuser, Basel, 1980).

    Google Scholar 

  12. K.B. Wolf, Canonical transforms. I. Complex linear transforms, J. Math. Phys. 15, 1295–1301 (1974).

    Google Scholar 

  13. V. Bargmann, Irreducible unitary representations of the Lorentz group. Ann. Math. 48, 568–640 (1947).

    Google Scholar 

  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. 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. 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. J.R. Klauder and E.C.G. Sudarshan, Fundamentals of Quantum Optics, (Benjamin, Reading, Mass., 1968).

    Google Scholar 

  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. K.B. Wolf, On time-dependent quadratic quantum Hamiltonians, SIAM J. Appl. Math. 40, 419–431 (1981).

    Google Scholar 

  20. J.W. Goodman, Introduction to Fourier Optics (Mc Graw-Hill, New York, 1968).

    Google Scholar 

  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. K.B. Wolf, Canonical transforms. II. Complex radial transforms. J. Math. Phys. 15, 2102–2111 (1974).

    Google Scholar 

  23. A. Weil, Sur certaines groups d'operateurs unitairs, Acta Math. 11, 143–211 (1963).

    Google Scholar 

  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. 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. 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. 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. 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. 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. K.B. Wolf, On self-reproducing functions under a class of integral transforms, J. Math. Phys. 18, 1046–1051 (1977).

    Google Scholar 

  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. 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. 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. 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. P.M. Woodward, Probability and Information Theory, with Applications to Radar, (Artech House, Dedham, Mass., 1980).

    Google Scholar 

  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. 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. 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. 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. W. Lassner, Symbol representations of noncommutative algebras. (submitted for publication, 1985).

    Google Scholar 

  41. M. Born and E. Wolf, Principles of Optics, (Pergamon Press, 6th Ed., 1980).

    Google Scholar 

  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. K.B. Wolf, A euclidean algebra of hamiltonian observables in Lie optics, Kinam 6, 141–156 (1985).

    Google Scholar 

  44. G.W. Mackey, A theorem of Stone and von Neumann, Duke Math. J. 16, 313–326 (1949).

    Google Scholar 

  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. 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 1 ≤ m ≤ n. J. Math. Phys. 15, 2096–2100 (1974).

    Google Scholar 

  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. 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. 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. 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 

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Authors
  1. Octavio Castaños
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  2. Enrique López-Moreno
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  3. Kurt Bernardo Wolf
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Editor information

J. Sánchez Mondragón K. B. Wolf

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© 1986 Springer-Verlag

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Castaños, O., López-Moreno, E., Wolf, K.B. (1986). Canonical transforms for paraxial wave optics. In: Sánchez Mondragón, J., Wolf, K.B. (eds) Lie Methods in Optics. Lecture Notes in Physics, vol 250. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-16471-5_5

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  • DOI: https://doi.org/10.1007/3-540-16471-5_5

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  • Print ISBN: 978-3-540-16471-5

  • Online ISBN: 978-3-540-39811-0

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