Abstract

We present a comprehensive normal-mode decomposition analysis for the recently introduced [ J. Opt. Soc. Am. A 10, 95 ( 1993)] class of twisted Gaussian Schell-model fields in partially coherent beam optics. The formal analogies to quantum mechanics in two dimensions are exploited. We also make effective use of a dynamical SU(2) symmetry of these fields to achieve the mode decomposition and to determine the spectrum. The twist phase is nonseparable in nature, rendering it nontrivially two dimensional. The consequences of this, resulting in the need to use Laguerre–Gaussian functions rather than products of Hermite–Gaussians, are carefully analyzed. An important identity involving these sets of special functions is established and is used in deriving the spectrum.

© 1993 Optical Society of America

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    [CrossRef]
  2. E. Wolf, “New spectral representation of random sources and of the partially coherent fields that they generate,” Opt. Commun. 38, 3–6 (1981).
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  3. E. Wolf, “New theory of partial coherence in the space-frequency domain. Part I: Spectra and cross-spectra of steady-state sources,”J. Opt. Soc. Am. 72, 343–351 (1982).
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  4. E. Wolf, “New theory of partial coherence in the space-frequency domain. Part II: Steady-state fields and higher order correlations,” J. Opt. Soc. Am. A 3, 76–85 (1986).
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    [CrossRef]
  6. R. Martinez-Herrero, P. M. Mejias, “Expansion of the cross-spectral density of general fields and its applications to beam characterization,” Opt. Commun. 94, 197–202 (1992).
    [CrossRef]
  7. F. Gori, “Mode propagation of the field generated by Collett–Wolf Schell model sources,” Opt. Commun. 46, 149–154 (1983).
    [CrossRef]
  8. F. Gori, R. Grella, “Shape invariant propagation of polychromatic fields,” Opt. Commun. 49, 173–177 (1984).
    [CrossRef]
  9. E. Wolf, G. S. Agrawal, “Coherence theory of laser resonator modes,” J. Opt. Soc. Am. A 1, 541–546 (1984).
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  10. P. S. Idell, J. W. Goodman, “Design of optimal imaging concentrators for partially coherent sources: absolute encircled energy criterion,” J. Opt. Soc. Am. A 3, 943–956 (1986); C. Pask, “Application of Wolf’s theory of coherence,” J. Opt. Soc. Am. A 3, 1097–1101 (1986).
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  11. R. Gase, “The multimode laser radiation as a Gaussian Schell-model beam,” J. Mod. Opt. 38, 1107–1115 (1991).
    [CrossRef]
  12. P. De Santis, F. Gori, G. Guattari, C. Palma, “Synthesis of partially coherent fields,” J. Opt. Soc. Am. A 3, 1258–1262 (1986).
    [CrossRef]
  13. F. Gori, G. Guattari, C. Padovani, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun. 64, 311–316 (1987).
    [CrossRef]
  14. F. Gori, G. Guattari, C. Palma, C. Padovani, “A class of shape invariant fields,” Opt. Commun. 66, 255–259 (1988).
    [CrossRef]
  15. F. Gori, G. Guattari, C. Palma, C. Padovani, “Specular cross-spectral density functions,” Opt. Commun. 68, 239–243 (1988).
    [CrossRef]
  16. G. Guattari, C. Palma, C. Padovani, “Cross-spectral densities with axial symmetry,” Opt. Commun. 73, 173–178 (1989).
    [CrossRef]
  17. W. H. Carter, M. Bertolotti, “An analysis of the far-field coherence and radiant intensity of light scattered from liquid crystals,”J. Opt. Soc. Am. 68, 329–333 (1978).
    [CrossRef]
  18. P. D. Santis, F. Gori, G. Guattari, C. Palma, “An example of a Collett–Wolf source,” Opt. Commun. 29, 256–260 (1979).
    [CrossRef]
  19. F. Gori, “Collett–Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
    [CrossRef]
  20. A. Gamliel, G. P. Agrawal, “Wolf effect in homogeneous and inhomogeneous media,” J. Opt. Soc. Am. A 7, 2184–2192 (1990); J. T. Foley, M. S. Zubairy, “The directionality of Gaussian Schell-model beams,” Opt. Commun. 26, 297–300 (1978); E. Collett, E. Wolf, “Is complete coherence necessary for the generation of highly directional light beams?” Opt. Lett. 2, 27–29 (1978).
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  21. H. Weber, “Wave optical analysis of the phase space analyser,” J. Mod. Opt. 39, 543–559 (1992); J. Serna, P. M. Mejias, R. Martinez-Herrero, “Beam quality dependence on the coherence length of Gaussian Schell-model fields propagating through ABCDoptical systems,” J. Mod. Opt. 39, 625–635 (1992).
    [CrossRef]
  22. B. E. A. Saleh, “Intensity distribution due to a partially coherent field and the Collett–Wolf equivalence theorem in the Fresnel zone,” Opt. Commun. 30, 135–138 (1979).
    [CrossRef]
  23. R. Simon, E. C. G. Sudarshan, N. Mukunda, “Generalized rays in first order optics: transformation properties of Gaussian Schell-model fields,” Phys. Rev. A 29, 3273–3279 (1984).
    [CrossRef]
  24. R. Simon, N. Mukunda, “Twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 10, 95–109 (1993).
    [CrossRef]
  25. N. Mukunda, “Algebraic aspects of the Wigner distribution in quantum mechanics,” Pramana-J. Phys. 11, 1–15 (1978); R. Simon, E. C. G. Sudarshan, N. Mukunda, “Gaussian Wigner distributions in quantum mechanics and optics,” Phys. Rev. A 36, 3868–3880 (1987), and references therein.
    [CrossRef] [PubMed]
  26. S. Pancharatnam, “Generalized theory of interference and its applications,” Proc. Ind. Acad. Sci. A 44, 247–262 (1956); reprinted in Collected Works of S. Pancharatnam (Oxford U. Press, Oxford, 1975).
  27. M. V. Berry, “The adiabatic phase and Pancharatnam’s phase for polarized light,” J. Mod. Opt. 34, 1401–1409 (1987).
    [CrossRef]
  28. T. H. Chyba, L. J. Wang, L. Mandel, R. Simon, “Measurement of the Pancharatnam phase for a light beam,” Opt. Lett. 13, 562–564 (1988).
    [CrossRef] [PubMed]
  29. R. Simon, H. J. Kimble, E. C. G. Sudarshan, “Evolving geometric phase and its dynamical manifestation as a frequency shift: an optical experiment,” Phys. Rev. Lett. 61, 19–22 (1988).
    [CrossRef] [PubMed]
  30. R. Bhandari, J. Samuel, “Observation of topological phase by use of a laser interferometer,” Phys. Rev. Lett. 60, 1211–1213 (1988).
    [CrossRef] [PubMed]
  31. H. Jiao, S. R. Wilkinson, R. Y. Chiao, H. Nathel, “Two topological phases in optics by means of a nonplanar Mach–Zehnder interferometer,” Phys. Rev. A 39, 3475–3486 (1989).
    [CrossRef] [PubMed]
  32. G. S. Agrawal, R. Simon, “Berry phase, interference of light beams, and the Hannay angle,” Phys. Rev. A 42, 6924–6927 (1990).
    [CrossRef]
  33. R. A. Campos, B. E. A. Saleh, M. C. Teich, “Quantum-mechanical lossless beam splitters: SU(2) symmetry and photon statistics,” Phys. Rev. A 40, 1371–1384 (1989).
    [CrossRef] [PubMed]
  34. B. Yurke, S. L. McCall, J. R. Klauder, “SU(2) and SU(1, 1) interferometers,” Phys. Rev. A 33, 4033–4054 (1986).
    [CrossRef] [PubMed]
  35. S. Danakas, P. K. Aravind, “Analogies between two optical systems (photon beam splitters and laser beams) and two quantum systems (the two-dimensional oscillator and the two-dimensional hydrogen atom),” Phys. Rev. A 45, 1973–1977 (1992).
    [CrossRef] [PubMed]
  36. See, for instance, L. C. Biedenharn, J. D. Louck, Angular Momentum in Quantum Physics: Theory and Application (Addison-Wesley, Reading, Mass., 1981), Chap. 4. More recently, Hamilton’s theory of turns has been generalized to the noncompact group SU(1, 1) in R. Simon, N. Mukunda, E. C. G. Sudarshan, “Hamilton’s theory of turns generalized to Sp(2, R),” Phys. Rev. Lett. 62, 1331–1334 (1989).
    [CrossRef] [PubMed]
  37. R. Simon, N. Mukunda, E. C. G. Sudarshan, “Hamilton’s theory of turns and a new geometrical representation for polarization optics,” Pramana 32, 769–792 (1989).
    [CrossRef]
  38. R. Simon, N. Mukunda, “Universal SU(2) gadget for polarization optics,” Phys. Lett. A 138, 474–480 (1989).
    [CrossRef]
  39. R. Simon, N. Mukunda, “Minimal three component SU(2) gadget for polarization optics,” Phys. Lett. A 143, 165–169 (1990).
    [CrossRef]
  40. K. Sundar, R. Simon, N. Mukunda, “Twisted Gaussian Schell-model beams: II. Spectrum analysis and propagation characteristics,” J. Opt. Soc. Am. A 10, 2017–2023 (1993).
    [CrossRef]
  41. R. Simon, E. C. G. Sundarshan, N. Mukunda, “Anisotropic Gaussian Schell-model beams: passage through optical systems and associated invariants,” Phys. Rev. A 31, 2419–2434 (1985), Appendix A in particular.
    [CrossRef] [PubMed]
  42. G. S. Agrawal, J. T. Foley, E. Wolf, “The radiance and phase-space representations of the cross-spectral density operator,” Opt. Commun. 62, 67–72 (1987).
    [CrossRef]
  43. J. Schwinger, “On angular momentum,” in Quantum Theory of Angular Momentum, L. C. Biedenharn, H. Van Dam, eds. (Academic, New York, 1965), pp. 229–279.
  44. R. Simon, E. C. G. Sudarshan, N. Mukunda, “Gaussian pure states in quantum mechanics and the symplectic group,” Phys. Rev. A 37, 2028–2038 (1988).
    [CrossRef]
  45. E. P. Wigner, Group Theory and Its Application to the Quantum Mechanics of Atomic Spectra (Academic, New York, 1959), p. 167.
  46. See, for example, S. Flügge, Practical Quantum Mechanics (Springer-Verlag, Berlin, 1971), Vol. 1, p. 107.
  47. I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980), p. 837, identity 7.374.8.

1993 (2)

1992 (3)

R. Martinez-Herrero, P. M. Mejias, “Expansion of the cross-spectral density of general fields and its applications to beam characterization,” Opt. Commun. 94, 197–202 (1992).
[CrossRef]

H. Weber, “Wave optical analysis of the phase space analyser,” J. Mod. Opt. 39, 543–559 (1992); J. Serna, P. M. Mejias, R. Martinez-Herrero, “Beam quality dependence on the coherence length of Gaussian Schell-model fields propagating through ABCDoptical systems,” J. Mod. Opt. 39, 625–635 (1992).
[CrossRef]

S. Danakas, P. K. Aravind, “Analogies between two optical systems (photon beam splitters and laser beams) and two quantum systems (the two-dimensional oscillator and the two-dimensional hydrogen atom),” Phys. Rev. A 45, 1973–1977 (1992).
[CrossRef] [PubMed]

1991 (1)

R. Gase, “The multimode laser radiation as a Gaussian Schell-model beam,” J. Mod. Opt. 38, 1107–1115 (1991).
[CrossRef]

1990 (3)

1989 (5)

R. A. Campos, B. E. A. Saleh, M. C. Teich, “Quantum-mechanical lossless beam splitters: SU(2) symmetry and photon statistics,” Phys. Rev. A 40, 1371–1384 (1989).
[CrossRef] [PubMed]

R. Simon, N. Mukunda, E. C. G. Sudarshan, “Hamilton’s theory of turns and a new geometrical representation for polarization optics,” Pramana 32, 769–792 (1989).
[CrossRef]

R. Simon, N. Mukunda, “Universal SU(2) gadget for polarization optics,” Phys. Lett. A 138, 474–480 (1989).
[CrossRef]

H. Jiao, S. R. Wilkinson, R. Y. Chiao, H. Nathel, “Two topological phases in optics by means of a nonplanar Mach–Zehnder interferometer,” Phys. Rev. A 39, 3475–3486 (1989).
[CrossRef] [PubMed]

G. Guattari, C. Palma, C. Padovani, “Cross-spectral densities with axial symmetry,” Opt. Commun. 73, 173–178 (1989).
[CrossRef]

1988 (6)

F. Gori, G. Guattari, C. Palma, C. Padovani, “A class of shape invariant fields,” Opt. Commun. 66, 255–259 (1988).
[CrossRef]

F. Gori, G. Guattari, C. Palma, C. Padovani, “Specular cross-spectral density functions,” Opt. Commun. 68, 239–243 (1988).
[CrossRef]

R. Simon, H. J. Kimble, E. C. G. Sudarshan, “Evolving geometric phase and its dynamical manifestation as a frequency shift: an optical experiment,” Phys. Rev. Lett. 61, 19–22 (1988).
[CrossRef] [PubMed]

R. Bhandari, J. Samuel, “Observation of topological phase by use of a laser interferometer,” Phys. Rev. Lett. 60, 1211–1213 (1988).
[CrossRef] [PubMed]

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Gaussian pure states in quantum mechanics and the symplectic group,” Phys. Rev. A 37, 2028–2038 (1988).
[CrossRef]

T. H. Chyba, L. J. Wang, L. Mandel, R. Simon, “Measurement of the Pancharatnam phase for a light beam,” Opt. Lett. 13, 562–564 (1988).
[CrossRef] [PubMed]

1987 (3)

G. S. Agrawal, J. T. Foley, E. Wolf, “The radiance and phase-space representations of the cross-spectral density operator,” Opt. Commun. 62, 67–72 (1987).
[CrossRef]

M. V. Berry, “The adiabatic phase and Pancharatnam’s phase for polarized light,” J. Mod. Opt. 34, 1401–1409 (1987).
[CrossRef]

F. Gori, G. Guattari, C. Padovani, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun. 64, 311–316 (1987).
[CrossRef]

1986 (4)

1985 (1)

R. Simon, E. C. G. Sundarshan, N. Mukunda, “Anisotropic Gaussian Schell-model beams: passage through optical systems and associated invariants,” Phys. Rev. A 31, 2419–2434 (1985), Appendix A in particular.
[CrossRef] [PubMed]

1984 (3)

E. Wolf, G. S. Agrawal, “Coherence theory of laser resonator modes,” J. Opt. Soc. Am. A 1, 541–546 (1984).
[CrossRef]

F. Gori, R. Grella, “Shape invariant propagation of polychromatic fields,” Opt. Commun. 49, 173–177 (1984).
[CrossRef]

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Generalized rays in first order optics: transformation properties of Gaussian Schell-model fields,” Phys. Rev. A 29, 3273–3279 (1984).
[CrossRef]

1983 (1)

F. Gori, “Mode propagation of the field generated by Collett–Wolf Schell model sources,” Opt. Commun. 46, 149–154 (1983).
[CrossRef]

1982 (2)

1981 (2)

E. Wolf, “A new description of second-order coherence phenomena in the space frequency domain,”AIP Conf. Proc. 65, 42–48 (1981).
[CrossRef]

E. Wolf, “New spectral representation of random sources and of the partially coherent fields that they generate,” Opt. Commun. 38, 3–6 (1981).
[CrossRef]

1980 (1)

F. Gori, “Collett–Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
[CrossRef]

1979 (2)

P. D. Santis, F. Gori, G. Guattari, C. Palma, “An example of a Collett–Wolf source,” Opt. Commun. 29, 256–260 (1979).
[CrossRef]

B. E. A. Saleh, “Intensity distribution due to a partially coherent field and the Collett–Wolf equivalence theorem in the Fresnel zone,” Opt. Commun. 30, 135–138 (1979).
[CrossRef]

1978 (2)

N. Mukunda, “Algebraic aspects of the Wigner distribution in quantum mechanics,” Pramana-J. Phys. 11, 1–15 (1978); R. Simon, E. C. G. Sudarshan, N. Mukunda, “Gaussian Wigner distributions in quantum mechanics and optics,” Phys. Rev. A 36, 3868–3880 (1987), and references therein.
[CrossRef] [PubMed]

W. H. Carter, M. Bertolotti, “An analysis of the far-field coherence and radiant intensity of light scattered from liquid crystals,”J. Opt. Soc. Am. 68, 329–333 (1978).
[CrossRef]

1956 (1)

S. Pancharatnam, “Generalized theory of interference and its applications,” Proc. Ind. Acad. Sci. A 44, 247–262 (1956); reprinted in Collected Works of S. Pancharatnam (Oxford U. Press, Oxford, 1975).

Agrawal, G. P.

Agrawal, G. S.

G. S. Agrawal, R. Simon, “Berry phase, interference of light beams, and the Hannay angle,” Phys. Rev. A 42, 6924–6927 (1990).
[CrossRef]

G. S. Agrawal, J. T. Foley, E. Wolf, “The radiance and phase-space representations of the cross-spectral density operator,” Opt. Commun. 62, 67–72 (1987).
[CrossRef]

E. Wolf, G. S. Agrawal, “Coherence theory of laser resonator modes,” J. Opt. Soc. Am. A 1, 541–546 (1984).
[CrossRef]

Aravind, P. K.

S. Danakas, P. K. Aravind, “Analogies between two optical systems (photon beam splitters and laser beams) and two quantum systems (the two-dimensional oscillator and the two-dimensional hydrogen atom),” Phys. Rev. A 45, 1973–1977 (1992).
[CrossRef] [PubMed]

Berry, M. V.

M. V. Berry, “The adiabatic phase and Pancharatnam’s phase for polarized light,” J. Mod. Opt. 34, 1401–1409 (1987).
[CrossRef]

Bertolotti, M.

Bhandari, R.

R. Bhandari, J. Samuel, “Observation of topological phase by use of a laser interferometer,” Phys. Rev. Lett. 60, 1211–1213 (1988).
[CrossRef] [PubMed]

Biedenharn, L. C.

See, for instance, L. C. Biedenharn, J. D. Louck, Angular Momentum in Quantum Physics: Theory and Application (Addison-Wesley, Reading, Mass., 1981), Chap. 4. More recently, Hamilton’s theory of turns has been generalized to the noncompact group SU(1, 1) in R. Simon, N. Mukunda, E. C. G. Sudarshan, “Hamilton’s theory of turns generalized to Sp(2, R),” Phys. Rev. Lett. 62, 1331–1334 (1989).
[CrossRef] [PubMed]

Campos, R. A.

R. A. Campos, B. E. A. Saleh, M. C. Teich, “Quantum-mechanical lossless beam splitters: SU(2) symmetry and photon statistics,” Phys. Rev. A 40, 1371–1384 (1989).
[CrossRef] [PubMed]

Carter, W. H.

Chiao, R. Y.

H. Jiao, S. R. Wilkinson, R. Y. Chiao, H. Nathel, “Two topological phases in optics by means of a nonplanar Mach–Zehnder interferometer,” Phys. Rev. A 39, 3475–3486 (1989).
[CrossRef] [PubMed]

Chyba, T. H.

Danakas, S.

S. Danakas, P. K. Aravind, “Analogies between two optical systems (photon beam splitters and laser beams) and two quantum systems (the two-dimensional oscillator and the two-dimensional hydrogen atom),” Phys. Rev. A 45, 1973–1977 (1992).
[CrossRef] [PubMed]

De Santis, P.

Flügge, S.

See, for example, S. Flügge, Practical Quantum Mechanics (Springer-Verlag, Berlin, 1971), Vol. 1, p. 107.

Foley, J. T.

G. S. Agrawal, J. T. Foley, E. Wolf, “The radiance and phase-space representations of the cross-spectral density operator,” Opt. Commun. 62, 67–72 (1987).
[CrossRef]

Gamliel, A.

Gase, R.

R. Gase, “The multimode laser radiation as a Gaussian Schell-model beam,” J. Mod. Opt. 38, 1107–1115 (1991).
[CrossRef]

Goodman, J. W.

Gori, F.

F. Gori, G. Guattari, C. Palma, C. Padovani, “Specular cross-spectral density functions,” Opt. Commun. 68, 239–243 (1988).
[CrossRef]

F. Gori, G. Guattari, C. Palma, C. Padovani, “A class of shape invariant fields,” Opt. Commun. 66, 255–259 (1988).
[CrossRef]

F. Gori, G. Guattari, C. Padovani, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun. 64, 311–316 (1987).
[CrossRef]

P. De Santis, F. Gori, G. Guattari, C. Palma, “Synthesis of partially coherent fields,” J. Opt. Soc. Am. A 3, 1258–1262 (1986).
[CrossRef]

F. Gori, R. Grella, “Shape invariant propagation of polychromatic fields,” Opt. Commun. 49, 173–177 (1984).
[CrossRef]

F. Gori, “Mode propagation of the field generated by Collett–Wolf Schell model sources,” Opt. Commun. 46, 149–154 (1983).
[CrossRef]

F. Gori, “Collett–Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
[CrossRef]

P. D. Santis, F. Gori, G. Guattari, C. Palma, “An example of a Collett–Wolf source,” Opt. Commun. 29, 256–260 (1979).
[CrossRef]

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980), p. 837, identity 7.374.8.

Grella, R.

F. Gori, R. Grella, “Shape invariant propagation of polychromatic fields,” Opt. Commun. 49, 173–177 (1984).
[CrossRef]

Guattari, G.

G. Guattari, C. Palma, C. Padovani, “Cross-spectral densities with axial symmetry,” Opt. Commun. 73, 173–178 (1989).
[CrossRef]

F. Gori, G. Guattari, C. Palma, C. Padovani, “A class of shape invariant fields,” Opt. Commun. 66, 255–259 (1988).
[CrossRef]

F. Gori, G. Guattari, C. Palma, C. Padovani, “Specular cross-spectral density functions,” Opt. Commun. 68, 239–243 (1988).
[CrossRef]

F. Gori, G. Guattari, C. Padovani, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun. 64, 311–316 (1987).
[CrossRef]

P. De Santis, F. Gori, G. Guattari, C. Palma, “Synthesis of partially coherent fields,” J. Opt. Soc. Am. A 3, 1258–1262 (1986).
[CrossRef]

P. D. Santis, F. Gori, G. Guattari, C. Palma, “An example of a Collett–Wolf source,” Opt. Commun. 29, 256–260 (1979).
[CrossRef]

Idell, P. S.

Jiao, H.

H. Jiao, S. R. Wilkinson, R. Y. Chiao, H. Nathel, “Two topological phases in optics by means of a nonplanar Mach–Zehnder interferometer,” Phys. Rev. A 39, 3475–3486 (1989).
[CrossRef] [PubMed]

Kimble, H. J.

R. Simon, H. J. Kimble, E. C. G. Sudarshan, “Evolving geometric phase and its dynamical manifestation as a frequency shift: an optical experiment,” Phys. Rev. Lett. 61, 19–22 (1988).
[CrossRef] [PubMed]

Klauder, J. R.

B. Yurke, S. L. McCall, J. R. Klauder, “SU(2) and SU(1, 1) interferometers,” Phys. Rev. A 33, 4033–4054 (1986).
[CrossRef] [PubMed]

Louck, J. D.

See, for instance, L. C. Biedenharn, J. D. Louck, Angular Momentum in Quantum Physics: Theory and Application (Addison-Wesley, Reading, Mass., 1981), Chap. 4. More recently, Hamilton’s theory of turns has been generalized to the noncompact group SU(1, 1) in R. Simon, N. Mukunda, E. C. G. Sudarshan, “Hamilton’s theory of turns generalized to Sp(2, R),” Phys. Rev. Lett. 62, 1331–1334 (1989).
[CrossRef] [PubMed]

Mandel, L.

Martinez-Herrero, R.

R. Martinez-Herrero, P. M. Mejias, “Expansion of the cross-spectral density of general fields and its applications to beam characterization,” Opt. Commun. 94, 197–202 (1992).
[CrossRef]

McCall, S. L.

B. Yurke, S. L. McCall, J. R. Klauder, “SU(2) and SU(1, 1) interferometers,” Phys. Rev. A 33, 4033–4054 (1986).
[CrossRef] [PubMed]

Mejias, P. M.

R. Martinez-Herrero, P. M. Mejias, “Expansion of the cross-spectral density of general fields and its applications to beam characterization,” Opt. Commun. 94, 197–202 (1992).
[CrossRef]

Mukunda, N.

R. Simon, N. Mukunda, “Twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 10, 95–109 (1993).
[CrossRef]

K. Sundar, R. Simon, N. Mukunda, “Twisted Gaussian Schell-model beams: II. Spectrum analysis and propagation characteristics,” J. Opt. Soc. Am. A 10, 2017–2023 (1993).
[CrossRef]

R. Simon, N. Mukunda, “Minimal three component SU(2) gadget for polarization optics,” Phys. Lett. A 143, 165–169 (1990).
[CrossRef]

R. Simon, N. Mukunda, E. C. G. Sudarshan, “Hamilton’s theory of turns and a new geometrical representation for polarization optics,” Pramana 32, 769–792 (1989).
[CrossRef]

R. Simon, N. Mukunda, “Universal SU(2) gadget for polarization optics,” Phys. Lett. A 138, 474–480 (1989).
[CrossRef]

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Gaussian pure states in quantum mechanics and the symplectic group,” Phys. Rev. A 37, 2028–2038 (1988).
[CrossRef]

R. Simon, E. C. G. Sundarshan, N. Mukunda, “Anisotropic Gaussian Schell-model beams: passage through optical systems and associated invariants,” Phys. Rev. A 31, 2419–2434 (1985), Appendix A in particular.
[CrossRef] [PubMed]

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Generalized rays in first order optics: transformation properties of Gaussian Schell-model fields,” Phys. Rev. A 29, 3273–3279 (1984).
[CrossRef]

N. Mukunda, “Algebraic aspects of the Wigner distribution in quantum mechanics,” Pramana-J. Phys. 11, 1–15 (1978); R. Simon, E. C. G. Sudarshan, N. Mukunda, “Gaussian Wigner distributions in quantum mechanics and optics,” Phys. Rev. A 36, 3868–3880 (1987), and references therein.
[CrossRef] [PubMed]

Nathel, H.

H. Jiao, S. R. Wilkinson, R. Y. Chiao, H. Nathel, “Two topological phases in optics by means of a nonplanar Mach–Zehnder interferometer,” Phys. Rev. A 39, 3475–3486 (1989).
[CrossRef] [PubMed]

Padovani, C.

G. Guattari, C. Palma, C. Padovani, “Cross-spectral densities with axial symmetry,” Opt. Commun. 73, 173–178 (1989).
[CrossRef]

F. Gori, G. Guattari, C. Palma, C. Padovani, “A class of shape invariant fields,” Opt. Commun. 66, 255–259 (1988).
[CrossRef]

F. Gori, G. Guattari, C. Palma, C. Padovani, “Specular cross-spectral density functions,” Opt. Commun. 68, 239–243 (1988).
[CrossRef]

F. Gori, G. Guattari, C. Padovani, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun. 64, 311–316 (1987).
[CrossRef]

Palma, C.

G. Guattari, C. Palma, C. Padovani, “Cross-spectral densities with axial symmetry,” Opt. Commun. 73, 173–178 (1989).
[CrossRef]

F. Gori, G. Guattari, C. Palma, C. Padovani, “A class of shape invariant fields,” Opt. Commun. 66, 255–259 (1988).
[CrossRef]

F. Gori, G. Guattari, C. Palma, C. Padovani, “Specular cross-spectral density functions,” Opt. Commun. 68, 239–243 (1988).
[CrossRef]

P. De Santis, F. Gori, G. Guattari, C. Palma, “Synthesis of partially coherent fields,” J. Opt. Soc. Am. A 3, 1258–1262 (1986).
[CrossRef]

P. D. Santis, F. Gori, G. Guattari, C. Palma, “An example of a Collett–Wolf source,” Opt. Commun. 29, 256–260 (1979).
[CrossRef]

Pancharatnam, S.

S. Pancharatnam, “Generalized theory of interference and its applications,” Proc. Ind. Acad. Sci. A 44, 247–262 (1956); reprinted in Collected Works of S. Pancharatnam (Oxford U. Press, Oxford, 1975).

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980), p. 837, identity 7.374.8.

Saleh, B. E. A.

R. A. Campos, B. E. A. Saleh, M. C. Teich, “Quantum-mechanical lossless beam splitters: SU(2) symmetry and photon statistics,” Phys. Rev. A 40, 1371–1384 (1989).
[CrossRef] [PubMed]

B. E. A. Saleh, “Intensity distribution due to a partially coherent field and the Collett–Wolf equivalence theorem in the Fresnel zone,” Opt. Commun. 30, 135–138 (1979).
[CrossRef]

Samuel, J.

R. Bhandari, J. Samuel, “Observation of topological phase by use of a laser interferometer,” Phys. Rev. Lett. 60, 1211–1213 (1988).
[CrossRef] [PubMed]

Santis, P. D.

P. D. Santis, F. Gori, G. Guattari, C. Palma, “An example of a Collett–Wolf source,” Opt. Commun. 29, 256–260 (1979).
[CrossRef]

Schwinger, J.

J. Schwinger, “On angular momentum,” in Quantum Theory of Angular Momentum, L. C. Biedenharn, H. Van Dam, eds. (Academic, New York, 1965), pp. 229–279.

Simon, R.

K. Sundar, R. Simon, N. Mukunda, “Twisted Gaussian Schell-model beams: II. Spectrum analysis and propagation characteristics,” J. Opt. Soc. Am. A 10, 2017–2023 (1993).
[CrossRef]

R. Simon, N. Mukunda, “Twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 10, 95–109 (1993).
[CrossRef]

G. S. Agrawal, R. Simon, “Berry phase, interference of light beams, and the Hannay angle,” Phys. Rev. A 42, 6924–6927 (1990).
[CrossRef]

R. Simon, N. Mukunda, “Minimal three component SU(2) gadget for polarization optics,” Phys. Lett. A 143, 165–169 (1990).
[CrossRef]

R. Simon, N. Mukunda, E. C. G. Sudarshan, “Hamilton’s theory of turns and a new geometrical representation for polarization optics,” Pramana 32, 769–792 (1989).
[CrossRef]

R. Simon, N. Mukunda, “Universal SU(2) gadget for polarization optics,” Phys. Lett. A 138, 474–480 (1989).
[CrossRef]

T. H. Chyba, L. J. Wang, L. Mandel, R. Simon, “Measurement of the Pancharatnam phase for a light beam,” Opt. Lett. 13, 562–564 (1988).
[CrossRef] [PubMed]

R. Simon, H. J. Kimble, E. C. G. Sudarshan, “Evolving geometric phase and its dynamical manifestation as a frequency shift: an optical experiment,” Phys. Rev. Lett. 61, 19–22 (1988).
[CrossRef] [PubMed]

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Gaussian pure states in quantum mechanics and the symplectic group,” Phys. Rev. A 37, 2028–2038 (1988).
[CrossRef]

R. Simon, E. C. G. Sundarshan, N. Mukunda, “Anisotropic Gaussian Schell-model beams: passage through optical systems and associated invariants,” Phys. Rev. A 31, 2419–2434 (1985), Appendix A in particular.
[CrossRef] [PubMed]

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Generalized rays in first order optics: transformation properties of Gaussian Schell-model fields,” Phys. Rev. A 29, 3273–3279 (1984).
[CrossRef]

Starikov, A.

Sudarshan, E. C. G.

R. Simon, N. Mukunda, E. C. G. Sudarshan, “Hamilton’s theory of turns and a new geometrical representation for polarization optics,” Pramana 32, 769–792 (1989).
[CrossRef]

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Gaussian pure states in quantum mechanics and the symplectic group,” Phys. Rev. A 37, 2028–2038 (1988).
[CrossRef]

R. Simon, H. J. Kimble, E. C. G. Sudarshan, “Evolving geometric phase and its dynamical manifestation as a frequency shift: an optical experiment,” Phys. Rev. Lett. 61, 19–22 (1988).
[CrossRef] [PubMed]

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Generalized rays in first order optics: transformation properties of Gaussian Schell-model fields,” Phys. Rev. A 29, 3273–3279 (1984).
[CrossRef]

Sundar, K.

Sundarshan, E. C. G.

R. Simon, E. C. G. Sundarshan, N. Mukunda, “Anisotropic Gaussian Schell-model beams: passage through optical systems and associated invariants,” Phys. Rev. A 31, 2419–2434 (1985), Appendix A in particular.
[CrossRef] [PubMed]

Teich, M. C.

R. A. Campos, B. E. A. Saleh, M. C. Teich, “Quantum-mechanical lossless beam splitters: SU(2) symmetry and photon statistics,” Phys. Rev. A 40, 1371–1384 (1989).
[CrossRef] [PubMed]

Wang, L. J.

Weber, H.

H. Weber, “Wave optical analysis of the phase space analyser,” J. Mod. Opt. 39, 543–559 (1992); J. Serna, P. M. Mejias, R. Martinez-Herrero, “Beam quality dependence on the coherence length of Gaussian Schell-model fields propagating through ABCDoptical systems,” J. Mod. Opt. 39, 625–635 (1992).
[CrossRef]

Wigner, E. P.

E. P. Wigner, Group Theory and Its Application to the Quantum Mechanics of Atomic Spectra (Academic, New York, 1959), p. 167.

Wilkinson, S. R.

H. Jiao, S. R. Wilkinson, R. Y. Chiao, H. Nathel, “Two topological phases in optics by means of a nonplanar Mach–Zehnder interferometer,” Phys. Rev. A 39, 3475–3486 (1989).
[CrossRef] [PubMed]

Wolf, E.

Yurke, B.

B. Yurke, S. L. McCall, J. R. Klauder, “SU(2) and SU(1, 1) interferometers,” Phys. Rev. A 33, 4033–4054 (1986).
[CrossRef] [PubMed]

AIP Conf. Proc. (1)

E. Wolf, “A new description of second-order coherence phenomena in the space frequency domain,”AIP Conf. Proc. 65, 42–48 (1981).
[CrossRef]

J. Mod. Opt. (3)

H. Weber, “Wave optical analysis of the phase space analyser,” J. Mod. Opt. 39, 543–559 (1992); J. Serna, P. M. Mejias, R. Martinez-Herrero, “Beam quality dependence on the coherence length of Gaussian Schell-model fields propagating through ABCDoptical systems,” J. Mod. Opt. 39, 625–635 (1992).
[CrossRef]

R. Gase, “The multimode laser radiation as a Gaussian Schell-model beam,” J. Mod. Opt. 38, 1107–1115 (1991).
[CrossRef]

M. V. Berry, “The adiabatic phase and Pancharatnam’s phase for polarized light,” J. Mod. Opt. 34, 1401–1409 (1987).
[CrossRef]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (7)

Opt. Commun. (12)

G. S. Agrawal, J. T. Foley, E. Wolf, “The radiance and phase-space representations of the cross-spectral density operator,” Opt. Commun. 62, 67–72 (1987).
[CrossRef]

P. D. Santis, F. Gori, G. Guattari, C. Palma, “An example of a Collett–Wolf source,” Opt. Commun. 29, 256–260 (1979).
[CrossRef]

F. Gori, “Collett–Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
[CrossRef]

B. E. A. Saleh, “Intensity distribution due to a partially coherent field and the Collett–Wolf equivalence theorem in the Fresnel zone,” Opt. Commun. 30, 135–138 (1979).
[CrossRef]

E. Wolf, “New spectral representation of random sources and of the partially coherent fields that they generate,” Opt. Commun. 38, 3–6 (1981).
[CrossRef]

R. Martinez-Herrero, P. M. Mejias, “Expansion of the cross-spectral density of general fields and its applications to beam characterization,” Opt. Commun. 94, 197–202 (1992).
[CrossRef]

F. Gori, “Mode propagation of the field generated by Collett–Wolf Schell model sources,” Opt. Commun. 46, 149–154 (1983).
[CrossRef]

F. Gori, R. Grella, “Shape invariant propagation of polychromatic fields,” Opt. Commun. 49, 173–177 (1984).
[CrossRef]

F. Gori, G. Guattari, C. Padovani, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun. 64, 311–316 (1987).
[CrossRef]

F. Gori, G. Guattari, C. Palma, C. Padovani, “A class of shape invariant fields,” Opt. Commun. 66, 255–259 (1988).
[CrossRef]

F. Gori, G. Guattari, C. Palma, C. Padovani, “Specular cross-spectral density functions,” Opt. Commun. 68, 239–243 (1988).
[CrossRef]

G. Guattari, C. Palma, C. Padovani, “Cross-spectral densities with axial symmetry,” Opt. Commun. 73, 173–178 (1989).
[CrossRef]

Opt. Lett. (1)

Phys. Lett. A (2)

R. Simon, N. Mukunda, “Universal SU(2) gadget for polarization optics,” Phys. Lett. A 138, 474–480 (1989).
[CrossRef]

R. Simon, N. Mukunda, “Minimal three component SU(2) gadget for polarization optics,” Phys. Lett. A 143, 165–169 (1990).
[CrossRef]

Phys. Rev. A (8)

R. Simon, E. C. G. Sundarshan, N. Mukunda, “Anisotropic Gaussian Schell-model beams: passage through optical systems and associated invariants,” Phys. Rev. A 31, 2419–2434 (1985), Appendix A in particular.
[CrossRef] [PubMed]

H. Jiao, S. R. Wilkinson, R. Y. Chiao, H. Nathel, “Two topological phases in optics by means of a nonplanar Mach–Zehnder interferometer,” Phys. Rev. A 39, 3475–3486 (1989).
[CrossRef] [PubMed]

G. S. Agrawal, R. Simon, “Berry phase, interference of light beams, and the Hannay angle,” Phys. Rev. A 42, 6924–6927 (1990).
[CrossRef]

R. A. Campos, B. E. A. Saleh, M. C. Teich, “Quantum-mechanical lossless beam splitters: SU(2) symmetry and photon statistics,” Phys. Rev. A 40, 1371–1384 (1989).
[CrossRef] [PubMed]

B. Yurke, S. L. McCall, J. R. Klauder, “SU(2) and SU(1, 1) interferometers,” Phys. Rev. A 33, 4033–4054 (1986).
[CrossRef] [PubMed]

S. Danakas, P. K. Aravind, “Analogies between two optical systems (photon beam splitters and laser beams) and two quantum systems (the two-dimensional oscillator and the two-dimensional hydrogen atom),” Phys. Rev. A 45, 1973–1977 (1992).
[CrossRef] [PubMed]

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Generalized rays in first order optics: transformation properties of Gaussian Schell-model fields,” Phys. Rev. A 29, 3273–3279 (1984).
[CrossRef]

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Gaussian pure states in quantum mechanics and the symplectic group,” Phys. Rev. A 37, 2028–2038 (1988).
[CrossRef]

Phys. Rev. Lett. (2)

R. Simon, H. J. Kimble, E. C. G. Sudarshan, “Evolving geometric phase and its dynamical manifestation as a frequency shift: an optical experiment,” Phys. Rev. Lett. 61, 19–22 (1988).
[CrossRef] [PubMed]

R. Bhandari, J. Samuel, “Observation of topological phase by use of a laser interferometer,” Phys. Rev. Lett. 60, 1211–1213 (1988).
[CrossRef] [PubMed]

Pramana (1)

R. Simon, N. Mukunda, E. C. G. Sudarshan, “Hamilton’s theory of turns and a new geometrical representation for polarization optics,” Pramana 32, 769–792 (1989).
[CrossRef]

Pramana-J. Phys. (1)

N. Mukunda, “Algebraic aspects of the Wigner distribution in quantum mechanics,” Pramana-J. Phys. 11, 1–15 (1978); R. Simon, E. C. G. Sudarshan, N. Mukunda, “Gaussian Wigner distributions in quantum mechanics and optics,” Phys. Rev. A 36, 3868–3880 (1987), and references therein.
[CrossRef] [PubMed]

Proc. Ind. Acad. Sci. A (1)

S. Pancharatnam, “Generalized theory of interference and its applications,” Proc. Ind. Acad. Sci. A 44, 247–262 (1956); reprinted in Collected Works of S. Pancharatnam (Oxford U. Press, Oxford, 1975).

Other (5)

See, for instance, L. C. Biedenharn, J. D. Louck, Angular Momentum in Quantum Physics: Theory and Application (Addison-Wesley, Reading, Mass., 1981), Chap. 4. More recently, Hamilton’s theory of turns has been generalized to the noncompact group SU(1, 1) in R. Simon, N. Mukunda, E. C. G. Sudarshan, “Hamilton’s theory of turns generalized to Sp(2, R),” Phys. Rev. Lett. 62, 1331–1334 (1989).
[CrossRef] [PubMed]

J. Schwinger, “On angular momentum,” in Quantum Theory of Angular Momentum, L. C. Biedenharn, H. Van Dam, eds. (Academic, New York, 1965), pp. 229–279.

E. P. Wigner, Group Theory and Its Application to the Quantum Mechanics of Atomic Spectra (Academic, New York, 1959), p. 167.

See, for example, S. Flügge, Practical Quantum Mechanics (Springer-Verlag, Berlin, 1971), Vol. 1, p. 107.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980), p. 837, identity 7.374.8.

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Equations (59)

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W ( ρ , ρ ; ν ) = n λ n ( ν ) φ n ( ρ ; ν ) φ n * ( ρ ; ν ) .
W ( ρ , ρ ; ν ) φ n ( ρ ; ν ) d 2 ρ = λ n ( ν ) φ n ( ρ ; ν ) .
[ x ^ , p ^ x ] = [ y ^ , p ^ y ] = i .
x ^ ρ = x ρ , y ^ ρ = y ρ .
ρ ρ = δ ( ρ - ρ ) ,             d 2 ρ ρ ρ = 1 ,
ρ x ^ ρ = x δ ( ρ - ρ ) , ρ y ^ ρ = y δ ( ρ - ρ ) ,
ρ p ^ x ρ = - i ( / x ) δ ( ρ - ρ ) , ρ p ^ y ρ = - i ( / y ) δ ( ρ - ρ ) .
ρ W ^ ( ν ) ρ = W ( ρ , ρ ; ν ) ,
W ^ ( ν ) = d 2 ρ d 2 ρ W ( ρ , ρ ; ν ) ρ ρ .
n ; ν = d 2 ρ φ n ( ρ ; ν ) } ρ ,
W ^ ( ν ) = n λ n ( ν ) n ; ν n ; ν ,
W ^ ( ν ) n ; ν = λ n ( ν ) n ; ν ,
n ; ν n ; ν = δ n n .
W ( ρ , ρ ; ν ) = I ( ν ) 2 π σ s ( ν ) 2 exp [ - 1 4 σ s ( ν ) 2 ( ρ 2 + ρ 2 ) ] × exp [ - 1 2 σ g ( ν ) 2 ( ρ - ρ ) 2 ] × exp [ - i u ( ν ) ρ ρ ] .
- σ g ( ν ) - 2 u ( ν ) σ g ( ν ) - 2 ,
a ( ν ) = 1 4 σ s ( ν ) 2 ,             b ( ν ) = 1 2 σ g ( ν ) 2 .
a ^ x ( c ) = c / 2 x ^ + i / 2 c p ^ x , a ^ y ( c ) = c / 2 y ^ + i / 2 c p ^ y , a ^ x ( c ) = c / 2 x ^ - i / 2 c p ^ x , a ^ y ( c ) = c / 2 y ^ - i / 2 c p ^ y .
[ a ^ x ( c ) , a ^ x ( c ) ] = [ a ^ y ( c ) , a ^ y ( c ) ] = 1 ,
J ^ 1 ( c ) = ( 1 / 2 ) [ a ^ x ( c ) a ^ x ( c ) - a ^ y ( c ) a ^ y ( c ) ] , J ^ 2 ( c ) = ( 1 / 2 ) [ a ^ y ( c ) a ^ x ( c ) + a ^ x ( c ) a ^ y ( c ) ] , J ^ 3 ( c ) = ( i / 2 ) [ a ^ y ( c ) a ^ x ( c ) - a ^ x ( c ) a ^ y ( c ) ] .
[ J ^ l ( c ) , J ^ m ( c ) ] = i l m n J ^ n ( c ) ,
J ^ 1 ( c ) = ( 1 / 4 ) [ c ( x ^ 2 - y ^ 2 ) + c - 1 ( p ^ x 2 - p ^ y 2 ) ] , J ^ 2 ( c ) = ( 1 / 2 ) [ c x ^ y ^ + c - 1 p ^ x p ^ y ] , J ^ 3 ( c ) = ( 1 / 2 ) [ x ^ p ^ y - y ^ p ^ x ] .
J ^ 0 ( c ) = ( 1 / 2 ) [ a ^ x ( c ) a ^ x ( c ) + a ^ y ( c ) a ^ y ( c ) ] = ( 1 / 4 ) [ c ( x ^ 2 + y ^ 2 ) + c - 1 ( p ^ x 2 + p ^ y 2 ) ] - ( 1 / 2 ) .
J ^ ( c ) 2 = J ^ 1 ( c ) 2 + J ^ 2 ( c ) 2 + J ^ 3 ( c ) 2 = J ^ 0 ( c ) [ J ^ 0 ( c ) + 1 ] .
[ J ^ l ( c ) , J ^ ( c ) 2 ] = 0 , [ J ^ 0 ( c ) , J ^ l ( c ) ] = 0 ,             l = 1 , 2 , 3.
[ J ^ k ( c ) , W ^ ( ν ) ] = 0 ,             k = 1 , 2 , 3 , [ J ^ 2 ( c ) , W ^ ( ν ) ] = 0 ,
c = 2 ( a 2 + 2 a b ) 1 / 2 .
c = 2 ( a 2 + 2 a b + u 2 / 4 ) 1 / 2 ,
[ J ^ 3 , W ^ ( ν ) ] = 0 , [ J ^ 1 2 + J ^ 2 2 , W ^ ( ν ) ] = 0 , [ J ^ 2 , W ^ ( ν ) ] = 0.
N ^ x = a ^ x a ^ x ,             N ^ y = a ^ y a ^ y ,
N ^ x n x = n x n x ,             n x = 0 , 1 , 2 , , φ n x ( x ) = x n x = ( 1 2 n x n x ! c π ) 1 / 2 exp [ - ( 1 / 2 ) c x 2 ] H n x ( c x ) ,
J ^ 0 = ( 1 / 2 ) ( N ^ x + N ^ y ) , J ^ 1 = ( 1 / 2 ) ( N ^ x - N ^ y ) .
n x , n y = n x n y , J ^ 0 n x , n y = ( 1 / 2 ) ( n x + n y ) n x , n y , J ^ 1 n x , n y = ( 1 / 2 ) ( n x - n y ) n x , n y .
j , m = n x , n y , J ^ 0 j , m = j j , m , J ^ 1 j , m = m j , m .
φ j , m ( ρ ) = ρ j , m = ρ n x , n y = [ 1 2 2 j ( j + m ) ! ( j - m ) ! c π ] 1 / 2 × exp [ - ( 1 / 2 ) c ( x 2 + y 2 ) ] H j + m ( c x ) H j - m ( c y ) , j = 0 , 1 / 2 , 1 , , m = j , j - 1 , , - j .
U ^ ( α , β , γ ) = exp ( - i α J ^ 1 ) exp ( - i β J ^ 3 ) exp ( - i γ J ^ 1 )
U ^ ( α , β , γ ) j , m = m = - j j D m , m j ( α , β , γ ) j , m , D m , m j ( α , β , γ ) = j , m } U ^ ( α , β , γ ) j , m = exp ( - i α m ) exp ( - i γ m ) d m , m j ( β ) , d m , m j ( β ) = μ ( - 1 ) μ - m + m × [ ( j + m ) ! ( j - m ) ! ( j + m ) ! ( j - m ) ! ] 1 / 2 ( j - m - μ ) ! ( j + m - μ ) ! μ ! ( m - m + μ ) ! × [ cos ( β / 2 ) ] 2 j + m - m - 2 μ [ sin ( β / 2 ) ] m - m + 2 μ .
J ^ 3 = U ^ 0 J ^ 1 U ^ 0 ,             U ^ 0 = exp [ i ( π / 2 ) J ^ 2 ] .
j , m = U ^ 0 j , m , J ^ 3 j , m = m j , m , J ^ 0 j , m = j j , m .
exp ( i θ J ^ 2 ) = exp [ - i ( π / 2 ) J ^ 1 ] exp ( - i θ J ^ 3 ) exp [ i ( π / 2 ) J ^ 1 ] ,
U ^ 0 = U ^ ( α , β , γ ) = U ^ ( π / 2 , π / 2 , - π / 2 ) .
j , m = U ^ 0 j , m = m = - j j Ω m , m j j , m , Ω m , m j = j m U ^ 0 j m = D m , m j ( π / 2 , π / 2 , - π / 2 ) = ( i ) m - m 1 2 j μ ( - 1 ) μ - m + m × [ ( j + m ) ! ( j - m ) ! ( j + m ) ! ( j - m ) ! ] 1 / 2 ( j - m - μ ) ! ( j + m - μ ) ! μ ! ( m - m + μ ) ! .
Φ j , m ( ρ ) = ρ j , m = m = - j j Ω m , m j φ j , m ( ρ ) = m = - j j Ω m , m j [ 2 2 j ( j + m ) ! ( j - m ) ! ] - 1 / 2 c / π × exp [ - ( 1 / 2 ) c ( x 2 + y 2 ) ] H j + m ( c x ) H j - m ( c y ) , j = 0 , 1 / 2 , 1 , , m = j , j - 1 , , - j .
Φ ˜ j , m ( ρ , θ ) = c π [ ( j - m ) ! ( j + m ) ! ] 1 / 2 ( c ρ ) 2 m L j - m 2 m ( c ρ 2 ) × exp [ - ( c / 2 ) ρ 2 ] exp ( i 2 m θ ) , J ^ 3 Φ ˜ j , m ( ρ , θ ) = - i 1 2 θ Φ ˜ j , m ( ρ , θ ) = m Φ ˜ j , m ( ρ , θ ) , J ^ 0 Φ ˜ j , m ( ρ , θ ) = [ - 1 4 c ( 2 ρ 2 + 1 ρ ρ - 4 m 2 ρ 2 ) + 1 4 c ρ 2 - 1 2 ] × Φ ˜ j , m ( ρ , θ ) = j Φ ˜ j , m ( ρ , θ ) , j = 0 , 1 / 2 , 1 , , m = j , j - 1 , , - j .
Φ j , m ( x , y ) = exp [ i χ ( j , m ) ] Φ ˜ j , m ( ρ , θ ) .
χ ( j , m ) = ( - π / 2 ) [ 2 ( j - m ) - ( j - m ) ] .
m = - j j Ω m m j c / π [ 2 2 j ( j + m ) ! ( j - m ) ! ] - 1 / 2 × exp [ - ( 1 / 2 ) c ( x 2 + y 2 ) H j + m ( c x ) H j - m ( c y ) = exp { - i π 2 [ 2 ( j - m ) - ( j - m ) ] } × c π [ ( j - m ) ! ( j + m ) ! ] 1 / 2 × ( c ρ ) 2 m L j - m 2 m ( c ρ 2 ) exp ( - c ρ 2 / 2 ) exp ( i 2 m θ ) ,
W ( ρ , ρ ; ν ) = 2 I a π exp [ - ( a + b ) ( x 2 + y 2 ) ] × exp [ - ( a + b ) ( x 2 + y 2 ) ] × exp [ ( 2 b x + i u y ) x + ( 2 b y - i u x ) y ] .
N m j = [ 2 2 j ( j + m ) ! ( j - m ) ! ] - 1 / 2 ( c / π ) 1 / 2 .
x = ( a + b + c / 2 ) 1 / 2 x , y = ( a + b + c / 2 ) 1 / 2 y .
- d x exp [ - ( x - α ) 2 ] H j + m ( β x ) = π 1 / 2 ( 1 - β 2 ) ( j + m ) / 2 H j + m [ α β ( 1 - β 2 ) 1 / 2 ] ,
α = 2 b x + i u y 2 ( a + b + c / 2 ) 1 / 2 ,             β = ( c a + b + c / 2 ) 1 / 2 .
α β ( 1 - β 2 ) 1 / 2 = c ( 2 b x + i u y ) ( 4 b 2 - u 2 ) 1 / 2 .
d x d y W ( ρ , ρ ; ν ) m Ω m m j N m j H j + m ( c x ) H j - m ( c y ) × exp [ - ( 1 / 2 ) c ( x 2 + y 2 ) ] = 2 a I a + b + c / 2 ( a + b - c / 2 a + b + c / 2 ) j m Ω m m j N m j , × H j + m [ c ( 2 b x + i u y ) ( 4 b 2 - u 2 ) 1 / 2 ] H j - m [ c ( 2 b y - i u x ) ( 4 b 2 - u 2 ) 1 / 2 ] × exp [ - ( 1 / 2 ) c ( x 2 + y 2 ) ] .
[ x y ] [ 2 b x + i u y ( 4 b 2 - u 2 ) 1 / 2 2 b y - i u x ( 4 b 2 - u 2 ) 1 / 2 ] = ( 4 b 2 - u 2 ) - 1 / 2 [ 2 b i u - i u 2 b ] [ x y ] .
[ ρ θ ] [ ρ θ + i ] ,
exp ( - 2 ) = 2 b + u 2 b - u .
exp ( 2 i m θ ) exp [ 2 i m ( θ + i ) ] = ( 2 b + u 2 b - u ) m exp ( 2 i m θ ) .
d 2 ρ W ( ρ , ρ ; ν ) Φ j , m ( ρ ) = λ j , m Φ j , m ( ρ ) ,
λ j , m = 2 a I a + b + ( c / 2 ) ( a + b - c / 2 a + b + c / 12 ) j ( 2 b + u 2 b - u ) m .

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