Abstract

The recently discovered twist phase is studied in the context of the full ten-parameter family of partially coherent general anisotropic Gaussian Schell-model beams. It is shown that the nonnegativity requirement on the cross-spectral density of the beam demands that the strength of the twist phase be bounded from above by the inverse of the transverse coherence area of the beam. The twist phase as a two-point function is shown to have the structure of the generalized Huygens kernel or Green’s function of a first-order system. The ray-transfer matrix of this system is exhibited. Wolf-type coherent-mode decomposition of the twist phase is carried out. Imposition of the twist phase on an otherwise untwisted beam is shown to result in a linear transformation in the ray phase space of the Wigner distribution. Though this transformation preserves the four-dimensional phase-space volume, it is not symplectic and hence it can, when impressed on a Wigner distribution, push it out of the convex set of all bona fide Wigner distributions unless the original Wigner distribution was sufficiently deep into the interior of the set.

© 1998 Optical Society of America

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    [CrossRef]
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  5. 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).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  26. 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.
  27. M. Selvadoray, M. Sanjay Kumar, R. Simon, “Photon distribution in two-mode squeezed coherent states with complex displacement and squeeze parameters,” Phys. Rev. A 49, 4957–4967 (1994).
    [CrossRef] [PubMed]
  28. 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]

1996 (1)

R. Simon, A. T. Friberg, E. Wolf, “Transfer of radiance by twisted Gaussian Schell-model beams in paraxial systems,” Pure Appl. Opt. 5, 331–343 (1996).
[CrossRef]

1995 (2)

K. Sundar, N. Mukunda, R. Simon, “Coherent-mode decomposition of general anisotropic Gaussian Schell-model beams,” J. Opt. Soc. Am. A 12, 560–569 (1995).
[CrossRef]

R. Gase, “Representation of Laguerre–Gaussian modes by the Wigner distribution function,” IEEE J. Quantum Electron. 31, 1811–1818 (1995).
[CrossRef]

1994 (6)

G. Nemes, A. E. Siegman, “Measurement of all ten second-order moments of an astigmatic beam by the use of rotating simple astigmatic (anamorphic) optics,” J. Opt. Soc. Am. A 11, 2257–2264 (1994).
[CrossRef]

M. Selvadoray, M. Sanjay Kumar, R. Simon, “Photon distribution in two-mode squeezed coherent states with complex displacement and squeeze parameters,” Phys. Rev. A 49, 4957–4967 (1994).
[CrossRef] [PubMed]

R. Simon, N. Mukunda, B. Dutta, “Quantum-noise matrix for multimode systems: U(n) invariance squeezing, and normal forms,” Phys. Rev. A 49, 1567–1583 (1994).
[CrossRef] [PubMed]

R. Gase, “Methods of quantum mechanics applied to partially coherent light beams,” J. Opt. Soc. Am. A 11, 2121–2129 (1994).
[CrossRef]

A. T. Friberg, E. Tervonen, J. Turunen, “Interpretation and experimental demonstration of twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 11, 1818–1826 (1994).
[CrossRef]

D. Ambrosini, V. Bagini, F. Gori, M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41, 1391–1399 (1994).
[CrossRef]

1993 (3)

1992 (2)

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]

M. J. Bastiaans, “ABCD law for partially coherent Gaussian light propagating through first-order optical systems,” Opt. Quantum Electron. 24, 1011–1019 (1992).
[CrossRef]

1988 (1)

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Gaussian pure states in quantum mechanics and the symplectic group,” Phys. Rev. A 37, 3028–3038 (1988).
[CrossRef] [PubMed]

1987 (1)

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Gaussian Wigner distributions in quantum mechanics and optics,” Phys. Rev. A 36, 3868–3880 (1987).
[CrossRef] [PubMed]

1985 (1)

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

1983 (1)

F. Gori, G. Guattari, “A new type of optical fields,” Opt. Commun. 48, 7–12 (1983).
[CrossRef]

1982 (3)

1981 (1)

H. Bacry, M. Cadilhac, “The metaplectic group and Fourier optics,” Phys. Rev. A 23, 2533–2536 (1981); V. Guillemin, S. Sternberg, Symplectic Techniques in Physics (Cambridge U. Press, Cambridge, UK, 1984); R. G. Littlejohn, “The semiclassical evolution of wave packets,” Phys. Rep. 138, 193–291 (1986); R. Simon, N. Mukunda, “The two-dimensional symplectic and metaplectic groups and their universal cover,” in Symmetries in Science, B. Gruber, ed. (Plenum, New York, 1993), Vol. VI, pp. 659–689; R. Simon, N. Mukunda, “Bargmann invariant and the geometry of the Gouy effect,” Phys. Rev. Lett. 70, 880–883 (1993).
[CrossRef] [PubMed]

1980 (1)

F. Gori, “Collett–Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980); R. Gase, “The multimode laser radiation as a Gaussian Schell-model beam,” J. Mod. Opt. 38, 1107–1115 (1991); 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]

1979 (1)

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); J. T. Foley, M. S. Zubairy, “The directionality of Gaussian Schell-model beams,” Opt. Commun. 26, 297–300 (1978); A. T. Friberg, R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41, 383–387 (1982); E. Wolf, E. Collett, “Partially coherent sources which produce the same intensity distribution as a laser,” Opt. Commun. 25, 293–296 (1978); 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); H. Weber, “Wave optical analysis of the phase space analyzer,” J. Mod. Opt. 39, 543–559 (1992).
[CrossRef]

1978 (1)

M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. 25, 26–30 (1978); M. J. Bastiaans, “Wigner distribution function and its application to first-order optics,” J. Opt. Soc. Am. 69, 1710–1716 (1979).
[CrossRef]

1932 (1)

E. P. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[CrossRef]

Ambrosini, D.

D. Ambrosini, V. Bagini, F. Gori, M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41, 1391–1399 (1994).
[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]

Bacry, H.

H. Bacry, M. Cadilhac, “The metaplectic group and Fourier optics,” Phys. Rev. A 23, 2533–2536 (1981); V. Guillemin, S. Sternberg, Symplectic Techniques in Physics (Cambridge U. Press, Cambridge, UK, 1984); R. G. Littlejohn, “The semiclassical evolution of wave packets,” Phys. Rep. 138, 193–291 (1986); R. Simon, N. Mukunda, “The two-dimensional symplectic and metaplectic groups and their universal cover,” in Symmetries in Science, B. Gruber, ed. (Plenum, New York, 1993), Vol. VI, pp. 659–689; R. Simon, N. Mukunda, “Bargmann invariant and the geometry of the Gouy effect,” Phys. Rev. Lett. 70, 880–883 (1993).
[CrossRef] [PubMed]

Bagini, V.

D. Ambrosini, V. Bagini, F. Gori, M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41, 1391–1399 (1994).
[CrossRef]

Bastiaans, M. J.

M. J. Bastiaans, “ABCD law for partially coherent Gaussian light propagating through first-order optical systems,” Opt. Quantum Electron. 24, 1011–1019 (1992).
[CrossRef]

M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. 25, 26–30 (1978); M. J. Bastiaans, “Wigner distribution function and its application to first-order optics,” J. Opt. Soc. Am. 69, 1710–1716 (1979).
[CrossRef]

Cadilhac, M.

H. Bacry, M. Cadilhac, “The metaplectic group and Fourier optics,” Phys. Rev. A 23, 2533–2536 (1981); V. Guillemin, S. Sternberg, Symplectic Techniques in Physics (Cambridge U. Press, Cambridge, UK, 1984); R. G. Littlejohn, “The semiclassical evolution of wave packets,” Phys. Rep. 138, 193–291 (1986); R. Simon, N. Mukunda, “The two-dimensional symplectic and metaplectic groups and their universal cover,” in Symmetries in Science, B. Gruber, ed. (Plenum, New York, 1993), Vol. VI, pp. 659–689; R. Simon, N. Mukunda, “Bargmann invariant and the geometry of the Gouy effect,” Phys. Rev. Lett. 70, 880–883 (1993).
[CrossRef] [PubMed]

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]

Dutta, B.

R. Simon, N. Mukunda, B. Dutta, “Quantum-noise matrix for multimode systems: U(n) invariance squeezing, and normal forms,” Phys. Rev. A 49, 1567–1583 (1994).
[CrossRef] [PubMed]

Friberg, A. T.

R. Simon, A. T. Friberg, E. Wolf, “Transfer of radiance by twisted Gaussian Schell-model beams in paraxial systems,” Pure Appl. Opt. 5, 331–343 (1996).
[CrossRef]

A. T. Friberg, E. Tervonen, J. Turunen, “Interpretation and experimental demonstration of twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 11, 1818–1826 (1994).
[CrossRef]

Gase, R.

R. Gase, “Representation of Laguerre–Gaussian modes by the Wigner distribution function,” IEEE J. Quantum Electron. 31, 1811–1818 (1995).
[CrossRef]

R. Gase, “Methods of quantum mechanics applied to partially coherent light beams,” J. Opt. Soc. Am. A 11, 2121–2129 (1994).
[CrossRef]

Gori, F.

D. Ambrosini, V. Bagini, F. Gori, M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41, 1391–1399 (1994).
[CrossRef]

F. Gori, G. Guattari, “A new type of optical fields,” Opt. Commun. 48, 7–12 (1983).
[CrossRef]

F. Gori, “Collett–Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980); R. Gase, “The multimode laser radiation as a Gaussian Schell-model beam,” J. Mod. Opt. 38, 1107–1115 (1991); 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]

Guattari, G.

F. Gori, G. Guattari, “A new type of optical fields,” Opt. Commun. 48, 7–12 (1983).
[CrossRef]

Li, Y.

Mandel, L.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995).

Mukunda, N.

K. Sundar, N. Mukunda, R. Simon, “Coherent-mode decomposition of general anisotropic Gaussian Schell-model beams,” J. Opt. Soc. Am. A 12, 560–569 (1995).
[CrossRef]

R. Simon, N. Mukunda, B. Dutta, “Quantum-noise matrix for multimode systems: U(n) invariance squeezing, and normal forms,” Phys. Rev. A 49, 1567–1583 (1994).
[CrossRef] [PubMed]

R. Simon, K. Sundar, N. Mukunda, “Twisted Gaussian Schell-model beams. I. Symmetry structure and normal-mode spectrum,” J. Opt. Soc. Am. A 10, 2008–2016 (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, “Twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 10, 95–109 (1993).
[CrossRef]

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Gaussian pure states in quantum mechanics and the symplectic group,” Phys. Rev. A 37, 3028–3038 (1988).
[CrossRef] [PubMed]

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Gaussian Wigner distributions in quantum mechanics and optics,” Phys. Rev. A 36, 3868–3880 (1987).
[CrossRef] [PubMed]

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

Nazarathy, M.

Nemes, G.

Saleh, B. E. A.

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); J. T. Foley, M. S. Zubairy, “The directionality of Gaussian Schell-model beams,” Opt. Commun. 26, 297–300 (1978); A. T. Friberg, R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41, 383–387 (1982); E. Wolf, E. Collett, “Partially coherent sources which produce the same intensity distribution as a laser,” Opt. Commun. 25, 293–296 (1978); 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); H. Weber, “Wave optical analysis of the phase space analyzer,” J. Mod. Opt. 39, 543–559 (1992).
[CrossRef]

Sanjay Kumar, M.

M. Selvadoray, M. Sanjay Kumar, R. Simon, “Photon distribution in two-mode squeezed coherent states with complex displacement and squeeze parameters,” Phys. Rev. A 49, 4957–4967 (1994).
[CrossRef] [PubMed]

Santarsiero, M.

D. Ambrosini, V. Bagini, F. Gori, M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41, 1391–1399 (1994).
[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.

Selvadoray, M.

M. Selvadoray, M. Sanjay Kumar, R. Simon, “Photon distribution in two-mode squeezed coherent states with complex displacement and squeeze parameters,” Phys. Rev. A 49, 4957–4967 (1994).
[CrossRef] [PubMed]

Shamir, J.

Siegman, A. E.

Simon, R.

R. Simon, A. T. Friberg, E. Wolf, “Transfer of radiance by twisted Gaussian Schell-model beams in paraxial systems,” Pure Appl. Opt. 5, 331–343 (1996).
[CrossRef]

K. Sundar, N. Mukunda, R. Simon, “Coherent-mode decomposition of general anisotropic Gaussian Schell-model beams,” J. Opt. Soc. Am. A 12, 560–569 (1995).
[CrossRef]

R. Simon, N. Mukunda, B. Dutta, “Quantum-noise matrix for multimode systems: U(n) invariance squeezing, and normal forms,” Phys. Rev. A 49, 1567–1583 (1994).
[CrossRef] [PubMed]

M. Selvadoray, M. Sanjay Kumar, R. Simon, “Photon distribution in two-mode squeezed coherent states with complex displacement and squeeze parameters,” Phys. Rev. A 49, 4957–4967 (1994).
[CrossRef] [PubMed]

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, K. Sundar, N. Mukunda, “Twisted Gaussian Schell-model beams. I. Symmetry structure and normal-mode spectrum,” J. Opt. Soc. Am. A 10, 2008–2016 (1993).
[CrossRef]

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

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Gaussian pure states in quantum mechanics and the symplectic group,” Phys. Rev. A 37, 3028–3038 (1988).
[CrossRef] [PubMed]

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Gaussian Wigner distributions in quantum mechanics and optics,” Phys. Rev. A 36, 3868–3880 (1987).
[CrossRef] [PubMed]

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

Sudarshan, E. C. G.

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Gaussian pure states in quantum mechanics and the symplectic group,” Phys. Rev. A 37, 3028–3038 (1988).
[CrossRef] [PubMed]

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Gaussian Wigner distributions in quantum mechanics and optics,” Phys. Rev. A 36, 3868–3880 (1987).
[CrossRef] [PubMed]

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

Sundar, K.

Tervonen, E.

Turunen, J.

Wigner, E. P.

E. P. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[CrossRef]

Wolf, E.

IEEE J. Quantum Electron. (1)

R. Gase, “Representation of Laguerre–Gaussian modes by the Wigner distribution function,” IEEE J. Quantum Electron. 31, 1811–1818 (1995).
[CrossRef]

J. Mod. Opt. (1)

D. Ambrosini, V. Bagini, F. Gori, M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41, 1391–1399 (1994).
[CrossRef]

J. Opt. Soc. Am. (2)

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

Opt. Commun. (4)

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); J. T. Foley, M. S. Zubairy, “The directionality of Gaussian Schell-model beams,” Opt. Commun. 26, 297–300 (1978); A. T. Friberg, R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41, 383–387 (1982); E. Wolf, E. Collett, “Partially coherent sources which produce the same intensity distribution as a laser,” Opt. Commun. 25, 293–296 (1978); 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); H. Weber, “Wave optical analysis of the phase space analyzer,” J. Mod. Opt. 39, 543–559 (1992).
[CrossRef]

M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. 25, 26–30 (1978); M. J. Bastiaans, “Wigner distribution function and its application to first-order optics,” J. Opt. Soc. Am. 69, 1710–1716 (1979).
[CrossRef]

F. Gori, G. Guattari, “A new type of optical fields,” Opt. Commun. 48, 7–12 (1983).
[CrossRef]

F. Gori, “Collett–Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980); R. Gase, “The multimode laser radiation as a Gaussian Schell-model beam,” J. Mod. Opt. 38, 1107–1115 (1991); 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]

Opt. Lett. (1)

Opt. Quantum Electron. (1)

M. J. Bastiaans, “ABCD law for partially coherent Gaussian light propagating through first-order optical systems,” Opt. Quantum Electron. 24, 1011–1019 (1992).
[CrossRef]

Phys. Rev. (1)

E. P. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[CrossRef]

Phys. Rev. A (7)

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Gaussian pure states in quantum mechanics and the symplectic group,” Phys. Rev. A 37, 3028–3038 (1988).
[CrossRef] [PubMed]

H. Bacry, M. Cadilhac, “The metaplectic group and Fourier optics,” Phys. Rev. A 23, 2533–2536 (1981); V. Guillemin, S. Sternberg, Symplectic Techniques in Physics (Cambridge U. Press, Cambridge, UK, 1984); R. G. Littlejohn, “The semiclassical evolution of wave packets,” Phys. Rep. 138, 193–291 (1986); R. Simon, N. Mukunda, “The two-dimensional symplectic and metaplectic groups and their universal cover,” in Symmetries in Science, B. Gruber, ed. (Plenum, New York, 1993), Vol. VI, pp. 659–689; R. Simon, N. Mukunda, “Bargmann invariant and the geometry of the Gouy effect,” Phys. Rev. Lett. 70, 880–883 (1993).
[CrossRef] [PubMed]

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

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Gaussian Wigner distributions in quantum mechanics and optics,” Phys. Rev. A 36, 3868–3880 (1987).
[CrossRef] [PubMed]

R. Simon, N. Mukunda, B. Dutta, “Quantum-noise matrix for multimode systems: U(n) invariance squeezing, and normal forms,” Phys. Rev. A 49, 1567–1583 (1994).
[CrossRef] [PubMed]

M. Selvadoray, M. Sanjay Kumar, R. Simon, “Photon distribution in two-mode squeezed coherent states with complex displacement and squeeze parameters,” Phys. Rev. A 49, 4957–4967 (1994).
[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]

Pure Appl. Opt. (1)

R. Simon, A. T. Friberg, E. Wolf, “Transfer of radiance by twisted Gaussian Schell-model beams in paraxial systems,” Pure Appl. Opt. 5, 331–343 (1996).
[CrossRef]

Other (2)

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995).

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.

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Equations (122)

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ΓL,M,K(ρ; ρ)=12π(det L)1/2×exp-14ρTLρ-14ρTLρ-12(ρ-ρ)TM(ρ-ρ)+i2ƛ(ρ-ρ)TK(ρ+ρ).
ρˆ1|ρ=ρ1|ρ,ρˆ2|ρ=ρ2|ρ,
ρ|ρ=δ(2)(ρ-ρ).
ΓL,M,K(ρ; ρ)=ρ|ΓˆL,M,K|ρ,
ΓˆL,M,K=ΓˆL,M,K,
tr ΓˆL,M,K=d2ρ ΓL,M,K(ρ; ρ)<  L>0.
ΓˆL,M,K0 
d2ρd2ρ[ψ(ρ)]*ΓL,M,K(ρ; ρ)ψ(ρ)0
 M0.
ξ=(ξa)=ρ1ρ2p1p2,a=1, 2, 3, 4.
WL,M,K(ξ)=(2πƛ)-2d2ρ exp(ipTρ/ƛ)×ρ-12ρ|ΓˆL,M,K|ρ+12ρ.
WL,M,K(ξ)<  L+4M>0,
WL,M,K(ξ)=(2π)-2[det G(L, M, K)]1/2×exp[-12ξTG(L, M, K)ξ],
G(L, M, K)=Q(-K)L00ƛ-2(14L+M)-1[Q(-K)]T,
Q(K)10K1.
d4ξ ξaWL,M,K(ξ)=0;
d4ξ ξaξbWL,M,K(ξ)Vab(L, M, K)
V(L, M, K)=Q(K)L-100ƛ2(14L+M) [Q(K)]T.
ΓˆL,M,K0  V(L, M, K)+iƛ2β0,
β=01-10.
K=Ks+Ka,KsT=Ks,
Ka=-KaT=-iuσ2=u0-110;
expi2ƛ(ρ-ρ)TK(ρ+ρ)
=expi2ƛρTKsρ-i2ƛρTKsρ×exp-iuƛρ1ρ2-ρ2ρ1;
V(L, M, K)=Q(Ks)V(L, M, Ka)[Q(Ks)]T.
Q(Ks)β[Q(Ks)]T=β.
ΓˆL,M,K0  V(L, M, Ka)+iƛ2β0;
Q(Ka)L-100ƛ2(14L+M)[Q(Ka)]T+iƛ2β0.
[Q(Ka)]-1=Q(-Ka).
ΓˆL,M,K0 
L-100ƛ2(14L+M)+ƛ20i-i2uσ20.
u2ƛ2 det M,
L=diag(σ1-2, σ2-2),M=diag(δ1-2, δ2-2).
Λσ120iƛ/200σ220iƛ/2-iƛ/20ƛ2γ1-2-iƛu0-iƛ/2iƛuƛ2γ2-20,
00iy10000iy2-iy100-iy30-iy2iy30-1,
y1=γ1/2σ1,y2=γ2/2σ2,y3=uγ1γ2/ƛ.
ω4-(y12+y22+y32)ω2+y12y22=0.
y32y12y22(y1-2-1)(y2-2-1).
u2ƛ2/(δ12δ22)=ƛ2 det M,
S: ΓL,M,K(ρ; ρ)ΓL,M,K(S-1ρ; S-1ρ)=ΓL,M,K(ρ; ρ),
L=(S-1)TLS-1,M=(S-1)TMS-1,
Ks=(S-1)TKsS-1,Ka=(det S)-1Ka=Ka.
S00(S-1)T  Sp(4, R).
det M=(δ1δ2)-2=det M.
Tu(ρ; ρ)=expuƛρTσ2ρ=exp-iuƛ(ρ1ρ2-ρ2ρ1).
ΓL,M,K(ρ; ρ)=ΓL,M,Ks(ρ; ρ)Tu(ρ; ρ)=expi2ƛρTKsρ-i2ƛρTKsρ×ΓL,M,0(ρ; ρ)Tu(ρ; ρ);
L=1σ2,M=1δ2,Ks=-1R,Ka=0.
δ: Γ(ρ; ρ)=ψ(ρ)[ψ(ρ)]*,
ψ(ρ)=(2πσ2)-1/2 exp-|ρ|24σ2-i|ρ|22ƛR.
L=diag(σ1-2, σ2-2),M=diag(δ1-2, δ2-2),
14σ14+1σ12δ12=14σ24+1σ22δ22,
L=1σ2,M=1δ2,
Ks=-1R,Ka=-iuσ2.
ΓTGSM(ρ; ρ)=ΓIGSM(ρ; ρ)Tu(ρ; ρ).
|u|ƛ/δ2.
[ρˆr, pˆs]=iƛδrs,
[ρˆr, ρˆs]=[pˆr, pˆs]=0,r, s=1, 2.
pˆr|p)=pr|p),
(p|p)=δ(2)(p-p),
ρ|p)=12πƛexp(iρTp/ƛ).
Tu(ρ; ρ)=exp-iuƛ(ρ1ρ2-ρ2ρ1)2πƛ|u|ρ|Tˆu|ρ.
Tˆu=Tˆu=Tˆu-1.
ρ|Tˆu|ρ=|u|ρ|-uρ2, uρ1),
Tˆu|ρ=|u||-uρ2, uρ1),
Tˆu|p)=|u|-1|u-1p2, -u-1p1.
Tˆu{ρ1, ρˆ2, pˆ1, pˆ2}Tˆu-1=1npˆ2, -1upˆ1, -uρˆ2, uρˆ1.
ξˆ=(ξˆa)=ρˆ1ρˆ2pˆ1pˆ2,
TˆuξˆTˆu-1=S(u)ξˆ,
S(u)=0001/u00-1/u00-u00u000Sp(4, R).
Lˆ=ρˆ1pˆ2-ρˆ2pˆ1,
Hˆ=12[pˆ12+pˆ22+u2(ρˆ12+ρˆ22)];
[Tˆu, Lˆ]=[Tˆu, Hˆ]=[Lˆ, Hˆ]=0.
exp(itHˆ/ƛ)ρˆ1pˆ1exp(-itHˆ/ƛ)
=cos utu-1 sin ut-u sin utcos utρˆ1pˆ1,
exp(iαLˆ/ƛ)ρˆ1ρˆ2exp(-iαLˆ/ƛ)=cos α-sin αsin αcos αρˆ1ρˆ2.
Tˆu=exp(iχ)expiπ2ƛ(u-1Hˆ-Lˆ).
aˆr=12ƛ|u|(|u|ρˆr+ipˆr),
aˆr=12ƛ|u|(|u|ρˆr-ipˆr),
[aˆr, aˆs]=δrs,[aˆr, aˆs]=[aˆr, aˆs]=0;
Lˆ=iƛ(aˆ2aˆ1-aˆ1aˆ2),Hˆ=ƛ|u|(aˆ1aˆ1+aˆ2aˆ2+1),
[Lˆ, aˆ1±iaˆ2]=±ƛ(aˆ1±iaˆ2)
Tˆu{aˆ1, aˆ2, aˆ1, aˆ2}Tˆu-1=iσ{-aˆ2, aˆ1, aˆ2, -aˆ1},
|J, m=12J(aˆ1+iaˆ2)J+m(aˆ1-iaˆ2)J-m(J+m)!(J-m)!|0, 0,
ar|0, 0=0,r=1, 2,Lˆ|J, m=2ƛm|J, m,
Hˆ|J, m=ƛ|u|(2J+1)|J, m,
J=0,1/2, 1,,m=J,J-1,, -J.
ψ0,0(ρ)exp-|u|2ƛ|ρ|2.
d2ρρ|Tˆu|ρψ0,0(ρ)=ψ0,0(ρ).
Tˆu=exp(-iσπ/2)expiσπ2ƛ(|u|-1Hˆ-σLˆ).
Tˆu|J, m=(-1)J-σm|J, m.
Tu(ρ; ρ)=2πƛuρ|Tˆu|ρ=2πƛ|u|J=0,1/2,1,m=-JJ(-1)J-σm×ρ|J, mJ, m|ρ.
ρ1=ρ cos θ,ρ2=ρ sin θ,
|J, mexp(2imθ)ρ2|m|×exp(-u|ρ2/2ƛ)LJ-|m|2|m|uƛρ2,
Jˆ1=12aˆσ1aˆ=12(aˆ1aˆ2+aˆ2aˆ1),
Jˆ2=12aˆσ2aˆ=i2(aˆ2aˆ1-aˆ1aˆ2)=12ƛLˆ,
Jˆ3=12aˆσ3aˆ=12(aˆ1aˆ1-aˆ2aˆ2);
[Jˆj, Jˆk]=ijklJˆl,[Jˆj, Hˆ]=0.
|J, m)=[(J+m)!(J-m)!]-1/2×(aˆ1)J+m(aˆ2)J-m|0, 0),
Jˆ3|J, m)=m|J, m),
Hˆ|J, m)=ƛ|u|(2J+1)|J, m),
Jˆ2|J, m)=j=13JˆjJˆj|J, m)=J(J+1)|J, m),
|J, m)exp(-|u||ρ|2/2ƛ)×HJ+m(|u/ƛ|ρ1)HJ-m(|u/ƛ|ρ2).
expiπ2Jˆaˆ1 exp-iπ2Jˆ1=12(aˆ1+iaˆ2),
expiπ2Jˆ1aˆ2 exp-iπ2Jˆ1=i2(aˆ1-iaˆ2),
|J, m=(-i)J-m expiπ2Jˆ1|J, m).
J=0, 1, 2, :spect Tˆu={1, -1, 1, -1, , 1},
J=1/2, 3/2, :spectTˆu={σ, -σ, , σ, -σ}.
Xˆ1=(ρˆ1-u-1pˆ2)/2,Pˆ1=(pˆ1+uρˆ2)/2,
Xˆ2=(ρˆ2-u-1pˆ1)/2,Pˆ2=(pˆ1+uρˆ2)/2.
Tˆu=expiσπƛ12|u|Pˆ12+|u|2Xˆ12-ƛ2,
Tˆu|n1, n2=(-1)n1|n1, n2,
|n1, n2|n1|n2,n1, n2=0, 1, 2,.
|ψ=Tˆu|ψ:
ψ(ρ)=d2ρρ|Tˆu|ρψ(ρ);
Γˆ=TˆuΓˆTˆu:
Γ(ρ; ρ)=d2ζd2ζρ|Tˆu|ζ×Γ(ζ, ζ)ζ|Tˆu|ρ.
Γˆ=TˆuΓˆTˆu  W(ξ)=W([S(u)]-1ξ).
Γ(ρ; ρ)=Γ(ρ; ρ)Tu0(ρ; ρ)=exp-iu0ƛ(ρ1ρ2-ρ2ρ1)Γ(ρ; ρ),
Tu0(ρ; ρ)ΓL,M,KΓL,M,K,
L=L,M=M,K=K-iu0σ2.
W(ξ)=W(Ω-1ξ),
Ω=100001000-u010u0001Sp(4, R).

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