Abstract

We present a complete solution to the problem of coherent-mode decomposition of the most general anisotropic Gaussian Schell-model (AGSM) beams, which constitute a ten-parameter family. Our approach is based on symmetry considerations. Concepts and techniques familiar from the context of quantum mechanics in the two-dimensional plane are used to exploit the Sp(4, R) dynamical symmetry underlying the AGSM problem. We take advantage of the fact that the symplectic group of first-order optical system acts unitarily through the metaplectic operators on the Hilbert space of wave amplitudes over the transverse plane, and, using the Iwasawa decomposition for the metaplectic operator and the classic theorem of Williamson on the normal forms of positive definite symmetric matrices under linear canonical transformations, we demonstrate the unitary equivalence of the AGSM problem to a separable problem earlier studied by Li and Wolf [ Opt. Lett. 7, 256 ( 1982)] and Gori and Guattari [ Opt. Commun. 48, 7 ( 1983)]. This connection enables one to write down, almost by inspection, the coherent-mode decomposition of the general AGSM beam. A universal feature of the eigenvalue spectrum of the AGSM family is noted.

© 1995 Optical Society of America

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  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).
    [CrossRef]
  2. P. D. Santis, F. Gori, G. Guattari, C. Palma, “An example of a Collett–Wolf source,” Opt. Commun. 29, 256–260 (1979).
    [CrossRef]
  3. 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]
  4. J. T. Foley, M. S. Zubairy, “The directionality of Gaussian Schell-model beams,” Opt. Commun. 26, 297–300 (1978).
    [CrossRef]
  5. A. T. Friberg, R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41, 383–387 (1982).
    [CrossRef]
  6. 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]
  7. H. Weber, “Wave optical analysis of the phase space analyzer,” J. Mod. Opt. 39, 543–559 (1992).
    [CrossRef]
  8. 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]
  9. A. Gamliel, G. P. Agrawal, “Wolf effect in homogeneous and inhomogeneous media,” J. Opt. Soc. Am. A 7, 2184–2192 (1990).
    [CrossRef]
  10. F. Gori, G. L. Marcopoli, M. Santarsiero, “Spectrum invariance on paraxial propagation,” Opt. Commun. 81, 123–130 (1991).
    [CrossRef]
  11. E. Wolf, “Invariance of the spectrum of light on propagation,” Phys. Rev. Lett. 56, 1370–1372 (1986).
    [CrossRef] [PubMed]
  12. E. Collett, E. Wolf, “Is complete coherence necessary for the generation of highly directional light beams?” Opt. Lett. 2, 27–29 (1978).
    [CrossRef] [PubMed]
  13. E. Wolf, E. Collett, “Partially coherent sources which produce the same intensity distribution as a laser,” Opt. Commun. 25, 293–296 (1978).
    [CrossRef]
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    [CrossRef] [PubMed]
  15. F. Gori, G. Guattari, “A new type of optical fields,” Opt. Commun. 48, 7–12 (1983).
    [CrossRef]
  16. 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]
  17. R. Simon, N. Mukunda, “Twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 10, 95–109 (1993).
    [CrossRef]
  18. E. Wolf, “New theory of partial coherence in the space–frequency domain. Part I: spectral and cross-spectra of steady state sources,” J. Opt. Soc. Am. 72, 343–351 (1982).
    [CrossRef]
  19. 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).
    [CrossRef]
  20. F. Gori, “Colett–Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
    [CrossRef]
  21. A. Starikov, E. Wolf, “Coherent-mode representation of Gaussian Schell-model sources and of their radiation fields,” J. Opt. Soc. Am. 72, 923–928 (1982).
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  22. F. Gori, “Mode propagation of the field generated by Collett–Wolf Schell model sources,” Opt. Commun. 46, 149–154 (1983).
    [CrossRef]
  23. E. Wolf, G. S. Agarwal, “Coherence theory of laser resonator modes,” J. Opt. Soc. Am. A 1, 541–546 (1984).
    [CrossRef]
  24. F. Gori, R. Grella, “Shape invariant propagation of polychromatic fields,” Opt. Commun. 49, 173–177 (1984).
    [CrossRef]
  25. 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).
    [CrossRef]
  26. C. Pask, “Application of Wolf’s theory of coherence,” J. Opt. Soc. Am. A 3, 1097–1101 (1986).
    [CrossRef]
  27. P. De Santis, F. Gori, G. Guattari, C. Palma, “Synthesis of partially coherent fields,” J. Opt. Soc. Am. A 3, 1258–1262 (1986).
    [CrossRef]
  28. F. Gori, G. Guattari, C. Padovani, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun. 64, 311–316 (1987).
    [CrossRef]
  29. F. Gori, G. Guattari, C. Palma, C. Padovani, “A class of shape invariant fields,” Opt. Commun. 66, 255–259 (1988).
    [CrossRef]
  30. R. Gase, “The multimode laser radiation as a Gaussian Schell-model beam,” J. Mod. Opt. 38, 1107–1115 (1991).
    [CrossRef]
  31. 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]
  32. 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]
  33. 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]
  34. H. Bacry, M. Cadilhac, “Metaplectic group and Fourier optics,” Phys. Rev. A 23, 2533–2536 (1981).
    [CrossRef]
  35. J. Williamson, “On the algebraic problem concerning the normal forms of linear dynamical systems,” Am. J. Math. 58, 141–163 (1936).
    [CrossRef]
  36. 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]
  37. S. Helgason, Differential Geometry, Lie Groups and Symmetric Species (Academic, New York, 1978), Chap. VI.
  38. 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.
  39. 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]
  40. A. E. Siegman, Lasers (Oxford U. Press, Oxford, 1986), p. 669.
  41. A. Perelomov, Generalized Coherent States and Their Application (Springer-Verlag, Berlin, 1986), Chap. 2.
    [CrossRef]
  42. 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]

1993 (3)

1992 (4)

H. Weber, “Wave optical analysis of the phase space analyzer,” J. Mod. Opt. 39, 543–559 (1992).
[CrossRef]

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. 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]

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 (2)

F. Gori, G. L. Marcopoli, M. Santarsiero, “Spectrum invariance on paraxial propagation,” Opt. Commun. 81, 123–130 (1991).
[CrossRef]

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

1990 (1)

1988 (2)

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

1987 (2)

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]

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

1986 (5)

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]

1984 (3)

E. Wolf, G. S. Agarwal, “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 (2)

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

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

1982 (4)

1981 (1)

H. Bacry, M. Cadilhac, “Metaplectic group and Fourier optics,” Phys. Rev. A 23, 2533–2536 (1981).
[CrossRef]

1980 (1)

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

1979 (2)

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]

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

1978 (4)

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]

J. T. Foley, M. S. Zubairy, “The directionality of Gaussian Schell-model beams,” Opt. Commun. 26, 297–300 (1978).
[CrossRef]

E. Collett, E. Wolf, “Is complete coherence necessary for the generation of highly directional light beams?” Opt. Lett. 2, 27–29 (1978).
[CrossRef] [PubMed]

E. Wolf, E. Collett, “Partially coherent sources which produce the same intensity distribution as a laser,” Opt. Commun. 25, 293–296 (1978).
[CrossRef]

1936 (1)

J. Williamson, “On the algebraic problem concerning the normal forms of linear dynamical systems,” Am. J. Math. 58, 141–163 (1936).
[CrossRef]

Agarwal, G. S.

Agrawal, G. P.

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, “Metaplectic group and Fourier optics,” Phys. Rev. A 23, 2533–2536 (1981).
[CrossRef]

Bertolotti, M.

Cadilhac, M.

H. Bacry, M. Cadilhac, “Metaplectic group and Fourier optics,” Phys. Rev. A 23, 2533–2536 (1981).
[CrossRef]

Carter, W. H.

Collett, E.

E. Collett, E. Wolf, “Is complete coherence necessary for the generation of highly directional light beams?” Opt. Lett. 2, 27–29 (1978).
[CrossRef] [PubMed]

E. Wolf, E. Collett, “Partially coherent sources which produce the same intensity distribution as a laser,” Opt. Commun. 25, 293–296 (1978).
[CrossRef]

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.

Foley, J. T.

J. T. Foley, M. S. Zubairy, “The directionality of Gaussian Schell-model beams,” Opt. Commun. 26, 297–300 (1978).
[CrossRef]

Friberg, A. T.

A. T. Friberg, R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41, 383–387 (1982).
[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. L. Marcopoli, M. Santarsiero, “Spectrum invariance on paraxial propagation,” Opt. Commun. 81, 123–130 (1991).
[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, G. Guattari, “A new type of optical fields,” Opt. Commun. 48, 7–12 (1983).
[CrossRef]

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

F. Gori, “Colett–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]

Grella, R.

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

Guattari, G.

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

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

Helgason, S.

S. Helgason, Differential Geometry, Lie Groups and Symmetric Species (Academic, New York, 1978), Chap. VI.

Idell, P. S.

Li, Y.

Marcopoli, G. L.

F. Gori, G. L. Marcopoli, M. Santarsiero, “Spectrum invariance on paraxial propagation,” Opt. Commun. 81, 123–130 (1991).
[CrossRef]

Martinez-Herrero, R.

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. 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]

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]

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]

Mukunda, N.

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, 2028–2038 (1988).
[CrossRef]

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]

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]

Padovani, C.

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]

Palma, C.

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

Pask, C.

Perelomov, A.

A. Perelomov, Generalized Coherent States and Their Application (Springer-Verlag, Berlin, 1986), Chap. 2.
[CrossRef]

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).
[CrossRef]

Santarsiero, M.

F. Gori, G. L. Marcopoli, M. Santarsiero, “Spectrum invariance on paraxial propagation,” Opt. Commun. 81, 123–130 (1991).
[CrossRef]

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.

Serna, J.

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]

Siegman, A. E.

A. E. Siegman, Lasers (Oxford U. Press, Oxford, 1986), p. 669.

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, 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, 2028–2038 (1988).
[CrossRef]

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]

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, 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. 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]

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]

Sudol, R. J.

A. T. Friberg, R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41, 383–387 (1982).
[CrossRef]

Sundar, K.

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]

Weber, H.

H. Weber, “Wave optical analysis of the phase space analyzer,” J. Mod. Opt. 39, 543–559 (1992).
[CrossRef]

Williamson, J.

J. Williamson, “On the algebraic problem concerning the normal forms of linear dynamical systems,” Am. J. Math. 58, 141–163 (1936).
[CrossRef]

Wolf, E.

Zubairy, M. S.

J. T. Foley, M. S. Zubairy, “The directionality of Gaussian Schell-model beams,” Opt. Commun. 26, 297–300 (1978).
[CrossRef]

Am. J. Math. (1)

J. Williamson, “On the algebraic problem concerning the normal forms of linear dynamical systems,” Am. J. Math. 58, 141–163 (1936).
[CrossRef]

J. Mod. Opt. (3)

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

H. Weber, “Wave optical analysis of the phase space analyzer,” J. Mod. Opt. 39, 543–559 (1992).
[CrossRef]

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]

J. Opt. Soc. Am A (1)

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]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (8)

Opt. Commun. (13)

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

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

E. Wolf, E. Collett, “Partially coherent sources which produce the same intensity distribution as a laser,” Opt. Commun. 25, 293–296 (1978).
[CrossRef]

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Equations (52)

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W ( ρ , ρ ; ν ) = I ( ν ) [ det L ( ν ) ] 1 / 2 2 π × exp { - 1 4 [ ρ T L ( ν ) ρ + ρ T L ( ν ) ρ ] - 1 2 ( ρ - ρ ) T M ( ν ) ( ρ - ρ ) + i 2 ( ρ - ρ ) T K ( ν ) ( ρ + ρ ) } ,
W ( ρ , p ; ν ) = ( 1 2 π ƛ ) 2 d 2 Δ ρ × W ( ρ - 1 2 Δ ρ , ρ + 1 2 Δ ρ ; ν ) exp ( i p · Δ ρ / ) .
ξ = [ x y p x p y ] = [ ρ p ] .
W ( ρ , ρ ; ν ) = W ( ξ ; ν ) = I ( ν ) 4 π 2 ( det V ) exp ( - 1 2 ξ T V ( ν ) - 1 ξ ) ,
V ( ν ) = [ V 1 ( ν ) V 3 ( ν ) V 3 T ( ν ) V 2 ( ν ) ] , V 1 ( ν ) = L ( ν ) - 1 , V 2 ( ν ) = ƛ 2 [ M ( ν ) + ¼ L ( ν ) ] + K ( ν ) L ( ν ) - 1 K ( ν ) T , V 3 ( ν ) = L ( ν ) - 1 K ( ν ) T .
I ( ν ) = d 4 ξ W ( ξ ; ν ) = d 2 ρ W ( ρ , ρ ; ν ) .
ξ α ξ β I ( ν ) - 1 d 4 ξ ξ α ξ β W ( ξ ; ν ) = [ V ( ν ) ] α β ,
S Sp ( 4 , R ) S T Ω S = Ω             Ω = [ O 2 × 2 I 2 × 2 - I 2 × 2 O 2 × 2 ] .
W ( ρ , ρ ; ν ) = U ^ ( S ) W ( ρ , ρ ; ν ) U ^ ( S ) .
W ( ξ ; ν ) = W ( S - 1 ξ ; ν ) .
V ( ν ) V ( ν ) = SV ( ν ) S T .
W c ( ρ , ρ ; ν ) = I ( ν ) W 1 ( ρ 1 , ρ 1 ; ν ) W 2 ( ρ 2 , ρ 2 ; ν ) , W j ( ρ j , ρ j ; ν ) = 1 2 π σ j ( ν ) exp [ - 1 4 σ j ( ν ) 2 ( ρ j 2 + ρ j 2 ) - 1 2 δ j ( ν ) 2 ( ρ j - ρ j ) 2 ] .
V 1 ( ν ) = [ σ 1 ( ν ) 2 0 0 σ 2 ( ν ) 2 ] , V 2 ( ν ) = [ ƛ 2 / γ 1 ( ν ) 2 0 0 ƛ 2 / γ 2 ( ν ) 2 ] ,
1 γ j ( ν ) 2 = 1 4 σ j ( ν ) 2 + 1 δ j ( ν ) 2 .
c ( ν ) 2 = 1 4 σ 1 ( ν ) 4 + 1 σ 1 ( ν ) 2 δ 1 ( ν ) 2 = 1 4 σ 2 ( ν ) 4 + 1 σ 2 ( ν ) 2 δ 2 ( ν ) 2 .
V c ( ν ) = [ α G 0 0 α - 1 G ] , G ( ν ) = ƛ [ g 1 0 0 g 2 ] , g j ( ν ) = 1 2 ( 1 + 4 σ j 2 δ j 2 ) 1 / 2 , α ( ν ) = ( c ƛ ) - 1 .
g j ( ν ) 1 / 2 ,             j = 1 , 2 ,
W j ( ρ j , ρ j ; ν ) = n = 0 q 0 j q j n φ n ( c ρ j ) φ n * ( c ρ j ) , φ n ( c ρ j ) = ( 1 2 n n ! c / π ) 1 / 2 exp ( - ½ c ρ j 2 ) H n ( c ρ j ) , q j = g j - ½ g j + ½ , q 0 j = ( n - 0 q j n ) - 1 = 1 g j + ½ .
W c ( ρ , ρ ; ν ) = n 1 , n 2 = 0 E n 1 , n 2 ( ν ) Ψ n 1 , n 2 ( ρ ; ν ) Ψ n 1 , n 2 * ( ρ ; ν ) , Ψ n 1 , n 2 ( ρ ) = φ n 1 ( c ρ 1 ) φ n 2 ( c ρ 2 ) , E n 1 , n 2 ( ν ) = 1 g 1 + ½ 1 g 2 + ½ × ( g 1 - ½ g 1 + ½ ) n 1 ( g 2 - ½ g 2 + ½ ) n 2 .
V 0 ( ν ) = [ κ 1 0 0 0 0 κ 2 0 0 0 0 κ 1 0 0 0 0 κ 2 ] .
V Ω V Ω T = S 0 V 0 2 S 0 - 1 .
tr ( V Ω V Ω T ) = 2 ( κ 1 2 + κ 2 2 ) ,
tr [ ( V Ω V Ω T ) 2 ] = 2 ( κ 1 4 + κ 2 4 ) .
S α = [ α - 1 / 2 0 0 0 0 α - 1 / 2 0 0 0 0 α 1 / 2 0 0 0 0 α 1 / 2 ]
V = S 0 α V 0 α S 0 α ,             S 0 α = S 0 S α , V 0 α = S α - 1 V 0 ( S α - 1 ) T = V 0 S α - 2 = [ α κ 1 0 0 0 0 α κ 2 0 0 0 0 α - 1 κ 1 0 0 0 0 α - 1 κ 2 ] .
S = [ A B C D ] .
S S = [ S O O S - 1 ] ,
U ^ ( S S ) ψ ( ρ ) = ( det S ) - 1 / 2 ψ ( S - 1 ρ ) .
S = [ I O - I ] ,
U ^ ( S ) ψ ( ρ ) = exp ( - i 2 ρ T ρ ) ψ ( ρ ) .
S X , Y = [ X - α Y α - 1 Y X ]
X X T + Y Y T = I ,             X Y T = Y X T .
U = X + i Y ,             U U = I .
S = [ A B C D ] = [ I O - I ] [ S O O S - 1 ] [ X - α Y α - 1 Y X ] .
S = ( A A T + α - 2 B B T ) - 1 / 2 , X + i Y = ( A A T + α - 2 B B T ) - 1 / 2 ( A - i α - 1 B ) , = - ( C A T + α - 2 D B T ) ( A A T + α - 2 B B T ) - 1 .
U ^ ( S ) = U ^ ( S ) U ^ ( S S ) U ^ ( S X , Y ) .
u 0 ( θ , φ ) = exp ( - i φ J 3 ) exp ( - i θ J 2 ) exp ( i φ J 3 ) = [ cos ( θ / 2 ) - exp ( - i φ ) sin ( θ / 2 ) exp ( i φ ) sin ( θ / 2 ) cos ( θ / 2 ) ] .
X + i Y = U ( θ , φ ; η , ζ ) = u 0 ( θ , φ ) exp ( - i η J 3 ) exp ( - i ζ J 0 ) , 0 θ π ,             0 φ < 2 π ,             0 η ,             ζ < 4 π .
a ^ j ( α ) = 1 2 α ƛ ρ ^ j + i α 2 ƛ ρ ^ j             j = 1 , 2 ,
T ^ 0 ( α ) = ( J 0 ) j k a ^ j ( α ) a ^ k ( a ) ,             T ^ a ( α ) = ( J a ) j k a ^ j ( α ) a ^ k ( α ) ,             a = 1 , 2 , 3 ,
T ^ 0 ( α ) = [ α - 1 ( ρ ^ 1 2 + ρ ^ 2 2 ) + α ( p ^ 1 2 + p ^ 2 2 ) ] / ( 4 ƛ ) - 1 / 2 , T ^ 3 ( α ) = [ α - 1 ( ρ ^ 1 2 - ρ ^ 2 2 ) + α ( p ^ 1 2 - p ^ 2 2 ) ] / ( ƛ 4 ) , T ^ 1 ( α ) = ( α - 1 ρ ^ 1 ρ ^ 2 + α p ^ 1 p ^ 2 ) / ( 2 ƛ ) , T ^ 2 ( α ) = ( ρ ^ 1 p ^ 2 - ρ ^ 2 p ^ 1 ) / ( 2 ƛ ) .
[ T ^ a ( α ) , T ^ b ( α ) ] = i a b c T ^ c ( α ) ,             [ T ^ a ( α ) , T ^ 0 ( α ) ] = 0.
U ^ ( S X , Y ) = U ^ ( θ , φ ; η , ζ ) = U ^ 0 ( θ , φ ) exp [ - i η T ^ 3 ( α ) ] exp [ - i ζ T ^ 0 ( α ) ] , U ^ 0 ( θ , φ ) = exp [ - i φ T ^ 3 ( α ) ] exp [ - i θ T ^ 2 ( α ) ] exp [ i φ T ^ 3 ( α ) ] .
Ψ j , m ( ρ ; ν ) = [ 1 2 2 j ( j + m ) ! ( j - m ) ! c π ] 1 / 2 × exp [ - c ( ρ 1 2 + ρ 2 2 ) / 2 ] H j + 1 ( c ρ 1 ) × H j - m ( c ρ 2 ) , T ^ 0 ( α ) Ψ j , m ( ρ ; ν ) = j Ψ j , m ( ρ ; ν ) ; T ^ 3 ( α ) Ψ j , m ( ρ ; ν ) = m Ψ j , m ( ρ ; ν ) ,             j = 0 , 1 / 2 , 1 , , m = - j , - j + 1 , , j .
Σ = { U ^ ( S X , Y ) Ψ j , m ( ρ , ν ) X + i Y U ( 2 ) } .
Σ = { Ψ j , m θ , φ ( ρ , ν ) = U ^ 0 ( θ , φ ) Ψ j , m ( ρ , ν ) ( θ , φ ) S 2 } .
Ψ j , m θ , φ ( ρ , ν ) = m = - j j D m , m j ( θ , φ ) Ψ j , m ( ρ , ν ) , D m , m j ( θ , φ ) = j , m U ^ 0 ( θ , φ ) j , m = exp [ - i φ ( m - m ) ] · d m , m j ( θ ) , d m , m j ( θ ) = j , m exp [ - i θ T ^ 2 ( α ) ] j , m = 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 ( θ / 2 ) ] 2 j + m - m - 2 μ [ sin ( θ / 2 ) ] m - m + 2 μ .
Ψ 1 , - 1 θ , φ ( x , y ; ν ) = c / 2 π exp [ - c ( x 2 + y 2 ) / 2 ] exp ( - i φ ) × [ ( 2 c y 2 - 1 ) cos 2 ( θ / 2 ) exp ( i φ ) - 2 c sin ( θ ) x y + ( 2 c x 2 - 1 ) sin 2 ( θ / 2 ) × exp ( - i φ ) ] , Ψ 1 , 0 θ , φ ( x , y ; ν ) = c / 4 π exp [ - c ( x 2 + y 2 ) / 2 ] × [ ( 2 c y 2 - 1 ) sin ( θ ) exp ( i φ ) + 4 c cos ( θ ) x y - ( 2 c x 2 - 1 ) sin ( θ ) × exp ( - i φ ) ] , Ψ 1 , 1 θ , φ ( x , y ; ν ) = c / 2 π exp [ - c / 2 ( x 2 + y 2 ) ] exp ( i φ ) × [ ( 2 c y 2 - 1 ) sin 2 ( θ / 2 ) exp ( i φ ) + 2 c sin ( θ ) x y + ( 2 c x 2 - 1 ) cos 2 ( θ / 2 ) × exp ( - i φ ) ] .
W V ( ρ , ρ ; ν ) = U ^ ( S 0 α ) W c ( ρ , ρ ; ν ) U ^ ( S 0 α ) .
W V ( ρ , ρ ; ν ) = j m E j , m ( ν ) Φ j , m ( ρ , ν ) Φ j , m * ( ρ , ν ) , E j , m ( ν ) = I ( ν ) ( g 1 + ½ ) ( g 2 + ½ ) ( g 1 - ½ g 1 + ½ ) j + m × ( g 2 - ½ g 2 + ½ ) j - m , Φ j , m ( ρ ) = U ^ ( S 0 α ) Ψ j , m ( ρ , ν ) .
Φ j , m ( ρ ) = 1 ( det S ) 1 / 2 exp ( - i ρ T ρ 2 ) Ψ j , m θ , φ ( S - 1 ρ ; ν ) .
Family 1 : κ 1 ( ν ) = κ 2 ( ν ) ; hence g 1 ( ν ) = g 2 ( ν ) , Family 2 : κ 1 ( ν ) κ 2 ( ν ) ; hence g 1 ( ν ) g 2 ( ν ) .

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