Abstract

We present a complete analysis of shape-invariant anisotropic Gaussian Schell-model beams, which generalizes the shape-invariant beams introduced earlier by Gori and Guattari [Opt. Commun. 48, 7 (1983)] and the recently discovered twisted Gaussian Schell-model beams. We show that the set of all shape-invariant Gaussian Schell-model beams forms a six-parameter family embedded within the ten-parameter family of all anisotropic Gaussian Schell-model beams. These shape-invariant beams are generically anisotropic and possess a saddlelike phase front in addition to a twist phase in such a way that the tendency of the latter to twist the beam in the course of propagation is exactly countered by the former. The propagation characteristics of these beams turn out to be surprisingly simple and are akin to those of coherent Gaussian beams. They are controlled by a single parameter that plays the role of the Rayleigh range; its value is determined by an interplay among the beam widths, transverse coherence lengths, and the strength of the twist parameter. The positivity requirement on the cross-spectral density is shown to be equivalent to an upper bound on the twist parameter. The entire analysis is carried out by use of the Wigner distribution, which reduces the problem to a purely algebraic one involving 4×4 matrices, thus rendering the complete solution immediately transparent.

© 1998 Optical Society of America

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  1. A. E. Siegman, Lasers (Oxford U. Press, Oxford, 1986), Chap. 19.
  2. H. Kogelnik, “Imaging of optical modes—resonators with internal lenses,” Bell Syst. Tech. J. 44, 455–494 (1965).
    [CrossRef]
  3. R. W. Boyd, “Intuitive explanation of the phase anomaly of focused light beams,” J. Opt. Soc. Am. 70, 877–880 (1980).
    [CrossRef]
  4. R. Simon, N. Mukunda, “Iwasawa decomposition for SU(1, 1) and the Gouy effect for squeezed states,” Opt. Commun. 95, 39–45 (1993); G. S. Agarwal, R. Simon, “An experiment for the study of the Gouy effect for the squeezed vacuum,” Opt. Commun. 100, 411–414 (1993).
    [CrossRef]
  5. R. Simon, N. Mukunda, “Bargmann invariant and the geometry of the Gouy effect,” Phys. Rev. Lett. 70, 880–883 (1993).
    [CrossRef] [PubMed]
  6. 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]
  7. P. D. Santis, F. Gori, G. Guattari, C. Palma, “An example of a Collett–Wolf source,” Opt. Commun. 29, 256–260 (1979).
    [CrossRef]
  8. 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]
  9. J. T. Foley, M. S. Zubairy, “The directionality of Gaussian Schell-model beams,” Opt. Commun. 26, 297–300 (1978).
    [CrossRef]
  10. A. T. Friberg, R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41, 383–387 (1982).
    [CrossRef]
  11. 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]
  12. H. Weber, “Wave optical analysis of the phase space analyzer,” J. Mod. Opt. 39, 543–559 (1992).
    [CrossRef]
  13. J. Serna, P. M. Mejias, R. Martinez-Herrero, “Beam quality dependence on the coherence length of Gaussian Schell-model fields propagating through ABCD optical systems,” J. Mod. Opt. 39, 625–635 (1992).
    [CrossRef]
  14. A. Gamliel, G. P. Agarwal, “Wolf effect in homogeneous and inhomogeneous media,” J. Opt. Soc. Am. A 7, 2184–2192 (1990).
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  15. F. Gori, G. L. Marcopoli, M. Santarsiero, “Spectrum invariance on paraxial propagation,” Opt. Commun. 81, 123–130 (1991).
    [CrossRef]
  16. E. Wolf, “Invariance of the spectrum of light on propagation,” Phys. Rev. Lett. 56, 1370–1372 (1986).
    [CrossRef] [PubMed]
  17. E. Collett, E. Wolf, “Is complete coherence necessary for the generation of highly directional light beams?” Opt. Lett. 2, 27–29 (1978).
    [CrossRef]
  18. E. Wolf, E. Collett, “Partially coherent sources which produce the same intensity distribution as a laser,” Opt. Commun. 25, 293–296 (1978).
    [CrossRef]
  19. G. Gori, R. Grella, “Shape invariant propagation of polychromatic fields,” Opt. Commun. 49, 173–177 (1984).
    [CrossRef]
  20. F. Gori, G. Guattari, C. Palma, C. Padovani, “A class of shape invariant fields,” Opt. Commun. 66, 255–259 (1988).
    [CrossRef]
  21. R. Gase, “The multimode laser radiation as a Gaussian Schell-model beam,” J. Mod. Opt. 38, 1107–1115 (1991).
    [CrossRef]
  22. 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); R. Gase, “Methods of quantum mechanics applied to partially coherent light beams,” J. Opt. Soc. Am. A 11, 2121–2129 (1994).
    [CrossRef]
  23. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, 1995).
  24. A. T. Friberg, J. Turunen, “Imaging of Gaussian Schell-model sources,” J. Opt. Soc. Am. A 5, 713–720 (1988).
    [CrossRef]
  25. R. Simon, N. Mukunda, E. C. G. Sudarshan, “Partially coherent beams and a generalized ABCD-law,” Opt. Commun. 65, 322–328 (1988).
    [CrossRef]
  26. R. Simon, N. Mukunda, “Twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 10, 95–109 (1993).
    [CrossRef]
  27. 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]
  28. 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]
  29. 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]
  30. E. Wolf, “New theory of partial coherence in the space-frequency domain: I. Spectra and cross-spectra of steady state sources,” J. Opt. Soc. Am. 72, 343–351 (1982).
    [CrossRef]
  31. D. Ambrosini, V. Bagini, F. Gori, M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41, 1391–1399 (1994).
    [CrossRef]
  32. Y. Li, E. Wolf, “Radiation from anisotropic Gaussian Schell-model sources,” Opt. Lett. 7, 256–258 (1982).
    [CrossRef] [PubMed]
  33. 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]
  34. F. Gori, G. Guattari, “A new type of optical fields,” Opt. Commun. 48, 7–12 (1983).
    [CrossRef]
  35. R. Simon, E. C. G. Sudarshan, N. Mukunda, “Gaussian Wigner distributions in quantum mechanics and optics,” Phys. Rev. A 36, 3868–3880 (1987); “Gaussian pure states in quantum mechanics and the symplectic group,” Phys. Rev. A 37, 2028–2038 (1988); R. Simon, N. Mukunda, B. Dutta, “Quantum-noise matrix, for multimode systems: U(n) invariance, squeezing, and normal forms,” Phys. Rev. A 49, 1567–1683 (1994).
    [CrossRef] [PubMed]
  36. 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]
  37. 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]
  38. G. Breitenbach, S. Schiller, T. Mylnek, “Measurement of the quantum states of squeezed light,” Nature (London) 387, 471–475 (1997).
    [CrossRef]
  39. J. Williamson, “On the algebraic problem concerning the normal forms of linear dynamical systems,” Am. J. Math. 58, 141–163 (1936).
    [CrossRef]
  40. E. C. G. Sudarshan, N. Mukunda, R. Simon, “Realization of first order optical systems using thin lenses,” Opt. Acta 32, 855–872 (1985).
    [CrossRef]

1997 (1)

G. Breitenbach, S. Schiller, T. Mylnek, “Measurement of the quantum states of squeezed light,” Nature (London) 387, 471–475 (1997).
[CrossRef]

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 (1)

1994 (2)

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

1993 (5)

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, N. Mukunda, “Iwasawa decomposition for SU(1, 1) and the Gouy effect for squeezed states,” Opt. Commun. 95, 39–45 (1993); G. S. Agarwal, R. Simon, “An experiment for the study of the Gouy effect for the squeezed vacuum,” Opt. Commun. 100, 411–414 (1993).
[CrossRef]

R. Simon, N. Mukunda, “Bargmann invariant and the geometry of the Gouy effect,” Phys. Rev. Lett. 70, 880–883 (1993).
[CrossRef] [PubMed]

1992 (3)

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

1991 (2)

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

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

1990 (1)

1988 (3)

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

A. T. Friberg, J. Turunen, “Imaging of Gaussian Schell-model sources,” J. Opt. Soc. Am. A 5, 713–720 (1988).
[CrossRef]

R. Simon, N. Mukunda, E. C. G. Sudarshan, “Partially coherent beams and a generalized ABCD-law,” Opt. Commun. 65, 322–328 (1988).
[CrossRef]

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); “Gaussian pure states in quantum mechanics and the symplectic group,” Phys. Rev. A 37, 2028–2038 (1988); R. Simon, N. Mukunda, B. Dutta, “Quantum-noise matrix, for multimode systems: U(n) invariance, squeezing, and normal forms,” Phys. Rev. A 49, 1567–1683 (1994).
[CrossRef] [PubMed]

1986 (1)

E. Wolf, “Invariance of the spectrum of light on propagation,” Phys. Rev. Lett. 56, 1370–1372 (1986).
[CrossRef] [PubMed]

1985 (2)

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]

E. C. G. Sudarshan, N. Mukunda, R. Simon, “Realization of first order optical systems using thin lenses,” Opt. Acta 32, 855–872 (1985).
[CrossRef]

1984 (2)

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

1982 (3)

1980 (1)

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]

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

1965 (1)

H. Kogelnik, “Imaging of optical modes—resonators with internal lenses,” Bell Syst. Tech. J. 44, 455–494 (1965).
[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. P.

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]

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]

Bertolotti, M.

Boyd, R. W.

Breitenbach, G.

G. Breitenbach, S. Schiller, T. Mylnek, “Measurement of the quantum states of squeezed light,” Nature (London) 387, 471–475 (1997).
[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]

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

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.

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]

A. T. Friberg, J. Turunen, “Imaging of Gaussian Schell-model sources,” J. Opt. Soc. Am. A 5, 713–720 (1988).
[CrossRef]

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]

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. 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, “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]

Gori, G.

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

Grella, R.

G. 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, “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]

Kogelnik, H.

H. Kogelnik, “Imaging of optical modes—resonators with internal lenses,” Bell Syst. Tech. J. 44, 455–494 (1965).
[CrossRef]

Li, Y.

Mandel, L.

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

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

J. Serna, P. M. Mejias, R. Martinez-Herrero, “Beam quality dependence on the coherence length of Gaussian Schell-model fields propagating through ABCD optical systems,” J. Mod. Opt. 39, 625–635 (1992).
[CrossRef]

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, “Iwasawa decomposition for SU(1, 1) and the Gouy effect for squeezed states,” Opt. Commun. 95, 39–45 (1993); G. S. Agarwal, R. Simon, “An experiment for the study of the Gouy effect for the squeezed vacuum,” Opt. Commun. 100, 411–414 (1993).
[CrossRef]

R. Simon, N. Mukunda, “Bargmann invariant and the geometry of the Gouy effect,” Phys. Rev. Lett. 70, 880–883 (1993).
[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, N. Mukunda, E. C. G. Sudarshan, “Partially coherent beams and a generalized ABCD-law,” Opt. Commun. 65, 322–328 (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); “Gaussian pure states in quantum mechanics and the symplectic group,” Phys. Rev. A 37, 2028–2038 (1988); R. Simon, N. Mukunda, B. Dutta, “Quantum-noise matrix, for multimode systems: U(n) invariance, squeezing, and normal forms,” Phys. Rev. A 49, 1567–1683 (1994).
[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]

E. C. G. Sudarshan, N. Mukunda, R. Simon, “Realization of first order optical systems using thin lenses,” Opt. Acta 32, 855–872 (1985).
[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]

Mylnek, T.

G. Breitenbach, S. Schiller, T. Mylnek, “Measurement of the quantum states of squeezed light,” Nature (London) 387, 471–475 (1997).
[CrossRef]

Padovani, C.

F. Gori, G. Guattari, C. Palma, C. Padovani, “A class of shape invariant fields,” Opt. Commun. 66, 255–259 (1988).
[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. D. Santis, F. Gori, G. Guattari, C. Palma, “An example of a Collett–Wolf source,” Opt. Commun. 29, 256–260 (1979).
[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.

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. 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]

Schiller, S.

G. Breitenbach, S. Schiller, T. Mylnek, “Measurement of the quantum states of squeezed light,” Nature (London) 387, 471–475 (1997).
[CrossRef]

Serna, J.

J. Serna, P. M. Mejias, R. Martinez-Herrero, “Beam quality dependence on the coherence length of Gaussian Schell-model fields propagating through ABCD optical systems,” J. Mod. Opt. 39, 625–635 (1992).
[CrossRef]

Siegman, A. E.

A. E. Siegman, Lasers (Oxford U. Press, Oxford, 1986), Chap. 19.

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, “Iwasawa decomposition for SU(1, 1) and the Gouy effect for squeezed states,” Opt. Commun. 95, 39–45 (1993); G. S. Agarwal, R. Simon, “An experiment for the study of the Gouy effect for the squeezed vacuum,” Opt. Commun. 100, 411–414 (1993).
[CrossRef]

R. Simon, N. Mukunda, “Bargmann invariant and the geometry of the Gouy effect,” Phys. Rev. Lett. 70, 880–883 (1993).
[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, N. Mukunda, “Twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 10, 95–109 (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, E. C. G. Sudarshan, “Partially coherent beams and a generalized ABCD-law,” Opt. Commun. 65, 322–328 (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); “Gaussian pure states in quantum mechanics and the symplectic group,” Phys. Rev. A 37, 2028–2038 (1988); R. Simon, N. Mukunda, B. Dutta, “Quantum-noise matrix, for multimode systems: U(n) invariance, squeezing, and normal forms,” Phys. Rev. A 49, 1567–1683 (1994).
[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]

E. C. G. Sudarshan, N. Mukunda, R. Simon, “Realization of first order optical systems using thin lenses,” Opt. Acta 32, 855–872 (1985).
[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]

Sudarshan, E. C. G.

R. Simon, N. Mukunda, E. C. G. Sudarshan, “Partially coherent beams and a generalized ABCD-law,” Opt. Commun. 65, 322–328 (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); “Gaussian pure states in quantum mechanics and the symplectic group,” Phys. Rev. A 37, 2028–2038 (1988); R. Simon, N. Mukunda, B. Dutta, “Quantum-noise matrix, for multimode systems: U(n) invariance, squeezing, and normal forms,” Phys. Rev. A 49, 1567–1683 (1994).
[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]

E. C. G. Sudarshan, N. Mukunda, R. Simon, “Realization of first order optical systems using thin lenses,” Opt. Acta 32, 855–872 (1985).
[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]

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.

Tervonen, E.

Turunen, J.

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.

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]

E. Wolf, “Invariance of the spectrum of light on propagation,” Phys. Rev. Lett. 56, 1370–1372 (1986).
[CrossRef] [PubMed]

E. Wolf, “New theory of partial coherence in the space-frequency domain: I. Spectra and cross-spectra of steady state sources,” J. Opt. Soc. Am. 72, 343–351 (1982).
[CrossRef]

Y. Li, E. Wolf, “Radiation from anisotropic Gaussian Schell-model sources,” Opt. Lett. 7, 256–258 (1982).
[CrossRef] [PubMed]

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

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

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

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]

Bell Syst. Tech. J. (1)

H. Kogelnik, “Imaging of optical modes—resonators with internal lenses,” Bell Syst. Tech. J. 44, 455–494 (1965).
[CrossRef]

J. Mod. Opt. (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 ABCD optical 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]

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. (3)

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

Nature (London) (1)

G. Breitenbach, S. Schiller, T. Mylnek, “Measurement of the quantum states of squeezed light,” Nature (London) 387, 471–475 (1997).
[CrossRef]

Opt. Acta (1)

E. C. G. Sudarshan, N. Mukunda, R. Simon, “Realization of first order optical systems using thin lenses,” Opt. Acta 32, 855–872 (1985).
[CrossRef]

Opt. Commun. (12)

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

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

G. Gori, R. Grella, “Shape invariant propagation of polychromatic fields,” Opt. Commun. 49, 173–177 (1984).
[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, “A new type of optical fields,” Opt. Commun. 48, 7–12 (1983).
[CrossRef]

R. Simon, N. Mukunda, E. C. G. Sudarshan, “Partially coherent beams and a generalized ABCD-law,” Opt. Commun. 65, 322–328 (1988).
[CrossRef]

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

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

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

R. Simon, N. Mukunda, “Iwasawa decomposition for SU(1, 1) and the Gouy effect for squeezed states,” Opt. Commun. 95, 39–45 (1993); G. S. Agarwal, R. Simon, “An experiment for the study of the Gouy effect for the squeezed vacuum,” Opt. Commun. 100, 411–414 (1993).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. A (3)

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); “Gaussian pure states in quantum mechanics and the symplectic group,” Phys. Rev. A 37, 2028–2038 (1988); R. Simon, N. Mukunda, B. Dutta, “Quantum-noise matrix, for multimode systems: U(n) invariance, squeezing, and normal forms,” Phys. Rev. A 49, 1567–1683 (1994).
[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]

Phys. Rev. Lett. (2)

R. Simon, N. Mukunda, “Bargmann invariant and the geometry of the Gouy effect,” Phys. Rev. Lett. 70, 880–883 (1993).
[CrossRef] [PubMed]

E. Wolf, “Invariance of the spectrum of light on propagation,” Phys. Rev. Lett. 56, 1370–1372 (1986).
[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, 1995).

A. E. Siegman, Lasers (Oxford U. Press, Oxford, 1986), Chap. 19.

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Equations (82)

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ρ=ρ1ρ2=xy,ρ=ρ1ρ2=xy.
W(ρ, ρ; ν)=I(ν) [det L(ν)]1/22π×exp-14[ρTL(ν)ρ+ρTL(ν)ρ]-12(ρ-ρ)TM(ν)(ρ-ρ)-i2ƛ(ρ-ρ)TK(ν)(ρ+ρ),
I(ν)=d2ρW(ρ, ρ; ν)=invariant.
W(ρ, p; ν)=12πƛ2×d2ΔρWρ-Δρ2, ρ+Δρ2; ν×exp(ip·Δρ/ƛ).
ξ=ρp=xypxpy,
W(ρ, p; ν)=W(ξ; ν)=[det V(ν)]-1/24π2exp[-½ξTV(ν)-1ξ],
V(ν)=A(ν)C(ν)C(ν)TB(ν),
B(ν)=ƛ2[¼L(ν)+M(ν)]+K(ν)L(ν)-1K(ν)T,
A(ν)=L(ν)-1,C(ν)=-L(ν)-1K(ν)T.
M(ν)=ƛ-2B(ν)-¼A(ν)-1-ƛ-2C(ν)TA(ν)-1C(ν),
L(ν)=A(ν)-1,K(ν)=-C(ν)TA(ν)-1.
ξa=ξaW(ξ; ν)d4ξ=0,
d4ξW(ξ; ν)=d2ρW(ρ, ρ; ν)=1.
ξaξb=ξaξbW(ξ; ν)d4ξVab(ν).
Vin(ν)Vout(ν)=SVin(ν)ST.
Eν(ρ)=W(ρ, ρ; ν)=d2pW(ρ, p; ν)=[det A(ν)]-1/22πexp-12ρTA(ν)-1ρ.
A(ν)=diag(σ1(ν)2, σ2(ν)2).
V0(ν)=A0(ν)C0(ν)C0(ν)TB0(ν).
S(z)=1z01Sp(4, R).
Vz(ν)=Az(ν)Cz(ν)Cz(ν)TBz(ν)=A0+z(C0+C0T)+z2B0C0+zB0C0T+zB0B0,
Az(ν)=A0(ν)+z[C0(ν)+C0(ν)T]+z2B0(ν)
Az(ν)=m(z; ν)2A0(ν),
C0(ν)+C0(ν)T=2f0-1A0(ν),
B0(ν)=f0-1z0-1A0(ν).
L(f)=10-f-11Sp(4, R)
V0(ν)L(f)V0(ν)L(f)T.
V0(ν)=A0(ν)Ω(ν)-1(ν)JΩ(ν)-1(ν)JTΩ(ν)-2A0(ν),
J=01-10=-JT=-J-1,
C0TA0-1C0=Ω-22det A0A0.
M0=ƛ-2Ω-2A0-14A0-1-ƛ-2Ω-22det A0A0,
L0=A0-1.
B0=ƛ2M0+14L0+Ω-22det A0A0,
L0(ν)[¼L0(ν)+M0(ν)]=g(ν),
L0(ν)=diag(σ1(ν)-2,σ2(ν)-2).
M0(ν)=diag(δ1(ν)-2,δ2(ν)-2).
14σ14+1σ12δ12=14σ24+1σ22δ22g.
K0(ν)=-C0(ν)TA0(ν)-1=u(ν)J+v(ν)P,
u(ν)=Ω(ν)-1(ν)[σ1(ν)2+σ2(ν)2]2σ1(ν)2σ2(ν)2,
v(ν)=Ω(ν)-1(ν)[σ1(ν)2-σ2(ν)2]2σ1(ν)2σ2(ν)2,
P=0110.
exp-i2ƛ(ρ-ρ)TK0(ν)(ρ+ρ)
=exp-iu(ν)ƛ(xy-yx)Φ(x, y)Φ(x, y)*,
Φ(x, y)=exp-iv(ν)2ƛx+y22-x-y22.
exp-i2 fƛ(x2-y2)
x+y2, x-y2
v/u=(σ12-σ22)/(σ12+σ22).
(ν)2=σ1(ν)2σ2(ν)2Ω(ν)2[u(ν)2-v(ν)2].
B0=ƛ2(M0+¼L0)+(u2-v2)A0.
Ω(ν)-2=ƛ2g(ν)+u(ν)2-v(ν)2.
VSVST=diag(κ1, κ2, κ3, κ4).
κ1κ3ƛ/2,κ2κ4ƛ/2.
V0(ν)=σ1200Ω-10σ22-Ω-100-Ω-1Ω-2σ120Ω-100Ω-2σ22
V˜0(ν)=R(θ)V0(ν)R(θ)T
=diag(a+b, a-b, Ω-2(a+b), Ω-2(a-b)),
R(θ)=cos θ200-Ω sin θ20cos θ2-Ω sin θ200Ω-1 sin θ2cos θ20Ω-1 sin θ200cos θ2,
a=σ12+σ222,b=σ12-σ2222+21/2,
tan θ2=2σ12-σ22.
a(ν)-b(ν)Ωƛ/2,
(ƛΩ)-2(a2-b2)+14(ƛΩ)-1a.
a2-b2=σ12σ22[1-Ω2(u2-v2)]=Ω2σ12σ22gƛ2,
σ12σ22g+142σ12+σ2222[g+ƛ-2(u2-v2)].
σ12σ22=σ4,σ12=σ2 exp(ξσ).
δ12δ22=δ4,δ12=δ2 exp(-ξδ),
sinh 2ξσ=4σ2δ2sinh(ξδ-ξσ).
14cosh 2ξσ+δ2σ2cosh(ξδ-ξσ)+142
cosh2 ξσ14cosh 2ξσ+σ2δ2cosh(ξδ-ξσ)+ƛ-2σ4(u2-v2),
δ2u2-v2 cosh ξσƛ.
δ1(ν)δ2(ν)|u(ν)|ƛ.
K(ν)=R(ν)-1+u(ν)J+v(ν)P
Vz(ν)=S(z)V0(ν)S(z)T=Az(ν)Cz(ν)Cz(ν)TBz(ν),
Az(ν)=(1+Ω-2z2)A0(ν),
Bz(ν)=Ω-2A0(ν)=Ω-21+Ω-2z2Az(ν),
Cz(ν)=Ω-2z1+Ω-2z2Az(ν)+Ω-1J.
Lz(ν)=Az(ν)-1=(1+Ω-2z2)-1L0(ν).
CzTAz-1Cz=Ω-4z2+u2-v21+Ω-2z2A0
Mz=ƛ-2Bz-14Az-1-ƛ-2CzTAz-1Cz=(1+Ω-2z2)-1gA0-14A0-1=(1+Ω-2z2)-1M0.
Kz=-CzTAz-1=Ω-2z1+Ω-2z2+11+Ω-2z2(uJ+vP).
Rz(ν)-1=Ω(ν)-2z1+Ω(ν)-2z2.
Lz(ν)-1Mz(ν)=L0(ν)-1M0(ν)
z-1Rz(ν)-1=Ω(ν)-21+Ω(ν)-2z2=Bz(ν)Az(ν)-1
Wz(ρ, ρ; ν)=m(z)-2W0(m(z)-1ρ, m(z)-1ρ; ν)×expi2ƛRz(ν)(ρTρ-ρTρ),
m(z)=(1+Ω-2z2)1/2,Rz=z+Ω2/z.

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