Abstract

We present the Iwasawa decomposition theorem for the group Sp(2, R) in a form particularly suited to first-order optics, and we exploit it to develop a uniform description of the shape-invariant propagation of several families of optical beams. Both coherent and partially coherent beams are considered. We analyze the Hermite–Gaussian beam as an example of the fully coherent case. For the partially coherent case, we treat the Gaussian Schell-model beams and the recently discovered twisted Gaussian Schell-model beams, both of which are axially symmetric, and also the axially nonsymmetric Gori–Guattari beams. The key observation is that by judicious choice of a free-scale parameter available in the Iwasawa decomposition, appropriately in each case, the one potentially nontrivial factor in the decomposition can be made to act trivially. Invariants of the propagation process are discussed. Shape-invariant propagation is shown to be equivalent to invariance under fractional Fourier transformation.

© 1998 Optical Society of America

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  1. A. E. Siegman, Lasers (Oxford U. Press, Oxford, UK, 1986), Chap. 17.
  2. H. Bacry, M. Cadilhac, “The metaplectic group and Fourier optics,” Phys. Rev. A 23, 2533–2536 (1981);M. Nazarathy, J. Shamir, “First order systems— a canonical operator representation: lossless systems,” J. Opt. Soc. Am. A 72, 356–364 (1982).
    [CrossRef]
  3. E. C. G. Sudarshan, N. Mukunda, R. Simon, “Realization of first order optical systems using thin lenses,” Opt. Acta 32, 855–872 (1985).
    [CrossRef]
  4. H. Kogelnik, “Imaging of optical modes—resonators with internal lenses,” Bell Syst. Tech. J. 44, 455–494 (1965).
    [CrossRef]
  5. R. Simon, N. Mukunda, “Bargmann invariant and the geometry of the Gouy effect,” Phys. Rev. Lett. 70, 880–883 (1993);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] [PubMed]
  6. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995), Chap. 5.
  7. B. E. A. Saleh, “Intensity distribution due to a partially co- herent field and the Collett–Wolf equivalence theorem in the Fresnel zone,” Opt. Commun. 30, 135–138 (1979).
    [CrossRef]
  8. P. D. Santis, F. Gori, G. Guattari, C. Palma, “An example of a Collett–Wolf source,” Opt. Commun. 29, 256–260 (1979);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. H. Weber, “Wave optical analysis of the phase space analyzer,” J. Mod. Opt. 39, 543–559 (1992).
    [CrossRef]
  12. 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]
  13. R. Gase, “The multimode laser radiation as a Gaussian Schell-model beam,” J. Mod. Opt. 38, 1107–1115 (1991).
    [CrossRef]
  14. 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]
  15. R. Gase, “Methods of quantum mechanics applied to partially coherent light beams,” J. Opt. Soc. Am. A 11, 2121–2129 (1994).
    [CrossRef]
  16. R. Simon, N. Mukunda, “Twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 10, 95–109 (1993).
    [CrossRef]
  17. 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]
  18. 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]
  19. 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]
  20. D. Ambrosini, V. Bagini, F. Gori, M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41, 1391–1399 (1994).
    [CrossRef]
  21. 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]
  22. F. Gori, G. Guattari, “A new type of optical fields,” Opt. Commun. 48, 7–12 (1983).
    [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. 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. S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces (Academic, New York, 1978), pp. 257–264, 401–407; S. Lang, SL2(R) (Addison-Wesley, Reading, Mass., 1975), p. 83.
  27. H. M. Ozaktas, D. Mendlovic, “Fourier transforms of fractional orders and their optical interpretation,” Opt. Commun. 101, 163–169 (1993).
    [CrossRef]
  28. D. Mendlovic, H. M. Ozaktas, “Fractional Fourier transforms and their optical implementation: I,” J. Opt. Soc. Am. A 10, 1875–1881 (1993);H. M. Ozaktas, D. Mendlovic, “Fractional Fourier transforms and their optical implementation: II,” J. Opt. Soc. Am. A 10, 2522–2531 (1993).
    [CrossRef]
  29. A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993).
    [CrossRef]
  30. G. S. Agarwal, R. Simon, “A simple realization of fractional Fourier transformation and relation to harmonic oscillator Green’s function,” Opt. Commun. 110, 23–26 (1994).
    [CrossRef]
  31. R. Simon, N. Mukunda, “The two-dimensional symplectic and metaplectic groups and their universal cover,” in Symmetries in Science VI, B. Gruber, ed. (Plenum, New York, 1993), pp. 659–689.
  32. D. Gloge, D. Marcuse, “Formal quantum theory of light rays,” J. Opt. Soc. Am. 59, 1629–1631 (1969).
    [CrossRef]
  33. D. Marcuse, Light Transmission Optics (Van NostrandReinhold, New York, 1972), Chap. 3.
  34. H. H. Arsenault, B. Macukow, “Factorization of the transfer matrix for symmetrical optical systems,” J. Opt. Soc. Am. 73, 1350–1359 (1983);B. Macukow, H. H. Arsenault, “Matrix decompositions for nonsymmetric optical systems,” J. Opt. Soc. Am. 73, 1360–1366 (1983).
    [CrossRef]
  35. S. Abe, J. T. Sheridan, “Optical operations on wave functions as the Abelian subgroups of the special affine Fourier transformation,” Opt. Lett. 19, 1801–1803 (1994);J. Shamir, N. Cohen, “Root and power transformations in optics,” J. Opt. Soc. Am. A 12, 2415–2423 (1995).
    [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. E. Collett, E. Wolf, “Is complete coherence necessary for the generation of highly directional light beams?” Opt. Lett. 2, 27–29 (1978).
    [CrossRef]
  38. E. Wolf, E. Collett, “Partially coherent sources which produce the same intensity distribution as a laser,” Opt. Commun. 25, 293–296 (1978).
    [CrossRef]
  39. Y. Li, E. Wolf, “Radiation from anisotropic Gaussian Schell-model sources,” Opt. Lett. 7, 256–258 (1982).
    [CrossRef] [PubMed]
  40. F. Gori, S. Vicalvi, M. Santarsiero, R. Borghi, “Shape-invariance range of a light beam,” Opt. Lett. 21, 1205–1207 (1996).
    [CrossRef] [PubMed]
  41. 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]

1996

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]

F. Gori, S. Vicalvi, M. Santarsiero, R. Borghi, “Shape-invariance range of a light beam,” Opt. Lett. 21, 1205–1207 (1996).
[CrossRef] [PubMed]

1995

1994

1993

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

H. M. Ozaktas, D. Mendlovic, “Fourier transforms of fractional orders and their optical interpretation,” Opt. Commun. 101, 163–169 (1993).
[CrossRef]

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

D. Mendlovic, H. M. Ozaktas, “Fractional Fourier transforms and their optical implementation: I,” J. Opt. Soc. Am. A 10, 1875–1881 (1993);H. M. Ozaktas, D. Mendlovic, “Fractional Fourier transforms and their optical implementation: II,” J. Opt. Soc. Am. A 10, 2522–2531 (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]

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]

A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993).
[CrossRef]

1992

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 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]

1991

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

1988

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

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

1985

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

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

1982

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

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

1981

H. Bacry, M. Cadilhac, “The metaplectic group and Fourier optics,” Phys. Rev. A 23, 2533–2536 (1981);M. Nazarathy, J. Shamir, “First order systems— a canonical operator representation: lossless systems,” J. Opt. Soc. Am. A 72, 356–364 (1982).
[CrossRef]

1979

B. E. A. Saleh, “Intensity distribution due to a partially co- herent 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);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]

1978

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

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

1969

1965

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

Abe, S.

Agarwal, G. S.

G. S. Agarwal, R. Simon, “A simple realization of fractional Fourier transformation and relation to harmonic oscillator Green’s function,” Opt. Commun. 110, 23–26 (1994).
[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]

Arsenault, H. H.

Bacry, H.

H. Bacry, M. Cadilhac, “The metaplectic group and Fourier optics,” Phys. Rev. A 23, 2533–2536 (1981);M. Nazarathy, J. Shamir, “First order systems— a canonical operator representation: lossless systems,” J. Opt. Soc. Am. A 72, 356–364 (1982).
[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]

Borghi, R.

Cadilhac, M.

H. Bacry, M. Cadilhac, “The metaplectic group and Fourier optics,” Phys. Rev. A 23, 2533–2536 (1981);M. Nazarathy, J. Shamir, “First order systems— a canonical operator representation: lossless systems,” J. Opt. Soc. Am. A 72, 356–364 (1982).
[CrossRef]

Collett, E.

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

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]

Gase, R.

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

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

Gloge, D.

Gori, F.

F. Gori, S. Vicalvi, M. Santarsiero, R. Borghi, “Shape-invariance range of a light beam,” Opt. Lett. 21, 1205–1207 (1996).
[CrossRef] [PubMed]

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]

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

Guattari, G.

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);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]

Helgason, S.

S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces (Academic, New York, 1978), pp. 257–264, 401–407; S. Lang, SL2(R) (Addison-Wesley, Reading, Mass., 1975), p. 83.

Kogelnik, H.

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

Li, Y.

Lohmann, A. W.

Macukow, B.

Mandel, L.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995), Chap. 5.

Marcuse, D.

D. Gloge, D. Marcuse, “Formal quantum theory of light rays,” J. Opt. Soc. Am. 59, 1629–1631 (1969).
[CrossRef]

D. Marcuse, Light Transmission Optics (Van NostrandReinhold, New York, 1972), Chap. 3.

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).
[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 ABCD optical systems,” J. Mod. Opt. 39, 625–635 (1992).
[CrossRef]

Mendlovic, D.

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

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

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, “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]

R. Simon, N. Mukunda, “The two-dimensional symplectic and metaplectic groups and their universal cover,” in Symmetries in Science VI, B. Gruber, ed. (Plenum, New York, 1993), pp. 659–689.

Ozaktas, H. M.

Palma, C.

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

Saleh, B. E. A.

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

Santarsiero, M.

F. Gori, S. Vicalvi, M. Santarsiero, R. Borghi, “Shape-invariance range of a light beam,” Opt. Lett. 21, 1205–1207 (1996).
[CrossRef] [PubMed]

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

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]

Sheridan, J. T.

Siegman, A. E.

A. E. Siegman, Lasers (Oxford U. Press, Oxford, UK, 1986), Chap. 17.

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]

G. S. Agarwal, R. Simon, “A simple realization of fractional Fourier transformation and relation to harmonic oscillator Green’s function,” Opt. Commun. 110, 23–26 (1994).
[CrossRef]

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

R. Simon, N. Mukunda, “Bargmann invariant and the geometry of the Gouy effect,” Phys. Rev. Lett. 70, 880–883 (1993);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] [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, E. C. G. Sudarshan, “Partially coherent beams and a generalized abcd-law,” Opt. Commun. 65, 322–328 (1988).
[CrossRef]

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, “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]

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R. Simon, N. Mukunda, E. C. G. Sudarshan, “Partially coherent beams and a generalized abcd-law,” Opt. Commun. 65, 322–328 (1988).
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E. C. G. Sudarshan, N. Mukunda, R. Simon, “Realization of first order optical systems using thin lenses,” Opt. Acta 32, 855–872 (1985).
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[CrossRef]

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

Tervonen, E.

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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).
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L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995), Chap. 5.

Zubairy, M. S.

J. T. Foley, M. S. Zubairy, “The directionality of Gaussian Schell-model beams,” Opt. Commun. 26, 297–300 (1978).
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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).
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R. Gase, “Methods of quantum mechanics applied to partially coherent light beams,” J. Opt. Soc. Am. A 11, 2121–2129 (1994).
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A. T. Friberg, J. Turunen, “Imaging of Gaussian Schell-model sources,” J. Opt. Soc. Am. A 5, 713–720 (1988).
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R. Simon, N. Mukunda, “Twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 10, 95–109 (1993).
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D. Mendlovic, H. M. Ozaktas, “Fractional Fourier transforms and their optical implementation: I,” J. Opt. Soc. Am. A 10, 1875–1881 (1993);H. M. Ozaktas, D. Mendlovic, “Fractional Fourier transforms and their optical implementation: II,” J. Opt. Soc. Am. A 10, 2522–2531 (1993).
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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).
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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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Opt. Acta

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

H. M. Ozaktas, D. Mendlovic, “Fourier transforms of fractional orders and their optical interpretation,” Opt. Commun. 101, 163–169 (1993).
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G. S. Agarwal, R. Simon, “A simple realization of fractional Fourier transformation and relation to harmonic oscillator Green’s function,” Opt. Commun. 110, 23–26 (1994).
[CrossRef]

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

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

F. Gori, G. Guattari, “A new type of optical fields,” Opt. Commun. 48, 7–12 (1983).
[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).
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B. E. A. Saleh, “Intensity distribution due to a partially co- herent field and the Collett–Wolf equivalence theorem in the Fresnel zone,” Opt. Commun. 30, 135–138 (1979).
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P. D. Santis, F. Gori, G. Guattari, C. Palma, “An example of a Collett–Wolf source,” Opt. Commun. 29, 256–260 (1979);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]

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

Opt. Lett.

Phys. Rev. A

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]

H. Bacry, M. Cadilhac, “The metaplectic group and Fourier optics,” Phys. Rev. A 23, 2533–2536 (1981);M. Nazarathy, J. Shamir, “First order systems— a canonical operator representation: lossless systems,” J. Opt. Soc. Am. A 72, 356–364 (1982).
[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]

Phys. Rev. Lett.

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

Pure Appl. Opt.

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

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995), Chap. 5.

A. E. Siegman, Lasers (Oxford U. Press, Oxford, UK, 1986), Chap. 17.

S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces (Academic, New York, 1978), pp. 257–264, 401–407; S. Lang, SL2(R) (Addison-Wesley, Reading, Mass., 1975), p. 83.

R. Simon, N. Mukunda, “The two-dimensional symplectic and metaplectic groups and their universal cover,” in Symmetries in Science VI, B. Gruber, ed. (Plenum, New York, 1993), pp. 659–689.

D. Marcuse, Light Transmission Optics (Van NostrandReinhold, New York, 1972), Chap. 3.

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Equations (109)

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x=x1x2,p=p1p2
xpxp=Sxp,
S=abcd,ad-bc=1.
J0=i20α-α-10,J1=i2100-1,
J2=-i20αα-10,
[J0, J1]=iJ2,
[J0, J2]=-iJ1,
[J1, J2]=-iJ0.
xˆk=xk,pˆk=-iƛ xk,
[xˆk, pˆl]=iƛδkl,k, l=1, 2.
Tˆ0=(α-1xˆ·xˆ+αpˆ·pˆ)/4ƛ,
Tˆ1=(xˆ·pˆ+pˆ·xˆ)/4ƛ,
Tˆ2=(α-1xˆ·xˆ-αpˆ·pˆ)/4ƛ.
[Tˆ0, Tˆ1]=iTˆ2,
[Tˆ0, Tˆ2]=-iTˆ1,
[Tˆ1, Tˆ2]=-iTˆ0.
S(β)=exp(-iβaJa)Sp(2, R).
Uˆ(β)=exp(-iβaTˆa)
ξˆ=xˆpˆ.
Uˆ(β)ξˆUˆ(β)=S(β)ξˆ,
Uˆ(β)ξˆνUˆ(β)=σ=12[S(β)]νσξˆσ,ν=1, 2, 3, 4.
[Uˆ(β)Uˆ(β)]ξˆ[Uˆ(β)Uˆ(β)]=[S(β)S(β)]ξˆ.
Ψ(x)=(Uˆ(S)Ψ)(x)
=d2x<x|Uˆ(S)|x>Ψ(x),
x|Uˆ(S)|x=-iλbexpi2bƛ(dx·x-2x·x+ax·x)forb0,
=|a|expi c2aƛx·xδ(2)(x-ax)
forb=0.
R(θ)=exp(-iθJ0)=cos θ/2α sin θ/2-α-1 sin θ/2cos θ/2,
M(ξ)=exp(-iξJ1)=eξ/200e-ξ/2,
L(g)=exp[-igα(J0+J2)]=10-g1;
ξ, gR, 0θ<4π.
S=abcd
=exp[-igα(J0+J2)]exp(-iξJ1)exp(-iθJ0)
=L(g)M(ξ)R(θ),
g=-(ac+α-2bd)/(a2+α-2b2),
eξ/2=(a2+α-2b2)1/2,
eiθ/2=(a+iα-1b)/(a2+α-2b2)1/2.
Uˆ(S)=Uˆ(L(g))Uˆ(M(ξ))Uˆ(R(θ)),
Uˆ(R(θ))=exp(-iθTˆ0)
=exp-iθ4ƛ(α-1xˆ·xˆ+αpˆ·pˆ),
Uˆ(M(ξ))=exp(-iξTˆ1)=exp-iξ4ƛ(xˆ·pˆ+pˆ·xˆ),
Uˆ(L(g))=exp[-igα(Tˆ0+Tˆ2)]=exp-ig2ƛxˆ·xˆ.
[Uˆ(L(g))Ψ](x)=exp-ig2λx·xΨ(x),
[Uˆ(M(ξ))Ψ](x)=m-1Ψ(m-1x),m=eξ/2.
S=R(π)=0α-α-10,
Ψ(x)=[Uˆ(R(π))Ψ](x)=-iαλd2x exp-i x·xƛαΨ(x)=-iαƛΨ˜xαƛ,
Ψn1,n2(x; 0)=A2πw021/2 exp-x12+x22w02×Hn12 x1w0Hn22 x2w0,
α=w022ƛ=πw02λzR,
Tˆ0Ψn1,n2(x; 0)=n1+n2+12Ψn1,n2(x; 0).
F(z)=1z01.
g(z)-1=-R(z)=-z[1+(zR/z)2],
m(z)=exp[ξ(z)/2]=[1+(z/zR)2]1/2,
θ=2 arctan(z/zR).
[Uˆ(R(θ))Ψn1,n2](x; 0)=[exp(-iθTˆ0)Ψn1,n2](x; 0)=exp[-iϕG(z)]Ψn1,n2(x; 0),
ϕG(z)=(n1+n2+1)arctan(z/zR),
Ψn1,n2(x; z)[Uˆ(F(z))Ψn1,n2](x; 0)=m(z)-1 exp[-iϕG(z)]exp-i g(z)2ƛx·x×Ψn1,n2(m(z)-1x; 0).
w(z)=m(z)w0=[1+(z/zR)2]1/2w0,
Ψn1,n2(x; z)=2πw(z)21/2 ×exp[-iϕG(z)]expix·x2ƛR(z)-x·xw(z)2×Hn1(2x1/w(z))Hn2(2x2/w(z)).
W(x, x; ν)W(x, x; ν)=Uˆ(S)W(x, x; ν)Uˆ(S),
W(x, x; ν)W(x, x; ν)
=Uˆ(R(θ))W(x, x; ν)Uˆ(R(θ))
=exp-i θ4αƛHˆW(x, x; ν),
Hˆ=x·x-x·x-ƛ2α2(2-2),
Uˆ(L(g))W(x, x; ν)Uˆ(L(g))
=exp-ig2ƛ(x·x-x·x)W(x, x; ν),
Uˆ(M(ξ))W(x, x; ν)Uˆ(M(ξ))
=m-2W(m-1x, m-1x; ν),
m=eξ/2.
W0(x, x; ν)=A exp-|x|2+|x|24σ0(ν)2-|(x-x)|22δ0(ν)2.
HˆW0(x, x; ν)=0.
α=4πσ0(ν)2λ1+4σ0(ν)2δ0(ν)2-1/2zR.
Uˆ(R(θ))W0(x, x; ν)U(R(θ))=W0(x, x; ν).
Wz(x, x; ν)=Uˆ(F(z))W0(x, x; ν)Uˆ(F(z))=m(z)-2 expi2ƛR(z)(x·x-x·x)×W0[m(z)-1x, m(z)-1x; ν],
m(z)=[1+(z/zR)2]1/2,
R(z)=z[1+(zR/z)2],
zR=4πσ0(ν)2λ1+4σ0(ν)2δ0(ν)2-1/2.
σ0(ν)σz(ν)=m(z)σ0(ν),
δ0(ν)δz(ν)=m(z)δ0(ν).
Δθ=limzσz(ν)z=σ0(ν)zR,
σ0(ν)zR=λ2π1δ0(ν)2+14σ0(ν)21/2
W0(x, x; ν)=A exp-|x|2+|x|24σ0(ν)2-|(x-x)|22δ0(ν)2-iu0(ν)x  x,
-δ0(ν)-2u0(ν)δ0(ν)-2.
η(ν)=u0(ν)δ0(ν)2,-1η(ν)1,
β(ν)=σ0(ν)/δ0(ν),0<β(ν)<.
α=4πσ0(ν)2λ(1+4β2+4η2β4)-1/2zR.
Wz(x, x; ν)=Uˆ(F(z))W0(x, x; ν)Uˆ(F(z))=m(z)-2 expi2ƛR(z)(x·x-x·x)×exp-|x|2+|x|24σz(ν)2-|(x-x)|22δz(ν)2-iuz(ν)x  x,
σz(ν)=m(z)σ0(ν),δz(ν)=m(z)δ0(ν),
uz(ν)=m(z)-2u0(ν),R(z)=z[1+(zR/z)2],
m(z)=[1+(z/zR)2]1/2,
zR=4πσ0(ν)2λ(1+4β2+4η2β4)-1/2.
Δθ=limzσz(ν)z=λ4πσ0(ν)(1+4β2+4η2β4)1/2.
W0(x, x; ν)=AW01(x1, x1; ν)W02(x2, x2; ν),
W0j(xj, xj; ν)=exp-14σ0j(ν)2(xj2+xj2)-12δ0j(ν)2(xj-xj)2,j=1, 2.
14σ01(ν)4+1σ01(ν)2δ01(ν)2
=14σ02(ν)4+1σ02(ν)2δ02(ν)2.
α=4πσ01(ν)2λ(1+4β12)-1/2
=4πσ02(ν)2λ(1+4β22)-1/2zR,
Wz(x, x; ν)=Uˆ(F(z))W0(x, x; ν)Uˆ(F(z))=A expi2ƛR(z)(|x|2-|x|2)×Wz1(x1, x1; ν)Wz2(x2, x2; ν),
Wzj(xj, xj; ν)=m(z)-1W0j(m(z)-1xj, m(z)-1xj; ν)=m(z)-1 exp-14σzj(ν)2(xj2+xj2)-12δzj(ν)2(xj-xj)2,
σzj(ν)=m(z)σ0j(ν),
δzj(ν)=m(z)δ0j(ν),
m(z)=[1+(z/zR)2]1/2,R(z)=z[1+(zR/z)2].
14σz1(ν)4+1σz1(ν)2δz1(ν)2
=14σz2(ν)4+1σz2(ν)2δz2(ν)2.
Uˆ(R(θ))=exp(-iθTˆ0)=exp-iθ4ƛ(α-1xˆ·xˆ+αpˆ·pˆ)
Uˆ(S)=Uˆ(L(g))Uˆ(M(ξ))Uˆ(R(θ)).
θ(z)=2 arctan(z/zR).
R(z)=z[1+(zR/z)2],
m(z)=[1+(z/zR)2]1/2,

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