Abstract

We introduce a new class of partially coherent axially symmetric Gaussian Schell-model (GSM) beams incorporating a new twist phase quadratic in configuration variables. This phase twists the beam about its axis during propagation and is shown to be bounded in strength because of the positive semidefiniteness of the cross-spectral density. Propagation characteristics and invariants for such beams are derived and interpreted, and two different geometric representations are developed. Direct effects of the twist phase on free propagation as well as in parabolic index fibers are demonstrated. Production of such twisted GSM beams, starting with Li–Wolf anisotropic GSM beams, is described.

© 1993 Optical Society of America

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  1. J. T. Foley, M. S. Zubairy, “The directionality of Gaussian Schell-model beams,” Opt. Commun. 26, 297–300 (1978).
    [CrossRef]
  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]
  3. P. De Santis, F. Gori, G. Guattari, C. Palma, “An example of a Collett–Wolf source,” Opt. Commun. 29, 256–260 (1979).
    [CrossRef]
  4. F. Gori, “Collett–Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
    [CrossRef]
  5. A. T. Friberg, R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41, 383–387 (1982).
    [CrossRef]
  6. F. Gori, “Mode propagation of the fields generated by Collett–Wolf Schell-model sources,” Opt. Commun. 46, 149–154 (1983).
    [CrossRef]
  7. F. Gori, G. Guattari, “A new type of optical fields,” Opt. Commun. 48, 7–12 (1983).
    [CrossRef]
  8. 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]
  9. This subject has been reviewed extensively in E. Wolf, “Coherence and radiometry,” J. Opt. Soc. Am. 68, 6–17 (1978),and in H. P. Baltes, J. Geist, A. Walther, “Radiometry and coherence,” in Inverse Source Problems in Optics, H. P. Baltes, ed. (Springer-Verlag, Berlin, 1978).
    [CrossRef]
  10. W. H. Carter, E. Wolf, “Coherence and radiometry with quasi-homogeneous planar sources,” J. Opt. Soc. Am. 67, 785–796 (1977).
    [CrossRef]
  11. E. Collett, E. Wolf, “Beams generated by Gaussian quasi-homogeneous sources,” Opt. Commun. 32, 27–31 (1980).
    [CrossRef]
  12. R. Simon, N. Mukunda, E. C. G. Sudarshan, “Partially coherent beams and a generalized abcd-law,” Opt. Commun. 65, 322–328 (1988).
    [CrossRef]
  13. H. Kogelnik, “Imaging of optical modes—resonators with internal lenses,” Bell Syst. Tech. J. 44, 455–494 (1965).
  14. 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]
  15. E. Collett, E. Wolf, “Is complete coherence necessary for the generation of highly directional light beams?” Opt. Lett. 2, 27–29 (1978).
    [CrossRef] [PubMed]
  16. E. Wolf, E. Collett, “Partially coherent sources which produce the same intensity distribution as a laser,” Opt. Commun. 25, 293–296 (1978).
    [CrossRef]
  17. E. Collett, E. Wolf, “New equivalence theorems for planar sources that generate the same distribution of radiant intensity,” J. Opt. Soc. Am. 69, 942–950 (1979).
    [CrossRef]
  18. A. Starikov, E. Wolf, “Coherent-mode representation of Gaussian Schell-model sources and their radiation fields,” J. Opt. Soc. Am. 72, 923–928 (1982).
    [CrossRef]
  19. Y. Li, E. Wolf, “Radiation from anisotropic Gaussian Schell-model sources,” Opt. Lett. 7, 256–258 (1982).
    [CrossRef] [PubMed]
  20. R. Simon, E. C. G. Sudarshan, N. Mukunda, “Anisotropic Gaussian Schell-model fields: passage through first order systems and associated invariants,” Phys. Rev. A 31, 2419–2434 (1985).
    [CrossRef] [PubMed]
  21. R. Simon, E. C. G. Sudarshan, N. Mukunda, “Gaussian Wigner distributions: a complete characterization,” Phys. Lett. A 124, 223–228 (1987).
    [CrossRef]
  22. 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]
  23. D. Gloge, D. Marcuse, “Formal quantum theory of light rays,” J. Opt. Soc. Am. 59, 1629–1631 (1969).
    [CrossRef]
  24. E. P. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
    [CrossRef]
  25. J. E. Moyal, “Quantum mechanics as a statistical theory,” Proc. Cambridge Philos. Soc. 45, 99–124 (1949).
    [CrossRef]
  26. H. Bacry, M. Cadilhac, “The metaplectic group and Fourier optics,” Phys. Rev. A 23, 2533–2536 (1981).
    [CrossRef]
  27. M. Nazarathy, J. Shamir, “First order optics—a canonical operator representation: lossless systems,” J. Opt. Soc. Am. 72, 356–364 (1982).
    [CrossRef]
  28. E. C. G. Sudarshan, N. Mukunda, R. Simon, “Realization of first order optical systems using thin lenses,” Opt. Acta 32, 855–872 (1985).
    [CrossRef]
  29. B. Macukov, H. H. Arsenault, “Matrix decompositions for nonsymmetric optical systems,” J. Opt. Soc. Am. 73, 1360–1366 (1983).
    [CrossRef]
  30. A. J. Dragt, E. Forest, K. B. Wolf, “Foundations of a Lie algebraic theory of geometrical optics,” in Lie Methods in Optics, J. Sanchez Mondragon, K. B. Wolf, eds. (Springer-Verlag, Berlin, 1986) p. 105.
    [CrossRef]
  31. H. A. Buchdahl, “Optical aberration coefficients. XIII. Theory of reversible semisymmetric systems,” J. Opt. Soc. Am. 57, 517–522 (1967).
    [CrossRef] [PubMed]
  32. W. Brouwer, E. L. O’Neill, A. Walther, “The role of eikonal and matrix methods in contrast transfer calculations,” Appl. Opt. 2, 1239–1245 (1963).
    [CrossRef]
  33. The abcdlaw for Gaussian Schell-model beams was derived first in Ref. 8and subsequently in J. Turunen, A. T. Friberg, “Matrix representation of Gaussian Schell-model beams in optical systems,” Opt. Laser Technol. 18, 259–267 (1986).It has been derived for an arbitrary partially coherent beam in Ref. 12.
    [CrossRef]
  34. J. A. Arnaud, Beam and Fiber Optics (Academic, New York, 1976), pp. 70 and 71.

1988

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

1987

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Gaussian Wigner distributions: a complete characterization,” Phys. Lett. A 124, 223–228 (1987).
[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]

1986

The abcdlaw for Gaussian Schell-model beams was derived first in Ref. 8and subsequently in J. Turunen, A. T. Friberg, “Matrix representation of Gaussian Schell-model beams in optical systems,” Opt. Laser Technol. 18, 259–267 (1986).It has been derived for an arbitrary partially coherent beam in Ref. 12.
[CrossRef]

1985

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 fields: passage through first order systems and associated invariants,” Phys. Rev. A 31, 2419–2434 (1985).
[CrossRef] [PubMed]

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

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

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

B. Macukov, H. H. Arsenault, “Matrix decompositions for nonsymmetric optical systems,” J. Opt. Soc. Am. 73, 1360–1366 (1983).
[CrossRef]

1982

1981

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

1980

E. Collett, E. Wolf, “Beams generated by Gaussian quasi-homogeneous sources,” Opt. Commun. 32, 27–31 (1980).
[CrossRef]

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

1979

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

E. Collett, E. Wolf, “New equivalence theorems for planar sources that generate the same distribution of radiant intensity,” J. Opt. Soc. Am. 69, 942–950 (1979).
[CrossRef]

1978

1977

1969

1967

1965

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

1963

1949

J. E. Moyal, “Quantum mechanics as a statistical theory,” Proc. Cambridge Philos. Soc. 45, 99–124 (1949).
[CrossRef]

1932

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

Arnaud, J. A.

J. A. Arnaud, Beam and Fiber Optics (Academic, New York, 1976), pp. 70 and 71.

Arsenault, H. H.

Bacry, H.

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

Bertolotti, M.

Brouwer, W.

Buchdahl, H. A.

Cadilhac, M.

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

Carter, W. H.

Collett, E.

E. Collett, E. Wolf, “Beams generated by Gaussian quasi-homogeneous sources,” Opt. Commun. 32, 27–31 (1980).
[CrossRef]

E. Collett, E. Wolf, “New equivalence theorems for planar sources that generate the same distribution of radiant intensity,” J. Opt. Soc. Am. 69, 942–950 (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]

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

De Santis, P.

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

Dragt, A. J.

A. J. Dragt, E. Forest, K. B. Wolf, “Foundations of a Lie algebraic theory of geometrical optics,” in Lie Methods in Optics, J. Sanchez Mondragon, K. B. Wolf, eds. (Springer-Verlag, Berlin, 1986) p. 105.
[CrossRef]

Foley, J. T.

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

Forest, E.

A. J. Dragt, E. Forest, K. B. Wolf, “Foundations of a Lie algebraic theory of geometrical optics,” in Lie Methods in Optics, J. Sanchez Mondragon, K. B. Wolf, eds. (Springer-Verlag, Berlin, 1986) p. 105.
[CrossRef]

Friberg, A. T.

The abcdlaw for Gaussian Schell-model beams was derived first in Ref. 8and subsequently in J. Turunen, A. T. Friberg, “Matrix representation of Gaussian Schell-model beams in optical systems,” Opt. Laser Technol. 18, 259–267 (1986).It has been derived for an arbitrary partially coherent beam in Ref. 12.
[CrossRef]

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

Gloge, D.

Gori, F.

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

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

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

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

Guattari, G.

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

P. De 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).

Li, Y.

Macukov, B.

Marcuse, D.

Moyal, J. E.

J. E. Moyal, “Quantum mechanics as a statistical theory,” Proc. Cambridge Philos. Soc. 45, 99–124 (1949).
[CrossRef]

Mukunda, N.

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).
[CrossRef] [PubMed]

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Gaussian Wigner distributions: a complete characterization,” Phys. Lett. A 124, 223–228 (1987).
[CrossRef]

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Anisotropic Gaussian Schell-model fields: passage through first order 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]

Nazarathy, M.

O’Neill, E. L.

Palma, C.

P. De 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]

Shamir, J.

Simon, R.

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).
[CrossRef] [PubMed]

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Gaussian Wigner distributions: a complete characterization,” Phys. Lett. A 124, 223–228 (1987).
[CrossRef]

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Anisotropic Gaussian Schell-model fields: passage through first order 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]

Starikov, A.

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: a complete characterization,” Phys. Lett. A 124, 223–228 (1987).
[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]

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 fields: passage through first order 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]

Turunen, J.

The abcdlaw for Gaussian Schell-model beams was derived first in Ref. 8and subsequently in J. Turunen, A. T. Friberg, “Matrix representation of Gaussian Schell-model beams in optical systems,” Opt. Laser Technol. 18, 259–267 (1986).It has been derived for an arbitrary partially coherent beam in Ref. 12.
[CrossRef]

Walther, A.

Wigner, E. P.

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

Wolf, E.

Wolf, K. B.

A. J. Dragt, E. Forest, K. B. Wolf, “Foundations of a Lie algebraic theory of geometrical optics,” in Lie Methods in Optics, J. Sanchez Mondragon, K. B. Wolf, eds. (Springer-Verlag, Berlin, 1986) p. 105.
[CrossRef]

Zubairy, M. S.

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

Appl. Opt.

Bell Syst. Tech. J.

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

J. Opt. Soc. Am.

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]

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

H. A. Buchdahl, “Optical aberration coefficients. XIII. Theory of reversible semisymmetric systems,” J. Opt. Soc. Am. 57, 517–522 (1967).
[CrossRef] [PubMed]

M. Nazarathy, J. Shamir, “First order optics—a canonical operator representation: lossless systems,” J. Opt. Soc. Am. 72, 356–364 (1982).
[CrossRef]

B. Macukov, H. H. Arsenault, “Matrix decompositions for nonsymmetric optical systems,” J. Opt. Soc. Am. 73, 1360–1366 (1983).
[CrossRef]

This subject has been reviewed extensively in E. Wolf, “Coherence and radiometry,” J. Opt. Soc. Am. 68, 6–17 (1978),and in H. P. Baltes, J. Geist, A. Walther, “Radiometry and coherence,” in Inverse Source Problems in Optics, H. P. Baltes, ed. (Springer-Verlag, Berlin, 1978).
[CrossRef]

W. H. Carter, E. Wolf, “Coherence and radiometry with quasi-homogeneous planar sources,” J. Opt. Soc. Am. 67, 785–796 (1977).
[CrossRef]

E. Collett, E. Wolf, “New equivalence theorems for planar sources that generate the same distribution of radiant intensity,” J. Opt. Soc. Am. 69, 942–950 (1979).
[CrossRef]

A. Starikov, E. Wolf, “Coherent-mode representation of Gaussian Schell-model sources and their radiation fields,” J. Opt. Soc. Am. 72, 923–928 (1982).
[CrossRef]

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).
[CrossRef]

Opt. Commun.

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, “Beams generated by Gaussian quasi-homogeneous sources,” Opt. Commun. 32, 27–31 (1980).
[CrossRef]

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

J. T. Foley, M. S. Zubairy, “The directionality of Gaussian Schell-model beams,” Opt. Commun. 26, 297–300 (1978).
[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. De Santis, F. Gori, G. Guattari, C. Palma, “An example of a Collett–Wolf source,” Opt. Commun. 29, 256–260 (1979).
[CrossRef]

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

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

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

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

Opt. Laser Technol.

The abcdlaw for Gaussian Schell-model beams was derived first in Ref. 8and subsequently in J. Turunen, A. T. Friberg, “Matrix representation of Gaussian Schell-model beams in optical systems,” Opt. Laser Technol. 18, 259–267 (1986).It has been derived for an arbitrary partially coherent beam in Ref. 12.
[CrossRef]

Opt. Lett.

Phys. Lett. A

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Gaussian Wigner distributions: a complete characterization,” Phys. Lett. A 124, 223–228 (1987).
[CrossRef]

Phys. Rev.

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

Phys. Rev. A

H. Bacry, M. Cadilhac, “The metaplectic group and Fourier optics,” Phys. Rev. A 23, 2533–2536 (1981).
[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 fields: passage through first order 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]

Proc. Cambridge Philos. Soc.

J. E. Moyal, “Quantum mechanics as a statistical theory,” Proc. Cambridge Philos. Soc. 45, 99–124 (1949).
[CrossRef]

Other

J. A. Arnaud, Beam and Fiber Optics (Academic, New York, 1976), pp. 70 and 71.

A. J. Dragt, E. Forest, K. B. Wolf, “Foundations of a Lie algebraic theory of geometrical optics,” in Lie Methods in Optics, J. Sanchez Mondragon, K. B. Wolf, eds. (Springer-Verlag, Berlin, 1986) p. 105.
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

Effect of the invariant twist phase on beam expansion, with β = 5, σs(0) = 103ƛ.

Fig. 2
Fig. 2

Twist phenomenon in free propagation, with β = 5.

Fig. 3
Fig. 3

Twist phenomenon in a parabolic-index fiber, with (a) Ω0Δ = 0.5, (b) Ω0Δ = 1, (c) Ω0Δ = 2. In all cases β = 5.

Equations (125)

Equations on this page are rendered with MathJax. Learn more.

= i σ 2 = [ 0 1 1 0 ] .
E ( ρ 1 , ρ 2 ) = α ρ 1 2 + β ρ 2 2 + γ ρ 1 ρ 2 + δ ρ 1 ρ 2 .
E ( ρ 1 , ρ 2 ) = a 1 ( ρ 1 2 + ρ 2 2 ) + γ ρ 1 ρ 2 + i a 2 ( ρ 1 2 ρ 2 2 ) + i a 3 ρ 1 ρ 2 = ( a 1 + γ / 2 ) ( ρ 1 2 + ρ 2 2 ) ( γ / 2 ) ( ρ 1 ρ 2 ) 2 + i a 2 ( ρ 1 2 ρ 2 2 ) + i a 3 ρ 1 ρ 2 ,
W z ( ρ 1 , ρ 2 ; ν ) = I ( ν ) 2 π σ s ( ν ) 2 × exp [ 1 4 σ s ( ν ) 2 ( ρ 1 2 + ρ 2 2 ) ( ρ 1 ρ 2 ) 2 2 σ g ( ν ) 2 × i 2 ƛ R ( ν ) ( ρ 1 2 ρ 2 2 ) i u ( ν ) ƛ ρ 1 ρ 2 ] .
I ( ν ) = R 2 d 2 ρ W z ( ρ , ρ ; ν ) .
W ( ρ , p ) = 1 ( 2 π ƛ ) 2 R 2 d 2 Δ ρ W ( ρ 1 2 Δ ρ , ρ + 1 2 Δ ρ ) × exp ( i p Δ ρ / ƛ ) .
I = R 2 W ( ρ , ρ ) d 2 ρ = R 4 d 2 p d 2 ρ W ( ρ , p ) .
ξ = ( ρ p ) ,
ξ out = S ξ in ,
S T K S = K , K = [ 0 1 1 0 ] ,
S ( z ) = [ 1 z 1 0 1 ] ;
S ( F ) = [ 1 0 F 1 ] ;
S ( M ) = [ M 0 0 ( M 1 ) T ] ;
S ( l 0 ) = [ 0 l 0 1 l 0 1 1 0 ] ,
S ( θ ) = [ cos θ sin θ 0 sin θ cos θ cos θ sin θ 0 sin θ cos θ ] .
S = [ a 1 b 1 c 1 d 1 ] , a d b c = 1
S = s 1 , s = [ a b c d ] Sp ( 2 , R ) ,
ψ ( ρ ) = d 2 ρ G ( ρ , ρ ) ψ ( ρ ) , G ( ρ , ρ ) = ( i / b λ ) exp [ i 2 b ƛ ( d ρ 2 2 ρ ρ + a ρ 2 ) ] ,
W ( ρ 1 , ρ 2 ) = d 2 ρ 1 d 2 ρ 2 G ( ρ 1 , ρ 1 ) * G ( ρ 2 , ρ 2 ) W ( ρ 1 , ρ 2 ) .
W ( ξ ) = W ( S 1 ξ ) ,
l = ( 4 σ s 2 ) 1 , m = ( σ g 2 ) 1
W ( ρ , p ) = I ( 2 π ) 3 ƛ 2 σ s 2 exp ( 2 l ρ 2 ) × R 2 d 2 Δ ρ exp [ 1 2 ( l + m ) Δ ρ 2 ] × exp [ i ƛ ( p + ρ R + u ρ ) Δ ρ ] .
R n d n x exp ( x T A x + i k T x ) = π n / 2 ( det A ) 1 / 2 exp ( 1 4 k T A 1 k ) ,
W ( ρ , p ) = I ( π ƛ ) 2 l l + m × exp [ 2 l ρ 2 1 2 ƛ 2 ( l + m ) ( p + ρ R + u ρ ) 2 ] .
ξ 1 = x = ξ 11 , ξ 2 = y = ξ 21 , ξ 3 = p x = ξ 12 , ξ 4 = p y = ξ 22 .
G a b G r α , s β = δ r s g α β + r s α β g 0 , g = [ 2 l 0 0 0 ] + 1 2 ƛ 2 ( l + m ) [ u 2 + 1 / R 2 1 / R 1 / R 1 ] , g 0 = u [ 2 ƛ 2 ( l + m ) ] 1 .
G 11 = G 22 = 2 l + ( u 2 + 1 / R 2 ) [ 2 ƛ 2 ( l + m ) ] 1 , G 33 = G 44 = [ 2 ƛ 2 ( l + m ) ] 1 , G 13 = G 24 = [ 2 ƛ 2 ( l + m ) R ] 1 , G 14 = G 23 = u [ 2 ƛ 2 ( l + m ) ] 1 .
det G = ( det g g 0 2 ) 2 = l 2 [ ƛ 2 ( l + m ) ] 2 ,
W ( ξ ) = I π 2 ( det G ) 1 / 2 exp ( ξ T G ξ ) .
V a b = ξ a ξ b = 1 I R 4 d 4 ξ ξ a ξ b W ( ξ ) .
R n d n x x a x b exp ( x T A x ) = π n / 2 2 ( det A ) 1 / 2 ( A 1 ) a b ,
V = ( 1 / 2 ) G 1 .
V a b V r α , s β = δ r s υ α β + r s α β υ 0 , υ = ½ [ det g / ( det g g 0 2 ) ] g 1 , = ½ [ 1 + u 2 / 4 ƛ 2 l ( l + m ) ] g 1 υ 0 = ½ g 0 / ( g 0 2 det g ) = u / 4 l .
V = 1 / 4 l [ 1 0 1 / R u 0 1 u ( 1 / R ) 1 / R u 4 ƛ 2 l ( l + m ) + u 2 + 1 / R 2 0 u 1 / R 2 0 4 ƛ 2 l ( l + m ) + u 2 + 1 / R 2 ] .
W ( ξ ) = I 4 π 2 ( det V ) 1 / 2 exp ( 1 2 ξ T V 1 ξ ) .
σ s 2 = V 11 = 1 / 4 l , 1 / R = V 13 / V 11 , u = V 14 / V 11 , σ g 2 = ƛ 2 V 11 / ( V 11 V 33 V 13 2 V 14 2 ƛ 2 / 4 ) .
V a b V a b = 1 I R 4 d 4 ξ ξ a ξ b W ( S 1 ξ ) .
V a b = 1 I R 4 d 4 ξ S a c ξ c S b d ξ d W ( ξ ) = S a c S b d V c d .
V V = S V S T .
G G = ( S 1 ) T G S 1 .
W ( x 1 , x 2 ) = I ( 2 π ) 1 / 2 1 σ s exp ( x 1 2 + x 2 2 4 σ s 2 ) exp ( ( x 1 x 2 ) 2 2 σ g 2 )
W ( ξ ) = ( I / 2 π ) ( det V ) 1 / 2 exp [ ( 1 / 2 ) ξ T V 1 ξ ] ,
V = [ σ s 2 0 0 ƛ 2 ( 1 / σ g 2 + 1 / 4 σ s 2 ) ] .
tr W = d x W ( x , x ) , tr W 2 = d x 1 d x 2 W ( x 1 , x 2 ) W ( x 2 , x 1 ) = d x 1 d x 2 | W ( x 1 , x 2 ) | 2
tr W 2 ( tr W ) 2 .
V 11 V 22 ƛ 2 4
Δ x Δ p x ƛ 2 .
S ( F = ( 4 l R ) 1 1 ) = [ 1 0 ( 4 l R ) 1 1 1 ] .
V = [ Ω 0 κ 0 0 u / 4 l 0 Ω 0 κ u / 4 l 0 0 u / 4 l Ω 0 1 κ 0 u / 4 l 0 0 Ω 0 1 κ ] ,
κ = ( 1 / 4 l ) ( 4 ƛ 2 l ( l + m ) + u 2 ) 1 / 2 , Ω 0 = ( 4 ƛ 2 l ( l + m ) + u 2 ) 1 / 2 .
R ( ϕ ) = [ cos ϕ 0 0 sin ϕ 0 cos ϕ sin ϕ 0 0 sin ϕ cos ϕ 0 sin ϕ 0 0 cos ϕ ] ,
S = S ( M = Ω 0 1 / 2 1 ) R ( ϕ = π / 4 ) S ( M = Ω 0 1 / 2 1 ) 1 = 1 2 [ 1 0 0 Ω 0 0 1 Ω 0 0 0 Ω 0 1 1 0 Ω 0 1 0 0 1 ] Sp ( 4 , R ) .
V = S V S T = [ Ω 0 ( κ u / 4 l ) 0 0 0 0 Ω 0 ( κ + u / 4 l ) 0 0 0 0 Ω 0 1 ( κ u / 4 l ) 0 0 0 0 Ω 0 1 ( κ + u / 4 l ) ] .
W ( ξ ) = I 4 π 2 ( det V ) 1 / 2 × exp [ Ω 0 1 x 2 + Ω 0 p x 2 2 ( κ u / 4 l ) Ω 0 1 y 2 + Ω 0 p y 2 2 ( κ + u / 4 l ) ] ,
| u | 4 l + ƛ 2 κ .
σ g 2 | u | ƛ .
V = υ 1 + υ 0 ,
υ = [ 1 / 4 l 1 / 4 l R 1 / 4 l R ( 4 ƛ 2 l ( l + m ) + u 2 + 1 / R 2 ) / 4 l ] .
V = S V S T = υ 1 + υ 0 ,
υ = s υ s T = ( 1 / 4 l 1 / 4 l R 1 / 4 l R ( 4 ƛ 2 l ( l + m ) + u 2 + 1 / R 2 ) / 4 l ) , υ 0 u / 4 l = υ 0 = u / 4 l .
det υ = ƛ 2 4 ( 1 + m l ) + ( u / 4 l ) 2 = κ 2 ,
C 1 = m / 4 l = σ s 2 / σ g 2 , C 2 = u / 4 l = σ s 2 u ,
υ ( z ) = [ 1 z 0 1 ] 1 4 l [ 1 1 / R 1 / R 1 / Ω 2 ] [ 1 0 z 1 ] = 1 4 l [ 1 2 z / R + z 2 / Ω 2 1 / R + z / Ω 2 1 / R + z / Ω 2 1 / Ω 2 ] , Ω = ( 4 ƛ 2 l ( l + m ) + u 2 + 1 / R 2 ) 1 / 2 .
σ s ( z ) 2 = σ s 2 ( 1 2 z / R + z 2 / Ω 2 ) .
z = z 0 = Ω 2 / R .
σ s ( z 0 ) = σ s ( 1 Ω 2 / R 2 1 / 2 .
υ ( 0 ) = [ σ s ( 0 ) 2 0 0 σ s ( 0 ) 2 / Ω ( 0 ) 2 ] , Ω ( 0 ) 2 = ( ƛ 2 / 4 ) σ s ( 0 ) 4 ( 1 + 4 σ s ( 0 ) 2 / σ g ( 0 ) 2 ) + u ( 0 ) 2 .
υ ( z ) = s ( z ) υ ( 0 ) s ( z ) T = [ σ s ( 0 ) 2 ( 1 + z 2 / Ω ( 0 ) 2 ) z σ s ( 0 ) 2 / Ω ( 0 ) 2 z σ s ( 0 ) 2 / Ω ( 0 ) 2 σ s ( 0 ) 2 / Ω ( 0 ) 2 ] .
σ s ( z ) = υ 11 ( z ) = σ s ( 0 ) ( 1 + z 2 / Ω ( 0 ) 2 ) 1 / 2 , R ( z ) = υ 11 ( z ) / υ 12 ( z ) = ( z + Ω ( 0 ) 2 / z ) .
σ g ( z ) = σ g ( 0 ) [ 1 + z 2 / Ω ( 0 ) 2 ] 1 / 2 , u ( z ) = u ( 0 ) / [ 1 + z 2 / Ω ( 0 ) 2 ] .
( d / d z ) R ( z ) = 0 , z = z R = Ω ( 0 ) , R ( z R ) = 2 Ω ( 0 ) .
σ s ( z R ) = 2 σ s ( 0 ) , σ g ( z R ) = 2 σ g ( 0 ) , u ( z R ) = u ( 0 ) / 2 .
C 3 = υ 22 = { ( ƛ 2 / 4 ) σ s ( z ) 4 [ 1 + 4 σ s ( z ) 2 / σ g ( z ) 2 ] + u ( z ) 2 + 1 / R ( z ) 2 } σ s ( z ) 2 = invariant .
C 3 = σ s ( 0 ) 2 / Ω ( 0 ) 2 .
( Δ θ ) 2 = ( angular divergence ) 2 = Lt z σ s ( z ) 2 / z 2 = σ s ( 0 ) 2 / Ω ( 0 ) 2 .
η = σ g ( z ) 2 u ( z ) / ƛ = σ g ( 0 ) 2 u ( 0 ) / ƛ .
β = σ s ( z ) / σ g ( z ) = σ s ( 0 ) / σ g ( 0 ) .
Ω ( 0 ) = ( 2 / ƛ ) σ s ( 0 ) 2 ( 1 + 4 β 2 + 4 η 2 β 4 ) 1 / 2 .
Ω ( 0 ) = ( 2 / ƛ ) σ s ( 0 ) 2 ( 1 + 4 β 2 ) 1 / 2 .
β eff = β ( 1 + β 2 η 2 ) 1 / 2 .
[ σ g ( 0 ) ] eff = σ g ( 0 ) / ( 1 + β 2 η 2 ) 1 / 2 .
( Δ θ ) twisted GSM ( Δ θ ) usual GSM = [ 1 + 4 η 2 β 4 / ( 1 + 4 β 2 ) ] 1 / 2 .
υ + i κ = [ υ 11 υ 12 + i κ υ 12 i κ υ 22 ] ,
υ + i κ = υ 22 [ q 1 ] [ q * 1 ] = υ 22 [ q q * q q * 1 ]
q = ( υ 12 + i κ ) / υ 22 = υ 11 / ( υ 12 i κ ) .
υ 22 = κ / Im ( q ) , υ 12 = υ 22 Re ( q ) , υ 11 = ( κ 2 + υ 12 2 ) / υ 22 ,
υ + i κ = s ( υ + i κ ) s T .
[ q 1 ] = μ s [ q 1 ] .
q = ( a q + b ) / ( c q + d ) ,
γ = [ 1 0 0 0 1 0 0 0 1 ] .
υ = [ x 0 + x 2 x 1 x 1 x 0 x 2 ] .
det υ = ( x 0 ) 2 ( x 1 ) 2 ( x 2 ) 2 = x T γ x = κ 2 > 0 , x 0 > 0 ,
x x = Λ ( s ) x , Λ ( s ) SO ( 2 , 1 ) ; Λ ( s 1 ) Λ ( s 2 ) = Λ ( s 1 s 2 ) .
Λ ( s ) = [ ½ ( a 2 + b 2 + c 2 + d 2 ) a b + c d ½ ( a 2 b 2 + c 2 d 2 ) a c + b d a d + b c a c b d ½ ( a 2 + b 2 c 2 d 2 ) a b c d ½ ( a 2 b 2 c 2 + d 2 ) ] .
S = [ m 0 0 0 0 m 1 m 1 0 0 0 0 0 0 0 0 m ] .
V ( 0 ) = [ Ω ( 0 ) κ m 2 0 0 m 2 υ 0 0 Ω ( 0 ) κ m 2 m 2 υ 0 0 0 m 2 υ 0 Ω ( 0 ) 1 κ m 2 0 m 2 υ 0 0 0 Ω ( 0 ) 1 κ m 2 ] [ A ( 0 ) C ( 0 ) C ( 0 ) T B ( 0 ) ] .
A ( 0 ) = [ Ω ( 0 ) κ m 2 0 0 Ω ( 0 ) κ m 2 ]
A ( 0 ) = Ω ( 0 ) κ ( 1 cosh μ + σ 3 sinh μ ) , B ( 0 ) = Ω ( 0 ) 1 κ ( 1 cosh μ σ 3 sinh μ ) , C ( 0 ) = υ 0 ( i σ 2 cosh μ + σ 1 sinh μ ) .
V ( z ) = [ 1 z 1 0 1 ] V ( 0 ) [ 1 0 z 1 1 ] [ A ( z ) C ( z ) C ( z ) T B ( z ) ] .
A ( z ) = Ω ( 0 ) κ ( 1 cosh μ + σ 3 sinh μ ) + 2 z υ 0 σ 1 sinh μ + z 2 Ω ( 0 ) 1 κ ( 1 cosh μ σ 3 sinh μ ) .
tan 2 θ ( z ) = 2 z υ 0 / Ω ( 0 ) κ [ 1 z 2 Ω ( 0 ) 2 ] .
[ w ± ( z ) ] 2 = Ω ( 0 ) κ [ 1 + z 2 Ω ( 0 ) 2 ] cosh μ ± { Ω ( 0 ) 2 κ 2 [ 1 z 2 Ω ( 0 ) 2 ] 2 + 4 z 2 υ 0 2 } 1 / 2 sinh μ .
θ ( z ) = π / 4 at z = Ω ( 0 ) .
Ω ( 0 ) = ( 2 / ƛ ) σ s ( 0 ) 2 ( 1 + 4 β 2 + 4 η 2 β 4 ) 1 / 2
υ 0 = η β 2 ƛ , κ = ( ƛ / 2 ) ( 1 + 4 β 2 + 4 η 2 β 4 ) 1 / 2 ,
tan 2 θ ( z ) = 4 η β 2 ( 1 + 4 β 2 + 4 η 2 β 4 ) 1 / 2 z Ω ( 0 ) 1 [ 1 z 2 / Ω ( 0 ) 2 ] .
S ( z ) = [ cos Δ z 1 Δ 1 sin Δ z 1 Δ sin Δ z 1 cos Δ z 1 ] .
V ( z ) = S ( z ) V ( 0 ) S ( z ) T [ A ( z ) C ( z ) C ( z ) T B ( z ) ] .
A ( z ) = A ( 0 ) cos 2 Δ z + B ( 0 ) Δ 2 sin 2 Δ z + ½ [ C ( 0 ) + C ( 0 ) T ] Δ 1 sin 2 Δ z = Ω ( 0 ) κ cosh μ [ cos 2 Δ z + Ω ( 0 ) 2 Δ 2 sin 2 Δ z ] 1 + υ 0 Δ 1 sinh μ sin 2 Δ z σ 1 + Ω ( 0 ) κ sinh μ [ cos 2 Δ z Ω ( 0 ) 2 Δ 2 sin 2 Δ z ] σ 3 .
tan 2 ψ ( z ) = υ 0 Δ 1 sin 2 Δ z Ω ( 0 ) κ ( cos 2 Δ z Ω ( 0 ) 2 Δ 2 sin 2 Δ z )
[ w ± ( z ) ] 2 = a ( z ) ± [ b ( z ) 2 + d ( z ) 2 ] 1 / 2 .
W ( ρ 1 , ρ 2 ) = I 2 π 1 σ s x σ s y exp ( x 1 2 + x 2 2 4 σ s x 2 y 1 2 + y 2 2 4 σ s y 2 ) × exp [ ( x 1 x 2 ) 2 2 σ g x 2 ( y 1 y 2 ) 2 2 σ g y 2 ] .
W ( ξ ) = I ( 2 π ) 2 ( det V ) 1 / 2 exp ( 1 2 ξ T V 1 ξ ) ,
V = [ σ s x 2 0 0 0 0 σ s y 2 0 0 0 0 ƛ 2 ( 1 / σ g x 2 + 1 / 4 σ s x 2 ) 0 0 0 0 ƛ 2 ( 1 / σ g y 2 + 1 / 4 σ s y 2 ) ] .
ƛ 2 ( 1 σ g x 2 + 1 4 σ s x 2 ) / σ s x 2 = ƛ 2 ( 1 σ g y 2 + 1 4 σ s y 2 ) / σ s y 2 .
Ω 0 2 = σ s x 2 / ƛ 2 ( 1 σ g x 2 + 1 4 σ s x 2 ) , κ = ƛ 2 [ ( 1 4 + σ s y 2 σ g y 2 ) 1 / 2 + ( 1 4 + σ s x 2 σ g x 2 ) 1 / 2 ] , υ 0 = ƛ 2 [ ( 1 4 + σ s y 2 σ g y 2 ) 1 / 2 ( 1 4 + σ s x 2 σ g x 2 ) 1 / 2 ] .
V = [ Ω 0 ( κ υ 0 ) 0 0 0 0 Ω 0 ( κ + υ 0 ) 0 0 0 0 Ω 0 1 ( κ υ 0 ) 0 0 0 0 Ω 0 1 ( κ + υ 0 ) ] .
S 0 = 1 2 [ 1 0 0 Ω 0 0 1 Ω 0 0 0 Ω 0 1 1 0 Ω 0 1 0 0 1 ] .
V = S 0 V S 0 T = [ Ω 0 κ 0 0 υ 0 0 Ω 0 κ υ 0 0 0 υ 0 Ω 0 1 κ 0 υ 0 0 0 Ω 0 1 κ ] .
x = [ 0 0 Ω 0 0 0 1 0 0 Ω 0 1 0 0 0 0 0 0 1 ] Sp ( 4 , R ) .
S 0 = x S ( θ = π / 4 ) x 1 ,
( x ) θ = S ( θ ) x S ( θ ) 1
V = [ S ( θ ) S 0 ) V ( S ( θ ) S 0 ] T .
V = S 0 V S 0 T , S 0 , = S ( θ = π / 4 ) S 0 = ( x ) π / 4 x 1 .
μ ( ρ 1 , ρ 2 ; ν ) = W ( ρ 1 , ρ 2 ; ν ) / [ W ( ρ 1 , ρ 1 ; ν ) W ( ρ 2 , ρ 2 ; ν ) ] 1 / 2

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