Abstract

For Gaussian–Schell beam propagation through complex matrix optical systems, it is shown that, in some particular cases, an A CD transformation law for the Wigner distribution function holds. For these situations, invariant quantities for the Gaussian–Schell beam propagation can be defined analogous to the real matrix case.

© 1995 Optical Society of America

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References

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  1. E. Wolf, E. Collett, “Partially coherent sources which produce the same far-field intensity distribution as a laser,” Opt. Commun. 25, 293–296 (1978).
    [Crossref]
  2. F. Gori, “Collett–Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
    [Crossref]
  3. A. Starikov, E. Wolf, “Coherent-mode representation of Gaussian Schell-model sources and of their radiation fields,” J. Opt. Soc. Am. 72, 923–928 (1982).
    [Crossref]
  4. A. T. Friberg, R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41, 383–387 (1982).
    [Crossref]
  5. A. T. Friberg, R. J. Sudol, “The spatial coherence properties of Gaussian Schell-model beams,” Opt. Acta 30, 1075–1097 (1983).
    [Crossref]
  6. F. Gori, “Mode propagation of the field generated by Collett–Wolf Schell-model sources,” Opt. Commun. 46, 149–154 (1983).
    [Crossref]
  7. 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]
  8. A. T. Friberg, J. Turunen, “Imaging of Gaussian–Schell-model sources,” J. Opt. Soc. Am. A 5, 713–720 (1988).
    [Crossref]
  9. M. Kauderer, “First-order sources in first-order systems: second-order correlations,” Appl. Opt. 30, 1025–1035 (1991); M. Kauderer, Appl. Opt., 30, 3788(E) (1991).
    [Crossref] [PubMed]
  10. M. Kauderer, “Gaussian–Schell model sources in one-dimensional first-order systems with loss or gain,” Appl. Opt. 32, 999–1017 (1993).
    [Crossref] [PubMed]
  11. E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
    [Crossref]
  12. T. A. C. M. Claasen, W. F. G. Meklenbrauker, “The Wigner distribution—a tool for time-frequency signal analysis. Part I: Continuous-time signals,” Philips J. Res. 35, 217–250 (1980); “Part II: Discrete-time signals,” 276–300; “Part III: Relation with other time-frequency signal transformations,” 372–389.
  13. M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. 25, 26–30 (1978).
    [Crossref]
  14. M. J. Bastiaans, “Wigner distribution function and its applications to first-order optics,” J. Opt. Soc. Am. 69, 1710–1716 (1979).
    [Crossref]
  15. M. J. Bastiaans, “Propagation law for the second-order moments of the Wigner distribution function in first-order optical systems,” Optik 82, 173–181 (1989).
  16. M. J. Bastiaans, “Second-order moments of the Wigner distribution function in first-order optical systems,” Optik 88, 163–168 (1991).
  17. D. Onciul, “Invariance properties of general astigmatic beams through first-order optical systems,” J. Opt. Soc. Am. A 10, 295–298 (1993).
    [Crossref]
  18. D. Dragoman, “Higher-order moments of the Wigner distribution function in first-order optical systems,” J. Opt. Soc. Am. A 11, 2643–2646 (1994).
    [Crossref]
  19. M. J. Bastiaans, “The Wigner distribution function of partially coherent light,” Opt. Acta 28, 1215–1224 (1981).
    [Crossref]
  20. M. J. Bastiaans, “Applications of the Wigner distribution function to partially coherent light,” J. Opt. Soc. Am. A 3, 1227–1237 (1986).
    [Crossref]
  21. V. Guillemin, S. Sternberg, Symplectic Techniques in Physics, (Cambridge U. Press, Cambridge, 1984).
  22. M. Nazarathy, J. Shamir, First-order optics: operator representation for systems with loss or gain,” J. Opt. Soc. Am. 72, 1398–1406 (1982).
    [Crossref]
  23. H. T. Yura, S. G. Hanson, “Optical beam wave propagation through complex optical systems,” J. Opt. Soc. Am. A 4, 1931–1948 (1987).
    [Crossref]
  24. D. Dragoman, “Wigner distribution function for a complex matrix optical system,” Optik (to be published).
  25. H. Kogelnik, “Imaging of optical modes-resonators with internal lenses,” Bell Syst. Tech. J. 44, 455–494 (1965).
  26. S. Barnett, Matrices: Methods and Applications (Clarendon, Oxford, 1990).
  27. J. Serna, R. Martinez-Herrero, P. M. Mejias, “Parametric characterization of general partially coherent beams propagation through ABCD optical systems,” J. Opt. Soc. Am. A 8, 1094–1098 (1991).
    [Crossref]
  28. J. A. Arnaud, Beam and Fiber Optics (Academic, New York, 1976), p. 70.
  29. J. Serna, P. M Mejias, R. Martinez-Herrero, “Beam quality in monomode diode lasers,” Opt. Quantum Electron. 24, 881–887 (1992).
    [Crossref]
  30. R. Martinez-Herrero, P. M. Mejias, J. L H. Neira, M. Sanchez, “Propagation invariance of laser beam parameters through optical systems,” in Eighth International Symposium on Gas Flow and Chemical Lasers, C. Domingo, J. M. Orza, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1397, 627–630 (1991).
  31. G. Piquero, P. M. Mejias, R. Martinez-Herrero, “Sharpness changes of Gaussian beams induced by spherically aberrated lenses,” Opt. Commun. 107, 179–183 (1994).
    [Crossref]

1994 (2)

D. Dragoman, “Higher-order moments of the Wigner distribution function in first-order optical systems,” J. Opt. Soc. Am. A 11, 2643–2646 (1994).
[Crossref]

G. Piquero, P. M. Mejias, R. Martinez-Herrero, “Sharpness changes of Gaussian beams induced by spherically aberrated lenses,” Opt. Commun. 107, 179–183 (1994).
[Crossref]

1993 (2)

1992 (1)

J. Serna, P. M Mejias, R. Martinez-Herrero, “Beam quality in monomode diode lasers,” Opt. Quantum Electron. 24, 881–887 (1992).
[Crossref]

1991 (3)

J. Serna, R. Martinez-Herrero, P. M. Mejias, “Parametric characterization of general partially coherent beams propagation through ABCD optical systems,” J. Opt. Soc. Am. A 8, 1094–1098 (1991).
[Crossref]

M. Kauderer, “First-order sources in first-order systems: second-order correlations,” Appl. Opt. 30, 1025–1035 (1991); M. Kauderer, Appl. Opt., 30, 3788(E) (1991).
[Crossref] [PubMed]

M. J. Bastiaans, “Second-order moments of the Wigner distribution function in first-order optical systems,” Optik 88, 163–168 (1991).

1989 (1)

M. J. Bastiaans, “Propagation law for the second-order moments of the Wigner distribution function in first-order optical systems,” Optik 82, 173–181 (1989).

1988 (1)

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

1987 (1)

H. T. Yura, S. G. Hanson, “Optical beam wave propagation through complex optical systems,” J. Opt. Soc. Am. A 4, 1931–1948 (1987).
[Crossref]

1986 (1)

M. J. Bastiaans, “Applications of the Wigner distribution function to partially coherent light,” J. Opt. Soc. Am. A 3, 1227–1237 (1986).
[Crossref]

1984 (1)

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Generalized rays in first-order optics: transformation properties of Gaussian Schell-model fields,” Phys. Rev. A 29, 3273–3279 (1984).
[Crossref]

1983 (2)

A. T. Friberg, R. J. Sudol, “The spatial coherence properties of Gaussian Schell-model beams,” Opt. Acta 30, 1075–1097 (1983).
[Crossref]

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

1982 (3)

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

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

M. Nazarathy, J. Shamir, First-order optics: operator representation for systems with loss or gain,” J. Opt. Soc. Am. 72, 1398–1406 (1982).
[Crossref]

1981 (1)

M. J. Bastiaans, “The Wigner distribution function of partially coherent light,” Opt. Acta 28, 1215–1224 (1981).
[Crossref]

1980 (2)

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

T. A. C. M. Claasen, W. F. G. Meklenbrauker, “The Wigner distribution—a tool for time-frequency signal analysis. Part I: Continuous-time signals,” Philips J. Res. 35, 217–250 (1980); “Part II: Discrete-time signals,” 276–300; “Part III: Relation with other time-frequency signal transformations,” 372–389.

1979 (1)

M. J. Bastiaans, “Wigner distribution function and its applications to first-order optics,” J. Opt. Soc. Am. 69, 1710–1716 (1979).
[Crossref]

1978 (2)

M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. 25, 26–30 (1978).
[Crossref]

E. Wolf, E. Collett, “Partially coherent sources which produce the same far-field 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).

1932 (1)

E. 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), p. 70.

Barnett, S.

S. Barnett, Matrices: Methods and Applications (Clarendon, Oxford, 1990).

Bastiaans, M. J.

M. J. Bastiaans, “Second-order moments of the Wigner distribution function in first-order optical systems,” Optik 88, 163–168 (1991).

M. J. Bastiaans, “Propagation law for the second-order moments of the Wigner distribution function in first-order optical systems,” Optik 82, 173–181 (1989).

M. J. Bastiaans, “Applications of the Wigner distribution function to partially coherent light,” J. Opt. Soc. Am. A 3, 1227–1237 (1986).
[Crossref]

M. J. Bastiaans, “The Wigner distribution function of partially coherent light,” Opt. Acta 28, 1215–1224 (1981).
[Crossref]

M. J. Bastiaans, “Wigner distribution function and its applications to first-order optics,” J. Opt. Soc. Am. 69, 1710–1716 (1979).
[Crossref]

M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. 25, 26–30 (1978).
[Crossref]

Claasen, T. A. C. M.

T. A. C. M. Claasen, W. F. G. Meklenbrauker, “The Wigner distribution—a tool for time-frequency signal analysis. Part I: Continuous-time signals,” Philips J. Res. 35, 217–250 (1980); “Part II: Discrete-time signals,” 276–300; “Part III: Relation with other time-frequency signal transformations,” 372–389.

Collett, E.

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

Dragoman, D.

D. Dragoman, “Higher-order moments of the Wigner distribution function in first-order optical systems,” J. Opt. Soc. Am. A 11, 2643–2646 (1994).
[Crossref]

D. Dragoman, “Wigner distribution function for a complex matrix optical system,” Optik (to be published).

Friberg, A. T.

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, “The spatial coherence properties of Gaussian Schell-model beams,” Opt. Acta 30, 1075–1097 (1983).
[Crossref]

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

Gori, F.

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

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

Guillemin, V.

V. Guillemin, S. Sternberg, Symplectic Techniques in Physics, (Cambridge U. Press, Cambridge, 1984).

Hanson, S. G.

H. T. Yura, S. G. Hanson, “Optical beam wave propagation through complex optical systems,” J. Opt. Soc. Am. A 4, 1931–1948 (1987).
[Crossref]

Kauderer, M.

M. Kauderer, “Gaussian–Schell model sources in one-dimensional first-order systems with loss or gain,” Appl. Opt. 32, 999–1017 (1993).
[Crossref] [PubMed]

M. Kauderer, “First-order sources in first-order systems: second-order correlations,” Appl. Opt. 30, 1025–1035 (1991); M. Kauderer, Appl. Opt., 30, 3788(E) (1991).
[Crossref] [PubMed]

Kogelnik, H.

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

Martinez-Herrero, R.

G. Piquero, P. M. Mejias, R. Martinez-Herrero, “Sharpness changes of Gaussian beams induced by spherically aberrated lenses,” Opt. Commun. 107, 179–183 (1994).
[Crossref]

J. Serna, P. M Mejias, R. Martinez-Herrero, “Beam quality in monomode diode lasers,” Opt. Quantum Electron. 24, 881–887 (1992).
[Crossref]

J. Serna, R. Martinez-Herrero, P. M. Mejias, “Parametric characterization of general partially coherent beams propagation through ABCD optical systems,” J. Opt. Soc. Am. A 8, 1094–1098 (1991).
[Crossref]

R. Martinez-Herrero, P. M. Mejias, J. L H. Neira, M. Sanchez, “Propagation invariance of laser beam parameters through optical systems,” in Eighth International Symposium on Gas Flow and Chemical Lasers, C. Domingo, J. M. Orza, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1397, 627–630 (1991).

Mejias, P. M

J. Serna, P. M Mejias, R. Martinez-Herrero, “Beam quality in monomode diode lasers,” Opt. Quantum Electron. 24, 881–887 (1992).
[Crossref]

Mejias, P. M.

G. Piquero, P. M. Mejias, R. Martinez-Herrero, “Sharpness changes of Gaussian beams induced by spherically aberrated lenses,” Opt. Commun. 107, 179–183 (1994).
[Crossref]

J. Serna, R. Martinez-Herrero, P. M. Mejias, “Parametric characterization of general partially coherent beams propagation through ABCD optical systems,” J. Opt. Soc. Am. A 8, 1094–1098 (1991).
[Crossref]

R. Martinez-Herrero, P. M. Mejias, J. L H. Neira, M. Sanchez, “Propagation invariance of laser beam parameters through optical systems,” in Eighth International Symposium on Gas Flow and Chemical Lasers, C. Domingo, J. M. Orza, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1397, 627–630 (1991).

Meklenbrauker, W. F. G.

T. A. C. M. Claasen, W. F. G. Meklenbrauker, “The Wigner distribution—a tool for time-frequency signal analysis. Part I: Continuous-time signals,” Philips J. Res. 35, 217–250 (1980); “Part II: Discrete-time signals,” 276–300; “Part III: Relation with other time-frequency signal transformations,” 372–389.

Mukunda, N.

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.

M. Nazarathy, J. Shamir, First-order optics: operator representation for systems with loss or gain,” J. Opt. Soc. Am. 72, 1398–1406 (1982).
[Crossref]

Neira, J. L H.

R. Martinez-Herrero, P. M. Mejias, J. L H. Neira, M. Sanchez, “Propagation invariance of laser beam parameters through optical systems,” in Eighth International Symposium on Gas Flow and Chemical Lasers, C. Domingo, J. M. Orza, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1397, 627–630 (1991).

Onciul, D.

Piquero, G.

G. Piquero, P. M. Mejias, R. Martinez-Herrero, “Sharpness changes of Gaussian beams induced by spherically aberrated lenses,” Opt. Commun. 107, 179–183 (1994).
[Crossref]

Sanchez, M.

R. Martinez-Herrero, P. M. Mejias, J. L H. Neira, M. Sanchez, “Propagation invariance of laser beam parameters through optical systems,” in Eighth International Symposium on Gas Flow and Chemical Lasers, C. Domingo, J. M. Orza, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1397, 627–630 (1991).

Serna, J.

Shamir, J.

M. Nazarathy, J. Shamir, First-order optics: operator representation for systems with loss or gain,” J. Opt. Soc. Am. 72, 1398–1406 (1982).
[Crossref]

Simon, R.

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.

Sternberg, S.

V. Guillemin, S. Sternberg, Symplectic Techniques in Physics, (Cambridge U. Press, Cambridge, 1984).

Sudarshan, E. C. G.

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, “The spatial coherence properties of Gaussian Schell-model beams,” Opt. Acta 30, 1075–1097 (1983).
[Crossref]

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

Turunen, J.

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

Wigner, E.

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

Wolf, E.

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

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

Yura, H. T.

H. T. Yura, S. G. Hanson, “Optical beam wave propagation through complex optical systems,” J. Opt. Soc. Am. A 4, 1931–1948 (1987).
[Crossref]

Appl. Opt. (1)

M. Kauderer, “First-order sources in first-order systems: second-order correlations,” Appl. Opt. 30, 1025–1035 (1991); M. Kauderer, Appl. Opt., 30, 3788(E) (1991).
[Crossref] [PubMed]

Appl. Opt. (1)

Bell Syst. Tech. J. (1)

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

J. Opt. Soc. Am. A (2)

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

M. J. Bastiaans, “Applications of the Wigner distribution function to partially coherent light,” J. Opt. Soc. Am. A 3, 1227–1237 (1986).
[Crossref]

J. Opt. Soc. Am. (1)

M. J. Bastiaans, “Wigner distribution function and its applications to first-order optics,” J. Opt. Soc. Am. 69, 1710–1716 (1979).
[Crossref]

J. Opt. Soc. Am. (1)

M. Nazarathy, J. Shamir, First-order optics: operator representation for systems with loss or gain,” J. Opt. Soc. Am. 72, 1398–1406 (1982).
[Crossref]

J. Opt. Soc. Am. A (1)

H. T. Yura, S. G. Hanson, “Optical beam wave propagation through complex optical systems,” J. Opt. Soc. Am. A 4, 1931–1948 (1987).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

Opt. Acta (2)

M. J. Bastiaans, “The Wigner distribution function of partially coherent light,” Opt. Acta 28, 1215–1224 (1981).
[Crossref]

A. T. Friberg, R. J. Sudol, “The spatial coherence properties of Gaussian Schell-model beams,” Opt. Acta 30, 1075–1097 (1983).
[Crossref]

Opt. Commun. (6)

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

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

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

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

M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. 25, 26–30 (1978).
[Crossref]

G. Piquero, P. M. Mejias, R. Martinez-Herrero, “Sharpness changes of Gaussian beams induced by spherically aberrated lenses,” Opt. Commun. 107, 179–183 (1994).
[Crossref]

Opt. Quantum Electron. (1)

J. Serna, P. M Mejias, R. Martinez-Herrero, “Beam quality in monomode diode lasers,” Opt. Quantum Electron. 24, 881–887 (1992).
[Crossref]

Optik (2)

M. J. Bastiaans, “Propagation law for the second-order moments of the Wigner distribution function in first-order optical systems,” Optik 82, 173–181 (1989).

M. J. Bastiaans, “Second-order moments of the Wigner distribution function in first-order optical systems,” Optik 88, 163–168 (1991).

Philips J. Res. (1)

T. A. C. M. Claasen, W. F. G. Meklenbrauker, “The Wigner distribution—a tool for time-frequency signal analysis. Part I: Continuous-time signals,” Philips J. Res. 35, 217–250 (1980); “Part II: Discrete-time signals,” 276–300; “Part III: Relation with other time-frequency signal transformations,” 372–389.

Phys. Rev. (1)

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

Phys. Rev. A (1)

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]

Other (5)

V. Guillemin, S. Sternberg, Symplectic Techniques in Physics, (Cambridge U. Press, Cambridge, 1984).

R. Martinez-Herrero, P. M. Mejias, J. L H. Neira, M. Sanchez, “Propagation invariance of laser beam parameters through optical systems,” in Eighth International Symposium on Gas Flow and Chemical Lasers, C. Domingo, J. M. Orza, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1397, 627–630 (1991).

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

D. Dragoman, “Wigner distribution function for a complex matrix optical system,” Optik (to be published).

S. Barnett, Matrices: Methods and Applications (Clarendon, Oxford, 1990).

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Equations (52)

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W ( x , p ) = Γ ( x + x 2 , x - x 2 ) exp ( - i k p x ) d x ,
W o ( x , p ) = W i ( D x - B p , - C x + A p ) ,
( x p ) o = [ A B C D ] ( x p ) i .
Γ ( x 1 , x 2 ) = exp [ - q 2 ( x 1 + x 2 ) 2 - m 2 ( x 1 - x 2 ) 2 + i k ( x 1 2 - x 2 2 ) / 2 R ] .
q o 2 = 1 Δ [ q i 2 - m i 2 2 - k C × A 4 - k C × B - A × D 4 R i - ( q i 2 + m i 2 ) ( C · B - A · D ) 2 - D × B ( k 4 R i 2 + 4 q i 2 m i 2 k ) ] ,
m o 2 = - 1 Δ [ q i 2 - m i 2 2 + k C × A 4 + k C × B - A × D 4 R i 2 + ( q i 2 + m i 2 ) ( C · B - A · D ) 2 + D × B ( k 4 R i 2 + 4 q i 2 m i 2 k ) ] ,
1 R o = 1 Δ [ C · A + A · D + C · B R i - 2 ( q i 2 + m i 2 ) ( A × D + C × B ) k + D · B ( 1 R i 2 + 16 q i 2 m i 2 k 2 ) ] ,
Δ = A + B ( 1 R i + i 2 q i 2 + m i 2 k ) 2 - 4 B 2 ( q i 2 - m i 2 ) 2 k 2 ,
Γ o ( x 1 , x 2 ) = 1 2 π W o [ ( x 1 + x 2 ) / 2 , p ] exp [ i k p ( x 1 - x 2 ) ] d p .
W o ( x o , p o ) = 1 2 π K ( x o , p o , x i , p i ) W i ( x i , p i ) d x i d p ,
K ( x o , p o , x i , p i ) = 2 π δ ( x o - A x i - B p i ) δ ( p o - C x i - D p i ) ,             A , B , C , D - real .
γ o = ( C + D γ i ) / ( A + B γ i )
W o ( x , p ) = W i ( D x - B p , - C x + A p ) .
γ = 1 / R + i 4 q 2 / k .
γ = 1 / R + i 4 q m / k .
γ = [ 1 R 2 - 4 ( q 2 - m 2 ) 2 k 2 ] 1 / 2 + i 2 q 2 + m 2 k .
[ A ( 1 + i ) / 2 B C D ( 1 - i ) / 2 ] ,
γ = ( ɛ 1 + ɛ 2 ) / 2 + i ( ɛ 1 - ɛ 2 ) / 2 ,
ɛ 1 = ( 1 R + 2 q 2 + m 2 k ) 2 - 8 ( q 2 - m 2 ) 2 k 2 , ɛ 2 = 1 R - 2 q 2 + m 2 k .
W ( x , p ) = π m exp [ - 4 q 2 x 2 - k 2 ( p - x / R ) 2 4 m 2 ] .
A = A r - A i a / b - B i b - B i a 2 / b ,
B = B r + B i a / b + A i / b ,
C = C r - C i a / b - D i b - D i a 2 / b ,
D = D r + D i a / b + C i / b ,
ξ ¯ = ξ W ( x , p ) d x d p / W ( x , p ) d x d p ,
x i ¯ p j ¯ = ( x - x ¯ ) i ( p - p ¯ ) j W ( x , p ) d x d p / W ( x , p ) d x d p ,
M j = ( x p ) ( x p ) ( x p ) ¯ j times ,
M j o = ( S S ) [ j / 2 ] times M j i ( S S ) T ( j - [ j / 2 ] ) times .
S = 1 ( A D - B C ) 1 / 2 [ A B C D ] ,
S T JS = J ,             where J = [ 0 - 1 1 0 ] ,
M 2 o = S M 2 i S T ,
det M 2 = 1 / 4 σ 2 = m 2 / 4 q 2 = const .
I 2 j = ω 2 j W ( x , p ) W ( x , p ) d x d p d x d p / W ( x , p ) W ( x , p ) d x d p d x d p ,
ω ( x , p , x , p ) = ( x p ) J ( x p ) .
det M 2 j = const . ,
det [ M 2 j + 1 T ( J J j times ) M 2 j + 1 ] = const . ,
det [ M 2 j + 1 M 2 ( j + 1 ) 1 M 2 j + 1 T M 2 j - T = const . ,
Tr [ M 2 j + 1 M 2 ( j + 1 ) - 1 M 2 j + 1 T M 2 j - T ] = const . ,
Q = x 2 ¯ p 2 ¯ - x p ¯ 2 = det M 2 = I 2 / 2.
a o = ( A r + a i B r - b i B i ) ( C r + a i D r - b i D i ) + ( C i + a i D i - b i D r ) ( A i - a i B i - b i B r ) ( A r + a i B r - b i B i ) 2 + ( A i + a i B i - b i B r ) 2 ,
b o = ( C i + a i D i + b i D r ) ( A r + a i B r - b i B i ) - ( C r + a i D r - b i D i ) ( A i + a i B i + b i B r ) ( A r + a i B r - b i B i ) 2 + ( A i + a i B i + b i B r ) 2 .
g 1 = 1 R ,
g 2 = 2 k ( q 2 + m 2 ) ,
r = 2 k ( q 2 - m 2 ) .
g 1 o = C · A + g 1 i ( A · D + C · B ) - g 2 i ( A × D + C × B ) + D · B ( g i · g i - r i 2 ) A + B g i 2 = B 2 r i 2 ,
g 2 o = - C × A + g 1 i ( C × B - A × D ) + g 2 i ( C · B - A · D ) + D × B ( g i · g i - r i 2 ) A + B g i 2 - B 2 r i 2 ,
r o = r i A + B g i 2 - B 2 r i 2 ,
a = g 1 ,             b = ± ( g 2 2 - r 2 ) 1 / 2 .
a = ± ( g 1 2 - r 2 ) 1 / 2 ,             b = g 2 .
a = 1 1 + α 2 { α ( α g 1 - g 2 ) ± [ ( α g 2 + g 1 ) 2 - r 2 ( 1 + α 2 ) ] 1 / 2 } , b = 1 1 + α 2 { g 2 - α g 1 ± α [ ( α g 2 + g 1 ) 2 - r 2 ( 1 + α 2 ) ] 1 / 2 } .
a = 1 1 + α 2 { g 1 + α g 2 α [ ( α g 1 - g 2 ) 2 - r 2 ( 1 + α 2 ) ] 1 / 2 } , b = 1 1 + α 2 { α ( g 1 + α g 2 ) ± [ ( α g 1 - g 2 ) 2 - r 2 ( 1 + α 2 ) ] 1 / 2 } .
a = 1 1 + Δ 2 { Δ ( g 2 + Δ g 1 ) ± [ ( g 1 - Δ g 2 ) 2 - r 2 ( 1 + Δ 2 ) ] 1 / 2 } , b = 1 1 + Δ 2 { g 2 + Δ g 1 Δ [ ( g 1 - Δ g 2 ) 2 - r 2 ( 1 + Δ 2 ) ] 1 / 2 } , Δ 1 + α β α - β .

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