Abstract

We present two six-parameter families of anisotropic Gaussian Schell-model beams that propagate in a shape-invariant manner, with the intensity distribution continuously twisting about the beam axis. The two families differ in the sense or helicity of this beam twist. The propagation characteristics of these shape-invariant beams are studied, and the restrictions on the beam parameters that arise from the optical uncertainty principle are brought out. Shape invariance is traced to a fundamental dynamical symmetry that underlies these beams. This symmetry is the product of spatial rotation and fractional Fourier transformation.

© 1999 Optical Society of America

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  1. A. E. Siegman, Lasers (Oxford U. Press, Oxford, UK, 1986), Chap. 19.
  2. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, 1995).
  3. F. Gori, G. Guattari, “A new type of optical fields,” Opt. Commun. 48, 7–12 (1983).
    [CrossRef]
  4. R. Simon, N. Mukunda, “Twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 10, 95–109 (1993).
    [CrossRef]
  5. 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]
  6. D. Ambrosini, V. Bagini, F. Gori, M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41, 1391–1399 (1994).
    [CrossRef]
  7. 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]
  8. 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]
  9. F. Gori, V. Bagini, M. Santarsiero, F. Frezza, G. Schettini, “Coherent and partially coherent twisting beams,” Opt. Rev. 1, 143–145 (1994).
    [CrossRef]
  10. G. Nemes, A. E. Siegman, “Measurement of all ten second-order moments of an astigmatic beam by the use of rotating simple astigmatic (anamorphic) optics,” J. Opt. Soc. Am. A 11, 2257–2264 (1994).
    [CrossRef]
  11. R. Gase, “Representation of Laguerre–Gaussian modes by the Wigner distribution function,” IEEE J. Quantum Electron. 31, 1811–1818 (1995).
    [CrossRef]
  12. 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]
  13. B. Eppich, A. T. Friberg, C. Gao, H. Weber, “Twist of coherent fields and beam quality,” in Third International Workshop on Laser Beam and Optics Characterization, M. Morin, A. Giesen, eds., Proc. SPIE2870, 260–267 (1996).
    [CrossRef]
  14. A. T. Friberg, C. Gao, B. Eppich, H. Weber, “Generation of partially coherent fields with twist,” in 10th Meeting on Optical Engineering in Israel, I. Shladov, S. R. Rotman, eds., Proc. SPIE3110, 317–328 (1997).
    [CrossRef]
  15. F. Gori, M. Santarsiero, V. Borghi, “Partially coherent sources with helicoidal modes,” J. Mod. Opt. 45, 539–554 (1998).
    [CrossRef]
  16. R. Simon, N. Mukunda, “Twist phase in Gaussian beam optics,” J. Opt. Soc. Am. A 15, 2373–2382 (1998).
    [CrossRef]
  17. F. Gori, S. Vicalvi, M. Santarsiero, R. Borghi, “Shape-invariance range of a light beam,” Opt. Lett. 21, 1205–1207 (1996).
    [CrossRef] [PubMed]
  18. R. Simon, N. Mukunda, “Iwasawa decomposition in first order optics: universal treatment of shape-invariant propagation for coherent and partially coherent beams,” J. Opt. Soc. Am. A 15, 2146–2155 (1998).
    [CrossRef]
  19. 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]
  20. G. Nemes, “Measuring and handling general astigmatic beams,” in Laser Beam Characterization, P. M. Mejias, H. Weber, R. Martı́nez-Herrero, A. González-Urena, eds. (Sociedad Española de Optica, Madrid, 1993), p. 325.
  21. G. Nemes, J. Serna, “Laser beam characterization with the use of second order moments: an overview,” in Diode Pumped Solid State Lasers: Applications and Issues, M. W. Dowley, ed., Vol. 17 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1998), p. 200.
  22. R. Simon, N. Mukunda, “Shape-invariant anisotropic Gaussian Schell-model beams: a complete characterization,” J. Opt. Soc. Am. A 15, 1361–1370 (1998).
    [CrossRef]
  23. E. Abramochkin, V. Volostnikov, “Spiral-type beams,” Opt. Commun. 102, 336–350 (1993).
    [CrossRef]
  24. E. Abramochkin, V. Volostnikov, “Spiral-type beams: optical and quantum aspects,” Opt. Commun. 125, 302–323 (1996).
    [CrossRef]
  25. 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]
  26. 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]
  27. 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]

1998 (4)

1996 (3)

E. Abramochkin, V. Volostnikov, “Spiral-type beams: optical and quantum aspects,” Opt. Commun. 125, 302–323 (1996).
[CrossRef]

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 (2)

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. Gase, “Representation of Laguerre–Gaussian modes by the Wigner distribution function,” IEEE J. Quantum Electron. 31, 1811–1818 (1995).
[CrossRef]

1994 (5)

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, V. Bagini, M. Santarsiero, F. Frezza, G. Schettini, “Coherent and partially coherent twisting beams,” Opt. Rev. 1, 143–145 (1994).
[CrossRef]

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]

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]

G. Nemes, A. E. Siegman, “Measurement of all ten second-order moments of an astigmatic beam by the use of rotating simple astigmatic (anamorphic) optics,” J. Opt. Soc. Am. A 11, 2257–2264 (1994).
[CrossRef]

1993 (4)

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).
[CrossRef] [PubMed]

1985 (1)

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Anisotropic Gaussian Schell-model beams: passage through optical systems and associated invariants,” Phys. Rev. A 31, 2419–2434 (1985).
[CrossRef] [PubMed]

1983 (1)

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

Abramochkin, E.

E. Abramochkin, V. Volostnikov, “Spiral-type beams: optical and quantum aspects,” Opt. Commun. 125, 302–323 (1996).
[CrossRef]

E. Abramochkin, V. Volostnikov, “Spiral-type beams,” Opt. Commun. 102, 336–350 (1993).
[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]

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]

F. Gori, V. Bagini, M. Santarsiero, F. Frezza, G. Schettini, “Coherent and partially coherent twisting beams,” Opt. Rev. 1, 143–145 (1994).
[CrossRef]

Borghi, R.

Borghi, V.

F. Gori, M. Santarsiero, V. Borghi, “Partially coherent sources with helicoidal modes,” J. Mod. Opt. 45, 539–554 (1998).
[CrossRef]

Dutta, B.

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]

Eppich, B.

B. Eppich, A. T. Friberg, C. Gao, H. Weber, “Twist of coherent fields and beam quality,” in Third International Workshop on Laser Beam and Optics Characterization, M. Morin, A. Giesen, eds., Proc. SPIE2870, 260–267 (1996).
[CrossRef]

A. T. Friberg, C. Gao, B. Eppich, H. Weber, “Generation of partially coherent fields with twist,” in 10th Meeting on Optical Engineering in Israel, I. Shladov, S. R. Rotman, eds., Proc. SPIE3110, 317–328 (1997).
[CrossRef]

Frezza, F.

F. Gori, V. Bagini, M. Santarsiero, F. Frezza, G. Schettini, “Coherent and partially coherent twisting beams,” Opt. Rev. 1, 143–145 (1994).
[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, C. Gao, B. Eppich, H. Weber, “Generation of partially coherent fields with twist,” in 10th Meeting on Optical Engineering in Israel, I. Shladov, S. R. Rotman, eds., Proc. SPIE3110, 317–328 (1997).
[CrossRef]

B. Eppich, A. T. Friberg, C. Gao, H. Weber, “Twist of coherent fields and beam quality,” in Third International Workshop on Laser Beam and Optics Characterization, M. Morin, A. Giesen, eds., Proc. SPIE2870, 260–267 (1996).
[CrossRef]

Gao, C.

B. Eppich, A. T. Friberg, C. Gao, H. Weber, “Twist of coherent fields and beam quality,” in Third International Workshop on Laser Beam and Optics Characterization, M. Morin, A. Giesen, eds., Proc. SPIE2870, 260–267 (1996).
[CrossRef]

A. T. Friberg, C. Gao, B. Eppich, H. Weber, “Generation of partially coherent fields with twist,” in 10th Meeting on Optical Engineering in Israel, I. Shladov, S. R. Rotman, eds., Proc. SPIE3110, 317–328 (1997).
[CrossRef]

Gase, R.

R. Gase, “Representation of Laguerre–Gaussian modes by the Wigner distribution function,” IEEE J. Quantum Electron. 31, 1811–1818 (1995).
[CrossRef]

Gori, F.

F. Gori, M. Santarsiero, V. Borghi, “Partially coherent sources with helicoidal modes,” J. Mod. Opt. 45, 539–554 (1998).
[CrossRef]

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, V. Bagini, M. Santarsiero, F. Frezza, G. Schettini, “Coherent and partially coherent twisting beams,” Opt. Rev. 1, 143–145 (1994).
[CrossRef]

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

Guattari, G.

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

Mandel, L.

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

Mukunda, N.

R. Simon, N. Mukunda, “Shape-invariant anisotropic Gaussian Schell-model beams: a complete characterization,” J. Opt. Soc. Am. A 15, 1361–1370 (1998).
[CrossRef]

R. Simon, N. Mukunda, “Twist phase in Gaussian beam optics,” J. Opt. Soc. Am. A 15, 2373–2382 (1998).
[CrossRef]

R. Simon, N. Mukunda, “Iwasawa decomposition in first order optics: universal treatment of shape-invariant propagation for coherent and partially coherent beams,” J. Opt. Soc. Am. A 15, 2146–2155 (1998).
[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, B. Dutta, “Quantum-noise matrix for multimode systems: U(n) invariance, squeezing, and normal forms,” Phys. Rev. A 49, 1567–1683 (1994).
[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, E. C. G. Sudarshan, N. Mukunda, “Gaussian Wigner distributions in quantum mechanics and optics,” Phys. Rev. A 36, 3868–3880 (1987).
[CrossRef] [PubMed]

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Anisotropic Gaussian Schell-model beams: passage through optical systems and associated invariants,” Phys. Rev. A 31, 2419–2434 (1985).
[CrossRef] [PubMed]

Nemes, G.

G. Nemes, A. E. Siegman, “Measurement of all ten second-order moments of an astigmatic beam by the use of rotating simple astigmatic (anamorphic) optics,” J. Opt. Soc. Am. A 11, 2257–2264 (1994).
[CrossRef]

G. Nemes, J. Serna, “Laser beam characterization with the use of second order moments: an overview,” in Diode Pumped Solid State Lasers: Applications and Issues, M. W. Dowley, ed., Vol. 17 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1998), p. 200.

G. Nemes, “Measuring and handling general astigmatic beams,” in Laser Beam Characterization, P. M. Mejias, H. Weber, R. Martı́nez-Herrero, A. González-Urena, eds. (Sociedad Española de Optica, Madrid, 1993), p. 325.

Santarsiero, M.

F. Gori, M. Santarsiero, V. Borghi, “Partially coherent sources with helicoidal modes,” J. Mod. Opt. 45, 539–554 (1998).
[CrossRef]

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, V. Bagini, M. Santarsiero, F. Frezza, G. Schettini, “Coherent and partially coherent twisting beams,” Opt. Rev. 1, 143–145 (1994).
[CrossRef]

Schettini, G.

F. Gori, V. Bagini, M. Santarsiero, F. Frezza, G. Schettini, “Coherent and partially coherent twisting beams,” Opt. Rev. 1, 143–145 (1994).
[CrossRef]

Serna, J.

G. Nemes, J. Serna, “Laser beam characterization with the use of second order moments: an overview,” in Diode Pumped Solid State Lasers: Applications and Issues, M. W. Dowley, ed., Vol. 17 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1998), p. 200.

Siegman, A. E.

Simon, R.

R. Simon, N. Mukunda, “Twist phase in Gaussian beam optics,” J. Opt. Soc. Am. A 15, 2373–2382 (1998).
[CrossRef]

R. Simon, N. Mukunda, “Iwasawa decomposition in first order optics: universal treatment of shape-invariant propagation for coherent and partially coherent beams,” J. Opt. Soc. Am. A 15, 2146–2155 (1998).
[CrossRef]

R. Simon, N. Mukunda, “Shape-invariant anisotropic Gaussian Schell-model beams: a complete characterization,” J. Opt. Soc. Am. A 15, 1361–1370 (1998).
[CrossRef]

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, B. Dutta, “Quantum-noise matrix for multimode systems: U(n) invariance, squeezing, and normal forms,” Phys. Rev. A 49, 1567–1683 (1994).
[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, E. C. G. Sudarshan, N. Mukunda, “Gaussian Wigner distributions in quantum mechanics and optics,” Phys. Rev. A 36, 3868–3880 (1987).
[CrossRef] [PubMed]

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Anisotropic Gaussian Schell-model beams: passage through optical systems and associated invariants,” Phys. Rev. A 31, 2419–2434 (1985).
[CrossRef] [PubMed]

Sudarshan, E. C. G.

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Gaussian Wigner distributions in quantum mechanics and optics,” Phys. Rev. A 36, 3868–3880 (1987).
[CrossRef] [PubMed]

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Anisotropic Gaussian Schell-model beams: passage through optical systems and associated invariants,” Phys. Rev. A 31, 2419–2434 (1985).
[CrossRef] [PubMed]

Sundar, K.

Tervonen, E.

Turunen, J.

Vicalvi, S.

Volostnikov, V.

E. Abramochkin, V. Volostnikov, “Spiral-type beams: optical and quantum aspects,” Opt. Commun. 125, 302–323 (1996).
[CrossRef]

E. Abramochkin, V. Volostnikov, “Spiral-type beams,” Opt. Commun. 102, 336–350 (1993).
[CrossRef]

Weber, H.

B. Eppich, A. T. Friberg, C. Gao, H. Weber, “Twist of coherent fields and beam quality,” in Third International Workshop on Laser Beam and Optics Characterization, M. Morin, A. Giesen, eds., Proc. SPIE2870, 260–267 (1996).
[CrossRef]

A. T. Friberg, C. Gao, B. Eppich, H. Weber, “Generation of partially coherent fields with twist,” in 10th Meeting on Optical Engineering in Israel, I. Shladov, S. R. Rotman, eds., Proc. SPIE3110, 317–328 (1997).
[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]

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

IEEE J. Quantum Electron. (1)

R. Gase, “Representation of Laguerre–Gaussian modes by the Wigner distribution function,” IEEE J. Quantum Electron. 31, 1811–1818 (1995).
[CrossRef]

J. Mod. Opt. (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]

F. Gori, M. Santarsiero, V. Borghi, “Partially coherent sources with helicoidal modes,” J. Mod. Opt. 45, 539–554 (1998).
[CrossRef]

J. Opt. Soc. Am. A (9)

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]

G. Nemes, A. E. Siegman, “Measurement of all ten second-order moments of an astigmatic beam by the use of rotating simple astigmatic (anamorphic) optics,” J. Opt. Soc. Am. A 11, 2257–2264 (1994).
[CrossRef]

R. Simon, N. Mukunda, “Shape-invariant anisotropic Gaussian Schell-model beams: a complete characterization,” J. Opt. Soc. Am. A 15, 1361–1370 (1998).
[CrossRef]

R. Simon, N. Mukunda, “Iwasawa decomposition in first order optics: universal treatment of shape-invariant propagation for coherent and partially coherent beams,” J. Opt. Soc. Am. A 15, 2146–2155 (1998).
[CrossRef]

R. Simon, N. Mukunda, “Twist phase in Gaussian beam optics,” J. Opt. Soc. Am. A 15, 2373–2382 (1998).
[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]

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]

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]

Opt. Commun. (3)

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

E. Abramochkin, V. Volostnikov, “Spiral-type beams,” Opt. Commun. 102, 336–350 (1993).
[CrossRef]

E. Abramochkin, V. Volostnikov, “Spiral-type beams: optical and quantum aspects,” Opt. Commun. 125, 302–323 (1996).
[CrossRef]

Opt. Lett. (1)

Opt. Rev. (1)

F. Gori, V. Bagini, M. Santarsiero, F. Frezza, G. Schettini, “Coherent and partially coherent twisting beams,” Opt. Rev. 1, 143–145 (1994).
[CrossRef]

Phys. Rev. A (3)

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, 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]

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 (6)

B. Eppich, A. T. Friberg, C. Gao, H. Weber, “Twist of coherent fields and beam quality,” in Third International Workshop on Laser Beam and Optics Characterization, M. Morin, A. Giesen, eds., Proc. SPIE2870, 260–267 (1996).
[CrossRef]

A. T. Friberg, C. Gao, B. Eppich, H. Weber, “Generation of partially coherent fields with twist,” in 10th Meeting on Optical Engineering in Israel, I. Shladov, S. R. Rotman, eds., Proc. SPIE3110, 317–328 (1997).
[CrossRef]

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

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

G. Nemes, “Measuring and handling general astigmatic beams,” in Laser Beam Characterization, P. M. Mejias, H. Weber, R. Martı́nez-Herrero, A. González-Urena, eds. (Sociedad Española de Optica, Madrid, 1993), p. 325.

G. Nemes, J. Serna, “Laser beam characterization with the use of second order moments: an overview,” in Diode Pumped Solid State Lasers: Applications and Issues, M. W. Dowley, ed., Vol. 17 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1998), p. 200.

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Figures (1)

Fig. 1
Fig. 1

The shaded region shows the physically acceptable values of (ƛΩ/2, ) for any choice of the other two waist plane parameters a0>a0α00. Point A corresponds to (ƛΩ/2, )=(a0-a0α02,a0α02).

Equations (158)

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W(ρ, ρ; z)=I2π [det L(z)]1/2 exp-14 ρTL(z)ρ-14 ρTL(z)ρ-12 (ρ-ρ)TM(z)(ρ-ρ)-i2ƛ (ρ-ρ)TK(z)(ρ+ρ).
τ1=100-1,τ2=0110,τ3=0-ii0.
K=R-112×2+v1τ1+v2τ2+iuτ3
=R-112×2+vτ+uJ,
J=iτ3=01-10,
W(ρ, p; z)=1λ2 d2Δρ exp(ipΔρ/ƛ)×Wρ-12 Δρ, ρ+12 Δρ; z.
ξ=ρp=(x y px py)T.
W(ξ; z)=I4π2 (det V)1/2 exp-12 ξTV-1ξ,
V=ACCTB;
A=L-1,C=-L-1KT,
B=ƛ2[M+L/4]+KL-1KT.
L=A-1,K=-CTA-1,
M=ƛ-2B-14 A-1-ƛ-2CTA-1C.
V+i2 ƛβ0,β=02×212×2-12×202×2.
A=a(12×2+ατ),B=b(12×2+βτ),
a, b>0,0|α|,|β|<1.
A=R(θ)A0R(θ)T,
R(θ)=cos θ-sin θsin θcos θ=exp(-θJ)=exp(-iθτ3),
A0=diag[a(1+α), a(1-α)],α|α|.
a(1+α)=σ12,a(1-α)=σ22,σ1σ2>0.
L=R(θ)L0R(θ)T,
L0=diag(1/σ12, 1/σ22).
tr M=1/δ12+1/δ22,det M=(δ12δ22)-1;
M=R(θ)M0R(θ)T,M0=diag(1/δ12, 1/δ22).
S(z)=12×2z12×202×212×2Sp(4, R).
V(z0+z)=S(z)V(z0)S(z)T,
A(z0+z)=A(z0)+z[C(z0)+C(z0)T]+z2B(z0),
B(z0+z)=B(z0),
C(z0+z)=C(z0)+zB(z0).
dAdz=C+CT,dBdz=0,dCdz=B,
L0=1σ2 12×2,M0=1δ2 12×2,K0=uJ,
A0=σ212×2,C0=σ2uJ,B0=b012×2,
V0=σ200σ2u0σ2-σ2u00-σ2ub00σ2u00b0;
b0=ƛ214σ2+1δ2+σ2u2.
|u|ƛ/δ2.
B0=A0/Ω2,
1Ω2=u2+ƛ2σ2 14σ2+1δ2,
A=(1+z2/Ω2)A0,B=B0,
C=C0+zB0;
L=(1+z2/Ω2)-1L0,M=(1+z2/Ω2)-1M0,
K=(1+z2/Ω2)-1K0-zΩ2 12×2.
A0=diag(σ12, σ22),σ1σ2>0,
B0=A0/Ω2=diag(σ12/Ω2, σ22/Ω2),
C0=Ω J,Ω>0,
V0=σ1200Ω0σ22-Ω00-Ωσ12Ω20Ω00σ22Ω2;
L0=diag(1/σ12, 1/σ22),
M0=diag(1/δ12, 1/δ22),
K0=vτ2+uJ,
1δ12=σ1ƛΩ21-σ1σ22-ƛΩ2σ122,
1δ22=σ2ƛΩ21-σ1σ22-ƛΩ2σ222,
v=(/2σ12σ22Ω)(σ12-σ22),
u=(/2σ12σ22Ω)(σ12+σ22).
||σ1σ21-ƛΩ2σ2221/2,
||σ1σ21-ƛΩ2σ121-ƛΩ2σ221/2.
1σ12δ12+14σ14=1σ22δ22+14σ24,
v=σ12-σ22σ12+σ22 u=αu.
δ1(σ2/σ1)δ2;
|u|ƛ/δ1δ2,
σ1=σ2=σ,=Ωσ2u,
1Ω2=ƛ2σ2 14σ2+1δ2+u2,
TGSMBeams(fourparameters)Strictlyshape-invariantGSMbeams(sixparameters)
AllAGSMbeams(tenparameters).
(V11)2(V33)2-(V13)2(ƛ/2)2,
(V2×2)2(V44)2-(V24)2(ƛ/2)2.
det V(ƛ/2)4.
SV:VV0=(SV)V(SV)T,
V0=diag(κ1Ω, κ2Ω, κ1/Ω, κ2/Ω),
κjƛ/2,j=1, 2
Cs=c12×2+γτ.
tr A2=r2(tr A)2,
tr A02=r2(tr A0)2,
tr (A0Cs0)=r2 tr A0 tr Cs0,
tr (A0B0+2Cs02)=r2[tr A0 tr B0+2(tr Cs0)2],
tr (B0Cs0)=r2 tr B0 tr Cs0,
tr B02=r2(tr B0)2.
|α|2 =|β|2=2r2-1α2,
αγ=α2c,
βγ=α2c,
ab(αβ-α2)=2(α2c2-|γ|2).
a0=12(σ12+σ22),b0=a0/Ω2;
α0=β0=α0(10)T,
α0=(2r2-1)1/2=(σ12-σ22)/(σ12+σ22),
c0=0,γ0=0.
α=α(cos 2θ, sin 2θ),-π/2<θ<π/2,
β=α(-1, 0),α=(2r2-1)1/2,
γ=(γ1, γ2).
γ1=-αc,γ2 sin θ cos θ=αc cos2 θ,
γ22=α2ab cos2 θ.
class(b):θ±π/2,Cs0foranyz.
γ1=-sαab sin θ,γ2=sαab cos θ,
c=sab sin θ,s=±1.
A=a[1+α(τ1 cos 2θ+τ2 sin 2θ)],
B=b(1-ατ1),
Cs=sab[sin θ12×2+α(-τ1 sin θ+τ2 cos θ)]:
dadz=2sab sin θ,ddz (sab sin θ)=b,
ddz (sab cos θ)=0,ddz (a cos 2θ)=-2sab sin θ,
ddz (a sin 2θ)=2sab cos θ.
A0=a0(1+α0τ1),B0=b0(1-α0τ1),
Cs0=sa0b0α0τ2.
a=a0+b0z2,
cos θ=1+b0a0 z2-1/2,
sin θ=sb0/a0z(1+b0z2/a0)-1/2.
a0(1+α0)=σ12,a0(1-α0)=σ22,
b0=a0/Ω2,Ca0=sΩ J.
A0=diag(σ12, σ22),σ1σ2>0,
B0=diag(σ22/Ω2, σ12/Ω2),
C0=Cs0+Ca0=sΩ (a0α0τ2+J).
a=a0(1+z2/Ω2),
exp[iθ(z)]=(1+isz/Ω)/(1+z2/Ω2)1/2.
A=(1+z2/Ω2)R(θ)A0R(θ)T,B=B0,
C=sΩ (a0α0τ2+J)+za0Ω2 (1-α0τ1).
M(z)=diag[(1+z2/Ω2)1/2, (1+z2/Ω2)-1/2]12×2,
R(z)=12×2R[θ(z)]=R[θ(z)]02×202×2R[θ(z)].
V0=A0C0C0TB0V1=A1C1C1TB1=M(z)R(z)V0R(z)TM(z)T;
A1=(1+z2/Ω2)R(θ)A0R(θ)T=A,
B1=(1+z2/Ω2)-1R(θ)B0R(θ)T
=a0Ω2+z2 1-α0Ω2+z2 [(Ω2-z2)τ1+2sΩzτ2],
C1=R(θ)C0R(θ)T=sΩ J+a0α0Ω2+z2 [(Ω2-z2)τ2-2sΩzτ1].
V1V2=A2C2C2TB2=L(g)V1L(g)T,
L(g)=12×202×2-g12×212×2;
A2=A1=A,C2=C1-gA,
B2=B1-g(C1+C1T)+g2A.
g(z)=-z/(Ω2+z2)
V=L[g(z)]M(z)R[θ(z)]V0R[θ(z)]TM(z)TL[g(z)]T.
FΩ(θ)=RΩ(θ)12×2,
RΩ(θ)=cos θ-Ω sin θΩ-1 sin θcos θ.
FΩ(θ)=R(θ)[FΩ(θ)R(-θ)].
L0=A0-1=diag(1/σ12, 1/σ22),
M0=diagσ22ƛ2Ω2-14σ12-a0α0-ƛΩσ22,×σ12ƛ2Ω2-14σ22-a0α0+ƛΩσ12,
a0α0=12(σ12-σ22),K0=vτ2+uJ,
(u, v)=sa0Ωσ12σ22 [-a0α02, α0(-a0)].
1σ22δ12+a0α0-ƛΩσ222=1σ12δ22+a0α0+ƛΩσ122=1(ƛΩ)2 1-ƛΩ2σ1σ22;
1σ22δ12+14σ12σ22=1(ƛΩ)2 1-a0α0-σ222,
1σ12δ22+14σ12σ22=1(ƛΩ)2 1-a0α0+σ122.
1σ22δ12-1σ12δ22+4uv/ƛ2=0,
(σ1/δ1)2+1/4(σ2/δ2)2+1/4=v-(σ12/σ22-1)uv-(1-σ22/σ12)u.
Ω=sα0α0u-v,=a0α0 u-α0vα0u-v,
V0=σ1200sΛ+Ω0σ22sΛ-Ω00sΛ-Ωσ22Ω20sΛ+Ω00σ12Ω2,Λ±a0α0±.
m1σ2/Ωσ1,m2σ1/Ωσ2,
S1=diag(m1, m2, 1/m1, 1/m2)Sp(4, R),
V1=S1V0S1T=σ1σ2Ω 100sΛ+σ1201sΛ-σ2200sΛ-σ2210sΛ+σ12001,
V0+i2 ƛβ0V1+i2 ƛβ0.
S2=12 100101100-110-1001Sp(4, R),
V2=S2V1S2T=σ1σ2Ω diag(1+μ, 1+ν, 1-ν, 1-μ),
μ=sΛ+/σ12,ν=sΛ-/σ22;
V0+i2 ƛβ0V2+i2 ƛβ0.
|Λ+|<σ12,|Λ-|<σ22.
-a0+2a0α0<<a0.
ƛΩ22(σ12+sΛ+)(σ22-sΛ-),
ƛΩ22(+sa0)2-2(s+1)a02α02,s=±1,
|a0-|ƛΩ/2,
|a0+|[(ƛΩ/2)2+4a02α02]1/2.
-a0+ƛΩ22+4a02α021/2a0-ƛΩ2,
ƛΩ2a0-a0α02=σ12σ22/a0.
L=(1+z2/Ω2)-1R(θ)L0R(θ)T,
M=(1+z2/Ω2)-1R(θ)M0R(θ)T,
K=(1+z2/Ω2)-1R(θ)K0R(θ)T-zΩ2+z2 12×2,
K=zΩ2+z2 12×2-2sΩ3zv(Ω2+z2)2 τ1+Ω2(Ω2-z2)v(Ω2+z2)2 τ2+uJ.

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