Abstract

We investigate second-harmonic generation by an astigmatic partially coherent beam. An explicit expression for the second-order correlation function of the second-harmonic field is obtained. The properties of the generated field and the conversion efficiency for second-harmonic generation are studied numerically. We find that using an astigmatic instead of a stigmatic partially coherent pump beam can increase the conversion efficiency of the second-harmonic generation.

©2007 Optical Society of America

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References

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  1. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).
  2. D. Kermisch, “Partially coherent image processing by laser scanning,” J. Opt. Soc. Am 65, 887–891 (1975).
    [Crossref]
  3. M. Von Waldkirch, P. Lukowicz, and G. Troster, “Effect of light coherence on depth of focus in head-mounted retinal projection displays,” Opt. Engineering 43, 1552–1560 (2004).
    [Crossref]
  4. Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, and C. Yamanaka, “Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression,” Phys. Rev. Lett. 53, 1057–1060 (1984).
    [Crossref]
  5. J. C. Ricklin and F. M. Davidson, “Atmospheric turbulence effects on a partially coherent Gaussian beam: implications for free-space laser communication,” J. Opt. Soc. Am. A 19, 1794–1802 (2002).
    [Crossref]
  6. Y. Cai and S. Zhu, “Ghost imaging with incoherent and partially coherent light radiation,” Phys. Rev. E 71, 056607 (2005).
    [Crossref]
  7. T. E. Gureyev, D. M. Paganin, A. W. Stevenson, S. C. Mayo, and S. W. Wilkin, “Generalized eikonal of partially coherent beams and its use in quantitative imaging,” Phys. Rev. Lett. 93, 068103 (2004).
    [Crossref] [PubMed]
  8. R. Simon, E. C. G. Sudarshan, and N. Mukunda, “Anisotropic Gaussian Schell-model beams: passage through optical systems and associated invariants,” Phys. Rev. A 31, 2419–2434 (1985).
    [Crossref] [PubMed]
  9. R. Simon and N. Mukunda, “Twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 10, 95–109 (1993).
    [Crossref]
  10. A. T. Friberg, E. Tervonen, and J. Turunen, “Interpretation and experimental demonstration of twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 11, 1818–1826 (1994).
    [Crossref]
  11. D. Ambrosini, V. Bagini, F. Gori, and M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41, 1391–1399 (1994).
    [Crossref]
  12. R. Simon, A. T. Friberg, and E. Wolf, “Transfer of radiance by twisted Gaussian Schell-model beams in paraxial systems,” Pure Appl. Opt. 5, 331–343 (1996).
    [Crossref]
  13. R. Simon and N. Mukunda, “Twist phase in Gaussian-beam optics,” J. Opt. Soc. Am. A 15, 2373–2382 (1998).
    [Crossref]
  14. R. Simon and N. Mukunda, “Optical phase space, Wigner representation, and invariant quality parameters,” J. Opt. Soc. Am. A 17, 2440–2463 (2000).
    [Crossref]
  15. S. A. Ponomarenko, “Twisted Gaussian Schell-mode solitons,” Phys. Rev. E 64, 036618 (2001).
    [Crossref]
  16. J. Serna and J. M. Movilla, “Orbital angular momentum of partially coherent beams,” Opt. Lett. 26, 405–406 (2001).
    [Crossref]
  17. Q. Lin and Y. Cai, “Tensor ABCD law for partially coherent twisted anisotropic Gaussian-Schell model beams,” Opt. Lett. 27, 216–218 (2002).
    [Crossref]
  18. Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89, 041117 (2006).
    [Crossref]
  19. H. Wang, X. Wang, A. Zeng, and K. Yang, “Effects of coherence on anisotropic electromagnetic Gaussian-Schell model beams on propagation,” Opt. Lett. 32, 2215–2217 (2007).
    [Crossref] [PubMed]
  20. P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118–119 (1961).
    [Crossref]
  21. R. W. Boyd, Nonlinear Optics (Academic, San Diego, Calif., 1992).
  22. J. E. Bjorkholm, “Optical second-harmonic generation using a focused Gaussian laser beam,” Phys. Rev. 142, 126–136 (1966).
    [Crossref]
  23. D. A. Kleinman, A. Ashkin, and G. D. Boyd, “Second-harmonic generation of light by focused laser beams,” Phys. Rev. 145, 338–379 (1966).
    [Crossref]
  24. T. Freegarde, J. Coutts, J. Walz, D. Leibfried, and T. W. Hänsch, “General analysis of type I second-harmonic generation with elliptical Gaussian beams,” J. Opt. Soc. Am. B 14, 2010–2016 (1997).
    [Crossref]
  25. G. S. Agrawal, “Second-harmonic generation with arbitrary pump-beam profiles,” Phys. Rev. A 23, 1863–1868 (1981).
    [Crossref]
  26. M. S. Zubairy and J. K. McIver, “Second-harmonic generation by a partially coherent beam,” Phys. Rev. A 36, 202–206 (1987).
    [Crossref] [PubMed]
  27. N. A. Ansari and M. S. Zubairy, “Second-harmonic generation by a Gaussian Schell-model source,” Opt. Commun. 59, 385–390 (1986).
    [Crossref]
  28. M. Zahid and M. S. Zubairy, “Coherence properties of second-harmonic beam generated by a partially coherent pump,” Opt. Commun. 76, 1–7 (1990).
    [Crossref]
  29. L. Wang and J. Xue, “Efficiency comparison analysis of second harmonic generation on flattened Gaussian and Gaussian beams through a crystal CsLiB6O10,” Jpn. J. Appl. Phys. 41, 7373–7376 (2002).
    [Crossref]
  30. R. Hanbury Brown, The Intensity Interferomenter (Taylor and Francis, London, 1974).
  31. B. E. A. Saleh, A. F. Abouraddy, A. V. Sirgienko, and M. C. Teich, “Duality between partial coherence and partial entanglement,” Phys. Rev. A 62, 043816 (2000).
    [Crossref]
  32. Y. Cai and S. Zhu, “Ghost interference with partially coherent radiation,” Opt. Lett. 29, 2716 (2004)
    [Crossref] [PubMed]
  33. F. Ferri, D. Magatti, A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “High-resolution ghost image and ghost diffraction experiments with thermal light,” Phys. Rev. Lett. 94, 183602 (2005).
    [Crossref] [PubMed]
  34. J. T. Foley and M.S. Zubairy, “The directionality of Gaussian Schell-model beams,” Opt. Commun. 26 ,297–300 (1978).
    [Crossref]
  35. F. Gori, “Collet-Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1978).
    [Crossref]
  36. A. T. Friberg and R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun 41, 383–387 (1982).
    [Crossref]

2007 (1)

2006 (1)

Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89, 041117 (2006).
[Crossref]

2005 (2)

Y. Cai and S. Zhu, “Ghost imaging with incoherent and partially coherent light radiation,” Phys. Rev. E 71, 056607 (2005).
[Crossref]

F. Ferri, D. Magatti, A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “High-resolution ghost image and ghost diffraction experiments with thermal light,” Phys. Rev. Lett. 94, 183602 (2005).
[Crossref] [PubMed]

2004 (3)

Y. Cai and S. Zhu, “Ghost interference with partially coherent radiation,” Opt. Lett. 29, 2716 (2004)
[Crossref] [PubMed]

T. E. Gureyev, D. M. Paganin, A. W. Stevenson, S. C. Mayo, and S. W. Wilkin, “Generalized eikonal of partially coherent beams and its use in quantitative imaging,” Phys. Rev. Lett. 93, 068103 (2004).
[Crossref] [PubMed]

M. Von Waldkirch, P. Lukowicz, and G. Troster, “Effect of light coherence on depth of focus in head-mounted retinal projection displays,” Opt. Engineering 43, 1552–1560 (2004).
[Crossref]

2002 (3)

2001 (2)

S. A. Ponomarenko, “Twisted Gaussian Schell-mode solitons,” Phys. Rev. E 64, 036618 (2001).
[Crossref]

J. Serna and J. M. Movilla, “Orbital angular momentum of partially coherent beams,” Opt. Lett. 26, 405–406 (2001).
[Crossref]

2000 (2)

R. Simon and N. Mukunda, “Optical phase space, Wigner representation, and invariant quality parameters,” J. Opt. Soc. Am. A 17, 2440–2463 (2000).
[Crossref]

B. E. A. Saleh, A. F. Abouraddy, A. V. Sirgienko, and M. C. Teich, “Duality between partial coherence and partial entanglement,” Phys. Rev. A 62, 043816 (2000).
[Crossref]

1998 (1)

1997 (1)

1996 (1)

R. Simon, A. T. Friberg, and E. Wolf, “Transfer of radiance by twisted Gaussian Schell-model beams in paraxial systems,” Pure Appl. Opt. 5, 331–343 (1996).
[Crossref]

1994 (2)

A. T. Friberg, E. Tervonen, and J. Turunen, “Interpretation and experimental demonstration of twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 11, 1818–1826 (1994).
[Crossref]

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

1993 (1)

1990 (1)

M. Zahid and M. S. Zubairy, “Coherence properties of second-harmonic beam generated by a partially coherent pump,” Opt. Commun. 76, 1–7 (1990).
[Crossref]

1987 (1)

M. S. Zubairy and J. K. McIver, “Second-harmonic generation by a partially coherent beam,” Phys. Rev. A 36, 202–206 (1987).
[Crossref] [PubMed]

1986 (1)

N. A. Ansari and M. S. Zubairy, “Second-harmonic generation by a Gaussian Schell-model source,” Opt. Commun. 59, 385–390 (1986).
[Crossref]

1985 (1)

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

1984 (1)

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, and C. Yamanaka, “Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression,” Phys. Rev. Lett. 53, 1057–1060 (1984).
[Crossref]

1982 (1)

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

1981 (1)

G. S. Agrawal, “Second-harmonic generation with arbitrary pump-beam profiles,” Phys. Rev. A 23, 1863–1868 (1981).
[Crossref]

1978 (2)

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

F. Gori, “Collet-Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1978).
[Crossref]

1975 (1)

D. Kermisch, “Partially coherent image processing by laser scanning,” J. Opt. Soc. Am 65, 887–891 (1975).
[Crossref]

1966 (2)

J. E. Bjorkholm, “Optical second-harmonic generation using a focused Gaussian laser beam,” Phys. Rev. 142, 126–136 (1966).
[Crossref]

D. A. Kleinman, A. Ashkin, and G. D. Boyd, “Second-harmonic generation of light by focused laser beams,” Phys. Rev. 145, 338–379 (1966).
[Crossref]

1961 (1)

P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118–119 (1961).
[Crossref]

Abouraddy, A. F.

B. E. A. Saleh, A. F. Abouraddy, A. V. Sirgienko, and M. C. Teich, “Duality between partial coherence and partial entanglement,” Phys. Rev. A 62, 043816 (2000).
[Crossref]

Agrawal, G. S.

G. S. Agrawal, “Second-harmonic generation with arbitrary pump-beam profiles,” Phys. Rev. A 23, 1863–1868 (1981).
[Crossref]

Ambrosini, D.

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

Ansari, N. A.

N. A. Ansari and M. S. Zubairy, “Second-harmonic generation by a Gaussian Schell-model source,” Opt. Commun. 59, 385–390 (1986).
[Crossref]

Arinaga, S.

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, and C. Yamanaka, “Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression,” Phys. Rev. Lett. 53, 1057–1060 (1984).
[Crossref]

Ashkin, A.

D. A. Kleinman, A. Ashkin, and G. D. Boyd, “Second-harmonic generation of light by focused laser beams,” Phys. Rev. 145, 338–379 (1966).
[Crossref]

Bache, M.

F. Ferri, D. Magatti, A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “High-resolution ghost image and ghost diffraction experiments with thermal light,” Phys. Rev. Lett. 94, 183602 (2005).
[Crossref] [PubMed]

Bagini, V.

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

Bjorkholm, J. E.

J. E. Bjorkholm, “Optical second-harmonic generation using a focused Gaussian laser beam,” Phys. Rev. 142, 126–136 (1966).
[Crossref]

Boyd, G. D.

D. A. Kleinman, A. Ashkin, and G. D. Boyd, “Second-harmonic generation of light by focused laser beams,” Phys. Rev. 145, 338–379 (1966).
[Crossref]

Boyd, R. W.

R. W. Boyd, Nonlinear Optics (Academic, San Diego, Calif., 1992).

Brambilla, E.

F. Ferri, D. Magatti, A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “High-resolution ghost image and ghost diffraction experiments with thermal light,” Phys. Rev. Lett. 94, 183602 (2005).
[Crossref] [PubMed]

Cai, Y.

Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89, 041117 (2006).
[Crossref]

Y. Cai and S. Zhu, “Ghost imaging with incoherent and partially coherent light radiation,” Phys. Rev. E 71, 056607 (2005).
[Crossref]

Y. Cai and S. Zhu, “Ghost interference with partially coherent radiation,” Opt. Lett. 29, 2716 (2004)
[Crossref] [PubMed]

Q. Lin and Y. Cai, “Tensor ABCD law for partially coherent twisted anisotropic Gaussian-Schell model beams,” Opt. Lett. 27, 216–218 (2002).
[Crossref]

Coutts, J.

Davidson, F. M.

Ferri, F.

F. Ferri, D. Magatti, A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “High-resolution ghost image and ghost diffraction experiments with thermal light,” Phys. Rev. Lett. 94, 183602 (2005).
[Crossref] [PubMed]

Foley, J. T.

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

Franken, P. A.

P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118–119 (1961).
[Crossref]

Freegarde, T.

Friberg, A. T.

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

Gatti, A.

F. Ferri, D. Magatti, A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “High-resolution ghost image and ghost diffraction experiments with thermal light,” Phys. Rev. Lett. 94, 183602 (2005).
[Crossref] [PubMed]

Gori, F.

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

F. Gori, “Collet-Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1978).
[Crossref]

Gureyev, T. E.

T. E. Gureyev, D. M. Paganin, A. W. Stevenson, S. C. Mayo, and S. W. Wilkin, “Generalized eikonal of partially coherent beams and its use in quantitative imaging,” Phys. Rev. Lett. 93, 068103 (2004).
[Crossref] [PubMed]

Hanbury Brown, R.

R. Hanbury Brown, The Intensity Interferomenter (Taylor and Francis, London, 1974).

Hänsch, T. W.

He, S.

Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89, 041117 (2006).
[Crossref]

Hill, A. E.

P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118–119 (1961).
[Crossref]

Kato, Y.

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, and C. Yamanaka, “Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression,” Phys. Rev. Lett. 53, 1057–1060 (1984).
[Crossref]

Kermisch, D.

D. Kermisch, “Partially coherent image processing by laser scanning,” J. Opt. Soc. Am 65, 887–891 (1975).
[Crossref]

Kitagawa, Y.

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, and C. Yamanaka, “Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression,” Phys. Rev. Lett. 53, 1057–1060 (1984).
[Crossref]

Kleinman, D. A.

D. A. Kleinman, A. Ashkin, and G. D. Boyd, “Second-harmonic generation of light by focused laser beams,” Phys. Rev. 145, 338–379 (1966).
[Crossref]

Leibfried, D.

Lin, Q.

Lugiato, L. A.

F. Ferri, D. Magatti, A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “High-resolution ghost image and ghost diffraction experiments with thermal light,” Phys. Rev. Lett. 94, 183602 (2005).
[Crossref] [PubMed]

Lukowicz, P.

M. Von Waldkirch, P. Lukowicz, and G. Troster, “Effect of light coherence on depth of focus in head-mounted retinal projection displays,” Opt. Engineering 43, 1552–1560 (2004).
[Crossref]

Magatti, D.

F. Ferri, D. Magatti, A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “High-resolution ghost image and ghost diffraction experiments with thermal light,” Phys. Rev. Lett. 94, 183602 (2005).
[Crossref] [PubMed]

Mandel, L.

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

Mayo, S. C.

T. E. Gureyev, D. M. Paganin, A. W. Stevenson, S. C. Mayo, and S. W. Wilkin, “Generalized eikonal of partially coherent beams and its use in quantitative imaging,” Phys. Rev. Lett. 93, 068103 (2004).
[Crossref] [PubMed]

McIver, J. K.

M. S. Zubairy and J. K. McIver, “Second-harmonic generation by a partially coherent beam,” Phys. Rev. A 36, 202–206 (1987).
[Crossref] [PubMed]

Mima, K.

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, and C. Yamanaka, “Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression,” Phys. Rev. Lett. 53, 1057–1060 (1984).
[Crossref]

Miyanaga, N.

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, and C. Yamanaka, “Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression,” Phys. Rev. Lett. 53, 1057–1060 (1984).
[Crossref]

Movilla, J. M.

Mukunda, N.

Nakatsuka, M.

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, and C. Yamanaka, “Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression,” Phys. Rev. Lett. 53, 1057–1060 (1984).
[Crossref]

Paganin, D. M.

T. E. Gureyev, D. M. Paganin, A. W. Stevenson, S. C. Mayo, and S. W. Wilkin, “Generalized eikonal of partially coherent beams and its use in quantitative imaging,” Phys. Rev. Lett. 93, 068103 (2004).
[Crossref] [PubMed]

Peters, C. W.

P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118–119 (1961).
[Crossref]

Ponomarenko, S. A.

S. A. Ponomarenko, “Twisted Gaussian Schell-mode solitons,” Phys. Rev. E 64, 036618 (2001).
[Crossref]

Ricklin, J. C.

Saleh, B. E. A.

B. E. A. Saleh, A. F. Abouraddy, A. V. Sirgienko, and M. C. Teich, “Duality between partial coherence and partial entanglement,” Phys. Rev. A 62, 043816 (2000).
[Crossref]

Santarsiero, M.

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

Serna, J.

Simon, R.

R. Simon and N. Mukunda, “Optical phase space, Wigner representation, and invariant quality parameters,” J. Opt. Soc. Am. A 17, 2440–2463 (2000).
[Crossref]

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

R. Simon, A. T. Friberg, and E. Wolf, “Transfer of radiance by twisted Gaussian Schell-model beams in paraxial systems,” Pure Appl. Opt. 5, 331–343 (1996).
[Crossref]

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

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

Sirgienko, A. V.

B. E. A. Saleh, A. F. Abouraddy, A. V. Sirgienko, and M. C. Teich, “Duality between partial coherence and partial entanglement,” Phys. Rev. A 62, 043816 (2000).
[Crossref]

Stevenson, A. W.

T. E. Gureyev, D. M. Paganin, A. W. Stevenson, S. C. Mayo, and S. W. Wilkin, “Generalized eikonal of partially coherent beams and its use in quantitative imaging,” Phys. Rev. Lett. 93, 068103 (2004).
[Crossref] [PubMed]

Sudarshan, E. C. G.

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

Sudol, R. J.

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

Teich, M. C.

B. E. A. Saleh, A. F. Abouraddy, A. V. Sirgienko, and M. C. Teich, “Duality between partial coherence and partial entanglement,” Phys. Rev. A 62, 043816 (2000).
[Crossref]

Tervonen, E.

Troster, G.

M. Von Waldkirch, P. Lukowicz, and G. Troster, “Effect of light coherence on depth of focus in head-mounted retinal projection displays,” Opt. Engineering 43, 1552–1560 (2004).
[Crossref]

Turunen, J.

Waldkirch, M. Von

M. Von Waldkirch, P. Lukowicz, and G. Troster, “Effect of light coherence on depth of focus in head-mounted retinal projection displays,” Opt. Engineering 43, 1552–1560 (2004).
[Crossref]

Walz, J.

Wang, H.

Wang, L.

L. Wang and J. Xue, “Efficiency comparison analysis of second harmonic generation on flattened Gaussian and Gaussian beams through a crystal CsLiB6O10,” Jpn. J. Appl. Phys. 41, 7373–7376 (2002).
[Crossref]

Wang, X.

Weinreich, G.

P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118–119 (1961).
[Crossref]

Wilkin, S. W.

T. E. Gureyev, D. M. Paganin, A. W. Stevenson, S. C. Mayo, and S. W. Wilkin, “Generalized eikonal of partially coherent beams and its use in quantitative imaging,” Phys. Rev. Lett. 93, 068103 (2004).
[Crossref] [PubMed]

Wolf, E.

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[Crossref]

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

Xue, J.

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[Crossref]

Yamanaka, C.

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, and C. Yamanaka, “Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression,” Phys. Rev. Lett. 53, 1057–1060 (1984).
[Crossref]

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[Crossref]

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M. Zahid and M. S. Zubairy, “Coherence properties of second-harmonic beam generated by a partially coherent pump,” Opt. Commun. 76, 1–7 (1990).
[Crossref]

M. S. Zubairy and J. K. McIver, “Second-harmonic generation by a partially coherent beam,” Phys. Rev. A 36, 202–206 (1987).
[Crossref] [PubMed]

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[Crossref]

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[Crossref]

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[Crossref]

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[Crossref]

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[Crossref]

J. Opt. Soc. Am. A (5)

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

Jpn. J. Appl. Phys. (1)

L. Wang and J. Xue, “Efficiency comparison analysis of second harmonic generation on flattened Gaussian and Gaussian beams through a crystal CsLiB6O10,” Jpn. J. Appl. Phys. 41, 7373–7376 (2002).
[Crossref]

Opt. Commun (1)

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[Crossref]

Opt. Commun. (4)

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[Crossref]

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[Crossref]

N. A. Ansari and M. S. Zubairy, “Second-harmonic generation by a Gaussian Schell-model source,” Opt. Commun. 59, 385–390 (1986).
[Crossref]

M. Zahid and M. S. Zubairy, “Coherence properties of second-harmonic beam generated by a partially coherent pump,” Opt. Commun. 76, 1–7 (1990).
[Crossref]

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[Crossref]

Opt. Lett. (4)

Phys. Rev. (2)

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[Crossref]

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[Crossref]

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[Crossref]

M. S. Zubairy and J. K. McIver, “Second-harmonic generation by a partially coherent beam,” Phys. Rev. A 36, 202–206 (1987).
[Crossref] [PubMed]

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[Crossref]

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Y. Cai and S. Zhu, “Ghost imaging with incoherent and partially coherent light radiation,” Phys. Rev. E 71, 056607 (2005).
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[Crossref]

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[Crossref] [PubMed]

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, and C. Yamanaka, “Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression,” Phys. Rev. Lett. 53, 1057–1060 (1984).
[Crossref]

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R. Simon, A. T. Friberg, and E. Wolf, “Transfer of radiance by twisted Gaussian Schell-model beams in paraxial systems,” Pure Appl. Opt. 5, 331–343 (1996).
[Crossref]

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R. W. Boyd, Nonlinear Optics (Academic, San Diego, Calif., 1992).

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

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Figures (6)
Fig. 1.
Fig. 1. Dependences of the transverse beam spot width σ Il and transverse coherence width σ gl of the second-harmonic field generated by a partially coherent TGSM beam on the crystal’s length l for different values of the initial coherence width σ g0 and of the twist factor μ 0

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Fig. 2.
Fig. 2. Normalized irradiance distribution (contour graph) of the second-harmonic field generated by a general astigmatic partially coherent beam for different values of crystal’s length l (a) l = 5mm , (b) l = 30mm , (c) l = 50mm , (d) l = 100mm , (e) l = 150mm , (f) pump beam

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Fig. 3.
Fig. 3. Normalized irradiance distribution (contour graph) of the second-harmonic field generated by a general astigmatic partially coherent beam for different values of the crystal’s length / and of the initial transverse coherence width matrix σ 2 g0 (a) l = 30mm , σ 2 g0 =0.01I(mm)2 , (b) l = 30mm , σ 2 g0 = 0.0025I(mm)2 , (c) l = 30mm , σ 2 g0 = 0.0001I(mm)2, (d) l = 30mm , σ 2 g0 = 0.000025I(mm)2 , (e) l = 300mm , σ 2 g0 = 0.000025I(mm)2

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Fig. 4.
Fig. 4. Normalized irradiance distribution (contour graph) of the second-harmonic field generated by a general astigmatic partially coherent beam for different values of the crystal’s length l and of the initial twist factor μ 0 (a) l = 5mm , μ 0 = 0.02mm -1 , (b) l = 30mm , μ 0 = 0.02mm -1 , (c) l = 150mm , μ 0 = 0.02mm -1 , (d) l = 30mm , μ 0 = -0.02mm -l , (e) l = 150mm , μ 0 = -0.02mm -1

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Fig. 5.
Fig. 5. Dependence of the relative conversion efficiency η = ηTGSM / ηGSM on the crystal’s length for different values of the initial twist factor μ 0 of the partially coherent TGSM beam

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Fig. 6.
Fig. 6. Dependence of the relative conversion efficiency η = ηTAGSM / ηGSM on the crystal’s length l for different values of the ratio σ I011 / σ I022 of th e partially coherent TAGSM beam

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Equations (40)

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∂ 2 ε w ∂ x 2 + ∂ 2 ε w ∂ y 2 − 2 i k 1 ∂ ε w ∂ z = − K 1 ε w * ε 2 w ,
∂ 2 ε 2 w ∂ x 2 + ∂ 2 ε 2 w ∂ y 2 − 2 i k 2 ∂ ε 2 w ∂ z = − K 2 ε w 2 ,
ε 2 w ( ρ , l ) = ∫∫ D ( ρ , l , s 1 , s 2 ) ε w ( s 1 , 0 ) ε w ( s 2 , 0 ) d s 1 d s 2 ,
D ( ρ , l , s 1 , s 2 ) = K 2 k 2 32 π 2 l exp ⁡ ( i k 2 l ) ∫ 0 l 1 z 1 exp ⁡ [ i k 2 2 l ρ 2 ] exp ⁡ [ i k 2 ( s 1 2 + s 2 2 ) 8 ( 1 l + 1 z 1 ) ]
exp ⁡ [ − i k 2 ( ρ s 1 + ρ s 2 ) 2 l ] exp ⁡ [ i k 2 s 1 s 2 4 ( 1 l − 1 z 1 ) ] d z 1 .
Γ ( 2 ) ( ρ 1 , ρ 2 , l ) = 〈 ε 2 w ( ρ 1 , l ) ε 2 w * ( ρ 2 , l ) 〉 = ( K 2 k 2 32 π 2 l ) 2
∫ … … ∫ D ( ρ 1 , s 1 , s 2 ) D * ( ρ 2 , s 3 , s 4 ) Γ ( 4 ) ( s 1 , s 2 , s 3 , s 4 ) d s 1 d s 2 d s 3 d s 4 d z 1 d z 2 ,
Γ ( 4 ) ( s 1 , s 2 , s 3 , s 4 ) = Γ ( 2 ) ( s 1 , s 3 , 0 ) Γ ( 2 ) ( s 2 , s 4 , 0 ) + Γ ( 2 ) ( s 1 , s 4 , 0 ) Γ ( 2 ) ( s 2 , s 3 , 0 ) .
Γ ( 2 ) ( ρ 1 , ρ 2 , l ) = ( K 2 k 2 32 π 2 l ) 2 exp ⁡ [ i k 2 2 l ρ 1 2 − i k 2 2 l ρ 2 2 ] ∫ 0 l ∫ 0 l 1 z 1 1 z 2 d z 1 d z 2
∫ [ Γ ( 2 ) ( s 1 , s 3 , 0 ) Γ ( 2 ) ( s 2 , s 4 , 0 ) + Γ ( 2 ) ( s 1 , s 4 , 0 ) Γ ( 2 ) ( s 2 , s 3 , 0 ) ] exp ⁡ [ i k 2 2 s ͂ T B ͂ − 1 s ͂ ] exp ⁡ [ − i k 2 s ͂ T D ͂ ρ ͂ ] d s , ͂
B ͂ − 1 = ( B ͂ l z 1 − 1 0 0 − B ͂ l z 2 − 1 ) , D ͂ = 1 2 l ( I ͂ 0 0 − I ͂ ) ,
B l z i − 1 = ( 1 4 ( 1 l + 1 z i ) I 1 4 ( 1 l − 1 z i ) I 1 4 ( 1 l − 1 z i ) I 1 4 ( 1 l + 1 z i ) I ) , ( i = 1,2 ) ,
Γ ( 2 ) ( s 1 , s 2 , 0 ) = G 0 exp ⁡ [ − 1 4 s 1 T ( σ I 0 2 ) − 1 s 1 − 1 4 s 2 T ( σ I 0 2 ) − 1 s 2 − 1 2 ( s 1 − s 2 ) T ( σ g 0 2 ) − 1 ( s 1 − s 2 ) ]
exp ⁡ [ − ik 2 ( s 1 − s 2 ) T ( R 0 − 1 + μ 0 J ) ( s 1 + s 2 ) ] ,
σ I 0 2 = ( σ I 011 2 σ I 012 2 σ I 012 2 σ I 022 2 ) , σ g 0 2 = ( σ g 011 2 σ g 012 2 σ g 012 2 σ g 022 2 ) , R 0 − 1 = ( R 011 − 1 R 012 − 1 R 021 − 1 R 022 − 1 ) ,
J = ( 0 1 − 1 0 ) .
Γ ( 2 ) ( s 1 , s 3 , 0 ) Γ ( 2 ) ( s 2 , s 4 , 0 ) = G 0 2 exp ⁡ [ − i k 1 2 s ͂ T M ͂ 1 − 1 s ͂ ] ,
Γ ( 2 ) ( s 1 , s 4 , 0 ) Γ ( 2 ) ( s 2 , s 3 , 0 ) = G 0 2 exp ⁡ [ − i k 1 2 s ͂ T M ͂ 2 − 1 s ͂ ] ,
M ͂ 1 − 1 = ( M ͂ 11 − 1 M ͂ 12 − 1 ( M ͂ 12 − 1 ) T ( − M ͂ 11 − 1 ) * ) , M ͂ 2 − 1 = ( M ͂ 11 − 1 M ͂ 21 − 1 ( M ͂ 21 − 1 ) T ( − M ͂ 11 − 1 ) * ) ,
M ͂ 11 − 1 = ( R 0 − 1 − i 2 k 1 ( σ I 0 2 ) − 1 − i k 1 ( σ g 0 2 ) − 1 0 0 R 0 − 1 − i 2 k 1 ( σ I 0 2 ) − 1 − i k 1 ( σ g 0 2 ) − 1 ) ,
M ͂ 12 − 1 = ( i k 1 ( σ g 0 2 ) − 1 + μ 0 J 0 0 i k 1 ( σ g 0 2 ) − 1 + μ 0 J ) , M ͂ 21 − 1 = ( 0 i k 1 ( σ g 0 2 ) − 1 + μ 0 J i k 1 ( σ g 0 2 ) − 1 + μ 0 J 0 ) ,
Γ ( 2 ) ( ρ 1 , ρ 2 , l ) = G 0 2 ( K 2 8 l k 2 ) 2 exp ⁡ [ i k 2 2 l ρ 1 2 − i k 2 2 l ρ 2 2 ] ∫ 0 l ∫ 0 l 1 z 1 1 z 2 d z 1 d z 2
( [ det ( M ͂ l 1 ) ] − 1 / 2 exp ⁡ [ i k 2 2 ρ ͂ T D ͂ T M ͂ l 1 − 1 D ͂ ρ ͂ ] + [ det ( M ͂ l 2 ) ] − 1 / 2 exp ⁡ [ i k 2 2 ρ ͂ T D ͂ T M ͂ l 2 − 1 D ͂ ρ ͂ ] ) ,
Γ ( 2 ) ( ρ 1 , ρ 2 , l ) = G 1 2 ( K 2 8 l k 2 ) 2 16 σ g 0 2 σ I 0 6 k 2 2 l 2 σ Il 2 ∫ 0 l 1 b ln [ 1 + bl a ] dz 1 exp ⁡ [ − ( ρ 1 2 + ρ 2 2 ) 4 σ Il 2 − ( ρ 1 − ρ 2 ) 2 2 σ gl 2 ] ,
a = σ g 0 2 σ I 0 4 k 2 2 + σ I 0 2 k 2 z 1 ( 2 i σ I 0 2 + i σ g 0 2 ) , b = ( 4 σ I 0 2 + σ g 0 2 ) z 1 − σ I 0 2 k 2 ( 2 i σ I 0 2 + i σ g 0 2 ) ,
σ Il 2 = 4 σ I 0 2 l 2 + σ g 0 2 l 2 + σ g 0 2 σ I 0 4 k 2 2 2 σ g 0 2 σ I 0 2 k 2 2 , σ gl 2 = 4 σ I 0 2 l 2 + σ g 0 2 l 2 + σ g 0 2 σ I 0 4 k 2 2 2 σ I 0 4 k 2 2 ,
σ Il σ gl = σ I 0 σ g 0 ,
σ I 0 2 = ( σ I 0 2 0 0 σ I 0 2 ) , σ g 0 2 = ( σ g 0 2 0 0 σ g 0 2 ) , R 0 − 1 = ( R 0 − 1 0 0 R 0 − 1 ) , μ ≠ 0 ,
Γ ( 2 ) ( ρ 1 , ρ 2 , l ) = G 1 2 ( K 2 8 l k 2 ) 2 16 σ g 0 2 σ I 0 6 k 2 2 l 2 R 0 2 σ Il 2 ∫ 0 l 1 b 1 ln ⁡ [ 1 + b 1 l a 1 ] d z 1
× exp ⁡ [ − ( ρ 1 2 + ρ 2 2 ) 4 σ Il 2 − ( ρ 1 − ρ 2 ) 2 2 σ gl 2 − i k 2 2 1 R l ( ρ 1 2 − ρ 2 2 ) − i k 2 μ l ρ 1 ∙ J ρ 2 ] ,
a 1 = σ g 0 2 σ I 0 4 k 2 2 R 0 2 + σ I 0 2 k 2 R 0 z 1 ( 2 i σ I 0 2 R 0 + i σ g 0 2 R 0 − σ g 0 2 σ I 0 2 k 2 ) ,
b 1 = ( 4 σ I 0 2 R 0 2 + σ g 0 2 R 0 2 + σ g 0 2 σ I 0 4 k 2 2 + σ g 0 2 σ I 0 4 k 2 2 R 0 2 μ 0 2 ) z 1 − σ I 0 2 k 2 R 0 ( 2 i σ I 0 2 R 0 + i σ g 0 2 R 0 + σ g 0 2 σ I 0 2 k 2 ) ,
σ Il 2 = 4 σ I 0 2 l 2 R 0 2 + σ g 0 2 l 2 R 0 2 + σ g 0 2 σ I 0 4 k 2 2 ( l − R 0 ) 2 + σ g 0 2 σ I 0 4 k 2 2 μ 0 2 l 2 R 0 2 2 σ g 0 2 σ I 0 2 k 2 2 R 0 2 ,
σ gl 2 = 4 σ I 0 2 l 2 R 0 2 + σ g 0 2 l 2 R 0 2 + σ g 0 2 σ I 0 4 k 2 2 ( l − R 0 ) 2 + σ g 0 2 σ I 0 4 k 2 2 μ 0 2 l 2 R 0 2 2 σ I 0 4 k 2 2 R 0 2 ,
R l = 4 σ I 0 2 l 2 R 0 2 + σ g 0 2 l 2 R 0 2 + σ g 0 2 σ I 0 4 k 2 2 ( l − R 0 ) 2 + σ g 0 2 σ I 0 4 k 2 2 μ 0 2 l 2 R 0 2 σ g 0 2 σ I 0 4 k 2 2 ( R 0 − l ) − 4 σ I 0 2 R 0 2 l − σ g 0 2 R 0 2 l − σ g 0 2 σ I 0 4 k 2 2 μ 0 2 R 0 2 l ,
μ l = σ I 0 4 σ g 0 2 k 2 2 R 0 2 μ 0 4 σ I 0 2 l 2 R 0 2 + σ g 0 2 l 2 R 0 2 + σ g 0 2 σ I 0 4 k 2 2 ( l − R 0 ) 2 + σ g 0 2 σ I 0 4 k 2 2 μ 0 2 l 2 R 0 2 ,
σ Il 2 σ gl 2 = σ I 0 2 σ g 0 2 , σ gl 2 μ l σ g 0 2 μ 0 = 1 2 , σ Il 2 μ l σ I 0 2 μ 0 = 1 2 .
η = ∫ − ∞ ∞ ∫ ∞ ∞ Γ ( 2 ) ( ρ , ρ , l ) d ρ x d ρ y ∫ − ∞ ∞ ∫ ∞ ∞ Γ ( 2 ) ( s , s , 0 ) d s x d s y ,
η GSM = K 2 2 σ g 0 2 σ I 0 2 8 π ∫ 0 l 1 b ln ⁡ [ 1 + bl a ] d z 1 ,
η TGSM = K 2 2 σ g 0 2 σ I 0 2 R 0 2 8 π ∫ 0 l 1 b 1 ln ⁡ [ 1 + b 1 l a 1 ] d z 1 ,

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