Abstract

We report the experimental observation of the fractional Fourier transform (FRT) for a partially coherent optical beam with Gaussian statistics [i.e., partially coherent Gaussian Schell-model (GSM) beam]. The intensity distribution (or beam width) and the modulus of the square of the spectral degree of coherence (or coherence width) of a partially coherent GSM beam in the FRT plane are measured, and the experimental results are analyzed and agree well with the theoretical results. The FRT optical system provides a convenient way to control the properties, e.g., the intensity distribution, beam width, spectral degree of coherence, and coherence width, of a partially coherent beam.

© 2007 Optical Society of America

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    [CrossRef]
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    [CrossRef]
  5. 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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    [CrossRef]
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    [CrossRef]
  9. S. Anand, B. K. Yadav, and H. C. Kandpal, "Experimental study of the phenomenon of 1×N spectral switch due to diffraction of partially coherent light," J. Opt. Soc. Am. A 19, 2223-2228 (2002).
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  15. Q. Lin and Y. Cai, "Tensor ABCD law for partially coherenttwisted anisotropic Gaussian Schell-model beams," Opt. Lett. 27, 216-218 (2002).
    [CrossRef]
  16. Y. Cai and Q. Lin, "Spectral shift of partially coherent twisted anisotropic Gaussian Schell-model beams in free space," Opt. Commun. 204, 17-23 (2002).
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  27. S. C. Pei, M. H. Yeh, and T. L. Luo, "Fractional Fourier series expansion for finite signals and dual extension to discrete-time fractional Fourier transform," IEEE Trans. Signal Process. 47, 2883-2888 (1999).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  33. C. Zheng, "Fractional Fourier transform for partially coherent off-axis Gaussian Schell-model beam," J. Opt. Soc. Am. A 23, 2161-2165 (2006).
    [CrossRef]
  34. Y. Cai, Q. Lin, and S. Zhu, "Coincidence fractional Fourier transform with entangled photon pairs and incoherent light," Appl. Phys. Lett. 86, 021112 (2005).
    [CrossRef]
  35. Y. Cai and S. Zhu, "Coincidence fractional Fourier transform with partially coherent light radiation," J. Opt. Soc. Am. A 22, 1798-1804 (2005).
    [CrossRef]
  36. Y. Cai, Q. Lin, and S. Zhu, "Coincidence subwavelength fractional Fourier transform," J. Opt. Soc. Am. A 23, 835-841 (2006).
    [CrossRef]
  37. Y. Cai and F. Wang, "Lensless optical implementation of the coincidence fractional Fourier transform," Opt. Lett. 31, 2278-2280 (2006).
    [CrossRef] [PubMed]
  38. F. Wang, Y. Cai, and S. He, "Experimental observation of coincidence fractional Fourier transform with a partially coherent beam," Opt. Express 14, 6999-7004 (2006).
    [CrossRef] [PubMed]
  39. R. Simon, E. C. G. Sudarshan, and N. Mukunda, "Generalized rays in first-order optics: transformation properties of Gaussian Schell-model fields," Phys. Rev. A 29, 3273-3279 (1984).
    [CrossRef]

2006

2005

Y. Cai, Q. Lin, and S. Zhu, "Coincidence fractional Fourier transform with entangled photon pairs and incoherent light," Appl. Phys. Lett. 86, 021112 (2005).
[CrossRef]

Y. Cai and S. Zhu, "Coincidence fractional Fourier transform with partially coherent light radiation," J. Opt. Soc. Am. A 22, 1798-1804 (2005).
[CrossRef]

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

2004

2003

2002

2001

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2001).

X. Xue, H. Q. Wei, and A. G. Kirk, "Beam analysis by fractional Fourier transform," Opt. Lett. 26, 1746-1748 (2001).
[CrossRef]

2000

1999

S. C. Pei, M. H. Yeh, and T. L. Luo, "Fractional Fourier series expansion for finite signals and dual extension to discrete-time fractional Fourier transform," IEEE Trans. Signal Process. 47, 2883-2888 (1999).
[CrossRef]

J. Pu, H. Zhang, and S. Nemoto, "Spectral shifts and spectral switches of partially coherent light passing through an aperture," Opt. Commun. 162, 57-63 (1999).
[CrossRef]

1998

Y. Zhang, B. Dong, B. Gu, and G. Yang, "Beam shaping in the fractional Fourier transform domain," J. Opt. Soc. Am. A 15, 1114-1120 (1998).
[CrossRef]

A. W. Lohmann, D. Medlovic, and Z. Zalevsky, "Fractional transformations in optics," in Progress in Optics, Vol. XXXVIII, E.Wolf, ed. (Elsevier, 1998).
[CrossRef]

1995

1993

1992

1988

Q. S. He, J. Turunen, and A. T. Friberg, "Propagation and imaging experiments with Gaussian Schell-model beams," Opt. Commun. 67, 245-250 (1988).
[CrossRef]

1987

E. Wolf, "Non-cosmological redshifts of spectral lines," Nature 326, 363-365 (1987).
[CrossRef]

1986

1985

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

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]

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

Anand, S.

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]

Bastiaans, M. J.

Belendez, A.

A. Belendez, L. Carretero, and A. Fimia, "The use of partially coherent light to reduce the efficiency of silver-halide noise gratings," Opt. Commun. 98, 236-240 (1993).
[CrossRef]

Bitran, Y.

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 L. Hu, "Propagation of partially coherent twisted anisotropic Gaussian Schell-model beams through an apertured astigmatic optical system," Opt. Lett. 31, 685-687 (2006).
[CrossRef] [PubMed]

Y. Cai, Q. Lin, and S. Zhu, "Coincidence subwavelength fractional Fourier transform," J. Opt. Soc. Am. A 23, 835-841 (2006).
[CrossRef]

Y. Cai and F. Wang, "Lensless optical implementation of the coincidence fractional Fourier transform," Opt. Lett. 31, 2278-2280 (2006).
[CrossRef] [PubMed]

F. Wang, Y. Cai, and S. He, "Experimental observation of coincidence fractional Fourier transform with a partially coherent beam," Opt. Express 14, 6999-7004 (2006).
[CrossRef] [PubMed]

Y. Cai, Q. Lin, and S. Zhu, "Coincidence fractional Fourier transform with entangled photon pairs and incoherent light," Appl. Phys. Lett. 86, 021112 (2005).
[CrossRef]

Y. Cai and S. Zhu, "Coincidence fractional Fourier transform with partially coherent light radiation," J. Opt. Soc. Am. A 22, 1798-1804 (2005).
[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, 2716-2718 (2004).
[CrossRef] [PubMed]

Y. Cai and Q. Lin, "Transformation and spectrum properties of partially coherent beams in the fractional Fourier transform plane," J. Opt. Soc. Am. A 20, 1528-1536 (2003).
[CrossRef]

Q. Lin and Y. Cai, "Fractional Fourier transform for partially coherent Gaussian-Schell model beams," Opt. Lett. 27, 1672-1674 (2002).
[CrossRef]

Q. Lin and Y. Cai, "Tensor ABCD law for partially coherenttwisted anisotropic Gaussian Schell-model beams," Opt. Lett. 27, 216-218 (2002).
[CrossRef]

Y. Cai and Q. Lin, "Spectral shift of partially coherent twisted anisotropic Gaussian Schell-model beams in free space," Opt. Commun. 204, 17-23 (2002).

Y. Cai and Q. Lin, "Propagation of partially coherent twisted anisotropic Gaussian Schell-model beams in dispersive and absorbing media," J. Opt. Soc. Am. A 19, 2036-2042 (2002).
[CrossRef]

Carretero, L.

A. Belendez, L. Carretero, and A. Fimia, "The use of partially coherent light to reduce the efficiency of silver-halide noise gratings," Opt. Commun. 98, 236-240 (1993).
[CrossRef]

Davidson, F. M.

Dong, B.

Dorsch, R. G.

Fimia, A.

A. Belendez, L. Carretero, and A. Fimia, "The use of partially coherent light to reduce the efficiency of silver-halide noise gratings," Opt. Commun. 98, 236-240 (1993).
[CrossRef]

Friberg, A. T.

E. Tervonen, A. T. Friberg, and J. Turunen, "Gaussian Schell-model beams generated with synthetic acousto-optic holograms," J. Opt. Soc. Am. A 9, 796-803 (1992).
[CrossRef]

Q. S. He, J. Turunen, and A. T. Friberg, "Propagation and imaging experiments with Gaussian Schell-model beams," Opt. Commun. 67, 245-250 (1988).
[CrossRef]

Gu, B.

He, Q. S.

Q. S. He, J. Turunen, and A. T. Friberg, "Propagation and imaging experiments with Gaussian Schell-model beams," Opt. Commun. 67, 245-250 (1988).
[CrossRef]

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]

F. Wang, Y. Cai, and S. He, "Experimental observation of coincidence fractional Fourier transform with a partially coherent beam," Opt. Express 14, 6999-7004 (2006).
[CrossRef] [PubMed]

Hu, L.

Kandpal, H. C.

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]

Kirk, A. G.

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]

Kutay, M. A.

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2001).

Lin, Q.

Liu, S.

Lohmann, A. W.

Luo, T. L.

S. C. Pei, M. H. Yeh, and T. L. Luo, "Fractional Fourier series expansion for finite signals and dual extension to discrete-time fractional Fourier transform," IEEE Trans. Signal Process. 47, 2883-2888 (1999).
[CrossRef]

Mandel, L.

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

Medlovic, D.

A. W. Lohmann, D. Medlovic, and Z. Zalevsky, "Fractional transformations in optics," in Progress in Optics, Vol. XXXVIII, E.Wolf, ed. (Elsevier, 1998).
[CrossRef]

Mendlovic, D.

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]

Mukunda, N.

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]

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

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]

Nemoto, S.

J. Pu, H. Zhang, and S. Nemoto, "Spectral shifts and spectral switches of partially coherent light passing through an aperture," Opt. Commun. 162, 57-63 (1999).
[CrossRef]

Ozaktas, H.

Ozaktas, H. M.

Pei, S. C.

S. C. Pei, M. H. Yeh, and T. L. Luo, "Fractional Fourier series expansion for finite signals and dual extension to discrete-time fractional Fourier transform," IEEE Trans. Signal Process. 47, 2883-2888 (1999).
[CrossRef]

Pu, J.

J. Pu, H. Zhang, and S. Nemoto, "Spectral shifts and spectral switches of partially coherent light passing through an aperture," Opt. Commun. 162, 57-63 (1999).
[CrossRef]

Ran, Q.

Ricklin, J. C.

Simon, R.

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]

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

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]

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

Tervonen, E.

Torre, A.

A. Torre, "The fractional Fourier transform and some of its applications to optics," in Progress in Optics, E.Wolf, ed. (Elsevier, 2002), Vol. 43.
[CrossRef]

Turunen, J.

E. Tervonen, A. T. Friberg, and J. Turunen, "Gaussian Schell-model beams generated with synthetic acousto-optic holograms," J. Opt. Soc. Am. A 9, 796-803 (1992).
[CrossRef]

Q. S. He, J. Turunen, and A. T. Friberg, "Propagation and imaging experiments with Gaussian Schell-model beams," Opt. Commun. 67, 245-250 (1988).
[CrossRef]

Wang, F.

Wei, H. Q.

Wolf, E.

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

E. Wolf, "Non-cosmological redshifts of spectral lines," Nature 326, 363-365 (1987).
[CrossRef]

Xue, X.

Yadav, B. K.

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]

Yang, G.

Yeh, M. H.

S. C. Pei, M. H. Yeh, and T. L. Luo, "Fractional Fourier series expansion for finite signals and dual extension to discrete-time fractional Fourier transform," IEEE Trans. Signal Process. 47, 2883-2888 (1999).
[CrossRef]

Zalevsky, Z.

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2001).

A. W. Lohmann, D. Medlovic, and Z. Zalevsky, "Fractional transformations in optics," in Progress in Optics, Vol. XXXVIII, E.Wolf, ed. (Elsevier, 1998).
[CrossRef]

D. Mendlovic, Z. Zalevsky, R. G. Dorsch, Y. Bitran, A. W. Lohmann, and H. Ozaktas, "New signal representation based on the fractional Fourier transform: definitions," J. Opt. Soc. Am. A 12, 2424-2431 (1995).
[CrossRef]

Zhang, H.

J. Pu, H. Zhang, and S. Nemoto, "Spectral shifts and spectral switches of partially coherent light passing through an aperture," Opt. Commun. 162, 57-63 (1999).
[CrossRef]

Zhang, Y.

Zheng, C.

Zhu, B.

Zhu, S.

Appl. Phys. Lett.

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, Q. Lin, and S. Zhu, "Coincidence fractional Fourier transform with entangled photon pairs and incoherent light," Appl. Phys. Lett. 86, 021112 (2005).
[CrossRef]

IEEE Trans. Signal Process.

S. C. Pei, M. H. Yeh, and T. L. Luo, "Fractional Fourier series expansion for finite signals and dual extension to discrete-time fractional Fourier transform," IEEE Trans. Signal Process. 47, 2883-2888 (1999).
[CrossRef]

J. Opt. Soc. Am. A

Y. Cai and Q. Lin, "Propagation of partially coherent twisted anisotropic Gaussian Schell-model beams in dispersive and absorbing media," J. Opt. Soc. Am. A 19, 2036-2042 (2002).
[CrossRef]

Y. Zhang, B. Dong, B. Gu, and G. Yang, "Beam shaping in the fractional Fourier transform domain," J. Opt. Soc. Am. A 15, 1114-1120 (1998).
[CrossRef]

Y. Cai and Q. Lin, "Transformation and spectrum properties of partially coherent beams in the fractional Fourier transform plane," J. Opt. Soc. Am. A 20, 1528-1536 (2003).
[CrossRef]

C. Zheng, "Fractional Fourier transform for partially coherent off-axis Gaussian Schell-model beam," J. Opt. Soc. Am. A 23, 2161-2165 (2006).
[CrossRef]

Y. Cai and S. Zhu, "Coincidence fractional Fourier transform with partially coherent light radiation," J. Opt. Soc. Am. A 22, 1798-1804 (2005).
[CrossRef]

Y. Cai, Q. Lin, and S. Zhu, "Coincidence subwavelength fractional Fourier transform," J. Opt. Soc. Am. A 23, 835-841 (2006).
[CrossRef]

D. Mendlovic, Z. Zalevsky, R. G. Dorsch, Y. Bitran, A. W. Lohmann, and H. Ozaktas, "New signal representation based on the fractional Fourier transform: definitions," J. Opt. Soc. Am. A 12, 2424-2431 (1995).
[CrossRef]

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

D. Mendlovic and H. M. Ozaktas, "Fractional Fourier transforms and their optical implementation: I," J. Opt. Soc. Am. A 10, 1875-1881 (1993).
[CrossRef]

H. M. Ozaktas and D. Mendlovic, "Fractional Fourier transforms and their optical implementation: II," J. Opt. Soc. Am. A 10, 2522-2531 (1993).
[CrossRef]

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

M. J. Bastiaans, "Application of the Wigner distribution function to partially coherent light," J. Opt. Soc. Am. A 3, 1227-1238 (1986).
[CrossRef]

E. Tervonen, A. T. Friberg, and J. Turunen, "Gaussian Schell-model beams generated with synthetic acousto-optic holograms," J. Opt. Soc. Am. A 9, 796-803 (1992).
[CrossRef]

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]

S. Anand, B. K. Yadav, and H. C. Kandpal, "Experimental study of the phenomenon of 1×N spectral switch due to diffraction of partially coherent light," J. Opt. Soc. Am. A 19, 2223-2228 (2002).
[CrossRef]

Nature

E. Wolf, "Non-cosmological redshifts of spectral lines," Nature 326, 363-365 (1987).
[CrossRef]

Opt. Commun.

J. Pu, H. Zhang, and S. Nemoto, "Spectral shifts and spectral switches of partially coherent light passing through an aperture," Opt. Commun. 162, 57-63 (1999).
[CrossRef]

A. Belendez, L. Carretero, and A. Fimia, "The use of partially coherent light to reduce the efficiency of silver-halide noise gratings," Opt. Commun. 98, 236-240 (1993).
[CrossRef]

Q. S. He, J. Turunen, and A. T. Friberg, "Propagation and imaging experiments with Gaussian Schell-model beams," Opt. Commun. 67, 245-250 (1988).
[CrossRef]

Y. Cai and Q. Lin, "Spectral shift of partially coherent twisted anisotropic Gaussian Schell-model beams in free space," Opt. Commun. 204, 17-23 (2002).

Opt. Express

Opt. Lett.

Phys. Rev. A

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]

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[CrossRef]

Phys. Rev. E

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

Phys. Rev. Lett.

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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[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Optical system with a thin lens for performing the FRT for an optical beam.

Fig. 2
Fig. 2

Experimental setup for observing the FRT for a partially coherent beam.

Fig. 3
Fig. 3

Experimental setup for measuring the coherence width of a partially coherent beam.

Fig. 4
Fig. 4

Experimental results of (a) the intensity distribution and (b) the corresponding normalized intensity distribution at cross line y = 0 in the input plane (dotted curve). The solid curve is a result of the Gaussian fit.

Fig. 5
Fig. 5

Experimental result of the modulus of the square of the spectral degree of coherence g 2 ( x 1 x 2 ) (along x 1 x 2 ) for the partially coherent beam in the input plane.

Fig. 6
Fig. 6

Experimental results of the intensity distributions and the corresponding normalized intensity distributions (dotted curves) at cross line v = 0 in the FRT plane for different fractional orders p (a) p = 0.44 , (b) p = 0.60 , (c) p = 0.68 , (d) p = 0.8 , (e) p = 1.0 . The solid curves are the corresponding results of the Gaussian fit for the experimental results, and the dashed curves are the corresponding theoretical results calculated by Eq. (14).

Fig. 7
Fig. 7

Experimental results of the modulus of the square of the spectral degree of coherence g p 2 ( u 1 u 2 ) (along u 1 u 2 ) for the partially coherent beam in the FRT plane for different fractional orders p (a) p = 0.44 , (b) p = 0.60 , (c) p = 0.68 , (d) p = 0.8 , (e) p = 1.0 . The solid curves are the corresponding results of the Gaussian fit for the experimental results, and the dashed curves are the corresponding theoretical results calculated by Eq. (16).

Fig. 8
Fig. 8

Experimental result (dotted curve) of the beam width σ I p of the partially coherent GSM beam in the Fourier plane ( p = 1 ) versus the initial coherence width in the input plane with σ I 0 2 mm . The solid curve is the corresponding theoretical result calculated by Eq. (15).

Equations (25)

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Γ p ( u 1 , u 2 ) = 1 ( λ f sin ϕ ) 2 Γ 0 ( r 1 , r 2 ) exp [ i π ( r 1 2 + u 1 2 r 2 2 u 2 2 ) λ f tan ϕ ] exp [ 2 π i ( r 1 u 1 r 2 u 2 ) λ f sin ϕ ] d r 1 d r 2 ,
Γ p ( u ̃ ) = 1 ( λ f sin ϕ ) 2 Γ 0 ( r ̃ ) exp [ i π λ R T N R ] d 2 r ̃ ,
N = [ N 11 N 12 N 12 N 11 ] ,
N 11 = 1 f tan ϕ [ I 0 0 I ] , N 12 = 1 f sin ϕ [ I 0 0 I ] ,
Γ 0 ( r 1 , r 2 ) = I ( r 1 ) I ( r 2 ) g ( r 1 r 2 ) ,
I ( r i ) = exp ( r i 2 2 σ I 0 2 ) ,
g ( r 1 r 2 ) = exp [ ( r 1 r 2 ) 2 2 σ g 0 2 ] ,
Γ 0 ( r ̃ ) = exp ( i k 2 r ̃ T M 0 1 r ̃ ) ,
M 0 1 = [ ( i 2 k σ I 0 2 i k σ g 0 2 ) I i k σ g 0 2 I i k σ g 0 2 I ( i 2 k σ I 0 2 i k σ g 0 2 ) I ] .
Γ p ( u ̃ ) = [ det ( A ¯ + B ¯ M 0 1 ) ] 1 2 exp [ i k 2 u ̃ T M p 1 u ̃ ] ,
M p 1 = ( C ¯ + D ¯ M 0 1 ) ( A ¯ + B ¯ M 0 1 ) 1 ,
A ¯ = cos ϕ [ I 0 0 I ] , B ¯ = f sin ϕ [ I 0 0 I ] ,
C ¯ = sin ϕ f [ I 0 0 I ] , D ¯ = cos ϕ [ I 0 0 I ] .
Γ p ( u 1 , u 2 ) = I p ( u 1 ) I p ( u 2 ) g p ( u 1 u 2 ) exp [ i k ψ p ( u 1 , u 2 ) ] = 1 cos 2 ϕ + f 2 k 2 σ I 0 2 ( 1 4 σ I 0 2 + 1 σ g 0 2 ) sin 2 ϕ exp [ u 1 2 + u 2 2 4 σ I 0 2 cos 2 ϕ + ( 1 k 2 σ I 0 2 + 4 k 2 σ g 0 2 ) f 2 sin 2 ϕ ] exp [ ( u 1 u 2 ) 2 2 σ g 0 2 cos 2 ϕ + ( σ g 0 2 2 k 2 σ I 0 2 + 2 k 2 σ I 0 2 ) f 2 sin 2 ϕ ] exp [ ( 4 i k σ I 2 f 2 + i k σ g 0 2 ( f 2 4 σ I 0 4 k 2 ) ) ( u 1 2 u 2 2 ) sin 2 ϕ 16 f σ g 0 2 σ I 0 4 k 2 cos 2 ϕ + ( 4 σ g 0 2 + 16 σ I 0 2 ) f 3 sin 2 ϕ ] ,
I p ( u i ) = 1 cos 2 ϕ + f 2 k 2 σ I 0 2 ( 1 4 σ I 0 2 + 1 σ g 0 2 ) sin 2 ϕ exp [ u i 2 2 σ I 0 2 cos 2 ϕ + 1 2 k 2 ( 1 σ I 0 2 + 4 σ g 0 2 ) f 2 sin 2 ϕ ] = 1 cos 2 ϕ + f 2 k 2 σ I 0 2 ( 1 4 σ I 0 2 + 1 σ g 0 2 ) sin 2 ϕ exp ( u i 2 2 σ I p 2 ) ,
σ I p = σ I 0 cos 2 ϕ + f 2 sin 2 ϕ k 2 σ I 0 2 ( 1 4 σ I 0 2 + 1 σ g 0 2 ) .
g p ( u 1 u 2 ) = exp [ ( u 1 u 2 ) 2 2 σ g 0 2 cos 2 ϕ + ( σ g 0 2 2 k 2 σ I 0 4 + 4 2 k 2 σ I 0 2 ) f 2 sin 2 ϕ ] = exp [ ( u 1 u 2 ) 2 2 σ g p 2 ] ,
σ g p = σ g 0 cos 2 ϕ + f 2 sin 2 ϕ k 2 σ I 0 2 ( 1 4 σ I 0 2 + 1 σ g 0 2 ) .
ψ p ( u 1 , u 2 ) = exp [ ( 4 σ I 2 f 2 + σ g 0 2 ( f 2 4 σ I 0 4 k 2 ) ) ( u 1 2 u 2 2 ) sin 2 ϕ 16 f σ g 0 2 σ I 0 2 k 2 cos 2 ϕ + ( 4 σ g 0 2 + 16 σ I 0 2 ) f 2 sin 2 ϕ ] .
G p ( 2 ) ( u 1 , u 2 ) = E p ( u 1 ) E p ( u 2 ) E p * ( u 2 ) E p * ( u 1 ) = I p ( u 1 ) I p ( u 2 ) + Γ p ( u 1 , u 2 ) 2 ,
I p ( u i ) = h i ( u i , x 1 ) h i * ( u i , x 2 ) Γ 0 ( x 1 , x 2 ) d x 1 d x 2 , i = 1 , 2 ,
Γ p ( u 1 , u 2 ) 2 = Γ 0 ( x 1 , x 2 ) h 1 ( u 1 , x 1 ) h 2 * ( u 2 , x 2 ) d x 1 d x 2 2 ,
g p 2 ( u 1 u 2 ) = exp [ ( u 1 u 2 ) 2 σ g p 2 ] = G p ( 2 ) ( u 1 , u 2 ) I p ( u 1 ) I p ( u 2 ) 1 ,
Δ σ ¯ I p = ( σ I p exp σ I p theor σ I p theor ) .
Δ σ ¯ g p = ( σ g p experimental σ g p theoretical σ g p theoretical ) .

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