Abstract

The fractional Fourier transform (FRT) is applied to partially coherent twisted anisotropic Gaussian–Schell model (GSM) beams based directly on the cross-spectral density. An analytical and concise formula is derived for the cross-spectral density of partially coherent twisted anisotropic GSM beams passing through a FRT system in terms of the tensor method. The corresponding tensor ABCD law for performing a FRT is obtained. The connection between the FRT formula and the generalized Collins formula for partially coherent beams is discussed. The formulas derived provide a powerful tool for analyzing and calculating the FRTs of partially coherent beams.

© 2002 Optical Society of America

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2002 (1)

2001 (1)

2000 (2)

R. Simon and N. Mukunda, J. Opt. Soc. Am. A 17, 2440 (2000).
[CrossRef]

Z. Zalevsky, Z. D. Medlovic, and H. M. Ozaktas, J. Opt. A 2, 83 (2000).
[CrossRef]

1998 (6)

1997 (1)

1996 (3)

1995 (2)

1994 (1)

1993 (3)

1985 (1)

R. Simon, E. C. G. Sudarshan, and N. Mukunda, Phys. Rev. A 31, 2419 (1985).
[CrossRef] [PubMed]

1984 (1)

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, and C. Yamanaka, Phys. Rev. Lett. 53, 1057 (1984).
[CrossRef]

1980 (1)

V. Namias, J. Inst. Math. Appl. 25, 241 (1980).
[CrossRef]

Arinaga, S.

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, and C. Yamanaka, Phys. Rev. Lett. 53, 1057 (1984).
[CrossRef]

Belendez, A.

A. Belendez, L. Carretero, and A. Fimia, Opt. Commun. 98, 236 (1993).
[CrossRef]

Bitran, Y.

Cai, Y.

Carretero, L.

A. Belendez, L. Carretero, and A. Fimia, Opt. Commun. 98, 236 (1993).
[CrossRef]

Chen, M.

Dong, B.

Dorsch, R. G.

Erden, M. F.

M. F. Erden, H. M. Ozaktas, and D. Mendlovic, J. Opt. Soc. Am. A 13, 1068 (1996).
[CrossRef]

M. F. Erden, H. M. Ozaktas, and D. Mendlovic, Opt. Commun. 125, 288 (1996).
[CrossRef]

Fan, D.

Z. Liu, X. Wu, and D. Fan, Opt. Commun. 155, 7 (1998).
[CrossRef]

Fimia, A.

A. Belendez, L. Carretero, and A. Fimia, Opt. Commun. 98, 236 (1993).
[CrossRef]

Gu, B.

Huang, M.

Huang, W.

Iwai, T.

H. Yoshimura and T. Iwai, Pure Appl. Opt. 7, 1133 (1998).
[CrossRef]

H. Yoshimura and T. Iwai, J. Opt. Soc. Am. A 14, 3388 (1997).
[CrossRef]

Jiang, Z.

Kato, Y.

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, and C. Yamanaka, Phys. Rev. Lett. 53, 1057 (1984).
[CrossRef]

Kitagawa, Y.

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, and C. Yamanaka, Phys. Rev. Lett. 53, 1057 (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, New York, 2000).

Lin, Q.

Liu, S.

Liu, Z.

Z. Liu, X. Wu, and D. Fan, Opt. Commun. 155, 7 (1998).
[CrossRef]

Lohmann, A. W.

Lu, Y.

Medlovic, Z. D.

Z. Zalevsky, Z. D. Medlovic, and H. M. Ozaktas, J. Opt. A 2, 83 (2000).
[CrossRef]

Mendlovic, D.

Mima, K.

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, and C. Yamanaka, Phys. Rev. Lett. 53, 1057 (1984).
[CrossRef]

Miyanaga, N.

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, and C. Yamanaka, Phys. Rev. Lett. 53, 1057 (1984).
[CrossRef]

Mukunda, N.

Nakatsuka, M.

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, and C. Yamanaka, Phys. Rev. Lett. 53, 1057 (1984).
[CrossRef]

Namias, V.

V. Namias, J. Inst. Math. Appl. 25, 241 (1980).
[CrossRef]

Ozaktas, H.

Ozaktas, H. M.

Z. Zalevsky, Z. D. Medlovic, and H. M. Ozaktas, J. Opt. A 2, 83 (2000).
[CrossRef]

M. F. Erden, H. M. Ozaktas, and D. Mendlovic, Opt. Commun. 125, 288 (1996).
[CrossRef]

M. F. Erden, H. M. Ozaktas, and D. Mendlovic, J. Opt. Soc. Am. A 13, 1068 (1996).
[CrossRef]

H. M. Ozaktas and D. Mendlovic, J. Opt. Soc. Am. A 10, 1875 (1993).
[CrossRef]

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, New York, 2000).

Pellat-Finet, P.

Simon, R.

Sudarshan, E. C. G.

R. Simon, E. C. G. Sudarshan, and N. Mukunda, Phys. Rev. A 31, 2419 (1985).
[CrossRef] [PubMed]

Tamura, S.

Tu, J. H.

Wu, X.

Z. Liu, X. Wu, and D. Fan, Opt. Commun. 155, 7 (1998).
[CrossRef]

Yamanaka, C.

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, and C. Yamanaka, Phys. Rev. Lett. 53, 1057 (1984).
[CrossRef]

Yang, G.

Yoshimura, H.

H. Yoshimura and T. Iwai, Pure Appl. Opt. 7, 1133 (1998).
[CrossRef]

H. Yoshimura and T. Iwai, J. Opt. Soc. Am. A 14, 3388 (1997).
[CrossRef]

Yu, L.

Zalevsky, Z.

Z. Zalevsky, Z. D. Medlovic, and H. M. Ozaktas, J. Opt. A 2, 83 (2000).
[CrossRef]

A. W. Lohmann, D. Mendlovic, and Z. Zalevsky, Opt. Lett. 21, 281 (1996).
[CrossRef] [PubMed]

D. Mendlovic, Z. Zalevsky, R. G. Dorsch, Y. Bitran, A. W. Lohmann, and H. Ozaktas, J. Opt. Soc. Am. A 12, 2424 (1995).
[CrossRef]

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, New York, 2000).

Zeng, X.

Zhang, Y.

Zhu, B.

Zhu, Z.

J. Inst. Math. Appl. (1)

V. Namias, J. Inst. Math. Appl. 25, 241 (1980).
[CrossRef]

J. Opt. A (1)

Z. Zalevsky, Z. D. Medlovic, and H. M. Ozaktas, J. Opt. A 2, 83 (2000).
[CrossRef]

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

Opt. Commun. (3)

A. Belendez, L. Carretero, and A. Fimia, Opt. Commun. 98, 236 (1993).
[CrossRef]

Z. Liu, X. Wu, and D. Fan, Opt. Commun. 155, 7 (1998).
[CrossRef]

M. F. Erden, H. M. Ozaktas, and D. Mendlovic, Opt. Commun. 125, 288 (1996).
[CrossRef]

Opt. Lett. (6)

Phys. Rev. A (1)

R. Simon, E. C. G. Sudarshan, and N. Mukunda, Phys. Rev. A 31, 2419 (1985).
[CrossRef] [PubMed]

Phys. Rev. Lett. (1)

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, and C. Yamanaka, Phys. Rev. Lett. 53, 1057 (1984).
[CrossRef]

Pure Appl. Opt. (1)

H. Yoshimura and T. Iwai, Pure Appl. Opt. 7, 1133 (1998).
[CrossRef]

Other (1)

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, New York, 2000).

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Equations (19)

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Epu=1iλf sin ϕ-Erexp-iπr2+u2λf tan ϕ×exp2πir·uλf sin ϕd2r,
Wor˜=Eor1Eo*r2,    Wpu˜=Epu1Ep*u2,
Wpu˜=1λf sin ϕ2-W0r˜×exp-iπr12+u12-r22-u22λf tan ϕ×exp2πir1·u1-r2·u2λf sin ϕdr1dr2.
Wpu˜=1λf sin ϕ2-W0r˜exp-iπλRT NRd2r˜,
N=N11-N12-N12N11,
N11=1f tan ϕI00-I, N12=1f sin ϕI00-I,
W0r˜=G0 exp-iπλr˜TMi-1r˜,
Mi-1=R-1-iλ4πσI2-1-iλ2πσg2-1iλ2πσg2-1+μJiλ2πσg2-1+μJT-R-1-iλ4πσI2-1-iλ2πσg2-1,
J=01-10.
Wpu˜=1λf sin ϕ2-exp-iπλLd2r˜,
L=r˜TN11+Mi-1r˜-2r˜TN12u˜+u˜TN11u˜=N11+Mi-11/2r˜-N11+Mi-11/2N12u˜2-u˜2TN12TN11+Mi-1-1N12u˜+u˜TN11u˜.
Wpu˜=G0f sin ϕ2detMi-1+N11-1/2×exp-iπλu˜TN11-N12TMi-1+N11-1N12u˜.
A¯=N12-1N11=cos ϕI00I, B¯=N12-1=f sin ϕI00-I, C¯=N11N12-1N11-N12T=sin ϕ/f-I00I, D¯=N11N12-1=cosI00I.
Wpu˜=G0detA˜+B˜Mi-1-1/2×exp-iπλu˜TMp-1u˜,
Mp-1=C˜+D˜Mi-1A˜+B˜Mi-1-1.
Mnp-1=C¯n+D¯nMi-1A¯n+B¯nMi-1-1,
A¯nB¯nC¯nD¯n=A¯B¯C¯D¯n.
N11=B¯-1A¯=D¯B¯-1,    N12=B¯-1=C¯-D¯B¯-1A¯.
Wpu˜=1λ2detB¯-1/2-Wor˜×exp-iπλr˜TB¯-1A¯r˜-2r˜TB¯-1u˜+u˜TD¯B¯-1u˜dr˜,

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