Abstract

An analytical and concise formula is derived for the fractional Fourier transform (FRT) of partially coherent beams that is based on the tensorial propagation formula of the cross-spectral density of partially coherent twisted anisotropic Gaussian–Schell-model (GSM) beams. The corresponding tensor ABCD law performing the FRT is obtained. The connections between the FRT formula and the generalized diffraction integral formulas for partially coherent beams passing through aligned optical systems and misaligned optical systems are discussed. With use of the derived formula, the transformation and spectrum properties of partially coherent GSM beams in the FRT plane are studied in detail. The results show that the fractional order of the FRT has strong effects on the transformation properties and the spectrum properties of partially coherent GSM beams. Our method provides a simple and convenient way to study the FRT of twisted anisotropic GSM beams.

© 2003 Optical Society of America

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    [CrossRef]

2002 (4)

H. C. Kandpal, S. Anand, J. S. Vaishva, “Experimental observation of the phenomenon of spectral switching for a class of partially coherent light,” IEEE J. Quantum Electron. 37, 336–339 (2002).
[CrossRef]

Y. Cai, Q. Lin, “Spectral shift of partially coherent twisted anisotropic Gaussian Schell-model beams in free space,” Opt. Commun. 204, 17–23 (2002).
[CrossRef]

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

Y. Cai, 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]

2001 (4)

B. Zhu, S. Liu, “Multifractional correlation,” Opt. Lett. 26, 578–580 (2001).
[CrossRef]

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

L. Pan, B. Lu, “The spectral switch of partially coherent light in Young’s experiment,” IEEE J. Quantum Electron. 37, 1377–1381 (2001).
[CrossRef]

S. Shinde, V. M. Gadre, “An uncertainty principle for real signals in the fractional Fourier transform domain,” IEEE Trans. Signal Process. 49, 2545–2548 (2001).
[CrossRef]

2000 (5)

Z. Zalevsky, D. Medlovic, H. M. Ozaktas, “Energetic efficient synthesis of general mutual intensity distribution,” J. Opt. A Pure Appl. Opt. 2, 83–87 (2000).
[CrossRef]

N. Nishi, T. Jitsuno, K. Tsubakimoto, S. Matsuoka, N. Miyanaga, M. Nakatsuka, “Two-dimensional multi-lens array with circular aperture spherical lens for flat-top irradiation of inertial confinement fusion target,” Opt. Rev. 7, 216–220 (2000).
[CrossRef]

C. Candan, M. A. Kutay, H. M. Ozaktas, “The discrete fractional Fourier transform,” IEEE Trans. Signal Process. 48, 1329–1337 (2000).
[CrossRef]

J. Pu, S. Nemoto, “Spectral shifts and spectral switches in diffraction of partially coherent light by a circular aperture,” IEEE J. Quantum Electron. 36, 1407–1411 (2000).
[CrossRef]

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

1999 (2)

S. C. Pei, M. H. Yeh, 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]

R. Simon, N. Mukunda, “Gaussian Schell-model beams and general shape invariance,” J. Opt. Soc. Am. A 16, 2465–2475 (1999).
[CrossRef]

1998 (10)

J. H. Tu, S. Tamura, “Analytic relation for recovering the mutual intensity by means of intensity information,” J. Opt. Soc. Am. A 15, 202–206 (1998).
[CrossRef]

Y. Zhang, B. Dong, B. Gu, G. Yang, “Beam shaping in the fractional Fourier transform domain,” J. Opt. Soc. Am. A 15, 1114–1120 (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, 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]

C. Palma, G. Cardone, G. Cincotti, “Spectral changes in Gaussian-cavity lasers,” IEEE J. Quantum Electron. 34, 1082–1088 (1998).
[CrossRef]

C. Palma, G. Cincotti, G. Guattari, “Spectral shift of a Gaussian Schell-model beam beyond a thin lens,” IEEE J. Quantum Electron. 34, 378–383 (1998).
[CrossRef]

C. J. R. Sheppard, K. G. Larkin, “Similarity theorems for fractional Fourier transforms and fractional Hankel transforms,” Opt. Commun. 154, 173–178 (1998).
[CrossRef]

H. Yoshimura, T. Iwai, “Effects of lens aperture on the average intensity in a fractional Fourier plane,” Pure Appl. Opt. 7, 1133–1141 (1998).
[CrossRef]

Z. Liu, X. Wu, D. Fan, “Collins formula in frequency-domain and fractional Fourier transforms,” Opt. Commun. 155, 7–11 (1998).
[CrossRef]

1997 (2)

1996 (3)

1995 (4)

1994 (2)

1993 (4)

1991 (1)

F. Gori, G. L. Marcopoli, M. Santarsiero, “Spectrum invariance on paraxial propagation,” Opt. Commun. 81, 123–130 (1991).
[CrossRef]

1990 (2)

Q. Lin, S. Wang, J. Alda, E. Bernabeu, “Transformation of non-symmetric Gaussian beam into symmetric one by means of tensor ABCD law,” Optik 85, 67–72 (1990).

A. Gamliel, “Mode analysis of spectral changes in light propagation from sources of any state of coherence,” J. Opt. Soc. Am. A 7, 1591–1597 (1990).
[CrossRef]

1987 (2)

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

A. C. McBride, F. H. Kerr, “On Namia’s fractional Fourier transforms,” IMA J. Appl. Math. 39, 159–175 (1987).
[CrossRef]

1986 (1)

E. Wolf, “Invariance of spectrum on propagation,” Phys. Rev. Lett. 56, 1370–1372 (1986).
[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, C. Palma, “Observation of optical redshifts and blueshifts produced by source correlation,” Opt. Acta 30, 1075–1097 (1983).

1973 (1)

V. Namias, “The fractional Fourier transform and its application in quantum mechanics,” J. Inst. Math. Appl. 25, 241–265 (1973).
[CrossRef]

1970 (1)

Alda, J.

Q. Lin, S. Wang, J. Alda, E. Bernabeu, “Transformation of non-symmetric Gaussian beam into symmetric one by means of tensor ABCD law,” Optik 85, 67–72 (1990).

Anand, S.

H. C. Kandpal, S. Anand, J. S. Vaishva, “Experimental observation of the phenomenon of spectral switching for a class of partially coherent light,” IEEE J. Quantum Electron. 37, 336–339 (2002).
[CrossRef]

Belendez, A.

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

Bernabeu, E.

Q. Lin, S. Wang, J. Alda, E. Bernabeu, “Transformation of non-symmetric Gaussian beam into symmetric one by means of tensor ABCD law,” Optik 85, 67–72 (1990).

Bitran, Y.

Cai, Y.

Candan, C.

C. Candan, M. A. Kutay, H. M. Ozaktas, “The discrete fractional Fourier transform,” IEEE Trans. Signal Process. 48, 1329–1337 (2000).
[CrossRef]

Cardone, G.

C. Palma, G. Cardone, G. Cincotti, “Spectral changes in Gaussian-cavity lasers,” IEEE J. Quantum Electron. 34, 1082–1088 (1998).
[CrossRef]

Carretero, L.

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

Cincotti, G.

C. Palma, G. Cardone, G. Cincotti, “Spectral changes in Gaussian-cavity lasers,” IEEE J. Quantum Electron. 34, 1082–1088 (1998).
[CrossRef]

C. Palma, G. Cincotti, G. Guattari, “Spectral shift of a Gaussian Schell-model beam beyond a thin lens,” IEEE J. Quantum Electron. 34, 378–383 (1998).
[CrossRef]

Collins, S. A.

Dong, B.

Dorsch, R. G.

Erden, M. F.

M. F. Erden, H. M. Ozaktas, D. Mendlovic, “Synthesis of mutual intensity distribution using the fractional Fourier transform,” Opt. Commun. 125, 288–301 (1996).
[CrossRef]

M. F. Erden, H. M. Ozaktas, D. Mendlovic, “Propagation of mutual intensity expressed in terms of the fractional Fourier transform,” J. Opt. Soc. Am. A 13, 1068–1071 (1996).
[CrossRef]

Fan, D.

Z. Liu, X. Wu, D. Fan, “Collins formula in frequency-domain and fractional Fourier transforms,” Opt. Commun. 155, 7–11 (1998).
[CrossRef]

Fimia, A.

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

Gadre, V. M.

S. Shinde, V. M. Gadre, “An uncertainty principle for real signals in the fractional Fourier transform domain,” IEEE Trans. Signal Process. 49, 2545–2548 (2001).
[CrossRef]

Gamliel, A.

Gori, F.

F. Gori, G. L. Marcopoli, M. Santarsiero, “Spectrum invariance on paraxial propagation,” Opt. Commun. 81, 123–130 (1991).
[CrossRef]

F. Gori, G. Guattari, C. Palma, “Observation of optical redshifts and blueshifts produced by source correlation,” Opt. Acta 30, 1075–1097 (1983).

Gu, B.

Guattari, G.

C. Palma, G. Cincotti, G. Guattari, “Spectral shift of a Gaussian Schell-model beam beyond a thin lens,” IEEE J. Quantum Electron. 34, 378–383 (1998).
[CrossRef]

F. Gori, G. Guattari, C. Palma, “Observation of optical redshifts and blueshifts produced by source correlation,” Opt. Acta 30, 1075–1097 (1983).

Hua, J.

Iwai, T.

H. Yoshimura, T. Iwai, “Effects of lens aperture on the average intensity in a fractional Fourier plane,” Pure Appl. Opt. 7, 1133–1141 (1998).
[CrossRef]

H. Yoshimura, T. Iwai, “Properties of the Gaussian Schell-Model source field in a fractional Fourier plane,” J. Opt. Soc. Am. A 14, 3388–3393 (1997).
[CrossRef]

Jiang, Z.

Jitsuno, T.

N. Nishi, T. Jitsuno, K. Tsubakimoto, S. Matsuoka, N. Miyanaga, M. Nakatsuka, “Two-dimensional multi-lens array with circular aperture spherical lens for flat-top irradiation of inertial confinement fusion target,” Opt. Rev. 7, 216–220 (2000).
[CrossRef]

Joshi, K. C.

H. C. Kandpal, J. S. Vaishya, K. Saxena, D. S. Mehta, K. C. Joshi, “Intensity distribution across a source from spectral measurements,” J. Mod. Opt. 42, 455–464 (1995).
[CrossRef]

Kandpal, H. C.

H. C. Kandpal, S. Anand, J. S. Vaishva, “Experimental observation of the phenomenon of spectral switching for a class of partially coherent light,” IEEE J. Quantum Electron. 37, 336–339 (2002).
[CrossRef]

H. C. Kandpal, J. S. Vaishya, K. Saxena, D. S. Mehta, K. C. Joshi, “Intensity distribution across a source from spectral measurements,” J. Mod. Opt. 42, 455–464 (1995).
[CrossRef]

Kerr, F. H.

A. C. McBride, F. H. Kerr, “On Namia’s fractional Fourier transforms,” IMA J. Appl. Math. 39, 159–175 (1987).
[CrossRef]

Kirk, A. G.

Kutay, M. A.

C. Candan, M. A. Kutay, H. M. Ozaktas, “The discrete fractional Fourier transform,” IEEE Trans. Signal Process. 48, 1329–1337 (2000).
[CrossRef]

Larkin, K. G.

C. J. R. Sheppard, K. G. Larkin, “Similarity theorems for fractional Fourier transforms and fractional Hankel transforms,” Opt. Commun. 154, 173–178 (1998).
[CrossRef]

Li, G.

Lin, Q.

Y. Cai, 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]

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

Y. Cai, Q. Lin, “Spectral shift of partially coherent twisted anisotropic Gaussian Schell-model beams in free space,” Opt. Commun. 204, 17–23 (2002).
[CrossRef]

Q. Lin, S. Wang, J. Alda, E. Bernabeu, “Transformation of non-symmetric Gaussian beam into symmetric one by means of tensor ABCD law,” Optik 85, 67–72 (1990).

Liu, L.

Liu, S.

Liu, Z.

Z. Liu, X. Wu, D. Fan, “Collins formula in frequency-domain and fractional Fourier transforms,” Opt. Commun. 155, 7–11 (1998).
[CrossRef]

Lohmann, A. W.

Lu, B.

L. Pan, B. Lu, “The spectral switch of partially coherent light in Young’s experiment,” IEEE J. Quantum Electron. 37, 1377–1381 (2001).
[CrossRef]

Luo, T. L.

S. C. Pei, M. H. Yeh, 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]

Marcopoli, G. L.

F. Gori, G. L. Marcopoli, M. Santarsiero, “Spectrum invariance on paraxial propagation,” Opt. Commun. 81, 123–130 (1991).
[CrossRef]

Matsuoka, S.

N. Nishi, T. Jitsuno, K. Tsubakimoto, S. Matsuoka, N. Miyanaga, M. Nakatsuka, “Two-dimensional multi-lens array with circular aperture spherical lens for flat-top irradiation of inertial confinement fusion target,” Opt. Rev. 7, 216–220 (2000).
[CrossRef]

McBride, A. C.

A. C. McBride, F. H. Kerr, “On Namia’s fractional Fourier transforms,” IMA J. Appl. Math. 39, 159–175 (1987).
[CrossRef]

Medlovic, D.

Z. Zalevsky, D. Medlovic, H. M. Ozaktas, “Energetic efficient synthesis of general mutual intensity distribution,” J. Opt. A Pure Appl. Opt. 2, 83–87 (2000).
[CrossRef]

Mehta, D. S.

H. C. Kandpal, J. S. Vaishya, K. Saxena, D. S. Mehta, K. C. Joshi, “Intensity distribution across a source from spectral measurements,” J. Mod. Opt. 42, 455–464 (1995).
[CrossRef]

Mendlovic, D.

Miyanaga, N.

N. Nishi, T. Jitsuno, K. Tsubakimoto, S. Matsuoka, N. Miyanaga, M. Nakatsuka, “Two-dimensional multi-lens array with circular aperture spherical lens for flat-top irradiation of inertial confinement fusion target,” Opt. Rev. 7, 216–220 (2000).
[CrossRef]

Mukunda, N.

Nakatsuka, M.

N. Nishi, T. Jitsuno, K. Tsubakimoto, S. Matsuoka, N. Miyanaga, M. Nakatsuka, “Two-dimensional multi-lens array with circular aperture spherical lens for flat-top irradiation of inertial confinement fusion target,” Opt. Rev. 7, 216–220 (2000).
[CrossRef]

Namias, V.

V. Namias, “The fractional Fourier transform and its application in quantum mechanics,” J. Inst. Math. Appl. 25, 241–265 (1973).
[CrossRef]

Nemoto, S.

J. Pu, S. Nemoto, “Spectral shifts and spectral switches in diffraction of partially coherent light by a circular aperture,” IEEE J. Quantum Electron. 36, 1407–1411 (2000).
[CrossRef]

Nishi, N.

N. Nishi, T. Jitsuno, K. Tsubakimoto, S. Matsuoka, N. Miyanaga, M. Nakatsuka, “Two-dimensional multi-lens array with circular aperture spherical lens for flat-top irradiation of inertial confinement fusion target,” Opt. Rev. 7, 216–220 (2000).
[CrossRef]

Ozaktas, H.

Ozaktas, H. M.

Z. Zalevsky, D. Medlovic, H. M. Ozaktas, “Energetic efficient synthesis of general mutual intensity distribution,” J. Opt. A Pure Appl. Opt. 2, 83–87 (2000).
[CrossRef]

C. Candan, M. A. Kutay, H. M. Ozaktas, “The discrete fractional Fourier transform,” IEEE Trans. Signal Process. 48, 1329–1337 (2000).
[CrossRef]

M. F. Erden, H. M. Ozaktas, D. Mendlovic, “Propagation of mutual intensity expressed in terms of the fractional Fourier transform,” J. Opt. Soc. Am. A 13, 1068–1071 (1996).
[CrossRef]

M. F. Erden, H. M. Ozaktas, D. Mendlovic, “Synthesis of mutual intensity distribution using the fractional Fourier transform,” Opt. Commun. 125, 288–301 (1996).
[CrossRef]

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

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

Palma, C.

C. Palma, G. Cardone, G. Cincotti, “Spectral changes in Gaussian-cavity lasers,” IEEE J. Quantum Electron. 34, 1082–1088 (1998).
[CrossRef]

C. Palma, G. Cincotti, G. Guattari, “Spectral shift of a Gaussian Schell-model beam beyond a thin lens,” IEEE J. Quantum Electron. 34, 378–383 (1998).
[CrossRef]

F. Gori, G. Guattari, C. Palma, “Observation of optical redshifts and blueshifts produced by source correlation,” Opt. Acta 30, 1075–1097 (1983).

Pan, L.

L. Pan, B. Lu, “The spectral switch of partially coherent light in Young’s experiment,” IEEE J. Quantum Electron. 37, 1377–1381 (2001).
[CrossRef]

Pei, S. C.

S. C. Pei, M. H. Yeh, 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]

Pellat-Finet, P.

Pu, J.

J. Pu, S. Nemoto, “Spectral shifts and spectral switches in diffraction of partially coherent light by a circular aperture,” IEEE J. Quantum Electron. 36, 1407–1411 (2000).
[CrossRef]

Ronchi, L.

S. Wang, L. Ronchi, “Principles and design of optical array,” in Progress in Optics, Vol. XXV, E. Wolf, ed. (Elsevier, Amsterdam, 1988), p. 279.

Santarsiero, M.

F. Gori, G. L. Marcopoli, M. Santarsiero, “Spectrum invariance on paraxial propagation,” Opt. Commun. 81, 123–130 (1991).
[CrossRef]

Saxena, K.

H. C. Kandpal, J. S. Vaishya, K. Saxena, D. S. Mehta, K. C. Joshi, “Intensity distribution across a source from spectral measurements,” J. Mod. Opt. 42, 455–464 (1995).
[CrossRef]

Sheppard, C. J. R.

C. J. R. Sheppard, K. G. Larkin, “Similarity theorems for fractional Fourier transforms and fractional Hankel transforms,” Opt. Commun. 154, 173–178 (1998).
[CrossRef]

Shinde, S.

S. Shinde, V. M. Gadre, “An uncertainty principle for real signals in the fractional Fourier transform domain,” IEEE Trans. Signal Process. 49, 2545–2548 (2001).
[CrossRef]

Simon, R.

Sudarshan, E. C. G.

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.

Tamura, S.

Tsubakimoto, K.

N. Nishi, T. Jitsuno, K. Tsubakimoto, S. Matsuoka, N. Miyanaga, M. Nakatsuka, “Two-dimensional multi-lens array with circular aperture spherical lens for flat-top irradiation of inertial confinement fusion target,” Opt. Rev. 7, 216–220 (2000).
[CrossRef]

Tu, J. H.

Vaishva, J. S.

H. C. Kandpal, S. Anand, J. S. Vaishva, “Experimental observation of the phenomenon of spectral switching for a class of partially coherent light,” IEEE J. Quantum Electron. 37, 336–339 (2002).
[CrossRef]

Vaishya, J. S.

H. C. Kandpal, J. S. Vaishya, K. Saxena, D. S. Mehta, K. C. Joshi, “Intensity distribution across a source from spectral measurements,” J. Mod. Opt. 42, 455–464 (1995).
[CrossRef]

Wang, S.

Q. Lin, S. Wang, J. Alda, E. Bernabeu, “Transformation of non-symmetric Gaussian beam into symmetric one by means of tensor ABCD law,” Optik 85, 67–72 (1990).

S. Wang, L. Ronchi, “Principles and design of optical array,” in Progress in Optics, Vol. XXV, E. Wolf, ed. (Elsevier, Amsterdam, 1988), p. 279.

Wei, H. Q.

Wolf, E.

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

E. Wolf, “Invariance of spectrum on propagation,” Phys. Rev. Lett. 56, 1370–1372 (1986).
[CrossRef] [PubMed]

Wu, X.

Z. Liu, X. Wu, D. Fan, “Collins formula in frequency-domain and fractional Fourier transforms,” Opt. Commun. 155, 7–11 (1998).
[CrossRef]

Xue, X.

Yang, G.

Yeh, M. H.

S. C. Pei, M. H. Yeh, 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]

Yoshimura, H.

H. Yoshimura, T. Iwai, “Effects of lens aperture on the average intensity in a fractional Fourier plane,” Pure Appl. Opt. 7, 1133–1141 (1998).
[CrossRef]

H. Yoshimura, T. Iwai, “Properties of the Gaussian Schell-Model source field in a fractional Fourier plane,” J. Opt. Soc. Am. A 14, 3388–3393 (1997).
[CrossRef]

Zalevsky, Z.

Zhang, Y.

Zhu, B.

Appl. Opt. (2)

IEEE J. Quantum Electron. (5)

C. Palma, G. Cardone, G. Cincotti, “Spectral changes in Gaussian-cavity lasers,” IEEE J. Quantum Electron. 34, 1082–1088 (1998).
[CrossRef]

C. Palma, G. Cincotti, G. Guattari, “Spectral shift of a Gaussian Schell-model beam beyond a thin lens,” IEEE J. Quantum Electron. 34, 378–383 (1998).
[CrossRef]

J. Pu, S. Nemoto, “Spectral shifts and spectral switches in diffraction of partially coherent light by a circular aperture,” IEEE J. Quantum Electron. 36, 1407–1411 (2000).
[CrossRef]

L. Pan, B. Lu, “The spectral switch of partially coherent light in Young’s experiment,” IEEE J. Quantum Electron. 37, 1377–1381 (2001).
[CrossRef]

H. C. Kandpal, S. Anand, J. S. Vaishva, “Experimental observation of the phenomenon of spectral switching for a class of partially coherent light,” IEEE J. Quantum Electron. 37, 336–339 (2002).
[CrossRef]

IEEE Trans. Signal Process. (3)

S. C. Pei, M. H. Yeh, 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]

S. Shinde, V. M. Gadre, “An uncertainty principle for real signals in the fractional Fourier transform domain,” IEEE Trans. Signal Process. 49, 2545–2548 (2001).
[CrossRef]

C. Candan, M. A. Kutay, H. M. Ozaktas, “The discrete fractional Fourier transform,” IEEE Trans. Signal Process. 48, 1329–1337 (2000).
[CrossRef]

IMA J. Appl. Math. (1)

A. C. McBride, F. H. Kerr, “On Namia’s fractional Fourier transforms,” IMA J. Appl. Math. 39, 159–175 (1987).
[CrossRef]

J. Inst. Math. Appl. (1)

V. Namias, “The fractional Fourier transform and its application in quantum mechanics,” J. Inst. Math. Appl. 25, 241–265 (1973).
[CrossRef]

J. Mod. Opt. (1)

H. C. Kandpal, J. S. Vaishya, K. Saxena, D. S. Mehta, K. C. Joshi, “Intensity distribution across a source from spectral measurements,” J. Mod. Opt. 42, 455–464 (1995).
[CrossRef]

J. Opt. A Pure Appl. Opt. (1)

Z. Zalevsky, D. Medlovic, H. M. Ozaktas, “Energetic efficient synthesis of general mutual intensity distribution,” J. Opt. A Pure Appl. Opt. 2, 83–87 (2000).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (16)

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

Y. Cai, 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]

M. F. Erden, H. M. Ozaktas, D. Mendlovic, “Propagation of mutual intensity expressed in terms of the fractional Fourier transform,” J. Opt. Soc. Am. A 13, 1068–1071 (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]

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

R. Simon, N. Mukunda, “Gaussian Schell-model beams and general shape invariance,” J. Opt. Soc. Am. A 16, 2465–2475 (1999).
[CrossRef]

J. H. Tu, S. Tamura, “Analytic relation for recovering the mutual intensity by means of intensity information,” J. Opt. Soc. Am. A 15, 202–206 (1998).
[CrossRef]

Y. Zhang, B. Dong, B. Gu, G. Yang, “Beam shaping in the fractional Fourier transform domain,” J. Opt. Soc. Am. A 15, 1114–1120 (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, 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]

H. Yoshimura, T. Iwai, “Properties of the Gaussian Schell-Model source field in a fractional Fourier plane,” J. Opt. Soc. Am. A 14, 3388–3393 (1997).
[CrossRef]

A. Gamliel, “Mode analysis of spectral changes in light propagation from sources of any state of coherence,” J. Opt. Soc. Am. A 7, 1591–1597 (1990).
[CrossRef]

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

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

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

Nature (1)

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

Opt. Acta (1)

F. Gori, G. Guattari, C. Palma, “Observation of optical redshifts and blueshifts produced by source correlation,” Opt. Acta 30, 1075–1097 (1983).

Opt. Commun. (6)

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

C. J. R. Sheppard, K. G. Larkin, “Similarity theorems for fractional Fourier transforms and fractional Hankel transforms,” Opt. Commun. 154, 173–178 (1998).
[CrossRef]

M. F. Erden, H. M. Ozaktas, D. Mendlovic, “Synthesis of mutual intensity distribution using the fractional Fourier transform,” Opt. Commun. 125, 288–301 (1996).
[CrossRef]

Z. Liu, X. Wu, D. Fan, “Collins formula in frequency-domain and fractional Fourier transforms,” Opt. Commun. 155, 7–11 (1998).
[CrossRef]

Y. Cai, Q. Lin, “Spectral shift of partially coherent twisted anisotropic Gaussian Schell-model beams in free space,” Opt. Commun. 204, 17–23 (2002).
[CrossRef]

F. Gori, G. L. Marcopoli, M. Santarsiero, “Spectrum invariance on paraxial propagation,” Opt. Commun. 81, 123–130 (1991).
[CrossRef]

Opt. Lett. (6)

Opt. Rev. (1)

N. Nishi, T. Jitsuno, K. Tsubakimoto, S. Matsuoka, N. Miyanaga, M. Nakatsuka, “Two-dimensional multi-lens array with circular aperture spherical lens for flat-top irradiation of inertial confinement fusion target,” Opt. Rev. 7, 216–220 (2000).
[CrossRef]

Optik (1)

Q. Lin, S. Wang, J. Alda, E. Bernabeu, “Transformation of non-symmetric Gaussian beam into symmetric one by means of tensor ABCD law,” Optik 85, 67–72 (1990).

Phys. Rev. A (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]

Phys. Rev. Lett. (1)

E. Wolf, “Invariance of spectrum on propagation,” Phys. Rev. Lett. 56, 1370–1372 (1986).
[CrossRef] [PubMed]

Pure Appl. Opt. (1)

H. Yoshimura, T. Iwai, “Effects of lens aperture on the average intensity in a fractional Fourier plane,” Pure Appl. Opt. 7, 1133–1141 (1998).
[CrossRef]

Other (1)

S. Wang, L. Ronchi, “Principles and design of optical array,” in Progress in Optics, Vol. XXV, E. Wolf, ed. (Elsevier, Amsterdam, 1988), p. 279.

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Figures (8)

Fig. 1
Fig. 1

Optical system for performing the FRT. (a) One-lens system, (b) two-lens system.

Fig. 2
Fig. 2

Misaligned optical system.

Fig. 3
Fig. 3

Contour graphs of the intensity distribution of the twisted anisotropic GSM beam in the FRT plane with different fractional orders. (a) p=0, (b) p=0.9, (c) p=1, (d) p=1.2.

Fig. 4
Fig. 4

Dependence of the transverse-spot-width matrix element σI112 of the twisted anisotropic GSM beam with different transverse-coherence-width matrix elements σg112 on the fractional order p in the FRT plane. Curve (a) σg112=0.1 mm2, curve (b) σg112=0.4 mm2, curve (c) σg112=2 mm2.

Fig. 5
Fig. 5

Dependence of the transverse-coherence-width matrix element σg112 of the twisted anisotropic GSM beam on the fractional order p in the FRT plane.

Fig. 6
Fig. 6

Normalized on-axis spectrum of the twisted anisotropic GSM beams in the FRT plane with different fractional orders p. Curve (a) Source spectrum, curve (b) p=0.5, curve (c) p=0.8, curve (d) p=1.

Fig. 7
Fig. 7

Dependence of the on-axis relative central-frequency shift of twisted anisotropic GSM beams with different transverse-coherence-width matrix elements σg112 on the fractional order p in the FRT plane. Curve (a) σg112=0.1 mm2, curve (b) σg112=0.6 mm2, curve (c) σg112=1 mm2.

Fig. 8
Fig. 8

Relative central-frequency shift of the twisted anisotrpic GSM in the FRT plane versus the transverse coordinate yp with different fractional orders. Curve (a) p=0.7, curve (b) p=0.8, curve (c) p=0.85.

Equations (63)

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Ep(u)=1iλf sin ϕ-E(r)exp-iπ(|r|2+|u|2)λftanϕ×exp2πiruλf sin ϕd2r,
Wo(r˜)=Eo(r1)Eo*(r2),Wp(u˜)=Ep(u1)Ep*(u2),
Wp(u˜)=ω2(2πcfsinϕ)2-W0(r˜)×exp-iω(|r1|2+|u1|2-|r2|2-|u2|2)2cftanϕ×expiω(r1u1-r2u2)cfsinϕdr1dr2.
Wp(u˜)=ω2(2πcfsinϕ)2-W0(r˜)exp-iω2cRTNRd2r˜,
N=N11-N12-N12N11.
N11=1ftanϕI00-I,N12=1fsinϕI00-I,
W0(r˜)=S0(ω)exp-iω2cr˜TMi-1r˜,
M-1=M11-1M12-1(M12-1)T(-M11-1)*=R-1-ic2ω (σI2)-1-icω (σg2)-1icω (σg2)-1+μJicω (σg2)-1+μJT-R-1-ic2ω (σI2)-1-icω (σg2)-1,
(σI2)-1=σI11-2σI12-2σI12-2σI22-2,(σg2)-1=σg11-2σg12-2σg12-2σg22-2,
R-1=R11-1R12-1R21-1R22-1,
J=01-10.
Wp(u˜)=S0(ω)(fsinϕ)2 [det(Mi-1+N11)]-1/2×exp-iω2cu˜T[N11-N12T(Mi-1+N11)-1N12]u˜.
A¯=N12-1N11=cos ϕI00I,
B¯=N12-1=fsinϕI00-I,
C¯=N11N12-1N11-N12T=sin ϕf-I00I,
D¯=N11N12-1=cos ϕI00I.
Wp(u˜)=S0(ω)[det(A¯+BM¯i-1)]-1/2exp-iω2cu˜TMp-1u˜,
Mp-1=(C¯+DM¯i-1)(A¯+BM¯i-1)-1.
N11=B¯-1A¯=D¯B¯-1,N12=B¯-1=D¯B¯-1A¯-C¯.
Wp(u˜)=S0(ω)λ2det[B¯]-1/2-W0(r˜)×exp-iπλ [r˜TB¯-1Ar¯˜-2r˜TB¯-1u˜+u˜TD¯B¯-1u˜]dr˜,
E2(ρ1x,ρ1y)=-iλbE1(x1, y1)×exp-iπλb [a(x12+y12)-2(x1ρ1x+y1ρ1y)+d(ρ1x2+ρ1y2)+ex1+fy1+gρ1x+hρ1y]dx1dy1.
e=2(αTεx+βTεx),
f=2(αTεy+βTεy),
g=2(bγT-dαT)εx+2(bδT-dβT)εx,
h=2(bγT-dαT)εy+2(bδT-dβT)εy,
αT=1-a,βT=l-b,γT=-c,
δT=±1-d,
E(ρ1)=-iλ[det(B)]1/2E(r1)×exp-iπλ (r1TB-1Ar1-2r1TB-1ρ1+ρ1TDB-1ρ1T)×exp-iπλ (r1TB-1ef+ρ1TB-1gh)dr1,
A=a00a,B=b00b,
C=c00c,D=d00d.
E(ρ1e)=-iλ[det(B)]1/2exp-iπλρ1eTDB-1ef+ρ1eTB-1gh+12efTB-1gh+14efTDB-1efE(r1)×exp-ik2 (r1TB-1Ar1-2r1TB-1ρ1e+ρ1eTDB-1ρ1eT)dr1.
W(r˜)=E(r1)E*(r2),W(ρ˜)=E(ρ1)E*(ρ2),
W(ρ˜)=k24π2[det(B¯)]1/2W(r˜)×exp-ik2 (r˜TB¯-1Ar¯˜-2r˜TB¯-1ρ˜+ρ˜TD¯B¯-1ρ˜)×exp-ik2 (r˜TB¯-1e¯f+ρ˜TB¯-1g¯h)dr˜,
A¯=A00A,B¯=B00-B,
C¯=C00-C,D¯=D00D.
ρ1e=ρ1-12ef,ρ2e=ρ2-12ef,
ρ˜e=ρ˜-12e¯f,
W(ρ˜e)=S0(ω)λ2[det(B¯)]1/2exp-iπλ (ρ˜eTD¯B¯-1ef+ρ˜eTB¯-1gh)W(r˜)exp-ik2 (r˜TB¯-1Ar¯˜-2r˜TB¯-1ρ˜e+ρ˜eTD¯B¯-1ρ˜eT)dr˜.
exp-iπλ (ρ˜eTD¯B¯-1ef+ρ˜eTB¯-1gh)
W(ρ˜e)=S0(ω)λ2[det(B¯)]1/2W(r˜)exp-ik2 (r˜TB¯-1A¯r˜-2r˜TB¯-1ρ˜e+ρ˜eTD¯B¯-1ρ˜eT)dr˜.
ρ1e=ρ1-12ef,ρ2e=ρ2-12ef,
ρ˜e=ρ˜-12e¯f.
(σI2)-1=10.20.20.5(mm)-2,
(σg2)-1=0.50.20.20.33(mm)-2,
R-1=0000(mm)-1,
μ=0.00001mm-1,f=2000mm.
σg122=5mm2,σg222=3mm2,λ=632.8 nm,
f=2000mm,(σI2)-1=10.20.20.5(mm)-2,
R-1=0000(mm)-1,μ=0.00001mm-1.
(σg2)-1=0.50.20.20.33(mm)-2,f=2000mm,
(σI2)-1=10.20.20.5(mm)-2,
R-1=0000(mm)-1,
μ=0.00001mm-1,λ=632.8 nm.
S(u1, ω)=W(u1=u2, ω).
S0(ω)=S0δ2(ω-ω0)2+δ2,
μ=0.00001mm-1,(σI2)-1=10.20.20.5(mm)-2,
(σg2)-1=10223.3(mm)-2,
R-1=0000(mm)-1,f=2000mm.
Δω/ω0=(ωm-ω0)/ω0.
(σ12)-1=10.20.20.5(mm)-2,
(σg2)-1=10223.3(mm)-2,
R-1=0000(mm)-1,f=2000mm,
xp=0.

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