Abstract

Based on the coherent optical theory, we extend the fractional Fourier transform of first-order correlation to a fractional Fourier transform of second-order correlation. An optical system for implementing a second-order fractional Fourier transform was designed. As a numerical example, we investigate the second-order fractional Fourier transform for a single slit.

© 2005 Optical Society of America

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References

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2004 (2)

A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, Phys. Rev. A 70, 013802 (2004).
[CrossRef]

J. Chen and S. Han, Phys. Rev. Lett. 92, 093903 (2004).
[CrossRef]

2002 (1)

2001 (3)

2000 (3)

B. Zhu, S. Liu, and Q. Ran, Opt. Lett. 25, 1159 (2000).
[CrossRef]

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, Phys. Rev. Lett. 85, 2733 (2000).
[CrossRef] [PubMed]

C. Candan, M. A. Kutay, and H. M. Ozaktas, IEEE Trans. Signal Process. 48, 1329 (2000).
[CrossRef]

1999 (1)

S. C. Pei, M. H. Yeh, and T. L. Luo, IEEE Trans. Signal Process. 47, 2883 (1999).
[CrossRef]

1998 (1)

C. J. R. Sheppard and K. G. Larkin, Opt. Commun. 154, 173 (1998).
[CrossRef]

1997 (2)

1996 (1)

1995 (1)

D. V. Stekalov, A. V. Sergienko, D. N. Klyshko, and Y. H. Shih, Phys. Rev. Lett. 74, 3600 (1995).
[CrossRef]

1993 (2)

1987 (1)

A. C. McBride and F. H. Kerr, IMA J. Appl. Math. 39, 159 (1987).
[CrossRef]

1980 (1)

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

1970 (1)

Abouraddy, A. F.

Abrams, D. S.

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, Phys. Rev. Lett. 85, 2733 (2000).
[CrossRef] [PubMed]

Bache, M.

A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, Phys. Rev. A 70, 013802 (2004).
[CrossRef]

Boto, A. N.

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, Phys. Rev. Lett. 85, 2733 (2000).
[CrossRef] [PubMed]

Brambilla, E.

A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, Phys. Rev. A 70, 013802 (2004).
[CrossRef]

Braunstein, S. L.

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, Phys. Rev. Lett. 85, 2733 (2000).
[CrossRef] [PubMed]

Cai, Y.

Candan, C.

C. Candan, M. A. Kutay, and H. M. Ozaktas, IEEE Trans. Signal Process. 48, 1329 (2000).
[CrossRef]

Chen, J.

J. Chen and S. Han, Phys. Rev. Lett. 92, 093903 (2004).
[CrossRef]

Collins, S. A.

Dowling, J. P.

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, Phys. Rev. Lett. 85, 2733 (2000).
[CrossRef] [PubMed]

Gatti, A.

A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, Phys. Rev. A 70, 013802 (2004).
[CrossRef]

Han, S.

J. Chen and S. Han, Phys. Rev. Lett. 92, 093903 (2004).
[CrossRef]

Hua, J.

Iwai, T.

Kerr, F. H.

A. C. McBride and F. H. Kerr, IMA J. Appl. Math. 39, 159 (1987).
[CrossRef]

Kirk, A. G.

Klyshko, D. N.

D. V. Stekalov, A. V. Sergienko, D. N. Klyshko, and Y. H. Shih, Phys. Rev. Lett. 74, 3600 (1995).
[CrossRef]

Kok, P.

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, Phys. Rev. Lett. 85, 2733 (2000).
[CrossRef] [PubMed]

Kutay, M. A.

C. Candan, M. A. Kutay, and H. M. Ozaktas, IEEE Trans. Signal Process. 48, 1329 (2000).
[CrossRef]

Larkin, K. G.

C. J. R. Sheppard and K. G. Larkin, Opt. Commun. 154, 173 (1998).
[CrossRef]

Li, G.

Lin, Q.

Liu, L.

Liu, S.

Lohmann, A. W.

Lugiato, L. A.

A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, Phys. Rev. A 70, 013802 (2004).
[CrossRef]

Luo, T. L.

S. C. Pei, M. H. Yeh, and T. L. Luo, IEEE Trans. Signal Process. 47, 2883 (1999).
[CrossRef]

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, New York, 1995).
[CrossRef]

McBride, A. C.

A. C. McBride and F. H. Kerr, IMA J. Appl. Math. 39, 159 (1987).
[CrossRef]

Mendlovic, D.

Namias, V.

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

Ozaktas, H. M.

C. Candan, M. A. Kutay, and H. M. Ozaktas, IEEE Trans. Signal Process. 48, 1329 (2000).
[CrossRef]

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

Pei, S. C.

S. C. Pei, M. H. Yeh, and T. L. Luo, IEEE Trans. Signal Process. 47, 2883 (1999).
[CrossRef]

Ran, Q.

Saleh, B. E. A.

Sergienko, A. V.

Sheppard, C. J. R.

C. J. R. Sheppard and K. G. Larkin, Opt. Commun. 154, 173 (1998).
[CrossRef]

Shih, Y. H.

D. V. Stekalov, A. V. Sergienko, D. N. Klyshko, and Y. H. Shih, Phys. Rev. Lett. 74, 3600 (1995).
[CrossRef]

Stekalov, D. V.

D. V. Stekalov, A. V. Sergienko, D. N. Klyshko, and Y. H. Shih, Phys. Rev. Lett. 74, 3600 (1995).
[CrossRef]

Teich, M. C.

Wei, H.

Williams, C. P.

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, Phys. Rev. Lett. 85, 2733 (2000).
[CrossRef] [PubMed]

Wolf, E.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, New York, 1995).
[CrossRef]

Xue, X.

Yeh, M. H.

S. C. Pei, M. H. Yeh, and T. L. Luo, IEEE Trans. Signal Process. 47, 2883 (1999).
[CrossRef]

Yoshimura, H.

Zalevsky, Z.

Zhu, B.

Appl. Opt. (1)

IEEE Trans. Signal Process. (2)

S. C. Pei, M. H. Yeh, and T. L. Luo, IEEE Trans. Signal Process. 47, 2883 (1999).
[CrossRef]

C. Candan, M. A. Kutay, and H. M. Ozaktas, IEEE Trans. Signal Process. 48, 1329 (2000).
[CrossRef]

IMA J. Appl. Math. (1)

A. C. McBride and F. H. Kerr, IMA J. Appl. Math. 39, 159 (1987).
[CrossRef]

J. Inst. Math. Appl. (1)

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

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

Opt. Commun. (1)

C. J. R. Sheppard and K. G. Larkin, Opt. Commun. 154, 173 (1998).
[CrossRef]

Opt. Express (1)

Opt. Lett. (5)

Phys. Rev. A (1)

A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, Phys. Rev. A 70, 013802 (2004).
[CrossRef]

Phys. Rev. Lett. (3)

J. Chen and S. Han, Phys. Rev. Lett. 92, 093903 (2004).
[CrossRef]

D. V. Stekalov, A. V. Sergienko, D. N. Klyshko, and Y. H. Shih, Phys. Rev. Lett. 74, 3600 (1995).
[CrossRef]

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, Phys. Rev. Lett. 85, 2733 (2000).
[CrossRef] [PubMed]

Other (1)

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, New York, 1995).
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

Optical system for a second-order FRT.

Fig. 2
Fig. 2

Optical system for another second-order FRT.

Fig. 3
Fig. 3

Second-order FRT for a single slit with slit width h=0.02 mm for three source’s transverse sizes, σI.

Equations (15)

Equations on this page are rendered with MathJax. Learn more.

G2u1,u2=Eu1Eu2E*u2E*u1=----h1x1,u1h1*x4,u1h2x2,u2×h2*x3,u2Ex1Ex2E*x3E*x4dx1dx2dx3dx4=Iu1Iu2+Γu1,u22,
Iui=--hix1,uihi*x2,ui×Ex1E*x2dx1dx2,    i=1,2,
Γu1,u2=--Ex1E*x2h1x1,u1×h2*x2,u2dx1dx2.
h1u1,x1=-1λ2zf11/2exp-iπλz1x12×-expiπλz12x1v1-v12+2iπλf1vu1Hv1dv1,
h2u2,x2=-iλb21/2×exp-iπλb2a2x22-2x2u2+d2u22,
a2b2c2d2=1l20110-1/f11l101=1-l2/fl1+l2-l1l2/f-1/f1-l1/f.
Ex1E*x2=Ix1δx1-x2,
Iu1=0=I0λf1-Hv12dv1, Iu2=I0λb2-dx1=.
Γu1=0,u2=I0iλ3zf1b21/2×--Hv1exp-iπλzx12-2x1v1+v12×expiπλb2a2x12-2x1u2+d2u22dx1dv1.
l1=z+fe tanϕ/2,    l2=fetanϕ/2, f=fe/sin ϕ
Γu1=0,u2=I0λ2f1fe sin ϕ1/2-Hv1×expiπλfe tan ϕv12+u22-2iπλfe sin ϕv1u2dv1.
Epx2=-E0x1expiπλfe tan ϕx12+x22-2iπλfe sin φx1x2dx1,
Ex1E*x2=exp-x12+x224σI2δx1-x2,
Iu1=0=2σI2π1/2λ2zf1--Hv12×exp-2σI2π2λ2z2v1-v22-iπλzv12+iπλzv22dv1dv2=const,
Γu1=0,u2=iπλ3zf1b2A11/2-Hv1×exp-π2λ2A1v1z-u2b22×exp-iπλzv12+id2πλb2u22dv1,

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