Abstract

The coincidence subwavelength fractional Fourier transforms (FRTs) with entangled photon pairs and incoherent light radiation are introduced as an extension of the recently introduced coincidence FRT. Optical systems for implementing the coincidence subwavelength FRTs are designed. The width of the coincidence subwavelength FRT pattern is two times narrower than the width of the coincidence FRT. The coincidence subwavelength FRT with partially coherent light radiation is also studied numerically. Differences between the coincidence subwavelength FRT with entangled photon pairs and that with incoherent light radiation are discussed.

© 2006 Optical Society of America

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2005 (7)

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

D. Cao, J. Xiong, and K. Wang, "Geometrical optics in correlated imaging systems," Phys. Rev. A 71, 013801 (2005).
[CrossRef]

A. Valencia, G. Scarcelli, M. D. Angelo, and Y. Shih, "Two-photon imaging with thermal light," Phys. Rev. Lett. 94, 063601 (2005).
[CrossRef] [PubMed]

J. Xiong, D. Cao, F. Huang, H. Li, X. Sun, and K. Wang, "Experimental observation of classical subwavelength interference with a pseudothermal light source" Phys. Rev. Lett. 94, 173601 (2005).
[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]

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 (6)

K. Wang and D. Cao, "Subwavelength coincidence interference with classical thermal light," Phys. Rev. A 70, 041801 (2004).
[CrossRef]

G. Scarcelli, A. Valencia, and Y. H. Shih, "Two-photon interference with thermal light," Europhys. Lett. 68, 618-624 (2004).
[CrossRef]

A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, "Ghost imaging with thermal light: comparing entanglement and classical correlation," Phys. Rev. Lett. 93, 093602 (2004).
[CrossRef] [PubMed]

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

J. Cheng and S. Han, "Incoherent coincidence imaging and its applicability in x-ray diffraction," Phys. Rev. Lett. 92, 093903 (2004).
[CrossRef] [PubMed]

Y. Cai and S. Zhu, "Ghost interference with partially coherent radiation," Opt. Lett. 29, 2716-2718 (2004).
[CrossRef] [PubMed]

2002 (1)

2001 (3)

2000 (2)

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, "Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit," Phys. Rev. Lett. 85, 2733-2736 (2000).
[CrossRef] [PubMed]

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

1998 (3)

1996 (2)

1995 (3)

R. G. Dorsch and A. W. Lohmann, "Fractional Fourier transform used for a lens-design problem," Appl. Opt. 34, 4111-4112 (1995).
[CrossRef] [PubMed]

T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, "Optical imaging by means of two-photon quantum entanglement," Phys. Rev. A 52, R3429-R3432 (1995).
[CrossRef] [PubMed]

D. V. Strekalov, A. V. Sergienko, D. N. Klyshko, and Y. H. Shih, "Observation of two-photon ghost interference and diffraction," Phys. Rev. Lett. 74, 3600-3603 (1995).
[CrossRef] [PubMed]

1994 (2)

1993 (3)

1976 (1)

J. T. Foley and M. S. Zubairy, "The directionality of Gaussian Schell-model beams," Opt. Commun. 26, 297-300 (1976).
[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, "Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit," Phys. Rev. Lett. 85, 2733-2736 (2000).
[CrossRef] [PubMed]

Angelo, M. D.

A. Valencia, G. Scarcelli, M. D. Angelo, and Y. Shih, "Two-photon imaging with thermal light," Phys. Rev. Lett. 94, 063601 (2005).
[CrossRef] [PubMed]

M. D. Angelo, M. V. Chekhova, and Y. H. Shih, "Two-photon diffraction and quantum lithography," Phys. Rev. Lett. 87, 013602 (2001).
[CrossRef] [PubMed]

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]

A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, "Ghost imaging with thermal light: comparing entanglement and classical correlation," Phys. Rev. Lett. 93, 093602 (2004).
[CrossRef] [PubMed]

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

Bernardo, L. M.

Bitran, Y.

Boto, A. N.

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, "Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit," Phys. Rev. Lett. 85, 2733-2736 (2000).
[CrossRef] [PubMed]

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]

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

A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, "Ghost imaging with thermal light: comparing entanglement and classical correlation," Phys. Rev. Lett. 93, 093602 (2004).
[CrossRef] [PubMed]

Braunstein, S. L.

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, "Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit," Phys. Rev. Lett. 85, 2733-2736 (2000).
[CrossRef] [PubMed]

Cai, Y.

Y. Cai and S. Zhu, "Ghost imaging with incoherent and partially coherent light radiation," Phys. Rev. A 71, 056607 (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 interference with partially coherent radiation," Opt. Lett. 29, 2716-2718 (2004).
[CrossRef] [PubMed]

Candan, C.

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

Cao, D.

J. Xiong, D. Cao, F. Huang, H. Li, X. Sun, and K. Wang, "Experimental observation of classical subwavelength interference with a pseudothermal light source" Phys. Rev. Lett. 94, 173601 (2005).
[CrossRef] [PubMed]

D. Cao, J. Xiong, and K. Wang, "Geometrical optics in correlated imaging systems," Phys. Rev. A 71, 013801 (2005).
[CrossRef]

K. Wang and D. Cao, "Subwavelength coincidence interference with classical thermal light," Phys. Rev. A 70, 041801 (2004).
[CrossRef]

Chekhova, M. V.

M. D. Angelo, M. V. Chekhova, and Y. H. Shih, "Two-photon diffraction and quantum lithography," Phys. Rev. Lett. 87, 013602 (2001).
[CrossRef] [PubMed]

Cheng, J.

J. Cheng and S. Han, "Incoherent coincidence imaging and its applicability in x-ray diffraction," Phys. Rev. Lett. 92, 093903 (2004).
[CrossRef] [PubMed]

Collins, S. A.

Dong, B.

Dorsch, R. G.

Dowling, J. P.

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, "Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit," Phys. Rev. Lett. 85, 2733-2736 (2000).
[CrossRef] [PubMed]

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 (1976).
[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]

A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, "Ghost imaging with thermal light: comparing entanglement and classical correlation," Phys. Rev. Lett. 93, 093602 (2004).
[CrossRef] [PubMed]

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

Gu, B.

Han, S.

J. Cheng and S. Han, "Incoherent coincidence imaging and its applicability in x-ray diffraction," Phys. Rev. Lett. 92, 093903 (2004).
[CrossRef] [PubMed]

Huang, F.

J. Xiong, D. Cao, F. Huang, H. Li, X. Sun, and K. Wang, "Experimental observation of classical subwavelength interference with a pseudothermal light source" Phys. Rev. Lett. 94, 173601 (2005).
[CrossRef] [PubMed]

Kirk, A. G.

Klyshko, D. N.

D. V. Strekalov, A. V. Sergienko, D. N. Klyshko, and Y. H. Shih, "Observation of two-photon ghost interference and diffraction," Phys. Rev. Lett. 74, 3600-3603 (1995).
[CrossRef] [PubMed]

Kok, P.

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, "Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit," Phys. Rev. Lett. 85, 2733-2736 (2000).
[CrossRef] [PubMed]

Kutay, M. A.

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

M. A. Kutay and H. M. Ozaktas, "Optimal image restoration with the fractional Fourier transform," J. Opt. Soc. Am. A 15, 825-833 (1998).
[CrossRef]

Larkin, K. G.

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

Li, H.

J. Xiong, D. Cao, F. Huang, H. Li, X. Sun, and K. Wang, "Experimental observation of classical subwavelength interference with a pseudothermal light source" Phys. Rev. Lett. 94, 173601 (2005).
[CrossRef] [PubMed]

Lin, Q.

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]

Lohmann, A. W.

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]

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

A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, "Ghost imaging with thermal light: comparing entanglement and classical correlation," Phys. Rev. Lett. 93, 093602 (2004).
[CrossRef] [PubMed]

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).

Mendlovic, D.

Ozaktas, H. M.

Pittman, T. B.

T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, "Optical imaging by means of two-photon quantum entanglement," Phys. Rev. A 52, R3429-R3432 (1995).
[CrossRef] [PubMed]

Saleh, B. E.

Scarcelli, G.

A. Valencia, G. Scarcelli, M. D. Angelo, and Y. Shih, "Two-photon imaging with thermal light," Phys. Rev. Lett. 94, 063601 (2005).
[CrossRef] [PubMed]

G. Scarcelli, A. Valencia, and Y. H. Shih, "Two-photon interference with thermal light," Europhys. Lett. 68, 618-624 (2004).
[CrossRef]

Sergienko, A. V.

A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, "Entangled-photon Fourier optics," J. Opt. Soc. Am. B 19, 1174-1184 (2002).
[CrossRef]

A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, "Quantum holography," Opt. Express 9, 498-505 (2001).
[CrossRef] [PubMed]

T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, "Optical imaging by means of two-photon quantum entanglement," Phys. Rev. A 52, R3429-R3432 (1995).
[CrossRef] [PubMed]

D. V. Strekalov, A. V. Sergienko, D. N. Klyshko, and Y. H. Shih, "Observation of two-photon ghost interference and diffraction," Phys. Rev. Lett. 74, 3600-3603 (1995).
[CrossRef] [PubMed]

Sheppard, C. J.

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

Shih, Y.

A. Valencia, G. Scarcelli, M. D. Angelo, and Y. Shih, "Two-photon imaging with thermal light," Phys. Rev. Lett. 94, 063601 (2005).
[CrossRef] [PubMed]

Shih, Y. H.

G. Scarcelli, A. Valencia, and Y. H. Shih, "Two-photon interference with thermal light," Europhys. Lett. 68, 618-624 (2004).
[CrossRef]

M. D. Angelo, M. V. Chekhova, and Y. H. Shih, "Two-photon diffraction and quantum lithography," Phys. Rev. Lett. 87, 013602 (2001).
[CrossRef] [PubMed]

D. V. Strekalov, A. V. Sergienko, D. N. Klyshko, and Y. H. Shih, "Observation of two-photon ghost interference and diffraction," Phys. Rev. Lett. 74, 3600-3603 (1995).
[CrossRef] [PubMed]

T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, "Optical imaging by means of two-photon quantum entanglement," Phys. Rev. A 52, R3429-R3432 (1995).
[CrossRef] [PubMed]

Soares, O. D.

Strekalov, D. V.

T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, "Optical imaging by means of two-photon quantum entanglement," Phys. Rev. A 52, R3429-R3432 (1995).
[CrossRef] [PubMed]

D. V. Strekalov, A. V. Sergienko, D. N. Klyshko, and Y. H. Shih, "Observation of two-photon ghost interference and diffraction," Phys. Rev. Lett. 74, 3600-3603 (1995).
[CrossRef] [PubMed]

Sun, X.

J. Xiong, D. Cao, F. Huang, H. Li, X. Sun, and K. Wang, "Experimental observation of classical subwavelength interference with a pseudothermal light source" Phys. Rev. Lett. 94, 173601 (2005).
[CrossRef] [PubMed]

Teich, M. C.

Valencia, A.

A. Valencia, G. Scarcelli, M. D. Angelo, and Y. Shih, "Two-photon imaging with thermal light," Phys. Rev. Lett. 94, 063601 (2005).
[CrossRef] [PubMed]

G. Scarcelli, A. Valencia, and Y. H. Shih, "Two-photon interference with thermal light," Europhys. Lett. 68, 618-624 (2004).
[CrossRef]

Wang, K.

D. Cao, J. Xiong, and K. Wang, "Geometrical optics in correlated imaging systems," Phys. Rev. A 71, 013801 (2005).
[CrossRef]

J. Xiong, D. Cao, F. Huang, H. Li, X. Sun, and K. Wang, "Experimental observation of classical subwavelength interference with a pseudothermal light source" Phys. Rev. Lett. 94, 173601 (2005).
[CrossRef] [PubMed]

K. Wang and D. Cao, "Subwavelength coincidence interference with classical thermal light," Phys. Rev. A 70, 041801 (2004).
[CrossRef]

Wei, H. Q.

Williams, C. P.

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, "Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit," Phys. Rev. Lett. 85, 2733-2736 (2000).
[CrossRef] [PubMed]

Wolf, E.

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

Xiong, J.

J. Xiong, D. Cao, F. Huang, H. Li, X. Sun, and K. Wang, "Experimental observation of classical subwavelength interference with a pseudothermal light source" Phys. Rev. Lett. 94, 173601 (2005).
[CrossRef] [PubMed]

D. Cao, J. Xiong, and K. Wang, "Geometrical optics in correlated imaging systems," Phys. Rev. A 71, 013801 (2005).
[CrossRef]

Xue, X.

Yang, G.

Zalevsky, Z.

Zhang, Y.

Zhu, S.

Y. Cai and S. Zhu, "Ghost imaging with incoherent and partially coherent light radiation," Phys. Rev. A 71, 056607 (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 interference with partially coherent radiation," Opt. Lett. 29, 2716-2718 (2004).
[CrossRef] [PubMed]

Zubairy, M. S.

J. T. Foley and M. S. Zubairy, "The directionality of Gaussian Schell-model beams," Opt. Commun. 26, 297-300 (1976).
[CrossRef]

Appl. Opt. (3)

Appl. Phys. Lett. (1)

Y. Cai, Q. Lin, and S. Zhu, "Coincidence fractional Fourier transform with entangled photon pairs and incoherent light," Appl. Phys. Lett. 86, 021112 (2005).
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Europhys. Lett. (1)

G. Scarcelli, A. Valencia, and Y. H. Shih, "Two-photon interference with thermal light," Europhys. Lett. 68, 618-624 (2004).
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IEEE Trans. Signal Process. (1)

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J. Opt. Soc. Am. (1)

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

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Opt. Express (1)

Opt. Lett. (3)

Phys. Rev. A (5)

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

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A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, "Correlated imaging, quantum and classical," Phys. Rev. A 70, 013802 (2004).
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K. Wang and D. Cao, "Subwavelength coincidence interference with classical thermal light," Phys. Rev. A 70, 041801 (2004).
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Phys. Rev. Lett. (8)

D. V. Strekalov, A. V. Sergienko, D. N. Klyshko, and Y. H. Shih, "Observation of two-photon ghost interference and diffraction," Phys. Rev. Lett. 74, 3600-3603 (1995).
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A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, "Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit," Phys. Rev. Lett. 85, 2733-2736 (2000).
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M. D. Angelo, M. V. Chekhova, and Y. H. Shih, "Two-photon diffraction and quantum lithography," Phys. Rev. Lett. 87, 013602 (2001).
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A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, "Ghost imaging with thermal light: comparing entanglement and classical correlation," Phys. Rev. Lett. 93, 093602 (2004).
[CrossRef] [PubMed]

J. Xiong, D. Cao, F. Huang, H. Li, X. Sun, and K. Wang, "Experimental observation of classical subwavelength interference with a pseudothermal light source" Phys. Rev. Lett. 94, 173601 (2005).
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J. Cheng and S. Han, "Incoherent coincidence imaging and its applicability in x-ray diffraction," Phys. Rev. Lett. 92, 093903 (2004).
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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).
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Figures (4)

Fig. 1
Fig. 1

Optical system for implementing the coincidence subwavelength FRT with entangled photon pairs. NLC, nonlinear crystal.

Fig. 2
Fig. 2

Optical system for implementing the coincidence subwavelength FRT with incoherent light radiation.

Fig. 3
Fig. 3

Coincidence subwavelength FRT pattern for a single slit with slit width h = 0.02 mm for different source transverse sizes with σ g = 0.000001 mm , p = 1 . (a) σ I = 1 mm , (b) σ I = 8 mm , (c) σ I = 10 mm , (d) σ I = 12 mm . v, visibility; Q, quality.

Fig. 4
Fig. 4

Coincidence subwavelength FRT for a single slit with slit width h = 0.02 mm for different source transverse coherence widths with σ I = 12 mm , p = 1 . (a) σ g = 0.000001 mm , (b) σ g = 0.00001 mm , (c) σ g = 0.0001 mm , (d) σ g = 0.001 mm .

Equations (33)

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G ( 2 ) ( u 1 , u 2 ) = Ψ E ( u 1 ) E ( u 2 ) E * ( u 2 ) E * ( u 1 ) Ψ = 0 , 0 E ( x 1 ) E ( x 2 ) Ψ 2 = ψ ( u 1 , u 2 ) 2 ,
ψ ( u 1 , u 2 ) = E p ( x ) h 1 ( u 1 , x ) h 2 ( u 2 , x ) d x ,
h 1 ( u 1 , x ) = ( 1 λ 2 z f e sin ϕ ) 1 2 H ( v ) exp [ i π λ z ( x 2 2 x v + v 2 ) ] exp [ i π λ f e tan ϕ ( v 2 + u 1 2 ) + 2 i π λ f e sin ϕ v u 1 ] d v ,
h 2 ( u 2 , x ) = ( i λ b 2 ) 1 2 exp [ i π λ b 2 ( a 2 x 2 2 x u 2 + d 2 u 2 2 ) ] ,
[ a 2 b 2 c 2 d 2 ] = [ 1 l 2 0 1 ] [ 1 0 1 f 1 ] [ 1 l 1 0 1 ] = [ 1 l 2 f l 1 + l 2 l 1 l 2 f 1 f 1 l 1 f ] .
E p ( x ) = exp [ x 2 2 σ I 2 ] ,
ψ ( u 1 , u 2 ) = ( i π λ 3 z f e sin ϕ A 1 b 2 ) 1 2 H ( v ) exp [ π 2 λ 2 A 1 ( v z + u 2 b 2 ) 2 i π λ z v 2 i π d 2 λ b 2 u 2 2 ] exp [ i π λ f e tan ϕ ( v 2 + u 1 2 ) + 2 i π λ f e sin ϕ v u 1 ] d v ,
l 1 = f e tan ϕ 2 z , l 2 = f e tan ϕ 2 , f = f e sin ϕ ,
ψ ( u 1 , u 2 ) = 1 λ f e sin ϕ H ( v ) exp [ i π λ f e tan ϕ ( v 2 + u 1 2 ) + 2 i π λ f e sin ϕ v u 1 ] exp [ i π λ f e tan ϕ ( v 2 + u 2 2 ) + 2 i π λ f e sin ϕ v u 2 ] d v .
ψ ( u 1 , u 1 ) = 1 λ f e sin ϕ H ( v ) exp [ 2 i π λ f e tan ϕ ( v 2 + u 1 2 ) + 4 i π λ f e sin ϕ v u 1 ] d v .
E f ( x 2 ) = E 0 ( x 1 ) exp [ i π λ f e tan ϕ ( x 1 2 + x 2 2 ) + 2 i π λ f e sin ϕ x 1 x 2 ] d x 1 ,
G ( 2 ) ( u 1 , u 2 ) = E ( u 1 ) E ( u 2 ) E * ( u 2 ) E * ( u 1 ) = I ( u 1 ) I ( u 2 ) + Γ ( u 1 , u 2 ) 2 ,
I ( u i ) = h i ( x 1 , u i ) h i * ( x 2 , u i ) E s ( x 1 ) E s * ( x 2 ) d x 1 d x 2 , i = 1 , 2 ,
Γ ( u 1 , u 2 ) = E s ( x 1 ) E s * ( x 2 ) h 1 ( x 1 , u 1 ) h 2 * ( x 2 , u 2 ) d x 1 d x 2 ,
E s ( x 1 ) E s * ( x 2 ) = I ( x 1 ) δ ( x 1 x 2 ) ,
h 1 ( u 1 , x 1 ) = ( 1 λ 2 b 1 f e sin ϕ ) 1 2 exp [ i π λ b 1 ( a 1 x 1 2 2 x 1 v + d 1 v 2 ) ] H ( v ) exp [ i π λ f e tan ϕ ( v 2 + u 1 2 ) + 2 i π λ f e sin ϕ v u 1 ] d v ,
h 2 ( u 2 , x 2 ) = ( i λ z ) 1 2 exp [ i π λ z ( x 2 2 2 x 2 u 2 + u 2 2 ) ] ,
[ a 1 b 1 c 1 d 1 ] = [ 1 l 2 0 1 ] [ 1 0 1 f 1 ] [ 1 l 1 0 1 ] = [ 1 l 2 f l 1 + l 2 l 1 l 2 f 1 f 1 l 1 f ] .
I ( u 1 ) I ( u 2 ) = ,
Γ ( u 1 , u 2 ) = I 0 ( i π λ 3 z f e b 1 A 1 sin ϕ ) 1 2 H ( v 1 ) exp [ i d 1 π λ b 1 v 2 + i π λ z u 2 2 ] exp [ i π b 1 z λ ( a 1 z b 1 ) ( v b 1 u 2 z ) 2 ] exp [ i π λ f e tan ϕ ( v 2 + u 1 2 ) + 2 i π λ f e sin ϕ v u 1 ] d x 1 d v 1 ,
A 1 = a 1 b 1 1 z .
l 1 = z + f e tan ϕ 2 , l 2 = f e tan ϕ 2 , f = f e sin ϕ ,
Γ ( u 1 , u 2 ) = 1 λ f e sin ϕ H ( v ) exp [ i π λ f e tan ϕ ( v 2 + u 2 2 ) 2 i π λ f e sin ϕ v u 2 ] exp [ i π λ f e tan ϕ ( v 2 + u 1 2 ) + 2 i π λ f e sin ϕ v u 1 ] d v .
Γ ( u 1 , u 1 ) = 1 λ f e sin ϕ H ( v ) exp [ 2 i π λ f e tan ϕ ( v 2 + u 1 2 ) + 4 i π λ f e sin ϕ v u 1 ] d v .
V = Γ ( u 1 , u 1 ) 2 max G ( 2 ) ( u 1 , u 1 ) max .
E s ( x 1 ) E s * ( x 2 ) = G 0 exp [ x 1 2 + x 2 2 4 σ I 2 ( x 1 x 2 ) 2 2 σ g 2 ] ,
I ( u 1 ) = π λ 2 b 1 f e sin ϕ A 1 A 2 H ( v 1 ) H * ( v 2 ) exp [ ( 2 i π λ b 1 v 2 i π A 1 σ g 2 λ b 1 v 1 ) 2 4 A 2 ] exp [ π 2 A 1 λ 2 b 1 2 v 2 2 ] exp [ i π λ f e tan ϕ ( v 1 2 + u 1 2 ) + 2 i π λ f e sin ϕ v 1 u 1 ] exp [ i π λ f e tan ϕ ( v 2 2 + u 1 2 ) 2 i π λ f e sin ϕ v 2 u 1 ] exp [ i π d 1 λ b 1 v 1 2 + i π d 1 λ b 1 v 2 2 ] d v 1 d v 2 ,
I ( u 2 ) = π λ z B 1 B 2 exp [ ( 2 i π λ z u 2 i π B 1 σ g 2 λ z u 2 ) 2 4 B 2 ] exp [ π 2 B 1 λ 2 z 2 u 2 2 ] ,
Γ ( u 1 , u 2 ) 2 = π 2 λ 3 z f e sin ϕ b 1 C 1 C 2 H ( v 1 ) exp [ 1 4 C 2 ( 2 i π λ z u 2 i π C 1 σ g 2 λ b 1 v 1 ) 2 ] exp [ π 2 C 1 λ 2 b 1 2 v 1 2 ] exp [ i π d 1 λ b 1 v 1 2 ] exp [ i π λ f e tan ϕ ( v 1 2 + u 1 2 ) + 2 i π λ f e sin ϕ v 1 u 1 ] d v 1 2 ,
A 1 = 1 4 σ I 2 + 1 2 σ g 2 + i π a 1 λ b 1 , A 2 = 1 4 σ I 2 + 1 2 σ g 2 i π a 1 λ b 1 1 4 A 1 σ g 4 ,
B 1 = 1 4 σ I 2 + 1 2 σ g 2 i π λ z , B 2 = 1 4 σ I 2 + 1 2 σ g 2 + i π λ z 1 4 B 1 σ g 4 ,
C 1 = 1 4 σ I 2 + 1 2 σ g 2 + i π a 1 λ b 1 , C 2 = 1 4 σ I 2 + 1 2 σ g 2 i π λ z 1 4 C 1 σ g 4 ,
Q = Γ ( u 1 , u 1 ) 2 Γ ( u 1 , u 1 ) max 2 Γ ( u 1 , u 1 ) σ g = 0 , σ I = 2 Γ ( u 1 , u 1 ) σ g = 0 , σ I = max 2 d u 1 Γ ( u 1 , u 1 ) σ g = 0 , σ I = 2 Γ ( u 1 , u 1 ) σ g = 0 , σ I = max 2 d u 1 . .

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