Abstract

Entangled photons, generated by spontaneous parametric downconversion from a second-order nonlinear crystal, present a rich potential for imaging and image-processing applications. Since this source is an example of a three-wave mixing process, there is more flexibility in the choices of illumination and detection wavelengths and in the placement of object(s) to be imaged. Moreover, this source is entangled, a fact that allows for imaging configurations and capabilities that cannot be achieved by use of classical sources of light. We examine a number of imaging and image-processing configurations that can be realized with this source. The formalism that we utilize facilitates the determination of the dependence of imaging resolution on the physical parameters of the optical arrangement.

© 2002 Optical Society of America

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2002 (1)

M. B. Nasr, A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Biphoton focusing for two-photon excitation,” Phys. Rev. A 65, 023816 (2002).
[CrossRef]

2001 (3)

A. F. Abouraddy, M. B. Nasr, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Demonstration of the complementarity of one- and two-photon interference,” Phys. Rev. A 63, 063803 (2001).
[CrossRef]

A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Role of entanglement in two-photon imaging,” Phys. Rev. Lett. 87, 123602 (2001).
[CrossRef] [PubMed]

A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Double-slit interference of biphotons generated in spontaneous parametric downconversion from a thick crystal,” J. Opt. B: Quantum Semiclass. Opt. 3, S50–S54 (2001).
[CrossRef]

2000 (2)

M. I. Kolobov and C. Fabre, “Quantum limits on optical resolution,” Phys. Rev. Lett. 85, 3789–3792 (2000).
[CrossRef] [PubMed]

B. E. A. Saleh, A. F. Abouraddy, A. V. Sergienko, and M. C. Teich, “Duality between partial coherence and partial entanglement,” Phys. Rev. A 62, 043816 (2000).
[CrossRef]

1999 (5)

M. I. Kolobov, “The spatial behavior of nonclassical light,” Rev. Mod. Phys. 71, 1539–1589 (1999).
[CrossRef]

A. Gatti, E. Brambilla, L. A. Lugiato, and M. I. Kolobov, “Quantum entangled images,” Phys. Rev. Lett. 83, 1763–1766 (1999).
[CrossRef]

A. Beranskis, W. Chinaglia, L. A. Lugiato, K.-H. Feller, and P. Di Trapani, “Spatial structures in optical parametric amplification,” Phys. Rev. A 60, 1626–1635 (1999).
[CrossRef]

S.-K. Choi, M. Vasilyev, and P. Kumar, “Noiseless optical amplification of images,” Phys. Rev. Lett. 83, 1938–1941 (1999).
[CrossRef]

I. V. Sokolov, M. I. Kolobov, and L. A. Lugiato, “Quantum fluctuations in traveling-wave amplification of optical images,” Phys. Rev. A 60, 2420–2430 (1999).
[CrossRef]

1998 (2)

C. H. Monken, P. H. S. Ribeiro, and S. Pádua, “Transfer of angular spectrum and image formation in spontaneous parametric down-conversion,” Phys. Rev. A 57, 3123–3126 (1998).
[CrossRef]

B. M. Jost, A. V. Sergienko, A. F. Abouraddy, B. E. A. Saleh, and M. C. Teich, “Spatial correlations of spontaneously down-converted photon pairs detected with a single-photon-sensitive CCD camera,” Opt. Express 3, 81–88 (1998).
[CrossRef] [PubMed]

1997 (2)

M. C. Teich and B. E. A. Saleh, “Entangled-photon microscopy,” Cesk. Cas. Fyz. 47, 3–8 (1997).

A. V. Burlakov, M. V. Chekhova, D. N. Klyshko, S. P. Kulik, A. N. Penin, Y. H. Shih, and D. V. Strekalov, “Interference effects in spontaneous two-photon parametric scattering from two macroscopic regions,” Phys. Rev. A 56, 3214–3225 (1997).
[CrossRef]

1996 (2)

A. Joobeur, B. E. A. Saleh, T. S. Larchuk, and M. C. Teich, “Coherence properties of entangled light beams generated by parametric downconversion: theory and experiment,” Phys. Rev. A 53, 4360–4371 (1996).
[CrossRef] [PubMed]

T. B. Pittman, D. V. Strekalov, D. N. Klyshko, M. H. Rubin, A. V. Sergienko, and Y. H. Shih, “Two-photon geometric optics,” Phys. Rev. A 53, 2804–2815 (1996).
[CrossRef] [PubMed]

1995 (2)

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 (3)

P. H. S. Ribeiro, S. Pádua, J. C. Machado da Silva, and G. A. Barbosa, “Controlling the degree of visibility of Young’s fringes with photon coincidence measurements,” Phys. Rev. A 49, 4176–4179 (1994).
[CrossRef] [PubMed]

A. Joobeur, B. E. A. Saleh, and M. C. Teich, “Spatiotemporal coherence properties of entangled light beams generated by parametric down-conversion,” Phys. Rev. A 50, 3349–3361 (1994).
[CrossRef] [PubMed]

A. V. Belinskii and D. N. Klyshko, “Two-photon optics: diffraction, holography, and transformation of two-dimensional signals,” Zh. Eksp. Teor. Fiz. 105, 487–493 (1994) [Sov. Phys. Dokl. 78, 259–262 (1994)].

1988 (1)

D. N. Klyshko, “Effect of focusing on photon correlation in parametric light scattering,” Zh. Eksp. Teor. Fiz. 94, 82–90 (1988) [Sov. Phys. JETP 67, 1131–1135 (1988)].

1985 (1)

A. A. Malygin, A. N. Penin, and A. V. Sergienko, “Spatiotemporal grouping of photons in spontaneous parametric scattering of light,”Dokl. Akad. Nauk SSSR 281, 308–313 (1985) [Sov. Phys. Dokl. 30, 227–229 (1985)].

1982 (1)

D. N. Klyshko, “Transverse photon bunching and two-photon processes in the field of parametrically scattered light,” Zh. Eksp. Teor. Fiz. 83, 1313–1323 (1982) [Sov. Phys. JETP 56, 753–758 (1982)].

1979 (1)

B. E. A. Saleh, “Optical bilinear transformations: general properties,” Opt. Acta 26, 777–799 (1979).
[CrossRef]

1963 (2)

R. J. Glauber, “The quantum theory of optical coherence,” Phys. Rev. 130, 2529–2539 (1963).
[CrossRef]

S. A. Akhmanov, A. I. Kovrigin, R. V. Khokhlov, and O. N. Chunaev, Zh. Eksp. Teor. Fiz. 45, 1336 (1963).

1962 (3)

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions among light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

D. A. Kleinman, “Nonlinear dielectric polarization in optical media,” Phys. Rev. 126, 1977–1979 (1962).
[CrossRef]

R. H. Kingston, “Parametric amplification and oscillation of optical frequencies,” Proc. IRE 50, 472 (1962).

Abouraddy, A. F.

M. B. Nasr, A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Biphoton focusing for two-photon excitation,” Phys. Rev. A 65, 023816 (2002).
[CrossRef]

A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Double-slit interference of biphotons generated in spontaneous parametric downconversion from a thick crystal,” J. Opt. B: Quantum Semiclass. Opt. 3, S50–S54 (2001).
[CrossRef]

A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Role of entanglement in two-photon imaging,” Phys. Rev. Lett. 87, 123602 (2001).
[CrossRef] [PubMed]

A. F. Abouraddy, M. B. Nasr, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Demonstration of the complementarity of one- and two-photon interference,” Phys. Rev. A 63, 063803 (2001).
[CrossRef]

B. E. A. Saleh, A. F. Abouraddy, A. V. Sergienko, and M. C. Teich, “Duality between partial coherence and partial entanglement,” Phys. Rev. A 62, 043816 (2000).
[CrossRef]

B. M. Jost, A. V. Sergienko, A. F. Abouraddy, B. E. A. Saleh, and M. C. Teich, “Spatial correlations of spontaneously down-converted photon pairs detected with a single-photon-sensitive CCD camera,” Opt. Express 3, 81–88 (1998).
[CrossRef] [PubMed]

Akhmanov, S. A.

S. A. Akhmanov, A. I. Kovrigin, R. V. Khokhlov, and O. N. Chunaev, Zh. Eksp. Teor. Fiz. 45, 1336 (1963).

Armstrong, J. A.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions among light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Barbosa, G. A.

P. H. S. Ribeiro, S. Pádua, J. C. Machado da Silva, and G. A. Barbosa, “Controlling the degree of visibility of Young’s fringes with photon coincidence measurements,” Phys. Rev. A 49, 4176–4179 (1994).
[CrossRef] [PubMed]

Belinskii, A. V.

A. V. Belinskii and D. N. Klyshko, “Two-photon optics: diffraction, holography, and transformation of two-dimensional signals,” Zh. Eksp. Teor. Fiz. 105, 487–493 (1994) [Sov. Phys. Dokl. 78, 259–262 (1994)].

Beranskis, A.

A. Beranskis, W. Chinaglia, L. A. Lugiato, K.-H. Feller, and P. Di Trapani, “Spatial structures in optical parametric amplification,” Phys. Rev. A 60, 1626–1635 (1999).
[CrossRef]

Bloembergen, N.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions among light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Brambilla, E.

A. Gatti, E. Brambilla, L. A. Lugiato, and M. I. Kolobov, “Quantum entangled images,” Phys. Rev. Lett. 83, 1763–1766 (1999).
[CrossRef]

Burlakov, A. V.

A. V. Burlakov, M. V. Chekhova, D. N. Klyshko, S. P. Kulik, A. N. Penin, Y. H. Shih, and D. V. Strekalov, “Interference effects in spontaneous two-photon parametric scattering from two macroscopic regions,” Phys. Rev. A 56, 3214–3225 (1997).
[CrossRef]

Chekhova, M. V.

A. V. Burlakov, M. V. Chekhova, D. N. Klyshko, S. P. Kulik, A. N. Penin, Y. H. Shih, and D. V. Strekalov, “Interference effects in spontaneous two-photon parametric scattering from two macroscopic regions,” Phys. Rev. A 56, 3214–3225 (1997).
[CrossRef]

Chinaglia, W.

A. Beranskis, W. Chinaglia, L. A. Lugiato, K.-H. Feller, and P. Di Trapani, “Spatial structures in optical parametric amplification,” Phys. Rev. A 60, 1626–1635 (1999).
[CrossRef]

Choi, S.-K.

S.-K. Choi, M. Vasilyev, and P. Kumar, “Noiseless optical amplification of images,” Phys. Rev. Lett. 83, 1938–1941 (1999).
[CrossRef]

Chunaev, O. N.

S. A. Akhmanov, A. I. Kovrigin, R. V. Khokhlov, and O. N. Chunaev, Zh. Eksp. Teor. Fiz. 45, 1336 (1963).

Di Trapani, P.

A. Beranskis, W. Chinaglia, L. A. Lugiato, K.-H. Feller, and P. Di Trapani, “Spatial structures in optical parametric amplification,” Phys. Rev. A 60, 1626–1635 (1999).
[CrossRef]

Ducuing, J.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions among light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Fabre, C.

M. I. Kolobov and C. Fabre, “Quantum limits on optical resolution,” Phys. Rev. Lett. 85, 3789–3792 (2000).
[CrossRef] [PubMed]

Feller, K.-H.

A. Beranskis, W. Chinaglia, L. A. Lugiato, K.-H. Feller, and P. Di Trapani, “Spatial structures in optical parametric amplification,” Phys. Rev. A 60, 1626–1635 (1999).
[CrossRef]

Gatti, A.

A. Gatti, E. Brambilla, L. A. Lugiato, and M. I. Kolobov, “Quantum entangled images,” Phys. Rev. Lett. 83, 1763–1766 (1999).
[CrossRef]

Glauber, R. J.

R. J. Glauber, “The quantum theory of optical coherence,” Phys. Rev. 130, 2529–2539 (1963).
[CrossRef]

Joobeur, A.

A. Joobeur, B. E. A. Saleh, T. S. Larchuk, and M. C. Teich, “Coherence properties of entangled light beams generated by parametric downconversion: theory and experiment,” Phys. Rev. A 53, 4360–4371 (1996).
[CrossRef] [PubMed]

A. Joobeur, B. E. A. Saleh, and M. C. Teich, “Spatiotemporal coherence properties of entangled light beams generated by parametric down-conversion,” Phys. Rev. A 50, 3349–3361 (1994).
[CrossRef] [PubMed]

Jost, B. M.

Khokhlov, R. V.

S. A. Akhmanov, A. I. Kovrigin, R. V. Khokhlov, and O. N. Chunaev, Zh. Eksp. Teor. Fiz. 45, 1336 (1963).

Kingston, R. H.

R. H. Kingston, “Parametric amplification and oscillation of optical frequencies,” Proc. IRE 50, 472 (1962).

Kleinman, D. A.

D. A. Kleinman, “Nonlinear dielectric polarization in optical media,” Phys. Rev. 126, 1977–1979 (1962).
[CrossRef]

Klyshko, D. N.

A. V. Burlakov, M. V. Chekhova, D. N. Klyshko, S. P. Kulik, A. N. Penin, Y. H. Shih, and D. V. Strekalov, “Interference effects in spontaneous two-photon parametric scattering from two macroscopic regions,” Phys. Rev. A 56, 3214–3225 (1997).
[CrossRef]

T. B. Pittman, D. V. Strekalov, D. N. Klyshko, M. H. Rubin, A. V. Sergienko, and Y. H. Shih, “Two-photon geometric optics,” Phys. Rev. A 53, 2804–2815 (1996).
[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]

A. V. Belinskii and D. N. Klyshko, “Two-photon optics: diffraction, holography, and transformation of two-dimensional signals,” Zh. Eksp. Teor. Fiz. 105, 487–493 (1994) [Sov. Phys. Dokl. 78, 259–262 (1994)].

D. N. Klyshko, “Effect of focusing on photon correlation in parametric light scattering,” Zh. Eksp. Teor. Fiz. 94, 82–90 (1988) [Sov. Phys. JETP 67, 1131–1135 (1988)].

D. N. Klyshko, “Transverse photon bunching and two-photon processes in the field of parametrically scattered light,” Zh. Eksp. Teor. Fiz. 83, 1313–1323 (1982) [Sov. Phys. JETP 56, 753–758 (1982)].

Kolobov, M. I.

M. I. Kolobov and C. Fabre, “Quantum limits on optical resolution,” Phys. Rev. Lett. 85, 3789–3792 (2000).
[CrossRef] [PubMed]

M. I. Kolobov, “The spatial behavior of nonclassical light,” Rev. Mod. Phys. 71, 1539–1589 (1999).
[CrossRef]

I. V. Sokolov, M. I. Kolobov, and L. A. Lugiato, “Quantum fluctuations in traveling-wave amplification of optical images,” Phys. Rev. A 60, 2420–2430 (1999).
[CrossRef]

A. Gatti, E. Brambilla, L. A. Lugiato, and M. I. Kolobov, “Quantum entangled images,” Phys. Rev. Lett. 83, 1763–1766 (1999).
[CrossRef]

Kovrigin, A. I.

S. A. Akhmanov, A. I. Kovrigin, R. V. Khokhlov, and O. N. Chunaev, Zh. Eksp. Teor. Fiz. 45, 1336 (1963).

Kulik, S. P.

A. V. Burlakov, M. V. Chekhova, D. N. Klyshko, S. P. Kulik, A. N. Penin, Y. H. Shih, and D. V. Strekalov, “Interference effects in spontaneous two-photon parametric scattering from two macroscopic regions,” Phys. Rev. A 56, 3214–3225 (1997).
[CrossRef]

Kumar, P.

S.-K. Choi, M. Vasilyev, and P. Kumar, “Noiseless optical amplification of images,” Phys. Rev. Lett. 83, 1938–1941 (1999).
[CrossRef]

Larchuk, T. S.

A. Joobeur, B. E. A. Saleh, T. S. Larchuk, and M. C. Teich, “Coherence properties of entangled light beams generated by parametric downconversion: theory and experiment,” Phys. Rev. A 53, 4360–4371 (1996).
[CrossRef] [PubMed]

Lugiato, L. A.

I. V. Sokolov, M. I. Kolobov, and L. A. Lugiato, “Quantum fluctuations in traveling-wave amplification of optical images,” Phys. Rev. A 60, 2420–2430 (1999).
[CrossRef]

A. Gatti, E. Brambilla, L. A. Lugiato, and M. I. Kolobov, “Quantum entangled images,” Phys. Rev. Lett. 83, 1763–1766 (1999).
[CrossRef]

A. Beranskis, W. Chinaglia, L. A. Lugiato, K.-H. Feller, and P. Di Trapani, “Spatial structures in optical parametric amplification,” Phys. Rev. A 60, 1626–1635 (1999).
[CrossRef]

Machado da Silva, J. C.

P. H. S. Ribeiro, S. Pádua, J. C. Machado da Silva, and G. A. Barbosa, “Controlling the degree of visibility of Young’s fringes with photon coincidence measurements,” Phys. Rev. A 49, 4176–4179 (1994).
[CrossRef] [PubMed]

Malygin, A. A.

A. A. Malygin, A. N. Penin, and A. V. Sergienko, “Spatiotemporal grouping of photons in spontaneous parametric scattering of light,”Dokl. Akad. Nauk SSSR 281, 308–313 (1985) [Sov. Phys. Dokl. 30, 227–229 (1985)].

Monken, C. H.

C. H. Monken, P. H. S. Ribeiro, and S. Pádua, “Transfer of angular spectrum and image formation in spontaneous parametric down-conversion,” Phys. Rev. A 57, 3123–3126 (1998).
[CrossRef]

Nasr, M. B.

M. B. Nasr, A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Biphoton focusing for two-photon excitation,” Phys. Rev. A 65, 023816 (2002).
[CrossRef]

A. F. Abouraddy, M. B. Nasr, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Demonstration of the complementarity of one- and two-photon interference,” Phys. Rev. A 63, 063803 (2001).
[CrossRef]

Pádua, S.

C. H. Monken, P. H. S. Ribeiro, and S. Pádua, “Transfer of angular spectrum and image formation in spontaneous parametric down-conversion,” Phys. Rev. A 57, 3123–3126 (1998).
[CrossRef]

P. H. S. Ribeiro, S. Pádua, J. C. Machado da Silva, and G. A. Barbosa, “Controlling the degree of visibility of Young’s fringes with photon coincidence measurements,” Phys. Rev. A 49, 4176–4179 (1994).
[CrossRef] [PubMed]

Penin, A. N.

A. V. Burlakov, M. V. Chekhova, D. N. Klyshko, S. P. Kulik, A. N. Penin, Y. H. Shih, and D. V. Strekalov, “Interference effects in spontaneous two-photon parametric scattering from two macroscopic regions,” Phys. Rev. A 56, 3214–3225 (1997).
[CrossRef]

A. A. Malygin, A. N. Penin, and A. V. Sergienko, “Spatiotemporal grouping of photons in spontaneous parametric scattering of light,”Dokl. Akad. Nauk SSSR 281, 308–313 (1985) [Sov. Phys. Dokl. 30, 227–229 (1985)].

Pershan, P. S.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions among light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Pittman, T. B.

T. B. Pittman, D. V. Strekalov, D. N. Klyshko, M. H. Rubin, A. V. Sergienko, and Y. H. Shih, “Two-photon geometric optics,” Phys. Rev. A 53, 2804–2815 (1996).
[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]

Ribeiro, P. H. S.

C. H. Monken, P. H. S. Ribeiro, and S. Pádua, “Transfer of angular spectrum and image formation in spontaneous parametric down-conversion,” Phys. Rev. A 57, 3123–3126 (1998).
[CrossRef]

P. H. S. Ribeiro, S. Pádua, J. C. Machado da Silva, and G. A. Barbosa, “Controlling the degree of visibility of Young’s fringes with photon coincidence measurements,” Phys. Rev. A 49, 4176–4179 (1994).
[CrossRef] [PubMed]

Rubin, M. H.

T. B. Pittman, D. V. Strekalov, D. N. Klyshko, M. H. Rubin, A. V. Sergienko, and Y. H. Shih, “Two-photon geometric optics,” Phys. Rev. A 53, 2804–2815 (1996).
[CrossRef] [PubMed]

Saleh, B. E. A.

M. B. Nasr, A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Biphoton focusing for two-photon excitation,” Phys. Rev. A 65, 023816 (2002).
[CrossRef]

A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Double-slit interference of biphotons generated in spontaneous parametric downconversion from a thick crystal,” J. Opt. B: Quantum Semiclass. Opt. 3, S50–S54 (2001).
[CrossRef]

A. F. Abouraddy, M. B. Nasr, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Demonstration of the complementarity of one- and two-photon interference,” Phys. Rev. A 63, 063803 (2001).
[CrossRef]

A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Role of entanglement in two-photon imaging,” Phys. Rev. Lett. 87, 123602 (2001).
[CrossRef] [PubMed]

B. E. A. Saleh, A. F. Abouraddy, A. V. Sergienko, and M. C. Teich, “Duality between partial coherence and partial entanglement,” Phys. Rev. A 62, 043816 (2000).
[CrossRef]

B. M. Jost, A. V. Sergienko, A. F. Abouraddy, B. E. A. Saleh, and M. C. Teich, “Spatial correlations of spontaneously down-converted photon pairs detected with a single-photon-sensitive CCD camera,” Opt. Express 3, 81–88 (1998).
[CrossRef] [PubMed]

M. C. Teich and B. E. A. Saleh, “Entangled-photon microscopy,” Cesk. Cas. Fyz. 47, 3–8 (1997).

A. Joobeur, B. E. A. Saleh, T. S. Larchuk, and M. C. Teich, “Coherence properties of entangled light beams generated by parametric downconversion: theory and experiment,” Phys. Rev. A 53, 4360–4371 (1996).
[CrossRef] [PubMed]

A. Joobeur, B. E. A. Saleh, and M. C. Teich, “Spatiotemporal coherence properties of entangled light beams generated by parametric down-conversion,” Phys. Rev. A 50, 3349–3361 (1994).
[CrossRef] [PubMed]

B. E. A. Saleh, “Optical bilinear transformations: general properties,” Opt. Acta 26, 777–799 (1979).
[CrossRef]

Sergienko, A. V.

M. B. Nasr, A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Biphoton focusing for two-photon excitation,” Phys. Rev. A 65, 023816 (2002).
[CrossRef]

A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Double-slit interference of biphotons generated in spontaneous parametric downconversion from a thick crystal,” J. Opt. B: Quantum Semiclass. Opt. 3, S50–S54 (2001).
[CrossRef]

A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Role of entanglement in two-photon imaging,” Phys. Rev. Lett. 87, 123602 (2001).
[CrossRef] [PubMed]

A. F. Abouraddy, M. B. Nasr, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Demonstration of the complementarity of one- and two-photon interference,” Phys. Rev. A 63, 063803 (2001).
[CrossRef]

B. E. A. Saleh, A. F. Abouraddy, A. V. Sergienko, and M. C. Teich, “Duality between partial coherence and partial entanglement,” Phys. Rev. A 62, 043816 (2000).
[CrossRef]

B. M. Jost, A. V. Sergienko, A. F. Abouraddy, B. E. A. Saleh, and M. C. Teich, “Spatial correlations of spontaneously down-converted photon pairs detected with a single-photon-sensitive CCD camera,” Opt. Express 3, 81–88 (1998).
[CrossRef] [PubMed]

T. B. Pittman, D. V. Strekalov, D. N. Klyshko, M. H. Rubin, A. V. Sergienko, and Y. H. Shih, “Two-photon geometric optics,” Phys. Rev. A 53, 2804–2815 (1996).
[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]

A. A. Malygin, A. N. Penin, and A. V. Sergienko, “Spatiotemporal grouping of photons in spontaneous parametric scattering of light,”Dokl. Akad. Nauk SSSR 281, 308–313 (1985) [Sov. Phys. Dokl. 30, 227–229 (1985)].

Shih, Y. H.

A. V. Burlakov, M. V. Chekhova, D. N. Klyshko, S. P. Kulik, A. N. Penin, Y. H. Shih, and D. V. Strekalov, “Interference effects in spontaneous two-photon parametric scattering from two macroscopic regions,” Phys. Rev. A 56, 3214–3225 (1997).
[CrossRef]

T. B. Pittman, D. V. Strekalov, D. N. Klyshko, M. H. Rubin, A. V. Sergienko, and Y. H. Shih, “Two-photon geometric optics,” Phys. Rev. A 53, 2804–2815 (1996).
[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]

Sokolov, I. V.

I. V. Sokolov, M. I. Kolobov, and L. A. Lugiato, “Quantum fluctuations in traveling-wave amplification of optical images,” Phys. Rev. A 60, 2420–2430 (1999).
[CrossRef]

Strekalov, D. V.

A. V. Burlakov, M. V. Chekhova, D. N. Klyshko, S. P. Kulik, A. N. Penin, Y. H. Shih, and D. V. Strekalov, “Interference effects in spontaneous two-photon parametric scattering from two macroscopic regions,” Phys. Rev. A 56, 3214–3225 (1997).
[CrossRef]

T. B. Pittman, D. V. Strekalov, D. N. Klyshko, M. H. Rubin, A. V. Sergienko, and Y. H. Shih, “Two-photon geometric optics,” Phys. Rev. A 53, 2804–2815 (1996).
[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]

Teich, M. C.

M. B. Nasr, A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Biphoton focusing for two-photon excitation,” Phys. Rev. A 65, 023816 (2002).
[CrossRef]

A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Double-slit interference of biphotons generated in spontaneous parametric downconversion from a thick crystal,” J. Opt. B: Quantum Semiclass. Opt. 3, S50–S54 (2001).
[CrossRef]

A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Role of entanglement in two-photon imaging,” Phys. Rev. Lett. 87, 123602 (2001).
[CrossRef] [PubMed]

A. F. Abouraddy, M. B. Nasr, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Demonstration of the complementarity of one- and two-photon interference,” Phys. Rev. A 63, 063803 (2001).
[CrossRef]

B. E. A. Saleh, A. F. Abouraddy, A. V. Sergienko, and M. C. Teich, “Duality between partial coherence and partial entanglement,” Phys. Rev. A 62, 043816 (2000).
[CrossRef]

B. M. Jost, A. V. Sergienko, A. F. Abouraddy, B. E. A. Saleh, and M. C. Teich, “Spatial correlations of spontaneously down-converted photon pairs detected with a single-photon-sensitive CCD camera,” Opt. Express 3, 81–88 (1998).
[CrossRef] [PubMed]

M. C. Teich and B. E. A. Saleh, “Entangled-photon microscopy,” Cesk. Cas. Fyz. 47, 3–8 (1997).

A. Joobeur, B. E. A. Saleh, T. S. Larchuk, and M. C. Teich, “Coherence properties of entangled light beams generated by parametric downconversion: theory and experiment,” Phys. Rev. A 53, 4360–4371 (1996).
[CrossRef] [PubMed]

A. Joobeur, B. E. A. Saleh, and M. C. Teich, “Spatiotemporal coherence properties of entangled light beams generated by parametric down-conversion,” Phys. Rev. A 50, 3349–3361 (1994).
[CrossRef] [PubMed]

Vasilyev, M.

S.-K. Choi, M. Vasilyev, and P. Kumar, “Noiseless optical amplification of images,” Phys. Rev. Lett. 83, 1938–1941 (1999).
[CrossRef]

Cesk. Cas. Fyz. (1)

M. C. Teich and B. E. A. Saleh, “Entangled-photon microscopy,” Cesk. Cas. Fyz. 47, 3–8 (1997).

J. Opt. B: Quantum Semiclass. Opt. (1)

A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Double-slit interference of biphotons generated in spontaneous parametric downconversion from a thick crystal,” J. Opt. B: Quantum Semiclass. Opt. 3, S50–S54 (2001).
[CrossRef]

Opt. Acta (1)

B. E. A. Saleh, “Optical bilinear transformations: general properties,” Opt. Acta 26, 777–799 (1979).
[CrossRef]

Opt. Express (1)

Phys. Rev. (3)

R. J. Glauber, “The quantum theory of optical coherence,” Phys. Rev. 130, 2529–2539 (1963).
[CrossRef]

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions among light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

D. A. Kleinman, “Nonlinear dielectric polarization in optical media,” Phys. Rev. 126, 1977–1979 (1962).
[CrossRef]

Phys. Rev. A (12)

A. Beranskis, W. Chinaglia, L. A. Lugiato, K.-H. Feller, and P. Di Trapani, “Spatial structures in optical parametric amplification,” Phys. Rev. A 60, 1626–1635 (1999).
[CrossRef]

I. V. Sokolov, M. I. Kolobov, and L. A. Lugiato, “Quantum fluctuations in traveling-wave amplification of optical images,” Phys. Rev. A 60, 2420–2430 (1999).
[CrossRef]

A. F. Abouraddy, M. B. Nasr, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Demonstration of the complementarity of one- and two-photon interference,” Phys. Rev. A 63, 063803 (2001).
[CrossRef]

P. H. S. Ribeiro, S. Pádua, J. C. Machado da Silva, and G. A. Barbosa, “Controlling the degree of visibility of Young’s fringes with photon coincidence measurements,” Phys. Rev. A 49, 4176–4179 (1994).
[CrossRef] [PubMed]

A. Joobeur, B. E. A. Saleh, and M. C. Teich, “Spatiotemporal coherence properties of entangled light beams generated by parametric down-conversion,” Phys. Rev. A 50, 3349–3361 (1994).
[CrossRef] [PubMed]

A. Joobeur, B. E. A. Saleh, T. S. Larchuk, and M. C. Teich, “Coherence properties of entangled light beams generated by parametric downconversion: theory and experiment,” Phys. Rev. A 53, 4360–4371 (1996).
[CrossRef] [PubMed]

A. V. Burlakov, M. V. Chekhova, D. N. Klyshko, S. P. Kulik, A. N. Penin, Y. H. Shih, and D. V. Strekalov, “Interference effects in spontaneous two-photon parametric scattering from two macroscopic regions,” Phys. Rev. A 56, 3214–3225 (1997).
[CrossRef]

C. H. Monken, P. H. S. Ribeiro, and S. Pádua, “Transfer of angular spectrum and image formation in spontaneous parametric down-conversion,” Phys. Rev. A 57, 3123–3126 (1998).
[CrossRef]

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]

T. B. Pittman, D. V. Strekalov, D. N. Klyshko, M. H. Rubin, A. V. Sergienko, and Y. H. Shih, “Two-photon geometric optics,” Phys. Rev. A 53, 2804–2815 (1996).
[CrossRef] [PubMed]

B. E. A. Saleh, A. F. Abouraddy, A. V. Sergienko, and M. C. Teich, “Duality between partial coherence and partial entanglement,” Phys. Rev. A 62, 043816 (2000).
[CrossRef]

M. B. Nasr, A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Biphoton focusing for two-photon excitation,” Phys. Rev. A 65, 023816 (2002).
[CrossRef]

Phys. Rev. Lett. (5)

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]

M. I. Kolobov and C. Fabre, “Quantum limits on optical resolution,” Phys. Rev. Lett. 85, 3789–3792 (2000).
[CrossRef] [PubMed]

A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Role of entanglement in two-photon imaging,” Phys. Rev. Lett. 87, 123602 (2001).
[CrossRef] [PubMed]

A. Gatti, E. Brambilla, L. A. Lugiato, and M. I. Kolobov, “Quantum entangled images,” Phys. Rev. Lett. 83, 1763–1766 (1999).
[CrossRef]

S.-K. Choi, M. Vasilyev, and P. Kumar, “Noiseless optical amplification of images,” Phys. Rev. Lett. 83, 1938–1941 (1999).
[CrossRef]

Proc. IRE (1)

R. H. Kingston, “Parametric amplification and oscillation of optical frequencies,” Proc. IRE 50, 472 (1962).

Rev. Mod. Phys. (1)

M. I. Kolobov, “The spatial behavior of nonclassical light,” Rev. Mod. Phys. 71, 1539–1589 (1999).
[CrossRef]

Sov. Phys. Dokl. (2)

A. V. Belinskii and D. N. Klyshko, “Two-photon optics: diffraction, holography, and transformation of two-dimensional signals,” Zh. Eksp. Teor. Fiz. 105, 487–493 (1994) [Sov. Phys. Dokl. 78, 259–262 (1994)].

A. A. Malygin, A. N. Penin, and A. V. Sergienko, “Spatiotemporal grouping of photons in spontaneous parametric scattering of light,”Dokl. Akad. Nauk SSSR 281, 308–313 (1985) [Sov. Phys. Dokl. 30, 227–229 (1985)].

Sov. Phys. JETP (2)

D. N. Klyshko, “Effect of focusing on photon correlation in parametric light scattering,” Zh. Eksp. Teor. Fiz. 94, 82–90 (1988) [Sov. Phys. JETP 67, 1131–1135 (1988)].

D. N. Klyshko, “Transverse photon bunching and two-photon processes in the field of parametrically scattered light,” Zh. Eksp. Teor. Fiz. 83, 1313–1323 (1982) [Sov. Phys. JETP 56, 753–758 (1982)].

Zh. Eksp. Teor. Fiz. (1)

S. A. Akhmanov, A. I. Kovrigin, R. V. Khokhlov, and O. N. Chunaev, Zh. Eksp. Teor. Fiz. 45, 1336 (1963).

Other (16)

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F. Zernike and J. E. Midwinter, Applied Nonlinear Optics (Wiley, New York, 1973).

A. Yariv, Quantum Electronics, 3rd ed. (Wiley, New York, 1988).

P. N. Butcher and D. Cotter, The Elements of Nonlinear Optics (Cambridge University, Cambridge, England, 1990).

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991).

D. L. Mills, Nonlinear Optics: Basic Concepts (Springer, Berlin, 1998).

D. N. Klyshko, Photons and Nonlinear Optics (Nauka, Moscow, 1980) [translation: Gordon and Breach, New York, 1988].

E. Schrödinger, “Die gegenwärtige Situation in der Quantenmechanik,” Naturwissenschaften 23, 807–849 (1935) [translation: J. D. Trimmer, “The present situation in quantum mechanics: a translation of Schrodinger’s ‘cat paradox’ paper,” Proc. Am. Phil. Soc. 124, 323–338 (1980); reprinted in Quantum Theory and Measurement, J. A. Wheeler and W. H. Zurek, eds. (Princeton University, Princeton, N.J., 1983)].
[CrossRef]

L. A. Lugiato, M. Brambilla, and A. Gatti, “Optical pattern formation,” in Advances in Atomic, Molecular, and Optical Physics, B. Bederson and H. Walther, eds. (Academic, Boston, 1999), Vol. 40, pp. 229–306.

B. E. A. Saleh, S. Popescu, and M. C. Teich, “Generalized entangled-photon imaging,” in Proceedings of the Ninth Annual Meeting of the IEEE Lasers and Electro-Optics Society, P. Zory, ed. (IEEE, Piscataway, N.J., 1996), Vol. 1, pp. 362–363.

M. Born and E. Wolf, Principles of Optics, 8th ed. (Cambridge University, New York, 1999).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

G. O. Reynolds, J. B. DeVelis, G. B. Parrent, Jr., and B. J. Thompson, The New Physical Optics Notebook: Tutorials in Fourier Optics (SPIE, Bellingham, Wash., 1989).

F. T. S. Yu and S. Jutamulia, Optical Signal Processing, Computing, and Neural Networks (Wiley, New York, 1992).

H. Stark, ed., Image Recovery: Theory and Application (Academic, London, 1987).

T. Wilson and C. Sheppard, Theory and Practice of Scanning Optical Microscopy (Academic, London, 1984).

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Figures (12)

Fig. 1
Fig. 1

Biphoton imaging using photon pairs generated by spontaneous parametric downconversion. NLC stands for nonlinear crystal; D1 and D2 are single-photon detectors at locations x1 and x2, respectively; G(2)(x1, x2) is the biphoton rate; hp(x, x), hs(x1, x), and hi(x2, x) are the impulse response functions of the optical systems placed in the paths of the pump, signal, and idler beams, respectively.

Fig. 2
Fig. 2

Object in the signal-beam configuration. Ep is the pump field at the entrance to the NLC; h1(x1, x) and h2(x, x) are the impulse-response functions of the optical systems placed in the signal beam; hi(x2, x) is the impulse-response function of the optical system placed in the idler beam; t(x) is the object to be imaged, placed in the signal beam.

Fig. 3
Fig. 3

Object in the signal-beam configuration of examples 1 and 2 displayed in an unfolded picture. Ep, h1(x1, x), and t(x) are the same as in Fig. 2; f is the focal length of a lens placed in the idler beam. See text for details.

Fig. 4
Fig. 4

Object in the signal-beam configuration of example 3 displayed in an unfolded mode. h1(x1, x) and t(x) are the same as in Fig. 2; a lens is placed in the pump beam and is represented here by the dotted lens of focal length f. See text for details.

Fig. 5
Fig. 5

Object in both signal- and idler-beam configuration. h1(x1, x), h2(x, x) are the impulse-response functions of the optical systems placed in the path of both the signal and idler beams, and t(x) is the object to be imaged.

Fig. 6
Fig. 6

Object in the pump-beam configuration. hs(x1, x), hi(x2, x) are as in Fig. 1, and t(x) is the object to be imaged placed in the pump beam.

Fig. 7
Fig. 7

Object is the detector configuration. h1(x1, x) is the impulse response of the optical system placed in the signal and idler paths, and D is a two-photon detector at location x1.

Fig. 8
Fig. 8

Configuration for triple correlation. h2s(x, x) and h1s(x1, x) are the impulse-response functions of the optical systems placed in the signal beam; h2i(x, x) and h1i(x2, x) are the impulse-response functions of the optical systems placed in the idler beam; tp(x), ts(x), and ti(x) are the three objects to be correlated.

Fig. 9
Fig. 9

Effect of NLC thickness l on the imaging resolution of object in the signal-beam configuration. Plots of normalized time-averaged marginal coincidence rate C(x2) versus detector D2 location x2 normalized with respect to xc=2λ0F#, for l=0.1, 1, and 10 mm; λ0=650 nm, F#=5.

Fig. 10
Fig. 10

Effect of the biphoton bandwidth Ω on the imaging resolution of object in the signal-beam configuration. Plots of normalized C(x2) versus x2 normalized with respect to xc=2λ0F# are shown for ρ=Ω/ωp=0.001, 0.01, and 0.02; λ0=650 nm, F#=5. The NLC is adjusted for collinear SPDC and is of thickness 1 mm.

Fig. 11
Fig. 11

Effect of the transverse width of the pump B on the imaging resolution of object in the signal-beam configuration. Plots of normalized C(x2) versus x2 normalized with respect to xc=2λ0F# are shown for B=2, 1, 0.5, and 0.1 mm; λ0=650 nm, F#=5.

Fig. 12
Fig. 12

Classical partially coherent imaging. The quantity G(1)(x, x) is the second-order correlation function of the optical field; t is the object to be imaged; h(x1, x) is the impulse response function of the imaging system; D is a detector placed at position x1 that records the intensity I(x1).

Equations (51)

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G(2)(x1, x2)=|ψ(x1, x2)|2,
ψ(x1, x2)=dxEp(x)hs(x1, x)hi(x2, x);
I(2)(x2)=dx1G(2)(x1, x2),
I0(2)(x2)=G(2)(0, x2).
ψ(x1, x2)=dxt(x)h1(x1, x)h3(x2, x),
h3(x2, x)=dxEp(x)hi(x2, x)h2(x, x).
I0(2)(x2)=dxt(x)h1(0, x)h3(x2, x)2.
I(2)(x2)=dx1dxt(x)h1(x1, x)h3(x2, x)2=dxdxt*(x)t(x)g(x, x)×h3*(x2, x)h3(x2, x).
g(x, x)=dx1h1*(x1, x)h1(x1, x)
I(2)(x2)=dx|t(x)|2|h3(x2, x)|2,
I(2)(x2)=dxt(x)f(x)h3(x2, x)2,
d1=di+dsλsλi.
h3(x2, x)=expj2πdsλs+di+fλiexp-j2πλifx2x.
I02(x2)=T2πλifx22,
1d1+1d2=1f,
h3(x2, x)=expj2πdsλs+di+d2λi×expjπx22λid21-1Mδ(x2-Mx),
1λsd1+1λid2=1λpR,
h3(x2, x)=expj2πd1λs+d2λi×expjπx22λid21-1Mδ(x2-Mx).
1d1+1d2=2R.
ψ(x1, x2)= dxdxt(x)t(x)ψc(x, x)×h1(x1, x)h1(x2, x),
ψc(x, x)=dxEp(x)h2(x, x)h2(x, x).
G(2)(x1, x2)=dxt(x)hs(x1, x)hi(x2, x)2,
I0(2)(x2)=dxt(x)hi(x2, x)2,
G(2)(x1, x1)=dxt(x)hs2(x1, x)2.
G(2)(x1, x2)=tx1λsfs+x2λififoλp2,
G(2)(x1, x1)=t2(x1)S(x1),
G(2)(x1, x2)=dxtp(x)Ts2πλsfs(x-x1)×Ti2πλifi(x-x2)2,
hi(x2, x)=hn(x-x2),n=1, 2, N.
Cn=G(2)(0, 0)=dxhs(-x)hn(x)2,
Cn=|hs(xn)|2,
G(2)(x1, t1; x2, t2)=|ψ(x1, t1; x2, t2)|2.
ψ(x1, t1; x2, t2)=exp(-iωpt1)×Ωdωs exp[-iωs(t1-t2)]ψ˜(x1, x2;ωs),
ψ˜(x1, x2; ωs)= dqsdqiΛ(qs, qi; ωs)×Hs(x1, qs; ωs)Hi(x2, qi; ωp-ωs),
Λ(qs, qi; ωs)=E˜p(qs+qi)ξ˜(qs, qi; ωs);
ξ˜(qs, qi; ωs)=l sincl2πΔrexp-jl2Δr;
C(x1, x2)=Ωdωs|ψ˜(x1, x2; ωs)|2,
C(x2)=dx1C(x1, x2),
C0(x2)=C(0, x2),
ψ˜(x1, x2; ωs)=dxt(x)h1(x1, x; ωs)h3(x2, x; ωs),
h3(x2, x; ωs)= dqsdqiΛ(qs, qi; ωs)×H2(x, qs; ωs)Hi(x2, qi; ωp-ωs).
C(x2)=Ωdωs|h3(x2, 0; ωs)|2g(ωs),
C0(x2)=Ωdωs|h3(x2, 0; ωs)|2g0(ωs),
h3(x2, 0; ωs)=dqsξ˜(qs, -qs; ωs)×H2(0, qs; ωs)Hi(x2, -qs; ωp-ωs).
H2(0, qs; ωs)=exp(jksds)exp-jdsqs22ks,
Hi(x2, -qs; ωp-ωs)=exp[jki(d1+d2)]×expjkix222d2exp-jdiqs22kiPgqs+kix2d2,
p(x)expjkix221d2-1f
d1=di+dsλsλi+leq,
leq=l2λiλsns+λini.
g(x1)= dxdf*(x)f(x)q(x1; x, x),
g(x1)=dxf(x)h(x1, x)2.
g(x1)=dx|f(x)|2|h(x1, x)|2,

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