Abstract

We propose and provide experimental evidence in support of a theory for the remote preparation of a complex spatial state of a single photon. An entangled two-photon source was obtained by spontaneous parametric down-conversion, and a double slit was placed in the path of the signal photon as a scattering object. The signal photon was detected after proper spatial filtering so that the idler photon was prepared in the corresponding single-photon state. By using a two-photon coincidence measurement, we obtained the Radon transform, at several longitudinal distances, of the single-photon Wigner distribution function modified by the double slit. The experimental results are consistent with the idler photon being in a pure state. An inverse Radon transformation can, in principle, be applied to the measured data to reconstruct the modified single-photon Wigner function, which is a complete representation of the amplitude and phase structure of the scattering object.

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2009

A. I. Lvovsky and M. G. Raymer, “Continuous-variable optical quantum-state tomography,” Rev. Mod. Phys. 81(1), 299–332 (2009).
[CrossRef]

B. Erkmen and J. Shapiro, “Signal-to-noise ratio of Gaussian-state ghost imaging,” Phys. Rev. A 79(2), 023833 (2009).
[CrossRef]

S. V. Polyakov and A. L. Migdall, “Quantum radiometry,” J. Mod. Opt. 56(9), 1045–1052 (2009).
[CrossRef]

D. Tasca, S. Walborn, P. Souto Ribeiro, F. Toscano, and P. Pellat-Finet, “Propagation of transverse intensity correlations of a two-photon state,” Phys. Rev. A 79(3), 033801 (2009).
[CrossRef]

2008

D. S. Tasca, S. P. Walborn, P. H. Souto Ribeiro, and F. Toscano, “Detection of transverse entanglement in phase space,” Phys. Rev. A 78(1), 010304 (2008).
[CrossRef]

J. H. Shapiro, “Computational ghost imaging,” Phys. Rev. A 78(6), 061802 (2008).
[CrossRef]

P. J. Mosley, J. S. Lundeen, B. J. Smith, P. Wasylczyk, A. B. U’Ren, C. Silberhorn, and I. A. Walmsley, “Heralded generation of ultrafast single photons in pure quantum States,” Phys. Rev. Lett. 100(13), 133601 (2008).
[CrossRef] [PubMed]

P. J. Mosley, J. S. Lundeen, B. J. Smith, and I. A. Walmsley, “Conditional preparation of single photons using parametric down-conversion: A recipe for purity,” N. J. Phys. 10(9), 093011 (2008).
[CrossRef]

2007

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79(1), 135–174 (2007).
[CrossRef]

W. Wasilewski, P. Kolenderski, and R. Frankowski, “Spectral density matrix of a single photon measured,” Phys. Rev. Lett. 99(12), 123601 (2007).
[CrossRef] [PubMed]

2006

2005

A. Valencia, G. Scarcelli, M. D’Angelo, and Y. Shih, “Two-photon imaging with thermal light,” Phys. Rev. Lett. 94(6), 063601 (2005).
[CrossRef] [PubMed]

Y. Cai and S. Zhu, “Coincidence fractional Fourier transform implemented with partially coherent light radiation,” J. Opt. Soc. Am. A 22(9), 1798–1804 (2005).
[CrossRef]

T. B. Pittman, B. C. Jacobs, and J. D. Franson, “Heralding single photons from pulsed parametric down-conversion,” Opt. Commun. 246(4-6), 545–550 (2005).
[CrossRef]

B. J. Smith, B. Killett, M. G. Raymer, I. A. Walmsley, and K. Banaszek, “Measurement of the transverse spatial quantum state of light at the single-photon level,” Opt. Lett. 30(24), 3365–3367 (2005).
[CrossRef]

2004

A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “Ghost imaging with thermal light: comparing entanglement and classical correlation,” Phys. Rev. Lett. 93(9), 093602 (2004).
[CrossRef] [PubMed]

M. D’Angelo, Y. H. Kim, S. P. Kulik, and Y. Shih, “Identifying entanglement using quantum ghost interference and imaging,” Phys. Rev. Lett. 92(23), 233601 (2004).
[CrossRef] [PubMed]

2003

A. Gatti, E. Brambilla, and L. A. Lugiato, “Entangled imaging and wave-particle duality: from the microscopic to the macroscopic realm,” Phys. Rev. Lett. 90(13), 133603 (2003).
[CrossRef] [PubMed]

2002

A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Entangled-photon Fourier optics,” J. Opt. Soc. Am. B 19(5), 1174–1184 (2002).
[CrossRef]

R. S. Bennink, S. J. Bentley, and R. W. Boyd, ““Two-photon” Coincidence Imaging with a Classical Source,” Phys. Rev. Lett. 89(11), 113601 (2002).
[CrossRef] [PubMed]

T. Aichele, A. I. Lvovsky, and S. Schiller, “Optical mode characterization of single photons prepared by means of conditional measurements on a biphoton state,” Eur. Phys. J. D 18(2), 237–245 (2002).
[CrossRef]

2001

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(12), 123602 (2001).
[CrossRef] [PubMed]

E. Knill, R. Laflamme, and G. J. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature 409(6816), 46–52 (2001).
[CrossRef] [PubMed]

A. I. Lvovsky, H. Hansen, T. Aichele, O. Benson, J. Mlynek, and S. Schiller, “Quantum state reconstruction of the single-photon Fock state,” Phys. Rev. Lett. 87(5), 050402 (2001).
[CrossRef] [PubMed]

P. Navez and E. Brambilla, “Spatial entanglement of twin quantum images,” Phys. Rev. A 65, 013813 (2001).
[CrossRef]

1998

C. Monken, P. Ribeiro, and S. Pádua, “Transfer of angular spectrum and image formation in spontaneous parametric down-conversion,” Phys. Rev. A 57(4), 3123–3126 (1998).
[CrossRef]

1997

M. G. Raymer, “Measuring the quantum mechanical wave function,” Contemp. Phys. 38(5), 343–355 (1997).
[CrossRef]

1996

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(4), 2804–2815 (1996).
[CrossRef] [PubMed]

Z. Zalevsky and D. Mendlovi, “Fractional Radon transform: definition,” Appl. Opt. 35(23), 4628–4631 (1996).
[CrossRef] [PubMed]

U. Leonhardt and M. Munroe, “Number of phases required to determine a quantum state in optical homodyne tomography,” Phys. Rev. A 54(4), 3682–3684 (1996).
[CrossRef] [PubMed]

1995

D. F. McAlister, M. Beck, L. Clarke, A. Mayer, and M. G. Raymer, “Optical phase retrieval by phase-space tomography and fractional-order Fourier transforms,” Opt. Lett. 20(10), 1181–1183 (1995).
[CrossRef] [PubMed]

B. Eppich and N. Reng, “Measurement of the Wigner distribution function based on the inverse Radon transformation,” Proc. SPIE 2375, 261–268 (1995).
[CrossRef]

H. M. Ozaktas and D. Mendlovic, “Fractional Fourier Optics,” J. Opt. Soc. Am. A 12(4), 743–751 (1995).
[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(5), 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(18), 3600–3603 (1995).
[CrossRef] [PubMed]

1994

M. G. Raymer, M. Beck, and D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72(8), 1137–1140 (1994).
[CrossRef] [PubMed]

1993

1935

A. Einstein, B. Podolsky, and N. Rosen, “Can Quantum-Mechanical Description of Physical Reality be considered complete?” Phys. Rev. 47(10), 777–780 (1935).
[CrossRef]

Abouraddy, A. F.

A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Entangled-photon Fourier optics,” J. Opt. Soc. Am. B 19(5), 1174–1184 (2002).
[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(12), 123602 (2001).
[CrossRef] [PubMed]

Aichele, T.

T. Aichele, A. I. Lvovsky, and S. Schiller, “Optical mode characterization of single photons prepared by means of conditional measurements on a biphoton state,” Eur. Phys. J. D 18(2), 237–245 (2002).
[CrossRef]

A. I. Lvovsky, H. Hansen, T. Aichele, O. Benson, J. Mlynek, and S. Schiller, “Quantum state reconstruction of the single-photon Fock state,” Phys. Rev. Lett. 87(5), 050402 (2001).
[CrossRef] [PubMed]

Bache, M.

A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “Ghost imaging with thermal light: comparing entanglement and classical correlation,” Phys. Rev. Lett. 93(9), 093602 (2004).
[CrossRef] [PubMed]

Banaszek, K.

Beck, M.

Bennink, R. S.

R. S. Bennink, S. J. Bentley, and R. W. Boyd, ““Two-photon” Coincidence Imaging with a Classical Source,” Phys. Rev. Lett. 89(11), 113601 (2002).
[CrossRef] [PubMed]

Benson, O.

A. I. Lvovsky, H. Hansen, T. Aichele, O. Benson, J. Mlynek, and S. Schiller, “Quantum state reconstruction of the single-photon Fock state,” Phys. Rev. Lett. 87(5), 050402 (2001).
[CrossRef] [PubMed]

Bentley, S. J.

R. S. Bennink, S. J. Bentley, and R. W. Boyd, ““Two-photon” Coincidence Imaging with a Classical Source,” Phys. Rev. Lett. 89(11), 113601 (2002).
[CrossRef] [PubMed]

Boyd, R. W.

R. S. Bennink, S. J. Bentley, and R. W. Boyd, ““Two-photon” Coincidence Imaging with a Classical Source,” Phys. Rev. Lett. 89(11), 113601 (2002).
[CrossRef] [PubMed]

Brambilla, E.

A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “Ghost imaging with thermal light: comparing entanglement and classical correlation,” Phys. Rev. Lett. 93(9), 093602 (2004).
[CrossRef] [PubMed]

A. Gatti, E. Brambilla, and L. A. Lugiato, “Entangled imaging and wave-particle duality: from the microscopic to the macroscopic realm,” Phys. Rev. Lett. 90(13), 133603 (2003).
[CrossRef] [PubMed]

P. Navez and E. Brambilla, “Spatial entanglement of twin quantum images,” Phys. Rev. A 65, 013813 (2001).
[CrossRef]

Cai, Y.

Clarke, L.

D’Angelo, M.

A. Valencia, G. Scarcelli, M. D’Angelo, and Y. Shih, “Two-photon imaging with thermal light,” Phys. Rev. Lett. 94(6), 063601 (2005).
[CrossRef] [PubMed]

M. D’Angelo, Y. H. Kim, S. P. Kulik, and Y. Shih, “Identifying entanglement using quantum ghost interference and imaging,” Phys. Rev. Lett. 92(23), 233601 (2004).
[CrossRef] [PubMed]

Dowling, J. P.

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79(1), 135–174 (2007).
[CrossRef]

Einstein, A.

A. Einstein, B. Podolsky, and N. Rosen, “Can Quantum-Mechanical Description of Physical Reality be considered complete?” Phys. Rev. 47(10), 777–780 (1935).
[CrossRef]

Eppich, B.

B. Eppich and N. Reng, “Measurement of the Wigner distribution function based on the inverse Radon transformation,” Proc. SPIE 2375, 261–268 (1995).
[CrossRef]

Erkmen, B.

B. Erkmen and J. Shapiro, “Signal-to-noise ratio of Gaussian-state ghost imaging,” Phys. Rev. A 79(2), 023833 (2009).
[CrossRef]

Frankowski, R.

W. Wasilewski, P. Kolenderski, and R. Frankowski, “Spectral density matrix of a single photon measured,” Phys. Rev. Lett. 99(12), 123601 (2007).
[CrossRef] [PubMed]

Franson, J. D.

T. B. Pittman, B. C. Jacobs, and J. D. Franson, “Heralding single photons from pulsed parametric down-conversion,” Opt. Commun. 246(4-6), 545–550 (2005).
[CrossRef]

Gatti, A.

A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “Ghost imaging with thermal light: comparing entanglement and classical correlation,” Phys. Rev. Lett. 93(9), 093602 (2004).
[CrossRef] [PubMed]

A. Gatti, E. Brambilla, and L. A. Lugiato, “Entangled imaging and wave-particle duality: from the microscopic to the macroscopic realm,” Phys. Rev. Lett. 90(13), 133603 (2003).
[CrossRef] [PubMed]

Hansen, H.

A. I. Lvovsky, H. Hansen, T. Aichele, O. Benson, J. Mlynek, and S. Schiller, “Quantum state reconstruction of the single-photon Fock state,” Phys. Rev. Lett. 87(5), 050402 (2001).
[CrossRef] [PubMed]

He, S.

Jacobs, B. C.

T. B. Pittman, B. C. Jacobs, and J. D. Franson, “Heralding single photons from pulsed parametric down-conversion,” Opt. Commun. 246(4-6), 545–550 (2005).
[CrossRef]

Killett, B.

Kim, Y. H.

M. D’Angelo, Y. H. Kim, S. P. Kulik, and Y. Shih, “Identifying entanglement using quantum ghost interference and imaging,” Phys. Rev. Lett. 92(23), 233601 (2004).
[CrossRef] [PubMed]

Klyshko, D. N.

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(4), 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(18), 3600–3603 (1995).
[CrossRef] [PubMed]

Knill, E.

E. Knill, R. Laflamme, and G. J. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature 409(6816), 46–52 (2001).
[CrossRef] [PubMed]

Kok, P.

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79(1), 135–174 (2007).
[CrossRef]

Kolenderski, P.

W. Wasilewski, P. Kolenderski, and R. Frankowski, “Spectral density matrix of a single photon measured,” Phys. Rev. Lett. 99(12), 123601 (2007).
[CrossRef] [PubMed]

Kulik, S. P.

M. D’Angelo, Y. H. Kim, S. P. Kulik, and Y. Shih, “Identifying entanglement using quantum ghost interference and imaging,” Phys. Rev. Lett. 92(23), 233601 (2004).
[CrossRef] [PubMed]

Laflamme, R.

E. Knill, R. Laflamme, and G. J. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature 409(6816), 46–52 (2001).
[CrossRef] [PubMed]

Leonhardt, U.

U. Leonhardt and M. Munroe, “Number of phases required to determine a quantum state in optical homodyne tomography,” Phys. Rev. A 54(4), 3682–3684 (1996).
[CrossRef] [PubMed]

Lohmann, A. W.

Lugiato, L. A.

A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “Ghost imaging with thermal light: comparing entanglement and classical correlation,” Phys. Rev. Lett. 93(9), 093602 (2004).
[CrossRef] [PubMed]

A. Gatti, E. Brambilla, and L. A. Lugiato, “Entangled imaging and wave-particle duality: from the microscopic to the macroscopic realm,” Phys. Rev. Lett. 90(13), 133603 (2003).
[CrossRef] [PubMed]

Lundeen, J. S.

P. J. Mosley, J. S. Lundeen, B. J. Smith, P. Wasylczyk, A. B. U’Ren, C. Silberhorn, and I. A. Walmsley, “Heralded generation of ultrafast single photons in pure quantum States,” Phys. Rev. Lett. 100(13), 133601 (2008).
[CrossRef] [PubMed]

P. J. Mosley, J. S. Lundeen, B. J. Smith, and I. A. Walmsley, “Conditional preparation of single photons using parametric down-conversion: A recipe for purity,” N. J. Phys. 10(9), 093011 (2008).
[CrossRef]

Lvovsky, A. I.

A. I. Lvovsky and M. G. Raymer, “Continuous-variable optical quantum-state tomography,” Rev. Mod. Phys. 81(1), 299–332 (2009).
[CrossRef]

T. Aichele, A. I. Lvovsky, and S. Schiller, “Optical mode characterization of single photons prepared by means of conditional measurements on a biphoton state,” Eur. Phys. J. D 18(2), 237–245 (2002).
[CrossRef]

A. I. Lvovsky, H. Hansen, T. Aichele, O. Benson, J. Mlynek, and S. Schiller, “Quantum state reconstruction of the single-photon Fock state,” Phys. Rev. Lett. 87(5), 050402 (2001).
[CrossRef] [PubMed]

Mayer, A.

McAlister, D. F.

Mendlovi, D.

Mendlovic, D.

Migdall, A. L.

S. V. Polyakov and A. L. Migdall, “Quantum radiometry,” J. Mod. Opt. 56(9), 1045–1052 (2009).
[CrossRef]

Milburn, G. J.

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79(1), 135–174 (2007).
[CrossRef]

E. Knill, R. Laflamme, and G. J. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature 409(6816), 46–52 (2001).
[CrossRef] [PubMed]

Mlynek, J.

A. I. Lvovsky, H. Hansen, T. Aichele, O. Benson, J. Mlynek, and S. Schiller, “Quantum state reconstruction of the single-photon Fock state,” Phys. Rev. Lett. 87(5), 050402 (2001).
[CrossRef] [PubMed]

Monken, C.

C. Monken, P. Ribeiro, and S. Pádua, “Transfer of angular spectrum and image formation in spontaneous parametric down-conversion,” Phys. Rev. A 57(4), 3123–3126 (1998).
[CrossRef]

Mosley, P. J.

P. J. Mosley, J. S. Lundeen, B. J. Smith, and I. A. Walmsley, “Conditional preparation of single photons using parametric down-conversion: A recipe for purity,” N. J. Phys. 10(9), 093011 (2008).
[CrossRef]

P. J. Mosley, J. S. Lundeen, B. J. Smith, P. Wasylczyk, A. B. U’Ren, C. Silberhorn, and I. A. Walmsley, “Heralded generation of ultrafast single photons in pure quantum States,” Phys. Rev. Lett. 100(13), 133601 (2008).
[CrossRef] [PubMed]

Munro, W. J.

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79(1), 135–174 (2007).
[CrossRef]

Munroe, M.

U. Leonhardt and M. Munroe, “Number of phases required to determine a quantum state in optical homodyne tomography,” Phys. Rev. A 54(4), 3682–3684 (1996).
[CrossRef] [PubMed]

Navez, P.

P. Navez and E. Brambilla, “Spatial entanglement of twin quantum images,” Phys. Rev. A 65, 013813 (2001).
[CrossRef]

Nemoto, K.

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79(1), 135–174 (2007).
[CrossRef]

Ozaktas, H. M.

Pádua, S.

C. Monken, P. Ribeiro, and S. Pádua, “Transfer of angular spectrum and image formation in spontaneous parametric down-conversion,” Phys. Rev. A 57(4), 3123–3126 (1998).
[CrossRef]

Pellat-Finet, P.

D. Tasca, S. Walborn, P. Souto Ribeiro, F. Toscano, and P. Pellat-Finet, “Propagation of transverse intensity correlations of a two-photon state,” Phys. Rev. A 79(3), 033801 (2009).
[CrossRef]

Pittman, T. B.

T. B. Pittman, B. C. Jacobs, and J. D. Franson, “Heralding single photons from pulsed parametric down-conversion,” Opt. Commun. 246(4-6), 545–550 (2005).
[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(4), 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(5), R3429–R3432 (1995).
[CrossRef] [PubMed]

Podolsky, B.

A. Einstein, B. Podolsky, and N. Rosen, “Can Quantum-Mechanical Description of Physical Reality be considered complete?” Phys. Rev. 47(10), 777–780 (1935).
[CrossRef]

Polyakov, S. V.

S. V. Polyakov and A. L. Migdall, “Quantum radiometry,” J. Mod. Opt. 56(9), 1045–1052 (2009).
[CrossRef]

Ralph, T. C.

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79(1), 135–174 (2007).
[CrossRef]

Raymer, M. G.

A. I. Lvovsky and M. G. Raymer, “Continuous-variable optical quantum-state tomography,” Rev. Mod. Phys. 81(1), 299–332 (2009).
[CrossRef]

B. J. Smith, B. Killett, M. G. Raymer, I. A. Walmsley, and K. Banaszek, “Measurement of the transverse spatial quantum state of light at the single-photon level,” Opt. Lett. 30(24), 3365–3367 (2005).
[CrossRef]

M. G. Raymer, “Measuring the quantum mechanical wave function,” Contemp. Phys. 38(5), 343–355 (1997).
[CrossRef]

D. F. McAlister, M. Beck, L. Clarke, A. Mayer, and M. G. Raymer, “Optical phase retrieval by phase-space tomography and fractional-order Fourier transforms,” Opt. Lett. 20(10), 1181–1183 (1995).
[CrossRef] [PubMed]

M. G. Raymer, M. Beck, and D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72(8), 1137–1140 (1994).
[CrossRef] [PubMed]

Reng, N.

B. Eppich and N. Reng, “Measurement of the Wigner distribution function based on the inverse Radon transformation,” Proc. SPIE 2375, 261–268 (1995).
[CrossRef]

Ribeiro, P.

C. Monken, P. Ribeiro, and S. Pádua, “Transfer of angular spectrum and image formation in spontaneous parametric down-conversion,” Phys. Rev. A 57(4), 3123–3126 (1998).
[CrossRef]

Rosen, N.

A. Einstein, B. Podolsky, and N. Rosen, “Can Quantum-Mechanical Description of Physical Reality be considered complete?” Phys. Rev. 47(10), 777–780 (1935).
[CrossRef]

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(4), 2804–2815 (1996).
[CrossRef] [PubMed]

Saleh, B. E. A.

A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Entangled-photon Fourier optics,” J. Opt. Soc. Am. B 19(5), 1174–1184 (2002).
[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(12), 123602 (2001).
[CrossRef] [PubMed]

Scarcelli, G.

A. Valencia, G. Scarcelli, M. D’Angelo, and Y. Shih, “Two-photon imaging with thermal light,” Phys. Rev. Lett. 94(6), 063601 (2005).
[CrossRef] [PubMed]

Schiller, S.

T. Aichele, A. I. Lvovsky, and S. Schiller, “Optical mode characterization of single photons prepared by means of conditional measurements on a biphoton state,” Eur. Phys. J. D 18(2), 237–245 (2002).
[CrossRef]

A. I. Lvovsky, H. Hansen, T. Aichele, O. Benson, J. Mlynek, and S. Schiller, “Quantum state reconstruction of the single-photon Fock state,” Phys. Rev. Lett. 87(5), 050402 (2001).
[CrossRef] [PubMed]

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(5), 1174–1184 (2002).
[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(12), 123602 (2001).
[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(4), 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(5), 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(18), 3600–3603 (1995).
[CrossRef] [PubMed]

Shapiro, J.

B. Erkmen and J. Shapiro, “Signal-to-noise ratio of Gaussian-state ghost imaging,” Phys. Rev. A 79(2), 023833 (2009).
[CrossRef]

Shapiro, J. H.

J. H. Shapiro, “Computational ghost imaging,” Phys. Rev. A 78(6), 061802 (2008).
[CrossRef]

Shih, Y.

A. Valencia, G. Scarcelli, M. D’Angelo, and Y. Shih, “Two-photon imaging with thermal light,” Phys. Rev. Lett. 94(6), 063601 (2005).
[CrossRef] [PubMed]

M. D’Angelo, Y. H. Kim, S. P. Kulik, and Y. Shih, “Identifying entanglement using quantum ghost interference and imaging,” Phys. Rev. Lett. 92(23), 233601 (2004).
[CrossRef] [PubMed]

Shih, Y. 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(4), 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(5), 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(18), 3600–3603 (1995).
[CrossRef] [PubMed]

Silberhorn, C.

P. J. Mosley, J. S. Lundeen, B. J. Smith, P. Wasylczyk, A. B. U’Ren, C. Silberhorn, and I. A. Walmsley, “Heralded generation of ultrafast single photons in pure quantum States,” Phys. Rev. Lett. 100(13), 133601 (2008).
[CrossRef] [PubMed]

Smith, B. J.

P. J. Mosley, J. S. Lundeen, B. J. Smith, P. Wasylczyk, A. B. U’Ren, C. Silberhorn, and I. A. Walmsley, “Heralded generation of ultrafast single photons in pure quantum States,” Phys. Rev. Lett. 100(13), 133601 (2008).
[CrossRef] [PubMed]

P. J. Mosley, J. S. Lundeen, B. J. Smith, and I. A. Walmsley, “Conditional preparation of single photons using parametric down-conversion: A recipe for purity,” N. J. Phys. 10(9), 093011 (2008).
[CrossRef]

B. J. Smith, B. Killett, M. G. Raymer, I. A. Walmsley, and K. Banaszek, “Measurement of the transverse spatial quantum state of light at the single-photon level,” Opt. Lett. 30(24), 3365–3367 (2005).
[CrossRef]

Souto Ribeiro, P.

D. Tasca, S. Walborn, P. Souto Ribeiro, F. Toscano, and P. Pellat-Finet, “Propagation of transverse intensity correlations of a two-photon state,” Phys. Rev. A 79(3), 033801 (2009).
[CrossRef]

Souto Ribeiro, P. H.

D. S. Tasca, S. P. Walborn, P. H. Souto Ribeiro, and F. Toscano, “Detection of transverse entanglement in phase space,” Phys. Rev. A 78(1), 010304 (2008).
[CrossRef]

Strekalov, D. V.

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(4), 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(5), 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(18), 3600–3603 (1995).
[CrossRef] [PubMed]

Tasca, D.

D. Tasca, S. Walborn, P. Souto Ribeiro, F. Toscano, and P. Pellat-Finet, “Propagation of transverse intensity correlations of a two-photon state,” Phys. Rev. A 79(3), 033801 (2009).
[CrossRef]

Tasca, D. S.

D. S. Tasca, S. P. Walborn, P. H. Souto Ribeiro, and F. Toscano, “Detection of transverse entanglement in phase space,” Phys. Rev. A 78(1), 010304 (2008).
[CrossRef]

Teich, M. C.

A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Entangled-photon Fourier optics,” J. Opt. Soc. Am. B 19(5), 1174–1184 (2002).
[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(12), 123602 (2001).
[CrossRef] [PubMed]

Toscano, F.

D. Tasca, S. Walborn, P. Souto Ribeiro, F. Toscano, and P. Pellat-Finet, “Propagation of transverse intensity correlations of a two-photon state,” Phys. Rev. A 79(3), 033801 (2009).
[CrossRef]

D. S. Tasca, S. P. Walborn, P. H. Souto Ribeiro, and F. Toscano, “Detection of transverse entanglement in phase space,” Phys. Rev. A 78(1), 010304 (2008).
[CrossRef]

U’Ren, A. B.

P. J. Mosley, J. S. Lundeen, B. J. Smith, P. Wasylczyk, A. B. U’Ren, C. Silberhorn, and I. A. Walmsley, “Heralded generation of ultrafast single photons in pure quantum States,” Phys. Rev. Lett. 100(13), 133601 (2008).
[CrossRef] [PubMed]

Valencia, A.

A. Valencia, G. Scarcelli, M. D’Angelo, and Y. Shih, “Two-photon imaging with thermal light,” Phys. Rev. Lett. 94(6), 063601 (2005).
[CrossRef] [PubMed]

Walborn, S.

D. Tasca, S. Walborn, P. Souto Ribeiro, F. Toscano, and P. Pellat-Finet, “Propagation of transverse intensity correlations of a two-photon state,” Phys. Rev. A 79(3), 033801 (2009).
[CrossRef]

Walborn, S. P.

D. S. Tasca, S. P. Walborn, P. H. Souto Ribeiro, and F. Toscano, “Detection of transverse entanglement in phase space,” Phys. Rev. A 78(1), 010304 (2008).
[CrossRef]

Walmsley, I. A.

P. J. Mosley, J. S. Lundeen, B. J. Smith, P. Wasylczyk, A. B. U’Ren, C. Silberhorn, and I. A. Walmsley, “Heralded generation of ultrafast single photons in pure quantum States,” Phys. Rev. Lett. 100(13), 133601 (2008).
[CrossRef] [PubMed]

P. J. Mosley, J. S. Lundeen, B. J. Smith, and I. A. Walmsley, “Conditional preparation of single photons using parametric down-conversion: A recipe for purity,” N. J. Phys. 10(9), 093011 (2008).
[CrossRef]

B. J. Smith, B. Killett, M. G. Raymer, I. A. Walmsley, and K. Banaszek, “Measurement of the transverse spatial quantum state of light at the single-photon level,” Opt. Lett. 30(24), 3365–3367 (2005).
[CrossRef]

Wang, F.

Wasilewski, W.

W. Wasilewski, P. Kolenderski, and R. Frankowski, “Spectral density matrix of a single photon measured,” Phys. Rev. Lett. 99(12), 123601 (2007).
[CrossRef] [PubMed]

Wasylczyk, P.

P. J. Mosley, J. S. Lundeen, B. J. Smith, P. Wasylczyk, A. B. U’Ren, C. Silberhorn, and I. A. Walmsley, “Heralded generation of ultrafast single photons in pure quantum States,” Phys. Rev. Lett. 100(13), 133601 (2008).
[CrossRef] [PubMed]

Zalevsky, Z.

Zhu, S.

Appl. Opt.

Contemp. Phys.

M. G. Raymer, “Measuring the quantum mechanical wave function,” Contemp. Phys. 38(5), 343–355 (1997).
[CrossRef]

Eur. Phys. J. D

T. Aichele, A. I. Lvovsky, and S. Schiller, “Optical mode characterization of single photons prepared by means of conditional measurements on a biphoton state,” Eur. Phys. J. D 18(2), 237–245 (2002).
[CrossRef]

J. Mod. Opt.

S. V. Polyakov and A. L. Migdall, “Quantum radiometry,” J. Mod. Opt. 56(9), 1045–1052 (2009).
[CrossRef]

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

N. J. Phys.

P. J. Mosley, J. S. Lundeen, B. J. Smith, and I. A. Walmsley, “Conditional preparation of single photons using parametric down-conversion: A recipe for purity,” N. J. Phys. 10(9), 093011 (2008).
[CrossRef]

Nature

E. Knill, R. Laflamme, and G. J. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature 409(6816), 46–52 (2001).
[CrossRef] [PubMed]

Opt. Commun.

T. B. Pittman, B. C. Jacobs, and J. D. Franson, “Heralding single photons from pulsed parametric down-conversion,” Opt. Commun. 246(4-6), 545–550 (2005).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev.

A. Einstein, B. Podolsky, and N. Rosen, “Can Quantum-Mechanical Description of Physical Reality be considered complete?” Phys. Rev. 47(10), 777–780 (1935).
[CrossRef]

Phys. Rev. A

C. Monken, P. Ribeiro, and S. Pádua, “Transfer of angular spectrum and image formation in spontaneous parametric down-conversion,” Phys. Rev. A 57(4), 3123–3126 (1998).
[CrossRef]

D. S. Tasca, S. P. Walborn, P. H. Souto Ribeiro, and F. Toscano, “Detection of transverse entanglement in phase space,” Phys. Rev. A 78(1), 010304 (2008).
[CrossRef]

D. Tasca, S. Walborn, P. Souto Ribeiro, F. Toscano, and P. Pellat-Finet, “Propagation of transverse intensity correlations of a two-photon state,” Phys. Rev. A 79(3), 033801 (2009).
[CrossRef]

P. Navez and E. Brambilla, “Spatial entanglement of twin quantum images,” Phys. Rev. A 65, 013813 (2001).
[CrossRef]

J. H. Shapiro, “Computational ghost imaging,” Phys. Rev. A 78(6), 061802 (2008).
[CrossRef]

B. Erkmen and J. Shapiro, “Signal-to-noise ratio of Gaussian-state ghost imaging,” Phys. Rev. A 79(2), 023833 (2009).
[CrossRef]

U. Leonhardt and M. Munroe, “Number of phases required to determine a quantum state in optical homodyne tomography,” Phys. Rev. A 54(4), 3682–3684 (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(4), 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(5), R3429–R3432 (1995).
[CrossRef] [PubMed]

Phys. Rev. Lett.

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(18), 3600–3603 (1995).
[CrossRef] [PubMed]

R. S. Bennink, S. J. Bentley, and R. W. Boyd, ““Two-photon” Coincidence Imaging with a Classical Source,” Phys. Rev. Lett. 89(11), 113601 (2002).
[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(12), 123602 (2001).
[CrossRef] [PubMed]

A. Gatti, E. Brambilla, and L. A. Lugiato, “Entangled imaging and wave-particle duality: from the microscopic to the macroscopic realm,” Phys. Rev. Lett. 90(13), 133603 (2003).
[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(9), 093602 (2004).
[CrossRef] [PubMed]

M. D’Angelo, Y. H. Kim, S. P. Kulik, and Y. Shih, “Identifying entanglement using quantum ghost interference and imaging,” Phys. Rev. Lett. 92(23), 233601 (2004).
[CrossRef] [PubMed]

P. J. Mosley, J. S. Lundeen, B. J. Smith, P. Wasylczyk, A. B. U’Ren, C. Silberhorn, and I. A. Walmsley, “Heralded generation of ultrafast single photons in pure quantum States,” Phys. Rev. Lett. 100(13), 133601 (2008).
[CrossRef] [PubMed]

A. I. Lvovsky, H. Hansen, T. Aichele, O. Benson, J. Mlynek, and S. Schiller, “Quantum state reconstruction of the single-photon Fock state,” Phys. Rev. Lett. 87(5), 050402 (2001).
[CrossRef] [PubMed]

A. Valencia, G. Scarcelli, M. D’Angelo, and Y. Shih, “Two-photon imaging with thermal light,” Phys. Rev. Lett. 94(6), 063601 (2005).
[CrossRef] [PubMed]

W. Wasilewski, P. Kolenderski, and R. Frankowski, “Spectral density matrix of a single photon measured,” Phys. Rev. Lett. 99(12), 123601 (2007).
[CrossRef] [PubMed]

M. G. Raymer, M. Beck, and D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72(8), 1137–1140 (1994).
[CrossRef] [PubMed]

Proc. SPIE

B. Eppich and N. Reng, “Measurement of the Wigner distribution function based on the inverse Radon transformation,” Proc. SPIE 2375, 261–268 (1995).
[CrossRef]

Rev. Mod. Phys.

A. I. Lvovsky and M. G. Raymer, “Continuous-variable optical quantum-state tomography,” Rev. Mod. Phys. 81(1), 299–332 (2009).
[CrossRef]

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79(1), 135–174 (2007).
[CrossRef]

Other

M. G. Raymer, M. Beck, and D. F. McAlister, “Spatial and temporal optical field reconstruction using phase-space tomography,” in Quantum Optics VI, D. F. Walls and J. D. Harvey, eds. (Springer, 1994).

T. Alieva, and M. J. Bastiaans, “Wigner distribution and fractional Fourier transform,” Signal Processing and its Applications, Sixth International Symposium on 2001, 1, 168–169 (2001).

D. N. Klyshko, Photons and Nonlinear Optics (Gordon and Breach, 1988).

M. Born, and E. Wolf, Principles of Optics, 6th Ed. (Cambridge University Press, 1993).

T. Anhut, B. Karamata, T. Lasser, M. G. Raymer, and L. Wenke, “Measurement of scattered light Wigner functions by phase space tomography and implications for parallel OCT,” in Coherence Domain Optical Methods and Optical Coherence Tomography in Biomedicine VII. Edited by Tuchin, Valery V.; Izatt, Joseph A.; Fujimoto, James G. Proceedings of the SPIE, Volume 4956, 120–128 (2003).

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Figures (9)

Fig. 1
Fig. 1

Relation between complex field distributions p ' s and  q ' s.

Fig. 2
Fig. 2

(a) The signal field generated from the pumped BBO crystal with specific κ-value κ 0 will be selected by a pinhole at the focal plane followed by a signal detector. The idler field will be collapsed to a single-photon state obtained from the original two-photon state function, as in Eq. (40). (b) The advanced wave picture, as in Eq. (33).

Fig. 3
Fig. 3

Experimental set-up. The type-I BBO is for the SHG and the type-II BBO is for SPDC. The signal and the idler beams are separated by the polarizing beam splitter (PBS). The lens with f = 300 mm is the collection lens and the lens with f = 200 mm is the imaging lens. APD-1 is moved to several longitudinal distances from the lens and then scanned along the transverse direction.

Fig. 4
Fig. 4

The assumed field distribution at the double slit, based on direct measurement of signal photons count rates at the detector APD-2 when one of the two slits was blocked alternatively.

Fig. 5
Fig. 5

Predicted Wigner distribution function calculated from the assumed field distribution in Fig. 4.

Fig. 6
Fig. 6

The Radon transform of the WDF in Fig. 5. Marginal distributions are calculated along the lines (a), (b),…,(f).

Fig. 8
Fig. 8

Experimental data and the theoretical curves for the marginal distributions at several longitudinal distances. The solid lines are theoretical curves and the filled dots are experimental data.

Fig. 7
Fig. 7

The FrFT angle (WDF rotation angle) as a function of the detector position d ' f .

Fig. 9
Fig. 9

The thin lens is represented by a two directional arrow.

Equations (62)

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

p 2 ( x ) = e i 2 π d / λ ( i λ d ) 1 / 2 exp [ i π ( x x ' ) 2 / λ d ] p 1 ( x ' ) d x ' = e i 2 π d / λ ( i λ d ) 1 / 2 exp [ i ( π / λ d ) ( x 2 2 x x ' + x ' 2 ) ] p 1 ( x ' ) d x ' .
p 1 ( x ' ) = q 1 ( x ' ) exp ( i π x ' 2 / λ R 1 )         ( R 1 < 0 )
p 2 ( x ) = q 2 ( x ) exp ( i π x 2 / λ R 2 )         ( R 2 > 0 ) .
q 2 ( x ) = e i 2 π d / λ ( i λ d ) 1 / 2 exp [ i π λ d ( x 2 ( 1 d R 2 ) 2 x x ' + x ' 2 ( 1 + d R 1 ) ) ] q 1 ( x ' ) d x ' .
g 1 1 + d / R 1 ,     g 2 1 d / R 2
q 2 ( ρ ) = e i 2 π d / λ s 1 ( i λ d ) 1 / 2 exp [ i π λ d ( g 2 s 2 2 ρ 2 2 s 1 s 2 ρ ρ ' + g 1 s 1 2 ρ ' 2 ) ] q 1 ( ρ ' ) d ρ '    .
g 1 s 1 2 λ d = cot φ ,     g 2 s 2 2 λ d = cot φ ,   and    s 1 s 2 λ d = csc φ    ,
q 2 ( ρ ) = e i 2 π d / λ s 1 ( i λ d ) 1 / 2 exp [ i π ( ρ 2 cot 2 φ 2 ρ ρ ' csc 2 φ + ρ ' 2 cot 2 φ ) ] q 1 ( ρ ' ) d ρ ' = { e i 2 π d / λ s 1 e ( π φ ^ / 4 φ / 2 ) | sin φ | 1 / 2 i λ d } × F a [ q 1 ( ρ ' ) ]    ,
F a [ q ( ρ ) ] e i ( π φ ^ / 4 φ / 2 ) | sin φ | 1 / 2 exp [ i π ( ρ 2 cot φ 2 ρ ρ ' csc φ + ρ ' 2 cot φ ) ] q ( ρ ' ) d ρ ' = B a ( ρ , ρ ' ) q ( ρ ' ) d ρ ' .
B a ( ρ , ρ ' ) = exp [ i ( π φ ^ / 4 φ / 2 ) ] | sin φ | 1 / 2 exp [ i π ( ρ 2 cot φ 2 ρ ρ ' csc φ + ρ ' 2 cot φ ) ]   
B a ( ρ , ρ ' ) = B a ( ρ ' , ρ )   (symmetric) F a 2 { F a 1 [ q ( ρ ' ) ] } = F a 2 + a 1 [ q ( ρ ' ) ]   (additive) or  d ρ B a 1 ( ρ ' , ρ ) B a 2 ( ρ , ρ ' ' ) = B a 1 + a 2 ( ρ ' , ρ ' ' )    .
| q 2 ( ρ ) | 2 = { s 1 2 | sin φ | λ d } × | F a [ q 1 ( ρ ' ) ] | 2 = | s 1 s 2 | | F a [ q 1 ( ρ ' ) ] | 2     .
| p 1 ( x ' ) | 2 = | q 1 ( x ' ) | 2 , | p 2 ( x ) | 2 = | q 2 ( x ) | 2
W p 1 ( x , k ) = p 1 ( x + s / 2 ) p 1 * ( x s / 2 ) e 2 π i s k d s .
g ( x ' , φ ) = [ f ( x , y ) ] = f ( x ' cos φ x ' ' sin φ , x ' sin φ + x ' ' cos φ ) d x ' ' ,
g p 1 ( x ' , φ ) = [ W p 1 ( x , k ) ] = W p 1 ( x ' cos φ k ' sin φ , x ' sin φ + k ' cos φ ) d k ' ,
| F a [ p 1 ( x ) ] | 2 = g p 1 ( x ' , φ = a π / 2 ) .
| Ψ = d κ s d κ i f ( κ s + κ i ) a ^ ( κ s ) a ^ ( κ i ) | 0 ,
E ^ ( x ) = E ^ ( + ) ( x ) + E ^ ( ) ( x ) ,
E ^ ( + ) ( x ) = d κ C E ( ω κ ) a ^ ( κ ) exp [ i κ x ]
E ^ ( ) ( x ) = d κ C E * ( ω κ ) a ^ ( κ ) exp [ i κ x ] ,
E ( + ) ( x ) = C E d κ   a ^ ( κ ) exp [ i κ x ] .
0 | E ^ s ( + ) ( x s ) E ^ i ( + ) ( x i ) | Ψ = 2 ( 2 π ) 2 C E 2 A ( x s ) δ ( x s x i )
0 | E ^ s ( + ) ( κ s ) E ^ i ( + ) ( κ i ) | Ψ = 2 C E 2 f ( κ s + κ i ) ,
E ^ ( + ) ( κ ) = C E a ^ ( κ ) and f ( κ ) = d x A ( x ) exp [ i κ x ]
E ( + ) ( ρ 1 ) = F a [ E ( + ) ( ρ 0 ) ] = d ρ 0 B a ( ρ 1 , ρ 0 ) E ( + ) ( ρ 0 ) ,
E ^ ( + ) ( ρ 1 ) = d ρ 0 B ( ρ 0 , ρ 1 ) E ^ ( + ) ( ρ 0 ) ,
E ^ ( + ) ( ρ 1 ) = d ρ 0 B ( ρ 0 , ρ 1 ) E ^ ( + ) ( ρ 0 ) .
E ^ s ( + ) ( ρ 1 ) = d ρ s B a 1 ( ρ s , ρ 1 ) E ^ s ( + ) ( ρ s ) ,
E ^ i ( + ) ( ρ 2 ) = d ρ i B a 2 ( ρ i , ρ 2 ) E ^ i ( + ) ( ρ i ) ,
0 | E ^ s ( + ) ( ρ 1 ) E ^ i ( + ) ( ρ 2 ) | Ψ = 0 | ( d ρ s B a 1 ( ρ s , ρ 1 ) E ^ s ( + ) ( ρ s ) ) ( d ρ i B a 2 ( ρ i , ρ 2 ) E ^ i ( + ) ( ρ i ) ) | Ψ = d ρ i d ρ s B a 1 ( ρ s , ρ 1 ) B a 2 ( ρ i , ρ 2 ) 0 | E ^ s ( + ) ( ρ s ) E ^ i ( + ) ( ρ i ) | Ψ .
0 | E ^ s ( + ) ( ρ s ) E ^ i ( + ) ( ρ i ) | Ψ = A ' δ ( ρ s ρ i ) , A ' = 2 ( 2 π ) 2 C E 2 × c o n s t . ,
0 | E ^ s ( + ) ( ρ 1 ) E ^ i ( + ) ( ρ 2 ) | Ψ = A ' d ρ i d ρ s B a 1 ( ρ s , ρ 1 ) B a 2 ( ρ i , ρ 2 ) δ ( ρ s ρ i ) = A ' d ρ i B a 1 ( ρ i , ρ 1 ) B a 2 ( ρ i , ρ 2 ) = A ' d ρ i B a 1 ( ρ 1 , ρ i ) B a 2 ( ρ i , ρ 2 ) = A ' B a 1 + a 2 ( ρ 1 , ρ 2 ) .
E ^ s ' ( ρ 1 ) = A ( ρ 1 ) E ^ s ( ρ 1 )
0 | E ^ s ' ( + ) ( ρ 1 ) E ^ i ( + ) ( ρ 2 ) | Ψ = A ( ρ 1 ) 0 | E ^ s ( + ) ( ρ 1 ) E ^ i ( + ) ( ρ 2 ) | Ψ .
0 | E ^ s ' ( + ) ( ρ 1 ) E ^ i ( + ) ( ρ 2 ) | Ψ = A ' A ( ρ 1 ) B a 1 + a 2 ( ρ 1 , ρ 2 ) .
E ^ s , D ( + ) ( ρ D ) = d ρ 1 E ^ s ( + ) ( ρ 1 ) exp [ i b ρ 1 ρ D ] , b 1 λ f .
0 | E ^ s , D ( + ) ( ρ D ) E ^ i ( + ) ( ρ 2 ) | Ψ = 0 | ( d ρ 1 E ^ s ( + ) ( ρ 1 ) exp [ i b ρ 1 ρ D ] ) E ^ i ( + ) ( ρ 2 ) | Ψ = d ρ 1 0 | E ^ s ( + ) ( ρ 1 ) E ^ i ( + ) ( ρ 2 ) | Ψ exp [ i b ρ 1 ρ D ] .
0 | E ^ s , D ( + ) ( ρ D ) E ^ i ( + ) ( ρ 2 ) | Ψ = d ρ 1 C E 2 A ( ρ 1 ) B a 1 + a 2 ( ρ 1 , ρ 2 ) exp [ i b ρ 1 ρ D ] .
0 | E ^ s , D ( + ) ( ρ D ) E ^ i ( + ) ( ρ 2 ) | Ψ = C E 2 d ρ 1 A ( ρ 1 ) δ ( ρ 1 + ρ 2 ) exp [ i b ρ 1 ρ D ] = C E 2 A ( ρ 2 ) exp [ i b ρ 2 ρ D ] .
| 0 | E ^ s , D ( + ) ( ρ D ) E ^ i ( + ) ( ρ 2 ) | Ψ | 2 = | C E | 4 | A ( ρ 2 ) | 2 ,
0 | E ^ s , D ( + ) ( 0 ) E ^ i ( + ) ( ρ 2 ) | Ψ = d ρ 1 C E 2 A ( ρ 1 ) B a 1 + a 2 ( ρ 1 , ρ 2 ) = C E 2 F a 1 + a 2 [ A ( ρ 1 ) ] .
| Ψ c o l l a p s e d | κ 0 κ 0 | d κ s d κ i f ( κ s + κ i ) a ^ ( κ s ) a ^ ( κ i ) | 0 = | κ 0 d κ i f ( κ 0 + κ i ) a ^ ( κ i ) | 0 .
μ 12 = ( 2 J 1 ( v ) v ) e i ψ .
v = 2 π λ ¯ ρ R ( X 1 X 2 ) 2 + ( Y 1 Y 2 ) 2 , ψ = 2 π λ ¯ [ ( X 1 2 + Y 1 2 ) ( X 2 2 + Y 2 2 ) 2 R ] .
cot φ o = s o 2 λ d o
cot φ ' = λ d o 2 s o 2 [ 1 d o + 1 d ' 1 f ] + s o 2 λ [ 1 d ' 1 f ] .
φ = tan 1 ( λ d s 1 2 ) = tan 1 [ λ ( z z 0 ) π w 2 ] = tan 1 [ ( z z 0 ) z R ] ,
s 2 2 = ( λ d ) 2 s 1 2 + s 1 2
s 2 2 s 1 2 = 1 + d 2 s 1 4 / λ 2 = 1 + ( z z 0 ) 2 z R 2 ,
s 2 = s 1 1 + ( z z 0 ) 2 z R 2 = s 1 cos φ ,
R 2 = ( z z 0 ) [ 1 + z R 2 ( z z 0 ) 2 ] ,
q + ( x ) = q ( x ) exp ( i π x 2 / λ f ) ,
p ( x ) = q ( x ) exp ( i π x 2 / λ R ) ,
p + ( x ) = p ( x ) exp ( i π x 2 / λ f ) = q ( x ) exp [ i π x 2 λ ( 1 R 1 f ) ] ,
1 R + = 1 R 1 f .
φ 0 = tan 1 ( λ d 0 s 0 2 ) .
s = ( λ d 0 ) 2 s 0 2 + s 0 2 ,
R = d 0 ( 1 + s 0 4 λ 2 d 0 2 ) ,
g + = 1 + d ' R +
cot φ ' = g + s + 2 λ d ' .
cot φ ' = λ d o 2 s o 2 [ 1 d o + 1 d ' 1 f ] + s o 2 λ [ 1 d ' 1 f ] .

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