Abstract

We report a theoretical study of spatial and spectral profiles of twin photons generated with a pump beam having a partially spatially coherent nature. These profiles are simulated for both type-I and type-II spontaneous parametric down conversion in collinear and non-collinear configurations and found to be highly dependent on the choice of beam waist and transverse correlation length of the pump beam. It is found that by suitable selection of coherence length and size of the pump beam, the asymmetry in spatial geometry of down-converted photons and the spectral width can be modified. Owing to the partially spatially coherent nature, these biphoton beams might offer more robustness against losses compared to their fully coherent counterparts, and thus could yield a high success rate in free-space quantum communication and in quantum key distribution.

© 2020 Optical Society of America

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References

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    [Crossref]
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  4. P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, “New high-intensity source of polarization-entangled photon pairs,” Phys. Rev. Lett. 75, 4337 (1995).
    [Crossref]
  5. V. V. na Hernández, H. Cruz-Ramírez, R. Ramírez-Alarcón, and A. B. U’Ren, “Classical to quantum transfer of optical vortices,” Opt. Express 22, 20027–20037 (2014).
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  12. P. G. Kwiat, E. Waks, A. G. White, I. Appelbaum, and P. H. Eberhard, “Ultrabright source of polarization-entangled photons,” Phys. Rev. A 60, R773–R776 (1999).
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    [Crossref]
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  28. H. Shun Poh, J. Lim, I. Marcikic, A. Lamas-Linares, and C. Kurtsiefer, “Eliminating spectral distinguishability in ultrafast spontaneous parametric down-conversion,” Phys. Rev. A 80, 043815 (2009).
  29. Y.-H. Kim and W. P. Grice, “Measurement of the spectral properties of the two-photon state generated via type-II spontaneous parametric downconversion,” Opt. Lett. 30, 908–910 (2005).
    [Crossref]
  30. 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]
  31. P. S. K. Lee, M. P. van Exter, and J. P. Woerdman, “How focused pumping affects type-II spontaneous parametric down-conversion,” Phys. Rev. A 72, 033803 (2005).
    [Crossref]
  32. Y. Jeronimo-Moreno and R. Jáuregui, “Type-I parametric down conversion of highly focused Gaussian beams in finite length crystals,” J. Opt. 16, 065201 (2014).
    [Crossref]
  33. J.-C. Lee and Y.-H. Kim, “Spatial and spectral properties of entangled photons from spontaneous parametric down-conversion with a focused pump,” Opt. Commun. 366, 442–450 (2016).
    [Crossref]
  34. R. Ramírez-Alarcón, H. Cruz-Ramírez, and A. B. U’Ren, “Effects of crystal length on the angular spectrum of spontaneous parametric downconversion photon pairs,” Laser Phys. 23, 055204 (2013).
    [Crossref]
  35. J. P. Torres, G. Molina-Terriza, and L. Torner, “The spatial shape of entangled photon states generated in non-collinear, walking parametric downconversion,” J. Opt. B 7, 235 (2005).
    [Crossref]
  36. R. S. Bennink, Y. Liu, D. D. Earl, and W. P. Grice, “Spatial distinguishability of photons produced by spontaneous parametric down-conversion with a focused pump,” Phys. Rev. A 74, 023802 (2006).
    [Crossref]
  37. H. Di Lorenzo Pires, F. M. G. J. Coppens, and M. P. van Exter, “Type-I spontaneous parametric down-conversion with a strongly focused pump,” Phys. Rev. A 83, 033837 (2011).
    [Crossref]
  38. V. G. Dmitriev, G. G. Gurzadyan, and D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals, (Springer, 2013), Vol. 64.
  39. J. T. Foley, “Effect of an aperture on the spectrum of partially coherent light,” J. Opt. Soc. Am. A 8, 1099–1105 (1991).
    [Crossref]
  40. S. Carrasco, J. P. Torres, L. Torner, A. Sergienko, B. E. A. Saleh, and M. C. Teich, “Spatial-to-spectral mapping in spontaneous parametric down-conversion,” Phys. Rev. A 70, 043817 (2004).
    [Crossref]

2019 (2)

2018 (1)

E. Giese, R. Fickler, W. Zhang, L. Chen, and R. W. Boyd, “Influence of pump coherence on the quantum properties of spontaneous parametric down-conversion,” Phys. Scripta 93, 084001 (2018).
[Crossref]

2016 (1)

J.-C. Lee and Y.-H. Kim, “Spatial and spectral properties of entangled photons from spontaneous parametric down-conversion with a focused pump,” Opt. Commun. 366, 442–450 (2016).
[Crossref]

2015 (1)

M. W. Hyde, S. Basu, D. G. Voelz, and X. Xiao, “Experimentally generating any desired partially coherent Schell-model source using phase-only control,” J. Appl. Phys. 118, 093102 (2015).
[Crossref]

2014 (3)

2013 (1)

R. Ramírez-Alarcón, H. Cruz-Ramírez, and A. B. U’Ren, “Effects of crystal length on the angular spectrum of spontaneous parametric downconversion photon pairs,” Laser Phys. 23, 055204 (2013).
[Crossref]

2011 (1)

H. Di Lorenzo Pires, F. M. G. J. Coppens, and M. P. van Exter, “Type-I spontaneous parametric down-conversion with a strongly focused pump,” Phys. Rev. A 83, 033837 (2011).
[Crossref]

2010 (1)

A. Jha and R. Boyd, “Spatial two-photon coherence of the entangled field produced by down-conversion using a partially spatially coherent pump beam,” Phys. Rev. A 81, 013828 (2010).
[Crossref]

2009 (2)

H. Shun Poh, J. Lim, I. Marcikic, A. Lamas-Linares, and C. Kurtsiefer, “Eliminating spectral distinguishability in ultrafast spontaneous parametric down-conversion,” Phys. Rev. A 80, 043815 (2009).

R. Rangarajan, M. Goggin, and P. Kwiat, “Optimizing type-I polarization-entangled photons,” Opt. Express 17, 18920–18933 (2009).
[Crossref]

2008 (2)

X. Ji, E. Zhang, and B. Lü, “Superimposed partially coherent beams propagating through atmospheric turbulence,” J. Opt. Soc. Am. B 25, 825–833 (2008).
[Crossref]

S.-Y. Baek and Y.-H. Kim, “Spectral properties of entangled photon pairs generated via frequency-degenerate type-I spontaneous parametric down-conversion,” Phys. Rev. A 77, 043807 (2008).
[Crossref]

2007 (2)

M. Fiorentino, S. M. Spillane, R. G. Beausoleil, T. D. Roberts, P. Battle, and M. W. Munro, “Spontaneous parametric down-conversion in periodically poled KTP waveguides and bulk crystals,” Opt. Express 15, 7479–7488 (2007).
[Crossref]

H. Shun Poh, C. Yang Lum, I. Marcikic, A. Lamas-Linares, and C. Kurtsiefer, “Joint spectrum mapping of polarization entanglement in spontaneous parametric down-conversion,” Phys. Rev. A 75, 043816 (2007).

2006 (1)

R. S. Bennink, Y. Liu, D. D. Earl, and W. P. Grice, “Spatial distinguishability of photons produced by spontaneous parametric down-conversion with a focused pump,” Phys. Rev. A 74, 023802 (2006).
[Crossref]

2005 (4)

J. P. Torres, G. Molina-Terriza, and L. Torner, “The spatial shape of entangled photon states generated in non-collinear, walking parametric downconversion,” J. Opt. B 7, 235 (2005).
[Crossref]

P. S. K. Lee, M. P. van Exter, and J. P. Woerdman, “How focused pumping affects type-II spontaneous parametric down-conversion,” Phys. Rev. A 72, 033803 (2005).
[Crossref]

Y.-H. Kim and W. P. Grice, “Measurement of the spectral properties of the two-photon state generated via type-II spontaneous parametric downconversion,” Opt. Lett. 30, 908–910 (2005).
[Crossref]

J. Kim, D. Miller, E. Kim, S. Oh, J. Oh, and T. E. Milner, “Optical coherence tomography speckle reduction by a partially spatially coherent source,” J. Biomed. Opt. 10, 064034 (2005).
[Crossref]

2004 (1)

S. Carrasco, J. P. Torres, L. Torner, A. Sergienko, B. E. A. Saleh, and M. C. Teich, “Spatial-to-spectral mapping in spontaneous parametric down-conversion,” Phys. Rev. A 70, 043817 (2004).
[Crossref]

2002 (1)

1999 (1)

P. G. Kwiat, E. Waks, A. G. White, I. Appelbaum, and P. H. Eberhard, “Ultrabright source of polarization-entangled photons,” Phys. Rev. A 60, R773–R776 (1999).
[Crossref]

1996 (2)

M. H. Rubin, “Transverse correlation in optical spontaneous parametric down-conversion,” Phys. Rev. A 54, 5349 (1996).
[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]

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]

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, “New high-intensity source of polarization-entangled photon pairs,” Phys. Rev. Lett. 75, 4337 (1995).
[Crossref]

1992 (1)

C. H. Bennett, G. Brassard, and N. D. Mermin, “Quantum cryptography without bell’s theorem,” Phys. Rev. Lett. 68, 557–559 (1992).
[Crossref]

1991 (2)

A. K. Ekert, “Quantum cryptography based on bell’s theorem,” Phys. Rev. Lett. 67, 661–663 (1991).
[Crossref]

J. T. Foley, “Effect of an aperture on the spectrum of partially coherent light,” J. Opt. Soc. Am. A 8, 1099–1105 (1991).
[Crossref]

1985 (1)

C. K. Hong and L. Mandel, “Theory of parametric frequency down conversion of light,” Phys. Rev. A 31, 2409–2418 (1985).
[Crossref]

1980 (1)

E. Collett and E. Wolf, “Beams generated by Gaussian quasi-homogeneous sources,” Opt. Comm. 32, 27–31 (1980).
[Crossref]

1970 (1)

T. Asakura, “Spatial coherence of laser light passed through rotating ground glass,” Opto-Electronics 2, 115–123 (1970).
[Crossref]

Appelbaum, I.

P. G. Kwiat, E. Waks, A. G. White, I. Appelbaum, and P. H. Eberhard, “Ultrabright source of polarization-entangled photons,” Phys. Rev. A 60, R773–R776 (1999).
[Crossref]

Asakura, T.

T. Asakura, “Spatial coherence of laser light passed through rotating ground glass,” Opto-Electronics 2, 115–123 (1970).
[Crossref]

Baek, S.-Y.

S.-Y. Baek and Y.-H. Kim, “Spectral properties of entangled photon pairs generated via frequency-degenerate type-I spontaneous parametric down-conversion,” Phys. Rev. A 77, 043807 (2008).
[Crossref]

Basu, S.

M. W. Hyde, S. Basu, D. G. Voelz, and X. Xiao, “Experimentally generating any desired partially coherent Schell-model source using phase-only control,” J. Appl. Phys. 118, 093102 (2015).
[Crossref]

Battle, P.

Beausoleil, R. G.

Bennett, C. H.

C. H. Bennett, G. Brassard, and N. D. Mermin, “Quantum cryptography without bell’s theorem,” Phys. Rev. Lett. 68, 557–559 (1992).
[Crossref]

Bennink, R. S.

R. S. Bennink, Y. Liu, D. D. Earl, and W. P. Grice, “Spatial distinguishability of photons produced by spontaneous parametric down-conversion with a focused pump,” Phys. Rev. A 74, 023802 (2006).
[Crossref]

Boyd, R.

A. Jha and R. Boyd, “Spatial two-photon coherence of the entangled field produced by down-conversion using a partially spatially coherent pump beam,” Phys. Rev. A 81, 013828 (2010).
[Crossref]

Boyd, R. W.

W. Zhang, R. Fickler, E. Giese, L. Chen, and R. W. Boyd, “Influence of pump coherence on the generation of position-momentum entanglement in optical parametric down-conversion,” Opt. Express 27, 20745–20753 (2019).
[Crossref]

E. Giese, R. Fickler, W. Zhang, L. Chen, and R. W. Boyd, “Influence of pump coherence on the quantum properties of spontaneous parametric down-conversion,” Phys. Scripta 93, 084001 (2018).
[Crossref]

Brassard, G.

C. H. Bennett, G. Brassard, and N. D. Mermin, “Quantum cryptography without bell’s theorem,” Phys. Rev. Lett. 68, 557–559 (1992).
[Crossref]

Carrasco, S.

S. Carrasco, J. P. Torres, L. Torner, A. Sergienko, B. E. A. Saleh, and M. C. Teich, “Spatial-to-spectral mapping in spontaneous parametric down-conversion,” Phys. Rev. A 70, 043817 (2004).
[Crossref]

Chen, L.

W. Zhang, R. Fickler, E. Giese, L. Chen, and R. W. Boyd, “Influence of pump coherence on the generation of position-momentum entanglement in optical parametric down-conversion,” Opt. Express 27, 20745–20753 (2019).
[Crossref]

E. Giese, R. Fickler, W. Zhang, L. Chen, and R. W. Boyd, “Influence of pump coherence on the quantum properties of spontaneous parametric down-conversion,” Phys. Scripta 93, 084001 (2018).
[Crossref]

Collett, E.

E. Collett and E. Wolf, “Beams generated by Gaussian quasi-homogeneous sources,” Opt. Comm. 32, 27–31 (1980).
[Crossref]

Coppens, F. M. G. J.

H. Di Lorenzo Pires, F. M. G. J. Coppens, and M. P. van Exter, “Type-I spontaneous parametric down-conversion with a strongly focused pump,” Phys. Rev. A 83, 033837 (2011).
[Crossref]

Cruz-Ramírez, H.

V. V. na Hernández, H. Cruz-Ramírez, R. Ramírez-Alarcón, and A. B. U’Ren, “Classical to quantum transfer of optical vortices,” Opt. Express 22, 20027–20037 (2014).
[Crossref]

R. Ramírez-Alarcón, H. Cruz-Ramírez, and A. B. U’Ren, “Effects of crystal length on the angular spectrum of spontaneous parametric downconversion photon pairs,” Laser Phys. 23, 055204 (2013).
[Crossref]

Defienne, H.

H. Defienne and S. Gigan, “Spatially entangled photon-pair generation using a partial spatially coherent pump beam,” Phys. Rev. A 99, 053831 (2019).
[Crossref]

Di Lorenzo Pires, H.

H. Di Lorenzo Pires, F. M. G. J. Coppens, and M. P. van Exter, “Type-I spontaneous parametric down-conversion with a strongly focused pump,” Phys. Rev. A 83, 033837 (2011).
[Crossref]

Dmitriev, V. G.

V. G. Dmitriev, G. G. Gurzadyan, and D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals, (Springer, 2013), Vol. 64.

Earl, D. D.

R. S. Bennink, Y. Liu, D. D. Earl, and W. P. Grice, “Spatial distinguishability of photons produced by spontaneous parametric down-conversion with a focused pump,” Phys. Rev. A 74, 023802 (2006).
[Crossref]

Eberhard, P. H.

P. G. Kwiat, E. Waks, A. G. White, I. Appelbaum, and P. H. Eberhard, “Ultrabright source of polarization-entangled photons,” Phys. Rev. A 60, R773–R776 (1999).
[Crossref]

Ekert, A. K.

A. K. Ekert, “Quantum cryptography based on bell’s theorem,” Phys. Rev. Lett. 67, 661–663 (1991).
[Crossref]

Fickler, R.

W. Zhang, R. Fickler, E. Giese, L. Chen, and R. W. Boyd, “Influence of pump coherence on the generation of position-momentum entanglement in optical parametric down-conversion,” Opt. Express 27, 20745–20753 (2019).
[Crossref]

E. Giese, R. Fickler, W. Zhang, L. Chen, and R. W. Boyd, “Influence of pump coherence on the quantum properties of spontaneous parametric down-conversion,” Phys. Scripta 93, 084001 (2018).
[Crossref]

Fiorentino, M.

Foley, J. T.

Fox, M.

M. Fox, Quantum Optics: An Introduction (OUP Oxford, 2006), Vol. 15.

Gbur, G.

Giese, E.

W. Zhang, R. Fickler, E. Giese, L. Chen, and R. W. Boyd, “Influence of pump coherence on the generation of position-momentum entanglement in optical parametric down-conversion,” Opt. Express 27, 20745–20753 (2019).
[Crossref]

E. Giese, R. Fickler, W. Zhang, L. Chen, and R. W. Boyd, “Influence of pump coherence on the quantum properties of spontaneous parametric down-conversion,” Phys. Scripta 93, 084001 (2018).
[Crossref]

Gigan, S.

H. Defienne and S. Gigan, “Spatially entangled photon-pair generation using a partial spatially coherent pump beam,” Phys. Rev. A 99, 053831 (2019).
[Crossref]

Goggin, M.

Grice, W. P.

R. S. Bennink, Y. Liu, D. D. Earl, and W. P. Grice, “Spatial distinguishability of photons produced by spontaneous parametric down-conversion with a focused pump,” Phys. Rev. A 74, 023802 (2006).
[Crossref]

Y.-H. Kim and W. P. Grice, “Measurement of the spectral properties of the two-photon state generated via type-II spontaneous parametric downconversion,” Opt. Lett. 30, 908–910 (2005).
[Crossref]

Gurzadyan, G. G.

V. G. Dmitriev, G. G. Gurzadyan, and D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals, (Springer, 2013), Vol. 64.

Hong, C. K.

C. K. Hong and L. Mandel, “Theory of parametric frequency down conversion of light,” Phys. Rev. A 31, 2409–2418 (1985).
[Crossref]

Hyde, M. W.

M. W. Hyde, S. Basu, D. G. Voelz, and X. Xiao, “Experimentally generating any desired partially coherent Schell-model source using phase-only control,” J. Appl. Phys. 118, 093102 (2015).
[Crossref]

Jáuregui, R.

Y. Jeronimo-Moreno and R. Jáuregui, “Type-I parametric down conversion of highly focused Gaussian beams in finite length crystals,” J. Opt. 16, 065201 (2014).
[Crossref]

Jeronimo-Moreno, Y.

Y. Jeronimo-Moreno and R. Jáuregui, “Type-I parametric down conversion of highly focused Gaussian beams in finite length crystals,” J. Opt. 16, 065201 (2014).
[Crossref]

Jha, A.

A. Jha and R. Boyd, “Spatial two-photon coherence of the entangled field produced by down-conversion using a partially spatially coherent pump beam,” Phys. Rev. A 81, 013828 (2010).
[Crossref]

Ji, X.

Kanseri, B.

B. Kanseri, Optical Coherence and Polarization: An Experimental Outlook (Lambert Academic Publishing, 2013).

Kim, E.

J. Kim, D. Miller, E. Kim, S. Oh, J. Oh, and T. E. Milner, “Optical coherence tomography speckle reduction by a partially spatially coherent source,” J. Biomed. Opt. 10, 064034 (2005).
[Crossref]

Kim, J.

J. Kim, D. Miller, E. Kim, S. Oh, J. Oh, and T. E. Milner, “Optical coherence tomography speckle reduction by a partially spatially coherent source,” J. Biomed. Opt. 10, 064034 (2005).
[Crossref]

Kim, Y.-H.

J.-C. Lee and Y.-H. Kim, “Spatial and spectral properties of entangled photons from spontaneous parametric down-conversion with a focused pump,” Opt. Commun. 366, 442–450 (2016).
[Crossref]

S.-Y. Baek and Y.-H. Kim, “Spectral properties of entangled photon pairs generated via frequency-degenerate type-I spontaneous parametric down-conversion,” Phys. Rev. A 77, 043807 (2008).
[Crossref]

Y.-H. Kim and W. P. Grice, “Measurement of the spectral properties of the two-photon state generated via type-II spontaneous parametric downconversion,” Opt. Lett. 30, 908–910 (2005).
[Crossref]

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, 2804–2815 (1996).
[Crossref]

Kurtsiefer, C.

H. Shun Poh, J. Lim, I. Marcikic, A. Lamas-Linares, and C. Kurtsiefer, “Eliminating spectral distinguishability in ultrafast spontaneous parametric down-conversion,” Phys. Rev. A 80, 043815 (2009).

H. Shun Poh, C. Yang Lum, I. Marcikic, A. Lamas-Linares, and C. Kurtsiefer, “Joint spectrum mapping of polarization entanglement in spontaneous parametric down-conversion,” Phys. Rev. A 75, 043816 (2007).

Kwiat, P.

Kwiat, P. G.

P. G. Kwiat, E. Waks, A. G. White, I. Appelbaum, and P. H. Eberhard, “Ultrabright source of polarization-entangled photons,” Phys. Rev. A 60, R773–R776 (1999).
[Crossref]

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, “New high-intensity source of polarization-entangled photon pairs,” Phys. Rev. Lett. 75, 4337 (1995).
[Crossref]

Lamas-Linares, A.

H. Shun Poh, J. Lim, I. Marcikic, A. Lamas-Linares, and C. Kurtsiefer, “Eliminating spectral distinguishability in ultrafast spontaneous parametric down-conversion,” Phys. Rev. A 80, 043815 (2009).

H. Shun Poh, C. Yang Lum, I. Marcikic, A. Lamas-Linares, and C. Kurtsiefer, “Joint spectrum mapping of polarization entanglement in spontaneous parametric down-conversion,” Phys. Rev. A 75, 043816 (2007).

Lee, J.-C.

J.-C. Lee and Y.-H. Kim, “Spatial and spectral properties of entangled photons from spontaneous parametric down-conversion with a focused pump,” Opt. Commun. 366, 442–450 (2016).
[Crossref]

Lee, P. S. K.

P. S. K. Lee, M. P. van Exter, and J. P. Woerdman, “How focused pumping affects type-II spontaneous parametric down-conversion,” Phys. Rev. A 72, 033803 (2005).
[Crossref]

Lim, J.

H. Shun Poh, J. Lim, I. Marcikic, A. Lamas-Linares, and C. Kurtsiefer, “Eliminating spectral distinguishability in ultrafast spontaneous parametric down-conversion,” Phys. Rev. A 80, 043815 (2009).

Liu, Y.

R. S. Bennink, Y. Liu, D. D. Earl, and W. P. Grice, “Spatial distinguishability of photons produced by spontaneous parametric down-conversion with a focused pump,” Phys. Rev. A 74, 023802 (2006).
[Crossref]

Lü, B.

Mandel, L.

C. K. Hong and L. Mandel, “Theory of parametric frequency down conversion of light,” Phys. Rev. A 31, 2409–2418 (1985).
[Crossref]

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

Marcikic, I.

H. Shun Poh, J. Lim, I. Marcikic, A. Lamas-Linares, and C. Kurtsiefer, “Eliminating spectral distinguishability in ultrafast spontaneous parametric down-conversion,” Phys. Rev. A 80, 043815 (2009).

H. Shun Poh, C. Yang Lum, I. Marcikic, A. Lamas-Linares, and C. Kurtsiefer, “Joint spectrum mapping of polarization entanglement in spontaneous parametric down-conversion,” Phys. Rev. A 75, 043816 (2007).

Mattle, K.

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, “New high-intensity source of polarization-entangled photon pairs,” Phys. Rev. Lett. 75, 4337 (1995).
[Crossref]

Mermin, N. D.

C. H. Bennett, G. Brassard, and N. D. Mermin, “Quantum cryptography without bell’s theorem,” Phys. Rev. Lett. 68, 557–559 (1992).
[Crossref]

Miller, D.

J. Kim, D. Miller, E. Kim, S. Oh, J. Oh, and T. E. Milner, “Optical coherence tomography speckle reduction by a partially spatially coherent source,” J. Biomed. Opt. 10, 064034 (2005).
[Crossref]

Milner, T. E.

J. Kim, D. Miller, E. Kim, S. Oh, J. Oh, and T. E. Milner, “Optical coherence tomography speckle reduction by a partially spatially coherent source,” J. Biomed. Opt. 10, 064034 (2005).
[Crossref]

Molina-Terriza, G.

J. P. Torres, G. Molina-Terriza, and L. Torner, “The spatial shape of entangled photon states generated in non-collinear, walking parametric downconversion,” J. Opt. B 7, 235 (2005).
[Crossref]

Munro, M. W.

na Hernández, V. V.

Nikogosyan, D. N.

V. G. Dmitriev, G. G. Gurzadyan, and D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals, (Springer, 2013), Vol. 64.

Oh, J.

J. Kim, D. Miller, E. Kim, S. Oh, J. Oh, and T. E. Milner, “Optical coherence tomography speckle reduction by a partially spatially coherent source,” J. Biomed. Opt. 10, 064034 (2005).
[Crossref]

Oh, S.

J. Kim, D. Miller, E. Kim, S. Oh, J. Oh, and T. E. Milner, “Optical coherence tomography speckle reduction by a partially spatially coherent source,” J. Biomed. Opt. 10, 064034 (2005).
[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]

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]

Ramírez-Alarcón, R.

V. V. na Hernández, H. Cruz-Ramírez, R. Ramírez-Alarcón, and A. B. U’Ren, “Classical to quantum transfer of optical vortices,” Opt. Express 22, 20027–20037 (2014).
[Crossref]

R. Ramírez-Alarcón, H. Cruz-Ramírez, and A. B. U’Ren, “Effects of crystal length on the angular spectrum of spontaneous parametric downconversion photon pairs,” Laser Phys. 23, 055204 (2013).
[Crossref]

Rangarajan, R.

Roberts, T. D.

Rubin, M. H.

M. H. Rubin, “Transverse correlation in optical spontaneous parametric down-conversion,” Phys. Rev. A 54, 5349 (1996).
[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]

Saleh, B. E. A.

S. Carrasco, J. P. Torres, L. Torner, A. Sergienko, B. E. A. Saleh, and M. C. Teich, “Spatial-to-spectral mapping in spontaneous parametric down-conversion,” Phys. Rev. A 70, 043817 (2004).
[Crossref]

Sergienko, A.

S. Carrasco, J. P. Torres, L. Torner, A. Sergienko, B. E. A. Saleh, and M. C. Teich, “Spatial-to-spectral mapping in spontaneous parametric down-conversion,” Phys. Rev. A 70, 043817 (2004).
[Crossref]

Sergienko, A. 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, 2804–2815 (1996).
[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]

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, “New high-intensity source of polarization-entangled photon pairs,” Phys. Rev. Lett. 75, 4337 (1995).
[Crossref]

Shih, Y.

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, “New high-intensity source of polarization-entangled photon pairs,” Phys. Rev. Lett. 75, 4337 (1995).
[Crossref]

Y. Shih, An Introduction to Quantum Optics: Photon and Biphoton Physics (Taylor & Francis, 2014).

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, 2804–2815 (1996).
[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]

Shun Poh, H.

H. Shun Poh, J. Lim, I. Marcikic, A. Lamas-Linares, and C. Kurtsiefer, “Eliminating spectral distinguishability in ultrafast spontaneous parametric down-conversion,” Phys. Rev. A 80, 043815 (2009).

H. Shun Poh, C. Yang Lum, I. Marcikic, A. Lamas-Linares, and C. Kurtsiefer, “Joint spectrum mapping of polarization entanglement in spontaneous parametric down-conversion,” Phys. Rev. A 75, 043816 (2007).

Spillane, S. M.

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, 2804–2815 (1996).
[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]

Teich, M. C.

S. Carrasco, J. P. Torres, L. Torner, A. Sergienko, B. E. A. Saleh, and M. C. Teich, “Spatial-to-spectral mapping in spontaneous parametric down-conversion,” Phys. Rev. A 70, 043817 (2004).
[Crossref]

Torner, L.

J. P. Torres, G. Molina-Terriza, and L. Torner, “The spatial shape of entangled photon states generated in non-collinear, walking parametric downconversion,” J. Opt. B 7, 235 (2005).
[Crossref]

S. Carrasco, J. P. Torres, L. Torner, A. Sergienko, B. E. A. Saleh, and M. C. Teich, “Spatial-to-spectral mapping in spontaneous parametric down-conversion,” Phys. Rev. A 70, 043817 (2004).
[Crossref]

Torres, J. P.

J. P. Torres, G. Molina-Terriza, and L. Torner, “The spatial shape of entangled photon states generated in non-collinear, walking parametric downconversion,” J. Opt. B 7, 235 (2005).
[Crossref]

S. Carrasco, J. P. Torres, L. Torner, A. Sergienko, B. E. A. Saleh, and M. C. Teich, “Spatial-to-spectral mapping in spontaneous parametric down-conversion,” Phys. Rev. A 70, 043817 (2004).
[Crossref]

U’Ren, A. B.

V. V. na Hernández, H. Cruz-Ramírez, R. Ramírez-Alarcón, and A. B. U’Ren, “Classical to quantum transfer of optical vortices,” Opt. Express 22, 20027–20037 (2014).
[Crossref]

R. Ramírez-Alarcón, H. Cruz-Ramírez, and A. B. U’Ren, “Effects of crystal length on the angular spectrum of spontaneous parametric downconversion photon pairs,” Laser Phys. 23, 055204 (2013).
[Crossref]

van Exter, M. P.

H. Di Lorenzo Pires, F. M. G. J. Coppens, and M. P. van Exter, “Type-I spontaneous parametric down-conversion with a strongly focused pump,” Phys. Rev. A 83, 033837 (2011).
[Crossref]

P. S. K. Lee, M. P. van Exter, and J. P. Woerdman, “How focused pumping affects type-II spontaneous parametric down-conversion,” Phys. Rev. A 72, 033803 (2005).
[Crossref]

Voelz, D. G.

M. W. Hyde, S. Basu, D. G. Voelz, and X. Xiao, “Experimentally generating any desired partially coherent Schell-model source using phase-only control,” J. Appl. Phys. 118, 093102 (2015).
[Crossref]

Waks, E.

P. G. Kwiat, E. Waks, A. G. White, I. Appelbaum, and P. H. Eberhard, “Ultrabright source of polarization-entangled photons,” Phys. Rev. A 60, R773–R776 (1999).
[Crossref]

Weinfurter, H.

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, “New high-intensity source of polarization-entangled photon pairs,” Phys. Rev. Lett. 75, 4337 (1995).
[Crossref]

White, A. G.

P. G. Kwiat, E. Waks, A. G. White, I. Appelbaum, and P. H. Eberhard, “Ultrabright source of polarization-entangled photons,” Phys. Rev. A 60, R773–R776 (1999).
[Crossref]

Woerdman, J. P.

P. S. K. Lee, M. P. van Exter, and J. P. Woerdman, “How focused pumping affects type-II spontaneous parametric down-conversion,” Phys. Rev. A 72, 033803 (2005).
[Crossref]

Wolf, E.

G. Gbur and E. Wolf, “Spreading of partially coherent beams in random media,” J. Opt. Soc. Am. A 19, 1592–1598 (2002).
[Crossref]

E. Collett and E. Wolf, “Beams generated by Gaussian quasi-homogeneous sources,” Opt. Comm. 32, 27–31 (1980).
[Crossref]

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

Xiao, X.

M. W. Hyde, S. Basu, D. G. Voelz, and X. Xiao, “Experimentally generating any desired partially coherent Schell-model source using phase-only control,” J. Appl. Phys. 118, 093102 (2015).
[Crossref]

Yang Lum, C.

H. Shun Poh, C. Yang Lum, I. Marcikic, A. Lamas-Linares, and C. Kurtsiefer, “Joint spectrum mapping of polarization entanglement in spontaneous parametric down-conversion,” Phys. Rev. A 75, 043816 (2007).

Zeilinger, A.

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, “New high-intensity source of polarization-entangled photon pairs,” Phys. Rev. Lett. 75, 4337 (1995).
[Crossref]

Zhang, E.

Zhang, W.

W. Zhang, R. Fickler, E. Giese, L. Chen, and R. W. Boyd, “Influence of pump coherence on the generation of position-momentum entanglement in optical parametric down-conversion,” Opt. Express 27, 20745–20753 (2019).
[Crossref]

E. Giese, R. Fickler, W. Zhang, L. Chen, and R. W. Boyd, “Influence of pump coherence on the quantum properties of spontaneous parametric down-conversion,” Phys. Scripta 93, 084001 (2018).
[Crossref]

J. Appl. Phys. (1)

M. W. Hyde, S. Basu, D. G. Voelz, and X. Xiao, “Experimentally generating any desired partially coherent Schell-model source using phase-only control,” J. Appl. Phys. 118, 093102 (2015).
[Crossref]

J. Biomed. Opt. (1)

J. Kim, D. Miller, E. Kim, S. Oh, J. Oh, and T. E. Milner, “Optical coherence tomography speckle reduction by a partially spatially coherent source,” J. Biomed. Opt. 10, 064034 (2005).
[Crossref]

J. Opt. (1)

Y. Jeronimo-Moreno and R. Jáuregui, “Type-I parametric down conversion of highly focused Gaussian beams in finite length crystals,” J. Opt. 16, 065201 (2014).
[Crossref]

J. Opt. B (1)

J. P. Torres, G. Molina-Terriza, and L. Torner, “The spatial shape of entangled photon states generated in non-collinear, walking parametric downconversion,” J. Opt. B 7, 235 (2005).
[Crossref]

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

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

Laser Phys. (1)

R. Ramírez-Alarcón, H. Cruz-Ramírez, and A. B. U’Ren, “Effects of crystal length on the angular spectrum of spontaneous parametric downconversion photon pairs,” Laser Phys. 23, 055204 (2013).
[Crossref]

Opt. Comm. (1)

E. Collett and E. Wolf, “Beams generated by Gaussian quasi-homogeneous sources,” Opt. Comm. 32, 27–31 (1980).
[Crossref]

Opt. Commun. (1)

J.-C. Lee and Y.-H. Kim, “Spatial and spectral properties of entangled photons from spontaneous parametric down-conversion with a focused pump,” Opt. Commun. 366, 442–450 (2016).
[Crossref]

Opt. Express (4)

Opt. Lett. (1)

Opto-Electronics (1)

T. Asakura, “Spatial coherence of laser light passed through rotating ground glass,” Opto-Electronics 2, 115–123 (1970).
[Crossref]

Phys. Rev. A (14)

P. G. Kwiat, E. Waks, A. G. White, I. Appelbaum, and P. H. Eberhard, “Ultrabright source of polarization-entangled photons,” Phys. Rev. A 60, R773–R776 (1999).
[Crossref]

A. Jha and R. Boyd, “Spatial two-photon coherence of the entangled field produced by down-conversion using a partially spatially coherent pump beam,” Phys. Rev. A 81, 013828 (2010).
[Crossref]

H. Defienne and S. Gigan, “Spatially entangled photon-pair generation using a partial spatially coherent pump beam,” Phys. Rev. A 99, 053831 (2019).
[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]

H. Shun Poh, C. Yang Lum, I. Marcikic, A. Lamas-Linares, and C. Kurtsiefer, “Joint spectrum mapping of polarization entanglement in spontaneous parametric down-conversion,” Phys. Rev. A 75, 043816 (2007).

H. Shun Poh, J. Lim, I. Marcikic, A. Lamas-Linares, and C. Kurtsiefer, “Eliminating spectral distinguishability in ultrafast spontaneous parametric down-conversion,” Phys. Rev. A 80, 043815 (2009).

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]

P. S. K. Lee, M. P. van Exter, and J. P. Woerdman, “How focused pumping affects type-II spontaneous parametric down-conversion,” Phys. Rev. A 72, 033803 (2005).
[Crossref]

S. Carrasco, J. P. Torres, L. Torner, A. Sergienko, B. E. A. Saleh, and M. C. Teich, “Spatial-to-spectral mapping in spontaneous parametric down-conversion,” Phys. Rev. A 70, 043817 (2004).
[Crossref]

R. S. Bennink, Y. Liu, D. D. Earl, and W. P. Grice, “Spatial distinguishability of photons produced by spontaneous parametric down-conversion with a focused pump,” Phys. Rev. A 74, 023802 (2006).
[Crossref]

H. Di Lorenzo Pires, F. M. G. J. Coppens, and M. P. van Exter, “Type-I spontaneous parametric down-conversion with a strongly focused pump,” Phys. Rev. A 83, 033837 (2011).
[Crossref]

C. K. Hong and L. Mandel, “Theory of parametric frequency down conversion of light,” Phys. Rev. A 31, 2409–2418 (1985).
[Crossref]

M. H. Rubin, “Transverse correlation in optical spontaneous parametric down-conversion,” Phys. Rev. A 54, 5349 (1996).
[Crossref]

S.-Y. Baek and Y.-H. Kim, “Spectral properties of entangled photon pairs generated via frequency-degenerate type-I spontaneous parametric down-conversion,” Phys. Rev. A 77, 043807 (2008).
[Crossref]

Phys. Rev. Lett. (3)

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, “New high-intensity source of polarization-entangled photon pairs,” Phys. Rev. Lett. 75, 4337 (1995).
[Crossref]

A. K. Ekert, “Quantum cryptography based on bell’s theorem,” Phys. Rev. Lett. 67, 661–663 (1991).
[Crossref]

C. H. Bennett, G. Brassard, and N. D. Mermin, “Quantum cryptography without bell’s theorem,” Phys. Rev. Lett. 68, 557–559 (1992).
[Crossref]

Phys. Scripta (1)

E. Giese, R. Fickler, W. Zhang, L. Chen, and R. W. Boyd, “Influence of pump coherence on the quantum properties of spontaneous parametric down-conversion,” Phys. Scripta 93, 084001 (2018).
[Crossref]

Other (5)

Y. Shih, An Introduction to Quantum Optics: Photon and Biphoton Physics (Taylor & Francis, 2014).

M. Fox, Quantum Optics: An Introduction (OUP Oxford, 2006), Vol. 15.

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

B. Kanseri, Optical Coherence and Polarization: An Experimental Outlook (Lambert Academic Publishing, 2013).

V. G. Dmitriev, G. G. Gurzadyan, and D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals, (Springer, 2013), Vol. 64.

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Figures (11)

Fig. 1.
Fig. 1. Non-collinear type-I SPDC in BBO crystal with pump of wavelength 405 nm as input. Notations $ {\phi _s} $, $ {\phi _{s \textrm out}} $, and $ {\phi _i} $, $ {\phi _{i \textrm out}} $ represent emission angles for signal and idler beams both inside and outside the crystal, respectively. $ {\rho _p} $ is walk-off angle of pump beam inside the crystal, $ f $ is focal length of lens used in $ 2f $ configuration, $ {D_1} $ and $ {D_2} $ are the single-photon detectors, and $ \theta $ is the angle between the wave vector of pump and optic axis (O.A) of BBO crystal.
Fig. 2.
Fig. 2. Spatial profile of the twin photons generated in degenerate 3° non-collinear type-I SPDC with pump parameters (a) $ {w_0} = 10\,\,\unicode{x00B5}\text{m} $, $ {l_c} = 10\,\,\unicode{x00B5}\text{m} $; (b) $ {w_0} = 10\,\,\unicode{x00B5}\text{m} $, $ {l_c} = 50\,\,\unicode{x00B5}\text{m} $; (c) $ {w_0} = 50\,\,\unicode{x00B5}\text{m} $, $ {l_c} = 10\,\,\unicode{x00B5}\text{m} $; and (d) $ {w_0} = 50\,\,\unicode{x00B5}\text{m} $, $ {l_c} = 50\,\,\unicode{x00B5}\text{m} $. The color bar represents the joint probability distribution normalized for each case from the maximum value of that particular case.
Fig. 3.
Fig. 3. Spatial profile of the twin photons generated in degenerate collinear type-I SPDC with pump parameters (a) $ {w_0} = 10\,\,\unicode{x00B5}\text{m} $, $ {l_c} = 10\,\,\unicode{x00B5}\text{m} $; (b) $ {w_0} = 10\,\,\unicode{x00B5}\text{m} $, $ {l_c} = 50\,\,\unicode{x00B5}\text{m} $; (c) $ {w_0} = 50\,\,\unicode{x00B5}\text{m} $, $ {l_c} = 10\,\,\unicode{x00B5}\text{m} $; and (d) $ {w_0} = 50\,\,\unicode{x00B5}\text{m} $, $ {l_c} = 50\,\,\unicode{x00B5}\text{m} $. The color bar represents the joint probability distribution normalized for each case from the maximum value of that particular case.
Fig. 4.
Fig. 4. Spatial profile of the twin photons generated in degenerate 3° non-collinear type-II SPDC with pump parameters (a) $ {w_0} = 10\,\,\unicode{x00B5}\text{m} $, $ {l_c} = 10\,\,\unicode{x00B5}\text{m} $; (b) $ {w_0} = 10\,\,\unicode{x00B5}\text{m} $, $ {l_c} = 50\,\,\unicode{x00B5}\text{m} $; (c) $ {w_0} = 50\,\,\unicode{x00B5}\text{m} $, $ {l_c} = 10\,\,\unicode{x00B5}\text{m} $; and (d) $ {w_0} = 50\,\,\unicode{x00B5}\text{m} $, $ {l_c} = 50\,\,\unicode{x00B5}\text{m} $. The color bar represents the joint probability distribution normalized for each case from the maximum value of that particular case.
Fig. 5.
Fig. 5. Spatial profile of the twin photons generated in degenerate collinear type-II SPDC with pump parameters (a) $ {w_0} = 10\,\,\unicode{x00B5}\text{m} $, $ {l_c} = 10\,\,\unicode{x00B5}\text{m} $; (b) $ {w_0} = 10\,\,\unicode{x00B5}\text{m} $, $ {l_c} = 50\,\,\unicode{x00B5}\text{m} $; (c) $ {w_0} = 50\,\,\unicode{x00B5}\text{m} $, $ {l_c} = 10\,\,\unicode{x00B5}\text{m} $; and (d) $ {w_0} = 50\,\,\unicode{x00B5}\text{m} $, $ {l_c} = 50\,\,\unicode{x00B5}\text{m} $. The color bar represents the joint probability distribution normalized for each case from the maximum value of that particular case.
Fig. 6.
Fig. 6. Plot of spectral width in type-I SPDC process with $ {l_c} $ (top subfigure) and with $ {w_0} $ (bottom subfigure) variation for collinear and 3° non-collinear configurations; dotted lines denote the spectral width without walk-off effect, and solid lines denote the spectral width with walk-off effect corresponding to respective parameters shown as legends.
Fig. 7.
Fig. 7. Joint spectrum of 3° non-collinear type-I SPDC with pump parameters (a) $ {w_0} = 100\,\,\unicode{x00B5}\text{m} $, $ {l_c} = 10\,\,\unicode{x00B5}\text{m} $; (b) $ {w_0} = 100\,\,\unicode{x00B5}\text{m} $, $ {l_c} = 1000\,\,\unicode{x00B5}\text{m} $; (c) $ {w_0} = 10\,\,\unicode{x00B5}\text{m} $, $ {l_c} = 100\,\,\unicode{x00B5}\text{m} $; and (d) $ {w_0} = 1000\,\,\unicode{x00B5}\text{m} $, $ {l_c} = 100\,\,\unicode{x00B5}\text{m} $. The color bar represents the joint probability distribution normalized for each case from the maximum value of that particular case.
Fig. 8.
Fig. 8. Joint spectrum of collinear type-I SPDC with pump parameters $ {w_0} = 100\,\,\unicode{x00B5}\text{m} $, $ {l_c} = 1000\,\,\unicode{x00B5}\text{m} $ and crystal length (a) $ L = 100\,\,\unicode{x00B5}\text{m} $; (b) $ L = 500\,\,\unicode{x00B5}\text{m} $; (c) $ L = 1000\,\,\unicode{x00B5}\text{m} $; and (d) $ L = 2000\,\,\unicode{x00B5}\text{m} $. The color bar represents the joint probability distribution normalized for each case from the maximum value of that particular case.
Fig. 9.
Fig. 9. Plot of spectral width in type-II SPDC process with $ {l_c} $ (top subfigure) and with $ {w_0} $ (bottom subfigure) variation for collinear and 3° non-collinear configurations; dotted lines denote the spectral width without walk-off effect, and solid lines denote the spectral width with walk-off effect corresponding to respective parameters. The scale of $y$ axis is changed as compared to Fig. 6 because of smaller spectral widths.
Fig. 10.
Fig. 10. Joint spectrum of 3° non-collinear type-II SPDC with pump parameters (a) $ {w_0} = 100\,\,\unicode{x00B5}\text{m} $, $ {l_c} = 10\,\,\unicode{x00B5}\text{m} $; (b) $ {w_0} = 100\,\,\unicode{x00B5}\text{m} $, $ {l_c} = 1000\,\,\unicode{x00B5}\text{m} $; (c) $ {w_0} = 10\,\,\unicode{x00B5}\text{m} $, $ {l_c} = 100\,\,\unicode{x00B5}\text{m} $; and (d) $ {w_0} = 1000\,\,\unicode{x00B5}\text{m} $, $ {l_c} = 100\,\,\unicode{x00B5}\text{m} $. The color bar represents the joint probability distribution normalized for each case from the maximum value of that particular case.
Fig. 11.
Fig. 11. Joint spectrum of collinear type-II SPDC with pump parameters (a) $ {w_0} = 100\,\,\unicode{x00B5}\text{m} $, $ {l_c} = 10\,\,\unicode{x00B5}\text{m} $; (b) $ {w_0} = 100\,\,\unicode{x00B5}\text{m} $, $ {l_c} = 1000\,\,\unicode{x00B5}\text{m} $; (c) $ {w_0} = 10\,\,\unicode{x00B5}\text{m} $, $ {l_c} = 100\,\,\unicode{x00B5}\text{m} $; and (d) $ {w_0} = 1000\,\,\unicode{x00B5}\text{m} $, $ {l_c} = 100\,\,\unicode{x00B5}\text{m} $. The color bar represents the joint probability distribution normalized for each case from the maximum value of that particular case.

Equations (13)

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

| ψ >= ι 0 t d t H | 00 > ,
H = ϵ 0 d 3 r χ ( 2 ) E p ( + ) ( r p , t ) E s ( ) ( r s , t ) E i ( ) ( r i , t ) + h.c. ,
E p ( + ) ( r p , t ) = d ω p d k p V ( ω p , k p ) exp ( i ( k p r p ω p t ) ) ,
E s ( ) ( r s , t ) = d ω s d k s a ^ s + ( k s , ω s ) exp ( i ( k s r s ω s t ) ) ,
E i ( ) ( r i , t ) = d ω i d k i a ^ i + ( k i , ω i ) exp ( i ( k i r i ω i t ) ) ,
| ψ 0 T d t d ω p d ω s d ω i d 3 r d k p d k s d k i × V ( ω p , k p ) exp ( i ( ( k p r p k s r s k i r i ) + k p x z tan ρ p ) ( ω p ω s ω i ) t ) ) | ω s , k s | ω i , k i .
ϕ ( ω s , k s , ω i , k i ) d 3 r d k p V ( ω s + ω i , k p ) × exp ( i ( k p . r p k s . r s k i r i ) + k p x z tan ρ p ) ) ,
ϕ ( ω s , k s , ω i , k i ) V ( ω s + ω i , Δ k x , Δ k y ) sin c ( Δ k z L / 2 ) ,
Γ = | ϕ ( ω s , k s , ω i , k i ) | 2 .
V ( r ) V ( r ) exp ( ( r + r ) 2 8 w 0 2 ) exp ( ( r r ) 2 2 δ 2 ) ,
Γ s = d 2 k i | ϕ ( ω s , k s , ω i , k i ) | 2 .
Γ s / i = d 2 k i / s V ( ω s + ω i , Δ k x , Δ k y ) × V ( ω s + ω i , Δ k x , Δ k y ) sin c ( Δ k z L / 2 ) × sin c ( Δ k z L / 2 ) .
Γ j = d ω j δ ( ω j ω f ) | ϕ ( ω j , k s , ω p ω j , k i ) | 2 ,