Abstract

We analyze the generation of single spatial mode, spectrally uncorrelated photon pairs via type II spontaneous parametric down-conversion in a Potassium Titanyl Phosphate (KTP) waveguide using real experimental parameters. We show that this source can be used as an efficient, heralded, pure-state single-photon source.

© 2010 OSA

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  1. W. P. Grice, A. B. U’Ren, and I. A. Walmsley, “Eliminating frequency and space-time correlations in multiphoton states,” Phys. Rev. A 64(6), 063815 (2001).
    [CrossRef]
  2. 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]
  3. D. Bouwmester, A. Ekert, and A. Zeilinger, “The physics of quantum information: Quantum cryptography, Quantum teleportation, Quantum computation,” (Springer, 2000).
  4. A. L. Migdall, D. Branning, and S. Castelletto, “Tailoring single-photon and multiphoton probabilities of a singlephoton on-demand source,” Phys. Rev. A 66(5), 053805 (2002).
    [CrossRef]
  5. S. Fasel, O. Alibart, S. Tanzilli, P. Baldi, A. Beveratos, N. Gisin, and A. Zavatta, “High-quality asynchronous heralded single-photon source at telecom wavelength,” N. J. Phys. 6, 163 (2004).
    [CrossRef]
  6. E. A. Goldschmidt, M. D. Eisaman, J. Fan, S. Polyakov, and A. Migdall, “Spectrally bright and broad fiber-based heralded single-photon source,” Phys. Rev. A 78(1), 013844 (2008).
    [CrossRef]
  7. A. R. McMillan, J. Fulconis, M. Halder, C. Xiong, J. G. Rarity, and W. J. Wadsworth, “Narrowband high-fidelity all-fibre source of heralded single photons at 1570 nm,” Opt. Express 17(8), 6156–6165 (2009).
    [CrossRef] [PubMed]
  8. A. B. U’ren, C. Silberhorn, K. Banaszek, I. A. Walmsley, R. Erdmann, W. P. Grice, and M. G. Raymer, “Generation of pure-state single-photon wavepackets by conditional preparation based on spontaneous parametric downconversion,” Laser Phys. 15, 146 (2005).
  9. 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]
  10. P. J. Mosley, J. S. Lundeen, B. J. Smith, and I. A. Walmsley, “Conditional preparation of single photons using parametric downconversion: a recipe for purity,” N. J. Phys. 10(9), 093011 (2008).
    [CrossRef]
  11. K. Garay-Palmett, H. J. McGuinness, O. Cohen, J. S. Lundeen, R. Rangel-Rojo, A. B. U’ren, M. G. Raymer, C. J. McKinstrie, S. Radic, and I. A. Walmsley, “Photon pair-state preparation with tailored spectral properties by spontaneous four-wave mixing in photonic-crystal fiber,” Opt. Express 15(22), 14870–14886 (2007).
    [CrossRef] [PubMed]
  12. O. Cohen, J. S. Lundeen, B. J. Smith, G. Puentes, P. J. Mosley, and I. A. Walmsley, “Tailored photon-pair generation in optical fibers,” Phys. Rev. Lett. 102(12), 123603 (2009).
    [CrossRef] [PubMed]
  13. M. Halder, J. Fulconis, B. Cemlyn, A. Clark, C. Xiong, W. J. Wadsworth, and J. G. Rarity, “Nonclassical 2-photon interference with separate intrinsically narrowband fibre sources,” Opt. Express 17(6), 4670–4676 (2009).
    [CrossRef] [PubMed]
  14. S. Tanzilli, H. de Riedmatten, W. Tittel, H. Zbinden, P. Baldi, M. De Micheli, D. B. Ostrowsky, and N. Gisin, “Highly efficient photon-pair source using periodically poled lithium niobate waveguide,” Electron. Lett. 37(1), 26–28 (2001).
    [CrossRef]
  15. A. B. U’Ren, C. Silberhorn, K. Banaszek, and I. A. Walmsley, “Efficient Conditional Preparation of High-Fidelity Single Photon States for Fiber-Optic Quantum Networks,” Phys. Rev. Lett. 93(9), 093601 (2004).
    [CrossRef] [PubMed]
  16. 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(12), 7479–7488 (2007).
    [CrossRef] [PubMed]
  17. T. Suhara, H. Okabe, and M. Fujimura, “Generation of polarization-entangled photons by type-II quasi-phasematched waveguide nonlinear-optic device,” IEEE Photon. Technol. Lett. 19(14), 1093–1095 (2007).
    [CrossRef]
  18. J. Chen, A. J. Pearlman, A. Ling, J. Fan, and A. L. Migdall, “A versatile waveguide source of photon pairs for chip-scale quantum information processing,” Opt. Express 17(8), 6727–6740 (2009).
    [CrossRef] [PubMed]
  19. A. Christ, K. Laiho, A. Eckstein, T. Lauckner, P. J. Mosley, and C. Silberhorn, “Spatial modes in waveguided parametric down-conversion,” Phys. Rev. A 80(3), 033829 (2009).
    [CrossRef]
  20. M. Avenhaus, M. V. Chekhova, L. A. Krivitsky, G. Leuchs, and C. Silberhorn, “Experimental verification of high spectral entanglement for pulsed waveguided spontaneous parametric down-conversion,” Phys. Rev. A 79(4), 043836 (2009).
    [CrossRef]
  21. T. Zhong, F. N. Wong, T. D. Roberts, and P. Battle, “High performance photon-pair source based on a fiber-coupled periodically poled KTiOPO4 waveguide,” Opt. Express 17(14), 12019–12030 (2009).
    [CrossRef] [PubMed]
  22. M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28(11), 2631–2654 (1992).
    [CrossRef]
  23. A. Peres, “Separability criteriod for density matrices,” Phys. Rev. Lett. 77(8), 1413–1415 (1996).
    [CrossRef] [PubMed]
  24. C. K. Law, I. A. Walmsley, and J. H. Eberly, “Continuous frequency entanglement: effective finite Hilbert space and entropy control,” Phys. Rev. Lett. 84(23), 5304–5307 (2000).
    [CrossRef] [PubMed]
  25. Private communication with T. D. Roberts of AdvR Inc, ( http://www.advr-inc.com/ ).
  26. J. D. Bierlein, A. Ferretti, L. H. Brixner, and W. Y. Hsu, “Fabrication and characterization of optical waveguides in KTiOPO4,” Appl. Phys. Lett. 50(18), 1216 (1987).
    [CrossRef]
  27. http://www.comsol.com .
  28. Certain trade names and company products are mentioned in the text or identified in an illustration in order to specify adequately the experimental procedure and equipment used. In no case does such identification imply recommendation or endorsement by the National Institute of Standards and Technology, nor does it necessarily imply that the products are the best available for the purpose.

2009 (7)

A. R. McMillan, J. Fulconis, M. Halder, C. Xiong, J. G. Rarity, and W. J. Wadsworth, “Narrowband high-fidelity all-fibre source of heralded single photons at 1570 nm,” Opt. Express 17(8), 6156–6165 (2009).
[CrossRef] [PubMed]

O. Cohen, J. S. Lundeen, B. J. Smith, G. Puentes, P. J. Mosley, and I. A. Walmsley, “Tailored photon-pair generation in optical fibers,” Phys. Rev. Lett. 102(12), 123603 (2009).
[CrossRef] [PubMed]

M. Halder, J. Fulconis, B. Cemlyn, A. Clark, C. Xiong, W. J. Wadsworth, and J. G. Rarity, “Nonclassical 2-photon interference with separate intrinsically narrowband fibre sources,” Opt. Express 17(6), 4670–4676 (2009).
[CrossRef] [PubMed]

J. Chen, A. J. Pearlman, A. Ling, J. Fan, and A. L. Migdall, “A versatile waveguide source of photon pairs for chip-scale quantum information processing,” Opt. Express 17(8), 6727–6740 (2009).
[CrossRef] [PubMed]

A. Christ, K. Laiho, A. Eckstein, T. Lauckner, P. J. Mosley, and C. Silberhorn, “Spatial modes in waveguided parametric down-conversion,” Phys. Rev. A 80(3), 033829 (2009).
[CrossRef]

M. Avenhaus, M. V. Chekhova, L. A. Krivitsky, G. Leuchs, and C. Silberhorn, “Experimental verification of high spectral entanglement for pulsed waveguided spontaneous parametric down-conversion,” Phys. Rev. A 79(4), 043836 (2009).
[CrossRef]

T. Zhong, F. N. Wong, T. D. Roberts, and P. Battle, “High performance photon-pair source based on a fiber-coupled periodically poled KTiOPO4 waveguide,” Opt. Express 17(14), 12019–12030 (2009).
[CrossRef] [PubMed]

2008 (3)

E. A. Goldschmidt, M. D. Eisaman, J. Fan, S. Polyakov, and A. Migdall, “Spectrally bright and broad fiber-based heralded single-photon source,” Phys. Rev. A 78(1), 013844 (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 downconversion: a recipe for purity,” N. J. Phys. 10(9), 093011 (2008).
[CrossRef]

2007 (3)

2005 (1)

A. B. U’ren, C. Silberhorn, K. Banaszek, I. A. Walmsley, R. Erdmann, W. P. Grice, and M. G. Raymer, “Generation of pure-state single-photon wavepackets by conditional preparation based on spontaneous parametric downconversion,” Laser Phys. 15, 146 (2005).

2004 (2)

S. Fasel, O. Alibart, S. Tanzilli, P. Baldi, A. Beveratos, N. Gisin, and A. Zavatta, “High-quality asynchronous heralded single-photon source at telecom wavelength,” N. J. Phys. 6, 163 (2004).
[CrossRef]

A. B. U’Ren, C. Silberhorn, K. Banaszek, and I. A. Walmsley, “Efficient Conditional Preparation of High-Fidelity Single Photon States for Fiber-Optic Quantum Networks,” Phys. Rev. Lett. 93(9), 093601 (2004).
[CrossRef] [PubMed]

2002 (1)

A. L. Migdall, D. Branning, and S. Castelletto, “Tailoring single-photon and multiphoton probabilities of a singlephoton on-demand source,” Phys. Rev. A 66(5), 053805 (2002).
[CrossRef]

2001 (3)

W. P. Grice, A. B. U’Ren, and I. A. Walmsley, “Eliminating frequency and space-time correlations in multiphoton states,” Phys. Rev. A 64(6), 063815 (2001).
[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]

S. Tanzilli, H. de Riedmatten, W. Tittel, H. Zbinden, P. Baldi, M. De Micheli, D. B. Ostrowsky, and N. Gisin, “Highly efficient photon-pair source using periodically poled lithium niobate waveguide,” Electron. Lett. 37(1), 26–28 (2001).
[CrossRef]

2000 (1)

C. K. Law, I. A. Walmsley, and J. H. Eberly, “Continuous frequency entanglement: effective finite Hilbert space and entropy control,” Phys. Rev. Lett. 84(23), 5304–5307 (2000).
[CrossRef] [PubMed]

1996 (1)

A. Peres, “Separability criteriod for density matrices,” Phys. Rev. Lett. 77(8), 1413–1415 (1996).
[CrossRef] [PubMed]

1992 (1)

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28(11), 2631–2654 (1992).
[CrossRef]

1987 (1)

J. D. Bierlein, A. Ferretti, L. H. Brixner, and W. Y. Hsu, “Fabrication and characterization of optical waveguides in KTiOPO4,” Appl. Phys. Lett. 50(18), 1216 (1987).
[CrossRef]

Alibart, O.

S. Fasel, O. Alibart, S. Tanzilli, P. Baldi, A. Beveratos, N. Gisin, and A. Zavatta, “High-quality asynchronous heralded single-photon source at telecom wavelength,” N. J. Phys. 6, 163 (2004).
[CrossRef]

Avenhaus, M.

M. Avenhaus, M. V. Chekhova, L. A. Krivitsky, G. Leuchs, and C. Silberhorn, “Experimental verification of high spectral entanglement for pulsed waveguided spontaneous parametric down-conversion,” Phys. Rev. A 79(4), 043836 (2009).
[CrossRef]

Baldi, P.

S. Fasel, O. Alibart, S. Tanzilli, P. Baldi, A. Beveratos, N. Gisin, and A. Zavatta, “High-quality asynchronous heralded single-photon source at telecom wavelength,” N. J. Phys. 6, 163 (2004).
[CrossRef]

S. Tanzilli, H. de Riedmatten, W. Tittel, H. Zbinden, P. Baldi, M. De Micheli, D. B. Ostrowsky, and N. Gisin, “Highly efficient photon-pair source using periodically poled lithium niobate waveguide,” Electron. Lett. 37(1), 26–28 (2001).
[CrossRef]

Banaszek, K.

A. B. U’ren, C. Silberhorn, K. Banaszek, I. A. Walmsley, R. Erdmann, W. P. Grice, and M. G. Raymer, “Generation of pure-state single-photon wavepackets by conditional preparation based on spontaneous parametric downconversion,” Laser Phys. 15, 146 (2005).

A. B. U’Ren, C. Silberhorn, K. Banaszek, and I. A. Walmsley, “Efficient Conditional Preparation of High-Fidelity Single Photon States for Fiber-Optic Quantum Networks,” Phys. Rev. Lett. 93(9), 093601 (2004).
[CrossRef] [PubMed]

Battle, P.

Beausoleil, R. G.

Beveratos, A.

S. Fasel, O. Alibart, S. Tanzilli, P. Baldi, A. Beveratos, N. Gisin, and A. Zavatta, “High-quality asynchronous heralded single-photon source at telecom wavelength,” N. J. Phys. 6, 163 (2004).
[CrossRef]

Bierlein, J. D.

J. D. Bierlein, A. Ferretti, L. H. Brixner, and W. Y. Hsu, “Fabrication and characterization of optical waveguides in KTiOPO4,” Appl. Phys. Lett. 50(18), 1216 (1987).
[CrossRef]

Branning, D.

A. L. Migdall, D. Branning, and S. Castelletto, “Tailoring single-photon and multiphoton probabilities of a singlephoton on-demand source,” Phys. Rev. A 66(5), 053805 (2002).
[CrossRef]

Brixner, L. H.

J. D. Bierlein, A. Ferretti, L. H. Brixner, and W. Y. Hsu, “Fabrication and characterization of optical waveguides in KTiOPO4,” Appl. Phys. Lett. 50(18), 1216 (1987).
[CrossRef]

Byer, R. L.

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28(11), 2631–2654 (1992).
[CrossRef]

Castelletto, S.

A. L. Migdall, D. Branning, and S. Castelletto, “Tailoring single-photon and multiphoton probabilities of a singlephoton on-demand source,” Phys. Rev. A 66(5), 053805 (2002).
[CrossRef]

Cemlyn, B.

Chekhova, M. V.

M. Avenhaus, M. V. Chekhova, L. A. Krivitsky, G. Leuchs, and C. Silberhorn, “Experimental verification of high spectral entanglement for pulsed waveguided spontaneous parametric down-conversion,” Phys. Rev. A 79(4), 043836 (2009).
[CrossRef]

Chen, J.

Christ, A.

A. Christ, K. Laiho, A. Eckstein, T. Lauckner, P. J. Mosley, and C. Silberhorn, “Spatial modes in waveguided parametric down-conversion,” Phys. Rev. A 80(3), 033829 (2009).
[CrossRef]

Clark, A.

Cohen, O.

De Micheli, M.

S. Tanzilli, H. de Riedmatten, W. Tittel, H. Zbinden, P. Baldi, M. De Micheli, D. B. Ostrowsky, and N. Gisin, “Highly efficient photon-pair source using periodically poled lithium niobate waveguide,” Electron. Lett. 37(1), 26–28 (2001).
[CrossRef]

de Riedmatten, H.

S. Tanzilli, H. de Riedmatten, W. Tittel, H. Zbinden, P. Baldi, M. De Micheli, D. B. Ostrowsky, and N. Gisin, “Highly efficient photon-pair source using periodically poled lithium niobate waveguide,” Electron. Lett. 37(1), 26–28 (2001).
[CrossRef]

Eberly, J. H.

C. K. Law, I. A. Walmsley, and J. H. Eberly, “Continuous frequency entanglement: effective finite Hilbert space and entropy control,” Phys. Rev. Lett. 84(23), 5304–5307 (2000).
[CrossRef] [PubMed]

Eckstein, A.

A. Christ, K. Laiho, A. Eckstein, T. Lauckner, P. J. Mosley, and C. Silberhorn, “Spatial modes in waveguided parametric down-conversion,” Phys. Rev. A 80(3), 033829 (2009).
[CrossRef]

Eisaman, M. D.

E. A. Goldschmidt, M. D. Eisaman, J. Fan, S. Polyakov, and A. Migdall, “Spectrally bright and broad fiber-based heralded single-photon source,” Phys. Rev. A 78(1), 013844 (2008).
[CrossRef]

Erdmann, R.

A. B. U’ren, C. Silberhorn, K. Banaszek, I. A. Walmsley, R. Erdmann, W. P. Grice, and M. G. Raymer, “Generation of pure-state single-photon wavepackets by conditional preparation based on spontaneous parametric downconversion,” Laser Phys. 15, 146 (2005).

Fan, J.

J. Chen, A. J. Pearlman, A. Ling, J. Fan, and A. L. Migdall, “A versatile waveguide source of photon pairs for chip-scale quantum information processing,” Opt. Express 17(8), 6727–6740 (2009).
[CrossRef] [PubMed]

E. A. Goldschmidt, M. D. Eisaman, J. Fan, S. Polyakov, and A. Migdall, “Spectrally bright and broad fiber-based heralded single-photon source,” Phys. Rev. A 78(1), 013844 (2008).
[CrossRef]

Fasel, S.

S. Fasel, O. Alibart, S. Tanzilli, P. Baldi, A. Beveratos, N. Gisin, and A. Zavatta, “High-quality asynchronous heralded single-photon source at telecom wavelength,” N. J. Phys. 6, 163 (2004).
[CrossRef]

Fejer, M. M.

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28(11), 2631–2654 (1992).
[CrossRef]

Ferretti, A.

J. D. Bierlein, A. Ferretti, L. H. Brixner, and W. Y. Hsu, “Fabrication and characterization of optical waveguides in KTiOPO4,” Appl. Phys. Lett. 50(18), 1216 (1987).
[CrossRef]

Fiorentino, M.

Fujimura, M.

T. Suhara, H. Okabe, and M. Fujimura, “Generation of polarization-entangled photons by type-II quasi-phasematched waveguide nonlinear-optic device,” IEEE Photon. Technol. Lett. 19(14), 1093–1095 (2007).
[CrossRef]

Fulconis, J.

Garay-Palmett, K.

Gisin, N.

S. Fasel, O. Alibart, S. Tanzilli, P. Baldi, A. Beveratos, N. Gisin, and A. Zavatta, “High-quality asynchronous heralded single-photon source at telecom wavelength,” N. J. Phys. 6, 163 (2004).
[CrossRef]

S. Tanzilli, H. de Riedmatten, W. Tittel, H. Zbinden, P. Baldi, M. De Micheli, D. B. Ostrowsky, and N. Gisin, “Highly efficient photon-pair source using periodically poled lithium niobate waveguide,” Electron. Lett. 37(1), 26–28 (2001).
[CrossRef]

Goldschmidt, E. A.

E. A. Goldschmidt, M. D. Eisaman, J. Fan, S. Polyakov, and A. Migdall, “Spectrally bright and broad fiber-based heralded single-photon source,” Phys. Rev. A 78(1), 013844 (2008).
[CrossRef]

Grice, W. P.

A. B. U’ren, C. Silberhorn, K. Banaszek, I. A. Walmsley, R. Erdmann, W. P. Grice, and M. G. Raymer, “Generation of pure-state single-photon wavepackets by conditional preparation based on spontaneous parametric downconversion,” Laser Phys. 15, 146 (2005).

W. P. Grice, A. B. U’Ren, and I. A. Walmsley, “Eliminating frequency and space-time correlations in multiphoton states,” Phys. Rev. A 64(6), 063815 (2001).
[CrossRef]

Halder, M.

Hsu, W. Y.

J. D. Bierlein, A. Ferretti, L. H. Brixner, and W. Y. Hsu, “Fabrication and characterization of optical waveguides in KTiOPO4,” Appl. Phys. Lett. 50(18), 1216 (1987).
[CrossRef]

Jundt, D. H.

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28(11), 2631–2654 (1992).
[CrossRef]

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]

Krivitsky, L. A.

M. Avenhaus, M. V. Chekhova, L. A. Krivitsky, G. Leuchs, and C. Silberhorn, “Experimental verification of high spectral entanglement for pulsed waveguided spontaneous parametric down-conversion,” Phys. Rev. A 79(4), 043836 (2009).
[CrossRef]

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]

Laiho, K.

A. Christ, K. Laiho, A. Eckstein, T. Lauckner, P. J. Mosley, and C. Silberhorn, “Spatial modes in waveguided parametric down-conversion,” Phys. Rev. A 80(3), 033829 (2009).
[CrossRef]

Lauckner, T.

A. Christ, K. Laiho, A. Eckstein, T. Lauckner, P. J. Mosley, and C. Silberhorn, “Spatial modes in waveguided parametric down-conversion,” Phys. Rev. A 80(3), 033829 (2009).
[CrossRef]

Law, C. K.

C. K. Law, I. A. Walmsley, and J. H. Eberly, “Continuous frequency entanglement: effective finite Hilbert space and entropy control,” Phys. Rev. Lett. 84(23), 5304–5307 (2000).
[CrossRef] [PubMed]

Leuchs, G.

M. Avenhaus, M. V. Chekhova, L. A. Krivitsky, G. Leuchs, and C. Silberhorn, “Experimental verification of high spectral entanglement for pulsed waveguided spontaneous parametric down-conversion,” Phys. Rev. A 79(4), 043836 (2009).
[CrossRef]

Ling, A.

Lundeen, J. S.

O. Cohen, J. S. Lundeen, B. J. Smith, G. Puentes, P. J. Mosley, and I. A. Walmsley, “Tailored photon-pair generation in optical fibers,” Phys. Rev. Lett. 102(12), 123603 (2009).
[CrossRef] [PubMed]

P. J. Mosley, J. S. Lundeen, B. J. Smith, and I. A. Walmsley, “Conditional preparation of single photons using parametric downconversion: 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]

K. Garay-Palmett, H. J. McGuinness, O. Cohen, J. S. Lundeen, R. Rangel-Rojo, A. B. U’ren, M. G. Raymer, C. J. McKinstrie, S. Radic, and I. A. Walmsley, “Photon pair-state preparation with tailored spectral properties by spontaneous four-wave mixing in photonic-crystal fiber,” Opt. Express 15(22), 14870–14886 (2007).
[CrossRef] [PubMed]

Magel, G. A.

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28(11), 2631–2654 (1992).
[CrossRef]

McGuinness, H. J.

McKinstrie, C. J.

McMillan, A. R.

Migdall, A.

E. A. Goldschmidt, M. D. Eisaman, J. Fan, S. Polyakov, and A. Migdall, “Spectrally bright and broad fiber-based heralded single-photon source,” Phys. Rev. A 78(1), 013844 (2008).
[CrossRef]

Migdall, A. L.

J. Chen, A. J. Pearlman, A. Ling, J. Fan, and A. L. Migdall, “A versatile waveguide source of photon pairs for chip-scale quantum information processing,” Opt. Express 17(8), 6727–6740 (2009).
[CrossRef] [PubMed]

A. L. Migdall, D. Branning, and S. Castelletto, “Tailoring single-photon and multiphoton probabilities of a singlephoton on-demand source,” Phys. Rev. A 66(5), 053805 (2002).
[CrossRef]

Milburn, G. J.

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]

Mosley, P. J.

O. Cohen, J. S. Lundeen, B. J. Smith, G. Puentes, P. J. Mosley, and I. A. Walmsley, “Tailored photon-pair generation in optical fibers,” Phys. Rev. Lett. 102(12), 123603 (2009).
[CrossRef] [PubMed]

A. Christ, K. Laiho, A. Eckstein, T. Lauckner, P. J. Mosley, and C. Silberhorn, “Spatial modes in waveguided parametric down-conversion,” Phys. Rev. A 80(3), 033829 (2009).
[CrossRef]

P. J. Mosley, J. S. Lundeen, B. J. Smith, and I. A. Walmsley, “Conditional preparation of single photons using parametric downconversion: 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, M. W.

Okabe, H.

T. Suhara, H. Okabe, and M. Fujimura, “Generation of polarization-entangled photons by type-II quasi-phasematched waveguide nonlinear-optic device,” IEEE Photon. Technol. Lett. 19(14), 1093–1095 (2007).
[CrossRef]

Ostrowsky, D. B.

S. Tanzilli, H. de Riedmatten, W. Tittel, H. Zbinden, P. Baldi, M. De Micheli, D. B. Ostrowsky, and N. Gisin, “Highly efficient photon-pair source using periodically poled lithium niobate waveguide,” Electron. Lett. 37(1), 26–28 (2001).
[CrossRef]

Pearlman, A. J.

Peres, A.

A. Peres, “Separability criteriod for density matrices,” Phys. Rev. Lett. 77(8), 1413–1415 (1996).
[CrossRef] [PubMed]

Polyakov, S.

E. A. Goldschmidt, M. D. Eisaman, J. Fan, S. Polyakov, and A. Migdall, “Spectrally bright and broad fiber-based heralded single-photon source,” Phys. Rev. A 78(1), 013844 (2008).
[CrossRef]

Puentes, G.

O. Cohen, J. S. Lundeen, B. J. Smith, G. Puentes, P. J. Mosley, and I. A. Walmsley, “Tailored photon-pair generation in optical fibers,” Phys. Rev. Lett. 102(12), 123603 (2009).
[CrossRef] [PubMed]

Radic, S.

Rangel-Rojo, R.

Rarity, J. G.

Raymer, M. G.

K. Garay-Palmett, H. J. McGuinness, O. Cohen, J. S. Lundeen, R. Rangel-Rojo, A. B. U’ren, M. G. Raymer, C. J. McKinstrie, S. Radic, and I. A. Walmsley, “Photon pair-state preparation with tailored spectral properties by spontaneous four-wave mixing in photonic-crystal fiber,” Opt. Express 15(22), 14870–14886 (2007).
[CrossRef] [PubMed]

A. B. U’ren, C. Silberhorn, K. Banaszek, I. A. Walmsley, R. Erdmann, W. P. Grice, and M. G. Raymer, “Generation of pure-state single-photon wavepackets by conditional preparation based on spontaneous parametric downconversion,” Laser Phys. 15, 146 (2005).

Roberts, T. D.

Silberhorn, C.

A. Christ, K. Laiho, A. Eckstein, T. Lauckner, P. J. Mosley, and C. Silberhorn, “Spatial modes in waveguided parametric down-conversion,” Phys. Rev. A 80(3), 033829 (2009).
[CrossRef]

M. Avenhaus, M. V. Chekhova, L. A. Krivitsky, G. Leuchs, and C. Silberhorn, “Experimental verification of high spectral entanglement for pulsed waveguided spontaneous parametric down-conversion,” Phys. Rev. A 79(4), 043836 (2009).
[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]

A. B. U’ren, C. Silberhorn, K. Banaszek, I. A. Walmsley, R. Erdmann, W. P. Grice, and M. G. Raymer, “Generation of pure-state single-photon wavepackets by conditional preparation based on spontaneous parametric downconversion,” Laser Phys. 15, 146 (2005).

A. B. U’Ren, C. Silberhorn, K. Banaszek, and I. A. Walmsley, “Efficient Conditional Preparation of High-Fidelity Single Photon States for Fiber-Optic Quantum Networks,” Phys. Rev. Lett. 93(9), 093601 (2004).
[CrossRef] [PubMed]

Smith, B. J.

O. Cohen, J. S. Lundeen, B. J. Smith, G. Puentes, P. J. Mosley, and I. A. Walmsley, “Tailored photon-pair generation in optical fibers,” Phys. Rev. Lett. 102(12), 123603 (2009).
[CrossRef] [PubMed]

P. J. Mosley, J. S. Lundeen, B. J. Smith, and I. A. Walmsley, “Conditional preparation of single photons using parametric downconversion: 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]

Spillane, S. M.

Suhara, T.

T. Suhara, H. Okabe, and M. Fujimura, “Generation of polarization-entangled photons by type-II quasi-phasematched waveguide nonlinear-optic device,” IEEE Photon. Technol. Lett. 19(14), 1093–1095 (2007).
[CrossRef]

Tanzilli, S.

S. Fasel, O. Alibart, S. Tanzilli, P. Baldi, A. Beveratos, N. Gisin, and A. Zavatta, “High-quality asynchronous heralded single-photon source at telecom wavelength,” N. J. Phys. 6, 163 (2004).
[CrossRef]

S. Tanzilli, H. de Riedmatten, W. Tittel, H. Zbinden, P. Baldi, M. De Micheli, D. B. Ostrowsky, and N. Gisin, “Highly efficient photon-pair source using periodically poled lithium niobate waveguide,” Electron. Lett. 37(1), 26–28 (2001).
[CrossRef]

Tittel, W.

S. Tanzilli, H. de Riedmatten, W. Tittel, H. Zbinden, P. Baldi, M. De Micheli, D. B. Ostrowsky, and N. Gisin, “Highly efficient photon-pair source using periodically poled lithium niobate waveguide,” Electron. Lett. 37(1), 26–28 (2001).
[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]

K. Garay-Palmett, H. J. McGuinness, O. Cohen, J. S. Lundeen, R. Rangel-Rojo, A. B. U’ren, M. G. Raymer, C. J. McKinstrie, S. Radic, and I. A. Walmsley, “Photon pair-state preparation with tailored spectral properties by spontaneous four-wave mixing in photonic-crystal fiber,” Opt. Express 15(22), 14870–14886 (2007).
[CrossRef] [PubMed]

A. B. U’ren, C. Silberhorn, K. Banaszek, I. A. Walmsley, R. Erdmann, W. P. Grice, and M. G. Raymer, “Generation of pure-state single-photon wavepackets by conditional preparation based on spontaneous parametric downconversion,” Laser Phys. 15, 146 (2005).

A. B. U’Ren, C. Silberhorn, K. Banaszek, and I. A. Walmsley, “Efficient Conditional Preparation of High-Fidelity Single Photon States for Fiber-Optic Quantum Networks,” Phys. Rev. Lett. 93(9), 093601 (2004).
[CrossRef] [PubMed]

W. P. Grice, A. B. U’Ren, and I. A. Walmsley, “Eliminating frequency and space-time correlations in multiphoton states,” Phys. Rev. A 64(6), 063815 (2001).
[CrossRef]

Wadsworth, W. J.

Walmsley, I. A.

O. Cohen, J. S. Lundeen, B. J. Smith, G. Puentes, P. J. Mosley, and I. A. Walmsley, “Tailored photon-pair generation in optical fibers,” Phys. Rev. Lett. 102(12), 123603 (2009).
[CrossRef] [PubMed]

P. J. Mosley, J. S. Lundeen, B. J. Smith, and I. A. Walmsley, “Conditional preparation of single photons using parametric downconversion: 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]

K. Garay-Palmett, H. J. McGuinness, O. Cohen, J. S. Lundeen, R. Rangel-Rojo, A. B. U’ren, M. G. Raymer, C. J. McKinstrie, S. Radic, and I. A. Walmsley, “Photon pair-state preparation with tailored spectral properties by spontaneous four-wave mixing in photonic-crystal fiber,” Opt. Express 15(22), 14870–14886 (2007).
[CrossRef] [PubMed]

A. B. U’ren, C. Silberhorn, K. Banaszek, I. A. Walmsley, R. Erdmann, W. P. Grice, and M. G. Raymer, “Generation of pure-state single-photon wavepackets by conditional preparation based on spontaneous parametric downconversion,” Laser Phys. 15, 146 (2005).

A. B. U’Ren, C. Silberhorn, K. Banaszek, and I. A. Walmsley, “Efficient Conditional Preparation of High-Fidelity Single Photon States for Fiber-Optic Quantum Networks,” Phys. Rev. Lett. 93(9), 093601 (2004).
[CrossRef] [PubMed]

W. P. Grice, A. B. U’Ren, and I. A. Walmsley, “Eliminating frequency and space-time correlations in multiphoton states,” Phys. Rev. A 64(6), 063815 (2001).
[CrossRef]

C. K. Law, I. A. Walmsley, and J. H. Eberly, “Continuous frequency entanglement: effective finite Hilbert space and entropy control,” Phys. Rev. Lett. 84(23), 5304–5307 (2000).
[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]

Wong, F. N.

Xiong, C.

Zavatta, A.

S. Fasel, O. Alibart, S. Tanzilli, P. Baldi, A. Beveratos, N. Gisin, and A. Zavatta, “High-quality asynchronous heralded single-photon source at telecom wavelength,” N. J. Phys. 6, 163 (2004).
[CrossRef]

Zbinden, H.

S. Tanzilli, H. de Riedmatten, W. Tittel, H. Zbinden, P. Baldi, M. De Micheli, D. B. Ostrowsky, and N. Gisin, “Highly efficient photon-pair source using periodically poled lithium niobate waveguide,” Electron. Lett. 37(1), 26–28 (2001).
[CrossRef]

Zhong, T.

Appl. Phys. Lett. (1)

J. D. Bierlein, A. Ferretti, L. H. Brixner, and W. Y. Hsu, “Fabrication and characterization of optical waveguides in KTiOPO4,” Appl. Phys. Lett. 50(18), 1216 (1987).
[CrossRef]

Electron. Lett. (1)

S. Tanzilli, H. de Riedmatten, W. Tittel, H. Zbinden, P. Baldi, M. De Micheli, D. B. Ostrowsky, and N. Gisin, “Highly efficient photon-pair source using periodically poled lithium niobate waveguide,” Electron. Lett. 37(1), 26–28 (2001).
[CrossRef]

IEEE J. Quantum Electron. (1)

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28(11), 2631–2654 (1992).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

T. Suhara, H. Okabe, and M. Fujimura, “Generation of polarization-entangled photons by type-II quasi-phasematched waveguide nonlinear-optic device,” IEEE Photon. Technol. Lett. 19(14), 1093–1095 (2007).
[CrossRef]

Laser Phys. (1)

A. B. U’ren, C. Silberhorn, K. Banaszek, I. A. Walmsley, R. Erdmann, W. P. Grice, and M. G. Raymer, “Generation of pure-state single-photon wavepackets by conditional preparation based on spontaneous parametric downconversion,” Laser Phys. 15, 146 (2005).

N. J. Phys. (2)

S. Fasel, O. Alibart, S. Tanzilli, P. Baldi, A. Beveratos, N. Gisin, and A. Zavatta, “High-quality asynchronous heralded single-photon source at telecom wavelength,” N. J. Phys. 6, 163 (2004).
[CrossRef]

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

Nature (1)

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. Express (6)

K. Garay-Palmett, H. J. McGuinness, O. Cohen, J. S. Lundeen, R. Rangel-Rojo, A. B. U’ren, M. G. Raymer, C. J. McKinstrie, S. Radic, and I. A. Walmsley, “Photon pair-state preparation with tailored spectral properties by spontaneous four-wave mixing in photonic-crystal fiber,” Opt. Express 15(22), 14870–14886 (2007).
[CrossRef] [PubMed]

A. R. McMillan, J. Fulconis, M. Halder, C. Xiong, J. G. Rarity, and W. J. Wadsworth, “Narrowband high-fidelity all-fibre source of heralded single photons at 1570 nm,” Opt. Express 17(8), 6156–6165 (2009).
[CrossRef] [PubMed]

M. Halder, J. Fulconis, B. Cemlyn, A. Clark, C. Xiong, W. J. Wadsworth, and J. G. Rarity, “Nonclassical 2-photon interference with separate intrinsically narrowband fibre sources,” Opt. Express 17(6), 4670–4676 (2009).
[CrossRef] [PubMed]

J. Chen, A. J. Pearlman, A. Ling, J. Fan, and A. L. Migdall, “A versatile waveguide source of photon pairs for chip-scale quantum information processing,” Opt. Express 17(8), 6727–6740 (2009).
[CrossRef] [PubMed]

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(12), 7479–7488 (2007).
[CrossRef] [PubMed]

T. Zhong, F. N. Wong, T. D. Roberts, and P. Battle, “High performance photon-pair source based on a fiber-coupled periodically poled KTiOPO4 waveguide,” Opt. Express 17(14), 12019–12030 (2009).
[CrossRef] [PubMed]

Phys. Rev. A (5)

A. Christ, K. Laiho, A. Eckstein, T. Lauckner, P. J. Mosley, and C. Silberhorn, “Spatial modes in waveguided parametric down-conversion,” Phys. Rev. A 80(3), 033829 (2009).
[CrossRef]

M. Avenhaus, M. V. Chekhova, L. A. Krivitsky, G. Leuchs, and C. Silberhorn, “Experimental verification of high spectral entanglement for pulsed waveguided spontaneous parametric down-conversion,” Phys. Rev. A 79(4), 043836 (2009).
[CrossRef]

W. P. Grice, A. B. U’Ren, and I. A. Walmsley, “Eliminating frequency and space-time correlations in multiphoton states,” Phys. Rev. A 64(6), 063815 (2001).
[CrossRef]

A. L. Migdall, D. Branning, and S. Castelletto, “Tailoring single-photon and multiphoton probabilities of a singlephoton on-demand source,” Phys. Rev. A 66(5), 053805 (2002).
[CrossRef]

E. A. Goldschmidt, M. D. Eisaman, J. Fan, S. Polyakov, and A. Migdall, “Spectrally bright and broad fiber-based heralded single-photon source,” Phys. Rev. A 78(1), 013844 (2008).
[CrossRef]

Phys. Rev. Lett. (5)

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]

O. Cohen, J. S. Lundeen, B. J. Smith, G. Puentes, P. J. Mosley, and I. A. Walmsley, “Tailored photon-pair generation in optical fibers,” Phys. Rev. Lett. 102(12), 123603 (2009).
[CrossRef] [PubMed]

A. B. U’Ren, C. Silberhorn, K. Banaszek, and I. A. Walmsley, “Efficient Conditional Preparation of High-Fidelity Single Photon States for Fiber-Optic Quantum Networks,” Phys. Rev. Lett. 93(9), 093601 (2004).
[CrossRef] [PubMed]

A. Peres, “Separability criteriod for density matrices,” Phys. Rev. Lett. 77(8), 1413–1415 (1996).
[CrossRef] [PubMed]

C. K. Law, I. A. Walmsley, and J. H. Eberly, “Continuous frequency entanglement: effective finite Hilbert space and entropy control,” Phys. Rev. Lett. 84(23), 5304–5307 (2000).
[CrossRef] [PubMed]

Other (4)

Private communication with T. D. Roberts of AdvR Inc, ( http://www.advr-inc.com/ ).

http://www.comsol.com .

Certain trade names and company products are mentioned in the text or identified in an illustration in order to specify adequately the experimental procedure and equipment used. In no case does such identification imply recommendation or endorsement by the National Institute of Standards and Technology, nor does it necessarily imply that the products are the best available for the purpose.

D. Bouwmester, A. Ekert, and A. Zeilinger, “The physics of quantum information: Quantum cryptography, Quantum teleportation, Quantum computation,” (Springer, 2000).

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Figures (6)

Fig. 1
Fig. 1

(a) Schema for a KTP crystal containing a waveguide. The waveguide dimensions and crystal axes are labeled. (b) Number of bound modes in the waveguide with polarization along z-axis.

Fig. 3
Fig. 3

(a) Effective refractive index of y- or z- polarized light beam propagating in bulk RTP and waveguide (WG). (b) Group velocity of y- or z- polarized optical modes in the waveguide. A selected group of wavelengths with group velocities satisfying Eq. (10) is shown as the red dots.

Fig. 2
Fig. 2

Normalized transverse electric field amplitude profile for fundamental modes, (a) λ p=764 nm, y-polarization; (b) λ i=1506 nm, y-polarization; (c) λ s=1550 nm, z-polarization. (d-e) Contour plots at FWHM of (a-c).

Fig. 4
Fig. 4

(a) Sample contours for phase-matched down-conversions with different groups of pump and signal wavelengths, each group has a certain spatial harmonic frequency. (b)Shaded spectral region of interest. The red dot corresponds to the phase-matched down-conversion at λ p = 764 nm, λ s = 1550 nm and λ i = 1506 nm. See text for detail.

Fig. 5
Fig. 5

Normalized spectral intensity profiles for the pump (a), phase-matching sinc-function (b), and two-photon joint spectrum (c), at the condition of the optimized pump bandwidth for the highest degree-of-factorization of two-photon joint spectrum. Insets in (c) are lineout across the center. (d) Degree-of-factorization of photon pairs as a function of the pump bandwidth. Inset in (d) is the mode distribution in the Schmidt decomposition with the optimized pump bandwidth.

Fig. 6
Fig. 6

(a) Degree-of-factorization of photon pairs as a function of spectral filter bandwidth, filtering only the signal, only the idler, or both. (b) Degree-of-factorization and two-photon transmittance as a function of collection bandwidth (labeled in the figure), filtering both the signal and the idler.

Equations (13)

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

| ψ = | 0 s , ​   0 i + χ d ω s d ω i f ( ω s , ω i ) a s + ( ω s ) a i + ( ω i ) | 0 s , 0 i ,
χ ~ A d y d z d 24 [ U s z ( y , z ) U i y ( y , z ) ] * U p y ( y , z ) ,
f ( ω s , ω i ) = p ( ω s , ω i ) sin c ( Δ κ L / 2 ) .
Δ κ = k s z + k i y k p y + 2 m π / Λ = 0 ,
Δ κ L = ( k s z + k i y k p y + 2 m π / Λ ) L = ( k s 0 z + k i 0 y k p 0 y + 2 m π / Λ ) L + [ k s ξ s + k i ξ i k p ( ξ s + ξ ) i ] L = ( k s k p ) L ξ s + ( k i k p ) L ξ i = τ s ξ s + τ i ξ i ,
f ( ω s , ω i ) = e ( ξ s + ξ ) i 2 / σ 2 γ ( τ s ξ s + τ i ξ i ) 2 / 4 = e ( 1 / σ 2 + γ τ s 2 ) ξ s 2 ( 1 / σ 2 + γ τ i 2 ) ξ i 2 ( 2 / σ 2 + γ τ s τ i / 2 ) ξ s ξ i ,
f ( ω s , ω i ) = f s ( ω s ) f i ( ω i ) = e ( 1 / σ 2 + γ τ s 2 ) ν s 2 e ( 1 / σ 2 + γ τ i 2 ) ν i 2 ,
4 / σ 2 + γ τ s τ i = 4 / σ 2 + γ ( k s k p ) ( k i k p ) L 2 = 0.
| τ s τ i | τ c o h e r e n c e 2 .
υ s υ p υ i , ​ ​   or ​   υ s υ p υ i .
f ( ω s ,   ​ ω i ) = j η j 1 / 2 | u j ( ω s ) | ν j ( ω i ) ,
K = 1 / j η j 2 .
R = 8 π 2 d 24 2 L 2 ε 0 P A d y d z [ U s z ( y , z ) U i y ( y , z ) ] * U p y ( y , z ) × s d s u i d i u d λ s d λ i { [ p ( 2 π c λ s , 2 π c λ i ) sin c ( Δ κ L / 2 ) f i l t s ( λ s ) f i l t i ( λ i ) ] 2 × λ p 2 λ s 3 λ i 3 n y W G ( λ p ) n z W G ( λ s ) n y W G ( λ i ) } ,

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