Abstract

A proposal is made for the generation of polarization-entangled photon pairs from a periodically poled crystal allowing for high collection efficiency, high entanglement, and stable operation. The theory is formulated for colinear propagation for application to waveguides. The key feature of the theory is the use of type II phase matching using both the +1 and −1 diffraction orders of the poling structure. Although these conditions are fairly restrictive in terms of operating parameters, practical operating conditions can be found. For example, we find that a HeNe pump laser may be used for a periodically poled rubidium-doped potassium titanyl phosphate (Rb:KTP) waveguide to yield single mode polarization-entangled pairs. Fidelities of 0.98 are possible under practical conditions.

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  13. Technically, the cm are frequency-dependent, however, because their frequency dependence is scaled to the band gap of (3.6 eV for KTP) whereas that of β(ωs, ωi) is scaled to the width of phase matching. For practical crystal lengths L ≳ 10 mm, this width is 1 meV or less, so the cm may be treated as constants.
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    [CrossRef]
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    [CrossRef]
  25. K. Frakin, A. Arie, A. Skliar, and G. Rosenman, “Tunable midinfrared source by difference frequency generation in bulk periodically poled ktiopo4,” Appl. Phys. Lett. 74, 914–916 (1999).
    [CrossRef]
  26. L. K. Cheng, L. T. Cheng, J. Galperin, P. A. M. Hotsenpiller, and J. D. Bierlein, “Crystal-growth and characterization of ktiopo4 isomorphs from the self-fluxes,” J. Cryst. Growth 137, 107 (1994).
    [CrossRef]
  27. http://www.comsol.com . Mention of commercial products is for information only; it does not imply recommendation or endorsement by NIST.
  28. J. D. Bierlein, A. Ferretti, L. H. Brixner, and W. Y. Hsu, “Fabrication and characterization of optical waveguides in ktiopo4,” Appl. Phys. Lett. 50, 1216 (1987).
    [CrossRef]
  29. S. Emanueli and A. Arie, “Temperature-dependent dispersion equations for ktiopo4 and ktioaso4,” Appl. Opt. 42, 6661 (2003).
    [CrossRef] [PubMed]
  30. T. Mikami, T. Okamoto, and K. Kato, “Sellmeier and thermo-optic dispersion formulas for rbtiopo4,” Appl. Opt. 42, 6661 (2003).
    [CrossRef]
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    [CrossRef]
  32. P. G. Kwiat, “Hyper-entangled states,” J. Mod. Opt. 44, 2173 (1997).
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    [CrossRef]

2009 (4)

M. P. Peloso, I. Gerhardt, C. Ho, A. Lamas-Linares, and C. Kurtsiefer, “Daylight operation of a free space, entanglement-based quantum key distribution system,” N. J. Phys. 11, 045007 (2009).
[CrossRef]

A. Rossi, G. Vallone, A. Chiuri, F. D. Martini, and P. Mataloni, “Multipath entanglement of two photons,” Phys. Rev. Lett. 102, 153902 (2009).
[CrossRef] [PubMed]

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

T. Zhong, F. N. C. Wong, T. D. Roberts, and P. Battle, “High performance photon-pair source based on a fiber-coupled periodically poled ktiopo4 waveguide,” Opt. Express 17, 12019–12029 (2009).
[CrossRef] [PubMed]

2008 (1)

2007 (2)

2006 (1)

T. Kim, M. Fiorentino, and F. N. C. Wong, “Phase-stable source of polarization-entangled photons using a polarization sagnac interferometer,” Phys. Rev. A 73, 012316 (2006).
[CrossRef]

2005 (1)

J. T. Barreiro, N. K. Langford, N. A. Peters, and P. G. Kwiat, “Generation of hyperentangled photon pairs,” Phys. Rev. Lett. 95, 260501 (2005).
[CrossRef]

2004 (2)

C. E. Kuklewicz, M. Fiorentino, G. Messin, F. N. C. Wong, and J. H. Shapiro, “High-flux source of polarization entangles photons from a periodically poled ktiopo4 parametric down-converter,” Phys. Rev. A 69, 013807 (2004).
[CrossRef]

F. König and F. N. C. Wong, “Extended phase matching of second-harmonic generation in periodically poled ktiopo4 with zero group-velocity mismatch,” Appl. Phys. Lett. 84, 1644–1646 (2004).
[CrossRef]

2003 (4)

M. Katz, D. Eger, H. Kim, L. Jankovic, G. Stegeman, S. Carrasco, and L. Torner, “Second harmonic generation tuning curves in quasiphase-matched potassium titanyl phosphate with narrow, high-intensity beams,” J. Appl. Phys. 93, 8852–8861 (2003).
[CrossRef]

F. Torabi-Goudarzi and E. Riis, “Efficient cw high-power frequency doubling in periodically poled ktp,” Opt. Commun. 227, 389–403 (2003).
[CrossRef]

S. Emanueli and A. Arie, “Temperature-dependent dispersion equations for ktiopo4 and ktioaso4,” Appl. Opt. 42, 6661 (2003).
[CrossRef] [PubMed]

T. Mikami, T. Okamoto, and K. Kato, “Sellmeier and thermo-optic dispersion formulas for rbtiopo4,” Appl. Opt. 42, 6661 (2003).
[CrossRef]

2000 (1)

J.-W. Pan, D. Bouwmeester, M. Daniell, H. Wienfurter, and A. Zellinger, “Experimental test of quantum nonlocality in three-photon greenberger-horne-zelinger entanglement,” Nature 403, 515–519 (2000).
[CrossRef] [PubMed]

1999 (2)

P. G. Kwiat, E. Waks, A. G. White, I. Appelbaum, and P. H. Eberhard, “Ultrabright source of polarization-entangled photons,” Phys. Rev. A 60, 773–776 (1999).
[CrossRef]

K. Frakin, A. Arie, A. Skliar, and G. Rosenman, “Tunable midinfrared source by difference frequency generation in bulk periodically poled ktiopo4,” Appl. Phys. Lett. 74, 914–916 (1999).
[CrossRef]

1997 (2)

P. G. Kwiat, “Hyper-entangled states,” J. Mod. Opt. 44, 2173 (1997).

D. Bouwmeester, J.-W. Pan, K. Mattle, M. Eibl, H. Wienfurter, and A. Zellinger, “Experimental quantum teleportation,” Nature 390, 575–579 (1997).
[CrossRef]

1995 (2)

P. G. Kwiat, K. Mattle, H. Weinfurter, and A. Zeilinger, “New high-intensity source of polarization-entangled photon pairs,” Phys. Rev. Lett. 75, 4337–4341 (1995).
[CrossRef] [PubMed]

P. Baldi, P. Aschieri, S. Nouh, M. D. Micheli, D. B. Ostrowsky, D. Delacourt, and M. Papuchon, “Modeling and experimental observation of parametric fluorescence in periodically poled lithium niobate waveguides,” IEEE J. Quantum Electron. 31, 997–1008 (1995).
[CrossRef]

1994 (3)

R. Jozsa, “Fidelity for mixed quantum states,” J. Mod. Opt. 41, 2315–2323 (1994).
[CrossRef]

L. K. Cheng, L. T. Cheng, J. Galperin, P. A. M. Hotsenpiller, and J. D. Bierlein, “Crystal-growth and characterization of ktiopo4 isomorphs from the self-fluxes,” J. Cryst. Growth 137, 107 (1994).
[CrossRef]

M. H. Rubin, D. N. Klyshko, Y. H. Shih, and A. V. Sergienko, “Theory of two-photon entanglement in type-ii parametric down-conversion,” Phys. Rev. A 50, 5122–5133 (1994).
[CrossRef] [PubMed]

1992 (2)

S. L. Braunstein, A. Mann, and M. Revzen, “Maximal violation of bell inequalities for mixed states,” Phys. Ref. Lett. 68, 1992 (1992).
[CrossRef]

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, 2631–2654 (1992).
[CrossRef]

1991 (1)

R. W. Risk, “Fabrication and characterization of planar ion-exchanged ktiopo4 waveguides for frequency doubling,” Appl. Phys. Lett. 58, 19–21 (1991).
[CrossRef]

1988 (1)

Z. Y. Ou and L. Mandel, “Violation of bell’s inequality and classical probability in a two-photon corrlelation experiment,” Phys. Rev. Lett. 61, 50–53 (1988).
[CrossRef] [PubMed]

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, 1216 (1987).
[CrossRef]

1964 (1)

R. C. Miller, “Optical second harmonic generation in piezoelectric crystals,” Appl. Phys. Lett. 5, 17–19 (1964).
[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, 773–776 (1999).
[CrossRef]

Arie, A.

S. Emanueli and A. Arie, “Temperature-dependent dispersion equations for ktiopo4 and ktioaso4,” Appl. Opt. 42, 6661 (2003).
[CrossRef] [PubMed]

K. Frakin, A. Arie, A. Skliar, and G. Rosenman, “Tunable midinfrared source by difference frequency generation in bulk periodically poled ktiopo4,” Appl. Phys. Lett. 74, 914–916 (1999).
[CrossRef]

Aschieri, P.

P. Baldi, P. Aschieri, S. Nouh, M. D. Micheli, D. B. Ostrowsky, D. Delacourt, and M. Papuchon, “Modeling and experimental observation of parametric fluorescence in periodically poled lithium niobate waveguides,” IEEE J. Quantum Electron. 31, 997–1008 (1995).
[CrossRef]

Baldi, P.

P. Baldi, P. Aschieri, S. Nouh, M. D. Micheli, D. B. Ostrowsky, D. Delacourt, and M. Papuchon, “Modeling and experimental observation of parametric fluorescence in periodically poled lithium niobate waveguides,” IEEE J. Quantum Electron. 31, 997–1008 (1995).
[CrossRef]

Barreiro, J. T.

J. T. Barreiro, N. K. Langford, N. A. Peters, and P. G. Kwiat, “Generation of hyperentangled photon pairs,” Phys. Rev. Lett. 95, 260501 (2005).
[CrossRef]

Battle, P.

Beausoleil, R. G.

Bierlein, J. D.

L. K. Cheng, L. T. Cheng, J. Galperin, P. A. M. Hotsenpiller, and J. D. Bierlein, “Crystal-growth and characterization of ktiopo4 isomorphs from the self-fluxes,” J. Cryst. Growth 137, 107 (1994).
[CrossRef]

J. D. Bierlein, A. Ferretti, L. H. Brixner, and W. Y. Hsu, “Fabrication and characterization of optical waveguides in ktiopo4,” Appl. Phys. Lett. 50, 1216 (1987).
[CrossRef]

Bouwmeester, D.

J.-W. Pan, D. Bouwmeester, M. Daniell, H. Wienfurter, and A. Zellinger, “Experimental test of quantum nonlocality in three-photon greenberger-horne-zelinger entanglement,” Nature 403, 515–519 (2000).
[CrossRef] [PubMed]

D. Bouwmeester, J.-W. Pan, K. Mattle, M. Eibl, H. Wienfurter, and A. Zellinger, “Experimental quantum teleportation,” Nature 390, 575–579 (1997).
[CrossRef]

Braunstein, S. L.

S. L. Braunstein, A. Mann, and M. Revzen, “Maximal violation of bell inequalities for mixed states,” Phys. Ref. Lett. 68, 1992 (1992).
[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, 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, 2631–2654 (1992).
[CrossRef]

Canalias, C.

Carrasco, S.

M. Katz, D. Eger, H. Kim, L. Jankovic, G. Stegeman, S. Carrasco, and L. Torner, “Second harmonic generation tuning curves in quasiphase-matched potassium titanyl phosphate with narrow, high-intensity beams,” J. Appl. Phys. 93, 8852–8861 (2003).
[CrossRef]

Chen, J.

Cheng, L. K.

L. K. Cheng, L. T. Cheng, J. Galperin, P. A. M. Hotsenpiller, and J. D. Bierlein, “Crystal-growth and characterization of ktiopo4 isomorphs from the self-fluxes,” J. Cryst. Growth 137, 107 (1994).
[CrossRef]

Cheng, L. T.

L. K. Cheng, L. T. Cheng, J. Galperin, P. A. M. Hotsenpiller, and J. D. Bierlein, “Crystal-growth and characterization of ktiopo4 isomorphs from the self-fluxes,” J. Cryst. Growth 137, 107 (1994).
[CrossRef]

Chiuri, A.

A. Rossi, G. Vallone, A. Chiuri, F. D. Martini, and P. Mataloni, “Multipath entanglement of two photons,” Phys. Rev. Lett. 102, 153902 (2009).
[CrossRef] [PubMed]

Daniell, M.

J.-W. Pan, D. Bouwmeester, M. Daniell, H. Wienfurter, and A. Zellinger, “Experimental test of quantum nonlocality in three-photon greenberger-horne-zelinger entanglement,” Nature 403, 515–519 (2000).
[CrossRef] [PubMed]

Delacourt, D.

P. Baldi, P. Aschieri, S. Nouh, M. D. Micheli, D. B. Ostrowsky, D. Delacourt, and M. Papuchon, “Modeling and experimental observation of parametric fluorescence in periodically poled lithium niobate waveguides,” IEEE J. Quantum Electron. 31, 997–1008 (1995).
[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, 773–776 (1999).
[CrossRef]

Ebrahim-Zadeh, M.

Eger, D.

M. Katz, D. Eger, H. Kim, L. Jankovic, G. Stegeman, S. Carrasco, and L. Torner, “Second harmonic generation tuning curves in quasiphase-matched potassium titanyl phosphate with narrow, high-intensity beams,” J. Appl. Phys. 93, 8852–8861 (2003).
[CrossRef]

Eibl, M.

D. Bouwmeester, J.-W. Pan, K. Mattle, M. Eibl, H. Wienfurter, and A. Zellinger, “Experimental quantum teleportation,” Nature 390, 575–579 (1997).
[CrossRef]

Emanueli, S.

Fan, J.

Fedrizzi, A.

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, 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, 1216 (1987).
[CrossRef]

Fiorentino, M.

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] [PubMed]

T. Kim, M. Fiorentino, and F. N. C. Wong, “Phase-stable source of polarization-entangled photons using a polarization sagnac interferometer,” Phys. Rev. A 73, 012316 (2006).
[CrossRef]

C. E. Kuklewicz, M. Fiorentino, G. Messin, F. N. C. Wong, and J. H. Shapiro, “High-flux source of polarization entangles photons from a periodically poled ktiopo4 parametric down-converter,” Phys. Rev. A 69, 013807 (2004).
[CrossRef]

Frakin, K.

K. Frakin, A. Arie, A. Skliar, and G. Rosenman, “Tunable midinfrared source by difference frequency generation in bulk periodically poled ktiopo4,” Appl. Phys. Lett. 74, 914–916 (1999).
[CrossRef]

Galperin, J.

L. K. Cheng, L. T. Cheng, J. Galperin, P. A. M. Hotsenpiller, and J. D. Bierlein, “Crystal-growth and characterization of ktiopo4 isomorphs from the self-fluxes,” J. Cryst. Growth 137, 107 (1994).
[CrossRef]

Gerhardt, I.

M. P. Peloso, I. Gerhardt, C. Ho, A. Lamas-Linares, and C. Kurtsiefer, “Daylight operation of a free space, entanglement-based quantum key distribution system,” N. J. Phys. 11, 045007 (2009).
[CrossRef]

Herbst, T.

Ho, C.

M. P. Peloso, I. Gerhardt, C. Ho, A. Lamas-Linares, and C. Kurtsiefer, “Daylight operation of a free space, entanglement-based quantum key distribution system,” N. J. Phys. 11, 045007 (2009).
[CrossRef]

Hotsenpiller, P. A. M.

L. K. Cheng, L. T. Cheng, J. Galperin, P. A. M. Hotsenpiller, and J. D. Bierlein, “Crystal-growth and characterization of ktiopo4 isomorphs from the self-fluxes,” J. Cryst. Growth 137, 107 (1994).
[CrossRef]

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, 1216 (1987).
[CrossRef]

Jankovic, L.

M. Katz, D. Eger, H. Kim, L. Jankovic, G. Stegeman, S. Carrasco, and L. Torner, “Second harmonic generation tuning curves in quasiphase-matched potassium titanyl phosphate with narrow, high-intensity beams,” J. Appl. Phys. 93, 8852–8861 (2003).
[CrossRef]

Jennewein, T.

Jozsa, R.

R. Jozsa, “Fidelity for mixed quantum states,” J. Mod. Opt. 41, 2315–2323 (1994).
[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, 2631–2654 (1992).
[CrossRef]

Kato, K.

Katz, M.

M. Katz, D. Eger, H. Kim, L. Jankovic, G. Stegeman, S. Carrasco, and L. Torner, “Second harmonic generation tuning curves in quasiphase-matched potassium titanyl phosphate with narrow, high-intensity beams,” J. Appl. Phys. 93, 8852–8861 (2003).
[CrossRef]

Kim, H.

M. Katz, D. Eger, H. Kim, L. Jankovic, G. Stegeman, S. Carrasco, and L. Torner, “Second harmonic generation tuning curves in quasiphase-matched potassium titanyl phosphate with narrow, high-intensity beams,” J. Appl. Phys. 93, 8852–8861 (2003).
[CrossRef]

Kim, T.

T. Kim, M. Fiorentino, and F. N. C. Wong, “Phase-stable source of polarization-entangled photons using a polarization sagnac interferometer,” Phys. Rev. A 73, 012316 (2006).
[CrossRef]

Klyshko, D. N.

M. H. Rubin, D. N. Klyshko, Y. H. Shih, and A. V. Sergienko, “Theory of two-photon entanglement in type-ii parametric down-conversion,” Phys. Rev. A 50, 5122–5133 (1994).
[CrossRef] [PubMed]

König, F.

F. König and F. N. C. Wong, “Extended phase matching of second-harmonic generation in periodically poled ktiopo4 with zero group-velocity mismatch,” Appl. Phys. Lett. 84, 1644–1646 (2004).
[CrossRef]

Kuklewicz, C. E.

C. E. Kuklewicz, M. Fiorentino, G. Messin, F. N. C. Wong, and J. H. Shapiro, “High-flux source of polarization entangles photons from a periodically poled ktiopo4 parametric down-converter,” Phys. Rev. A 69, 013807 (2004).
[CrossRef]

Kurtsiefer, C.

M. P. Peloso, I. Gerhardt, C. Ho, A. Lamas-Linares, and C. Kurtsiefer, “Daylight operation of a free space, entanglement-based quantum key distribution system,” N. J. Phys. 11, 045007 (2009).
[CrossRef]

Kwiat, P. G.

J. T. Barreiro, N. K. Langford, N. A. Peters, and P. G. Kwiat, “Generation of hyperentangled photon pairs,” Phys. Rev. Lett. 95, 260501 (2005).
[CrossRef]

P. G. Kwiat, E. Waks, A. G. White, I. Appelbaum, and P. H. Eberhard, “Ultrabright source of polarization-entangled photons,” Phys. Rev. A 60, 773–776 (1999).
[CrossRef]

P. G. Kwiat, “Hyper-entangled states,” J. Mod. Opt. 44, 2173 (1997).

P. G. Kwiat, K. Mattle, H. Weinfurter, and A. Zeilinger, “New high-intensity source of polarization-entangled photon pairs,” Phys. Rev. Lett. 75, 4337–4341 (1995).
[CrossRef] [PubMed]

Lamas-Linares, A.

M. P. Peloso, I. Gerhardt, C. Ho, A. Lamas-Linares, and C. Kurtsiefer, “Daylight operation of a free space, entanglement-based quantum key distribution system,” N. J. Phys. 11, 045007 (2009).
[CrossRef]

Langford, N. K.

J. T. Barreiro, N. K. Langford, N. A. Peters, and P. G. Kwiat, “Generation of hyperentangled photon pairs,” Phys. Rev. Lett. 95, 260501 (2005).
[CrossRef]

Laurell, F.

Ling, A.

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, 2631–2654 (1992).
[CrossRef]

Mandel, L.

Z. Y. Ou and L. Mandel, “Violation of bell’s inequality and classical probability in a two-photon corrlelation experiment,” Phys. Rev. Lett. 61, 50–53 (1988).
[CrossRef] [PubMed]

Mann, A.

S. L. Braunstein, A. Mann, and M. Revzen, “Maximal violation of bell inequalities for mixed states,” Phys. Ref. Lett. 68, 1992 (1992).
[CrossRef]

Martini, F. D.

A. Rossi, G. Vallone, A. Chiuri, F. D. Martini, and P. Mataloni, “Multipath entanglement of two photons,” Phys. Rev. Lett. 102, 153902 (2009).
[CrossRef] [PubMed]

Mataloni, P.

A. Rossi, G. Vallone, A. Chiuri, F. D. Martini, and P. Mataloni, “Multipath entanglement of two photons,” Phys. Rev. Lett. 102, 153902 (2009).
[CrossRef] [PubMed]

Mathew, S. C.

Mattle, K.

D. Bouwmeester, J.-W. Pan, K. Mattle, M. Eibl, H. Wienfurter, and A. Zellinger, “Experimental quantum teleportation,” Nature 390, 575–579 (1997).
[CrossRef]

P. G. Kwiat, K. Mattle, H. Weinfurter, and A. Zeilinger, “New high-intensity source of polarization-entangled photon pairs,” Phys. Rev. Lett. 75, 4337–4341 (1995).
[CrossRef] [PubMed]

Messin, G.

C. E. Kuklewicz, M. Fiorentino, G. Messin, F. N. C. Wong, and J. H. Shapiro, “High-flux source of polarization entangles photons from a periodically poled ktiopo4 parametric down-converter,” Phys. Rev. A 69, 013807 (2004).
[CrossRef]

Micheli, M. D.

P. Baldi, P. Aschieri, S. Nouh, M. D. Micheli, D. B. Ostrowsky, D. Delacourt, and M. Papuchon, “Modeling and experimental observation of parametric fluorescence in periodically poled lithium niobate waveguides,” IEEE J. Quantum Electron. 31, 997–1008 (1995).
[CrossRef]

Migdall, A.

Mikami, T.

Miller, R. C.

R. C. Miller, “Optical second harmonic generation in piezoelectric crystals,” Appl. Phys. Lett. 5, 17–19 (1964).
[CrossRef]

Munro, M. W.

Nouh, S.

P. Baldi, P. Aschieri, S. Nouh, M. D. Micheli, D. B. Ostrowsky, D. Delacourt, and M. Papuchon, “Modeling and experimental observation of parametric fluorescence in periodically poled lithium niobate waveguides,” IEEE J. Quantum Electron. 31, 997–1008 (1995).
[CrossRef]

Okamoto, T.

Ostrowsky, D. B.

P. Baldi, P. Aschieri, S. Nouh, M. D. Micheli, D. B. Ostrowsky, D. Delacourt, and M. Papuchon, “Modeling and experimental observation of parametric fluorescence in periodically poled lithium niobate waveguides,” IEEE J. Quantum Electron. 31, 997–1008 (1995).
[CrossRef]

Ou, Z. Y.

Z. Y. Ou and L. Mandel, “Violation of bell’s inequality and classical probability in a two-photon corrlelation experiment,” Phys. Rev. Lett. 61, 50–53 (1988).
[CrossRef] [PubMed]

Pan, J.-W.

J.-W. Pan, D. Bouwmeester, M. Daniell, H. Wienfurter, and A. Zellinger, “Experimental test of quantum nonlocality in three-photon greenberger-horne-zelinger entanglement,” Nature 403, 515–519 (2000).
[CrossRef] [PubMed]

D. Bouwmeester, J.-W. Pan, K. Mattle, M. Eibl, H. Wienfurter, and A. Zellinger, “Experimental quantum teleportation,” Nature 390, 575–579 (1997).
[CrossRef]

Papuchon, M.

P. Baldi, P. Aschieri, S. Nouh, M. D. Micheli, D. B. Ostrowsky, D. Delacourt, and M. Papuchon, “Modeling and experimental observation of parametric fluorescence in periodically poled lithium niobate waveguides,” IEEE J. Quantum Electron. 31, 997–1008 (1995).
[CrossRef]

Pasiskevicius, V.

Pearlmand, A. J.

Peloso, M. P.

M. P. Peloso, I. Gerhardt, C. Ho, A. Lamas-Linares, and C. Kurtsiefer, “Daylight operation of a free space, entanglement-based quantum key distribution system,” N. J. Phys. 11, 045007 (2009).
[CrossRef]

Peters, N. A.

J. T. Barreiro, N. K. Langford, N. A. Peters, and P. G. Kwiat, “Generation of hyperentangled photon pairs,” Phys. Rev. Lett. 95, 260501 (2005).
[CrossRef]

Poppe, A.

Revzen, M.

S. L. Braunstein, A. Mann, and M. Revzen, “Maximal violation of bell inequalities for mixed states,” Phys. Ref. Lett. 68, 1992 (1992).
[CrossRef]

Riis, E.

F. Torabi-Goudarzi and E. Riis, “Efficient cw high-power frequency doubling in periodically poled ktp,” Opt. Commun. 227, 389–403 (2003).
[CrossRef]

Risk, R. W.

R. W. Risk, “Fabrication and characterization of planar ion-exchanged ktiopo4 waveguides for frequency doubling,” Appl. Phys. Lett. 58, 19–21 (1991).
[CrossRef]

Roberts, T. D.

Rosenman, G.

K. Frakin, A. Arie, A. Skliar, and G. Rosenman, “Tunable midinfrared source by difference frequency generation in bulk periodically poled ktiopo4,” Appl. Phys. Lett. 74, 914–916 (1999).
[CrossRef]

Rossi, A.

A. Rossi, G. Vallone, A. Chiuri, F. D. Martini, and P. Mataloni, “Multipath entanglement of two photons,” Phys. Rev. Lett. 102, 153902 (2009).
[CrossRef] [PubMed]

Rubin, M. H.

M. H. Rubin, D. N. Klyshko, Y. H. Shih, and A. V. Sergienko, “Theory of two-photon entanglement in type-ii parametric down-conversion,” Phys. Rev. A 50, 5122–5133 (1994).
[CrossRef] [PubMed]

Samanta, G. K.

Sergienko, A. V.

M. H. Rubin, D. N. Klyshko, Y. H. Shih, and A. V. Sergienko, “Theory of two-photon entanglement in type-ii parametric down-conversion,” Phys. Rev. A 50, 5122–5133 (1994).
[CrossRef] [PubMed]

Shapiro, J. H.

C. E. Kuklewicz, M. Fiorentino, G. Messin, F. N. C. Wong, and J. H. Shapiro, “High-flux source of polarization entangles photons from a periodically poled ktiopo4 parametric down-converter,” Phys. Rev. A 69, 013807 (2004).
[CrossRef]

Shih, Y. H.

M. H. Rubin, D. N. Klyshko, Y. H. Shih, and A. V. Sergienko, “Theory of two-photon entanglement in type-ii parametric down-conversion,” Phys. Rev. A 50, 5122–5133 (1994).
[CrossRef] [PubMed]

Skliar, A.

K. Frakin, A. Arie, A. Skliar, and G. Rosenman, “Tunable midinfrared source by difference frequency generation in bulk periodically poled ktiopo4,” Appl. Phys. Lett. 74, 914–916 (1999).
[CrossRef]

Spillane, S. M.

Stegeman, G.

M. Katz, D. Eger, H. Kim, L. Jankovic, G. Stegeman, S. Carrasco, and L. Torner, “Second harmonic generation tuning curves in quasiphase-matched potassium titanyl phosphate with narrow, high-intensity beams,” J. Appl. Phys. 93, 8852–8861 (2003).
[CrossRef]

Torabi-Goudarzi, F.

F. Torabi-Goudarzi and E. Riis, “Efficient cw high-power frequency doubling in periodically poled ktp,” Opt. Commun. 227, 389–403 (2003).
[CrossRef]

Torner, L.

M. Katz, D. Eger, H. Kim, L. Jankovic, G. Stegeman, S. Carrasco, and L. Torner, “Second harmonic generation tuning curves in quasiphase-matched potassium titanyl phosphate with narrow, high-intensity beams,” J. Appl. Phys. 93, 8852–8861 (2003).
[CrossRef]

Vallone, G.

A. Rossi, G. Vallone, A. Chiuri, F. D. Martini, and P. Mataloni, “Multipath entanglement of two photons,” Phys. Rev. Lett. 102, 153902 (2009).
[CrossRef] [PubMed]

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, 773–776 (1999).
[CrossRef]

Weinfurter, H.

P. G. Kwiat, K. Mattle, H. Weinfurter, and A. Zeilinger, “New high-intensity source of polarization-entangled photon pairs,” Phys. Rev. Lett. 75, 4337–4341 (1995).
[CrossRef] [PubMed]

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, 773–776 (1999).
[CrossRef]

Wienfurter, H.

J.-W. Pan, D. Bouwmeester, M. Daniell, H. Wienfurter, and A. Zellinger, “Experimental test of quantum nonlocality in three-photon greenberger-horne-zelinger entanglement,” Nature 403, 515–519 (2000).
[CrossRef] [PubMed]

D. Bouwmeester, J.-W. Pan, K. Mattle, M. Eibl, H. Wienfurter, and A. Zellinger, “Experimental quantum teleportation,” Nature 390, 575–579 (1997).
[CrossRef]

Wong, F. N. C.

T. Zhong, F. N. C. Wong, T. D. Roberts, and P. Battle, “High performance photon-pair source based on a fiber-coupled periodically poled ktiopo4 waveguide,” Opt. Express 17, 12019–12029 (2009).
[CrossRef] [PubMed]

T. Kim, M. Fiorentino, and F. N. C. Wong, “Phase-stable source of polarization-entangled photons using a polarization sagnac interferometer,” Phys. Rev. A 73, 012316 (2006).
[CrossRef]

C. E. Kuklewicz, M. Fiorentino, G. Messin, F. N. C. Wong, and J. H. Shapiro, “High-flux source of polarization entangles photons from a periodically poled ktiopo4 parametric down-converter,” Phys. Rev. A 69, 013807 (2004).
[CrossRef]

F. König and F. N. C. Wong, “Extended phase matching of second-harmonic generation in periodically poled ktiopo4 with zero group-velocity mismatch,” Appl. Phys. Lett. 84, 1644–1646 (2004).
[CrossRef]

Zeilinger, A.

A. Fedrizzi, T. Herbst, A. Poppe, T. Jennewein, and A. Zeilinger, “A wavelength-tunable fiber-coupled csource of narrowband engtangled photons,” Opt. Express 15, 15377–15386 (2007).
[CrossRef] [PubMed]

P. G. Kwiat, K. Mattle, H. Weinfurter, and A. Zeilinger, “New high-intensity source of polarization-entangled photon pairs,” Phys. Rev. Lett. 75, 4337–4341 (1995).
[CrossRef] [PubMed]

Zellinger, A.

J.-W. Pan, D. Bouwmeester, M. Daniell, H. Wienfurter, and A. Zellinger, “Experimental test of quantum nonlocality in three-photon greenberger-horne-zelinger entanglement,” Nature 403, 515–519 (2000).
[CrossRef] [PubMed]

D. Bouwmeester, J.-W. Pan, K. Mattle, M. Eibl, H. Wienfurter, and A. Zellinger, “Experimental quantum teleportation,” Nature 390, 575–579 (1997).
[CrossRef]

Zhong, T.

Appl. Opt. (2)

Appl. Phys. Lett. (5)

J. D. Bierlein, A. Ferretti, L. H. Brixner, and W. Y. Hsu, “Fabrication and characterization of optical waveguides in ktiopo4,” Appl. Phys. Lett. 50, 1216 (1987).
[CrossRef]

R. C. Miller, “Optical second harmonic generation in piezoelectric crystals,” Appl. Phys. Lett. 5, 17–19 (1964).
[CrossRef]

R. W. Risk, “Fabrication and characterization of planar ion-exchanged ktiopo4 waveguides for frequency doubling,” Appl. Phys. Lett. 58, 19–21 (1991).
[CrossRef]

F. König and F. N. C. Wong, “Extended phase matching of second-harmonic generation in periodically poled ktiopo4 with zero group-velocity mismatch,” Appl. Phys. Lett. 84, 1644–1646 (2004).
[CrossRef]

K. Frakin, A. Arie, A. Skliar, and G. Rosenman, “Tunable midinfrared source by difference frequency generation in bulk periodically poled ktiopo4,” Appl. Phys. Lett. 74, 914–916 (1999).
[CrossRef]

IEEE J. Quantum Electron. (2)

P. Baldi, P. Aschieri, S. Nouh, M. D. Micheli, D. B. Ostrowsky, D. Delacourt, and M. Papuchon, “Modeling and experimental observation of parametric fluorescence in periodically poled lithium niobate waveguides,” IEEE J. Quantum Electron. 31, 997–1008 (1995).
[CrossRef]

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, 2631–2654 (1992).
[CrossRef]

J. Appl. Phys. (1)

M. Katz, D. Eger, H. Kim, L. Jankovic, G. Stegeman, S. Carrasco, and L. Torner, “Second harmonic generation tuning curves in quasiphase-matched potassium titanyl phosphate with narrow, high-intensity beams,” J. Appl. Phys. 93, 8852–8861 (2003).
[CrossRef]

J. Cryst. Growth (1)

L. K. Cheng, L. T. Cheng, J. Galperin, P. A. M. Hotsenpiller, and J. D. Bierlein, “Crystal-growth and characterization of ktiopo4 isomorphs from the self-fluxes,” J. Cryst. Growth 137, 107 (1994).
[CrossRef]

J. Mod. Opt. (2)

P. G. Kwiat, “Hyper-entangled states,” J. Mod. Opt. 44, 2173 (1997).

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[CrossRef]

N. J. Phys. (1)

M. P. Peloso, I. Gerhardt, C. Ho, A. Lamas-Linares, and C. Kurtsiefer, “Daylight operation of a free space, entanglement-based quantum key distribution system,” N. J. Phys. 11, 045007 (2009).
[CrossRef]

Nature (2)

J.-W. Pan, D. Bouwmeester, M. Daniell, H. Wienfurter, and A. Zellinger, “Experimental test of quantum nonlocality in three-photon greenberger-horne-zelinger entanglement,” Nature 403, 515–519 (2000).
[CrossRef] [PubMed]

D. Bouwmeester, J.-W. Pan, K. Mattle, M. Eibl, H. Wienfurter, and A. Zellinger, “Experimental quantum teleportation,” Nature 390, 575–579 (1997).
[CrossRef]

Opt. Commun. (1)

F. Torabi-Goudarzi and E. Riis, “Efficient cw high-power frequency doubling in periodically poled ktp,” Opt. Commun. 227, 389–403 (2003).
[CrossRef]

Opt. Express (4)

Opt. Lett. (1)

Phys. Ref. Lett. (1)

S. L. Braunstein, A. Mann, and M. Revzen, “Maximal violation of bell inequalities for mixed states,” Phys. Ref. Lett. 68, 1992 (1992).
[CrossRef]

Phys. Rev. A (4)

C. E. Kuklewicz, M. Fiorentino, G. Messin, F. N. C. Wong, and J. H. Shapiro, “High-flux source of polarization entangles photons from a periodically poled ktiopo4 parametric down-converter,” Phys. Rev. A 69, 013807 (2004).
[CrossRef]

M. H. Rubin, D. N. Klyshko, Y. H. Shih, and A. V. Sergienko, “Theory of two-photon entanglement in type-ii parametric down-conversion,” Phys. Rev. A 50, 5122–5133 (1994).
[CrossRef] [PubMed]

T. Kim, M. Fiorentino, and F. N. C. Wong, “Phase-stable source of polarization-entangled photons using a polarization sagnac interferometer,” Phys. Rev. A 73, 012316 (2006).
[CrossRef]

P. G. Kwiat, E. Waks, A. G. White, I. Appelbaum, and P. H. Eberhard, “Ultrabright source of polarization-entangled photons,” Phys. Rev. A 60, 773–776 (1999).
[CrossRef]

Phys. Rev. Lett. (4)

A. Rossi, G. Vallone, A. Chiuri, F. D. Martini, and P. Mataloni, “Multipath entanglement of two photons,” Phys. Rev. Lett. 102, 153902 (2009).
[CrossRef] [PubMed]

J. T. Barreiro, N. K. Langford, N. A. Peters, and P. G. Kwiat, “Generation of hyperentangled photon pairs,” Phys. Rev. Lett. 95, 260501 (2005).
[CrossRef]

Z. Y. Ou and L. Mandel, “Violation of bell’s inequality and classical probability in a two-photon corrlelation experiment,” Phys. Rev. Lett. 61, 50–53 (1988).
[CrossRef] [PubMed]

P. G. Kwiat, K. Mattle, H. Weinfurter, and A. Zeilinger, “New high-intensity source of polarization-entangled photon pairs,” Phys. Rev. Lett. 75, 4337–4341 (1995).
[CrossRef] [PubMed]

Other (2)

Technically, the cm are frequency-dependent, however, because their frequency dependence is scaled to the band gap of (3.6 eV for KTP) whereas that of β(ωs, ωi) is scaled to the width of phase matching. For practical crystal lengths L ≳ 10 mm, this width is 1 meV or less, so the cm may be treated as constants.

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Figures (7)

Fig. 1
Fig. 1

Optical system for the generation of polarization-entangled photons. (a) A waveguide in KTP, showing the crystal axes X, Y, and Z. The crystal axes are aligned along spatial z, x, and y, respectively. The region of high Rb is shown. (b) An illustration of two different ways to achieve type-II phase matching. (c) The optical system for a two-photon interference experiment. The first dichroic mirror prevents parametric down conversion in the temporal KTP compensator. The beams enter a bulk KTP crystal whose length is 0.4844 of the length of the waveguide and whose orientation is rotated 90° about the optic axis with respect to the KTP waveguide, i.e., matching the spatial axes (x,y,z) to the crystal axes (–Z,Y,X). The beams are then separated with a dichroic mirror and filtered to remove down converted photons from higher harmonics of the poling period. The mirror on the idler branch may have a polarization dependent reflectivity which may enhance the fidelity as discussed in the text. The beams are coupled into unpolarized single mode optical fibers.

Fig. 2
Fig. 2

Contours of Δkm (ωs , ωi ) = 0 (vertical, red) and Δkm (ωi , ωs ) = 0 (horizontal, blue) for Λ = 1 mm and m = ±1,±3, as defined in Eq. (6). The axes are labeled by frequencies ν = ω/2π as well as by the free space wavelengths λ = c/ν. The line with negative slope obeys ωs + ωi = ωp for λp = 632.58 nm. The line with positive slope indicates ωs = ωi . The large black dots represent the pairs ( ω s ( 1 ) , ω i ( 1 ) )=( ω i ( 1 ) , ω s ( 1 ) ) and ( ω i ( 1 ) , ω s ( 1 ) )=( ω s ( 1 ) , ω i ( 1 ) ) as defined in Eq. (7). The small black dots represent the pairs of daughter photons generated at the frequencies pairs ( ω s ( ± 3 ) , ω i ( ± 3 ) ) and ( ω i ( ± 3 ) , ω s ( ± 3 ) ) . Additional frequency pairs will be generated for additional pairs of odd ±m, but these are not in the range of the figure.

Fig. 3
Fig. 3

The index of refraction for bulk RTP and KTP is given as a function of the free space wavelength for polarization along the crystal Y and Z axes. The effective index of refraction for the fundamental mode is given for an Rb:KTP waveguide (WG) with parameters w = 3.5 μm and Z 0 = 4.5 μm.

Fig. 4
Fig. 4

(top) The simulation domain, including a 3.5 μm wide strip of Rb:KTP, two quarter circles of KTP of radius 20 μm, and an air cap with a height of 4 μm. The light blue region, a 4.5 μm × 5.5 μm rectangle, is expanded in the panels below. (bottom left) The index of refraction, with white for air (1), blue for KTP (1.74), and the Rb:KTP strip ranging from the pure RTP value of 1.765 to 1.74. Specifically nY (λ = 1265.4 nm) is shown. (bottom middle) The Z polarized mode at λ = 1265.4 nm. (bottom right) The Y-polarized mode at λ = 1265.4 nm. In the bottom three figures, horizontal lines are drawn at a depth of 1 μm, 2 μm, and 3 μm. The inner vertical lines are 3.5 μm apart and mark the edge of the Rb-doped region.

Fig. 5
Fig. 5

Poling period as a function of degenerate SPDC wavelength (λs = λi = 2λp ) for type-0 and type-II phase matching in both bulk KTP and in a Rb:KTP waveguide (WG) with parameters from Fig. 3. The wavelengths for unpoled type-II SHG are indicated. The curves for RTP are similar to KTP, but shifted to longer wavelengths including a shift of the the unpoled wavelength to 1139 nm. The diffraction order m as defined in Eq. (6) is indicated on the graph; the type 0 phase matching always uses m = 1, but type II phase matching uses m = 1 for shorter wavelengths and m = +1 for longer wavelengths.

Fig. 6
Fig. 6

The signal, idler, and pump wavelengths (with polarizations along the crystal axes Z, Y, and Y, respectively) as a function of the poling period for bulk KTP and for a waveguide (WG) using the parameters from Fig. 3. The curves for bulk RTP are between the two families shown here. The values for limΛ → ∞ are reported in Table 2.

Fig. 7
Fig. 7

The fidelity, according to Eq. (25) for bulk KTP, bulk RTP, and the Rb:KTP waveguide (WG) using the parameters from Fig. 3. (A single mode is assumed for bulk KTP and bulk RTP which give these a model character.) The spatial mode effect is estimated to reduce the fidelity by a factor of 0.97 for Λ ≥ 0.2 mm, falling to 0.89 for Λ = 0.1 mm. However, the spatial mode effect can be largely eliminated, as discussed in the text.

Tables (2)

Tables Icon

Table 1 Comparison of Present Sellmeier Equation to Experimental Results for Type I Second Harmonic Generation, which is the Inverse of Degenerate SPDC

Tables Icon

Table 2 Pump and Degenerate SPDC Wavelengths (nm) in Unpoled Rb:KTP Waveguides and Bulk RTP and KTP a

Equations (44)

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| 2 ¯ = 4 d ω s d ω i d r s d r i β ( ω s , ω i ) U Y ( r s , ω s ) U Z ( r i , ω i ) a Y ( r s , ω s ) a Z ( r i , ω i ) | 0 ,
| 2 ¯ = 4 d ω s d ω i β ( ω s , ω i ) a Y ( ω s ) a Z ( ω i ) | 0 ,
β ( ω s , ω i ) = α ( ω s + ω i ) m = c m sin c [ Δ k m ( ω s , ω i ) ] .
α ( ω p ) = 1 ( 2 π ) 1 / 4 σ p 1 / 2 exp ( ( ω p ω p ( 1 ) ) 2 4 σ p 2 ) ,
lim σ p 0 [ α ( ω p ) ] 2 = δ ( ω p ω p ( 1 ) ) .
Δ k m ( ω s , ω i ) = k Z ( ω s ) + k Y ( ω i ) k Y ( ω p ) + 2 π m Λ
Δ k 1 ( ω s , ω i ) = k Z ( ω s ( 1 ) ) + k Y ( ω i ( 1 ) ) k Y ( ω p ( 1 ) ) + 2 π Λ = 0 Δ k 1 ( ω i ( 1 ) , ω s ( 1 ) ) = k Z ( ω i ( 1 ) ) + k Y ( ω s ( 1 ) ) k Y ( ω p ( 1 ) ) 2 π Λ = 0 ,
Δ k ± 1 ( ω s , ω i ) = k z ( ω s ) + k y ( ω i ) k y ( ω p ) ± 2 π Λ k z ( ω s ( ± 1 ) ) + k z ( ω s ( ± 1 ) ) ξ s + k y ( ω i ( ± 1 ) ) + k y ( ω i ( ± 1 ) ) ξ i k y ( ω p ( ± 1 ) ) k y ( ω p ( ± 1 ) ) ξ p ± 2 π Λ ,
Δ k ± 1 ( 0 ) ( ξ s , ξ i ) k z ( ω s ( ± 1 ) ) ξ s + k y ( ω i ( ± 1 ) ) ξ i k y ( ω p ( ± 1 ) ) ( ξ s + ξ i )
Δ k 1 ( 0 ) ( ξ i , ξ s ) k z ( ω 1 ( 1 ) ) ξ i + k y ( ω s ( 1 ) ) ξ s k y ( ω p ( 1 ) ) ( ξ s + ξ i )
τ 1 s = ( k z ( ω s ( 1 ) ) k y ( ω p ( 1 ) ) ) , τ 1 i = ( k y ( ω i ( 1 ) ) k y ( ω p ( 1 ) ) ) , τ 1 s = ( k y ( ω s ( 1 ) ) k y ( ω p ( 1 ) ) ) , and τ 1 i = ( k z ( ω i ( 1 ) ) k y ( ω p ( 1 ) ) ) ,
Δ k 1 ( 0 ) ( ξ s , ξ i ) τ 1 s ξ s + τ 1 i ξ i Δ k 1 ( 0 ) ( ξ i , ξ s ) τ 1 s ξ s + τ 1 i ξ i .
| 2 ¯ = 2 d ω s d ω i [ β ( ω s , ω i ) a Y ( ω s ) a Z ( ω i ) + β ( ω i , ω s ) a Y ( ω i ) a Z ( ω s ) ] | 0 .
| 2 ¯ ± = d ω s d ω i [ β ( ω s , ω i ) ± β ( ω i , ω s ) ] [ a Y ( ω s ) a Z ( ω i ) ± a Y ( ω i ) a Z ( ω s ) ] | 0 ,
| 2 ¯ = ± | 2 ¯ ±
| 2 = | 2 ¯ 2 ¯ | 2 ¯ 1 / 2
2 ¯ | 2 ¯ = ± 2 ¯ ± | 2 ¯ ± .
F = | 2 + | 2 | 2 = 2 ¯ + | 2 ¯ + 2 | 2
0 | [ a Z ( ω ¯ i ) a Y ( ω ¯ s ) ± a Z ( ω ¯ s ) a Y ( ω ¯ i ) ] [ a Y ( ω s ) a Z ( ω i ) ± a Y ( ω i ) a Z ( ω s ) ] | 0 = 2 δ ( ω ¯ s ω s ) δ ( ω ¯ i ω i ) ± 2 δ ( ω ¯ s ω i ) δ ( ω ¯ i ω s )
2 ¯ ± | 2 ¯ ± = d ω ¯ s d ω ¯ i d ω s d ω i [ β ( ω ¯ s , ω ¯ i ) ± β ( ω ¯ i , ω ¯ s ) ] * [ β ( ω s , ω i ) ± β ( ω i , ω s ) ] × [ 2 δ ( ω ¯ s ω s ) δ ( ω ¯ i ω i ) ± 2 δ ( ω ¯ s ω i ) δ ( ω ¯ i ω s ) ] .
I = d ω s d ω i | β ( ω s , ω i ) | 2
J = d ω s d ω i β * ( ω i , ω s ) β ( ω s , ω i ) .
2 ¯ ± | 2 ¯ ± = 8 I ± 8 Re J .
F = 1 2 ( 1 + Re J I ) .
F = 1 | Δ τ + Δ τ | 2 ( Δ τ + + Δ τ )
2 ¯ ± | 2 ¯ ± = 2 d ω ¯ s d ω ¯ i d ω s d ω i d r ¯ s d r ¯ i β * ( ω ¯ s , ω ¯ i ) β ( ω s , ω i ) ± [ U Y * ( r ¯ s , ω ¯ s ) U Z * ( r ¯ i , ω ¯ i ) ± U Y * ( r ¯ i , ω i ) U Z * ( r ¯ s , ω s ) ] [ U Y ( r s , ω s ) U Z ( r i , ω i ) ± U Y ( r i , ω i ) U Z ( r s , ω s ) ] [ 2 δ ( ω ¯ ω s ) δ ( ω ¯ i ω i ) δ ( r ¯ s r s ) δ ( r ¯ i r i ) ± 2 δ ( ω ¯ s ω i ) δ ( ω ¯ i ω s ) δ ( r ¯ s r i ) δ ( r ¯ i r s ) ] = 8 I ± 8 J d r s d r i U Y * ( r s , ω s ) U Z * ( r i , ω i ) U Y ( r i , ω i ) U Z ( r s , ω s )
2 ¯ ± | 2 ¯ ± = 8 I ± 8 J | d r U Y ( r ) * U Z ( r ) | 2 .
d ( x , y ) = n = + d n ( x ) e i n K z
f | H I | i = d h ¯ ( 2 𝒫 p ω s ω i ) 1 / 2 ( ε 0 cn s 2 n i 2 n p ) 1 / 2 L sinc ( Δ k L 2 ) A dxdy U p ( x , y ) U i * ( x , y ) U s * ( x , y )
c 1 c 1 = d YZY ( ω p , ω s , ω i ) d YZY ( ω p , ω i , ω s ) n Z ( ω i ) n Z ( ω s ) n Y ( ω s ) n Y ( ω i ) G ( ω s , ω s , ω i ) G ( ω p , ω i , ω s )
G ( ω p , ω s , ω i ) = A dxdy U Y ( x , y ; ω p ) U Z * ( x , y ; ω s ) U Y * ( x , y ; ω i )
d YZY ( ω p , ω s , ω i ) δ YZY χ YY ( ω p ) χ ZZ ( ω s ) χ ZZ ( ω i )
c 1 c 1 = n Z 2 ( ω s ) 1 n Z 2 ( ω i ) 1 n Y 2 ( ω i ) 1 n Y 2 ( ω s ) 1 n Z ( ω i ) n Z ( ω s ) n Y ( ω s ) n Y ( ω i ) G ( ω p , ω s , ω i ) G ( ω p , ω i , ω s ) .
I = d ω s d ω i [ α ( ω s + ω i ) ] 2 { ± sinc [ Δ k ± 1 ( ω s , ω i ) ] } 2 .
I = I + + + I + I + + I + .
1 | τ ± 1 s τ ± 1 i | | ω s ( 1 ) ω i ( 1 ) |
I + + = d ξ s d ξ i [ α ( 0 ) ( ξ s + ξ i ) ] 2 sinc 2 [ Δ k 1 ( 0 ) ( ξ s , ξ i ) ] d ξ s d ξ i [ α ( 0 ) ( ξ s + ξ i ) ] 2 sinc 2 ( τ 1 s ξ s + τ 1 i ξ i ) .
I = d ξ i d ξ s [ α ( 0 ) ( ξ i + ξ s ) ] 2 sinc 2 [ Δ k 1 ( 0 ) ( ξ i , ξ s ) ] d ξ s d ξ i [ α ( 0 ) ( ξ s + ξ i ) ] 2 sinc 2 ( τ 1 s ξ s + τ 1 i ξ i ) .
J = d ω s d ω i [ α ( ω s + ω i ) ] 2 ± { sinc [ Δ k ± 1 ( ω s , ω i ) ] sinc [ Δ k ± 1 ( ω i , ω s ) ] + sinc [ Δ k ± 1 ( ω s , ω i ) ] sinc [ Δ k 1 ( ω i , ω s ) ] } .
J = J + + + J + J + + J + .
J + = d ξ s d ξ i [ α ( 0 ) ( ξ s + ξ i ) ] 2 sinc [ Δ k 1 ( 0 ) ( ξ s , ξ i ) ] sinc [ Δ k 1 ( 0 ) ( ξ i , ξ s ) ] d ξ s d ξ i [ α ( 0 ) ( ξ s + ξ i ) ] 2 sinc ( τ 1 s ξ s + τ 1 i ξ i ) sinc ( τ 1 s ξ s + τ 1 i ξ i ) .
I ± ± = d ξ sinc 2 ( δ τ ± ξ ) = π Δ τ ± ,
J + = d ξ sinc ( δ τ + ξ ) sinc ( δ τ ξ ) = π Δ τ + + Δ τ | Δ τ + Δ τ | 2 Δ τ + Δ τ .
I ± I ± ± = π Δ τ + + Δ τ Δ τ + Δ τ

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