Abstract

We present equations for the power generated via spontaneous (quantum) and stimulated (classical) nonlinear optical processes in integrated devices. Equations for the same structure and same order process are derived from the same Hamiltonian, allowing for direct and easy comparison including the ability to estimate the efficiency of a quantum process based solely on experimental data from a classical process in the same device. We show that, in the CW limit and under the undepleted pump approximation, the average energy of a generated photon divided by a characteristic time plays the role of the classical “seed” signal in a quantum process, and that extending the length of a structure or taking advantage of a resonant cavity does not enhance spontaneous processes the same way as stimulated processes.

© 2012 Optical Society of America

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  3. S. Tanzilli, W. Tittel, H. De Riedmatten, H. Zbinden, P. Baldi, M. DeMicheli, D. B. Ostrowsky, and N. Gisin, “PPLN waveguide for quantum communication,” Eur. Phys. J. D 18, 155–160 (2002).
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    [CrossRef]
  6. E. Y. Zhu, Z. Tang, L. Qian, L. G. Helt, M. Liscidini, J. E. Sipe, C. Corbari, A. Canagasabey, M. Ibsen, and P. G. Kazansky, “Direct generation of polarization-entangled photon pairs in a poled fiber,” Phys. Rev. Lett. 108, 213902 (2012).
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    [CrossRef]
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    [CrossRef]
  24. Y. S. Kim and M. E. Noz, Phase Space Picture of Quantum Mechanics (World Scientific, 1991).

2012

E. Y. Zhu, Z. Tang, L. Qian, L. G. Helt, M. Liscidini, J. E. Sipe, C. Corbari, A. Canagasabey, M. Ibsen, and P. G. Kazansky, “Direct generation of polarization-entangled photon pairs in a poled fiber,” Phys. Rev. Lett. 108, 213902 (2012).

R. Horn, P. Abolghasem, B. J. Bijlani, D. Kang, A. S. Helmy, and G. Weihs, “Monolithic source of photon pairs,” Phys Rev. Lett. 108, 153605 (2012).

2011

C. Xiong, G. D. Marshall, A. Peruzzo, M. Lobino, A. S. Clark, D.-Y. Choi, S. J. Madden, C. M. Natarajan, M. G. Tanner, R. H. Hadfield, S. N. Dorenbox, T. Zijlstra, V. Zsiller, M. G. Thompson, J. G. Rarity, M. J. Steel, B. Luther-Davies, B. J. Eggleton, and J. L. O’Brien, “Generation of correlated photon pairs in a chalcogenide As2S3 waveguide,” Appl. Phys. Lett. 98, 051101 (2011).
[CrossRef]

2010

2009

2008

M. B. Nasr, S. Carrasco1, B. E. A. Saleh, A. V. Sergienko, M. C. Teich, J. P. Torres, L. Torner, D. S. Hum, and M. M. Fejer, “Ultrabroadband biphotons generated via chirped quasi-phase-matched optical parametric down-conversion,” Phys. Rev. Lett. 100, 183601 (2008).
[CrossRef]

M. Ferrera, L. Razzari1, D. Duchesne, R. Morandotti, Z. Yang, M. Liscidini, J. E. Sipe, S. Chu, B. E. Little, and D. J. Moss, “Low-power continuous-wave nonlinear optics in doped silica glass integrated waveguide structures,” Nat. Photon. 2, 737–740 (2008).
[CrossRef]

A. C. Turner, M. A. Foster, A. L. Gaeta, and M. Lipson, “Ultra-low power parametric frequency conversion in a silicon microring resonator,” Opt. Express 16, 4881–4887 (2008).
[CrossRef]

Z. Yang, M. Liscidini, and J. E. Sipe, “Spontaneous parametric down-conversion in waveguides: a backward Heisenberg picture approach,” Phys. Rev. A 77, 033808 (2008).
[CrossRef]

2007

2006

2004

C. K. Law and J. H. Eberly, “Analysis and interpretation of high transverse entanglement in optical parametric down conversion,” Phys. Rev. Lett. 92, 127903 (2004).
[CrossRef]

2002

S. Tanzilli, W. Tittel, H. De Riedmatten, H. Zbinden, P. Baldi, M. DeMicheli, D. B. Ostrowsky, and N. Gisin, “PPLN waveguide for quantum communication,” Eur. Phys. J. D 18, 155–160 (2002).

2001

A. De Rossi, V. Berger, M. Calligaro, G. Leo, V. Ortiz, and X. Marcadet, “Parametric fluorescence in oxidized gallium arsenide waveguides,” Appl. Phys. Lett. 79, 3758–3760 (2001).
[CrossRef]

2000

1961

P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118–119 (1961).
[CrossRef]

Abolghasem, P.

R. Horn, P. Abolghasem, B. J. Bijlani, D. Kang, A. S. Helmy, and G. Weihs, “Monolithic source of photon pairs,” Phys Rev. Lett. 108, 153605 (2012).

Absil, P. P.

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Elsevier, 2006).

Aitchison, J. S.

Baldi, P.

S. Tanzilli, W. Tittel, H. De Riedmatten, H. Zbinden, P. Baldi, M. DeMicheli, D. B. Ostrowsky, and N. Gisin, “PPLN waveguide for quantum communication,” Eur. Phys. J. D 18, 155–160 (2002).

Berger, V.

A. De Rossi, V. Berger, M. Calligaro, G. Leo, V. Ortiz, and X. Marcadet, “Parametric fluorescence in oxidized gallium arsenide waveguides,” Appl. Phys. Lett. 79, 3758–3760 (2001).
[CrossRef]

Bijlani, B. J.

R. Horn, P. Abolghasem, B. J. Bijlani, D. Kang, A. S. Helmy, and G. Weihs, “Monolithic source of photon pairs,” Phys Rev. Lett. 108, 153605 (2012).

Brainis, E.

E. Brainis, “Four-photon scattering in birefringent fibers,” Phys. Rev. A 79, 023840 (2009).
[CrossRef]

Bristow, A. D.

Calligaro, M.

A. De Rossi, V. Berger, M. Calligaro, G. Leo, V. Ortiz, and X. Marcadet, “Parametric fluorescence in oxidized gallium arsenide waveguides,” Appl. Phys. Lett. 79, 3758–3760 (2001).
[CrossRef]

Canagasabey, A.

E. Y. Zhu, Z. Tang, L. Qian, L. G. Helt, M. Liscidini, J. E. Sipe, C. Corbari, A. Canagasabey, M. Ibsen, and P. G. Kazansky, “Direct generation of polarization-entangled photon pairs in a poled fiber,” Phys. Rev. Lett. 108, 213902 (2012).

Carrasco1, S.

M. B. Nasr, S. Carrasco1, B. E. A. Saleh, A. V. Sergienko, M. C. Teich, J. P. Torres, L. Torner, D. S. Hum, and M. M. Fejer, “Ultrabroadband biphotons generated via chirped quasi-phase-matched optical parametric down-conversion,” Phys. Rev. Lett. 100, 183601 (2008).
[CrossRef]

Chak, P.

Cho, P. S.

Chu, S.

M. Ferrera, L. Razzari1, D. Duchesne, R. Morandotti, Z. Yang, M. Liscidini, J. E. Sipe, S. Chu, B. E. Little, and D. J. Moss, “Low-power continuous-wave nonlinear optics in doped silica glass integrated waveguide structures,” Nat. Photon. 2, 737–740 (2008).
[CrossRef]

Clark, A.

C. Monat, A. Clark, C. Xiong, C. Grillet, G. Marshall, M. J. Steel, J. Li, L. O’Faolain, T. F. Krauss, and B. J. Eggleton, “Correlated photon-pair generation in an ultra-compact silicon photonic crystal waveguide,” in CLEO:2011, Baltimore (2011), paper PDPC4.

Corbari, C.

E. Y. Zhu, Z. Tang, L. Qian, L. G. Helt, M. Liscidini, J. E. Sipe, C. Corbari, A. Canagasabey, M. Ibsen, and P. G. Kazansky, “Direct generation of polarization-entangled photon pairs in a poled fiber,” Phys. Rev. Lett. 108, 213902 (2012).

De Riedmatten, H.

S. Tanzilli, W. Tittel, H. De Riedmatten, H. Zbinden, P. Baldi, M. DeMicheli, D. B. Ostrowsky, and N. Gisin, “PPLN waveguide for quantum communication,” Eur. Phys. J. D 18, 155–160 (2002).

De Rossi, A.

A. De Rossi, V. Berger, M. Calligaro, G. Leo, V. Ortiz, and X. Marcadet, “Parametric fluorescence in oxidized gallium arsenide waveguides,” Appl. Phys. Lett. 79, 3758–3760 (2001).
[CrossRef]

DeMicheli, M.

S. Tanzilli, W. Tittel, H. De Riedmatten, H. Zbinden, P. Baldi, M. DeMicheli, D. B. Ostrowsky, and N. Gisin, “PPLN waveguide for quantum communication,” Eur. Phys. J. D 18, 155–160 (2002).

Duchesne, D.

M. Ferrera, L. Razzari1, D. Duchesne, R. Morandotti, Z. Yang, M. Liscidini, J. E. Sipe, S. Chu, B. E. Little, and D. J. Moss, “Low-power continuous-wave nonlinear optics in doped silica glass integrated waveguide structures,” Nat. Photon. 2, 737–740 (2008).
[CrossRef]

Eberly, J. H.

C. K. Law and J. H. Eberly, “Analysis and interpretation of high transverse entanglement in optical parametric down conversion,” Phys. Rev. Lett. 92, 127903 (2004).
[CrossRef]

Eggleton, B. J.

C. Monat, A. Clark, C. Xiong, C. Grillet, G. Marshall, M. J. Steel, J. Li, L. O’Faolain, T. F. Krauss, and B. J. Eggleton, “Correlated photon-pair generation in an ultra-compact silicon photonic crystal waveguide,” in CLEO:2011, Baltimore (2011), paper PDPC4.

Fejer, M. M.

M. B. Nasr, S. Carrasco1, B. E. A. Saleh, A. V. Sergienko, M. C. Teich, J. P. Torres, L. Torner, D. S. Hum, and M. M. Fejer, “Ultrabroadband biphotons generated via chirped quasi-phase-matched optical parametric down-conversion,” Phys. Rev. Lett. 100, 183601 (2008).
[CrossRef]

Ferrera, M.

M. Ferrera, L. Razzari1, D. Duchesne, R. Morandotti, Z. Yang, M. Liscidini, J. E. Sipe, S. Chu, B. E. Little, and D. J. Moss, “Low-power continuous-wave nonlinear optics in doped silica glass integrated waveguide structures,” Nat. Photon. 2, 737–740 (2008).
[CrossRef]

Foster, M. A.

Franken, P. A.

P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118–119 (1961).
[CrossRef]

Gaeta, A. L.

Gisin, N.

S. Tanzilli, W. Tittel, H. De Riedmatten, H. Zbinden, P. Baldi, M. DeMicheli, D. B. Ostrowsky, and N. Gisin, “PPLN waveguide for quantum communication,” Eur. Phys. J. D 18, 155–160 (2002).

Grillet, C.

C. Monat, A. Clark, C. Xiong, C. Grillet, G. Marshall, M. J. Steel, J. Li, L. O’Faolain, T. F. Krauss, and B. J. Eggleton, “Correlated photon-pair generation in an ultra-compact silicon photonic crystal waveguide,” in CLEO:2011, Baltimore (2011), paper PDPC4.

Grover, R.

J. E. Heebner, R. Grover, and T. Ibrahim, Optical Microresonators: Theory, Fabrication, and Applications (Springer, 2008).

Heebner, J. E.

J. E. Heebner, R. Grover, and T. Ibrahim, Optical Microresonators: Theory, Fabrication, and Applications (Springer, 2008).

Helmy, A. S.

R. Horn, P. Abolghasem, B. J. Bijlani, D. Kang, A. S. Helmy, and G. Weihs, “Monolithic source of photon pairs,” Phys Rev. Lett. 108, 153605 (2012).

Helt, L. G.

Hill, A. E.

P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118–119 (1961).
[CrossRef]

Ho, P.-T.

Horn, R.

R. Horn, P. Abolghasem, B. J. Bijlani, D. Kang, A. S. Helmy, and G. Weihs, “Monolithic source of photon pairs,” Phys Rev. Lett. 108, 153605 (2012).

Hryniewicz, J. V.

Hum, D. S.

M. B. Nasr, S. Carrasco1, B. E. A. Saleh, A. V. Sergienko, M. C. Teich, J. P. Torres, L. Torner, D. S. Hum, and M. M. Fejer, “Ultrabroadband biphotons generated via chirped quasi-phase-matched optical parametric down-conversion,” Phys. Rev. Lett. 100, 183601 (2008).
[CrossRef]

Ibrahim, T.

J. E. Heebner, R. Grover, and T. Ibrahim, Optical Microresonators: Theory, Fabrication, and Applications (Springer, 2008).

Ibsen, M.

E. Y. Zhu, Z. Tang, L. Qian, L. G. Helt, M. Liscidini, J. E. Sipe, C. Corbari, A. Canagasabey, M. Ibsen, and P. G. Kazansky, “Direct generation of polarization-entangled photon pairs in a poled fiber,” Phys. Rev. Lett. 108, 213902 (2012).

Iyer, R.

Joneckis, L. G.

Kang, D.

R. Horn, P. Abolghasem, B. J. Bijlani, D. Kang, A. S. Helmy, and G. Weihs, “Monolithic source of photon pairs,” Phys Rev. Lett. 108, 153605 (2012).

Kazansky, P. G.

E. Y. Zhu, Z. Tang, L. Qian, L. G. Helt, M. Liscidini, J. E. Sipe, C. Corbari, A. Canagasabey, M. Ibsen, and P. G. Kazansky, “Direct generation of polarization-entangled photon pairs in a poled fiber,” Phys. Rev. Lett. 108, 213902 (2012).

Kim, Y. S.

Y. S. Kim and M. E. Noz, Phase Space Picture of Quantum Mechanics (World Scientific, 1991).

Krauss, T. F.

C. Monat, A. Clark, C. Xiong, C. Grillet, G. Marshall, M. J. Steel, J. Li, L. O’Faolain, T. F. Krauss, and B. J. Eggleton, “Correlated photon-pair generation in an ultra-compact silicon photonic crystal waveguide,” in CLEO:2011, Baltimore (2011), paper PDPC4.

Kumar, P.

Law, C. K.

C. K. Law and J. H. Eberly, “Analysis and interpretation of high transverse entanglement in optical parametric down conversion,” Phys. Rev. Lett. 92, 127903 (2004).
[CrossRef]

Lee, K. F.

Leo, G.

A. De Rossi, V. Berger, M. Calligaro, G. Leo, V. Ortiz, and X. Marcadet, “Parametric fluorescence in oxidized gallium arsenide waveguides,” Appl. Phys. Lett. 79, 3758–3760 (2001).
[CrossRef]

Li, J.

C. Monat, A. Clark, C. Xiong, C. Grillet, G. Marshall, M. J. Steel, J. Li, L. O’Faolain, T. F. Krauss, and B. J. Eggleton, “Correlated photon-pair generation in an ultra-compact silicon photonic crystal waveguide,” in CLEO:2011, Baltimore (2011), paper PDPC4.

Liang, C.

Lipson, M.

Liscidini, M.

E. Y. Zhu, Z. Tang, L. Qian, L. G. Helt, M. Liscidini, J. E. Sipe, C. Corbari, A. Canagasabey, M. Ibsen, and P. G. Kazansky, “Direct generation of polarization-entangled photon pairs in a poled fiber,” Phys. Rev. Lett. 108, 213902 (2012).

L. G. Helt, Z. Yang, M. Liscidini, and J. E. Sipe, “Spontaneous four-wave mixing in microring resonators,” Opt. Lett. 35, 3006–3008 (2010).
[CrossRef]

L. G. Helt, E. Y. Zhu, M. Liscidini, L. Qian, and J. E. Sipe, “Proposal for in-fiber generation of telecom-band polarization-entangled photon pairs using a periodically poled fiber,” Opt. Lett. 34, 2138–2140 (2009).
[CrossRef]

Z. Yang, M. Liscidini, and J. E. Sipe, “Spontaneous parametric down-conversion in waveguides: a backward Heisenberg picture approach,” Phys. Rev. A 77, 033808 (2008).
[CrossRef]

M. Ferrera, L. Razzari1, D. Duchesne, R. Morandotti, Z. Yang, M. Liscidini, J. E. Sipe, S. Chu, B. E. Little, and D. J. Moss, “Low-power continuous-wave nonlinear optics in doped silica glass integrated waveguide structures,” Nat. Photon. 2, 737–740 (2008).
[CrossRef]

Little, B. E.

M. Ferrera, L. Razzari1, D. Duchesne, R. Morandotti, Z. Yang, M. Liscidini, J. E. Sipe, S. Chu, B. E. Little, and D. J. Moss, “Low-power continuous-wave nonlinear optics in doped silica glass integrated waveguide structures,” Nat. Photon. 2, 737–740 (2008).
[CrossRef]

P. P. Absil, J. V. Hryniewicz, B. E. Little, P. S. Cho, R. A. Wilson, L. G. Joneckis, and P.-T. Ho, “Wavelength conversion in GaAs micro-ring resonators,” Opt. Lett. 25, 554–556 (2000).
[CrossRef]

Marcadet, X.

A. De Rossi, V. Berger, M. Calligaro, G. Leo, V. Ortiz, and X. Marcadet, “Parametric fluorescence in oxidized gallium arsenide waveguides,” Appl. Phys. Lett. 79, 3758–3760 (2001).
[CrossRef]

Marshall, G.

C. Monat, A. Clark, C. Xiong, C. Grillet, G. Marshall, M. J. Steel, J. Li, L. O’Faolain, T. F. Krauss, and B. J. Eggleton, “Correlated photon-pair generation in an ultra-compact silicon photonic crystal waveguide,” in CLEO:2011, Baltimore (2011), paper PDPC4.

Medic, M.

Monat, C.

C. Monat, A. Clark, C. Xiong, C. Grillet, G. Marshall, M. J. Steel, J. Li, L. O’Faolain, T. F. Krauss, and B. J. Eggleton, “Correlated photon-pair generation in an ultra-compact silicon photonic crystal waveguide,” in CLEO:2011, Baltimore (2011), paper PDPC4.

Morandotti, R.

M. Ferrera, L. Razzari1, D. Duchesne, R. Morandotti, Z. Yang, M. Liscidini, J. E. Sipe, S. Chu, B. E. Little, and D. J. Moss, “Low-power continuous-wave nonlinear optics in doped silica glass integrated waveguide structures,” Nat. Photon. 2, 737–740 (2008).
[CrossRef]

Moss, D. J.

M. Ferrera, L. Razzari1, D. Duchesne, R. Morandotti, Z. Yang, M. Liscidini, J. E. Sipe, S. Chu, B. E. Little, and D. J. Moss, “Low-power continuous-wave nonlinear optics in doped silica glass integrated waveguide structures,” Nat. Photon. 2, 737–740 (2008).
[CrossRef]

Nasr, M. B.

M. B. Nasr, S. Carrasco1, B. E. A. Saleh, A. V. Sergienko, M. C. Teich, J. P. Torres, L. Torner, D. S. Hum, and M. M. Fejer, “Ultrabroadband biphotons generated via chirped quasi-phase-matched optical parametric down-conversion,” Phys. Rev. Lett. 100, 183601 (2008).
[CrossRef]

Noz, M. E.

Y. S. Kim and M. E. Noz, Phase Space Picture of Quantum Mechanics (World Scientific, 1991).

O’Faolain, L.

C. Monat, A. Clark, C. Xiong, C. Grillet, G. Marshall, M. J. Steel, J. Li, L. O’Faolain, T. F. Krauss, and B. J. Eggleton, “Correlated photon-pair generation in an ultra-compact silicon photonic crystal waveguide,” in CLEO:2011, Baltimore (2011), paper PDPC4.

Ortiz, V.

A. De Rossi, V. Berger, M. Calligaro, G. Leo, V. Ortiz, and X. Marcadet, “Parametric fluorescence in oxidized gallium arsenide waveguides,” Appl. Phys. Lett. 79, 3758–3760 (2001).
[CrossRef]

Ostrowsky, D. B.

S. Tanzilli, W. Tittel, H. De Riedmatten, H. Zbinden, P. Baldi, M. DeMicheli, D. B. Ostrowsky, and N. Gisin, “PPLN waveguide for quantum communication,” Eur. Phys. J. D 18, 155–160 (2002).

Peters, C. W.

P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118–119 (1961).
[CrossRef]

Qian, L.

E. Y. Zhu, Z. Tang, L. Qian, L. G. Helt, M. Liscidini, J. E. Sipe, C. Corbari, A. Canagasabey, M. Ibsen, and P. G. Kazansky, “Direct generation of polarization-entangled photon pairs in a poled fiber,” Phys. Rev. Lett. 108, 213902 (2012).

L. G. Helt, E. Y. Zhu, M. Liscidini, L. Qian, and J. E. Sipe, “Proposal for in-fiber generation of telecom-band polarization-entangled photon pairs using a periodically poled fiber,” Opt. Lett. 34, 2138–2140 (2009).
[CrossRef]

Razzari1, L.

M. Ferrera, L. Razzari1, D. Duchesne, R. Morandotti, Z. Yang, M. Liscidini, J. E. Sipe, S. Chu, B. E. Little, and D. J. Moss, “Low-power continuous-wave nonlinear optics in doped silica glass integrated waveguide structures,” Nat. Photon. 2, 737–740 (2008).
[CrossRef]

Saleh, B. E. A.

M. B. Nasr, S. Carrasco1, B. E. A. Saleh, A. V. Sergienko, M. C. Teich, J. P. Torres, L. Torner, D. S. Hum, and M. M. Fejer, “Ultrabroadband biphotons generated via chirped quasi-phase-matched optical parametric down-conversion,” Phys. Rev. Lett. 100, 183601 (2008).
[CrossRef]

Schmidt, B. S.

Sergienko, A. V.

M. B. Nasr, S. Carrasco1, B. E. A. Saleh, A. V. Sergienko, M. C. Teich, J. P. Torres, L. Torner, D. S. Hum, and M. M. Fejer, “Ultrabroadband biphotons generated via chirped quasi-phase-matched optical parametric down-conversion,” Phys. Rev. Lett. 100, 183601 (2008).
[CrossRef]

Sharping, J. E.

Sipe, J. E.

E. Y. Zhu, Z. Tang, L. Qian, L. G. Helt, M. Liscidini, J. E. Sipe, C. Corbari, A. Canagasabey, M. Ibsen, and P. G. Kazansky, “Direct generation of polarization-entangled photon pairs in a poled fiber,” Phys. Rev. Lett. 108, 213902 (2012).

L. G. Helt, Z. Yang, M. Liscidini, and J. E. Sipe, “Spontaneous four-wave mixing in microring resonators,” Opt. Lett. 35, 3006–3008 (2010).
[CrossRef]

L. G. Helt, E. Y. Zhu, M. Liscidini, L. Qian, and J. E. Sipe, “Proposal for in-fiber generation of telecom-band polarization-entangled photon pairs using a periodically poled fiber,” Opt. Lett. 34, 2138–2140 (2009).
[CrossRef]

M. Ferrera, L. Razzari1, D. Duchesne, R. Morandotti, Z. Yang, M. Liscidini, J. E. Sipe, S. Chu, B. E. Little, and D. J. Moss, “Low-power continuous-wave nonlinear optics in doped silica glass integrated waveguide structures,” Nat. Photon. 2, 737–740 (2008).
[CrossRef]

Z. Yang, M. Liscidini, and J. E. Sipe, “Spontaneous parametric down-conversion in waveguides: a backward Heisenberg picture approach,” Phys. Rev. A 77, 033808 (2008).
[CrossRef]

Z. Yang, and J. E. Sipe, “Generating entangled photons via enhanced spontaneous parametric downconversion in AlGaAs microring resonators,” Opt. Lett. 32, 3296–3298 (2007).
[CrossRef]

Z. Yang, P. Chak, A. D. Bristow, H. M. van Driel, R. Iyer, J. S. Aitchison, A. L. Smirl, and J. E. Sipe, “Enhanced second-harmonic generation in AlGaAs microring resonators,” Opt. Lett. 32, 826–828 (2007).
[CrossRef]

Smirl, A. L.

Steel, M. J.

C. Monat, A. Clark, C. Xiong, C. Grillet, G. Marshall, M. J. Steel, J. Li, L. O’Faolain, T. F. Krauss, and B. J. Eggleton, “Correlated photon-pair generation in an ultra-compact silicon photonic crystal waveguide,” in CLEO:2011, Baltimore (2011), paper PDPC4.

Tang, Z.

E. Y. Zhu, Z. Tang, L. Qian, L. G. Helt, M. Liscidini, J. E. Sipe, C. Corbari, A. Canagasabey, M. Ibsen, and P. G. Kazansky, “Direct generation of polarization-entangled photon pairs in a poled fiber,” Phys. Rev. Lett. 108, 213902 (2012).

Tanzilli, S.

S. Tanzilli, W. Tittel, H. De Riedmatten, H. Zbinden, P. Baldi, M. DeMicheli, D. B. Ostrowsky, and N. Gisin, “PPLN waveguide for quantum communication,” Eur. Phys. J. D 18, 155–160 (2002).

Teich, M. C.

M. B. Nasr, S. Carrasco1, B. E. A. Saleh, A. V. Sergienko, M. C. Teich, J. P. Torres, L. Torner, D. S. Hum, and M. M. Fejer, “Ultrabroadband biphotons generated via chirped quasi-phase-matched optical parametric down-conversion,” Phys. Rev. Lett. 100, 183601 (2008).
[CrossRef]

Tittel, W.

S. Tanzilli, W. Tittel, H. De Riedmatten, H. Zbinden, P. Baldi, M. DeMicheli, D. B. Ostrowsky, and N. Gisin, “PPLN waveguide for quantum communication,” Eur. Phys. J. D 18, 155–160 (2002).

Torner, L.

M. B. Nasr, S. Carrasco1, B. E. A. Saleh, A. V. Sergienko, M. C. Teich, J. P. Torres, L. Torner, D. S. Hum, and M. M. Fejer, “Ultrabroadband biphotons generated via chirped quasi-phase-matched optical parametric down-conversion,” Phys. Rev. Lett. 100, 183601 (2008).
[CrossRef]

Torres, J. P.

M. B. Nasr, S. Carrasco1, B. E. A. Saleh, A. V. Sergienko, M. C. Teich, J. P. Torres, L. Torner, D. S. Hum, and M. M. Fejer, “Ultrabroadband biphotons generated via chirped quasi-phase-matched optical parametric down-conversion,” Phys. Rev. Lett. 100, 183601 (2008).
[CrossRef]

Turner, A. C.

van Driel, H. M.

Weihs, G.

R. Horn, P. Abolghasem, B. J. Bijlani, D. Kang, A. S. Helmy, and G. Weihs, “Monolithic source of photon pairs,” Phys Rev. Lett. 108, 153605 (2012).

Weinreich, G.

P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118–119 (1961).
[CrossRef]

Wilson, R. A.

Xiong, C.

C. Monat, A. Clark, C. Xiong, C. Grillet, G. Marshall, M. J. Steel, J. Li, L. O’Faolain, T. F. Krauss, and B. J. Eggleton, “Correlated photon-pair generation in an ultra-compact silicon photonic crystal waveguide,” in CLEO:2011, Baltimore (2011), paper PDPC4.

Yang, Z.

L. G. Helt, Z. Yang, M. Liscidini, and J. E. Sipe, “Spontaneous four-wave mixing in microring resonators,” Opt. Lett. 35, 3006–3008 (2010).
[CrossRef]

Z. Yang, M. Liscidini, and J. E. Sipe, “Spontaneous parametric down-conversion in waveguides: a backward Heisenberg picture approach,” Phys. Rev. A 77, 033808 (2008).
[CrossRef]

M. Ferrera, L. Razzari1, D. Duchesne, R. Morandotti, Z. Yang, M. Liscidini, J. E. Sipe, S. Chu, B. E. Little, and D. J. Moss, “Low-power continuous-wave nonlinear optics in doped silica glass integrated waveguide structures,” Nat. Photon. 2, 737–740 (2008).
[CrossRef]

Z. Yang, P. Chak, A. D. Bristow, H. M. van Driel, R. Iyer, J. S. Aitchison, A. L. Smirl, and J. E. Sipe, “Enhanced second-harmonic generation in AlGaAs microring resonators,” Opt. Lett. 32, 826–828 (2007).
[CrossRef]

Z. Yang, and J. E. Sipe, “Generating entangled photons via enhanced spontaneous parametric downconversion in AlGaAs microring resonators,” Opt. Lett. 32, 3296–3298 (2007).
[CrossRef]

Yariv, A.

A. Yariv, Quantum Electronics, 2nd ed. (Wiley, 1975).

Zbinden, H.

S. Tanzilli, W. Tittel, H. De Riedmatten, H. Zbinden, P. Baldi, M. DeMicheli, D. B. Ostrowsky, and N. Gisin, “PPLN waveguide for quantum communication,” Eur. Phys. J. D 18, 155–160 (2002).

Zhu, E. Y.

E. Y. Zhu, Z. Tang, L. Qian, L. G. Helt, M. Liscidini, J. E. Sipe, C. Corbari, A. Canagasabey, M. Ibsen, and P. G. Kazansky, “Direct generation of polarization-entangled photon pairs in a poled fiber,” Phys. Rev. Lett. 108, 213902 (2012).

L. G. Helt, E. Y. Zhu, M. Liscidini, L. Qian, and J. E. Sipe, “Proposal for in-fiber generation of telecom-band polarization-entangled photon pairs using a periodically poled fiber,” Opt. Lett. 34, 2138–2140 (2009).
[CrossRef]

Appl. Phys. Lett.

C. Xiong, G. D. Marshall, A. Peruzzo, M. Lobino, A. S. Clark, D.-Y. Choi, S. J. Madden, C. M. Natarajan, M. G. Tanner, R. H. Hadfield, S. N. Dorenbox, T. Zijlstra, V. Zsiller, M. G. Thompson, J. G. Rarity, M. J. Steel, B. Luther-Davies, B. J. Eggleton, and J. L. O’Brien, “Generation of correlated photon pairs in a chalcogenide As2S3 waveguide,” Appl. Phys. Lett. 98, 051101 (2011).
[CrossRef]

A. De Rossi, V. Berger, M. Calligaro, G. Leo, V. Ortiz, and X. Marcadet, “Parametric fluorescence in oxidized gallium arsenide waveguides,” Appl. Phys. Lett. 79, 3758–3760 (2001).
[CrossRef]

Eur. Phys. J. D

S. Tanzilli, W. Tittel, H. De Riedmatten, H. Zbinden, P. Baldi, M. DeMicheli, D. B. Ostrowsky, and N. Gisin, “PPLN waveguide for quantum communication,” Eur. Phys. J. D 18, 155–160 (2002).

Nat. Photon.

M. Ferrera, L. Razzari1, D. Duchesne, R. Morandotti, Z. Yang, M. Liscidini, J. E. Sipe, S. Chu, B. E. Little, and D. J. Moss, “Low-power continuous-wave nonlinear optics in doped silica glass integrated waveguide structures,” Nat. Photon. 2, 737–740 (2008).
[CrossRef]

Opt. Express

Opt. Lett.

Phys Rev. Lett.

R. Horn, P. Abolghasem, B. J. Bijlani, D. Kang, A. S. Helmy, and G. Weihs, “Monolithic source of photon pairs,” Phys Rev. Lett. 108, 153605 (2012).

Phys. Rev. A

Z. Yang, M. Liscidini, and J. E. Sipe, “Spontaneous parametric down-conversion in waveguides: a backward Heisenberg picture approach,” Phys. Rev. A 77, 033808 (2008).
[CrossRef]

E. Brainis, “Four-photon scattering in birefringent fibers,” Phys. Rev. A 79, 023840 (2009).
[CrossRef]

Phys. Rev. Lett.

C. K. Law and J. H. Eberly, “Analysis and interpretation of high transverse entanglement in optical parametric down conversion,” Phys. Rev. Lett. 92, 127903 (2004).
[CrossRef]

M. B. Nasr, S. Carrasco1, B. E. A. Saleh, A. V. Sergienko, M. C. Teich, J. P. Torres, L. Torner, D. S. Hum, and M. M. Fejer, “Ultrabroadband biphotons generated via chirped quasi-phase-matched optical parametric down-conversion,” Phys. Rev. Lett. 100, 183601 (2008).
[CrossRef]

E. Y. Zhu, Z. Tang, L. Qian, L. G. Helt, M. Liscidini, J. E. Sipe, C. Corbari, A. Canagasabey, M. Ibsen, and P. G. Kazansky, “Direct generation of polarization-entangled photon pairs in a poled fiber,” Phys. Rev. Lett. 108, 213902 (2012).

P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118–119 (1961).
[CrossRef]

Other

C. Monat, A. Clark, C. Xiong, C. Grillet, G. Marshall, M. J. Steel, J. Li, L. O’Faolain, T. F. Krauss, and B. J. Eggleton, “Correlated photon-pair generation in an ultra-compact silicon photonic crystal waveguide,” in CLEO:2011, Baltimore (2011), paper PDPC4.

A. Yariv, Quantum Electronics, 2nd ed. (Wiley, 1975).

G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Elsevier, 2006).

J. E. Heebner, R. Grover, and T. Ibrahim, Optical Microresonators: Theory, Fabrication, and Applications (Springer, 2008).

Y. S. Kim and M. E. Noz, Phase Space Picture of Quantum Mechanics (World Scientific, 1991).

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Figures (2)

Fig. 1.
Fig. 1.

Schematic of frequency spacings of second- and third-order nonlinear optical processes.

Fig. 2.
Fig. 2.

Schematic of frequencies generated by second- and third-order nonlinear optical processes, organized according to spontaneous, stimulated, and reverse processes, with corresponding processes occupying the first two columns.

Tables (2)

Tables Icon

Table 1. Expressions for the Power Generated in Corresponding Nonlinear Optical Processes in Channel Waveguides

Tables Icon

Table 2. Expressions for the Power Generated in Corresponding Nonlinear Optical Processes in Microring Resonators

Equations (138)

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H=HL+HNL,
HL=dkωFkaFkaFk+dkωSHkaSHkaSHk,
HNL=dk1dk2dkS(k1,k2,k)aFk1aFk2aSHk+H.c.,
|ψgen=exp(βCIIH.c.)|vac,
CII=12dω1dω2ϕ(ω1,ω2)aFω1aFω2,
ϕ(ω1,ω2)=iLdkF(ω1)dω1dkF(ω2)dω2(dkSH(ω)dω)ω=ω1+ω2×ω1ω2(χ¯2)2NP8πε0|β|2n¯6(ω1+ω2)A[kF(ω1),kF(ω2),kSH(ω1+ω2)]×ϕP(ω1+ω2)sinc{[kF(ω1)+kF(ω2)kSH(ω1+ω2)]L/2}.
A[kF(ω1),kF(ω2),kSH(ω1+ω2)]=|dxdyn¯3χ2ijk(x,y)[dFω1i(x,y)]*[dFω2j(x,y)]*dSH(ω1+ω2)k(x,y)χ¯2ε03/2n2(x,y;ω1)n2(x,y;ω2)n2(x,y;ω1+ω2)|2
dω1dω2|ϕ(ω1,ω2)|2=1,
|ψgen|vac+βCII|vac,
ND=2ωPNPL2PAT,
T=2π0ωP/2dΩ[1(2Ω/ωP)2]sinc2{[kF(ωP/2+Ω)+kF(ωP/2Ω)kSH(ωP)]L/2},
PI=ωFTPPL2PA.
Gm(z,t)=(ωmvm/(2π))1/2dkeikzamkei(ωmtkmz),
HL=m=P,S,IdzGmGm/vm+i2ωmdz(GmzGmGmGmz),
H2NL=χ¯2n¯32ε0vSHvF2AdzGPGSGIei(kS+kIkP)z+H.c.,
1vSHGPt+GPz=2iPAGSGIei(kS+kIkP)z,
1vFGSt+GSz=iPAGIGPei(kS+kIkP)z,
1vFGIt+GIz=iPAGSGPei(kS+kIkP)z,
PI=PSPPL2PAsinc2{[kS(ωS)+kI(ωPωS)kP(ωP)]L2}.
PSH=PF2L2PAsinc2{[kSH(2ωF)2kF(ωF)]L2},
T2π0ωP/2dΩsinc2{[kF(ωP/2Ω)+kF(ωP/2+Ω)kSH(ωP)]L/2}.
km(ω)=km(ωm0)+1vm(ωωm0)+β2(ωm0)2(ωωm0)2,
T322π|β2(ωP/2)|L,
PI=ωF322π|β2(ωP/2)|LPPL2PA.
T(Ω)(Bsinc2{[kF(ωP/2Ω)+kF(ωP/2+Ω)kSH(ωP)]L/2})1,
PI(Ω)=ωFB1PPL2PAsinc2{[kF(ωP/2Ω)+kF(ωP/2+Ω)kSH(ωP)]L2}
PI=ωF32τFPPL2PA,
H3=HL+HNL,
HL=dkωkakak,
HNL=dk1dk2dk3dk4S(k1,k2,k3,k4)ak1ak2ak3ak4+H.c.,
PI=ωPT(γPPL)2,
T=2π0ωPdΩ[1(Ω/ωP)2]sinc2{[2k(ωP)k(ωP+Ω)k(ωPΩ)]L/2},
PI=PS(γPPL)2sinc2{[2kP(ωP)kS(ωS)kI(2ωPωS)]L/2}.
k(ω)=k(ωP)+1v(ωωP)+β2(ωP)2(ωωP)2,
T322π|β2(ωP)|L.
PI=ωP322π|β2(ωP)|L(γPPL)2.
T(Ω)(Bsinc2{[2k(ωP)k(ωP+Ω)k(ωPΩ)]L/2})1.
PI(Ω)=ωPB1(γPPL)2sinc2{[2k(ωP)k(ωP+Ω)k(ωPΩ)]L2},
PI=ωP32τP(γPPL)2,
H=HL+HNL,
HL=Hch+Hr+Hcp,
HL=μ1.μ2[Sμ1μ2μSHbμSHbμ1b2+H.c.],
Hch=μ[ωμdzψμ(z)ψμ(z)+i2vμdz(ψμ(z)zψμ(z)H.c.)],
Hcp=2πμ[cμbμψμ(0)+H.c.],
HR=μωμbμbμ,
kμ·R=N,
PI=ωμFTPPL2PA|FμP(ωμP)|2.
Fμ(ω)=i[2(1σμ)]1/2(1σμ)i(ωωμ)L/vμ,
Fμ(ωμ)=2iQμvμωμL.
T=2πdω|FμF(ωμP/2ω)|2|FμF(ωμP/2+ω)|2.
PI=PSPPL2PA|FμS(ωS)|2|FμI(ωPωS)|2|FμP(ωP)|2,
PSH=PF2L2PA|FμF(ωF)|4|FμSH(2ωF)|2.
T=|FμF(ωμF)|4L3(2ωμFωμP)2+16vμF2L8vμF3|FμF(ωμF)|2,
T=2LvμF|FμF(ωμF)|2=ωμFL22vμF2QμF,
PI=ωμFvμF2L|FμF(ωμF)|2PPL2PA|FμP(ωμP)|2|FμF(ωμF)|4=ωμFQμF16πτμFPPL2PAQμPvμPωμPL(QμFvμFωμFL)2,
PI(Ω)=ωμFB1PPL2PA|FμP(ωμP)|2|FμF(ωμP/2Ω)|2|FμF(ωμP/2+Ω)|2,
NNF=R[ωμF/(2|ΞF|QμF)]1/2,
PI=ωμPT(γPPL)2|FμP(ωμP)|4,
T=2πdω|FμP(ωμPω)|2|FμP(ωμP+ω)|2.
PI=PS(γPPL)2|FμP(ωP)|4|FμS(ωS)|2|FμI(2ωPωS)|2,
T=2LvμP|FμP(ωμP)|2=ωμPL22vμP2QμP,
PI=ωμPvμP2L|FμP(ωμP)|2(γPPL)2|FμP(ωμP)|8=ωμPQμP64πτμP(γPPL)2(QμPvμPωμPL)4,
PI(Ω)=ωμPB1(γPPL)2|FμP(ωμP)|4|FμP(ωμPΩ)|2|FμP(ωμP+Ω)|2,
NNP=R[ωμP/(2|ΞP|QμP)]1/2,
S(k1,k2,k3,k4)=32(ωk1)(ωk2)(ωk3)(ωk4)(4π)4ε02χ¯3n¯4L/2L/2dzei(k3+k4k1k2)zeiϕ(k1,k2,k3,k4)A(k1,k2,k3,k4),
eiϕ(k1,k2,k3,k4)A(k1,k2,k3,k4)=dxdyn¯4χ3ijkl(x,y)[dk1i(x,y)]*[dk2j(x,y)]*dk3k(x,y)dk4l(x,y)χ¯3ε02n2(x,y;ωk1)n2(x,y;ωk2)n2(x,y;ωk3)n2(x,y;ωk4)
H0=HL=dkωkakak,
V^(t)=U(t1,t)eiH0t/HNLeiH0t/U(t1,t)=dk1dk2dk3dk4S(k1,k2,k3,k4;t)a¯k1(t)a¯k2(t)a¯k3(t)a¯k4(t),
U(t,t)=eiH0t/eiH(tt)/eiH0t/,
O¯(t)=U(t1,t)OU(t1,t),
O¯(t1)=O,
S(k1,k2,k3,k4;t)=S(k1,k2,k3,k4)ei(ωk1+ωk2ωk3ωk4)t,
iO¯(t)t=[O¯(t),V^(t)].
|ψin=e(αAPH.c.)|vac,
AP=dkϕP(k)ak,
dk|ϕP(k)|2=1,
|ψout=eαA¯P(t0)H.c.|vac,
A¯P(t0)=dkϕP(k)a¯k(t0),
ida¯k(t)dt=2dk1dk2dk3S(k1,k2,k3,k;t)a¯k1(t)a¯k2(t)a¯k3(t).
(a¯k(t))0=a¯k(t1)=ak,
a¯k(t)=ak+2idk1dk2dk3[t1t0dtS(k1,k2,k3,k;t)]ak1ak2ak3=ak+2idk1dk2dk3[t0t1dtS(k1,k2,k3,k;t)]ak1ak2ak3=ak+4πidk1dk2dk3S(k1,k2,k3,k)ak1ak2ak3δ(ωk1+ωk2ωk3ωk),
eA+B=eAe12[A,B]+higher order termseB,
[S1,[S2,[Sn1,Sn]]],Si=1,2,n=AorB,
A=αdkϕP(k)akH.c.,B=4πiαdkdk1dk2dk3ϕP(k)S(k1,k2,k3,k)ak1ak2ak3H.c.,
12[A,B]=2πiα2dkdk1dk2dk3ϕP(k3)ϕP(k)S(k1,k2,k3,k)ak1ak2H.c.+8πi|α|2dkdk1dk2dk3ϕP*(k2)ϕP(k)S(k1,k2,k3,k)ak1ak3H.c..
eB|vac=|vac,
|ψout=e(αAP+βCII)H.c.|vac,
CII=12dk1dk2ϕ(k1,k2)ak1ak2,
ϕ(k1,k2)=22πα2βidkdkϕP(k)ϕP3(k)×S(k1,k2,k,k)δ(ωk+ωkωk1ωk2),
dk1dk2|ϕ(k1,k2)|2=1.
|ψgen=eβCIIH.c.|vac|vac+β|II+,
|II=CII|vac=12dk1dk2ϕ(k1,k2)ak1ak2|vac
dkakak|II=2|II,II.|II=1,
ND=2|β|2.
a˜ω=dk(ω)dωak(ω),
ϕ˜P(ω)=dk(ω)dωϕP(k(ω)),
ϕ˜(ω1,ω2)=dk(ω1)dω1dk(ω2)dω2ϕ(k(ω1),k(ω2)),
dk(ω1)dω1=(dk(ω)dω)ω=ω1,
0dω|ϕ˜P(ω)|2=1,
0dω10dω2|ϕ˜(ω1,ω2)|2=1.
ϕ˜(ω1,ω2)=22πα2βidk(ω1)dω1dk(ω2)dω20dω0dω{dk(ω)dωdk(ω)dωϕP(ω)ϕP(ω)S[k(ω1),k(ω2),k(ω),k(ω)]δ(ω1+ω2ωω)},
ND=NP2(χ¯3)2n¯89K64π2ε02,
K=0ω1dk(ω1)dω10ω2dk(ω2)dω2J(ω1,ω2)dω1dω2,
J(ω1,ω2)=|0ω1+ω2dωdk(ω)dωdk(ω1+ω2ω)d(ω1+ω2ω)ϕ˜P(ω)ϕ˜P(ω1+ω2ω)ω(ω1+ω2ω)×L/2L/2dzei(k(ω)+k(ω1+ω2ω)k(ω1)k(ω2))zeiϕ(k(ω1),k(ω2),k(ω),k(ω1+ω2ω))A[k(ω1),k(ω2),k(ω),k(ω1+ω2ω)]|2.
k(ω)=k(ωP)+1v(ωωP)+β2(ωP)2(ωωP)2,
ϕ˜(ω1,ω2)=iLv22ω1ω2KA20ω1+ω2dωsin[(ωωP)Δt/2](ωωP)πΔt/2sin[(ω1+ω2ωωP)Δt/2](ω1+ω2ωωP)πΔt/2×ω(ω1+ω2ω)sinc{β2(ωP)2[(ωω1+ω22)2(ω1ω22)2]L},
K=2L2v4A20dω10dω2ω1ω2|0ω1+ω2dωsin[(ωωP)Δt/2](ωωP)πΔt/2sin[(ω1+ω2ωωP)Δt/2](ω1+ω2ωωP)πΔt/2ω(ω1+ω2ω)sinc{β2(ωP)2[(ωω1+ω22)2(ω1ω22)2]L}|2,
ϕ˜P(ω)=sin[(ωωP)Δt/2](ωωP)πΔt/2.
K=42L2ωP2πv4A2Δt0dωtδ(ωt2ωP)012ωtdωr(14ωt2ωr2)sinc2(β2(ωP)ωr2L/2)=42L2ωP2πv4A2Δt0ωPdωr(ωP2ωr2)sinc2(β2(ωP)ωr2L/2)=82L2ωP4π2v4A2ΔtT,
γ=3χ¯3ωP4ε0v2n¯4A,
ND=2(γωPNPL)2ΔtT.
PI=ωPT(γPPL)2,
ωmk=ωm+vm(kkm)+,
ωm=(ωmk)k=km,
vm=(ωmkk)k=km
ωmkωm,
dmki(x,y)dmi(x,y)
gm(z,t)=dk2πamkei(kkm)z,
[gS(z,t),gI(z,t)]=[gS(z,t),gP(z,t)]=[gI(z,t),gP(z,t)]=0.
HL=m=P,S,I[ωmdzgmgm+i2vmdz(gmzgmgmgmz)],
HNL=3ε0(ωP)2(ωS)(ωI)24A2χ¯3n¯4dzei(2kPkSkI)zgSgIgP2+H.c.,
A=|dxdyn¯4χ3ijkl(x,y)χ¯3[dSi(x,y)]*[dIj(x,y)]*dPk(x,y)dPl(x,y)ε02n2(x,y;ωS)n2(x,y;ωI)n4(x,y;ωP)|1,
gPt=i[gP,H]=iωPgPvPgPz6iε0(ωP)2(ωS)(ωI)24A2χ¯3n¯4ei(2kPkSkI)zgPgSgI,
gSt=i[gS,H3]=iωSgSvSgSz3iε0(ωP)2(ωS)(ωI)24A2χ¯3n¯4ei(2kPkSkI)zgIgP2,
gIt=i[gI,H3]=iωIgIvIgIz3iε0(ωP)2(ωS)(ωI)24A2χ¯3n¯4ei(2kPkSkI)zgSgP2.
gm=g˜meiωmt,
2ωP=ωS+ωI
Gm=ωmvmg˜m,
1vPGPt+GPz=6iε0ωP224A2vSvIvP2χ¯3n¯4ei(2kPkSkI)zGPGSGI,
1vSGSt+GSz=3iε0ωS224A2vSvIvP2χ¯3n¯4ei(2kPkSkI)zgIgP2,
1vIGIt+GIz=3iε0ωI224A2vSvIvP2χ¯3n¯4ei(2kPkSkI)zGSGP2.
GPz=0,
GSz=0,
GI=3iε0ωI224A2vSvIvP2χ¯3n¯4GSGP2Lsinc[(2kPkSkI)L2]
γ=n2ω0cAeff=3χ3xxxx4ε0cn¯2ω0cAeff,
γ=3ωI4ε0vPvSvIχ¯3n¯41A,
GI=iγGSGP2Lsinc[(2kPkSkI)L/2],
PI=PS(γPPL)2sinc2{2kP(ωP)kS(ωS)kI(2ωPωS)L/2},

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