Abstract

We show that highly nonlinear chalcogenide glass nanowire waveguides with near-zero anomalous dispersion should be capable of generating correlated photon-pairs by spontaneous four-wave mixing at frequencies detuned by over 17 THz from the pump where Raman noise is absent. In this region we predict a photon pair correlation of >100, a figure of merit >10 and brightness of ~8×108 pairs/s over a bandwidth of >15 THz in nanowires with group velocity dispersion of <5 ps∙km−1nm−1. We present designs for double-clad Ge11.5As24Se64.5 glass nanowires with realistic tolerance to fabrication errors that achieve near-zero anomalous dispersion at a 1420 nm pump wavelength. This structure has a fabrication tolerance of 80–170 nm in the waveguide width and utilizes a SiO2/Al2O3 layer deposited by atomic layer deposition to compensate the fabrication errors in the film thickness.

© 2012 OSA

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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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2011 (2)

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. Dorenbos, T. Zijlstra, V. Zwiller, 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(5), 051101 (2011).
[CrossRef]

X. Gai, D.-Y. Choi, S. Madden, and B. Luther-Davies, “Interplay between Raman scattering and four-wave mixing in As2S3 chalcogenide glass waveguides,” J. Opt. Soc. Am. B 28(11), 2777–2784 (2011).
[CrossRef]

2010 (4)

2009 (2)

2008 (3)

2007 (2)

2006 (2)

2004 (1)

2003 (1)

N. Yoran and B. Reznik, “Deterministic linear optics quantum computation with single photon qubits,” Phys. Rev. Lett. 91(3), 037903 (2003).
[CrossRef] [PubMed]

2002 (1)

N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quamtum cryptography,” Rev. Mod. Phys. 74(1), 145–195 (2002).
[CrossRef]

1994 (1)

P. Lüsse, P. Stuwe, J. Schüle, and H. G. Unger, “Analysis of vectorial mode fields in optical waveguides by a new finite difference method,” J. Lightwave Technol. 12(3), 487–494 (1994).
[CrossRef]

Agrawal, G. P.

Bristow, A. D.

A. D. Bristow, N. Rotenberg, and H. M. van Driel, “Two-photon absorption and Kerr coefficients of silicon for 850–2200 nm,” Appl. Phys. Lett. 90(19), 191104 (2007).
[CrossRef]

Bulla, D.

Chen, J.

Choi, D.-Y.

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. Dorenbos, T. Zijlstra, V. Zwiller, 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(5), 051101 (2011).
[CrossRef]

X. Gai, D.-Y. Choi, S. Madden, and B. Luther-Davies, “Interplay between Raman scattering and four-wave mixing in As2S3 chalcogenide glass waveguides,” J. Opt. Soc. Am. B 28(11), 2777–2784 (2011).
[CrossRef]

X. Gai, S. Madden, D.-Y. Choi, D. Bulla, and B. Luther-Davies, “Dispersion engineered Ge11.5As24Se64.5 nanowires with a nonlinear parameter of 136 W⁻¹m⁻¹ at 1550 nm,” Opt. Express 18(18), 18866–18874 (2010).
[CrossRef] [PubMed]

X. Gai, T. Han, A. Prasad, S. Madden, D.-Y. Choi, R. Wang, D. Bulla, and B. Luther-Davies, “Progress in optical waveguides fabricated from chalcogenide glasses,” Opt. Express 18(25), 26635–26646 (2010).
[CrossRef] [PubMed]

F. Luan, M. D. Pelusi, M. R. E. Lamont, D.-Y. Choi, S. Madden, B. Luther-Davies, and B. J. Eggleton, “Dispersion engineered As2S3 planar waveguides for broadband four-wave mixing based wavelength conversion of 40 Gb/s signals,” Opt. Express 17(5), 3514–3520 (2009).
[CrossRef] [PubMed]

M. R. Lamont, B. Luther-Davies, D.-Y. Choi, S. Madden, X. Gai, and B. J. Eggleton, “Net-gain from a parametric amplifier on a chalcogenide optical chip,” Opt. Express 16(25), 20374–20381 (2008).
[CrossRef] [PubMed]

Clark, A. S.

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. Dorenbos, T. Zijlstra, V. Zwiller, 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(5), 051101 (2011).
[CrossRef]

Coen, S.

Dorenbos, S. N.

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. Dorenbos, T. Zijlstra, V. Zwiller, 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(5), 051101 (2011).
[CrossRef]

Eggleton, B. J.

Fallahkhair, A. B.

Fauchet, P. M.

Foster, M. A.

Gaeta, A. L.

Gai, X.

George, S. M.

S. M. George, “Atomic Layer Deposition: An Overview,” Chem. Rev. 110(1), 111–131 (2010).
[CrossRef] [PubMed]

Gisin, N.

N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quamtum cryptography,” Rev. Mod. Phys. 74(1), 145–195 (2002).
[CrossRef]

Hadfield, R. H.

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. Dorenbos, T. Zijlstra, V. Zwiller, 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(5), 051101 (2011).
[CrossRef]

Han, T.

Harvey, J. D.

Helt, L. G.

Hsieh, A. S. Y.

Judge, A. C.

Kumar, P.

Lamont, M. R.

Lamont, M. R. E.

Lee, K. F.

Leonhardt, R.

Li, K. S.

Li, X.

Lin, Q.

Lipson, M.

Lobino, M.

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. Dorenbos, T. Zijlstra, V. Zwiller, 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(5), 051101 (2011).
[CrossRef]

Luan, F.

Lüsse, P.

P. Lüsse, P. Stuwe, J. Schüle, and H. G. Unger, “Analysis of vectorial mode fields in optical waveguides by a new finite difference method,” J. Lightwave Technol. 12(3), 487–494 (1994).
[CrossRef]

Luther-Davies, B.

X. Gai, D.-Y. Choi, S. Madden, and B. Luther-Davies, “Interplay between Raman scattering and four-wave mixing in As2S3 chalcogenide glass waveguides,” J. Opt. Soc. Am. B 28(11), 2777–2784 (2011).
[CrossRef]

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. Dorenbos, T. Zijlstra, V. Zwiller, 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(5), 051101 (2011).
[CrossRef]

X. Gai, T. Han, A. Prasad, S. Madden, D.-Y. Choi, R. Wang, D. Bulla, and B. Luther-Davies, “Progress in optical waveguides fabricated from chalcogenide glasses,” Opt. Express 18(25), 26635–26646 (2010).
[CrossRef] [PubMed]

X. Gai, S. Madden, D.-Y. Choi, D. Bulla, and B. Luther-Davies, “Dispersion engineered Ge11.5As24Se64.5 nanowires with a nonlinear parameter of 136 W⁻¹m⁻¹ at 1550 nm,” Opt. Express 18(18), 18866–18874 (2010).
[CrossRef] [PubMed]

F. Luan, M. D. Pelusi, M. R. E. Lamont, D.-Y. Choi, S. Madden, B. Luther-Davies, and B. J. Eggleton, “Dispersion engineered As2S3 planar waveguides for broadband four-wave mixing based wavelength conversion of 40 Gb/s signals,” Opt. Express 17(5), 3514–3520 (2009).
[CrossRef] [PubMed]

M. R. Lamont, B. Luther-Davies, D.-Y. Choi, S. Madden, X. Gai, and B. J. Eggleton, “Net-gain from a parametric amplifier on a chalcogenide optical chip,” Opt. Express 16(25), 20374–20381 (2008).
[CrossRef] [PubMed]

A. Prasad, C.-J. Zha, R.-P. Wang, A. Smith, S. Madden, and B. Luther-Davies, “Properties of GexAsySe1-x-y glasses for all-optical signal processing,” Opt. Express 16(4), 2804–2815 (2008).
[CrossRef] [PubMed]

Madden, S.

Madden, S. J.

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. Dorenbos, T. Zijlstra, V. Zwiller, 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(5), 051101 (2011).
[CrossRef]

Marshall, G. D.

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. Dorenbos, T. Zijlstra, V. Zwiller, 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(5), 051101 (2011).
[CrossRef]

C. Xiong, L. G. Helt, A. C. Judge, G. D. Marshall, M. J. Steel, J. E. Sipe, and B. J. Eggleton, “Quantum-correlated photon pair generation in chalcogenide As2S3 waveguides,” Opt. Express 18(15), 16206–16216 (2010).
[CrossRef] [PubMed]

Matthews, J. C. F.

J. C. F. Matthews, A. Politi, A. Stefanov, and J. L. O’Brien, “Manipulating multi-photon entanglement in waveguide quantum circuits,” Nat. Photonics 3(6), 346–350 (2009).
[CrossRef]

Murdoch, S. G.

Murphy, T. E.

Natarajan, C. M.

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. Dorenbos, T. Zijlstra, V. Zwiller, 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(5), 051101 (2011).
[CrossRef]

O’Brien, J. L.

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. Dorenbos, T. Zijlstra, V. Zwiller, 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(5), 051101 (2011).
[CrossRef]

J. C. F. Matthews, A. Politi, A. Stefanov, and J. L. O’Brien, “Manipulating multi-photon entanglement in waveguide quantum circuits,” Nat. Photonics 3(6), 346–350 (2009).
[CrossRef]

Pelusi, M. D.

Peruzzo, A.

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. Dorenbos, T. Zijlstra, V. Zwiller, 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(5), 051101 (2011).
[CrossRef]

Politi, A.

J. C. F. Matthews, A. Politi, A. Stefanov, and J. L. O’Brien, “Manipulating multi-photon entanglement in waveguide quantum circuits,” Nat. Photonics 3(6), 346–350 (2009).
[CrossRef]

Prasad, A.

Rarity, J. G.

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. Dorenbos, T. Zijlstra, V. Zwiller, 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(5), 051101 (2011).
[CrossRef]

Reznik, B.

N. Yoran and B. Reznik, “Deterministic linear optics quantum computation with single photon qubits,” Phys. Rev. Lett. 91(3), 037903 (2003).
[CrossRef] [PubMed]

Ribordy, G.

N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quamtum cryptography,” Rev. Mod. Phys. 74(1), 145–195 (2002).
[CrossRef]

Rotenberg, N.

A. D. Bristow, N. Rotenberg, and H. M. van Driel, “Two-photon absorption and Kerr coefficients of silicon for 850–2200 nm,” Appl. Phys. Lett. 90(19), 191104 (2007).
[CrossRef]

Schmidt, B. S.

Schüle, J.

P. Lüsse, P. Stuwe, J. Schüle, and H. G. Unger, “Analysis of vectorial mode fields in optical waveguides by a new finite difference method,” J. Lightwave Technol. 12(3), 487–494 (1994).
[CrossRef]

Sharping, J.

Sharping, J. E.

Sipe, J. E.

Smith, A.

Steel, M. J.

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. Dorenbos, T. Zijlstra, V. Zwiller, 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(5), 051101 (2011).
[CrossRef]

C. Xiong, L. G. Helt, A. C. Judge, G. D. Marshall, M. J. Steel, J. E. Sipe, and B. J. Eggleton, “Quantum-correlated photon pair generation in chalcogenide As2S3 waveguides,” Opt. Express 18(15), 16206–16216 (2010).
[CrossRef] [PubMed]

Stefanov, A.

J. C. F. Matthews, A. Politi, A. Stefanov, and J. L. O’Brien, “Manipulating multi-photon entanglement in waveguide quantum circuits,” Nat. Photonics 3(6), 346–350 (2009).
[CrossRef]

Stuwe, P.

P. Lüsse, P. Stuwe, J. Schüle, and H. G. Unger, “Analysis of vectorial mode fields in optical waveguides by a new finite difference method,” J. Lightwave Technol. 12(3), 487–494 (1994).
[CrossRef]

Tanner, M. G.

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. Dorenbos, T. Zijlstra, V. Zwiller, 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(5), 051101 (2011).
[CrossRef]

Thompson, M. G.

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. Dorenbos, T. Zijlstra, V. Zwiller, 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(5), 051101 (2011).
[CrossRef]

Tittel, W.

N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quamtum cryptography,” Rev. Mod. Phys. 74(1), 145–195 (2002).
[CrossRef]

Turner, A. C.

Unger, H. G.

P. Lüsse, P. Stuwe, J. Schüle, and H. G. Unger, “Analysis of vectorial mode fields in optical waveguides by a new finite difference method,” J. Lightwave Technol. 12(3), 487–494 (1994).
[CrossRef]

van Driel, H. M.

A. D. Bristow, N. Rotenberg, and H. M. van Driel, “Two-photon absorption and Kerr coefficients of silicon for 850–2200 nm,” Appl. Phys. Lett. 90(19), 191104 (2007).
[CrossRef]

Vanholsbeeck, F.

Voss, P.

Wang, R.

Wang, R.-P.

Wong, G. K. L.

Xiong, C.

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. Dorenbos, T. Zijlstra, V. Zwiller, 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(5), 051101 (2011).
[CrossRef]

C. Xiong, L. G. Helt, A. C. Judge, G. D. Marshall, M. J. Steel, J. E. Sipe, and B. J. Eggleton, “Quantum-correlated photon pair generation in chalcogenide As2S3 waveguides,” Opt. Express 18(15), 16206–16216 (2010).
[CrossRef] [PubMed]

Yoran, N.

N. Yoran and B. Reznik, “Deterministic linear optics quantum computation with single photon qubits,” Phys. Rev. Lett. 91(3), 037903 (2003).
[CrossRef] [PubMed]

Zbinden, H.

N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quamtum cryptography,” Rev. Mod. Phys. 74(1), 145–195 (2002).
[CrossRef]

Zha, C.-J.

Zhang, J.

Zijlstra, T.

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Figures (6)

Fig. 1
Fig. 1

Fourier transform of the Raman response function hR of Ge11.5, As2S3 and SiO2. (a) The imaginary part Im[hR(Ω)]. (b) The real part Re[hR(Ω)].

Fig. 2
Fig. 2

Photon pair generation rate with γPL = 0.1 and GVD of 15, 5 and 1.5 ps∙km−1nm−1Blue curve is for SFWM including Re[hR(Ω)]. Red curve is for pure SFWM. The frequency detuning is for pump-idler detuning ωip = ωps. G/s is for 1 × 109 pairs per second.

Fig. 3
Fig. 3

Criteria for correlated photon pair generation for γPL = 0.07, 0.1 and 0.15. (a) The photon pair generation rate of SFWM and SpRS. (b) Figure of merit F = SSFWM/SSpRS. (c) Photon pair correlation. G/s is for 1 × 109 pairs per second.

Fig. 4
Fig. 4

(a) A standard single clad Ge11.5 waveguide structure. (b) The GVD for a 0.625 µm wide waveguide. Dark blue curve is the contour for GVD of 7 ps∙km−1nm−1; light blue curve is the zero-dispersion contour; the black line shows the pump wavelength at 1.42 µm. (c) the GVD for a 0.58 µm film thickness and 0.61, 0.625 and 0.64 µm waveguide width.

Fig. 5
Fig. 5

(a) GVD at 1.42 µm as a function of film thickness and waveguide width. (b) GVD as a function of film thickness in region I. (c) GVD as a function of waveguide width in region II.

Fig. 6
Fig. 6

(a) Waveguide with an inserted layer of SiO2. (b) GVD as a function of SiO2 layer thickness and waveguide width with 0.64 µm initial film thickness. (c) GVD as a function of waveguide width with 0.64µm initial film thickness. (d) Waveguide with an inserted layer of Al2O3. (e) GVD as a function of Al2O3 layer thickness and waveguide width with 0.61 µm initial film thickness. (f) GVD as a function of waveguide width with 0.61 µm initial film thickness.

Equations (8)

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A p z + i 2 β 2 2 A p t 2 + α 2 A p =iγ[ | A p | 2 +(2+ f R (Re[ h ˜ R (Ω) ]1)) | A s | 2 +(2+ f R (Re[ h ˜ R (Ω) ]1)) | A i | 2 ] A p γIm[ h ˜ R (Ω) ] f R | A s | 2 A p γIm[ h ˜ R (Ω) ] f R | A i | 2 A p ,
A s z + i 2 β 2 2 A s t 2 + α 2 A s =iγ[ | A s | 2 +(2+ f R (Re[ h ˜ R (Ω) ]1)) | A p | 2 ] A s +iγ[ 1+ f R (Re[ h ˜ R (Ω) ]1) ] A p 2 A i * γIm[ h ˜ R (Ω) ] f R | A p | 2 A s γIm[ h ˜ R (Ω)] f R A p 2 A i * ,
A i z + i 2 β 2 2 A i t 2 + α 2 A i =iγ[ | A i | 2 +(2+ f R (Re[ h ˜ R (Ω) ]1)) | A p | 2 ] A i +iγ[ 1+ f R (Re[ h ˜ R (Ω) ]1) ] A p 2 A s * γIm[ h ˜ R (Ω) ] f R | A p | 2 A i γIm[ h ˜ R (Ω)] f R A p 2 A s * ,
4γ P 0 4γ P 0 ƒ(Re[ h R (ω)]1)<2 β 2 Δ ω 2 < 0 ,
G i ( υ )= P i (z) P s (z) = | q Kq(iR)/tanh(rRPL) | 2 ,
f SpRS =PL| g R (υ) |( n th +1),
F= S FWM S SpRS .
ρ Raman = [γRe(η)] 2 + | g R ( n th +1/2) | 2 [ | γη | 2 PL+| g R ( n th +1) |][ | γη | 2 PL+| g R | n th ] ,

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