Abstract

Wavelength conversion based on four-wave mixing (FWM) in a silicon–organic hybrid slot waveguide is theoretically investigated in the telecommunication bands. Compared with vertical slot waveguides, the horizontal slot waveguide structure exhibits much flatter dispersion. The maximum nonlinearity coefficient γ of 1.5×107W1km1 and the minimum effective mode area Aeff of 0.065μm2 are obtained in a horizontal slot waveguide with a 20-nm-thick optically nonlinear layer by controlling the geometric parameters. The wavelength conversion efficiency of 7.45 dB with a pump power of 100 mW in a 4-mm-long horizontal slot waveguide is obtained. This low power on-chip wavelength convertor will have potential applications in highly integrated optical circuits.

© 2014 Optical Society of America

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2012 (3)

2011 (2)

Z. Wang, H. Liu, N. Huang, Q. Sun, and J. Wen, “Influence of spectral broadening on femtosecond wavelength conversion based on four-wave mixing in silicon waveguides,” Appl. Opt. 50, 5430–5436 (2011).
[CrossRef]

Z. Wang, H. Liu, N. Huang, Q. Sun, and J. Wen, “Impact of dispersion profiles of silicon waveguides on optical parametric amplification in the femtosecond regime,” Opt. Express 19, 24732–24737 (2011).

2010 (2)

2009 (3)

2008 (2)

2007 (2)

2006 (3)

2004 (3)

S. Polyakov, F. Yoshino, M. Liu, and G. Stegeman, “Nonlinear refraction and multiphoton absorption in polydiacetylenes from 1200 to 2200  nm,” Phys. Rev. B 69, 115421 (2004).

V. R. Almeida, Q. Xu, C. A. Barrios, and M. Lipson, “Guiding and confining light in void nanostructure,” Opt. Lett. 29, 1290–1291 (2004).

V. Raghunathan, R. Claps, D. Dimitropoulos, and B. Jalali, “Wavelength conversion in silicon using Raman-induced four-wave mixing,” Appl. Phys. Lett. 85, 34–36 (2004).
[CrossRef]

2002 (1)

H. K. Tsang, C. S. Wong, T. K. Liang, I. E. Day, S. W. Roberts, A. Harpin, J. Drake, and M. Asghari, “Optical dispersion, two-photon absorption and self-phase modulation in silicon waveguides at 1.5  μm wavelength,” Appl. Phys. Lett. 80, 416–418 (2002).
[CrossRef]

2001 (1)

1989 (1)

G. J. Blanchard, J. P. Heritage, A. C. V. Lehmen, M. K. Kelly, G. L. Baker, and S. Etemad, “Excitonic and phonon-mediated optical Stark effects in a conjugated polymer,” Phys. Rev. Lett. 63, 887–890 (1989).
[CrossRef]

1982 (1)

D. N. Batchelder and D. Bloor, “An investigation of the electronic excited state of a polydiacetylene by resonance Raman spectroscopy,” J. Phys. C 15, 3005–3018 (1982).

1977 (1)

Z. Iqbal, R. R. Chance, and R. H. Baughman, “Electronic structure change at a phase transition in a polydiacetylene crystal,” J. Chem. Phys. 66, 5520–5525 (1977).
[CrossRef]

Agrawal, G. P.

Alloatti, L.

Almeida, V. R.

V. R. Almeida, Q. Xu, C. A. Barrios, and M. Lipson, “Guiding and confining light in void nanostructure,” Opt. Lett. 29, 1290–1291 (2004).

Asghari, M.

H. K. Tsang, C. S. Wong, T. K. Liang, I. E. Day, S. W. Roberts, A. Harpin, J. Drake, and M. Asghari, “Optical dispersion, two-photon absorption and self-phase modulation in silicon waveguides at 1.5  μm wavelength,” Appl. Phys. Lett. 80, 416–418 (2002).
[CrossRef]

Baets, R.

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon–organic hybrid slot waveguides,” Nat. Photonics 3, 216–219 (2009).
[CrossRef]

T. Vallaitis, S. Bogatscher, L. Alloatti, P. Dumon, R. Baets, M. L. Scimeca, I. Biaggio, F. Diederich, C. Koos, W. Freude, and J. Leuthold, “Optical properties of highly nonlinear silicon organic hybrid (SOH) waveguide geometries,” Opt. Express 17, 17357–17368 (2009).
[CrossRef]

Baker, G. L.

G. J. Blanchard, J. P. Heritage, A. C. V. Lehmen, M. K. Kelly, G. L. Baker, and S. Etemad, “Excitonic and phonon-mediated optical Stark effects in a conjugated polymer,” Phys. Rev. Lett. 63, 887–890 (1989).
[CrossRef]

Ballesteros, G. C.

Barrios, C. A.

V. R. Almeida, Q. Xu, C. A. Barrios, and M. Lipson, “Guiding and confining light in void nanostructure,” Opt. Lett. 29, 1290–1291 (2004).

Batchelder, D. N.

D. N. Batchelder and D. Bloor, “An investigation of the electronic excited state of a polydiacetylene by resonance Raman spectroscopy,” J. Phys. C 15, 3005–3018 (1982).

Baughman, R. H.

Z. Iqbal, R. R. Chance, and R. H. Baughman, “Electronic structure change at a phase transition in a polydiacetylene crystal,” J. Chem. Phys. 66, 5520–5525 (1977).
[CrossRef]

Beausoleil, R. G.

Bhowmik, A. K.

Biaggio, I.

T. Vallaitis, S. Bogatscher, L. Alloatti, P. Dumon, R. Baets, M. L. Scimeca, I. Biaggio, F. Diederich, C. Koos, W. Freude, and J. Leuthold, “Optical properties of highly nonlinear silicon organic hybrid (SOH) waveguide geometries,” Opt. Express 17, 17357–17368 (2009).
[CrossRef]

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon–organic hybrid slot waveguides,” Nat. Photonics 3, 216–219 (2009).
[CrossRef]

Blanchard, G. J.

G. J. Blanchard, J. P. Heritage, A. C. V. Lehmen, M. K. Kelly, G. L. Baker, and S. Etemad, “Excitonic and phonon-mediated optical Stark effects in a conjugated polymer,” Phys. Rev. Lett. 63, 887–890 (1989).
[CrossRef]

Blasco, J.

Bloor, D.

D. N. Batchelder and D. Bloor, “An investigation of the electronic excited state of a polydiacetylene by resonance Raman spectroscopy,” J. Phys. C 15, 3005–3018 (1982).

Bogaerts, W.

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon–organic hybrid slot waveguides,” Nat. Photonics 3, 216–219 (2009).
[CrossRef]

Bogatscher, S.

Boyraz, O.

Chance, R. R.

Z. Iqbal, R. R. Chance, and R. H. Baughman, “Electronic structure change at a phase transition in a polydiacetylene crystal,” J. Chem. Phys. 66, 5520–5525 (1977).
[CrossRef]

Claps, R.

V. Raghunathan, R. Claps, D. Dimitropoulos, and B. Jalali, “Wavelength conversion in silicon using Raman-induced four-wave mixing,” Appl. Phys. Lett. 85, 34–36 (2004).
[CrossRef]

Cohen, O.

Cristiani, I.

Day, I. E.

H. K. Tsang, C. S. Wong, T. K. Liang, I. E. Day, S. W. Roberts, A. Harpin, J. Drake, and M. Asghari, “Optical dispersion, two-photon absorption and self-phase modulation in silicon waveguides at 1.5  μm wavelength,” Appl. Phys. Lett. 80, 416–418 (2002).
[CrossRef]

Dekker, R.

Diederich, F.

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon–organic hybrid slot waveguides,” Nat. Photonics 3, 216–219 (2009).
[CrossRef]

T. Vallaitis, S. Bogatscher, L. Alloatti, P. Dumon, R. Baets, M. L. Scimeca, I. Biaggio, F. Diederich, C. Koos, W. Freude, and J. Leuthold, “Optical properties of highly nonlinear silicon organic hybrid (SOH) waveguide geometries,” Opt. Express 17, 17357–17368 (2009).
[CrossRef]

Dimitropoulos, D.

V. Raghunathan, R. Claps, D. Dimitropoulos, and B. Jalali, “Wavelength conversion in silicon using Raman-induced four-wave mixing,” Appl. Phys. Lett. 85, 34–36 (2004).
[CrossRef]

Drake, J.

H. K. Tsang, C. S. Wong, T. K. Liang, I. E. Day, S. W. Roberts, A. Harpin, J. Drake, and M. Asghari, “Optical dispersion, two-photon absorption and self-phase modulation in silicon waveguides at 1.5  μm wavelength,” Appl. Phys. Lett. 80, 416–418 (2002).
[CrossRef]

Driessen, A.

Dumon, P.

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon–organic hybrid slot waveguides,” Nat. Photonics 3, 216–219 (2009).
[CrossRef]

T. Vallaitis, S. Bogatscher, L. Alloatti, P. Dumon, R. Baets, M. L. Scimeca, I. Biaggio, F. Diederich, C. Koos, W. Freude, and J. Leuthold, “Optical properties of highly nonlinear silicon organic hybrid (SOH) waveguide geometries,” Opt. Express 17, 17357–17368 (2009).
[CrossRef]

Esembeson, B.

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon–organic hybrid slot waveguides,” Nat. Photonics 3, 216–219 (2009).
[CrossRef]

Etemad, S.

G. J. Blanchard, J. P. Heritage, A. C. V. Lehmen, M. K. Kelly, G. L. Baker, and S. Etemad, “Excitonic and phonon-mediated optical Stark effects in a conjugated polymer,” Phys. Rev. Lett. 63, 887–890 (1989).
[CrossRef]

Fauchet, P. M.

Fédéli, J. M.

Först, M.

Freude, W.

J. Leuthold, C. Koos, and W. Freude, “Nonlinear silicon photonics,” Nat. Photonics 4, 535–544 (2010).
[CrossRef]

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon–organic hybrid slot waveguides,” Nat. Photonics 3, 216–219 (2009).
[CrossRef]

T. Vallaitis, S. Bogatscher, L. Alloatti, P. Dumon, R. Baets, M. L. Scimeca, I. Biaggio, F. Diederich, C. Koos, W. Freude, and J. Leuthold, “Optical properties of highly nonlinear silicon organic hybrid (SOH) waveguide geometries,” Opt. Express 17, 17357–17368 (2009).
[CrossRef]

C. Koos, L. Jacome, C. Poulton, J. Leuthold, and W. Freude, “Nonlinear silicon-on-insulator waveguides for all-optical signal processing,” Opt. Express 15, 5976–5990 (2007).
[CrossRef]

C. Koos, L. Jacome, C. Poulton, J. Leuthold, and W. Freude, “Nonlinear high index-contrast waveguides with optimum geometry,” in Nonlinear Photonics, OSA Technical Digest (CD) (Optical Society of America, 2007), paper JWA2.

Hainberger, R.

Harpin, A.

H. K. Tsang, C. S. Wong, T. K. Liang, I. E. Day, S. W. Roberts, A. Harpin, J. Drake, and M. Asghari, “Optical dispersion, two-photon absorption and self-phase modulation in silicon waveguides at 1.5  μm wavelength,” Appl. Phys. Lett. 80, 416–418 (2002).
[CrossRef]

Heritage, J. P.

G. J. Blanchard, J. P. Heritage, A. C. V. Lehmen, M. K. Kelly, G. L. Baker, and S. Etemad, “Excitonic and phonon-mediated optical Stark effects in a conjugated polymer,” Phys. Rev. Lett. 63, 887–890 (1989).
[CrossRef]

Huang, N.

Iqbal, Z.

Z. Iqbal, R. R. Chance, and R. H. Baughman, “Electronic structure change at a phase transition in a polydiacetylene crystal,” J. Chem. Phys. 66, 5520–5525 (1977).
[CrossRef]

Jacome, L.

C. Koos, L. Jacome, C. Poulton, J. Leuthold, and W. Freude, “Nonlinear silicon-on-insulator waveguides for all-optical signal processing,” Opt. Express 15, 5976–5990 (2007).
[CrossRef]

C. Koos, L. Jacome, C. Poulton, J. Leuthold, and W. Freude, “Nonlinear high index-contrast waveguides with optimum geometry,” in Nonlinear Photonics, OSA Technical Digest (CD) (Optical Society of America, 2007), paper JWA2.

Jalali, B.

V. Raghunathan, R. Claps, D. Dimitropoulos, and B. Jalali, “Wavelength conversion in silicon using Raman-induced four-wave mixing,” Appl. Phys. Lett. 85, 34–36 (2004).
[CrossRef]

Jones, R.

Kelly, M. K.

G. J. Blanchard, J. P. Heritage, A. C. V. Lehmen, M. K. Kelly, G. L. Baker, and S. Etemad, “Excitonic and phonon-mediated optical Stark effects in a conjugated polymer,” Phys. Rev. Lett. 63, 887–890 (1989).
[CrossRef]

Koos, C.

J. Leuthold, C. Koos, and W. Freude, “Nonlinear silicon photonics,” Nat. Photonics 4, 535–544 (2010).
[CrossRef]

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon–organic hybrid slot waveguides,” Nat. Photonics 3, 216–219 (2009).
[CrossRef]

T. Vallaitis, S. Bogatscher, L. Alloatti, P. Dumon, R. Baets, M. L. Scimeca, I. Biaggio, F. Diederich, C. Koos, W. Freude, and J. Leuthold, “Optical properties of highly nonlinear silicon organic hybrid (SOH) waveguide geometries,” Opt. Express 17, 17357–17368 (2009).
[CrossRef]

C. Koos, L. Jacome, C. Poulton, J. Leuthold, and W. Freude, “Nonlinear silicon-on-insulator waveguides for all-optical signal processing,” Opt. Express 15, 5976–5990 (2007).
[CrossRef]

C. Koos, L. Jacome, C. Poulton, J. Leuthold, and W. Freude, “Nonlinear high index-contrast waveguides with optimum geometry,” in Nonlinear Photonics, OSA Technical Digest (CD) (Optical Society of America, 2007), paper JWA2.

Kuo, Y. H.

Lacava, C.

Lehmen, A. C. V.

G. J. Blanchard, J. P. Heritage, A. C. V. Lehmen, M. K. Kelly, G. L. Baker, and S. Etemad, “Excitonic and phonon-mediated optical Stark effects in a conjugated polymer,” Phys. Rev. Lett. 63, 887–890 (1989).
[CrossRef]

Leuthold, J.

J. Leuthold, C. Koos, and W. Freude, “Nonlinear silicon photonics,” Nat. Photonics 4, 535–544 (2010).
[CrossRef]

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon–organic hybrid slot waveguides,” Nat. Photonics 3, 216–219 (2009).
[CrossRef]

T. Vallaitis, S. Bogatscher, L. Alloatti, P. Dumon, R. Baets, M. L. Scimeca, I. Biaggio, F. Diederich, C. Koos, W. Freude, and J. Leuthold, “Optical properties of highly nonlinear silicon organic hybrid (SOH) waveguide geometries,” Opt. Express 17, 17357–17368 (2009).
[CrossRef]

C. Koos, L. Jacome, C. Poulton, J. Leuthold, and W. Freude, “Nonlinear silicon-on-insulator waveguides for all-optical signal processing,” Opt. Express 15, 5976–5990 (2007).
[CrossRef]

C. Koos, L. Jacome, C. Poulton, J. Leuthold, and W. Freude, “Nonlinear high index-contrast waveguides with optimum geometry,” in Nonlinear Photonics, OSA Technical Digest (CD) (Optical Society of America, 2007), paper JWA2.

Li, X.

Li, Y. X.

Liang, T. K.

H. K. Tsang, C. S. Wong, T. K. Liang, I. E. Day, S. W. Roberts, A. Harpin, J. Drake, and M. Asghari, “Optical dispersion, two-photon absorption and self-phase modulation in silicon waveguides at 1.5  μm wavelength,” Appl. Phys. Lett. 80, 416–418 (2002).
[CrossRef]

Lin, Q.

Lipson, M.

V. R. Almeida, Q. Xu, C. A. Barrios, and M. Lipson, “Guiding and confining light in void nanostructure,” Opt. Lett. 29, 1290–1291 (2004).

Liu, A.

Liu, H.

Liu, M.

S. Polyakov, F. Yoshino, M. Liu, and G. Stegeman, “Nonlinear refraction and multiphoton absorption in polydiacetylenes from 1200 to 2200  nm,” Phys. Rev. B 69, 115421 (2004).

Liu, X.

X. Liu, Nonlinear Applications Using Silicon Nanophotonic Wires (Columbia University, 2011).

Marti, J.

Martí, J.

Martinez, A.

Mathlouthi, W.

Matres, J.

Michinobu, T.

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon–organic hybrid slot waveguides,” Nat. Photonics 3, 216–219 (2009).
[CrossRef]

Minzioni, P.

Moormann, C.

Muellner, P.

Niehusmann, J.

Oton, C. J.

Palik, E. D.

E. D. Palik, Handbook of Optical Constants of Solids (Academic, 1998).

Paniccia, M.

Polyakov, S.

S. Polyakov, F. Yoshino, M. Liu, and G. Stegeman, “Nonlinear refraction and multiphoton absorption in polydiacetylenes from 1200 to 2200  nm,” Phys. Rev. B 69, 115421 (2004).

Poulton, C.

C. Koos, L. Jacome, C. Poulton, J. Leuthold, and W. Freude, “Nonlinear silicon-on-insulator waveguides for all-optical signal processing,” Opt. Express 15, 5976–5990 (2007).
[CrossRef]

C. Koos, L. Jacome, C. Poulton, J. Leuthold, and W. Freude, “Nonlinear high index-contrast waveguides with optimum geometry,” in Nonlinear Photonics, OSA Technical Digest (CD) (Optical Society of America, 2007), paper JWA2.

Premaratne, M.

Raday, O.

Raghunathan, V.

V. Raghunathan, R. Claps, D. Dimitropoulos, and B. Jalali, “Wavelength conversion in silicon using Raman-induced four-wave mixing,” Appl. Phys. Lett. 85, 34–36 (2004).
[CrossRef]

Roberts, S. W.

H. K. Tsang, C. S. Wong, T. K. Liang, I. E. Day, S. W. Roberts, A. Harpin, J. Drake, and M. Asghari, “Optical dispersion, two-photon absorption and self-phase modulation in silicon waveguides at 1.5  μm wavelength,” Appl. Phys. Lett. 80, 416–418 (2002).
[CrossRef]

Rong, H.

Rukhlenko, I. D.

Sanchis, P.

Sang, X.

Scimeca, M. L.

Stegeman, G.

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[CrossRef]

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C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon–organic hybrid slot waveguides,” Nat. Photonics 3, 216–219 (2009).
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Appl. Opt. (1)

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[CrossRef]

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P. Muellner, M. Wellenzohn, and R. Hainberger, “Nonlinearity of optimized silicon photonic slot waveguides,” Opt. Express 17, 9282–9287 (2009).
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C. Koos, L. Jacome, C. Poulton, J. Leuthold, and W. Freude, “Nonlinear silicon-on-insulator waveguides for all-optical signal processing,” Opt. Express 15, 5976–5990 (2007).
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Z. Wang, H. Liu, N. Huang, Q. Sun, and J. Wen, “Impact of dispersion profiles of silicon waveguides on optical parametric amplification in the femtosecond regime,” Opt. Express 19, 24732–24737 (2011).

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Phys. Rev. B (1)

S. Polyakov, F. Yoshino, M. Liu, and G. Stegeman, “Nonlinear refraction and multiphoton absorption in polydiacetylenes from 1200 to 2200  nm,” Phys. Rev. B 69, 115421 (2004).

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Figures (9)

Fig. 1.
Fig. 1.

(a) Schemes of horizontal and (b) vertical SOH slot waveguides.

Fig. 2.
Fig. 2.

Dispersion distribution of (a) horizontal SOH waveguide for different values of silicon wires length w with constant h=200nm and s=20nm and (b) vertical SOH waveguide for different values of silicon wires length w with constant h=300nm and s=50nm.

Fig. 3.
Fig. 3.

(a) TM-mode of electric mode and (b) Ex and (c) Ey electric field components of the horizontal SOH waveguide at 1550 nm. (d) The TE-mode of electric mode and (e) Ex and (f) Ey electric field components of the vertical SOH waveguide at 1550 nm.

Fig. 4.
Fig. 4.

Effective mode area and nonlinear coefficient of the (a) horizontal and (b) vertical SOH waveguide.

Fig. 5.
Fig. 5.

(a) Linear phase mismatch with different pump wavelengths. (b) Phase mismatch with different pump power.

Fig. 6.
Fig. 6.

(a) Conversion efficiency and (b) pulse energy as a function of waveguide length.

Fig. 7.
Fig. 7.

Conversion efficiency as a function of signal wavelengths for four different pulse widths in a 4-mm waveguide.

Fig. 8.
Fig. 8.

Output temporal profiles from the input pulse widths of (a) 0.1 ps, (b) 0.5 ps, (c) 1 ps, and (d) 10 ps.

Fig. 9.
Fig. 9.

(a) Idler conversion efficiency, (b) signal gain, and (c) output pump power as a function of pump power.

Equations (8)

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nSi2=11.6858+0.939816/λ2+8.10461×1.10712/(λ21.10712),
nPTS2=2.9971+0.6193λ2/(λ20.4044)0.2732λ2.
Aeff=(|F(x,y)|2dxdy)2|F(x,y)|4dxdy,
γe=2πλn2(x,y)|F(x,y)|4dxdy(|F(x,y)|2dxdy)2,
ΔK=Δβ+2γPppump,
Apz+iβ2p22ApT2β3p63ApT3=12αpAp+iγp(1+iωpT)|Ap|2Ap+2iγp(|As|2+|Ai)|2)Ap+2iγpAsAiAp*exp(iΔβz),
Asz+dsAsT+iβ2s22AsT2β3s63AsT3=12αsAs+iγs(1+iωsT)|As|2As+2iγs(|Ap|2+|Ai)|2)As+iγsAp2Ai*exp(iΔβz),
Aiz+diAiT+iβ2i22AiT2β3i63AiT3=12αiAi+iγi(1+iωiT)|Ai|2Ai+2iγi(|Ap|2+|As)|2)Ai+iγiAp2As*exp(iΔβz),

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