Theory of nonlinear pulse propagation in silicon-nanocrystal waveguides

Ivan D. Rukhlenko

doi:10.1364/OE.21.002832

Theory of nonlinear pulse propagation in silicon-nanocrystal waveguides

Ivan D. Rukhlenko

Optics Express
Vol. 21,
Issue 3,
pp. 2832-2846
(2013)
•https://doi.org/10.1364/OE.21.002832

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Ivan D. Rukhlenko, "Theory of nonlinear pulse propagation in silicon-nanocrystal waveguides," Opt. Express 21, 2832-2846 (2013)

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Related Topics
Table of Contents Category
- Nonlinear Optics
Optics & Photonics Topics
?

The topics in this list come from the OSA Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Nonlinear optical materials (160.4330)
- Nonlinear optics (190.0190)
- Nonlinear optics, materials (190.4400)
- Effective medium theory (260.2065)

About this Article
History
- Original Manuscript: December 14, 2012
- Revised Manuscript: January 15, 2013
- Manuscript Accepted: January 16, 2013
- Published: January 29, 2013

Accessible

Open Access

Abstract

We develop a comprehensive theory of the nonlinear propagation of optical pulses through silica waveguides doped with highly nonlinear silicon nanocrystals. Our theory describes the dynamics of arbitrarily polarized pump and Stokes fields by a system of four generalized nonlinear Schrödinger equations for the slowly varying field amplitudes, coupled to the rate equation for the number density of free carriers. In deriving these equations, we use an analytic expression for the third-order effective susceptibility of the waveguide with randomly oriented nanocrystals, which takes into account both the weakening of the nonlinear optical response of silicon nanocrystals due to their embedment in fused silica and the change in the tensor properties of the response due to the modification of light interaction with electrons and phonons inside the silicon-nanocrystal waveguide. In order to facilitate the use of our theory by experimentalists, and for reasons of methodology, we provide a great deal of detail on the mathematical treatment throughout the paper, even though the derivation of the coupled-amplitude equations is quite straightforward. The developed theory can be applied for the solving of a wide variety of specific problems that require modeling of nonlinear optical phenomena in silicon-nanocrystal waveguides.

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I. D. Rukhlenko, I. L. Garanovich, M. Premaratne, A. A. Sukhorukov, G. P. Agrawal, and Y. S. Kivshar, “Polarization rotation in silicon waveguides: Analytical modeling and applications,” IEEE Photonics J. 2, 423–435 (2010).
[Crossref]

2012 (8)

I. D. Rukhlenko and M. Premaratne, “Optimization of nonlinear performance of silicon-nanocrystal cylindrical nanowires,” IEEE Photonics J. 4, 952–959 (2012).
[Crossref]

T. Nikitin, R. Velagapudi, J. Sainio, J. Lahtinen, M. Räsänen, S. Novikov, and L. Khriachtchev, “Optical and structural properties of SiOx films grown by molecular beam deposition: Effect of the Si concentration and annealing temperature,” J. Appl. Phys. 112, 094316–094316 (2012).
[Crossref]

L. Sirleto, M. A. Ferrara, T. Nikitin, S. Novikov, and L. Khriachtchev, “Giant Raman gain in silicon nanocrystals,” Nat. Commun. 3, 1220 (2012).
[Crossref] [PubMed]

S. N. Volkov, J. J. Saarinen, and J. E. Sipe, “Effective medium theory for 2D disordered structures: A comparison to numerical simulations,” J. Mod. Opt. 59, 954–961 (2012).
[Crossref]

M. A. Ferrara, I. Rendina, S. N. Basu, L. D. Negro, and L. Sirleto, “Raman amplifier based on amorphous silicon nanoparticles,” Int. J. Photoenergy 2012, 254946 (2012).
[Crossref]

K. Imakita, M. Ito, R. Naruiwa, M. Fujii, and S. Hayashi, “Enhancement of ultrafast nonlinear optical response of silicon nanocrystals by boron-doping,” Opt. Lett. 37, 1877–1879 (2012).
[Crossref] [PubMed]

I. D. Rukhlenko, M. Premaratne, and G. P. Agrawal, “Effective mode area and its optimization in silicon-nanocrystal waveguides,” Opt. Lett. 37, 2295–2297 (2012).
[Crossref] [PubMed]

I. D. Rukhlenko, W. Zhu, M. Premaratne, and G. P. Agrawal, “Effective third-order susceptibility of silicon-nanocrystal-doped silica,” Opt. Express 20, 26275–26284 (2012).
[Crossref] [PubMed]

2011 (4)

I. D. Rukhlenko, M. Premaratne, and G. P. Agrawal, “Nonlinear propagation in silicon-based plasmonic waveguides from the standpoint of applications,” Opt. Express 19, 206–217 (2011).
[Crossref] [PubMed]

J. Wei, J. Price, T. Wang, C. Hessel, and M. C. Downer, “Size-dependent optical properties of Si nanocrystals embedded in amorphous SiO2 measured by spectroscopic ellipsometry,” J. Vac. Sci. Technol. B 29, 04D112 (2011).
[Crossref]

T. Nikitin, K. Aitola, S. Novikov, M. Räsänen, R. Velagapudi, J. Sainio, J. Lahtinen, K. Mizohata, T. Ahlgren, and L. Khriachtchev, “Optical and structural properties of silicon-rich silicon oxide films: Comparison of ion implantation and molecular beam deposition methods,” Phys. Status Solidi (a) 208, 2176–2181 (2011).
[Crossref]

F. D. Leonardis and V. M. N. Passaro, “Dispersion engineered silicon nanocrystal slot waveguides for soliton ultrafast optical processing,” Adv. OptoElectron. 2011, 751498 (2011).

2010 (14)

M. Krause, H. Renner, and E. Brinkmeyer, “Silicon Raman amplifiers with ring-resonator-enhanced pump power,” IEEE J. Sel. Top. Quantum Electron. 16, 216–225 (2010).
[Crossref]

D. Liang and J. E. Bowers, “Recent progress in lasers on silicon,” Nat. Photonics 4, 511–517 (2010).
[Crossref]

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

A. Martinez, J. Blasco, P. Sanchis, J. V. Galan, J. Garcia-Ruperez, E. Jordana, P. Gautier, Y. Lebour, S. Hernandez, R. Spano, R. Guider, N. Daldosso, B. Garrido, J. M. Fedeli, L. Pavesi, and J. Marti, “Ultrafast all-optical switching in a silicon-nanocrystal-based silicon slot waveguide at telecom wavelengths,” Nano Lett. 10, 1506–1511 (2010).
[Crossref] [PubMed]

M. Paniccia, “Integrating silicon photonics,” Nat. Photonics 4, 498–499 (2010).
[Crossref]

R. J. Kashtiban, U. Bangert, I. F. Crowe, M. Halsall, A. J. Harvey, and M. Gass, “Study of erbium-doped silicon nanocrystals in silica,” J. Phys.: Conference Series 241, 012097 (2010).
[Crossref]

I. D. Rukhlenko, C. Dissanayake, M. Premaratne, and G. P. Agrawal, “Optimization of Raman amplification in silicon waveguides with finite facet reflectivities,” IEEE J. Sel. Top. Quantum Electron. 16, 226–233 (2010).
[Crossref]

I. D. Rukhlenko, M. Premaratne, and G. P. Agrawal, “Analytical study of optical bistability in silicon ring resonators,” Opt. Lett. 35, 55–57 (2010).
[Crossref] [PubMed]

B. A. Daniel and G. P. Agrawal, “Vectorial nonlinear propagation in silicon nanowire waveguides: Polarization effects,” J. Opt. Soc. Am. B 27, 956–965 (2010).
[Crossref]

I. D. Rukhlenko, I. Udagedara, M. Premaratne, and G. P. Agrawal, “Effect of free carriers on pump-to-signal noise transfer in silicon Raman amplifiers,” Opt. Lett. 35, 2343–2345 (2010).
[Crossref] [PubMed]

I. D. Rukhlenko, M. Premaratne, I. L. Garanovich, A. A. Sukhorukov, and G. P. Agrawal, “Analytical study of pulse amplification in silicon Raman amplifiers,” Opt. Express 18, 18324–18338 (2010).
[Crossref] [PubMed]

C. M. Dissanayake, M. Premaratne, I. D. Rukhlenko, and G. P. Agrawal, “FDTD modeling of anisotropic nonlinear optical phenomena in silicon waveguides,” Opt. Express 18, 21427–21448 (2010).
[Crossref] [PubMed]

I. D. Rukhlenko, M. Premaratne, and G. P. Agrawal, “Nonlinear silicon photonics: Analytical tools,” IEEE J. Sel. Top. Quantum Electron. 16, 200–215 (2010).
[Crossref]

I. D. Rukhlenko, I. L. Garanovich, M. Premaratne, A. A. Sukhorukov, G. P. Agrawal, and Y. S. Kivshar, “Polarization rotation in silicon waveguides: Analytical modeling and applications,” IEEE Photonics J. 2, 423–435 (2010).
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2009 (12)

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M. Krause, H. Renner, and E. Brinkmeyer, “Silicon Raman amplifiers with ring-resonator-enhanced pump power,” IEEE J. Sel. Top. Quantum Electron. 16, 216–225 (2010).
[Crossref]

M. Krause, H. Renner, S. Fathpour, B. Jalali, and E. Brinkmeyer, “Gain enhancement in cladding-pumped silicon Raman amplifiers,” IEEE J. Quantum Electron. 44, 692–704 (2008).
[Crossref]

M. Krause, H. Renner, and E. Brinkmeyer, “Analysis of Raman lasing characteristics in silicon-on-insulator waveguides,” Opt. Express 12, 5703–5710 (2004).
[Crossref] [PubMed]

Lahtinen, J.

Lebour, Y.

Lee, M.

H. Rong, S. Xu, O. Cohen, O. Raday, M. Lee, V. Sih, and M. Paniccia, “A cascaded silicon Raman laser,” Nat. Photonics 2, 170–174 (2008).
[Crossref]

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Leonardis, F. D.

F. D. Leonardis and V. M. N. Passaro, “Dispersion engineered silicon nanocrystal slot waveguides for soliton ultrafast optical processing,” Adv. OptoElectron. 2011, 751498 (2011).

Leuthold, J.

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

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H. Richter, Z. P. Wang, and L. Ley, “The one phonon Raman spectrum in microcrystalline silicon,” Solid State Commun. 39, 625–629 (1981).
[Crossref]

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D. Liang and J. E. Bowers, “Recent progress in lasers on silicon,” Nat. Photonics 4, 511–517 (2010).
[Crossref]

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Q. Lin, O. J. Painter, and G. P. Agrawal, “Nonlinear optical phenomena in silicon waveguides: Modeling and applications,” Opt. Express 15, 16604–16644 (2007).
[Crossref] [PubMed]

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M. A. Foster, A. C. Turner, M. Lipson, and A. L. Gaeta, “Nonlinear optics in photonic nanowires,” Opt. Express 16, 1300–1320 (2008).
[Crossref] [PubMed]

C. Manolatou and M. Lipson, “All-optical silicon modulators based on carrier injection by two-photon absorption,” J. Lightwave Technol. 24, 1433–1439 (2006).
[Crossref]

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D. J. Moss, L. Fu, I. Littler, and B. J. Eggleton, “Ultrafast all-optical modulation via two-photon absorption in silicon-on-insulator waveguides,” Electron. Lett. 41, 320–321 (2005).
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H. K. Tsang and Y. Liu, “Nonlinear optical properties of silicon waveguides,” Semicond. Sci. Technol. 23, 064007 (2008).
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L. Ding, T. P. Chen, Y. Liu, C. Y. Ng, and S. Fung, “Optical properties of silicon nanocrystals embedded in a SiO2 matrix,” Phys. Rev. B 72, 125419 (2005).
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V. A. Belyakov, V. A. Burdov, R. Lockwood, and A. Meldrum, “Silicon nanocrystals: Fundamental theory and implications for stimulated emission,” Adv. Opt. Technol. 2008, 279502 (2008).

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R. Soref and J. Lorenzo, “All-silicon active and passive guided-wave components for λ = 1.3 and 1.6 μm,” IEEE J. Quantum Electron. 22, 873–879 (1986).
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C. Manolatou and M. Lipson, “All-optical silicon modulators based on carrier injection by two-photon absorption,” J. Lightwave Technol. 24, 1433–1439 (2006).
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[Crossref] [PubMed]

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J. I. Dadap, R. L. Espinola, R. M. Osgood, S. J. McNab, and Y. A. Vlasov, “Spontaneous Raman scattering in ultrasmall silicon waveguides,” Opt. Lett. 29, 2755–2757 (2004).
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V. A. Belyakov, V. A. Burdov, R. Lockwood, and A. Meldrum, “Silicon nanocrystals: Fundamental theory and implications for stimulated emission,” Adv. Opt. Technol. 2008, 279502 (2008).

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Monro, T. M.

Moss, D. J.

D. J. Moss, L. Fu, I. Littler, and B. J. Eggleton, “Ultrafast all-optical modulation via two-photon absorption in silicon-on-insulator waveguides,” Electron. Lett. 41, 320–321 (2005).
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L. Cao, B. Nabet, and J. E. Spanier, “Enhanced Raman scattering from individual semiconductor nanocones and nanowires,” Phys. Rev. Lett. 96, 157402 (2006).
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L. Ding, T. P. Chen, Y. Liu, C. Y. Ng, and S. Fung, “Optical properties of silicon nanocrystals embedded in a SiO2 matrix,” Phys. Rev. B 72, 125419 (2005).
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L. Sirleto, M. A. Ferrara, T. Nikitin, S. Novikov, and L. Khriachtchev, “Giant Raman gain in silicon nanocrystals,” Nat. Commun. 3, 1220 (2012).
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F. Trani, D. Ninno, and G. Iadonisi, “Role of local fields in the optical properties of silicon nanocrystals using the tight binding approach,” Phys. Rev. B 75, 033312 (2007).
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Osgood, R.

R. Espinola, J. Dadap, R. Osgood, S. McNab, and Y. Vlasov, “C-band wavelength conversion in silicon photonic wire waveguides,” Opt. Express 13, 4341–4349 (2005).
[Crossref] [PubMed]

Osgood, R. M.

X. Chen, N. C. Panoiu, and R. M. Osgood, “Theory of Raman-mediated pulsed amplification in silicon-wire waveguides,” IEEE J. Quantum Electron. 42, 160–170 (2006).
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J. I. Dadap, R. L. Espinola, R. M. Osgood, S. J. McNab, and Y. A. Vlasov, “Spontaneous Raman scattering in ultrasmall silicon waveguides,” Opt. Lett. 29, 2755–2757 (2004).
[Crossref] [PubMed]

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Painter, O. J.

Q. Lin, O. J. Painter, and G. P. Agrawal, “Nonlinear optical phenomena in silicon waveguides: Modeling and applications,” Opt. Express 15, 16604–16644 (2007).
[Crossref] [PubMed]

Paniccia, M.

M. Paniccia, “Integrating silicon photonics,” Nat. Photonics 4, 498–499 (2010).
[Crossref]

H. Rong, S. Xu, O. Cohen, O. Raday, M. Lee, V. Sih, and M. Paniccia, “A cascaded silicon Raman laser,” Nat. Photonics 2, 170–174 (2008).
[Crossref]

Panoiu, N. C.

X. Chen, N. C. Panoiu, and R. M. Osgood, “Theory of Raman-mediated pulsed amplification in silicon-wire waveguides,” IEEE J. Quantum Electron. 42, 160–170 (2006).
[Crossref]

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F. D. Leonardis and V. M. N. Passaro, “Dispersion engineered silicon nanocrystal slot waveguides for soliton ultrafast optical processing,” Adv. OptoElectron. 2011, 751498 (2011).

Pavesi, L.

Pelant, I.

Poulton, C.

Premaratne, M.

I. D. Rukhlenko, W. Zhu, M. Premaratne, and G. P. Agrawal, “Effective third-order susceptibility of silicon-nanocrystal-doped silica,” Opt. Express 20, 26275–26284 (2012).
[Crossref] [PubMed]

I. D. Rukhlenko and M. Premaratne, “Optimization of nonlinear performance of silicon-nanocrystal cylindrical nanowires,” IEEE Photonics J. 4, 952–959 (2012).
[Crossref]

I. D. Rukhlenko, M. Premaratne, and G. P. Agrawal, “Effective mode area and its optimization in silicon-nanocrystal waveguides,” Opt. Lett. 37, 2295–2297 (2012).
[Crossref] [PubMed]

I. D. Rukhlenko, M. Premaratne, and G. P. Agrawal, “Nonlinear silicon photonics: Analytical tools,” IEEE J. Sel. Top. Quantum Electron. 16, 200–215 (2010).
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I. D. Rukhlenko, M. Premaratne, and G. P. Agrawal, “Analytical study of optical bistability in silicon ring resonators,” Opt. Lett. 35, 55–57 (2010).
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I. D. Rukhlenko, M. Premaratne, and G. P. Agrawal, “Analytical study of optical bistability in silicon-waveguide resonators,” Opt. Express 17, 22124–22137 (2009).
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Pulci, O.

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M. Dinu, F. Quochi, and H. Garcia, “Third-order nonlinearities in silicon at telecom wavelengths,” Appl. Phys. Lett. 82, 2954–2956 (2003).
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H. Rong, S. Xu, O. Cohen, O. Raday, M. Lee, V. Sih, and M. Paniccia, “A cascaded silicon Raman laser,” Nat. Photonics 2, 170–174 (2008).
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M. A. Ferrara, I. Rendina, S. N. Basu, L. D. Negro, and L. Sirleto, “Raman amplifier based on amorphous silicon nanoparticles,” Int. J. Photoenergy 2012, 254946 (2012).
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M. Krause, H. Renner, and E. Brinkmeyer, “Silicon Raman amplifiers with ring-resonator-enhanced pump power,” IEEE J. Sel. Top. Quantum Electron. 16, 216–225 (2010).
[Crossref]

M. Krause, H. Renner, S. Fathpour, B. Jalali, and E. Brinkmeyer, “Gain enhancement in cladding-pumped silicon Raman amplifiers,” IEEE J. Quantum Electron. 44, 692–704 (2008).
[Crossref]

M. Krause, H. Renner, and E. Brinkmeyer, “Analysis of Raman lasing characteristics in silicon-on-insulator waveguides,” Opt. Express 12, 5703–5710 (2004).
[Crossref] [PubMed]

Richter, H.

H. Richter, Z. P. Wang, and L. Ley, “The one phonon Raman spectrum in microcrystalline silicon,” Solid State Commun. 39, 625–629 (1981).
[Crossref]

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H. Rong, S. Xu, O. Cohen, O. Raday, M. Lee, V. Sih, and M. Paniccia, “A cascaded silicon Raman laser,” Nat. Photonics 2, 170–174 (2008).
[Crossref]

Rukhlenko, I. D.

I. D. Rukhlenko, W. Zhu, M. Premaratne, and G. P. Agrawal, “Effective third-order susceptibility of silicon-nanocrystal-doped silica,” Opt. Express 20, 26275–26284 (2012).
[Crossref] [PubMed]

I. D. Rukhlenko, M. Premaratne, and G. P. Agrawal, “Effective mode area and its optimization in silicon-nanocrystal waveguides,” Opt. Lett. 37, 2295–2297 (2012).
[Crossref] [PubMed]

I. D. Rukhlenko and M. Premaratne, “Optimization of nonlinear performance of silicon-nanocrystal cylindrical nanowires,” IEEE Photonics J. 4, 952–959 (2012).
[Crossref]

I. D. Rukhlenko, M. Premaratne, and G. P. Agrawal, “Analytical study of optical bistability in silicon ring resonators,” Opt. Lett. 35, 55–57 (2010).
[Crossref] [PubMed]

I. D. Rukhlenko, M. Premaratne, and G. P. Agrawal, “Nonlinear silicon photonics: Analytical tools,” IEEE J. Sel. Top. Quantum Electron. 16, 200–215 (2010).
[Crossref]

I. D. Rukhlenko, M. Premaratne, and G. P. Agrawal, “Analytical study of optical bistability in silicon-waveguide resonators,” Opt. Express 17, 22124–22137 (2009).
[Crossref] [PubMed]

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S. N. Volkov, J. J. Saarinen, and J. E. Sipe, “Effective medium theory for 2D disordered structures: A comparison to numerical simulations,” J. Mod. Opt. 59, 954–961 (2012).
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Sih, V.

H. Rong, S. Xu, O. Cohen, O. Raday, M. Lee, V. Sih, and M. Paniccia, “A cascaded silicon Raman laser,” Nat. Photonics 2, 170–174 (2008).
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S. N. Volkov, J. J. Saarinen, and J. E. Sipe, “Effective medium theory for 2D disordered structures: A comparison to numerical simulations,” J. Mod. Opt. 59, 954–961 (2012).
[Crossref]

Sirleto, L.

M. A. Ferrara, I. Rendina, S. N. Basu, L. D. Negro, and L. Sirleto, “Raman amplifier based on amorphous silicon nanoparticles,” Int. J. Photoenergy 2012, 254946 (2012).
[Crossref]

L. Sirleto, M. A. Ferrara, T. Nikitin, S. Novikov, and L. Khriachtchev, “Giant Raman gain in silicon nanocrystals,” Nat. Commun. 3, 1220 (2012).
[Crossref] [PubMed]

Soref, R.

R. Soref and J. Lorenzo, “All-silicon active and passive guided-wave components for λ = 1.3 and 1.6 μm,” IEEE J. Quantum Electron. 22, 873–879 (1986).
[Crossref]

Soref, R. A.

R. A. Soref, “The past, present, and future of silicon photonics,” IEEE J. Sel. Top. Quantum Electron. 12, 1678–1687 (2006).
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L. Cao, B. Nabet, and J. E. Spanier, “Enhanced Raman scattering from individual semiconductor nanocones and nanowires,” Phys. Rev. Lett. 96, 157402 (2006).
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Spano, R.

Spector, S. J.

Stroud, D.

X. C. Zeng, D. J. Bergman, P. M. Hui, and D. Stroud, “Effective-medium theory for weakly nonlinear composites,” Phys. Rev. B 38, 10970–10973 (1988).
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Tartara, L.

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R. Hillenbrand, T. Taubner, and F. Keilmann, “Phonon-enhanced light-matter interaction at the nanometre scale,” Nature 418, 159–162 (2002).
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F. Trani, D. Ninno, and G. Iadonisi, “Role of local fields in the optical properties of silicon nanocrystals using the tight binding approach,” Phys. Rev. B 75, 033312 (2007).
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Trojanek, F.

Tsang, H. K.

H. K. Tsang and Y. Liu, “Nonlinear optical properties of silicon waveguides,” Semicond. Sci. Technol. 23, 064007 (2008).
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M. A. Foster, A. C. Turner, M. Lipson, and A. L. Gaeta, “Nonlinear optics in photonic nanowires,” Opt. Express 16, 1300–1320 (2008).
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Turner, M. D.

Udagedara, I.

Velagapudi, R.

Vlasov, Y.

R. Espinola, J. Dadap, R. Osgood, S. McNab, and Y. Vlasov, “C-band wavelength conversion in silicon photonic wire waveguides,” Opt. Express 13, 4341–4349 (2005).
[Crossref] [PubMed]

Vlasov, Y. A.

J. I. Dadap, R. L. Espinola, R. M. Osgood, S. J. McNab, and Y. A. Vlasov, “Spontaneous Raman scattering in ultrasmall silicon waveguides,” Opt. Lett. 29, 2755–2757 (2004).
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Volkov, S. N.

S. N. Volkov, J. J. Saarinen, and J. E. Sipe, “Effective medium theory for 2D disordered structures: A comparison to numerical simulations,” J. Mod. Opt. 59, 954–961 (2012).
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R. J. Walters, G. I. Bourianoff, and H. A. Atwater, “Field-effect electroluminescence in silicon nanocrystals,” Nat. Mater. 4, 143–146 (2005).
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H. Richter, Z. P. Wang, and L. Ley, “The one phonon Raman spectrum in microcrystalline silicon,” Solid State Commun. 39, 625–629 (1981).
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H. Rong, S. Xu, O. Cohen, O. Raday, M. Lee, V. Sih, and M. Paniccia, “A cascaded silicon Raman laser,” Nat. Photonics 2, 170–174 (2008).
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L. Yin, J. Zhang, P. M. Fauchet, and G. P. Agrawal, “Optical switching using nonlinear polarization rotation inside silicon waveguides,” Opt. Lett. 34, 476–478 (2009).
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Yu, J.

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X. C. Zeng, D. J. Bergman, P. M. Hui, and D. Stroud, “Effective-medium theory for weakly nonlinear composites,” Phys. Rev. B 38, 10970–10973 (1988).
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L. Yin, J. Zhang, P. M. Fauchet, and G. P. Agrawal, “Optical switching using nonlinear polarization rotation inside silicon waveguides,” Opt. Lett. 34, 476–478 (2009).
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I. D. Rukhlenko, W. Zhu, M. Premaratne, and G. P. Agrawal, “Effective third-order susceptibility of silicon-nanocrystal-doped silica,” Opt. Express 20, 26275–26284 (2012).
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Zidek, K.

Adv. Opt. Photonics (1)

Adv. Opt. Technol. (1)

V. A. Belyakov, V. A. Burdov, R. Lockwood, and A. Meldrum, “Silicon nanocrystals: Fundamental theory and implications for stimulated emission,” Adv. Opt. Technol. 2008, 279502 (2008).

Adv. OptoElectron. (1)

F. D. Leonardis and V. M. N. Passaro, “Dispersion engineered silicon nanocrystal slot waveguides for soliton ultrafast optical processing,” Adv. OptoElectron. 2011, 751498 (2011).

Appl. Phys. Lett. (1)

M. Dinu, F. Quochi, and H. Garcia, “Third-order nonlinearities in silicon at telecom wavelengths,” Appl. Phys. Lett. 82, 2954–2956 (2003).
[Crossref]

Electron. Lett. (1)

D. J. Moss, L. Fu, I. Littler, and B. J. Eggleton, “Ultrafast all-optical modulation via two-photon absorption in silicon-on-insulator waveguides,” Electron. Lett. 41, 320–321 (2005).
[Crossref]

IEEE J. Quantum Electron. (3)

M. Krause, H. Renner, S. Fathpour, B. Jalali, and E. Brinkmeyer, “Gain enhancement in cladding-pumped silicon Raman amplifiers,” IEEE J. Quantum Electron. 44, 692–704 (2008).
[Crossref]

R. Soref and J. Lorenzo, “All-silicon active and passive guided-wave components for λ = 1.3 and 1.6 μm,” IEEE J. Quantum Electron. 22, 873–879 (1986).
[Crossref]

X. Chen, N. C. Panoiu, and R. M. Osgood, “Theory of Raman-mediated pulsed amplification in silicon-wire waveguides,” IEEE J. Quantum Electron. 42, 160–170 (2006).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (4)

I. D. Rukhlenko, M. Premaratne, and G. P. Agrawal, “Nonlinear silicon photonics: Analytical tools,” IEEE J. Sel. Top. Quantum Electron. 16, 200–215 (2010).
[Crossref]

R. A. Soref, “The past, present, and future of silicon photonics,” IEEE J. Sel. Top. Quantum Electron. 12, 1678–1687 (2006).
[Crossref]

M. Krause, H. Renner, and E. Brinkmeyer, “Silicon Raman amplifiers with ring-resonator-enhanced pump power,” IEEE J. Sel. Top. Quantum Electron. 16, 216–225 (2010).
[Crossref]

IEEE Photon. Technol. Lett. (2)

IEEE Photonics J. (2)

I. D. Rukhlenko and M. Premaratne, “Optimization of nonlinear performance of silicon-nanocrystal cylindrical nanowires,” IEEE Photonics J. 4, 952–959 (2012).
[Crossref]

IEEE Photonics Technol. Lett. (1)

Int. J. Photoenergy (1)

M. A. Ferrara, I. Rendina, S. N. Basu, L. D. Negro, and L. Sirleto, “Raman amplifier based on amorphous silicon nanoparticles,” Int. J. Photoenergy 2012, 254946 (2012).
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J. Appl. Phys. (1)

J. Lightwave Technol. (2)

C. Manolatou and M. Lipson, “All-optical silicon modulators based on carrier injection by two-photon absorption,” J. Lightwave Technol. 24, 1433–1439 (2006).
[Crossref]

J. Mod. Opt. (1)

S. N. Volkov, J. J. Saarinen, and J. E. Sipe, “Effective medium theory for 2D disordered structures: A comparison to numerical simulations,” J. Mod. Opt. 59, 954–961 (2012).
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J. Opt. Soc. Am. B (1)

B. A. Daniel and G. P. Agrawal, “Vectorial nonlinear propagation in silicon nanowire waveguides: Polarization effects,” J. Opt. Soc. Am. B 27, 956–965 (2010).
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J. Phys.: Conference Series (1)

R. J. Kashtiban, U. Bangert, I. F. Crowe, M. Halsall, A. J. Harvey, and M. Gass, “Study of erbium-doped silicon nanocrystals in silica,” J. Phys.: Conference Series 241, 012097 (2010).
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J. Vac. Sci. Technol. B (1)

Nano Lett. (1)

Nat. Commun. (1)

L. Sirleto, M. A. Ferrara, T. Nikitin, S. Novikov, and L. Khriachtchev, “Giant Raman gain in silicon nanocrystals,” Nat. Commun. 3, 1220 (2012).
[Crossref] [PubMed]

Nat. Mater. (1)

R. J. Walters, G. I. Bourianoff, and H. A. Atwater, “Field-effect electroluminescence in silicon nanocrystals,” Nat. Mater. 4, 143–146 (2005).
[Crossref] [PubMed]

Nat. Photonics (4)

D. Liang and J. E. Bowers, “Recent progress in lasers on silicon,” Nat. Photonics 4, 511–517 (2010).
[Crossref]

H. Rong, S. Xu, O. Cohen, O. Raday, M. Lee, V. Sih, and M. Paniccia, “A cascaded silicon Raman laser,” Nat. Photonics 2, 170–174 (2008).
[Crossref]

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

M. Paniccia, “Integrating silicon photonics,” Nat. Photonics 4, 498–499 (2010).
[Crossref]

Nature (2)

R. Hillenbrand, T. Taubner, and F. Keilmann, “Phonon-enhanced light-matter interaction at the nanometre scale,” Nature 418, 159–162 (2002).
[Crossref] [PubMed]

Opt. Express (19)

O. Boyraz and B. Jalali, “Demonstration of a silicon Raman laser,” Opt. Express 12, 5269–5273 (2004).
[Crossref] [PubMed]

M. Krause, H. Renner, and E. Brinkmeyer, “Analysis of Raman lasing characteristics in silicon-on-insulator waveguides,” Opt. Express 12, 5703–5710 (2004).
[Crossref] [PubMed]

I. D. Rukhlenko, M. Premaratne, and G. P. Agrawal, “Analytical study of optical bistability in silicon-waveguide resonators,” Opt. Express 17, 22124–22137 (2009).
[Crossref] [PubMed]

R. Espinola, J. Dadap, R. Osgood, S. McNab, and Y. Vlasov, “C-band wavelength conversion in silicon photonic wire waveguides,” Opt. Express 13, 4341–4349 (2005).
[Crossref] [PubMed]

Q. Lin, O. J. Painter, and G. P. Agrawal, “Nonlinear optical phenomena in silicon waveguides: Modeling and applications,” Opt. Express 15, 16604–16644 (2007).
[Crossref] [PubMed]

M. A. Foster, A. C. Turner, M. Lipson, and A. L. Gaeta, “Nonlinear optics in photonic nanowires,” Opt. Express 16, 1300–1320 (2008).
[Crossref] [PubMed]

I. D. Rukhlenko, W. Zhu, M. Premaratne, and G. P. Agrawal, “Effective third-order susceptibility of silicon-nanocrystal-doped silica,” Opt. Express 20, 26275–26284 (2012).
[Crossref] [PubMed]

Opt. Express. (1)

Opt. Lett. (8)

L. Yin, J. Zhang, P. M. Fauchet, and G. P. Agrawal, “Optical switching using nonlinear polarization rotation inside silicon waveguides,” Opt. Lett. 34, 476–478 (2009).
[Crossref] [PubMed]

I. D. Rukhlenko, M. Premaratne, and G. P. Agrawal, “Analytical study of optical bistability in silicon ring resonators,” Opt. Lett. 35, 55–57 (2010).
[Crossref] [PubMed]

J. I. Dadap, R. L. Espinola, R. M. Osgood, S. J. McNab, and Y. A. Vlasov, “Spontaneous Raman scattering in ultrasmall silicon waveguides,” Opt. Lett. 29, 2755–2757 (2004).
[Crossref] [PubMed]

I. D. Rukhlenko, M. Premaratne, and G. P. Agrawal, “Effective mode area and its optimization in silicon-nanocrystal waveguides,” Opt. Lett. 37, 2295–2297 (2012).
[Crossref] [PubMed]

D. Dimitropoulos, B. Houshmand, R. Claps, and B. Jalali, “Coupled-mode theory of Raman effect in silicon-on-insulator waveguides,” Opt. Lett. 28, 1954–1956 (2003).
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Phys. Rev. B (4)

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Equations (52)

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\nabla \times {\tilde{E}}_{μ}^{(0)} (r, ω) = i ω μ_{0} {\tilde{H}}_{μ}^{(0)} (r, ω)

\nabla \times {\tilde{H}}_{μ}^{(0)} (r, ω) = - i ω ε_{0} ε_{L} (r_{⊥}, ω) {\tilde{E}}_{μ}^{(0)} (r, ω),

{\tilde{E}}_{μ} (r, ω) = \sum_{ν} {\tilde{a}}_{μ ν} (z, ω - ω_{μ}) \frac{e_{μ ν} (r_{⊥}, ω_{μ})}{\sqrt{N_{μ ν}}} e^{i β_{μ ν} z}

{\tilde{H}}_{μ} (r, ω) = \sum_{ν} {\tilde{a}}_{μ ν} (z, ω - ω_{μ}) \frac{h_{μ ν} (r_{⊥}, ω_{μ})}{\sqrt{N_{μ ν}}} e^{i β_{μ ν} z},

\iint (e_{μ ν}^{*} \times h_{μ ν^{'}} + c . c .) d r_{⊥} = δ_{ν ν^{'}} \iint (e_{μ ν}^{*} \times h_{μ ν} + c . c .) d r_{⊥} = 4 N_{μ ν},

P_{μ} = \frac{1}{2} \iint Re ({\tilde{E}}_{μ} \times {\tilde{H}}_{μ}^{*}) d r_{⊥} = \sum_{ν} {| {\tilde{a}}_{μ ν} |}^{2} .

\nabla \times {\tilde{E}}_{μ} (r, ω) = i ω μ_{0} {\tilde{H}}_{μ} (r, ω)

\nabla \times {\tilde{H}}_{μ} (r, ω) = - i ω ε_{0} ε_{L} (r_{⊥}, ω) {\tilde{E}}_{μ} (r, ω) - i ω {\tilde{P}}_{μ}^{NL} (r, ω) .

\frac{\partial}{\partial z} \iint ({\tilde{E}}_{μ}^{(0)} \times {\tilde{H}}_{μ}^{*} + {\tilde{E}}_{μ}^{*} \times {\tilde{H}}_{μ}^{(0)}) d r_{⊥} = \iint \nabla ({\tilde{E}}_{μ}^{(0)} \times {\tilde{H}}_{μ}^{*} + {\tilde{E}}_{μ}^{*} \times {\tilde{H}}_{μ}^{(0)}) d r_{⊥} .

\nabla ({\tilde{E}}_{μ}^{(0)} \times {\tilde{H}}_{μ}^{*} + {\tilde{E}}_{μ}^{*} \times {\tilde{H}}_{μ}^{(0)}) = - i ω ({\tilde{E}}_{μ}^{(0)} \cdot {\tilde{P}}_{μ}^{N L_{*}}) .

{\tilde{E}}_{μ}^{(0)} = \frac{e_{μ ν} (r_{⊥}, ω_{μ})}{\sqrt{N_{μ ν}}} e^{i β_{ν} (ω) z} and {\tilde{H}}_{μ}^{(0)} = \frac{h_{μ ν} (r_{⊥}, ω_{μ})}{\sqrt{N_{μ ν}}} e^{i β_{ν} (ω) z}

\begin{array}{l} \frac{\partial}{\partial z} {{\tilde{a}}_{μ ν} (z, ω - ω_{μ}) e^{i [β_{ν} (ω_{μ}) - β_{ν} (ω)] z}} \\ = \frac{i ω}{4 \sqrt{N_{μ ν}}} \iint e_{μ ν}^{*} (r_{⊥}, ω_{μ}) \cdot {\tilde{P}}_{μ}^{NL} (r, ω) e^{- i β_{ν} (ω) z} d r_{⊥} . \end{array}

\begin{array}{l} (\frac{\partial}{\partial z} + i \sum_{n = 1}^{\infty} \frac{1}{n!} {\frac{\partial β_{ν}}{\partial ω} |}_{ω_{μ}} {(ω - ω_{μ})}^{n}) {\tilde{a}}_{μ ν} (z, ω - ω_{μ}) \\ = \frac{i ω}{4 \sqrt{N_{μ ν}}} \iint e_{μ ν}^{*} (r_{⊥}, ω_{μ}) \cdot {\tilde{P}}_{μ}^{NL} (r, ω) e^{- i β_{ν} (ω_{μ}) z} d r_{⊥} . \end{array}

\begin{array}{l} (\frac{\partial}{\partial z} + \sum_{n = 1}^{\infty} \frac{i^{n + 1}}{n!} {\frac{\partial β_{ν}}{\partial ω} |}_{ω_{μ}} \frac{\partial^{n}}{\partial t^{n}}) a_{μ ν} (z, t) \\ = - \frac{e^{- i β_{μ ν} z}}{4 \sqrt{N_{μ ν}}} \iint e_{μ ν}^{*} (r_{⊥}, ω_{μ}) \frac{\partial P_{μ}^{NL} (r, t)}{\partial t} e^{i ω_{μ} t} d r_{⊥}, \end{array}

a_{μ ν} (z, t) = \frac{1}{2 π} \int_{- \infty}^{+ \infty} {\tilde{a}}_{μ ν} (z, ω) e^{- i ω t} d ω and P_{μ}^{NL} (r, t) = \frac{1}{2 π} \int_{- \infty}^{+ \infty} {\tilde{P}}_{μ}^{NL} (r, ω) e^{- i ω t} d ω .

\frac{\partial a_{μ ν}}{\partial z} + \sum_{n = 1}^{\infty} \frac{i^{n + 1} β_{μ ν}^{(n)}}{n!} \frac{\partial^{n} a_{μ ν}}{\partial t^{n}} = \frac{i ω_{μ}}{4 \sqrt{N_{μ ν}}} (1 + \frac{i}{ω_{μ}} \frac{\partial}{\partial t}) e^{- i β_{μ ν} z} \iint (e_{μ ν}^{*} \cdot P_{ω_{μ}}^{NL}) d r_{⊥},

P_{ω_{μ}}^{NL} (r, t) = P_{ω_{μ}}^{K} (r, t) + P_{ω_{μ}}^{R} (r, t) + P_{ω_{μ}}^{FC} (r, t),

P_{ω_{μ}}^{K} (r, t) = ε_{0} χ_{K}^{(3)} (ω_{μ}; ω_{μ}, - ω_{μ}, ω_{μ}) E_{ω_{μ}} (r, t) E_{ω_{μ}}^{*} (r, t) E_{ω_{μ}} (r, t),

χ_{K}^{(3)} (ω_{μ}; ω_{μ}, - ω_{μ}, ω_{μ}) = χ_{μ} (\frac{8 + 7 ρ}{45} (δ_{k l} δ_{m n} + δ_{k m} δ_{\ln} + δ_{k n} δ_{l m}) + \frac{1 - ρ}{9} δ_{k l} δ_{l m} δ_{m n}),

χ_{μ} = c ε_{0} ε_{eff} [n_{2} + i β_{TPA} / (2 k_{μ})] ξ,

ξ = \frac{1}{f} {(\frac{\partial ε_{eff}}{\partial ε_{1}})}^{2} = \frac{{[(3 f - 1) ε_{eff} + ε_{2}]}^{2}}{f (u^{2} + 8 ε_{1} ε_{2})}

P_{ω_{μ}}^{K} = ε_{0} χ_{μ} [\frac{8 + 7 ρ}{45} (2 {| E_{ω_{μ}} |}^{2} E_{ω_{μ}} + E_{ω_{μ}}^{2} E_{ω_{μ}}^{*}) + \frac{1 - ρ}{9} \sum_{η} E_{η}^{2} E_{η}^{*} \hat{η}],

E_{ω_{μ}} = \sum_{ν = x, y} a_{μ ν} \frac{e_{μ ν}}{\sqrt{N_{μ ν}}} e^{i β_{μ ν} z},

\begin{array}{l} \frac{e^{- i β_{μ ν} z}}{\sqrt{N_{μ ν}}} \iint (e_{μ ν}^{*} \cdot P_{ω_{μ}}^{K}) d r_{⊥} \\ = ε_{0} χ_{μ} [\frac{8 + 7 ρ}{45} (2 a_{μ ν} \sum_{ν^{'}} Γ_{ν ν^{'}}^{(μ)} {| a_{μ ν^{'}} |}^{2} + a_{μ ν}^{*} \sum_{ν^{'}} a_{μ ν^{'}}^{2} Λ_{ν ν^{'}}^{(μ)} e^{2 i (β_{μ ν^{'}} - β_{μ ν}) z}) \\ + \frac{1 - ρ}{9} Γ_{ν ν}^{(μ)} a_{μ ν} {| a_{μ ν} |}^{2}], \end{array}

Γ_{ν ν^{'}}^{(μ)} = \frac{1}{N_{μ ν} N_{μ ν^{'}}} \iint {| e_{μ ν} |}^{2} {| e_{μ ν^{'}} |}^{2} d r_{⊥}

Λ_{ν ν^{'}}^{(μ)} = \frac{1}{N_{μ ν} N_{μ ν^{'}}} \iint e_{μ ν}^{* 2} e_{μ ν^{'}}^{2} d r_{⊥} .

N_{μ ν} = \frac{β_{μ ν}}{2 μ_{0} ω_{μ}} \iint {| e_{μ ν} |}^{2} d r_{⊥} .

\begin{array}{l} P_{ω_{μ}}^{R} (r, t) = e^{i ω_{μ} t} \int_{- \infty}^{t} d t_{1} \int_{- \infty}^{t} d t_{2} \int_{- \infty}^{t} d t_{3} ε_{0} χ_{R}^{(3)} (t - t_{1}, t - t_{2}, t - t_{3}) \\ \times (E_{ω_{μ^{'}}} (r, t_{1}) E_{ω_{μ^{'}}}^{*} (r, t_{2}) E_{ω_{μ}} (r, t_{3}) e^{- i (ω_{μ^{'}} t_{1} - ω_{μ^{'}} t_{2} + ω_{μ} t_{3})} \\ + E_{ω_{μ}} (r, t_{1}) E_{ω_{μ^{'}}}^{*} (r, t_{2}) E_{ω_{μ^{'}}} (r, t_{3}) e^{- i (ω_{μ} t_{1} - ω_{μ^{'}} t_{2} + ω_{μ^{'}} t_{3})}), \end{array}

χ_{R}^{(3)} (t_{1}, t_{2}, t_{3}) = \frac{1}{2} [δ (t_{1} - t_{2}) δ (t_{3}) ℛ_{k l m n} + δ (t_{1}) δ (t_{2} - t_{3}) ℛ_{k n m l}] ξ H (t_{2}),

ℛ_{k l m n} = \frac{29}{45} (δ_{k m} δ_{\ln} + δ_{k n} δ_{l m}) - \frac{16}{45} δ_{k l} δ_{m n} - \frac{2}{9} δ_{k l} δ_{l m} δ_{m n},

H (t) = 2 χ_{R} \frac{Γ_{R} Ω_{R}}{{(Ω_{R}^{2} - Γ_{R}^{2})}^{1 / 2}} e^{- t / τ_{2}} sin (t / τ_{1}),

P_{ω_{μ}}^{R} (r, t) = ε_{0} ξ \int_{- \infty}^{t} d t_{1} H (t - t_{1}) ℛ_{k l m n} E_{ω_{μ}} (r, t_{1}) E_{ω_{μ^{'}}}^{*} (r, t_{1}) E_{ω_{μ^{'}}} (r, t) e^{i (ω_{μ} - ω_{μ^{'}}) (t - t_{1})} .

\begin{array}{l} \frac{e^{- i β_{μ ν} z}}{ε_{0} ξ \sqrt{N_{μ ν}}} \iint (e_{μ ν}^{*} \cdot P_{ω_{μ}}^{R}) d r_{⊥} \\ = \frac{29}{45} \sum_{ν^{'}} Λ_{ν ν^{'}}^{μ μ^{'}} exp (i β_{μ ν μ^{'} ν}^{μ ν^{'} μ^{'} ν^{'}} z) a_{μ^{'} ν^{'}} (t) \int_{- \infty}^{t} a_{μ^{'} ν}^{*} (t_{1}) a_{μ ν^{'}} (t_{1}) H (t - t_{1}) e^{i ω_{μ μ^{'}} (t - t_{1})} d t_{1} \\ + \frac{29}{45} \sum_{ν^{'}} Ψ_{ν ν^{'}}^{μ μ^{'}} exp (i β_{μ ν μ^{'} ν^{'}}^{μ ν^{'} μ^{'} ν} z) a_{μ^{'} ν} (t) \int_{- \infty}^{t} a_{μ^{'} ν^{'}}^{*} (t_{1}) a_{μ ν} (t_{1}) H (t - t_{1}) e^{i ω_{μ μ^{'}} (t - t_{1})} d t_{1} \\ - \frac{16}{45} \sum_{ν^{'}} Γ_{ν ν^{'}}^{μ μ^{'}} a_{μ^{'} ν^{'}} (t) \int_{- \infty}^{t} a_{μ^{'} ν^{'}}^{*} (t_{1}) a_{μ ν} (t_{1}) H (t - t_{1}) e^{i ω_{μ μ^{'}} (t - t_{1})} d t_{1} \\ - \frac{2}{9} Λ_{ν ν}^{μ μ^{'}} a_{μ^{'} ν} t \int_{- \infty}^{t} a_{μ^{'} ν}^{*} (t_{1}) a_{μ ν} (t_{1}) H (t - t_{1}) e^{i ω_{μ μ^{'}} (t - t_{1})} d t_{1}, \end{array}

Λ_{ν ν^{'}}^{μ μ^{'}} = \iint \frac{(e_{μ ν}^{*} e_{μ^{'} ν}^{*}) (e_{μ ν^{'}} e_{μ^{'} ν^{'}})}{\sqrt{N_{μ ν} N_{μ^{'} ν} N_{μ ν^{'}} N_{μ^{'} ν^{'}}}} d r_{⊥},

Ψ_{ν ν^{'}}^{μ μ^{'}} = \iint \frac{(e_{μ ν}^{*} e_{μ^{'} ν}) (e_{μ ν^{'}} e_{μ^{'} ν^{'}}^{*})}{\sqrt{N_{μ ν} N_{μ^{'} ν} N_{μ ν^{'}} N_{μ^{'} ν^{'}}}} d r_{⊥},

Γ_{ν ν^{'}}^{μ μ^{'}} = \frac{1}{N_{μ ν} N_{μ^{'} ν^{'}}} \iint {| e_{μ ν} |}^{2} {| e_{μ^{'} ν^{'}} |}^{2} d r_{⊥} .

P_{ω_{μ}}^{FC} (r, t) = 2 ζ ε_{0} n_{eff} [Δ n_{FC} + i c / (2 ω_{μ}) Δ α_{FC}] E_{ω_{μ}} (r, t),

ζ = \frac{\partial n_{eff}}{\partial n_{1}} = {(\frac{ε_{1}}{ε_{eff}})}^{1 / 2} \frac{(3 f - 1) ε_{eff} + ε_{2}}{\sqrt{u^{2} + 8 ε_{1} ε_{2}}}

Δ n_{FC} = - σ_{n} {(ω_{0} / ω_{μ})}^{2} (1 + ς N^{0.2}) N^{0.8} and Δ α_{FC} = σ_{α} {(ω_{0} / ω_{μ})}^{2} N,

\frac{e^{- i β_{μ ν} z}}{\sqrt{N_{μ ν}}} \iint (e_{μ ν}^{*} \cdot P_{ω_{μ}}^{FC}) d r_{⊥} = 4 (ζ / c) (n_{eff} / n_{μ ν}) [Δ n_{FC} + i c / (2 ω_{μ}) Δ α_{FC}] a_{μ ν},

\frac{\partial N}{\partial t} = - \frac{N}{τ_{c}} - \sum_{μ} \frac{1}{2 \bar{h} ω_{μ} A_{eff}} \frac{\partial P_{μ}}{\partial z},

\begin{array}{l} \frac{\partial P_{μ}}{\partial z} & = \sum_{ν} (a_{μ ν} \frac{\partial a_{μ ν}^{*}}{\partial z} + a_{μ ν}^{*} \frac{\partial a_{μ ν}}{\partial z}) \\ = - \frac{1}{4} ξ c^{2} ε_{0}^{2} ε_{eff} β_{TPA} \sum_{ν} (\frac{8 + 7 ρ}{45} \sum_{ν^{'}} 2 Γ_{ν ν^{'}}^{(μ)} {| a_{μ ν} |}^{2} {| a_{μ ν^{'}} |}^{2} + \frac{13 + 2 ρ}{45} Γ_{ν ν}^{(μ)} {| a_{μ ν} |}^{4}) . \end{array}

A_{eff}^{μ ν} = {(\iint {| e_{μ ν} |}^{2} d r_{⊥})}^{2} / \iint {| e_{μ ν} |}^{4} d r_{⊥} .

A_{eff} = \prod_{μ, ν} {(A_{eff}^{μ ν})}^{1 / 4} .

𝒜_{eff}^{μ ν} = 𝒜_{NL} \iint_{\infty} {| e_{μ ν} |}^{2} d r_{⊥} / \iint_{NL} {| e_{μ ν} |}^{2} d r_{⊥},

\begin{array}{l} \frac{\partial a_{μ ν}}{\partial z} + \sum_{n = 1}^{\infty} \frac{i^{n + 1} β_{μ ν}^{(n)}}{n!} \frac{\partial^{n} a_{μ ν}}{\partial t^{n}} + \frac{α_{μ ν}}{2} a_{μ ν} \\ = - \frac{1}{8} ξ c^{2} ε_{0}^{2} ε_{eff} (β_{TPA} - 2 i n_{2} k_{μ}) (\frac{8 + 7 ρ}{45} 2 Γ_{ν ν^{'}}^{(μ)} {| a_{μ ν^{'}} |}^{2} + \frac{29 + 16 ρ}{45} Γ_{ν ν}^{(μ)} {| a_{μ ν} |}^{2}) a_{μ ν} \\ + \frac{8 i ε_{0} ω_{μ}}{45} ξ Γ_{ν ν}^{μ μ^{'}} a_{μ^{'} ν} (t) \int_{- \infty}^{t} a_{μ^{'} ν}^{*} (t_{1}) a_{μ ν} (t_{1}) H (t - t_{1}) e^{i ω_{μ μ^{'}} (t - t_{1})} d t_{1} \\ - \frac{4 i ε_{0} ω_{μ}}{45} ξ Γ_{ν ν^{'}}^{μ μ^{'}} a_{μ^{'} ν^{'}} (t) \int_{- \infty}^{t} a_{μ^{'} ν^{'}}^{*} (t_{1}) a_{μ ν} (t_{1}) H (t - t_{1}) e^{i ω_{μ μ^{'}} (t - t_{1})} d t_{1} \\ - ζ \frac{n_{eff}}{n_{μ ν}} {(\frac{ω_{0}}{ω_{μ}})}^{2} (i σ_{n} k_{μ} (1 + ς N^{0.2}) + \frac{σ_{α}}{2} N^{0.2}) N^{0.8} a_{μ ν}, \end{array}

\frac{\partial N}{\partial t} = - \frac{N}{τ_{c}} + \frac{ξ c^{2} ε_{0}^{2} ε_{eff}}{4 A_{eff}} \sum_{μ, ν} \frac{β_{TPA}}{2 \bar{h} ω_{μ}} (\frac{8 + 7 ρ}{45} 2 Γ_{ν ν^{'}}^{(μ)} {| a_{μ ν^{'}} |}^{2} + \frac{29 + 16 ρ}{45} Γ_{ν ν}^{(μ)} {| a_{μ ν} |}^{2}) {| a_{μ ν} |}^{2} .

\frac{4 i ε_{0} ω_{μ}}{45} ξ \tilde{H} (ω_{μ μ^{'}}) (2 Γ_{ν ν}^{μ μ^{'}} {| a_{μ^{'} ν} |}^{2} - Γ_{ν ν}^{μ μ^{'}} {| a_{μ^{'} ν^{'}} |}^{2}) a_{μ ν},

\tilde{H} (ω) = \int_{0}^{\infty} H (t) e^{i ω t} d t = \frac{2 χ_{R} Γ_{R} Ω_{R}}{Ω_{R}^{2} + 2 i Γ_{R} ω - ω^{2}}

\begin{array}{l} \frac{\partial ln a_{μ ν}}{\partial z} + \frac{α_{μ ν}}{2} = - ξ (\frac{β_{TPA}}{2} - i n_{2} k_{μ}) (\frac{8 + 7 ρ}{45} 2 γ_{ν ν^{'}}^{(μ)} I_{μ ν^{'}} + \frac{29 + 16 ρ}{45} γ_{ν ν}^{(μ)} I_{μ ν}) \\ + \frac{32}{45} ξ {\tilde{g}}_{R} (ω_{μ μ^{'}}) (2 γ_{ν ν}^{μ μ^{'}} I_{μ^{'} ν} - γ_{ν ν^{'}}^{μ μ^{'}} I_{μ^{'} ν^{'}}) - ζ ξ τ_{c} \frac{n_{eff}}{n_{μ ν}} {(\frac{ω_{0}}{ω_{μ}})}^{2} (\frac{σ_{α}}{2} + i {\bar{σ}}_{n} k_{μ}) \\ \times \sum_{μ, ν} \frac{β_{TPA}}{2 \bar{h} ω_{μ}} (\frac{8 + 7 ρ}{45} 2 γ_{ν ν^{'}}^{(μ)} I_{μ ν^{'}} + \frac{29 + 16 ρ}{45} γ_{ν ν}^{(μ)} I_{μ ν}) I_{μ ν}, \end{array}

γ_{ν ν^{'}}^{μ μ^{'}} = \frac{n_{eff}^{2}}{n_{μ ν} n_{μ^{'} ν^{'}}} \frac{A_{eff} \iint {| e_{μ ν} |}^{2} {| e_{μ^{'} ν^{'}} |}^{2} d r_{⊥}}{\iint {| e_{μ ν} |}^{2} d r_{⊥} \iint {| e_{μ^{'} ν^{'}} |}^{2} d r_{⊥}},

{\tilde{g}}_{R} (ω) = \frac{2 i g_{R} Γ_{R} Ω_{R}}{Ω_{R}^{2} + 2 i Γ_{R} ω - ω^{2}} .