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)
•doi: 10.1364/OE.21.002832

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

Export Citation
- BibTex
- Endnote (RIS)
- HTML
- Plain Text
Get PDF (924 KB)
Set citation alerts
Save article

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.

Full Article | PDF Article

References

View by:
Article Order
|
Year
|
Author
|
Publication

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]
J. Leuthold, C. Koos, and W. Freude, “Nonlinear silicon photonics,” Nat. Photonics 4, 535–544 (2010).
[Crossref]
R. M. Osgood, N. C. Panoiu, J. I. Dadap, X. Liu, X. Chen, I.-W. Hsieh, E. Dulkeith, W. M. Green, and Y. A. Vlasov, “Engineering nonlinearities in nanoscale optical systems: Physics and applications in dispersion-engineered silicon nanophotonic wires,” Adv. Opt. Photonics 1, 162–235 (2009).
[Crossref]
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]
R. A. Soref, “The past, present, and future of silicon photonics,” IEEE J. Sel. Top. Quantum Electron. 12, 1678–1687 (2006).
[Crossref]
G. T. Reed and A. P. Knights, Silicon Photonics: An Introduction (John Wiley, Hoboken, 2004).
[Crossref]
L. Pavesi and D. Lockwood, eds., Silicon Photonics, vol. 94 of Topics in Applied Physics (Springer-Verlag, Berlin, 2004).
M. Paniccia, “Integrating silicon photonics,” Nat. Photonics 4, 498–499 (2010).
[Crossref]
J. I. Dadap, N. C. Panoiu, X. Chen, I.-W. Hsieh, X. Liu, C.-Y. Chou, E. Dulkeith, S. J. McNab, F. Xia, W. M. J. Green, L. Sekaric, Y. A. Vlasov, and J. R. M. Osgood, “Nonlinear-optical phase modification in dispersion-engineered Si photonic wires,” Opt. Express 16, 1280–1299 (2008).
[Crossref] [PubMed]
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] [PubMed]
X. Chen, N. C. Panoiu, I. Hsieh, J. I. Dadap, and R. M. Osgood, “Third-order dispersion and ultrafast-pulse propagation in silicon wire waveguides,” IEEE Photon. Technol. Lett. 18, 2617–2619 (2006).
[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]
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]
R. Jones, A. Liu, H. Rong, M. Paniccia, O. Cohen, and D. Hak, “Lossless optical modulation in a silicon waveguide using stimulated Raman scattering,” Opt. Express 13, 1716–1723 (2005).
[Crossref] [PubMed]
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]
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]
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]
I. D. Rukhlenko, C. Dissanayake, M. Premaratne, and G. P. Agrawal, “Maximization of net optical gain in silicon-waveguide Raman amplifiers,” Opt. Express 17, 5807–5814 (2009).
[Crossref] [PubMed]
I. D. Rukhlenko, M. Premaratne, C. Dissanayake, and G. P. Agrawal, “Continuous-wave Raman amplification in silicon waveguides: Beyond the undepleted pump approximation,” Opt. Lett. 34, 536–538 (2009).
[Crossref] [PubMed]
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. A. Foster, A. C. Turner, J. E. Sharping, B. S. Schmidt, M. Lipson, and A. L. Gaeta, “Broad-band optical parametric gain on a silicon photonic chip,” Nature 441, 960–963 (2006).
[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]
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]
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]
O. Boyraz and B. Jalali, “Demonstration of a silicon Raman laser,” Opt. Express 12, 5269–5273 (2004).
[Crossref] [PubMed]
M. W. Geis, S. J. Spector, M. E. Grein, J. U. Yoon, D. M. Lennon, and T. M. Lyszczarz, “Silicon waveguide infrared photodiodes with > 35 GHz bandwidth and phototransistors with 50 AW-1 response,” Opt. Express 17, 5193–5204 (2009).
[Crossref] [PubMed]
W. Astar, J. B. Driscoll, X. Liu, J. I. Dadap, W. M. J. Green, Y. A. Vlasov, G. M. Carter, and R. M. Osgood, “Conversion of 10 Gb/s NRZ-OOK to RZ-OOK utilizing XPM in a Si nanowire,” Opt. Express 17, 12987–12999 (2009).
[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]
H. K. Tsang and Y. Liu, “Nonlinear optical properties of silicon waveguides,” Semicond. Sci. Technol. 23, 064007 (2008).
[Crossref]
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.-W. Hsieh, X. Chen, X. Liu, J. I. Dadap, N. C. Panoiu, C.-Y. Chou, F. Xia, W. M. Green, Y. A. Vlasov, and R. M. Osgood, “Supercontinuum generation in silicon photonic wires,” Opt. Express 15, 15242–15249 (2007).
[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]
M. W. Geis, S. J. Spector, R. C. Williamson, and T. M. Lyszczarz, “Submicrosecond submilliwatt silicon-on-insulator thermooptic switch,” IEEE Photon. Technol. Lett. 16, 2514–2516 (2004).
[Crossref]
M. Dinu, F. Quochi, and H. Garcia, “Third-order nonlinearities in silicon at telecom wavelengths,” Appl. Phys. Lett. 82, 2954–2956 (2003).
[Crossref]
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]
L. Pavesi and R. Turan, eds., Silicon Nanocrystals: Fundamentals, Synthesis and Applications (WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim, 2010).
L. Khriachtchev, ed., Silicon Nanophotonics: Basic Principles, Present Status and Perspectives (Pan Stanford, Singapore, 2009).
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).
F. Iori, E. Degoli, R. Magri, I. Marri, G. Cantele, D. Ninno, F. Trani, O. Pulci, and S. Ossicini, “Engineering silicon nanocrystals: Theoretical study of the effect of codoping with boron and phosphorus,” Phys. Rev. B 76, 085302 (2007).
[Crossref]
I. D. Rukhlenko and M. Premaratne, “Optimization of nonlinear performance of silicon-nanocrystal cylindrical nanowires,” IEEE Photonics J. 4, 952–959 (2012).
[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).
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]
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]
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]
R. J. Walters, G. I. Bourianoff, and H. A. Atwater, “Field-effect electroluminescence in silicon nanocrystals,” Nat. Mater. 4, 143–146 (2005).
[Crossref] [PubMed]
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]
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]
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).
[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]
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).
[Crossref]
C. M. Dissanayake, I. D. Rukhlenko, M. Premaratne, and G. P. Agrawal, “Raman-mediated nonlinear interactions in silicon waveguides: Copropagating and counterpropagating pulses,” IEEE Photonics Technol. Lett. 21, 1372–1374 (2009).
[Crossref]
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]
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, 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, 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]
M. D. Turner, T. M. Monro, and S. Afshar V., “A full vectorial model for pulse propagation in emerging waveguides with subwavelength structures part II: Stimulated Raman scattering,” Opt. Express 17, 11565–11581 (2009).
[Crossref] [PubMed]
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, C. Dissanayake, and G. P. Agrawal, “Nonlinear pulse evolution in silicon waveguides: An approximate analytic approach,” J. Lightwave Technol. 27, 3241–3248 (2009).
[Crossref]
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).
[Crossref] [PubMed]
S. Afshar V. and T. M. Monro, “A full vectorial model for pulse propagation in emerging waveguides with subwavelength structures part I: Kerr nonlinearity,” Opt. Express 17, 2298–2318 (2009).
[Crossref] [PubMed]
R. W. Boyd, Nonlinear Optics, 3rd ed. (Academic Press, San Diego, 2008).
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]
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]
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).
[Crossref]
W. Cai and V. Shalaev, Optical Metamaterials: Fundamentals and Applications (Springer, New York, 2010).
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]
F. Trojanek, K. Neudert, K. Zidek, K. Dohnalova, I. Pelant, and P. Maly, “Femtosecond photoluminescence spectroscopy of silicon nanocrystals,” Physica Status Solidi (c) 3, 3873–3876 (2006).
[Crossref]
R. Spano, N. Daldosso, M. Cazzanelli, L. Ferraioli, L. Tartara, J. Yu, V. Degiorgio, E. Giordana, J. M. Fedeli, and L. Pavesi, “Bound electronic and free carrier nonlinearities in silicon nanocrystals at 1550 nm,” Opt. Express 17, 3941–3950 (2009).
[Crossref] [PubMed]
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]
M. A. Ferrara, I. Rendina, and L. Sirleto, “Stimulated Raman scattering in quantum dots and nanocomposite silicon based materials,” in “Nonlinear Optics,” N. Kamanina, ed. (InTech, Rijeka, 2012), pp. 53–70.
H. Richter, Z. P. Wang, and L. Ley, “The one phonon Raman spectrum in microcrystalline silicon,” Solid State Commun. 39, 625–629 (1981).
[Crossref]
I. H. Campbell and P. M. Fauchet, “The effects of microcrystal size and shape on the one phonon Raman spectra of crystalline semiconductors,” Solid State Commun. 58, 739–741 (1986).
[Crossref]
L. Cao, B. Nabet, and J. E. Spanier, “Enhanced Raman scattering from individual semiconductor nanocones and nanowires,” Phys. Rev. Lett. 96, 157402 (2006).
[Crossref] [PubMed]
R. Hillenbrand, T. Taubner, and F. Keilmann, “Phonon-enhanced light-matter interaction at the nanometre scale,” Nature 418, 159–162 (2002).
[Crossref] [PubMed]
G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, San Diego, 2007).
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).
[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).
[Crossref]

2009 (12)

S. Afshar V. and T. M. Monro, “A full vectorial model for pulse propagation in emerging waveguides with subwavelength structures part I: Kerr nonlinearity,” Opt. Express 17, 2298–2318 (2009).
[Crossref] [PubMed]

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, C. Dissanayake, and G. P. Agrawal, “Continuous-wave Raman amplification in silicon waveguides: Beyond the undepleted pump approximation,” Opt. Lett. 34, 536–538 (2009).
[Crossref] [PubMed]

R. Spano, N. Daldosso, M. Cazzanelli, L. Ferraioli, L. Tartara, J. Yu, V. Degiorgio, E. Giordana, J. M. Fedeli, and L. Pavesi, “Bound electronic and free carrier nonlinearities in silicon nanocrystals at 1550 nm,” Opt. Express 17, 3941–3950 (2009).
[Crossref] [PubMed]

M. W. Geis, S. J. Spector, M. E. Grein, J. U. Yoon, D. M. Lennon, and T. M. Lyszczarz, “Silicon waveguide infrared photodiodes with > 35 GHz bandwidth and phototransistors with 50 AW-1 response,” Opt. Express 17, 5193–5204 (2009).
[Crossref] [PubMed]

I. D. Rukhlenko, C. Dissanayake, M. Premaratne, and G. P. Agrawal, “Maximization of net optical gain in silicon-waveguide Raman amplifiers,” Opt. Express 17, 5807–5814 (2009).
[Crossref] [PubMed]

M. D. Turner, T. M. Monro, and S. Afshar V., “A full vectorial model for pulse propagation in emerging waveguides with subwavelength structures part II: Stimulated Raman scattering,” Opt. Express 17, 11565–11581 (2009).
[Crossref] [PubMed]

W. Astar, J. B. Driscoll, X. Liu, J. I. Dadap, W. M. J. Green, Y. A. Vlasov, G. M. Carter, and R. M. Osgood, “Conversion of 10 Gb/s NRZ-OOK to RZ-OOK utilizing XPM in a Si nanowire,” Opt. Express 17, 12987–12999 (2009).
[Crossref] [PubMed]

I. D. Rukhlenko, M. Premaratne, C. Dissanayake, and G. P. Agrawal, “Nonlinear pulse evolution in silicon waveguides: An approximate analytic approach,” J. Lightwave Technol. 27, 3241–3248 (2009).
[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]

C. M. Dissanayake, I. D. Rukhlenko, M. Premaratne, and G. P. Agrawal, “Raman-mediated nonlinear interactions in silicon waveguides: Copropagating and counterpropagating pulses,” IEEE Photonics Technol. Lett. 21, 1372–1374 (2009).
[Crossref]

R. M. Osgood, N. C. Panoiu, J. I. Dadap, X. Liu, X. Chen, I.-W. Hsieh, E. Dulkeith, W. M. Green, and Y. A. Vlasov, “Engineering nonlinearities in nanoscale optical systems: Physics and applications in dispersion-engineered silicon nanophotonic wires,” Adv. Opt. Photonics 1, 162–235 (2009).
[Crossref]

2008 (6)

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]

H. K. Tsang and Y. Liu, “Nonlinear optical properties of silicon waveguides,” Semicond. Sci. Technol. 23, 064007 (2008).
[Crossref]

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).

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]

J. I. Dadap, N. C. Panoiu, X. Chen, I.-W. Hsieh, X. Liu, C.-Y. Chou, E. Dulkeith, S. J. McNab, F. Xia, W. M. J. Green, L. Sekaric, Y. A. Vlasov, and J. R. M. Osgood, “Nonlinear-optical phase modification in dispersion-engineered Si photonic wires,” Opt. Express 16, 1280–1299 (2008).
[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]

2007 (5)

I.-W. Hsieh, X. Chen, X. Liu, J. I. Dadap, N. C. Panoiu, C.-Y. Chou, F. Xia, W. M. Green, Y. A. Vlasov, and R. M. Osgood, “Supercontinuum generation in silicon photonic wires,” Opt. Express 15, 15242–15249 (2007).
[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]

F. Iori, E. Degoli, R. Magri, I. Marri, G. Cantele, D. Ninno, F. Trani, O. Pulci, and S. Ossicini, “Engineering silicon nanocrystals: Theoretical study of the effect of codoping with boron and phosphorus,” Phys. Rev. B 76, 085302 (2007).
[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] [PubMed]

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).
[Crossref]

2006 (7)

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]

X. Chen, N. C. Panoiu, I. Hsieh, J. I. Dadap, and R. M. Osgood, “Third-order dispersion and ultrafast-pulse propagation in silicon wire waveguides,” IEEE Photon. Technol. Lett. 18, 2617–2619 (2006).
[Crossref]

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

M. A. Foster, A. C. Turner, J. E. Sharping, B. S. Schmidt, M. Lipson, and A. L. Gaeta, “Broad-band optical parametric gain on a silicon photonic chip,” Nature 441, 960–963 (2006).
[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]

F. Trojanek, K. Neudert, K. Zidek, K. Dohnalova, I. Pelant, and P. Maly, “Femtosecond photoluminescence spectroscopy of silicon nanocrystals,” Physica Status Solidi (c) 3, 3873–3876 (2006).
[Crossref]

L. Cao, B. Nabet, and J. E. Spanier, “Enhanced Raman scattering from individual semiconductor nanocones and nanowires,” Phys. Rev. Lett. 96, 157402 (2006).
[Crossref] [PubMed]

2005 (5)

R. Jones, A. Liu, H. Rong, M. Paniccia, O. Cohen, and D. Hak, “Lossless optical modulation in a silicon waveguide using stimulated Raman scattering,” Opt. Express 13, 1716–1723 (2005).
[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]

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]

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

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).
[Crossref]

2004 (4)

M. W. Geis, S. J. Spector, R. C. Williamson, and T. M. Lyszczarz, “Submicrosecond submilliwatt silicon-on-insulator thermooptic switch,” IEEE Photon. Technol. Lett. 16, 2514–2516 (2004).
[Crossref]

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]

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]

2003 (2)

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).
[Crossref] [PubMed]

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

2002 (1)

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

1988 (1)

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).
[Crossref]

1986 (2)

I. H. Campbell and P. M. Fauchet, “The effects of microcrystal size and shape on the one phonon Raman spectra of crystalline semiconductors,” Solid State Commun. 58, 739–741 (1986).
[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]

1981 (1)

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

Afshar V., S.

Agrawal, G. P.

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]

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

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-waveguide resonators,” Opt. Express 17, 22124–22137 (2009).
[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]

G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, San Diego, 2007).

Ahlgren, T.

Aitola, K.

Astar, W.

Atwater, H. A.

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

Bangert, U.

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]

Basu, S. N.

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]

Belyakov, V. A.

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).

Bergman, D. J.

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).
[Crossref]

Blasco, J.

Bourianoff, G. I.

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

Bowers, J. E.

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

Boyd, R. W.

R. W. Boyd, Nonlinear Optics, 3rd ed. (Academic Press, San Diego, 2008).

Boyraz, O.

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

Brinkmeyer, E.

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]

Burdov, V. A.

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).

Cai, W.

W. Cai and V. Shalaev, Optical Metamaterials: Fundamentals and Applications (Springer, New York, 2010).

Campbell, I. H.

I. H. Campbell and P. M. Fauchet, “The effects of microcrystal size and shape on the one phonon Raman spectra of crystalline semiconductors,” Solid State Commun. 58, 739–741 (1986).
[Crossref]

Cantele, G.

Cao, L.

L. Cao, B. Nabet, and J. E. Spanier, “Enhanced Raman scattering from individual semiconductor nanocones and nanowires,” Phys. Rev. Lett. 96, 157402 (2006).
[Crossref] [PubMed]

Carter, G. M.

Cazzanelli, M.

Chen, T. P.

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).
[Crossref]

Chen, X.

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]

Chou, C.-Y.

Claps, R.

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).
[Crossref] [PubMed]

Cohen, O.

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]

Crowe, I. F.

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]

Dadap, J.

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]

Dadap, J. I.

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]

Daldosso, N.

Daniel, B. A.

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]

Degiorgio, V.

Degoli, E.

Dimitropoulos, D.

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).
[Crossref] [PubMed]

Ding, L.

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).
[Crossref]

Dinu, M.

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

Dissanayake, C.

Dissanayake, C. M.

Dohnalova, K.

Downer, M. C.

Driscoll, J. B.

Dulkeith, E.

Eggleton, B. 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).
[Crossref]

Espinola, 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]

Espinola, R. L.

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]

Fathpour, S.

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]

Fauchet, P. M.

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. H. Campbell and P. M. Fauchet, “The effects of microcrystal size and shape on the one phonon Raman spectra of crystalline semiconductors,” Solid State Commun. 58, 739–741 (1986).
[Crossref]

Fedeli, J. M.

Ferraioli, L.

Ferrara, M. A.

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]

M. A. Ferrara, I. Rendina, and L. Sirleto, “Stimulated Raman scattering in quantum dots and nanocomposite silicon based materials,” in “Nonlinear Optics,” N. Kamanina, ed. (InTech, Rijeka, 2012), pp. 53–70.

Foster, M. A.

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]

Freude, W.

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

Fu, L.

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]

Fujii, M.

Fung, S.

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).
[Crossref]

Gaeta, A. L.

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]

Galan, J. V.

Garanovich, I. L.

Garcia, H.

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

Garcia-Ruperez, J.

Garrido, B.

Gass, M.

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]

Gautier, P.

Geis, M. W.

Giordana, E.

Green, W. M.

Green, W. M. J.

Grein, M. E.

Guider, R.

Hak, D.

Halsall, M.

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]

Harvey, A. J.

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]

Hayashi, S.

Hernandez, S.

Hessel, C.

Hillenbrand, R.

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

Houshmand, B.

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).
[Crossref] [PubMed]

Hsieh, I.

Hsieh, I.-W.

Hui, P. M.

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).
[Crossref]

Iadonisi, G.

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).
[Crossref]

Imakita, K.

Iori, F.

Ito, M.

Jacome, L.

Jalali, B.

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]

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

Jones, R.

Jordana, E.

Kashtiban, R. J.

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]

Keilmann, F.

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

Khriachtchev, L.

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]

Kivshar, Y. S.

Knights, A. P.

G. T. Reed and A. P. Knights, Silicon Photonics: An Introduction (John Wiley, Hoboken, 2004).
[Crossref]

Koos, C.

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

Krause, M.

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]

Lennon, D. M.

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]

Ley, L.

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

Liang, D.

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

Lin, Q.

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]

Lipson, M.

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]

Littler, I.

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]

Liu, A.

Liu, X.

Liu, Y.

H. K. Tsang and Y. Liu, “Nonlinear optical properties of silicon waveguides,” Semicond. Sci. Technol. 23, 064007 (2008).
[Crossref]

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).
[Crossref]

Lockwood, R.

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).

Lorenzo, J.

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]

Lyszczarz, T. M.

Magri, R.

Maly, P.

Manolatou, C.

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]

Marri, I.

Marti, J.

Martinez, A.

McNab, S.

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]

McNab, S. J.

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]

Meldrum, A.

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).

Mizohata, K.

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).
[Crossref]

Nabet, B.

L. Cao, B. Nabet, and J. E. Spanier, “Enhanced Raman scattering from individual semiconductor nanocones and nanowires,” Phys. Rev. Lett. 96, 157402 (2006).
[Crossref] [PubMed]

Naruiwa, R.

Negro, L. D.

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]

Neudert, K.

Ng, C. Y.

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).
[Crossref]

Nikitin, T.

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]

Ninno, D.

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).
[Crossref]

Novikov, S.

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]

Osgood, J. R. M.

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).
[Crossref]

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]

Ossicini, S.

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]

Passaro, V. M. N.

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 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, 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, “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]

Price, J.

Pulci, O.

Quochi, F.

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

Raday, O.

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]

Räsänen, M.

Reed, G. T.

G. T. Reed and A. P. Knights, Silicon Photonics: An Introduction (John Wiley, Hoboken, 2004).
[Crossref]

Rendina, I.

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]

Renner, H.

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]

Rong, H.

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 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, 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, “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]

Saarinen, J. J.

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]

Sainio, J.

Sanchis, P.

Schmidt, B. S.

Sekaric, L.

Shalaev, V.

W. Cai and V. Shalaev, Optical Metamaterials: Fundamentals and Applications (Springer, New York, 2010).

Sharping, J. E.

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).
[Crossref]

Sipe, J. E.

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).
[Crossref]

Spanier, J. E.

L. Cao, B. Nabet, and J. E. Spanier, “Enhanced Raman scattering from individual semiconductor nanocones and nanowires,” Phys. Rev. Lett. 96, 157402 (2006).
[Crossref] [PubMed]

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).
[Crossref]

Sukhorukov, A. A.

Tartara, L.

Taubner, T.

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

Trani, F.

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).
[Crossref]

Trojanek, F.

Tsang, H. K.

H. K. Tsang and Y. Liu, “Nonlinear optical properties of silicon waveguides,” Semicond. Sci. Technol. 23, 064007 (2008).
[Crossref]

Turner, A. C.

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]

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).
[Crossref] [PubMed]

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).
[Crossref]

Walters, R. J.

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

Wang, T.

Wang, Z. P.

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

Wei, J.

Williamson, R. C.

Xia, F.

Xu, S.

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]

Yin, L.

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]

Yoon, J. U.

Yu, J.

Zeng, X. C.

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).
[Crossref]

Zhang, J.

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]

Zhu, W.

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]

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)

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]

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]

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)

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]

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]

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).
[Crossref]

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).
[Crossref]

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).
[Crossref]

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).
[Crossref]

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)

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

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

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]

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)

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]

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]

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]

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]

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

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]

Opt. Express. (1)

Opt. Lett. (8)

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]

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]

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).
[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]

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]

Phys. Rev. B (4)

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).
[Crossref]

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).
[Crossref]

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).
[Crossref]

Phys. Rev. Lett. (1)

L. Cao, B. Nabet, and J. E. Spanier, “Enhanced Raman scattering from individual semiconductor nanocones and nanowires,” Phys. Rev. Lett. 96, 157402 (2006).
[Crossref] [PubMed]

Phys. Status Solidi (a) (1)

Physica Status Solidi (c) (1)

Semicond. Sci. Technol. (1)

H. K. Tsang and Y. Liu, “Nonlinear optical properties of silicon waveguides,” Semicond. Sci. Technol. 23, 064007 (2008).
[Crossref]

Solid State Commun. (2)

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

I. H. Campbell and P. M. Fauchet, “The effects of microcrystal size and shape on the one phonon Raman spectra of crystalline semiconductors,” Solid State Commun. 58, 739–741 (1986).
[Crossref]

Other (8)

G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, San Diego, 2007).

L. Pavesi and R. Turan, eds., Silicon Nanocrystals: Fundamentals, Synthesis and Applications (WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim, 2010).

L. Khriachtchev, ed., Silicon Nanophotonics: Basic Principles, Present Status and Perspectives (Pan Stanford, Singapore, 2009).

G. T. Reed and A. P. Knights, Silicon Photonics: An Introduction (John Wiley, Hoboken, 2004).
[Crossref]

L. Pavesi and D. Lockwood, eds., Silicon Photonics, vol. 94 of Topics in Applied Physics (Springer-Verlag, Berlin, 2004).

W. Cai and V. Shalaev, Optical Metamaterials: Fundamentals and Applications (Springer, New York, 2010).

R. W. Boyd, Nonlinear Optics, 3rd ed. (Academic Press, San Diego, 2008).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.

Click here to see a list of articles that cite this paper

Equations (52)

Equations on this page are rendered with MathJax. Learn more.

\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}} .