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.

© 2013 OSA

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2012 (8)

I. D. Rukhlenko, W. Zhu, M. Premaratne, and G. P. Agrawal, “Effective third-order susceptibility of silicon-nanocrystal-doped silica,” Opt. Express20, 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]

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]

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]

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]

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. Photoenergy2012, 254946 (2012).
[CrossRef]

2011 (4)

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. B29, 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]

I. D. Rukhlenko, M. Premaratne, and G. P. Agrawal, “Nonlinear propagation in silicon-based plasmonic waveguides from the standpoint of applications,” Opt. Express19, 206–217 (2011).
[CrossRef] [PubMed]

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)

D. Liang and J. E. Bowers, “Recent progress in lasers on silicon,” Nat. Photonics4, 511–517 (2010).
[CrossRef]

J. Leuthold, C. Koos, and W. Freude, “Nonlinear silicon photonics,” Nat. Photonics4, 535–544 (2010).
[CrossRef]

M. Paniccia, “Integrating silicon photonics,” Nat. Photonics4, 498–499 (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]

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. Express18, 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, 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. B27, 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]

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 Series241, 012097 (2010).
[CrossRef]

C. M. Dissanayake, M. Premaratne, I. D. Rukhlenko, and G. P. Agrawal, “FDTD modeling of anisotropic nonlinear optical phenomena in silicon waveguides,” Opt. Express18, 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)

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. Express17, 3941–3950 (2009).
[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. Express17, 2298–2318 (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).
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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. Express17, 11565–11581 (2009).
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Equations (52)

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× E ˜ μ ( 0 ) ( r , ω ) = i ω μ 0 H ˜ μ ( 0 ) ( r , ω )
× H ˜ μ ( 0 ) ( r , ω ) = i ω ε 0 ε L ( r , ω ) E ˜ μ ( 0 ) ( r , ω ) ,
E ˜ μ ( r , ω ) = ν a ˜ μ ν ( z , ω ω μ ) e μ ν ( r , ω μ ) N μ ν e i β μ ν z
H ˜ μ ( r , ω ) = ν a ˜ μ ν ( z , ω ω μ ) h μ ν ( r , ω μ ) N μ ν e i β μ ν z ,
( e μ ν * × h μ ν + c . c . ) d r = δ ν ν ( e μ ν * × h μ ν + c . c . ) d r = 4 N μ ν ,
P μ = 1 2 Re ( E ˜ μ × H ˜ μ * ) d r = ν | a ˜ μ ν | 2 .
× E ˜ μ ( r , ω ) = i ω μ 0 H ˜ μ ( r , ω )
× H ˜ μ ( r , ω ) = i ω ε 0 ε L ( r , ω ) E ˜ μ ( r , ω ) i ω P ˜ μ NL ( r , ω ) .
z ( E ˜ μ ( 0 ) × H ˜ μ * + E ˜ μ * × H ˜ μ ( 0 ) ) d r = ( E ˜ μ ( 0 ) × H ˜ μ * + E ˜ μ * × H ˜ μ ( 0 ) ) d r .
( E ˜ μ ( 0 ) × H ˜ μ * + E ˜ μ * × H ˜ μ ( 0 ) ) = i ω ( E ˜ μ ( 0 ) P ˜ μ N L * ) .
E ˜ μ ( 0 ) = e μ ν ( r , ω μ ) N μ ν e i β ν ( ω ) z and H ˜ μ ( 0 ) = h μ ν ( r , ω μ ) N μ ν e i β ν ( ω ) z
z { a ˜ μ ν ( z , ω ω μ ) e i [ β ν ( ω μ ) β ν ( ω ) ] z } = i ω 4 N μ ν e μ ν * ( r , ω μ ) P ˜ μ NL ( r , ω ) e i β ν ( ω ) z d r .
( z + i n = 1 1 n ! β ν ω | ω μ ( ω ω μ ) n ) a ˜ μ ν ( z , ω ω μ ) = i ω 4 N μ ν e μ ν * ( r , ω μ ) P ˜ μ NL ( r , ω ) e i β ν ( ω μ ) z d r .
( z + n = 1 i n + 1 n ! β ν ω | ω μ n t n ) a μ ν ( z , t ) = e i β μ ν z 4 N μ ν e μ ν * ( r , ω μ ) P μ NL ( r , t ) t e i ω μ t d r ,
a μ ν ( z , t ) = 1 2 π + a ˜ μ ν ( z , ω ) e i ω t d ω and P μ NL ( r , t ) = 1 2 π + P ˜ μ NL ( r , ω ) e i ω t d ω .
a μ ν z + n = 1 i n + 1 β μ ν ( n ) n ! n a μ ν t n = i ω μ 4 N μ ν ( 1 + i ω μ t ) e i β μ ν z ( e μ ν * 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 ( 8 + 7 ρ 45 ( δ k l δ m n + δ k m δ ln + δ k n δ l m ) + 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 = 1 f ( ε eff ε 1 ) 2 = [ ( 3 f 1 ) ε eff + ε 2 ] 2 f ( u 2 + 8 ε 1 ε 2 )
P ω μ K = ε 0 χ μ [ 8 + 7 ρ 45 ( 2 | E ω μ | 2 E ω μ + E ω μ 2 E ω μ * ) + 1 ρ 9 η E η 2 E η * η ^ ] ,
E ω μ = ν = x , y a μ ν e μ ν N μ ν e i β μ ν z ,
e i β μ ν z N μ ν ( e μ ν * P ω μ K ) d r = ε 0 χ μ [ 8 + 7 ρ 45 ( 2 a μ ν ν Γ ν ν ( μ ) | a μ ν | 2 + a μ ν * ν a μ ν 2 Λ ν ν ( μ ) e 2 i ( β μ ν β μ ν ) z ) + 1 ρ 9 Γ ν ν ( μ ) a μ ν | a μ ν | 2 ] ,
Γ ν ν ( μ ) = 1 N μ ν N μ ν | e μ ν | 2 | e μ ν | 2 d r
Λ ν ν ( μ ) = 1 N μ ν N μ ν e μ ν * 2 e μ ν 2 d r .
N μ ν = β μ ν 2 μ 0 ω μ | e μ ν | 2 d r .
P ω μ R ( r , t ) = e i ω μ t t d t 1 t d t 2 t d t 3 ε 0 χ R ( 3 ) ( t t 1 , t t 2 , t 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 ) + 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 ) ) ,
χ R ( 3 ) ( t 1 , t 2 , t 3 ) = 1 2 [ δ ( t 1 t 2 ) δ ( t 3 ) k l m n + δ ( t 1 ) δ ( t 2 t 3 ) k n m l ] ξ H ( t 2 ) ,
k l m n = 29 45 ( δ k m δ ln + δ k n δ l m ) 16 45 δ k l δ m n 2 9 δ k l δ l m δ m n ,
H ( t ) = 2 χ R 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 t d t 1 H ( t t 1 ) 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 ) .
e i β μ ν z ε 0 ξ N μ ν ( e μ ν * P ω μ R ) d r = 29 45 ν Λ ν ν μ μ exp ( i β μ ν μ ν μ ν μ ν z ) a μ ν ( t ) 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 + 29 45 ν Ψ ν ν μ μ exp ( i β μ ν μ ν μ ν μ ν z ) a μ ν ( t ) 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 16 45 ν Γ ν ν μ μ a μ ν ( t ) 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 2 9 Λ ν ν μ μ a μ ν t 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 ,
Λ ν ν μ μ = ( e μ ν * e μ ν * ) ( e μ ν e μ ν ) N μ ν N μ ν N μ ν N μ ν d r ,
Ψ ν ν μ μ = ( e μ ν * e μ ν ) ( e μ ν e μ ν * ) N μ ν N μ ν N μ ν N μ ν d r ,
Γ ν ν μ μ = 1 N μ ν N μ ν | 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 = n eff n 1 = ( ε 1 ε eff ) 1 / 2 ( 3 f 1 ) ε eff + ε 2 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 ,
e i β μ ν z N μ ν ( e μ ν * 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 ,
N t = N τ c μ 1 2 h ¯ ω μ A eff P μ z ,
P μ z = ν ( a μ ν a μ ν * z + a μ ν * a μ ν z ) = 1 4 ξ c 2 ε 0 2 ε eff β TPA ν ( 8 + 7 ρ 45 ν 2 Γ ν ν ( μ ) | a μ ν | 2 | a μ ν | 2 + 13 + 2 ρ 45 Γ ν ν ( μ ) | a μ ν | 4 ) .
A eff μ ν = ( | e μ ν | 2 d r ) 2 / | e μ ν | 4 d r .
A eff = μ , ν ( A eff μ ν ) 1 / 4 .
𝒜 eff μ ν = 𝒜 NL | e μ ν | 2 d r / NL | e μ ν | 2 d r ,
a μ ν z + n = 1 i n + 1 β μ ν ( n ) n ! n a μ ν t n + α μ ν 2 a μ ν = 1 8 ξ c 2 ε 0 2 ε eff ( β TPA 2 i n 2 k μ ) ( 8 + 7 ρ 45 2 Γ ν ν ( μ ) | a μ ν | 2 + 29 + 16 ρ 45 Γ ν ν ( μ ) | a μ ν | 2 ) a μ ν + 8 i ε 0 ω μ 45 ξ Γ ν ν μ μ a μ ν ( t ) 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 4 i ε 0 ω μ 45 ξ Γ ν ν μ μ a μ ν ( t ) 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 n eff n μ ν ( ω 0 ω μ ) 2 ( i σ n k μ ( 1 + ς N 0.2 ) + σ α 2 N 0.2 ) N 0.8 a μ ν ,
N t = N τ c + ξ c 2 ε 0 2 ε eff 4 A eff μ , ν β TPA 2 h ¯ ω μ ( 8 + 7 ρ 45 2 Γ ν ν ( μ ) | a μ ν | 2 + 29 + 16 ρ 45 Γ ν ν ( μ ) | a μ ν | 2 ) | a μ ν | 2 .
4 i ε 0 ω μ 45 ξ 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 ,
H ˜ ( ω ) = 0 H ( t ) e i ω t d t = 2 χ R Γ R Ω R Ω R 2 + 2 i Γ R ω ω 2
ln a μ ν z + α μ ν 2 = ξ ( β TPA 2 i n 2 k μ ) ( 8 + 7 ρ 45 2 γ ν ν ( μ ) I μ ν + 29 + 16 ρ 45 γ ν ν ( μ ) I μ ν ) + 32 45 ξ 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 n eff n μ ν ( ω 0 ω μ ) 2 ( σ α 2 + i σ ¯ n k μ ) × μ , ν β TPA 2 h ¯ ω μ ( 8 + 7 ρ 45 2 γ ν ν ( μ ) I μ ν + 29 + 16 ρ 45 γ ν ν ( μ ) I μ ν ) I μ ν ,
γ ν ν μ μ = n eff 2 n μ ν n μ ν A eff | e μ ν | 2 | e μ ν | 2 d r | e μ ν | 2 d r | e μ ν | 2 d r ,
g ˜ R ( ω ) = 2 i g R Γ R Ω R Ω R 2 + 2 i Γ R ω ω 2 .

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