## Theory of nonlinear pulse propagation in silicon-nanocrystal waveguides

Optics Express, Vol. 21, Issue 3, pp. 2832-2846 (2013)

http://dx.doi.org/10.1364/OE.21.002832

Acrobat PDF (924 KB)

### 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

## 1. Introduction

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

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

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

9. 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]

11. 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]

12. 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]

15. 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]

16. 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]

22. 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]

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

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

27. 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]

28. 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]

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

_{2}matrix [36

36. 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]

40. 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]

_{2}composite may be anywhere between 1.45 and 2.2 depending on the excess of silicon, this composite can confine and guide light better than the pure silica fibers do [41

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

_{2}composite may exceed those of silicon by factors of 100 and 10000 [37, 43

43. 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]

44. 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]

^{3+}ions doping the SiO

_{2}matrix, one can realize the desired emission from the NCs and thus create from them a promising material for light emitting devices [39, 45

45. 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]

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

_{2}composite will eventually replace the silicon-on-insulator (SOI) technology in creating the key elements of silicon photonics.

_{2}composite over the past years [12

12. 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]

43. 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]

47. T. Nikitin, R. Velagapudi, J. Sainio, J. Lahtinen, M. Räsänen, S. Novikov, and L. Khriachtchev, “Optical and structural properties of SiO* _{x}* films grown by molecular beam deposition: Effect of the Si concentration and annealing temperature,” J. Appl. Phys.

**112**, 094316–094316 (2012). [CrossRef]

50. L. Ding, T. P. Chen, Y. Liu, C. Y. Ng, and S. Fung, “Optical properties of silicon nanocrystals embedded in a SiO_{2} matrix,” Phys. Rev. B **72**, 125419 (2005). [CrossRef]

36. 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]

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

51. 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]

52. 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]

3. R. M. Osgood Jr., 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]

4. 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]

53. 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]

54. 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]

62. 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]

_{2}composite. The demand for such a platform is even higher than it was for silicon, as it is extremely computationally challenging to solve Maxwell’s equations for a macroscopic sample of the composite with hundreds of thousands of tiny NCs treated individually.

_{2}composite [36

36. 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]

_{2}composite. The free-carrier contribution depends on the generation rate of electron–hole pairs inside the NCs due to the effect of two-photon absorption, whose efficiency is governed by field intensity and thus depends on the effective mode area (EMA) of the waveguide. The ambiguity in defining EMA is emphasized in Section 4, where we provide two alternative definitions of EMA and explain the often ignored difference between them. The results of Sections 2–4 are merged in Section 5 to give the desired coupled amplitude equations governing the nonlinear propagation of the two pulses. These equations are then simplified for the case of continuous waves, before Section 6 summarizes our results and concludes the paper.

## 2. Nonlinear propagation equations

*μ*=

*p*) and Stokes (

*μ*=

*s*) pulses, whose spectra are centered around frequencies

*ω*and

_{p}*ω*. In this case, the unperturbed propagating modes of the linear waveguide are the solutions to Maxwell’s equations for the pump and Stokes fields and where

_{s}*ε*(

_{L}**r**

_{⊥},

*ω*) is the transverse profile of linear permittivity and

**r**

_{⊥}is the two-dimensional radius vector perpendicular to the waveguide axis

*z*.

**e**

*(*

_{μν}**r**

_{⊥},

*ω*)

_{μ}*e*and

^{iβμνz}**h**

*(*

_{μν}**r**

_{⊥},

*ω*)

_{μ}*e*satisfying Eq. (1) as and where

^{iβμνz}*β*≡

_{μν}*β*(

_{ν}*ω*) is the real propagation constant of mode

_{μ}*ν*evaluated at the carrier frequency

*ω*of field

_{μ}*μ*and we assume the modal profiles

**e**

*and*

_{μν}**h**

*to be normalized according to the condition where the integration is taken over the entire area of the*

_{μν}*xy*plane and the constants

*N*are implicitly defined. With this normalization, one finds that the total power carried by field

_{μν}*μ*is given by As in Eq. (3), the double integral here is evaluated over the entire transverse plane.

**f**×

**g**) =

**g**· (∇ ×

**f**) −

**f**· (∇ ×

**g**), together with Eqs. (1) and (5), it is possible to express the gradient in Eq. (6) by means of the scalar product By introducing this expression into Eq. (6), substituting in the resulting relation both the unperturbed solutions and the perturbed ones given in Eq. (2), we find using Eq. (3) that The evaluation of the derivative in this equation gives

*e*

^{−i(ω−ωμ)t}and integrating with respect to

*ω*, we arrive at the following time-domain equation: in which we have set

*n*th-order dispersion parameter at the frequency

*ω*. The term with the time derivative on the right-hand side of this equation accounts for the effect of self steepening, which is significant for ultrashort optical pulses [64].

_{μ}*a*of pump and Stokes fields is due to the nonlinear material polarization of Si-NCs/SiO

_{μν}_{2}waveguide. This equation is similar to those derived in Refs. [59

59. 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]

63. 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]

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

## 3. Nonlinear polarization of Si-NCs/SiO_{2} waveguide

_{2}waveguide may originate from the non-linearities of both silicon crystallites and silica matrix. Since the nonlinear effects in Si NCs are typically much stronger than those in SiO

_{2}, one may safely neglect the third-order susceptibility of silica, provided the volume fraction of the NCs is larger than 0.1% [36

**20**, 26275–26284 (2012). [CrossRef] [PubMed]

3. R. M. Osgood Jr., 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]

4. 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]

53. 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]

69. 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]

12. 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]

17. 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]

19. 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]

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

*via*TPA also change the refractive index of the NCs, which is usually referred to as the effect of free-carrier dispersion (FCD), and result in the free-carrier absorption (FCA) of the pump and Stokes fields [19

19. 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]

29. 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]

### 3.1. Electronic contribution

_{2}waveguide with randomly oriented nanocrystals is given by the tensor product where

**E**

_{ωμ}(

**r**,

*t*) is the slowly varying amplitude of the electric field, which may be calculated by Fourier-transforming Eq. (2), and the third-order susceptibility tensor is of the form [36

**20**, 26275–26284 (2012). [CrossRef] [PubMed]

*k*,

*l*,

*m*} = {

*x*,

*y*,

*z*},

*δ*is the Kronecker delta, and

_{ij}*ρ*is the nonlinear anisotropy factor (

*ρ*≈ 1.27 near the 1.55

*μ*m wavelength). Owing to a uniform distribution of Si-NC orientations in space, the Kerr tensor of the Si-NCs/SiO

_{2}composite is no longer described with respect to the crystallographic basis, but rather is given in the reference frame associated with the nonlinear waveguide.

4. 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]

56. 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]

69. 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]

*n*

_{2}and

*β*

_{TPA}are the nonlinear Kerr parameter and TPA coefficient of a Si NC,

*k*=

_{μ}*ω*/

_{μ}*c*, and is the third-order susceptibility attenuation factor [36

**20**, 26275–26284 (2012). [CrossRef] [PubMed]

*E*is the

_{η}*η*th component of the vector

**E**

_{ωμ}and

*η*̂ is the unit vector in the direction of the

*η*th Cartesian axis.

**e**

*and*

_{μ}_{x}**e**

*) in the expansion of Eq. (2), assuming them to be polarized along the*

_{μ}_{y}*x*and

*y*axes. In this case, the orthogonality relations

### 3.2. Raman contribution

_{2}waveguide with randomly oriented nanocrystals can be represented as

*μ*,

*μ*′} = {

*p*,

*s*},

*μ*≠

*μ*′, and the effective susceptibility tensor is given by the expressions [3

3. R. M. Osgood Jr., 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]

**15**, 16604–16644 (2007). [CrossRef] [PubMed]

**20**, 26275–26284 (2012). [CrossRef] [PubMed]

69. 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]

*χ*is the peak value of the Raman susceptibility of an individual Si NC, 2Γ

_{R}*= 2/*

_{R}*τ*

_{2}is the amplification bandwidth, and

_{2}composite have shown [43

43. 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]

^{−17}m

^{2}/V

^{2}) [3

**1**, 162–235 (2009). [CrossRef]

*ξ*in the composite, this gives us an estimate

*ξχ*∼ 10

_{R}^{−13}m

^{2}/V

^{2}. The peak position and bandwidth of gain spectrum in Si NCs also change significantly with respect to their values 2Γ

*≈ 105 GHz and Ω*

_{R}*= 15.6 THz in bulk silicon. In particular, for spherical NCs of 2 nm in diameter, the gain peak broadens up to 2 THz and red shifts by about 0.6 THz [72*

_{R}72. 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]

*et al.*[74

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

75. 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]

76. 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]

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

*reststrahlen*band of phonon-polariton dispersion and is analogous to the plasmon-induced field enhancement in the vicinity of metallic nanoparticles.

*μ*≠

*μ*′,

*ω*

_{μμ}_{′}=

*ω*−

_{μ}*ω*

_{μ}_{′}, and

### 3.3. Free-carrier effects

*n*=

_{μν}*β*/

_{μν}*k*.

_{μ}*∂P*/

_{μ}*∂z*is the rate of mode

*μ*power dissipation due to the TPA and

*τ*is the effective free-carrier lifetime. The same average effective mode area (EMA),

_{c}*A*

_{eff}, for all modes have been assumed in this equation for the sake of simplicity. As will be discussed below, this assumption has to be abandoned if one needs to write the propagation equations in terms of the average field intensities inside the Si-NCs/SiO

_{2}composite.

*n*≠

_{μν}*n*

_{μν}_{′}) and do not contribute to the TPA-induced power dissipation.

## 4. Effective mode area

*ν*at

*ω*in the common fashion as [4

_{μ}**15**, 16604–16644 (2007). [CrossRef] [PubMed]

10. 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]

*a*|

_{μν}^{2}/

*A*

_{eff}are not modal intensities. In order to treat the nonlinear propagation in terms of the average mode intensities inside the nonlinear composite, one needs to use a different EMA [51

51. 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]

_{NL}is the cross section area of the nonlinear core of the waveguide (filled with Si-NCs/SiO

_{2}composite) and the symbols NL and ∞ denote integrations over the nonlinear core and the entire transverse plane, respectively. It is easy to see that

*μ*at

*ω*, i.e., the ratio of the power carried by this field through the waveguide core to the core area 𝒜

_{ν}_{NL}.

## 5. Coupled amplitude equations

*a*and free-carrier density

_{μν}*N*. These equations may be simplified by neglecting the effects of self steepening and assuming that no phase matching occurs between the waves of different frequencies and polarizations. In this case, the coupled amplitude equations acquire the form of the generalized nonlinear Schrödinger equation

*ν*=

*ν*′ and explicitly written out the remaining terms with

*ν*′ ≠

*ν*. We have also employed the identity

*α*/2, which account for waveguide losses through the linear absorption coefficients

_{μν}a_{μν}*a*.

_{μν}**1**, 162–235 (2009). [CrossRef]

56. 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]

59. 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]

63. 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]

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

79. 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]

*ξ*and

*ζ*entering Eqs. (38) and (39) are responsible for the weakening of the nonlinear effects of Si NCs due to their embedment in fused silica. For small filling factors (

*f*≲ 0.01), the extent of weakening may be estimated using the approximate expressions

*ξ*≈ (3

*ε*

_{2})

^{4}

*f*/(

*ε*

_{1}+ 2

*ε*

_{2})

^{4}≈ 0.022

*f*and

*β*

_{TPA},

*n*

_{2},

*χ*,

_{R}*σ*, and

_{n}*σ*are the characteristics of individual nanocrystals, whereas the products

_{α}*ξβ*

_{TPA},

*ξn*

_{2},

*ξχ*,

_{R}*ζσ*, and

_{n}*ζσ*characterize the Si-NCs/SiO

_{α}_{2}composite as a whole. Second, the strengths of self-phase modulation and TPA are reduced by about 45/(8 + 7

*ρ*) ≈ 2.7 times, while the effects of cross-phase modulation and cross-TPA are enhanced by a factor of (26+16

*ρ*)/45 ≈ 1.1, as compared to their values in SOI waveguides fabricated along the [01̄1] direction [3

**1**, 162–235 (2009). [CrossRef]

**15**, 16604–16644 (2007). [CrossRef] [PubMed]

79. 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]

80. 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]

_{2}composite and is completely absent in the equations governing the nonlinear propagation through silicon. It should be also emphasized that the coupled amplitude equations written in a coordinate basis aligned with the waveguide edges are uniform, as the information on crystallographic directions in silicon is completely lost due to the random orientation of the crystallites in the composite.

_{2}waveguides and constitute the main result of this paper.

### 5.1. A continuous-wave regime

*n*

_{FC}≈ −

*σ̄*(

_{n}*ω*

_{0}/

*ω*)

_{μ}^{2}

*N*, with

*σ*̄

*= 5.3 × 10*

_{n}^{−27}m

^{3}[4

**15**, 16604–16644 (2007). [CrossRef] [PubMed]

^{22}to 10

^{23}m

^{−3}[the measurements of Spano

*et al.*[71

71. 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]

*ζσ̄*= (1.2 ± 0.3) × 10

_{n}^{−28}m

^{3}for

*f*= 8%]. Some algebra shows that Eqs. (38) and (39) in this case yield

*I*= |

_{μν}*a*|

_{μν}^{2}/

*A*

_{eff},

*ω*−

_{p}*ω*= Ω

_{s}*, the last expression gives*

_{R}*g*̃

*(Ω*

_{R}*) =*

_{R}*g*.

_{R}*I*entering Eq. (42) do not represent modal intensities, as they are defined through the average EMA of the waveguide. One may rewrite the propagation equations in terms of the average field intensities ℐ

_{μν}*, by simply replacing*

_{μν}*A*

_{eff}with

_{2}waveguides.

## 6. Conclusions

## Acknowledgments

## References and links

1. | R. Soref and J. Lorenzo, “All-silicon active and passive guided-wave components for |

2. | J. Leuthold, C. Koos, and W. Freude, “Nonlinear silicon photonics,” Nat. Photonics |

3. | R. M. Osgood Jr., 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 |

4. | Q. Lin, O. J. Painter, and G. P. Agrawal, “Nonlinear optical phenomena in silicon waveguides: Modeling and applications,” Opt. Express |

5. | R. A. Soref, “The past, present, and future of silicon photonics,” IEEE J. Sel. Top. Quantum Electron. |

6. | G. T. Reed and A. P. Knights, |

7. | L. Pavesi and D. Lockwood, eds., |

8. | M. Paniccia, “Integrating silicon photonics,” Nat. Photonics |

9. | 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 |

10. | C. Koos, L. Jacome, C. Poulton, J. Leuthold, and W. Freude, “Nonlinear silicon-on-insulator waveguides for all-optical signal processing,” Opt. Express. |

11. | 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. |

12. | 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. |

13. | C. Manolatou and M. Lipson, “All-optical silicon modulators based on carrier injection by two-photon absorption,” J. Lightwave Technol. |

14. | 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 |

15. | 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. |

16. | 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 |

17. | M. Krause, H. Renner, and E. Brinkmeyer, “Silicon Raman amplifiers with ring-resonator-enhanced pump power,” IEEE J. Sel. Top. Quantum Electron. |

18. | I. D. Rukhlenko, C. Dissanayake, M. Premaratne, and G. P. Agrawal, “Maximization of net optical gain in silicon-waveguide Raman amplifiers,” Opt. Express |

19. | 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. |

20. | M. Krause, H. Renner, S. Fathpour, B. Jalali, and E. Brinkmeyer, “Gain enhancement in cladding-pumped silicon Raman amplifiers,” IEEE J. Quantum Electron. |

21. | 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 |

22. | 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. |

23. | D. Liang and J. E. Bowers, “Recent progress in lasers on silicon,” Nat. Photonics |

24. | H. Rong, S. Xu, O. Cohen, O. Raday, M. Lee, V. Sih, and M. Paniccia, “A cascaded silicon Raman laser,” Nat. Photonics |

25. | M. Krause, H. Renner, and E. Brinkmeyer, “Analysis of Raman lasing characteristics in silicon-on-insulator waveguides,” Opt. Express |

26. | O. Boyraz and B. Jalali, “Demonstration of a silicon Raman laser,” Opt. Express |

27. | M. W. Geis, S. J. Spector, M. E. Grein, J. U. Yoon, D. M. Lennon, and T. M. Lyszczarz, “Silicon waveguide infrared photodiodes with |

28. | 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 |

29. | I. D. Rukhlenko, M. Premaratne, and G. P. Agrawal, “Analytical study of optical bistability in silicon-waveguide resonators,” Opt. Express |

30. | H. K. Tsang and Y. Liu, “Nonlinear optical properties of silicon waveguides,” Semicond. Sci. Technol. |

31. | M. A. Foster, A. C. Turner, M. Lipson, and A. L. Gaeta, “Nonlinear optics in photonic nanowires,” Opt. Express |

32. | 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 |

33. | R. Espinola, J. Dadap, R. Osgood Jr., S. McNab, and Y. Vlasov, “C-band wavelength conversion in silicon photonic wire waveguides,” Opt. Express |

34. | M. W. Geis, S. J. Spector, R. C. Williamson, and T. M. Lyszczarz, “Submicrosecond submilliwatt silicon-on-insulator thermooptic switch,” IEEE Photon. Technol. Lett. |

35. | M. Dinu, F. Quochi, and H. Garcia, “Third-order nonlinearities in silicon at telecom wavelengths,” Appl. Phys. Lett. |

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

37. | L. Pavesi and R. Turan, eds., |

38. | L. Khriachtchev, ed., |

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

40. | 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 |

41. | I. D. Rukhlenko and M. Premaratne, “Optimization of nonlinear performance of silicon-nanocrystal cylindrical nanowires,” IEEE Photonics J. |

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

43. | L. Sirleto, M. A. Ferrara, T. Nikitin, S. Novikov, and L. Khriachtchev, “Giant Raman gain in silicon nanocrystals,” Nat. Commun. |

44. | 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. |

45. | 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 |

46. | R. J. Walters, G. I. Bourianoff, and H. A. Atwater, “Field-effect electroluminescence in silicon nanocrystals,” Nat. Mater. |

47. | T. Nikitin, R. Velagapudi, J. Sainio, J. Lahtinen, M. Räsänen, S. Novikov, and L. Khriachtchev, “Optical and structural properties of SiO 112, 094316–094316 (2012). [CrossRef] |

48. | J. Wei, J. Price, T. Wang, C. Hessel, and M. C. Downer, “Size-dependent optical properties of Si nanocrystals embedded in amorphous SiO |

49. | 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) |

50. | L. Ding, T. P. Chen, Y. Liu, C. Y. Ng, and S. Fung, “Optical properties of silicon nanocrystals embedded in a SiO |

51. | I. D. Rukhlenko, M. Premaratne, and G. P. Agrawal, “Effective mode area and its optimization in silicon-nanocrystal waveguides,” Opt. Lett. |

52. | 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 |

53. | 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. |

54. | I. D. Rukhlenko, M. Premaratne, and G. P. Agrawal, “Nonlinear propagation in silicon-based plasmonic waveguides from the standpoint of applications,” Opt. Express |

55. | I. D. Rukhlenko, M. Premaratne, and G. P. Agrawal, “Analytical study of optical bistability in silicon ring resonators,” Opt. Lett. |

56. | B. A. Daniel and G. P. Agrawal, “Vectorial nonlinear propagation in silicon nanowire waveguides: Polarization effects,” J. Opt. Soc. Am. B |

57. | 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. |

58. | 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. |

59. | 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 |

60. | L. Yin, J. Zhang, P. M. Fauchet, and G. P. Agrawal, “Optical switching using nonlinear polarization rotation inside silicon waveguides,” Opt. Lett. |

61. | I. D. Rukhlenko, M. Premaratne, C. Dissanayake, and G. P. Agrawal, “Nonlinear pulse evolution in silicon waveguides: An approximate analytic approach,” J. Lightwave Technol. |

62. | D. Dimitropoulos, B. Houshmand, R. Claps, and B. Jalali, “Coupled-mode theory of Raman effect in silicon-on-insulator waveguides,” Opt. Lett. |

63. | 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 |

64. | R. W. Boyd, |

65. | X. Chen, N. C. Panoiu, and R. M. Osgood Jr., “Theory of Raman-mediated pulsed amplification in silicon-wire waveguides,” IEEE J. Quantum Electron. |

66. | 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. |

67. | X. C. Zeng, D. J. Bergman, P. M. Hui, and D. Stroud, “Effective-medium theory for weakly nonlinear composites,” Phys. Rev. B |

68. | W. Cai and V. Shalaev, |

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

70. | F. Trojanek, K. Neudert, K. Zidek, K. Dohnalova, I. Pelant, and P. Maly, “Femtosecond photoluminescence spectroscopy of silicon nanocrystals,” Physica Status Solidi (c) |

71. | 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 |

72. | M. A. Ferrara, I. Rendina, S. N. Basu, L. D. Negro, and L. Sirleto, “Raman amplifier based on amorphous silicon nanoparticles,” Int. J. Photoenergy |

73. | M. A. Ferrara, I. Rendina, and L. Sirleto, “Stimulated Raman scattering in quantum dots and nanocomposite silicon based materials,” in “ |

74. | H. Richter, Z. P. Wang, and L. Ley, “The one phonon Raman spectrum in microcrystalline silicon,” Solid State Commun. |

75. | 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. |

76. | L. Cao, B. Nabet, and J. E. Spanier, “Enhanced Raman scattering from individual semiconductor nanocones and nanowires,” Phys. Rev. Lett. |

77. | R. Hillenbrand, T. Taubner, and F. Keilmann, “Phonon-enhanced light-matter interaction at the nanometre scale,” Nature |

78. | G. P. Agrawal, |

79. | I. D. Rukhlenko, M. Premaratne, and G. P. Agrawal, “Nonlinear silicon photonics: Analytical tools,” IEEE J. Sel. Top. Quantum Electron. |

80. | 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. |

**OCIS Codes**

(160.4330) Materials : Nonlinear optical materials

(190.0190) Nonlinear optics : Nonlinear optics

(190.4400) Nonlinear optics : Nonlinear optics, materials

(260.2065) Physical optics : Effective medium theory

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: December 14, 2012

Revised Manuscript: January 15, 2013

Manuscript Accepted: January 16, 2013

Published: January 29, 2013

**Citation**

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

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-3-2832

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### References

- 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. Photonics4, 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. Photonics1, 162–235 (2009). [CrossRef]
- Q. Lin, O. J. Painter, and G. P. Agrawal, “Nonlinear optical phenomena in silicon waveguides: Modeling and applications,” Opt. Express15, 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. Photonics4, 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. Express16, 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. Express13, 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. 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, C. Dissanayake, M. Premaratne, and G. P. Agrawal, “Maximization of net optical gain in silicon-waveguide Raman amplifiers,” Opt. Express17, 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,” Nature441, 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. Photonics4, 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. Photonics2, 170–174 (2008). [CrossRef]
- M. Krause, H. Renner, and E. Brinkmeyer, “Analysis of Raman lasing characteristics in silicon-on-insulator waveguides,” Opt. Express12, 5703–5710 (2004). [CrossRef] [PubMed]
- O. Boyraz and B. Jalali, “Demonstration of a silicon Raman laser,” Opt. Express12, 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. Express17, 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. Express17, 12987–12999 (2009). [CrossRef] [PubMed]
- I. D. Rukhlenko, M. Premaratne, and G. P. Agrawal, “Analytical study of optical bistability in silicon-waveguide resonators,” Opt. Express17, 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. Express16, 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. Express15, 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. Express13, 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. Express20, 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. B76, 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 Series241, 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. 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]
- 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. B72, 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. B75, 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. Express19, 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. 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]
- 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). [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. Express17, 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. B38, 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. Express18, 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. Express17, 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. Photoenergy2012, 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,” Nature418, 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]

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