## Nonlinear optical phenomena in silicon waveguides: Modeling and applications

Optics Express, Vol. 15, Issue 25, pp. 16604-16644 (2007)

http://dx.doi.org/10.1364/OE.15.016604

Acrobat PDF (921 KB)

### Abstract

Several kinds of nonlinear optical effects have been observed in recent years using silicon waveguides, and their device applications are attracting considerable attention. In this review, we provide a unified theoretical platform that not only can be used for understanding the underlying physics but should also provide guidance toward new and useful applications. We begin with a description of the third-order nonlinearity of silicon and consider the tensorial nature of both the electronic and Raman contributions. The generation of free carriers through two-photon absorption and their impact on various nonlinear phenomena is included fully within the theory presented here. We derive a general propagation equation in the frequency domain and show how it leads to a generalized nonlinear Schrödinger equation when it is converted to the time domain. We use this equation to study propagation of ultrashort optical pulses in the presence of self-phase modulation and show the possibility of soliton formation and supercontinuum generation. The nonlinear phenomena of cross-phase modulation and stimulated Raman scattering are discussed next with emphasis on the impact of free carriers on Raman amplification and lasing. We also consider the four-wave mixing process for both continuous-wave and pulsed pumping and discuss the conditions under which parametric amplification and wavelength conversion can be realized with net gain in the telecommunication band.

© 2007 Optical Society of America

## 1. Introduction

3. R. A. Soref, “The Past, Present, and Future of Silicon Photonics,” IEEE J. Sel. Top. Quantum Electron. **12**, 1678–1687 (2006). [CrossRef]

*µ*m and exhibit optical properties that are useful for a variety of applications [4

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

6. R. Claps, D. Dimitropoulos, V. Raghunathan, Y. Han, and B. Jalali, “Observation of stimulated Raman amplification in silicon waveguides,” Opt. Express **11**, 1731–1739 (2003). [CrossRef] [PubMed]

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

23. 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 Jr., “Supercontinuum generation in silicon photonic wires,” Opt. Express **15**, 15242–15248 (2007). [CrossRef] [PubMed]

14. R. Dekker, A. Driessen, T. Wahlbrink, C. Moormann, J. Niehusmann, and M. Först, “Ultrafast Kerr-induced all-optical wavelength conversion in silicon waveguides using 1.55 *µ*m femtosecond pulses,” Opt. Express **14**, 8336–8346 (2006). [CrossRef] [PubMed]

24. A. Hache and M. Bourgeois, “Ultrafast all-optical switching in a silicon-based photonic crystal,” Appl. Phys. Lett. **77**, 4089–4091 (2000). [CrossRef]

26. I-W. Hsieh, X. Chen, J. I. Dadap, N. C. Panoiu, R. M. Osgood Jr., S. J. McNab, and Y. A. Vlasov, “Cross-phase modulation-induced spectral and temporal effects on co-propagating femtosecond pulses in silicon photonic wires,” Opt. Express **15**, 1135–1146 (2007). [CrossRef] [PubMed]

27. R. Claps, D. Dimitropoulos, Y. Han, and B. Jalali, “Observation of Raman emission in silicon waveguide at 1.54 *µ*m,” Opt. Express **10**, 1305–1313 (2002). [PubMed]

60. V. Raghunathan, D. Borlaug, R. R. Rice, and B. Jalali, “Demonstration of a mid-infrared silicon Raman amplifier,” Opt. Express **15**, 14355–14362 (2007). [CrossRef] [PubMed]

61. R. Claps, V. Raghunathan, D. Dimitropoulos, and B. Jalali, “Anti-Sotkes Raman conversion in silicon waveguides,” Opt. Express **11**, 2862–2872 (2003). [CrossRef] [PubMed]

79. M. A. Foster, A. C. Turner, R. Salem, M. Lipson, and A. L. Gaeta, “Broad-band continuous-wave parametric wavelength conversion in silicon nanowaveguides,” Opt. Express **15**, 12949–12958 (2007). [CrossRef] [PubMed]

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

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

81. R. A. Soref and B. R. Bennett, “Electrooptical effects in silicon,” IEEE J. Quantum Electron. **23**, 123–129 (1987). [CrossRef]

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

82. V. Raghunathan, R. Shori, O. M. Stafsudd, and B. Jalali, “Nonlinear absorption in silicon and the prospects of mid-infrared silicon Raman lasers,” Physica Status Solidi A **203**, R38–R40 (2006). [CrossRef]

87. J. Zhang, Q. Lin, G. Piredda, R. W. Boyd, G. P. Agrawal, and P. M. Fauchet, “Anisotropic nonlinear response of silicon in the near-infrared region,” Appl. Phys. Lett. **90**, 071113 (2007). [CrossRef]

8. O. Boyraz, T. Indukuri, and B. Jalali, “Self-phase-modulation induced spectral broadening in silicon waveguides,” Opt. Express **12**, 829–834 (2004). [CrossRef] [PubMed]

25. Ö. Boyraz, P. Koonath, V. Raghunathan, and B. Jalali, “All optical switching and continuum generation in silicon waveguides,” Opt. Express **12**, 4094–4102 (2004). [CrossRef] [PubMed]

88. P. E. Barclay, K. Srinivasan, and O. Painter, “Nonlinear response of silicon photonic crystal microresonators excited via an integrated waveguide and fiber taper”, Opt. Express **13**, 801–820 (2005). [CrossRef] [PubMed]

89. T. J. Johnson, M. Borselli, and O. Painter, “Self-induced optical modulation of the transmission through a high-Q silicon microdisk resonator”, Opt. Express **14**, 817–831 (2006). [CrossRef] [PubMed]

90. V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, “All-optical control of light on a silicon chip,” Nature **431**, 1081–1084 (2004). [CrossRef] [PubMed]

100. T. K. Liang, L. R. Nunes, M. Tsuchiya, K. S. Abedin, T. Miyazaki, D. Van Thourhout, W. Bogaerts, P. Dumon, R. Baets, and H. K. Tsang, “High speed logic gate using two-photon absorption in silicon waveguides,” Opt. Commun. **265**, 171–174 (2006). [CrossRef]

25. Ö. Boyraz, P. Koonath, V. Raghunathan, and B. Jalali, “All optical switching and continuum generation in silicon waveguides,” Opt. Express **12**, 4094–4102 (2004). [CrossRef] [PubMed]

32. A. Liu, H. Rong, M. Paniccia, O. Cohen, and D. Hak, “Net optical gain in a low loss silicon-on-insulator waveguide by stimulated Raman scattering,” Opt. Express **12**, 4261–4268 (2004). [CrossRef] [PubMed]

33. H. Rong, A. Liu, R. Nicolaescu, M. Paniccia, O. Cohen, and D. Hak, “Raman gain and nonlinear optical absorption measurement in a low-loss silicon waveguide,” Appl. Phys. Lett. **85**, 2196–2198 (2004). [CrossRef]

35. T. K. Liang and H. K. Tsang, “Role of free carriers from two-photon absorption in Raman amplification in silicon-on-insulator waveguides,” Appl. Phys. Lett. **84**, 2745–2747 (2004). [CrossRef]

37. T. K. Liang and H. K. Tsang, “Nonlinear absorption and Raman scattering in silicon-on-insulator optical waveguides,” IEEE J. Quantum Electron. **10**, 1149–1153 (2004). [CrossRef]

39. 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]

65. H. Fukuda, K. Yamada, T. Shoji, M. Takahashi, T. Tsuchizawa, T. Watanabe, J. Takahashi, and S. Itabashi, “Four-wave mixing in silicon wire waveguides,” Opt. Express **13**, 4629–4637 (2005). [CrossRef] [PubMed]

68. Q. Lin, J. Zhang, P. M. Fauchet, and G. P. Agrawal, “Ultrabroadband parametric generation and wavelength conversion in silicon waveguides,” Opt. Express **14**, 4786–4799 (2006). [CrossRef] [PubMed]

89. T. J. Johnson, M. Borselli, and O. Painter, “Self-induced optical modulation of the transmission through a high-Q silicon microdisk resonator”, Opt. Express **14**, 817–831 (2006). [CrossRef] [PubMed]

13. L. Yin, Q. Lin, and G. P. Agrawal, “Dispersion tailoring and soliton propagation in silicon waveguides,” Opt. Lett. **31**, 1295–1297 (2006). [CrossRef] [PubMed]

16. L. Yin, Q. Lin, and G. P. Agrawal, “Soliton fission and supercontinuum generation in silicon waveguides,” Opt. Lett. **32**, 391–393 (2007). [CrossRef] [PubMed]

20. R. Dekker, N. Usechak, M. Först, and A. Driessen, “Ultrafast nonlinear all-optical processes in silicon-on-insulator waveguides,” J. Phys. D: Appl. Phys. **40**, R249–R271 (2007). [CrossRef]

22. N. Suzuki, “FDTD analysis of two-photon absorption and free-carrier absorption in Si high-index-contrast waveguides,” J. Lightwave Technol. **25**, 2495–2501 (2007). [CrossRef]

26. I-W. Hsieh, X. Chen, J. I. Dadap, N. C. Panoiu, R. M. Osgood Jr., S. J. McNab, and Y. A. Vlasov, “Cross-phase modulation-induced spectral and temporal effects on co-propagating femtosecond pulses in silicon photonic wires,” Opt. Express **15**, 1135–1146 (2007). [CrossRef] [PubMed]

28. D. Dimitropoulos, B. Houshmand, R. Claps, and B. Jalali, “Coupled-mode theory of the Raman effect in silicon-on-insulator waveguides,” Opt. Lett. **28**, 1954–1956 (2003). [CrossRef] [PubMed]

46. X. Yang and C. W. Wong, “Design of photonic band gap nanocavities for stimulated Raman amplification and lasing in monolithic silicon,” Opt. Express **13**, 4723–4730 (2005). [CrossRef] [PubMed]

48. V. M. N. Passaro and F. D. Leonardis, “Space-time modeling of Raman pulses in silicon-on-insulator optical waveguides,” IEEE J. Lightwave Technol. **24**, 2920–2931 (2006). [CrossRef]

50. S. Blair and K. Zheng, “Intensity-tunable group delay using stimulated Raman scattering in silicon slow-light waveguides,” Opt. Express **14**, 1064–1069 (2006). [CrossRef] [PubMed]

55. X. Yang and C. W. Wong, “Coupled-mode theory for stimulated Raman scattering in high-Q/*V*_{m} silicon photonic band gap defect cavity lasers,” Opt. Express **15**, 4763–4780 (2007). [CrossRef] [PubMed]

58. F. De Leonardis and V. M. N. Passaro, “Modelling of Raman amplification in silicon-on-insulator optical microcavities,” New J. Phys. **9**, 25 (2007). [CrossRef]

59. F. De Leonardis and V. M. N. Passaro, “Modeling and performance of a guided-wave optical angular-velocity sensor based on Raman effect in SOI,” IEEE J. Lightwave Technol. **25**, 2352–2366 (2007). [CrossRef]

62. D. Dimitropoulos, V. Raghunathan, R. Claps, and B. Jalali, “Phase-matching and nonlinear optical processes in silicon waveguides,” Opt. Express **12**, 149–160 (2003). [CrossRef]

66. V. Raghunathan, R. Claps, D. Dimitropoulos, and B. Jalali, “Parametric Raman wavelength conversion in scaled silicon waveguides,” IEEE J. Lightwave Technol. **23**, 2094–2102 (2005). [CrossRef]

68. Q. Lin, J. Zhang, P. M. Fauchet, and G. P. Agrawal, “Ultrabroadband parametric generation and wavelength conversion in silicon waveguides,” Opt. Express **14**, 4786–4799 (2006). [CrossRef] [PubMed]

70. D. Dimitropoulos, D. R. Solli, R. Claps, and B. Jalali, “Noise figure and photon statistics in coherent anti-Stokes Raman scattering,” Opt. Express **14**, 11418–11432 (2006). [CrossRef] [PubMed]

73. Q. Lin and G. P. Agrawal, “Silicon waveguides for creating quantum-correlated photon pairs,” Opt. Lett. **31**, 3140–3142 (2006). [CrossRef] [PubMed]

74. N. C. Panoiu, X. Chen, and R. M. Osgood Jr., “Modulation instability in silicon photonic nanowires,” Opt. Lett. **31**, 3609–3611 (2006). [CrossRef] [PubMed]

76. N. Vermeulen, C. Debaes, and H. Thienpont, “Mitigating heat dissipation in near- and mid-infrared silicon-based Raman lasers using CARS,” IEEE J. Sel. Top. Quantum Electron. **13**, 770–787 (2007). [CrossRef]

88. P. E. Barclay, K. Srinivasan, and O. Painter, “Nonlinear response of silicon photonic crystal microresonators excited via an integrated waveguide and fiber taper”, Opt. Express **13**, 801–820 (2005). [CrossRef] [PubMed]

96. K. Ikeda and Y. Fainman, “Nonlinear Fabry-Perot resonator with a silicon photonic crystal waveguide,” Opt. Lett. **31**, 3486–3488 (2006). [CrossRef] [PubMed]

## 2. General formalism

101. J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. **78**, 1135–1184 (2006). [CrossRef]

### 2.1. Third-order susceptibility of silicon

*χ*

^{(3)}. When an optical field

**E**(

**r**,

*t*) propagates inside a silicon crystal or waveguide, the induced nonlinear polarization can be written in the frequency domain in the general form [102, 103]

*ω*≡

_{l}*ω*+

_{i}*ω*-

_{k}*ω*, and we have adopted the notation of Ref. [102]. Here,

_{j}*i*,

*j*,

*k*, and

*l*take values

*x*,

*y*, and

*z*and

*E*̃

*(*

_{i}**,**

*r**ω*) is the Fourier transform of the

*i*th component

*E*(

_{i}**,**

*r**t*) of the electric field defined as

*B*̃(

**,**

*r**ω*)=∫

^{∞}

_{-∞}

*B*(

**,**

*r**t*)exp(

*iωt*)

*dt*. In Eq. (1), we have excluded the sum-frequency generation assuming that this process is not phase-matched.

*χ*

^{(3)}

*=*

_{ijkl}*χ*

^{e}*+*

_{ijkl}*χ*

^{R}*, where the second term represents the Raman contribution involving optical phonons. These two terms have quite different dispersion and polarization characteristics.*

_{ijkl}104. Y. R. Shen and N. Bloembergen, “Theory of stimulated Brillouin and Raman scattering,” Phys. Rev. **137**, A1787–A1805 (1965). [CrossRef]

*H*̃

*(Ω), is the same for three normal modes. Unlike silica glass which has a very broad Raman spectrum [107*

_{R}107. R. H. Stolen, J. P. Gordon, W. J. Tomlinson, and H. A. Haus, “Raman response function of silica-core fibers,” J. Opt. Soc. Am. B **6**, 1159–1166 (1989). [CrossRef]

*/2*

_{R}*π*=15.6 THz at room temperature [108

108. P. A. Temple and C. E. Hathaway, “Multiphonon Raman spectrum of silicon,” Phys. Rev. B **7**, 3685–3697 (1973). [CrossRef]

110. A. Zwick and R. Carles, “Multiple-order Raman scattering in crystalline and amorphous silicon,” Phys. Rev. B **48**, 6024–6032 (1993). [CrossRef]

104. Y. R. Shen and N. Bloembergen, “Theory of stimulated Brillouin and Raman scattering,” Phys. Rev. **137**, A1787–A1805 (1965). [CrossRef]

*v*

*describes polarization dependence of Raman scattering. As the three phonon modes belong to the Γ*

_{ij}_{25}′ symmetry group and are polarized along a crystallographic axis (indicated by the superscript of ℜ), they interact resonantly only with optical waves polarized orthogonal to their own axes. For this reason, the three Raman tensors have the form [111

111. R. Loudon, “The Raman effect in crystals,” Adv. Phys. **50**, 813–864 (2001). [CrossRef]

*x*,

*y*, and

*z*axes are oriented along the crystallographic axes. Substituting Eq. (4) in Eq. (2), we obtain the following compact form for the Raman susceptibility:

*δ*equals 1 only when

_{ijkl}*i*=

*j*=

*k*=

*l*and is 0 otherwise.

108. P. A. Temple and C. E. Hathaway, “Multiphonon Raman spectrum of silicon,” Phys. Rev. B **7**, 3685–3697 (1973). [CrossRef]

110. A. Zwick and R. Carles, “Multiple-order Raman scattering in crystalline and amorphous silicon,” Phys. Rev. B **48**, 6024–6032 (1993). [CrossRef]

108. P. A. Temple and C. E. Hathaway, “Multiphonon Raman spectrum of silicon,” Phys. Rev. B **7**, 3685–3697 (1973). [CrossRef]

110. A. Zwick and R. Carles, “Multiple-order Raman scattering in crystalline and amorphous silicon,” Phys. Rev. B **48**, 6024–6032 (1993). [CrossRef]

112. J. R. Sandercock, “Brillouin-scattering measurements on silicon and germanium,” Phys. Rev. Lett. **28**, 237–240 (1972). [CrossRef]

*χ*(-

^{e}_{ijkl}*ω*;

_{i}*ω*,-

_{j}*ω*,

_{k}*ω*). Fortunately, as a silicon crystal belongs to the

_{l}*m*3

*m*point-group symmetry, its electronic nonlinear response has only four independent components [103, 117

117. R. W. Hellwarth, “Third-order optical susceptibilities of liquids and solids,” Prog. Quantum Electron. **5**, 1–68 (1977). [CrossRef]

118. P. D. Maker and R. W. Terhune, “Study of optical effects due to an induced polarization third order in the electric field strength,” Phys. Rev. **137**, A801–A818 (1965). [CrossRef]

*χ*≡

^{e}_{d}*χ*

^{e}_{1111}-

*χ*

^{e}_{1122}-

*χ*

^{e}_{1212}-

*χ*

^{e}_{1221}represents the nonlinearity anisotropy. In practice, the most relevant electronic nonlinearity is the one involving only one frequency, i.e.,

*χ*(-

^{e}_{ijkl}*ω*;

*ω*,-

*ω*,

*ω*). In this case, Eq. (6) is simplified considerably owing to the intrinsic permutation symmetry of

*χ*

^{e}_{1122}(-

*ω*;

*ω*,-

*ω*,

*ω*)=

*χ*

^{e}_{1221}(-

*ω*;

*ω*,-

*ω*,

*ω*). Moreover, for photon energies

*h*̄

*ω*well below

*E*,

_{g}*χ*

^{e}_{1212}(-

*ω*;

*ω*,-

*ω*,

*ω*)≈

*χ*

^{e}_{1122}(-

*ω*;

*ω*,-

*ω*,

*ω*) [105

105. M. D. Lvenson and N. Bloembergen, “Dispersion of the nonlinear optical susceptibility tensor in centrosymmetric media,” Phys. Rev. B **10**, 4447–4463 (1974). [CrossRef]

118. P. D. Maker and R. W. Terhune, “Study of optical effects due to an induced polarization third order in the electric field strength,” Phys. Rev. **137**, A801–A818 (1965). [CrossRef]

121. R. Buhleier, G. Lüpke, G. Marowsky, Z. Gogolak, and J. Kuhl, “Anisotropic interference of degenerate four-wave mixing in crystalline silicon,” Phys. Rev. B **50**, 2425–2431 (1994). [CrossRef]

*ρ*≡3

*χ*

^{e}_{1122}/

*χ*

^{e}_{1111}characterizes the nonlinear anisotropy at the degenerate frequency

*ω*. Note that Eq. (7) remains valid for third harmonic generation [86

86. D. J. Moss, H. M. van Driel, and J. E. Sipe, “Dispersion in the anisotropy of optical third-harmonic generation in silicon,” Opt. Lett. **14**, 57–59 (1989). [CrossRef] [PubMed]

118. P. D. Maker and R. W. Terhune, “Study of optical effects due to an induced polarization third order in the electric field strength,” Phys. Rev. **137**, A801–A818 (1965). [CrossRef]

119. S. S. Jha and N. Bloembergen, “Nonlinear optical susceptibilities in group-IV and III–V semiconductors,” Phys. Rev. **171**, 891–898 (1968). [CrossRef]

122. W. K. Burns and N. Bloembergen, “Third-harmonic generation in absorbing media of cubic or isotropic symmetry,” Phys. Rev. B **4**, 3437–3450 (1971). [CrossRef]

125. D. J. Moss, E. Ghahramani, J. E. Sipe, and H. M. van Driel, “Band-structure calculation of dispersion and anisotropy in *χ** ^{(3)}* for third-harmonic generation in Si, Ge, and GaAs,” Phys. Rev. B

**41**, 1542–1560 (1990). [CrossRef]

*ρ*can be complex in general.

*n*

_{0}(

*ω*) is the linear refractive index of silicon at the frequency

*ω*. Extensive measurements have been carried out to characterize

*n*

_{2}and

*β*over a wide frequency range [5

_{T}**82**, 2954–2956 (2003). [CrossRef]

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

9. G. W. Rieger, K. S. Virk, and J. F. Yong, “Nonlinear propagation of ultrafast 1.5 *µ*m pulses in high-index-contrast silicon-on-insulator waveguides,” Appl. Phys. Lett. **84**, 900–902 (2004). [CrossRef]

11. H. Yamada, M. Shirane, T. Chu, H. Yokoyama, S. Ishida, and Y. Arakawa, “Nonlinear-optic silicon-nanowire waveguides,” Jap. J. Appl. Phys. **44**, 6541–6545 (2005). [CrossRef]

12. E. Dulkeith, Y. A. Vlasov, X. Chen, N. C. Panoiu, and R. M. Osgood Jr., “Self-phase-modulation in submicron silicon-on-insulator photonic wires,” Opt. Express **14**, 5524–5534 (2006). [CrossRef] [PubMed]

24. A. Hache and M. Bourgeois, “Ultrafast all-optical switching in a silicon-based photonic crystal,” Appl. Phys. Lett. **77**, 4089–4091 (2000). [CrossRef]

65. H. Fukuda, K. Yamada, T. Shoji, M. Takahashi, T. Tsuchizawa, T. Watanabe, J. Takahashi, and S. Itabashi, “Four-wave mixing in silicon wire waveguides,” Opt. Express **13**, 4629–4637 (2005). [CrossRef] [PubMed]

82. V. Raghunathan, R. Shori, O. M. Stafsudd, and B. Jalali, “Nonlinear absorption in silicon and the prospects of mid-infrared silicon Raman lasers,” Physica Status Solidi A **203**, R38–R40 (2006). [CrossRef]

93. T. G. Eusera and W. L. Vos, “Spatial homogeneity of optically switched semiconductor photonic crystals and of bulk semiconductors,” J. Appl. Phys. **97**, 043102 (2005). [CrossRef]

126. J. F. Reintjes and J. C. McGroddy, “Indirect two-photon transition in Si at 1.06 *µ*m”, Phys. Rev. Lett. **30**, 901–903 (1973). [CrossRef]

*n*

_{2}for silicon is found to be more than 100 times larger in the 1.55-

*µ*m region than that of fused silica. However, TPA is also quite large in this spectral region. The relative magnitudes of the Kerr and TPA coefficients are often characterized by a nonlinear figure of merit (NFOM) [127

127. V. Mizrahi, K. W. DeLong, G. I. Stegeman, M. A. Saifi, and M. J. Andrejco, “Two-photon absorption as a limitation to all-optical switching,” Opt. Lett. **14**, 1140–1142 (1989). [CrossRef] [PubMed]

*F*=

_{n}*n*

_{2}/(

*λβ*), where

_{T}*λ*≡2

*πc*/

*ω*is the optical wavelength in vacuum. This NFOM is quite small for silicon (only about 0.3) in the 1.55-

*µ*m spectral region [5

**82**, 2954–2956 (2003). [CrossRef]

84. Q. Lin, J. Zhang, G. Piredda, R. W. Boyd, P. M. Fauchet, and G. P. Agrawal, “Dispersion of silicon nonlinearities in the near-infrared region,” Appl. Phys. Lett. **90**, 021111 (2007). [CrossRef]

**82**, 2954–2956 (2003). [CrossRef]

82. V. Raghunathan, R. Shori, O. M. Stafsudd, and B. Jalali, “Nonlinear absorption in silicon and the prospects of mid-infrared silicon Raman lasers,” Physica Status Solidi A **203**, R38–R40 (2006). [CrossRef]

*n*

_{2}and

*β*also vary considerably with

_{T}*λ*in the spectral region extending from 1.1

*µ*m to 2.2

*µ*m (close to half band gap), with

*n*

_{2}peaking around 1.8–1.9

*µ*m. Since TPA decreases quickly to zero when the wavelength exceeds 1.7

*µ*m, the NFOM increases considerably for

*λ*near or beyond 2

*µ*m[84

84. Q. Lin, J. Zhang, G. Piredda, R. W. Boyd, P. M. Fauchet, and G. P. Agrawal, “Dispersion of silicon nonlinearities in the near-infrared region,” Appl. Phys. Lett. **90**, 021111 (2007). [CrossRef]

*n*

_{2}changes its sign at photon energies around 0.7

*E*[128

_{g}128. M. Sheik-Bahae, D. C. Hutchings, D. J. Hagan, and E. W. Van Stryland, “Dispersion of bound electronic nonlinear refraction in solids,” IEEE J. Quantum Electron. **27**, 1296–1309 (1991). [CrossRef]

*n*

_{2}for silicon is always positive for all photon energies below

*E*[83

_{g}83. A. D. Bristow, N. Rotenberg, and H. M. van Driel, “Two-photon absorption and Kerr coefficients of silicon for 850–2200 nm,” Appl. Phys. Lett. **90**, 191104 (2007). [CrossRef]

84. Q. Lin, J. Zhang, G. Piredda, R. W. Boyd, P. M. Fauchet, and G. P. Agrawal, “Dispersion of silicon nonlinearities in the near-infrared region,” Appl. Phys. Lett. **90**, 021111 (2007). [CrossRef]

113. M. Dinu, “Dispersion of phonon-assisted nonresonant third-order nonlinearities,” IEEE J. Quantum Electron. **39**, 1498–1503 (2003). [CrossRef]

114. H. Garcia and R. Kalyanaraman, “Phonon-assisted two-photon absorption in the presence of a dc-field: the nonlinear Franz-Keldysh effect in indirect gap semiconductor,” J. Phys. B **39**, 2737–2746 (2006). [CrossRef]

### 2.2. Free-carrier effects

*µ*and

_{e}*µ*, in silicon are in the range of 100–1000 cm

_{h}^{2}/(V·s) for densities of up to 10

^{18}cm

^{-3}[131], the momentum relaxation times [

*τ*=

_{v}*µ*

_{v}m^{*}

*v*/

*q*(

*v*=

*e*,

*h*), where

*m*

^{*}

*is the effective mass and*

_{v}*q*is electron’s charge] lies in subpicosecond regime, much longer than the duration of an optical cycle [102]. As a result, free carriers can follow oscillations of an optical wave almost instantaneously and affect its propagation right after their creation. The dynamics of free carriers are well described by the Drude model [81

81. R. A. Soref and B. R. Bennett, “Electrooptical effects in silicon,” IEEE J. Quantum Electron. **23**, 123–129 (1987). [CrossRef]

132. A. Othonos, “Probing ultrafast carrier and phonon dynamics in semiconductors,” J. Appl. Phys. **83**, 1789–1830 (1998), and references therein. [CrossRef]

*N*and

_{e}*N*are densities of free electrons and holes, respectively. In this equation, 〈

_{h}*p*〉 with

^{v}_{i}*v*=

*e*,

*h*is the statistically averaged response of a single electron or hole to the electric field. According to the Drude model, it takes a simple form in the frequency domain, 〈

*p*̃

*(*

^{v}_{i}**,**

*r**ω*)=

*ε*

_{0}

*γ*(

_{v}*ω*)

*E*̃

*(*

_{i}**,**

*r**ω*), where the carrier polarizability γ

*is given by [102]*

_{v}*χ*̃

^{f}is defined as

*N*̃

*(*

_{v}*v*=

*e*,

*h*) is the Fourier transform of the carrier density

*N*. Equation (11) shows that the free-carrier response has a linear and isotropic nature, because of the cubic rotational symmetry of a silicon crystal [102].

_{v}*ω*, each of which has a limited bandwidth. On the other hand, carrier densities vary on a time scale longer than that associated with an optical pulse,

_{u}*i.e.*, they vary much slower than the carrier dipole moment oscillating at optical frequencies. As a result, the time-domain induced polarization consists of a few terms, each of which adiabatically follows variations of carrier densities, while oscillating around a specific carrier frequency

*ω*. More specifically,

_{u}*E*(

_{i}**,**

*r**ω*,

_{u}*t*) is the optical field at the carrier frequency

*ω*and the induced susceptibility is given by

_{u}*χ*

^{f}is complex in general [see Eq. (10)], we can relate its real and imaginary parts to changes in the refractive index and the absorption coefficient induced by free carriers using the relation,

*n*

_{f}is the free-carrier index (FCI) change and α

_{f}governs free-carrier absorption (FCA). Ideally, according to the Drude model of Eq. (10), they vary with optical frequency as (

*ωτ*≫1) [81

_{v}81. R. A. Soref and B. R. Bennett, “Electrooptical effects in silicon,” IEEE J. Quantum Electron. **23**, 123–129 (1987). [CrossRef]

*=2*

_{r}*πc*/

*ω*=1550 nm, it is common to employ the following empirical formulas [81

_{r}**23**, 123–129 (1987). [CrossRef]

*N*and

_{e}*N*have units of cm

_{h}^{-3}and α

_{f}is expressed in units of cm

^{-1}.

*N*=

_{e}*N*≡

_{h}*N*. In this case, it is more convenient to write

*n*

_{f}and α

_{f}as

*=1.45×10*

_{a}^{-17}(

*ω*/

_{r}*ω*)

^{2}(in units of cm

^{2}) and σ

*=ς(*

_{n}*ω*/

_{r}*ω*)

^{2}. The value of ς depends on the density region because of the (

*N*)

_{h}^{0.8}dependence for holes in Eq. (18). For example, Eq. (18) shows that the contribution of holes is about 3.8 and 6.1 times larger than that of electrons for a density of 10

^{17}and 10

^{16}cm

^{-3}, respectively. As the carrier density created by TPA is typically in this range, we assume that the hole contribution to FCI is 5 times that of electrons, and use ς≈-5.3×10

^{-21}(in units of cm

^{3}) in this paper.

132. A. Othonos, “Probing ultrafast carrier and phonon dynamics in semiconductors,” J. Appl. Phys. **83**, 1789–1830 (1998), and references therein. [CrossRef]

133. A. J. Sabbah and D. M. Riffe, “Femtosecond pump-probe reflectivity study of silicon carrier dynamics,” Phys. Rev. B **66**, 165217 (2002). [CrossRef]

132. A. Othonos, “Probing ultrafast carrier and phonon dynamics in semiconductors,” J. Appl. Phys. **83**, 1789–1830 (1998), and references therein. [CrossRef]

133. A. J. Sabbah and D. M. Riffe, “Femtosecond pump-probe reflectivity study of silicon carrier dynamics,” Phys. Rev. B **66**, 165217 (2002). [CrossRef]

### 2.3. General frequency-domain wave equation

*F*̃

*(*

_{i}*x*,

*y*,

*ω*) governs the mode profile in the plane transverse to the propagation direction

*z*. Substituting Eq. (22) into Eq. (21), multiplying by

*F*̃*

*, and integrating over the transverse plane, we obtain*

_{i}*β*(

_{i}*ω*) is the propagation constant given by

*T*denotes the transverse part of the Laplacian operator. The linear dispersion curve of a silicon waveguide is obtained from Eq. (24) after finding the transverse mode profile under specific boundary conditions set by the waveguide geometry. Note that the material refractive index

*n*

_{0}is generally different for the core and cladding layers.

*∂A*̃

*/*

_{i}*∂z*≈

*iβ*(

_{i}*ω*)

*A*̃

*even when the free-carrier and nonlinear effects are included, where we have assumed that the incident optical field propagates along the +*

_{i}*z*direction. Assuming that small perturbations do not reflect light and the backward wave can thus be ignored, we make the slowly varying envelope approximation and use

*A*(

*z*,

*t*)|

^{2}has units of power. The nonlinear parameter

*γ*in Eq. (27) is defined as

_{ijkl}*η*is the mode-overlap factor defined as

_{ijkl}*a*is over the whole transverse plane, but the dominant contribution to other integrals in Eqs. (29) and (30) comes from the silicon core layer if the third-order susceptibility is negligible for cladding layers.

_{v}*χ*

^{(3)}

*, the nonlinear parameter*

_{ijkl}*γ*=

_{ijkl}*γ*+

^{e}_{ijkl}*γ*is also composed of the electronic and Raman parts. Its expression in Eq. (28) includes wavelength dependence of the nonlinear parameters, effective mode area, mode overlap, and modal refractive index. If only one wave at the frequency

^{R}_{ijkl}*ω*is involved,

_{i}*η*=1,

_{iiii}*a*̄=

*a*, and Eq. (28) reduces to the conventional nonlinear parameter [80]. A detailed analysis shows that the fundamental modes of a straight waveguide overlap well with each other, leading to

_{i}*η*≈1. But this is not so for a microdisk or microring resonator because of the curved nature of device geometry [137].

_{ijkl}*β*̃

^{f}

*given by*

_{i}*χ*̃

^{f}is given in Eq. (12). In general, free carriers have specific transverse density distributions inside the waveguide, and

*β*̃

^{f}

*includes the effect of a partial overlap between the charge distribution and the mode profile [138*

_{i}138. M. J. Adams, S. Ritchie, and M. J. Robertson, “Optimum overlap of electric and optical fields in semiconductor waveguide devices,” Appl. Phys. Lett. **18**, 820–822 (1986). [CrossRef]

*χ*̃

^{f}is linearly proportional to the carrier densities [see Eqs. (9) through (20)], Eq. (31) is simplified, resulting in the following expression:

*ω*. By using Eqs. (13) and (14), it is easily to show that the free-carrier effects on each wave manifest in the time domain as a perturbation to the propagation constant through FCI and FCA. From Eqs. (13), (15), and (32), this perturbation is given by

_{u}*n*

_{f}and α

_{f}are given by Eqs. (16) and (17) [or Eqs. (18)–(20)] after replacing

*N*and

_{e}*N*with their averaged values defined in Eq. (32). Equation (33) shows that, for the same distribution of carrier densities, free-carrier effects are enhanced in a waveguide compared with bulk material as the modal refractive index is smaller because of mode confinement. The free carriers may also introduce extra birefringence or polarization-dependent losses because

_{h}*n*can be quite different for different polarization modes. Such effects are negligible for waveguides with a relatively large mode area, but they can become significant inside a nano-size waveguide. As an example, for an air-clad SOI waveguide with a cross section of 600×300 nm

_{i}^{2},

*n*is 2.76 and 2.32 at λ=1.55

_{i}*µ*m for the fundamental TE and TM modes, respectively. These values are much smaller than the material index (

*n*

_{0}=3.48) and indicate that free-carrier effects would be enhanced by 26 and 50% for the TE and TM modes, respectively.

*E*_{dc}. In general, the dynamics of carrier density is governed by the continuity equation [131]

*v*=

*e*for electrons,

*v*=

*h*for holes,

*s*=1,

_{h}*s*=-1,

_{e}*D*is the diffusion coefficient,

_{v}*τ*′

*is the carrier lifetime, and*

_{v}*µ*is the mobility. The generation rate

_{v}*G*is a function of optical field, if free carriers are generated through optical excitation like TPA.

*N*̄

*. By noting that the diffusion and drift of carriers away from the waveguide core reduces carrier density inside it, just as recombination of carriers does, we write the spatially averaged terms in Eq. (34) as*

_{v}*τ*

^{*}

*represents the effective lifetime associated with thermal diffusion and field-induced drift of free carriers. As a result, the dynamics of averaged carrier densities are governed by a simple equation of the form*

_{v}*G*̄is the generation rate averaged over the optical mode profile and

*τ*

_{0}ࣕ

*τ*′

_{v}*τ*

^{*}

*/(*

_{v}*τ*′

*+*

_{v}*τ*

^{*}

*) is an effective carrier lifetime that includes all the effects of recombination, diffusion, and drift; we have assumed it to be the same for electrons and holes.*

_{v}139. D. Dimitropoulos, R. Jhaveri, R. Claps, J. C. S. Woo, and B. Jalali, “Lifetime of photogenerated carriers in silicon-on-insulator rib waveguides,” Appl. Phys. Lett. **86**, 071115 (2005). [CrossRef]

42. H. Rong, R. Jones, A. Liu, O. Cohen, D. Hak, A. Fang, and M. Paniccia, “A continuous-wave Raman silicon laser,” Nature **433**, 725–728 (2005). [CrossRef] [PubMed]

67. H. Rong, Y. Kuo, A. Liu, M. Paniccia, and O. Cohen, “High efficiency wavelength conversion of 10 Gb/s data in silicon waveguides,” Opt. Express **14**, 1182–1188 (2006). [CrossRef] [PubMed]

140. Y. Liu and H. K. Tsang, “Nonlinear absorption and Raman gain in helium-ion-implanted silicon waveguides,” Opt. Lett. **31**, 1714–1716 (2006). [CrossRef] [PubMed]

143. T. Tanabe, K. Nishiguchi, A. Shinya, E. Kuramochi, H. Inokawa, and M. Notomi, “Fast all-optical switching using ion-implanted silicon photonic crystal nanocavities,” Appl. Phys. Lett. **90**, 031115 (2007). [CrossRef]

144. D. Dimitropoulos, S. Fathpour, and B. Jalali, “Limitations of active carrier removal in silicon Raman amplifiers and lasers,” Appl. Phys. Lett. **87**, 261108 (2005). [CrossRef]

54. H. Rong, S. Xu, Y. Kuo, V. Sih, O. Cohen, O. Raday, and M. Paniccia, “Low-threshold continuous-wave Raman silicon laser,” Nature Photon. **1**, 232–237 (2007). [CrossRef]

90. V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, “All-optical control of light on a silicon chip,” Nature **431**, 1081–1084 (2004). [CrossRef] [PubMed]

141. Y. Liu and H. K. Tsang, “Time dependent density of free carriers generated by two photon absorption in silicon waveguides,” Appl. Phys. Lett. **90**, 211105 (2007). [CrossRef]

41. H. Rong, A. Liu, R. Jones, O. Cohen, D. Hak, R. Nicolaescu, A. Fang, and M. Paniccia, “An all-silicon Raman laser,” Nature **433**, 292–294 (2005). [CrossRef] [PubMed]

42. H. Rong, R. Jones, A. Liu, O. Cohen, D. Hak, A. Fang, and M. Paniccia, “A continuous-wave Raman silicon laser,” Nature **433**, 725–728 (2005). [CrossRef] [PubMed]

*β*(

*ω*) is treated as a complex quantity. This frequency-domainwave equation can be used to investigate nonlinear interactions inside silicon waveguides for optical fields with arbitrary spectra. It can be simplified considerably if we notice that both the free-carrier and the third-order electronic nonlinear effects are only weakly dependent on optical frequency across a pulse spectrum. Often, the dispersion induced by FCI is negligible compared with the material and waveguide dispersion because |

*n*

_{f}|<10

^{-3}even at carrier densities up to 10

^{17}cm

^{-3}[81

**23**, 123–129 (1987). [CrossRef]

_{f}∝1/

*ω*

^{2}∝λ

^{2}[see Eqs. (16) and (17)], the magnitude of FCA changes by <14% even when the wavelength changes over 100 nm in the 1.55-

*µ*m spectral band. For similar changes in the wavelength, the electronic nonlinearity of silicon varies by ~20% in this spectral region [83

83. A. D. Bristow, N. Rotenberg, and H. M. van Driel, “Two-photon absorption and Kerr coefficients of silicon for 850–2200 nm,” Appl. Phys. Lett. **90**, 191104 (2007). [CrossRef]

*γ*(-

^{e}_{ijkl}*ω*;

_{i}*ω*,-

_{j}*ω*,

_{k}*ω*)≈

_{l}*γ*(-

^{e}_{ijkl}*ω*;

_{i}*ω*,-

_{i}*ω*,

_{i}*ω*) and use Eq. (7). The use of this approximation simplifies the theory considerably.

_{i}### 2.4. Time-domain description

*ω*

_{0}propagates along the waveguide, we can expand the propagation constant

*β*(

*ω*) in a Taylor series around

*ω*

_{0}. The time-domain description of the nonlinear process is then realized by replacing

*ω*-

*ω*

_{0}with the derivative

*i*(

*∂*/

*∂t*). Transferring Eq. (27) into time domain with this approach, we obtain the following equation for the field amplitude

*A*(

_{i}*z*,

*t*):

*β*is the

_{im}*m*th-order dispersion parameter defined as

*β*=(

_{im}*d*

^{m}*β*/

_{i}*dω*)|

^{m}_{ω=ω0}. These parameters also include the linear loss and its dispersion when

*β*(

*ω*) is a complex quantity. In the following discussion, we assume that free carriers are generated only optically so that

*N*̄

*=*

_{e}*N*̄

*=*

_{h}*N*̄, and write the

*β*

^{f}

_{i}term as

*β*

^{f}

*(*

_{i}*ω*

_{0},

*N*̂).

*P*(

^{NL}_{i}*z*,

*t*) has the following compact form in the time domain:

*(*

_{e}*ω*

_{0}) is defined as

*γ*

_{0}=

*ω*

_{0}

*n*

_{2}/(

*ca*̄) is the nonlinear Kerr parameter and

*β*′

*=*

_{T}*β*/

_{T}*a*̄ is the TPA coefficient normalized by the effective mode area.

_{2}=1/Γ

*and τ*

_{R}_{1}=1/(Ω

^{2}

*-Γ*

_{R}^{2}

*)*

_{R}^{1/2}≈1/Ω

*. The Raman-gain bandwidth of 105 GHz in silicon corresponds to a response time of τ*

_{R}_{2}≈3 ps. Similarly, the Raman shift of 15.6 THz corresponds to τ

_{1}≈10 fs.

*E*in a silicon waveguide contains only a relatively small fraction of incident power, particularly in the case of the fundamental TE and TM modes. As a result, the nonlinear effects are dominated by the transverse polarization components of the electric field. For this reason, most problems can be simplified by neglecting the

_{z}*E*component, as far as the nonlinear effects are concerned.

_{z}*R*′

^{(3)}

*=*

_{ijkl}*R*

^{(3)}

_{qrst}*M*, where

_{qi}M_{rj}M_{sk}M_{tl}*M*is a rotation matrix. By noting that all terms in Eq. (39) except those involving

_{uv}*δ*are rotation invariant, the nonlinear response function in the rotated coordinate system is found to be

_{ijkl}*x*axis, as shown in Fig. 1, and the rotation matrix is given by

*x*′ and

*y*′ axis, respectively, and those in Eq. (45) govern the nonlinear coupling between these two orthogonal polarizations. Two important conclusions can be drawn from Eq. (44) for SOI waveguides fabricated along the [0 1̄ 1] direction. First, the Raman contribution is absent for the quasi-TM modes polarized along the

*x*′ axis. Second, the electronic contribution is enhanced for the quasi-TE modes by a factor (about 14%) having its origin in the nonlinear anisotropy [87

87. J. Zhang, Q. Lin, G. Piredda, R. W. Boyd, G. P. Agrawal, and P. M. Fauchet, “Anisotropic nonlinear response of silicon in the near-infrared region,” Appl. Phys. Lett. **90**, 071113 (2007). [CrossRef]

28. D. Dimitropoulos, B. Houshmand, R. Claps, and B. Jalali, “Coupled-mode theory of the Raman effect in silicon-on-insulator waveguides,” Opt. Lett. **28**, 1954–1956 (2003). [CrossRef] [PubMed]

34. T. K. Liang and H. K. Tsang, “Efficient Raman amplificationin silicon-on-insulator waveguides,” Appl. Phys. Lett. **85**, 3343–3345 (2004). [CrossRef]

52. A. Liu, H. Rong, R. Jones, O. Cohen, D. Hak, and M. Paniccia, “Optical amplification and lasing by stimulated Raman scattering in silicon waveguides,” IEEE J. Lightwave Technol. **24**, 1440–1455 (2006). [CrossRef]

## 3. SPM effects on short optical pulses

### 3.1. Dispersion engineering

*µ*m beyond which GVD becomes anomalous. In contrast, silicon has significant normal dispersion over its transparent spectral region beyond 1.2

*µ*m, as seen in Fig. 2. However, it is well known that mode confinement provided by waveguide geometry introduces significant dispersion, which can be used to compensate for the material dispersion. This feature has been utilized in silica fibers to tailor their dispersion over a broad near-infrared region by either changing the size of the fiber core [147

147. T. A. Birks, W. J. Wadsworth, and P. St. J. Russell, “Supercontinuum generation in tapered fibers,” Opt. Lett. **25**, 1415–1416 (2000). [CrossRef]

148. P. St. J. Russell, “Photonic crystal fibers,” IEEE J. Lightwave Technol. **24**, 4729–4749 (2006). [CrossRef]

101. J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. **78**, 1135–1184 (2006). [CrossRef]

13. L. Yin, Q. Lin, and G. P. Agrawal, “Dispersion tailoring and soliton propagation in silicon waveguides,” Opt. Lett. **31**, 1295–1297 (2006). [CrossRef] [PubMed]

149. A. C. Turner, C. Manolatou, B. S. Schmidt, M. Lipson, M. A. Foster, J. E. Sharping, and A. L. Gaeta, “Tailored anomalous group-velocity dispersion in silicon channel waveguides,” Opt. Express **14**, 4357–4362 (2006). [CrossRef] [PubMed]

*β*

_{2}, obtained with the finite-element method, for some rectangular-shapewaveguides (see the inset). In each case, the height is fixed at 400 nm, but the width

*W*varies from 0.6 to 1.75

*µ*m. As seen there, the waveguide dispersion changes dramatically the magnitude of GVD, and the ZDWL for the fundamental TE mode can be engineered to lie anywhere from 1.2 to beyond 3

*µ*m simply by changing the waveguide width. More specifically, an SOI waveguide with a cross-section of 0.6×0.4

*µ*m

^{2}exhibits its ZDWLnear 1328 nm for the fundamental TE mode, resulting in a deeply anomalous GVD in the telecommunication band (about -1.52 ps

^{2}/m at 1550 nm). For the same mode, ZDWL can be tailored to near 1577 nm simply by increasing the waveguide width to 1.0

*µ*m. The ZDWL shifts dramatically to 2322 nm in the mid-infrared region with a further increase of the width to 1.75

*µ*m, resulting in a normal dispersion of 0.59 ps

^{2}/m at λ=1550 nm. In contrast, the GVD curves are similar in shape for the fundamental TM mode (dashed curves) of these three waveguide geometries, because of a fixed waveguide height used. Clearly, the tight mode confinement inside an SOI waveguide provides a significant degree of freedom for engineering GVD.

148. P. St. J. Russell, “Photonic crystal fibers,” IEEE J. Lightwave Technol. **24**, 4729–4749 (2006). [CrossRef]

68. Q. Lin, J. Zhang, P. M. Fauchet, and G. P. Agrawal, “Ultrabroadband parametric generation and wavelength conversion in silicon waveguides,” Opt. Express **14**, 4786–4799 (2006). [CrossRef] [PubMed]

*L*=

_{d}*T*

^{2}

_{0}/|

*β*

_{2}| [80], where

*T*

_{0}is a measure of the width of input pulses. Although the magnitude of GVD can be engineered considerably in SOI waveguides, the dispersion length is still often larger than the waveguide length (

*L*~1 cm). For example, for a GVD of |

*β*

_{2}|=2 ps

^{2}/m,

*L*=50 cm for a pulse with

_{d}*T*

_{0}=1 ps. Therefore, GVD-induced pulse broadening is relatively small until input pulses become shorter than 100 fs.

### 3.2. Relative magnitudes of the nonlinear and free-carrier effects

*µ*m wavelength,” Appl. Phys. Lett. **80**, 416–418 (2002). [CrossRef]

12. E. Dulkeith, Y. A. Vlasov, X. Chen, N. C. Panoiu, and R. M. Osgood Jr., “Self-phase-modulation in submicron silicon-on-insulator photonic wires,” Opt. Express **14**, 5524–5534 (2006). [CrossRef] [PubMed]

14. R. Dekker, A. Driessen, T. Wahlbrink, C. Moormann, J. Niehusmann, and M. Först, “Ultrafast Kerr-induced all-optical wavelength conversion in silicon waveguides using 1.55 *µ*m femtosecond pulses,” Opt. Express **14**, 8336–8346 (2006). [CrossRef] [PubMed]

15. I-W. Hsieh, X. Chen, J. I. Dadap, N. C. Panoiu, R. M. Osgood Jr., S. J. McNab, and Y. A. Vlasov, “Ultrafast-pulse self-phase modulation and third-order dispersion in Si photonic wire-waveguides,” Opt. Express **14**, 12380–12387 (2006). [CrossRef] [PubMed]

17. J. Zhang, Q. Lin, G. Piredda, R. W. Boyd, G. P. Agrawal, and P. M. Fauchet, “Optical solitons in a silicon waveguide,” Opt. Express **15**, 7682–7688 (2007). [CrossRef] [PubMed]

23. 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 Jr., “Supercontinuum generation in silicon photonic wires,” Opt. Express **15**, 15242–15248 (2007). [CrossRef] [PubMed]

25. Ö. Boyraz, P. Koonath, V. Raghunathan, and B. Jalali, “All optical switching and continuum generation in silicon waveguides,” Opt. Express **12**, 4094–4102 (2004). [CrossRef] [PubMed]

*L*=(

_{N}*γ*

_{0}

*P*

_{0})

^{-1}, where

*P*

_{0}is the peak power of input pulses. Because of a relatively large value of

*γ*

_{0}in silicon waveguides,

*L*can easily become 1 mm or less at moderate peak power levels. As a result, the SPM-induced spectral broadening is frequently observed in short silicon waveguides [7

_{N}*µ*m wavelength,” Appl. Phys. Lett. **80**, 416–418 (2002). [CrossRef]

12. E. Dulkeith, Y. A. Vlasov, X. Chen, N. C. Panoiu, and R. M. Osgood Jr., “Self-phase-modulation in submicron silicon-on-insulator photonic wires,” Opt. Express **14**, 5524–5534 (2006). [CrossRef] [PubMed]

10. A. R. Cowan, G. W. Rieger, and J. F. Young, “Nonlinear transmission of 1.5 *µ*m pulses through single-mode silicon-on-insulator waveguide structures,” Opt. Express **12**, 1611–1621 (2004). [CrossRef] [PubMed]

19. R. Salem, M. A. Foster, A. C. Turner, D. F. Geraghty, M. Lipson, and A. L. Gaeta, “All-optical regeneration on a silicon chip,” Opt. Express **15**, 7802–7809 (2007). [CrossRef] [PubMed]

18. P. Koonath, D. R. Solli, and B. Jalali, “Continuum generation and carving on a silicon chip,” Appl. Phys. Lett. **91**, 061111 (2007). [CrossRef]

8. O. Boyraz, T. Indukuri, and B. Jalali, “Self-phase-modulation induced spectral broadening in silicon waveguides,” Opt. Express **12**, 829–834 (2004). [CrossRef] [PubMed]

10. A. R. Cowan, G. W. Rieger, and J. F. Young, “Nonlinear transmission of 1.5 *µ*m pulses through single-mode silicon-on-insulator waveguide structures,” Opt. Express **12**, 1611–1621 (2004). [CrossRef] [PubMed]

**14**, 5524–5534 (2006). [CrossRef] [PubMed]

14. R. Dekker, A. Driessen, T. Wahlbrink, C. Moormann, J. Niehusmann, and M. Först, “Ultrafast Kerr-induced all-optical wavelength conversion in silicon waveguides using 1.55 *µ*m femtosecond pulses,” Opt. Express **14**, 8336–8346 (2006). [CrossRef] [PubMed]

18. P. Koonath, D. R. Solli, and B. Jalali, “Continuum generation and carving on a silicon chip,” Appl. Phys. Lett. **91**, 061111 (2007). [CrossRef]

19. R. Salem, M. A. Foster, A. C. Turner, D. F. Geraghty, M. Lipson, and A. L. Gaeta, “All-optical regeneration on a silicon chip,” Opt. Express **15**, 7802–7809 (2007). [CrossRef] [PubMed]

**12**, 4094–4102 (2004). [CrossRef] [PubMed]

9. G. W. Rieger, K. S. Virk, and J. F. Yong, “Nonlinear propagation of ultrafast 1.5 *µ*m pulses in high-index-contrast silicon-on-insulator waveguides,” Appl. Phys. Lett. **84**, 900–902 (2004). [CrossRef]

*µ*m femtosecond pulses,” Opt. Express **14**, 8336–8346 (2006). [CrossRef] [PubMed]

20. R. Dekker, N. Usechak, M. Först, and A. Driessen, “Ultrafast nonlinear all-optical processes in silicon-on-insulator waveguides,” J. Phys. D: Appl. Phys. **40**, R249–R271 (2007). [CrossRef]

22. N. Suzuki, “FDTD analysis of two-photon absorption and free-carrier absorption in Si high-index-contrast waveguides,” J. Lightwave Technol. **25**, 2495–2501 (2007). [CrossRef]

**12**, 4094–4102 (2004). [CrossRef] [PubMed]

*i*from

*A*. A further simplification occurs if we note that the Raman scattering is not important in this case because of a large Raman shift and a narrowband nature of the Raman-gain spectrum.

_{i}*τ*=

*t*-

*β*

_{1}

*z*so that

*τ*=0 corresponds to the pulse center.

*in Eq. (40), the power-loss rate introduced by TPA is given by*

_{e}*∂P*/

*∂z*=-

*β*|

_{T}*A*|

^{4}/

*a*̄. Accordingly, the carrier generation rate in Eq. (36) becomes

*A*(0,

*τ*)|

^{2}=

*P*

_{0}exp(-

*τ*

^{2}/

*T*

^{2}

_{0}). In this case,

*N*̄

*becomes*

_{m}*of the FCA parameter α*

_{fm}*. As the maximum TPA is governed by α*

_{f}*=*

_{Tm}*βT*

*P*

_{0}/

*ā*(at the pulse peak), the ratio of these two provides a criterion for testing the relative magnitudes of FCA and TPA through the dimensionless parameter

*𝓔*=√

_{p}*π*

*P*

_{0}

*T*

_{0}is the input energy of a Gaussian pulse. Note that

*n*

_{0}and

*n*are, respectively, the material and modal refractive indices at the carrier frequency

*ω*

_{0}of the pulse.

*and Φ*

_{K}*are the phase shifts induced by Kerr nonlinearity and free carriers, respectively. Both of them depend on pulse’s temporal profile and thus chirp the pulse. Such frequency chirps are more relevant in practice than the phase shifts themselves since they indicate the extent of spectral broadening [80]. Noting that the frequency shift varies as*

_{f}*δω*=-

*∂*Φ/

*∂τ*, the two frequency chirps satisfy

*T*

_{0}/

*τ*

_{0}, and is thus negligible compared with the first term when

*T*

_{0}≪

*τ*

_{0}. As a result, the growth rate of FCC from Eq. (54) is approximately given by

*is negative [see Eqs. (18) and (20)], it follows that FCC always causes the pulse spectrum to be blue-shifted. The maximum chirp occurs at the pulse center and is given by*

_{n}*∂*(

*δω*)/

_{Km}*∂z*|≈

*γ*

_{0}

*P*

_{0}/

*T*

_{0}at two temporal locations around the pulse center [80]. Therefore, the ratio of the FCC and KIC for Gaussian pulses is given by

*r*, this ratio depends only on the pulse energy (rather than on pulse width or peak power alone). The FCC and KIC become comparable (

_{a}*r*=1) for

_{c}*𝓔*/

_{p}*ā*≈4 mJ/cm

^{2}, or at a pulse energy of

*𝓔*≈20 pJ for a waveguide with

_{p}*ā*=0.5

*µ*m

^{2}. For pulse energies larger than this value, FCC becomes significant. This condition is easy to meet in practice. For example, the experiments in Refs. [8

8. O. Boyraz, T. Indukuri, and B. Jalali, “Self-phase-modulation induced spectral broadening in silicon waveguides,” Opt. Express **12**, 829–834 (2004). [CrossRef] [PubMed]

10. A. R. Cowan, G. W. Rieger, and J. F. Young, “Nonlinear transmission of 1.5 *µ*m pulses through single-mode silicon-on-insulator waveguide structures,” Opt. Express **12**, 1611–1621 (2004). [CrossRef] [PubMed]

**14**, 5524–5534 (2006). [CrossRef] [PubMed]

*µ*m femtosecond pulses,” Opt. Express **14**, 8336–8346 (2006). [CrossRef] [PubMed]

18. P. Koonath, D. R. Solli, and B. Jalali, “Continuum generation and carving on a silicon chip,” Appl. Phys. Lett. **91**, 061111 (2007). [CrossRef]

19. R. Salem, M. A. Foster, A. C. Turner, D. F. Geraghty, M. Lipson, and A. L. Gaeta, “All-optical regeneration on a silicon chip,” Opt. Express **15**, 7802–7809 (2007). [CrossRef] [PubMed]

15. I-W. Hsieh, X. Chen, J. I. Dadap, N. C. Panoiu, R. M. Osgood Jr., S. J. McNab, and Y. A. Vlasov, “Ultrafast-pulse self-phase modulation and third-order dispersion in Si photonic wire-waveguides,” Opt. Express **14**, 12380–12387 (2006). [CrossRef] [PubMed]

17. J. Zhang, Q. Lin, G. Piredda, R. W. Boyd, G. P. Agrawal, and P. M. Fauchet, “Optical solitons in a silicon waveguide,” Opt. Express **15**, 7682–7688 (2007). [CrossRef] [PubMed]

91. T. Tanabe, M. Notomi, S. Mitsugi, A. Shinya, and E. Kuramochi, “All-optical switches on a silicon chip realized using photonic crystal nanocavities,” Appl. Phys. Lett. **87**, 151112 (2005). [CrossRef]

96. K. Ikeda and Y. Fainman, “Nonlinear Fabry-Perot resonator with a silicon photonic crystal waveguide,” Opt. Lett. **31**, 3486–3488 (2006). [CrossRef] [PubMed]

*T*

_{0}=10 ps. The device length

*L*and peak power

*P*

_{0}of pulses are chosen such that

*ϕ*=

_{K}*γ*

_{0}

*P*

_{0}

*L*=15.5

*π*in the absence of TPA and the resulting free-carrier effects. Red curves neglect both FCA and FCC, assuming negligible density of free carriers. black curves include FCA but neglect FCC, while the green curves include both effects. The blue shift induced by FCC is seen clearly. The free carriers also introduce considerable spectral asymmetry because they are created by the leading edge of the pulse and affect mainly its trailing portion. Notice how the nonlinear phase shift changes from positive to negative values in the tail part of the pulses because of the free-carrier effects.

*r*≪1 in Eq. (58), free-carrier effects become negligible. In this case, we can set

_{c}*β*

^{f}=0 in Eq. (47), resulting in

*=*

_{K}*γ*

_{0}∫

^{L}_{0}

*P*(

*z*,

*t*)

*dz*, is given by [21

21. L. Yin and G. P. Agrawal, “Impact of two-photon absorption on self-phase modulation in silicon waveguides,” Opt. Lett. **32**, 2031–2033 (2007). [CrossRef] [PubMed]

*on the peak power to a logarithmic one. As a result, it reduces the value of Φ*

_{K}*by a factor that increases with increasing peak power and sets the fundamental limit on the extent of SPM-induced spectral broadening in silicon waveguides.*

_{K}### 3.3. Ultrashort pulse propagation and soliton formation

*γ*is a complex parameter in view of the TPA effects that cannot be ignored for silicon waveguides. Since linear losses and TPA reduce the peak power continuously along the waveguide length, an ideal soliton cannot form inside silicon waveguides. However, a soliton-like behavior can still be observed if the SPM effects are made strong enough initially that the dispersion-induced pulse broadening is negligible at the output. Indeed, we have found that if the input pulse is launched with an appropriate peak power, a solitary wave can form with a relatively small pulse broadening [13

_{e}13. L. Yin, Q. Lin, and G. P. Agrawal, “Dispersion tailoring and soliton propagation in silicon waveguides,” Opt. Lett. **31**, 1295–1297 (2006). [CrossRef] [PubMed]

*γ*

_{0}

*P*̄

_{0}

*L*=1, where

_{d}*P*̄

_{0}=

*L*

^{-1}∫

*L*

_{0}

*P*

_{0}(

*z*)

*dz*is the average peak power along the waveguide. Under such conditions, an input pulse preserves its temporal and spectral shape reasonably well at the output end [13

**31**, 1295–1297 (2006). [CrossRef] [PubMed]

17. J. Zhang, Q. Lin, G. Piredda, R. W. Boyd, G. P. Agrawal, and P. M. Fauchet, “Optical solitons in a silicon waveguide,” Opt. Express **15**, 7682–7688 (2007). [CrossRef] [PubMed]

**15**, 7682–7688 (2007). [CrossRef] [PubMed]

### 3.4. Soliton fission and supercontinuum generation

**91**, 061111 (2007). [CrossRef]

*µ*m wavelength,” Appl. Phys. Lett. **80**, 416–418 (2002). [CrossRef]

*µ*m pulses through single-mode silicon-on-insulator waveguide structures,” Opt. Express **12**, 1611–1621 (2004). [CrossRef] [PubMed]

**14**, 5524–5534 (2006). [CrossRef] [PubMed]

*µ*m femtosecond pulses,” Opt. Express **14**, 8336–8346 (2006). [CrossRef] [PubMed]

**12**, 4094–4102 (2004). [CrossRef] [PubMed]

101. J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. **78**, 1135–1184 (2006). [CrossRef]

*β*(

_{i}*ω*) should be accurately accounted for across the entire broadened spectral region because the soliton fission process is very sensitive to high-order dispersion [101

**78**, 1135–1184 (2006). [CrossRef]

16. L. Yin, Q. Lin, and G. P. Agrawal, “Soliton fission and supercontinuum generation in silicon waveguides,” Opt. Lett. **32**, 391–393 (2007). [CrossRef] [PubMed]

15. I-W. Hsieh, X. Chen, J. I. Dadap, N. C. Panoiu, R. M. Osgood Jr., S. J. McNab, and Y. A. Vlasov, “Ultrafast-pulse self-phase modulation and third-order dispersion in Si photonic wire-waveguides,” Opt. Express **14**, 12380–12387 (2006). [CrossRef] [PubMed]

23. 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 Jr., “Supercontinuum generation in silicon photonic wires,” Opt. Express **15**, 15242–15248 (2007). [CrossRef] [PubMed]

## 4. XPM and Raman interactions involving two waves

*A*̃

*(*

_{i}*z*,

*ω*)=

*A*̃

*(*

_{p}*ω*-

*ω*)+

_{p}*A*̃

*(*

_{s}*ω*-

*ω*), where the subscripts

_{s}*p*and

*s*denote the pump and signal waves, respectively, decomposing the equation into equations for individual waves, and converting the resulting equations to time domain, we obtain the following two coupled NLS equations:

*η*

_{uv}≡

*η*

_{uvvu}is given by Eq. (30). Note that

*g*scales linearly with optical frequency. When

_{R}*ω*=

_{u}*ω*,

_{v}*γ*

^{R}_{uv}and

*g*(

_{R}*ω*) reduce to the single-wave case discussed in Section 3. For the TE mode polarized along the [0 1̄ 1] direction, the electronic nonlinear parameter is given by

_{u}*γ*

^{R}_{uv}scales with Γ

*/Ω*

_{R}*, its value is only a few percent of the electronic part*

_{R}*γ*

^{e}_{uv}. As a result,

*γ*

_{uv}(0)≈

*γ*

^{e}_{uv}.

### 4.1. XPM and cross two-photon absorption

*h*(

_{R}*t*) in Eqs. (63) and (64) are irrelevant. For picosecond optical pulses, dispersion length is typically much longer than the waveguide length, and the dispersion terms (

*m*>1) are also negligible in these equations, resulting in

*γ*

^{R}_{uv}because of its small magnitude and used

*γ*

_{uv}(0)≈

*γ*

^{e}_{uv}. We also neglected linear losses and removed the trivial constant phase factor

*µ*m femtosecond pulses,” Opt. Express **14**, 8336–8346 (2006). [CrossRef] [PubMed]

26. I-W. Hsieh, X. Chen, J. I. Dadap, N. C. Panoiu, R. M. Osgood Jr., S. J. McNab, and Y. A. Vlasov, “Cross-phase modulation-induced spectral and temporal effects on co-propagating femtosecond pulses in silicon photonic wires,” Opt. Express **15**, 1135–1146 (2007). [CrossRef] [PubMed]

*µ*m femtosecond pulses,” Opt. Express **14**, 8336–8346 (2006). [CrossRef] [PubMed]

**12**, 4094–4102 (2004). [CrossRef] [PubMed]

*P*=|

_{p}*A*|

_{p}^{2}and

*P*=|

_{s}*A*|

_{s}^{2}, change along the waveguide length. These powers satisfy the following set of two coupled equations:

*β*′

_{Tuv}≡2Im(

*γ*

^{e}_{uv})≡

*β*

_{Tuv}/

*ā*

_{uv}. These equations show that TPA can occur in three ways: by absorbing two pump photons, two signal photons, or one pump and one signal photon (cross-TPA). The factor of 2 for the last process arises from the interference effects and the instantaneous nature of the TPA process [150

150. N. Bloembergen and P. Lallemand, “Complex intensity-dependent index of refraction, frequency broadening of stimulated Raman scattering, and stimulated Rayleigh scattering,” Phys. Rev. Lett. **16**, 81–84 (1966). [CrossRef]

*χ*

^{e}_{1111}(-

*ω*;

_{p}*ω*,-

_{s}*ω*,

_{s}*ω*)=

_{p}*χ*

^{e}_{1111}(-

*ω*;

_{s}*ω*,-

_{p}*ω*,

_{p}*ω*), from Eqs. (28) and (66), we find that

_{s}*β*

_{Tpp}and

*β*

_{Tss}are given in Eq. (8) for degenerate frequency

*ω*and

_{p}*ω*, respectively. The cross-TPA coefficient

_{s}*β*

_{Tps}is measurable in principle but it has not yet been measured for silicon. A good approximation is to estimate it at the mean frequency

*ω*̄=(

*ω*+

_{p}*ω*)/2 as

_{s}*β*

_{Tps}≈

*β*

_{T}(

*ω*̄), because the TPA process is most sensitive to the total energy of the two absorbed photons [128

128. M. Sheik-Bahae, D. C. Hutchings, D. J. Hagan, and E. W. Van Stryland, “Dispersion of bound electronic nonlinear refraction in solids,” IEEE J. Quantum Electron. **27**, 1296–1309 (1991). [CrossRef]

*E*[83

_{g}83. A. D. Bristow, N. Rotenberg, and H. M. van Driel, “Two-photon absorption and Kerr coefficients of silicon for 850–2200 nm,” Appl. Phys. Lett. **90**, 191104 (2007). [CrossRef]

**90**, 021111 (2007). [CrossRef]

*β*

_{Tpp}≈

*β*

_{Tss}≈

*β*

_{Tps}when frequencies

*ω*and

_{p}*ω*are close to each other.

_{s}151. K. Kikuchi, “Highly sensitive interferometric autocorrelator using Si avalanche photodiode as two-photon absorber,” Electron. Lett. **34**, 123–125 (1998). [CrossRef]

156. M. Dinu, D. C. Kilper, and H. R. Stuart, “Optical performance monitoring using data stream intensity autocorrelation,” IEEE J. Lightwave Technol. **24**, 1194–1202 (2006). [CrossRef]

98. T. K. Liang, L. R. Nunes, T. Sakamoto, K. Sasagawa, T. Kawanishi, M. Tsuchiya, G. R. A. Priem, D. Van Thourhout, P. Dumon, R. Baets, and H. K. Tsang, “Ultrafast all-optical switching by cross-absorption modulation in silicon wire waveguides,” Opt. Express **13**, 7298–7303 (2005). [CrossRef] [PubMed]

100. T. K. Liang, L. R. Nunes, M. Tsuchiya, K. S. Abedin, T. Miyazaki, D. Van Thourhout, W. Bogaerts, P. Dumon, R. Baets, and H. K. Tsang, “High speed logic gate using two-photon absorption in silicon waveguides,” Opt. Commun. **265**, 171–174 (2006). [CrossRef]

157. K. Taira, Y. Fukuchi, R. Ohta, K. Katoh, and K. Kikuchi, “Background-free intensity autocorrelator employing Si avalanche photodiode as two-photon absorber,” Electron. Lett. **38**, 1465–1466 (2002). [CrossRef]

*P*≪

_{s}*P*) satisfy

_{p}*∂*Φ

*/*

_{Km}*∂z*=2Re(

*γ*

^{e}_{sp})

*P*

_{0}at the pulse center. For a Gaussian pump pulse much shorter than the carrier lifetime, the maximum rate of carrier-induced XPM can be estimated by using Eq. (51) in Eq. (73) and is given by

*r*is given in Eq. (58) and the last approximation is valid when the pump and signal frequencies are not too far from each other. Clearly, the discussion about

_{c}*r*

*in the preceding section applies to*

_{c}*r*as well. As this ratio is typically much larger than 1 for moderate pump-pulse energies, free-carrier-induced XPM provides a more efficient way for optical switching, particularly when an interferometric resonance inside a micro-resonator is used to enhance such effects [90

_{x}90. V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, “All-optical control of light on a silicon chip,” Nature **431**, 1081–1084 (2004). [CrossRef] [PubMed]

92. S. F. Preble, Q. Xu, B. S. Schmidt, and M. Lipson, “Ultrafast all-optical modulation on a silicon chip,” Opt. Lett. **30**, 2891–2893 (2005). [CrossRef] [PubMed]

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

92. S. F. Preble, Q. Xu, B. S. Schmidt, and M. Lipson, “Ultrafast all-optical modulation on a silicon chip,” Opt. Lett. **30**, 2891–2893 (2005). [CrossRef] [PubMed]

### 4.2. Raman amplification and lasing

31. Q. Xu, V. R. Almeida, and M. Lipson, “Time-resolved study of Raman gain in highly confined silicon-on-insulator waveguides,” Opt. Express **12**, 4437–4442 (2004). [CrossRef] [PubMed]

34. T. K. Liang and H. K. Tsang, “Efficient Raman amplificationin silicon-on-insulator waveguides,” Appl. Phys. Lett. **85**, 3343–3345 (2004). [CrossRef]

47. 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]

47. 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]

48. V. M. N. Passaro and F. D. Leonardis, “Space-time modeling of Raman pulses in silicon-on-insulator optical waveguides,” IEEE J. Lightwave Technol. **24**, 2920–2931 (2006). [CrossRef]

32. A. Liu, H. Rong, M. Paniccia, O. Cohen, and D. Hak, “Net optical gain in a low loss silicon-on-insulator waveguide by stimulated Raman scattering,” Opt. Express **12**, 4261–4268 (2004). [CrossRef] [PubMed]

40. Q. Xu, V. R. Almeida, and M. Lipson, “Demonstration of high Raman gain in a submicrometer-size silicon-on-insulator waveguide,” Opt. Lett. **30**, 35–37 (2005). [CrossRef] [PubMed]

32. A. Liu, H. Rong, M. Paniccia, O. Cohen, and D. Hak, “Net optical gain in a low loss silicon-on-insulator waveguide by stimulated Raman scattering,” Opt. Express **12**, 4261–4268 (2004). [CrossRef] [PubMed]

40. Q. Xu, V. R. Almeida, and M. Lipson, “Demonstration of high Raman gain in a submicrometer-size silicon-on-insulator waveguide,” Opt. Lett. **30**, 35–37 (2005). [CrossRef] [PubMed]

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

43. O. Boyraz and B. Jalali, “Demonstration of directly modulated silicon Raman laser,” Opt. Express **13**, 796–800 (2005). [CrossRef] [PubMed]

*P*and

_{p}*P*, evolve along the waveguide length. These powers satisfy the following coupled equations:

_{s}*H*̃

*(Ω) is given in Eq. (3). When the signal is located at the Stokes side of the pump with a frequency detuning of the Raman shift, Ω*

_{R}_{sp}=-Ω

*, Eq. (3) and Eq. (79) result in a peak Raman gain of*

_{R}*g*(

_{R}*ω*)/

_{s}*ā*

_{sp}where

*g*(

_{R}*ω*) is given in Eq. (65).

_{s}**12**, 4261–4268 (2004). [CrossRef] [PubMed]

36. R. Claps, V. Raghunathan, D. Dimitropoulos, and B. Jalali, “Influence of nonlinear absorption on Raman amplification in silicon waveguides,” Opt. Express **12**, 2774–2780 (2004). [CrossRef] [PubMed]

*β*

_{Tsp}≈

*β*

_{Tpp}and

*ā*

_{sp}≈

*ā*

_{pp}, we find from Eq. (81) that

*r*>1 at a moderate power level of

_{a}*P*/

_{p}*ā*

_{sp}>35MW/cm

^{2}for an SOI waveguide with τ

_{0}=1 ns. To realize a Raman amplification, we need

*g*/

_{R}P_{p}L*ā*

_{sp}>1, which leads to the requirement

*P*/

_{p}*ā*

_{sp}>50 MW/cm

^{2}. Clearly, FCA is the major obstacle for Raman amplification.

*P*is a real positive quantity lead to the following condition on the carrier lifetime:

_{p}*β*

_{Tsp}≈

*β*

_{Tpp}, the carrier lifetime should be<60 ns for a waveguide with a low linear loss of 0.2 dB/cm. Indeed, positive Raman amplification with CW pumping was observed inside a waveguide with a linear loss of 0.22 dB/cm and a carrier lifetime of 25 ns [33

33. H. Rong, A. Liu, R. Nicolaescu, M. Paniccia, O. Cohen, and D. Hak, “Raman gain and nonlinear optical absorption measurement in a low-loss silicon waveguide,” Appl. Phys. Lett. **85**, 2196–2198 (2004). [CrossRef]

_{0}are required with increasing linear losses, Eq. (83) is hard to satisfy for practical silicon waveguides because a low loss available for large waveguides also leads to longer carrier lifetimes. Decreasing the waveguide cross section helps reduce the carrier lifetime [139

139. D. Dimitropoulos, R. Jhaveri, R. Claps, J. C. S. Woo, and B. Jalali, “Lifetime of photogenerated carriers in silicon-on-insulator rib waveguides,” Appl. Phys. Lett. **86**, 071115 (2005). [CrossRef]

140. Y. Liu and H. K. Tsang, “Nonlinear absorption and Raman gain in helium-ion-implanted silicon waveguides,” Opt. Lett. **31**, 1714–1716 (2006). [CrossRef] [PubMed]

41. H. Rong, A. Liu, R. Jones, O. Cohen, D. Hak, R. Nicolaescu, A. Fang, and M. Paniccia, “An all-silicon Raman laser,” Nature **433**, 292–294 (2005). [CrossRef] [PubMed]

42. H. Rong, R. Jones, A. Liu, O. Cohen, D. Hak, A. Fang, and M. Paniccia, “A continuous-wave Raman silicon laser,” Nature **433**, 725–728 (2005). [CrossRef] [PubMed]

45. 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]

54. H. Rong, S. Xu, Y. Kuo, V. Sih, O. Cohen, O. Raday, and M. Paniccia, “Low-threshold continuous-wave Raman silicon laser,” Nature Photon. **1**, 232–237 (2007). [CrossRef]

56. V. Sih, S. Xu, Y. Kuo, H. Rong, M. Paniccia, O. Cohen, and O. Raday, “Raman amplification of 40 Gb/s data in low-loss silicon waveguides,” Opt. Express **15**, 357–362 (2007). [CrossRef] [PubMed]

41. H. Rong, A. Liu, R. Jones, O. Cohen, D. Hak, R. Nicolaescu, A. Fang, and M. Paniccia, “An all-silicon Raman laser,” Nature **433**, 292–294 (2005). [CrossRef] [PubMed]

**433**, 725–728 (2005). [CrossRef] [PubMed]

54. H. Rong, S. Xu, Y. Kuo, V. Sih, O. Cohen, O. Raday, and M. Paniccia, “Low-threshold continuous-wave Raman silicon laser,” Nature Photon. **1**, 232–237 (2007). [CrossRef]

_{ls}should be replaced by total distributed cavity losses α

_{ts}that include coupling losses and reflection from cavity mirrors. It turns out that there is an upper limit on the carrier lifetime above which the Raman laser would not be able to operate, no matter how large the pump power is [39

39. 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]

_{ls}with α

_{ts}. Equation (82) cannot be satisfied for any pump power, if τ

_{0}is larger than τ

_{th}given in Eq. (83). If cavity losses are small (

*e.g.*, in the case of a microcavity), the lasing threshold is found from Eq. (82) to be

*τ*

_{0}<

*τ*

_{th}. If the carrier lifetime is small enough that

*τ*

_{0}≪

*τ*

_{th}, Eq. (85) leads to

158. T. J. Kippenberg, S. M. Spillane, B. Min, and K. J. Vahala, “Theoretical and Experimental Study of Stimulated and Cascaded Raman Scattering in Ultrahigh-Q Optical Microcavities,” IEEE J. Sel. Top. Quantum Electron. **10**, 1219–1228 (2004). [CrossRef]

_{th}, Eq. (85) shows that the lasing threshold becomes twice of that given in Eq. (86). Of course, as the pump itself starts to experience significant losses, the real threshold would more than double in practice [39

39. 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]

158. T. J. Kippenberg, S. M. Spillane, B. Min, and K. J. Vahala, “Theoretical and Experimental Study of Stimulated and Cascaded Raman Scattering in Ultrahigh-Q Optical Microcavities,” IEEE J. Sel. Top. Quantum Electron. **10**, 1219–1228 (2004). [CrossRef]

*P*

_{m}→∞ when

*τ*

_{0}≪

*τ*

_{th}, and the lasing saturation is negligible. However, as

*τ*

_{0}becomes close to

*τ*

_{th}, the maximum allowed pump power

*P*

_{m}approaches

*P*

_{th}. Eventually, the power range to obtain positive intracavity gain becomes so small that Raman lasing is quenched.

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59. F. De Leonardis and V. M. N. Passaro, “Modeling and performance of a guided-wave optical angular-velocity sensor based on Raman effect in SOI,” IEEE J. Lightwave Technol. **25**, 2352–2366 (2007). [CrossRef]

52. A. Liu, H. Rong, R. Jones, O. Cohen, D. Hak, and M. Paniccia, “Optical amplification and lasing by stimulated Raman scattering in silicon waveguides,” IEEE J. Lightwave Technol. **24**, 1440–1455 (2006). [CrossRef]

53. B. Jalali, V. Raghunathan, D. Dimitropoulos, and O. Boyraz, “Raman-based silicon photonics,” IEEE J. Sel. Top. Quantum Electron. **12**, 412–421 (2006). [CrossRef]

50. S. Blair and K. Zheng, “Intensity-tunable group delay using stimulated Raman scattering in silicon slow-light waveguides,” Opt. Express **14**, 1064–1069 (2006). [CrossRef] [PubMed]

51. Y. Okawachi, M. A. Foster, J. E. Sharping, A. L. Gaeta, Q. Xu, and M. Lipson, “All-optical slow-light on a photonic chip,” Opt. Express **14**, 2317–2322 (2006). [CrossRef] [PubMed]

51. Y. Okawachi, M. A. Foster, J. E. Sharping, A. L. Gaeta, Q. Xu, and M. Lipson, “All-optical slow-light on a photonic chip,” Opt. Express **14**, 2317–2322 (2006). [CrossRef] [PubMed]

## 5. FWM and its applications

### 5.1. Free-carrier effects on FWM

160. S. Fathpour, K. K. Tsia, and B. Jalali, “Energy harvesting in silicon Raman amplifiers,” Appl. Phys. Lett. **89**, 061109 (2006). [CrossRef]

161. K. K. Tsia, S. Fathpour, and B. Jalali, “Energy harvesting in silicon wavelength converters,” Opt. Express **14**, 12327–12333 (2006). [CrossRef] [PubMed]

*γ*is defined as

_{f}*A*(

*z*,

*t*) is composed of three waves such that

*A*is much more intense than the signal

_{p}*A*and idler

_{s}*A*. We substitute this expression into Eq. (90) and obtain the following three equations at individual frequencies:

_{i}_{sp}=Ω

_{pi}and kept only the first-order terms in

*A*and

_{s}*A*. Equations (92)–(94) show clearly that the fifth-order nonlinear effect associated with free carriers also introduce SPM, XPM, and FWM-like processes, some of which (SPM and XPM) have been discussed in preceding sections.

_{i}*=*

_{uv}*ω*-

_{u}*ω*(

_{v}*u*,

*v*=

*p*,

*s*,

*i*) and

*β*=

_{v}*β*

_{0}(

*ω*) is the propagation constant at the carrier frequency

_{v}*ω*. Note that Ω

_{v}_{ps}=Ω

_{ip}for a FWM process. In Eqs. (95)–(97), we have neglected linear losses and assumed pulse widths to be wide enough that all dispersive effects are negligible for individual waves.

*γ*stem from the Kerr nonlinearity, but the terms containing

_{e}*γ*result from the TPA-generated free carriers. Comparing the nonlinear effects induced by these two mechanisms, we find that the efficiency of Kerr-inducedFWMscales as

_{f}*γ*

_{e}*A*

^{2}

*, but that of free-carrier-induced FWM scales as 2*

_{p}*γ*

_{f}*τ*

_{0}|

*A*|

_{p}^{4}/(1+

*i*Ω

_{ps}

*τ*

_{0}). The relative importance of these two FWM processes is governed by the ratio

*γf*)≫Im(

*γf*) from Eqs. (20) and (91), and using

*γ*=

_{e}*β*′

*(2*

_{T}*πF*+

_{n}*i*/2)≈2

*πF*′

_{n}β*for*

_{T}*F*>0.2, where

_{n}*F*is the NFOM introduced earlier, Eq. (98) can be approximated by

_{n}*γ*)|

_{e}*A*|

_{p}^{2}

*L*~

*π*. In a typical silicon waveguide, this condition requires a pump intensity of |

*A*|

_{p}^{2}/

*ā*≈0.4 GW/cm

^{2}inside a 5-cm-longwaveguide. At such power levels, |

*r*

_{FWM}|≪1 when the pump-signal detuning Ω

_{sp}/2

*π*≪60 GHz, since the NFOM is

*F*≈0.3 in the telecom band. The magnitude of

_{n}*r*

_{FWM}becomes even smaller at lower pump powers. As a result, FWM induced by free carriers is negligible in most practical situations of parametric generation and wavelength conversion. However, one should keep in mind that this process could become quite efficient when pump-signal detuning becomes relatively small. For example, for a carrier lifetime of 1 ns, |

*r*

_{FWM}| can be larger than 250 when |Ω

_{sp}|

*τ*

_{0}<1, a condition satisfied for a pump-signal detuning of <160 MHz.

*κ*=2Re(

_{K}*γ*)|

_{e}*A*|

_{p}^{2}, the ratio between the two is found to be

*r*

_{FWM}applies to

*r*as well. In other words,

_{κ}*r*is negligible in most practical situations of parametric generation. Physically speaking, if the pump-signal detuning is not too small, free carriers impose nearly identical phase modulations on the three waves, leading to negligible phase mismatch among them, even though the absolute magnitude of such phase modulations could be large. Note that Eqs. (95)–(97) neglect the wavelength dependence of

_{κ}*γ*by assuming that it has the same value for all three waves. A more detailed analysis shows that free carriers do introduce some phase mismatch; this effect is discussed later. In practice, as long as the pump-signal detuning is not too small, the effects of free carriers on both FWM efficiency and phase-matching condition are negligible in the quasi-CW case, and their dominant effect comes from FCA [68

_{f}**14**, 4786–4799 (2006). [CrossRef] [PubMed]

*T*

_{0}is assumed to be the same for the three waves, and dispersive effects are neglected assuming that the dispersion length is longer than the waveguide length.

^{2}pumping level for a typical silicon waveguide. Clearly, free-carrier effects will be quite significant in this case. Note that the real part of

*γ*is negative because σ

_{f}*<0 (free carriers reduce the refractive index). As a result, with increasing pump power, free-carrier effects tends to cancel those induced by the Kerr nonlinearity, leading to a net decrease in the FWM efficiency. Moreover, the parametric bandwidth also decreases with increasing pump power. This may explain the gain saturation of parametric amplification observed experimentally with 3.5-ps pump pulses [69*

_{n}69. 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]

### 5.2. Broadband parametric generation and wavelength conversion

162. T. Torounidis and P. Andrekson, “Broadband single-pumped fiber-optic parametric amplifiers,” IEEE Photon. Technol. Lett. **19**, 650–652 (2007). [CrossRef]

163. J. M. Chavez Boggio, J. D. Marconi, S. R. Bickham, and H. L. Fragnito, “Spectrally flat and broadband doublepumped fiber optical parametric amplifiers,” Opt. Express **15**, 5288–5309 (2007). [CrossRef] [PubMed]

*P*=|

_{j}*A*|

_{j}^{2}for

*j*=

*s*,

*i*,

*p*and the nonlinear parameters appearing in the FWM terms are obtained from Eq. (28). For example,

*γ*

^{e}_{pspi}is given by

*γ*

^{e}_{spip}and

*γ*

^{e}_{ipsp}can be obtained from this equation by exchanging the subscripts. We have neglected FWM induced by free carriers because of its negligible magnitude in the CW regime.

*β*=

*β*+

_{s}*β*-2

_{i}*β*, represnts linear phase mismatch, the second term, Δ

_{p}*β*=Re(

_{f}*β*

^{f}

*+*

_{s}*β*

^{f}

*-2*

_{i}*β*

^{f}

*), is the phase mismatch induced by free carriers, and the last term is the nonlinear phase mismatch induced by the Kerr nonlinearity. From Eqs. (20) and (33), we find that*

_{p}*β*

^{f}(

*ω*,

*N*̄) is linearly proportional to

*n*

_{0}(

*ω*)/

*β*(

*ω*). As a result, the free-carrier-induced phase mismatch is given by

*=[*

_{m}*∂*(

^{m}*n*

_{0}

*β*

^{-1})/

*ω*]

^{m}_{ω=ωp}. As

*σ*<0, Eq. (118) shows that FCI-induced phase mismatch acts like adding negative (anomalous) second- and higher-order dispersions. The magnitudes of such equivalent GVD and fourth-order dispersion (FOD) are given by

_{np}_{2}~8×10

^{-13}m·ps

^{2}and ζ

_{4}~2×10

^{-17}m·ps

^{4}for a typical silicon waveguide. As a result, at a pumping level of 0.4GW/cm

^{2}in the telecom band, the FCI-inducedGVD and FOD are about

*β*

_{2}≈-0.02 ps

^{2}/m and

*β*

_{4}≈-5×10

^{-7}ps

^{4}/m, respectively. Both are much smaller than the GVD and FOD resulting from the waveguide confinement. Therefore, the phase-matching condition for FWM inside a silicon waveguide is dominated by the linear phase mismatch Δ

*β*and the nonlinear contribution resulting from the Kerr effect.

*κL*|<

*π*/2. As is known from FWM in silica fibers [80, 159

159. J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron. **8**, 506–520 (2002). [CrossRef]

*L*~1 cm) is much shorter than that used for fibers. Figure 7 shows examples of the signal gain (a) and wavelength-conversion efficiency (b) for three choices of pump wavelengths for a waveguide designed with a ZDWL at 1551 nm [68

**14**, 4786–4799 (2006). [CrossRef] [PubMed]

79. M. A. Foster, A. C. Turner, R. Salem, M. Lipson, and A. L. Gaeta, “Broad-band continuous-wave parametric wavelength conversion in silicon nanowaveguides,” Opt. Express **15**, 12949–12958 (2007). [CrossRef] [PubMed]

**14**, 4786–4799 (2006). [CrossRef] [PubMed]

**14**, 4786–4799 (2006). [CrossRef] [PubMed]

**14**, 4786–4799 (2006). [CrossRef] [PubMed]

*γ*

^{e}_{spip}≈

*γ*

^{e}_{p},

*β*

_{Tsp}≈

*β*

_{Tpp}, and

*ā*

_{sp}≈

*ā*

_{pp}in the last approximation, assuming that these quantities do not change much with the signal frequency. Equation (121) is similar to Eq. (83) obtained for Raman amplification. However, as the Kerr parameter is about 10 times smaller than the Raman gain coefficient, Eq. (121) imposes a much more stringent limit on the carrier lifetime. A detailed analysis shows that it is not possible to obtain positive parametric gain unless the carrier lifetime is reduced to below 100 ps [68

**14**, 4786–4799 (2006). [CrossRef] [PubMed]

### 5.3. Coherent anti-Stokes Raman scattering

*n*

^{R}_{2}≈

*cgR*/(4

*ω*)~1.2×10

_{s}^{-4}cm

^{2}/GW, which is much larger than that produced by the Kerr nonlinearity. As a result, when SRS contributes to theFWM process, it makes FWM much more efficient than that possible from the Kerr nonlinearity alone. This regime of FWM is known as the coherent anti-Stokes Raman scattering (CARS) process [103,164

164. M. D. Levenson, C. Flytzanis, and N. Bloembergen, “Interference of resonant and nonresonant three-wave mixing in diamond,” Phys. Rev. B **6**, 3962–3965 (1972). [CrossRef]

**14**, 4786–4799 (2006). [CrossRef] [PubMed]

*i.e.*,

*γ*

_{sp}(Ω

_{sp})=

*γ*

^{e}_{sp}+

*γ*

^{R}_{sp}(Ω

_{sp}) and

*γ*

_{spip}(Ω

_{sp})=

*γ*

^{e}_{spip}+

*γ*

^{R}_{spip}(Ω

_{sp}).

61. R. Claps, V. Raghunathan, D. Dimitropoulos, and B. Jalali, “Anti-Sotkes Raman conversion in silicon waveguides,” Opt. Express **11**, 2862–2872 (2003). [CrossRef] [PubMed]

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

66. V. Raghunathan, R. Claps, D. Dimitropoulos, and B. Jalali, “Parametric Raman wavelength conversion in scaled silicon waveguides,” IEEE J. Lightwave Technol. **23**, 2094–2102 (2005). [CrossRef]

**14**, 4786–4799 (2006). [CrossRef] [PubMed]

**14**, 4786–4799 (2006). [CrossRef] [PubMed]

### 5.4. Highly tunable parametric generation from the telecom band to the mid-infrared

*µ*m. As the pump wavelength then falls below the half-band gap of silicon, TPA vanishes [83

**90**, 191104 (2007). [CrossRef]

**90**, 021111 (2007). [CrossRef]

*µ*m

^{2}, parametric amplification becomes very efficient when pump wavelength is located in the vicinity of ZDWL because no TPA can occur. For example, a broadband gain spectrum with positive gain is obtained at a pump wavelength of 2.45

*µ*m falling in the anomalous-GVD regime. Even when the pump wavelength falls in the normal-GVD regime, higher-order dispersive effects can assist in satisfying the phase-matching condition to provide gain in spectral regions far from the pump. In this regime, phase matching becomes very sensitive to the pump wavelength. The main point to note is that efficient parametric amplification is possible at signal wavelengths extending from 1.5

*µ*m to mid infrared by simply tuning the pump wavelength within 100 nm. Tunability of the signal is more than 5 times larger than the pump itself. Although it was proposed recently to use silicon Raman laser in mid infrared [166

166. B. Jalali, V. Raghunathan, R. Shori, S. Fathpour, D. Dimitropoulos, and O. Stafsudd, “Prospects for silicon Mid-IR Raman Lasers,” IEEE J. Sel. Top. Quantum Electron. **12**, 1618–1627 (2006). [CrossRef]

*χ*

^{(2)}materials such as a periodically poled lithium-niobate waveguide.

### 5.5. Photon pair generation by FWM

73. Q. Lin and G. P. Agrawal, “Silicon waveguides for creating quantum-correlated photon pairs,” Opt. Lett. **31**, 3140–3142 (2006). [CrossRef] [PubMed]

*s*and

*i*. The four noise operators (last 4 terms) in Eq. (125) are associated with scattering losses, FCA, TPA, and Raman gain/loss, respectively. They obey a commutation relation of the form

*D*(

_{j}*j*=

*l*,

*f*,

*T*,

*R*) stands for

*α*(

_{l}*ω*),

_{u}*α*

_{f}(

*z*

_{1},

*ω*), 2

_{u}*β*′

_{Tup}, and

*gR*(Ω

_{up})/

*ā*

_{up}for the four noise sources, respectively. In the case of SpRS, the photon frequency

*ω*in Eq. (126) is replaced with the phonon frequency Ω=

*ω*-

*ω*.

_{p}## Acknowledgements

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73. | Q. Lin and G. P. Agrawal, “Silicon waveguides for creating quantum-correlated photon pairs,” Opt. Lett. |

74. | N. C. Panoiu, X. Chen, and R. M. Osgood Jr., “Modulation instability in silicon photonic nanowires,” Opt. Lett. |

75. | J. E. Sharping, K. F. Lee, M. A. Foster, A. C. Turner, B. S. Schmidt, M. Lipson, A. L. Gaeta, and P. Kumar, “Generation of correlated photons in nanoscale silicon waveguides,” Opt. Express |

76. | N. Vermeulen, C. Debaes, and H. Thienpont, “Mitigating heat dissipation in near- and mid-infrared silicon-based Raman lasers using CARS,” IEEE J. Sel. Top. Quantum Electron. |

77. | A. C. Turner, M. A. Foster, A. L Gaeda, and M. Lipson, “Ultra-low power frequency conversion in silicon microring resonators,” Proc. Conf. Lasers Electro-Optics (OSA, Washington, DC, 2007), paper CPDA3. |

78. | S. Ayotte, H. Rong, S. Xu, O. Cohen, and M. Paniccia, “Multichannel dispersion compensation using a silicon waveguide-based optical phase conjugator,” Opt. Lett. |

79. | M. A. Foster, A. C. Turner, R. Salem, M. Lipson, and A. L. Gaeta, “Broad-band continuous-wave parametric wavelength conversion in silicon nanowaveguides,” Opt. Express |

80. | G. P. Agrawal, |

81. | R. A. Soref and B. R. Bennett, “Electrooptical effects in silicon,” IEEE J. Quantum Electron. |

82. | V. Raghunathan, R. Shori, O. M. Stafsudd, and B. Jalali, “Nonlinear absorption in silicon and the prospects of mid-infrared silicon Raman lasers,” Physica Status Solidi A |

83. | A. D. Bristow, N. Rotenberg, and H. M. van Driel, “Two-photon absorption and Kerr coefficients of silicon for 850–2200 nm,” Appl. Phys. Lett. |

84. | Q. Lin, J. Zhang, G. Piredda, R. W. Boyd, P. M. Fauchet, and G. P. Agrawal, “Dispersion of silicon nonlinearities in the near-infrared region,” Appl. Phys. Lett. |

85. | M. Foster and A. L. Gaeta, “Wavelength dependence of the ultrafast third-order nonlinearity of Silicon,” Proc. Conf. Lasers Electro-Optics (OSA, Washington, DC, 2007), Paper CTuY5. |

86. | D. J. Moss, H. M. van Driel, and J. E. Sipe, “Dispersion in the anisotropy of optical third-harmonic generation in silicon,” Opt. Lett. |

87. | J. Zhang, Q. Lin, G. Piredda, R. W. Boyd, G. P. Agrawal, and P. M. Fauchet, “Anisotropic nonlinear response of silicon in the near-infrared region,” Appl. Phys. Lett. |

88. | P. E. Barclay, K. Srinivasan, and O. Painter, “Nonlinear response of silicon photonic crystal microresonators excited via an integrated waveguide and fiber taper”, Opt. Express |

89. | T. J. Johnson, M. Borselli, and O. Painter, “Self-induced optical modulation of the transmission through a high-Q silicon microdisk resonator”, Opt. Express |

90. | V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, “All-optical control of light on a silicon chip,” Nature |

91. | T. Tanabe, M. Notomi, S. Mitsugi, A. Shinya, and E. Kuramochi, “All-optical switches on a silicon chip realized using photonic crystal nanocavities,” Appl. Phys. Lett. |

92. | S. F. Preble, Q. Xu, B. S. Schmidt, and M. Lipson, “Ultrafast all-optical modulation on a silicon chip,” Opt. Lett. |

93. | T. G. Eusera and W. L. Vos, “Spatial homogeneity of optically switched semiconductor photonic crystals and of bulk semiconductors,” J. Appl. Phys. |

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

95. | F. Gan, F. J. Grawert, J. M. Schley, S. Akiyama, J. Michel, K. Wada, L. C. Kimerling, and F. X. Kärtner, “Design of all-optical switches based on carrier injection in Si/SiO2 split-ridge waveguides (SRWs),” IEEE J. Lightwave Technol. |

96. | K. Ikeda and Y. Fainman, “Nonlinear Fabry-Perot resonator with a silicon photonic crystal waveguide,” Opt. Lett. |

97. | E. Tien, N. S. Yuksek, F. Qian, and O. Boyraz, “Pulse compression and modelocking by using TPA in silicon waveguides,” Opt. Express |

98. | T. K. Liang, L. R. Nunes, T. Sakamoto, K. Sasagawa, T. Kawanishi, M. Tsuchiya, G. R. A. Priem, D. Van Thourhout, P. Dumon, R. Baets, and H. K. Tsang, “Ultrafast all-optical switching by cross-absorption modulation in silicon wire waveguides,” Opt. Express |

99. | 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. |

100. | T. K. Liang, L. R. Nunes, M. Tsuchiya, K. S. Abedin, T. Miyazaki, D. Van Thourhout, W. Bogaerts, P. Dumon, R. Baets, and H. K. Tsang, “High speed logic gate using two-photon absorption in silicon waveguides,” Opt. Commun. |

101. | J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. |

102. | P. N. Butcher and D. Cotter, |

103. | R. W. Boyd, |

104. | Y. R. Shen and N. Bloembergen, “Theory of stimulated Brillouin and Raman scattering,” Phys. Rev. |

105. | M. D. Lvenson and N. Bloembergen, “Dispersion of the nonlinear optical susceptibility tensor in centrosymmetric media,” Phys. Rev. B |

106. | M. Cardona, “Resonance phenomena,” in |

107. | R. H. Stolen, J. P. Gordon, W. J. Tomlinson, and H. A. Haus, “Raman response function of silica-core fibers,” J. Opt. Soc. Am. B |

108. | P. A. Temple and C. E. Hathaway, “Multiphonon Raman spectrum of silicon,” Phys. Rev. B |

109. | T. R. Hart, R. L. Aggarwal, and B. Lax, “Temperature dependence of Raman scattering in silicon,” Phys. Rev. B |

110. | A. Zwick and R. Carles, “Multiple-order Raman scattering in crystalline and amorphous silicon,” Phys. Rev. B |

111. | R. Loudon, “The Raman effect in crystals,” Adv. Phys. |

112. | J. R. Sandercock, “Brillouin-scattering measurements on silicon and germanium,” Phys. Rev. Lett. |

113. | M. Dinu, “Dispersion of phonon-assisted nonresonant third-order nonlinearities,” IEEE J. Quantum Electron. |

114. | H. Garcia and R. Kalyanaraman, “Phonon-assisted two-photon absorption in the presence of a dc-field: the nonlinear Franz-Keldysh effect in indirect gap semiconductor,” J. Phys. B |

115. | M. Sheik-Bahae and E. W. Van Stryland, “Optical nonlinearities in the transparency region of bulk semiconductors,” in |

116. | G. P. Agrawal, |

117. | R. W. Hellwarth, “Third-order optical susceptibilities of liquids and solids,” Prog. Quantum Electron. |

118. | P. D. Maker and R. W. Terhune, “Study of optical effects due to an induced polarization third order in the electric field strength,” Phys. Rev. |

119. | S. S. Jha and N. Bloembergen, “Nonlinear optical susceptibilities in group-IV and III–V semiconductors,” Phys. Rev. |

120. | J. J. Wynne, “Optical third-order mixing in GaAs, Ge, Si, and InAs,” Phys. Rev. |

121. | R. Buhleier, G. Lüpke, G. Marowsky, Z. Gogolak, and J. Kuhl, “Anisotropic interference of degenerate four-wave mixing in crystalline silicon,” Phys. Rev. B |

122. | W. K. Burns and N. Bloembergen, “Third-harmonic generation in absorbing media of cubic or isotropic symmetry,” Phys. Rev. B |

123. | D. J. Moss, H. M. van Driel, and J. E. Sipe, “Third harmonic generation as a structure diagonostic of ion-implanted amorphous and crystalline silicon,” Appl. Phys. Lett. |

124. | C. C. Wang, J. Bomback, W. T. Donlon, C. R. Huo, and J. V. James, “Optical third-harmonic generation in reflection from crystalline and amorphous samples of silicon,” Phys. Rev. Lett. |

125. | D. J. Moss, E. Ghahramani, J. E. Sipe, and H. M. van Driel, “Band-structure calculation of dispersion and anisotropy in 41, 1542–1560 (1990). [CrossRef] |

126. | J. F. Reintjes and J. C. McGroddy, “Indirect two-photon transition in Si at 1.06 |

127. | V. Mizrahi, K. W. DeLong, G. I. Stegeman, M. A. Saifi, and M. J. Andrejco, “Two-photon absorption as a limitation to all-optical switching,” Opt. Lett. |

128. | M. Sheik-Bahae, D. C. Hutchings, D. J. Hagan, and E. W. Van Stryland, “Dispersion of bound electronic nonlinear refraction in solids,” IEEE J. Quantum Electron. |

129. | R. Salem and T. E. Murphy, “Polarization-insensitive cross correlation using two-photon absorption in a silicon photodiode,” Opt. Lett. |

130. | T. Kagawa and S. Ooami, “Polarization dependence of two-photon absorption in Si avalanche photodiodes,” Jap. J. Appl. Phys. |

131. | S. M. Sze and K. K. Ng, |

132. | A. Othonos, “Probing ultrafast carrier and phonon dynamics in semiconductors,” J. Appl. Phys. |

133. | A. J. Sabbah and D. M. Riffe, “Femtosecond pump-probe reflectivity study of silicon carrier dynamics,” Phys. Rev. B |

134. | A. Kost, “Resonant optical nonlinearities in semiconductors,” in |

135. | R. A. Soref and B. R. Bennett, “Kramers-Kronig analysis of electro-optical switching in silicon,” Proc. SPIE |

136. | D. S. Chemla, “Ultrafast transient nonlinear optical processes in semiconductors,” in |

137. | Q. Lin, T. Johnson, R. Perahia, C. Michael, and O. J. Painter, “Highly tunable optical parametric oscillation in silicon micro-resonators,” submitted for publication. |

138. | M. J. Adams, S. Ritchie, and M. J. Robertson, “Optimum overlap of electric and optical fields in semiconductor waveguide devices,” Appl. Phys. Lett. |

139. | D. Dimitropoulos, R. Jhaveri, R. Claps, J. C. S. Woo, and B. Jalali, “Lifetime of photogenerated carriers in silicon-on-insulator rib waveguides,” Appl. Phys. Lett. |

140. | Y. Liu and H. K. Tsang, “Nonlinear absorption and Raman gain in helium-ion-implanted silicon waveguides,” Opt. Lett. |

141. | Y. Liu and H. K. Tsang, “Time dependent density of free carriers generated by two photon absorption in silicon waveguides,” Appl. Phys. Lett. |

142. | M. Först, J. Niehusmann, T. Plötzing, J. Bolten, T. Wahlbrink, C. Moormann, and H. Kurz, “High-speed all-optical switching in ion-implanted silicon-on-insulator microring resonators,” Opt. Lett. |

143. | T. Tanabe, K. Nishiguchi, A. Shinya, E. Kuramochi, H. Inokawa, and M. Notomi, “Fast all-optical switching using ion-implanted silicon photonic crystal nanocavities,” Appl. Phys. Lett. |

144. | D. Dimitropoulos, S. Fathpour, and B. Jalali, “Limitations of active carrier removal in silicon Raman amplifiers and lasers,” Appl. Phys. Lett. |

145. | J. M. Ralston and R. K. Chang, “Spontaneous-Raman-scattering efficiency and stimulated scattering in silicon”, Phys. Rev. B |

146. | J. B. Renucci, R. N. Tyte, and M. Cardona, “Resonant Raman scattering in silicon”, Phys. Rev. B |

147. | T. A. Birks, W. J. Wadsworth, and P. St. J. Russell, “Supercontinuum generation in tapered fibers,” Opt. Lett. |

148. | P. St. J. Russell, “Photonic crystal fibers,” IEEE J. Lightwave Technol. |

149. | A. C. Turner, C. Manolatou, B. S. Schmidt, M. Lipson, M. A. Foster, J. E. Sharping, and A. L. Gaeta, “Tailored anomalous group-velocity dispersion in silicon channel waveguides,” Opt. Express |

150. | N. Bloembergen and P. Lallemand, “Complex intensity-dependent index of refraction, frequency broadening of stimulated Raman scattering, and stimulated Rayleigh scattering,” Phys. Rev. Lett. |

151. | K. Kikuchi, “Highly sensitive interferometric autocorrelator using Si avalanche photodiode as two-photon absorber,” Electron. Lett. |

152. | C. Xu, J. M. Roth, W. H. Knox, and K. Bergman, “Ultra-sensitive autocorrelation of 1.5 |

153. | T. K. Liang, H. K. Tsang, T. E. Day, J. Drake, A. P. Knights, and M. Asghari, “Silicon waveguide two-photon absorption detector at 1.5 |

154. | D. Panasenko and Y. Fainman, “Single-shot sonogram generation for femtosecond laser pulse diagnostics by use of two-photon absorption in a silicon CCD camera,” Opt. Lett. |

155. | R. Salem, G. E. Tudury, T. U. Horton, G. M. Carter, and T. E. Murphy, “Polarization-insensitive optical clock recovery at 80 Gb/s using a silicon photodiode,” IEEE Photon. Technol. Lett. |

156. | M. Dinu, D. C. Kilper, and H. R. Stuart, “Optical performance monitoring using data stream intensity autocorrelation,” IEEE J. Lightwave Technol. |

157. | K. Taira, Y. Fukuchi, R. Ohta, K. Katoh, and K. Kikuchi, “Background-free intensity autocorrelator employing Si avalanche photodiode as two-photon absorber,” Electron. Lett. |

158. | T. J. Kippenberg, S. M. Spillane, B. Min, and K. J. Vahala, “Theoretical and Experimental Study of Stimulated and Cascaded Raman Scattering in Ultrahigh-Q Optical Microcavities,” IEEE J. Sel. Top. Quantum Electron. |

159. | J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron. |

160. | S. Fathpour, K. K. Tsia, and B. Jalali, “Energy harvesting in silicon Raman amplifiers,” Appl. Phys. Lett. |

161. | K. K. Tsia, S. Fathpour, and B. Jalali, “Energy harvesting in silicon wavelength converters,” Opt. Express |

162. | T. Torounidis and P. Andrekson, “Broadband single-pumped fiber-optic parametric amplifiers,” IEEE Photon. Technol. Lett. |

163. | J. M. Chavez Boggio, J. D. Marconi, S. R. Bickham, and H. L. Fragnito, “Spectrally flat and broadband doublepumped fiber optical parametric amplifiers,” Opt. Express |

164. | M. D. Levenson, C. Flytzanis, and N. Bloembergen, “Interference of resonant and nonresonant three-wave mixing in diamond,” Phys. Rev. B |

165. | M. D. Levenson and S. Kano, |

166. | B. Jalali, V. Raghunathan, R. Shori, S. Fathpour, D. Dimitropoulos, and O. Stafsudd, “Prospects for silicon Mid-IR Raman Lasers,” IEEE J. Sel. Top. Quantum Electron. |

167. | H. Takesue and K. Inoue, “Generation of polarization-entangled photon pairs and violation of Bell’s inequality using spontaneous four-wave mixing in a fiber loop,” Phys. Rev. A |

168. | X. Li, P. L. Voss, J. E. Sharping, and P. Kumar, “Optical-Fiber Source of Polarization-Entangled Photons in the 1550 nm Telecom Band,” Phys. Rev. Lett. |

169. | J. Fulconis, O. Alibart, W. J. Wadsworth, P. St. J. Russell, and J. G. Rarity, “High brightness single mode source of correlated photon pairs using a photonic crystal fiber,” Opt. Express |

170. | J. Fan, A. Migdall, and L. J. Wang, “Efficient generation of correlated photon pairs in a microstructure fiber,” Opt. Lett. |

171. | X. Li, J. Chen, P. Voss, J. Sharping, and P. Kumar, “All-fiber photon-pair source for quantum communications: Improved generation of correlated photons,” Opt. Express |

172. | Q. Lin, F. Yaman, and G. P. Agrawal, “Photon-pair generation by four-wave mixing in optical fibers,” Opt. Lett. |

173. | Q. Lin, F. Yaman, and G. P. Agrawal, “Photon-pair generation in optical fibers through four-wave mixing: Role of Raman scattering and pump polarization,” Phys. Rev. A |

174. | H. Takesue and K. Inoue, “1.5- |

175. | K. F. Lee, J. Chen, C. Liang, X. Li, P. L. Voss, and P. Kumar, “Generation of high-purity telecom-band entangled photon pairs in dispersion-shifted fiber,” Opt. Lett. |

**OCIS Codes**

(130.3120) Integrated optics : Integrated optics devices

(190.5970) Nonlinear optics : Semiconductor nonlinear optics including MQW

(320.7110) Ultrafast optics : Ultrafast nonlinear optics

(250.4390) Optoelectronics : Nonlinear optics, integrated optics

**ToC Category:**

Nonlinear Optics for Functional Devices and Applications

**History**

Original Manuscript: October 9, 2007

Revised Manuscript: November 22, 2007

Manuscript Accepted: November 25, 2007

Published: November 29, 2007

**Virtual Issues**

Focus Serial: Frontiers of Nonlinear Optics (2007) *Optics Express*

**Citation**

Q. Lin, Oskar J. Painter, and Govind P. Agrawal, "Nonlinear optical phenomena in silicon waveguides: modeling and applications," Opt. Express **15**, 16604-16644 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-25-16604

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