## 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

Silicon photonics has attracted much attention recently because of its potential applications in the spectral region extending from near- to mid-infrared [1–3]. Silicon crystals, with a band gap near 1.12 eV, become nearly transparent in the spectral region beyond 1.1 *µ*m and exhibit optical properties that are useful for a variety of applications [4]. The high refractive index of silicon allows for a tight confinement of optical waves to a sub-micron region using the silicon-on-insulator (SOI) technology [1]. Moreover, silicon exhibits a large third-order nonlinearity, with a Kerr coefficient more than 100 times larger [5] and a Raman gain coefficient more than 1000 times larger [6] than those of silica glass in the telecommunication band. These features enable efficient nonlinear interaction of optical waves at relatively low power levels inside a short SOI waveguide (<5 cm long). For this reason, considerable effort has been directed in recent years toward investigating the nonlinear phenomena such as self-phase modulation (SPM) [7–23], cross-phase modulation (XPM) [14,24–26], stimulated Raman scattering (SRS) [27–60], and four-wave mixing (FWM) [61–79]. All of these nonlinear effects are currently being explored to realize a variety of optical functions on the chip scale.

Although the third-order nonlinear effects have been studied extensively for silica fibers [80], and these studies can be used as a guidance for SOI waveguides, it is important to remember that silicon is a semiconductor crystal exhibiting unique features such as two-photon absorption (TPA) [5, 7], free-carrier dispersion [81], and anisotropic and dispersive third-order nonlinearity [5, 82–87]. The interplay among various dispersive and nonlinear effects leads to many interesting features [8–25, 88, 89] that provide new functionalities on the one hand [90–100], but may become obstacles in some cases on the other [25,32,33,35–37,39,65–68,89]. Therefore, it is important to have a unified theoretical platform that not only can be used for understanding the underlying physics but also provides guidance toward new and useful applications. Indeed, considerable efforts have been made in the past few years to develop a theoretical approach for understanding a specific nonlinear phenomenon [13–16, 20–22, 26, 28, 46–48, 50, 55, 58, 59, 62, 66, 68, 70, 73, 74, 76, 88, 96]. In this paper, we review recent progress realized in modeling nonlinear phenomena inside SOI waveguides and develop a unified theoretical platform. We then apply it to investigate various nonlinear effects occurring inside silicon waveguides with a keen eye toward their applications.

The paper is organized as follows. We begin in Section 2 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 in Section 3 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 in Section 4 with emphasis on the impact of free carriers on Raman amplification and lasing. Section 5 focuses on the FWM process and its applications. We consider first the impact of free carriers and show that, although index changes induced by them have a negligible impact on FWM, free-carrier absorption limits the FWM efficiency so much that a net positive gain is difficult to be realized with CW pumping in the telecommunication band. However, this problem can be solved by pumping at wavelengths beyond 2.2 *µ*m because TPA-induced free carriers are then absent. We also show that FWM can occur over a wide bandwidth (>300 nm), with a proper choice of the pump wavelength, because of much smaller waveguide lengths employed compared with those required for silica fibers. We discuss briefly the use of FWM in silicon waveguides for generating correlated photon pairs that are useful for quantum applications.

## 2. General formalism

The nonlinear interactions of optical waves inside silica fibers are well understood owing to extensive investigations over the past few decades [80]. The so-called generalized nonlinear Schrödinger (NLS) equation provides a fairly accurate description, even for ultrashort pulses creating an octave-spanning supercontinuum [101]. As the origin of third-order nonlinearity in silicon waveguides is quite similar to that for silica fibers, a similar theoretical formalism can be used for silicon waveguides, with suitable modifications to account for the features unique to silicon. In this section, we develop a general formalism that can be used to describe various nonlinear effects inside silicon waveguides.

#### 2.1. Third-order susceptibility of silicon

As silicon crystals exhibit an inversion symmetry, the lowest-order nonlinear effects stem from the third-order susceptibility *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]

where *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.

The third-order susceptibility of silicon has two dominant contributions, one from bound electrons and the other from optical phonons, and it is useful to write it as *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}Consider the Raman contribution first. Raman scattering in silicon is dominated by optical phonons near the Brillouin-zone center. As these phonons have three degenerate normal modes (with the same phonon energy), the Raman susceptibility is given by [104–106]

where the spectral response of Raman scattering, *H*̃* _{R}*(Ω), is the same for three normal modes. Unlike silica glass which has a very broad Raman spectrum [107], optical phonons at the Brillouin-zone center of silicon have a well-defined frequency of Ω

*/2*

_{R}*π*=15.6 THz at room temperature [108–110], resulting in a relatively narrow Raman-gain spectrum, with a Lorentzian shape of the form [104–106]

Here Γ* _{R}* is related inversely to the phonon lifetime (about 3 ps), resulting in a full width at half-maximum (FWHM) of the Raman-gain spectrum of Γ

*/*

_{R}*π*≈105 GHz at room temperature [108]. In general, both Ω

*and Γ*

_{R}*are sensitive to temperature [37, 108, 109]. As an example, Ω*

_{R}*/2*

_{R}*π*decreases by ~40 GHz, while Γ

*/*

_{R}*π*increases by ~10 GHz, if temperature increases by 50°C. The parameter g′ in Eq. (2) is related to the Raman polarizability associated with the Brillouin-zone-center phonons and is nearly independent of optical frequency [102, 104, 106]. It determines the peak value of the Raman gain coefficient that is discussed later in Section 2.4.

In Eq. (2), the Raman tensor ℜ*v*
* _{ij}* describes polarization dependence of Raman scattering. As the three phonon modes belong to the Γ

_{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]

where we have assumed that *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:

$$+g\prime {\tilde{H}}_{R}\left({\omega}_{j}-{\omega}_{k}\right)\left({\delta}_{\mathrm{ik}}{\delta}_{\mathrm{jl}}+{\delta}_{\mathrm{ij}}{\delta}_{\mathrm{kl}}-2{\delta}_{\mathrm{ijkl}}\right),$$

where *δ _{ijkl}* equals 1 only when

*i*=

*j*=

*k*=

*l*and is 0 otherwise.

The Raman susceptibility in Eq. (5) describes Raman scattering involving a single phonon. Apart from this dominant contribution, Raman scattering can also involve multiple phonons simultaneously, a process known as higher-order Raman scattering [106, 108, 110]. In the case of silicon, the second-order Raman scattering exhibits a broadband peak at a frequency near 29 THz resulting from two transverse optical phonons and another peak around 9 THz associated with two transverse acoustic phonons. However, as their amplitudes are more than 50 and 200 times smaller, respectively, compared with the dominant first-order Raman peak [108,110], higher-order Raman effects are negligible in most practical situations.

Optical waves can also interact with a single acoustic phonon and experience Brillouin scattering. However, the relative strength of this process in silicon is nearly two orders of magnitude smaller than that of first-order Raman scattering [112]. As a result, Brillouin scattering is negligible in most cases. This is in strong contrast to silica glass, where Brillouin scattering is three orders of magnitude stronger than Raman scattering [80]. In this paper, we neglect the effects of Brillouin scattering.

We next focus on the electronic contribution stemming from oscillations of bound electrons and leading to the optical Kerr effect through intensity-dependent changes in the refractive index. It also leads to two-photon absorption (TPA) whenever the energy of photons exceeds the half band gap *E _{g}*/2, where

*E*≈1.12 eV for silicon corresponds to a wavelength of 1.1

_{g}*µ*m. Bound electrons in the valence band can be excited to the conduction band through TPA by absorbing two photons with total energy exceeding

*E*. This process has to be assisted by phonons to conserve momentum because of the indirect nature of the band gap in silicon [113,114]. The response time of both the Kerr effect and the TPA process are extremely short. As an estimate, the uncertainty principle implies that non-resonant electronic transition occurs on a time scale of |

_{g}*ω*-

*E*/

_{g}*h*̄|

^{-1}. For frequencies well below the band gap, the response time is below 10 fs [102, 115]. Such a nearly instantaneous nature of electronic nonlinear response leads to many potential applications related to high-speed optical signal processing [116].

Equation (1) shows that a complete description of third-order nonlinear effects requires knowledge of the tensorial and dispersive properties of *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, 118]:

where *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, 118–121]. As a result, Eq. (6) reduces to

where *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 [86,118,119,122–125]. Also note that the value of

*ρ*can be complex in general.

To characterize the electronic nonlinearity, one needs to measure *χ ^{e}*

_{1111}(-

*ω*;

*ω*,-

*ω*,

*ω*). The real and imaginary parts are related to the Kerr coefficient

*n*

_{2}and the TPA coefficient

*β*as [80, 102]

_{T}where *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 _{T}* over a wide frequency range [5–7, 9, 11, 12, 24, 65,82–85, 93, 126]. The value of

*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] defined as

*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, 84]. Recent experiments [5, 82–85] show that

*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]. Unlike a direct band-gap semiconductor for which

*n*

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

*E*[128],

_{g}*n*

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

*E*[83, 84]. This is a consequence of the phonon-assisted nature of electronic nonlinearity [113, 114].

_{g}The magnitude of nonlinear anisotropy is related to the band structure of a silicon crystal. It has been measured in the past four decades through third-harmonic generation [86, 122–125], degenerate FWM [120, 121], and the z-scan technique [87]. The magnitude of *ρ* is quite dispersive in the opaque region (*h*̄*ω*>*E _{g}*) [86], but it becomes nearly constant for

*h*̄

*ω*<

*E*[86, 87, 120]. The measured value of

_{g}*ρ*is close to 1.27 in the telecom band, with a negligible imaginary part [87]. The absence of an imaginary part indicates that anisotropy is the same for both the Kerr and TPA coefficients. From the standpoint of applications, nonlinear anisotropy affects the polarization dependence of nonlinear phenomena inside silicon waveguides. Recently, photocurrent induced by TPA inside a silicon avalanche photodiode (APD) was used to characterize the anisotropy of TPA [129, 130]. In contrast to the optical characterization, TPA-induced photocurrents indicated that the TPA process was isotropic inside a silicon APD. This discrepancy between the optical and electrical measurements remains an open issue at this time. One possible reason may be related to the nonlinear Franz-Keldysh effect [114] introduced by the high dc field inside an APD.

#### 2.2. Free-carrier effects

The TPA process may generate a considerable number of free electrons and holes, depending on the peak power associated with the incident optical field. These excessive carriers not only absorb light but also affect the nature of wave propagation by changing the refractive index. As the electron and hole mobilities, *µ _{e}* and

*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, 102, 132–134], and the induced polarization varies linearly with the carrier densities as

where *N _{e}* and

*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}The carrier-induced polarization in the frequency domain is thus given by

where the spectral response *χ*̃^{f} is defined as

Here *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}Equation (10) indicates how the spectral response of free carriers varies with optical frequency. However, in most cases of nonlinear interactions, the optical field consists of only a few waves at specific carrier frequencies *ω _{u}*, 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,

*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}where *E _{i}*(

**,**

*r**ω*,

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

*ω*and the induced susceptibility is given by

_{u}As the susceptibility *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,

where *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 _{v}*≫1) [81, 102, 134]

In practice, their magnitudes for silicon are larger than those predicted by Eqs. (16) and (17). At a specific wavelength of λ* _{r}*=2

*πc*/

*ω*=1550 nm, it is common to employ the following empirical formulas [81, 135]

_{r}where *N _{e}* and

*N*have units of cm

_{h}^{-3}and α

_{f}is expressed in units of cm

^{-1}.

In the case of nonlinear optical interactions, free carriers are created through TPA with equal densities, i.e., *N _{e}*=

*N*≡

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

*n*

_{f}and α

_{f}as

where σ* _{a}*=1.45×10

^{-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.

Equations (16), (17), and (20) show that FCI and FCA changes with time reflect precisely temporal variations of free carriers. This feature has been widely used to investigate the carrier dynamics in semiconductors by probing the transient FCI and FCA excited by an ultrashort pump pulse [132, 133]. Note that the model presented in this section for free-carrier effects is valid only when free carriers reach a certain thermal quasi-equilibrium. This is the case for time scales >50 fs because optically excited free carriers can be thermalized over such time intervals by various scattering processes [132, 133, 136]. In the context of nonlinear silicon photonics, almost all practical applications fall in this regime. However, a quantum-mechanical description is necessary for accurate modeling of free-carrier dynamics for ultrashort pulses containing only of a few optical cycles [136].

Apart from density changes, optical excitation of free carriers between nonparabolic conduction and valence bands may also modify the effective masses of electrons and holes and thus introduce additional nonlinear effects associated with free carriers [134]. However, this effect is minor compared with the changes in the carrier density and is neglected in this paper.

#### 2.3. General frequency-domain wave equation

In this subsection we derive a general equation describing propagation of an optical field inside a silicon waveguide. As usual, the starting point is the Maxwell wave equation. After including the induced polarizations associated with the third-order nonlinearity and free carriers, the wave equation in the frequency domain takes the form

In general, the induced polarizations are only small perturbations to the linear wave equation. To the first order, we can assume that the waveguide modes are not affected by them and write the electric field in the form

where *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}where *β _{i}*(

*ω*) is the propagation constant given by

and the subscript *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.

The general solution of Eq. (23), in the absence of the free-carrier and nonlinear effects, consists of the forward and backward propagatingwaves with the phase factors ${e}^{\pm i{\beta}_{i}\left(\omega \right)z}$. This phase factor varies in a length scale of optical wavelength, much shorter than the length scale in which the small free-carrier and third-order nonlinear effects evolve. As a result, *∂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

As a result, Eq. (23) reduces to

By substituting Eq. (22) into Eqs. (1) and (11) and using them in Eq. (26), we obtain the following equation for the field amplitude:

$$+\frac{i}{4{\pi}^{2}}\iint {\gamma}_{\mathrm{ijkl}}(-\omega ;{\omega}_{j},-{\omega}_{k},{\omega}_{l}){A}_{j}(z,{\omega}_{j}){A}_{k}^{*}(z,{\omega}_{k}){A}_{l}(z,{\omega}_{l})d{\omega}_{j}d{\omega}_{k},$$

where we have normalized the field amplitude such that the corresponding temporal profile |*A*(*z*,*t*)|^{2} has units of power. The nonlinear parameter *γ _{ijkl}* in Eq. (27) is defined as

where *n _{v}*(

*ω*)=

_{v}*cβ*(

_{v}*ω*)/

_{v}*ω*(

_{v}*v*=

*i*,

*j*,

*k*,

*l*) is the modal refractive index at

*ω*,

_{v}*a*̄ is the

*average*effective mode area related to that at individual frequencies as

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

The integral in the numerator of *a _{v}* 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.

Similar to *χ*
^{(3)}
* _{ijkl}*, the nonlinear parameter

*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}In Eq. (27), the second term on the right side represents the effect of free carriers with *β*̃^{f}
* _{i}* given by

where *χ*̃^{f} is given in Eq. (12). In general, free carriers have specific transverse density distributions inside the waveguide, and *β*̃^{f}
* _{i}* includes the effect of a partial overlap between the charge distribution and the mode profile [138]. If we assume that

*χ*̃

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

The physical meaning of free-carrier effects can be seen more clearly in the time domain. The general form of the free-carrier term in Eq. (27) can be simplified considerably if we notice that nonlinear interactions typically involve only a few optical waves, each with a limited bandwidth around a carrier frequency *ω _{u}*. 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

where *n*
_{f} and α_{f} are given by Eqs. (16) and (17) [or Eqs. (18)–(20)] after replacing *N _{e}* and

*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.

Free electrons and holes can be generated either optically inside the waveguide or electrically through current injection from outside. After being created, they can diffuse to a low-density area through thermal motion, or drift away by an external dc electric field *E*_{dc}. In general, the dynamics of carrier density is governed by the continuity equation [131]

where *v*=*e* for electrons, *v*=*h* for holes, *s _{h}*=1,

*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.

What we are interested in is not a detailed density distribution across the waveguide but its effect on the optical field. Thus, we can average Eq. (34) over the transverse coordinates to obtain a dynamic equation for the average density *N*̄* _{v}*. 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

where *τ*
^{*}
* _{v}* 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

where *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}Note from Eq. (35) that the effective lifetime in general depends on the waveguide geometry. Indeed, it can be reduced considerably by increasing the surface-to-volume ratio of the waveguide [139]. It can also be reduced by applying an external dc field [42, 67] or introducing non-radiative centers through ion implantation [140–143]. However, as nonlinear interactions require relatively high intensity inside the waveguide, generated carriers may screen the applied dc field and thus limit the reduction in effective carrier lifetime [144]. For SOI waveguides commonly used for photonic applications, the effective lifetime varies from sub-nanosecond [54, 90, 141] to tens of nanoseconds [41, 42].

Equation (27) is quite general as it includes all dispersive, nonlinear, and polarization effects introduced by waveguide confinement, free carriers and third-order nonlinearity. The first term on its right side governs the dispersive effects. It can also include linear absorptive and scattering losses, if *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]. As α_{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 [83–85]. Thus, if the incident and generated optical fields have a bandwidth much smaller than 100 nm, we can employ the approximation *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

In many practical situations, the input field is either a continuous wave (CW) or is in the form of a pulse train. If only one wave at the carrier frequency *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*):

where *β _{im}* is the

*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*̂).

The nonlinear polarization *P ^{NL}_{i}* (

*z*,

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

where the third-order nonlinear response function is given by

$$+{\gamma}_{R}{h}_{R}\left(\tau \right)\left({\delta}_{\mathrm{ik}}{\delta}_{\mathrm{jl}}+{\delta}_{\mathrm{il}}{\delta}_{\mathrm{jk}}-2{\delta}_{\mathrm{ijkl}}\right),$$

The electronic nonlinear parameter γ* _{e}*(

*ω*

_{0}) is defined as

where *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.

The Raman nonlinear parameter *γ _{R}* is related to the Raman gain coefficient

*g*as

_{R}*γ*(

_{R}*ω*

_{0})=

*g*(

_{R}*ω*

_{0})Γ

*/(*

_{R}*a*̄Ω

*), where*

_{R}*g*is obtained from Eqs. (3) and (28) and is given by

_{R}*g*(

_{R}*ω*

_{0})=3

*ω*

_{0}

*g*′Ω

*/(2*

_{R}*ε*

_{0}

*c*

^{2}

*n*

^{2}Γ

*). Note that*

_{R}*g*is linearly proportional to

_{R}*ω*

_{0}. In the 1.55-

*µ*m spectral band, its values are in the range of 4.3 to 76 cm/GW, based on the experimental measurements of spontaneous and stimulated Raman scattering [6, 27, 29, 30, 32, 34, 41, 145, 146]. Thus,

*g*is more than 1000 times larger than that found for silica [80]. It is also nearly 10 times larger than the electronic nonlinear parameter of silicon. However, as

_{R}*γ*depends linearly on Γ

_{R}*/Ω*

_{R}*, the magnitude of*

_{R}*γ*is only a few percent of

_{R}*γ*

_{0}and is negligible unless the pulse spectrum falls within the Raman gain bandwidth. The Raman response function

*h*

*(*

_{R}*τ*) in Eq. (39) is the inverse Fourier transform of

*H*̃

*(Ω). Using Eq. (3), it is found to be [*

_{R}*h*(

_{R}*t*)=0 for

*t*<0]

where τ_{2}=1/Γ* _{R}* and τ

_{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.

Although Eq. (39) shows that the third-order nonlinearity involves all three components of the electric field, the longitudinal field component *E _{z}* 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

*E*component, as far as the nonlinear effects are concerned.

_{z}As discussed in Section 2.1, Eq. (39) is written in a coordinate basis aligned along the crystallographic axes. For commonly used (1 0 0) silicon wafers, this amounts to assuming that the waveguide is fabricated along the [0 1 0] or [0 0 1] direction. If that is not the case, the nonlinear response in other Cartesian coordinate systems can be found by a suitable rotation of the basis, resulting in *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}$$+{h}_{R}\left(\tau \right)\left({\delta}_{\mathrm{ik}}{\delta}_{\mathrm{jl}}+{\delta}_{\mathrm{il}}{\delta}_{\mathrm{jk}}-2\sum _{s}{M}_{\mathrm{si}}{M}_{\mathrm{sj}}{M}_{\mathrm{sk}}{M}_{\mathrm{sl}}\right).$$

As an example, consider an SOI waveguide fabricated along the [0 1̄ 1] direction because of cleaving convenience. In this case, the new coordinate system is obtained by a 45 ° rotation along the *x* axis, as shown in Fig. 1, and the rotation matrix is given by

In this case, it is easy to show that

The two components in Eq. (44) represent the nonlinear response of an optical field polarized linearly along the *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]. Note that Raman scattering can occur for TE modes, but not for TM modes. In contrast, Raman coupling between the pump and signal waves, polarized orthogonally along the TE and TM modes, occurs with the same magnitude, [28, 34, 52] but the Kerr nonlinearity is reduced roughly by one third for such a polarization configuration [80].

In Eq. (37), ξ is related to the frequency dependence of the nonlinear parameter, and is given by

where *χ _{e}*(

*ω*

_{0})≡

*χ*

^{e}_{1111}(-

*ω*

_{0};

*ω*

_{0},-

*ω*

_{0},

*ω*

_{0}). In the 1.55-

*µ*m spectral region, the contribution of the second term in Eq. (46), related to the dispersion of silicon electronic nonlinearity [5, 83, 84], dominates, in contrast with silica fibers for which the first term dominates because of a negligible dispersion of

*n*

_{2}. In practice, the ξ term becomes important only when the spectrum of input pulses broadens by a large factor during their propagation inside a silicon waveguide; for example, in the case of supercontinuum generation [16]. In many cases of practical interest, ξ≪1, and its contribution can be neglected.

Equation (27) in the frequency domain and Eq. (37) in the time domain provide a general theoretical basis for studying the nonlinear effects in silicon waveguides. We use them in the following sections to discuss a variety of nonlinear phenomena that can be employed for a multitude of practical application of silicon waveguides.

## 3. SPM effects on short optical pulses

In this section, we apply the general theory to the simplest case in which short optical pulses at a specific carrier wavelength are launched inside a silicon waveguide. The propagation of such pulses is affected considerably by the nonlinear phenomenon of SPM, especially when the dispersive effects cannot be ignored. As Eq. (37) is quite similar to the generalized NLS equation governing the nonlinear effects inside silica fibers [80], we would expect similar phenomena to occur inside a silicon waveguide. In particular, the possibility of soliton formation and supercontinuum generation exists. Of course, the differences unique to silicon waveguides, such as TPA and free-carrier effects, will have a significant impact on these nonlinear phenomena. This section focuses on such differences.

#### 3.1. Dispersion engineering

Before we discuss the SPM effects, we should consider the nature of group-velocity dispersion (GVD) inside SOI waveguides, as it has a profound impact on nonlinear pulse propagation [80]. Fused silica has a zero-dispersion wavelength (ZDWL) around 1.276 *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], or by introducing a photonic-crystal cladding [148]. Optical pulses exhibit rich nonlinear dynamics inside such fibers with engineered dispersion [101]. The idea of dispersion engineering can be transferred directly to silicon waveguides. Moreover, as the refractive index of silicon (around 3.48) is much larger than silica glass, mode confinement is naturally stronger in silicon waveguides, particularly those with an air cladding. This feature enables efficient dispersion engineering simply by changing the size and aspect ratio of a rectangular waveguide [13, 149].

In general, the GVD for an optical mode is dominantly set by the boundaries where the electric field is discontinuous. As a result, the quasi-TE modes tend to be more sensitive to the two sidewalls and the waveguide width. In contrast, the quasi-TM modes are more sensitive to the top and bottom interfaces and the waveguide height. This is illustrated clearly in Fig. 2, which shows the wavelength dependence of the GVD parameter *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.

Further dispersion engineering can be realized by introducing a photonic-crystal cladding, again similar to the case of silica fibers [148]. Such dispersion engineering enables observation of nonlinear phenomena inside a short waveguide at moderate power levels to an extent that is not possible for silica fibers. Moreover, an appropriate design of the waveguide also allows one to obtain nearly identical GVD curves for the two polarizations [68]. The extent of dispersive effects on pulse propagation is characterized by the dispersion length, defined as *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

As mentioned earlier, the free-carrier effects have a significant impact on pulse propagation. Thus, our first task is to estimate the relative importance of FCA and FCI compared with the nonlinear phenomena of SPM and TPA. The free-carrier effects depend, in general, on the repetition rate of input pulses. In this section, we consider the simplest case and assume that the repetition rate is low enough (<100 MHz) that free carriers produced by one pulse have enough time to recombine before the next pulse arrives. This simplification allows us to focus on a single pulse. Many experiments carried out in the past few years have employed a repetition rate low enough that the single-pulse assumption holds for them [7–12, 14, 15, 17, 23, 25].

The importance of the nonlinear effects is governed by a parameter known as the nonlinear length [80] and defined as *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 [7–12]. Such spectral broadening has already been used to realize useful functions such as optical gating [10], regeneration [19] and multichannel spectral carving [18]. In many cases, a shift of the broadened spectrum toward shorter wavelengths was observed [8–10, 12, 14, 18, 19, 25]. Detailed numerical simulations show that this blue shift is caused by a free-carrier-induced chirp (FCC) that affects pulse dynamics on a different time scale [9, 14, 20–22, 25].

_{N}Equation (37) can be used to describe pulse propagation inside SOI waveguides under quite general conditions. For simplifying the following discussion, we do not consider the polarization effects by assuming that the input pulse is polarized along the fundamental TE or TM mode of the waveguide. In this case, the pulse maintains its initial state of polarization, and a scalar approach can be employed in Eq. (37) by dropping the subscript *i* from *A _{i}*. 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.

Even with these simplifications, the situation is more complicated than that encountered inside silica fibers. Equation (37) shows that both the electronic nonlinearity and free carriers would induce nonlinear phase shifts and extra losses on the pulse. Here we provide a simple analytical approach to estimate their relative magnitudes. Assuming that the dispersion length is so much longer than the waveguide length that dispersive effects are negligible, we neglect all dispersion terms in Eq. (37) and obtain the simple equation

where we have also neglected linear scattering losses and changed the time reference frame to *τ*=*t*-*β*
_{1}
*z* so that *τ*=0 corresponds to the pulse center.

Consider first the relative importance of the FCA and TPA effects. Noting that the TPA originates from the imaginary part of γ* _{e}* in Eq. (40), the power-loss rate introduced by TPA is given by

*∂P*/

*∂z*=-

*β*|

_{T}*A*|

^{4}/

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

The use of this expression in Eq. (36) provides a formal solution for the carrier density as

Equation (49) shows clearly that carriers accumulate over the pulse duration through TPA, but they also disappear because of recombination. The upper limit of the carrier density is thus the value accumulated over the whole pulse without recombination, and is given by

To be specific, consider a Gaussian input pulse with the power profile |*A*(0,*τ*)|^{2}=*P*
_{0} exp(-*τ*
^{2}/*T*
^{2}
_{0}). In this case, *N*̄* _{m}* becomes

If we substitute this expression into Eq. (20) and use Eq. (33), we obtain the upper limit α* _{fm}* of the FCA parameter α

*. 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

where *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.

It follows from Eq. (52) that the FCA effects are relatively small when *r _{a}*≪1. Using typical parameter values at wavelengths near 1550 nm, Eq. (52) leads to a condition,

*𝓔*/

_{p}*ā*≪25 mJ/cm

^{2}. For example, pulse energy should satisfy

*𝓔*

_{p}≪125 pJ for a waveguide with

*ā*=0.5

*µ*m

^{2}. This condition is often satisfied in practice for pulses shorter than 0.1 ns. As a result, TPA dominates for ultrashort pulses propagating inside a silicon waveguide. Moreover,

*α*is related to

_{fm}*r*as

_{a}*α*=

_{fm}*r*

_{a}β_{T}P_{0}/

*ā*. With

*r*≪1 and

_{a}*β*≈0.5 cm/GW,

_{T}*α*≪0.4 dB/cm at a peak intensity of 0.2 GW/cm

_{fm}^{2}. Clearly, FCA is negligible for ultrashort pulses of modest energies. However, it can become significant when pulse widths become longer than the carrier lifetime and the peak powers become relatively high. In this case, free carriers created by the leading edge of a pulse would absorb light in the pulse tail. As a result, FCA will dampen the pulse tail but leave intact the leading part of a pulse. This feature has been recently used to generate short pulses by removing their trailing part through FCA [97].

We next focus on the role played by index changes induced by free carriers (FCI). Both the Kerr nonlinearity and free carriers change the refractive index and impose SPM on the pulse. From Eqs. (20), (33), and (47), these phase changes evolve as

where Φ* _{K}* and Φ

*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

¿From Eqs. (36), (48) and (49), we find that changes in the carrier density evolve as

Optical pulses used for investigatingSPM generally have widths shorter than the carrier lifetime to prevent severe FCA. For such pulses, the second term in Eq. (55) is of the order of *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

Noting that σ* _{n}* 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

On the other hand, the Kerr-induced chirp (KIC) has a maximum growth rate of |*∂*(*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

Similar to the case of *r _{a}*, this ratio depends only on the pulse energy (rather than on pulse width or peak power alone). The FCC and KIC become comparable (

*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. [8–10, 12, 14, 18, 19] fall in this regime and result in a blue spectral shift. However, the experiments of Refs. [15, 17] fall outside of this regime and thus do not show such a blue shift. Clearly, compared with FCA, FCC has a more significant effect on pulse propagation, especially when the pulse width is much shorter than the carrier lifetime. Moreover, such FCC can be significantly enhanced inside a high-quality microcavity through cavity resonance, leading to much less requirement of input pulse energy [91, 96].

Figure 3 shows numerical examples of SPM-induced spectral broadening and phase shifts for Gaussian pulses with *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.

For relatively low-energy pulses such that *r _{c}*≪1 in Eq. (58), free-carrier effects become negligible. In this case, we can set

*β*

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

where we have added in the term representing linear losses. This equation can be easily solved to obtain the following solution of pulse power:

Accordingly, the Kerr-induced nonlinear phase shift, Φ* _{K}*=

*γ*

_{0}∫

^{L}_{0}

*P*(

*z*,

*t*)

*dz*, is given by [21]

where ${L}_{\mathrm{eff}}=\left(1-{e}^{-{\alpha}_{l}L}\right)\u2044{\alpha}_{l}$ is the effective length. TPA converts linear dependence of Φ* _{K}* on the peak power to a logarithmic one. As a result, it reduces the value of Φ

*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

The preceding analysis neglected the dispersive effects. However, if the dispersion length becomes smaller or comparable to the waveguide length, GVD effects must be considered. If we keep the GVD term, neglect the high-order dispersive and nonlinear effects, and ignore the impact of free carriers, Eq. (37) reduces to the following simpler NLS equation:

The main difference compared with optical fibers is that *γ _{e}* 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]. Such a solitary wave corresponds to a path-averaged soliton [80] for which SPM and GVD do not balance each other locally. Rather, they are balanced on average along the whole device length such that

*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].

More interestingly, if we launch a Gaussian pulse with an appropriate power in the anomalous GVD regime of an SOI waveguide, the pulse becomes quite resistant to the dispersion-induced broadening [17]. It evolves itself into a sech-like shape, as shown clearly in the numerical example of Fig. 4. In this case, the pulse spectrum narrows down since the Gaussian-shape input spectrum is too broad to hold a solitary wave. This feature is in strong contrast to the case of nondispersive SPM (discussed earlier) that resulted in spectral broadening.

Spectral narrowing suggesting the formation of a path-averaged soliton was observed in a recent experiment in which 120-fs Gaussian pulses were launched in the anomalous-GVD regime of an SOI waveguide [17]. As shown in Fig. 5, the 27.8-nm-wide (FWHM) spectrum of input pulses is narrowed by about 33% at the output end. The shape of the output spectrum is best described by a sech shape, rather than a Gaussian. Numerically simulations performed with the experimental parameters fit the experimental spectrum reasonably well, as seen in Fig. 5(b).

#### 3.4. Soliton fission and supercontinuum generation

Spectral broadening introduced by SPM can be used for applications such as multichannel spectral carving [18]. This technique may find applications in the context of optical interconnects based on the scheme of wavelength-division multiplexing. However, as discussed above, spectral broadening introduced by the sole effect of SPM is quite limited because TPA imposes a fundamental limit on the extent of such broadening. Current experiments only show spectral broadening of tens of nanometers [7–10, 12, 14, 25].

To overcome the barrier set by TPA, an optical pulse needs to broaden its spectrum by a large factor before TPA reduces its peak power. This can be done by employing the concept of soliton fission and supercontinuum generation, two processes that have been extensively studied for silica fibers in the past few years [101]. Spectral broadening is enhanced dramatically by forming a higher-order soliton at the input end of the waveguide that splits into multiple fundamental solitons through a fission process. In general, Eq. (27), which includes all possible linear and nonlinear effects and wave-mixing processes, should be used to simulate the dramatic spectral broadening during supercontinuum generation. In particular, the spectral dependence of the propagation constant *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]. In practice, the generalized time-domain NLS equation (37) provides a fairly accurate description of such spectral broadening if correctly including in higher-order linear dispersion terms together with the nonlinear dispersion of Eq. (46).

Numerical simulations typically make use of the split-step Fourier method [80] that deals with dispersion in the frequency domain and treats SPM in the time domain. It is thus relatively easy to include higher-order dispersive effects. The results show that the soliton fission process can generate a supercontinuum that extends over 400 nm even for a 3-mm-long SOI waveguide [16]. Figure 6 shows the supercontinuum created when a 50-fs pulse is launched to excite a third-order soliton inside the waveguide. Although TPA and free carriers reduce the total bandwidth, a 400-nm-wide spectrum could be generated in spite of them. Such wide spectra have not yet been observed experimentally, but recent experiments have shown evidence of the emission of dispersive waves from an optical pulse under the impact of third-order dispersion [15, 23]. This process enhances spectral broadening, and it also provides a simple way to characterize the third-order dispersion.

## 4. XPM and Raman interactions involving two waves

The situation becomes considerably more complicated when two optical waves with different wavelengths, and possibly different states of polarization, copropagate inside a silicon waveguide. For example, SRS may occur if the wavelength difference matches the Raman shift. Even when the Raman effects are not important, XPM couples the two waves and results in a variety of new effects [80]. To include the possibility of Raman scattering, we consider two waves copolarized along the TE mode of a waveguide, fabricated along the [0 1̄ 1] direction.

The starting point in this case is the frequency-domain NLS equation (27). By writing the total field in the form *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:

$$+i{\gamma}_{\mathrm{ps}}^{R}{A}_{s}{\int}_{-\infty}^{t}{h}_{R}\left(t-t\prime \right){e}^{i{\Omega}_{\mathrm{ps}}\left(t-t\prime \right)}{A}_{s}^{*}(z,t\prime ){A}_{p}(z,t\prime )dt\prime ,$$

$$+i{\gamma}_{\mathrm{sp}}^{R}{A}_{p}{\int}_{-\infty}^{t}{h}_{R}\left(t-t\prime \right){e}^{i{\Omega}_{\mathrm{sp}}\left(t-t\prime \right)}{A}_{p}^{*}(z,t\prime ){A}_{s}(z,t\prime )dt\prime ,$$

where *α*
_{lv}(*v*=*p*,*s*) is linear loss at the carrier frequency *ω _{v}*,

*β*=

_{mv}*β*(

_{m}*ω*) accounts for various orders of dispersion,

_{v}*β*

^{f}

*≡*

_{v}*β*

^{f}(

*ω*,

_{v}*N*̄) governs the free-carrier effects, and Ω

_{uv}≡

*ω*-

_{u}*ω*represents the frequency detuning. We have assumed that the pulses are long enough that the higher-order nonlinear terms can be neglected. In Eqs. (63) and (64), the nonlinear parameter γ

_{v}_{uv}includes both electronic and Raman contributions as γ

_{uv}(Ω)≡

*γ*

^{e}_{uv}+

*γ*

^{R}_{uv}

*H*̃

*(Ω). Similar to the case of single-pulse propagation, the Raman nonlinear parameter is related to the Raman gain coefficient as*

_{R}*γ*

^{R}_{uv}=

*gR*(

*ω*)Γ

_{u}*/(*

_{R}*ā*

_{uv}Ω

*), where the effective mode area ${\overline{a}}_{\mathrm{uv}}=\sqrt{{a}_{u}{a}_{v}}$ is obtained from Eq. (29). The Raman gain coefficient is obtained from Eqs. (2) and (28) and is given by*

_{R}where the mode overlap factor *η*
_{uv}≡*η*
_{uvvu} is given by Eq. (30). Note that *g _{R}* scales linearly with optical frequency. When

*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}As discussed previously, since *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

If the frequency detuning between the two waves is far from the Raman frequency shift, the terms involving the Raman response function *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

where we neglected the terms associated with *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 ${e}^{i{\beta}_{0v}z}\left(v=p,s\right)$. These two equations are similar to those describing the XPM effects in silica fibers, except for the TPA and free-carrier terms [80]. As a result, we may expect that XPM inside a silicon waveguide should exhibit similar spectral characteristics. This indeed was observed recently by using femtosecond pulses [14, 26]. In particular, the XPM-induced spectral asymmetry because of a pump-probe delay was seen clearly, and its features agreed with theory. Similar to the case of silica fibers [116], the XPM-induced phase shift and its associated chirp can be used for optical switching and wavelength conversion [14, 25].

The TPA effects can be deduced from Eqs. (67) and (68) by studying how the powers, *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:

where the TPA coefficient is defined as *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]. Noting that

*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}irrespective of the pump and signal frequencies. Physically, this equality indicates that cross-TPA annihilates one pump and one signal photon simultaneously. Taking into account the three TPA processes, the free-carrier generation rate in Eq. (36) is given by

where the self-TPA coefficient *β*
_{Tpp} and *β*
_{Tss} are given in Eq. (8) for degenerate frequency *ω _{p}* and

*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]. As self-TPA coefficient does not show sharp dependence on photon energy below

*E*[83, 84],

_{g}*β*

_{Tpp}≈

*β*

_{Tss}≈

*β*

_{Tps}when frequencies

*ω*and

_{p}*ω*are close to each other.

_{s}Equation (72) shows that cross-TPA-induced carriers and the associated photocurrent are generated only when both pulses are present simultaneously. This feature enables the use of cross-TPA for measuring the temporal characteristics of ultrashort optical pulses [151–156] and employing it for ultrafast optical switching [98–100]. However, as the cross-TPA generation rate is only four times larger than that of self-TPA, each pulse by itself also generates a considerable number of carriers [the first two terms in Eq. (72)]. As a result, the autocorrelation based on cross-TPA is generally not background free, unless a specific technique is used to remove the background [157].

Apart from the XPM induced by the Kerr nonlinearity, TPA-generated free carriers induce additional phase modulation through index changes produced by free carriers. From Eqs. (20), (33), and (68), these phase shifts for a pump-probe configuration (*P _{s}*≪

*P*) satisfy

_{p}The Kerr-induced XPM has a maximum growth rate of *∂*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

The relative ratio of the growth rates of these two phase modulations is given by

where *r _{c}* 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

*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, 92]. However, this carrier-injection scheme has a speed limited by the carrier lifetime (typically ~1 ns) [94], and it responds much slower compared with a scheme based on Kerr-induced XPM or TPA. The modulation speed can be increased considerably by reducing the effective carrier lifetime. This is often realized by applying an external field to swipe away the free carriers [92].

_{x}#### 4.2. Raman amplification and lasing

When the frequency detuning between the pump and signal becomes close to the Raman shift, SRS begins to affect the pulse propagation. As the Raman response time is about 3 ps, SRS is only effective for pulses longer than this. Through a pump-probe scheme, 6-dB and 6.8-dB gains were observed in experiments employing 3.5-ps and 6.6-ps-wide pump pulses, respectively [31, 34]. Numerical simulations show that a signal pulse can grow even from noise [47]. However, for such short pump pulses, SRS process is affected by the walk-off between the pump and signal pulses [47, 48] because the group-velocity mismatch is not negligible for a Raman shift of 15.6 THz. This walk-off problem can be solved by launching a CW signal [32, 40, 44]. Indeed, a net Raman gain of 2 [32], 3.1 [40] and 13 dB [44] was observed in three such experiments. Raman lasing has also been realized through pulsed pumping [38, 43].

However, practical applications generally require CW pumping. In this case, both the TPA and FCA affect the SRS process considerably. To study their impact, we employ Eqs. (63) and (64). The time integral in these equations is readily performed for CW fields, resulting in

We are mainly interested in studying how the pump and signal powers, *P _{p}* and

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

_{s}where *H*̃* _{R}*(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

_{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}As expected, apart from the gain or loss provided by the SRS process, the two waves suffer losses from the TPA and FCA processes. In the following analysis, we assume that the pump is much more intense than the signal so that free carriers are dominantly generated from the pump-induced TPA. In this case, Eqs. (20), (33), and (49) show that FCA coefficient is given by

In the CW pumping case, it turns out that FCA is the dominant loss mechanism rather than TPA [32–36]. This can be seen from the loss ratio defined in Eq. (52). In the steady state with CW pumping, Eq. (52) becomes

Using *β*
_{Tsp}≈*β*
_{Tpp} and *ā*
_{sp}≈*ā*
_{pp}, we find from Eq. (81) that *r _{a}*>1 at a moderate power level of

*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.

As a rough estimate of the combined effects of SRS, FCA, and TPA, assume that the signal is located at Raman gain peak, Eq. (79) together with Eq. (80) shows that net amplification of the signal through SRS is possible only when

This quadratic relation and the requirement that *P _{p}* is a real positive quantity lead to the following condition on the carrier lifetime:

This condition imposes a stringent limitation on the carrier lifetime for net Raman amplification. For example, using typical parameter values with *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]. However, as smaller values of τ_{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], but it also increases scattering losses. The real situation is worse than the rough estimate in Eq. (83) because FCA and TPA also reduce the pump power along the waveguide that was assumed to remain constant. In this case, Eq. (82) should be replaced with

As a relatively low effective lifetime of free carriers in SOI waveguides is critical for practical application of Raman amplification, significant efforts have bee made to reduce its magnitude. For example, the effective lifetime could be reduced in one experiment from 100 ns to 1.9 ns through helium-ion implantation, thereby enabling positive Raman amplification [140]. In another approach, free carriers are quickly swept away from the core area by applying an external dc field [41, 42, 45, 54, 56]. Indeed, efficient Raman amplification realized with this approach has been used to make CW silicon Raman lasers with a threshold as low as 20 mW [41, 42, 54].

The lasing threshold can be estimated from Eq. (84) by setting its left side equal to zero. In this case, the linear loss factor α_{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]. This can be seen clearly from Eqs. (82) and (83) after replacing α_{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

where *V _{m}*=

*ā*

_{sp}

*L*is the effective mode volume,

*n*

_{gv}is the group index at

*ω*(

_{v}*v*=

*p*,

*s*), and

*Q*

_{ev}and

*Q*

_{tv}are the external and loaded quality factors for the cavity, respectively. They are related to linear loss as

*Q*

_{uv}=

*n*

_{gv}

*ω*/(

_{v}*cα*

_{uv}) (

*u*=

*e*,

*t*), where α

_{ev}is linear coupling loss at

*ω*.

_{v}Equation (85) shows clearly that lasing is possible only when *τ*
_{0}<*τ*
_{th}. If the carrier lifetime is small enough that *τ*
_{0}≪*τ*
_{th}, Eq. (85) leads to

This expression reduces to the case of microcavities when TPA is negligible [158]. However, if the carrier lifetime approaches τ_{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]. Similar to the case of Raman lasing in microcavities [158], Eq. (85) shows that the lasing threshold is linearly proportional to effective mode volume and inversely proportional to the square of the cavity Q factor. As a result, increasing the Q factor or reducing the mode volume would significantly lower the lasing threshold in Raman lasers built using SOI waveguides.

It is important to note that the inequalities in Eqs. (82) and (84) can only be satisfied for a certain range of pump powers. In practical terms, FCA would always cause the Raman amplification/ lasing to saturate and thus limit the maximum available output power. The upper limit of pump power for Raman lasers can be estimated from Eq. (82) and is given by:

This equation shows clearly that *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.

The preceding simple analytical theory explains qualitatively the observed features of Raman silicon lasers [39, 41, 42, 54]. A similar discussion can be found in [58]. A quantitative description of such lasers requires numerical simulations based on Eqs. (78) and (79). Note that the lasing in general can occur in both the forward and backward directions. Thus, an accurate quantitative description of silicon Raman lasers requires numerical simulations of three equations [39], or even four equations if the pump wave also propagates along both directions [59]. More technical details about Raman lasers can be found in two recent reviews [52, 53].

Apart from the power amplification, SRS is a resonant process that also changes the refractive index of silicon itself. From Eqs. (3), (76) and (77), the Raman-induced perturbation to the propagation constant is given by

This quantity is positive (negative) when the pump-signal detuning is smaller (larger) than the Raman frequency shift. Physically, this stems from the fact of a standard resonant system that stimulated lattice vibrations follow the pump-signal beating in phase (out of phase) when the beating frequency is smaller (larger) than the resonant frequency. As the Raman gain spectrum is relatively narrow, such changes of propagation constant within the Raman gain spectrum lead to dramatic changes in the group velocity. In particular, the group delay at the center of the Raman gain spectrum is given by

The magnitude of Raman-induced group delay is directly proportional to the pump power. This feature provides tunable delays simply by changing the pump power [50,51]. Interestingly, such Raman-induced group delays exist even when no net Raman amplification occurs (TPA and FCA depend on frequency only weakly and introduce a negligible change in group velocity). This scheme was recently used to demonstrate a tunable group delay of up to 4 ps [51].

## 5. FWM and its applications

The situation becomes more complicated in the case of FWM because three or four waves co-propagate inside an SOI waveguide simultaneously. Only three waves are involved in the case of degenerate FWM. These are called the pump, signal, and idler waves, and their photon energies satisfy the energy-conservation relation 2*h̄ω _{p}*=

*h̄ω*+

_{s}*h̄ω*. As this process is a four-photon, elastic scattering process originating from the Kerr nonlinearity, its instantaneous nature makes it useful for a variety of applications related to ultrafast optical signal processing using silica fibers [159]. The possibility of using FWM in silicon waveguides for similar applications has excited considerable interest recently [61–75, 77–79, 137].

_{i}#### 5.1. Free-carrier effects on FWM

As seen in preceding sections, free carriers introduce considerable loss and phase modulation, and thus impact any nonlinear interaction inside a silicon waveguide. Indeed, the density of TPA-created free carriers becomes large enough in some cases that the reuse of their energy was proposed recently [160, 161]. Clearly, free carriers would affect the FWM process as well. Before discussing the details of FWM, we first look into the impact of free carriers.

It turns out that most free-carrier effects on FWM can be deduced from Eq. (37). For simplicity, we first assume that all optical waves are polarized along the TM mode so that Raman scattering is absent. By using Eqs. (20), (33) and (49) in Eq. (37), we obtain

where the parameter *γ _{f}* is defined as

Equation (90) shows clearly that TPA-induced free carriers produce a fifth-order nonlinear effect that is accumulative in nature [115].

To describe the FWM process, we assume that *A*(*z*,*t*) is composed of three waves such that $A(z,t)={A}_{p}(z,t){e}^{-i{\omega}_{p}t}+{A}_{s}(z,t){e}^{-i{\omega}_{s}t}+{A}_{i}(z,t){e}^{-i{\omega}_{i}t}$, among which the pump field *A _{p}* is much more intense than the signal

*A*and idler

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

_{i}$$+2i{\gamma}_{f}{A}_{p}{\int}_{-\infty}^{t}{e}^{-\left(t-t\prime \right)\u2044{\tau}_{0}}{e}^{i{\Omega}_{\mathrm{sp}}\left(t-t\prime \right)}{\mid {A}_{p}\mid}^{2}\left({A}_{p}^{*}{A}_{s}+{A}_{p}{A}_{i}^{*}\right)\mathrm{dt}\prime ,$$

$$+2i{\gamma}_{f}{A}_{p}{\int}_{-\infty}^{t}{e}^{-\left(t-t\prime \right)\u2044{\tau}_{0}}{e}^{i{\Omega}_{\mathrm{ip}}\left(t-t\prime \right)}{\mid {A}_{p}\mid}^{2}\left({A}_{p}^{*}{A}_{i}+{A}_{p}{A}_{s}^{*}\right)\mathrm{dt}\prime ,$$

where we have used Ω_{sp}=Ω_{pi} and kept only the first-order terms in *A _{s}* and

*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}We now focus on a CW or quasi-CW case and assume that the pulse widths associated with the three fields are much longer than the carrier lifetime. In this case, the field amplitudes inside the integrals in Eqs. (92)–(94) can be treated as constant, and these equations reduce to

$$+i{\gamma}_{f}{\tau}_{0}{\mid {A}_{p}\mid}^{2}\left[{\mid {A}_{p}\mid}^{2}{A}_{s}+\frac{2}{1+i{\Omega}_{\mathrm{ps}}{\tau}_{0}}\left({\mid {A}_{p}\mid}^{2}{A}_{s}+{A}_{p}^{2}{A}_{i}^{*}\right)\right],$$

$$+i{\gamma}_{f}{\tau}_{0}{\mid {A}_{p}\mid}^{2}\left[{\mid {A}_{p}\mid}^{2}{A}_{i}+\frac{2}{1+i{\Omega}_{\mathrm{pi}}{\tau}_{0}}\left({\mid {A}_{p}\mid}^{2}{A}_{i}+{A}_{p}^{2}{A}_{s}^{*}\right)\right],$$

where Ω* _{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.

The physical origin of various nonlinear terms in Eqs. (95)–(97) is clear. All terms involving *γ _{e}* stem from the Kerr nonlinearity, but the terms containing

*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

Noting that Re(*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}As we know from nonlinear fiber optics [80], efficient Kerr-induced FWM requires Re(*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.

Free carriers can affect a FWM process in another way because FWM is a coherent process and requires phase matching among the interacting waves. As free carriers introduce considerable phase modulations, they would affect the phase-matching condition of FWM. From Eqs. (95)–(97), we find that the phase mismatch induced by free carriers is given by

As the Kerr-induced phase mismatch is given by Δ*κ _{K}*=2Re(

*γ*)|

_{e}*A*|

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

This expression is quite similar to Eq. (98). As a result, preceding discussion about the magnitude of *r*
_{FWM} applies to *r _{κ}* as well. In other words,

*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}The situation becomes quite different in the pulsed regime in which pulse widths involved become much smaller than the carrier lifetime. Although FCA is negligible in this case (as discussed earlier), transient index changes produced by free carriers can have significant impact on the FMW process. Mathematically, all the field amplitudes inside the integrals in Eqs. (92)–(94) can be considered as an ultrashort impulse, resulting in

where the pulse width *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.

Following a reasoning similar to the quasi-CW case, we can find the relative impact of free carriers on the FWM efficiency and the phase mismatch through two ratios that now become

It turns out that the magnitude of these two ratios can become quite large in some cases. For example, it is ~4 for 10-ps pulses at a 0.4 GW/cm^{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 _{f}* is negative because σ

*<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]. However, free-carrier effects decrease for shorter pump pulses, and may becomes negligible when the width of pump pulses is reduced to below 1 ps.*

_{n}In concluding this subsection, index changes produced by free carriers have a significant impact on the FWM process for pump and signal pulses in the picosecond regime. Their impact is negligible in the quasi-CW regime in which pulse widths are much longer than the carrier lifetime. As CW pumping is required in most practical applications, we focus on this case in the following section dealing with broadband parametric generation.

#### 5.2. Broadband parametric generation and wavelength conversion

One advantage of the FWM process is that its instantaneous nature supports parametric generation over a broad bandwidth. In silica fibers, parametric gain bandwidths of up to 100 nm have been demonstrated [162, 163]. In this section, we focus on FWM inside a silicon waveguide and show that, with an appropriate control of waveguide dispersion, parametric gain may be realized over a bandwidth larger than that possible for silica fibers. For a complete description of the broadband FWM process, we return to Eq. (27) rather than using Eq. (37) or (90). For simplicity, we first assume that Raman scattering does not play a significant role.

As before, the optical field is composed of three CW waves at different frequencies. substituting this form in Eq. (27), we obtain the following three equations for the pump, signal, and idler fields:

where *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

and *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.

Because of the time-reversal symmetry, the third-order susceptibilities satisfy the relation

As a result, we find from Eq. (109) that

Therefore, as far as the FWM process is concerned, the last terms in Eqs. (106)–(108) lead to

This is the well-known Manley-Rowe relation for the FWM process, indicating the conservation of photon numbers among the three interacting waves [102, 103].

Equations (106)–(108) are quite general and include all effects related to linear loss and dispersion, nonlinear loss TPA and free carriers produced by it, various nonlinear effects (SPM, XPM, and FWM), and pump depletion. The TPA processes now become more complicated than the cases discussed earlier. The involvement of three waves leads to six possible types of TPA processes. As a result, the carrier generation rate in Eq. (36) is given by

where ${\overline{a}}_{\mathrm{uv}}\equiv \sqrt{{a}_{u}{a}_{v}}$ is obtained from Eq. (29).

In most practical situations, the pump is much more intense than the signal and idler fields. In this case we can ignore pump depletion as well as the SPM and XPM effects produced by the signal and idler, and Eqs. (106)–(108) reduce to

As mentioned earlier, FWM requires phase matching among the interacting waves (related to momentum conservation among the four photons involved in the FWM process). From Eqs. (114)–(116), we find that the total phase mismatch is given by

where the first term, Δ*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

where ζ* _{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}However, these quantities are relatively small. For example, ζ_{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.

The bandwidth of parametric gain is determined by the condition |*κL*|<*π*/2. As is known from FWM in silica fibers [80, 159], it would be maximized when the pump wavelength falls close to the ZDWL of a silicon waveguide. However, the parametric bandwidth is enhanced dramatically for SOI waveguides because a typical waveguide length (*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]. With a proper choice of the pump wavelength, such parametric amplifiers can cover the entire so-called S, C, L, and U telecommunications bands. Similarly, if the idler is used for wavelength conversion, such wavelength converters can operate over a 300-nm bandwidth or more. Further extension of the parametric bandwidth can be realized by engineering the waveguide to reduce the impact of fourth-order dispersion. This scheme was used in a recent experiment to demonstrate wavelength conversion over a bandwidth of about 150 nm [79].

Unfortunately, although the bandwidth can be made very broad through dispersion engineering, the efficiency of the FWM process is severely limited by FCA, and it is difficult to realize a net amplification with CW pumping [68]. Similar to the case of Raman amplification/lasing in Section 4.2, we can estimate the required effective carrier lifetime for parametric amplification to occur through FWM. Equations (114)–(116) show that net amplification is possible only when [68]

where we have used Eq. (80) for the FCA coefficient. This inequality imposes the following upper limit on the carrier lifetime [68] :

where we used *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]. Such a stringent requirement on carrier lifetime imposes a serious challenge to practical applications of FWM in the telecommunication band.

#### 5.3. Coherent anti-Stokes Raman scattering

Although efficient FWM with CW pumping is difficult to realize through the Kerr nonlinearity, it turns out that amplification of the signal can occur if we take advantage of SRS. We have seen in Section 4.2 that the SRS process in silicon waveguides not only provides gain but it also introduces significant changes in the refractive index, when the pump-signal detuning lies within the Raman-gain bandwidth. Equation (88) shows that SRS introduces a maximum nonlinear index change of *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] and is used widely for molecular spectroscopy [165]. Of course, the phase-matching condition should be maintained over the spectral region covering the Raman-gain spectrum [68].

As SRS needs to be present for this process, a specific polarization mode is required for the CARS. In the case of silicon waveguides fabricated along the [0 1̄ 1] direction, the fundamental TE mode is employed. In this situation, Eqs. (114)–(116) are replaced with the following equations to account for the Raman-induced coupling:

where the nonlinear parameters now include the Raman contributions, *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}).

The CARS process in silicon waveguides has been employed for wavelength conversion [61–64,66], but its efficiency was relatively low. The reason turns out to be the phase-matching condition [68]: CARS can become efficient only when phase matching is realized for a signal separated from the pump by the Raman shift of 15.6 THz. This can be realized by choosing the pump wavelength appropriately with respect to the ZDWL of an SOI waveguide [68]. Figure 8 shows the signal gain and wavelength-conversion efficiency as a function of signal wavelength under conditions similar to those used for Figure 7, except that the Raman-induced enhancement of the FWM process is included. As seen there, by choosing the pump wavelength in the normal-dispersion region of the waveguide, the phase matching condition can be satisfied for the Stokes and anti-Stokes waves, leading to a positive gain of about 9 dB, even with CW pumping.

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

Although the CARS process enables positive parametric gain, its spectral location is fixed within the Raman gain bandwidth. To take advantage of the instantaneous nature of FWM outside this spectral region, we should go back to the Kerr nonlinearity and ask what limits the FWM efficiency in the CW regime used for Figure 7. As we have seen, it is the presence of losses induced by the free carriers (FCA). Recalling that free carriers are created through TPA, it is immediately obvious that the FWM efficiency can be improved drastically by choosing the pump wavelength beyond 2.2 *µ*m. As the pump wavelength then falls below the half-band gap of silicon, TPA vanishes [83, 84], and efficient parametric generation becomes possible. Note that the signal wavelength can still be well above the half band gap, if the SOI waveguide is designed properly to satisfy the phase-matching condition over a wide bandwidth [137].

Figure 9 shows an example of such a FWM configuration. By tailoring the ZDWL of the fundamental TE mode to 2349 nm using a waveguide cross section of 1.8×0.4 *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], its lasing frequency is fixed with respect to the pump. In contrast, the FWM-based parametric amplification discussed here can provide a much larger bandwidth and broader tunability, features comparable with those available in *χ*
^{(2)} materials such as a periodically poled lithium-niobate waveguide.

#### 5.5. Photon pair generation by FWM

So far, we have focused on the traditional regime in which a high-power pump is used to amplify a signal and simultaneously generate an idler beam. However, as FWM is a four-photon elastic scattering process, it conserves physical quantities, such as energy and momentum, among the four interacting photons. As a result, if no signal is initially present so that FWM is initiated from vacuum noise, and the pump power is relatively low so that stimulated FWM does not occur, it is possible to create only one pair of signal and idler photons at a time (within the coherence time of the pump) that are correlated quantum mechanically in multiple dimensions. Such correlated photon pairs are useful for applications in quantum information processing. Indeed, spontaneous FWM in silica fibers have been used for this purpose successfully in the past few years [167–170].

Unfortunately, pump photons also interact with thermal phonons and generate uncorrelated signal/idler photons through spontaneous Raman scattering (SpRS). This process is found to impose a severe limit on the degree of the quantum correlation between the signal and idler photons [171–173]. Because of the broadband nature of SpRS in silica fibers, high-quality correlated photon pairs can be generated only at frequencies far from the pump (>25 THz) [169, 170]. They can also be generated within the Raman-gain bandwidth by cooling the fiber with liquid nitrogen [174,175]. The SpRS problem can also be solved using silicon waveguides. As we have seen earlier, Raman scattering in a silicon crystal occurs only for specific polarizations within a very narrow (105-GHz wide) band and can easily be avoided. For this reason, if spontaneous FWM inside a silicon waveguide is used to create correlated photon pairs, photon pairs should have a very high quality [73].

In the low-power regime, pump depletion can be neglected completely, and the pump wave is described classically by Eq. (122). However, the signal and idler waves must be treated quantum mechanically. Equations (123) and (124) should thus be replaced by the corresponding operator equations, as was done first in a 2006 study [73], resulting in the following signal equation:

$$+{\widehat{m}}_{l}(z,{\omega}_{s})+{\widehat{m}}_{f}(z,{\omega}_{s})+{\widehat{m}}_{T}(z,{\omega}_{s}){A}_{p}+i{\widehat{m}}_{R}(z,{\Omega}_{\text{sp}}){A}_{p}.$$

The idler equation can be obtained by exchanging the subscripts *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

where *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}Equation (125) together with Eqs. (122) and (126) can be used to find the photon generation rate and the quantum correlation between the signal and idler photons [73]. An example is shown in Fig. 10. A detailed analysis shows that FCA and TPA play minor roles because of the low pump power used. Thus, FWM in silicon waveguides is quite efficient for photon-pair generation. Such a silicon-base scheme not only provides a pair correlation close to the fundamental limit set by a pure FWM process but also exhibits a spectral brightness that is comparable to other photon-pair sources. This scheme has recently been used to generate correlated photon pairs [75], and a maximum coincidence-to-accidental ratio of 25 was obtained at a photon production rate of 0.05 per pulse. This value is still far below the predicted theoretical value, indicating room for future improvement.

## 6. Summary

Several kinds of nonlinear optical effects have been observed in silicon waveguides in recent years, and their device applications are attracting considerably attention. In this review, we have provided a unified theoretical platform that can be used not only for understanding the underlying physics but also to provide guidance toward new and useful applications. We began in Section 2 with a description of the third-order nonlinearity of silicon and considered the tensorial nature of both the electronic and Raman contributions. The generation of free carriers through TPA and their impact on various nonlinear phenomena is included fully in the theory presented here. We derived a general propagation equation in the frequency domain and showed how it leads to a generalized nonlinear Schrödinger equation when it is converted to the time domain. We used this equation to study propagation of ultrashort optical pulses in the presence of SPM and showed the possibility of soliton formation and supercontinuum generation. The nonlinear phenomena of XPM and SRS were discussed next in a separate section, with emphasis on the impact of free carriers on Raman amplification and lasing. Section 5 focused on the FWM process and its applications. We considered first the impact of free carriers and showed that, although index changes induced by them have a negligible impact on FWM, FCA limits the FWM efficiency so much that a net positive gain cannot be realized in the telecommunication band in the case of CW pumping. However, this problem can be solved by pumping at wavelengths beyond 2.2 *µ*m because TPA-induced free carriers are then absent. We also showed that FWM can occur over a wide bandwidth, with a proper choice of the pump wavelength, because of much smaller waveguide lengths employed compared with those required for silica fibers. We also discussed briefly the use of FWM in silicon waveguides for generating correlated photon pairs that are useful for quantum applications.

## Acknowledgements

The authors thank P. M. Fauchet, R. W. Boyd, T. E Murphy, J. Zhang, G. Piredda, L. Yin, T. J. Johnson, R. Perahia, and C. P. Michael for helpful discussions. This work was supported in part by the National Science Foundation, Air Force Research Office, and DARPA EPIC program.

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