Abstract

A deep insight into the inherent anisotropic optical properties of silicon is required to improve the performance of silicon-waveguide-based photonic devices. It may also lead to novel device concepts and substantially extend the capabilities of silicon photonics in the future. In this paper, for the first time to the best of our knowledge, we present a three-dimensional finite-difference time-domain (FDTD) method for modeling optical phenomena in silicon waveguides, which takes into account fully the anisotropy of the third-order electronic and Raman susceptibilities. We show that, under certain realistic conditions that prevent generation of the longitudinal optical field inside the waveguide, this model is considerably simplified and can be represented by a computationally efficient algorithm, suitable for numerical analysis of complex polarization effects. To demonstrate the versatility of our model, we study polarization dependence for several nonlinear effects, including self-phase modulation, cross-phase modulation, and stimulated Raman scattering. Our FDTD model provides a basis for a full-blown numerical simulator that is restricted neither by the single-mode assumption nor by the slowly varying envelope approximation.

© 2010 Optical Society of America

1. Introduction

The microelectronics industry has thrived on the unique electrical and mechanical properties of silicon, which have enabled significant miniaturization and mass-scale integration of electronic components on a single silicon chip [1, 2]. Unfortunately, this miniaturization process cannot be continued much beyond the current level because of the inherent heat generation and the metal-interconnects bottleneck effect [3]. A promising way to overcome these fundamental limitations is to use photons, instead of electrons, as information carriers. This would lead to a gradual replacement of electronic integrated circuits (ICs) by photonic ICs. Silicon is well suited for realizing photonic ICs, owing to the recent advances in the fabrication technologies and the ease of integrating electronics with silicon photonics [49]. A relatively high refractive-index contrast provided by the silicon-on-insulator (SOI) technology allows one to confine the optical mode to subwavelength dimensions and to realize nanoscale optical waveguides (often called photonic nanowires), for on-chip optoelectronic applications [8].

The strong optical nonlinearities of silicon give rise to a variety of nonlinear effects, such as stimulated Raman scattering (SRS) [10, 11], self-phase modulation (SPM) [12, 13], cross-phase modulation (XPM) [14], cross-absorption modulation (XAM) [15], four-wave mixing (FWM) [1618], and coherent anti-Stokes Raman scattering (CARS) [9, 19, 20]. These effects have been successfully utilized in a number of chip-scale photonic devices, including lasers [21, 22], amplifiers [23, 24], switchers [15, 25], modulators [2628], broad-band frequency converters [20, 29], detectors [30], all-optical logic gates [25], continuum generators [31, 32], and pulse compressors [33]. Since the Kerr effect, two-photon absorption (TPA), and Raman scattering are anisotropic in the case of silicon, the state of polarization of the optical field and waveguide orientation could be tailored to take advantage of optical anisotropy for making novel optical devices. Even though there is a strong potential for anisotropic effects to be used in new devices, none of the above-listed devices utilizes optical anisotropy. Meanwhile, recent theoretical works have demonstrated that the anisotropy exhibited by silicon waveguides can be used to realize fast optical switching [3436] and power equalization [36].

To study the impact of anisotropic nonlinear effects on light propagation in silicon, a complex numerical simulator that takes into account the tensorial nature of third-order susceptibility is required. The finite-difference time-domain (FDTD) scheme is the best choice for sophisticated modeling, since it allows Maxwell's equations to be exactly solved in three dimensions with few assumptions [37]. This scheme is suitable for analyzing complex waveguide geometries with few-cycle pulses, which is beyond the capability of other popular modeling techniques. In this paper, we present an FDTD model of optical phenomena in silicon waveguides, which incorporates the anisotropy of silicon nonlinearities and the vectorial nature of the electromagnetic field. In Section 2, we describe the structure of the general FDTD algorithm for nonlinear anisotropic media. Starting with Maxwell's equations, we outline the steps required for obtaining the update equations for the electromagnetic field and set up notations employed in the rest of the paper. In Section 3, we consider the relation between the material polarization and the electric field inside a silicon waveguide and derive the contributions of different optical effects with emphasis on the underlying physics. The full material polarization is used in Section 4 to derive the update equations for the electric field in the three-dimensional case. We then simplify these equations to obtain a computationally efficient algorithm that can be used to simulate SOI-based photonic devices. In Section 5, we apply the simplified algorithm to investigate the impact of optical anisotropy on pulse propagation in silicon waveguides. We summarize our study and conclude the paper in Section 6.

2. Three-dimensional FDTD scheme for nonlinear anisotropic media

Classical electromagnetic theory is based on Maxwell's equations, which admit exact analytical solutions only for a limited number of relatively simple problems [38]. In modern nonlinear optics, these equations are usually solved numerically using different computational schemes. The FDTD method is one of the most widely used methods for this purpose, due to the simplicity of its implementation, its applicability to arbitrary scattering media (e.g., media exhibiting anisotropic and/or nonlinear responses), and the high accuracy of the generated results [37].

In order to establish the notations used in the following sections, we sketch the main steps involved in the FDTD algorithm that solves the source-free Maxwell's equations in their differential form [38]

B(r,t)t=×E(r,t),D(r,t)t=×H(r,t),

where B(r, t) and D(r, t) are the magnetic and electric inductions, and E(r, t) and H(r, t) are the electric and magnetic fields. For nonmagnetic media, the constitutive relations between the four vector fields in Eq. (1) are given by

H(r,t)=B(r,t)μ0,E(r,t)=D(r,t)P(r,t)ɛ0,

where μ 0 and ε 0 are, respectively, the permeability and permittivity of vacuum. The response of an inhomogeneous anisotropic medium to an external electric field is taken into account by the material polarization vector P(r, t).

An arbitrary vector field V = {B, D, H, E, P} can be split into its Cartesian components Vξ along the axis ξ = {α, β, γ} of the FDTD coordinate system as

V(r,t)=α^Vα(r,t)+β^Vβ(r,t)+γ^Vγ(r,t),

where α^ , β^, and γ^ are the unit vectors. In the FDTD scheme, each of these components are discretized in time and space domains according to the recipe of Yee [37, 39].

The discretization scheme proposed by Yee can be visualized using the cubic unit cell and the time line shown in Fig. 1. The electric and magnetic fields are defined on different points of the unit cell and at alternative time steps. Mathematically, each component Vξ (r, t) becomes a function of four discrete arguments,

Vξ|i+d1,j+d2,k+d3n+d4Vξ[(i+d1)Δξ,(j+d2)Δξ,(k+d3)Δξ,(n+d4)Δt],

where {i, j, k, n} ∈ ℕ, and Δξ and Δt are the space and time increments. The values of dμ (μ = 1, 2, 3, 4), corresponding to the arrangement adopted in Fig. 1, are given in Table 1. It should be noted that not all components shown in Fig. 1 belong to the same unit cell.

 

Fig. 1 Staggered space arrangement (left) and leapfrog time ordering (right) of discretized electromagnetic-field components using a cubic Yee cell.

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Tables Icon

Table 1. Values of dμ specifying relative positions of components Vξ inside a four-dimensional unit cell.

In order to obtain the update equations for the discrete electromagnetic field, one needs to replace the time and space derivatives in Eq. (1) by finite differences. With this approach, the update equations for the vectors B, H, and D can be readily derived from Eqs. (1) and (2). For example, for α components the update equations are:

Bα|i1/2,j1/2,k1/2n+1/2=Bα|n1/2Sc(Eγ|i,j+1/2,knEγ|i,j1/2,knEβ|i,j,k+1/2n+Eβ|i,j,k1/2n),Hα|i1/2,j1/2,k1/2n+1/2=Hα|n1/2+1μ0(Bα|n+1/2Bα|n1/2),Dα|ijkn+1=Dα|ijkn+Sc(Hγ|i,j+1/2,kn+1/2Hγ|i,j1/2,kn+1/2Hβ|i,j,k+1/2n+1/2+Hβ|i,j,k1/2n+1/2),

where S = cΔtξ is the Courant stability number and c is the speed of light in vacuum. For brevity, from here on we omit space indexes on the right side of update equations that coincide with those on the left side. With these and similar equations for other field components, we can calculate the values of Bξ, Hξ, and Dξ at any time step, if their values and the values of Eξ are known at the previous time step.

To update the electric field component Eξ (ξ = α, β, γ) at the time step n + 1, we need to know the components Dξ and Pξ at the time steps n + 1 and n. According to Eq. (2), the update equation takes the form

Eξ|i+d1,j+d2,k+d3n+1=Eξ|n+1ɛ0(Dξ|n+1Dξ|nPξ|n+1+Pξ|n).

In turn, the material polarization can be calculated if the electric field inside the medium is known. This polarization includes contributions from different physical phenomena, which may be either isotropic (i) or anisotropic (a), i.e.,

P(r,t)=iP(i)+aP(a).

The relations between the electric field and the induced polarizations P ( i ) and P ( a ) are available either in the time or the frequency domain, depending on the specific models of the dielectric response [4042]. The frequency-domain relations can be transformed into the time domain either by using the standard replacement → −/∂t or by employing the higher-order Fourier transforms. The resulting time-domain relations can be converted into the update equations for material polarization using finite differences for the derivatives.

If the material response is isotropic, then the ξ component of the polarization vector depends only on the ξ component of the electric field, and we can write symbolically

Pξ(i)|i+d1,j+d2,k+d3n+1=fξ(i)(Pξ(i)|n,Pξ(i)|n1,,Eξ|n+1,Eξ|n,Eξ|n1,),

where fξ(i)() denotes the functional dependence.

In the case of the anisotropic material response, the value of Pξ(a) is determined by all three components of the electric field such that

Pξ(a)|i+d1,j+d2,k+d3n+1=fξ(a)(Pξ(a)|n,Pξ(a)|n1,,Eα|n+1,Eβ|n+1,Eγ|n+1,Eα|n,Eβ|n).

This equation assumes that the material response is local and that the values of three components of the electric field are to be taken at the same grid node as the polarization component. However, only the component Eξ coincides in space with Pξ(a) (see Table 1); the other two components of the electric field are defined at the adjacent nodes. It is common to calculate the values of these two components at the position of Pξ(a) by averaging their values over the four nearest nodes surrounding Pξ(a) [4345]. For example, the components Eβ and Eγ are projected onto the position of the component Pα(a) using the relations (see Fig. 2)

 

Fig. 2 Positions of electric field components in the FDTD grid. The values of components Eβ and Eγ at the node (i, j, k) are calculated by averaging their four values over the nodes (i ± 1/2, j ± 1/2, k) and (i ± 1/2, j, k ± 1/2), respectively.

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Eβ|ijkn=14(Eβ|i1/2,j1/2,kn+Eβ|i1/2,j+1/2,kn+Eβ|i+1/2,j+1/2,kn+Eβ|i+1/2,j1/2,kn),Eγ|ijkn=14(Eγ|i1/2,j,k1/2n+Eγ|i1/2,j,k+1/2n+Eγ|i+1/2,j,k1/2n+Eγ|i+1/2,j,k+1/2n).

Equations (3)(5) are solved iteratively for a fixed value of D until the desired accuracy is achieved.

3. Optical phenomena in silicon waveguides

As we have seen, both the isotropic and anisotropic contributions of the optical response of a dielectric medium can be easily handled with the FDTD scheme. However, its implementation requires establishing a functional relation between the material polarization induced in a silicon waveguide and the electric field of the propagating mode inside the waveguide.

A number of linear and nonlinear optical phenomena should be taken into account for accurate modeling of silicon-based photonic devices. The major linear effects are dispersion, absorption, and scattering losses. Linear absorption in silicon is isotropic and much stronger than that in fibers; however, its effect on optical propagation is usually weak as opposed to the impact from nonlinear losses. Similar to optical fibers, linear dispersion plays a significant role in silicon waveguides when the spectrum of a propagating pulse is sufficiently broad. Due to a high refractive-index contrast, the dispersive properties of a SOI waveguide depend strongly on its geometry; this dependence leads to a relatively large waveguide dispersion. In particular, large modal birefringence (∼ 10−3) can arise in SOI waveguides with asymmetric cross sections [46], even though the intrinsic optical anisotropy of silicon is relatively low (∼ 10−6) [47, 48]. Modal birefringence and waveguide dispersion are automatically included in the FDTD scheme when the linear optical response of silicon is characterized by a scalar susceptibility, χ˜ (1)(ω).

If there is no inhomogeneous mechanical stress breaking the inversion symmetry of the crystal field, the second-order electro-optic effect is absent in silicon [4951]. In this paper, we only consider silicon waveguides that are free from mechanical stresses and ignore all second-order nonlinear effects.

The nonlinear interaction of an optical field with bound electrons of silicon atoms results in the Kerr effect, two-photon absorption (TPA), and Raman scattering [5254]. These effects are of the third order with respect to the applied field. They have anisotropic character and are described by two fourth-rank tensors, χ˜κλμνe and χ˜κλμνR [49, 53, 55].

The process of TPA may generate a considerable number of free carriers, which absorb light and reduce the effective refractive index of a silicon waveguide. In addition, TPA generates heat and raises the waveguide temperature leading to an increase in the refractive index; this phenomenon is referred to as the thermo-optic effect (TOE) in the literature [5659]. The impact of TOE on pulse propagation is typically much stronger than that due to free-carrier dispersion (FCD) [56]. Free-carrier effects and TOE are isotropic, but strongly nonlinear; their strength grows in proportion to the fifth power of the electric field and can be approximately described by an intensity-dependent susceptibility, χ˜ FC(|E|4, ω).

With the preceding nonlinear processes in mind, the polarization κ(r, ω) induced inside a silicon waveguide by the electric field κ(r, ω) can be represented in the form [49, 53]

P˜κ(r,ω)=ɛ0χ˜(1)(ω)E˜κ(r,ω)+ɛ0(2π)2λμν+dω1+dω2χ˜κλμν(3)(ω;ω1,ω2,ω3)E˜λ(ω1)E˜μ(ω2)E˜ν(ω3)+ɛ0χ˜FC(|E|4,ω)E˜κ(r,ω),

where χ˜κλμν(3)=χ˜κλμνe+χ˜κλμνR, ω 3 = ωω 1ω 2, and we used the following convention for the Fourier transform:

V˜(ω)=F[V(t)]+V(t)eiωtdt,V(t)=F1[V˜(ω)]12π+V˜(ω)eiωtdω.

Equation (6) is written in the FDTD coordinate system, whose axes do not generally coincide with the crystallographic axes of the waveguide (see Fig. 3). Therefore before using this equation, it is necessary to transform the third-order susceptibility tensor from the crystallographic coordinates (x, y, and z) into the FDTD coordinates (α, β, and γ). The general transformation for a fourth-rank tensor is given by [60]:

χκλμν(3)=klmnakκalλamμanνχklmn(3),

where the Latin indices denote the crystallographic axes and a is the transformation matrix.

 

Fig. 3 Relative orientation of FDTD (α, β, γ) and crystallographic (x, y, z) axes adopted in the paper. For ϑ=π4, the FDTD axes α, β, and γ coincide, respectively, with the [110], [1̄10], and [001] crystallographic directions. The inset shows the TM and TE polarizations and an arbitrary linear polarization determined by the angle φ.

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It is convenient to choose the FDTD axes along the edges of the silicon waveguide. Without loss of generality, we select the α axis along the waveguide length and take the other two axes to form a right-handed coordinate system. Since silicon waveguides are often fabricated on the (001) plane, we assume that the γ axis coincides with the [001] direction (see Fig. 3). In this case, the FDTD axes α and β are obtained by rotating the crystallographic axes x and y about the z axis. If the rotation angle is ϑ and the rotation is performed as shown in Fig. 3, then the transformation matrix is of the form

a=(cosϑsinϑ0sinϑcosϑ0001).

As an example, ϑ = π/4 corresponds to the widely used waveguides fabricated along the [110] direction [53, 61].

3.1. Linear absorption and linear dispersion

Linear optical properties of silicon waveguide are described by the complex-valued dielectric function

ɛ˜(ω)ɛ˜(ω)+iɛ˜(ω)=1+χ˜(1)(ω).

The real part of this function accounts for dispersion of the linear refractive index n(ω)ɛ˜(ω), while the imaginary part is responsible for linear losses, ε˜″(ω) ≈ cn 0 αL/(2ω), where αL is the linear-loss coefficient and n 0n(ω 0) is the refractive index at the absolute maximum of the input field spectrum located at frequency ω 0. In practice, αL includes scattering losses occurring at waveguide interfaces.

Dispersion of the refractive index has been measured for silicon crystals over a broad spectral range. The experimental data is often approximated with an appropriately chosen analytical formula. One of the simplest formulae for n 2(ω) is given by the 2s-pole Sellmeier equation [6264]

n2(ω)=1+j=1sajωj2ωj2ω2,

where aj and ωj are the fitting parameters. In Section 5, we use this equation in the specific case s = 2 with a 1 = 9.733, a 2 = 0.936, ω 1 = 1032.49 THz, and ω 2 = 817.28 THz. These values provide an accurate fit of the refractive-index data for silicon in the wavelength range from 1.2 to 2 μm [65].

3.2. The Kerr effect and two-photon absorption

At high optical intensities, electrons residing at the outer shells of silicon atoms start moving anharmonically. This leads to an increase by Δn NL of the refractive index in proportion to the local intensity I of light. This phenomena, referred to as the Kerr effect, is characterized by the nonlinear coefficient n 2 = Δn NL/I ≈ 6 × 10−5 cm2/GW. The Kerr effect is responsible for self-phase modulation (SPM), soliton formation, supercontinuum generation, and modulation instability, among other things [12, 64, 66, 67].

In the telecommunication band around 1.55 μm, energy of infrared photons (about 0.8 eV) exceeds one half of the indirect band gap in silicon (0.56 eV). As a result, silicon exhibits two-photon absorption (TPA) accompanied with the emission of transverse optical (TO) phonons [68, 69] and generation of free carriers. Despite the fact that TPA is a nonlinear-loss mechanism and is undesirable in majority of photonic devises, it has been successfully employed to create ultrafast optical modulators and switches [2527, 33].

The general form of the susceptibility χ˜κλμνe(ω;ω1,ω2,ω3) required to calculate the electronic part of the nonlinear polarization, caused by an optical beam with an arbitrary spectrum, is not known. Fortunately, the response time of bound electrons is comparable to the duration of an optical cycle and can be considered to be negligible for a relatively narrow field spectrum [49, 55]. In this case, dispersion of the electronic susceptibility can be ignored, and the Kleinman's symmetry can be used to factorize the third-order susceptibility as [49]

χ˜κλμνe(ω;ω1,ω2,ω3)χ˜ααααe(ω)κλμν,

where χ˜ααααe(ω)=ɛ2ηTPA/(iω), ɛ2=ɛ0n02cn2, and ηTPA=12ɛ0c2n02βTPA. The real and imaginary parts of χ˜ααααe(ω) represent the Kerr effect and TPA, respectively. All anisotropic effects are included through the anisotropy tensor ℰκλμν.

In the crystallographic coordinate system, the anisotropy tensor ℰklmn has the following well-known form [49, 53]:

klmn=(ρ/3)(δklδmn+δkmδln+δknδlm)+(1ρ)δklδlmδmn,

where δkl is the Kronecker delta function and the real number ρ characterizes anisotropy of the electronic susceptibility (ρ ≈ 1.27 near λ = 1.55 μm). Transforming ℰklmn according to Eqs. (7) and (8), we obtain the following nonzero elements of the anisotropy tensor in the FDTD coordinates:

αααα=ββββ=1+Aϑ,γγγγ=1,αααβ=ααβα=αβαα=βααα=Bϑ,βββα=ββαβ=βαββ=αβββ=Bϑ,ααββ=αββα=ββαα=αβαβ=βαβα=βααβ=ρ/3Aϑ,ααγγ=αγγα=γγαα=αγαγ=γαγα=γααγ=ρ/3,ββγγ=βγγβ=γγββ=βγβγ=γβγβ=γββγ=ρ/3,

where Aϑ=12(ρ1)sin2(2ϑ) and Bϑ=14(ρ1)sin(4ϑ).

Using the preceding results and Eq. (6), we obtain the Kerr-induced material polarization in the time domain in the form

PκK(r,t)=ɛ0ɛ2λμνκλμνEλ(r,t)Eμ(r,t)Eν(r,t)=ɛ0ɛ2SκK(r,t)Eκ(r,t),

where the auxiliary functions SκK are given by

SαK=(1+Aϑ)Eα2+(ρ3Aϑ)Eβ2+ρEγ2+3BϑEαEβBϑEβ3/Eα,
SβK=(1+Aϑ)Eβ2+(ρ3Aϑ)Eα2+ρEγ23BϑEαEβ+BϑEα3/Eβ,
SγK=ρEα2+ρEβ2+Eγ2.

In the approximation of instantaneous electronic response, the anisotropy of TPA is identical to that of the Kerr effect. Owing to this fact, the TPA-induced polarization can be written similar to Eq. (9),

P˜κTPA(r,ω)=ɛ0ηTPAiωF[SκK(r,t)Eκ(r,t)].

In Section 4, we convert this equation into the time domain and use it together with Eqs. (9) and (10) to incorporate the Kerr effect and TPA into the FDTD model.

3.3. Raman scattering

If an intense optical field at frequency ωp propagates through a silicon waveguide, it generates another two fields at frequencies ωp ± ΩR through a phenomenon known as the spontaneous Raman scattering [55,70]. These fields are induced by the spontaneous emission and absorption of optical phonons of frequency ΩR, the so-called Raman shift. The process of Raman scattering is substantially intensified (stimulated) if a second optical field of frequency ωs = ωp − Ωr is launched into the waveguide. In this case, one says that the energy is transferred between the fields via the process of stimulated Raman scattering (SRS) [5355].

In silicon, Raman scattering is produced by optical phonons near the Brillouin zone center [71]. The three phonon modes (almost dispersionless) have the same energy of about 65 meV. This energy determines the population of the phonon modes and relative intensities of red-shifted (Stokes) and blue-shifted (anti-Stokes) lines in the optical spectrum. At room temperature (T ≈ 25 meV), the Stokes line is nearly 13.5 times stronger than the anti-Stokes line. Even though the processes of Stokes and anti-Stokes scattering substantially differ in strength, it is important for the accuracy of numerical data that both are easily accounted for in the FDTD scheme.

The dispersion of Raman scattering is adequately described by treating silicon cores as classical oscillators [49,54]. With this model, one can write the Raman susceptibility in the form [53]

χ~κλμνR(ω;ω1,ω2,ω3)=12[H~(ω1+ω2)κλμν+H~(ω2+ω3)κνμλ],

where the dimensionless tensor ℛκλμν describes the anisotropy of the Raman scattering and the function (ω) represents the Raman gain profile,

H~(ω)=2ξRΩRΓRΩR22iωΓRω2.

Here, ξR = 2ε 0 n 0 c 2 gR(ω)/ω, gR(ω) is the Raman gain coefficient, ΩR = 15.6 THz, and 2ΓR ≈ 100 GHz is the gain bandwidth [72]. The ratio gR(ω)/ω is almost independent of ω since gR(ω) ∝ ω [53]. In the 1.55-μm region, the values of gR reported by different experimental groups lie in the range from 4.3 to 76 cm/GW [73]. The fact that these values are more than 450 and 7500 times larger than the Raman gain coefficient in optical fibers (∼ 0.01 cm/GW), enables net gain from SRS even in the presence of strong nonlinear absorption [21, 74, 75].

In the crystallographic coordinate system, the anisotropy tensor for silicon has the form

klmn=δkmδln+δknδlm2δklδlmδmn.

This expression shows, for example, that the Raman scattering does not occur between two optical beams polarized along the same crystallographic axis (ℛkjjj = δkj + δkj − 2δkj = 0). Therefore, if a single pulse polarized along any principal axis propagates through a silicon waveguide, the SRS-induced redistribution of energy within the pulse bandwidth does not occur.

As before, we transform ℛklmn into the FDTD coordinate system using Eqs. (7) and (8). It is easy to see that the 24 nonzero elements of the anisotropy tensor are given by

αααα=ββββ=ααββ=ββαα=Cϑ,αααβ=ααβα=αβαα=αβββ=Dϑ,βββα=ββαβ=βαββ=βααα=Dϑ,αβαβ=βαβα=αββα=βααβ=1Cϑ,αγαγ=γαγα=αγγα=γααγ=βγβγ=γβγβ=βγγβ=γββγ=1,

where C ϑ=sin2(2ϑ) and Dϑ=12sin(4ϑ).

Using Eq. (12) and introducing a new function H (t) = F −1 [(ω)], we arrive at the following time-domain expression for the Raman susceptibility:

χκλμνR(t1,t2,t3)=1(2π)3+dω1+dω2+dω3χκλμνR(ω;ω1,ω2,ω3)ei(ω1t1+ω2t2+ω3t3)=12[δ(t1t2)δ(t3)κλμν+δ(t1)δ(t2t3)κνμλ]H(t2).

With this result, the Raman-induced material polarization can be written in the following form suitable for the FDTD implementation:

PκR(r,t)=ɛ0λμνtdt1tdt2tdt3χκλμνR(tt1,tt2,tt3)Eλ(r,t1)Eμ(r,t2)Eν(r,t3)=ɛ0λμνκλμνEλ(t)tH(tt1)Eμ(t1)Eν(t1)dt1=ɛ0λΞκλR(r,t)Eλ(r,t),

where

ΞκλR(r,t)=tH(tt1)SκλR(r,t1)dt1,SααR=SββR=Cϑ(Eα2Eβ2)+2DϑEαEβ,SγγR=0,SαβR=SβαR=Dϑ(Eα2Eβ2)+2(1Cϑ)EαEβ,SκγR=SγκR=2EκEγ,(κ=α,β).

3.4. Free-carrier and thermo-optic effects

At high optical powers, TPA produces a large number of free electrons and holes, which give rise to another optical-loss mechanism known as free-carrier absorption (FCA). Besides, free carriers reduce the effective refractive index of silicon waveguides; this effect is referred to as free-carrier dispersion (FCD). The strengths of FCD and FCA depend on the free-carrier density N and are proportional to the real and imaginary parts of the susceptibility as

χ~FC(ω)=n0ΔnFC(N)cn02iωΔαFC(N),

where Δn FC(N) and Δα FC(N) stand, respectively, for the free-carrier-induced change in the refractive index and for the loss coefficient. We assume that carriers are neither injected into the waveguide nor moved out of it, and the densities of electrons and holes are equal. In this case, it is common to use the following empirical formulas [53]:

ΔnFC(N)=ζ(ωr/ω0)2N,ΔαFC(N)=σ(ωr/ω0)2N,ωr=2πc/(1.55μm),ζ=5.3×1027m3,σ=1.45×1021m2.

The evolution of free-carrier density is governed by the interplay between two processes: carrier generation via TPA and carrier recombination through all possible nonradiative channels. These processes are described by the rate equation

N(r,t)t=N(r,t)τc+βTPA2ω0I2(r,t),

where τc is the free-carrier lifetime and I(r,t)=12cɛ0n0|E(r,t)|2 is the optical intensity.

Since silicon is nondirect gap semiconductor, TPA is accompanied by the emission of phonons. In addition, phonons are emitted during the electron-hole recombination. As a result of these two processes, the temperature and refractive index of a silicon waveguide increase. In silicon, the effect of thermal changes in the refractive index—the thermo-optic effect (TOE)—is much stronger than FCD [56]. This is a consequence of the large value of thermo-optic coefficient κ = 1.86 × 10−4 K−1, which relates the nonlinear change in the refractive index to device temperature. The fact that all energy dissipated through TPA is eventually converted to heat, allows one to account for TOE by reducing FCD coefficient ζ by the amount

ζTOE=2ωκθCτcρ(ω0ωr)2,

where θ ≈ 1 μs is the thermal dissipation time, C ≈ 0.7 J/(g×K) is the thermal capacity, and ρ = 2.3 g/cm3 is the density of silicon. Generally, both FCD and TOE should be included in the FDTD simulator because both effects are crucial for phase-sensitive problems.

4. Update equations for the electric field in silicon waveguides

The general FDTD scheme discussed in Section 2 will now be applied to silicon by employing the various susceptibilities specified in the preceding section. This involves discretization of the second constitutive relation in Eq. (2) and derivation of algebraic equations for updating the electric field components.

4.1. Three-dimensional implementation

We start by considering the general situation in which dimensions of the silicon waveguide and the polarization state of the input field are arbitrary. In this case, all six components of the electromagnetic field should be accounted for in the FDTD scheme. In the frequency domain, the electric field is related to the material polarization of the silicon waveguide as

E~κ(r,ω)=(1/ɛ0)[D~κ(ω)P~κLD(ω)P~κK(ω)P~κR(ω)]+ηTPAiωF[SκK(r,t)Eκ(r,t)]n0ΔnFCE~κ(ω)+cn0iωαL+ΔαFC2Eκ~(ω),

where κ = α, β and γ. Multiplying both sides of this equation by and using Eqs. (9) and (14), the standard replacement of the factor − with the finite-difference operator in the time domain yields:

Eκ|n+1Eκ|nΔt=Dκ|n+1Dκ|nɛ0ΔtPκLD|n+1PκLD|nɛ0Δtɛ2SκK|n+1Eκ|n+1SκK|nEκ|nΔt+ηTPASκK|n+1Eκ|n+1+SκK|nEκ|n2λΞκλR|n+1Eλ|n+1ΞκλR|nEλ|nΔtn0(ΔnFC|n+1/2Eκ|n+1Eκ|nΔt+Eκ|n+1+Eκ|n2ΔnFC|n+1ΔnFC|nΔt)cn0αL+ΔαFC|n+1/22Eκ|n+1+Eκ|n2,

where, for compactness, we have omitted the space subscripts (i + d 1,J + d 2,k + d 3) because they are common to all functions. We continue to use this notational simplification in what follows.

To incorporate linear dispersion into the preceding equation, we calculate the value of PκLD at the time step n + 1, by independently updating its s components that correspond to different terms in the Sellmeier equation, i.e.,

PκLD|n+1=j=1sPκjLD|n+1.

Conversion into the time domain of the equation for the jth component, (ωj2ω2)P~κjLD=ɛ0ajωjE~κ, gives us the differential equation

2PκjLDt2+ωj2PκjLD=ɛ0ajωj2Eκ.

This equation is readily discretized with the central difference scheme to obtain

PκjLD|n+1=42+ωj2Δt2PκjLD|nPκjLD|n1+2ωj2Δt22+ωj2Δt2ɛ0ajEκ|n.

In a similar fashion, we use Eq. (13) to find that the evolution of nine auxiliary variables ΞκλR in Eq. (14) is governed by the differential equation

2ΞκλRt2+2ΓRΞκλRt+ΩR2ΞκλR=2ΓRΩRξRSκλR,

whose discretized version takes the form

ΞκλR|n+1=42+2ΓRΔt+ΩR2Δt2ΞκλR|n22ΓRΔt+ΩR2Δt22+2ΓRΔt+ΩR2Δt2ΞκλR|n1+4ΓRΩRΔt22+2ΓRΔt+ΩR2Δt2ξRSκλR|n.

The relations between the functions SκλR and the electric field components are given in Section 3.3.

Since Eq. (15) is the first-order differential equation, the functions Δα FC(N) and Δn FC(N) are defined at the non-integer time steps. The carrier density at the time step n+12 can be found using its value at the time step n12 and the stored components Eκ. The updated values of the functions Δα FC and Δn FC are thus calculated using the equation

N|n+1/2=2τcΔt2τc+ΔtN|n1/2+τcΔt2τc+Δtɛ0ηTPA2ω0(Eα2|n+Eβ2|n+Eγ2|n)2.

For stability and accuracy of the FDTD simulations, time step Δt should be much less than the duration of a single optical cycle. Therefore, we can simplify the algorithm in Eq. (16) with the approximation

ΔnFC|n+1ΔnFC|nΔnFC|n+1/2ΔnFC|n1/2,

without noticeably affecting the FDTD results. Rearranging the terms in Eq. (16), we finally arrive at the following coupled update equations for the electric field components:

Eκ|n+1=κκ+Eκ|n+DκNL|n+1DκNL|nɛ0κ+νκΞκνR|n+1Eν|n+1ΞκνR|nEν|nκ+,

where DκNL=DκPκLD and

κ±=1+ΞκκR|n+1/2±1/2+(ɛ2±ηTPAΔt/2)SκK|n+1/2±1/2±n0cΔt4(αL+ΔαFC|n+1/2)±n02[(1±2)ΔnFC|n+1/2ΔnFC|n1/2].

Notice that each of the auxiliary functions SκK depends polynomially on all three components of the electric field through Eq. (10). For this reason, Eq. (20) cannot be solved analytically for the unknown component Eκ|n +1 (κ = α, β, γ). Rather, values of the functions PκjLD,ΞκλR, and N are updated using Eqs. (17)(19) and the electric field components calculated at the previous time step. With these values, Eq. (20) is then solved iteratively to update the electric field.

4.2. A simplified FDTD model for silicon waveguides

Tight confinement of optical mode inside SOI waveguides results in strong variations of the magnetic field over the mode profile. If the incident beam is polarized perpendicular to the waveguide axis α, such variations generate a relatively large longitudinal component Eα of the electric field as an optical beam propagates through the waveguide [35, 76]. Generally, Eα cannot be ignored even in a single-mode waveguide for which spatial variations of the magnetic field are relatively weak. The reason for this is the SRS-induced coupling between the electric field components, which is expressed by the sum in Eq. (20).

However, the buildup of the longitudinal electric field in the vicinity of the mode center can be ignored when the spectrum of the incident pulse is much narrower than the Raman shift of 15.6 THz. This condition is satisfied for optical pulses of widths > 100 fs. For shorter pulses, the adopted models of the Kerr effect and TPA may need be modified to allow for a finite response time of the bound electrons. Indeed, as can be seen from Eq. (14), the Raman polarization PαR is not produced by the transverse components Eβ and Eγ when the waveguide is fabricated along the crystallographic direction determined by the angle ϑj=π4j, where j = 0,1,2,3 (see Fig. 3), resulting in Dϑj = 0.

In this section we consider a one-dimensional FDTD domain in which the transverse spatial distributions of the TE and TM modes are ignored, and Maxwell's equations are solved only along the waveguide length (the axis α). It amounts to assuming that the incident field is polarized in the βγ plane and the silicon waveguide is fabricated along one of the directions ϑj. Further, the waveguide dimensions are assumed to be large enough that the axial field components remain small all along the waveguide length, i.e., the optical field preserves its transverse nature along the mode center. The effects of waveguide dimensions are then included only through the effective mode index n 0 and the effective mode area. In practice, the results of such a simplified FDTD model should be applicable for square-shape waveguides with dimensions larger than the optical wavelength inside silicon (λ/n 0).

With these limitations in mind, the update equation (20) can be simplified considerably near the mode center and takes the form

Eκ|n+1=κκ+Eκ|n+DκNL|n+1DκNL|nɛ0κ+ΞκνR|n+1Eν|n+1ΞκνR|nEν|nκ+,

where κ = β or γ, ν ≠ κ, and the functions κ± should be used with Eα = 0. These equations show that the preceding two assumptions simplify the general three-dimensional FDTD model substantially and reduce it to a pair of one-dimensional FDTD models [37, 43]. These two one-dimensional models describe propagation of the β-polarized and γ-polarized transverse electromagnetic modes, which are coupled through the Kerr effect, TPA, and SRS. Free-carrier effects do not couple Eqs. (21), as they depend on the electric field at the previous time steps, n and n − 1. The one-dimensional nature of the coupled FDTD models is also evident by their Yee cells shown in Fig. 4.

 

Fig. 4 Three one-dimensional Yee cells corresponding to the simplified FDTD model. The electric and magnetic fields along the propagation direction α are ignored.

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It is important to keep in mind that the reduced implementation of the FDTD scheme cannot be used to analyze anisotropy stemming from the asymmetry of the waveguide cross section (e.g., modal birefringence, FCA-induced polarization rotation, etc. [35]). However, it allows one to examine the effects of intrinsic anisotropy of silicon nonlinearities without too much computational effort. Since the maximum intensity is achieved at the center of a single-mode waveguide, the one-dimensional implementation also allows one to estimate the strongest impact of nonlinear optical phenomena on the field propagation.

5. Numerical examples and discussion

The FDTD algorithm developed here allows one to simulate the propagation of a short optical pulse through a silicon waveguide under quite general conditions. In this section we use the simplified FDTD algorithm to analyze polarization-dependent optical phenomena in silicon waveguides. We consider propagation of optical pulses through waveguides fabricated along the [110] direction on the (001) surface (ϑ = π/4) . We include all nonlinear effects but neglect thermal effects. In the case of XPM, we assume that the two pulses at different wavelengths are launched at the same end of the waveguide. The parameter values are chosen to be: ε 2 = 1.72 × 10−19 m2/V2, β TPA= 0.9 cm/GW, n 0 = 3.17, and αL = 1 dB/cm.

5.1. Input–output characteristics

It is well known that the output power in silicon waveguides saturates with incident power, due to the nonlinear absorption resulting from TPA and FCA [52,77,78]. For relatively short optical pulses, TPA dominates FCA, and the output intensity exhibits dependence on input polarization. Figure 5 shows the input–output characteristics for three silicon waveguides, pumped by Gaussian pulses with a full width at half maximum (FWHM) of 1.4 ps polarized along the TE and TM directions, by plotting the output peak intensity as a function of the input peak intensity. It can be seen that the output intensity is higher for TM pulses because TPA is larger for the TE mode than for the TM mode by a factor of 12(1+ρ)1.14. More generally, Eqs. (10b) and (10c) show that the anisotropy of TPA depends on the waveguide orientation specified by the angle ϑ as 1+12(ρ1) sin2(2ϑ). Since the effects of TPA and FCA accumulate with propagation, the peak output intensity is lower in longer waveguides.

 

Fig. 5 Input–output characteristics for 100-, 200-, and 500-μm-long silicon waveguides pumped by TM- and TE-polarized, 1.4-ps Gaussian pulses. Other parameter values are given in the text.

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5.2. Nonlinear polarization rotation

Aside from different rates of TPA, TE and TM modes in silicon waveguides exhibit different nonlinear phase shifts imposed by SPM and XPM. Both the TPA and XPM modify the state of polarization (SOP) of a sufficiently strong optical field [35]. The efficiency of the nonlinear polarization rotation can be characterized by the transmittance of a linearly polarized input pulse through the analyzer that blocks the pulse for low input intensities [36]. Figure 6 shows such a transmittance as a function of the input polarization angle φ for a 1.4-ps Gaussian pulse; φ = 0 and φ = π/2 correspond to TE and TM modes, respectively (see Fig. 3). We define the transmittance as a ratio of the energy flux, +I(t)dt at the output and input ends of the waveguide.

 

Fig. 6 Efficiency of SPM-induced polarization rotation for different input SOPs of a 1.4-ps Gaussian pulse. The output polarizer is perpendicular to the input polarization state of the pulse. We choose τc = 0.8 ns; other parameters are given in the text.

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One can see from Fig. 6 that the pulse preserves its initial SOP not only when it propagates as either a pure TE or TM mode but also when the input SOP corresponds to φ0=tan1(1/2)35°. Interestingly, the value of φ 0 does not depend on the material parameters or pulse characteristics. For an arbitrary fabrication direction ϑ, it can be found by noting that conservation of SOP requires SβK=SγK at any point along the waveguide. Using Eqs. (10b) and (10c), this condition yields

φ0=tan1112sin2(2ϑ).

Thus, φ 0 reaches its maximum value of π/4 when the waveguide is fabricated along the principal axis [100] or [010]. The relative nonlinear phase shift between the TE and TM polarization components increases monotonically with input pulse intensity I 0 and waveguide length L. However, due to nonlinear losses, there are optimal values of I 0 and L that maximize the pulse transmittance [36].

5.3. Polarization variations along the pulse

Since the processes of SPM and TPA depend on optical intensity, different parts of a pulse exhibit different polarization changes. It is useful to plot variations of the SOP along the output pulse on the Poincaré sphere. The instantaneous SOP of the pulse can be characterized by the polarization ellipse inscribed by the tip of the electric field vector [79]. The time evolution of SOP at the waveguide output is then represented by a closed trace on the Poincaré sphere. Figure 7 shows how SOP changes for the most intense pulse in Fig. 6(a). Each trajectory on the Poincaré sphere starting from the equator corresponds to a specific linear SOP of the input pulse. Consider a pulse with φ = π/10 portrayed by the trace AB. As intensity builds along the leading edge of the pulse, anisotropic nonlinear effects come into force and start increasing the ellipticity and azimuth of the instantaneous SOP (arc AA′). After the ellipticity and azimuth peak at the pulse center with maximum intensity (point B), the efficiencies of the TPA and SPM processes start to decrease (arc BB′), and the SOP returns to the initial linear SOP at the trailing edge of the pulse (point A).

 

Fig. 7 Poincaré-sphere representation of SOP variations along a 1.4-ps Gaussian pulse at the output of a 0.5-mm-long silicon waveguide. Numbers near the traces staring from the equator show φ values for the input SOPs (see Fig. 3); φ 0 ≈ 35°. Traces starting from the poles correspond to circularly polarized input pulses. Here, I 0 = 400 GW/cm2; other parameters are the same as in Fig. 6(a).

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It is interesting to note that the arcs AA′B and BB′A on the Poincaré sphere nearly coincide. A detailed analysis shows that this feature reflects the fact that pulse polarization predominantly changes in response to cross-intensity modulation of the TM and TE modes caused by the degenerate four-wave mixing (FWM), rather than by the SPM-induced relative phase shift between the two modes. The relative nonlinear phase shift between the two polarization components is negligible for picosecond pulses in waveguides shorter than 1 mm, but it becomes important in the quasi-CW regimes or for longer waveguides.

Visualization of the SOP on the Poincaré sphere also reveals the significance of polarization angle φ 0 in Eq. (22). When φ < φ 0, the intensity of the TE mode builds up more slowly than the intensity of the TM mode at the leading edge of the output pulse owing to cross-intensity modulation; this results in the development of the right-handed elliptic polarization. In contrast, when φ > φ 0, energy transfer between the two polarization components results in steeper growth of the TM mode and in the development of the left-handed elliptic polarization. The TM and TE modes excited by an optical pulse whose linear SOP is characterized by the angle φ 0 do not exhibit degenerate FWM and, therefore, do not interchange any energy.

5.4. Nonlinear switching through polarization changes

Nonlinear polarization rotation (NPR) in silicon waveguides can be used for optical switching [34, 36]. Switching of a linearly polarized CW signal is realized by launching an intense pump into the waveguide, which changes the SOP of the CW signal and results in its partial transmittance through the output polarizer (called the analyzer). If the signal is in the form of a pulse, its intensity must be small enough to avoid nonlinear effects in the absence of the pump. Otherwise, different parts of the signal experience different amounts of NPR, and the signal cannot be completely blocked by the analyzer. This restriction on signal intensity can, however, be lifted, if the signal is in the form of a CW beam.

To analyze the switching scenario in detail, we consider switching of a 200-THz CW beam (wavelength 1.5 μm) by 350-fs Gaussian pulses whose spectrum peaks at 210 THz. The intensity of the CW beam is assumed to be 1 GW/cm2 so that it produces almost no nonlinear effects. Figures 8(a)–8(c) show the analyzer transmittance or switching efficiency defined as the ratio of the output signal energy (per unit area near the mode center) to the energy of a 350-fs Gaussian pulse with a peak intensity of 1 GW/cm2. The analyzer is aligned perpendicular to the linear SOP of the input CW beam, since modal birefringence is absent and signal intensity is too low to cause significant NPR in the absence of the pump. It is evident that switching efficiency depends strongly on polarizations of the pump and signal. For example, when the pump excites either the TE or TM mode, the maximum switching efficiency is achieved for the signal that excites both of them equally (i.e., for φ = π/4); if the pump is initially polarized along the direction φ = π/4 (TE-TM case), maximum transmittance occurs for φ = 0 and φ = π/2 [see Fig. 8(a)].

 

Fig. 8 Switching efficiency of a CW beam for a 350-fs Gaussian pump pulse as a function of (a) input polarization angle φ of the CW beam, (b) waveguide length L, and (c) pump's peak intensity Ip 0. Panel (d) shows switching windows for three different waveguide lengths; dashed curve shows the Gaussian pulse that is used in the definition of transmittance. In panels (b)–(d), φ = π/4. In all panels input CW intensity is 1 GW/cm2 and τc = 0.8 ns; other parameters are given in the text.

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Large values of signal transmittance in Fig. 8 are a consequence of using subpicosecond pump pulses and short waveguides. For a given free-carrier lifetime, the transmittance can be maximized by varying the waveguide length L and the pump intensity Ip 0 [36]. The existence of optimal values for L and Ip 0 is evidenced by the maxima seen in Figs. 8(b) and 8(c). Signal profiles at the output of 0.1-, 0.2-, and 0.4-mm-long waveguides are shown in Fig. 8(d). It is seen that the signal is similar in shape to the pump (although slightly asymmetric) in the case of short waveguides. The asymmetry is produced by FCA, which predominantly attenuates the trailing edge of the pump pulse and reduces the NPR that it causes. In longer waveguides, the signal profile changes dramatically. For example, the profile exhibits a local minimum near the point where the pump intensity peaks for L = 0.4 mm. This minimum in signal transmittance is due to excessive NPR leading to partial restoration of the initial signal SOP.

5.5. Polarization dependence of Raman amplification

As a final example, we use our FDTD simulator to illustrate how the SOPs of two pulses, separated in their carrier frequencies by the Raman shift, affect the efficiency of the SRS process that is responsible for Raman amplification. Both the signal and pump pulses are assumed to be 1.4-ps Gaussian pulses with peak intensities of 1 and 100 GW/cm2, respectively. Figure 9 shows the evolution of the peak intensity of the signal pulse along the waveguide length for different SOPs of the input pulses. As can be seen there, no SRS occurs, if both the pump and signal pulses propagate in the form of TM modes, a well-known result for silicon waveguides fabricated along the [110] direction [53, 54, 75].

 

Fig. 9 Peak intensity of TM- and TE-polarized signals amplified via SRS by TM- and TE-polarized pumps. Both input signal and input pump are assumed to be 1.4-ps Gaussian pulses with peak intensities of 1 and 100 GW/cm2, respectively; carrier frequencies are 200 and 215.6 THz; gR = 76 cm/GW; other parameters are given in the text.

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The different evolutions of the signal peak intensity along the waveguide length for three other polarization combinations (the blue, green, and red curves) are due to the anisotropy of the TPA and SRS phenomena. The Raman amplification is considerably stronger when the pump and signal are launched with orthogonal SOPs, a result that does not appear to have been fully appreciated in the literature on silicon Raman amplifiers. It should also be noted that the reduction of peak signal intensity with increasing waveguide length is caused not only by the cumulative nature of FCA, but also by FWM and CARS [19, 63]. Calculation of the output signal intensity for other possible polarizations of the pump and signal is required to optimize the performance of silicon Raman amplifiers.

6. Conclusions

We have presented a comprehensive three-dimensional FDTD model for studying nonlinear optical phenomena in silicon waveguides, which allows—for the first time to our knowledge—for anisotropy of the Kerr effect, two-photon absorption, and stimulated Raman scattering. Based on our model, we developed a computationally efficient FDTD algorithm suitable for simulating polarization-dependent propagation of optical pulses through silicon waveguides. We applied this algorithm to examine several polarization effects that are most favorable for applications. In particular, we demonstrated that the developed algorithm can be used for the optimization of Kerr shutters and silicon Raman amplifiers. Wherever possible, we compared our findings with available experimental data and theoretical results obtained within the framework of the slowly varying envelope approximation. The comparison revealed a high accuracy of our FDTD simulator within its applicability domain. Owing to its generality in handling complex waveguide geometries and short optical pulses, our FDTD simulator is very useful for testing the suitability of anisotropic nonlinearities for different silicon-based photonic devices. Also, we envision using it as a testing vehicle for nonlinear differential equations that describe specific anisotropic phenomena in silicon waveguides.

Acknowledgments

This work was funded by the Australian Research Council through its Discovery Grant scheme under grant DP0877232. The work of Govind Agrawal is also supported by the NSF award ECCS-0801772.

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41. P. M. Goorjian, A. Taflove, R. M. Joseph, and S. C. Hagness, “Computational modeling of femtosecond optical solitons from Maxwell's equations,” IEEE J. Quantum Electron. 28, 2416–2422 (1992). [CrossRef]  

42. F. L. Teixeira, “Time-domain finite-difference and finite-element methods for Maxwell equations in complex media,” IEEE Trans. Antennas Propag. 56, 2150–2166 (2008). [CrossRef]  

43. J. Schneider and S. Hudson, “A finite-difference time-domain method applied to anisotropic material,” IEEE Trans. Antennas Propag. 41, 994–999 (1993). [CrossRef]  

44. R. Holland, “Finite-difference solution of Maxwell's equations in generalized nonorthogonal coordinated,” IEEE Trans. Nucl. Sci. NS-30, 4589–4591 (1983). [CrossRef]  

45. L. X. Dou and A. R. Sebak, “3D FDTD method for arbitrary anisotropic materials,” Microwave Opt. Technol. Lett. 48, 2083–2090 (2006). [CrossRef]  

46. D.-X. Xu, P. Cheben, D. Dalacu, A. Delâge, S. Janz, B. Lamontagne, M.-J. Picard, and W. N. Ye, “Eliminating the birefringence in silicon-on-insulator ridge waveguides by use of cladding stress,” Opt. Lett. 29, 2384–2386 (2004). [CrossRef]   [PubMed]  

47. J. Pastrnak and K. Vedam, “Optical anisotropy of silicon single crystals,” Phys. Rev. B 3, 2567–2571 (1971). [CrossRef]  

48. T. Chu, M. Yamada, J. Donecker, M. Rossberg, V. Alex, and H. Riemann, “Optical anisotropy in dislocation-free silicon single crystals,” Microelectron. Eng. 66, 327–332 (2003). [CrossRef]  

49. R. W. Boyd, Nonlinear Optics, 2nd ed. (Academic Press, Boston, 2003).

50. E. B. Graham and R. E. Raab, “Light propagation in cubic and other anisotropic crystals,” Proc. R. Soc. London , Ser. A 430, 593–614 (1990). [CrossRef]  

51. P. N. Butcher and D. Cotter, The Elements of Nonlinear Optics (Cambridge U. Press, New York, 1991).

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

53. Q. Lin, O. J. Painter, and G. P. Agrawal, “Nonlinear optical phenomena in silicon waveguides: Modeling and applications,” Opt. Express 15, 16604–16644 (2007). [CrossRef]   [PubMed]  

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

55. G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic Press, Boston, 2007).

56. I. D. Rukhlenko, M. Premaratne, and G. P. Agrawal, “Analytical study of optical bistability in silicon-waveguide resonators,” Opt. Express 17, 22124–22137 (2009). [CrossRef]   [PubMed]  

57. I. D. Rukhlenko, M. Premaratne, and G. P. Agrawal, “Analytical study of optical bistability in silicon ring resonators,” Opt. Lett. 35, 55–57 (2010). [CrossRef]   [PubMed]  

58. V. R. Almeida and M. Lipson, “Optical bistability on a silicon chip,” Opt. Lett. 29, 2387–2389 (2004). [CrossRef]   [PubMed]  

59. Q. Xu and M. Lipson, “Carrier-induced optical bistability in silicon ring resonators,” Opt. Lett. 31, 341–343 (2006). [CrossRef]   [PubMed]  

60. R. F. Tinder, Tensor Properties of Solids (Morgan & Claypool Publishers, San Rafael, 2008).

61. J. Zhang, Q. Lin, G. Piredda, R. W. Boyd, G. P. Agrawal, and P. M. Fauchet, “Anisotropic nonlinear response of silicon in the near-infrared region,” Appl. Phys. Lett. 91, 071113(1–3) (2007). [CrossRef]  

62. B. Tatian, “Fitting refractive-index data with the Sellmeier dispersion formula,” Appl. Opt. 23, 4477–4485 (1984). [CrossRef]   [PubMed]  

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

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

65. H. H. Li, “Refractive index of silicon and germanium and its wavelength and temperature derivatives,” J. Phys. Chem. Ref. Data 9, 561–658 (1980). [CrossRef]  

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

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

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

69. P. H. Wendland and M. Chester, “Electric field effects on indirect optical transitions in silicon,” Phys. Rev. 140, A1384–A1390 (1965). [CrossRef]  

70. M. Cardona and G. Güntherodt, Light Scattering in Solids IIM. Cardona and G. Güntherodt, Eds. (Springer-Verlag, New York, 1982).

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

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

73. H. K. Tsang and Y. Liu, “Nonlinear optical properties of silicon waveguides,” Semicond. Sci. Technol. 23, 064007(1–9) (2008). [CrossRef]  

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

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

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

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

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

79. I. D. Rukhlenko, C. Dissanayake, and M. Premaratne, “Visualization of electromagnetic-wave polarization evolution using the Poincaré sphere,” Opt. Lett. 35, 2221–2223 (2010). [CrossRef]   [PubMed]  

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  54. B. Jalali, V. Raghunathan, D. Dimitropoulos, and ¨O. Boyraz, "Raman-based silicon photonics," IEEE J. Sel. Top. Quantum Electron. 12, 412-421 (2006).
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  57. I. D. Rukhlenko, M. Premaratne, and G. P. Agrawal, "Analytical study of optical bistability in silicon ring resonators," Opt. Lett. 35, 55-57 (2010).
    [CrossRef] [PubMed]
  58. V. R. Almeida and M. Lipson, "Optical bistability on a silicon chip," Opt. Lett. 29, 2387-2389 (2004).
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    [CrossRef]
  64. L. Yin, Q. Lin, and G. P. Agrawal, "Soliton fission and supercontinuum generation in silicon waveguides," Opt. Lett. 32, 391-393 (2007).
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    [CrossRef]
  72. Y. Okawachi, M. A. Foster, J. E. Sharping, A. L. Gaeta, Q. Xu, and M. Lipson, "All-optical slow-light on a photonic chip," Opt. Express 14, 2317-2322 (2006).
    [CrossRef] [PubMed]
  73. H. K. Tsang and Y. Liu, "Nonlinear optical properties of silicon waveguides," Semicond. Sci. Technol.  23, 064007(1-9) (2008).
    [CrossRef]
  74. R. Claps, V. Raghunathan, D. Dimitropoulos, and B. Jalali, "Influence of nonlinear absorption on Raman amplification in silicon-on-insulator waveguides," Opt. Express 12, 2774-2780 (2004).
    [CrossRef] [PubMed]
  75. A. Liu, H. Rong, R. Jones, O. Cohen, D. Hak, and M. Paniccia, "Optical amplification and lasing by stimulated Raman scattering in silicon waveguides," J. Lightwave Technol. 24, 1440-1455 (2006).
    [CrossRef]
  76. S. Afshar and T.M. Monro, "A full vectorial model for pulse propagation in emerging waveguides with subwavelength structures part I: Kerr nonlinearity," Opt. Express 17, 2298-2318 (2009).
    [CrossRef]
  77. G. W. Rieger, K. S. Virk, and J. F. Young, "Nonlinear propagation of ultrafast 1.5 μm pulses in high-indexconstrast silicon-on-insulator waveguides," Appl. Phys. Lett. 84, 900-902 (2004).
    [CrossRef]
  78. N. Suzuki, "FDTD analysis of two-photon absorption and free-carrier absorption in Si high-contrast waveguides," J. Lightwave Technol. 25, 2495-2501 (2007).
    [CrossRef]
  79. I. D. Rukhlenko, C. Dissanayake, and M. Premaratne, "Visualization of electromagnetic-wave polarization evolution using the Poincar’e sphere," Opt. Lett. 35, 2221-2223 (2010).
    [CrossRef] [PubMed]

2010 (4)

2009 (5)

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

L. Yin, J. Zhang, P. M. Fauchet, and G. P. Agrawal, "Optical switching using nonlinear polarization rotation inside silicon waveguides," Opt. Lett. 34, 476-478 (2009).
[CrossRef] [PubMed]

I. D. Rukhlenko, M. Premaratne, and G. P. Agrawal, "Analytical study of optical bistability in silicon-waveguide resonators," Opt. Express 17, 22124-22137 (2009).
[CrossRef] [PubMed]

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

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

2008 (3)

H. K. Tsang and Y. Liu, "Nonlinear optical properties of silicon waveguides," Semicond. Sci. Technol.  23, 064007(1-9) (2008).
[CrossRef]

F. L. Teixeira, "Time-domain finite-difference and finite-element methods for Maxwell equations in complex media," IEEE Trans. Antennas Propag. 56, 2150-2166 (2008).
[CrossRef]

B. Jalali, "Can silicon change photonics?" Phys. Status Solidi(a) 2, 213-224 (2008).
[CrossRef]

2007 (8)

J. Zhang, Q. Lin, G. Piredda, R. W. Boyd, G. P. Agrawal, and P. M. Fauchet, "Anisotropic nonlinear response of silicon in the near-infrared region," Appl. Phys. Lett.  91, 071113(1-3) (2007).
[CrossRef]

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

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

E. K. Tien, N. S. Yuksek, F. Qian, and O. Boyraz, "Pulse compression and modelocking by using TPA in silicon waveguides," Opt. Express 15, 6500-6506 (2007).
[CrossRef] [PubMed]

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

N. Suzuki, "FDTD analysis of two-photon absorption and free-carrier absorption in Si high-contrast waveguides," J. Lightwave Technol. 25, 2495-2501 (2007).
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I.-W. Hsieh, X. Chen, X. Liu, J. I. Dadap, N. C. Panoiu, C.-Y. Chou, F. Xia, W. M. Green, Y. A. Vlasov, and R. M. Osgood, "Supercontinuum generation in silicon photonic wires," Opt. Express 15, 15242-15249 (2007).
[CrossRef] [PubMed]

Q. Lin, O. J. Painter, and G. P. Agrawal, "Nonlinear optical phenomena in silicon waveguides: Modeling and applications," Opt. Express 15, 16604-16644 (2007).
[CrossRef] [PubMed]

2006 (18)

B. Jalali and S. Fathpour, "Silicon Photonics," J. Lightwave Technol. 24, 4600-4615 (2006).
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Q. Xu and M. Lipson, "Carrier-induced optical bistability in silicon ring resonators," Opt. Lett. 31, 341-343 (2006).
[CrossRef] [PubMed]

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

C. Manolatou and M. Lipson, "All-optical silicon modulators based on carrier injection by two-photon absorption," J. Lightwave Technol. 24, 1433-1439 (2006).
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A. Liu, H. Rong, R. Jones, O. Cohen, D. Hak, and M. Paniccia, "Optical amplification and lasing by stimulated Raman scattering in silicon waveguides," J. Lightwave Technol. 24, 1440-1455 (2006).
[CrossRef]

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

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

R. Dekker, A. Driessen, T. Wahlbrink, C. Moormann, J. Niehusmann, and M. F¨orst, "Ultrafast Kerr-induced all-optical wavelength conversion in silicon waveguides using 1.55 μm femtosecond pulses," Opt. Express 14, 8336-8346 (2006).
[CrossRef] [PubMed]

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

Y.-H Kuo, H. Rong, V. Sih, S. Xu, M. Paniccia, and O. Cohen, "Demonstration of wavelength conversion at 40 Gb/s data rate in silicon waveguides," Opt. Express 14, 11721-11726 (2006).
[CrossRef] [PubMed]

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

X. Chen, N. C. Panoiu, and R. M. Osgood, Jr., "Theory of Raman-mediated pulsed amplification in silicon-wire waveguides," IEEE J. Quantum Electron. 42, 160-170 (2006).
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H. Garcia and R. Kalyanaraman, "Phonon-assisted two-photon absorption in the presence of a dc-field: The nonlinear Franz-Keldysh effect in indirect gap semiconductors," J. Phys. B 39, 2737-2746 (2006).
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L. X. Dou and A. R. Sebak, "3D FDTD method for arbitrary anisotropic materials," Microwave Opt. Technol. Lett. 48, 2083-2090 (2006).
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B. Jalali, V. Raghunathan, D. Dimitropoulos, and ¨O. Boyraz, "Raman-based silicon photonics," IEEE J. Sel. Top. Quantum Electron. 12, 412-421 (2006).
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R. A. Soref, "The past, present, and future of silicon photonics," IEEE J. Sel. Top. Quantum Electron. 12, 1678-1687 (2006).
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S. E. Thompson and S. Parthasarathy, "Moore’s law: The future of Si microelectronics," Mater. Today 9, 20-25 (2006).
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T. K. Liang, L. R. Nunes, M. Tsuchiya, K. S. Abedin, T. Miyazaki, D. Van Thourhout, W. Bogaerts, P. Dumon, R. Baets, and H. K. Tsang, "High speed logic gate using two-photon absorption in silicon waveguides," Opt. Commun. 265, 171-174 (2006).
[CrossRef]

2005 (8)

D. J. Moss, L. Fu, I. Littler, and B. J. Eggleton, "Ultrafast all-optical modulation via two-photon absorption in silicon-on-insulator waveguides," Electron. Lett. 41, 320-321 (2005).
[CrossRef]

R. Espinola, J. Dadap, R. Osgood, Jr., S. McNab, and Y. Vlasov, "C-band wavelength conversion in silicon photonic wire waveguides," Opt. Express 13, 8336-8346 (2005).
[CrossRef]

R. Salem, G. E. Tudury, T. U. Horton, G. M. Carter, and T. E. Murphy, "Polarization-insensitive optical clock recovery at 80 Gb/s using a silicon photodiode," IEEE Photon. Technol. Lett. 17, 1968-1970 (2005).
[CrossRef]

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

Q. Xu, V. R. Almeida, and M. Lipson, "Demonstration of high Raman gain in a submicrometer-size silicon-oninsulator waveguide," Opt. Lett. 30, 35-37 (2005).
[CrossRef] [PubMed]

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

V. Raghunathan, R. Claps, D. Dimitropoulos, and B. Jalali, "Parametric Raman wavelength conversion in scaled silicon waveguides," J. Lightwave Technol. 23, 2094-2102 (2005).
[CrossRef]

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

2004 (6)

2003 (4)

T. Chu, M. Yamada, J. Donecker, M. Rossberg, V. Alex, and H. Riemann, "Optical anisotropy in dislocation-free silicon single crystals," Microelectron. Eng. 66, 327-332 (2003).
[CrossRef]

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

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

R. Claps, V. Raghunathan, D. Dimitropoulos, and B. Jalali, "Anti-Stokes Raman conversion in silicon waveguides," Opt. Express 11, 2862-2872 (2003).
[CrossRef] [PubMed]

2002 (2)

H. Iwai and S. Ohmi, "Silicon integrated circuit technology from past to future," Microelectron. Reliab. 42, 465-491 (2002).
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R. Claps, D. Dimitropoulos, and B. Jalali, "Stimulated Raman scattering in silicon waveguides," Electron. Lett. 38, 1352-1354 (2002).
[CrossRef]

1999 (1)

M. Schulz, "The end of the road for silicon?" Nature 399, 729-730 (1999).
[CrossRef]

1998 (1)

B. Jalali, S. Yegnanarayanan, T. Yoon, T. Yoshimoto, I. Rendina, and F. Coppinger, "Advances in silicon-oninsulator optoelectronics," IEEE J. Sel. Top. Quantum Electron. 4, 938-947 (1998).
[CrossRef]

1997 (1)

R. M. Joseph and A. Taflove, "FDTD Maxwell’s equations models for nonlinear electrodynamics and optics," IEEE Trans. Antennas Propag. 45, 364-374 (1997).
[CrossRef]

1993 (1)

J. Schneider and S. Hudson, "A finite-difference time-domain method applied to anisotropic material," IEEE Trans. Antennas Propag. 41, 994-999 (1993).
[CrossRef]

1992 (1)

P. M. Goorjian, A. Taflove, R. M. Joseph, and S. C. Hagness, "Computational modeling of femtosecond optical solitons from Maxwell’s equations," IEEE J. Quantum Electron. 28, 2416-2422 (1992).
[CrossRef]

1990 (1)

E. B. Graham and R. E. Raab, "Light propagation in cubic and other anisotropic crystals," Proc. R. Soc. London, Ser. A 430, 593-614 (1990).
[CrossRef]

1984 (1)

1983 (1)

R. Holland, "Finite-difference solution ofMaxwell’s equations in generalized nonorthogonal coordinated," IEEE Trans. Nucl. Sci. NS-30, 4589-4591 (1983).
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1980 (1)

H. H. Li, "Refractive index of silicon and germanium and its wavelength and temperature derivatives," J. Phys. Chem. Ref. Data 9, 561-658 (1980).
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1971 (1)

J. Pastrnak and K. Vedam, "Optical anisotropy of silicon single crystals," Phys. Rev. B 3, 2567-2571 (1971).
[CrossRef]

1966 (1)

K. S. Yee, "Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media," IEEE Trans. Antennas Propag. 14, 302-307 (1966).
[CrossRef]

1965 (1)

P. H. Wendland and M. Chester, "Electric field effects on indirect optical transitions in silicon," Phys. Rev. 140, A1384-A1390 (1965).
[CrossRef]

Abedin, K. S.

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

Afshar, S.

Agrawal, G. P.

I. D. Rukhlenko, M. Premaratne, and G. P. Agrawal, "Analytical study of optical bistability in silicon ring resonators," Opt. Lett. 35, 55-57 (2010).
[CrossRef] [PubMed]

B. A. Daniel and G. P. Agrawal, "Vectorial nonlinear propagation in silicon nanowire waveguides: Polarization effects," J. Opt. Soc. Am. B 27, 956-965 (2010).
[CrossRef]

I. D. Rukhlenko, I. L. Garanovich, M. Premaratne, A. A. Sukhorukov, G. P. Agrawal, and Yu. S. Kivshar, "Polarization rotation in silicon waveguides: Analytical modeling and applications," IEEE Photon. J. 2, 423-435 (2010).
[CrossRef]

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

L. Yin, J. Zhang, P. M. Fauchet, and G. P. Agrawal, "Optical switching using nonlinear polarization rotation inside silicon waveguides," Opt. Lett. 34, 476-478 (2009).
[CrossRef] [PubMed]

I. D. Rukhlenko, M. Premaratne, and G. P. Agrawal, "Analytical study of optical bistability in silicon-waveguide resonators," Opt. Express 17, 22124-22137 (2009).
[CrossRef] [PubMed]

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

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

J. Zhang, Q. Lin, G. Piredda, R. W. Boyd, G. P. Agrawal, and P. M. Fauchet, "Anisotropic nonlinear response of silicon in the near-infrared region," Appl. Phys. Lett.  91, 071113(1-3) (2007).
[CrossRef]

Q. Lin, O. J. Painter, and G. P. Agrawal, "Nonlinear optical phenomena in silicon waveguides: Modeling and applications," Opt. Express 15, 16604-16644 (2007).
[CrossRef] [PubMed]

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

Alex, V.

T. Chu, M. Yamada, J. Donecker, M. Rossberg, V. Alex, and H. Riemann, "Optical anisotropy in dislocation-free silicon single crystals," Microelectron. Eng. 66, 327-332 (2003).
[CrossRef]

Almeida, V. R.

Baets, R.

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

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

Bogaerts, W.

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

Boyd, R. W.

J. Zhang, Q. Lin, G. Piredda, R. W. Boyd, G. P. Agrawal, and P. M. Fauchet, "Anisotropic nonlinear response of silicon in the near-infrared region," Appl. Phys. Lett.  91, 071113(1-3) (2007).
[CrossRef]

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

Boyraz, ¨O.

Boyraz, O.

Carter, G. M.

R. Salem, G. E. Tudury, T. U. Horton, G. M. Carter, and T. E. Murphy, "Polarization-insensitive optical clock recovery at 80 Gb/s using a silicon photodiode," IEEE Photon. Technol. Lett. 17, 1968-1970 (2005).
[CrossRef]

Cheben, P.

Chen, X.

Cheng, X.

Chester, M.

P. H. Wendland and M. Chester, "Electric field effects on indirect optical transitions in silicon," Phys. Rev. 140, A1384-A1390 (1965).
[CrossRef]

Chou, C.-Y.

Chu, T.

T. Chu, M. Yamada, J. Donecker, M. Rossberg, V. Alex, and H. Riemann, "Optical anisotropy in dislocation-free silicon single crystals," Microelectron. Eng. 66, 327-332 (2003).
[CrossRef]

Claps, R.

Cohen, O.

Coppinger, F.

B. Jalali, S. Yegnanarayanan, T. Yoon, T. Yoshimoto, I. Rendina, and F. Coppinger, "Advances in silicon-oninsulator optoelectronics," IEEE J. Sel. Top. Quantum Electron. 4, 938-947 (1998).
[CrossRef]

Dadap, J.

R. Espinola, J. Dadap, R. Osgood, Jr., S. McNab, and Y. Vlasov, "C-band wavelength conversion in silicon photonic wire waveguides," Opt. Express 13, 8336-8346 (2005).
[CrossRef]

Dadap, J. I.

Dalacu, D.

Daniel, B. A.

Dekker, R.

Dimitropoulos, D.

Dinu, M.

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

Dissanayake, C.

I. D. Rukhlenko, C. Dissanayake, and M. Premaratne, "Visualization of electromagnetic-wave polarization evolution using the Poincar’e sphere," Opt. Lett. 35, 2221-2223 (2010).
[CrossRef] [PubMed]

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

Donecker, J.

T. Chu, M. Yamada, J. Donecker, M. Rossberg, V. Alex, and H. Riemann, "Optical anisotropy in dislocation-free silicon single crystals," Microelectron. Eng. 66, 327-332 (2003).
[CrossRef]

Dou, L. X.

L. X. Dou and A. R. Sebak, "3D FDTD method for arbitrary anisotropic materials," Microwave Opt. Technol. Lett. 48, 2083-2090 (2006).
[CrossRef]

Driessen, A.

Dulkeith, E.

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

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

Dumon, P.

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

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

Eggleton, B. J.

D. J. Moss, L. Fu, I. Littler, and B. J. Eggleton, "Ultrafast all-optical modulation via two-photon absorption in silicon-on-insulator waveguides," Electron. Lett. 41, 320-321 (2005).
[CrossRef]

Espinola, R.

R. Espinola, J. Dadap, R. Osgood, Jr., S. McNab, and Y. Vlasov, "C-band wavelength conversion in silicon photonic wire waveguides," Opt. Express 13, 8336-8346 (2005).
[CrossRef]

F¨orst, M.

Fang, A.

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

Fathpour, S.

Fauchet, P. M.

Foster, M. A.

Fu, L.

D. J. Moss, L. Fu, I. Littler, and B. J. Eggleton, "Ultrafast all-optical modulation via two-photon absorption in silicon-on-insulator waveguides," Electron. Lett. 41, 320-321 (2005).
[CrossRef]

Fukuda, H.

Gaeta, A. L.

Garanovich, I. L.

I. D. Rukhlenko, I. L. Garanovich, M. Premaratne, A. A. Sukhorukov, G. P. Agrawal, and Yu. S. Kivshar, "Polarization rotation in silicon waveguides: Analytical modeling and applications," IEEE Photon. J. 2, 423-435 (2010).
[CrossRef]

Garcia, H.

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

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

Goorjian, P. M.

P. M. Goorjian, A. Taflove, R. M. Joseph, and S. C. Hagness, "Computational modeling of femtosecond optical solitons from Maxwell’s equations," IEEE J. Quantum Electron. 28, 2416-2422 (1992).
[CrossRef]

Graham, E. B.

E. B. Graham and R. E. Raab, "Light propagation in cubic and other anisotropic crystals," Proc. R. Soc. London, Ser. A 430, 593-614 (1990).
[CrossRef]

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Vlasov, Y.

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I.-W. Hsieh, X. Chen, X. Liu, J. I. Dadap, N. C. Panoiu, C.-Y. Chou, F. Xia, W. M. Green, Y. A. Vlasov, and R. M. Osgood, "Supercontinuum generation in silicon photonic wires," Opt. Express 15, 15242-15249 (2007).
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Figures (9)

Fig. 1
Fig. 1

Staggered space arrangement (left) and leapfrog time ordering (right) of discretized electromagnetic-field components using a cubic Yee cell.

Fig. 2
Fig. 2

Positions of electric field components in the FDTD grid. The values of components Eβ and Eγ at the node (i, j, k) are calculated by averaging their four values over the nodes (i ± 1/2, j ± 1/2, k) and (i ± 1/2, j, k ± 1/2), respectively.

Fig. 3
Fig. 3

Relative orientation of FDTD (α, β, γ) and crystallographic (x, y, z) axes adopted in the paper. For ϑ = π 4 , the FDTD axes α, β, and γ coincide, respectively, with the [110], [1̄10], and [001] crystallographic directions. The inset shows the TM and TE polarizations and an arbitrary linear polarization determined by the angle φ.

Fig. 4
Fig. 4

Three one-dimensional Yee cells corresponding to the simplified FDTD model. The electric and magnetic fields along the propagation direction α are ignored.

Fig. 5
Fig. 5

Input–output characteristics for 100-, 200-, and 500-μm-long silicon waveguides pumped by TM- and TE-polarized, 1.4-ps Gaussian pulses. Other parameter values are given in the text.

Fig. 6
Fig. 6

Efficiency of SPM-induced polarization rotation for different input SOPs of a 1.4-ps Gaussian pulse. The output polarizer is perpendicular to the input polarization state of the pulse. We choose τc = 0.8 ns; other parameters are given in the text.

Fig. 7
Fig. 7

Poincaré-sphere representation of SOP variations along a 1.4-ps Gaussian pulse at the output of a 0.5-mm-long silicon waveguide. Numbers near the traces staring from the equator show φ values for the input SOPs (see Fig. 3); φ 0 ≈ 35°. Traces starting from the poles correspond to circularly polarized input pulses. Here, I 0 = 400 GW/cm2; other parameters are the same as in Fig. 6(a).

Fig. 8
Fig. 8

Switching efficiency of a CW beam for a 350-fs Gaussian pump pulse as a function of (a) input polarization angle φ of the CW beam, (b) waveguide length L, and (c) pump's peak intensity Ip 0. Panel (d) shows switching windows for three different waveguide lengths; dashed curve shows the Gaussian pulse that is used in the definition of transmittance. In panels (b)–(d), φ = π/4. In all panels input CW intensity is 1 GW/cm2 and τ c = 0.8 ns; other parameters are given in the text.

Fig. 9
Fig. 9

Peak intensity of TM- and TE-polarized signals amplified via SRS by TM- and TE-polarized pumps. Both input signal and input pump are assumed to be 1.4-ps Gaussian pulses with peak intensities of 1 and 100 GW/cm2, respectively; carrier frequencies are 200 and 215.6 THz; gR = 76 cm/GW; other parameters are given in the text.

Tables (1)

Tables Icon

Table 1 Values of dμ specifying relative positions of components Vξ inside a four-dimensional unit cell.

Equations (48)

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B ( r , t ) t = × E ( r , t ) , D ( r , t ) t = × H ( r , t ) ,
H ( r , t ) = B ( r , t ) μ 0 , E ( r , t ) = D ( r , t ) P ( r , t ) ɛ 0 ,
V ( r , t ) = α ^ V α ( r , t ) + β ^ V β ( r , t ) + γ ^ V γ ( r , t ) ,
V ξ | i + d 1 , j + d 2 , k + d 3 n + d 4 V ξ [ ( i + d 1 ) Δ ξ , ( j + d 2 ) Δ ξ , ( k + d 3 ) Δ ξ , ( n + d 4 ) Δ t ] ,
B α | i 1 / 2 , j 1 / 2 , k 1 / 2 n + 1 / 2 = B α | n 1 / 2 S c ( E γ | i , j + 1 / 2 , k n E γ | i , j 1 / 2 , k n E β | i , j , k + 1 / 2 n + E β | i , j , k 1 / 2 n ) , H α | i 1 / 2 , j 1 / 2 , k 1 / 2 n + 1 / 2 = H α | n 1 / 2 + 1 μ 0 ( B α | n + 1 / 2 B α | n 1 / 2 ) , D α | ijk n + 1 = D α | ijk n + S c ( H γ | i , j + 1 / 2 , k n + 1 / 2 H γ | i , j 1 / 2 , k n + 1 / 2 H β | i , j , k + 1 / 2 n + 1 / 2 + H β | i , j , k 1 / 2 n + 1 / 2 ) ,
E ξ | i + d 1 , j + d 2 , k + d 3 n + 1 = E ξ | n + 1 ɛ 0 ( D ξ | n + 1 D ξ | n P ξ | n + 1 + P ξ | n ) .
P ( r , t ) = i P ( i ) + a P ( a ) .
P ξ ( i ) | i + d 1 , j + d 2 , k + d 3 n + 1 = f ξ ( i ) ( P ξ ( i ) | n , P ξ ( i ) | n 1 , , E ξ | n + 1 , E ξ | n , E ξ | n 1 , ) ,
P ξ ( a ) | i + d 1 , j + d 2 , k + d 3 n + 1 = f ξ ( a ) ( P ξ ( a ) | n , P ξ ( a ) | n 1 , , E α | n + 1 , E β | n + 1 , E γ | n + 1 , E α | n , E β | n ) .
E β | ijk n = 1 4 ( E β | i 1 / 2 , j 1 / 2 , k n + E β | i 1 / 2 , j + 1 / 2 , k n + E β | i + 1 / 2 , j + 1 / 2 , k n + E β | i + 1 / 2 , j 1 / 2 , k n ) , E γ | ijk n = 1 4 ( E γ | i 1 / 2 , j , k 1 / 2 n + E γ | i 1 / 2 , j , k + 1 / 2 n + E γ | i + 1 / 2 , j , k 1 / 2 n + E γ | i + 1 / 2 , j , k + 1 / 2 n ) .
P ˜ κ ( r , ω ) = ɛ 0 χ ˜ ( 1 ) ( ω ) E ˜ κ ( r , ω ) + ɛ 0 ( 2 π ) 2 λ μ ν + d ω 1 + d ω 2 χ ˜ κ λ μ ν ( 3 ) ( ω ; ω 1 , ω 2 , ω 3 ) E ˜ λ ( ω 1 ) E ˜ μ ( ω 2 ) E ˜ ν ( ω 3 ) + ɛ 0 χ ˜ FC ( | E | 4 , ω ) E ˜ κ ( r , ω ) ,
V ˜ ( ω ) = F [ V ( t ) ] + V ( t ) e i ω t d t , V ( t ) = F 1 [ V ˜ ( ω ) ] 1 2 π + V ˜ ( ω ) e i ω t d ω .
χ κ λ μ ν ( 3 ) = klmn a k κ a l λ a m μ a n ν χ klmn ( 3 ) ,
a = ( cos ϑ sin ϑ 0 sin ϑ cos ϑ 0 0 0 1 ) .
ɛ ˜ ( ω ) ɛ ˜ ( ω ) + i ɛ ˜ ( ω ) = 1 + χ ˜ ( 1 ) ( ω ) .
n 2 ( ω ) = 1 + j = 1 s a j ω j 2 ω j 2 ω 2 ,
χ ˜ κ λ μ ν e ( ω ; ω 1 , ω 2 , ω 3 ) χ ˜ α α α α e ( ω ) κ λ μ ν ,
klmn = ( ρ / 3 ) ( δ kl δ mn + δ km δ ln + δ kn δ lm ) + ( 1 ρ ) δ kl δ lm δ mn ,
α α α α = β β β β = 1 + A ϑ , γ γ γ γ = 1 , α α α β = α α β α = α β α α = β α α α = B ϑ , β β β α = β β α β = β α β β = α β β β = B ϑ , α α β β = α β β α = β β α α = α β α β = β α β α = β α α β = ρ / 3 A ϑ , α α γ γ = α γ γ α = γ γ α α = α γ α γ = γ α γ α = γ α α γ = ρ / 3 , β β γ γ = β γ γ β = γ γ β β = β γ β γ = γ β γ β = γ β β γ = ρ / 3 ,
P κ K ( r , t ) = ɛ 0 ɛ 2 λ μ ν κ λ μ ν E λ ( r , t ) E μ ( r , t ) E ν ( r , t ) = ɛ 0 ɛ 2 S κ K ( r , t ) E κ ( r , t ) ,
S α K = ( 1 + A ϑ ) E α 2 + ( ρ 3 A ϑ ) E β 2 + ρ E γ 2 + 3 B ϑ E α E β B ϑ E β 3 / E α ,
S β K = ( 1 + A ϑ ) E β 2 + ( ρ 3 A ϑ ) E α 2 + ρ E γ 2 3 B ϑ E α E β + B ϑ E α 3 / E β ,
S γ K = ρ E α 2 + ρ E β 2 + E γ 2 .
P ˜ κ TPA ( r , ω ) = ɛ 0 η TPA i ω F [ S κ K ( r , t ) E κ ( r , t ) ] .
χ ~ κ λ μ ν R ( ω ; ω 1 , ω 2 , ω 3 ) = 1 2 [ H ~ ( ω 1 + ω 2 ) κ λ μ ν + H ~ ( ω 2 + ω 3 ) κ ν μ λ ] ,
H ~ ( ω ) = 2 ξ R Ω R Γ R Ω R 2 2 i ω Γ R ω 2 .
klmn = δ km δ ln + δ kn δ lm 2 δ kl δ lm δ mn .
α α α α = β β β β = α α β β = β β α α = C ϑ , α α α β = α α β α = α β α α = α β β β = D ϑ , β β β α = β β α β = β α β β = β α α α = D ϑ , α β α β = β α β α = α β β α = β α α β = 1 C ϑ , α γ α γ = γ α γ α = α γ γ α = γ α α γ = β γ β γ = γ β γ β = β γ γ β = γ β β γ = 1 ,
χ κ λ μ ν R ( t 1 , t 2 , t 3 ) = 1 ( 2 π ) 3 + d ω 1 + d ω 2 + d ω 3 χ κ λ μ ν R ( ω ; ω 1 , ω 2 , ω 3 ) e i ( ω 1 t 1 + ω 2 t 2 + ω 3 t 3 ) = 1 2 [ δ ( t 1 t 2 ) δ ( t 3 ) κ λ μ ν + δ ( t 1 ) δ ( t 2 t 3 ) κ ν μ λ ] H ( t 2 ) .
P κ R ( r , t ) = ɛ 0 λ μ ν t d t 1 t d t 2 t d t 3 χ κ λ μ ν R ( t t 1 , t t 2 , t t 3 ) E λ ( r , t 1 ) E μ ( r , t 2 ) E ν ( r , t 3 ) = ɛ 0 λ μ ν κ λ μ ν E λ ( t ) t H ( t t 1 ) E μ ( t 1 ) E ν ( t 1 ) d t 1 = ɛ 0 λ Ξ κ λ R ( r , t ) E λ ( r , t ) ,
Ξ κ λ R ( r , t ) = t H ( t t 1 ) S κ λ R ( r , t 1 ) d t 1 , S α α R = S β β R = C ϑ ( E α 2 E β 2 ) + 2 D ϑ E α E β , S γ γ R = 0 , S α β R = S β α R = D ϑ ( E α 2 E β 2 ) + 2 ( 1 C ϑ ) E α E β , S κ γ R = S γ κ R = 2 E κ E γ , ( κ = α , β ) .
χ ~ F C ( ω ) = n 0 Δ n FC ( N ) c n 0 2 i ω Δ α FC ( N ) ,
Δ n FC ( N ) = ζ ( ω r / ω 0 ) 2 N , Δ α FC ( N ) = σ ( ω r / ω 0 ) 2 N , ω r = 2 π c / ( 1.55 μ m ) , ζ = 5.3 × 10 27 m 3 , σ = 1.45 × 10 21 m 2 .
N ( r , t ) t = N ( r , t ) τ c + β TPA 2 ω 0 I 2 ( r , t ) ,
ζ TOE = 2 ω κ θ C τ c ρ ( ω 0 ω r ) 2 ,
E ~ κ ( r , ω ) = ( 1 / ɛ 0 ) [ D ~ κ ( ω ) P ~ κ LD ( ω ) P ~ κ K ( ω ) P ~ κ R ( ω ) ] + η TPA i ω F [ S κ K ( r , t ) E κ ( r , t ) ] n 0 Δ n FC E ~ κ ( ω ) + c n 0 i ω α L + Δ α FC 2 E κ ~ ( ω ) ,
E κ | n + 1 E κ | n Δ t = D κ | n + 1 D κ | n ɛ 0 Δ t P κ LD | n + 1 P κ LD | n ɛ 0 Δ t ɛ 2 S κ K | n + 1 E κ | n + 1 S κ K | n E κ | n Δ t + η TPA S κ K | n + 1 E κ | n + 1 + S κ K | n E κ | n 2 λ Ξ κ λ R | n + 1 E λ | n + 1 Ξ κ λ R | n E λ | n Δ t n 0 ( Δ n FC | n + 1 / 2 E κ | n + 1 E κ | n Δ t + E κ | n + 1 + E κ | n 2 Δ n FC | n + 1 Δ n FC | n Δ t ) c n 0 α L + Δ α FC | n + 1 / 2 2 E κ | n + 1 + E κ | n 2 ,
P κ LD | n + 1 = j = 1 s P κ j LD | n + 1 .
2 P κ j LD t 2 + ω j 2 P κ j LD = ɛ 0 a j ω j 2 E κ .
P κ j LD | n + 1 = 4 2 + ω j 2 Δ t 2 P κ j LD | n P κ j LD | n 1 + 2 ω j 2 Δ t 2 2 + ω j 2 Δ t 2 ɛ 0 a j E κ | n .
2 Ξ κ λ R t 2 + 2 Γ R Ξ κ λ R t + Ω R 2 Ξ κ λ R = 2 Γ R Ω R ξ R S κ λ R ,
Ξ κ λ R | n + 1 = 4 2 + 2 Γ R Δ t + Ω R 2 Δ t 2 Ξ κ λ R | n 2 2 Γ R Δ t + Ω R 2 Δ t 2 2 + 2 Γ R Δ t + Ω R 2 Δ t 2 Ξ κ λ R | n 1 + 4 Γ R Ω R Δ t 2 2 + 2 Γ R Δ t + Ω R 2 Δ t 2 ξ R S κ λ R | n .
N | n + 1 / 2 = 2 τ c Δ t 2 τ c + Δ t N | n 1 / 2 + τ c Δ t 2 τ c + Δ t ɛ 0 η TPA 2 ω 0 ( E α 2 | n + E β 2 | n + E γ 2 | n ) 2 .
Δ n FC | n + 1 Δ n FC | n Δ n FC | n + 1 / 2 Δ n FC | n 1 / 2 ,
E κ | n + 1 = κ κ + E κ | n + D κ NL | n + 1 D κ NL | n ɛ 0 κ + ν κ Ξ κ ν R | n + 1 E ν | n + 1 Ξ κ ν R | n E ν | n κ + ,
κ ± = 1 + Ξ κ κ R | n + 1 / 2 ± 1 / 2 + ( ɛ 2 ± η TPA Δ t / 2 ) S κ K | n + 1 / 2 ± 1 / 2 ± n 0 c Δ t 4 ( α L + Δ α FC | n + 1 / 2 ) ± n 0 2 [ ( 1 ± 2 ) Δ n FC | n + 1 / 2 Δ n FC | n 1 / 2 ] .
E κ | n + 1 = κ κ + E κ | n + D κ NL | n + 1 D κ NL | n ɛ 0 κ + Ξ κ ν R | n + 1 E ν | n + 1 Ξ κ ν R | n E ν | n κ + ,
φ 0 = tan 1 1 1 2 sin 2 ( 2 ϑ ) .

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