## Abstract

By performing time-resolved experiments and power-dependent measurements using femtosecond pulses inside submicron cross-section Si photonic-wire waveguides, we demonstrate strong cross-phase modulation (XPM) effects. We find that XPM in Si wires can be significant even for low peak pump powers, i.e., ∼15 mW for π phase shift. Our experimental data closely match numerical simulations using a rigorous coupled-wave theoretical treatment. Our results suggest that XPM is a potentially useful approach for all-optical control of photonic devices in Si wires.

©2007 Optical Society of America

## 1. Introduction

Within the past few years, the field of silicon photonics has received increasing attention due to major advances in the design, fabrication, and applications of silicon photonics devices. Because of its favorable linear and nonlinear optical properties coupled with advanced CMOS fabrication technology, Si fabricated on Silicon-on-Insulator (SOI) material system, in particular, has the potential to become the major deeply scaled integrated optical platform. Its relatively large refractive index contrast has permitted miniaturization of SOI waveguides to submicron cross-section dimensions and allowed structures such as sharp bends of micron-sized turning radii, with minimal optical loss [1]. This decrease in the cross-section of Si waveguides has resulted in three major advantages. First, the linear and nonlinear optical parameters are renormalized, making dispersion engineering possible. For silicon-wire waveguides (Si-WWGs), having a transverse dimension of < 1 μm, their dispersion properties are governed chiefly by the *waveguide* dispersion [2–7]. Thus by carefully designing the transverse waveguide dimensions, one can *tailor* these dispersion properties. For example, one can design the zero-GVD wavelength to be around 1550 nm or the C-band to be in the anomalous dispersion regime [6]. Second, Si-WWGs have intrinsically low carrier lifetimes. At the relevant telecommunications wavelengths and high powers, free-carriers can be generated by two-photon absorption (TPA). For relatively large cross-section waveguides, these carriers have relatively long lifetimes (≫ 1 ns) and serve as a loss mechanism [8]. With the use of smaller cross-section waveguides, the carrier lifetime is intrinsically lower (< 1 ns), brought about by the reduced diffusion time and recombination at the boundary of the waveguide [9]. Further reduction in carrier lifetime has been demonstrated by sweeping out the carriers by application of electric field through *p*-*i*-*n* structures [10,11], thus further minimizing the waveguide loss. Third, because of high optical confinement in Si-WWGs, large power densities are available, which enhances the nonlinear optical effects. Having a centrosymmetric bulk medium, silicon has a negligible second-order optical nonlinearity. It has, however, a relatively large third-order optical nonlinearity, and in conjunction with the strong confinement from silicon wire waveguides, this has resulted in demonstration of various nonlinear optical devices that are only a few mm in length, in contrast, for example, to km-long fiber optical devices. SOI has been used to demonstrate a wide variety of optical active functionalities including all-optical modulators, wavelength converters, Raman lasers, Raman amplifiers, and thermo-optic switches [10–23].

Dispersion engineering together with the strong nonlinear optical properties of Si-WWGs, has brought about important active functionalities that have been proposed or demonstrated, such as broadband optical parametric gain [13–17], temporal pulse compression and soliton generation in the anomalous dispersion regime [6,7]. Thus it is crucial to investigate the importance of nonlinear effects such as self-phase modulation (SPM) [24–27] and cross-phase modulation (XPM) [28–30] in Si wire waveguides as a means to implement nonlinear photonic devices. Of these nonlinear effects, XPM is particularly important since it allows control of one pulse via a nonlinear phase shift using a second “pump” pulse at a different wavelength. Several groups have reported XPM in Si waveguides. These include one observation of XPM in 2 μm^{2} scale waveguides [28], for which free-carrier effects are important, while a second study examined XPM in Si wires using very high intensity 300 fs pulses [30]. In the case of high intensity fs-pulses several nonlinear phenomena and pulse-dispersion effects can enter into the interpretation of the results and thus it is essential to match experiments with a full theoretical treatment.

In this paper, we present an experimental and theoretical study of XPM in Si-WWGs operating in a distinct pulse-propagation and dispersion regime, i.e., the interaction (or the walk-off) length between the pump and the probe pulses is a fraction of the waveguide length. In this work, we use ∼200 fs pulses and a dimension-tailored waveguide such that the group velocity dispersion (GVD), third-order dispersion (TOD), and nonlinear lengths are all comparable to the waveguide length. Such a system, in which these various length scales are comparable, yields complex but rich information on pulse propagation and pulse distortion in Si-wires. In addition, we use a rigorous numerical simulation of the nonlinear coupled-equations linking the pump and probe field envelopes and the carrier density, which allows us to interpret accurately the various nonlinear processes that are observed, and which agree well with the experimental observations. Our observations show clearly that Si-wire waveguides have the potential to form a “fiber-on-a-chip” system allowing for nonlinear optical control of on-chip functions or integrated photonic circuits.

The outline of this paper is as follows: In Section 2, we discuss the parameters of the waveguide and the experimental setup. In Section 3, we establish the theoretical background for XPM, which includes a discussion of the linear optical dispersion properties of the silicon waveguide and the pulse propagation model. In Section 4, we present and discuss the experimental and numerical results for the power-dependence of the temporal and spectral profile of the pump, determined by the SPM, as well as a comprehensive time-resolved study of the XPM effects induced by the pump pulse on a co-propagating probe. In addition, applications of our results to future optoelectronic devices are also discussed.

## 2. Experimental setup

Our experiments use single-mode Si wire waveguides (Si-WWG) having a cross-section of *A*
_{0} = *w*×*h* =445×220 nm^{2} and length *L*=4.7 mm fabricated on Unibond SOI with a 1-μm-thick oxide layer and aligned along the [110] crystallographic direction. Each end of the waveguides has an inverse-taper mode-converter, which allows efficient coupling. The devices were fabricated using the CMOS production line at the IBM T. J. Watson Research Center [1].

The laser source is an ultrafast mode-locked Er-doped fiber-laser having a pulse repetition rate of 37 MHz and a bandwidth of 80 nm. A beam splitter separates the laser into pump and probe beams. The two beams then pass through bandpass filters, which spectrally separate the pump and probe beams, such that the center wavelengths are *λ _{p}*=1527 nm and

*λ*=1590 nm, respectively. The resulting pump has a pulse width of

_{s}*T*≈ 204 fs whereas the pulse width of the much weaker probe is

_{p}*T*≈ 170 fs; the pulse widths were measured by autocorrelation and cross-correlation, respectively. A delay line is used to vary the temporal spacing between the two pulses, which are then coupled into the waveguide with a free-space objective. The polarization directions of the two beams are aligned along the TE field direction of the waveguide. The output was collected by a tapered fiber, and is characterized by an optical spectrum analyzer (OSA) and power meter. Free-space coupling instead of tapered fiber coupling is employed to rule out SPM in the input fiber, but at the expense of a larger coupling loss between the lens and the waveguide of ∼30 dB. The coupled power, however, is strong enough for the pump beam to modulate the probe through XPM. The coupled peak pump and probe pulse powers are estimated to be 20 mW and 10 μW, respectively. In addition, the propagation loss inside the waveguide has been previously characterized to be ∼3.5 dB/cm [1].

_{s}## 3. Simulation model and XPM theory

#### 3.1 Dispersion properties of the silicon photonic wire

To describe the dynamics of the pulse propagation, we first determined the waveguide dispersion properties, *viz*. effective index *n _{eff}*, group index

*n*, group-velocity dispersion coefficient, β

_{g}_{2}, and third-order dispersion coefficient, β

_{3}; these quantities are defined by

*n*=β

_{g}_{1}

*c*and β

_{m}= d

^{m}β

_{0}/dω

^{m}, where β

_{0}=

*n*(ω)ω/

_{eff}*c*and ω is the carrier frequency. We calculated

*n*using the RSoft BeamPROP software [31] based on a full vectorial beam propagation method, and the result was crosschecked with a finite-element method (FEM) calculation and experimental data. We then fit the values of

_{eff}*n*with a 7

_{eff}^{th}-order polynomial and took numerical derivatives of this polynomial to obtain

*n*and β

_{g}_{2}. The dispersion coefficients, up to the second order, that are obtained by using this method agree with FEM calculations results within 0.1% of each other and with experimental results as well [3]. Notice that for the wavelength range used in our experiments, our waveguide exhibits anomalous dispersion (β

_{2}<0). The GVD coefficients are β

_{2,p}= -3.89 ps

^{2}/m at 1527 nm and β

_{2,s}= -4.26 ps

^{2}/m at 1590 nm. With regard to the calculation of β

_{3}, this is a more difficult problem because the accumulated and enlarged errors from each numerical derivative step prevent ready determination of the consecutive numerical derivatives above the second order; see Ref. [27] for a more complete discussion of this point. As a result, we only show the TOD coefficients calculated using this approach in Fig. 1(b) for completeness. As will be discussed in a later section, we have used a new approach to determine the TOD coefficients more accurately at several wavelengths; this method can be crosschecked by comparing simulation and experimental measurements

#### 3.2 Simulation model

In Si wire waveguides, the pulse dynamics are governed by the interplay of nonlinear and dispersion effects whose relative strengths are determined by several characteristic lengths, namely the 2^{nd}- and 3^{rd}-order dispersion lengths, defined as *L _{D}* =

*T*

_{p}^{2}/|β

_{2P}| and

*L*′ =

_{D}*T*

_{p}^{3}/|β

_{3p}|, respectively, and the nonlinear length, defined as

*L*= ε

_{NL}_{0}

*v*

_{g,p}^{2}/3ω

*Γ*

_{p}P_{p}_{p}′, where

*v*and

_{g,p}*v*are the group velocities of the pump and probe pulses, respectively; to define these characteristic lengths, we used the parameters of the pump pulse with the appropriate material and waveguide parameters. The nonlinear effects described by our model, i.e., SPM, XPM, and TPA, are characterized by a set of complex effective third-order nonlinear coefficients of the Si-WWG, Γ

_{g,s}_{s,p}= Γ′

_{s,p}+

*i*Γ″

_{S,P}and Γ

_{sp,ps}= Γ′

_{sP,PS}+

*i*Γ″

_{sP,PS,}which are defined as the overlap integral of the bulk third-order susceptibility tensor of silicon, χ

^{(3)}, and the waveguide modes at the pump and probe (signal) frequencies; prime and double-prime designate the real and imaginary part, respectively. According to these definitions, the real parts of the quantities Γ

_{S,P}and Γ

_{sP,PS}describe the SPM and XPM effects, respectively, and are directly related to the nonlinear refractive index,

*n*

_{2}, whereas their imaginary parts describe TPA processes, which are quantified by the TPA coefficient β. In addition, we take into account the dependence of the pulse dynamics on free carriers (FC) generated through TPA, such as free-carrier absorption (FCA) and FC-induced dispersion. Under these conditions, the dynamics of pulse propagation in the Si-WWG, described by the slowly varying envelopes

*u*(

_{p,s}*z, t*) of the co-propagating pulses, is governed by the following system of coupled nonlinear differential equations [2,5]:

where *P _{p,s}* are the pulse peak-powers,

*α*is the intrinsic loss,

_{in}*α*is the FCA coefficient, and

_{FC}^{p,s}*δn*is the FC-induced change of the refractive index. The parameters Γ

_{FC}^{p,s}_{S,P}and Γ

_{sp,ps}are defined as Γ

_{j}=

*A*

_{0}∫

**e**

^{*}

_{j}∙χ

^{(3)}⋮

**e**

_{j}

**e**

_{j}

^{ *}

_{j}

**e**

_{j}

*dA*/J

_{j}

^{2}, Γ

_{jl}=

*A*

_{0}∫

**e**

^{*}

_{l}∙ χ

^{(3)}⋮

**e**

_{j}

**e**

^{*}

_{j}

**e**

_{l}

*dA*/

*(*J

_{j}J

_{l}

*)*, (

*j,l*=

*p,s*), where J

_{p,s}= ∫

*n*

^{2}

*(*

**r**˔

*)*|

**e**

_{p,s}|

^{2}

*dA*and

**e**

_{p,s}=

**e**

_{p,s}

*(*ω

_{p,s};

**r**

*)*are the waveguide modes. To compute these parameters we used [2,32] χ

^{(3)}

_{1111}=(2.20+

*i*0.27)×10

^{-1 9}m

^{2}/V

^{2}and

*χ*

^{(3)}

_{1122}= (5.60 +1.82)×10

^{-20}m

^{2}/V

^{2}. Based on a Drude model,

*δn*and

_{FC}*α*are given by [33]

_{FC}*δn*= -

_{FC}*e*

^{2}(

*N*/

*m*

^{*}

_{ce}+

*N*

^{0.8}/

*m*

^{*}

_{ch})/2ε

_{0}

*nω*

^{2}and

*α*=

_{FC}*e*

^{3}

*N*(1/

*μ*

_{e}m^{*2}

_{ce}+1/

*μ*

_{h}m^{*2}

_{ch})/ε

_{0}

*cnω*

^{2}, where

*N*is the free-carrier density (in cm

^{-3});

*m*

^{*}

_{ce}=0.26

*m*

_{0}(

*m*

^{*}

_{ce}= 0.39

*m*

_{0}) is the effective mass of the electrons (holes) with

*m*

_{0}as the mass of the electron; and μ

_{e}(

*μ*) is the electron (hole) mobility.

_{h}For our XPM experiment, the pulse width is in the fs domain, so that the characteristic lengths *L _{D}*,

*L*′, and

_{D}*L*are comparable at peak powers of just a few mW. Specifically, for the pump used in our experiment,

_{NL}*T*=204 fs,

_{p}*L*∼10.7 mm and

_{D}*L*′ ∼11.6 mm, whereas the length

_{D}*L*, which depends on the peak power, has a similar value,

_{NL}*L*, = 9 mm, if

_{NL}*P*

_{0}= 5 mW. Consequently, at or above

*P*

_{0}∼ 5 mW, all GVD, TOD, and SPM effects must be incorporated in a complete description of the dynamics of the co-propagating pulses.

#### 3.3 Nonlinearity-induced phase shift

One of the main effects induced by the optical nonlinearity is a gradual shift of the optical phase of the pulse, i.e., the various frequency components of the pulse undergo a frequency shift, which leads to an overall broadening of the pulse. In the more general case of co-propagation of the two optical pulses, the overall nonlinear phase shift is the result of two cumulative effects: the SPM-induced phase shift, which originates from the nonlinear change of the refractive index induced by the pulse itself, and the XPM contribution, which is related to the nonlinear change of the refractive index induced by the co-propagating pulse. In the general case described by the system of equations (1.a–c) the nonlinearly induced phase shift can only be found numerically; however, in the case, in which the SPM and XPM terms in Eqs. (1.a,b) are the dominant ones, one can derive [34, 35] an analytic formula for the nonlinear-induced phase shift of the probe, ϕ_{s,} which we will use in interpreting a portion of our experimental results. Thus, by neglecting for the moment the TPA effects, the nonlinear phase shift ϕ_{s}(*z,T*) can be written as:

Here, *T* = *t*-*z*/*v _{g,p}* is the time in the reference frame of the pump pulse,

*γ*= 3ω

_{s}_{s}Γ

_{s}/

*ε*

_{0}

*A*

_{0}

*v*

_{g,s}^{2}and

*γ*= 3

_{ps}*ω*Γ

_{s}_{ps}/

*ε*

_{0}

*A*

_{0}

*v*are the SPM and XPM coefficients, respectively, and ∆ = 1/

_{g,P}v_{g,S}*v*-1/

_{g,s}*v*is the temporal walk-off The calculated XPM coefficient,

_{g,p}*γ*= 2.18.10

_{ps}^{4}W

^{-1}m

^{-1}, is more than six orders of magnitude larger than the XPM coefficient of optical fibers,

*γ*=

_{fiber}*n*

_{2}ω/

*cA*∼3.10

_{eff}^{- 3}W

^{-1}m

^{-1}, so that a much stronger XPM interaction is expected in Si-WWGs. Note also that the nonlinear coefficients

*γ*have different expressions for fibers and Si-WWG, a fact that must be taken into account when the nonlinear properties of optical fibers and Si-WWG are compared; this difference can ultimately be traced to the large waveguide dispersion in Si wires and to their tensor nonlinear-optical properties [2]. Furthermore, this orders-of-magnitude difference in the effective nonlinearity is due both to the fact that the parameter

*γ*is inversely proportional to the modal area, which in Si wires is ∼ 10

^{3}× smaller than the modal area of optical fibers (∼ 0.1 μm

^{2}as compared to ∼ 100 μm

^{2}), as well as due to the much stronger third-order susceptibility of Si as compared to that of silica. In our experiment,

*v*= 6.63×10

_{g,p}^{7}m/s and

*v*= 6.54×10

_{g,s}^{7}m/s, which leads to a temporal walk-off of ∆ = 207.6 fs/mm. More importantly, the corresponding walk-off length is

*L*=

_{w}*T*/|∆| ∼ 1 mm, which is several times smaller than the waveguide length. As a result, the propagation length is large enough for the two pulses to pass through each other and thus experience a strong mutual interaction. Furthermore, in our experiment, the probe peak power is < 1 mW, and as a result, the SPM-induced phase shift of the probe can be neglected. Under these circumstances, the nonlinear phase shift of the probe is chiefly determined by XPM, i.e., by the second term in Eq. (2). This equation shows that XPM-induced phase shift is proportional to the peak power of the pump

_{p}*P*and the effective nonlinear coefficient of the waveguide, Γ

_{p}_{ps}. Straightforward calculations show that, if we assume that both pump and probe are Gaussian pulses, namely

*u*(0,

_{p}*T*) = exp[-(

*T*-

*T*)

_{d}^{2}/2

*T*

_{p}^{2}] and

*u*(0,

_{s}*T*)=exp(-

*T*

^{2}/2

*T*

_{s}

^{2}), then the XPM-induced phase shift in Eq. (2) can be written as

where τ= *T*/*T _{p}*, τ

_{d}=

*T*/

_{d}*T*, and δ=

_{p}*z*∆/

*T*.

_{p}*T*is defined as the temporal separation between the maximum intensity points of these two pulses prior to their entry into the waveguide. In the convention we use here, this time delay is positive (negative) when the probe leads (trails) the pump. For our experiment, we are interested in extracting the phase information at the end of the waveguide, i.e., ϕ

_{d}_{s}(

*L*,τ). Henceforth, δ is evaluated at

*z*=

*L*, i.e., ϕ=

*L∆*/

*T*=

_{P}*L*/

*L*= 4.78. Finally, from Eq. (3) we can derive the frequency shift of the probe due to the XPM induced by the pump:

_{w}## 4. Experimental results and analysis

#### 4.1 SPM of pump

Before presenting the experimental results regarding the XPM in Si-WWGs, we first show the SPM-induced effect on the pump. This step has its own importance, since it allows us to determine physical parameters of the Si-WWG, namely its TOD coefficient, β_{3}, which then will be used in numerical simulations. In addition, when the pump is strong enough to induce SPM on itself, it also has enough optical power to induce a nonlinear phase shift onto the probe. Figure 2 shows the pump spectra normalized to the peak spectral power, measured for several pump powers. We observe that as we increase the pump power, new spectral components are generated, which represents a signature of SPM. For our XPM studies, we adjusted the coupled peak pump power inside the waveguide to be ∼ 20 mW so as to avoid spectral overlap between the pump and the probe due to this spectral broadening of the pump. This choice also ensures that the coupled-mode equations (1.a–c) remain valid. For example, if the spectra of the pump and probe pulses begin to overlap during the propagation, then the assumption that the optical field can be separated in two distinct pulses breaks down; in this scenario, the dynamics of the optical field is described instead by a single equation similar to Eq. (1.a).

At 20 mW coupled peak pump power, the SPM-induced phase shift of the pump is estimated to be 1.5π, a result obtained by counting the number of peaks in the output spectrum using the relation ϕ_{max} ≈ (*M*-1/2)π, where *M* is the number of peaks in the spectrum note that the second peak starts to form at *P _{p}*=20 mW). To simulate SPM of the pump using our model, we first calculated numerically the coefficients β

_{1}, β

_{2}, and Γ and then determined the value of the TOD coefficient β

_{3}that led to the best fit between the experimental and numerical results that correspond to the propagation of a Gaussian input pulse with

*T*= 204 fs (for details, see [27]). The best agreement between the simulation and experiment was obtained for β

_{p}_{3,p}= -0.43 ± 0.05 ps

^{3}/m, as shown in Fig. 2(b). This result is comparable to the value of β

_{3}= -0.73 ± 0.05 ps

^{3}/m at 1537 nm [27].

#### 4.2 XPM pump/probe power dependence

One of the most distinctive aspects of XPM is that the phase across the probe pulse varies with the pump power. As described by Eq. (3), the induced phase shift is proportional to the peak pump power. Figure 3 shows experimental results for the effect of pump power on XPM by varying the pump power for a delay *T _{d}* ≈ 200 fs. As shown in Fig. 3, as we increase the peak pump power, the newly generated spectral components become stronger, which indicates that XPM increases with higher pump power.

Another key signature of XPM is the lack of dependence of the normalized probe spectrum on its peak power. To verify that the observed spectral reshaping of the output probe pulse is due to XPM, we investigate whether the normalized probe output is independent of peak probe power. Thus, as shown in Fig. 4, as we vary the probe power from 100% to 30% of 10 μW, the normalized probe output spectrum remained relatively unaltered. In fact, as long as the probe power was much smaller than the pump power, and thus insufficient to cause SPM, the effect of XPM was independent on the peak power of the probe.

In order to compare our experimental results with simulations, the simulated output spectra of the probe are shown in Fig. 5(a) for a set of peak-pump power values, all corresponding to an initial time delay *T _{d}* = 0. The spectra of the output pulses clearly show significant spectral broadening as the pump power is increased. Using Eq. (3), the maximum XPM-induced phase shift for peak pump power of 10 mW, 50 mW, and 100 mW are 0.76π, 3.8π, and 7.6π, or approximately 15 mW for a π phase shift. Note that this exact calculation agrees relatively well with the less rigorous estimate based on counting the number of peak splits in the output spectrum of the probe using the relation

*ϕ*≈ (

_{max}*M*–1/2)π. Moreover, as discussed earlier, the normalized output probe spectrum did not change with the probe power. As shown in Fig. 5(b), for a pump peak power of 100 mW, the probe spectrum is unchanged when the power of the probe is increased from 1 μW to 10 μW.

#### 4.3 Dependence of the XPM on the pump-probe time-delay

Our results above examine the spectral variations in XPM, but even more insight can be obtained by investigating the temporal evolution the XPM spectra as the delay between the pump and probe pulses prior to injection into the waveguide, is varied. In particular, XPM occurs because the probe pulse experiences a change in refractive index induced by the co-propagating pump pulse. This effect is evident when the probe and the pump present a temporal overlap during the propagation inside the Si-WWG. As mentioned, because of the difference in their wavelength, the pump and the probe have different group velocities inside the waveguide, which leads to a temporal walk-off. The total time delay corresponding to the propagation in the waveguide is *T _{L}* =

*L*∆ = 975.7 fs.

Figure 6(a) shows the experimental result of the probe spectrum for various time delays between the pump and probe. To demonstrate XPM experimentally, both pump and probe pulses are injected into the waveguide and a delay line is used to control the time delay between them. The peak power of the probe is 10 μW, a value that is clearly insufficient to cause SPM. On the other hand, the peak pump power is 20 mW, which is strong enough to generate SPM on itself and therefore an even stronger XPM on the probe.

At the extreme values of the temporal delay, e.g., at *T _{d}* = −1300 fs or

*T*= 2000 fs, the probe spectrum is unaffected by the pump pulse due to the lack of overlap between the two pulses. Indeed, in both cases |

_{d}*T*| >

_{d}*T*and therefore the pulses do not overlap during the propagation in the Si-WWG. As the time delay decreases from its maximum value of

_{L}*T*= 2000 fs, the probe develops a series of spectral modulations and shows an increased spectral asymmetry, both effects representing a clear signature of the XPM. This spectral distortions of the probe reach a maximum at

_{d}*T*∼ 1000 fs (in normalized units this corresponds to τ

_{d}_{d}∼ δ, where δ is calculated for

*z*=

*L*). Moreover, as the time delay further decreases these spectral modulations also decrease and almost vanish for

*T*∼ 500 fs, i.e., for τ

_{d}_{d}∼ δ/2. As the time delay further decreases we observe a reverse scenario, namely the spectral modulations again increases, reaching a maximum at

*T*∼ 0 and then decrease to zero as the time delay reaches large negative values. These results can be easily understood if we consider the frequency shift of the probe described by the Eq. (4). Thus, if τ

_{d}_{d}= δ most of the probe will experience a negative (positive) frequency (wavelength) shift whereas if τ

*=0 the probe experiences a positive (negative) frequency (wavelength) shift, with both these cases corresponding to a maximum absolute value of the frequency shift. In the intermediate case, i.e., when τ*

_{d}*= δ/2, the probe experiences a minimum frequency shift.*

_{d}As we have mention before, in order to derive the Eq. (4) we have neglected the TPA and dispersion effects. Therefore, to obtain a more complete understanding of the observed effects we have employed a rigorous numerical analysis of the dynamics of the co-propagating pulses, based on the numerical integration of the system (1.a-c). Among other parameters, such a rigorous study requires the knowledge of the exact temporal profile of both the pump and the probe pulses. In the case of the pump, our measurements show that it has a Gaussian profile (see Fig. 2) with a measured pulse width *T _{p}* = 204 fs. However, as our measurement for the probe spectrum suggests, the probe is not really a simple Gaussian. More exactly, Fig. 7 shows that the spectrum of the probe is somewhat asymmetric and its temporal profile deviates from that of a Gaussian pulse. In order to determine the exact characteristics of the probe we model our temporal profile of the pulse as a sum of two Gaussians, with the dominant one possessing a small linear chirp

and fit the parameters of the input pulse so as to obtain a good agreement between the measurements and the results of our numerical simulations. Here, *T _{s}* is the pulse width of the probe,

*A*is the amplitude of the perturbation of the Gaussian, and α is related to a frequency shift from the center frequency. The values of the fitted parameters are

*T*=170 fs,

_{s}*A*= 0.099, and α = 6.

The results of these numerical simulations are summarized in Fig. 6(b). A comparison between Fig. 6(a) and (b) illustrates that our numerical simulations agree very well with the general trends observed in experiments (only some details of the spectral profiles of the probe pulse are somewhat different). Note that as in the experiments, the maximum pulse-probe interaction is observed for τ_{d} = δ and τ_{d} = 0, whereas the minimum XPM-induced modulations of the probe is observed for large absolute values of the time delay, |τ_{d}| ≫ 1, and for τ_{d} = δ/2.

In addition to the agreement between experiments and simulation as shown in Fig. 6(a) and (b), respectively, both results show that the co-propagation of the strong pump and the much weaker probe pulse does not lead to the depletion of the probe pulse energy. This important result suggests that absorption due to the free carriers generated by the pump pulse does not play a significant role. This finding can be explained by the facts that (1) the small cross section of the waveguide quenches the amount of carriers faster due to surface recombination, yielding an estimated free-carrier lifetime of ∼ 0.5 ns, and (2) the pulse width of the pump is very small and therefore the pump pulse energy (∼ 15 fJ) can generate only a low concentration of free carriers. In addition, since the time interval between the pump pulses (27 ns) is much larger than the free carrier lifetime (∼ 0.5 ns) the FC accumulation is negligible so as it does not influence the pulse dynamics.

Finally, we have investigated both experimentally and theoretically the dependence of the nonlinear frequency shift on the physical properties of the interacting pulses. The central results are presented in Fig. 8. There are several important conclusions illustrated by this figure. First, there is a good agreement between theoretical analysis and experimental results, as well as a confirmation of the results presented in Fig. 6 (a, b). Thus, for large absolute values of the time delay or near *T _{d}*=

*T*/2 no wavelength shift is observed, whereas at

_{L}*T*= 0 and

_{d}*T*= 850 fs (close to

_{d}*T*=

_{d}*T*) the XPM interaction induces a large nonlinear wavelength shift (more than 1 nm). In addition, the sign of this nonlinear wavelength shift (∆λ < 0 at τ

_{L}_{d}= 0 and (∆λ > 0 at τ

_{d}= δ) agrees with our analysis based on Eq. (4). Second, Fig. 8 shows that, unlike in optical-fiber experiments, the maximum value of the wavelength shift at τ

_{d}= 0 is larger than the maximum wavelength shift at τ

_{d}=δ. This result is explained by the fact that in the case in which τ

_{d}= 0 the pump-probe interaction takes place mostly near the input facet of the waveguide whereas when τ

_{d}= δ the pump interacts with the probe mostly near the output of the Si-WWG, i.e., after the pump has lost part of its optical power due to intrinsic and TPA losses. Another important observation is that the center wavelength is almost the same for

*T*= ±2000 fs, despite the fact that

_{d}*T*= 2000 fs is much less than the FC lifetime of 0.5 ns [27].

_{d}This result again shows that FC generation from the pump is negligible although TPA is still significant. Finally, note that the peak near τ_{d} = 0 is narrower than the one near τ_{d} = δ, an effect that is attributable to the temporal broadening of both the pump and probe pulses due to dispersion and pump SPM.

As demonstrated above, a potential application of XPM is nonlinear frequency (wavelength) shifting, which has important use in wavelength-channel dropping functionality. As we have mentioned, the amount of wavelength shift scales with the pump power. Indeed, the work of Dekker *et al*. clearly demonstrates that Kerr-induced wavelength shifts of as much as >10 nm can be obtained, which are comparable to the spectral width of the input probe pulses [30]. Therefore, an XPM-induced frequency shift can be employed in designing ultra-fast all-optical switches, which can be used to switch-off pulses as short as a few hundred femtoseconds.

## 5. Conclusion

We have presented a careful comparative experimental and theoretical study of XPM for ultrafast pulses in silicon photonic-wire waveguides. Our complete simulation model allows us to show clearly the importance of waveguide dispersion in controlling the shape and interaction of pump and probe pulses; inclusion of accurate values of dispersion allows us to match closely the experimental results with simulation. Both the wavelength shift and pump/probe delay behavior follow the calculated behavior for our waveguide system. Our experiments show that XPM in Si wires can be significant even for low peak pump power, i.e., ∼ 15 mW for π phase shift, suggesting that XPM is a potentially useful approach for all-optical control of photonic devices in Si wires. This operating power level can be further reduced if, as illustrated by our theoretical model, one uses photonic structures employing slow-light modes, such as photonic crystal-based waveguides [36] or coupled resonant structures [37,38]. Indeed, Eqs. (1.a,b) show that the effective waveguide nonlinearity is inversely proportional to the mode group velocity, which suggests that, similar to the Raman amplification [39], by employing slow-light modes whose group velocity is much smaller than the speed of light, the effects reported here can potentially be observed even in sub-millimeter-long photonic structures.

## Acknowledgments

This research was supported by the DoD STTR, Contract No. FA9550-05-C-1954, and by the AFOSR Grant FA9550-05-1-0428. The IBM part of this work as supported by Grant No. N00014-07-C-0105 ONR/DARPA. We thank Cheng-Yun Chou for his help with the computational work.

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