## Abstract

The influence of three-photon absorption (3PA) on cross-phase modulation (XPM) effect in the mid-infrared (IR) region is theoretically investigated in silicon-on-sapphire (SOS) waveguides. It is found that the 3PA-induced nonlinear losses in the SOS waveguide will be considerable for the pulse propagation in the wavelength region of 2300 nm-3300 nm when the pump peak intensity is high enough. For the XPM process, the 3PA and 3PA-induced free-carrier effects can affect the spectrum and temporal profiles of the pump and signal pulses for sufficiently high pump peak intensities. Moreover, the XPM-induced frequency shift of signal spectrum is also discussed with different pump peak intensities, and the XPM-induced blue and red shifts are reduced due to 3PA.

© 2013 OSA

## 1. Introduction

In recent years, the operating wavelength range has been extended to the mid- and long-wave-IR regions for silicon photonics [1–4]. There are many potential mid-IR applications for silicon photonics including biochemical detection, environment monitoring, and free-space communications [1]. Although the nonlinear silicon photonic for near-IR applications are well known and are attracting great interest [5–15], there are few studies investigating nonlinear silicon photonics in the mid-IR range [16–25]. In detail, Liu *et al* have realized mid-IR optical parametric amplification in silicon nanophotonic waveguides [16], and Zlatanovic *et al* have achieved mid-IR wavelength conversion in silicon waveguides [17]. The pumps of the above studies used operated near 2200 nm, corresponding to the wavelength spectrum beyond the onset of two photon absorption (TPA). Despite these progresses, there is still a strong motivation to investigate the influence of 3PA on nonlinear silicon photonics in the mid-IR region ranging from 2300 nm to 3300 nm, where the 3PA is the main source of nonlinear loss [26]. Since some nonlinear processes need high pump peak intensities for mid-IR applications, the investigating of 3PA and 3PA-induced free-carrier effects in silicon waveguides becomes significant and crucial.

The XPM effect in silicon waveguides has many potential applications for making nonlinear optical devices such as pulse compressor [27], optical switch [8, 28, 29], and wavelength converter [30], since it allows to use a pump pulse to control a signal pulse at a different wavelength. Within the past few years, the XPM effect has been explored theoretically and experimentally in silicon waveguides-typically in the near-IR region [28–34]. In detail, Dekker *et al* have reported ultrafast all-optical wavelength conversion based on XPM in silicon waveguides using 1.55 μm femtosecond pulses, and they observed large Kerr-induced red shift of 9 nm and blue shift of 15 nm [30]. Hsieh *et al* have investigated XPM-induced spectral and temporal effects with near-IR pump and signal pulses [31]. Astar *et al* have demonstrated the conversion of 10 Gb/s non-return-to-zero ON-OFF keying (NRZ-OOK) to RZ-OOK in the C-band using XPM effect in a silicon nanowire [32]. Until now, there are few reports to investigate the XPM effect in the mid-IR for silicon waveguides, thus it will be very interesting to investigate the spectral and temporal effects induced by the XPM effect in silicon waveguides by taking into account 3PA and 3PA-induced free-carrier effects in the mid-IR wavelength range of 2300-3300 nm.

In this paper, we investigate the XPM effect in a SOS waveguide with mid-IR pump and signal pulses. This is a detailed report of 3PA and 3PA-induced free-carrier effects for mid-IR nonlinear effects in silicon waveguides. The influence of 3PA and 3PA-induced free-carrier effects on the spectral and temporal effects for the co-propagating mid-IR pump and signal pulses are investigated with different pump peak intensities. Furthermore, the XPM-induced frequency shift is also discussed for different signal center wavelengths.

## 2. Simulation model and XPM theory for 2300 nm-3300 nm

The silicon waveguide for the mid-IR applications is designed as a 1-cm-long straight SOS rib waveguide, since sapphire (Al_{2}O_{3}) is transparent from visible to 5.5 μm [3]. Figure 1
shows the dimension of the SOS rib waveguide with width *W* = 800 nm, height *H* = 700 nm, and etch depth *h* = 400 nm. The SOS waveguide is a multimode waveguide, and mode bifurcation may occur [35]. In the following analysis, we only consider the fundamental TM mode in the SOS waveguide, because the high-order modes do not play an important role in the parametric process and inverse taper used in each end of the waveguide can ensure only the fundamental mode excited [29, 36]. To determine the performance of the waveguide, we need to simulate the mode profiles at mid-IR wavelengths. For the mid-IR XPM process, the pump pulse is assumed as 2300 nm and the signal pulse is tuned from 2400 nm to 2700 nm. Both of the pump and signal pulses are TM polarization, thus stimulated Raman scattering (SRS) cannot occur [37, 38]. The fundamental TM mode profile of the SOS waveguide at the wavelength of 2300 nm is simulated using a finite-difference mode solver [39], and the E_{y} and E_{x} electric field components are shown in Fig. 1. It is clear that the SOS waveguide can support 90% confinement in silicon region at the wavelength of 2300 nm.

To describe the dynamics of the ultrashort pulse propagation, we first determined the waveguide dispersion properties. For the SOS rib waveguide, the TM mode effective indices *n _{eff}* are calculated using the finite-difference mode solver. The dispersion relation is then calculated from

*β*(

*ω*)

*= n*(

_{eff}*ω*)

*ω/c*, where

*ω*is the carrier frequency and

*c*is the speed of light in vacuum. Higher order dispersion is finally calculated via numerical differentiation from

*β*

_{n}

*= d*. The dispersion parameters

^{n}β/dω^{n}*n*, group index

_{eff}*n*

_{g}, group-velocity dispersion (GVD) coefficient

*β*

_{2}, and third-order dispersion (TOD) coefficient

*β*

_{3}of the SOS waveguide are shown in Fig. 2 , where

*n*

_{g}=

*β*

_{1}

*c*. It is shown that a large anomalous dispersion region from 2200 nm to 2700 nm is obtained, and this dispersion profile is not suitable for phase matching because the pump wavelength is far from the zero-dispersion wavelength [40]. Therefore, the in-band four-wave mixing effect can be neglected in the pulses propagation process in our analysis. In addition, if the wavelength spacing of the pump and signal is larger, which is not considered in our analysis, the phase matching of out-of-band discrete FWM should be investigated according to [3].

To describe the nonlinear optical interaction of the pump and signal in the SOS waveguide, we use the formulism described in [20, 27] and take into account the effects of 3PA, free-carrier absorption (FCA), and free-carrier dispersion (FCD). The XPM process can be described by the following coupling equations:

*A*is the slowly varying amplitude (

_{j}*j = p, s*), and

*z*is the propagation distance.

*T = t-z/v*is the time in the reference frame of the pump pulse traveling at speed

_{gp}*v*,

_{gp}*d = β*

_{1}

_{s}-β_{1}

*is the temporal walk-off parameter, and*

_{p}*β*

_{3PA}is the coefficient of 3PA. The nonlinear coefficient

*γ*

_{j}= ω_{j}n_{2}

*/cA*is the effective nonlinearity of the waveguide, where

_{eff}*A*is the effective area of the propagating mode and

_{eff}*n*

_{2}is the nonlinear index coefficient. Here, we assume

*n*

_{2}= 5 × 10

^{−18}m

^{2}W

^{−1}[3, 41],

*β*

_{3PA}= 2.5 × 10

^{−26}m

^{3}W

^{−2}[26], and the

*A*is calculated as 0.45 μm

_{eff}^{2}. The parameter α

*represents the linear propagation losses of the waveguide, which is assumed as 1 dB/cm for 2300 nm-2700 nm [42]. The α*

_{l}*and Δn*

_{fcj}*represent the absorption and index change induced by free carriers, respectively. Based on a Drude model, the α*

_{fcj}*and Δn*

_{fc}*are given by*

_{fc}*α*

_{fc}= q^{3}

*N*(

_{c}*1/μ*

_{e}m*^{2}

_{e}+ 1/μ_{h}m*^{2}

*)*

_{h}*/ε*and

_{0}cnω^{2}*Δn*

_{fc}= -q^{2}

*N*(

_{c}*1/m**)

_{e}+ 1/m*_{h}*/*2

*ε*

_{0}

*nω*

^{2}, where

*m**= 0.26 m

_{e}_{0}(

*m**= 0.39 m

_{h}_{0}) is the effective mass of the electrons (holes), and

*μ*(

_{e}*μ*) is the electron (hole) mobility [38]. The free-carrier density

_{h}*N*caused by 3PA can be obtained by the following equation [43]:

_{c}*h*is Planck’s constant,

*υ*is the pump frequency, and the carrier lifetime is

_{p}*τ*5 ns [30]. Here, free carriers induced by the signal are negligible compared with that induced by the pump as the pump intensity is assumed much larger than that of the signal in this paper.

_{c}≈The XPM process is numerically studied by injecting pump pulses centered at 2300 nm and tunable signal pulses with same pulse width *T _{FWHM}* = 2 ps (

*T*=

_{0}*T*/1.665) and same repetition rate. Here, we assume that the repetition rate is very low (<100 MHz), which means that single pump and signal pulses can be focused on, because free carriers generated during a pulse have sufficient time to recombine before the next pulse arrives. Both of the pulses are Gaussian pulses, which are shown as [27]:

_{FWHM}*P*and

_{p}*P*are the peak power of the input pump and signal pulses, and

_{s}*T*is the initial time delay of the pump and signal. Then the XPM-induced phase shift of the signal pulse can be written as the following equation [27, 31]:

_{d}*τ = T/T*, and

_{0}, τ_{d}= T_{d}/T_{0}, δ = dL/T_{0}*L*is the waveguide length. And then, we can get the pump-induced frequency chirp imposed on the signal pulse:

_{d}= 0 and τ

_{d}= δ are shown in Fig. 3 , where the value of the chirp has been normalized. The input signal center wavelength is 2400 nm, and the walk-off parameter

*d = β*

_{1}

_{s}-β_{1}

*= 140 fs/mm, where*

_{p}*β*

_{1}

*= 1.355 × 10*

_{s}^{4}ps/m, and

*β*

_{1}

*= 1.341 × 10*

_{p}^{4}ps/m calculated from Fig. 2. The walk-off length

*L*= 8.58 mm, and

_{w}= T_{0}/d*δ*= 1.1655.

## 3. Results and discussion

In simulations, the signal peak intensity coupled inside the 1-cm-long SOS waveguide is kept constant at 2.22 MW/cm^{2}, while the input pump peak intensity is much larger than the signal peak intensity. Thus the pump pulse is mainly affected by the self-phase modulation (SPM) effect, and the XPM effect induced by the signal pulse imposed on the pump can be neglected. Figure 4
shows the output pump peak intensity versus input pump peak intensity. In order to guarantee the veracity for researching the 3PA effect with input and output pump peak intensity, pulse distortion (compression or broadening), which can be induced by SPM and dispersion effects, should be avoided. Thus, the GVD and TOD are not considered in the simulation. From Fig. 4, it is clear that the output pump peak intensity scales linearly with the input pump peak intensity when the 3PA effect is neglected. However, when the 3PA effect is included in the simulation, the output pump peak intensity appears saturation effect for higher input pump peak intensity (> 2 GW/cm^{2}). After considering FCA, the output pump peak intensity is only reduced a little, which indicates that the optical-limiting is mainly attributed to 3PA not FCA. This validates that the 3PA effect is the leading-order nonlinear loss for mid-IR applications when the input pump peak intensity is high enough [18]. In the following part, we will investigate the impact of 3PA and 3PA-induced free-carrier effects on the spectral and temporal effects in this mid-IR XPM process by taking into account GVD and TOD.

Figure 5
depicts the output spectra of the pump pulse with corresponding temporal profiles for different pump peak intensities, and the spectra are largely broadened due to SPM when the 3PA is ignored. When the pump peak intensity is 2.22 GW/cm^{2}, the 3PA only induces a little change of the pump spectrum, and the FCA and FCD effects can be ignored as illustrated in Fig. 5. This indicates that the 3PA does not play an important role and the nonlinear losses induced by 3PA and FCA can be neglected when the pump peak intensity is less than 2.22 GW/cm^{2}. With the increase of the pump peak intensity, the 3PA will gradually become the leading-order source of the nonlinear loss according to Fig. 4. It is clear that the spectral width can be reduced due to 3PA effect, while the free-carrier effects (including FCA and FCD) induce spectrum asymmetric for *I*_{0} = 6.67 GW/cm^{2}. It is also found that a little blue shift is generated due to FCD [44]. Corresponding to the temporal profile, the loss of the pulse in the trailing edge is larger than the leading edge due to the accumulation of the free carriers along the time. When the pump peak intensity is increased to 8.89 GW/cm^{2}, the spectrum asymmetric induced by free-carrier effects become larger, and the FCD-induced blue shift is distinct. Therefore, when the pump peak intensity is high enough, the 3PA and 3PA-induced free-carrier effects in the Mid-IR region have the same impact on the spectrum and temporal profiles of the pump as the TPA and TPA induced free-carrier effects in the near-IR region with a relatively low intensity [44].

The center wavelength of the signal pulse is assumed longer than 2300 nm, hence the pump pulse travels faster than the signal pulse due to smaller n_{g} for pump from Fig. 2. According to Eq. (6) and Fig. 3, the XPM induces a blue shift of the center wavelength on the signal spectrum for *τ _{d}* = 0, and induces a red shift of the center wavelength on the signal spectrum for

*τ*= δ. Figure 6 shows the output spectra and temporal profiles of the signal pulse for

_{d}*I*

_{0}= 6.67 GW/cm

^{2}, when the input signal center wavelength is 2400 nm. From Fig. 6, it is clear that the 3PA-induced nonlinear loss can be ignored for the signal pulses due to the low intensity of the signal pulse. When

*τ*= 0, the free-carrier effects are relatively large for the signal pulse, because the slow-moving signal pulse interacts mainly with the trailing edge of the pump pulse that means the signal pulse affected by a relatively high carrier density. However, when

_{d}*τ*= δ, the signal pulse interacts mainly with the leading edge of the pump pulse, which means low free-carrier effects and the FCA and FCD are negligible. After considering 3PA, the red and blue shifts are all reduced for the two case of

_{d}*τ*= 0 and

_{d}*τ*= δ as shown in Fig. 6, because the 3PA-induced nonlinear losses reduce the intensity of the pump, thus lower the XPM effect. When all the effects are included, the blue shift is reduced to 14 nm for

_{d}*τ*= 0 and the red shift is reduced to 12 nm for

_{d}*τ*= δ, that is because the FCD-induced blue shift also supply a contribution to the frequency shift for

_{d}*τ*= 0 as shown in Fig. 6. In conclusion, the XPM-induced frequency shift is reduced due to 3PA for a relatively high pump intensity.

_{d}After taking into account the 3PA, FCA and FCD effects in the XPM process, the output spectra of the signal pulse with different pump peak intensities are shown in Fig. 7
for τ_{d} = 0 and τ_{d} = δ, respectively. It is found that the frequency shift of the center wavelength (2400 nm) become larger as the pump intensity increases. However, the frequency shift appears saturation when the pump peak intensity exceeds 4.44 GW/cm^{2}. Hence the increasing nonlinear losses induced by 3PA imposed on pump pulse suppress the XPM-induced frequency shift as the pump peak intensity increases, because larger nonlinear losses induced by the 3PA reduce the pump intensity, and then lower the pump-induced XPM effect. Furthermore, the FCD-induced blue shift enhance the XPM-induced blue shift for τ_{d} = 0 as illustrated in Fig. 7, where the maximal blue shift is 18 nm for τ_{d} = 0, while the maximal red shift is only 16 nm for τ_{d} = δ for *I*_{0} = 11.11 GW/cm^{2}.

When the input signal center wavelengths are 2400 nm, 2500 nm, 2600 nm and 2700 nm, the output spectra are shown in Fig. 8
with the input pump peak intensity of 6.67 GW/cm^{2}, respectively. The spectra are shifted to the blue part due to τ_{d} = 0, and the blue shifts are 14 nm, 10 nm, 7 nm and 5 nm, respectively. It is clear that the blue shift and the number of peaks of the output signal spectrum are reduced as the input signal center wavelength increases. According to Fig. 2, n_{g} is increased with the increase of the signal wavelength, which means a large difference of group velocity between the pump and signal pulses. Thus, the walk-off length become shorter as the wavelength difference between the pump and signal increases, which means that the interaction time between the pump and signal decreases and the XPM effect becomes weaker. Therefore, the frequency shift is reduced with the increase of signal center wavelength due to pulse walk-off.

## 4. Conclusion

The complete simulation model in the mid-IR region by taking into account 3PA, FCA and FCD allows us to show clearly the importance of 3PA effect for the XPM in a SOS waveguide. For a sufficiently high pump intensity, the 3PA and 3PA-induced free-carrier effects in the Mid-IR region have the same impact on the spectrum and temporal profiles of the pump pulse as the TPA and TPA-induced free-carrier effects with a relatively low pump peak intensity in the near-IR. Moreover, the XPM-induced frequency shift of the signal spectrum is reduced due to 3PA.

## Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grant 61078029, 61178023 and 61275134.

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