1. Introduction

Cross-phase modulation (XPM) is a nonlinear process in which the phase of one wave is modulated by the intensity of another wave through the optical Kerr effect. The XPM manifests itself as spectral broadening similar to that occurring for self-phase modulation (SPM) on pulsed waves. In pure Kerr media, the ratio of broadening of XPM to SPM is well known to be a factor of two [1]. One area in which control of the XPM process is critical is all-optical signal processing. In this field, XPM leads to channel crosstalk through pulse distortion and spectral broadening. While different methods have been proposed to compensate the XPM-induced nonlinear distortions, XPM remains a key contributor to the nonlinear Shannon limit of optical fiber communication systems [2–5]. Separately, engineered utilization of XPM provides various optical functions including: wavelength conversion [6, 7], Mamyshev 2R regeneration [8], mode-locked lasing [9], radio frequency analyzers [10], squeezed light [11], and photon number measurement [12].

Simultaneous to these developments in large scale all-optical signal processing in glass fibers, the miniaturization and integration of optical devices onto silicon chips is progressing rapidly [13]. A key property of silicon in the telecommunication band of 1.5 μm is the presence of two-photon absorption (TPA) which restricts the Kerr effect through nonlinear attenuation of optical power. The TPA-induced free carriers cause free-carrier absorption (FCA) and free-carrier dispersion (FCD), further reducing the desired Kerr nonlinearity and introducing nonlinear distortions. Both TPA and free carriers complicate the analysis of phase modulation in silicon compared to pure Kerr media.

We briefly summarize progress in understanding non-parametric nonlinear effects on single and multiple pulses in silicon. SPM in silicon has been investigated numerically and experimentally with specific examination of the roles of TPA and free carriers on the SPM-induced spectral broadening [14–19]. In the multiple pulse regime, outside of an initial XPM demonstration [17] and recent work on cross two-photon absorption [20], most work involving silicon has focused on utilizing this effect for applications [6,7,10]. In the context of all-optical signal processing it is clear that understanding multiple wave interactions such as XPM is a crucial step to developing multiple-channel systems and mitigating detrimental crosstalk in silicon photonic chips [1,3,14]. At present a detailed quantitative analysis of TPA and free carriers on XPM-induced spectral broadening is still missing.

In this work, we theoretically and experimentally investigate XPM-induced spectral broadening in silicon waveguides. Here we define the broadening ratio (ρ_υ) of XPM to SPM as ${\rho }_{\upsilon }=\frac{\mathrm{\Delta }{\upsilon }_{\text{XPM}}}{\mathrm{\Delta }{\upsilon }_{\text{SPM}}}$, with the spectral broadening factors for SPM (Δυ_SPM) and XPM (Δυ_XPM). In contrast to the canonical case, we surprisingly find ρ_υ ≠ 2 in silicon waveguides in theory, since TPA and free carriers modify the pulse shape and phase. We support the theoretical analysis with experimental observations in silicon photonic crystal waveguides.

2. Theory

The power and phase evolutions of a strong pump and a weak probe in silicon waveguides have been presented in our recent work [20], while the single beam SPM case is highlighted in Ref. [14]. Here we quote the existing results for detailed discussion. We first discuss the case with TPA only. If there is negligible dispersion, the output power of the pump (P₁) and probe (P₂) can be solved analytically:

Figure 1(a) shows the output pump and probe temporal pulse shapes in a 200 μm-long photonic crystal waveguide with parameters given below in the experiment. The pulse shapes are plotted using the analytic solution Eq. (1). The input pump and probe are Gaussian pulses with a full-width half maximum (FWHM) of t_FWHM=7 ps. At a low input power of 2 W (left), the pump and probe maintain the input shapes. At a high input power of 10 W (right), both the probe and pump distort from the Gaussian shape. The most noticeable feature is the big dip in the center of the probe pulse due to the strong cross-two-photon absorption (XTPA) when it overlaps with the peak of the pump. This differential absorption in the pulse peak and wings will play an important role in the spectral broadening analysis that follows.

 

Fig. 1 (a) Output pulse shapes of the pump and probe with TPA-only calculated from analytic solution Eq. (1) at the input power of 2 W (left) and 10 W (right). (b) Spectral broadening factor obtained from ${\left(1+0.77{\varphi }_{\text{max}}^2\right)}^{1/2}$ and numerical calculation (NLSE) with and without free carriers (FC).

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We now compare the effects of TPA/XTPA on the spectral broadening due to XPM and SPM. The root mean square of the spectral width is numerically calculated using a second-order moment Δυ² = 〈(υ − υ_avg)²〉 − 〈(υ − υ_avg)〉², where υ_avg is the first-order average frequency and the brackets denote the average over the spectrum [1, 21]. This spectral width can be numerically calculated by solving the Nonlinear Schrödinger Equation (NLSE) with a split-step Fourier method [1]. In addition, for an unchirped input Gaussian pulse, the spectral broadening factor is estimated by $\frac{\mathrm{\Delta }{\upsilon }_{\text{out}}}{\mathrm{\Delta }{\upsilon }_{\text{in}}}={\left(1+0.77{\varphi }_{\text{max}}^2\right)}^{1/2}$ [21], where ϕ_max is the maxima of phases (max[ϕ_1,2]) given in Eq. (2) and Δυ_in, Δυ_out are the input and output spectral widths, respectively. We note this is only for TPA and we apply the classic formulation for Kerr here since it gives a clear and simple estimate of the spectral broadening. For arbitrary pulse shapes, the spectral broadening can be derived as in Ref. [21]. Figure 1(b) compares the broadening factors obtained from the approximate equation ( ${\left(1+0.77{\varphi }_{\text{max}}^2\right)}^{1/2}$) and numerical calculations using the NLSE. In general, the results of the approximate equation and numerical calculation (NLSE (TPA)) agree very well. The difference between the two approaches is due to the assumption of the broadening equation that the input pulses maintain a Gaussian shape along propagation. When there is TPA, the pulse shape is slightly modified from the Gaussian shape (see Fig. 1(a) and Eq. (1a)). However, the broadening equation still gives a reasonable estimate. For instance, the error is less than 10% and 8% for the probe and pump compared to the numerical calculation at an input power of 10 W.

When free carriers are included, there is no simple analytic solution, and we use numerical solutions for the output powers [20],

We now discuss the role of free carriers in the spectral broadening. The thick lines in Fig. 1(b) are the simulated broadening factors ( $\frac{\mathrm{\Delta }{\upsilon }_{\text{out}}}{\mathrm{\Delta }{\upsilon }_{\text{in}}}$) with free carriers using Eq. (4). The broadening factors in both the pump and probe increase in the presence of free carrier accumulation that is asymmetric in both pulse shape and phase. This asymmetry causes the blue shift of frequencies [14, 18]. Since the free carriers are mainly generated by the pump, they impact the pump and probe equally. Importantly, the relative impact of free carriers on the pump versus the probe is very different with important implications on the pulse dynamics and broadening ratio. This is illustrated in the present figure by the larger net change of the broadening factor of the pump compared to the the probe. We describe the physical underpinning of this effect below. With further simulations (not shown) we found that the exact broadening factor is dependent on the pulse widths and FCA and FCD coefficients.

Now that we have established the role of TPA and free carriers in the SPM and XPM pulse broadening, we analyze the evolution of the broadening ratio ${\rho }_{\upsilon }=\frac{\mathrm{\Delta }{\upsilon }_{\text{XPM}}}{\mathrm{\Delta }{\upsilon }_{\text{SPM}}}$ due to each of these effects as a function of power. We show the results of this numerical analysis in Fig. 2(a). For a baseline, we see ρ_υ approaches two in the pure Kerr case (dashed line) as expected. After we introduce TPA, we notice ρ_υ approaches 2.06, slightly larger than two at an input power of 20 W. The reason that ρ_υ is larger than two is because the probe pulse is distorted from a Gaussian shape much more than the pump (see Eq. (1a) and right plot in Fig. 1(a)).

 

Fig. 2 (a) Numerically calculated ${\rho }_{\upsilon }=\frac{\mathrm{\Delta }{\upsilon }_{\text{XPM}}}{\mathrm{\Delta }{\upsilon }_{\text{SPM}}}$ with various nonlinear effects. (b) Chirp rates of pump (solid) and probe (dashed) under different conditions at an input power of 10 W.

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When we include the TPA-generated free carriers, either FCA or FCD reduces ρ_υ < 2. We first consider the case of FCA only. Although in this case both the pump and probe are distorted and attenuated, ρ_υ approaches 1.9 at the pump power of 20 W. The reason that ρ_υ is still close to two is because FCA only slightly modifies the pulse shape [16, 20], while the phase contribution of XPM is still twice that of SPM since we have not yet included FCD. When we now include FCD, the trend of ρ_υ experiences a striking modification. This can be explained by the competition between symmetric Kerr redshift and asymmetric FCD blueshift (see Eq. (4)). The broadening ratio ρ_υ increases with the pump power at low power levels and reaches its maxima around 3 W before the free carriers start to play an important role. Note that the specific power level depends on free-carrier density and thus pulse duration, since longer pulses have more time to accumulate free-carrier effects (N_c ∼ t_FWHM) [14]. After that, ρ_υ reduces gradually to 1.2 because the strong FCD overcomes the Kerr effect. The reason that ρ_υ saturates at 1.2 instead of one is due to competition with TPA. Using the method to distinguish between the loss contributions due to TPA versus FCA proposed in [20], we find TPA absorbs 18% and 29% of the pump and probe powers at 20 W, respectively. Increasing FCD with longer pulses and thus reducing relative TPA can push ρ_υ close to one.

The competition between FCD and Kerr and their impact on spectral broadening is best illustrated by analyzing the chirp rate ( $\frac{1}{2\pi }\frac{d{\varphi }^2}{d{t}^2}$). We use chirp rate instead of instantaneous frequency as the broadening is proportional to the second derivative of the phase (or chirp rate) for a Gaussian pulse [1]. In addition, the chirp rate is a measure of the magnitude of the chirp, and a chirp does not necessarily result in spectral broadening while a non-constant chirp-rate does. Figure 2(b) shows the chirp rate for different conditions at an input pump power of 10 W. Since FCA affects mainly the tail of the pulses, the chirp rates become asymmetric (middle plot) compared to the case with TPA only (left plot). When FCD is considered, FCD significantly increases the chirp rates in both the pump and the probe waves. However, the net change in the pump is larger than the probe, i.e., the maximum pump chirp rate changes from half of the probe chirp rate (middle plot) to be nearly the same as the probe (right plot). Therefore, due to the strong FCD, the broadening ratio tends towards one in Fig. 2(a) but TPA limits how close it gets to. These results explain the different evolution of the pump and probe shown in Fig. 1(b).

3. Experimental results

Now we experimentally investigate the XPM using a pump-probe scheme in silicon waveguides. The experimental setup and silicon photonic crystal waveguide properties were discussed in [20]. We inject pump and probe waves centered at 1555 nm and 1560 nm with a pulse width of t_1,2FWHM = 7 ps. We here use the pulse width of 7 ps to minimize the free carrier effects as explained before. To observe the cross effects, we record the output spectra of the pump and probe as a function of delay. Figure 3(a) shows the probe spectrum at an input peak power of 3.8 W of the pump and 10 mW of the probe. Here the negative delay means the probe is ahead of the pump. The arrows indicate the regions of free carriers and TPA. The variation of the probe intensity is due to XTPA and FCA as explained in [20].

 

Fig. 3 (a) Output probe spectrum as a function of delay at an input peak pump power of 3.8 W. The arrows indicate the regions of free carriers (FC) and TPA. (b) Probe spectral width as a function of time at different pump power levels.

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Figure 3(b) summarizes the evolution of the probe spectral width in Fig. 3(a) as a function of delay at various pump power levels. The probe broadening is determined by XPM and FCD caused by temporal overlap with the strong pump. Therefore, the probe spectral width curve is very similar to the pump temporal shape. The maximum spectral width occurs around 0 ps, which is also the delay at which the maximum XTPA occurs [20]. The spectral broadening disappears at 10 ps, because 10 ps is not long enough for recombination of the free carriers with a lifetime of 1 ns [14] and the probe experiences a constant phase due to FCD. Recall that a dynamic change in phase is required to observe spectral broadening, $\mathrm{\Delta }\omega =-\frac{\partial \varphi }{\partial t}$. In addition, the shape of the spectral width curve distorts from symmetric (zero delay) to asymmetric (−2 ps delay) due to FCA as the pump power increases. The spectrogram generated by this approach could be used to in an XPM frequency-resolved optical gating (FROG) scheme for on-chip pulse characterization [22].

Figure 4(a) shows a comparison of the experimental and simulated spectral broadening ratios. As expected, the experimental broadening ratio ρ_υ (markers) varies with the input power, and the probe experiences more broadening than the pump. In general, the numerical modelling with unchirped Gaussian inputs (NLSE) shows a reasonable trend with experimental results. In the numerical calculation, the parameters are properly scaled to the slow-light factor S = n_g/n_si of the photonic crystal waveguide with group index n_g = 32 and silicon nanowire index n_si = 3.45 [20]. At λ = 1.55 μm, we have α = S × 13 dB/cm, n₂ = S² × 6 × 10⁻¹⁸ m²/W, α_TPA = S² × 10 × 10⁻¹² m/W, A_eff = 0.5 μm², n_FC = −S × 2 × 10⁻²⁷ m³, σ = S × 1.45 × 10⁻²¹ m², t_1,2FWHM = 7 ps, and P_2,in = 10 mW. The dispersion is negligible for picosecond pulses in our waveguide as the dispersion length is 50 times longer than the waveguide length. The temporal information of the input pump pulse was measured with a FROG based on second-harmonic generation. However, due to the initial chirp from the optical amplifier in experiment, the experimental ρ_υ is slightly smaller than the simulation results. When the measured initial chirp is considered, the numerical calculation (NLSE+chirp) shows an excellent agreement with experimental observation. The nice matching in Fig. 4(a) is confirmed by the agreement between the simulated and experimental spectra in Fig. 4(b). Our analysis is consistent with Ref. [15] which shows that the initial chirp can modify the output spectral width.

 

Fig. 4 (a) Comparison of spectral broadening ratio of XPM and SPM (ρ_υ) obtained from experimental results and numerical modelling (NLSE) with and without chirp. (b) Experimental and simulated spectra at three input pump powers.

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4. Conclusion

In summary, we analytically and experimentally characterized cross-phase modulation (XPM) in silicon waveguides. We found the well-known ratio of two between XPM-induced and self-phase modulation-induced spectral broadening in pure Kerr media does not survive with either TPA or free carriers in silicon. In the case with TPA-only the pulse shape is distorted from an ideal Gaussian. Although this ratio is not rigorously two we can nonetheless approximate this ratio as the known value for TPA. Separately, free carriers modify the ratio to be less than two and it varies with power due to the competition between TPA-limited Kerr and free-carrier effects. We emphasize that although it is known that free carriers can be minimized using PIN junctions [23], our results show explicitly that PIN junctions are required in silicon (and all semiconductor) waveguides for large pump powers or long pulses to control the phase and associated spectral broadening. These results further our understanding of XPM-induced crosstalk for all-optical signal processing in silicon photonic chips. The conclusion and analysis can be extended to materials limited by three-photon absorption, such as silicon beyond 2.2 μm [24] and wide-gap semiconductors [25].