## Abstract

We theoretically and experimentally investigate the nonlinear evolution of two optical pulses in a silicon waveguide. We provide an analytic solution for the weak probe wave undergoing non-degenerate two-photon absorption (TPA) from the strong pump. At larger pump intensities, we employ a numerical solution to study the interplay between TPA and photo-generated free carriers. We develop a simple and powerful approach to extract and separate out the distinct loss contributions of TPA and free-carrier absorption from readily available experimental data. Our analysis accounts accurately for experimental results in silicon photonic crystal waveguides.

© 2015 Optical Society of America

## 1. Introduction

Optical nonlinear Kerr effects in silicon waveguides have been exploited for a variety of processes involving multiple wave interactions, including parametric amplification [1], signal regeneration [2], and phase sensitive amplification [3,4]. An important drawback of silicon is that at telecommunication wavelengths, two-photon absorption (TPA) significantly restricts the desirable Kerr effect through nonlinear attenuation of optical power [5–7]. The TPA-induced free carriers cause free-carrier absorption (FCA) and free-carrier dispersion (FCD), further reducing the efficiency of the desired Kerr nonlinearity.

For the TPA process, the two absorbed photons can share the same frequency (degenerate TPA) or have different frequencies (non-degenerated TPA) [8]. For simplicity, to distinguish from degenerate TPA we will denote the non-degenerate TPA as XTPA for the remainder of the article. Although TPA has been thoroughly understood in theory and experiment [1,6,7,9,10], little attention has been paid to the inevitable XTPA in multiple wave interactions. For instance, four-wave mixing (FWM) in silicon has been extensively investigated since the first demonstration in 2005 [1–3,7,11], and it is well known that TPA in the pump limits the FWM conversion efficiency. However, only one theoretical report discusses the effect of XTPA between the pump and signal on FWM [12]. Thus far, neither detailed experimental characterization nor comprehensive analytic solutions of XTPA have been reported.

Although TPA has been considered as a fundamental limitation for nonlinear silicon photonic devices, engineered utilization of cross nonlinear absorption (XTPA and FCA) provides various all-optical functions, including ultrafast optical modulation and switching [13–17], pulse shaping [18], monocycle pulse generation [19], and mid-infrared detection [8]. Characterizing the XTPA and FCA is a critical step to fully understand the nonlinear processes and optimize these optical nonlinear functions. However, it is not easy to analyse the contributions of TPA, XTPA and FCA as they are intrinsically coupled.

In this work, we theoretically and experimentally investigate cross nonlinear absorption (XTPA and FCA) in silicon waveguides. We provide an analytic solution for the pulse evolution in the TPA-only case and a numerical extension to take into account free carriers. These solutions give a clear, concise and general picture of how the XTPA and FCA affect the probe power. In addition, the effects of XTPA, TPA and FCA on power are experimentally extracted using our simple method. We obtain a good agreement between theory and experimental results.

## 2. Theory

The interaction between pump and probe waves can be described by two coupled mode equations [10]. To obtain an approximate analytic solution, we make the following assumptions. First, the pump power is strong compared to the weak probe wave. It is reasonable to ignore the probe contribution to self-phase modulation (SPM), cross phase modulation (XPM) and generation of free carriers. Second, the dispersion is negligible for pulses with durations of tens of picoseconds as typically the dispersion length is much larger than the waveguide length. Third, the frequencies of the pump and probe are so close that the XTPA and TPA coefficients are the same, since the TPA coefficient varies slowly with frequency [20]. Based on the above assumptions, we get the following coupled mode equations,

*A*

_{1,2}are the slowly varying electric field envelopes of the pump (

*A*

_{1}) and probe (

*A*

_{2}) waves, and

*z*is the propagation distance. The SPM coefficient is defined as

*γ*=

*k*

_{0}

*n*

_{2}

*/A*

_{eff}, where the wavenumber

*k*

_{0}= 2

*π/λ*,

*n*

_{2}is the nonlinear Kerr coefficient and

*A*

_{eff}is the effective mode area. The bulk TPA coefficient

*α*

_{TPA}is related to

*γ*

_{TPA}=

*α*

_{TPA}

*/A*

_{eff}. The linear propagation loss is denoted by

*α*. The last term contributes to absorption and dispersion from free carriers with a density of

*N*, where

_{c}*σ*and

*n*

_{FC}are the FCA and FCD coefficients [21], respectively. The factor 2 in Eq. (1b) is from the cross-term contribution of the nonlinear polarization. The free carriers seen by the probe in Eq. (1b) are mostly generated by the pump.

Substituting
${A}_{j}=\sqrt{{P}_{i}}\mathrm{exp}(i{\varphi}_{j})$ into Eq. (1) yields a set of coupled differential equations for the temporal power *P*(*z,t*) and the phase *φ* (*z,t*). When there are no free carriers, e.g. *N _{c}* = 0, the probe output power and phase can be analytically solved from the differential equations,

*P*

_{1,2,in}(

*t*) =

*P*

_{1,2}(0

*,t*) are the temporal power distributions of the input waves,

*z*

_{eff}= (1 − e

*) /*

^{−αz}*α*is the effective length. The analytic solution of probe wave has not been presented − in the literature to our knowledge, while the pump evolution has been derived elsewhere [5,22],

In Eq. (2), we see that the scaling with the input pump power is completely different for the pump and probe waves. Due to XTPA, the probe output power decreases inverse quadratically with the input pump power, while the pump output power increases then saturates with the input pump power due to TPA. Also, the probe phase is twice that of the pump because of XPM. In amorphous silicon with a small TPA coefficient and a short free carrier lifetime, Eq. (2) is capable of predicting and describing the power and phase evolutions of the pump and probe [15].

More generally, there is no exact analytic solution in the presence of free carriers. For low repetition rate pulses, where the pulse separation is longer than the carrier recombination time, an approximate analytic solution of the pump has been given in [12, 22]. Here we integrate Eq. (1) involving the powers and phases with boundary conditions to arrive at,

Equation (3) can be solved numerically. Once *P*_{1}(*z,t*) is obtained in Eq. (3a), the free carrier density is calculated from
${N}_{\mathrm{c}}(z,t)=\frac{{\gamma}_{\mathrm{TPA}}}{2h\nu {A}_{\mathrm{eff}}}{\displaystyle {\int}_{-\infty}^{t}{\mathrm{e}}^{-\frac{t-\tau}{{\tau}_{\mathrm{c}}}}}{P}_{1}^{2}(z,\tau )d\tau $ with the photon energy *hν* and the free carrier lifetime in silicon *τ _{c}* = 1 ns [22]. Equations (3a) and (3c) clearly separate the different loss effects on the power with individual exponential terms. The probe and pump experience the same linear loss and FCA while the XTPA in the probe is twice of TPA in the pump. Later we will use these expressions to extract the amount of loss from TPA, XTPA, and FCA. It is easy to verify that Eq. (3) can be simplified to Eq. (2) in the absence of free carriers.

Now we explore the pulse evolution of the probe and pump by numerically solving Eq. (3). Figs. 1(a) and (b) show the output powers and phases in a 196 *μ*m-long photonic crystal (PhC) waveguide with parameters given below in the analysis of the experiment. The pump and probe input are both Gaussian pulses with a full-width half maximum (FWHM) *t*_{FWHM} = 7 ps. The pump input peak power is 4.5 W and the probe peak power is 10 mW. In Fig. 1, the strong FCA and FCD induce asymmetry to the power and phase profiles, respectively [10, 21, 22]. As expected from Eq. (3), the probe experiences more nonlinear loss while gaining larger positive phase shift than the pump.

Figure 1(c) summarizes the normalized output power of the pump and probe with (Eq. (3)) and without (Eq. (2)) free carriers as a function of input pump power. Although the output powers of the pump and probe reduce in the presence of free carriers, compared to the case without free carriers, the pump experiences more net change (2.5 dB) compared to the probe (1 dB). To see the evolution of absolute powers, Fig. 1(d) shows the output powers in Fig. 1(c) in a linear scale. The pump output increases gradually with the increase of the input pump power, while the probe output decreases due to cross nonlinear absorption. Both the pump and probe outputs saturate at high input powers and FCA shifts the saturation threshold down to lower powers. In addition, the gap between the probe power with and without free carriers is relatively unchanged above 5 W. As we will see later in the experiment part, this pump power is where TPA and XTPA starts to saturate.

## 3. Experimental setup

Figure 2(a) shows the experimental setup. The pump and probe waves are spectrally sliced from a low repetition rate (38 MHz) broadband (40 nm) mode-locked laser (MLL) using a spectral pulse shaper (SPS1, Finisar Waveshaper). The resulting pump and probe are both *t*_{FWHM} = 7 ps Gaussian pulses centred at wavelengths of 1555 nm and 1560 nm. The probe is attenuated by 30 dB compared to pump in SPS1 before passing through the erbium-doped fiber amplifier (EDFA). The second spectral pulse shaper (SPS2) is used to adjust the pulse peak powers and pulse spectral widths. The polarization of the light is aligned to the TE slow-light mode of the waveguide by a polarization controller (PC). The input and output on-chip powers were monitored by a power meter (PM) and the output spectra were recorded with an optical spectrum analyzer (OSA).

Figure 2(b) shows the measured linear transmission loss (including coupling loss) and group index of the TE-mode of the silicon PhC waveguide. The detailed design and measurement method of this device were given in [3, 23]. The dispersion engineered waveguide has a slow-light bandwidth of *~*14 nm and a group index around *n _{g}* ~ 32 at the wavelengths of interest. The slow-light factor is defined as

*S*=

*n*with

_{g}/n_{si}*n*= 3.5 the silicon refractive index. The input and output coupling losses are estimated to be 8/7.5 dB with a linear propagation loss of 120 dB/cm with the slow-light effect at the pump center wavelength of 1555 nm. Figure 2(b) also indicates the positions of the pump and probe. In order to inhibit FWM, we locate the pump close to the slow-light edge to suppress phase-matching condition by taking advantage of large higher-order dispersion and strong walk-off in the idler. The idler around 1550 nm has

_{si}*ng*∼ 24, giving a 6 ps walk off with respect to the pump.

## 4. Results and discussions

To investigate the cross nonlinear absorption, we measured the on-chip probe attenuation by varying the probe delay at SPS2 at a fixed pump power. Figure 3(a) shows the output spectra of the probe only and probe with pump on at zero delay and 10 ps delay (i.e. probe after pump). For more than 10 ps delay, the pump and probe separate further and they do not catch up with each other. The input peak power of the pump is 4.4 W and the probe is 10 mW. We clearly observe the output spectra intensity of the probe varies at different delays. The delay-dependent probe intensity is a strong sign of nonlinear cross absorption. The spectra also highlight several features unique to two-wave interaction in silicon. Firstly, we observe strong asymmetric SPM on the pump and asymmetric XPM on the probe at zero delay. The asymmetry and the frequency blue shift are caused by the strong free carrier effects [10]. Our calculation of the average center frequency confirms that the pump and probe almost obtain a similar blue shift of −56 GHz and −60 GHz. However this probe blue shift disappears at 10 ps delay, since the whole probe pulse temporally experiences a flat FCD-induced phase. Secondly, the probe output powers with pump on at zero delay and 10 ps delay are lower than the probe power without the pump. In addition, the generated idler at 1555 nm is due to FWM. The FWM conversion efficiency is estimated to be less than −20 dB from the spectrum and thus can be neglected.

Figure 3(b) summarizes the experimental total loss of the probe as a function of delay. We normalize the probe output to the input by integrating the spectra $\frac{{\displaystyle \int {P}_{2}(\mathrm{out})(\lambda )}}{{\displaystyle \int {P}_{2}(\mathrm{out}-\mathrm{nopump})(\lambda )}}$ and considering linear propagation loss. As expected, the probe experiences distinct losses at different delay regimes. When the probe is far ahead of the pump, e.g. −15 ps, since there is no temporal overlap between the pump and probe, the probe only undergoes linear loss (~ 2 dB). When the two waves partially overlap, the probe experiences nonlinear absorption. The maximum sum of nonlinear absorptions (FCA+XTPA) is observed when the pump and probe are perfectly overlapped with −1 ps delay. Once the probe is far behind the pump, e.g. 10 ps, it only experiences FCA (~ 3.1 dB). The dashed curve in Fig. 3(b) indicates the numerically extracted XTPA using Eq. (3c) without the FCA term in the probe but with FCA still on the pump, i.e. ${P}_{2}={P}_{2,\mathrm{in}}{\mathrm{e}}^{-\alpha z}{\mathrm{e}}^{-{\gamma}_{\mathrm{TPA}}{\displaystyle \int {P}_{1}dz}}$. The simulation parameters will be given in the next paragraph. Note the delay origin of the simulated XTPA curve is shifted because of the slightly walk off (1 ps) between the probe and pump, and the uncertainty in experiment. We obtained a maximum XTPA of ~ 4 dB at a delay of −2 ps. The different delays of the maximum XTPA and maximum XTPA+FCA (at −1 ps) is due to the competition between FCA and XTPA. Also, because the free carriers shift the pump peak [21], the maximum XTPA occurs slightly earlier. The widths of the nonlinear loss curves are around 10 ps, which roughly agrees with the convolution width of the probe and pump.

We confirm the measurement results with our numerical solution. To take into account the slow-light enhancement, the linear loss and the free-carrier coefficients are proportionally scaled to the slow light factor *S* while the Kerr and TPA coefficients are scaled to *S*^{2} [3, 23]. The parameters used in this calculation at *λ* = 1.55 *μ*m are *α* = *S* 13 dB/cm, *n*_{2} = *S*^{2} *×* 6 *×* 10^{−18} m^{2}/W, *α*^{TPA} = *S*^{2} *×* 10 *×* 10^{−12} m/W, *n*_{FC} = −*S ×* 2 10^{−27} m^{3}, *A*_{eff} = 0.5 *μ*m^{2}, *σ* = *S ×* 1.45 *×* 10^{−21} m^{2}, *t*_{1,2FWHM} = 7 ps, *P*_{1,in} = 4.4 W and *P*_{2,in} = 10 mW With these considerations, our analysis using Eq. (3) shows excellent agreement with the experimental results. This agreement confirms that the probe intensity created by FWM is negligible, as we mentioned before.

To further explore the dynamics of the cross absorption process, we repeated the measurement in Fig. 3 as a function of input pump power. Figure 4(a) summarizes the normalized pump and probe output as a function of input power at −1 ps delay and at 10 ps delay (the transmission curve of the pump at 10 ps delay is not shown here since it is similar to the one at −1 ps delay because of the negligible effects of XTPA and FWM on the pump). Again, our experimental observation is verified by our numerical solution (solid lines). All the three curves converge to the linear loss of −2 dB at a very low input pump power of 0.1 W. Our experimental results confirm that the pump with TPA goes through less nonlinear absorption than the probe which experiences XTPA, as theoretically discussed before. However, above 5 W the FCA becomes dominant and the relative gap between the pump and probe loss only increases slowly. The interplay between FCA and XTPA can be explained by the ratio of FCA to XTPA
$(\frac{\sigma {\displaystyle \int {N}_{c}}}{2{\gamma}_{\mathrm{TPA}}{\displaystyle \int {P}_{1}}}\phantom{\rule{0.2em}{0ex}}\text{indB})$ from Eq. (3c). This ratio is approximately proportional to exp(*P*_{1}) since
${N}_{c}\propto {P}_{1}^{2}$. The strong FCA also can be seen from the decrease of the probe output at 10 ps delay.

Here we introduce a simple approximate method to separate and extract the different nonlinear loss terms (TPA, XTPA and FCA) from experimental data. This method is useful because these effects often occur together and are in general difficult to analyze as separate contributions. Taking the ratio of the outputs in Eqs. (3a) and (3c), we can approximate the TPA in the pump. Since FCA and linear loss are the same for the pump and probe, these two losses cancel out. The XTPA is twice of the TPA (twice in dB and square in linear scale) and can also be extracted. Combining with Eq. (3), the FCA can be obtained from the following TPA and XTPA expressions in linear scale,

*P*

_{1}and

*P*

_{2}are the output power of the pump and probe at −1 ps delay when the probe has the minimum output. We can easily get the linear loss term e

*from the probe*

^{−αz}*P*

_{out}vs

*P*

_{in}curve, which is not shown here. Therefore, our method in Eq. (4) is capable of effectively extracting the effects of the mutually coupled TPA, XTPA and FCA on powers from experimental data. This formula also works in the case without free carriers. The assumption of this method is that the maximum XTPA and maximum nonlinear absorption of the probe occur at the same delay. Although the two maxima have slightly different delays (see −2 ps and −1 ps in Fig. 3(b)), Eq. (4) gives less than 10% error in the XTPA up to the power of 7 W compared to our modelling results. Note that since the TPA coefficient varies only slightly across C-band [20], the analytic solution works at telecommunication wavelengths that we are interested in. This method also works when the linear loss is wavelength-dependent and the XTPA coefficient is not equal to TPA coefficient. More details are discussed in Appendix.

In Fig. 4(b), the numerically obtained XTPA and TPA (solid lines) is compared with the experimentally extracted XTPA and TPA (markers) using Eq. (4). As we can see the XTPA of the probe is twice of the TPA of the pump. Combined with Fig. 4(a), we can easily extract different nonlinear absorptions, e.g., the TPA is 2 dB, XTPA is 4 dB and FCA is 2.2 dB at 5 W at a delay of −1 ps. The dashed lines in Fig. 4(b) are the XTPA and TPA obtained with the analytic solution without free carriers in Eq. (2). In general, this analytic solution gives a reasonable estimation of the experimental data. The analytic estimation is valid up to 5 W with an error of around 20%. The reason that the analytic estimation of XTPA works well is because FCA does not affect the TPA at the front part of the pulse. Mathematically, in Eq. (4a) and (4b), when there are free carriers, the ratio of the pump and probe output goes down slightly, although the two outputs both decrease.

We now discuss the effect of free carriers on the analytic solution of Eq. (2). In the presence of free carriers, for low repetition rate pulses (*τ _{c}R_{e}* ≪ 1), where

*R*is the pulse repetition rate, Eq. (2) can still be used to estimate the XTPA effect, as discussed in Fig. 4(b). In silicon, the analytic solution works up to 100 MHz pulses, since the free carrier lifetime is around 1 ns. However, the analytically estimated XTPA is valid for powers below a certain threshold, which is dependent on the pulse width. To assess the accuracy of our analytic solution when there are free carriers, we calculate the difference of XTPA between the analytic solution (

_{e}*P*in dB) and NLSE (

_{a}*P*in dB) as

_{n}*δ*= |

*P*−

_{a}*P*|/

_{n}*P*Here the waveguide parameters are the same as in Fig. 4. Figure 5 shows

_{a}*δ*as a function of input pump power and pulse width. As we can see,

*δ*increases at high power levels for all pulse widths due to free carriers. The dashed line indicates the case

*δ*= 10% up to where the analytic solution gives an accurate estimation. Any area below this curve implies that FCA is negligible. Since longer pulses generate more free carriers than short pulses, the shorter the pulse implies a larger acceptable pump powers. For instance, the acceptable pump power is around 1.5 W for a 10 ps pulse, while the pump peak power can be higher than 100 W for any pulse shorter than 100 fs. However, the dispersion must be included for sub-picoseconds pump and probe pulses [22]. In addition, when there is free carriers accumulation in high repetition pulses (

*τ*≫ 1), e.g. for any pulse strains faster than 10 GHz or continuous waves, since FCA is the dominant loss mechanism, TPA is negligible and the analytic solution loses its validity. However, the output power can be solved from the steady state [22]. We emphasize that the analytic solution is always valid if the free carriers are minimized using a P-I-N junction [4].

_{c}R_{e}## 5. Conclusion

In summary, we characterized the cross nonlinear absorption in silicon waveguides using a pump-probe configuration. We provided an analytic solution to include the TPA-only and a numerical solution to take into account free carriers. We found XTPA and FCA compete with each other in the probe power. We derived a simple approach to extract and separate the XTPA and FCA contributions from experimental data. Our proposed method and experimental results in PhC waveguides were confirmed by our numerical solution. We found, even in the presence of free carriers, the TPA-only analytic solution can be used to effectively estimate the XTPA. Our analysis is helpful for understanding and optimizing optical functions based on cross nonlinear absorptions. In addition, our solutions can be further extended to any multiple wave interactions to consider the cross nonlinear absorption, e.g. FWM and parametric amplifications, in all-optical signal processing.

## Appendix

When the linear loss and the TPA coefficient are wavelength-dependent, our method to extract each nonlinear absorption still works. Replacing *α* in Eq. (3a) and (3b) with *α*_{1} and *α*_{2}, respectively, and defining *η* as the ratio between XTPA and TPA coefficient *γ*_{XTPA} = *ηγ*_{TPA} in (3b), we arrive at a modified version of Eq. (4),

*P*

_{out}vs

*P*

_{in}. As long as we know the exact values of the TPA and XTPA coefficients, we can separate out TPA, XTPA and FCA. It is easy to verify that Eq. (5) can be simplified to Eq. (4) at

*η*= 1 when

*α*=

*α*

_{1}=

*α*

_{2}and

*γ*

_{XTPA}=

*γ*

_{TPA}.

## Acknowledgments

This work was supported by the Australian Research Councils, Center of Excellence CUDOS ( CE110001018), Laureate Fellowship ( FL120100029), Discovery Early Career Researcher ( DE120101329, DE120102069) schemes, and EPSRC of U.K. under Grant EP/J01771X/1 (Structured Light).

## References and links

**1. **M. A. Foster, A. C. Turner, J. E. Sharping, B. S. Schmidt, M. Lipson, and A. L. Gaeta, “Broad-band optical parametric gain on a silicon photonic chip,” Nature **441**, 960–963 (2006). [CrossRef] [PubMed]

**2. **R. Salem, M. A. Foster, A. C. Turner, D. F. Geraghty, M. Lipson, and A. L. Gaeta, “Signal regeneration using low-power four-wave mixing on silicon chip,” Nature Photon. **2**, 35–38 (2008). [CrossRef]

**3. **Y. Zhang, C. Husko, J. Schröder, S. Lefrancois, I. H. Rey, T. F. Krauss, and B. J. Eggleton, “Phase-sensitive amplification in silicon photonic crystal waveguides,” Opt. Lett. **39**, 363–366 (2014). [CrossRef] [PubMed]

**4. **F. Da Ros, D. Vukovic, A. Gajda, K. Dalgaard, L. Zimmermann, B. Tillack, M. Galili, K. Petermann, and C. Peucheret, “Phase regeneration of DPSK signals in a silicon waveguide with reverse-biased P-I-N junction,” Opt. Express **22**, 5029–5036 (2014). [CrossRef] [PubMed]

**5. **L. Yin and G. P. Agrawal, “Impact of two-photon absorption on self-phase modulation in silicon waveguides,” Opt. Lett. **32**, 2031–2033 (2007). [CrossRef] [PubMed]

**6. **H. Tsang, C. Wong, T. Liang, I. Day, S. Roberts, A. Harpin, J. Drake, and M. Asghari, “Optical dispersion, two-photon absorption and self-phase modulation in silicon waveguides at 1.5 μm wavelength,” Appl. Phys. Lett. **80**, 416–418 (2002). [CrossRef]

**7. **C. Monat, M. Ebnali-Heidari, C. Grillet, B. Corcoran, B. Eggleton, T. White, L. OFaolain, J. Li, and T. Krauss, “Four-wave mixing in slow light engineered silicon photonic crystal waveguides,” Opt. Express **18**, 22915–22927 (2010). [CrossRef] [PubMed]

**8. **D. A. Fishman, C. M. Cirloganu, S. Webster, L. A. Padilha, M. Monroe, D. J. Hagan, and E. W. Van Stryland, “Sensitive mid-infrared detection in wide-bandgap semiconductors using extreme non-degenerate two-photon absorption,” Nature Photon. **5**, 561–565 (2011). [CrossRef]

**9. **A. D. Bristow, N. Rotenberg, and H. M. Van Driel, “Two-photon absorption and kerr coefficients of silicon for 850–2200,” Appl. Phys. Lett. **90**, 191104 (2007). [CrossRef]

**10. **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]

**11. **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]

**12. **P. Kanakis, T. Kamalakis, and T. Sphicopoulos, “Approximate expressions for estimation of four-wave mixing efficiency in slow-light photonic crystal waveguides,” J. Opt. Soc. Am. B **31**, 366–375 (2014). [CrossRef]

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

**14. **X. Sang, E.-K. Tien, and O. Boyraz, “Applications of two photon absorption in silicon,” J. Optoelectron. Adv. Mater. **11**, 15–25 (2009).

**15. **Y. Shoji, T. Ogasawara, T. Kamei, Y. Sakakibara, S. Suda, K. Kintaka, H. Kawashima, M. Okano, T. Hasama, H. Ishikawa, and M. Mori, “Ultrafast nonlinear effects in hydrogenated amorphous silicon wire waveguide,” Opt. Express **18**, 5668–5673 (2010). [CrossRef] [PubMed]

**16. **P. Mehta, N. Healy, T. Day, J. Sparks, P. Sazio, J. Badding, and A. Peacock, “All-optical modulation using two-photon absorption in silicon core optical fibers,” Opt. Express **19**, 19078–19083 (2011). [CrossRef] [PubMed]

**17. **L. Shen, N. Healy, C. J. Mitchell, J. SolerPenades, M. Nedeljkovic, G. Z. Mashanovich, and A. C. Peacock, “Two-photon absorption and all-optical modulation in germanium-on-silicon waveguides for the mid-infrared,” Opt. Lett. **40**, 2213–2216 (2015). [CrossRef]

**18. **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]

**19. **Y. Yue, H. Huang, L. Zhang, J. Wang, J.-Y. Yang, O. F. Yilmaz, J. S. Levy, M. Lipson, and A. E. Willner, “UWB monocycle pulse generation using two-photon absorption in a silicon waveguide,” Opt. Lett. **37**, 551–553 (2012). [CrossRef] [PubMed]

**20. **M. Sheik-Bahae, D. C. Hutchings, D. J. Hagan, and E. W. Van Stryland, “Dispersion of bound electron nonlinear refraction in solids,” IEEE J. Quantum Electron. **27**, 1296–1309 (1991). [CrossRef]

**21. **S. Lefrancois, C. Husko, A. Blanco-Redondo, and B. J. Eggleton, “Nonlinear silicon photonics analyzed with the moment method,” J. Opt. Soc. Am. B **32**, 218–226 (2015). [CrossRef]

**22. **Y. Zhang, C. Husko, J. Schröder, and B. J. Eggleton, “Pulse evolution and phase-sensitive amplification in silicon waveguides,” Opt. Lett. **39**, 5329–5332 (2014). [CrossRef]

**23. **J. Li, L. O’Faolain, I. H. Rey, and T. F. Krauss, “Four-wave mixing in photonic crystal waveguides: slow light enhancement and limitations,” Opt. Express **19**, 4458–4463 (2011). [CrossRef] [PubMed]