Abstract

We analytically and experimentally investigate cross-phase modulation (XPM) in silicon waveguides. In contrast to the well known result in pure Kerr media, the spectral broadening ratio of XPM to self-phase modulation is not two in the presence of either two-photon absorption (TPA) or free carriers. The physical origin of this change is different for each effect. In the case of TPA, this nonlinear absorption attenuates and slightly modifies the pulse shape due to differential absorption in the pulse peak and wings. When free carriers are present two different mechanisms modify the dynamics. First, free-carrier absorption performs a similar role to TPA, but is additionally asymmetric due to the delayed free-carrier response. Second, free-carrier dispersion induces an asymmetric blue phase shift which competes directly with the symmetric Kerr-induced XPM red shift. We confirm this analysis with pump-probe experiments in a silicon photonic crystal waveguide.

© 2016 Optical Society of America

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 (ρυ) of XPM to SPM as ρυ=ΔυXPMΔυSPM, with the spectral broadening factors for SPM (ΔυSPM) and XPM (ΔυXPM). In contrast to the canonical case, we surprisingly find ρυ ≠ 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:

P1(z,t)=P1,in(t)eαz1+P1,in(t)zeffγTPA,
P2(z,t)=P2,in(t)eαz(1+P1,in(t)zeffγTPA)2,
with the corresponding output phases
ϕ1(z,t)=γγTPAln[1+γTPAP1,in(t)zeff],
ϕ2(z,t)=2ϕ1=2γγTPAln[1+γTPAP1,in(t)zeff],
where P1,2,in(z, t) = P1,2(0, t) = P1,2,in(t) are the temporal power shapes of the input waves, and zeff = (1 − eαz)/α is the effective distance with linear loss α. The bulk TPA coefficient αTPA is related to γTPA = αTPA/Aeff. The SPM coefficient is defined as γ = k0n2/Aeff with wavenumber k0 = 2π/λ, nonlinear Kerr coefficient n2 and effective mode area Aeff. We see in Eq. (1) that both the pump and probe pulses are slightly modified from the input pulse by the TPA, with a larger inverse quadratic scaling for the probe. In terms of the phase in Eq. (2), the probe phase at each temporal point is twice that of the pump due to XPM as in Kerr media. However, the TPA saturates the XPM phase logarithmically.

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 tFWHM=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 (1+0.77ϕmax2)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 = 〈(υυavg)2〉 − 〈(υυavg)〉2, where υavg 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 ΔυoutΔυin=(1+0.77ϕmax2)1/2 [21], where ϕmax is the maxima of phases (max[ϕ1,2]) given in Eq. (2) and Δυin, Δυout 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 ( (1+0.77ϕmax2)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],

P1(z,t)=P1,ineαzeγTPAP1dzeσNcdz,
P2(z,t)=P2,ineαze2γTPAP1dzeσNcdz,
and output phases,
ϕ1(z,t)=γP1(z,t)dz+k0nFCNc(z,t)dz,
ϕ2(z,t)=2γP1(z,t)dz+k0nFCNc(z,t)dz,
where σ is the FCA coefficient, P1(z, t) in Eq. (4) is solved from Eq. (3), and the carrier density Nc determined by a separate rate equation [14,16,20], with nFC the free-carrier refractive index coefficient. Critically, the FCD effect is generated by free carriers accumulated by the pulse, and is therefore asymmetric from the front to the tail of the pulse. Further, nFC < 0 such that FCD causes a blue phase shift that varies across the pulse. The net result is an asymmetric (blue) broadening of the spectrum. This contrasts to the symmetric spectral broadening of pure Kerr, or even Kerr and TPA [14,18]. The relative balance of these effects determines the overall shape of the spectral components and will play a key role in the broadening ratio ρυ as we explain below.

We now discuss the role of free carriers in the spectral broadening. The thick lines in Fig. 1(b) are the simulated broadening factors ( ΔυoutΔυin) 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 ρυ=ΔυXPMΔυSPM 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 ρυ approaches two in the pure Kerr case (dashed line) as expected. After we introduce TPA, we notice ρυ approaches 2.06, slightly larger than two at an input power of 20 W. The reason that ρυ 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 ρυ=ΔυXPMΔυSPM 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 ρυ < 2. We first consider the case of FCA only. Although in this case both the pump and probe are distorted and attenuated, ρυ approaches 1.9 at the pump power of 20 W. The reason that ρυ 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 ρυ 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 ρυ 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 (NctFWHM) [14]. After that, ρυ reduces gradually to 1.2 because the strong FCD overcomes the Kerr effect. The reason that ρυ 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 ρυ close to one.

The competition between FCD and Kerr and their impact on spectral broadening is best illustrated by analyzing the chirp rate ( 12πdϕ2dt2). 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 t1,2FWHM = 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, Δω=ϕt. 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 ρυ (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 = ng/nsi of the photonic crystal waveguide with group index ng = 32 and silicon nanowire index nsi = 3.45 [20]. At λ = 1.55 μm, we have α = S × 13 dB/cm, n2 = S2 × 6 × 10−18 m2/W, αTPA = S2 × 10 × 10−12 m/W, Aeff = 0.5 μm2, nFC = −S × 2 × 10−27 m3, σ = S × 1.45 × 10−21 m2, t1,2FWHM = 7 ps, and P2,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 ρυ 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 (ρυ) 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].

Acknowledgments

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

References and links

1. G. P. Agrawal, Nonlinear Fiber Optics (Springer, 2000).

2. A. D. Ellis, J. Zhao, and D. Cotter, “Approaching the nonlinear Shannon limit,” J. Lightwave Technol. 28, 423–433 (2010). [CrossRef]  

3. D. Dimitropoulos and B. Jalali, “Optical information capacity of silicon,” OFC, 2–25 (2015).

4. A. V. Cartaxo, “Cross-phase modulation in intensity modulation-direct detection WDM systems with multiple optical amplifiers and dispersion compensators,” J. Lightwave Technol. 17, 178–190 (1999). [CrossRef]  

5. R.-J. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel, “Capacity limits of optical fiber networks,” J. Lightwave Technol. 28, 662–701 (2010). [CrossRef]  

6. H. Hu, J. D. Andersen, A. Rasmussen, B. M. Sørensen, K. Dalgaard, M. Galili, M. Pu, K. Yvind, K. J. Larsen, S. Forchhammer, and L. K. Oxenløwe, “Forward error correction supported 150 Gbit/s error-free wavelength conversion based on cross phase modulation in silicon,” Opt. Express 21, 3152–3160 (2013). [CrossRef]   [PubMed]  

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

8. H.-S. Hsieh, K.-M. Feng, and M.-C. M. Lee, “Study of cross-phase modulation and free-carrier dispersion in silicon photonic wires for Mamyshev signal regenerators,” Opt. Express 18, 9613–9621 (2010). [CrossRef]   [PubMed]  

9. M. Hofer, M. E. Fermann, F. Haberl, M. Ober, and A. Schmidt, “Mode locking with cross-phase and self-phase modulation,” Opt. Lett. 16, 502–504 (1991). [CrossRef]   [PubMed]  

10. B. Corcoran, T. D. Vo, M. D. Pelusi, C. Monat, D.-X. Xu, A. Densmore, R. Ma, S. Janz, D. J. Moss, and B. J. Eggleton, “Silicon nanowire based radio-frequency spectrum analyzer,” Opt. Express 18, 20190–20200 (2010). [CrossRef]   [PubMed]  

11. M. Margalit, C. Yu, E. Ippen, and H. Haus, “Cross phase modulation squeezing in optical fibers,” Opt. Express 2, 72–76 (1998). [CrossRef]   [PubMed]  

12. V. Venkataraman, K. Saha, and A. L. Gaeta, “Phase modulation at the few-photon level for weak-nonlinearity-based quantum computing,” Nature Photon. 7, 138–141 (2013). [CrossRef]  

13. J. Leuthold, C. Koos, and W. Freude, “Nonlinear silicon photonics,” Nature Photon. 4, 535–544 (2010). [CrossRef]  

14. 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]  

15. O. Boyraz, T. Indukuri, and B. Jalali, “Self-phase-modulation induced spectral broadening in silicon waveguides,” Opt. Express 12, 829–834 (2004). [CrossRef]   [PubMed]  

16. 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]  

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

18. P. Koonath, D. R. Solli, and B. Jalali, “Limiting nature of continuum generation in silicon,” Appl. Phys. Lett. 93, 091114 (2008). [CrossRef]  

19. D. Castelló-Lurbe, E. Silvestre, and P. Andrés, “Spectral broadening enhancement in silicon waveguides through pulse shaping,” Opt. Lett. 37, 2757–2759 (2012). [CrossRef]   [PubMed]  

20. Y. Zhang, C. Husko, S. Lefrancois, I. H. Rey, T. F. Krauss, J. Schröder, and B. J. Eggleton, “Nondegenerate two-photon absorption in silicon waveguides: analytical and experimental study,” Opt. Express 23, 17101–17110 (2015). [CrossRef]   [PubMed]  

21. S. C. Pinault and M. Potasek, “Frequency broadening by self-phase modulation in optical fibers,” J. Opt. Soc. Am. B 2, 1318–1319 (1985). [CrossRef]  

22. E.-K. Tien, X.-Z. Sang, F. Qing, Q. Song, and O. Boyraz, “Ultrafast pulse characterization using cross phase modulation in silicon,” Appl. Phys. Lett. 95, 051101 (2009). [CrossRef]  

23. A. C. Turner-Foster, M. A. Foster, J. S. Levy, C. B. Poitras, R. Salem, A. L. Gaeta, and M. Lipson, “Ultrashort free-carrier lifetime in low-loss silicon nanowaveguides,” Opt. Express 18, 3582–3591 (2010). [CrossRef]   [PubMed]  

24. Z. Wang, H. Liu, N. Huang, Q. Sun, J. Wen, and X. Li, “Influence of three-photon absorption on mid-infrared cross-phase modulation in silicon-on-sapphire waveguides,” Opt. Express 21, 1840–1848 (2013). [CrossRef]   [PubMed]  

25. C. Husko, P. Colman, S. Combrié, A. De Rossi, and C. W. Wong, “Effect of multiphoton absorption and free carriers in slow-light photonic crystal waveguides,” Opt. Lett. 36, 2239–2241 (2011). [CrossRef]   [PubMed]  

References

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  1. G. P. Agrawal, Nonlinear Fiber Optics (Springer, 2000).
  2. A. D. Ellis, J. Zhao, and D. Cotter, “Approaching the nonlinear Shannon limit,” J. Lightwave Technol. 28, 423–433 (2010).
    [Crossref]
  3. D. Dimitropoulos and B. Jalali, “Optical information capacity of silicon,” OFC, 2–25 (2015).
  4. A. V. Cartaxo, “Cross-phase modulation in intensity modulation-direct detection WDM systems with multiple optical amplifiers and dispersion compensators,” J. Lightwave Technol. 17, 178–190 (1999).
    [Crossref]
  5. R.-J. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel, “Capacity limits of optical fiber networks,” J. Lightwave Technol. 28, 662–701 (2010).
    [Crossref]
  6. H. Hu, J. D. Andersen, A. Rasmussen, B. M. Sørensen, K. Dalgaard, M. Galili, M. Pu, K. Yvind, K. J. Larsen, S. Forchhammer, and L. K. Oxenløwe, “Forward error correction supported 150 Gbit/s error-free wavelength conversion based on cross phase modulation in silicon,” Opt. Express 21, 3152–3160 (2013).
    [Crossref] [PubMed]
  7. R. Dekker, A. Driessen, T. Wahlbrink, C. Moormann, J. Niehusmann, and M. Först, “Ultrafast Kerr-induced all-optical wavelength conversion in silicon waveguides using 1.55 μm femtosecond pulses,” Opt. Express 14, 8336–8346 (2006).
    [Crossref] [PubMed]
  8. H.-S. Hsieh, K.-M. Feng, and M.-C. M. Lee, “Study of cross-phase modulation and free-carrier dispersion in silicon photonic wires for Mamyshev signal regenerators,” Opt. Express 18, 9613–9621 (2010).
    [Crossref] [PubMed]
  9. M. Hofer, M. E. Fermann, F. Haberl, M. Ober, and A. Schmidt, “Mode locking with cross-phase and self-phase modulation,” Opt. Lett. 16, 502–504 (1991).
    [Crossref] [PubMed]
  10. B. Corcoran, T. D. Vo, M. D. Pelusi, C. Monat, D.-X. Xu, A. Densmore, R. Ma, S. Janz, D. J. Moss, and B. J. Eggleton, “Silicon nanowire based radio-frequency spectrum analyzer,” Opt. Express 18, 20190–20200 (2010).
    [Crossref] [PubMed]
  11. M. Margalit, C. Yu, E. Ippen, and H. Haus, “Cross phase modulation squeezing in optical fibers,” Opt. Express 2, 72–76 (1998).
    [Crossref] [PubMed]
  12. V. Venkataraman, K. Saha, and A. L. Gaeta, “Phase modulation at the few-photon level for weak-nonlinearity-based quantum computing,” Nature Photon. 7, 138–141 (2013).
    [Crossref]
  13. J. Leuthold, C. Koos, and W. Freude, “Nonlinear silicon photonics,” Nature Photon. 4, 535–544 (2010).
    [Crossref]
  14. 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]
  15. O. Boyraz, T. Indukuri, and B. Jalali, “Self-phase-modulation induced spectral broadening in silicon waveguides,” Opt. Express 12, 829–834 (2004).
    [Crossref] [PubMed]
  16. 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]
  17. I.-W. Hsieh, X. Chen, J. I. Dadap, N. C. Panoiu, R. M. Osgood, S. J. McNab, and Y. A. Vlasov, “Cross-phase modulation-induced spectral and temporal effects on co-propagating femtosecond pulses in silicon photonic wires,” Opt. Express 15, 1135–1146 (2007).
    [Crossref] [PubMed]
  18. P. Koonath, D. R. Solli, and B. Jalali, “Limiting nature of continuum generation in silicon,” Appl. Phys. Lett. 93, 091114 (2008).
    [Crossref]
  19. D. Castelló-Lurbe, E. Silvestre, and P. Andrés, “Spectral broadening enhancement in silicon waveguides through pulse shaping,” Opt. Lett. 37, 2757–2759 (2012).
    [Crossref] [PubMed]
  20. Y. Zhang, C. Husko, S. Lefrancois, I. H. Rey, T. F. Krauss, J. Schröder, and B. J. Eggleton, “Nondegenerate two-photon absorption in silicon waveguides: analytical and experimental study,” Opt. Express 23, 17101–17110 (2015).
    [Crossref] [PubMed]
  21. S. C. Pinault and M. Potasek, “Frequency broadening by self-phase modulation in optical fibers,” J. Opt. Soc. Am. B 2, 1318–1319 (1985).
    [Crossref]
  22. E.-K. Tien, X.-Z. Sang, F. Qing, Q. Song, and O. Boyraz, “Ultrafast pulse characterization using cross phase modulation in silicon,” Appl. Phys. Lett. 95, 051101 (2009).
    [Crossref]
  23. A. C. Turner-Foster, M. A. Foster, J. S. Levy, C. B. Poitras, R. Salem, A. L. Gaeta, and M. Lipson, “Ultrashort free-carrier lifetime in low-loss silicon nanowaveguides,” Opt. Express 18, 3582–3591 (2010).
    [Crossref] [PubMed]
  24. Z. Wang, H. Liu, N. Huang, Q. Sun, J. Wen, and X. Li, “Influence of three-photon absorption on mid-infrared cross-phase modulation in silicon-on-sapphire waveguides,” Opt. Express 21, 1840–1848 (2013).
    [Crossref] [PubMed]
  25. C. Husko, P. Colman, S. Combrié, A. De Rossi, and C. W. Wong, “Effect of multiphoton absorption and free carriers in slow-light photonic crystal waveguides,” Opt. Lett. 36, 2239–2241 (2011).
    [Crossref] [PubMed]

2015 (1)

2014 (1)

2013 (3)

2012 (1)

2011 (1)

2010 (6)

2009 (1)

E.-K. Tien, X.-Z. Sang, F. Qing, Q. Song, and O. Boyraz, “Ultrafast pulse characterization using cross phase modulation in silicon,” Appl. Phys. Lett. 95, 051101 (2009).
[Crossref]

2008 (1)

P. Koonath, D. R. Solli, and B. Jalali, “Limiting nature of continuum generation in silicon,” Appl. Phys. Lett. 93, 091114 (2008).
[Crossref]

2007 (2)

2006 (1)

2004 (1)

1999 (1)

1998 (1)

1991 (1)

1985 (1)

Agrawal, G. P.

Andersen, J. D.

Andrés, P.

Boyraz, O.

E.-K. Tien, X.-Z. Sang, F. Qing, Q. Song, and O. Boyraz, “Ultrafast pulse characterization using cross phase modulation in silicon,” Appl. Phys. Lett. 95, 051101 (2009).
[Crossref]

O. Boyraz, T. Indukuri, and B. Jalali, “Self-phase-modulation induced spectral broadening in silicon waveguides,” Opt. Express 12, 829–834 (2004).
[Crossref] [PubMed]

Cartaxo, A. V.

Castelló-Lurbe, D.

Chen, X.

Colman, P.

Combrié, S.

Corcoran, B.

Cotter, D.

Dadap, J. I.

Dalgaard, K.

De Rossi, A.

Dekker, R.

Densmore, A.

Dimitropoulos, D.

D. Dimitropoulos and B. Jalali, “Optical information capacity of silicon,” OFC, 2–25 (2015).

Driessen, A.

Eggleton, B. J.

Ellis, A. D.

Essiambre, R.-J.

Feng, K.-M.

Fermann, M. E.

Forchhammer, S.

Först, M.

Foschini, G. J.

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Freude, W.

J. Leuthold, C. Koos, and W. Freude, “Nonlinear silicon photonics,” Nature Photon. 4, 535–544 (2010).
[Crossref]

Gaeta, A. L.

V. Venkataraman, K. Saha, and A. L. Gaeta, “Phase modulation at the few-photon level for weak-nonlinearity-based quantum computing,” Nature Photon. 7, 138–141 (2013).
[Crossref]

A. C. Turner-Foster, M. A. Foster, J. S. Levy, C. B. Poitras, R. Salem, A. L. Gaeta, and M. Lipson, “Ultrashort free-carrier lifetime in low-loss silicon nanowaveguides,” Opt. Express 18, 3582–3591 (2010).
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Galili, M.

Goebel, B.

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Haus, H.

Hofer, M.

Hsieh, H.-S.

Hsieh, I.-W.

Hu, H.

Huang, N.

Husko, C.

Indukuri, T.

Ippen, E.

Jalali, B.

P. Koonath, D. R. Solli, and B. Jalali, “Limiting nature of continuum generation in silicon,” Appl. Phys. Lett. 93, 091114 (2008).
[Crossref]

O. Boyraz, T. Indukuri, and B. Jalali, “Self-phase-modulation induced spectral broadening in silicon waveguides,” Opt. Express 12, 829–834 (2004).
[Crossref] [PubMed]

D. Dimitropoulos and B. Jalali, “Optical information capacity of silicon,” OFC, 2–25 (2015).

Janz, S.

Koonath, P.

P. Koonath, D. R. Solli, and B. Jalali, “Limiting nature of continuum generation in silicon,” Appl. Phys. Lett. 93, 091114 (2008).
[Crossref]

Koos, C.

J. Leuthold, C. Koos, and W. Freude, “Nonlinear silicon photonics,” Nature Photon. 4, 535–544 (2010).
[Crossref]

Kramer, G.

Krauss, T. F.

Larsen, K. J.

Lee, M.-C. M.

Lefrancois, S.

Leuthold, J.

J. Leuthold, C. Koos, and W. Freude, “Nonlinear silicon photonics,” Nature Photon. 4, 535–544 (2010).
[Crossref]

Levy, J. S.

Li, X.

Lipson, M.

Liu, H.

Ma, R.

Margalit, M.

McNab, S. J.

Monat, C.

Moormann, C.

Moss, D. J.

Niehusmann, J.

Ober, M.

Osgood, R. M.

Oxenløwe, L. K.

Panoiu, N. C.

Pelusi, M. D.

Pinault, S. C.

Poitras, C. B.

Potasek, M.

Pu, M.

Qing, F.

E.-K. Tien, X.-Z. Sang, F. Qing, Q. Song, and O. Boyraz, “Ultrafast pulse characterization using cross phase modulation in silicon,” Appl. Phys. Lett. 95, 051101 (2009).
[Crossref]

Rasmussen, A.

Rey, I. H.

Saha, K.

V. Venkataraman, K. Saha, and A. L. Gaeta, “Phase modulation at the few-photon level for weak-nonlinearity-based quantum computing,” Nature Photon. 7, 138–141 (2013).
[Crossref]

Salem, R.

Sang, X.-Z.

E.-K. Tien, X.-Z. Sang, F. Qing, Q. Song, and O. Boyraz, “Ultrafast pulse characterization using cross phase modulation in silicon,” Appl. Phys. Lett. 95, 051101 (2009).
[Crossref]

Schmidt, A.

Schröder, J.

Silvestre, E.

Solli, D. R.

P. Koonath, D. R. Solli, and B. Jalali, “Limiting nature of continuum generation in silicon,” Appl. Phys. Lett. 93, 091114 (2008).
[Crossref]

Song, Q.

E.-K. Tien, X.-Z. Sang, F. Qing, Q. Song, and O. Boyraz, “Ultrafast pulse characterization using cross phase modulation in silicon,” Appl. Phys. Lett. 95, 051101 (2009).
[Crossref]

Sørensen, B. M.

Sun, Q.

Tien, E.-K.

E.-K. Tien, X.-Z. Sang, F. Qing, Q. Song, and O. Boyraz, “Ultrafast pulse characterization using cross phase modulation in silicon,” Appl. Phys. Lett. 95, 051101 (2009).
[Crossref]

Turner-Foster, A. C.

Venkataraman, V.

V. Venkataraman, K. Saha, and A. L. Gaeta, “Phase modulation at the few-photon level for weak-nonlinearity-based quantum computing,” Nature Photon. 7, 138–141 (2013).
[Crossref]

Vlasov, Y. A.

Vo, T. D.

Wahlbrink, T.

Wang, Z.

Wen, J.

Winzer, P. J.

Wong, C. W.

Xu, D.-X.

Yin, L.

Yu, C.

Yvind, K.

Zhang, Y.

Zhao, J.

Appl. Phys. Lett. (2)

P. Koonath, D. R. Solli, and B. Jalali, “Limiting nature of continuum generation in silicon,” Appl. Phys. Lett. 93, 091114 (2008).
[Crossref]

E.-K. Tien, X.-Z. Sang, F. Qing, Q. Song, and O. Boyraz, “Ultrafast pulse characterization using cross phase modulation in silicon,” Appl. Phys. Lett. 95, 051101 (2009).
[Crossref]

J. Lightwave Technol. (3)

J. Opt. Soc. Am. B (1)

Nature Photon. (2)

V. Venkataraman, K. Saha, and A. L. Gaeta, “Phase modulation at the few-photon level for weak-nonlinearity-based quantum computing,” Nature Photon. 7, 138–141 (2013).
[Crossref]

J. Leuthold, C. Koos, and W. Freude, “Nonlinear silicon photonics,” Nature Photon. 4, 535–544 (2010).
[Crossref]

Opt. Express (10)

O. Boyraz, T. Indukuri, and B. Jalali, “Self-phase-modulation induced spectral broadening in silicon waveguides,” Opt. Express 12, 829–834 (2004).
[Crossref] [PubMed]

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

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M. Margalit, C. Yu, E. Ippen, and H. Haus, “Cross phase modulation squeezing in optical fibers,” Opt. Express 2, 72–76 (1998).
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H. Hu, J. D. Andersen, A. Rasmussen, B. M. Sørensen, K. Dalgaard, M. Galili, M. Pu, K. Yvind, K. J. Larsen, S. Forchhammer, and L. K. Oxenløwe, “Forward error correction supported 150 Gbit/s error-free wavelength conversion based on cross phase modulation in silicon,” Opt. Express 21, 3152–3160 (2013).
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R. Dekker, A. Driessen, T. Wahlbrink, C. Moormann, J. Niehusmann, and M. Först, “Ultrafast Kerr-induced all-optical wavelength conversion in silicon waveguides using 1.55 μm femtosecond pulses,” Opt. Express 14, 8336–8346 (2006).
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H.-S. Hsieh, K.-M. Feng, and M.-C. M. Lee, “Study of cross-phase modulation and free-carrier dispersion in silicon photonic wires for Mamyshev signal regenerators,” Opt. Express 18, 9613–9621 (2010).
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Y. Zhang, C. Husko, S. Lefrancois, I. H. Rey, T. F. Krauss, J. Schröder, and B. J. Eggleton, “Nondegenerate two-photon absorption in silicon waveguides: analytical and experimental study,” Opt. Express 23, 17101–17110 (2015).
[Crossref] [PubMed]

A. C. Turner-Foster, M. A. Foster, J. S. Levy, C. B. Poitras, R. Salem, A. L. Gaeta, and M. Lipson, “Ultrashort free-carrier lifetime in low-loss silicon nanowaveguides,” Opt. Express 18, 3582–3591 (2010).
[Crossref] [PubMed]

Z. Wang, H. Liu, N. Huang, Q. Sun, J. Wen, and X. Li, “Influence of three-photon absorption on mid-infrared cross-phase modulation in silicon-on-sapphire waveguides,” Opt. Express 21, 1840–1848 (2013).
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Opt. Lett. (5)

Other (2)

G. P. Agrawal, Nonlinear Fiber Optics (Springer, 2000).

D. Dimitropoulos and B. Jalali, “Optical information capacity of silicon,” OFC, 2–25 (2015).

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Figures (4)

Fig. 1
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 ( 1 + 0.77 ϕ max 2 ) 1 / 2 and numerical calculation (NLSE) with and without free carriers (FC).
Fig. 2
Fig. 2 (a) Numerically calculated ρ υ = Δ υ XPM Δ υ SPM with various nonlinear effects. (b) Chirp rates of pump (solid) and probe (dashed) under different conditions at an input power of 10 W.
Fig. 3
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.
Fig. 4
Fig. 4 (a) Comparison of spectral broadening ratio of XPM and SPM (ρυ) obtained from experimental results and numerical modelling (NLSE) with and without chirp. (b) Experimental and simulated spectra at three input pump powers.

Equations (8)

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P 1 ( z , t ) = P 1 , in ( t ) e α z 1 + P 1 , in ( t ) z eff γ TPA ,
P 2 ( z , t ) = P 2 , in ( t ) e α z ( 1 + P 1 , in ( t ) z eff γ TPA ) 2 ,
ϕ 1 ( z , t ) = γ γ TPA ln [ 1 + γ TPA P 1 , in ( t ) z eff ] ,
ϕ 2 ( z , t ) = 2 ϕ 1 = 2 γ γ TPA ln [ 1 + γ TPA P 1 , in ( t ) z eff ] ,
P 1 ( z , t ) = P 1 , in e α z e γ TPA P 1 d z e σ N c d z ,
P 2 ( z , t ) = P 2 , in e α z e 2 γ TPA P 1 d z e σ N c d z ,
ϕ 1 ( z , t ) = γ P 1 ( z , t ) d z + k 0 n FC N c ( z , t ) d z ,
ϕ 2 ( z , t ) = 2 γ P 1 ( z , t ) d z + k 0 n FC N c ( z , t ) d z ,

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