## Abstract

We observe spectral broadening of more than 350 nm, i.e., a 3/10-octave span, upon propagation of ultrashort 1.3-μm-wavelength optical pulses in a 4.7-mm-long silicon-photonic-wire waveguide. We measure the wavelength dependence of the spectral features and relate it to waveguide dispersion and input power. The spectral characteristics of the output pulses are shown to be consistent, in part, with higher-order soliton radiative effects.

©2007 Optical Society of America

## 1. Introduction

Supercontinuum generation is a device functionality that has important applications in many areas of photonic integrated circuits [1]. For example, in the case of wavelength-division multiplexing applications, it is often beneficial to use a single broadband laser source, select out filter-specific wavelength channels, and then modulate these channels, instead of using a separate laser for each wavelength channel. Use of a single source with continuum generation reduces both the complexity of on- or off-chip multiple laser integration and its concomitant power dissipation. These are important considerations in telecommunication applications, such as optical transceivers, or in emerging on-chip optical networks for multi-processor chips. In addition, continuum generation is important in other non-communication applications. These include, for example, optical coherence tomography (OCT) where a low power Si supercontinuum source can enable measurement of axial features in a sample at optimum wavelengths, i.e., ~1.3 – 1.5 μm, [2] for imaging in nontransparent biological tissues.

There have been many theoretical and experimental reports of supercontinuum generation in various guided-wave structures, such as single-mode fibers [3], photonic-crystal fibers (PCF) [1,4–9], silica nanowires [10], and LiNbO_{3} [11]. These previous studies suggest that it is possible to achieve supercontinuum generation at low optical power and over short propagation distances provided that the guiding medium exhibits high nonlinear response and tunable dispersion properties. In the case of PCFs, the combination of control over the geometry of the core profile and lattice parameters and the large index difference between the silica core and air cladding provide dispersion tunability and a higher nonlinear response than is available in conventional fiber. In a typical PCF-based supercontinuum source, the effective mode area of the PCF is ~ 1 μm^{2}, with a typical PCF length of the order of several meters [4]. By using further dimensional reduction through the use of silica nanowires, achieved by tapering the micro-structured core, the effective optical nonlinearity can be further increased [10]. Although these previous studies have demonstrated efficient generation of supercontinuum, their use in on-chip integration applications is made difficult by the large propagation length required for inducing large spectral broadening.

A promising alternative solution is provided by Si waveguide sources [12], which have the advantage of employing an emerging SOI integrated-photonics platform. The advantages of Si waveguides may be taken to their limit by the use of Si-wire waveguides (Si-WWGs) [13,14], which have several unique properties that can be employed to achieve on-chip supercontinuum generation. Because of the small transverse dimensions (< 1 μm) of Si-WWGs, their dispersion properties are governed chiefly by the *waveguide* dispersion. As a result, by carefully designing the transverse waveguide dimensions, one can *tailor* these dispersion properties [14–16], e.g., the wavelength corresponding to the zero-group-velocity dispersion (ZGVD) can be tuned over hundreds of nanometers. Furthermore, because of high optical confinement in Si-WWGs due to the high index contrast between silicon and the surrounding media (air or SiO_{2}), large optical intensities are obtained with even a modest input power, e.g., 1 W of peak power yields 1 GW/cm^{2} in the guide, and therefore nonlinear optical effects are strongly enhanced. In this connection, our group has recently demonstrated the effects of strong optical nonlinearity in Si-WWGs, such as self-phase modulation (SPM) [17,18], cross-phase modulation (XPM) [19], and solitonic phenomena [20]. Finally, as reported in our previous work [14,18], the effective nonlinear coefficient of Si-WWGs is several orders of magnitude higher than that of optical fibers or PCFs, due to the higher material nonlinearity of Si and the more favorable modal overlap integral, than in a low-confinement fiber.

The combination of these effects makes it reasonable to observe particularly efficient supercontinuum generation in such optical devices. Moreover, it is well known [1,3–10] that this efficiency is greatly enhanced if the input pulse is launched in the anomalous dispersion regime, near the ZGVD point. This property, in conjunction with the fact that by proper design of waveguide dimensions Si-WWGs can exhibit anomalous dispersion over a broad range of wavelengths, led us to consider continuum generation in mm-long silicon photonic wires. In fact, most recently, Yin, *et al*. have also theoretically investigated the possibility of continuum generation in Si-WWGs [21], but an experimental study of this phenomenon is still missing. Here we report on an experimental demonstration of continuum generation in Si-WWGs of more than 350 nm. In the work below, to characterize the spectral width of our output spectrum, we have adopted the approach used in previous PCF literature [1,4–6], namely the spectral width is determined from the intersection of the signal with the detection floor. In our case, the floor is set by the noise level of our optical spectrum analyzer (OSA). In our experiment, this measured spectral width is one third of an octave, hence we use the term “supercontinuum” to describe our results.

## 2. Experimental setup

Our experiments use single-mode Si-WWGs having a cross-section of *A*
_{0} = *w*×*h* = 520 × 220 nm^{2} and a length *L* = 4.7 mm, fabricated on Unibond SOI with a 1-μm-thick oxide layer and aligned along the [110] crystallographic direction. Each end of the waveguides has an inverse-taper mode-converter, which allows efficient coupling of light. The devices were fabricated using the CMOS production line at the IBM T. J. Watson Research Center [13].

The laser source is an optical parametric amplifier (OPA) pumped by a regeneratively amplified Ti: Sapphire laser system. The OPA has a pulse repetition rate of 250 kHz and a pulse width of *T _{p}* ≈ 100 fs. It produces wavelengths, λ

_{0}, ranging from 1300 nm to 1600 nm with half-power bandwidths of ~30 nm. The pulse is coupled into the waveguide with a free-space objective. The polarization direction of the pump is chosen in such a way that the TE waveguide mode is efficiently excited. The output is collected by a tapered-fiber, and is characterized by an optical spectrum analyzer (OSA) and power meter. Free-space coupling instead of tapered fiber coupling is employed to rule out SPM in the input fiber, but at the expense of a larger coupling loss, ~ 30 dB, between the lens and the waveguide. The coupled peak pump power is estimated to be 1 W. In addition, the propagation loss inside the waveguide has been characterized to be ~ 2.5 dB/cm. Note also that earlier measurements [13] show that the optical loss in the waveguides increases in going from 1.50 to 1.30 μm. For example, for the waveguide in Ref. [13], loss increased from 3.5 dB/cm at 1.50 μm to 6.8 dB/cm at 1.30 μm. Similarly, optical absorption in Si becomes significant at the short-wavelength limit of our measurements, i.e., at 1.1 μm the loss in undoped Si is ~10 dB/cm. Finally, note that our waveguide takes on an increasingly multimode character in going to shorter wavelengths, e.g., the second-order TE

_{1}mode is cutoff at 1400 nm; this fact will make a propagation analysis somewhat more complex. However, since the input polarization is almost purely along the

*x*-axis, this minimizes coupling into the TE

_{1}mode that has a relative large

*E*component relative to a TE

_{y}_{0}mode.

## 3. Results

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

Since the dynamics of supercontinuum generation is strongly influenced by the linear dispersion properties of the photonic wire, it is necessary to have a complete description of the optical parameters that characterize the optical dispersion of the wire, *viz*. effective index *n*
_{eff}, group index *n*
_{g}, GVD coefficient, *β*
_{2}, and third-order dispersion (TOD) coefficient, *β*
_{3}. These quantities are defined as *n*
_{g}=*β*
_{1}
*c* and *β _{m}* = d

^{m}*β*

_{0}/d

*ω*, (where

^{m}*m*= 1,2,3) where

*β*

_{0}=

*n*

_{eff}(

*ω*)

*ω*/

*c*is the mode propagation constant and

*ω*is the carrier frequency. We calculated

*n*

_{eff}using the RSoft BeamPROP software [15] based on a full vectorial beam propagation method, and the result was crosschecked with a finite-element method (FEM) calculation and experimental data. We then fit the values of

*n*

_{eff}with a 7

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

*n*

_{g}and

*β*

_{2}. The dispersion coefficients, up to the second order, that result from using this method agree with FEM calculations results to within 0.1%; good agreement with experimental results is also observed [15]. Notice that for most of the wavelength range used in our experiments, our waveguide exhibits anomalous dispersion (

*β*

_{2}< 0). In addition, as illustrated in Fig. 1, our numerical calculations show that

*β*

_{3}> 0 (

*β*

_{3}< 0) for λ < 1730 nm (λ > 1730 nm). With regard to the calculation of

*β*

_{3}, this is less straightforward than for the lower-dispersion coefficients because the numerical errors accrued at each step, at which we calculate the derivative, prevent a rigorous determination of these numerical derivatives beyond the second order; see Ref. [18] for a more complete discussion of this point. We have recently developed a new procedure for experimentally measuring the TOD by using the properties of TOD-induced soliton-emitted radiation [18]. In our case, we used the results of this measurement to obtain

*β*

_{3}for the waveguide used here.

#### 3.2 Nonlinearity-induced phase shift

Figure 2(a) shows the dependence of the output spectrum as a function of the in-coupled peak pump power at pump center wavelength of λ0 = 1310 nm. At the lowest observable pump power of *P*
_{0} ~ 10 mW, the spectral width is ~80 nm. As the pump power increases the spectral width increases until at the highest power, 1 W, the spectral width has increased to more than 350 nm. Recall, as mentioned above, that the spectral width is determined from the intersection of the signal with the noise level of our OSA. This is a significant degree of spectral broadening, i.e., 3/10 of an octave, particularly since the pump pulse has propagated only 4.7 mm in the wire waveguide. For comparison, with current technology, a pulse with an optical intensity of 1 GW/cm^{2} needs to propagate several meters in photonic crystal fibers to achieve a comparable level of spectral broadening. Furthermore, the data presented in Fig. 2(a) show that, as a result of increased nonlinear interaction, several spectral peaks develop at high optical powers. The relevant nonlinear optical effects for the supercontinuum generation include cascaded self-phase modulation, TOD-induced soliton radiation, and soliton fission [1,21]. A more detailed discussion on the nonlinear processes leading to supercontinuum generation in Si-WWGs, as well as the exact origin of the spectral features seen in Fig. 2(a) will be presented in a following section.

Finally, note that two-photon absorption (TPA) plays an important role in determining the upper limit to the broadening process. This effect is important for the short-pulse excitation used here and its presence is readily seen in the optical-limiting effects previously discussed in conjunction with SPM [17,18]. In an approximate sense, two-photon absorption clamps the maximum propagating power in an optical waveguide and thus it inhibits spectral broadening above a certain level. This effect can be seen in Fig. 2(b), which plots spectral width versus input power. Despite this effect, our results here show that significant broadening can be achieved prior to reaching power limiting due to TPA. In contrast, optical loss due to free-carrier absorption (FCA) generated by TPA is negligible for the case of ultrashort pulses; this point has been discussed in more detail in Ref. [18]. Using the pulse parameter defined in the previous section and *P*
_{0} = 1 W, we calculated the maximum change in refractive index and loss due to FCA to be 4.5×10^{-4} and 1.77 dB/cm, respectively. However, these maximum values correspond to the conditions where the pulse has just entered the waveguide, and they decrease exponentially along the Si-WWG due to linear power loss. The total loss due to FCA after propagating through a Si-WWG of 4.7 mm is 0.028 dB, and therefore is negligible. Using the same model, which takes into account all nonlinear and dispersion effects including FCA, we also simulated using a sech-pulse input the spectral broadening as a function of coupled peak power, as denoted by the blue line in Fig. 2(b). In the simulation, the spectral broadening is defined as the 30 dB bandwidth of the output spectrum. The simulation shows a good agreement with our experiment, and the differences may be explained by the fact that the input pulse shape in our experiment is not exactly hyperbolic secant.

#### 3.3 Wavelength dependence of supercontinuum generation

The next series of measurements examined the wavelength dependence of the output-pulse spectrum. To investigate this dependence, we varied the central wavelength of the input pulse from 1310 to 1570 nm, with all other parameters of the input pulses remaining constant; the results of these measurements are shown in Fig 3. The output spectra show that the broadening increases (see the inset in Fig. 3) as the central wavelength λ_{0} of the input pulse approaches the ZGVD point at 1290 nm (see Fig. 1). This behavior is expected, since near the ZGVD point, the linear dispersion is small and thus temporal pulse broadening is reduced; consequently, strong nonlinear interaction is maintained over a long propagation distance. Another important fact shown in Fig. 3 is that the long-wavelength region of the spectra shows several spectral peaks, which are somewhat more evident at longer wavelengths, λ_{0}. In addition, the short-wavelength region of the spectra shows a well-formed spectral peak (see black arrows in Fig. 3) whose amplitude decreases with shorter λ_{0}. As we will discuss in detail in the next section, these effects can be considered, respectively, to the TOD-induced spectral separation of the solitons contained in the input pulse and the soliton-emitted radiation [21]. Finally as mentioned in the experimental section above, optical losses in the waveguide increase toward the short-wavelength range of our experiments. The most significant of these is due to intrinsic band-to-band optical absorption by Si starting at λ ~ 1000 nm. In general, absorption or scattering losses are sufficiently small for λ > 1050 nm, such that they do not significantly affect the shape of the spectra in Fig. 3; however, if λ^{0} were reduced to 1200 nm, intrinsic Si absorption would clearly set the short-wavelength limit.

#### 4. Discussion

The origin of continuum or supercontinuum radiation in guided-wave structures has been previously investigated by many groups. Generally speaking, the strong spectral broadening observed in the process of generation of white-light (supercontinuum radiation) is attributable to the onset of (cascaded) nonlinear effects, with the particular details of the evolution from the input-pulse to the output-pulse spectrum being strongly dependent on the specific pulse parameters such as pulse width, pulse peak power, pulse chirp, carrier frequency, as well as the linear and nonlinear optical properties of the corresponding optical medium [1]. In order to increase the efficiency of the supercontinuum-generation process, input pulses are launched near the ZGVD point so that the optical dispersion is small and thus minimizing temporal pulse broadening, which reduces the strength of the nonlinear effects. In this connection, depending on the dispersion properties of the optical medium, namely the sign of the GVD coefficient *β*
_{2}, two different scenarios can be identified. In the first case, when the pulse propagates in the *normal* dispersion regime, i.e., *β*
_{2} > 0, the main nonlinear processes that contribute to the supercontinuum radiation are four-wave mixing (FWM), intra-pulse Raman scattering, and, to a smaller extent, SPM and modulation instability (MI). However, in this normal dispersion regime, FWM processes have small efficiency [9] due to the poor phase-matching characteristics and become even less efficient as the peak power increases. In addition, if femtosecond pulses are used, as is the case in this work, the Raman interaction in Si is rather weak, as the Raman response time in Si is in the range of a few picoseconds. Moreover, Raman Stokes frequency shift in silicon (~15.6 THz) is much larger spontaneous Raman spectrum width (~105 GHz), and thus the intra-pulse Raman scattering can normally be ignored. As a result, spectral broadening of pulses propagating in the normal dispersion regime is expected to be small. By contrast, for pulses propagating in the *anomalous* dispersion regime, *β*
_{2} < 0, both FWM and MI can be strongly phase matched, and thus both nonlinear optical processes become efficient in generating new optical frequencies.

In addition, a different effect that contributes to the pulse spectral broadening, which can be dominant in the initial stages of the generation of the supercontinuum radiation, is the higher-order-soliton fission [22–27]. Thus, under the influence of perturbative effects such as finite-time response and frequency dispersion of the optical nonlinearity, as well as higher-order linear dispersion, the initial pulse splits into a series of solitons, the frequency shift of each soliton, as well as the dependence of this frequency shift on the propagation distance, being strongly dependent on the optical power contained in each soliton [22]. Through this mechanism, an incipient spectral broadening builds up. Furthermore, solitons generated in this initial stage emit radiation at frequencies at which the soliton and the cw wavevectors are in resonance [24,25], contributing to a further increase in the spectral broadening. Finally, FWM processes between the newly formed single solitons and the corresponding soliton-generated spectral components become phase-matched [4, 5], leading to the generation of additional spectral components.

The general theoretical model we used to analyze the nonlinear processes in Si-WWGs can be found in our previous publications [14, 17–19]. As we pointed out earlier, the output-pulse spectrum is strongly dependent on the specific pulse parameters (pulse shape, pulse width, pulse peak power, and chirp across the pulse). Due to the cascaded nature of the pulse dynamics, a slight change in the initial pulse shape and chirp will result in a significant difference in the output spectrum. Nevertheless, the theoretical model still gives us some instructive insights. Our analysis suggests that this general description of the supercontinuum generation is consistent with the data presented in Fig. 3. Thus, the spectral peaks seen in the long-wavelength region of the spectra can be interpreted as solitons emitted from the optical pulse, due to the influence of the TOD effects (best seen for λ_{0} = 1320 nm). In this connection, we introduce the soliton number *N*, defined as *N* = (γ*P*
_{0}
*T*
_{0}
^{2}/∣*β*
_{2}∣)^{1/2}, where γ=*ε*
_{0}
*A*
_{0}/3 *ω*/*β*
_{1}
^{2}Γ’ is the effective nonlinear coefficient of the Si WWG [17,18]. Using γ= 6×10^{4} W^{-1}m^{-1}, *P*
_{0} = 1 W, *T*
_{0} = 100 fs, and the values of *β*
_{2} extracted from Fig. 1, we determined the soliton number for λ_{0} = 1320 nm and λ_{0} = 1570 nm to be *N* = 32 and *N* = 14, respectively. It is important to note that although the soliton number is higher for pump wavelength near the ZGVD point, the corresponding emitted solitons are not as distinct as those at longer pump wavelengths. This apparent contradiction is explained by the fact that the dispersion length, *L*
_{D} = *T*
_{0}
^{2}/∣*β*
_{2}∣, for λ_{0} = 1320 nm (*L*
_{D} = 17.1 mm) is significantly larger than the propagation length of 4.7 mm, whereas the dispersion length for λ_{0} = 1570 nm (*L*
_{D} = 3.6 mm) is comparable with the waveguide length; therefore, it is expected that the soliton fission is more evident in the latter case. Furthermore, as previously reported [18], the spectral peak seen in the short-wavelength region of the spectra corresponds to the radiation emitted by the generated solitons. For our waveguides, the soliton-induced emission of radiation is generated towards the blue side of the spectrum, due to the fact that *β*
_{3} is positive in the wavelength range used in our experiment (see Fig. 1). To verify this fact, we performed simulations based on the model described in Ref [18], and confirmed that for positive values of *β*
_{3}, soliton-induced radiation features would develop in the blue side of the output spectrum. However, a further study of the simulation is made more difficult than in Ref [18] due to the large soliton number *N* involved in this experiment, and thus resulting in more complex nonlinear processes leading to spectral broadening of the soliton peaks.

In addition in our experiment, the spectral broadening due to soliton-induced emission of radiation is inherently limited since the radiation peak is generated towards the blue side of the spectrum, near the absorption band of Si. This drawback can be easily overcome by tuning the dispersion of the Si-WWG. Thus, through proper design of the waveguide geometry, the dispersion curve can be shifted towards lower wavelengths, such that its *β*
_{3}<0 section is brought near 1550 nm. As a result, the soliton-emitted radiation peak will be generated within the red side of the spectrum, and therefore a much broader spectrum can be obtained. As an additional advantage, this spectral region is characterized by reduced two-photon absorption, which translates to a weak suppression of the nonlinear interactions. In summary, while a more extensive study is required for a clear interpretation of the observed results and for assessing the importance of other known nonlinear optical processes in Si wires, such as modulation instability [28], it is apparent that soliton effects, as recently observed in Si wires [18] and previously known to be important in fiber optics [4,5,7,9,24–27], can serve as the source of significant spectral broadening.

## 5. Conclusions

In this experimental study of supercontinuum generation in Si-WWG, we demonstrate a more than 3/10 octave broadening of the output spectrum with a peak coupled-input-power of 1 W and a short propagation distance of 4.7 mm. This degree of broadening is reached prior to the onset of optical limiting of the spectral broadening due to two-photon absorption. Our analysis is consistent with the importance of soliton fission in forming the supercontinuum radiation in our Si-WWGs. The measured wavelength response of our continuum spectra also suggests that proper dispersion engineering of waveguide geometry can be used to extend the supercontinuum generation into a longer wavelength region than shown here.

## Acknowledgments

This research was supported by the DoD STTR, Contract No. FA9550-05-C-1954, and by the AFOSR Grant FA9550-05-1-0428. The IBM part of this work as supported by Grant No. N00014-07-C-0105 ONR/DARPA.

## References and Links

**01. **J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. **78**, 1135–1184 (2006). [CrossRef]

**02. **W. Drexler, “Ultrahigh-resolution optical coherence tomography,” J. Biomed. Opt. **9**, 47–74 (2004) [CrossRef] [PubMed]

**03. **P. L. Baldeck and R. R. Alfano, “Intensity effects on the stimulated four photon spectra generated by picosecond pulses in optical fibers,” J. Lightwave Technol. **5**, 1712–1715 (1987). [CrossRef]

**04. **A. V. Husakou and J. Herrmann, “Supercontinuum generation of higher-order solitons by fission in photonic crystal fibers,” Phys. Rev. Lett. **87**, 203901–203904 (2001). [CrossRef] [PubMed]

**05. **J. Herrmann, U. Griebner, N. Zhavoronkov, A. Husakou, D. Nickel, J. C. Knight, W. J. Wadsworth, and P. St. J. Russell, “Experimental evidence for supercontinuum generation by fission of higher-order solitons in photonic fibers,” Phys. Rev. Lett. **88**, 173901–173904 (2002). [CrossRef] [PubMed]

**06. **A. L. Gaeta, “Nonlinear propagation and continuum generation in microstructured optical fibers,” Opt. Lett. **27**, 924–926 (2002). [CrossRef]

**07. **K. M. Hilligsoe, H. N. Paulsen, J. Thogersen, S. R. Keiding, and J. J. Larsen, “Initial steps of supercontinuum generation in photonic crystal fibers,” J. Opt. Soc. Am. B **20**, 1887–1893 (2003). [CrossRef]

**08. **W. J. Wadsworth, N. Joly, J. C. Knight, T. A. Birks, F. Biancalana, and P. St. J. Russell, “Supercontinuum and four-wave mixing with Q-switched pulses in endlessly single-mode photonic crystal fibres,” Opt. Express **12**, 299–309 (2004). [CrossRef] [PubMed]

**09. **A. Demircan and U. Bandelow, “Analysis of the interplay between soliton fission and modulation instability in supercontinuum generation,” Appl. Phys. B **86**, 31–39 (2007). [CrossRef]

**10. **M. A. Foster, J. M. Dudley, B. Kibler, Q. Cao, D. Lee, R. Trebino, and A. L. Gaeta, “Nonlinear pulse propagation and supercontinuum generation in photonic nanowires: experiment and simulation,” Appl. Phys. B **81**, 363–367 (2005). [CrossRef]

**11. **C. Langrock, M. M. Fejer, I. Hartl, and M. E. Fermann, “Octave-level spectral broadening in RPE PPLN Waveguides,” in *Conference on Lasers and Electro-Optics*, (Optical Society of America, 2007), paper CTuK2.

**12. **Ö. Boyraz, P. Koonath, V. Raghunathan, and B. Jalali, “All optical switching and continuum generation in silicon waveguides,” Opt. Express **12**, 4094–4102 (2004). [CrossRef] [PubMed]

**13. **Y. A. Vlasov and S. J. McNab, “Losses in single-mode silicon-on-insulator strip waveguides and bends,” Opt. Express **12**, 1622–1631 (2004). [CrossRef] [PubMed]

**14. **X. Chen, N. C. Panoiu, and R. M. Osgood, “Theory of Raman-mediated pulsed amplification in silicon-wire waveguides,” IEEE J. Quantum Electron. **42**, 160–170 (2006). [CrossRef]

**15. **E. Dulkeith, F. Xia, L. Schares, W. M. J. Green, and Y. A. Vlasov, “Group index and group velocity dispersion in silicon-on-insulator photonic wires,” Opt. Express **14**, 3853–3863 (2906). [CrossRef]

**16. **M. A. Foster, A. C. Turner, R. Salem, M. Lipson, and A. L. Gaeta, “Broad-band continuous-wave parametric wavelength conversion in silicon nanowaveguides,” Opt. Express **15**, 12949–12958 (2007) [CrossRef] [PubMed]

**17. **E. Dulkeith, Y. A. Vlasov, X. Chen, N. C. Panoiu, and R. M. Osgood Jr., “Self-phase-modulation in submicron silicon-on-insulator photonic wires,” Opt. Express **14**, 5524–5534 (2006). [CrossRef] [PubMed]

**18. **I. -W. Hsieh, X. Chen, J. I. Dadap, N. C. Panoiu, R. M. Osgood, S. McNab, and Y. A. Vlasov, “Ultrafast-pulse self-phase modulation and third-order dispersion in Si photonic wire-waveguides,” Opt. Express **14**, 12380–12387 (2006). [CrossRef] [PubMed]

**19. **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]

**20. **X. Chen, N. Panoiu, I. Hsieh, J. I. Dadap, and R. M. Osgood Jr., “Third-order Dispersion and Ultrafast Pulse Propagation in Silicon Wire Waveguides,” IEEE Photon. Technol. Lett., **18**, 2617–2619 (2006). [CrossRef]

**21. **L. Yin, Q. Lin, and G. P. Agrawal, “Soliton fission and supercontinuum generation in silicon waveguides,” Opt. Lett. **32**, 391–393 (2007). [CrossRef] [PubMed]

**22. **Y. S. Kivshar and B. A. Malomed, “Dynamics of solitons in nearly integrable systems,” Rev. Mod. Phys. **61**, 763–916 (1989). [CrossRef]

**23. **Y. Kodama and A. Hasegawa, “Nonlinear pulse propagation in a monomode dielectric guide,” IEEE J. Quantum Electron. **23**, 510–524 (1987). [CrossRef]

**24. **P. K. A. Wai, C. R. Menyuk, Y. C. Lee, and H. H. Chen, “Nonlinear pulse propagation in the neighborhood of the zero-dispersion wavelength of monomode optical fibers,” Opt. Lett. **11**, 464–466 (1986). [CrossRef] [PubMed]

**25. **N. Akhmediev and M. Karlsson, “Cherenkov radiation emitted by solitons in optical fibers,” Phys. Rev. A **51**, 2602–2607 (1995). [CrossRef] [PubMed]

**26. **J. N. Elgin, T. Brabec, and S. M. J. Kelly, “A perturbative theory of soliton propagation in the presence of third order dispersion,” Opt. Commun. **114**, 321–328 (1995). [CrossRef]

**27. **N. C. Panoiu, D. Mihalache, D. Mazilu, I. V. Melnikov, J. S. Aitchison, F. Lederer, and R. M. Osgood Jr., “Dynamics of dual-frequency solitons under the influence of frequency-sliding filters, third-order dispersion, and intrapulse Raman scattering” IEEE J. Sel. Top. Quantum Electron. **10**, 885–892 (2004). [CrossRef]

**28. **N. C. Panoiu, X. Chen, and R. M. Osgood, “Modulation instability in silicon photonic nanowires,” Opt. Lett. **31**, 3609–3611 (2006). [CrossRef] [PubMed]