## Cross-phase modulation-induced spectral and temporal effects on co- propagating femtosecond pulses in silicon photonic wires

Optics Express, Vol. 15, Issue 3, pp. 1135-1146 (2007)

http://dx.doi.org/10.1364/OE.15.001135

Acrobat PDF (251 KB)

### Abstract

By performing time-resolved experiments and power-dependent measurements using femtosecond pulses inside submicron cross-section Si photonic-wire waveguides, we demonstrate strong cross-phase modulation (XPM) effects. We find that XPM in Si wires can be significant even for low peak pump powers, i.e., ∼15 mW for π phase shift. Our experimental data closely match numerical simulations using a rigorous coupled-wave theoretical treatment. Our results suggest that XPM is a potentially useful approach for all-optical control of photonic devices in Si wires.

© 2007 Optical Society of America

## 1. Introduction

1. 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]

*waveguide*dispersion [2–7

2. 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]

*tailor*these dispersion properties. For example, one can design the zero-GVD wavelength to be around 1550 nm or the C-band to be in the anomalous dispersion regime [6

6. 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]

8. H. K. Tsang, C. S. Wong, T. K. Liang, I. E. Day, S. Roberts, A. Harpin, J. Drake, and M. Asghari, “Optical dispersion, TPA and SPM in Si waveguides at 1.5 μm wavelength,” Appl. Phys. Lett **80**,416–418 (2002). [CrossRef]

9. D. Dimitripoulos, R. Jhaveri, R. Claps, J. C. S. Woo, and B. Jalali, “Lifetime of photogenerated carriers in silicon-on-insulator rib waveguides,” Appl. Phys. Lett **86**,071115 (2005) [CrossRef]

*p*-

*i*-

*n*structures [10

10. H. Rong, A. Liu, R. Jones, O. Cohen, D. Hak, R. Nicolaescu, A. Fang, and M. Paniccia, “An all-silicon Raman laser,” Nature **433**,294–296 (2005). [CrossRef]

11. Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature **435**,325–327 (2005). [CrossRef] [PubMed]

10. H. Rong, A. Liu, R. Jones, O. Cohen, D. Hak, R. Nicolaescu, A. Fang, and M. Paniccia, “An all-silicon Raman laser,” Nature **433**,294–296 (2005). [CrossRef]

13. R. Espinola, J. I. Dadap, R. Osgood Jr., S. McNab, and Y. Vlasov, “C-band wavelength conversion in silicon photonic wire waveguides,” Opt. Express **13**,4341–4349 (2005). [CrossRef] [PubMed]

6. 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]

7. L. Yin, Q. Lin, and G. P. Agrawal, “Dispersion tailoring and soliton propagation in silicon waveguides,” Opt. Lett **31**,1295–1297 (2006). [CrossRef] [PubMed]

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

28. O. 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]

^{2}scale waveguides [28

28. O. 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]

30. 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]

## 2. Experimental setup

*A*

_{0}=

*w*×

*h*=445×220 nm

^{2}and 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. The devices were fabricated using the CMOS production line at the IBM T. J. Watson Research Center [1

1. 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]

*λ*=1527 nm and

_{p}*λ*=1590 nm, respectively. The resulting pump has a pulse width of

_{s}*T*≈ 204 fs whereas the pulse width of the much weaker probe is

_{p}*T*≈ 170 fs; the pulse widths were measured by autocorrelation and cross-correlation, respectively. A delay line is used to vary the temporal spacing between the two pulses, which are then coupled into the waveguide with a free-space objective. The polarization directions of the two beams are aligned along the TE field direction of the waveguide. The output was 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 between the lens and the waveguide of ∼30 dB. The coupled power, however, is strong enough for the pump beam to modulate the probe through XPM. The coupled peak pump and probe pulse powers are estimated to be 20 mW and 10 μW, respectively. In addition, the propagation loss inside the waveguide has been previously characterized to be ∼3.5 dB/cm [1

_{s}1. 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]

## 3. Simulation model and XPM theory

### 3.1 Dispersion properties of the silicon photonic wire

*viz*. effective index

*n*, group index

_{eff}*n*, group-velocity dispersion coefficient, β

_{g}_{2}, and third-order dispersion coefficient, β

_{3}; these quantities are defined by

*n*=β

_{g}_{1}

*c*and β

_{m}= d

^{m}β

_{0}/dω

^{m}, where β

_{0}=

*n*(ω)ω/

_{eff}*c*and ω is the carrier frequency. We calculated

*n*using the RSoft BeamPROP software [31

_{eff}31. “BeamProp,” RSoftDesign, Inc., http://www.rsoftdesign.com.

*n*with a 7

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

*n*and β

_{g}_{2}. The dispersion coefficients, up to the second order, that are obtained by using this method agree with FEM calculations results within 0.1% of each other and with experimental results as well [3

3. 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 (2006). [CrossRef]

_{2}<0). The GVD coefficients are β

_{2,p}= -3.89 ps

^{2}/m at 1527 nm and β

_{2,s}= -4.26 ps

^{2}/m at 1590 nm. With regard to the calculation of β

_{3}, this is a more difficult problem because the accumulated and enlarged errors from each numerical derivative step prevent ready determination of the consecutive numerical derivatives above the second order; see Ref. [27

27. I. 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]

### 3.2 Simulation model

*P*are the pulse peak-powers,

_{p,s}*α*is the intrinsic loss,

_{in}*α*is the FCA coefficient, and

_{FC}^{p,s}*δn*is the FC-induced change of the refractive index. The parameters Γ

_{FC}^{p,s}_{S,P}and Γ

_{sp,ps}are defined as Γ

_{j}=

*A*

_{0}∫

**e**

^{*}

_{j}∙χ

^{(3)}⋮

**e**

_{j}

**e**

_{j}

^{ *}

_{j}

**e**

_{j}

*dA*/J

_{j}

^{2}, Γ

_{jl}=

*A*

_{0}∫

**e**

^{*}

_{l}∙ χ

^{(3)}⋮

**e**

_{j}

**e**

^{*}

_{j}

**e**

_{l}

*dA*/

*(*J

_{j}J

_{l}

*)*, (

*j,l*=

*p,s*), where J

_{p,s}= ∫

*n*

^{2}

*(*

**r**˔

*)*|

**e**

_{p,s}|

^{2}

*dA*and

**e**

_{p,s}=

**e**

_{p,s}

*(*ω

_{p,s};

**r**

*)*are the waveguide modes. To compute these parameters we used [2

2. 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]

32. M. Dinu, F. Quochi, and H. Garcia, “Third-order nonlinearities in silicon at telecom wavelengths,” Appl. Phys. Lett **82**,2954–2956 (2003). [CrossRef]

^{(3)}

_{1111}=(2.20+

*i*0.27)×10

^{-1 9}m

^{2}/V

^{2}and

*χ*

^{(3)}

_{1122}= (5.60 +1.82)×10

^{-20}m

^{2}/V

^{2}. Based on a Drude model,

*δn*and

_{FC}*α*are given by [33

_{FC}33. R. Soref and B. Bennett, “Electrooptical effects of silicon,” IEEE J. Quantum Electron **QE-23**123–129 (1987). [CrossRef]

*δn*= -

_{FC}*e*

^{2}(

*N*/

*m*

^{*}

_{ce}+

*N*

^{0.8}/

*m*

^{*}

_{ch})/2ε

_{0}

*nω*

^{2}and

*α*=

_{FC}*e*

^{3}

*N*(1/

*μ*

_{e}m^{*2}

_{ce}+1/

*μ*

_{h}m^{*2}

_{ch})/ε

_{0}

*cnω*

^{2}, where

*N*is the free-carrier density (in cm

^{-3});

*m*

^{*}

_{ce}=0.26

*m*

_{0}(

*m*

^{*}

_{ce}= 0.39

*m*

_{0}) is the effective mass of the electrons (holes) with

*m*

_{0}as the mass of the electron; and μ

_{e}(

*μ*) is the electron (hole) mobility.

_{h}*L*,

_{D}*L*′, and

_{D}*L*are comparable at peak powers of just a few mW. Specifically, for the pump used in our experiment,

_{NL}*T*=204 fs,

_{p}*L*∼10.7 mm and

_{D}*L*′ ∼11.6 mm, whereas the length

_{D}*L*, which depends on the peak power, has a similar value,

_{NL}*L*, = 9 mm, if

_{NL}*P*

_{0}= 5 mW. Consequently, at or above

*P*

_{0}∼ 5 mW, all GVD, TOD, and SPM effects must be incorporated in a complete description of the dynamics of the co-propagating pulses.

### 3.3 Nonlinearity-induced phase shift

*T*=

*t*-

*z*/

*v*is the time in the reference frame of the pump pulse,

_{g,p}*γ*= 3ω

_{s}_{s}Γ

_{s}/

*ε*

_{0}

*A*

_{0}

*v*

_{g,s}^{2}and

*γ*= 3

_{ps}*ω*Γ

_{s}_{ps}/

*ε*

_{0}

*A*

_{0}

*v*are the SPM and XPM coefficients, respectively, and ∆ = 1/

_{g,P}v_{g,S}*v*-1/

_{g,s}*v*is the temporal walk-off The calculated XPM coefficient,

_{g,p}*γ*= 2.18.10

_{ps}^{4}W

^{-1}m

^{-1}, is more than six orders of magnitude larger than the XPM coefficient of optical fibers,

*γ*=

_{fiber}*n*

_{2}ω/

*cA*∼3.10

_{eff}^{- 3}W

^{-1}m

^{-1}, so that a much stronger XPM interaction is expected in Si-WWGs. Note also that the nonlinear coefficients

*γ*have different expressions for fibers and Si-WWG, a fact that must be taken into account when the nonlinear properties of optical fibers and Si-WWG are compared; this difference can ultimately be traced to the large waveguide dispersion in Si wires and to their tensor nonlinear-optical properties [2

2. 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]

*γ*is inversely proportional to the modal area, which in Si wires is ∼ 10

^{3}× smaller than the modal area of optical fibers (∼ 0.1 μm

^{2}as compared to ∼ 100 μm

^{2}), as well as due to the much stronger third-order susceptibility of Si as compared to that of silica. In our experiment,

*v*= 6.63×10

_{g,p}^{7}m/s and

*v*= 6.54×10

_{g,s}^{7}m/s, which leads to a temporal walk-off of ∆ = 207.6 fs/mm. More importantly, the corresponding walk-off length is

*L*=

_{w}*T*/|∆| ∼ 1 mm, which is several times smaller than the waveguide length. As a result, the propagation length is large enough for the two pulses to pass through each other and thus experience a strong mutual interaction. Furthermore, in our experiment, the probe peak power is < 1 mW, and as a result, the SPM-induced phase shift of the probe can be neglected. Under these circumstances, the nonlinear phase shift of the probe is chiefly determined by XPM, i.e., by the second term in Eq. (2). This equation shows that XPM-induced phase shift is proportional to the peak power of the pump

_{p}*P*and the effective nonlinear coefficient of the waveguide, Γ

_{p}_{ps}. Straightforward calculations show that, if we assume that both pump and probe are Gaussian pulses, namely

*u*(0,

_{p}*T*) = exp[-(

*T*-

*T*)

_{d}^{2}/2

*T*

_{p}^{2}] and

*u*(0,

_{s}*T*)=exp(-

*T*

^{2}/2

*T*

_{s}

^{2}), then the XPM-induced phase shift in Eq. (2) can be written as

*T*/

*T*, τ

_{p}_{d}=

*T*/

_{d}*T*, and δ=

_{p}*z*∆/

*T*.

_{p}*T*is defined as the temporal separation between the maximum intensity points of these two pulses prior to their entry into the waveguide. In the convention we use here, this time delay is positive (negative) when the probe leads (trails) the pump. For our experiment, we are interested in extracting the phase information at the end of the waveguide, i.e., ϕ

_{d}_{s}(

*L*,τ). Henceforth, δ is evaluated at

*z*=

*L*, i.e., ϕ=

*L∆*/

*T*=

_{P}*L*/

*L*= 4.78. Finally, from Eq. (3) we can derive the frequency shift of the probe due to the XPM induced by the pump:

_{w}## 4. Experimental results and analysis

### 4.1 SPM of pump

_{3}, which then will be used in numerical simulations. In addition, when the pump is strong enough to induce SPM on itself, it also has enough optical power to induce a nonlinear phase shift onto the probe. Figure 2 shows the pump spectra normalized to the peak spectral power, measured for several pump powers. We observe that as we increase the pump power, new spectral components are generated, which represents a signature of SPM. For our XPM studies, we adjusted the coupled peak pump power inside the waveguide to be ∼ 20 mW so as to avoid spectral overlap between the pump and the probe due to this spectral broadening of the pump. This choice also ensures that the coupled-mode equations (1.a–c) remain valid. For example, if the spectra of the pump and probe pulses begin to overlap during the propagation, then the assumption that the optical field can be separated in two distinct pulses breaks down; in this scenario, the dynamics of the optical field is described instead by a single equation similar to Eq. (1.a).

_{max}≈ (

*M*-1/2)π, where

*M*is the number of peaks in the spectrum note that the second peak starts to form at

*P*=20 mW). To simulate SPM of the pump using our model, we first calculated numerically the coefficients β

_{p}_{1}, β

_{2}, and Γ and then determined the value of the TOD coefficient β

_{3}that led to the best fit between the experimental and numerical results that correspond to the propagation of a Gaussian input pulse with

*T*= 204 fs (for details, see [27

_{p}27. I. 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]

_{3,p}= -0.43 ± 0.05 ps

^{3}/m, as shown in Fig. 2(b). This result is comparable to the value of β

_{3}= -0.73 ± 0.05 ps

^{3}/m at 1537 nm [27

27. I. 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]

### 4.2 XPM pump/probe power dependence

*T*≈ 200 fs. As shown in Fig. 3, as we increase the peak pump power, the newly generated spectral components become stronger, which indicates that XPM increases with higher pump power.

_{d}*T*= 0. The spectra of the output pulses clearly show significant spectral broadening as the pump power is increased. Using Eq. (3), the maximum XPM-induced phase shift for peak pump power of 10 mW, 50 mW, and 100 mW are 0.76π, 3.8π, and 7.6π, or approximately 15 mW for a π phase shift. Note that this exact calculation agrees relatively well with the less rigorous estimate based on counting the number of peak splits in the output spectrum of the probe using the relation

_{d}*ϕ*≈ (

_{max}*M*–1/2)π. Moreover, as discussed earlier, the normalized output probe spectrum did not change with the probe power. As shown in Fig. 5(b), for a pump peak power of 100 mW, the probe spectrum is unchanged when the power of the probe is increased from 1 μW to 10 μW.

### 4.3 Dependence of the XPM on the pump-probe time-delay

*T*=

_{L}*L*∆ = 975.7 fs.

*T*= −1300 fs or

_{d}*T*= 2000 fs, the probe spectrum is unaffected by the pump pulse due to the lack of overlap between the two pulses. Indeed, in both cases |

_{d}*T*| >

_{d}*T*and therefore the pulses do not overlap during the propagation in the Si-WWG. As the time delay decreases from its maximum value of

_{L}*T*= 2000 fs, the probe develops a series of spectral modulations and shows an increased spectral asymmetry, both effects representing a clear signature of the XPM. This spectral distortions of the probe reach a maximum at

_{d}*T*∼ 1000 fs (in normalized units this corresponds to τ

_{d}_{d}∼ δ, where δ is calculated for

*z*=

*L*). Moreover, as the time delay further decreases these spectral modulations also decrease and almost vanish for

*T*∼ 500 fs, i.e., for τ

_{d}_{d}∼ δ/2. As the time delay further decreases we observe a reverse scenario, namely the spectral modulations again increases, reaching a maximum at

*T*∼ 0 and then decrease to zero as the time delay reaches large negative values. These results can be easily understood if we consider the frequency shift of the probe described by the Eq. (4). Thus, if τ

_{d}_{d}= δ most of the probe will experience a negative (positive) frequency (wavelength) shift whereas if τ

*=0 the probe experiences a positive (negative) frequency (wavelength) shift, with both these cases corresponding to a maximum absolute value of the frequency shift. In the intermediate case, i.e., when τ*

_{d}*= δ/2, the probe experiences a minimum frequency shift.*

_{d}*T*= 204 fs. However, as our measurement for the probe spectrum suggests, the probe is not really a simple Gaussian. More exactly, Fig. 7 shows that the spectrum of the probe is somewhat asymmetric and its temporal profile deviates from that of a Gaussian pulse. In order to determine the exact characteristics of the probe we model our temporal profile of the pulse as a sum of two Gaussians, with the dominant one possessing a small linear chirp

_{p}_{d}= δ and τ

_{d}= 0, whereas the minimum XPM-induced modulations of the probe is observed for large absolute values of the time delay, |τ

_{d}| ≫ 1, and for τ

_{d}= δ/2.

*T*=

_{d}*T*/2 no wavelength shift is observed, whereas at

_{L}*T*= 0 and

_{d}*T*= 850 fs (close to

_{d}*T*=

_{d}*T*) the XPM interaction induces a large nonlinear wavelength shift (more than 1 nm). In addition, the sign of this nonlinear wavelength shift (∆λ < 0 at τ

_{L}_{d}= 0 and (∆λ > 0 at τ

_{d}= δ) agrees with our analysis based on Eq. (4). Second, Fig. 8 shows that, unlike in optical-fiber experiments, the maximum value of the wavelength shift at τ

_{d}= 0 is larger than the maximum wavelength shift at τ

_{d}=δ. This result is explained by the fact that in the case in which τ

_{d}= 0 the pump-probe interaction takes place mostly near the input facet of the waveguide whereas when τ

_{d}= δ the pump interacts with the probe mostly near the output of the Si-WWG, i.e., after the pump has lost part of its optical power due to intrinsic and TPA losses. Another important observation is that the center wavelength is almost the same for

*T*= ±2000 fs, despite the fact that

_{d}*T*= 2000 fs is much less than the FC lifetime of 0.5 ns [27

_{d}**14**,12380–12387 (2006). [CrossRef] [PubMed]

_{d}= 0 is narrower than the one near τ

_{d}= δ, an effect that is attributable to the temporal broadening of both the pump and probe pulses due to dispersion and pump SPM.

*et al*. clearly demonstrates that Kerr-induced wavelength shifts of as much as >10 nm can be obtained, which are comparable to the spectral width of the input probe pulses [30

30. 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]

## 5. Conclusion

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37. A. Yariv, Y. Xu, R. K. Lee, and A scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett **24**,711–713 (1999). [CrossRef]

38. F. Xia, L. Sekaric, and Y. A. Vlasov, “Mode conversion losses in silicon-on-insulator photonic wire based racetrack resonators,” Opt. Express **14**,3872–3886 (2006) [CrossRef] [PubMed]

39. J. E. McMillan, X. D. Yang, N. C. Panoiu, R. M. Osgood, and C. W. Wong, “Enhanced stimulated Raman scattering in slow-light photonic crystal waveguides,” Opt. Lett **31**,1235–1237 (2006). [CrossRef] [PubMed]

## Acknowledgments

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19. | A. W. Fang, H. Park, O. Cohen, R. Jones, M. J. Paniccia, and J. E. Bowers, "“Electrically pumped hybrid AlGaInAs-silicon evanescent laser,” Opt. Express |

20. | R. Espinola, J. I. Dadap, R. Osgood Jr., S. J. McNab, and Y. A. Vlasov, “Raman amplification in ultrasmall silicon-on-insulator wire waveguides,” Opt. Express |

21. | J. I. Dadap, R. L. Espinola, R. M. Osgood, S. J. McNab, and Y. A. Vlasov, “Spontaneous Raman scattering in ultrasmall silicon waveguides,” Opt. Lett |

22. | R. L. Espinola, M.-C. Tai, J. T. Yardley, and R. M. Osgood, “Fast and low-power thermooptic switch on thin silicon-on-insulator,” IEEE Photon. Technol. Lett |

23. | M. W. Geis, S. J. Spector, R. C. Williamson, and T. M. Lyszczarz, “Submicrosecond, submilliwatt, silicon on insulator thermooptic switch,” IEEE Photon. Technol. Lett |

24. | O. Boyraz, T. Indukuri, and B. Jalali, “Self-phase-modulation induced spectral broadening in silicon waveguides,” Opt. Express |

25. | G. W. Rieger, K. S. Virk, and J. F. Young, “Nonlinear propagation of ultrafast 1.5 μm pulses in high-index-contrast silicon-on-insulator waveguides,” Appl. Phys. Lett |

26. | 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 |

27. | I. 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 |

28. | O. Boyraz, P. Koonath, V. Raghunathan, and B. Jalali, “All optical switching and continuum generation in silicon waveguides,” Opt. Express |

29. | I. Hsieh, X. Chen, J. I. Dadap, N. C. Panoiu, S. McNab, Y. A. Vlasov, and R. M. Osgood, “Cross-Phase Modulation in Si Photonic Wire Waveguides,” |

30. | 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 |

31. | “BeamProp,” RSoftDesign, Inc., http://www.rsoftdesign.com. |

32. | M. Dinu, F. Quochi, and H. Garcia, “Third-order nonlinearities in silicon at telecom wavelengths,” Appl. Phys. Lett |

33. | R. Soref and B. Bennett, “Electrooptical effects of silicon,” IEEE J. Quantum Electron |

34. | G. P. Agrawal, |

35. | G. P. Agrawal, P. L. Baldeck, and R. R. Alfano, “Temporal and spectral effects of cross-phase modulation on copropagating ultrashort pulses in optical fibers,” Phys. Rev. A |

36. | Y. A. Vlasov, M. O’Boyle, H. F. Hamann, and S. J. McNab, “Active control of slow light on a chip with photonic crystal waveguides,” Nature |

37. | A. Yariv, Y. Xu, R. K. Lee, and A scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett |

38. | F. Xia, L. Sekaric, and Y. A. Vlasov, “Mode conversion losses in silicon-on-insulator photonic wire based racetrack resonators,” Opt. Express |

39. | J. E. McMillan, X. D. Yang, N. C. Panoiu, R. M. Osgood, and C. W. Wong, “Enhanced stimulated Raman scattering in slow-light photonic crystal waveguides,” Opt. Lett |

**OCIS Codes**

(060.5060) Fiber optics and optical communications : Phase modulation

(190.4390) Nonlinear optics : Nonlinear optics, integrated optics

(230.7370) Optical devices : Waveguides

(320.7110) Ultrafast optics : Ultrafast nonlinear optics

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: December 7, 2006

Revised Manuscript: January 17, 2007

Manuscript Accepted: January 21, 2007

Published: February 5, 2007

**Citation**

I-Wei Hsieh, Xiaogang Chen, Jerry I. Dadap, Nicolae C. Panoiu, Richard M. Osgood, Jr., Sharee J. McNab, and Yurii 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)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-3-1135

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### References

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