Influence of three-photon absorption on Mid-infrared cross-phase modulation in silicon-on-sapphire waveguides

doi:10.1364/OE.21.001840

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Influence of three-photon absorption on Mid-infrared cross-phase modulation in silicon-on-sapphire waveguides

Zhaolu Wang, Hongjun Liu, Nan Huang, Qibing Sun, Jin Wen, and Xuefeng Li »View Author Affiliations

Optics Express, Vol. 21, Issue 2, pp. 1840-1848 (2013)
http://dx.doi.org/10.1364/OE.21.001840

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▸ Article Outline
– Abstract
1. Introduction
2. Simulation model and XPM theory for 2300 nm-3300 nm
3. Results and discussion
4. Conclusion
– References and links

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Abstract

The influence of three-photon absorption (3PA) on cross-phase modulation (XPM) effect in the mid-infrared (IR) region is theoretically investigated in silicon-on-sapphire (SOS) waveguides. It is found that the 3PA-induced nonlinear losses in the SOS waveguide will be considerable for the pulse propagation in the wavelength region of 2300 nm-3300 nm when the pump peak intensity is high enough. For the XPM process, the 3PA and 3PA-induced free-carrier effects can affect the spectrum and temporal profiles of the pump and signal pulses for sufficiently high pump peak intensities. Moreover, the XPM-induced frequency shift of signal spectrum is also discussed with different pump peak intensities, and the XPM-induced blue and red shifts are reduced due to 3PA.

1. Introduction

In recent years, the operating wavelength range has been extended to the mid- and long-wave-IR regions for silicon photonics [1

R. Soref, “Mid-infrared photonics in silicon and germanium,” Nat. Photonics 4(8), 495–497 (2010). [CrossRef]

–4

Z. Wang, H. Liu, N. Huang, Q. Sun, and J. Wen, “Efficient terahertz-wave generation via four-wave mixing in silicon membrane waveguides,” Opt. Express 20(8), 8920–8928 (2012). [CrossRef] [PubMed]

]. There are many potential mid-IR applications for silicon photonics including biochemical detection, environment monitoring, and free-space communications [1

R. Soref, “Mid-infrared photonics in silicon and germanium,” Nat. Photonics 4(8), 495–497 (2010). [CrossRef]

]. Although the nonlinear silicon photonic for near-IR applications are well known and are attracting great interest [5

J. Leuthold, C. Koos, and W. Freude, “Nonlinear silicon photonics,” Nat. Photonics 4(8), 535–544 (2010). [CrossRef]

–15

M. Zhu, H. Liu, X. Li, N. Huang, Q. Sun, J. Wen, and Z. Wang, “Ultrabroadband flat dispersion tailoring of dual-slot silicon waveguides,” Opt. Express 20(14), 15899–15907 (2012). [CrossRef] [PubMed]

], there are few studies investigating nonlinear silicon photonics in the mid-IR range [16

X. Liu, R. M. Osgood Jr, Y. A. Vlasov, and W. M. J. Green, “Mid-infrared optical parametric amplifier using silicon nanophotonic waveguides,” Nat. Photonics 4(8), 557–560 (2010). [CrossRef]

–25

O. Boyraz, Y. Huang, and X. Sang, “Silicon on sapphire and SOI photonic devices for mid-infrared and near-IR wavelengths,” Proc. SPIE 8431, 843104, 843104-12 (2012). [CrossRef]

]. In detail, Liu et al have realized mid-IR optical parametric amplification in silicon nanophotonic waveguides [16

X. Liu, R. M. Osgood Jr, Y. A. Vlasov, and W. M. J. Green, “Mid-infrared optical parametric amplifier using silicon nanophotonic waveguides,” Nat. Photonics 4(8), 557–560 (2010). [CrossRef]

], and Zlatanovic et al have achieved mid-IR wavelength conversion in silicon waveguides [17

S. Zlatanovic, J. S. Park, S. Moro, J. M. C. Boggio, I. B. Divliansky, N. Alic, S. Mookherjea, and S. Radic, “Mid-infrared wavelength conversion in silicon waveguides using ultracompact telecom-band-derived pump source,” Nat. Photonics 4(8), 561–564 (2010). [CrossRef]

]. The pumps of the above studies used operated near 2200 nm, corresponding to the wavelength spectrum beyond the onset of two photon absorption (TPA). Despite these progresses, there is still a strong motivation to investigate the influence of 3PA on nonlinear silicon photonics in the mid-IR region ranging from 2300 nm to 3300 nm, where the 3PA is the main source of nonlinear loss [26

S. Pearl, N. Rotenberg, and H. M. Driel, “Three photon absorption in silicon for 2300-3300 nm,” Appl. Phys. Lett. 93(13), 131102 (2008). [CrossRef]

]. Since some nonlinear processes need high pump peak intensities for mid-IR applications, the investigating of 3PA and 3PA-induced free-carrier effects in silicon waveguides becomes significant and crucial.

The XPM effect in silicon waveguides has many potential applications for making nonlinear optical devices such as pulse compressor [27

G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, 2007).

], optical switch [8

T. K. Liang, L. Nunes, T. Sakamoto, K. Sasagawa, T. Kawanishi, M. Tsuchiya, G. Priem, D. Van Thourhout, P. Dumon, R. Baets, and H. Tsang, “Ultrafast all-optical switching by cross-absorption modulation in silicon wire waveguides,” Opt. Express 13(19), 7298–7303 (2005). [CrossRef] [PubMed]

, 28

L. Yin, J. Zhang, P. M. Fauchet, and G. P. Agrawal, “Optical switching using nonlinear polarization rotation inside silicon waveguides,” Opt. Lett. 34(4), 476–478 (2009). [CrossRef] [PubMed]

, 29

J. Y. Lee, L. Yin, G. P. Agrawal, and P. M. Fauchet, “Ultrafast optical switching based on nonlinear polarization rotation in silicon waveguides,” Opt. Express 18(11), 11514–11523 (2010). [CrossRef] [PubMed]

], and wavelength converter [30

R. Dekker, A. Driessen, T. Wahlbrink, C. Moormann, J. Niehusmann, and M. Forst, “Ultrafast Kerr-induced all-optical wavelength conversion in silicon waveguides using 1.55 μm,” Opt. Express 14(18), 8336–8346 (2006). [CrossRef]

], since it allows to use a pump pulse to control a signal pulse at a different wavelength. Within the past few years, the XPM effect has been explored theoretically and experimentally in silicon waveguides-typically in the near-IR region [28

L. Yin, J. Zhang, P. M. Fauchet, and G. P. Agrawal, “Optical switching using nonlinear polarization rotation inside silicon waveguides,” Opt. Lett. 34(4), 476–478 (2009). [CrossRef] [PubMed]

–34

Y. Shoji, T. Ogasawara, T. Kamei, Y. Sakakibara, S. Suda, K. Kintaka, H. Kawashima, M. Okano, T. Hasama, H. Ishikawa, and M. Mori, “Ultrafast nonlinear effects in hydrogenated amorphous silicon wire waveguide,” Opt. Express 18(6), 5668–5673 (2010). [CrossRef] [PubMed]

]. In detail, Dekker et al have reported ultrafast all-optical wavelength conversion based on XPM in silicon waveguides using 1.55 μm femtosecond pulses, and they observed large Kerr-induced red shift of 9 nm and blue shift of 15 nm [30

]. Hsieh et al have investigated XPM-induced spectral and temporal effects with near-IR pump and signal pulses [31

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(3), 1135–1146 (2007). [CrossRef] [PubMed]

]. Astar et al have demonstrated the conversion of 10 Gb/s non-return-to-zero ON-OFF keying (NRZ-OOK) to RZ-OOK in the C-band using XPM effect in a silicon nanowire [32

W. Astar, J. B. Driscoll, X. Liu, J. I. Dadap, W. M. J. Green, Y. A. Vlasov, G. M. Carter, and R. M. Osgood, “Conversion of 10 Gb/s NRZ-OOK to RZ-OOK utilizing XPM in a Si nanowire,” Opt. Express 17(15), 12987–12999 (2009). [CrossRef] [PubMed]

]. Until now, there are few reports to investigate the XPM effect in the mid-IR for silicon waveguides, thus it will be very interesting to investigate the spectral and temporal effects induced by the XPM effect in silicon waveguides by taking into account 3PA and 3PA-induced free-carrier effects in the mid-IR wavelength range of 2300-3300 nm.

In this paper, we investigate the XPM effect in a SOS waveguide with mid-IR pump and signal pulses. This is a detailed report of 3PA and 3PA-induced free-carrier effects for mid-IR nonlinear effects in silicon waveguides. The influence of 3PA and 3PA-induced free-carrier effects on the spectral and temporal effects for the co-propagating mid-IR pump and signal pulses are investigated with different pump peak intensities. Furthermore, the XPM-induced frequency shift is also discussed for different signal center wavelengths.

2. Simulation model and XPM theory for 2300 nm-3300 nm

The silicon waveguide for the mid-IR applications is designed as a 1-cm-long straight SOS rib waveguide, since sapphire (Al₂O₃) is transparent from visible to 5.5 μm [3

E. K. Tien, Y. Huang, S. Gao, Q. Song, F. Qian, S. K. Kalyoncu, and O. Boyraz, “Discrete parametric band conversion in silicon for mid-infrared applications,” Opt. Express 18(21), 21981–21989 (2010). [CrossRef] [PubMed]

]. Figure 1 shows the dimension of the SOS rib waveguide with width W = 800 nm, height H = 700 nm, and etch depth h = 400 nm. The SOS waveguide is a multimode waveguide, and mode bifurcation may occur [35

A. R. Davoyan, I. V. Shadrivov, and Y. S. Kivshar, “Nonlinear plasmonic slot waveguides,” Opt. Express 16(26), 21209–21214 (2008). [CrossRef] [PubMed]

]. In the following analysis, we only consider the fundamental TM mode in the SOS waveguide, because the high-order modes do not play an important role in the parametric process and inverse taper used in each end of the waveguide can ensure only the fundamental mode excited [29

, 36

A. C. Turner, C. Manolatou, B. S. Schmidt, M. Lipson, M. A. Foster, J. E. Sharping, and A. L. Gaeta, “Tailored anomalous group-velocity dispersion in silicon channel waveguides,” Opt. Express 14(10), 4357–4362 (2006). [CrossRef] [PubMed]

]. To determine the performance of the waveguide, we need to simulate the mode profiles at mid-IR wavelengths. For the mid-IR XPM process, the pump pulse is assumed as 2300 nm and the signal pulse is tuned from 2400 nm to 2700 nm. Both of the pump and signal pulses are TM polarization, thus stimulated Raman scattering (SRS) cannot occur [37

L. Yin, Q. Lin, and G. P. Agrawal, “Soliton fission and supercontinuum generation in silicon waveguides,” Opt. Lett. 32(4), 391–393 (2007). [CrossRef] [PubMed]

, 38

Q. Lin, O. J. Painter, and G. P. Agrawal, “Nonlinear optical phenomena in silicon waveguides: Modeling and applications,” Opt. Express 15(25), 16604–16644 (2007). [CrossRef] [PubMed]

]. The fundamental TM mode profile of the SOS waveguide at the wavelength of 2300 nm is simulated using a finite-difference mode solver [39

T. E. Murphy, software available at http://www.photonics.umd.edu.

], and the E_y and E_x electric field components are shown in Fig. 1. It is clear that the SOS waveguide can support 90% confinement in silicon region at the wavelength of 2300 nm.

Fig. 1 Left: SOS rib waveguide structure; Right: Ey and Ex electric field components of the SOS waveguide at the wavelength of 2300 nm.

To describe the dynamics of the ultrashort pulse propagation, we first determined the waveguide dispersion properties. For the SOS rib waveguide, the TM mode effective indices n_eff are calculated using the finite-difference mode solver. The dispersion relation is then calculated from β(ω) = n_eff(ω)ω/c, where ω is the carrier frequency and c is the speed of light in vacuum. Higher order dispersion is finally calculated via numerical differentiation from β_n = dⁿβ/dωⁿ. The dispersion parameters n_eff, group index n_g, group-velocity dispersion (GVD) coefficient β₂, and third-order dispersion (TOD) coefficient β₃ of the SOS waveguide are shown in Fig. 2 , where n_g = β₁c. It is shown that a large anomalous dispersion region from 2200 nm to 2700 nm is obtained, and this dispersion profile is not suitable for phase matching because the pump wavelength is far from the zero-dispersion wavelength [40

Q. Lin, T. J. Johnson, R. Perahia, C. P. Michael, and O. J. Painter, “A proposal for highly tunable optical parametric oscillation in silicon micro-resonators,” Opt. Express 16(14), 10596–10610 (2008). [CrossRef] [PubMed]

]. Therefore, the in-band four-wave mixing effect can be neglected in the pulses propagation process in our analysis. In addition, if the wavelength spacing of the pump and signal is larger, which is not considered in our analysis, the phase matching of out-of-band discrete FWM should be investigated according to [3

Fig. 2 (a) Effective and group indices and (b) second and third-order dispersion of the SOS waveguide as a function of wavelength.

To describe the nonlinear optical interaction of the pump and signal in the SOS waveguide, we use the formulism described in [20

A. C. Peacock, “Mid-IR soliton compression in silicon optical fibers and fiber tapers,” Opt. Lett. 37(5), 818–820 (2012). [CrossRef] [PubMed]

, 27

G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, 2007).

] and take into account the effects of 3PA, free-carrier absorption (FCA), and free-carrier dispersion (FCD). The XPM process can be described by the following coupling equations:

\begin{matrix} \frac{\partial A_{p}}{\partial z} + \frac{i β_{2 p}}{2} \frac{\partial^{2} A_{p}}{\partial T^{2}} - \frac{β_{3 p}}{6} \frac{\partial^{3} A_{p}}{\partial T^{3}} = i γ_{p} ({| A_{p} |}^{2} + 2 {| A_{s} |}^{2}) A_{p} - \frac{β_{3 P A}}{2 A_{e f f}^{2}} {| A_{p} |}^{4} A_{p} \\ - \frac{1}{2} (α_{l} + α_{f c p}) A_{p} + i \frac{2 π}{λ_{p}} Δ n_{f c p} A_{p}, \end{matrix}

(1)

\begin{matrix} \frac{\partial A_{s}}{\partial z} + d \frac{\partial A_{s}}{\partial T} + \frac{i β_{2 s}}{2} \frac{\partial^{2} A_{s}}{\partial T^{2}} - \frac{β_{3 s}}{6} \frac{\partial^{3} A_{s}}{\partial T^{3}} = i γ_{s} ({| A_{s} |}^{2} + 2 {| A_{p} |}^{2}) A_{s} - \frac{β_{3 P A}}{2 A_{e f f}^{2}} {| A_{s} |}^{4} A_{s} \\ - \frac{1}{2} (α_{l} + α_{f c s}) A_{s} + i \frac{2 π}{λ_{s}} Δ n_{f c s} A_{s}, \end{matrix}

(2)

where A_j is the slowly varying amplitude (j = p, s), and z is the propagation distance. T = t-z/v_gp is the time in the reference frame of the pump pulse traveling at speed v_gp, d = β₁_s-β₁_p is the temporal walk-off parameter, and β_3PA is the coefficient of 3PA. The nonlinear coefficient γ_j = ω_jn₂/cA_eff is the effective nonlinearity of the waveguide, where A_eff is the effective area of the propagating mode and n₂ is the nonlinear index coefficient. Here, we assume n₂ = 5 × 10⁻¹⁸ m²W⁻¹ [3

, 41

A. D. Bristow, N. Rotenberg, and H. M. van Driel, “Two-photon absorption and Kerr coefficients of silicon for 850-2200 nm,” Appl. Phys. Lett. 90(19), 191104 (2007). [CrossRef]

], β_3PA = 2.5 × 10⁻²⁶ m³W⁻² [26

S. Pearl, N. Rotenberg, and H. M. Driel, “Three photon absorption in silicon for 2300-3300 nm,” Appl. Phys. Lett. 93(13), 131102 (2008). [CrossRef]

], and the A_eff is calculated as 0.45 μm². The parameter α_l represents the linear propagation losses of the waveguide, which is assumed as 1 dB/cm for 2300 nm-2700 nm [42

F. Li, S. D. Jackson, C. Grillet, E. Magi, D. Hudson, S. J. Madden, Y. Moghe, C. O’Brien, A. Read, S. G. Duvall, P. Atanackovic, B. J. Eggleton, and D. J. Moss, “Low propagation loss silicon-on-sapphire waveguides for the mid-infrared,” Opt. Express 19(16), 15212–15220 (2011). [CrossRef] [PubMed]

]. The α_fcj and Δn_fcj represent the absorption and index change induced by free carriers, respectively. Based on a Drude model, the α_fc and Δn_fc are given by α_fc = q³N_c(1/μ_em*²_e + 1/μ_hm*²_h)/ε₀cnω² and Δn_fc = -q²N_c(1/m*_e + 1/m*_h)/ 2ε₀nω², where m*_e = 0.26 m₀ (m*_h = 0.39 m₀) is the effective mass of the electrons (holes), and μ_e (μ_h) is the electron (hole) mobility [38

Q. Lin, O. J. Painter, and G. P. Agrawal, “Nonlinear optical phenomena in silicon waveguides: Modeling and applications,” Opt. Express 15(25), 16604–16644 (2007). [CrossRef] [PubMed]

]. The free-carrier density N_c caused by 3PA can be obtained by the following equation [43

P. Colman, C. Husko, S. Combrie, I. Sagnes, C. W. Wong, and A. D. Rossi, “Temporal solitons and pulse compression in photonic crystal waveguides,” Nat. Photonics 4(12), 862–868 (2010). [CrossRef]

\frac{\partial N_{c}}{\partial t} = \frac{β_{3 P A}}{3 h υ_{p}} \frac{{| A_{p} |}^{6}}{A_{_{}^{}}_{e f f}^{3}} - \frac{N_{c}}{τ_{c}},

(3)

where h is Planck’s constant, υ_p is the pump frequency, and the carrier lifetime is τ_c≈5 ns [30

]. Here, free carriers induced by the signal are negligible compared with that induced by the pump as the pump intensity is assumed much larger than that of the signal in this paper.

The XPM process is numerically studied by injecting pump pulses centered at 2300 nm and tunable signal pulses with same pulse width T_FWHM = 2 ps (T₀ = T_FWHM/1.665) and same repetition rate. Here, we assume that the repetition rate is very low (<100 MHz), which means that single pump and signal pulses can be focused on, because free carriers generated during a pulse have sufficient time to recombine before the next pulse arrives. Both of the pulses are Gaussian pulses, which are shown as [27

G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, 2007).

A_{p} (0, T) = \sqrt{P_{p}} \exp [- \frac{{(T - T_{d})}^{2}}{2 T_{0}^{2}}], A_{s} (0, T) = \sqrt{P_{s}} \exp [- \frac{T^{2}}{2 T_{0}^{2}}],

(4)

where P_p and P_s are the peak power of the input pump and signal pulses, and T_d is the initial time delay of the pump and signal. Then the XPM-induced phase shift of the signal pulse can be written as the following equation [27

G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, 2007).

, 31

ϕ_{s} (L, τ) = \frac{γ_{s} P_{p} L \sqrt{π}}{δ} [e r f (τ - τ_{d} + δ) - e r f (τ - τ_{d})],

(5)

where τ = T/T₀, τ_d = T_d/T₀, δ = dL/T₀, and L is the waveguide length. And then, we can get the pump-induced frequency chirp imposed on the signal pulse:

δ υ_{s} (L, τ) = - \frac{1}{2 π} \frac{\partial ϕ_{s}}{\partial T} = - \frac{γ_{s} P_{p} L}{π T_{0} δ} {\exp [- {(τ - τ_{d} + δ)}^{2}] - \exp [- {(τ - τ_{d})}^{2}]} .

(6)

According to Eq. (6), the XPM-induced chirps for τ_d = 0 and τ_d = δ are shown in Fig. 3 , where the value of the chirp has been normalized. The input signal center wavelength is 2400 nm, and the walk-off parameter d = β₁_s-β₁_p = 140 fs/mm, where β₁_s = 1.355 × 10⁴ ps/m, and β₁_p = 1.341 × 10⁴ ps/m calculated from Fig. 2. The walk-off length L_w = T₀/d = 8.58 mm, and δ = 1.1655.

Fig. 3 The XPM-induced frequency chirp for a signal pulse with τ_d = 0 and τ_d = δ, respectively. The input center wavelength of the signal pulse is 2400 nm, and signal pulse shape is shown by a dashed line.

3. Results and discussion

In simulations, the signal peak intensity coupled inside the 1-cm-long SOS waveguide is kept constant at 2.22 MW/cm², while the input pump peak intensity is much larger than the signal peak intensity. Thus the pump pulse is mainly affected by the self-phase modulation (SPM) effect, and the XPM effect induced by the signal pulse imposed on the pump can be neglected. Figure 4 shows the output pump peak intensity versus input pump peak intensity. In order to guarantee the veracity for researching the 3PA effect with input and output pump peak intensity, pulse distortion (compression or broadening), which can be induced by SPM and dispersion effects, should be avoided. Thus, the GVD and TOD are not considered in the simulation. From Fig. 4, it is clear that the output pump peak intensity scales linearly with the input pump peak intensity when the 3PA effect is neglected. However, when the 3PA effect is included in the simulation, the output pump peak intensity appears saturation effect for higher input pump peak intensity (> 2 GW/cm²). After considering FCA, the output pump peak intensity is only reduced a little, which indicates that the optical-limiting is mainly attributed to 3PA not FCA. This validates that the 3PA effect is the leading-order nonlinear loss for mid-IR applications when the input pump peak intensity is high enough [18

X. Liu, J. B. Driscoll, J. I. Dadap, R. M. Osgood Jr, S. Assefa, Y. A. Vlasov, and W. M. Green, “Self-phase modulation and nonlinear loss in silicon nanophotonic wires near the mid-infrared two-photon absorption edge,” Opt. Express 19(8), 7778–7789 (2011). [CrossRef] [PubMed]

]. In the following part, we will investigate the impact of 3PA and 3PA-induced free-carrier effects on the spectral and temporal effects in this mid-IR XPM process by taking into account GVD and TOD.

Fig. 4 Output pump peak intensity versus input pump peak intensity.

Figure 5 depicts the output spectra of the pump pulse with corresponding temporal profiles for different pump peak intensities, and the spectra are largely broadened due to SPM when the 3PA is ignored. When the pump peak intensity is 2.22 GW/cm², the 3PA only induces a little change of the pump spectrum, and the FCA and FCD effects can be ignored as illustrated in Fig. 5. This indicates that the 3PA does not play an important role and the nonlinear losses induced by 3PA and FCA can be neglected when the pump peak intensity is less than 2.22 GW/cm². With the increase of the pump peak intensity, the 3PA will gradually become the leading-order source of the nonlinear loss according to Fig. 4. It is clear that the spectral width can be reduced due to 3PA effect, while the free-carrier effects (including FCA and FCD) induce spectrum asymmetric for I₀ = 6.67 GW/cm². It is also found that a little blue shift is generated due to FCD [44

L. Yin and G. P. Agrawal, “Impact of two-photon absorption on self-phase modulation in silicon waveguides,” Opt. Lett. 32(14), 2031–2033 (2007). [CrossRef] [PubMed]

]. Corresponding to the temporal profile, the loss of the pulse in the trailing edge is larger than the leading edge due to the accumulation of the free carriers along the time. When the pump peak intensity is increased to 8.89 GW/cm², the spectrum asymmetric induced by free-carrier effects become larger, and the FCD-induced blue shift is distinct. Therefore, when the pump peak intensity is high enough, the 3PA and 3PA-induced free-carrier effects in the Mid-IR region have the same impact on the spectrum and temporal profiles of the pump as the TPA and TPA induced free-carrier effects in the near-IR region with a relatively low intensity [44

L. Yin and G. P. Agrawal, “Impact of two-photon absorption on self-phase modulation in silicon waveguides,” Opt. Lett. 32(14), 2031–2033 (2007). [CrossRef] [PubMed]

Fig. 5 Output spectra of the pump pulse with corresponding temporal profiles for different pump peak intensities under different conditions: 3PA = 0, 3PA only, 3PA + FCA, and full (3PA + FCA + FCD).

The center wavelength of the signal pulse is assumed longer than 2300 nm, hence the pump pulse travels faster than the signal pulse due to smaller n_g for pump from Fig. 2. According to Eq. (6) and Fig. 3, the XPM induces a blue shift of the center wavelength on the signal spectrum for τ_d = 0, and induces a red shift of the center wavelength on the signal spectrum for τ_d = δ. Figure 6 shows the output spectra and temporal profiles of the signal pulse for I₀ = 6.67 GW/cm², when the input signal center wavelength is 2400 nm. From Fig. 6, it is clear that the 3PA-induced nonlinear loss can be ignored for the signal pulses due to the low intensity of the signal pulse. When τ_d = 0, the free-carrier effects are relatively large for the signal pulse, because the slow-moving signal pulse interacts mainly with the trailing edge of the pump pulse that means the signal pulse affected by a relatively high carrier density. However, when τ_d = δ, the signal pulse interacts mainly with the leading edge of the pump pulse, which means low free-carrier effects and the FCA and FCD are negligible. After considering 3PA, the red and blue shifts are all reduced for the two case of τ_d = 0 and τ_d = δ as shown in Fig. 6, because the 3PA-induced nonlinear losses reduce the intensity of the pump, thus lower the XPM effect. When all the effects are included, the blue shift is reduced to 14 nm for τ_d = 0 and the red shift is reduced to 12 nm for τ_d = δ, that is because the FCD-induced blue shift also supply a contribution to the frequency shift for τ_d = 0 as shown in Fig. 6. In conclusion, the XPM-induced frequency shift is reduced due to 3PA for a relatively high pump intensity.

Fig. 6 Output spectra and temporal profiles of the signal pulse under different conditions for τ_d = 0 and τ_d = δ, respectively.

After taking into account the 3PA, FCA and FCD effects in the XPM process, the output spectra of the signal pulse with different pump peak intensities are shown in Fig. 7 for τ_d = 0 and τ_d = δ, respectively. It is found that the frequency shift of the center wavelength (2400 nm) become larger as the pump intensity increases. However, the frequency shift appears saturation when the pump peak intensity exceeds 4.44 GW/cm². Hence the increasing nonlinear losses induced by 3PA imposed on pump pulse suppress the XPM-induced frequency shift as the pump peak intensity increases, because larger nonlinear losses induced by the 3PA reduce the pump intensity, and then lower the pump-induced XPM effect. Furthermore, the FCD-induced blue shift enhance the XPM-induced blue shift for τ_d = 0 as illustrated in Fig. 7, where the maximal blue shift is 18 nm for τ_d = 0, while the maximal red shift is only 16 nm for τ_d = δ for I₀ = 11.11 GW/cm².

Fig. 7 Output spectra with different pump peak intensities for τ_d = 0 and τ_d = δ, respectively.

When the input signal center wavelengths are 2400 nm, 2500 nm, 2600 nm and 2700 nm, the output spectra are shown in Fig. 8 with the input pump peak intensity of 6.67 GW/cm², respectively. The spectra are shifted to the blue part due to τ_d = 0, and the blue shifts are 14 nm, 10 nm, 7 nm and 5 nm, respectively. It is clear that the blue shift and the number of peaks of the output signal spectrum are reduced as the input signal center wavelength increases. According to Fig. 2, n_g is increased with the increase of the signal wavelength, which means a large difference of group velocity between the pump and signal pulses. Thus, the walk-off length become shorter as the wavelength difference between the pump and signal increases, which means that the interaction time between the pump and signal decreases and the XPM effect becomes weaker. Therefore, the frequency shift is reduced with the increase of signal center wavelength due to pulse walk-off.

Fig. 8 Output spectra with different input signal center wavelengths when the pump peak intensity is 6.67 GW/cm².

4. Conclusion

The complete simulation model in the mid-IR region by taking into account 3PA, FCA and FCD allows us to show clearly the importance of 3PA effect for the XPM in a SOS waveguide. For a sufficiently high pump intensity, the 3PA and 3PA-induced free-carrier effects in the Mid-IR region have the same impact on the spectrum and temporal profiles of the pump pulse as the TPA and TPA-induced free-carrier effects with a relatively low pump peak intensity in the near-IR. Moreover, the XPM-induced frequency shift of the signal spectrum is reduced due to 3PA.

Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grant 61078029, 61178023 and 61275134.

References and links

1.	R. Soref, “Mid-infrared photonics in silicon and germanium,” Nat. Photonics 4(8), 495–497 (2010). [CrossRef]
2.	R. A. Soref, S. J. Emelett, and W. R. Buchwald, “Silicon waveguided components for the long-wave infrared region,” J. Opt. A, Pure Appl. Opt. 8(10), 840–848 (2006). [CrossRef]
3.	E. K. Tien, Y. Huang, S. Gao, Q. Song, F. Qian, S. K. Kalyoncu, and O. Boyraz, “Discrete parametric band conversion in silicon for mid-infrared applications,” Opt. Express 18(21), 21981–21989 (2010). [CrossRef] [PubMed]
4.	Z. Wang, H. Liu, N. Huang, Q. Sun, and J. Wen, “Efficient terahertz-wave generation via four-wave mixing in silicon membrane waveguides,” Opt. Express 20(8), 8920–8928 (2012). [CrossRef] [PubMed]
5.	J. Leuthold, C. Koos, and W. Freude, “Nonlinear silicon photonics,” Nat. Photonics 4(8), 535–544 (2010). [CrossRef]
6.	H. Rong, R. Jones, A. Liu, O. Cohen, D. Hak, A. Fang, and M. Paniccia, “A continuous-wave Raman silicon laser,” Nature 433(7027), 725–728 (2005). [CrossRef] [PubMed]
7.	V. Raghunathan, R. Claps, D. Dimitropoulos, and B. Jalali, “Parametric Raman wavelength conversion in scaled silicon waveguides,” J. Lightwave Technol. 23(6), 2094–2102 (2005). [CrossRef]
8.	T. K. Liang, L. Nunes, T. Sakamoto, K. Sasagawa, T. Kawanishi, M. Tsuchiya, G. Priem, D. Van Thourhout, P. Dumon, R. Baets, and H. Tsang, “Ultrafast all-optical switching by cross-absorption modulation in silicon wire waveguides,” Opt. Express 13(19), 7298–7303 (2005). [CrossRef] [PubMed]
9.	I.-W. Hsieh, X. Chen, J. I. Dadap, N. C. Panoiu, R. M. Osgood Jr, S. J. McNab, and Y. A. Vlasov, “Ultrafast-pulse self-phase modulation and third-order dispersion in Si photonic wire-waveguides,” Opt. Express 14(25), 12380–12387 (2006). [CrossRef] [PubMed]
10.	M. A. Foster, A. C. Turner, J. E. Sharping, B. S. Schmidt, M. Lipson, and A. L. Gaeta, “Broad-band optical parametric gain on a silicon photonic chip,” Nature 441(7096), 960–963 (2006). [CrossRef] [PubMed]
11.	J. Wen, H. Liu, N. Huang, Q. Sun, and W. Zhao, “Influence of the initial chirp on the supercontinuum generation in silicon-on-insulator waveguide,” Appl. Phys. B 104(4), 867–871 (2011). [CrossRef]
12.	Z. Wang, H. Liu, N. Huang, Q. Sun, and J. Wen, “Influence of spectral broadening on femtosecond wavelength conversion based on four-wave mixing in silicon waveguides,” Appl. Opt. 50(28), 5430–5436 (2011). [CrossRef] [PubMed]
13.	Z. Wang, H. Liu, N. Huang, Q. Sun, and J. Wen, “Impact of dispersion profiles of silicon waveguides on optical parametric amplification in the femtosecond regime,” Opt. Express 19(24), 24730–24737 (2011). [CrossRef] [PubMed]
14.	J. Wen, H. Liu, N. Huang, Q. Sun, and W. Zhao, “Widely tunable femtosecond optical parametric oscillator based on silicon-on-insulator waveguide,” Opt. Express 20, 3490–3498 (2012).
15.	M. Zhu, H. Liu, X. Li, N. Huang, Q. Sun, J. Wen, and Z. Wang, “Ultrabroadband flat dispersion tailoring of dual-slot silicon waveguides,” Opt. Express 20(14), 15899–15907 (2012). [CrossRef] [PubMed]
16.	X. Liu, R. M. Osgood Jr, Y. A. Vlasov, and W. M. J. Green, “Mid-infrared optical parametric amplifier using silicon nanophotonic waveguides,” Nat. Photonics 4(8), 557–560 (2010). [CrossRef]
17.	S. Zlatanovic, J. S. Park, S. Moro, J. M. C. Boggio, I. B. Divliansky, N. Alic, S. Mookherjea, and S. Radic, “Mid-infrared wavelength conversion in silicon waveguides using ultracompact telecom-band-derived pump source,” Nat. Photonics 4(8), 561–564 (2010). [CrossRef]
18.	X. Liu, J. B. Driscoll, J. I. Dadap, R. M. Osgood Jr, S. Assefa, Y. A. Vlasov, and W. M. Green, “Self-phase modulation and nonlinear loss in silicon nanophotonic wires near the mid-infrared two-photon absorption edge,” Opt. Express 19(8), 7778–7789 (2011). [CrossRef] [PubMed]
19.	B. Kuyken, X. Liu, R. M. Osgood Jr, R. Baets, G. Roelkens, and W. M. Green, “Mid-infrared to telecom-band supercontinuum generation in highly nonlinear silicon-on-insulator wire waveguides,” Opt. Express 19(21), 20172–20181 (2011). [CrossRef] [PubMed]
20.	A. C. Peacock, “Mid-IR soliton compression in silicon optical fibers and fiber tapers,” Opt. Lett. 37(5), 818–820 (2012). [CrossRef] [PubMed]
21.	F. Li, S. D. Jackson, C. Grillet, E. Magi, D. Hudson, S. J. Madden, Y. Moghe, C. O’Brien, A. Read, S. G. Duvall, P. Atanackovic, B. J. Eggleton, and D. J. Moss, “Low propagation loss silicon-on-sapphire waveguides for the mid-infrared,” Opt. Express 19(16), 15212–15220 (2011). [CrossRef] [PubMed]
22.	T. Baehr-Jones, A. Spott, R. Ilic, A. Spott, B. Penkov, W. Asher, and M. Hochberg, “Silicon-on-sapphire integrated waveguides for the mid-infrared,” Opt. Express 18(12), 12127–12135 (2010). [CrossRef] [PubMed]
23.	Y. Huang, E. K. Tien, S. Gao, S. K. Kalyoncu, Q. Song, F. Qian, E. Adas, D. Yildirim, and O. Boyraz, “Electrical signal-to-noise ratio improvement in indirect detection of mid-IR signals by wavelength conversion in silicon-on-sapphire waveguides,” Appl. Phys. Lett. 99(18), 181122 (2011). [CrossRef]
24.	Z. Cheng, X. Chen, C. Y. Wong, K. Xu, C. K. Y. Fung, Y. M. Chen, and H. K. Tsang, “Mid-Infrared Grating Couplers for Silicon-on-Sapphire Waveguides,” IEEE Photon. J. 4, 104–113 (2012).
25.	O. Boyraz, Y. Huang, and X. Sang, “Silicon on sapphire and SOI photonic devices for mid-infrared and near-IR wavelengths,” Proc. SPIE 8431, 843104, 843104-12 (2012). [CrossRef]
26.	S. Pearl, N. Rotenberg, and H. M. Driel, “Three photon absorption in silicon for 2300-3300 nm,” Appl. Phys. Lett. 93(13), 131102 (2008). [CrossRef]
27.	G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, 2007).
28.	L. Yin, J. Zhang, P. M. Fauchet, and G. P. Agrawal, “Optical switching using nonlinear polarization rotation inside silicon waveguides,” Opt. Lett. 34(4), 476–478 (2009). [CrossRef] [PubMed]
29.	J. Y. Lee, L. Yin, G. P. Agrawal, and P. M. Fauchet, “Ultrafast optical switching based on nonlinear polarization rotation in silicon waveguides,” Opt. Express 18(11), 11514–11523 (2010). [CrossRef] [PubMed]
30.	R. Dekker, A. Driessen, T. Wahlbrink, C. Moormann, J. Niehusmann, and M. Forst, “Ultrafast Kerr-induced all-optical wavelength conversion in silicon waveguides using 1.55 μm,” Opt. Express 14(18), 8336–8346 (2006). [CrossRef]
31.	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(3), 1135–1146 (2007). [CrossRef] [PubMed]
32.	W. Astar, J. B. Driscoll, X. Liu, J. I. Dadap, W. M. J. Green, Y. A. Vlasov, G. M. Carter, and R. M. Osgood, “Conversion of 10 Gb/s NRZ-OOK to RZ-OOK utilizing XPM in a Si nanowire,” Opt. Express 17(15), 12987–12999 (2009). [CrossRef] [PubMed]
33.	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(9), 9613–9621 (2010). [CrossRef] [PubMed]
34.	Y. Shoji, T. Ogasawara, T. Kamei, Y. Sakakibara, S. Suda, K. Kintaka, H. Kawashima, M. Okano, T. Hasama, H. Ishikawa, and M. Mori, “Ultrafast nonlinear effects in hydrogenated amorphous silicon wire waveguide,” Opt. Express 18(6), 5668–5673 (2010). [CrossRef] [PubMed]
35.	A. R. Davoyan, I. V. Shadrivov, and Y. S. Kivshar, “Nonlinear plasmonic slot waveguides,” Opt. Express 16(26), 21209–21214 (2008). [CrossRef] [PubMed]
36.	A. C. Turner, C. Manolatou, B. S. Schmidt, M. Lipson, M. A. Foster, J. E. Sharping, and A. L. Gaeta, “Tailored anomalous group-velocity dispersion in silicon channel waveguides,” Opt. Express 14(10), 4357–4362 (2006). [CrossRef] [PubMed]
37.	L. Yin, Q. Lin, and G. P. Agrawal, “Soliton fission and supercontinuum generation in silicon waveguides,” Opt. Lett. 32(4), 391–393 (2007). [CrossRef] [PubMed]
38.	Q. Lin, O. J. Painter, and G. P. Agrawal, “Nonlinear optical phenomena in silicon waveguides: Modeling and applications,” Opt. Express 15(25), 16604–16644 (2007). [CrossRef] [PubMed]
39.	T. E. Murphy, software available at http://www.photonics.umd.edu.
40.	Q. Lin, T. J. Johnson, R. Perahia, C. P. Michael, and O. J. Painter, “A proposal for highly tunable optical parametric oscillation in silicon micro-resonators,” Opt. Express 16(14), 10596–10610 (2008). [CrossRef] [PubMed]
41.	A. D. Bristow, N. Rotenberg, and H. M. van Driel, “Two-photon absorption and Kerr coefficients of silicon for 850-2200 nm,” Appl. Phys. Lett. 90(19), 191104 (2007). [CrossRef]
42.	F. Li, S. D. Jackson, C. Grillet, E. Magi, D. Hudson, S. J. Madden, Y. Moghe, C. O’Brien, A. Read, S. G. Duvall, P. Atanackovic, B. J. Eggleton, and D. J. Moss, “Low propagation loss silicon-on-sapphire waveguides for the mid-infrared,” Opt. Express 19(16), 15212–15220 (2011). [CrossRef] [PubMed]
43.	P. Colman, C. Husko, S. Combrie, I. Sagnes, C. W. Wong, and A. D. Rossi, “Temporal solitons and pulse compression in photonic crystal waveguides,” Nat. Photonics 4(12), 862–868 (2010). [CrossRef]
44.	L. Yin and G. P. Agrawal, “Impact of two-photon absorption on self-phase modulation in silicon waveguides,” Opt. Lett. 32(14), 2031–2033 (2007). [CrossRef] [PubMed]

OCIS Codes
(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: September 27, 2012
Revised Manuscript: December 11, 2012
Manuscript Accepted: December 30, 2012
Published: January 17, 2013

Citation
Zhaolu Wang, Hongjun Liu, Nan Huang, Qibing Sun, Jin Wen, and Xuefeng Li, "Influence of three-photon absorption on Mid-infrared cross-phase modulation in silicon-on-sapphire waveguides," Opt. Express 21, 1840-1848 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-2-1840

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References

R. Soref, “Mid-infrared photonics in silicon and germanium,” Nat. Photonics4(8), 495–497 (2010). [CrossRef]
R. A. Soref, S. J. Emelett, and W. R. Buchwald, “Silicon waveguided components for the long-wave infrared region,” J. Opt. A, Pure Appl. Opt.8(10), 840–848 (2006). [CrossRef]
E. K. Tien, Y. Huang, S. Gao, Q. Song, F. Qian, S. K. Kalyoncu, and O. Boyraz, “Discrete parametric band conversion in silicon for mid-infrared applications,” Opt. Express18(21), 21981–21989 (2010). [CrossRef] [PubMed]
Z. Wang, H. Liu, N. Huang, Q. Sun, and J. Wen, “Efficient terahertz-wave generation via four-wave mixing in silicon membrane waveguides,” Opt. Express20(8), 8920–8928 (2012). [CrossRef] [PubMed]
J. Leuthold, C. Koos, and W. Freude, “Nonlinear silicon photonics,” Nat. Photonics4(8), 535–544 (2010). [CrossRef]
H. Rong, R. Jones, A. Liu, O. Cohen, D. Hak, A. Fang, and M. Paniccia, “A continuous-wave Raman silicon laser,” Nature433(7027), 725–728 (2005). [CrossRef] [PubMed]
V. Raghunathan, R. Claps, D. Dimitropoulos, and B. Jalali, “Parametric Raman wavelength conversion in scaled silicon waveguides,” J. Lightwave Technol.23(6), 2094–2102 (2005). [CrossRef]
T. K. Liang, L. Nunes, T. Sakamoto, K. Sasagawa, T. Kawanishi, M. Tsuchiya, G. Priem, D. Van Thourhout, P. Dumon, R. Baets, and H. Tsang, “Ultrafast all-optical switching by cross-absorption modulation in silicon wire waveguides,” Opt. Express13(19), 7298–7303 (2005). [CrossRef] [PubMed]
I.-W. Hsieh, X. Chen, J. I. Dadap, N. C. Panoiu, R. M. Osgood, S. J. McNab, and Y. A. Vlasov, “Ultrafast-pulse self-phase modulation and third-order dispersion in Si photonic wire-waveguides,” Opt. Express14(25), 12380–12387 (2006). [CrossRef] [PubMed]
M. A. Foster, A. C. Turner, J. E. Sharping, B. S. Schmidt, M. Lipson, and A. L. Gaeta, “Broad-band optical parametric gain on a silicon photonic chip,” Nature441(7096), 960–963 (2006). [CrossRef] [PubMed]
J. Wen, H. Liu, N. Huang, Q. Sun, and W. Zhao, “Influence of the initial chirp on the supercontinuum generation in silicon-on-insulator waveguide,” Appl. Phys. B104(4), 867–871 (2011). [CrossRef]
Z. Wang, H. Liu, N. Huang, Q. Sun, and J. Wen, “Influence of spectral broadening on femtosecond wavelength conversion based on four-wave mixing in silicon waveguides,” Appl. Opt.50(28), 5430–5436 (2011). [CrossRef] [PubMed]
Z. Wang, H. Liu, N. Huang, Q. Sun, and J. Wen, “Impact of dispersion profiles of silicon waveguides on optical parametric amplification in the femtosecond regime,” Opt. Express19(24), 24730–24737 (2011). [CrossRef] [PubMed]
J. Wen, H. Liu, N. Huang, Q. Sun, and W. Zhao, “Widely tunable femtosecond optical parametric oscillator based on silicon-on-insulator waveguide,” Opt. Express20, 3490–3498 (2012).
M. Zhu, H. Liu, X. Li, N. Huang, Q. Sun, J. Wen, and Z. Wang, “Ultrabroadband flat dispersion tailoring of dual-slot silicon waveguides,” Opt. Express20(14), 15899–15907 (2012). [CrossRef] [PubMed]
X. Liu, R. M. Osgood, Y. A. Vlasov, and W. M. J. Green, “Mid-infrared optical parametric amplifier using silicon nanophotonic waveguides,” Nat. Photonics4(8), 557–560 (2010). [CrossRef]
S. Zlatanovic, J. S. Park, S. Moro, J. M. C. Boggio, I. B. Divliansky, N. Alic, S. Mookherjea, and S. Radic, “Mid-infrared wavelength conversion in silicon waveguides using ultracompact telecom-band-derived pump source,” Nat. Photonics4(8), 561–564 (2010). [CrossRef]
X. Liu, J. B. Driscoll, J. I. Dadap, R. M. Osgood, S. Assefa, Y. A. Vlasov, and W. M. Green, “Self-phase modulation and nonlinear loss in silicon nanophotonic wires near the mid-infrared two-photon absorption edge,” Opt. Express19(8), 7778–7789 (2011). [CrossRef] [PubMed]
B. Kuyken, X. Liu, R. M. Osgood, R. Baets, G. Roelkens, and W. M. Green, “Mid-infrared to telecom-band supercontinuum generation in highly nonlinear silicon-on-insulator wire waveguides,” Opt. Express19(21), 20172–20181 (2011). [CrossRef] [PubMed]
A. C. Peacock, “Mid-IR soliton compression in silicon optical fibers and fiber tapers,” Opt. Lett.37(5), 818–820 (2012). [CrossRef] [PubMed]
F. Li, S. D. Jackson, C. Grillet, E. Magi, D. Hudson, S. J. Madden, Y. Moghe, C. O’Brien, A. Read, S. G. Duvall, P. Atanackovic, B. J. Eggleton, and D. J. Moss, “Low propagation loss silicon-on-sapphire waveguides for the mid-infrared,” Opt. Express19(16), 15212–15220 (2011). [CrossRef] [PubMed]
T. Baehr-Jones, A. Spott, R. Ilic, A. Spott, B. Penkov, W. Asher, and M. Hochberg, “Silicon-on-sapphire integrated waveguides for the mid-infrared,” Opt. Express18(12), 12127–12135 (2010). [CrossRef] [PubMed]
Y. Huang, E. K. Tien, S. Gao, S. K. Kalyoncu, Q. Song, F. Qian, E. Adas, D. Yildirim, and O. Boyraz, “Electrical signal-to-noise ratio improvement in indirect detection of mid-IR signals by wavelength conversion in silicon-on-sapphire waveguides,” Appl. Phys. Lett.99(18), 181122 (2011). [CrossRef]
Z. Cheng, X. Chen, C. Y. Wong, K. Xu, C. K. Y. Fung, Y. M. Chen, and H. K. Tsang, “Mid-Infrared Grating Couplers for Silicon-on-Sapphire Waveguides,” IEEE Photon. J.4, 104–113 (2012).
O. Boyraz, Y. Huang, and X. Sang, “Silicon on sapphire and SOI photonic devices for mid-infrared and near-IR wavelengths,” Proc. SPIE8431, 843104, 843104-12 (2012). [CrossRef]
S. Pearl, N. Rotenberg, and H. M. Driel, “Three photon absorption in silicon for 2300-3300 nm,” Appl. Phys. Lett.93(13), 131102 (2008). [CrossRef]
G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, 2007).
L. Yin, J. Zhang, P. M. Fauchet, and G. P. Agrawal, “Optical switching using nonlinear polarization rotation inside silicon waveguides,” Opt. Lett.34(4), 476–478 (2009). [CrossRef] [PubMed]
J. Y. Lee, L. Yin, G. P. Agrawal, and P. M. Fauchet, “Ultrafast optical switching based on nonlinear polarization rotation in silicon waveguides,” Opt. Express18(11), 11514–11523 (2010). [CrossRef] [PubMed]
R. Dekker, A. Driessen, T. Wahlbrink, C. Moormann, J. Niehusmann, and M. Forst, “Ultrafast Kerr-induced all-optical wavelength conversion in silicon waveguides using 1.55 μm,” Opt. Express14(18), 8336–8346 (2006). [CrossRef]
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. Express15(3), 1135–1146 (2007). [CrossRef] [PubMed]
W. Astar, J. B. Driscoll, X. Liu, J. I. Dadap, W. M. J. Green, Y. A. Vlasov, G. M. Carter, and R. M. Osgood, “Conversion of 10 Gb/s NRZ-OOK to RZ-OOK utilizing XPM in a Si nanowire,” Opt. Express17(15), 12987–12999 (2009). [CrossRef] [PubMed]
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. Express18(9), 9613–9621 (2010). [CrossRef] [PubMed]
Y. Shoji, T. Ogasawara, T. Kamei, Y. Sakakibara, S. Suda, K. Kintaka, H. Kawashima, M. Okano, T. Hasama, H. Ishikawa, and M. Mori, “Ultrafast nonlinear effects in hydrogenated amorphous silicon wire waveguide,” Opt. Express18(6), 5668–5673 (2010). [CrossRef] [PubMed]
A. R. Davoyan, I. V. Shadrivov, and Y. S. Kivshar, “Nonlinear plasmonic slot waveguides,” Opt. Express16(26), 21209–21214 (2008). [CrossRef] [PubMed]
A. C. Turner, C. Manolatou, B. S. Schmidt, M. Lipson, M. A. Foster, J. E. Sharping, and A. L. Gaeta, “Tailored anomalous group-velocity dispersion in silicon channel waveguides,” Opt. Express14(10), 4357–4362 (2006). [CrossRef] [PubMed]
L. Yin, Q. Lin, and G. P. Agrawal, “Soliton fission and supercontinuum generation in silicon waveguides,” Opt. Lett.32(4), 391–393 (2007). [CrossRef] [PubMed]
Q. Lin, O. J. Painter, and G. P. Agrawal, “Nonlinear optical phenomena in silicon waveguides: Modeling and applications,” Opt. Express15(25), 16604–16644 (2007). [CrossRef] [PubMed]
T. E. Murphy, software available at http://www.photonics.umd.edu .
Q. Lin, T. J. Johnson, R. Perahia, C. P. Michael, and O. J. Painter, “A proposal for highly tunable optical parametric oscillation in silicon micro-resonators,” Opt. Express16(14), 10596–10610 (2008). [CrossRef] [PubMed]
A. D. Bristow, N. Rotenberg, and H. M. van Driel, “Two-photon absorption and Kerr coefficients of silicon for 850-2200 nm,” Appl. Phys. Lett.90(19), 191104 (2007). [CrossRef]
F. Li, S. D. Jackson, C. Grillet, E. Magi, D. Hudson, S. J. Madden, Y. Moghe, C. O’Brien, A. Read, S. G. Duvall, P. Atanackovic, B. J. Eggleton, and D. J. Moss, “Low propagation loss silicon-on-sapphire waveguides for the mid-infrared,” Opt. Express19(16), 15212–15220 (2011). [CrossRef] [PubMed]
P. Colman, C. Husko, S. Combrie, I. Sagnes, C. W. Wong, and A. D. Rossi, “Temporal solitons and pulse compression in photonic crystal waveguides,” Nat. Photonics4(12), 862–868 (2010). [CrossRef]
L. Yin and G. P. Agrawal, “Impact of two-photon absorption on self-phase modulation in silicon waveguides,” Opt. Lett.32(14), 2031–2033 (2007). [CrossRef] [PubMed]

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