We show that stimulated Raman scattering is a sufficiently strong effect in silicon waveguides to allow the intensity of a pump input to control the group delay of a weak Stokes-shifted signal. While the fractional change in group delay is minimal in a straight waveguide, microresonator enhancement can produce a fractional increase of 50% with respect to the linear group delay of the resonator.
© 2006 Optical Society of America
Stimulated scattering mechanisms in optical fiber have been demonstrated Ref. [1, 2, 3] as a means to control the group delay of a signal pulse via gain induced by the intensity of a pump pulse. Stimulated scattering mechanisms can also be employed for chip-based tunable optical delay lines that may have applications in pulse retiming and pulse position modulation, for example. Silicon waveguides Ref. [4, 5, 6, 7] have been shown to possess very large Raman gain coefficients as compared to silica, by roughly a factor of 104. In addition, the bandwidth of the Raman gain is about 100-times smaller than in silica. Both of these factors lead to ~ 106 increase in group delay for comparable pump intensities, but this increase is offset by the detrimental effects of free-carrier absorption. Another factor is that silicon waveguides produce stronger confinement than silica, increasing the intensity for a given launch power. In this paper, we numerically study the change in group delay on a weak signal pulse produced by a strong pump in silicon rectangular waveguides and cascaded microring resonators Ref. [8, 9]. We show that the group delay can be increased by a factor of 1.5 over linear propagation.
Figure 1 shows the generic CMRR slow-light waveguide geometry which consists of a sequence of microring resonators coupled to a common bus waveguide Ref. . The linear transfer function of the cascade is the product of the transfer functions of the individual rings. We will only consider systems consisting of identical rings, such that the overall transfer function can be written Ref. 
where , r is the coupling coefficient from the waveguide to the ring, a = exp(-αL/2) is the single-pass attenuation within the ring, L is the circumference of the ring, ϕ = kL is the single-pass phase within the ring, k = 2nπ/λ, and n is the effective refractive index of the ring. Critical coupling occurs for (i.e. t = a), which is the condition under which all incident light is dissipated within the ring on resonance.
In order to describe stimulated Raman scattering, we use the following coupled nonlinear differential equations for the pump and signal waves Ref. 
where Ip = ε 0 cn|Ep|2 /2, n 2 = 4 × 10-14 cm2/W is the nonlinear Kerr index, β 2 = 7 × 10-10 cm/W is the two-photon absorption coefficient, gR is the Raman gain coefficient, σ = 1.5 × 10-17 cm2 relates free-carrier density to Δα, ρ = - 1 × 10-21 cm3 relates free-carrier density to Δn, and Ω = ωp - ωs is the Stokes downshift. These nonlinear parameters are consistent with experimental measurements for silicon Ref. [11, 12]. In silicon, free carriers are generated via two-photon absorption, where, in steady-state, the free-carrier density is given by N = τβ 2 I 2/2h̄ω, with τ the free-carrier lifetime and I = ε 0 cn|Ep + Es|2 /2. We assume τ = 1 ns, which can be accomplished by sweeping carriers from the optical mode volume by an applied field Ref. .
The Raman gain and nonlinear refraction coefficients are calculated from the imaginary and real parts of the Raman susceptibility function Ref. 
We use the parameters n(ω) ~ 3.4, Υf = 2π × 15.9 rad/ps, γ = 2π × 0.08 rad/ps, and peak Raman gain gR = 25 cm/GW measured near λ = 1.55 μm. Note that there are considerable differences in the reported value of gR Ref. [12, 5, 6]; we chose to employ a moderate value consistent with Ref. . From these parameters, we can determine gR and nR for all Stokes shifts Ω, as plotted in Fig. 2 for the signal wave.
Equations 2 and 3 are used to relate the fields at the input (i.e. Ep(0) and Es(0)) of a straight waveguide or resonator to the fields at the output (i.e. Ep(L) and Es(L)). Under resonance conditions in a microring, Ep and Es are increased, leading to increased effective nonlinearity. This calculation is iterated within each ring sequentially Ref. , taking into account the coupling of light between the ring and the bus waveguide. At all pump input intensities, a constant input ratio Ep/Es = 100 is maintained, such that the maximum intensity amplification factor is roughly 104. The peak of the Raman gain in silicon is downshifted from the pump by about 16 THz, so that in our simulations, the pump and signal frequencies are separated by 9 FSR for 50 μm ring circumference, with resonant wavelengths λp = 1.428 μm and λs = 1.545 μm.
3. Raman-mediated tunable slow-light
In the absence of attenuation, the nonlinear phase shift induced on a weak signal beam by a strong pump beam is given by
where L is the waveguide or resonator length. Because of the variation of nR with signal frequency for fixed ωp, the group delay of the signal takes on a nonlinear dependence, given by
where it was assumed that dnR/dω is the dominant term. Figure 2 plots t NL gd/IpL, which is group delay per unit intensity per unit length; within the region of Raman gain, the induced group delay is positive, while outside of that region, negative group delay is induced. It is important to note that the bandwidth of the Raman-induced group delay is less than that of the ~80 GHz Raman gain, such that significant resonator enhancement can be employed.
We first start by studying Raman-induced group delay in straight rectangular waveguides of length 5.75 mm (comparable to the effective path-length of a five-resonator cascade with 50 GHz ring bandwidth, as studied in the following section) with attenuation of 1 cm-1. Figure 3 plots the amplification and group delay as a function of signal frequency (normalized to a microresonator FSR of 1.77 THz) and normalized pump intensity. At low pump intensity, the total group delay is 65.2 ps, while at an intensity of n 2 Ip = 1 × 10-4, the group delay increases to 69.3 ps, a factor of 1.06; the bandwidth of the induced group delay is roughly 40 GHz. There are also regions of decreased total group delay in the wings of the central peak. At the higher pump intensities, the moderate Raman gain beings to decrease, indicating that the induced group delay increases more slowly and is reaching a point of saturation due to intensity-dependent loss mechanisms on the pump.
If the length of the waveguide is increased, the total change in group delay is also increased, but the fractional change essentially remains constant. Figure 4 plots the Raman gain and group delay for a waveguide of 1.18 cm total length (comparable to five-resonator cascade with 25 GHz ring bandwidth). The linear group delay is 133.7 ps, while the maximum delay including the Raman effect is 139.2 ps, a factor of 1.05 increase.
These results show that significant change in group delay cannot be achived via the Raman effect in straight waveguides.
4. Microresonator-enhanced tunable slow-light
A microresonator can provide significant resonance-enhancement of the Raman effect through increase of the intra-resonator pump intensity and increase in the effective path length. We first study the enhancement from single side-coupled resonators of 50 GHz and 25 GHz resonance bandwidths. Figure 5 plots the Raman-induced gain and group delay for a single resonator with L = 50 μm circumference and coupling coefficient r = 0.399, which results in a 50 GHz bandwidth, group delay of 13.0 ps and effective path length 1.15 mm. For normalized pump intensity n 2 Ip = 3×10-6, a maximum group delay of 18.2 ps is obtained, with a bandwidth of 33.6 GHz. The Raman-induced increase by a factor of 1.4 is considerably greater than obtained in a straight waveguide. However, refractive detuning of the resonator relative to the peak of the Raman gain blue-shifts the peak of the group delay by 18.5 GHz. The blue shift of the resonator serves as a saturating mechanism and arises from the negative chage in refractive index due to the generation of free carriers.
If the resonator bandwidth is reduced to 25 GHz (using a coupling coefficient of r = 0.284), the linear group delay for a single resonator is 26.7 ps with effective length 2.35 mm. As a function of pump intensity, the group delay increases to 39.8 ps (increase by a factor of 1.5), with a bandwidth of 15.7 GHz and blue-shift of 18 GHz.
The factor 1.4 to 1.5 change in group delay can be integrated across multiple resonators in sequence. This is shown in Fig. 7 for a sequence of 5 microresonators with pump input intensity n 2 Ip = 1×10-6. The change in group delay with respect to the linear case follows a nearly constant proportionality factor. The line represents the group delay of a straight waveguide of the same effective length and same pump intensity, indicating that little to no change occurs.
In conclusion, we have shown that cascaded microring resonators can be used to enhance the efficiency of Raman-induced group delay on a chip as compared to straight waveguides. Group delay change of up to 50% is prediced for silicon resonators. Similar results might be expected for GaAs/AlGaAs microresonator structures Ref.  where the Raman response is greater than in silica and has narrower bandwidth Ref. . In comparison to silicon, GaAs may have an advantage because of the shorter carrier lifetimes and reduced ratio β 2/n 2, but this advantage is offset by a lower ratio gR/n 2.
The authors would like to acknowledge J. Lowell and J. Sipe for useful discussions relevant to this problem.
References and links
1. K. Y. Song, M. G. Herraez, and L. Thevenaz “Observation of pulse delaying and advancement in optical fibers using stimulated Brillouin scattering,” Opt. Express 13, 82–88 (2005), http://www.opticsexpress.org/abstract.cfm?URI=oe-13-1-82. [CrossRef]
2. Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. Zhu, A. Schweinsberg, D. J. Gauthier, R. W. Boyd, and A. L. Gaeta “Tunable all-optical delays via Brillouin slow light in an optical fiber,” Phys. Rev. Lett. 69, 153902 (2005). [CrossRef]
3. J. E. Sharping, Y. Okawachi, and A. L. Gaeta “Wide bandwidth slow light using a Raman fiber amplifier,” Opt. Express 13, 6092–6098 (2005), http://www.opticsexpress.org/abstract.cfm?URI=oe-13-16-6092. [CrossRef]
4. R. Claps, D. Dimitropoulos, V. Raghunathan, Y. Han, and B. Jalali “Observation of stimulated Raman amplification in silicon waveguides,” Opt. Express 11, 1731–1739 (2003), http://www.opticsexpress.org/abstract.cfm?URI=oe-11-15-1731. [CrossRef]
5. H. Rong, A. Liu, R. Nicolaescu, M. Paniccia, O. Cohen, and D. Hak “Raman gain and nonlinear optical absorption measurements in low-loss silicon waveguide,” Appl. Phys. Lett. 85, 2196–2198 (2004). [CrossRef]
6. R. L. Espinola, J. I. Dadap, J. R. M. Osgood, S. J. McNab, and Y. A. Vlasov “Raman amplification in ultrasmall silicon-on-insulator waveguides,” Opt. Express 12, 3713–3718 (2004), http://www.opticsexpress.org/abstract.cfm?URI=oe-12-16-3713. [CrossRef]
7. Q. Xu, V. R. Almeida, and M. Lipson “Time-resolved study of Raman gain in highly confined silicon-on-insulator waveguides,” Opt. Express 12, 4437–4442 (2004), http://www.opticsexpress.org/abstract.cfm?URI=oe-12-19-4437. [CrossRef]
8. J. Heebner, R. W. Boyd, and Q.-H. Park “SCISSOR solitons and other novel propagation effects in microresonator-modified waveguides,” J. Opt. Soc. Am. B 19, 722–731 (2002). [CrossRef]
9. Y. Chen and S. Blair “Nonlinear phase shift of cascaded microring resonators,” J. Opt. Soc. Am. B 20, 2125–2132 (2003). [CrossRef]
10. S. Blair Optical Soliton-Based Logic Gates. PhD thesis University of Colorado1998, http://photonics.ece.utah.edu/files/blair_thesis_98.pdf.
11. R. A. Soref and B. R. Bennett “Electrooptical effects in silicon,” IEEE J. Quantum Electron. QE-23, 123–129 (1987). [CrossRef]
12. R. Claps, V. Raghunathan, D. Dimitropoulos, and B. Jalali “Influence of nonlinear absorption on Raman amplification in Silicon waveguides,” Opt. Express 12, 2774–2780 (2004), http://www.opticsxpress.org/abstract.cfm?URI=oe-12-12-2774. [CrossRef]
14. D. Rafizadeh, J. P. Zhang, S. C. Hagness, A. Taflove, K. A. Stair, S. T. Ho, and R. C. Tiberio “Waveguide-coupled AlGaAs/GaAs microcavity ring and disk resonators with high finesse and 21.6-nm free spectral range,” Opt. Lett. 22, 1244–1246 (1997). [CrossRef]
15. Y.-H. Kao, M. N. Islam, J. M. Saylor, R. E. Slusher, and W. S. Hobson “Raman effect in AlGaAs waveguides for subpicosecond pulses,” J. Appl. Phys. 78, 2198–2203 (1995). [CrossRef]