Strongly asymmetric wavelength dependence of optical gain in nanocavity-based Raman silicon lasers : supplementary material

While the realization of silicon lasers using interband transitions is still technically problematic, utilization of Raman scattering processes seems to be the most feasible alternative. Raman silicon lasers based on photonic crystal nanocavities provide sub-microwatt thresholds and CMOS compatibility. Therefore, this type of laser is suitable for dense integration in Si photonic circuits. However, details of the gain mechanism, which are important for improving laser performance, have rarely been discussed due to the lack of a suitable characterization technique. Here, we report on the excitation-wavelength dependence of optical gain in a high-quality nanocavity-based Raman silicon laser. For this, we employ a so-called stimulated-Raman-scattering excitation (SRE) spectroscopy, which allows us to reveal the range of excitation wavelengths enabling laser operation, the excitation condition for maximum output, shift of the gain peak, and enhancement of Raman gain including nonlinear optical losses. In particular, we find that laser output remarkably decreases in the long-wavelength region of cavity resonance as excitation power increases. Numerical simulations suggest that optical loss due to free-carrier absorption induced by two-photon absorption grows substantially above a certain threshold.

This document provides supplementary information to "Strongly asymmetric wavelength dependence of optical gain in nanocavity-based Raman silicon lasers," https://doi.org/10.1364/OPTICA.5.001256. Figure S1 shows the Raman scattering spectra that were obtained under excitation of the pump nanocavity mode [ Fig. 1(e)]. The spectra were resolved by a monochromator with a focal length of 500 mm and detected by a liquid-nitrogen-cooled InGaAs array. The incident laser was not chopped for this measurement. Each Raman scattering spectrum was measured by adjusting the excitation wavelength (λin) to the optimum value that maximizes the Stokes emission intensity. This data was taken several months after the measurements of the stimulated-Raman-scattering excitation (SRE) spectra. The details of the experimental method are provided in previous publications [1,2].

S1. Raman scattering spectra
It is noted that the Raman scattered light is emitted through the Stokes nanocavity mode even at excitation powers significantly below the threshold Pth (the broad background by the spontaneous Raman scattering is hardly visible since it is very small). Therefore, the linewidth of the emission was approximately 1 pm. The resolution limit of the monochromator is about 100 pm Pth. The vertical axis represents the intensity detected by the liquid-nitrogen-cooled InGaAs array. and thus we were not able to detect the expected linewidth narrowing during lasing. By increasing the input power Pinput, the peak gradually redshifted mainly due to the adjustment of the λin. Figure S2 shows the Raman emission intensity as a function of Pinput. The intensities were normalized by the intensity at Pinput = 2.2 × 10 − 3 Pth. The Raman output power increased linearly in the low pump-power regime. As Pinput increased further, the Raman output power started to change nonlinearly and rapidly increased by more than two-orders of magnitude around Pinput = Pth.

S2. Calculated results using coupled mode theory
Here, we describe the calculation method and the original numerical data used to draw the spectra shown in Fig. 4. We employed a calculation framework based on coupled mode theory [3−5]. Figure S3 shows an overview of the calculation model. We used the following cavity parameters: Qp = 170,000, QS = 1,600,000, and Δf = 15.614 THz. Although these values are slightly different from those of the samples measured in the main text, no significant differences appeared in the calculation results. The details of the other parameters are explained in our previous paper [5].
The time evolution of the number of photons confined in the two nanocavity modes can be calculated with the rate equations for the amplitude of the pump light (ap) [3], The refractive index of silicon (Si) for the pump mode and Stokes mode is ni (i = p, S is the Raman gain coefficient that accounts for the detuning between the peak of the spontaneous Raman scattering and the resonance peak of the Stokes mode [see Eq. (S15)]. Further, 1/τp, in is the coupling strength between the propagating light in the pump excitation waveguide and the pump nanocavity mode as shown in Fig. S3. Finally, Pinput is the power of the excitation light in the pump waveguide, and ωin = 2πc/λin. The temporal evolution of the free-carrier density in the nanocavity, NC, is given by the following rate equation.
Here, τcarrier is the dissipation time of the free carriers in the nanocavity, determined by nonradiative recombination and carrier diffusion. Gcarrier is the generation rate of free carriers, i.e., the carriers generated by the TPA process [see Eq. (S19)]. The energy of the light absorbed by TPA and FCA is ultimately converted into heat through three different processes [see Eqs.
(S20)−(S23)] and this induces a temperature rise of the cavity material. In order to properly evaluate the temperature change of the nanocavity, we considered that the heat generated in the cavity diffuses to the surrounding photonic crystal (PC). The changes in cavity temperature and surrounding temperature are denoted as ΔTcav and ΔTsurround, respectively. They were evaluated using   Pinput, and the thermal reservoir, an oscillating behavior may be observed during Raman lasing [5], but in the present condition no oscillation appears. Figure S4(b) shows the relative shifts of the resonance peak wavelengths for the pump nanocavity mode (Δλp) and for the Stokes mode (ΔλS) with respect to the corresponding intrinsic values. Initially, a blueshift of the resonance peak positions is observed due to the plasma effect, but then the resonance peaks gradually redshift due to the increase of the cavity temperature, which is shown in Fig. S4(e). Figure S4(c) shows the frequency spacing between the two nanocavity modes, Δf. As shown in Fig. S4(b), the magnitudes of Δλp and ΔλS are similar. Therefore, the induced shift Δf is very small. Figure S4(d) shows the carrier density that is generated by TPA, NC. The carrier density NC rapidly increases within a short time after the start of the excitation, but for later times it gradually decreases because of the detuning between the λin and λp [the latter changes as shown in Fig. S4(b)]. Figure S4(e) presents the temperature change of the cavity, ΔTcav. The calculation predicts that the emission converges to a stable level within several μs after start of the excitation. The spectral data in Fig. 4 were obtained from the converged values.

S3. SRE spectra without modulation of the pump laser
As shown in Fig. 2 in the main text, we modulated the pump laser light by placing a mechanical chopper in the excitation path. Such a setup allows us to avoid the hysteresis response of the SRE spectrum that is observed by changing the λin sweeping direction.
The hysteresis occurs due to the accumulation of heat generated by the TPA carriers. However, by chopping the excitation beam (which has a wavelength λin that is redshifted with respect to λp, 0), the cavity resonance wavelength changes periodically between the cold-equilibrium wavelength λp, 0 (beam blocked) to the warmequilibrium wavelength (beam open), while passing through a non-equilibrium state. It is well known that the emissions from high-Q microcavities often oscillate in such a non-equilibrium state [5−7]. However, it has been demonstrated that the Raman laser signal converges to a stable operation condition within a few microseconds of continuous laser excitation [5]. Accordingly, the results shown in Figs. 3 and 5 should reflect the shape of the gain in actual devices.
On the other hand, for the application of Raman lasers, the responses without pump modulation can also be useful. Figure S5 presents the resonance spectra of the pump mode (upper panel) and the corresponding SRE spectra (lower panel) that were observed when the mechanical chopper is placed in the collection path just before the photodiode in Fig. 2. Although an hysteresis response was clearly observed for high excitation (4.0 × Pth and 8.0 × Pth), the spectral shapes of the SRE data for the upward sweeping direction were similar to those shown in Fig. 3(b). These results suggest that optical bistable switching with a switching contrast larger than 30dB can be obtained using the nanocavity-based Raman Si laser [8].
Finally, we comment on the reproducibility of the results. We confirmed that the setup shown in Fig. 2 reproduces the SRE spectra shown in Fig. 3 with good accuracy. In addition, similar results were obtained even for samples with different Q and Δf. On the other hand, the reproducibility is not high when the pump laser is not modulated, probably due to the strong thermo-optic effect. Here, the spectral shapes for the upward and downward scans for 4.0 × Pth and 8.0 × Pth were different in each measurement. Fluctuations in the heat diffusion from the cavity to the surrounding PC, the high-QS value, and the carrier lifetime due to the occupation of surface states may be responsible for this complex behavior. These effects will be important issues for improving the laser performance. The differences between the two measurements suggests that the temperature drift of the nanocavity does not completely stop even within 0.5 milliseconds of continuous laser excitation with high excitation powers.

Appendix
Below we present the equations employed to derive the calculation results presented in Section S2.
The 1/τi, total (i = p, S) in Eqs. (S1) and (S2) is the rate of the energy loss in the cavity. It has several contributions as shown in Eq. (S7).
Here, 1/τi, in and 1/τi, v represent the rates of energy transfer into the waveguide (in-plane direction) and into free space (vertical direction), respectively. 1/τi, TPA is defined as the loss rate due to TPA (proportional to the energy stored in the cavity where the first terms represent the TPA process involving either two pump photons or two Stokes photons. The second terms represent the simulataneous absorption process involving one pump photon and one Stokes photon. βSi denotes the TPA coefficient of bulk silicon. The effective mode volume for TPA, Vi, TPA , is [3] ( ) (S10) Vo, TPA describes the spatial overlap between the pump mode and the Stokes mode, and we consider Vo, TPA = VR . The latter is provided in Eq. (S14). Then we have where bulk R g denotes the Raman gain coefficient of bulk Si. VR is the effective modal volume for Raman scattering, which determines the spatial overlap between the pump and Stokes modes. It can be calculated using the following expression. (S14) Here, i ε is the dielectric constant in Si and Ei is the electric field.
The Δ is the full width at half-maximum of the spontaneous Raman scattering in Si and ΔfR is the Raman shift of the Si nanocavity. The partial shifts of the refractive index induced by the carrierplasma effect and the thermo-optic effect in Eq. (S3) are defined as [3] is the cavity volume for the free carriers and is defined as the product (cavity length) ⋅ (distance between the air holes nearest to the center of the cavity in the y direction) ⋅ (slab thickness) [9]. The heat generation rate Gheat in Eq. (S5) is expressed as a sum of three terms: Gheat, TPA − relax denotes the heat generation rate due to relaxation of TPA carriers to the bottom of the conduction band, Gheat, FCA − relax is the heat generation rate due to relaxation of carriers generated by FCA, and the heat generation rate for nonradiative interband recombination of the carriers is represented as Gheat, recomb. These rates are calculated using the following three equations.
Finally, Eg is the bandgap energy of Si. To simplify the calculations, we assume that all carriers generated by TPA recombine in the surrounding PC region.