1. Introduction

Integrated optical circuits based on silicon that combine thin waveguides, small resonators, short optical modulators and photo-detectors are expected to be useful in information processing [1]. For example, optical transceivers for short-distance optical communication [2], all-optical routing for efficient high-speed networks [3], and quantum optical signal processing for large-scale quantum computing [4] have been intensively studied. Silicon-based laser sources using stimulated Raman scattering (SRS) and optical amplifiers are also attractive elements, because they can enhance the functionality of such optical circuits [5–13].

Raman silicon lasers that are based on high-quality factor (high-Q) photonic crystal (PC) nanocavities have resonator lengths on the order of several micrometers and lasing thresholds on the order of one microwatt [14], and it has been shown that continuous-wave laser operation with an excitation wavelength in the S-band, the O-band, or the T-band is also possible [15,16]. Moreover, the fabrication of such lasers by a CMOS-compatible approach has been demonstrated [17]. Such lasers can even be used to detect ionized air [18,19]. However, the device design used in these previous studies has a significant limitation: the power emitted from the edge of an adjacent waveguide (P_edge) is several times smaller than the surface emission power from the nanocavity (P_surface) [20]. On the other hand, for the applications described above, edge-emitting Raman silicon lasers can often be advantageous.

It has been shown that photonic mirrors that reflect photons by a mode-gap difference can be used to efficiently emit photons from the high-Q nanocavity to an adjacent waveguide [21]. Efficient edge emission has been demonstrated using devices that consist of a nanocavity, an input waveguide with a side mirror, and an output waveguide with another mirror [22,23]. Photonic mirrors have also been utilized in studies related to optical buffer memories for silicon optical circuits [24–26]. In these previous studies, a single high-Q resonant mode in the nanocavity was used, and thus the input waveguide and the output waveguide had the same design. In contrast, a Raman laser uses two high-Q resonant modes (to confine the pump photons and the Stokes Raman-scattered photons). Since the wavelength of the Stokes photons is about 100 nm longer than that of the pump photons, the design of the output waveguide with the photonic mirror needs to be different from that of the input waveguide. Numerical simulations for a Raman laser with such waveguides have indicated a possible enhancement of P_edge [27].

Here, we experimentally and numerically investigate the increase in P_edge of a Raman silicon nanocavity laser that can be achieved by introducing heterointerface mirrors (HMs). We call this type of laser a HM Raman laser. In Section 2, the device configuration of our laser is explained. In Section 3, experimental results for HM and conventional Raman lasers are compared; the average P_edge of the HM Raman lasers is 4.3 times higher. Section 4 describes the used calculation framework, and the calculation results are described in Section 5. This numerical analysis can explain the experimentally observed increase in P_edge. In Section 6, we discuss how P_edge can be further increased.

2. Sample structure

Figure 1 clarifies the HM Raman laser device design. We consider the use of a silicon-on-insulator (SOI) substrate where the thickness of the top silicon layer is 220 nm and that of the buried oxide (BOX) layer is 3 µm. Figures 1(a) and (c) show the top view of the silicon layer, and the x-axis is along the [100] direction of the silicon layer. The basic PC consists of air holes arranged in a triangular lattice with a lattice constant (a) of 410 nm. The air hole radius (r) is about 127 nm. Such PC patterns can be formed by electron beam (EB) lithography and inductively coupled plasma (ICP) etching. To obtain a free-standing PC slab, the BOX layer below the PC pattern needs to be removed (for example, by using an 48% hydrofluoric acid solution). Regarding the samples used in this work, the waveguide edge was fabricated by manual cleaving.

 

Fig. 1. Device structure of the HM Raman laser. (a) Schematic of the heterostructure nanocavity and the two adjacent waveguides. (b) Energy diagram of the heterostructure nanocavity. (c) Entire HM-laser device layout. (d) and (e) clarify the structures of the photonic mirrors with a heterointerface for reflection in the cases of pump and Stokes waveguides, respectively.

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Figure 1(a) is used to explain the core-region design of the laser. The heterostructure nanocavity consists of 27 missing air holes in the PC pattern. To create the heterostructure, the lattice constant a is slightly adapted in the x-direction near the center of the cavity: a is increased from 410 nm to 420 nm in steps of 5 nm [28]. This slight increase in a shifts the propagation bands to lower frequencies, as shown in Fig. 1(b). The cavity width remains constant, $\sqrt 3 a$ (W₁ = 710 nm). The resonant mode formed by the 2nd propagation band and the 2nd mode gap is the so-called pump mode (with resonance wavelength λ_p). The other cavity mode at lower energy is the Stokes mode (with resonance wavelength λ_S). The pump and Stokes modes are used to confine the pump light and the Stokes Raman-scattered light, respectively. The design Q values of the pump mode and the Stokes mode are Q_{p_design} = 6.14 × 10⁵ and Q_{S_design} = 5.18 × 10⁶, respectively. It has been shown that high quality factors help to enhance both spontaneous and stimulated Raman scattering [11,12,29]. Note that the above design Q values were determined by three-dimensional (3D) finite-difference time-domain (FDTD) calculations without waveguides adjacent to the cavity.

To enhance SRS in this device, the frequency spacing between these two modes (Δf) should be close to the Raman shift of silicon (15.606 THz) [30]. In addition, according to previous results that clarify the relationship between the crystal direction and the effective modal volume for SRS, the cavity should extend along the [100] direction of silicon [14,31]. It is known that Δf almost linearly increases with r [32]. Therefore, we fabricated several samples with slightly different values of r and determined the sample that shows laser oscillation with the lowest threshold. Furthermore, we used an SOI wafer where the crystal direction of the top silicon layer is rotated by 45 degrees with respect to the supporting Si substrate. Therefore, the [100] direction of the top silicon layer and the [110] direction of the supporting substrate are parallel [20]. This allows us to fabricate the Raman laser fabricated along the [100] direction while we can cleave the sample vertically to the waveguide, which is important for optical circuits.

The line defect below the nanocavity in Fig. 1(a) is the pump waveguide for the excitation of the pump mode. The upper waveguide is the Stokes waveguide for the extraction of Stokes light. The widths of the two waveguides are 0.88 W₁ and 1.10 W₁. The pump mode of the cavity couples to the guided mode in the 1st propagation band of the pump waveguide, and the Stokes mode of the cavity couples to the guided mode in the 1st propagation band of the Stokes waveguide. This coupling leads to lower quality factors of the pump and Stokes modes. We use the term Q_total to denote the calculated Q value including the coupling to the waveguides:

In an actual sample, the positions and radii of the air holes exhibit small random deviations (in the sub-nanometer range) with respect to the ideal values due to fluctuations during the air-hole fabrication process. These deviations can induce additional losses by optical scattering, leading to the factor 1/Q_scat [33]. Furthermore, the absorption losses related to defects in the silicon nanocavity and surface water need to be considered (1/Q_abs) [34,35]. Thus, the experimental Q value (Q_exp) is smaller than Q_total, which is expressed by the following relation:

Figure 1(c) shows the entire HM-laser device layout. Between the two waveguides, ten nanocavities with the same r are placed with a spacing of about 20 µm. This sample design enables an efficient measurement of the individual laser samples [36]: Both the λ_p and λ_S values are expected to be slightly different for each of the ten cavities due to the above-mentioned air-hole fabrication fluctuations. The magnitudes of the variation in λ_p and λ_S are usually more than one order of magnitude larger than the full widths at half-maximum of the pump and Stokes modes [32]. Therefore, in a structure that contains only a few nanocavities, it is unlikely that even one of the resonance modes interferes with any of the other resonance modes [24]. This means that we are usually able to characterize the individual resonance peaks of the nanocavities even if we only use one set of waveguides. In our sample, the waveguide edge was formed by cleaving the sample. Note that we did not add any sophisticated structure that would allow us to control the radiation pattern of the edge emission, the reflection, and the degree of scattering.

The positions of the two HMs used to reflect the light propagating in the waveguides are indicated by the blue and red rectangles in Fig. 1(c) for the pump light and the Stokes light, respectively. The lattice constant of the holes in the colored areas in Figs. 1(d) and (e) is 380 nm in the x-direction (that in the y-direction is 710 nm to maintain structural uniformity). The propagation bands in the colored area shift to higher frequencies as shown in the diagrams at the bottom of Figs. 1(d) and (e). The mode-gap differences at the heterointerfaces result in the reflection of light at λ_p and λ_S.

For the evaluation of the performance increase that can be achieved by the HMs, ten conventional Raman laser cavities were also fabricated on the same chip. The design of the core region is the same as that shown in Fig. 1(a). The HM Raman laser array and conventional Raman laser array are displaced in the y-direction by 100 µm. Therefore, the magnitudes of Q_scat and Q_abs should be similar for all cavities.

3. Experimental results

We used a tunable laser to determine the following properties of each cavity: the resonance wavelengths λ_p and λ_S, the frequency spacing Δf, Q_{p_exp} and Q_{S_exp}, the threshold for laser oscillation (I_th), and the maximum P_edge (P_{edge_max}). The details of the measurement procedure are described in Appendix A1. The measurement was performed at 24 °C in air. Table 1 summarizes the results obtained from the four HM Raman laser cavities in which laser oscillation was achieved. These four cavities are hereafter referred to as cavities #1–#4. Table 2 summarizes the results obtained from the five conventional Raman laser cavities in which laser oscillation occurred. These cavities are referred to as cavities #5–#9.

Table 1. Measurement results for four HM Raman lasers (λ_p, λ_S: resonance wavelengths, Δf: frequency spacing, Q_{p_exp} and Q_{S_exp}: actual Q values, I_th: threshold, P_{edge_max}: maximum P_edge)

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Figure 2 shows more details of the experimental results of cavity #1, which provides the largest P_{edge_max}. In Figure 2(a), the resonance and transmission spectra of the pump mode are shown by the closed and open circles, respectively. To obtain these spectra, the output of the tunable laser was focused at the input port indicated in Fig. 1(c) to excite the pump mode. As illustrated in the inset of Fig. 2(a), the surface emission from the cavity in the vertical direction of the slab (+z direction) was measured, as well as the transmitted intensity at the waveguide edge opposite to the input port. Figure 2(b) shows the spectra for the Stokes mode. For this measurement, the tunable laser irradiated the output port indicated in Fig. 1(c).

 

Fig. 2. Details of the results of HM Raman laser cavity #1. (a) Resonance spectrum of the pump mode (closed circles) and the corresponding transmission spectrum (open circles). The solid curve is the fitting result. (b) The resonance and transmission spectra of the Stokes mode. (c) P_edge and P_surface as a function of I. (d)−(g) NIR camera images for I = 1.2 I_th obtained by using a long-pass filter to remove the pump light. The images show (d) the edge emission at the output port, (e) the scattered light at the HM of the Stokes waveguide, (f) the scattered light at the output port, and (f) the surface emission from the nanocavity.

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Due to the reflection at the heterointerfaces shown in Figs. 1(d) and 1(e), the transmitted intensities shown in Figs. 2(a) and 2(b) are very small. In each figure, the solid curve shows the fit of the resonance spectrum to a Lorentzian function, from which the resonance wavelength (λ) and the full width at half-maximum (Δλ) are evaluated. The experimental Q value is calculated using the relation Q_exp = λ/Δλ, and the obtained Q_{p_exp} and Q_{S_exp} values are 1.49 × 10⁵ and 7.87 × 10⁵, respectively. This Q_{S_exp} is smaller than the values reported in previous studies due to the smaller d_S (see Appendix A2 regarding the relation between Q_exp and I_th). The measured Δf is 15.602 THz, and thus the detuning from the Raman shift (|Δf_det.| = |Δf − 15.606 THz|) is only 0.004 THz [30,32]. These optical characteristics allow Raman laser oscillation.

Figure 2(c) shows the input–output characteristics of this HM Raman laser. The orange points represent P_edge (the Stokes light emitted via the output port), while the black points represent P_surface (the Stokes light emitted from the cavity into free space). The pump power I is the power coupled into the pump mode of the nanocavity (to clearly distinguish between the pump power and the output characteristics, the symbol I is used to denote the pump power while P is used for the output power). P_edge, P_surface, and I were estimated from the intensities detected by using photodiodes and taking into account the loss of the optical components (details are described in Appendix A1).

As shown in Fig. 2(c), P_edge and P_surface increase steeply when I starts to exceed the threshold I_th = 1200 nW, but the Raman laser output saturates as I increases further. This is due to free-carrier absorption (FCA) by the carriers generated by two photon absorption (TPA) [37]. FCA is responsible for an additional (nonlinear) loss term, 1/Q_{i_FCA}, that needs to be added to Eq. (2) in the case of strong excitation. It has been shown that the Q_{p_exp} and Q_{S_exp} values under lasing conditions are smaller [38]. The Q_{i_exp} values presented in Table 1 and 2 were measured with low pump powers where TPA effects are negligible.

The maximum P_edge (8.8 nW) is obtained at I = 4580 nW and the experimentally obtained ratio of P_edge/P_surface is 0.83. Note that our experimental setup leads to an underestimation of this ratio, because the numerical aperture (NA) of the objective lens that collects the surface emission is 0.7 and this value is larger than that of the lens that collects the edge emission, NA = 0.4 (see Appendix A1). We calculated the angular distribution of the edge emission from a PC waveguide edge using 3D FDTD simulations. The intensity emitted at angles larger than 23.6 degrees (with respect to the waveguide axis), cannot be collected by a lens with NA = 0.4, but this intensity can be as much as the intensity emitted at angles smaller than 23.6 degrees. Since the radiation pattern changes depending on the shape of the air holes at the edge, the difference in the NA for P_edge and P_surface is not considered in the analysis.

Figures 2(d)−(g) present different near-infrared (NIR) camera images obtained during laser oscillation at I = 1.2 I_th. Figure 2(d) is an image of the edge emission at the output port (view direction: parallel to the x-axis shown in Fig. 1). A clear emission spot is obtained. Figure 2(e) is an NIR camera image at the HM of the Stokes waveguide (top view). The scattered light at the heterointerface is very small (we consider that this scattered light are Stokes photons with TM polarization [39]). This small intensity of the scattered light and the small intensity of the transmitted light in Fig. 2(b) indicates that the reflectivity of this HM is close to 100%.

Figure 2(f) is the top view of the silicon slab at the position of the output port. A spot with strong scattering to the z-direction is observed. The intensity of this emission spot is 0.80 × P_surface at I = 4580 nW (determined by a photodiode), and thus it is almost the same as P_{edge_max}. A similar amount of scattered light is probably also radiated into the direction of the silicon substrate (−z direction). If a sophisticated structure, for example a spot size converter (SSC), had been added to the output port to reduce the scattering and to control the radiation pattern, the P_{edge_max} of cavity #1 would have exceeded maybe even 30 nW (methods to increase P_edge are discussed in Section 6).

The above results are now compared with those of cavity #5, the cavity with the largest P_{edge_max} among the five conventional Raman lasers. Figures 3(a) and 3(b) indicate a Q_{p_exp} and a Q_{S_exp} of 1.59 × 10⁵ and 6.89 × 10⁵, respectively. The Δf is 15.588 THz, and thus |Δf_det.| = 0.018 THz. These values are equivalent to those of cavity #1.

 

Fig. 3. Details of the results of conventional Raman laser cavity #5. (a) Resonance spectrum of the pump mode (closed circles) and the corresponding transmission spectrum (open circles). The solid curve is the fitting result. (b) The resonance and transmission spectra of the Stokes mode. (c) Input–output characteristics of the laser. (d)−(f) NIR camera images of (d) the edge emission at the output port, (e) the scattered light at output port, and (f) the surface emission from nanocavity obtained at I = 1.2 I_th.

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Figure 3(c) shows the input–output characteristics of this Raman laser and indicates an I_th of about 1600 nW. The saturation of the Stokes power due to TPA-induced FCA is also observed. We find that P_{edge_max} = 2.0 nW at I = 4520 nW and P_edge/P_surface = 0.11. The latter value is only 13% of the P_edge/P_surface of cavity #1. On the other hand, the sum P_edge + P_surface is almost the same (20 nW for cavity #5 and 19.5 nW for cavity #1).

Figures 3(d)−(f) present the NIR camera images obtained at I = 1.2 I_th. The intensities of the emission spots in Figs. 3(d) and 3(e) are much smaller than those in Fig. 3(f). The power of the light scattered at the output port into the z-direction is also small, 0.08 × P_surface at I = 4520 nW. This is 10 times smaller than that of cavity #1. These results strongly suggest that the conventional Raman laser emits most of the laser light from the cavity into free space.

Figure 4(a) shows the normalized input–output characteristics of the four HM Raman lasers presented in Table 1 (we normalized the pump power to the threshold power for each sample). The black and orange curves correspond to P_surface and P_edge, respectively. Figure 4(b) shows the normalized input–output characteristics of the five conventional Raman lasers shown in Table 2. These figures clearly show the tendency of a larger P_edge for Raman lasers with HMs. The average P_{edge_max} of the four HM Raman lasers is 4.3 times larger than that of the five conventional Raman lasers.

Table 2. Measurement results for five conventional Raman lasers (λ_p, λ_S: resonance wavelengths, Δf: frequency spacing, Q_{p_exp} and Q_{S_exp}: actual Q values, I_th: threshold, P_{edge_max}: maximum P_edge)

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Fig. 4. (a) Normalized input–output characteristics of four HM Raman samples (cavities #1∼#4), (b) Normalized input–output characteristics of five conventional Raman lasers (cavities #5∼#9).

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On the other hand, the sum of P_edge and P_surface is similar for both types of lasers except for cavity #7. Furthermore, the average Q_p and Q_S of the four HM Raman lasers are equivalent to those of the five conventional Raman lasers as shown in Tables 1 and 2. Therefore, it can be concluded that the HMs lead to an increase in the P_edge of a Raman laser.

It is noted that the P_{surface_max} of cavity #4 is much larger than its P_{edge_max}. This suggests that even in a Raman laser with HMs, the increase in P_edge is sometimes not as large as would be expected from simple considerations. The coefficient of variation of P_{edge_max} (the ratio of the standard deviation to the mean) for the HM Raman laser is 0.78, while that for the conventional Raman laser is 0.55. Although the number of measured samples is small, the P_edge of a HM Raman laser may be subject to larger changes. These characteristics are analyzed in Section 5 by numerical simulations.

4. Calculation method

This section presents the model based on coupled-mode theory used to describe the devices and to investigate the P_edge and P_surface of Raman silicon nanocavity lasers with HMs. Figure 5 clarifies the physical meaning of the parameters used in the calculation model. Port 1 and Port 4 correspond to the input and output ports in Fig. 1(c), respectively. The HMs are placed at Port 2 and at Port 3. A reflectivity of 100% is assumed. To simplify the calculations, the reflectivities at the input and output waveguide edges are assumed to be 0%. In the actually measured samples, each pump waveguide is used to excite ten nanocavities, but the calculation considers only one nanocavity.

 

Fig. 5. Calculation model of the Raman silicon nanocavity laser with two HMs.

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The nanocavity has a pump mode and a Stokes mode characterized by the angular frequency ω_p and ω_S, respectively. The photons from the excitation source (photons at ω = ω_p in the case of resonant excitation) enter the system at Port 1, and the Stokes Raman scattered photons at ω_S leave the system from Port 4. Furthermore, the nanocavity also emits photons at ω_p and ω_S in the direction vertical to the PC slab. The value of 1/Q_{p_in} describes the strength of the coupling between the pump mode and the pump waveguide, and 1/Q_{S_in} is the strength of the coupling between the Stokes mode and the Stokes waveguide. The photons at ω_p do not couple to the Stokes waveguide and similarly, the photons at ω_S do not couple to the pump waveguide. The strength of the coupling between the nanocavity and free space is denoted as 1/Q_{i_v}. In an ideal nanocavity, Q_{i_v} is equal to Q_{i_design}. On the other hand, actual nanocavities include scattering and absorption losses. Accordingly, Q_{i_v} should be expressed in terms of Q_{i_design}, Q_{i_scat}, and Q_{i_abs} as shown in the following relation:

The amplitudes of light propagating from each port to the cavity are denoted as S _+ k (k = 1, 2, 3, 4) and the amplitudes of light propagating from the cavity to each port are denoted S_-k. d₁–d₄ are the distances between the cavity and Ports 1–4, respectively. In the case that S₋₁ and S _+ 2 interfere destructively, the number of photons at ω_p that are trapped in the cavity (N_p) increases [21,22], and if they interfere constructively, N_p decreases. Furthermore, if S _+ 3 and S₋₄ interfere constructively, the magnitude of S₋₄ increases, and thus P_edge becomes larger. θ_in and θ_out denote the round-trip phase shifts of the light reflected at the HMs. The degrees of interference in the pump and Stokes waveguides depend on θ_in and the θ_out, respectively.

For the numerical simulations, we use Eqs. (4)−(10) provided below. The coupled-mode theory used to derive these equations is described in Appendix A3.

5. Calculation results

For the calculations in this section, we use Q_{p_v} = 2.27 × 10⁵, Q_{p_in} = 5.32 × 10⁵, Q_{S_v} = 1.02 × 10⁶, and Q_{S_in} = 2.12 × 10⁶. These values result in Q_{i_exp} values equal to those of cavity #5 (the estimation method is described in Appendix A4). Figures 6(a) and (b) show the calculated input–output characteristics of the HM Raman laser and the corresponding conventional laser, respectively, for (θ_in, θ_out) = (0, 0). The x-axis represents the excitation power at Port 1, |S _+ 1|². It should be noted that this definition is different from that for the x-axis in Figs. 2(c) or 3(c). In the case of the conventional Raman laser, we can roughly estimate that the power coupled into the pump mode of the nanocavity is about 0.4 × |S _+ 1|². In Fig. 6, the number of pump photons in the cavity, N_p, is shown by the dashed line. The orange curve is P_edge and the black curve is P_surface.

 

Fig. 6. Theoretical input–output characteristics for (θ_in, θ_out) = (0, 0). (a) P_edge, P_surface, and N_p as a function of |S _+ 1|² for a HM Raman laser and (b) the results for a conventional Raman laser.

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Similar to Figs. 2(c) or 3(c), P_edge and P_surface in Figs. 6(a) and (b) rapidly increase as |S _+ 1|² becomes larger than the threshold (I_{th_cal}). N_p increases linearly in the region |S _+ 1|² < I_{th_cal} and becomes constant in the region |S _+ 1|² > I_{th_cal}. The calculated values of P_edge and P_surface are much larger than the experimentally measured values in Fig. 4. This is probably due to the fact that the calculations ignore TPA-induced FCA and, in the experiments, it was only possible to measure a part of P_edge. The difference between the P_edge values in Figs. 6(a) and (b) is discussed after the explanation of the threshold behavior.

By comparing Figs. 6(a) and (b), we find that the I_{th_cal} of the HM Raman laser is about half of that of the conventional Raman laser. In a HM Raman laser with θ_in = 0, S _+ 2 and S₋₁ interfere destructively and the excitation efficiency of the pump mode is improved. Therefore, the rate of increase in N_p in Fig. 6(a) is larger than that for Fig. 6(b) in the region |S _+ 1|² < I_{th_cal}.

The number of pump photons in the cavity at I_{th_cal}, N_{p_th}, in the case of the HM Raman laser is 1890, which is larger than that of the conventional laser. As explained below, this is due to the smaller Q_{S_total} of the HM Raman laser. Instead of Eq. (1), the Q_{i_total} of the HM laser is determined by the following equation:

Now we consider the influence of the HMs on P_edge. In a HM Raman laser with θ_out = 0, S _+ 3 and S₋₄ interfere constructively and thus the P_edge increases: the P_edge of the HM Raman laser in Fig. 6 is 938 nW at 3 I_{th_cal} while that of the conventional Raman laser is 197 nW. The ratio is 4.8, which is close to the ratio of the P_{edge_max} values of cavities #1 and #5 shown in Figs. 2(c) and 3(c). The calculation results in Fig. 6 are qualitatively consistent with the input–output characteristics of cavities #1 and #5.

To investigate the dependence of P_edge and P_surface on the round-trip phase shifts θ_in and θ_out, we calculated the input–output characteristics of the HM Raman laser for various sets of (θ_in, θ_out). Figure 7(a) corresponds to (θ_in, θ_out) = (0, 0) [the same graph as in Fig. 6(a)]. Figures 7(b)−(d) show the calculation results for (θ_in, θ_out) = (0, π), (π, 0), and (π, π), respectively.

 

Fig. 7. (a)−(d) Calculated input-output characteristics of the HM Raman laser for (θ_in, θ_out) = (0, 0), (0, π), (π, 0), and (π, π), respectively. (e)−(h) Dependence of P_edge on the (θ_in, θ_out) for four different values of |S _+ 1|² (0.1 × I_{th_cal_θ=0}, 1.0 × I_{th_cal_θ=0}, 1.5 × I_{th_cal_θ=0}, and 3.0 × I_{th_cal_θ=0}). (i) (θ_in, θ_out) dependence of P_surface at 3.0 × I_{th_cal_θ=0}. (j) Total Stokes output at 3.0 × I_{th_cal_θ=0}.

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The HM Raman lasers with θ_out = π have a ${Q^{\prime}_{{{\rm S}\_{\rm in}}}} = \infty $, that is, no coupling between the Stokes mode and the Stokes waveguide (P_edge is zero). Therefore, the P_surface in Fig. 7(b) is larger than that in Fig. 7(a). Since P_surface is also emitted towards the substrate, the P_surface does not increase as rapidly as the P_edge in Fig. 7(a). The values of I_{th_cal} and N_{p_th} in Fig. 7(b) are smaller than those in Fig. 7(a) because the Q_{S_total} for Fig. 7(b) is 1.02 × 10⁶, which is larger than that for Fig. 7(a), 5.20 × 10⁵. The larger Q_{S_total} leads to a decrease in N_{p_th}, but the rate of increase in N_p below I_{th_cal} is identical with that in Fig. 7(a) since the ${Q^{\prime}_{{\rm p}\_{\rm in}}}$ values are the same for both calculations. The calculation results shown in Fig. 7(b) with (θ_in, θ_out) = (0, π) exhibit the lowest I_{th_cal} and the largest P_surface.

In the case of θ_in = π, the excitation efficiency of the pump mode is zero because ${Q^{\prime}_{{\rm p}\_{\rm in}}} = \infty $. Therefore, no laser oscillation occurs in Figs. 7(c) and (d). The results of Figs. 7(a)−(d) indicate that the threshold and the P_edge of a HM Raman laser strongly depend on (θ_in, θ_out). Cavities #1−#4 in Fig. 4(a) probably exhibit a large variation in P_edge because the (θ_in, θ_out) values have a strong sample dependence.

The images in Figs. 7(e)−(h) show the (θ_in, θ_out) dependence of P_edge for different input powers. In each image, the red region shows the combinations of (θ_in, θ_out) that are possible to achieve a strong P_edge. The results of P_edge are a function of θ_in and θ_out with a period of 2π. Therefore, we hereafter discuss P_edge in the range −π < θ_in < π and −π < θ_out < π. We define I_{th_cal_θ=0} as the threshold power for (θ_in, θ_out) = (0, 0) [the threshold in Fig. 7(a)]. Figures 7(e)−(h) are the results for |S _+ 1|² = 0.1 × I_{th_cal_θ=0}, 1.0 × I_{th_cal_θ=0}, 1.5 × I_{th_cal_θ=0}, and 3.0 × I_{th_cal_θ=0}, respectively.

For |S _+ 1|² = 0.1 × I_{th_cal_θ=0}, P_edge exhibits a maximum at (θ_in, θ_out) = (0, 0). The maximum value is only 16 pW because the input power is low. It is known that the excitation efficiency of a cavity with a HM is maximized under the condition ${Q_{{\rm p}\_{\rm v}}} = {Q^{\prime}_{{\rm p}\_{\rm in}}}$ if the excitation power is small enough to be able to ignore the nonlinear effects of SRS or TPA [21]. Here, this condition is not realized for any θ_in since Q_{p_in} (5.32 × 10⁵) is more than two times larger than Q_{p_v} (2.27 × 10⁵). The highest excitation efficiency is realized for θ_in = 0, because here the minimum ${Q^{\prime}_{{\rm p}\_{\rm in}}}$ (Q_{p_in}/2) is obtained. The coupling between the Stokes mode and the Stokes waveguide is maximized at θ_out = 0, where ${Q^{\prime}_{{{\rm S}\_{\rm in}}}}$ reaches its minimum. As a result, P_edge reaches its maximum at (θ_in, θ_out) = (0, 0) in Fig. 7(e).

For |S _+ 1|² = 1.0 × I_{th_cal_θ=0}, the maximum P_edge is obtained at (θ_in, θ_out) = (0, ±π/2). This is because the threshold for the HM Raman laser with (θ_in, θ_out) = (0, ±π/2) is smaller than I_{th_cal_θ=0}: as θ_out approaches π or −π from 0, the value of ${Q^{\prime}_{{{\rm S}\_{\rm in}}}}$ increases according to Eq. (7) and Q_{S_total} increases according to Eq. (11), and thus N_{p_th} decreases. The lowest threshold is obtained at (θ_in, θ_out) = (0, π) and (0, −π), but P_edge becomes zero under such a condition as shown in Fig. 7(b). Therefore, in Fig. 7(f), the maximum P_edge is obtained at (θ_in, θ_out) = (0, ±π/2).

In Fig. 7(g), the value of (θ_in, θ_out) that is advantageous for P_edge shifts again to the region around (0, 0): as the excitation intensity become higher than 1.0 × I_{th_cal_θ=0}, strong coupling with the Stokes waveguide due to a smaller ${Q^{\prime}_{{{\rm S}\_{\rm in}}}}$ is more advantageous than a lower threshold. In Fig. 7(h), which is for 3.0 × I_{th_cal_θ=0}, a maximum P_edge of about 900 nW is obtained at (θ_in, θ_out) = (0, 0).

In the experimental results shown in Figs. 2(c) and 3(c), the maximum P_edge appears at excitation powers near 3.0 × I_th. Thus, the calculation condition of Fig. 7(h) should be considered in discussions on further increases in P_edge. Although (θ_in, θ_out) = (0, 0) does not minimize the threshold, these round-trip phase shifts are the most desirable values to increase the P_edge of HM Raman lasers. This feature is maintained even if the four Q values change, which is demonstrated in Section 6. Since cavity #1 has the highest I_th and the largest P_edge among the fabricated HM Raman lasers, the calculation result in Fig. 7(h) suggests that the (θ_in, θ_out) of cavity #1 is close to (0, 0).

Figure 7(i) shows the (θ_in, θ_out) dependence of P_surface at 3.0 × I_{th_cal_θ=0}. The maximum P_surface is obtained at (θ_in, θ_out) = (0, ±π). This clarifies that the HM laser with (θ_in, θ_out) = (0, ±π) realizes the lowest threshold and the largest P_surface. Cavity #4 exhibits the largest P_surface as shown in Table 1, while its P_edge is the lowest. Furthermore, this cavity has the highest Q_{S_exp} and the lowest I_th. Based on these features, we consider that cavity #4 has a (θ_in, θ_out) close to (0, ±π).

Figure 7(j) shows the (θ_in, θ_out) dependence of the total output (P_edge + 2 × P_surface) at 3.0 × I_{th_cal_θ=0}. In the case of θ_in = 0, the Stokes output is larger than 1900 nW, independent of θ_out. On the other hand, the intensities in the range 21π/25 ≤ θ_in ≤ π are less than 10 pW since laser oscillation does not occur due to the low excitation efficiency. This means that laser oscillation may not occur even if the Q_exp values are sufficiently high and the Δf is in principle appropriate for laser oscillation.

To clarify the impact of a change in either θ_in or θ_out, we summarize the I_{th_cal} and the N_{p_th} for (θ_in, θ_out) = (0, 0), (0, π/3), (0, 2π/3), (0, 5π/6), (π/3, 0), (2π/3, 0), and (5π/6, 0) in Table 3. The impact of θ_out on Q_{S_total} is described by Eqs. (7) and (11), and thus a variation of θ_out changes N_{p_th} and I_{th_cal}. The impact of θ_in on the excitation efficiency is a result of a change in Q_{p_total} according to Eqs. (6) and (11). As a result, a variation of θ_in changes only I_{th_cal}.

Table 3. I_{th_cal} and N_{p_th} for the seven input–output relations displayed in Fig. 8

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Table 4. Parameters used in the calculations

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Figure 8(a) shows the normalized input–output characteristics for (θ_in, θ_out) = (0, 0), (0, π/3), (0, 2π/3), and (0, 5π/6) (we normalized the input power to the threshold for each sample). P_edge and P_surface are shown by the orange and black curves, respectively. Similarly, Fig. 8(b) shows the characteristics for (θ_in, θ_out) = (0, 0), (π/3, 0), (2π/3, 0), (5π/6, 0). Figure 8 shows that P_edge varies relatively strongly with (θ_in, θ_out). This is probably the reason why the variation in P_edge for the HM Raman lasers in Fig. 4 is 1.42 times larger than that for the conventional Raman lasers. Although the results in Fig. 4(a) are for cavities with different Q values and different Δf values, the degree of consistency with the results in Fig. 8 is good. The (θ_in, θ_out) values of the four HM Raman lasers shown in Table 1 are probably different from each other. The control of (θ_in, θ_out) is thus considered important to reduce the variation in P_edge.

 

Fig. 8. (a) Calculated normalized input–output characteristics of HM Raman lasers for θ_in = 0 and seven different values of θ_out. (b) The input–output characteristics for θ_out = 0 and seven different values of θ_in.

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The 2D plot in Fig. 9 shows the ratio of the P_edge of the HM Raman laser at 3.0 × I_{th_cal_θ=0} [Fig. 7(h)] to the P_edge of the conventional Raman laser at 3.0 × I_{th_cal} [Fig. 6(b)]. The maximum value is 4.76, which is obtained at (θ_in, θ_out) = (0,0). The red region describes the (θ_in, θ_out) range where the P_edge of the HM Raman laser is higher than that of the conventional laser (P_{edge_HM} / P_{edge_conv} > 1.0). The area of this region is 56% of the total area. The area of the region where P_{edge_HM} / P_{edge_conv} > 3.0 is 22% of the total area. Cavities #1–#3 have a larger P_{edge_max} than the conventional laser cavities #5–#9, and thus their (θ_in, θ_out) pairs may be located in the red region. On the other hand, the (θ_in, θ_out) pair of cavity #4 may be located in the blue region. We conclude that it is important to choose a HM Raman laser design with a (θ_in, θ_out) close to (0, 0) and to fabricate the device accurately.

 

Fig. 9. (θ_in, θ_out) dependence of P_{edge_HM} / P_{edge_conv}. To prepare this plot, we used the P_{edge_HM} values for 3.0 × I_{th_cal_θ=0} and the P_{edge_conv} values for 3.0 × I_{th_cal}. The red regions show conditions where P_{edge_HM} / P_{edge_conv} > 1.0.

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6. Discussion

The two types of Raman silicon nanocavity lasers investigated in Section 3 have the same nanocavity design and they were fabricated on the same chip with a separation of 100 µm. They have similar average Q values as shown in Tables 1 and 2, while the average P_{edge_max} of the HM Raman lasers is 4.3 times larger than that of the conventional Raman lasers. On the other hand, the average P_{surface_max} of the conventional Raman lasers is 1.84 times larger than that of the HM Raman lasers. The input–output characteristics presented in Figs. 6(a) and (b), which were calculated using Q values that are consistent with the experimental Q values of cavity #5, are in good agreement with the experimental results shown in Figs. 2(c) and 3(c). It is apparent that adding HMs to a Raman silicon nanocavity laser can lead to an increase in P_edge.

As shown in Fig. 4(a), the P_edge of the HM Raman laser varies significantly from sample to sample. This is most likely due to the difference in the round-trip phase shift (θ_in, θ_out), as explained in Figs. 7 − 9. Figure 9 clarified that the values of (θ_in, θ_out) should be close to (0,0) in order to obtain a high P_edge. To this end, numerical investigations of the relation between (θ_in, θ_out) and the distances between the cavity and the ports, d_k, would be useful. Furthermore, the influence of the reflection at ports 1 and 4 should be studied. Such studies will also be useful for the design of a SSC for the output and input ports. In addition, we need to consider that the positions and radii of the air holes randomly deviate from the design values due to fabrication imperfections [32,33]. These variations can affect θ_in and θ_out. It will be important to perform device simulations by 3D FDTD calculations including random deviations of the air hole pattern.

The experimentally determined maximum value of P_edge shown in Fig. 2(c) is about two orders of magnitude smaller than the calculated one shown in Fig. 6(a). We can consider three possible causes for this result: Firstly, only a part of the total P_edge was measured. Secondly, the actual absorption losses in the fabricated sample (1/Q_{i_abs}) are significantly large although they were assumed to be zero in the calculation. In the fabricated sample, the magnitude of Q_{S_abs} is expected to be larger than 5.0 × 10⁶. On the other hand, the Q_{p_abs} can be less than 5.0 × 10⁵. This is comparable to Q_{p_v}, and thus it is possible that about half of the pump photons were lost due to absorption. The third possible cause is that the Q_{S_v} in the experiment decreased by TPA-induced FCA. A Q_{S_FCA} of about 1 million has been previously reported in the high-excitation power regime [38], which is also comparable to Q_{S_v}. Therefore, it is possible that about half of the Stokes photons were lost due to FCA.

To increase the edge emission, not only the round-trip phase shifts θ_in and θ_out are important, but also Q_{p_v}, Q_{p_in}, Q_{S_v} , and Q_{S_in}. The values of Q_{p_v}, Q_{p_in}, Q_{S_v}, and Q_{S_in} used in Section 5 are 2.27 × 10⁵, 5.32 × 10⁵, 1.02 × 10⁶, and 2.12 × 10⁶, respectively. This set of Q values belongs to the regime where Q_{p_v} < Q_{p_in} and Q_{S_v} < Q_{S_in}. We define this as regime 1. Figures 10(a) and (b) are the results of calculations in which the first two values are exchanged (resulting in regime 2 with Q_{p_v} < Q_{p_in} and Q_{S_v} > Q_{S_in}), while Figs. 10(c) and (d) are for the calculations in which the last two values are exchanged (resulting in regime 3 with Q_{p_v} > Q_{p_in} and Q_{S_v} < Q_{S_in}). Figures 10(e) and (f) are the results of calculations in which both the first two and the last two values are exchanged, resulting in regime 4 with Q_{p_v} > Q_{p_in} and Q_{S_v} > Q_{S_in}.

 

Fig. 10. Calculation results for different Q-value sets. The left column shows the dependence of P_edge on (θ_in, θ_out) at 3.0 × I_{th_cal_θ=0} for (a) regime 2, (c) regime 3, and (e) regime 4. The right column shows P_edge, P_surface, and N_p as a function of |S _+ 1|² for (b) regime 2, (d) regime 3, and (f) regime 4. Here, we used the condition (θ_in, θ_out) = (0, 0).

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Figures 10(a), (c), and (e) show the (θ_in, θ_out) dependence of P_edge at |S _+ 1|² = 3.0 × I_{th_cal_θ=0} for the sets of Q values in regimes 2 − 4, respectively (the dependence for other excitation intensities are shown in Figs. 12 − 14 in Appendix A5). These three figures and Fig. 7(h) clearly indicate that the condition (θ_in, θ_out) = (0, 0) maximizes P_edge in all four regimes.

Figures 10(b), (d), and (f) show the input–output characteristics for the regimes 2 − 4 under the condition (θ_in, θ_out) = (0, 0). In Fig. 10(b), the P_edge at 3.0 × I_{th_cal} is 1970 nW. This is 2.1 times larger than that in regime 1, because the condition Q_{S_v} > Q_{S_in} increases the output efficiency to the Stokes waveguide. In Fig. 10(d), the P_edge at 3.0 × I_{th_cal} is 1220 nW, which is 1.3 times larger than that in regime 1, because the excitation efficiency of the pump mode increases under the condition Q_{p_v} > Q_{p_in}. The enhancement of P_edge in regime 2 is larger than that in regime 3, indicating that the condition Q_{S_v} > Q_{S_in} is more important for P_edge. In Fig. 10(f), the P_edge at 3.0 × I_{th_cal} is 2540 nW, which is 2.7 times larger than that of the regime 1. The P_edge in regime 4 is the largest among the values in the four regimes.

The above shows that we should fabricate a HM Raman laser with (θ_in, θ_out) = (0, 0), Q_{p_v} > Q_{p_in}, and Q_{S_v} > Q_{S_in} in order to increase P_edge. Raman lasers with Q_{p_exp} > 1.0 × 10⁶ and Q_{S_exp} > 3.0 × 10⁶ have already been reported [41]. The Q_{i_in} can be controlled by the width of the excitation waveguide or the distance between the waveguide and the nanocavity, d_i. Therefore, it is possible to fabricate a HM Raman laser with Q values in regime 4.

Finally, we comment on the remaining challenges regarding the fabrication process. The Q_{p_v} and Q_{S_v}, which are expressed in terms of Q_design, Q_scat, and Q_abs in Eq. (3), should be as high as possible. The Q_{i_design} can be further increased by using machine learning [42], and thus it is important to improve the fabrication process in such a way that 1/Q_scat and 1/Q_abs become smaller. This will be important not only for Raman lasers fabricated using electron-beam lithography, but also for samples fabricated by CMOS-compatible processes [43]. Since a reverse-biased diode structure adjacent to the nanocavity can shorten the lifetime of TPA carriers, the P_edge in the high-excitation regime can be improved by adding such a structure [10,11,44]. Recently, the combination of a p–i–n diode and a nanocavity, where a Q value larger than one million was maintained, has been reported [26]. Raman silicon nanocavity lasers with a reverse-biased diode can be developed. The addition of SSCs to the waveguide edges can further improve P_edge by several times [45].

7. Summary

We investigated the increase in the laser emission from the edge of the Stokes waveguide of a Raman silicon nanocavity laser that can be achieved by adding HMs. The average intensity of the edge emission of our Raman lasers with HMs is 4.3 times stronger than that of our conventional Raman lasers. Numerical calculations based on coupled-mode theory reproduce the experimentally observed increase in the edge emission and its variation well. The calculations reveal that the round-trip phase shifts (θ_in, θ_out) should be close to (0, 0) and the Q_{i_v} values should be larger than the Q_{i_in} values.

Appendix

A1. Experimental method

Figure 11 shows the measurement system used to obtain the results presented in Section 3. The figure illustrates the situation in which laser oscillation occurs. A continuous-wave tunable laser (Santec TSL-510) was used as the light source. The wavelength of the tunable laser was accurately measured by a wavelength meter (Agilent 86122A). The excitation light (Fig. 11; blue beam) was focused at the edge of the excitation waveguide (Fig. 1; input port) by an objective lens with a numerical aperture (NA) of 0.4. The surface emission was collected by an objective lens placed above the silicon slab (NA = 0.7) and detected by an InGaAs photodiode (Fig. 11; photodiode 1) with a lock-in amplifier. The edge emission was collected by the objective lens on the left-hand side of the sample (NA = 0.4) in Fig. 11 and detected by photodiode 2. Because of the limitations of the lens diameter and the working distance, the NA of the lens that collects the edge emission is smaller than that for the surface emission. As indicated in Fig. 11, the excitation light and the light emitted from the waveguide edge was passed through polarizers that transmit the TE component.

 

Fig. 11. Measurement system.

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The resonance spectra in Figs. 2 and 3 were obtained by measuring the wavelength of the tunable laser and the surface and edge emission while changing the wavelength gradually from short to long wavelengths. The pump power used in these measurements was small enough to be able to neglect the effect of TPA in the analysis. The long-pass filters (LPFs) were not used when we measured the resonance spectra.

TPA-induced carriers can lead to a shift of λ_p mainly through the thermo-optic effect and the carrier plasma effect [38,40]. When we measured the input–output characteristics of the Raman lasers [Figs. 2(c) and 3(c)], we used the excitation wavelength at which the laser intensity is maximized (in other words, we always adjusted the wavelength of the tunable laser to the shifted λ_p [40]). The LPFs with a cutoff wavelength of 1500 nm were inserted into the detection paths to remove the pump light. To obtain the NIR camera images, switching mirrors were inserted into the optical paths. The pump power was estimated from the surface emission at λ_p in the weak-excitation regime by assuming that the power coupled into the pump mode is linearly proportional to the power of the pump laser. This estimation method includes an uncertainty due to the variation in the collection efficiency of the emitted light (since the emission pattern is slightly different for each sample) [33]. Therefore, the estimated I_th should have a relative error equal to that of the collection efficiency.

A2. Relation between the threshold and the Q values

In the case that TPA-induced FCA loss can be ignored, the laser threshold I_th is inversely proportional to the product Q_{p_exp}·Q_{S_exp} and the Raman gain coefficient of the nanocavity, $g_\textrm{R}^{\textrm{cav}}$ [12]:

(12)$${I_{\textrm{th}}} \propto {({g_\textrm{R}^{\textrm{cav}}{Q_{{{\rm p}\_{\rm{exp}}}}}{Q_{\rm S}\_{\rm{exp}}}} )^{ - 1}}.$$

According to this relation, the I_th values presented in Tables 1 and 2 are larger than in the previous studies.

A3. Coupled-mode theory and calculation parameters

In Fig. 5, the time evolution of the amplitudes of the pump light confined in the nanocavity (a_p) and the Stokes light (a_S) can be calculated using the following rate equations [12,13]:

(13)$$\begin{array}{l} \frac{{d{a_\textrm{p}}}}{{dt}} = (j{\omega _\textrm{p}} - \frac{{{\omega _\textrm{p}}}}{{2{Q_{{\rm v}\_{\rm p}}}}} - \frac{{{\omega _\textrm{p}}}}{{2{Q_{{\rm{in}}\_{\rm p}}}}}){a_\textrm{p}} - g_\textrm{R}^{\textrm{cav}}({N_\textrm{S}} + 1){a_\textrm{p}}\\ + \sqrt {\frac{{{\omega _\textrm{p}}}}{{2{Q_{{\rm{in}}\_{\rm p}}}}}} {e^{ - j\beta {d_1}}}{S_{ + 1}} + \sqrt {\frac{{{\omega _\textrm{p}}}}{{2{Q_{{\rm{in}}\_{\rm p}}}}}} {e^{ - j\beta {d_2}}}{S_{ + 2}}, \end{array}$$

(14)$$\begin{array}{l} \frac{{d{a_\textrm{S}}}}{{dt}} = (j{\omega _\textrm{s}} - \frac{{{\omega _\textrm{s}}}}{{2{Q_{{\rm v}\_{\rm S}}}}} - \frac{{{\omega _\textrm{s}}}}{{2{Q_{{\rm{in}}\_{\rm S}}}}}){a_\textrm{s}} + g_\textrm{R}^{\textrm{cav}}{N_\textrm{P}}{a_\textrm{s}}\\ + \sqrt {\frac{{{\omega _\textrm{s}}}}{{2{Q_{{\rm{in}}\_{\rm S}}}}}} {e^{ - j\beta {d_3}}}{S_{ + 3}} + \sqrt {\frac{{{\omega _\textrm{s}}}}{{2{Q_{{\rm{in}}\_{\rm S}}}}}} {e^{ - j\beta {d_4}}}{S_{ + 4}}. \end{array}$$

Here, j is the imaginary unit and β is the propagation constant. N_i with i = p and S describes the number of pump photons and Stokes photons in the nanocavity, respectively. N_i has the following relationship with a_i:

(15)$${N_i} = \frac{{{{|{{a_i}} |}^2}}}{{\hbar {\omega _i}}}\textrm{(}i\textrm{ = p,S)}\textrm{.}$$

The Raman gain coefficient of the silicon nanocavity can be written as

(16)$$g_\textrm{R}^{\textrm{cav}} = \frac{{\hbar {\omega _\textrm{p}}{c^2}{g_{{\rm R}\_{\rm{Si}}}}}}{{2{n_\textrm{p}}{n_\textrm{S}}{V_\textrm{R}}}}$$

Here, n_i is the refractive index, ω_p is the angular frequency of the pump light, c is the speed of light, and g_{R_Si} is the Raman gain coefficient of bulk silicon. V_R is the effective mode volume for Raman scattering. V_R is determined by the electromagnetic field distributions of the two nanocavity modes and the Raman selection law of crystalline silicon. Our Raman silicon nanocavity laser extends along the [100] direction of silicon to reduce V_R [31]. The amplitudes of the light propagating in the waveguides, S_+k and S_−k, are given by

(17)$${S_{ - 1}} = {e^{ - j\beta ({d_1} + {d_2})}}{S_{ + 2}} - {e^{ - j\beta {d_1}}}\sqrt {\frac{{{\omega _\textrm{p}}}}{{2{Q_{{\rm{in}}\_{\rm p}}}}}} {a_\textrm{p}},$$

(18)$${S_{ - 2}} = {e^{ - j\beta ({d_1} + {d_2})}}{S_{ + 1}} - {e^{ - j\beta {d_2}}}\sqrt {\frac{{{\omega _\textrm{p}}}}{{2{Q_{{\rm{in}}\_{\rm p}}}}}} {a_\textrm{p}},$$

(19)$${S_{ - 3}} = {e^{ - j\beta ({d_3} + {d_4})}}{S_{ + 4}} - {e^{ - j\beta {d_3}}}\sqrt {\frac{{{\omega _\textrm{s}}}}{{2{Q_{{\rm{in}}\_{\rm S}}}}}} {a_\textrm{s}},$$

(20)$${S_{ - 4}} = {e^{ - j\beta ({d_3} + {d_4})}}{S_{ + 3}} - {e^{ - j\beta {d_4}}}\sqrt {\frac{{{\omega _\textrm{s}}}}{{2{Q_{{\rm{in}}\_{\rm S}}}}}} {a_\textrm{s}}.$$

In our model, the amplitudes S _+ 2 and S _+ 3 are the result of HMs with 100% reflectivity:

(21)$${S_{ + 2}} = {S_{ - 2}}{e^{ - j{\Delta _{\textrm{in}}}}},$$

(22)$${S_{ + 3}} = {S_{ - 3}}{e^{ - j{\Delta _{\textrm{out}}}}}.$$

The parameters Δ_in and Δ_out are the phase shifts due to the reflection at the HMs. The reflection at Port 4 is zero and thus S _+ 4 = 0. The round-trip phase shifts θ_in and θ_out are defined by the following two equations:

(23)$${\theta _{\textrm{in}}} = 2\beta {d_2} + {\Delta _{\textrm{in}}}.$$

(24)$${\theta _{\textrm{out}}} = 2\beta {d_3} + {\Delta _{\textrm{out}}}.$$

A4. Estimation of the four Q values used in Section 5

In Section 5, we used Q_{p_v} = 2.27 × 10⁵, Q_{p_in} = 5.32 × 10⁵, Q_{S_v} = 1.02 × 10⁶, and Q_{S_in} = 2.12 × 10⁶. The experimental Q_{i_exp} values have the following relationship with Q_{i_in} and Q_{i_v} (i = p, S):

(25)$$\frac{1}{{{Q_{i{\_{\rm{exp}}}}}}} = \frac{1}{{{Q_{i{\_{\rm v}}}}}} + \frac{1}{{{Q_{i{\_{\rm{in}}}}}}}.$$

Furthermore, according to coupled-mode theory, Q_{i_v} has the following relationship with Q_{i_exp} [47]:

(26)$$\frac{1}{{{Q_{i{\_{\rm v}}}}}} = \frac{{{Q_{i{\_{\rm{exp}}} }}}}{{\sqrt T }}$$

Here, T is the transmittance at either λ_p or λ_S of the transmission spectrum. From Fig. 3(b), we obtained Q_{S_exp} = 6.89 × 10⁵ and T = 0.48. These values were used for the estimation of Q_{S_v} = 1.02 × 10⁶ and Q_{S_in} = 2.12 × 10⁶. On the other hand, the transmittance for the pump mode cannot be correctly estimated from Fig. 3(a). Therefore, we derived the Q_{p_in} of 5.32 × 10⁵ from Eq. (1) with Q_{p_design} = 6.14 × 10⁵ and Q_{p_total} = 2.85 × 10⁵ obtained by FDTD calculations. Then, the Q_{p_v} of 2.27 × 10⁵ was estimated from Eq. (25) with Q_{p_exp} of 1.59 × 10⁵ and the Q_{p_in} estimated above.

A5. (θ_in, θ_out) dependence of P_edge in regimes 2–4 for various input powers

Figures 12(a)−(d) present the (θ_in, θ_out) dependence of P_edge in regime 2 for |S _+ 1|² = 0.1 × I_{th_cal_θ=0}, 1.0 × I_{th_cal_θ=0}, 1.5 × I_{th_cal_θ=0}, and 3.0 × I_{th_cal_θ=0}, respectively. The images in Figs. 13 and 14 correspond to the results in regimes 3 and 4, respectively. The calculation results for regime 1 are shown in Figs. 7(e)−(h).

 

Fig. 12. (a)−(d) Dependence of P_edge on (θ_in, θ_out) in regime 2 for 0.1 × I_{th_cal_θ=0}, 1.0 × I_{th_cal_θ=0}, 1.5 × I_{th_cal_θ=0}, and 3.0 × I_{th_cal_θ=0}, respectively.

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Fig. 13. (a)−(d) Dependence of P_edge on (θ_in, θ_out) in regime 3 for 0.1 × I_{th_cal_θ=0}, 1.0 × I_{th_cal_θ=0}, 1.5 × I_{th_cal_θ=0}, and 3.0 × I_{th_cal_θ=0}, respectively.

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Fig. 14. (a)−(d) Dependence of P_edge on (θ_in, θ_out) in regime 4 for 0.1 × I_{th_cal_θ=0}, 1.0 × I_{th_cal_θ=0}, 1.5 × I_{th_cal_θ=0}, and 3.0 × I_{th_cal_θ=0}, respectively.

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We can confirm that the images of regime 2 are similar to those of regime1. On the other hand, the images for 0.1 × I_{th_cal_θ=0} to 1.5 × I_{th_cal_θ=0} in regimes 3 and 4 differ significantly from those in regime1 [Figs. 7(e)−(g)]. This is because Q_{p_v} is larger than Q_{p_in} in regimes 3 and 4, whereas Q_{p_v} is smaller than Q_{p_in} in regime 1. As mentioned in the explanation of Fig. 7(e), in the case that Q_{p_v} = ${Q^{\prime}_{{\rm p}\_{\rm in}}}$, the excitation efficiency of the pump mode is maximized [21]. In regimes 1 and 2, this matching condition cannot be satisfied with the given set of Q values. On the other hand, in regimes 3 and 4, Q_{p_v} is equal to ${Q^{\prime}_{{\rm p}\_{\rm in}}}$ at about θ_in = 25π/36. Therefore, P_edge reaches the maximum near (θ_in, θ_out) = (25/36π, 0) at 0.1 ×, 1.0 ×, and 1.5 × I_{th_cal_θ=0}. However, at 3.0 × I_{th_cal_θ=0}, P_edge reaches its maximum under the condition (θ_in, θ_out) = (0, 0).

Funding

Program for Creating STart-ups from Advanced Research and Technology (JPMJST2032, JPMJST2111); Japan Society for the Promotion of Science (18H01479, 21H01373, 22H01988).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

References

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