1. Introduction

Silicon is a suitable material for cavities with high quality (Q)-factors and small values of the mode volume (V) due to the availability of high-purity substrates and well-established microfabrication technologies. Since the interaction strength between light and matter in a material is proportional to the ratio Q/V of a cavity, stimulated Raman scattering can be enhanced by using high-Q silicon microcavities. Therefore, the development of Raman silicon lasers has attracted attention [1–9]. Continuous-wave (cw) operation of a Raman silicon laser at room temperature was first reported using a rib-waveguide [10]. A decrease of the laser threshold and cascaded Raman laser operation were achieved by using high-Q ring cavities based on rib-waveguides [11,12].

In 2013, a Raman silicon laser with a threshold of 1 µW was reported [13]. This device was based on a high-Q photonic crystal (PC) heterostructure nanocavity whose V is about four orders of magnitude smaller than that of rib-waveguide-based resonators. So far, important mechanisms related to the performance of such Raman silicon nanocavity lasers have been clarified (for example, mechanisms such as the initial lasing process and the wavelength dependence of the Raman gain) [14,15]. In addition, laser operation in the 1.3-µm band and fabrication by a complementary metal–oxide–semiconductor (CMOS)-compatible process have been demonstrated [16,17]. The study of methods that enable a further decrease of the threshold, is considered important for applications and is also interesting from the viewpoint of fundamental physics.

A Raman silicon nanocavity laser utilizes two high-Q resonance modes to confine the pump light and the Stokes Raman scattering light into the nanocavity. These two modes are hereafter referred to as the pump mode and the Stokes mode. In order to reduce the laser threshold, the increase of the Q factor of the pump mode (Q_p) and that of the Stokes mode (Q_S) should be considered, because the theoretically predicted threshold is inversely proportional to the product Q_p × Q_S if nonlinear optical losses are ignored [7]. We note that in all our previous studies on Raman silicon nanocavity lasers, we used the same nanocavity design. In other words, in these previous studies, we investigated devices whose theoretical Q factors (Q_design) were almost the same.

It has been demonstrated that the Q_design of a PC nanocavity can be improved by appropriately shifting the air holes in the PC slab [18–21]. For example, a visualization of the leaky components of the electric field enables us to identify the air holes whose positions should be optimized to improve the Q_design [22,23]. In Refs. [22,23], the nanocavities were manually designed by repeating the cycle of three-dimensional (3D) finite-difference time-domain (FDTD) calculation and hole-position adjustment. However, it is difficult to simultaneously increase the Q_{p_design} and the Q_{S_design} of a Raman laser by such a manual method. This is because the pump mode and the Stokes mode have different spatial symmetries, and furthermore, their resonance wavelengths are separated by more than 100 nm. Recently, several types of automated optimization methods for nanocavity structures have been demonstrated to enable simultaneous optimization of a larger number of structural parameters than in manual optimization methods [24–26]. Because such automated optimization methods can search designs that are very complex, it is expected that a nanocavity design that simultaneously exhibits a higher Q_{p_design} and a higher Q_{S_design}, can be obtained by using such a method.

Here we use an automated optimization method based on machine learning to obtain a new nanocavity design with a Q-factor product Q_{p_design} × Q_{S_design} that is 17.0 times higher than that of the conventional Raman silicon nanocavity laser design. We fabricate and measure ten samples for each of these two cavity designs. It is shown that the average experimental value of Q_p × Q_S for the new design is more than 2.5 times larger than that for the conventional design. This is the first report of an input–output characteristic of a Raman silicon laser with a sub-100-nW threshold. Furthermore, a threshold of 40 nW is observed for one of the cavities with the new design. This threshold is three times smaller than the previously reported lowest threshold for a Raman silicon laser [14].

2. Nanocavity design

The cavities used in this work are basically formed by a line defect consisting of 27 missing holes in a PC slab (the conventional heterostructure nanocavity design used for the reference samples is described in Appendix A1). Figure 1(a) shows the new nanocavity structure whose air hole positions near the line defect were optimized by an automated method based on machine learning. Here, we employed a neural network (NN) to increase the product of Q_{p_design} and Q_{S_design}, and therefore this structure is hereafter referred to as the NN cavity (the details of the optimization method are provided in Appendix A2). The arrows in Fig. 1(a) indicate the displacements of the 62 holes that were considered in the optimization, with respect to their original positions before the automated optimization (the numerical values of the displacements are provided in Appendix A3). The arrows are magnified by a factor of 1000 to make the displacements visible. The structure before the automated optimization corresponds to the conventional cavity design, which is a multi-heterostructure nanocavity. The Q_{p_design} and Q_{S_design} for the conventional design, calculated by 3D FDTD, are 6.1 × 10⁵ and 5.2 × 10⁶, respectively. The Q_{p_design} and Q_{S_design} for the NN cavity are 5.4 × 10⁶ and 1.0 × 10⁷, respectively. A comparison of the optimization method described in Appendix A2 with other automated optimization approaches, such as the genetic algorithm [20] and the particle-swarm optimization [21], can be found in the literature [24,27,28].

 

Fig. 1. (a) Schematic of the heterostructure nanocavity designed by a method based on machine learning. (b) Band diagram of the nanocavity. (c) The x- and y-components of the electric field distribution for the pump nanocavity mode, E_{x_pump} and E_{y_pump}, and (d) the components for the Stokes nanocavity mode, E_{x_Stokes} and E_{y_Stokes}.

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Compared to the Q factors of the conventional design, the small displacements of the air holes shown in Table 2 in the Appendix A3 induce an increase of Q_{p_design} by 8.9 times and an increase of Q_{S_design} by 1.9 times, leading to an increase in the Q-factor product by 17.0 times. Obviously, the extent of the increase in Q_{p_design} is larger than that for Q_{S_design}. The reason for the relatively small enhancement of the latter may be that the conventional nanocavity has been well optimized with respect to Q_S. Since it is difficult to fabricate nanocavities with Q_S values larger than 10⁷, the larger increase in Q_{p_design} is desirable.

To achieve a low threshold, it is important to consider that the frequency spacing (Δf) between the resonance wavelengths of the pump and Stokes modes (λ_p and λ_S, respectively) should match the Raman shift of silicon well, that is, Δf should be close to 15.606 THz [29]. Since the calculated λ_p (λ_S) of the NN cavity and the λ_p (λ_S) of the conventional cavity only differs by about 1 nm, the Δf for the NN cavity is almost the same as that for the conventional design.

If nonlinear optical losses in the nanocavity are ignored, the laser threshold (I_th) is inversely proportional to the product Q_p·Q_S and the Raman gain coefficient of the nanocavity, $g_\textrm{R}^{\textrm{cav}}$ [7]:

Therefore, a lower I_th can be also obtained by increasing $g_\textrm{R}^{\textrm{cav}}$, which obeys the following relation [14]:

The variables ${g_{\textrm{R\_Si}}}$ and Δ are the Raman gain coefficient of bulk silicon and the Raman gain width, respectively. These values depend on the crystallinity of silicon, and thus it is difficult to improve these values. The variable Δf_det. denotes the detuning of the actual Δf from 15.606 THz. The value of Δf_det. can be controlled by the radius of the air holes [30] and we confirmed that there is no difference between the controllability of the Δf_det. of the NN cavity and that of the conventional cavity. V_R is the effective modal volume for Raman scattering where the spatial overlap between the pump mode and the Stokes mode is important. Here, the Raman selection rule of crystal silicon needs to be considered [13].

Figures 1(c) and (d) show the calculated electric field distributions of the pump and Stokes modes for the NN cavity, respectively. When the nanocavity is fabricated along to the [100] crystal direction of (100) silicon, the overlap between E_{x_pump} (E_{y_pump}) and E_{y_Stokes} (E_{x_Stokes}) determines V_R according to [31]

Therefore, the overlap integral is also an important factor for the laser threshold. It is noted that the optimization procedure described in Appendix A2 does not consider the overlap integral in the optimization process, and thus we need to assess the overlap in the NN cavity and that in the conventional cavity. For comparison, we calculated the overlap values for the Raman gain by using the electric field distribution at the time instant when the electric energy reaches its maximum in the FDTD simulation. The overlap of the NN cavity estimated by this method is 84% of that of the conventional cavity. Since the increases in Q_{p_design} and Q_{S_design} are much larger than the decrease in the overlap integral, a significant decrease in I_th can still be expected for the NN cavity. Therefore, we discuss the results in the following sections from the viewpoint of the Q values.

3. Device design and fabrication

We fabricated twenty Raman silicon nanocavity lasers (ten devices based on the NN cavity and ten devices based on the conventional cavity design) on the same chip. We used a silicon-on-insulator (SOI) wafer with a 45-degree-rotated top silicon layer in order to match the cleavable direction of the SOI substrate, i.e. [110], with the direction [100] at which the Raman lasers were fabricated [32]. The thickness of the top Si layer and the buried oxide (BOX) layer of the SOI wafer were 220 nm and 3 µm, respectively. The employed fabrication process was the same as that explained in our previous reports [32,33]. The PC patterns were defined by electron beam (EB) lithography and dry etching. All twenty cavities were drawn within an area of 200 µm × 150 µm. The BOX layer underneath the PC pattern was removed to form an air-bridge structure.

Figure 2(a) shows a scanning electron microscope (SEM) image of one fabricated NN cavity. From this image it is not possible to confirm the air hole shifts presented in Table 2 in Appendix A3 because the number of pixels is insufficient. The two line defects above and below the nanocavity are the waveguides used for excitation of the pump mode and the Stokes mode, respectively. The widths of the two waveguides are $0.88 \times \sqrt 3 {a_1}$ and $1.10 \times \sqrt 3 {a_1}$. To enable an efficient measurement of the two cavity types, we prepared two sets of excitation waveguides (one set consist of one pump and one Stokes waveguide) and placed ten nanocavities with a spacing of 20 µm along each waveguide set [25].

 

Fig. 2. (a) SEM image of one fabricated NN cavity. (b) The theoretically predicted Q_{p_total} (closed symbols) and Q_{p_in} (open symbols) for the pump mode as a function of d_p. The red data is for the NN cavity, and the black data is for the conventional cavity. (c) The theoretically predicted Q_{S_total} and Q_{S_in} as a function of d_S.

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Figure 2(a) also shows that the distance between the cavity and the pump waveguide, d_p, is 4 rows. That cavity–waveguide distance used for the Stokes waveguide is d_S = 8 rows. It is important to consider that the Q factor of a cavity decreases by coupling to a waveguide [34]. We use the term Q_total to denote the calculated Q factor including the load of the waveguide (Q_in):

In our experiment, it is advantageous if the pump mode is excited efficiently. Because the excitation efficiency of a nanocavity mode is maximized when Q_in is equal to Q_design, we used d_p = 4, that is, the pump mode of the conventional cavity is efficiently excited. Regarding d_S, we used a value of 8 rows for both cavities, because it is difficult to measure the light emission from the Stokes mode if d_S > 8 rows. The Stokes waveguide is added to investigate the Q and λ of the Stokes nanocavity mode but is not essential for the laser operation. In fact, adding the Stokes waveguide decreases the quality factor [see Eq. (4)], and thus results in an increase of the laser threshold. On the other hand, the Stokes waveguide is an indispensable element when extracting the Raman laser output to a waveguide. Note that device structures with a heterointerface mirrors have been studied with the aim of an efficient in-plane operation of nanocavity Raman lasers [35]. In such a device configuration, a high Q_{S_design} is important for a high efficiency.

In the configuration shown in Fig. 2(a), the Q_{p_total} and the Q_{S_total} for the NN cavity are 9.9 × 10⁵ and 8.7 × 10⁶, respectively. The values for the conventional cavity are 2.8 × 10⁵ and 4.7 × 10⁶, respectively. The Q_{p_total} for the NN cavity is five times smaller than the Q_{p_design} since the d_p is set to 4 rows for comparison with the conventional cavity. As a result, compared to the conventional cavity, the Q_{p_total} of the NN cavity is 3.5 times larger, and the Q_S is 1.9 times larger. The product Q_{p_total} × Q_{S_total} of the NN cavity including waveguides is thus 6.5 times larger than that of the conventional cavity. Note that the Q_in values of the fabricated samples are randomly above or below the predicted value due to structural imperfections of the hole positions and radii (see Appendix A4). The experimental Q_exp is determined by Q_design, Q_in, and Q_imperfect, where the latter is mainly determined by the scattering loss and the absorption loss [19]. If Q_in is much smaller than Q_imperfect, an experimental Q_exp value that is larger than the predicted Q_total can be obtained. In our sample, the average Q_imperfect is estimated to be higher than several millions [19], and thus Q_{p_exp} can be larger than the Q_{p_total} for d_p = 4 rows. On the other hand, Q_{S_exp} should be always smaller than the Q_{S_total} for d_S = 8 rows because Q_{S_in} is much larger than Q_imperfect.

4. Experimental results

The characteristics of the fabricated samples were investigated by measuring the emission intensities of the pump and Stokes modes under different excitation conditions. The experimental setup and the estimation of the threshold power are described in Appendix A5. The resonance spectra of the pump mode and the Stokes mode for the NN cavity with the lowest laser threshold (I_th) of 40 nW are presented in Figs. 3(a) and (b), respectively. The open circles are the experimental data and the solid curves show the fits to a Lorentzian function. The pump mode has a resonance peak wavelength (λ_p) of 1424.128 nm and the corresponding full width at half-maximum (Δλ_p) is 1.1 pm. According to the relation Q_exp = λ/Δλ, the estimated value of Q_{p_exp} is 1.3 × 10⁶. This value is more than two times higher than the Q_{p_design} of the conventional cavity.

 

Fig. 3. Resonance spectra of (a) the pump mode and (b) the Stokes mode for the nanocavity that exhibited the lowest threshold (i.e. NN cavity #3).

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From Fig. 3(b), we obtained λ_S= 1538.168 nm. We repeated the measurement several times and confirmed a significant variation of the obtained Δλ_S values with a range Δλ_S < 0.5 pm. Although the sample temperature was stabilized by a Peltier controller (see Fig. 8 in Appendix A5), the sample temperature was able to slowly drift within a range of 0.01 K while measuring the spectrum. As λ_S changes with a rate of 81.2 pm/K [13] and Δλ_S is less than 0.5 pm, the error in the estimation of Q_exp is large. Therefore, here we can only estimate that the Q_{S_exp} of this sample is larger than 3.0 × 10⁶ (to accurately measure the Q value, time-domain measurements are necessary [36]). If we assume that the Q values corresponding to the losses due to random fabrication imperfections (which lead to scattering) and due to absorption are similar to those in the previous study that reported a Q_exp of about 10 million (i.e. 2.1 × 10⁷ and 1.7 × 10⁷, respectively) [33], we can estimate an upper limit of 4.5 × 10⁶ for the Q_exp of this sample [= (1/8.7 + 1/21 + 1/17)^-1 × 10⁶]. The Δf between the two cavity modes is 15.603 THz, which closely matches the Raman shift of silicon, 15.606 THz [29].

The Q-factor product Q_{p_exp} × Q_{S_exp} estimated from Fig. 3 is four times larger than that of our previously reported cavity with I_th = 120 nW, which is the lowest threshold value in previous report [14]. Therefore, we conclude that the significantly smaller I_th of 40 nW obtained for this NN cavity is due to the achieved increase in Q_exp.

Table 1 shows the summary of the Q_exp values for the twenty cavities. The results of NN cavity #3 were shown in Fig. 3. NN cavity #5 and #10 were not measured due to the presence of contamination generated during fabrication. The variations in Q_exp that can be confirmed in Table 1, are due to the fluctuations in the hole radius and position [37,38]. Several cavities exhibit a Q_{p_exp} that is even higher than the calculated Q_{p_total} due to the variation of Q_in as described in Section 3. The range of Q_{p_exp} values for the NN cavities is relatively large (1.2 × 10⁵ to 1.3 × 10⁶) since the Q_{p_in} for this cavity design is much smaller than the Q_{p_design}. Among the NN cavities, the Q_{S_exp} values of four samples were estimated to be larger than 3.0 × 10⁶. The corresponding average Q_{S_exp} is larger than 2.6 × 10⁶. This value is high enough to consider the present fabrication accuracy of the air holes similar to the fabrication accuracy in Ref. [33].

Table 1. The experimental Q values of the 20 samples (10 samples based on the NN cavity design and 10 based on the conventional cavity design)

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The average Q_{p_exp} for the NN cavities is 1.61 times higher than that for the conventional design, and the average Q_{S_exp} is at least 1.53 times higher. The average Q-factor product Q_{p_exp} × Q_{S_exp} is thus more than 2.5 times higher than that for the conventional design. Because the twenty cavities were fabricated on the same chip, the increase of Q_exp is a result of the higher Q_design of the NN cavity.

Figure 4 shows the dependence of I_th on the inverse of Q_{p_exp} × Q_{S_exp} for the seven samples in which cw laser oscillation was observed (#1, #3, #7, #8 for NN cavity and #4, #6, #7 for conventional cavity). Although the magnitude of I_th also depends on the degree of detuning of Δf from the Raman shift of silicon, 15.606 THz [30], this graph demonstrates that I_th decreases as the product Q_{p_exp} × Q_{S_exp} increases. The sample with the highest threshold of 2,000 nW has a relatively large Δf of 15.623 THz while the Δf for the other six samples are 15.595∼15.603 THz. The impact of the detuning of Δf from 15.606 THz (Δf_det.) can be estimated by considering Eq. (2). Because the Raman gain width of silicon (Δ) is approximately 0.08 THz [39], we consider that the Raman gain of the sample with Δf = 15.623 THz is reduced by 15% from the maximum. Since the (nonlinear) free-carrier absorption (FCA) loss caused by two photon absorption (TPA) can be considerably large at excitation intensities near the threshold [14,15], the threshold can increase significantly even if the decrease in the Raman gain is only 15%. Note that an increase of the Q-factor product is not expected to reduce the influence of FCA, because TPA is more likely to occur as Q increases. A development of a nondetrimental surface modification that enables us to shorten the carrier lifetime, will be important in reducing the impact of the TPA-induced FCA loss [40].

 

Fig. 4. Plot of the laser threshold values of the cavities that exhibited cw operation, as a function of 1/(Q_{p_exp} × Q_{S_exp}). The red circles are for the NN cavity, and the black triangles are for the conventional cavity.

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Because we had to investigate the Q_p, Q_S, and the I_th for many samples, the detailed input–output relation was not measured on the first day of measurement. An important aspect of the above result is that I_th = 40 nW was obtained on the first day of measurement, and 60 days later we obtained I_th = 90 nW when we measured the input–output relation of the same sample. We confirmed that the product Q_{p_exp} × Q_{S_exp} decreased to about half of the initial value within 60 days after the first measurement. This decrease may be related to the change of the surface condition [19,33]. Previously, we reported Q_exp values larger than five million for heterostructure nanocavities that were placed in a chamber filled with dry air immediately after a hydrofluoric acid treatment [19,33]. On the other hand, all measurements in this study were performed in ambient air with a relative humidity of 33∼36%. Furthermore, the sample was stored in a low-humidity desiccator together with unfinished samples. These samples were still coated with the EB resist and had contaminations on the surface that had been generated during the fabrication process. When the Raman laser sample was stored separately from them, the decrease in Q_exp seemed to stop.

Figure 5 shows the Raman laser output power as a function of the excitation power coupled into nanocavity. This data was measured when the Q product had decreased to half of the initial value. Although the threshold shown in Fig. 5 is slightly larger than on the first day of measurement (due to the abovementioned decrease in Q_{p_exp} × Q_{S_exp}), a sub-100-nW threshold of 90 nW can be confirmed. At higher excitation powers, the slope of the increase in the Raman laser output power gradually decreases due to the FCA loss caused by TPA [15,41].

 

Fig. 5. Laser output power as a function of the pump power coupled into the NN cavity #3 measured 60 days later. The inset is a camera image of the laser emission.

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These results show that a cavity optimization by a machine-learning method can help to reduce the threshold of a Raman silicon nanocavity laser. Further decreases in I_th should be possible because the calculated Q_{p_total} of an NN cavity with d_p = 5 rows is 3.4 times larger than that of the NN cavity design used in this study (d_p = 4 rows). We also consider that it is important to improve the fabrication process. An average Q_exp of 1.9 × 10⁶ has been reported for a heterostructure nanocavity fabricated using a CMOS-compatible process [42]. The mass production of devices based on the NN cavity design using a CMOS-compatible process is considered possible.

Regarding the potential of optimization methods based on machine learning for Raman silicon lasers, it may be interesting to study whether such a method can be used to design a cavity in which the effect of air hole fluctuations on Q_{p_exp} × Q_{S_exp} is less detrimental. It should be also clarified whether such a method can be used to design a cavity that has a larger overlap integral between E_{x_pump} (E_{y_pump}) and E_{y_Stokes} (E_{x_Stokes}) for the Raman gain. Machine-learning-based methods may be useful to design a Raman laser that uses a broadband light source for excitation [43–45].

5. Conclusion

We have presented a new nanocavity structure whose product of the pump-mode Q-factor and the Stokes-mode Q-factor was improved by using a machine-learning method. The positions of the air holes were optimized by a convolutional deep neural network model with four layers and two output nodes to predict the Q_p and Q_S. The obtained Q-factor product Q_{p_design} × Q_{S_design} was 17.0 times larger than the corresponding Q-factor product of the conventional cavity. The average of Q_{p_exp} × Q_{S_exp} for the NN cavity was more than 2.5 times larger than that of the conventional cavity. We have demonstrated that the threshold I_th decreases with an increase in Q_{p_exp} × Q_{S_exp} by measuring the threshold of the seven cavities that exhibited cw operation. The lowest threshold obtained in our experiments was 40 nW for the NN cavity with the highest Q-factor product of Q_{p_exp} × Q_{S_exp} ≥ 3.9 × 10¹². The previously reported lowest threshold for a Raman silicon laser was 120 nW, which was measured in a conventional cavity with Q_{p_exp} × Q_{S_exp} ≅ 9.4 × 10¹¹ [14]. Optimizing cavity structures by methods based on machine learning can help to improve the optical properties of Raman silicon nanocavity lasers.

Appendix

A1. Conventional heterostructure nanocavity design

As shown in Fig. 1(a), a multi-heterostructure nanocavity consisting of 27 missing holes in a PC with a triangular lattice was used as the starting structure for the machine-learning-based optimization [46]. The base PC considered here, comprises a triangular lattice of air holes (with lattice constant a₁) formed in a 220-nm-thick silicon slab. The conventional heterostructure nanocavity is obtained by symmetrically changing the lattice constant in three steps; the lattice constant in the x-direction is changed in steps of 5 nm where the largest lattice constant is at the center of the cavity [Fig. 1(a); a₁ = 410 nm, a₂ = 415 nm, a₃ = 420 nm]. Therefore, the pump nanocavity mode and the Stokes nanocavity mode are formed by the mode gap confinement as shown in Fig. 1(b). The E_{x_pump} (E_{y_pump}) and E_{x_Stokes} (E_{y_Stokes}) distributions in the conventional heterostructure have different parities because they belong to different propagation bands. When the radii of the air holes are about 127 nm, the frequency spacing between the two confined modes is close to the Raman shift of silicon, which is 15.606 THz [29].

A2. Framework of our optimization method using neural networks

The optimization of the cavity structure was performed using the machine-learning-based procedure presented in our previous reports [24,27]. Note that the conditions and procedures were modified to solve the new optimization problem, which is to increase the Q-factor product Q_{p_design} × Q_{S_design}. The conditions and procedures are outlined below:

• 1. The starting structure is a multi-heterostructure cavity with larger lattice constants in the x-direction near the center of the cavity (conventional cavity design explained in Appendix A1). The three different lattice constants, the air hole radius, and the silicon slab thickness are shown in Fig. 1(a).
• 2. The positions of 62 air holes near the center of the defect line of the cavity [the 62 holes in Fig. 1(a) where arrows are attached] are selected for the structural optimization. During the optimization, the hole positions are adjusted with the constraint that the mirror symmetries about x- and y-axes are maintained. Therefore, the number of free parameters in the structural optimization is 31 ( = 15 × 2 + 1, since the hole on the x-axis shifts only in the y-direction).
• 3. 1000 cavity structures are generated by randomly displacing the 62 air holes of the starting structure according to the condition described in Step 2. The standard deviation of the random displacements in x- and y-directions is set to 1/1000 a₁.
• 4. The Q_p and Q_S values of the cavities prepared in Step 3 are determined by 3D-FDTD calculations.
• 5. To predict the Q_p and Q_S values from the cavity structure defined by the 31 parameters described in Step 2, a convolutional deep NN model with four layers is used. This NN has two output nodes to predict log(Q_p) and log(Q_S), separately (see Fig. 6).
• 6. The NN prepared in Step 5 is trained using the dataset prepared in Step 4. For the training, we used the following loss function L:

  (A1)$$L = {|{\textrm{Out}1 - {{\log }_{10}}({Q_{\textrm{p}\_\textrm{FDTD}}})} |^2} + {|{\textrm{Out2} - {{\log }_{10}}({Q_{\textrm{S}\_\textrm{FDTD}}})} |^2} + \frac{1}{2}\lambda \sum\limits_i {w_i^2}$$

  The first and second terms represent the deviations of the predictions from the exact solutions. The third term penalizes large connection weights w_i in the network, and the summation is taken over all weights in the network. λ is the control hyper parameter [24].

• 7. A new candidate structure that is expected to exhibit a lager Q_p × Q_S value is searched based on the gradient method, where the gradient of log(Q_p × Q_S) (= Out1 + Out2) with respect to the 31 structural parameters is predicted by the trained NN using the backpropagation method. Here, we used the loss L’ defined by

  (A2)$$\begin{array}{c} L^{\prime} = {|{\textrm{lo}{\textrm{g}_{10}}{Q_{\textrm{p}\_\textrm{NN}}} - {{\log }_{10}}{Q_{\textrm{p}\_\textrm{target}}}} |^2} + {|{\textrm{lo}{\textrm{g}_{10}}{Q_{\textrm{S}\_\textrm{NN}}} - {{\log }_{10}}{Q_{\textrm{S}\_\textrm{target}}}} |^2}\\ + \frac{1}{2}\lambda ^{\prime}{\sum\limits_{i,j} {|{{{\vec{d}}_{i,j}}} |} ^2} \end{array}$$

  The first and second terms represent the deviations of the predictions from the target Q factors for the pump and Stokes modes (the target values in our present calculations were 10¹⁰). The third term penalizes large displacements of air holes, and the summation is taken over all air holes in the target region shown in Fig. 1(a). λ’ is the control hyper parameter. d_i,j is the displacement vector of the air hole at the position defined by i and j, which are the X and Y indices, respectively [24].

• 8. Several candidate structures are obtained by changing the initial structure and searching conditions.
• 9. The actual Q_p and Q_S values of the candidate structures are determined by 3D FDTD calculations.
• 10. The structure that exhibits the best Q_p × Q_S is selected, and then the procedures explained in Steps 3 to 9 are repeated one more time, but this time by using the presently selected best structure as the starting structure (because the procedure consisting of Steps 3 to 9 is performed twice, in total 2000 cavity structures were used for this optimization).
• 11. The structure that exhibits the best Q_p × Q_S after Step 10 is the final new cavity structure.

 

Fig. 6. Configuration of the neural network (NN) prepared to learn the relationship between the displacements of the air holes and the Q_p and Q_S factors.

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A3. Displacements of the air holes

Table 2 summarizes the displacements of the air holes for the NN cavity, where the X and Y indices correspond to the coordinates shown in Fig. 1(a). This table only shows the data of the shifted air holes in the first quadrant (due to the symmetry of the nanocavity), and the displacements are discretized in units of 0.125 nm considering the resolution of the EB lithography system used for fabrication.

Table 2. Displacement data for the shifted holes indicated in Fig. 1(a)

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A4. Calculation results of Q_total including air hole fluctuations

We additionally calculated the Q values that can be expected for the NN cavity with d_p = 4 rows and d_S = 8 rows in the presence of air hole fluctuations. In these 3D FDTD simulations, random offsets were added to the ideal air-hole positions and radii, and the magnitude of these random variations is represented by the standard deviation σ_hole. Figures 7(a) and (b) show the Q_{p_total_fluc} and Q_{S_total_fluc} values for thirty different fluctuation patterns with σ_hole = 0.33 nm (i.e. the standard deviations for the random offsets δr, δx, and δy are 0.33 nm). This magnitude is the same as that estimated from the experimental results in [38]. The horizontal dashed lines in Figs. 7(a) and (b) show the Q_{i_total} values (that is, those without the random variations), which are 9.9 × 10⁵ and 8.7 × 10⁶, respectively. It is noted that the Q_{i_total_fluc} values in some patterns are larger than Q_{i_total}.

 

Fig. 7. Calculated Q factors of the NN cavity shown in Fig. 2(a) for 30 fluctuation patterns with σ_hole = 0.33 nm. (a) Results for the pump mode. The circles depicted on the right-hand side show the considered variations in the air hole radius and position. δx and δy represent the changes in the hole position, and δr represents the change in the radius. (b) Results for the Stokes mode.

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A5. Experimental methods

Figure 8 shows the experimental setup used in this study. The experimental methods used to obtain the results shown in Figs. 3 and 5 are the same as those used in [32]. A cw tunable laser (Santec TSL-510) was used for excitation. We used a beam splitter to divide the laser light into a weak (10%) and a strong (90%) beam. The weak beam was used to analyze the laser wavelength by a high-precision wavelength meter (Agilent 86122A). The strong beam was modulated by a mechanical chopper and was focused on either the facet of the pump excitation waveguide or that of the Stokes excitation waveguide. The coupling efficiency from the cw laser to the excitation waveguides can be lower than 1% since we did not introduce the spot size converter to reduce the coupling loss [32]. The coupling efficiency from the pump waveguide to the pump mode can be larger than 40% from the values of Q_{p_total} and Q_{p_in} presented in Fig. 2(b). The light emitted from the cavity in the direction perpendicular to the slab was detected by an InGaAs photodiode with a lock-in amplifier system (NF Corporation LI5630). By sweeping the excitation wavelength from short to long wavelengths in steps of about 0.15 pm, we obtained the resonance spectra shown in Figs. 3(a) and 3(b). To avoid a decrease of Q_exp due to TPA, the spectra were measured with an excitation intensity well below the threshold. A Peltier device was used to keep the sample temperature stable at room temperature.

To measure the Raman laser light (the Stokes emission), we used a stronger excitation of the pump mode and a long-pass filter (LPF) with a cutoff wavelength of 1500 nm in the detection path to block the pump light. The threshold I_th was estimated from the pump-mode emission in the low excitation power regime by assuming that the power coupled into the pump mode is linearly proportional to the power of the cw excitation laser. Because the collection efficiency of the light emitted from the pump mode varies for each sample (due to the random variations in the air-hole position and radius) [37], the estimated thresholds shown in Fig. 4 should include some error.

 

Fig. 8. The experimental setup used in this work. The LPF in the detection path was employed when investigating the Raman laser properties. NA: numerical aperture.

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Funding

New Energy and Industrial Technology Development Organization (JPNP13004); Japan Society for the Promotion of Science (18H01479, 19H02629, 21H01373).

Acknowledgments

T. Kawakatsu was supported by a fellowship from the ICOM Foundation.

Disclosures

The authors declare no conflicts of interest.

References

1. R. Claps, D. Dimitropoulos, V. Raghunathan, Y. Han, and B. Jalali, “Observation of stimulated Raman amplification in silicon waveguides,” Opt. Express 11(15), 1731–1739 (2003). [CrossRef]  

2. R. L. Espinola, J. I. Dadap, R. M. Osgood, S. J. McNab, and Y. A. Vlasov, “Raman amplification in ultrasmall silicon-on-insulator wire waveguides,” Opt. Express 12(16), 3713–3718 (2004). [CrossRef]  

3. A. Liu, H. Rong, M. Paniccia, O. Cohen, and D. Hak, “Net optical gain in a low loss silicon-on-insulator waveguide by stimulated Raman scattering,” Opt. Express 12(18), 4261–4268 (2004). [CrossRef]  

4. Q. Xu, V. R. Almeida, and M. Lipson, “Time-resolved study of Raman gain in highly confined silicon-on-insulator waveguides,” Opt. Express 12(19), 4437–4442 (2004). [CrossRef]  

5. O. Boyraz and B. Jalali, “Demonstration of a silicon Raman laser,” Opt. Express 12(21), 5269–5273 (2004). [CrossRef]  

6. M. Krause, H. Renner, and E. Brinkmeyer, “Analysis of Raman lasing characteristics in silicon-on-insulator waveguides,” Opt. Express 12(23), 5703–5710 (2004). [CrossRef]  

7. X. Yang and C. W. Wong, “Coupled-mode theory for stimulated Raman scattering in high-Q/Vm silicon photonic band gap defect cavity lasers,” Opt. Express 15(8), 4763–4780 (2007). [CrossRef]  

8. X. Checoury, Z. Han, and P. Boucaud, “Stimulated Raman scattering in silicon photonic crystal waveguides under continuous excitation,” Phys. Rev. B 82(4), 041308 (2010). [CrossRef]  

9. Y. Takahashi, R. Terawaki, M. Chihara, T. Asano, and S. Noda, “First observation of Raman scattering emission from silicon high-Q photonic crystal nanocavities,” in Proc. of Conference on Lasers and Electro-Optics (CLEO) (2011), paper QWC3.

10. 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]  

11. H. Rong, S. Xu, Y. Kuo, V. Sih, O. Cohen, O. Raday, and M. Paniccia, “Low-threshold continuous-wave Raman silicon laser,” Nat. Photonics 1(4), 232–237 (2007). [CrossRef]  

12. H. Rong, S. Xu, O. Cohen, O. Raday, M. Lee, V. Sih, and M. Paniccia, “A cascaded silicon Raman laser,” Nat. Photonics 2(3), 170–174 (2008). [CrossRef]  

13. Y. Takahashi, Y. Inui, M. Chihara, T. Asano, R. Terawaki, and S. Noda, “A micrometre-scale Raman silicon laser with a microwatt threshold,” Nature 498(7455), 470–474 (2013). [CrossRef]  

14. D. Yamashita, Y. Takahashi, J. Kurihara, T. Asano, and S. Noda, “Lasing dynamics of optically-pumped ultralow-threshold Raman silicon nanocavity lasers,” Phys. Rev. Appl. 10(2), 024039 (2018). [CrossRef]  

15. D. Yamashita, T. Asano, S. Noda, and Y. Takahashi, “Strongly asymmetric wavelength dependence of optical gain in nanocavity-based Raman silicon lasers,” Optica 5(10), 1256–1263 (2018). [CrossRef]  

16. M. Kuwabara, S. Noda, and Y. Takahashi, “Ultrahigh-Q photonic nanocavity devices on a dual thickness SOI substrate operating at both 1.31- and 1.55-(m telecommunication wavelength bands,” Laser Photon. Rev. 13(2), 1800258 (2019). [CrossRef]  

17. T. Yasuda, M. Okano, M. Ohtsuka, M. Seki, N. Yokoyama, and Y. Takahashi, “Raman silicon laser based on a nanocavity fabricated by photolithography,” OSA Continuum 3(4), 814–823 (2020). [CrossRef]  

18. Y. Akahane, T. Asano, B. S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature (London) 425(6961), 944–947 (2003). [CrossRef]  

19. H. Sekoguchi, Y. Takahashi, T. Asano, and S. Noda, “Photonic crystal nanocavity with a Q-factor of ∼9 million,” Opt. Express 22(1), 916–924 (2014). [CrossRef]  

20. M. Minkov and V. Savona, “Automated optimization of photonic crystal slab cavities,” Sci. Rep. 4(1), 5124 (2015). [CrossRef]  

21. M. Minkov, V. Savona, and D. Gerace, “Photonic crystal slab cavity simultaneously optimized for ultra-high Q/V and vertical radiation coupling,” Appl. Phys. Lett. 111(13), 131104 (2017). [CrossRef]  

22. T. Nakamura, Y. Takahashi, T. Asano, and S. Noda, “Improvement in the quality factors for photonic crystal nanocavities via visualization of the leaky components,” Opt. Express 24(9), 9541–9549 (2016). [CrossRef]  

23. K. Maeno, Y. Takahashi, T. Nakamura, T. Asano, and S. Noda, “Analysis of high-Q photonic crystal L3 nanocavities designed by visualization of the leaky components,” Opt. Express 25(1), 367–376 (2017). [CrossRef]  

24. T. Asano and S. Noda, “Optimization of photonic crystal nanocavities based on deep learning,” Opt. Express 26(25), 32704–32716 (2018). [CrossRef]  

25. M. Nakadai, K. Tanaka, T. Asano, Y. Takahashi, and S. Noda, “Statistical evaluation of Q factors of fabricated photonic crystal nanocavities designed by using a deep neural network,” Appl. Phys. Express 13(1), 012002 (2020). [CrossRef]  

26. R. Abe, T. Takeda, R. Shiratori, S. Shirakawa, S. Saito, and T. Baba, “Optimization of an H0 photonic crystal nanocavity using machine learning,” Opt. Lett. 45(2), 319–322 (2020). [CrossRef]  

27. T. Asano and S. Noda, “Iterative optimization of photonic crystal nanocavity designs by using deep neural networks,” Nanophotonics 8(12), 2243–2256 (2019). [CrossRef]  

28. T. Shibata, T. Asano, and S. Noda, “Fabrication and characterization of an L3 nanocavity designed by an iterative machine-learning method,” APL Photonics 6(3), 036113 (2021). [CrossRef]  

29. D. Yamashita, Y. Takahashi, T. Asano, and S. Noda, “Raman shift and strain effect in high-Q photonic crystal silicon nanocavity,” Opt. Express 23(4), 3951–3959 (2015). [CrossRef]  

30. J. Kurihara, D. Yamashita, N. Tanaka, T. Asano, S. Noda, and Y. Takahashi, “Detrimental fluctuation of frequency spacing between the two high-quality resonant modes in a Raman silicon nanocavity laser,” IEEE J. Select. Topics Quantum Electron. 26(2), 1–12 (2020). [CrossRef]  

31. Y. Takahashi, Y. Inui, M. Chihara, T. Asano, R. Terawaki, and S. Noda, “High-Q resonant modes in a photonic crystal heterostructure nanocavity and applicability to a Raman silicon laser,” Phys. Rev. B 88(23), 235313 (2013). [CrossRef]  

32. Y. Yamauchi, M. Okano, H. Shishido, S. Noda, and Y. Takahashi, “Implementing a Raman silicon nanocavity laser for integrated optical circuits by using a (100) SOI wafer with a 45-degree-rotated silicon top layer,” OSA Continuum 2(7), 2098–2112 (2019). [CrossRef]  

33. T. Asano, Y. Ochi, Y. Takahashi, K. Kishimoto, and S. Noda, “Photonic crystal nanocavity with a Q factor exceeding eleven million,” Opt. Express 25(3), 1769–1777 (2017). [CrossRef]  

34. S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and H. A. Haus, “Channel drop tunneling through localized states,” Phys. Rev. Lett. 80(5), 960–963 (1998). [CrossRef]  

35. H. Takano, B. S. Song, T. Asano, and S. Noda, “Highly effective in-plane channel-drop filters in two-dimensional heterostructure photonic-crystal slab,” Jpn. J. Appl. Phys. 45(8A), 6078–6086 (2006). [CrossRef]  

36. Y. Takahashi, Y. Tanaka, H. Hagino, T. Sugiya, Y. Sato, T. Asano, and S. Noda, “Design and demonstration of high-Q photonic heterostructure nanocavities suitable for integration,” Opt. Express 17(20), 18093–18102 (2009). [CrossRef]  

37. H. Hagino, Y. Takahashi, Y. Tanaka, T. Asano, and S. Noda, “Effects of fluctuation in air hole radii and positions on optical characteristics in photonic crystal heterostructure nanocavities,” Phys. Rev. B 79(8), 085112 (2009). [CrossRef]  

38. Y. Taguchi, Y. Takahashi, Y. Sato, T. Asano, and S. Noda, “Statistical studies of photonic heterostructure nanocavities with an average Q factor of three million,” Opt. Express 19(12), 11916–11921 (2011). [CrossRef]  

39. V. Sih, S. Xu, Y. H. Kuo, H. Rong, M. Paniccia, O. Cohen, and O. Raday, “Raman amplification of 40 Gb/s data in low-loss silicon waveguides,” Opt. Express 15(2), 357 − 362 (2007). [CrossRef]  

40. T. Ito, K. Ashida, K. Kinoshita, R. Moriya, T. Machida, K. Maeno, T. Endo, K. Yamada, M. Okano, and Y. Takahashi, “Nondetrimental Surface Modification of Ultrahigh-Q Photonic Crystal Silicon Nanocavities,” in Proc. of Conference on Lasers and Electro-Optics Pacific Rim (CLEO-PR) (2018), paper W4D.3.

41. T. K. Liang and H. K. Tsang, “Nonlinear absorption and Raman scattering in silicon-on-insulator optical waveguides,” IEEE J. Select. Topics Quantum Electron. 10(5), 1149–1153 (2004). [CrossRef]  

42. K. Ashida, M. Okano, T. Yasuda, M. Ohtsuka, M. Seki, N. Yokoyama, K. Koshino, K. Yamada, and Y. Takahashi, “Photonic crystal nanocavities with an average Q factor of 1.9 million fabricated on a 300-mm-wide SOI wafer using a CMOS-compatible process,” J. Lightwave Technol. 36(20), 4774–4782 (2018). [CrossRef]  

43. T. Datta and M. Sen, “LED pumped micron-scale all-silicon Raman amplifier,” Superlattices Microstruct. 110, 273–280 (2017). [CrossRef]  

44. R. Shiozaki, T. Ito, and Y. Takahashi, “Utilizing broadband light from a superluminescent diode for excitation of photonic crystal high-Q nanocavities,” J. Lightwave Technol. 37(10), 2458–2466 (2019). [CrossRef]  

45. T. Kawakatsu, D. Yamashita, T. Asano, S. Noda, and Y. Takahashi, “Raman scattering emission from a silicon photonic nanocavity excited by a superluminescent diode,” in Proc. of Conference on Lasers and Electro-Optics Pacific Rim (CLEO-PR) (2020), paper C8H_3.

46. B. S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nat. Mater. 4(3), 207–210 (2005). [CrossRef]