1. Introduction

Two-dimensional photonic crystal (PC) nanocavities have attracted attention due to their small modal volumes and high quality factors. Various types of PC nanocavities have so far been studied, for example, nanocavities formed by a single missing air hole [1], nanocavities formed by three missing air holes (the so-called L3 nanocavity design) [2], and nanocavities formed by a line-defect with tens of missing air holes and a photonic heterostructure [3]. These nanocavities are expected to be used in devices such as multi-channel wavelength filters [4,5], highly sensitive sensors [6–8], exotic light sources [9–12], and optical switching devices [13–15]. Since a high quality-factor (Q) is important for such applications, the fabrication process has been improved [16–18] and the methods used to design high-Q nanocavities have also been improved.

In 2003, an important design rule concerning the increase of the design Q (Q_design) was reported [2], and it was shown that Q_design values larger than 100,000 can be achieved in an L3 nanocavity by shifting the positions of the two air holes nearest to the PC nanocavity. Subsequent research indicated that the Q_design can be increased further if the positions of more holes are optimized [19]. Therefore, various methods for hole-pattern optimization have been developed, such as the use of genetic algorithms [20], the visualization of leaky components [21], and particle swarm optimization [22]. As a result, experimental Q (Q_exp) values of about two million have been achieved by optimizing the L3 nanocavity design [23,24].

In 2018 and in 2019, machine learning based on neural networks was used to design a symmetric heterostructure nanocavity with a Q_design larger than one billion and a symmetric L3 nanocavity with a Q_design larger than ten million, respectively [25,26]. It has been demonstrated that the Q_exp can be increased by using these nanocavity designs [27,28]. Furthermore, machine learning is able to simultaneously increase the two Q_design values of the resonance modes used in a Raman Si nanocavity laser [29]. Machine learning can also be used to simultaneously increase the Q values and the mutual coupling constants of a coupled nanocavity system, which is for example required for electrically controlled on-demand photon transfer [15]. Machine learning has made it possible to study nanocavities with air-hole shift patterns that cannot be explored manually.

For applications, it is important not only to increase Q_exp but also to decrease the sample-to-sample variation in Q_exp. Because the positions and radii of the air holes in fabricated nanocavities deviate randomly from their design values (by values on the order of nanometers or sub-nanometers), the Q_exp varies from sample to sample even if nanocavities with the same design are fabricated [30,31]. Therefore, it is important to investigate whether machine learning can be used to explore nanocavity designs whose Q is less affected by small random changes in the air-hole pattern.

In the nanocavities reported previously in studies concerning the increase of Q_exp, the air-hole patterns exhibit mirror symmetry about the two symmetry axes of the nanocavity (the x- and y-axes) [2,3,23,24,27–29]. Such symmetric cavity designs were used because the mirror-symmetry-induced suppression of the radiation loss effectively improves Q [2,3]. However, when fabricating nanocavities with a symmetric design, the mirror symmetry about the x- and y-axes only holds approximately due to inevitable air-hole fabrication fluctuations. Therefore, in terms of the width of the Q-factor variation, it is uncertain whether nanocavity designs with mirror symmetry have the best tolerance to such fabrication fluctuations. Considering this fundamental issue, we recently investigated the variation in Q for an L3 nanocavity design in which the air-hole pattern has no mirror symmetry. Preliminary results of this approach have been reported recently [32]. Here we designed an asymmetric L3 nanocavity with a Q_design of 250,000 by machine learning using neural networks and found that the fabricated asymmetric L3 cavities have a smaller variation in Q_exp than the symmetric L3 nanocavities with a similar Q_design. In this work, after providing a detailed explanation of these two cavity designs and the measured sample characteristics, the advantages of such asymmetric cavity designs for applications are discussed with the aid of simulation results in which the air-hole positions and radii are randomly varied.

2. Asymmetric and symmetric L3 cavity designs

Figure 1(a) shows the initial L3 nanocavity structure that was used to design the asymmetric L3 cavity by machine learning [2]. The considered silicon slab thickness (t) is 220 nm, the radius of the air holes (r) is 102.5 nm, and the PC lattice constant (a) is 410 nm. The Q_design determined by the three-dimensional (3D) finite-difference time-domain (FDTD) method is about 7,500, and the design resonance-wavelength (λ_design) is 1563.4 nm. By machine learning using neural networks, the positions of the 78 air holes enclosed by the dashed green line in Fig. 1(a) were optimized with respect to Q_design. The used optimization procedure was almost the same as that in [26] except that we did not introduce a procedure that ensures mirror symmetry. The number of parameters that had to be optimized was thus about four times larger than those in [26]. Note that the waveguide that is required to excite the nanocavity was not considered in the optimization process. The details of the optimization are described in Appendix A1.

 

Fig. 1. (a) The base structure used for the automated optimization of the air-hole pattern. The dashed rectangle indicates the holes to be optimized. (b) The asymmetric L3 cavity structure designed by machine learning using neural networks. The distribution of the y-component of the electric field is shown by the color map. (c) The symmetric L3 cavity structure with four shifted air-holes.

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Figure 1(b) shows the obtained new nanocavity structure with a Q_design of 250,000 and the distribution of the y-component of the electromagnetic field, E_y (the quality factor including the effect of coupling to the waveguide shown in Section 3 is 246,200). The resonance wavelength of this structure is λ_design = 1569.4 nm. The cavities fabricated using this design are hereafter referred to as the asymmetric cavities. The arrows in Fig. 1(b) indicate the displacements of the 78 holes with respect to their original positions (the numerical values of the displacements are provided in Table 1 in Appendix A1). It is apparent that the mirror symmetry about the x- and y-axes is not maintained. Nevertheless, the Q_design is about 30 times larger than that of the initial structure.

Figure 1(c) shows the cavity design that is used for comparison. The Q_design is 264,000 (the loaded quality factor is 259,800) and the λ_design is 1576.9 nm. The cavities fabricated using this design are hereafter referred to as the symmetric cavities. The positions of the 1st and 3rd neighboring holes of the L3 cavity [the four holes with the arrows in Fig. 1(c)] were optimized with respect to their original positions shown in Fig. 1(a) while maintaining mirror symmetry (the values of the shifts are provided in Table 2 in Appendix A2). Compared to the resonance wavelength of the initial L3 cavity, the λ_design of the optimized structure is longer, which is mainly due to the shift of the 1st neighboring holes [2,24]. We used a manual optimization method to obtain the symmetric cavity design in Fig. 1(c), and the optimization strategy and the mechanism that leads to the increase in Q is described in [21]. Appendix A2 shows how the dependence of Q_design on the shift values. In Sections 4 and 5, the variation in Q due to air-hole fabrication fluctuations is investigated for both types of cavities.

3. Fabrication procedure and device design including excitation waveguides

The 50 asymmetric cavities with the design shown in Fig. 1(b) and the 50 symmetric cavities with the design of Fig. 1(c) were fabricated on a silicon-on-insulator (SOI) chip. In the device used for the measurements, all 100 cavities are located within an area of 300 µm × 1000 µm. The employed fabrication process was the same as that explained in our previous report on the fabrication using an SOI wafer with a 45-degree-rotated silicon top layer [33]. The PC air-hole patterns were defined by electron beam (EB) lithography with a positional resolution of 0.125 nm. Therefore, the shifts shown in Tables 1 and 2 are discretized in units of 0.125 nm. After immersing the sample in the developer solution, the air holes were formed in the top silicon layer of the SOI chip by dry etching. The surface residues were removed by standard silicon processes, and the 3-µm-thick SiO₂ layer under the top silicon layer was removed with 48% hydrofluoric acid to form an air bridge structure. Finally, to adjust r to a value close to the design value of 102.5 nm, we used thermal oxidation of the Si surface with subsequent removal of the oxide film by 1% hydrofluoric acid.

Figure 2(a) shows a scanning electron microscope (SEM) image of one of the fabricated asymmetric cavities. The positions of the air holes enclosed by the dashed line agree well with those presented in Fig. 1(b). Figure 2(b) shows a SEM image of one of the fabricated symmetric cavities. The air-hole radius estimated from the image is 100.0 nm for both cavities (there should be an error of ±2 nm in the estimated value due to the used estimation technique). By using a thickness monitor, the thickness of the PC slab was estimated to be 221.4 nm. Based on previous studies, the magnitude of the random deviations of the air-hole positions and radii with respect to the design values (σ_hole) is expected to be less than 0.5 nm in terms of the standard deviation (note that σ_hole cannot be accurately estimated from the SEM images) [18].

 

Fig. 2. (a) SEM image of a fabricated asymmetric L3 cavity. (b) SEM image of a fabricated symmetric L3 cavity.

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The waveguides used to excite the nanocavities are separated from the cavities by twelve rows of air holes and are 5% wider than the cavity (1.05 W1 = $1.05\sqrt 3 a$). Due to this configuration, the Q value that corresponds to the inverse of the coupling strength between the cavity and the waveguide (Q_in) is larger than 10,000,000. The values of Q_design and λ_design for the asymmetric cavity including the excitation waveguide are 268,400 and 1575.4 nm, respectively, and those for the symmetric cavity are 253,800 and 1583.0 nm, respectively. These values are slightly different from those in Section 2 mainly because of the difference in r. In Appendix A3, the dependence of Q_design on r is shown for both cavity types.

4. Experimental results

To determine the Q_exp and the experimental resonance wavelength (λ_exp) values of the 50 asymmetric cavities and the 50 symmetric cavities, we measured the resonance spectra by using the measurement system described in Appendix A4. Figure 3(a) shows the resonance spectrum of the nanocavity with the highest Q_exp among the 50 asymmetric cavities. The open circles are the experimental data and the solid curve is a fit using a Lorentzian line shape; λ_exp = 1573.235 nm and the corresponding full width at half-maximum (Δλ) is 6.2 pm. By using the relation Q = λ/Δλ, we estimate a Q_exp of 253,700, which is close to the Q_design of 268,400.

 

Fig. 3. The data on the left-hand side of the figure represent (a) the resonance spectrum of the asymmetric cavity with Q_exp = 253,700, (c) an image of the actual emission from this cavity, (d) the corresponding calculated emission pattern, and (g), (i) the measured and calculated polarization characteristics for this cavity, respectively. The data on the right-hand side represent (b) the resonance spectrum of the symmetric cavity with Q_exp = 282,300, (e), (f) the measured and calculated emission pattern, respectively, and (h), (j) the measured and calculated polarization characteristics, respectively.

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Figure 3(b) shows the resonance spectrum of the nanocavity with the highest Q_exp among the 50 symmetric cavities. The λ_exp is 1580.671 nm, which is 7.4 nm longer than that of the asymmetric cavity. This difference is in good agreement with the difference between the calculated values. The Q_exp estimated using Δλ = 5.6 pm is 282,300, which is higher than the Q_design of 253,700. The precision of the Q_exp measurement for these cavities is 1,850 in terms of the standard deviation, which was determined by measuring the same nanocavity several tens of times.

Figure 3(c) is an image of the actual emission from the asymmetric cavity with Q_exp = 253,700. The experimentally determined emission pattern exhibits multiple emission spots, which agree well with the emission pattern calculated by FDTD as shown in Fig. 3(d). Figure 3(e) shown an image of the actual emission from the asymmetric cavity with Q_exp = 282,300. The observed single spot is in good agreement with the result in [24] and with the calculated emission pattern in Fig. 3(f).

Figures 3(g) and (h) show the experimentally determined polarization characteristics of the emission for the asymmetric cavity with Q_exp = 253,700 and the symmetric cavity Q_exp = 282,300, respectively. In the former case, the maximum intensity is obtained at 105° (azimuthal angle), whereas in the latter case, the maximum intensity is obtained at 90°. Figures 3(i) and 3(j) show the calculated polarization characteristics of the asymmetric and symmetric cavities, respectively, and they are in good agreement with the experimental results.

The results in Fig. 3 demonstrate that the Q_exp, the actual emission pattern, and the polarization characteristics of the asymmetric cavity can be predicted by 3D FDTD simulations. Because the polarization characteristics and the emission pattern of the asymmetric cavity are different from those of the symmetric cavity, the approach of designing asymmetric cavities by machine learning may be useful for flexible control of polarization characteristics and emission patterns of nanocavities.

Figure 4(a) shows the histogram of the Q_exp values of the 50 asymmetric cavities. The Q_exp values randomly vary due to random differences in the air-hole patterns [30,31]. The vertical dashed line indicates the value of Q_design, which is larger than the average Q_exp (232,776). Figure 4(b) shows the histogram of the Q_exp values for 49 symmetric cavities (one cavity had a very low Q_exp of 147,000 due to contamination, and thus the data of this cavity is excluded from the analysis). Here, the average Q_exp is 234,121, and only one of the cavities exhibits a Q_exp larger than the corresponding Q_design. The average Q_exp of the asymmetric cavities is slightly smaller than that of the symmetric cavities, which is opposite to the relation between the Q_design values. This is probably due to an error in the value of r estimated from the SEM images. The hole radius is probably slightly larger than 100 nm (see Appendix A3).

 

Fig. 4. (a) Histogram of the Q_exp values of the 50 asymmetric cavities. (b) Histogram of the Q_exp values of 49 symmetric cavities. (c) Histogram of the resonance wavelengths of the 50 asymmetric cavities. (d) Histogram of the resonance wavelengths of 49 symmetric cavities.

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From Figs. 4(a) and (b), we find that the standard deviation of the Q_exp values (σ_Qexp) for the asymmetric cavities is 6,404, and that for the symmetric cavities is 10,477. Since the average Q_exp is not the same, we compare the degrees of the variability of Q_exp using the coefficient of variation (the ratio of σ_Qexp to the average Q_exp): the coefficient of variation for the asymmetric cavities is 0.02751, while that for the symmetric cavities is 0.04475. In other words, the Q_exp variation for the asymmetric cavity design is 39% smaller than that for the symmetric cavity design. This indicates that the Q value of the asymmetric cavity design has a higher tolerance to air-hole fabrication fluctuations.

Figure 4(c) shows the histogram of the λ_exp values for the 50 asymmetric cavities. The λ_exp values also vary randomly due to the air-hole fabrication fluctuations [30,31]. The vertical dashed line indicates the value of λ_design. The average λ_exp is 1573.7 nm and the standard deviation is 0.68 nm. Figure 4(d) shows the corresponding histogram for the 49 symmetric cavities. The average λ_exp is 1582.1 nm and the standard deviation is 0.76 nm, and thus the standard deviation of the λ_exp values for the asymmetric cavities is slightly smaller than that for the symmetric cavities. Note that the SOI substrate used in this study has a thickness fluctuation with a standard deviation of about 0.2 nm [34], which also causes a random variation in λ_exp with a standard deviation of about 0.24 nm [35]. Therefore, it is still unclear whether the λ of the asymmetric cavity design has a higher tolerance to air-hole fabrication fluctuations than the λ of the symmetric cavity design. This issue should be investigated in the future.

5. FDTD simulations of nanocavities with air-hole fabrication fluctuations

In this section, we theoretically estimate the magnitude of the variation in Q due to fluctuations during the air-hole fabrication. 200 patterns of random deviations from the ideal air-hole positions and radii are used in the 3D FDTD simulations. The FDTD cell size is about a/10 in both the x- and y-directions. Furthermore, we employ a sub-cell size of about a/4000, and the dielectric constant of each cell is determined by averaging over its sub-cells. Therefore, even geometry deviations on the order of a/4000 can be resolved in the FDTD simulation. The details of the calculation method are described in [26,30]. The cavity structures used in the calculations are the same as those considered in Section 3, and thus include waveguides. Figures 5(a) and 5(b) show the histograms of the calculated Q_FDTD values for the asymmetric and symmetric cavities, respectively. The average Q_FDTD of the 200 asymmetric cavities is 257,300 (29 cavities have values larger than Q_design). The average Q_FDTD of the 200 symmetric cavities is 243,100 (64 cavities have values larger than Q_design).

 

Fig. 5. (a) Histogram of the calculated Q_FDTD values of 200 asymmetric cavities in the case of air-hole fabrication fluctuations with σ_hole = 0.5 nm. The red curve is a fit to a Gaussian function. (b) The histogram of the Q_FDTD values for 200 symmetric cavities in the case of σ_hole = 0.5 nm. (c), (d) Schematic of the variation in the air-hole radius and position, respectively. δr, δx, and δy represent the deviations from the ideal values.

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Figures 5(c) and 5(d) illustrate the considered fluctuation in the radius and the position, respectively. The variables r, x, and y are the ideal values for a given hole, and δr, δx, and δy are the deviations from the ideal values. The three fluctuation parameters (δr, δx, and δy) were varied according to a normal distribution with a standard deviation of σ_hole = 0.5 nm, and the generated individual deviation values were added to the corresponding design values of the 101 × 44 air holes around the nanocavity in the FDTD simulation. We applied the same deviation values to the asymmetric and the symmetric cavity designs, and repeated the generation of the deviation values to obtain 200 deviation patterns.

The theoretically estimated standard deviation (σ_{Q_FDTD}) for the asymmetric cavities is 9,850, and that for the symmetric cavities is 20,598. Hence, the coefficient of variation for the asymmetric cavities is 0.03828, while that for the symmetric cavities is 0.08473. The Q variation for the asymmetric cavity design is 55% smaller than that for the symmetric cavity design. As explained in Appendix A5, the Q variation of the asymmetric cavity design is smaller than that of the symmetric cavity design even when different radii are used in the calculations. Similar results are also obtained if we use σ_hole = 1.0 nm. Hence, in agreement with the experiments, the FDTD simulations also indicate that the Q value of the asymmetric cavity design has a higher tolerance to fabrication fluctuations.

Note that the σ_{Q_FDTD} of the asymmetric cavities with σ_hole = 0.5 nm is 1.54 times larger than the σ_Qexp shown in Fig. 4(a). Table 4 shows that the magnitude of σ_{Q_FDTD} is proportional to σ_hole. Therefore, it can be considered that the σ_hole of the fabricated nanocavities is less than 0.5 nm. In previous studies on high-Q heterostructure nanocavities with a Q_design of more than 10 million, we estimated the magnitude of σ_hole by using scattering theory [17,18,31]. However, we consider that this estimation method cannot be applied to this study since the Q_design values of the L3 cavities are too low. Furthermore, for both cavity types, the average Q_exp is slightly smaller than the average Q calculated with σ_hole = 0.5 nm. This is probably because the fabricated cavities include additional absorption losses, which lead to a decrease in Q_exp [17,36].

6. Discussion and future prospects

The hole pattern of the L3 cavity design shown in Fig. 1(b) has no mirror symmetry. This asymmetric cavity can be fabricated while maintaining a good circular structure of the air holes as shown in Fig. 2(a). The air holes near the cavity seem to be randomly displaced, but this cavity design provides Q_exp values as high as 250,000 as shown in Fig. 3(a). This value is almost equal to Q_design. Furthermore, the experimental results in Fig. 3 demonstrate that the λ_exp, the emission pattern, and the polarization characteristics of such L3 cavities are in good agreement with the theoretical prediction (we confirmed that the emission patterns and the polarization characteristics of the other 49 asymmetric cavities are similar to those shown in Fig. 3). We consider that it is rather difficult to design such asymmetric nanocavities by the previously reported manual methods.

It should be emphasized that the variation in Q_exp for the asymmetric cavity design is 1.63 times smaller than that for the symmetric cavity design (Fig. 4). This suggests that the Q value of the asymmetric cavity design has a higher tolerance to air-hole fabrication fluctuations. The FDTD simulations including the air-hole fabrication fluctuations in Fig. 5 and Appendix A5 support this interpretation. So far, most of the studies on high-Q nanocavities have focused on suppressing the radiation loss by maintaining perfect mirror symmetry. This approach can be the best choice if we aim at increasing Q_exp as much as possible. On the other hand, it cannot be the best choice if we need to suppress the Q_exp variation due to air-hole fabrication fluctuations.

Nanocavities with Q_exp values smaller than a few hundred thousand can be useful for many applications [4,6–10,13,14], and also the possibility of mass-production of high-Q PC nanocavities using a CMOS-compatible process has been investigated [34]. Concerning applications, the suppression of the Q_exp variation of mass-produced nanocavities will be important. For example, suppressing the Q_exp variation will improve the reproducibility of systems for refractive-index-change sensing and can reduce the cost [37,38]. Figures 4 and 5 (and Appendix A5) suggest that asymmetric cavities can be superior to symmetric cavities in some applications. We believe that it will be possible to fabricate the asymmetric nanocavity in Fig. 1(b) by using CMOS-compatible process [39].

We consider that the larger variation in Q_exp for the symmetric cavity design is due to the fact that this design is relatively sensitive to mirror-symmetry breaking. When a nanocavity based on a design with a perfect mirror symmetry of the air-hole locations is fabricated, the symmetry of the actual sample is not perfect due to the finite accuracy of the PC hole fabrication. On the other hand, an asymmetric cavity design may be less sensitive to air-hole fabrication fluctuations because its symmetry is inherently imperfect. The periodic structure found in butterfly scales is frequently cited as an analog for a PC. Appendix A6 shows a SEM image of a butterfly scale (Fig. 11). While the air holes are formed periodically, the structural properties (position and shape) do not have perfect mirror symmetry. Breaking mirror symmetry while maintaining a certain degree of periodicity may slightly improve the tolerance to external fluctuations. At least, the approach of using asymmetric cavities instead of symmetric cavities allows us to explore a wider range of candidate structures during the machine-learning-based design process. Detailed studies on the merits of asymmetric cavities should be performed in the future.

In this study, only a single asymmetric L3 cavity design [Fig. 1(b)] was investigated. It is possible to design other asymmetric L3 cavities with different patterns of air holes using machine learning. Among these asymmetric cavities, there may be a cavity design where the variation of Q_exp is even narrower than that shown in Fig. 4(a). It will be interesting to study neural-network-learning methods that can intentionally design such nanocavities. Furthermore, the asymmetric cavity may be useful to simultaneously optimize multiple properties (for example the polarization and the emission pattern) while maintaining a high Q. Studies on neural-network-learning methods that are able to design such nanocavities are also considered useful for applications.

The concept of the asymmetric cavity design can also be applied to other PC nanocavity structures besides the L3 cavity (L3 cavities have been widely studied and thus we also used them for the demonstration in this study). Two issues need to be clarified in that regard: whether the suppression of the variation in Q_exp can be achieved in the other types of nanocavities, and whether the reduction of the average Q_exp (relative to Q_design) due to a deviation of the air-hole positions and radii can be decreased by using asymmetric designs [30]. In particular, we consider that the application of this concept to heterostructure nanocavities with a Q_design ≥ 10 million can be important [25,27]. The FDTD simulation results shown in Fig. 5 indicate that the reduction of Q_exp is slightly smaller in the case of the asymmetric cavity. The Q value used to describe the additional loss caused by the air-hole fabrication fluctuations with σ_hole = 0.5 nm is about 6,222,000 for the asymmetric cavity, while it is 5,766,000 for the symmetric cavity. However, our experimental data is still inconclusive regarding this fluctuation-induced reduction of Q_exp due to the relatively low Q_design of the L3 cavities. If the fluctuation-induced reduction of the Q_exp of a heterostructure nanocavity is smaller in the case of an asymmetric cavity design, this will lead to improved Raman Si nanocavity lasers [29,35] and the realization of a longer time for photon manipulation [15,40].

In this study, the variation in Q_in due to the air-hole fabrication fluctuations is ignored since Q_in is larger than 10 million. In the experiment shown in Fig. 3, the radiation efficiency of the L3 cavity would be maximized if we used a geometry with a Q_in of about 250,000 [4]. Therefore, depending on the application, it may be also important to reduce the variation in Q_in. Whether the asymmetric cavity design can reduce the sample-to-sample variation in Q_in should be investigated in the future.

Finally, the influence of the air-hole fabrication fluctuations becomes more significant as the λ of the nanocavity becomes shorter, because the fabrication becomes more difficult and the impact of the fluctuations increases if they stay at the same magnitude (σ_hole). Asymmetric cavity designs will be useful for the development of nanocavity devices for visible- and near-infrared bands [41–45].

Appendix

A1. Details of the optimization method using neural networks

To prepare the initial dataset consisting of 1,000 slightly different hole patterns, the positions of the 78 air holes enclosed by the dashed line in Fig. 1(a) are randomly displaced with a magnitude equal to or less than 0.1 a. Accordingly, all cavities of the initial dataset are asymmetric cavities, while the initial cavities in Refs. [25,26] were symmetric cavities. Figure 6(a) shows the histogram of the Q values of the obtained nanocavity structures determined by FDTD simulations. There are about 90 nanocavities with a Q_FDTD larger than the Q_FDTD of the initial L3 nanocavity (7,500). Then, the positions of the air holes are optimized by an automated method using a neural network (NN) to increase the Q_design. The following Steps 1 to 6 were repeated 205 times to design the nanocavity in Fig. 1(b):

• 1. A NN is trained using the training dataset. The NN architecture and training methodology is the same as that used in our previous study [26].
• 2. A new candidate structure that is expected to exhibit a lager Q_design is searched based on the gradient method starting from one chosen structure. The gradient of Q with respect to the structural parameters is calculated by the NN using the backpropagation method. The air-hole pattern of each candidate structure is asymmetric (in our previous studies, asymmetric candidate structures were transformed into symmetric cavities by calculating the average of a given hole position and the corresponding hole positions of the left–right- and upside–down-flipped structures and the 180-degree-rotated structure [25,26]).
• 3. Steps 1 and 2 are repeated five times to obtain five candidate nanocavity structures. These five structures are obtained by changing the starting structure and the searching conditions.
• 4. The Q_FDTD values of these five new candidate structures are calculated. These structures and the Q_FDTD values are added to the dataset.
• 5. New nanocavity structures are generated by adding random shifts to the air-hole positions of the best structure (i.e., the structure with the highest Q_FDTD) in the dataset.
• 6. The Q_FDTD values of the nanocavities generated in Step 5 are calculated and they are added to the dataset.

 

Fig. 6. (a) Histogram of the calculated Q factors for the 1,000 nanocavities used as the initial dataset. (b) Relationship between the number of optimization iterations and the Q_FDTD of the best structure generated in the corresponding cycle. (c), (d) Correlation between the calculated Q_FDTD and the Q_NN predicted by the neural network for the test and training datasets, respectively.

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Figure 6(b) plots the Q_FDTD value of the best structure among the candidate structures generated in each optimization cycle (in Steps 3 and 5) as a function of the optimization-cycle number. The structure in Fig. 1(b) was obtained at the 170th cycle. Figures 6(c) and 6(d) show the relationship between the calculated Q_FDTD and the Q predicted by the NN (Q_NN) for the initial dataset. Figure 6(c) is the relationship for the test data (the 100 structures among the initial dataset that are not used for training). Figure 6(d) shows the relationship for the training data (the remaining 900 structures). The general meaning of these figures is explained in [25].

Table 1 summarizes the displacements of the air holes of the cavity structure presented in Fig. 1(b) (the X and Y indices are also shown in the figure). The displacements are discretized in units of 0.125 nm considering the resolution of the EB lithography system used for fabrication.

Table 1. Displacement data for the shifted holes of the asymmetric cavity structure presented in Fig. 1(b)

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A2. Details of the optimization of the symmetric L3 cavity

Figure 7 shows the Q_design of the symmetric L3 cavity as functions of the absolute shifts of the 1st and 3rd neighboring holes (the shifts occur along the x-direction). In the structure shown in Fig. 1(c), the positions of the 1st and 3rd neighboring holes are shifted by 0.21 a and 0.26 a, respectively. Table 2 summarizes the displacements of the air holes of the symmetric cavity structure presented in Fig. 1(c).

 

Fig. 7. Relationship between the shifts of the 1st and 3rd neighboring holes and the Q_design of the symmetric L3 cavity. The pairs of row and column indices (X index, Y index) that define the positions of the 1st and 3rd neighboring holes are (±4, 0) and (±8, 0), respectively.

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

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A3. Dependence of Q_design on the air-hole radius

Figure 8 shows the dependence of Q_design on the air-hole radius for the asymmetric and symmetric cavities. These calculations consider the excitation waveguide.

 

Fig. 8. Dependence of the calculated Q on the air-hole radius for the asymmetric cavity design (open circles) and the symmetric cavity design (open triangles).

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A4. Experimental setup

Figure 9 shows the system used to evaluate the Q_exp and λ_exp values of the fabricated nanocavities. The measurement method is the same as that reported in our previous work [44]. The power of the light coupled into the nanocavity was less than 10 nW to avoid the influence of two-photon absorption [46]. Note that the resonance wavelength of these samples redshifts with a rate of 80 pm/K [46]. To reduce the temperature changes during the measurement of the resonance spectrum (approximately 1 minute), we enclosed the sample by plates to block any air flow. The polarizer in the detection path in Fig. 8 was only present during the measurements of the data shown in Figs. 3(g) and 3(h).

 

Fig. 9. Experimental setup used in this work. NA: numerical aperture, CW: continuous wave.

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A5. σ_{Q_FDTD} for different values of the hole radius and σ_hole

Figure 10 shows the results of calculations similar to those for Fig. 5 (we used the same random hole patterns) but for different values of the hole radius and σ_hole. Figures 10(a) and 10(b) show the histograms of the Q_FDTD values in the case of r = 97.5 nm for the asymmetric and symmetric cavities, respectively. Figures 10(c) and 10(d) show the results in the case of r = 107.5 nm. Table 3 summarizes the average Q_FDTD values for σ_hole = 0.5 and 1.0 nm. Table 4 summarizes the corresponding σ_{Q_FDTD} values.

 

Fig. 10. (a) Histogram of the calculated Q values of 200 asymmetric cavities in the case of r = 97.5 nm and σ_hole = 0.5 nm. (b) The corresponding histogram for 200 symmetric cavities. (c) Histogram of the calculated Q values of 200 asymmetric cavities in the case of r = 107.5 nm and σ_hole = 0.5 nm. (d) The corresponding histogram for 200 symmetric cavities.

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Table 3. Hole-radius dependence of the average Q_FDTD of the asymmetric cavities for σ_hole = 0.5 nm and 1.0 nm (2nd and 3rd column), and that of the symmetric cavities (4th and 5th column)

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Table 4. Hole-radius dependence of σ_{Q_FDTD} of the asymmetric cavities for σ_hole = 0.5 nm and 1.0 nm (2nd and 3rd column), and that of the symmetric cavities (4th and 5th column)

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A6. A SEM image of a butterfly scale

 

Fig. 11. SEM image of a wing scale of a butterfly (Lycaenidae). The photograph on the left upper side shows the pale grass blue used for the measurement.

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Funding

Program for Creating STart-ups from Advanced Research and Technology (JPMJST2111); Japan Society for the Promotion of Science (19H02629, 21H01373, 22H01988).

Acknowledgments

Akari Fukuda and Taro Kawakatsu were supported by a fellowship from the ICOM Foundation.

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

1. C. Reese, B. Gayral, B. D. Gerardot, A. Imamoǧlu, P. M. Petroff, and E. Hu, “High-Q photonic crystal microcavities fabricated in a thin GaAs membrane,” J. Vac. Sci. Technol., B: Microelectron. Process. Phenom. 19(6), 2749–2752 (2001). [CrossRef]  

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

3. 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]  

4. Y. Takahashi, T. Asano, D. Yamashita, and S. Noda, “Ultra-compact 32-channel drop filter with 100 GHz spacing,” Opt. Express 22(4), 4692–4698 (2014). [CrossRef]  

5. E. Kuramochi, K. Nozaki, A. Shinya, K. Takeda, T. Sato, S. Matsuo, H. Taniyama, H. Sumikura, and M. Notomi, “Large-scale integration of wavelength-addressable all-optical memories on a photonic crystal chip,” Nat. Photonics 8(6), 474–481 (2014). [CrossRef]  

6. A. K. Goyal, H. S. Dutta, and S. Pal, “Recent advances and progress in photonic crystal-based gas sensors,” J. Phys. D: Appl. Phys. 50(20), 203001 (2017). [CrossRef]  

7. T. Baba, “Photonic and iontronic sensing in GaInAsP semiconductor photonic crystal nanolasers,” Photonics 6(2), 65 (2019). [CrossRef]  

8. Y. Takahashi, M. Fujimoto, K. Kikunaga, and Y. Takahashi, “Detection of ionized air using a photonic-crystal nanocavity excited by broadband light from a superluminescent diode,” Opt. Express 30(7), 10694–10708 (2022). [CrossRef]  

9. M. Nomura, N. Kumagai, S. Iwamoto, Y. Ota, and Y. Arakawa, “Laser oscillation in a strongly coupled single quantum dot-nanocavity system,” Nat. Phys. 6(4), 279–283 (2010). [CrossRef]  

10. S. Matsuo, K. Takeda, T. Sato, M. Notomi, A. Shinya, K. Nozaki, H. Taniyama, K. Hasebe, and T. Kakitsuka, “Room-temperature continuous-wave operation of lateral current injection wavelength-scale embedded active-region photonic-crystal laser,” Opt. Express 20(4), 3773–3780 (2012). [CrossRef]  

11. T. Ihara, Y. Takahashi, S. Noda, and Y. Kanemitsu, “Enhanced radiative recombination rate for electron-hole droplets in a silicon photonic crystal nanocavity,” Phys. Rev. B 96(3), 035303 (2017). [CrossRef]  

12. 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]  

13. T. Tanabe, K. Nishiguchi, A. Shinya, E. Kuramochi, H. Inokawa, M. Notomi, K. Yamada, T. Tsuchizawa, T. Watanabe, H. Fukuda, H. Shinojima, and S. Itabashi, “Fast all-optical switching using ion-implanted silicon photonic crystal nanocavities,” Appl. Phys. Lett. 90(3), 031115 (2007). [CrossRef]  

14. K. Nozaki, E. Kuramochi, A. Shinya, and M. Notomi, “25-channel all-optical gate switches realized by integrating silicon photonic crystal nanocavities,” Opt. Express 22(12), 14263–14274 (2014). [CrossRef]  

15. M. Nakadai, T. Asano, and S. Noda, “Electrically controlled on-demand photon transfer between high-Q photonic crystal nanocavities on a silicon chip,” Nat. Photonics 16(2), 113–118 (2022). [CrossRef]  

16. Y. Takahashi, H. Hagino, Y. Tanaka, B. S. Song, T. Asano, and S. Noda, “High-Q nanocavity with a 2-ns photon lifetime,” Opt. Express 15(25), 17206–17213 (2007). [CrossRef]  

17. 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]  

18. 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]  

19. Y. Akahane, T. Asano, B. S. Song, and S. Noda, “Fine-tuned high-Q photonic-crystal nanocavity,” Opt. Express 13(4), 1202–1214 (2005). [CrossRef]  

20. M. Minkov and V. Savana, “Automated optimization of photonic crystal slab cavities,” Sci. Rep. 4(1), 05124 (2014). [CrossRef]  

21. 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]  

22. J. P. Vasco and V. Savona, “Global optimization of an encapsulated Si/SiO₂ L3 cavity with a 43 million quality factor,” Sci. Rep. 11(1), 10121 (2021). [CrossRef]  

23. Y. Lai, S. Pirotta, G. Urbinati, D. Gerace, M. Minkov, V. Savona, A. Badolato, and M. Galli, “Genetically designed L3 photonic crystal nanocavities with measured quality factor exceeding one million,” Appl. Phys. Lett. 104(24), 241101 (2014). [CrossRef]  

24. 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]  

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

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

27. 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]  

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. T. Kawakatsu, T. Asano, S. Noda, and Y. Takahashi, “Sub-100-nW-threshold Raman silicon laser designed by a machine-learning method that optimizes the product of the cavity Q-factors,” Opt. Express 29(11), 17053–17068 (2021). [CrossRef]  

30. 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]  

31. 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]  

32. A. Fukuda, T. Asano, Y. Takahashi, and S. Noda, “High-Q photonic crystal nanocavity with an air-hole pattern without mirror symmetry about the x-and y-axes,” in 13th International Conference on Optics-photonics Design & Fabrication (2022), paper P_OTh_40.

33. 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]  

34. K. Ashida, M. Okano, M. Ohtsuka, M. Seki, N. Yokoyama, K. Koshino, M. Mori, T. Asano, S. Noda, and Y. Takahashi, “Ultrahigh-Q photonic crystal nanocavities fabricated by CMOS process technologies,” Opt. Express 25(15), 18165–18174 (2017). [CrossRef]  

35. 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. Topic Quantum Electron. 26(2), 8300112 (2020). [CrossRef]  

36. T. Asano, B.-S. Song, and S. Noda, “Analysis of the experimental Q factors (∼1 million) of photonic crystal nanocavities,” Opt. Express 14(5), 1996–2002 (2006). [CrossRef]  

37. S. Kita, K. Nozaki, and T. Baba, “Refractive index sensing utilizing a cw photonic crystal nanolaser and its array configuration,” Opt. Express 16(11), 8174–8180 (2008). [CrossRef]  

38. 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]  

39. 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]  

40. Y. Sato, Y. Tanaka, J. Upham, Y. Takahashi, T. Asano, and S. Noda, “Strong coupling between distant photonic nanocavities and its dynamic control,” Nat. Photonics 6(1), 56–61 (2012). [CrossRef]  

41. 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-µm telecommunication wavelength bands,” Laser Photonics Rev. 13(2), 1800258 (2019). [CrossRef]  

42. B. S. Song, T. Asano, S. Jeon, H. Kim, C. Chen, D. D. Kang, and S. Noda, “Ultrahigh-Q photonic crystal nanocavities based on 4 H silicon carbide,” Optica 6(8), 991–995 (2019). [CrossRef]  

43. K. Kuruma, Y. Ota, M. Kakuda, S. Iwamoto, and Y. Arakawa, “Surface-passivated high-Q GaAs photonic crystal nanocavity with quantum dots,” APL Photonics 5(4), 046106 (2020). [CrossRef]  

44. H. Okada, M. Fujimoto, N. Tanaka, Y. Saito, T. Asano, S. Noda, and Y. Takahashi, “1.2-µm-band ultrahigh-Q photonic crystal nanocavities and their potential for Raman silicon lasers,” Opt. Express 29(15), 24396–24410 (2021). [CrossRef]  

45. S. Ichikawa, Y. Sasaki, T. Iwaya, M. Murakami, M. Ashida, D. Timmerman, J. Tatebayashi, and Y. Fujiwara, “Enhanced red emission of Eu, O-codoped GaN embedded in a photonic crystal nanocavity with hexagonal air holes,” Phys. Rev. Appl. 15(3), 034086 (2021). [CrossRef]  

46. 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]