Fabrication and characterization of an L3 nanocavity designed by an iterative machine-learning method

Optical nanocavities that are based on artificial defects in a two-dimensional (2D) photonic crystal (PC) slab have attracted much attention as device elements for light control. This is because they can realize high quality-factor (Q) values while maintaining a very small modal volume V. The experimental Q values were initially limited to values of less than one thousand but increased to more than ten million over time owing to advances in fabrication accuracy, material purity, and cavity design.^1–4 The ratio Q/V is a performance index that describes the strength of the light–mater interaction in the cavity material, as well as the product between the photon storage time and the capability of the cavity for dense integration.⁵ Therefore, by increasing Q/V, a higher performance can be delivered in applications such as nanolasers,^6–9 bio-/chemical sensing devices,^8,9 bistable memories,¹⁰ second-harmonic generation,¹¹ quantum information processing,^12,13 Si Raman lasers,¹⁴ and dynamic control of photons.^15–18 

The Q value of a given PC nanocavity design [design Q value (Q_design)], as well as V, depends on the cavity structure. It is known that the former, in particular, is not only determined by the cavity type; Q_design can be improved by several orders of magnitude by fine-tuning the positions of the air holes that define the cavity,^{2–5,19–24} while V does not change so strongly except in cavity types based on electric-field concentration, such as those using slits,²⁵ and in cavity types based on the direct design of the confinement potential.^22–24 The Q value of a fabricated cavity [experimental Q value (Q_exp)] is lower than Q_design due to factors such as the deviations of the actual structure from the designed structure or light absorption by the material.²⁶ However, since the achievable maximum of Q_exp of a cavity is restricted by its Q_design, the optimization of the cavity design (e.g., the hole pattern) is important. In Table I, we summarize some optimization results of Q_design values that have been obtained for different 2D PC cavity types, such as the L3 cavity,^2,27–30 the heterostructure cavity,^3,4,26 the H0 cavity,^27,31 the L4/3 cavity,³² the locally width-modulated line-defect cavity (A1 cavity),^22,33 and the effective bichromatic potential cavity,^23,24 together with the theoretical V values in units of (λ³/n³) (here, λ is the resonant wavelength of the cavity and n is the refractive index of the cavity material).³⁴ It can be confirmed that Q_design increases by optimizing the cavity structure parameters. Several different methods have so far been proposed for the optimization of the hole positions in the PC, e.g., the consideration of the envelope function of the electromagnetic field distribution,^2,3,21–24 the use of a genetic algorithm,^27,28,31 the visualization of the leaky-mode distribution,^4,29,30 particle swarm optimization,³² and the use of machine learning.³⁵ 

Due to the problems of the cost of manual tuning, and even computation time in the case of automatic tuning, the above-mentioned methods, except for machine learning, can usually only exploit a part of the high degree of freedom that is provided by the hole positions of a 2D PC cavity structure (while there are 20–30 degrees of freedom under symmetry constraints, the number of parameters that could be adjusted simultaneously was about 10 or smaller). To make sufficient use of the high degree of the structural freedom of a 2D PC nanocavity, machine learning can be used to generate a regression function (in the parameter space) that predicts Q_design based on the hole pattern that forms the cavity.³⁵ The regression function can then be used to calculate the gradient of the Q value with respect to the hole positions and updates the hole positions of a considered candidate pattern in such a way that the Q value increases.³⁵ By applying this method to the heterostructure cavity, we were able to improve Q_design from 140 × 10⁶ [with V = 1.5 (λ³/n³)] to 1600 × 10⁶ [with V = 1.5 (λ³/n³)],³⁵ and in the experiments, we succeeded in improving the Q_exp distribution.³⁶ After the publication of these results, attempts to optimize cavities with the help of machine learning have increased significantly.³⁷ Although even tuning of a few parameters can lead to a high Q_design,^22–24 optimization techniques that enable tuning of a large number of structural parameters are important because such a tuning capability is inevitable for more versatile purposes, such as simultaneous improvement of different target parameters of a cavity.^32,38

However, the number of tuning parameters is not the only important factor in optimization. For example, because the prediction in the above-mentioned machine-learning method becomes more incorrect as the new pattern departs from the vicinity of the parameter space region of the initial training dataset, it has not been possible to apply this method to optimizations where large shifts in the hole positions are required to find a significantly improved cavity (In the case of the above-mentioned results, the largest amount of shift was about 2%–3% of the lattice constant.). To solve this issue, a method has been proposed that additionally determines the Q_design values of the final candidate structures by a first-principles method, adds them to the training dataset, and then iterates these steps (i.e., generates a regression function, improves the candidates, evaluates their Q_design values by first-principles, and then adds these values to the training dataset).³⁹ The advantage of this iterative optimization method is the combination of a relatively high number of parameters and the automatic expansion of the training dataset based on the experience of the neural network to reach the parameter space region of the more optimized structure. This can significantly reduce the size of the training dataset required to reach a given degree of optimization. By applying this method to optimize 25 cavity parameters of an L3 cavity with a low initial Q_design of seven thousand, we obtained a pattern with a Q_design of 29 × 10⁶ and a small V of 0.7 (λ³/n³).⁴⁰ The maximum shift in the hole positions that was introduced in this case is 20% of the lattice constant. This Q_design is much larger than 4 × 10⁶ obtained in the case of the optimization by the genetic algorithm and the leaky-mode visualization method.^27,30 However, an experimental verification with respect to this iterative optimization technique has not yet been performed.

Although there are cavity designs that exhibit even higher Q_design values, they usually also exhibit an increased V (see Table I). For example, while a work using particle swarm reported a much larger Q_design of 190 × 10⁶, the obtained V was 1.1 (λ³/n³).⁴¹ However, because the experimental Q factors are limited to values below 10 × 10⁶ with the present fabrication technologies, it is important to realize cavity designs with Q_design ≫ 10 × 10⁶ while maintaining a small V. An optimized L4/3 cavity with a Q_design of 21 × 10⁶ and a V of 0.32 (λ³/n³) has been reported,³² but this design has not yet been verified experimentally. Therefore, the experimental verification our L3 cavity design⁴⁰ optimized by the iterative optimization method is significant for further development of devices.

In this work, we fabricate samples with the L3 cavity structure designed by our iterative optimization method based on machine learning and verify the effectiveness of this optimization method experimentally. We obtain one cavity with an unloaded Q_exp of 4.3 × 10⁶, and the median unloaded Q_exp of the cavities with the largest cavity–waveguide distance and hole radii within 102.5 ± 2 nm is 3.7 × 10⁶. These values significantly exceed the corresponding values of the previously reported L3 cavities, i.e., about 2.1 × 10⁶ and 1.1 × 10⁶, respectively (see Table I).

In Fig. 1(a), we show the designed cavity structure (air-hole pattern) of the L3 nanocavity investigated in this work. The basic PC structure consists of air holes located at the sites of a triangular lattice on the Si slab with a refractive index n of 3.46 and a slab thickness t of 220 nm. The hole radius is r = 102.5 nm (= 0.25 a) for all air holes, where a is the basic lattice constant of 410 nm. The L3 nanocavity is formed by removing three holes from this basic PC structure. In addition, the positions of the air holes within the red broken rectangle (extension: 21 columns × 5 rows) in the vicinity of the cavity have been adjusted according to the iterative optimization method based on machine learning.⁴⁰ In particular, for this optimization, we employed all three types of artificial losses described in Sec. 3.3 of Ref. 39. In Fig. 1(a), the black arrows indicate the changes in the hole positions with respect to the hole positions before optimization. A discussion of the characteristics of this cavity structure is given in  Appendix A. For the present fabrication procedure, we discretized the air-hole shifts at the resolution of the electron beam lithography (EBL) system (0.125 nm). The details of the hole position shifts after discretization are shown in Table II. Moreover, we also confirmed Q_design of the cavity with the discretized hole positions by three-dimensional finite-difference time-domain (FDTD) analysis. There was almost no difference compared to the Q value before discretization, and we obtained Q_design = 29 × 10⁶ and V = 0.70 (λ³/n³).

Prior to fabrication, we also examined the impact of the expected imperfections of the real device on the Q value. Three types of losses are considered: fabrication errors in the hole pattern, absorption by the material, and coupling to the excitation waveguide.

First, to fabricate such a nanostructure pattern, an EBL process and a transfer of the pattern by dry-etching are used. Here, small fluctuations in the fabricated hole shape are unavoidable. With respect to PC nanocavities, it is known that the influences of these fluctuations can be characterized by converting them to fluctuations in the air-hole radius and position.^26,42,43 In our presently used fabrication conditions, the empirical standard deviation of these parameters is less than 0.5 nm.^4,36,43 To determine the resulting loss, 1/Q_scat, we generated 100 random air-hole patterns based on the structure shown in Fig. 1(a) and calculated their cavity Q values. The random patterns included fluctuations in the hole radius and position with a standard deviation of 0.41 nm (one thousandth of the lattice constant a), and the resulting loss 1/Q_scat = 1/(7.7 × 10⁶) was obtained by subtracting the inverse of the above-mentioned Q_design of the ideal structure from the average of the inverse of the Q values of these random structures. Note that this value is almost the same as the loss of 1/(7.9 × 10⁶) that was obtained for a heterostructure cavity when we used the same standard deviation for the calculation of 1/Q_scat.^4,44 Thus, we confirmed that the present optimized L3 cavity is not particularly weak with respect to structural imperfections. The explanation for this result is provided in  Appendix B.

We also examined the impact of the average deviation of the air-hole radius from the optimum radius. This is necessary because the average value can change by a few nanometers.³⁰ The reason for this is that the overall hole radius depends on the etching conditions. On the other hand, the average deviation of the hole position defined by the electron beam is, in principle, sufficiently small because the EBL accuracy is sufficiently stable owing to the laser interferometer stage in the EBL system. We also examined the effect of the thickness variation of the top Si slab of the silicon-on-insulator (SOI) substrate used in the fabrication because the slab thickness varies by about 1–2 nm depending on the position in the wafer. The result in Fig. 1(b) evidences that Q_design of the presently considered cavity has a small dependence on the slab thickness, but Q_design decreases to one third if the hole radius deviates by ±2 nm from the optimum value. Such a behavior has been also reported for an L3 cavity that has been optimized by the visualization of the leaky modes.³⁰ The reason for this behavior is discussed in  Appendix B. Here, it should be mentioned that, even if the average air-hole radius is deviated by about ±2 nm, Q_design would still be about two times larger than Q_design of the L3 cavities that have been optimized by conventional methods and have optimum hole radii.^27,30

Second, it is known that a fabricated cavity has an additional loss that is caused by light absorption due to both the material and imperfections of its surface.^4,45 The magnitude of this loss, 1/Q_abs, can be changed by applying surface treatments, but from our past experience it is expected that it is at least on the order of 1/(12 × 10⁶).⁴ By considering the additional losses (1/Q_scat and 1/Q_abs), the theoretical Q value (:Q_theory) becomes about 4 × 10⁶ in the case of the optimum average hole radius, and in the case that the average hole radius deviates by ±2 nm, it is expected that Q_thoeory becomes about 3.2 × 10⁶.

Third, in order to actually measure the value of Q_exp, it is necessary to add a waveguide for excitation in the vicinity of the cavity. A design with an appropriate coupling strength (several times larger than the Q value of the cavity alone) is required because a reduction in Q_exp occurs due to the coupling to the waveguide.⁴⁶ We decided to use a width of 1.1 × $3$a for the excitation waveguide. To clarify the impact of the waveguide, we changed the distance between the cavity and the waveguide, d, from 7 to 9 rows, and determined the changes in the Q values. The waveguide-coupling losses, 1/Q_wg, for d = 7, 8, and 9 rows determined by three-dimensional FDTD analysis were 1/(0.84 × 10⁶), 1/(4.9 × 10⁶), and 1/(12 × 10⁶), respectively. Hereafter, we denote Q_exp including the waveguide-coupling loss 1/Q_wg as the loaded Q_exp or$Qexploaded$⁠, and Q_exp excluding 1/Q_wg as the unloaded Q_exp.

Regarding the fabrication of the nanocavities, here we used a SOI substrate, and after the formation of the PC pattern in the top Si layer, we fabricated an air-bridge structure by removing the buried oxide (BOX) layer by hydrofluoric acid (HF). The average thicknesses of the top Si layer and the BOX layer of the SOI substrate were 220 and 3000 nm, respectively. For the formation of the PC pattern, we employed EBL and inductively coupled plasma etching with SF₆-based gas. As mentioned above, the resolution of the EBL system was 0.125 nm. Even if all electron-beam-written radii are exactly equal, the radii of the formed holes change depending on the etching conditions. To experimentally verify the relation between Q_exp and the average air-hole radius, we fabricated several cavities with different average hole radii by slightly changing the hole radius used in each EBL process. We fabricated nine types of cavities: Their average air-hole radii after the etching process were in the range from 99 to 113 nm in equally spaced intervals. For each of the nine sample types, we fabricated one long and thin PC pattern (total length 2.4 mm) and placed a long waveguide for the excitation at the center of the PC pattern. In the vicinity of the waveguide, the cavities were placed with a sufficient distance between them in the direction of the waveguide. In this way, we placed several cavities with different values of d along a single excitation waveguide. For each cavity-parameter set, two cavities were prepared.

A SEM image of one of the fabricated cavities is shown in Fig. 1(c). The fabricated structure well reflects the designed hole pattern. The air-hole radii can be evaluated from such SEM images by assessing the ratio of the air-hole radius (measured from the center of the air hole to the white ring at the hole edge) to the lattice constant and multiplying the result by the known lattice constant of the PC design. The lattice constant was measured in a location that contains no intentional shift in the hole positions. Note that the white ring at the hole edge has a certain linewidth, and we evaluated the hole radius by measuring the distance from the center of the hole to the centerline of the broad ring. The threshold used to identify the white ring is approximately set to the highest brightness observed at the flat region of the slab. The accuracy of the evaluated value is about one fifth of the width of the white ring and is about ±2 nm in our case. Moreover, to avoid a possible pollution of the sample surface due to the SEM characterization, the characterization of the hole radii was performed on a simultaneously fabricated reference sample. Here, we characterized the two hole radii of the structures with the largest and smallest holes and estimated the radii in between by using a linear approximation.

For the optical characterization of the fabricated cavities, we first measured the resonance wavelength λ₀ by using a wavelength-tunable laser. Then, we determined the photon lifetime in the cavity, τ, from a time-resolved measurement and calculated $Qexploaded$ using the relation $Qexploaded=2πcλ0τ$ (here, c is the speed of light in vacuum).⁴⁷ We used the time-correlated single photon counting (TCPSC) method to measure the decay of the light emitted from the cavity after resonant excitation using an optical pulse with a width of 10 ns. In Fig. 2(a), we show the correlation between $Qexploaded$ and the average air-hole radius (note that the cavities with r = 101.2 nm and 109.7 nm were not measured). The colors of the data points are used to clarify the dependence on the distance d between the cavity and the waveguide. The purple, green, and orange dots are the data for d = 7, 8, and 9 rows, respectively. (The number of data points for d = 7, 8, and 9 rows is not equal because not all fabricated cavities were characterized optically.) From Fig. 2(a), we find that $Qexploaded$ exhibits a distribution with a peak at a hole radius of about 104.6 nm. Moreover, the larger the value of d is, the higher $Qexploaded$ becomes. The highest $Qexploaded$ among these cavities is 2.7 × 10⁶ obtained for the cavity with d = 9. The broken curves in Fig. 2(a) are theoretical predictions $(Qtheoryloaded)$ that consider the scattering loss (1/Q_scat) due to the fluctuations in the hole radius and hole position, the absorption loss (1/Q_abs), the loss due to the coupling to the waveguide (1/Q_wg), and the influence of the deviation of the average hole radius from the optimum value, as explained above. By comparing the experimental and the theoretically predicted values, it is found that the experimental values are overall shifted toward larger hole radii by about 3 nm and that the dependence on d is slightly weaker. To explain the shift by about 3 nm, we can consider the measurement error in the hole-radius assessment by SEM (about ±2 nm) and the deviation between the structures of the sample used for the hole-radius assessment and that actually used in the optical measurements. Note that the theory and experiment agreed when the hole-radius evaluation criterion in the SEM images was changed to the distance from the center of the hole to a position closer to the inner edge of the white ring by 15% of the width of the white ring. The relation between the $Qexploaded$ value and the hole radius evaluated with this corrected criterion is shown in Fig. 2(b). Regarding the dependence on d, a good agreement with the experimental results was obtained when we assumed 1/Q_wg = 1/(2 × 10⁶), 1/(4 × 10⁶), and 1/(9 × 10⁶) for d = 7, 8, and 9 rows, respectively [$Qtheoryloaded′$⁠; the purple, green, and orange solid curves in Fig. 2(b)]. The 1/Q_wg values derived in the above-mentioned FDTD calculation are 1/(0.84 × 10⁶), 1/(4.9 × 10⁶), 1/(12 × 10⁶), respectively, but we believe that this degree of difference is reasonable. When we analyze the overall distribution of the experimental results for d = 9 rows in Fig. 2(b), we find that an $Qexploaded$ of about 2 × 10⁶ can be obtained even if the hole radius deviates from the optimum value by ±2 nm.

As the intervals between the nine different average radii of the samples fabricated in this work were about 1.6 nm, there is the possibility that the experimental optimum radius is not among the fabricated average radii. To find the optimum radius experimentally, we tried to adjust the hole diameter by using a post-process. Note that the interval of 1.6 nm is not the accuracy limit of the EBL system, and thus, a post-process is not necessary if it is more feasible to prepare additional samples with different design hole radii. For the adjustment, we used a method that consists of forming a thin oxide layer on the sample surface by high-temperature oxidation and subsequently removing the surface oxide layer using diluted hydrofluoric acid (DHF). While the degree of diameter-change depends on the details of the oxidation conditions, it has been reported that about less than 1 nm of material is removed from the Si surface during each cycle of this process.⁴ In this process, the size of the hole increases, and at the same time, the slab thickness decreases. However, as shown in Fig. 1(b), the impact of the slab-thickness decrease on the Q value is small. We can accurately determine the amount of the removed Si, i.e., the change in the average air-hole radii, from the change in the cavity resonance wavelength due to the post-process. With the conditions used in this work, λ₀ was reduced by 3.3–3.4 nm during each cycle of the post-process. By estimating the amount of removed material that corresponds to this shift toward shorter wavelengths using FDTD calculations, we found that this decrease in λ₀ is equivalent to uniform removal of 0.7 nm of the material (Si) from all surfaces of the sample, that is, the top and bottom surfaces of the slab and at the hole walls. Based on this result, we determined the average hole radii after the post-process by adding 0.7 nm for each post-process cycle to the hole radii that was determined from the SEM image by using the criterion defined for Fig. 2(b).

The relations between the $Qexploaded$ value and the hole radius before post-processing, after one post-process cycle, and after two post-process cycles for the samples with d = 9 rows are shown in Fig. 3 by the orange, blue, and red data points, respectively. Moreover, to make the changes clearer, each change in the $Qexploaded$ value of a cavity due to a post-process cycle is indicated by a green arrow (note that not all samples have been measured at each processing step). From Fig. 3, it is found that the $Qexploaded$ values of the cavities that had an initial hole radius smaller than 102.5 nm increase due to the post-processing and the $Qexploaded$ values of the cavities that had larger radii decrease. This trend agrees well with the theoretical prediction (orange solid curve). Moreover, $Qexploaded$ of the cavity with the highest initial $Qexploaded$ among our samples (2.7 × 10⁶) increased further due to the post-processing. With this, we confirmed that this cavity had a structure with an average hole radius slightly (about 0.7 nm) below the optimum hole radius. The lifetime measurement result of this cavity after one post-process cycle is shown in Fig. 4. From Fig. 4, a photon lifetime of 2.36 ns is obtained, which corresponds to a $Qexploaded$ of 2.9 × 10⁶. The unloaded Q_exp of this cavity is 4.3 × 10⁶, which is obtained by subtracting the waveguide coupling loss evaluated in Fig. 2(b) [1/Q_wg = 1/(9 × 10⁶)].

The median $Qexploaded$ of the fabricated cavities with d = 9 rows and appropriate hole radii (within 102.5 ± 2 nm) is 2.25 × 10⁶ before post-processing and 2.6 × 10⁶ after the second post-process cycle. The corresponding median unloaded Q_exp values are 3.0 and 3.7 × 10⁶, respectively. When we define the yield as the ratio of the number of cavities with $Qexploaded$ > 2 × 10⁶ to the number of cavities with average hole radii within 102.5 ± 2 nm, the yield was 75% before post-processing and 100% after the first and second post-process cycles. These values demonstrate the robustness of this L3 cavity to fabrication-dependent fluctuations.

Figure 4 shows that it was possible to fabricate an L3 cavity with an unloaded Q_exp of 4.3 × 10⁶. This value significantly exceeds the previously reported top experimental L3-cavity unloaded Q values of 1.96 × 10⁶ (the directly measured Q value of the cavity alone)²⁸ and 2.1 × 10⁶ (the Q value of the cavity alone obtained by removing the loss due to the coupling to the waveguide from $Qexploaded$⁠).³⁰ Additionally, the modal volume of our cavity design, V = 0.70 (λ³/n³), is smaller than those of the cavity designs used in these two previous investigations, i.e., V = 0.96 (λ³/n³)²⁸ and 0.75 (λ³/n³).²⁹ In our present work, the median unloaded Q_exp after the second post-process cycle was 3.7 × 10⁶ (obtained by evaluating two cavities), which significantly exceeds the median value that has been reported for an L3 cavity optimized by a genetic algorithm (1.1 × 10⁶, obtained by evaluating 21 cavities).²⁸ It is plausible that a large part of this improvement is due to an increase in Q_design from 4 × 10⁶ (previous designs)^28,30 to 29 × 10⁶ (the present design). This explanation is supported by the fact that the predicted unloaded Q_exp would become 2.2 × 10⁶ (which is similar to the values in Ref. 28) if Q_design of the present structure decreased to 4 × 10⁶. This result indicates that the recently proposed iterative optimization based on machine learning can result in a clear positive effect even in the presence of imperfections such as fabrication-induced fluctuations in the hole radius and hole position.

For comparison with values of other investigations, it should be mentioned that the loss due to the coupling to the waveguide, 1/Q_wg, is usually evaluated experimentally from the waveguide transmission spectrum.⁴⁶ However, because it was difficult to measure a reliable transmission spectrum in the presently employed structure, wherein many cavities are coupled to the single, long (2.4 mm) waveguide, we were not able to estimate this coupling loss experimentally. Instead, we evaluated 1/Q_wg by fitting the $Qexploaded$ values for samples with different waveguide–cavity distances d. As explained in Sec. III, the obtained value of Q_wg for d = 9 rows was comparable to the value obtained from the 3D-FDTD simulation without structural fluctuation (9 × 10⁶ and 12 × 10⁶, respectively). For the estimation of the degree of the uncertainty in Q_wg in the presence of structural fluctuations, we performed one hundred 3D-FDTD simulations of the cavity pattern for d = 9 rows including random fluctuations in the hole radius and position with a standard deviation of 0.41 nm. We obtained an average Q_wg value of 9.9 × 10⁶, and the corresponding standard deviation was 1.1 ×10⁶. When we consider a Q_wg of 9.9 × 10⁶ ± 1.1 × 10⁶, the unloaded Q_exp of the cavity with $Qexploaded$ = 2.9 × 10⁶ (Fig. 4) lies in the range of 3.9–4.3 × 10⁶. When we assume that the predicted standard deviation of 1.1 × 10⁶ applies to the experimentally fitted Q_wg of 9 × 10⁶, the estimated range of the unloaded Q_exp value is 4.1–4.6 × 10⁶. Therefore, we consider that the unloaded Q_exp of the cavity with $Qexploaded$= 2.9 × 10⁶ lies within a range of 3.9–4.6 × 10⁶.

An additional aspect of the fabrication is that we fabricated nine types of cavities with different average hole radii in order to confirm the results of the theoretical prediction. A result that is worth noting is that the dependence of the Q_exp on the hole radius almost coincides with the theoretical prediction if the air-hole radius is appropriately estimated from the SEM image. Therefore, it is considered that by adding a few cavities with different average hole radii as a set of cavities for adjustment on one chip in addition to the main cavity-set, the necessity of an adjustment of the hole radius of the main cavity-set can be judged from the measurement of the Q values of the cavity set for adjustment.

We have reported on the fabrication and performance characterization of an L3 cavity with a design Q value of 29 × 10⁶. The design was obtained in a previous study by optimizing 25 degrees of freedom of the L3 cavity using an iterative optimization method based on machine learning. In this work, we obtained experimental results that well reflect the theoretical relationship between the Q value and the average hole radius, and thus, we have confirmed that a cavity with a complex hole pattern design can be actually fabricated. The ratio of cavities with an unloaded Q_exp > 2.6 × 10⁶ (⁠$Qexploaded$ > 2 × 10⁶) to the cavities with d = 9 rows and appropriate average hole radii (within 102.5 ± 2 nm) was 75% before post-processing and 100% after the first and second post-process cycles. The obtained maximum unloaded and loaded experimental Q values were 4.3 × 10⁶ (3.9–4.6 × 10⁶ when considering the uncertainty in the waveguide coupling loss) and 2.9 × 10⁶, respectively. These values largely exceed the previously reported unloaded experimental Q value of 2.1 × 10⁶. Moreover, the obtained median unloaded and loaded experimental Q values were 3.7 and 2.6 × 10⁶, respectively, both being significantly larger than the median unloaded experimental Q value of 1.1 × 10⁶ reported previously. We consider that a large portion of this improvement can be attributed to an increase in the design Q value from 4 × 10⁶ of the previous design to 29 × 10⁶ of the present design. Our experimental results have shown that the iterative optimization method based on machine learning is effective for the improvement of cavity Q values. We believe that exploiting the high degree of freedom in the design of 2D-PC nanocavities by machine learning will play an important role in optimization of not only single Q factor values but also other parameters of nanocavities.

This work was partially supported by KAKENHI Grant No. 19H02629 and received funding from the New Energy and Industrial Technology Development Organization (NEDO) under Grant No. JPNP13004.

The data that support the findings of this study are available from the corresponding author upon reasonable request.

With respect to our present optimized cavity design, we noticed that many air holes tend to shift toward the center of the cavity along the direction normal to the contour of the electric field, although exceptions exist, as shown in Fig. 1(a). It is also interesting that the cavity-field envelope function in the plane of the x axis (y = 0) is more similar to a sinc function than to a Gaussian function, as shown in Fig. 5. Because the Fourier transform of a sinc function is a rectangular function while that of a Gaussian function is another Gaussian function, a sinc envelope function (in real space) is theoretically more appropriate to reduce leaky components.²⁰ These results indicate that our iterative machine learning method automatically detected a cavity design that had a mode field better than the Gaussian confinement.

First, we consider the reason why the scattering loss of the present optimized L3 cavity is comparable to that of the heterostructure in Ref. 4. It is known that the magnitude of the scattering loss is determined by $ΔP2=ϵΔV×E2$⁠,⁴⁸ where ΔV is the change in the dielectric volume due to a fluctuation, ϵ is the dielectric constant, and E is the electric field in that volume. Therefore, in the case that the magnitude of the fluctuation is independent of the hole size, the scattering loss due to small fluctuations in the air-hole radius and position is determined by the optical energy that is distributed at the edges of the air holes. The distribution of the optical energy at the air-hole edge in an L3 nanocavity is considered to be slightly larger than that in a heterostructure nanocavity because of the existence of air holes in the center row of the L3 cavity [e.g., see Figs. 1(a) and 6(a)]. However, the air-hole radius of the optimized L3 cavity used in this work (102.5 nm) is smaller than that of the heterostructure cavity used for comparison (110 nm), which partly compensates (by about 13%) the effect of the field distribution. As a result, the scattering loss of an optimized L3 cavity with r = 102.5 nm can be comparable to that of a heterostructure nanocavity with r = 110 nm. Although a difference between the tolerances to structural fluctuations of partially optimized L3 and heterostructure cavities has been reported,⁴⁹ it should be noted the structures investigated in this previous work are different from ours.

Second, we found that the dependence of Q_design on the average hole radius is weaker for the optimized heterostructure cavity, as shown in Fig. 6(b). This different tendency for different cavity types is attributed to a difference in the intensity of the radiation field that occurs if the air-hole positions are not tuned in accordance with the hole size. In other words, we consider that the hole shifts introduced during the optimization process lead to a subtle cancellation of the radiation field, and this cancellation can be destroyed by a change in the air-hole radius. Figure 7 shows the electric field distributions and the leaky component distributions²⁹ of three L3 cavities with the same optimized hole position pattern (optimized for r = 0.25a), but with three different air-hole radii. We can confirm that the electric field distributions are almost the same (Fig. 7, upper panels), while the intensities of the leaky components are very different (Fig. 7, lower panels). This result supports the explanation that a finely established far-field cancellation can be easily disturbed by a change in the air-hole radius. In such a situation, the radiation intensity can reach levels equivalent to the case without tuning. Such a degradation in Q_design can be considered more severe for cavities with a larger ratio between the initial and optimized Q_design values. The Q_design values of the present L3 nanocavity before and after our optimization are seven thousand and 29 × 10⁶,⁴⁰ respectively, which differ by four orders of magnitude. On the other hand, the Q_design values before and after our optimization of the heterostructure nanocavity are 20 × 10⁶ and 1600 × 10⁶,^35,36 respectively, which differ by only two orders of magnitude. Therefore, the degradation of the Q factor is more severe for L3 cavities.