1. Introduction

The small mode volumes and high quality (Q) factors of microresonators have made them excellent tools to confine light and steer up high nonlinear effects with low input power threshold [1]. One example is four-wave mixing (FWM), which leads to the formation of optical frequency combs [2]. These combs have shown great applications [3], mainly in the fields of spectroscopy [4], sensing [5], telecommunications [6], and waveform generation [7], due to their broad spectrum with uniform spacings. Sufficiently good overlaps of the equidistant FWM generated sidebands and the respective resonance frequencies of the microresonator are needed to generate broadband frequency combs. Thus, the comb formation is strongly influenced by dispersion, which leads to uneven spacings between the resonance frequencies. These deviations in resonances from the equidistant positions are evaluated via the integrated dispersion ${D_{\textrm{int}}}$, which can be written as

Here, ${\omega _m}$ is the angular resonance frequency of the ${m^{th}}$ mode with respect to the pump mode ${m_0}$ at angular pump frequency ${\omega _0}$. ${\raise0.7ex\hbox{${{D_1}}$} \!\mathord{/ {\vphantom {{{D_1}} {2\pi }}}}\!\lower0.7ex\hbox{${2\pi }$}}$ is the free spectral range (FSR) around the pump mode. Resonator dispersion also plays an important role in the temporal intracavity soliton dynamics. Out of many applications, the generation of bright Kerr frequency combs occurs in presence of anomalous dispersion [8], whereas, normal dispersion [9] favors the formation of dark pulse solitons as shown in Fig. 1(a), where ${D_{\textrm{int}}}$ has been plotted as a function of wavelength for silicon nitride (Si₃N₄) ring resonators with similar radii but different core heights and widths. Recently, it has been demonstrated that dark-bright soliton bound states can be generated in a microresonator crossing different dispersion regimes [10], leading to light states with close to constant output power but resembling a comb in the frequency domain. In addition, resonators with zero dispersion not only exhibit different soliton dynamics, but also are desirable for ultra-broadband comb generation [11,12]. Therefore, proper engineering of the dispersion becomes vital [13,14]. Dispersion calculations can be carried out via simulations and experiments. A quick analysis of dispersion is not possible by the conventional simulation techniques due to their iterative nature leading to high time-complexity. Most importantly, this simulation or measurement is a unidirectional process, i.e., from the structure, one can get an idea about the dispersion but not the other way around. Methods of the inverse design of microresonators [15–17] usually follow the approach of starting with a structure from empirical knowledge and then iteratively modifying the geometries based on the deviation from the desired dispersion profile. Moreover, some inverse design methods [18] require additional quality factor engineering. Because of the highly nonlinear relation between the geometrical specifications of a microresonator structure and its dispersion profile, a simple method like linear regression for the generation of structures is not feasible. Advanced optimization methods such as evolutionary algorithms can help us in that aspect, however, extensive numerical simulations are required in each step of the process, making the process slow and time-consuming [19].

 

Fig. 1. (a) Influence of microresonator geometry on the integrated dispersion. Inset: The structural dimensions of the resonator core cross-section are shown. (b) Flowchart for simulating the resonance frequencies for a given structure and material. After completion of this iterative process, the data is used to calculate the dispersion parameter. Part of the dataset is used to train the regression model and the rest is used to test the prediction of the model.

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Machine learning (ML) is highly successful to find inherently complex nonlinear relationships between datasets in domains like, signal processing [20], natural language processing [21], computer vision [22] and data mining [23]. Due to the vast success, applications of ML transcend the boundaries of computer science to fields like classical and quantum photonics as well [24,25]. The propagation of pulses through a nonlinear medium has been effectively modelled by a recurrent neural network [26]. ML algorithms have been utilized for predicting dispersion relations of photonic crystals [27], estimating effective refractive index, mode area, dispersion and other parameters of photonic crystal fibers [28], as well as for developing photonic crystal based gas sensors [29]. In the problem of photonic inverse designing, various ML approaches show impressive efficiencies [30]. A deep learning (DL) based bidirectional model has successfully predicted subwavelength geometry of nanoparticles from far-field optical response [19]. For a desired scattering profile, the geometries of the nanoparticles have been predicted using an artificial neural network [31]. A generative adversarial network based model has been used to predict metasurface geometries [32].

In this work, we propose a novel method to predict the structures of the microresonators based on the dispersion parameter ${D_{\textrm{int}}}$ using ML. For the inverse design, a regression ML model has been trained by creating a dataset simulating dispersion variation over wavelength for different microresonator geometries. The simulations have been carried out using the finite element method (FEM) in COMSOL. After the validation of the model with the test data, this ML technique is used to estimate the heights, widths, and radii of the fabricated resonators from the experimentally obtained dispersion plots. Among different dispersion measurement techniques available [13], we chose frequency comb assisted diode laser spectroscopy [33].

2. Experimental procedures and simulations

The dispersion simulations are performed with structural and material parameters as input as shown in Fig. 1(b). To initiate the FEM solver, an approximate resonance frequency is needed which is given by

The variation of FSR over frequency is illustrated in Fig. 2(a) for both normal and anomalous dispersion regime. To verify the regression model, trained from the simulation dataset, it is important to check whether comparable structural details are acquired when dispersion parameters obtained from experiments are fed as input. Figure 2(b) shows the experimental setup to collect the transmission spectrum as well as the calibration point for dispersion estimation [33]. The SEM image for the 50-µm-radius resonator is portrayed in Fig. 2(c). The external diode cavity laser (ECDL) is swept from 1520 nm to 1615 nm and a microresonator transmission spectrum is recorded by an oscilloscope. Simultaneously, frequency calibration points are obtained by beating the sweeping diode laser with a stable commercial frequency comb (repetition rate 100 MHz), filtered through two bandpass filters (central frequencies at 35 MHz and 45 MHz with 2% and 1% filter bandwidths respectively). The frequency calibration points are simultaneously recorded by the oscilloscope.

 

Fig. 2. (a) Variation of FSR with frequency for different kinds of dispersion. (b) The experimental setup used to measure the dispersion of the various microresonators. ECDL: External cavity diode laser, PC: Polarization controller, PD: Photodiode, BPF: Bandpass filter. (c) Scanning electron microscope (SEM) image of the 50-µm-radius resonator.

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3. Results and discussions

3.1 Simulation results

The performance of the geometrical parameter prediction model on the simulated dataset using Random Forest (RF) and Decision Tree (DT) regressors are presented in this section. Given the small size of the dataset, ML models have been chosen over DL models. The RF model has empirically outperformed other ML algorithms such as support vector machine [34] and artificial neural networks [35]. Both RF and DT algorithms have been implemented using the “Scikit-learn” library in python. The performances are evaluated in terms of mean absolute percentage error (MAPE). It is defined as,

Table 1. Performance of regressors for inverse design

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It can be seen that the DT model yields a consistent yet poor prediction accuracy. However, when using RF, the performance improves drastically. The optimal model of RF regressor with not normalized data has 200 different DTs and can be seen that the MAPE for all the structural parameters are less than or around 15% for the worst case. Normalization of the dataset affects the performance of ML algorithms. Therefore, we performed the training step for RF regressor again with max-min normalized data and the results are presented in Table 1. It can be observed that, the hyperparameters of the RF model are different for normalized and not-normalized data samples, as for the two cases we have performed the grid search separately and noted the best set of hyperparameters in each case. From a comparison of the columns in Table 1, it can be seen that normalization of input and output dataset deteriorates the performance of the RF model in terms of radius prediction. Error in radius affects the result significantly, since change in radius affects the FSR of the resonator itself notably. One of the main concerns for our model is that overfitting may affect the performance of the model due to the small size of the training data. Overfitting is encountered when a regressor increases nonlinearity to a great extent such that it fits to all of the datapoints in the training dataset, thereby reducing the performance on the test dataset. We followed cross-validation to tackle this problem. 4-fold cross-validation has been used for all of the above regressors. In this process, the total training dataset is divided into four parts in each training step, the training is done with three parts and the model is validated with the remaining section. Figure 3 shows a histogram that illustrates the overall performance of the prediction model on the test-segment of the simulated data. The performance of the model on the test-data shows that the median of the errors in all three design parameters are less or around 10%. In the inset, it can be seen that with increasing number of trees in an RF model, the negative mean absolute error (NMAE) decreases initially, but saturates afterwards. The performance of our model has motivated us to validate the model using experimental data. Therefore, we have applied the ML model, trained on simulated data, to predict structural parameters from experimental measurements. In order to check the effect of the prediction error (on average 10%) on the resulting dispersion, we calculated the change in dispersion induced by increasing each structural parameter (radius, height, width) separately by 10%. The corresponding MAPE for the integrated dispersion across 1500 nm to 1600 nm (experimental range) are found to be 0.01%, 0.04% and 0.05% respectively, which is not very significant. The simulation for dataset generation has been carried on a computer with an Intel Xeon W-1290P CPU @ 3.70 GHz and 64 GB RAM. For resonators of radii 150 µm. and 50 µm, the elapsed times for the simulation of eigen frequencies between 1 µm and 2 µm are around 197 minutes and 71 minutes, respectively. In evolutionary approaches of optimization, which may be considered as an advanced optimization method, huge number of simulations (often more than hundreds) are required for every structure design task [19]. ML significantly speeds up the prediction of structural parameters, independent of the resonator size. A trained RF model (with specifications: max. depth = 20, estimators = 200) predicts structure from a dispersion profile in an average of 0.013 seconds.

 

Fig. 3. Histogram showing the prediction performance of the optimized RF model on simulated data. The dashed lines indicate the median values of prediction error for each parameter. Inset: Negative mean absolute error of the RF model versus number of trees in the model.

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3.2 Experimental results

To validate the accuracy of our prediction model, we fabricated two $\textrm{S}{\textrm{i}_3}{\textrm{N}_4}$ ring microresonators, measured their dispersion, then fed the dispersion values to the RF model and finally, compared the predicted structural details with the original designs. The resonators under test have different radii and waveguide heights but same waveguide width. The transmission profiles of these two resonators with radii of 50 µm and 100 µm are shown in Fig. 4(a) and 4(c) respectively, along with their corresponding ${D_{\textrm{int}}}$ plots in Fig. 4(b) and 4(d). The pump wavelength is considered to be around 1557 nm. Since the transmission plots contain multiple mode families, the one with higher Q-factor has been used for dispersion measurement in each case. The resonators have loaded Q-factors around $5 \times {10^5}$ (around 0.3 GHz linewidth). The dimensions of the actual and the predicted structures for both experiments have been listed in Fig. 4(e) and 4(f). The simulated dispersion profiles for both the predicted structure and the SEM observed structure are also portrayed in both Fig. 4(b) and 4(d). In the second case, the height of the microresonator is chosen to be slightly outside the training range to check whether the model can still predict in this range. The model predicted the radius, height and width within the 15% error margin. The MAPE error for the experimental case is similar to that found in Table 1 for the simulated training dataset. There are fabrication uncertainties regarding the accuracy of the structural parameters (around 10 nm for the best case) and also there can be additional impurities or inclinations of the core walls, which alter the dispersion curve and can contribute to the deviation between the actual and predicted geometrical parameters. In some cases, different resonator geometries can have similar dispersion profiles. Moreover, the avoided mode crossing (AMX) [36], as seen prominently in Fig. 4(d) around 1560 nm, changes slightly the quadratic fit of the dispersion profile. The sensitive ML models are influenced by such minute deviations. These kinds of crossings are not encountered during the simulations as the walls of the core are considered to be exactly perpendicular which is very difficult to maintain during fabrication of the resonators. The effect of these crossings can be negated if the FSR is small or the scanning range is broader so that we can have more and more modes and thus the fitting becomes easier. More importantly, if the dataset can be increased along with decreasing the intervals between the chosen structural parameters, the prediction the proposed ML model can improve, additionally opening the possibilities of application of DL models for the discussed inverse design problem.

 

Fig. 4. Microresonator resonance spectra and corresponding dispersions used to predict the structural parameters of the microresonator. (a) Transmission spectrum of a 50-µm-radius $\textrm{S}{\textrm{i}_3}{\textrm{N}_4}$ resonator with its corresponding fundamental mode family being highlighted. (b) Integrated dispersion for the fundamental mode along with a quadratic fit for the spectrum in panel (a). (c), (d) The spectrum and dispersion for a 100-µm-radius resonator. The simulation of the predicted dimensions and the SEM observed dimensions (after fabrication) have been plotted in both (b) and (d). (e), (f) Dimensions of the resonators predicted by the RF model from the experimentally obtained integrated dispersion values and comparison to the actual structures.

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4. Conclusion and outlook

We have trained a machine learning model on simulated data for predicting the structure of integrated ring resonators from their dispersion profiles generated by varying the height, radius and width of microresonators in finite element simulations. Comparing “decision tree” and “random forest” algorithms, we concluded that “random forest” yields better performance for the generation of microresonator structures. The optimized and tuned model has been used for structure prediction based on experimentally measured dispersion profiles from silicon nitride resonators. Increasing the size and spectral width of the training dataset and subsequent applications of DL could further increase the accuracy of the inverse design performance. With the ever-growing possibilities in fabrication of photonic integrated circuits, machine learning can play an important role to precisely engineer dispersion for a large number of different applications. In particular, this enables the generation of spectrally tailored optical soliton frequency combs in chip-integrated microresonators. In the future, this work can be extended to more complex geometries with organically changing resonator dimensions, which enables even more broadband control of microresonator mode structures.