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Abstract
Soliton microcombs generated by the third-order nonlinearity of microresonators exhibit high coherence, low noise, and stable spectra envelopes, which can be designed for many applications. However, conventional dispersion engineering based design methods require iteratively solving Maxwell's equations through time-consuming electromagnetic field simulations until a local optimum is obtained. Moreover, the overall inverse design from soliton microcomb to the microcavity geometry has not been systematically investigated. In this paper, we propose a high accuracy microcomb-to-geometry inverse design method based on the genetic algorithm (GA) and deep neural network (DNN), which effectively optimizes dispersive wave position and power. The method uses the Lugiato-Lefever equation and GA (LLE-GA) to obtain second- and higher-order dispersions from a target microcomb, and it utilizes a pre-trained forward DNN combined with GA (FDNN-GA) to obtain microcavity geometry. The results show that the dispersive wave position deviations of the inverse designed MgF2 and Si3N4 microresonators are less than 0.5%, and the power deviations are less than 5 dB, which demonstrates good versatility and effectiveness of our method for various materials and structures.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
Corrections
14 September 2023: A typographical correction was made to the author affiliations.
1. Introduction
Microresonator has naturally become a platform for chip-scale optical frequency comb (microcomb) due to its high quality-factor Q, extremely small mode volume Veff, and material's optical nonlinearity properties. In recent years, dissipative Kerr soliton (DKS) microcombs, based on balancing the intra-cavity dispersion and nonlinearity as well as gain and dissipation, have been demonstrated to have high coherence, low phase noise, and stable spectra envelope [1,2]. In addition, microcombs based on various soliton dynamics have been experimentally proposed, such as dark soliton [3], Stokes soliton [4], Raman soliton [5], soliton crystal [6], Dirac soliton [7], Pockels soliton [8], and mechanical soliton [9]. Researchers have also produced soliton microcombs with various material platform, such as MgF2 [10], Si3N4 [11], AlN [12], SiO2 [13], etc.
Designing the soliton microcomb with a specific envelope is of great significance for practical applications, such as astronomical spectral calibration [14], LiDAR ranging [15], coherent optical communication [16], etc. In particular, for frequency metrology and spectroscopy applications, the dispersive wave (DW) generated by soliton Cherenkov radiation in the presence of higher-order dispersion [17] has attracted considerable attention because of the specific comb-line power increment. Several studies have reported utilizing DW to achieve f-2f self-referencing [18], repetition rate noise reduction [19], and chip-scale atomic clocks [20]. Although the DW position can be estimated through the phase-matching condition [21], the frequency and power of DW are usually limited because of nonlinearity-induced frequency shift. In addition, previous microresonator design methods aiming for desired DW require high computational costs due to iterative electromagnetic field simulations [22,23]. Therefore, an efficient and versatile high-accuracy automated design method for microcomb applications is highly desired.
In recent years, with the rapid development of deep learning, photonic inverse design has been effectively applied to metasurfaces [24], photonic crystals [25], and silicon photonic devices [26], etc. Several research groups have also achieved photonic inverse design for microcomb applications. Ahn et al. experimentally demonstrated an agile dispersion engineering approach for on-chip microresonators based on partially reflective elements [27]. Lucas et al. theoretically proposed an optimization method based on LLE-GA for the envelope-to-dispersion inverse design of the microcomb, which obtained the near Gaussian pulses, specific bandwidth, and specific DW [28]. Wang et al. used an inverse neural network to directly inverse design the group velocity dispersion (GVD) of As2S3 and Si3N4 microresonators [29]. However, none of the previous approaches considered the overall inverse design method from the target microcomb to the geometry of microresonator with various materials and structures. Furthermore, the proposed DNN-based method [29] is difficult to achieve a direct inverse design from microcombs to geometry, because the design space of most resonant devices is non-convex [30], which limits the accuracy of gradient descent based methods.
In this paper, a microcomb-to-geometry inverse design method of soliton microcomb based on GA and FDNN is proposed. Each order of dispersion was globally optimized to meet the desired DW position and power. For optimizing the target dispersion, the symmetric split-step Fourier (SSF) method was implemented to solve the dimensionless LLE combined with GA. The finite element method (FEM) obtains the geometry-dispersion dataset in the specified material microresonator by parametric sweeping. Subsequently, the corresponding relationship between the geometry and dispersion was obtained by training FDNN. Finally, the target geometry was obtained using FDNN-GA. Moreover, FDNN-GA introduces a physics loss to improve the robustness of global optimization, which is implemented with additional FEM and LLE processes.
2. Principle
The proposed soliton microcomb inverse design method consists of three main steps: the desired comb spectra to the dispersion, the FDNN pre-training, and the dispersion to microresonator geometry, as shown in Fig. 1.
Fig. 1. (a) Workflow of microcomb-to-geometry inverse design method of soliton microcomb. (b) Workflow of FDNN-GA based dispersion-to-geometry inverse design method.
2.1 LLE-based evolution
The input parameters of LLE-based evolution include pump power, laser detuning, total mode numbers, initial comb spectra, target DW position, DW power, and GA parameters (population size, chromosome length, mutation rate, crossover rate, and total generation). For generality of the approach, the theoretical model is based on the dimensionless LLE in this study. The dynamic equation of the dimensionless intra-cavity field can be expressed as [21,31]:
- $T = \frac{\kappa }{2}t$ is dimensionless time ($\kappa = {\kappa _0} + {\kappa _{ex}} = {{{\omega _0}} / Q}$ is the total loss rate of microresonator),
- $\Theta $ is the angular coordinate that co-propagates with the group velocity of soliton pulse, and $\theta = \frac{{2\delta \omega }}{\kappa }$ is the dimensionless angular frequency detuning,
- ${d_k}$ is the dimensionless kth-order dispersion, ${d_{{\mathop{\rm int}} }} = \sum\limits_{k \ge 2} {{\mu ^k}{d_k}}$ is the integrated dispersion, $\mu $ is the relative mode number,
- $p = \sqrt {\frac{{{P_{in}}}}{{{P_{thres}}}}} = \sqrt {\frac{{8\eta g{P_{in}}}}{{{\kappa ^2}\hbar {\omega _0}}}}$ is the dimensionless driving pump field ($g = {{\hbar {\omega _0}^2c{n_2}} / {{n^2}{V_{eff}}}}$ is the nonlinearity, $\eta$ is the coupling rate).
Assuming that the step size dt of the propagation of $A({T,\Theta } )$ is small enough, the dispersion and nonlinearity can be regarded as acting on $A({T,\Theta } )$ independently. Thus, Eq. (1) can be written as:
Equation (2) is solved using the symmetric SSF method as follows [32]:
According to the Baker-Hausdorff formula [33], the error term of Eq. (3) is in the order of dt3, which shows a higher local accuracy than the conventional SSF method. Therefore, after 20-30 photon lifetimes, a stable microcomb spectra envelope ${\cal F}\{A \}$ can be obtained, where ${\cal F}$ is the Fourier transform operator.
Since LLE-GA is used for envelope inverse design, the proper initial dispersion can compress the design space of GA global optimization. According to the phase-matching condition of DW, the initial guess of integrated dispersion $d_{{\mathop{\rm int}} }^{guess}$ can be estimated as follows [21]:
The initial comb is then obtained by substituting the estimated dispersion $d_{{\mathop{\rm int}} }^{guess}$ into Eq. (1), in which the GA iteratively optimizes. The GA first obtains a genetic operator by binary encoding each order dispersion ${d_k}$. The relation between ${d_k}$ and N-bit binary genetic operator can be written as follows:
Then the GA globally optimizes the genetic operator through mutation, crossover, and selection. Depending on the desired application, the fitness function can be set to flatness, specific comb power, and mean squared error (MSE) between design and target comb envelopes, etc. Since the optimization targets are DW position ${\mu _{tgt}}$ and power ${P_{tgt}}$ in this study, the fitness function FLLE-GA is set as:
When solving the LLE-GA, the pump power p2 and detuning θ are fixed to $[{{p^\textrm{2}},\theta } ]= [{8,8} ]$, ensuring that the soliton comb solution is stable [34]. Once the iteration reaches its maximum generation or the fitness achieves its desired value, the optimized dimensionless dispersion $d_k^{tgt}$ is output and sent to the FDNN-GA process.
2.2 FEM simulation and FDNN pre-training
The second primary step of inverse design is pre-training FDNN using a geometry-dispersion dataset. First, a series of FEM calculated fundamental mode fields is obtained by sweeping the geometry G and resonance frequency ω0 of the microresonator. The resonance frequency ω0 is included in the dataset, since the integrated dispersion is also affected by the pump laser frequency. Furthermore, the scale of geometry-dispersion dataset is enlarged by adding the center resonance frequency ω0, which can expand the dataset for deep learning. Note that ω0 is chosen within the frequency tuning range of the actual laser diode.
Second, the azimuthal quantum number m of the fundamental mode under different geometries and frequencies is obtained through mode analysis. According to the relationship between the propagation constant, angular frequency ω, and azimuthal quantum number m, the actual dispersion Dk can be calculated as [22,35]:
Third, the FDNN is pre-trained to predict dispersion through the input matrix of geometry and resonance frequency $[{D_2},D{}_3,\ldots ,D{}_k,{\omega _0}]$. FDNN is implemented based on PyTorch framework and trained by RTX3060 GPU; the dataset is min-max normalized to 0-1 range and divided into training and testing datasets at the ratio of 80/20 to improve generality; the number of neural network layers and the number of neurons in each layer are adjusted according to the geometric complexity of the microresonator; the optimizer is set to be Adam algorithm with a learning rate of 1E-4 and 800 epochs; the activation function adopts leaky-ReLU [36] with a slope factor of 0.01 to prevent dead neurons.
2.3 FDNN-based evolution
The last step of our inverse design is using FDNN-GA to obtain the target geometry from the optimized dimensionless dispersion $d_k^{tgt}$. Note that the input parameters of FDNN-based evolution include target dispersion, target comb spectra, typical microcavity quality factor, geometric parameter range, pretrained FDNN, and GA parameters. The target dispersion $D_k^{tgt}$ is obtained by calculating the LLE-GA optimized dimensionless dispersion $d_k^{tgt}$ according to the typical Q value of the fabricated microresonators:
Then the target dispersion $D_k^{tgt}$ is binary-encoded and substituted into the FDNN-GA process, as shown in Fig. 1(b). During the FDNN-GA process, the fitness function FFDNN-GA is set as the MSE between the GA optimized dispersion and the target dispersion $D_k^{tgt}$. Since the magnitude difference between each order dispersion is usually large, the fitness function FFDNN-GA can be written as:
The binary-coded geometry value is globally optimized to reach maximum fitness through crossover, mutation, and selection. Note that the design space of the FDNN-GA is within the range of geometry dataset to ensure high accuracy. Finally, to prevent the singular point of global optimization [37], we introduce the physics loss based on the FEM and LLE. The output geometry Gopt is sent back to the FEM and calculated using the LLE and Eq. (6) to compare the FDNN-GA-optimized comb with the LLE-GA-optimized comb, which verifies the optimization results. Finally, the geometry Gopt of the microresonator corresponding to the target dispersion $D_k^{tgt}$ is obtained, which can be utilized for further fabrication.
3. Results
3.1 LLE-GA based microcomb-to-dispersion inverse design
For the microcomb application with a specific DW, it is essential to increase the comb power around the desired mode number µ. We assumed that the target mode number was μ=1200, and the normalized target comb line power was -30 dB. The initial dispersion was set to ${d_{ini}} = \{{d_\textrm{2}^{ini},d_\textrm{3}^{ini}} \}= \{{\textrm{ - 1}\textrm{.20E - 3, 1}\textrm{.00E - 6}} \}$ according to Eq. (6). As shown in Fig. 2(b), the nonlinearity-induced frequency shift at a large mode number µ causes DW to deviate from the target position. Therefore, the fitness function of LLE-GA is a combination of the frequency and power deviation of DW as Eq. (6). The population size, chromosome length, mutation rate, crossover rate, and total generation of the LLE-based GA were set as 10, 300, 0.05, 0.5, and 100, respectively. After 100 iterations of the GA, the DW position was optimized to µ=1200, and the DW normalized power was increased to -37 dB. The optimized dispersion was ${d_{opt}} = \{{d_\textrm{2}^{opt},d_3^{opt}} \}= \{{\textrm{ - 1}\textrm{.26E - 4, 1}\textrm{.19E - 7}} \}$. The optimized integrated dispersion was flatter than the initial integrated dispersion, as shown in Fig. 2(c), which led to a broader microcomb bandwidth and higher DW power.
Fig. 2. (a) The convergence curve of LLE-GA. (b) The LLE-GA optimized envelope of single DW soliton microcomb, where target DW position µ=1200, target DW normalized power P = -30 dB. (c) The LLE-GA optimized integrated dispersion of single DW soliton microcomb.
3.2 FDNN-GA based dispersion-to-geometry inverse design
Dispersion engineering has been implemented for various microresonators. For example, in crystal microcavities (e.g., MgF2, CaF2), dispersion can be controlled by mechanical grinding and polishing to adjust the cross-sectional shape [38]; in on-chip microresonators (e.g., Si3N4, AlN), dispersion engineering can also be managed by dual-cavity coupling structures, changing cladding materials, and adjusting cladding thickness [39–43], etc. For simplicity, we first investigated a MgF2 cavity with an elliptical cross-sectional shape. The relative structures and FEM-simulated TE fundamental mode are shown in Fig. 3.
Fig. 3. (a) The structure and FEM-simulated TE fundamental mode of MgF2 microresonator with elliptical cross-sectional shape. (b) The comparison of predicted and actual dispersion of MgF2 microresonator. (c) The learning curve of MgF2 microresonator based FDNN.
Before the FDNN-GA process, the dataset of MgF2 was generated using FEM solver and Eq. (6). It is worth noting that the area of the FEM simulation region can be reduced by exploiting the axial symmetry of the geometry, doubling the calculation speed.
In a MgF2 microcavity with an elliptical cross-sectional shape, the dispersion can be coarsely controlled by the main radius R of the microcavity and finely adjusted by the long and short axes of the ellipse. The design space of the MgF2 microresonator is ${G_1} = \{{R,{R_x},{R_y}} \}$ where $R({mm} )\in [{0.2,5} ],{R_x}({\mu m} )\in [{100,1000} ],{R_y}({\mu m} )\in [{5,80} ]$, which is fabrication-compatible.
During the training process of MgF2 microresonator based FDNN, the geometry matrix combined with various center frequencies was set as the input dataset $\{{G,{\omega_0}|{{\omega_0}} ({THz} )\in [{191,195} ]} \}$ with a scale of over 20000, which was generated at a time cost of 48 h, and the calculated dispersion matrix was set as the output matrix. Since the tunable dimension of MgF2 geometry is 3, the number of hidden layers was set to 3, and the number of neurons per layer was set to 40, 30, and 20, respectively. To validate the accuracy of the pre-trained FDNN, we predicted the dispersion in the test dataset, which showed good agreement between the predicted dispersion and FEM simulation results, as shown in Fig. 3(b). As shown in Fig. 3(c), the MSE was reduced from 0.5 to 1.6E-5 during the training process. The learning curves show that our model was well-fitted after 800 epochs of training, demonstrating good generalizability.
Based on the optimized dispersion of the LLE-GA process, the target dispersion $D_k^{tgt}$ of the MgF2 microresonator was obtained according to the Q-factor of 5e8, which is a typical Q-factor of fabricated MgF2 resonator [10,44,45]. The GA parameters of the FDNN-GA, such as the population size, chromosome length, mutation rate, crossover rate, and total generation of the LLE-based GA, were set as 50, 100, 0.01, 0.5, and 500, respectively. In addition, the maximum iteration number of physics-loss-based verification process was set as 10. The optimized result of MgF2-based soliton comb is shown in Fig. 4. The FDNN-GA optimized MgF2 microresonator geometry is ${G_1} = \{{R({mm} )= 3.65,{R_x}({\mu m} )= 53.56,{R_y}({\mu m} )= 301.08} \}$ with an optimized center frequency of 194.15 THz. These geometry values are not included in the training dataset of the pre-trained FDNN, indicating that our inverse design method also has excellent generalization ability. The relative mode number deviation of DW of the MgF2-based soliton comb was 0.42%, and the power deviation of the DW was only 1 dB, which manifests the effectiveness of the inverse design. These deviations are probably due to error propagation from the FDNN-predicted dispersion to the FDNN-GA optimized geometry, which can be minimized by further increasing the accuracy of the FDNN.
Fig. 4. The comparison of initial comb, LLE-GA optimized and FDNN-GA optimized single DW comb of MgF2 microresonator.
4. Discussions
In order to verify the versatility of the dispersion-to-geometry inverse design in microresonators with different structures and materials, the inverse design from a soliton comb with dual DWs (2-DWs) to a Si3N4 microresonator with a dual-cavity coupling scheme was performed as shown in Fig. 5. The target 2-DWs’ positions were set to -400 and 500, and the normalized target power was set to -20 dB. The Q-factor of Si3N4 resonator was set as 1e6. Note that the fourth-order dispersion D4 was included in the model according to published results [20,46]. The initial dispersion was set to $\{{d_\textrm{2}^{ini},d_\textrm{3}^{ini},d_4^{ini}} \}= \{{\textrm{ - 2E - 3, 1E - 6, 1E - 8}} \}$, and optimized to $\{{d_\textrm{2}^{opt},d_\textrm{3}^{opt},d_4^{opt}} \}= \{{\textrm{ - 9}\textrm{.34E - 4, - 6}\textrm{.42E - 7, 4}\textrm{.9E - 9}} \}$ through the LLE-GA process as shown in Fig. 5.
Fig. 5. (a) The structure and FEM-simulated TE fundamental mode of Si3N4 microresonator with dual-cavity coupling for dual-DWs comb. (b) The comparison of predicted and actual dispersion of Si3N4 microresonator. (c) The learning curve of Si3N4 microresonator based FDNN.
In the Si3N4 microresonator with a dual-cavity coupling scheme, the dispersion can be fine-tuned by adjusting the coupling gap and geometry of coupled ring waveguides. Using FEM simulation and (6), the Si3N4 dataset with a scale of 40000 was obtained at a time cost of nearly 50 h. The training settings of the Si3N4 microresonator based FDNN are similar to MgF2-based FDNN. Besides, the number of hidden layers was set to 4, and the number of neurons per layer was set to 60, 50, 40, and 30, which is more than the MgF2-based FDNN due to the higher structure complexity. As shown in Figs. 5(b) and (c), the accuracy and loss convergence of the Si3N4-based FDNN are similar to those of the MgF2-based FDNN, which shows the strong transferability of our model. The fabrication-compatible design space and FDNN-GA optimized geometry of the Si3N4 resonator are listed in Table 1.
Table 1. Design space and optimized results of structure in Figure 5
As shown in Fig. 6, the inverse design results also exhibit good performance because the FDNN-GA optimized 2-DWs comb is close to the target comb. The relative mode number deviations of the 2-DWs were 0.4% and 7.6%, respectively, and the power deviations of the 2-DWs were both within 5 dB. The larger mode-number deviation of the short-wavelength DW was mainly due to the inability to obtain the corresponding geometry of the target dispersion, resulting in a trade-off between the position and power of the DW. In addition, since the free spectral range (FSR) of the optimized Si3N4 microresonator is 150.5 GHz, the frequency range between 2-DWs exceeds an octave, suggesting that our inverse designed microcomb can be applied to f-2f self-reference applications. Note that, for other applications such as chip-scale atomic clocks, where DWs frequencies are most desired, the model should consider D1, which represents the FSR of the resonator.
Fig. 6. The comparison of initial comb, LLE-GA optimized and FDNN-GA optimized dual-DWs comb of Si3N4 microresonator.
The mechanism of our proposed inverse design method is that the relationship between the microcavity geometry and the integrated dispersion can be expressed as a function $f({G,{\omega_0}} )= {d_{{\mathop{\rm int}} }}$. As shown in Fig. 7, the numerical FEM results show a clear relationship between MgF2 geometry G and the integrated dispersion ${d_{{\mathop{\rm int}} }}$. The major radius of microresonator divides the dispersion parameter space into several clusters, and dispersion value of each order generally increases with the total radius, as shown in Fig. 7(a). In addition, within each cluster, the long and short axes of the ellipse shape fine-tune the dispersion, as shown in Fig. 7(b). The mechanism is that the geometric parameters change the intracavity field constraints, thereby changing the mode distribution, resulting in the variation of each order dispersion. Thus, the function $f({G,{\omega_0}} )= {d_{{\mathop{\rm int}} }}$ can be numerically approximated through the nonlinear mapping between layers.
Fig. 7. (a) The dispersion parameter space of MgF2 microresonator, which shows each order dispersion mainly increases with the total radius R and each order dispersion is fine-tuned by cross-sectional geometry Rx and Ry; (b) the dispersion parameter space in the anomalous dispersion regime shows obvious multi roots phenomenon.
As shown in Fig. 7(b), the parameter space near the anomalous dispersion regime (D2 > 0) shows an apparent multi-roots phenomenon, which leads to multiple local optima in the solution space. Therefore, a straightforward DNN-based inverse design based on gradient descent cannot usually obtain the correct geometric parameters without restricting the parameter space [47]. Our proposed FDNN-GA based on a random search algorithm can effectively solve this “one-to-many” problem. In this method, the FDNN plays the role of a high-speed FEM solver, which reduces the calculation time of the dispersion simulation from several hundred seconds to only several microseconds while ensuring a highly accurate solution.
Since FDNN is a data-driven process like other deep neural networks, the deviation between the FDNN prediction and the actual FEM results is primarily due to the data-splitting process. Note that the FDNN loss is equivalent to the deviation in terms of its physical meaning. As shown in Fig. 8, the deviation of FDNN rises when training data ratio falls below 40%, since less data leads to less learnable information for the FDNN. Additionally, the training data ratio from 50% to 80% has the same level deviation, indicating that our FDNN model maintains high accuracy in sparse geometric parameter space.
Fig. 8. The deviation between the FDNN prediction and real FEM results at various data splitting ratios.
However, due to the limited complexity of the microcavity cross-section geometry, the dispersion parameter space is relatively sparse, hindering actual agile dispersion engineering. In addition, the increased complexity of the cross-sectional shape results in additional scattering losses, which reduce the microresonator's Q-factor and increase the fabrication difficulty. Therefore, by incorporating the influence of other modes, nonlinear effects, and high-order dispersion [48] in future work, our proposed method expects better microcomb inverse design flexibility.
5. Summary
In this study, an overall microcomb inverse design method from a target comb to geometry using the GA combined with LLE and FDNN is proposed. This method can optimize the desired DW position and power, as well as predict the microresonator geometry of various materials. The dimensionless parameterization of the GA-LLE makes this method universal. By training the dispersion prediction network of microresonators with various materials and structures, inverse design of various microcombs can be achieved. The inverse design results show the high accuracy of the designed microresonators with the generated DW position and power deviation are of 0.42% and 5 dB, respectively. Our inverse design method has the advantages of low complexity, low computational cost, good versatility, and strong transferability. In addition, this method can be further improved by optimizing the DNN configuration, the meta-heuristic algorithm optimization accuracy, and dispersion engineering precision.
Funding
National Key Research and Development Program of China (2022YFF0608304); National Natural Science Foundation of China (62075206, 62205324).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper may be obtained from the authors upon reasonable request.
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