The advent of photonic integrated circuits (PICs) is expected to revolutionize many industries such as optical interconnects [1–3], optical sensing [3–6], and optical computing [7,8]. The progress of PICs demands compact photonic devices with a variety of properties to increasingly integrate more functionality on a chip. The property of a photonic device is highly determined by its geometry, which is usually designed and optimized by an experienced photonic designer based on intuition and numerical simulations. This traditional method is limited by designers’ experience and usually only applies to the designs with simple geometric structures. However, complex geometries are often required for realizing ultra-compact photonic devices with challenging properties. The design of these complex structures with a large number of parameters is beyond the capability of human intuition and demands more efficient methods. The brute force optimization method is impractical, as the numerical simulation algorithms, such as the finite-difference time-domain (FDTD) method, are very computation intensive and time consuming. Traditional regression-based optimization methods face difficulties in convergence due to the high-dimensional parameter space. A variety of inverse design methods such as level set methods, adjoint method, local-optimization techniques, and direct-binary search [9–19] have been developed to tackle this issue. These methods expand the design space and realize many photonic devices with complex and nonintuitive structures that present extraordinary properties. Fundamentally, these algorithms are rule-based approaches containing iterative searching steps in a case-by-case manner, often relying on numerical simulations in each step to produce intermediate results that help to modify the searching strategy. Such stochastic algorithms are limited by their random-search nature and are insufficient for complex designs in a multi-constrained problem [20–22].

This challenge may be addressed by deep learning (DL) and more specifically deep neutral networks (DNNs), which use many processing layers to learn the representations of data with multiple levels of abstraction [23]. DNN has been revolutionizing many fields, including image analysis [24], natural language processing [25], materials science [26], quantum physics [27], and so on. Neural network (NN)-based methods have previously been adopted in studying photonic devices such as phase delay structures [28], photonic crystal nanocavities [29], and nanophotonic particles [30]. However, applying a DNN to photonic device design faces a practical challenge of preparing a large training dataset [20]. For example, Ma et al. used an NN model trained on 30,000 device designs for a chiral metamaterial inverse design [31]; Tahersima et al. used an NN model trained on 20,000 designs to realize a power splitter with specific splitting ratios [32]. These large amounts of design data are generated by high-resolution 3D FDTD simulations and may take days to complete on a typical workstation.

In this paper, we present a genetic-algorithm-based deep neural network (GDNN) method that requires an order of magnitude less simulation data (less than 3000 in the presented cases) for training. By using this approach, we have designed several silicon photonics devices including power splitters with uncommon splitting ratios, a TE mode converter, and a broadband power splitter. Besides high efficiency, the algorithm exhibits great flexibility and ability in dealing with multiple design constrains. For example, the broadband power splitter is designed for as many as 11 optimization targets, and the optimal design exhibits a maximum insertion loss of 0.47 dB and good flatness over 400 nm wavelength range with a $2.5\mathrm{\mu m}\times 1.5\mathrm{\mu m}$ footprint. In addition, these devices are free of the features beyond the capability of photolithography and generally in compliance with silicon photonics fabrication design rules.

The GDNN-based inverse design algorithm developed in this work includes encoding, selection, and DNN-based forward and inverse design processes. The workflow of the algorithm is exhibited in Fig. 1. First, the device models and numerical simulation results of the initial device designs are encoded into a dataset called population (step A of Fig. 1). In mimicking the natural selection process, the population is selected with respect to an objective function (step B of Fig. 1). The whole population is also used to generate the weight parameters in the DNN models of both the forward design and the inverse design [step C (1) of Fig. 1]. The selection process produces an adaptive (good performance) generation as the parents to reproduce their offspring through the inverse design process [step C (2) of Fig. 1]. These offspring are then added to the initial population (step D of Fig. 1) and update the weights in the DNN models to start the next iteration. This process is repeated until the DNN model meets the required optimization target to the investigated design problem.

The encoding process aims to map a physical problem, e.g., the output optical power of an optical device with multiple degrees of freedom in its physical structures, to a mathematical data structure. An encoded data structure represents a solution to the problem and affects the optimization process. In this work, each individual of the initial population is described with two sets of parameters. One set represents the encoded device physical structure that will be genetically optimized, and the other represents the investigated optical performance. The encoding process of a power splitter device is schematically shown in Fig. 2. An arbitrarily shaped one-by-two power splitter includes one input waveguide, two output waveguides, and a core that is described by its border line. By considering the device design rules and its symmetry, we adopt the polar coordinate system with its origin located on the center line and use two sets of polar vectors ($R_i^1=\{R_0^1,R_1^1,\dots ,R_n^1\},R_i^2=\{R_0^2,R_1^2,\dots ,R_n^2\}$) to discretize the top (within [$0,\pi$]) and the bottom (within [$\pi ,2\pi$]) part of the border line, respectively. The complete device structure is formed by smoothly connecting these two sets of polar vectors and combining the input and the two output ports. Practically, a foundry generally imposes design rules to disallow very small or sharp structures in a layout. Thus, we also introduce a set of gradient constraints $\alpha$ ($S=\{R_0^1,R_0^2,{\alpha }_1^1,{\alpha }_1^2,\dots ,{\alpha }_n^1,{\alpha }_n^2,d_0\}$) between the radii of the two adjacent polar vectors. With this encoding process, an individual device is completely described by a design parameter vector $S=\{R_0^1,R_0^2,{\alpha }_1^1,{\alpha }_1^2,\dots ,{\alpha }_n^1,{\alpha }_n^2,d_0\}$, where $d_0$ is the distance of the output branch from the origin as shown in Fig. 2. The investigated performance or the optimization objective is represented by a figure of merit (FOM), which is defined as the optical power in the zeroth transverse electric (TE0) mode of an output waveguide. If the transverse magnetic (TM) component is negligible, which is validated by simulation results, the FOM can be calculated by $$\mathrm{FOM}=\frac{1}{4}\frac{{|{\int }_{\sigma }(E\times H_0^{*}+E_0^{*}\times H)·\mathrm{d}\sigma |}^2}{{\int }_{\sigma }\mathrm{Re}(E_0\times H_0^{*})·\mathrm{d}\sigma },$$(1)where $\sigma$ is the cross section of the waveguide, $E_0$ ($H_0$) is the electric (magnetic) field distribution of the TE0 mode in the waveguide, and $E$ ($H$) is the output electric (magnetic) field distribution from the device FDTD simulation. We believe that using the polar coordinate system with carefully designed constraints can effectively describe a sufficiently flexible device geometry while avoiding the violation of foundry design rules. This encoding approach simplifies the complexity of the GDNN inverse design algorithm and can be generalized for other optical device structures beyond the several examples shown in this work.

The population selection process is a prerequisite for DNN-based inversed design. It evaluates the FOM of each individual in the current population and selects a subset of the population with good values to be the parents that “reproduce” the next generation of device designs. Due to the “genetic” correlation between the parents and the offspring connected by the neural networks, additional factors can be added to the selection process to intentionally guide the direction of certain features in the evolution. For example, a punishment factor on the device area can be added to guide the algorithm to favor more compact designs.

A genetic algorithm mainly includes three operations: selection, crossover, and mutation. Crossover and mutation are the key steps to produce excellent offspring. In the GDNN algorithm, we use the DNN model to replace the crossover and mutation operations of the genetic algorithm. The DNN model can be inferred in both forward and inverse ways. The forward design process can obtain the device FOM and network weight parameters in the form of a given encoded design parameter vector. The network weight parameters are the key settings for enabling the reverse design process. The inverse design process after obtaining the network weight parameters utilizes the FOM data associated with wavelength as the input to produce offspring device designs that are correlated but different than the parent. The forward and the inverse design processes are schematically shown in Fig. 3. A fully connected multilayer DNN is used with the sigmoid activation function and back-propagation method for weight updates.

In the inverse design process, the cost function is represented by the error E of the FOM of a specific design ($S=\{x_1,\dots ,x_n\}$, $Y=\{{\mathrm{FOM}}_1,{\mathrm{FOM}}_2\}$) from its ideal value $T$, i.e., $E={\sum }_{i=1}^m{({\mathrm{FOM}}_i-T_i)}^2$. This error is back-propagated by the equation $${\delta }_i^k=\frac{\partial E}{\partial Z_i^k}={\sum }_{j=1}^N(\frac{\partial E}{\partial Z_j^{k+1}}·\frac{\partial Z_j^{k+1}}{\partial Z_i^k})={\sum }_{j=1}^N({\delta }_j^{k+1}·\frac{\partial Z_j^{k+1}}{\partial Z_i^k}),$$(2)where $Z_i^k$ is the weight of the neurons in the $k$th layer, i.e., $Z_i^k={\sum }_{j=1}^{N}f(W_{j,i}^{k-1}+Z_j^{k-1})$, where $f$ is the activation function. By repeating this step, the error eventually propagated to the input layer gives the offspring device design variation $$\mathrm{\Delta }{x}_i={\sum }_{j=1}^N({\delta }_j^1·\frac{\partial Z_j^1}{\partial x_j})={\sum }_{j=1}^N[{\delta }_j^1·f^{\prime }(x_j)],$$(3)where $f^{\prime }(x_j)$ is the derivative of the activation function.

In order to study its capability and performance, we applied the GDNN algorithm to design several very compact Si photonics devices as examples, including low-loss power splitters with specific unequal output powers, a TE mode converter, and a broadband power splitter. All of these devices are designed based on common Si photonics platforms with silicon-on-insulator (SOI) substrates with 220 nm device silicon and a 2–3 μm buried oxide (BOX) layer. The waveguide is 500 nm in width and operated at (or around) 1550 nm wavelength. All the devices are designed according to the process design rule that no line or space feature size is smaller than 0.2 μm.

We take a power splitter with splitting ratio of 2:3 as an example to quantitatively analyze the design process. The initial population is composed of 1000 device structures randomly generated according to the encoding process stated earlier. The distribution of total optical transmission of a typical random population is shown in Fig. 4(d). In this case, the majority of the population show very poor transmission, and only 1.5% transmit $>80\%$ light. The FDTD simulation result of the best device in this initial population is shown in Fig. 4(b). Then the DNN-based inverse and forward design processes are iterated with 50 offspring designs produced in each cycle. The evolution of the population proportion that exceeds a certain optical performance target, e.g., transmission $>90\%$, with respect to the number of design iterations (generations) is shown in Fig. 4(a). The evolution plot shows the growth of performance with iterations until it saturates to about 80%. Another way to study the convergence is the cost function mentioned earlier, i.e., the root-mean-square error (RMSE) of the FOM with respect to a target value. The RMSE as a function of generation is also shown in Fig. 4(a) and clearly exhibits a convergence behavior. The algorithm generates a robust high-percentile good performance as well as low variations beyond about 14 generations. The FDTD simulation result of the final device design generated from this algorithm is shown in Fig. 4(c). The device has a size of only $2\mathrm{\mu m}\times 1.7\mathrm{\mu m}$, 0.27 dB insertion loss, and 2:00:3.08 splitting ratio.

Similar design processes are adopted to optimize the other optical devices including a 1:2 power splitter with 0.36 dB insertion loss and $2.2\mathrm{\mu m}\times 1.8\mathrm{\mu m}$ footprint [Fig. 5(a)], a 1:1 power splitter with 0.18 dB insertion loss and $2\mathrm{\mu m}\times 1.5\mathrm{\mu m}$ footprint [Fig. 5(b)], a TE mode converter with 77% TE0-to-TE1 mode conversion efficiency and $3\mathrm{\mu m}\times 1.5\mathrm{\mu m}$ footprint [Fig. 5(c)], and a broadband power splitter operating over a 400 nm range [Fig. 5(d)]. The TE mode converter is an order of magnitude smaller than a conventional directional coupler-based design [33,34]. The successful design of the broadband power splitter shows the capability of multi-objective optimization in our algorithm. The broadband power splitter is optimized from the 1:1 power splitter shown earlier by introducing optimization objectives at 11 wavelengths evenly distributed between 1350 and 1550 nm. The multi-objective GDNN model optimizes the power splitter design [Fig. 5(d)] with a maximum insertion loss of 0.47 dB and a maximum loss variation of 0.36 dB within 400 nm (1250–1650 nm) and $2.5\mathrm{\mu m}\times 1.5\mathrm{\mu m}$ footprint. More importantly, the maximum insertion loss is 0.2 dB and the maximum loss variation is 0.09 dB within the 200 nm (1350–1550 nm) design range. In comparison, the 1:1 power splitter shown in Fig. 5(b) shows more than double insertion loss and loss variation. The designed broadband power splitter presents good performance from the O band to the C band and is useful in applications demanding wide-band operations.

In this work, the DNN was implemented by using PyTorch, an open source machine learning framework. The same GDNN architecture is used for all the designs. The architecture includes 10 hidden layers with the dimension of 100. There are 1000 trained data for the initial population and 50 reproduced offspring for each iteration. The algorithm is implemented on a workstation with an AMD Ryzen 3700X CPU and a Nvidia GeForce RTX 2070 GPU. It takes about 6–10 h to complete the whole design including the FDTD simulation of the 1000 initial population data and the whole genetic optimization process.

DNN has been proved to be a powerful tool for designing optical devices using an inverse design process. Previous DNN-based inverse design algorithms usually require 20,000 to 40,000 training data [31,32], which take the majority of the design time due to the computation intensive high-resolution FDTD simulations. The use of genetic optimization reduces the number of the training data by an order of magnitude. In addition, unlike many inverse design algorithms, our GDNN algorithm does not require the initial data set to include a large amount of good performance designs because the quality of the population is iteratively improved during the evolution. In fact, we intentionally include some device designs, violating design rules that serve for regulating the constraints of the inverse design. As shown in Table 1, the GDNN uses 1000 sets of data for the initial training of the model. After that, 50 sets of offspring will be added to participate in the training of the model. It is worth mentioning that the model design with similar structural design uses the same set of initial datasets (1000 sets of data), for instance, power splitter (1:1) and broadband splitter, power splitter (1:2) and power splitter (2:3). Our GDNN algorithm completes a design task with less than 3000 design data including the 1000 initial population and the offspring of all iterations. This is an order of magnitude less than the training data for a typical previous inverse design algorithm.Table 1.

Training Data Summary of the Designs in This Work

In comparison, we have investigated using an efficiency-focused generative adversarial network (GAN) model to design the same photonic devices (2:3 power splitter). The results (Fig. 6) show that greater than 10,000 device design training data are required to achieve designs with comparable performance.

In conclusion, by combining DNN with genetic evolution, our GDNN method exhibits great flexibility and efficiency in designing ultra-compact photonic devices with challenging properties. Compared with previous optimization algorithms, the GDNN algorithm is more capable of complex multi-constraint and multi-objective optimization. Without relying on pixelized features in a predefined geometry and grid, our method allows design of more flexible device structures while obeying fabrication design rules. The genetic evolution nature of the algorithm greatly increases computation efficiency and reduces FDTD training data by an order of magnitude compared with previous DNN-based inverse design methods. In addition, the genetic selection process allows introduction of biases to guide certain design directions such as a small footprint. To exhibit the capability of the method, we design several Si photonics devices including power splitters with uncommon splitting ratios, a TE mode converter, and a broadband power splitter. These devices are free of the features beyond the capability of photolithography and generally in compliance with silicon photonics fabrication design rules. We believe the presented GDNN algorithm can be widely implemented in designing many complex micro- and nanophotonic structures that could not be easily realized before.

Acknowledgment. The authors thank Dr. Shiyue Hua for helpful discussion.