Inverse design of planar optical components using deep learning

[N.J. Dinsdale et al., arxiv, 2009.11810, (2020)]

Photonic integrable circuits are basically waveguide structures that allow performing useful operations, such as mode or wavelength multiplexing/demultiplexing in the case of telecommunication applications. For many operations, we can find quite easy solutions, where the shape of the structure imposes certain boundary conditions that force light to behave the way we want. However, for an arbitrary operation, it is not always possible to find a trivial solution. Non-trivial solutions, where the link between the geometry of the structure and its function is not direct, should then be considered. In the present paper, the authors use deep learning to find geometrical configurations for planar photonic circuits that look like disordered waveguides but actually perform a previously chosen linear operation. These configurations lead experimentally to robust, high throughput, and accurate behaviors.

The idea to use numerical optimizations to find non-trivial, seemingly random, structures that perform any desired operation is not new. In this paper [Jiao, Fan and Miller, Opt. Lett., 30 (2005)] for instance, the authors use numerical simulations to iteratively change the position of diffusers in a 2D waveguide to achieve the desired mode conversion. However, performing the correct operation is not everything, other requirements need to be considered for real-life applications:

• The optical system should have a high transmission, to limit losses,
• the optical system should be robust to fabrication imperfections,
• the optimization procedure should be scalable to unlock high-rank operations.

The optical system considered in the paper is composed of a planar waveguide with single or multiple inputs and outputs. The index of refraction of the waveguide can be modified by locally etching the material. In essence, the 2D structure of the waveguide is like a 2D array of pixels where the index of refraction can take two different values (see Fig 2.a). The goal is to design and fabricate a structure that has the desired transmission matrix.

Principle

The optimization procedure is based on a deep learning framework and has lots of technical tricks that I will not detail here. I will simply highlight the main ideas and the challenges such an approach faces. If you only want to see the results, I invite you to skip the following paragraphs.

The starting point of training an artificial neural network is a good training set. To generate the initial dataset, the authors use FDTD simulations for a set of random positions of the etched areas. One problem is that random perturbations statistically lead to small transmissions. Because we want to find solutions with high throughput, the system needs to learn on configurations with a high transmission. Numerically, an optimization of the total transmission is done to obtain random configurations with high transmissions. The pairs of configurations of etched sites and the corresponding transmission matrices with a high transmission represent the actual training set of the system.

The cost function consists of the mean square error between the target transmission matrix and the one obtained for a considered configuration. To enforce robustness to fabrication, the authors add another term to the cost function (Kullback-Leibler-divergence term), that forces the network to "learn a smooth and continuous compressed representation", so that small changes in the configuration do not lead to totally different behaviors.

The model consists of two networks of encoder-decoder type. One that does the physics predictions, i.e. that simulates, for a given configuration of the etched sites, the corresponding transmission matrix. The second one that does the inverse problem, i.e. that infers the geometrical configuration of the waveguide for a given transmission matrix. That is, of course, the last one we want to use when everything is done training. The two networks are trained simultaneously against the cost function (with a fair amount of tricks and subtleties).

Results

The first task the system is tried on, is to generate a 1 to 2 splitter with a controlled ratio between the two outputs. The system gives good results, as shown in Fig. 1. That is not a complicated problem so far, but the important part here is that the solution is robust enough to be experimentally fabricated, leading to results very similar to the numerical predictions (see Fig. 2).

Figure 1. Numerical simulations of three different configurations obtained using deep learning for a 1 to 2 splitter. (left) The intensity map inside the device, and (right) the expected (orange) vs simulated output cross-section intensity (green).

Figure 2. Experimental results for an optimized configuration. (a) The experimental geometry, (b) the measured intensity map inside the device, and (c) the FDTD simulations.

In a second experiment, the authors try to find configurations for a 3 ports to 3 ports linear system represented a by 3 by 3 transmission matrix. For a given target transmission matrix, the model outputs a configuration. We show in Fig. 3 two examples of such configurations. The simulations confirm that the system does have a transmission matrix very close to the one expected.

Figure 3. Two solutions found by the model for two target transmission matrices. The graphs represent the intensity map inside the system for the injection in each of the input port (FDTD simulations). In inset is shown the output intensity cross-section for the expected behavior (blue), the simulation (green), and the predictions from the model (orange).

In the following, the authors show that it is possible to control, not just the amplitude of the coefficient of the transmission matrix, which represents the portion of energy that goes from each input port to each output port, but also their relative phase. It then allows the generation of any possible transmission matrix.

Again, it is not the first attempt at solving the problem of inverse-design for photonic waveguide structures, but it does bring a lot of hope for the robustness to fabrication. Moreover, the deep learning design allow envisioning better scalability for higher transmission matrix sizes compared to previous techniques, such as brute force optimizations.