In recent years, integrated Silicon photonics has shown great promise in the context of next-generation communications systems and data interconnects. Functionalities such as on-chip light generation, modulation, guiding and detection of light have been demonstrated. Silicon waveguides constitute one of the building blocks of these functionalities and their geometric shape plays a crucial role in the way they operate. Moreover, a system of two parallel waveguides shows a coupling effect when the waveguides are close enough to one another, so that photons can move from one waveguide to the other. Such a system of coupled waveguides follows the physics of a SU(2) system and their dynamics can be described by a 2x2 Hamiltonian. The coupling is evanescent and therefore is highly dependent on the geometry of the system. It follows that when the geometry of the waveguides changes along the propagation axis, the coefficients of the Hamiltonian become length dependent, analogous to time dependent 2x2 Hamiltonians. Typically, such waveguide devices are buried under an oxide layer and therefore precisely recovering the geometry once the device is fabricated is a major challenge.

This can be solved by measuring the electric field distribution, which can be used to recover the waveguides’ geometry. However, while the electric field response for a given geometry and fixed initial conditions can be obtained by solving Maxwell’s equations for the system, the inverse problem, i.e. inferring the geometry from the electric field distribution is highly nonlinear and deemed hard to solve.

Recently, machine learning, and in particular deep learning, has been used to tackle such inverse problems in nanophotonics [1-2]. In this work, we harness the power of deep learning to address the challenge of inferring the waveguide geometry from the field distribution. We formulate the problem as an image-to-image translation problem. The input is a map of the spatial distribution of the electric field intensity measured above the oxide layer for two different frequencies, while the output is a spatial map of the structures’ dielectric constant. We employ a deep learning general adversarial neural network (GAN) based on pix2pix [3]. The direct problem, that is the computation of the electric field from the geometry, was solved with Lumerical MODE and used therefore to create the learning dataset.

Image-to-image translation has been defined [3] as the task of translating one possible representation of a scene to another. The pix2pix design uses a conditional GAN and an additional L1/L2 loss to translate one image to another, achieving a general purpose model that has been shown to perform well in a variety of tasks. These include synthesizing photos from label maps and reconstructing objects from edge maps. Its adaptability to different domains makes it a good candidate for our task of reconstructing the unknown function that translates the electrical field intensity distribution to the underlying geometry.

In this work we considered two types of geometries: one composed of segments with varying widths and one composed of tapered pieces, where the thickness follows a Gaussian random walk. From our results, the architecture we propose greatly outperforms k-nearest neighbors (kNN). The mean squared error (MSE) measured on the predicted dielectric constant map is one order of magnitude lower than the same score measured with kNN. The advantage of the neural network is particularly evident for mixed datasets that contain samples with different types of geometries, where the model correctly identifies the right type of geometry. Moreover, predictions obtained by the model are more accurate, and the results of these predictions better resemble the original samples.

[1] I. Malkiel, M. Mrejen, A. Nagler, U. Arieli, L. Wolf, L. & H. Suchowski, “Plasmonic nanostructure design and characterization via Deep Learning”. Light: Science and Applications, 7(1), 60 (2018).
[2] M. H. Tahersima, K. Kojima, T. Koike-Akino, D. Jha, B. Wang, C. Lin, K. Parsons, “Deep Neural Network Inverse Design of Integrated Photonic Power Splitters”, Scientific Reports, Nature Publishing Group, 9 (2019).
[3] Isola, P., Zhu, J.-Y., Zhou, T., & Efros, A. A. (2016). Image-to-Image Translation with Conditional Adversarial Networks. Retrieved from http://arxiv.org/abs/1611.07004