Deep neural networks for the prediction of the optical properties and the free-form inverse design of metamaterials

Many phenomena in physics, including light, water waves, and sound, are described by wave equations. Given their coefficients, wave equations can be solved to high accuracy, but the presence of the wavelength scale often leads to large computer simulations for anything beyond the simplest geometries. The inverse problem, determining the coefficients from a field on a boundary, is even more demanding, since traditional optimization requires a large number of forward problems to be solved sequentially. Here we show that the free-form inverse problem of wave equations can be solved with machine learning. First we show that deep neural networks can be used to predict the optical properties of nanostructured materials such as metasurfaces. Then we demonstrate the free-form inverse design of such nanostructures and show that constraints imposed by experimental feasibility can be taken into account. Our neural networks promise automated design in several technologies based on the wave equation.

Figure 1

The forward model, which is schematically depicted in this figure, approximates the $\mathcal{S}$ parameters of a given meta-atom. (a) Rendered illustration of a meta-atom. (b) Grid representation (2D cross section) of the meta-atom. (c) Schematic architecture of the forward model. (d) Output of the forward model. (e) Comparison of $\mathcal{S}$ parameters predicted by the forward model (circles) and calculated by full-wave simulations (crosses).

Figure 2

The conditional generative adversarial network model. The generator creates meta-atom grids from noise and $\mathcal{S}$ parameters, while the encoder prevents mode collapse. Two expert networks (forward and classifier model) are integrated inside the discriminator.

Figure 3

Comparison of the $\mathcal{S}$ parameters as requested from the CGAN (black dots), predicted by the forward model (blue and red dots), and calculated by full-wave simulations (green and purple crosses). (a) A meta-atom from the validation set (MSE=0.0004; NMSE=0.001). (b) A meta-atom with a resonance obtained by requesting a desired transmission amplitude and phase at a single frequency from the CGAN (MSE=0.007; NMSE=0.02). (c) A meta-atom obtained from the CGAN by requesting a constant transmission amplitude and linear transmission phase (MSE=0.0003; NMSE=0.0008).

Figure 4

Different CGAN output grids for the same $\mathcal{S}$ parameters, dependent on 2D noise. From the different designs we can select the best suitable grid in terms of how close it approximates the desired $\mathcal{S}$ parameters and the ease of fabrication.

Figure 5

Cylindrical lens constructed from meta-atoms with predetermined phase generated by the CGAN. (a) Top view of the central section of the lens. (b) Optical intensity of the beam focused by the ANN-designed lens. The designed focal distance was 50 wavelengths away from the metasurface.