Gradient-based optimization of scatterer arrangements based on the T-Matrix method

TL;DR

This work tackles the challenge of high-dimensional inverse design in nanophotonics by introducing a differentiable multiscattering framework based on the T-matrix that yields exact gradients with respect to scatterer geometry in both finite clusters and periodic metasurfaces. The method integrates the T-matrix formalism with automatic differentiation (via JAX) to compute gradients in a single backward pass, enabling efficient gradient-based optimization of scatterer radii and positions, including arbitrarily shaped particles when their T-matrices are differentiable. The authors demonstrate two Kerker-type designs: a finite cluster optimized for a high forward-to-backward scattering ratio and a metasurface unit cell optimized for polarization-dependent reflectance, with both numerical results and microwave experiments validating the approach. The framework is released as open-source and shown to support multi-objective optimization, offering a mechanism-agnostic, high-fidelity path to inverse design of complex nanophotonic architectures.

Abstract

The demand for inverse design is increasing as the ability to fabricate sub-10 nm features expands the design space by orders of magnitude. Efficient inverse design benefits from differentiable models of light-structure interaction. While traditional full-wave solvers based on finite differences, finite elements, or Fourier modal methods have already been presented for that purpose, a dedicated tool adapted for performing multiple scattering simulations is still lacking. To overcome this limitation, we provide a multiple-scattering framework compatible to automatic differentiation, suitable for treating periodic and non-periodic arrangements of scatterers. It yields exact gradients regarding geometric and positional parameters in finite clusters and infinite metasurfaces. In this work, we use spheres as the elementary building blocks to demonstrate the framework's capabilities as a standalone tool. However, the framework is adaptable to arbitrarily shaped scatterers, provided the individual T-matrices are calculated using differentiable full-wave Maxwell solvers. Since the gradients are obtained simultaneously in a single backward pass, the framework is well-suited for moderately dimensional problems. It is also possible to combine multiple performance goals into a single objective function. The versatility of our method is illustrated in proof-of-concept examples that focus on various aspects of Kerker-type physics. In the first example, a finite cluster of scatterers is optimized in order to reach a high forward-to-backward scattering ratio, and we show experimental feasibility of the designs. In the second example, a metasurface made from multiple scatterers in each unit cell is designed to maximize the reflectance contrast between orthogonal linear polarizations of the incident light. We make the framework publicly available at https://github.com/tfp-photonics/dreams.

Figures (12)

• Figure 1: (a) Convergence of the F/B ratio during optimization at the design wavelength of 800 nm when optimizing the sample from a given initial configuration, as explained in the main text. (b) 3D view of the cluster, with the z = 0 plane shown in grey and spheres on the plane rendered with lower transparency. The initial configuration is indicated in red, and the optimized structure in blue. (c) F/B ratio spectrum shows an enhancement at the design wavelength (dashed line). (d) Total scattering cross-section of the initial and final arrangement.
• Figure 2: Top: Polar plots of the radiation pattern in the $XZ$ (a) and $YZ$ planes (b). Bottom: Multipole contributions to the scattered cross-section of the initial (c) and final arrangement (d). For visual clarity, bars representing electric multipole contributions are shown in magenta, while magnetic multipoles are in cyan. All results are shown at the design wavelength.
• Figure 3: Fabricated samples using 3D-printing and foam hosting: (a) the initial cluster and (b) optimized cluster. The ratio of directly forward (0$^\circ$) to directly backward (180$^\circ$) scattering under $y$-polarized plane wave for both structures -- initial (solid blue) and optimized (solid red), and their quotient (optimized/initial, dash-dotted): (c) experimental measurements and (d) CST numerical simulations.
• Figure 4: Initial (red) and optimized (blue) constituents of the metasurface unit cell targeting reflection peak for one linear polarization of the incident light and a low reflection for the other one at (a) 950 nm, (b) 1050 nm, and (c) simultaneously 950 nm and 1050 nm. The transparency of each sphere decreases with its proximity to the viewer. Panels (d--f) show corresponding reflection spectra for $x$-polarized (blue curves) and $y$-polarized (orange curves) incident light. Vertical red dashed lines indicate the target wavelength(s).
• Figure 5: Phasor diagram of contributions of electric and magnetic multipoles to the backward-scattered field coefficients. Panels (a, c) show the results for a non-diffractive metasurface optimized at 950 nm, while panels (b, d) demonstrate the results of optimization at 1050 nm. Starting from 0, the vectors build up as $E_1$, $M_1$, $E_2$, $M_2$, and so on, where $E$ stands for electric and $M$ for magnetic multipoles, which is followed by the multipole order. Each panel shows the phasor build-up of $x$- and $y$-polarized scattered field components separately. However, the cross-polarized components are negligible. For both optimized structures illuminated with $x$-polarized light, in a) and c), the electric and magnetic multipole contributions cancel each other, whereas in b) and d), under $y$-illumination, they sum up to a high value of reflectance.