Time Reversal Differentiation of FDTD for Photonic Inverse Design

Abstract

Differentiable models enable the efficient computation of parameter gradients for continuous functions, greatly expediting the optimization of highdimensional systems. This makes them an asset for the design of nanostructured metasurfaces. The adjoint variable method (AVM) is the workhorse for photonic gradient computation but can be challenging to implement with the finite difference time domain (FDTD) electromagnetic simulation method for certain optimization problems. Automatic differentiation (AD) platforms remove the need for manual constructions while retaining favorable computational scaling, but high memory consumption limits their application to small systems. Here, we introduce a method of gradient calculation based on the direct differentiation of the FDTD update equations by leveraging the time-reversible nature of Maxwell's equations. We support open and closed systems by recording the time-dependent fields at lossy boundaries and playing them back during the time-reversed FDTD simulation. The method is generally applicable without the high memory consumption of AD by eliminating redundant memory operations performed at each time step. We demonstrate this architecture in a 3D FDTD simulation. Its computational cost is comparable to the adjoint method, and it reduces memory requirements by 98% compared to an equivalent AD calculation for calculating a 900-element gradient vector. The differentiable simulator is applied to design two systems: a color sorter with frequency-domain behavior and a resonant nanostructure array with time-domain behavior. This approach to differentiate grid-based simulators is applicable to a broad range of physics simulators, thereby broadening the scope of inverse design topology optimization across fields.

Towards subwavelength pixels: nanophotonic color routers for ultra-compact high-efficiency CMOS image sensors

“…(d) A schematic and the designed free-form color router based on a time-reversal differentiation of FDTD. Reprinted with permission from [114]. Copyright © 2023 American Chemical Society.…”

“…The fabricated device exhibited the measured color routing efficiencies of 42%, 52%, and 33% at wavelengths of 660 nm, 530 nm, and 455 nm, respectively. Recently, another approach, the direct differentiation (DD) method, has been introduced, which aims to solve the problem of large memory requirements in AD by directly differentiating the FDTD update equations by exploiting the time reversibility of Maxwell's equations [114]. The method is applicable to both open and closed systems and reduces the memory requirement by 98% compared to the AD method.…”

“…Figure 6(d) shows the inverse design of a multifunctional nanophotonic device for a frequency domain system using DD. Through frequency domain optimization, they designed a TiO 2 -based color router that separates light incident at two different wavelengths (λ 1 = 488 nm and λ 2 = 633 nm) into two separate spatial regions [114]. The optimized device was successfully designed to operate effectively at both wavelengths by maximizing the overall objective function and concurrently enhancing the minimum of the objective function for each wavelength, resulting in a clear Gaussian field profile for the transmitted electric field at the intended wavelengths.…”

Efficient Waveguide Taper Design with Rapid and Data-Driven Eigenmode Expansion

“…In these approaches, a design space consisting of geometrical parameters of the device is sampled and searched for the most suitable optical response [10,11]. These approaches are typically coupled with FDFD [12] or FDTD [13] simulations that estimate individual devices' performance during this optimization process. While these specific electromagnetic simulations are known to be physically accurate, their computationally demanding nature results in design tasks that can be computationally *corresponding author: esmagden@ku.edu.tr prohibitive, especially when a large number of iterations are required in order to achieve the required device performance.…”