Large-scale simulations of light propagation in complex structured media are essential in many fields of optics. There are many classical methods, including finite-difference time-domain (FDTD), finite-difference frequency-domain (FDFD), modified Born series (MBS), and finite element method (FEM), which can conduct rigorous simulations by solving Maxwell Equations. For large-scale simulations with dimensions in the scale of thousands of wavelengths, such as metasurface design, biology imaging, and lithography simulations, the memory requirement is the bottleneck for all numerical simulators mentioned above. Therefore, it is essential to reduce memory consumption to ensure the feasibility of large-scale simulations.

Memory bottleneck in large-scale simulation

In most situations of interest, one usually needs to simulate a non-periodic domain with scatter inside. In these scenarios, absorbing boundary conditions (ABC) are applied to surround the region of interest with the aim of absorbing the outgoing light. For qualified ABCs, it is necessary to achieve both low reflection and transmission simultaneously. Low reflection is attainable when the variation of the refractive index in the boundary region is smooth enough. Provided that the RI changes smoothly, a thick boundary is required to get low transmission. In other words, a large amount of memory is often occupied by the ABCs.

The most well-known and widely applied ABC is the perfectly matched layer (PML). Although the PML is more powerful than other contemporary methods, it still requires significant thickness to achieve accurate simulations. Moreover, the PML introduces non-uniform permeability, which exacerbates the computational requirement and is unnecessary for many applications. We propose a virtual boundary condition (VBC) to reduce the memory usage of the ABCs [1]. The memory usage with the VBC will be compressed to a negligible level compared with that of traditional ABCs. Combined with the Fourier transforms-based modified Born series, memory usage can be reduced to a level close to the theoretical limit.

Virtual boundary condition

Fig 1. Schematic diagram of the virtual boundary condition. (a) The pseudo propagation. (b) The basic idea of the virtual boundary condition. (c) The amplitude of field updated by the angular spectrum method.

The VBC is inspired by the parallel strategy of the FDTD and the pseudo propagation in the MBS method. In the FDTD method, the field is updated iteratively based on the adjacent field of the last time step. If there is any error in the simulation domain, the error propagation is limited by the number of iterations. The MBS method is a frequency domain method and does not have physical propagation in the time domain. However, there is a pseudo propagation shown in Fig 1a, which results from the localized dyadic Green's function in the MBS method [2].

Based on the concept of pseudo propagation, the VBC uses the angular spectrum method (ASM) to eliminate the boundary artifacts caused by insufficient absorption. The field in the absorbing boundary region, which is generated by pseudo propagation and may contain errors, is recalculated with the angular spectrum method (ASM). Both the absorption in the boundary region and the localization in the Green's function are utilized to achieve zero reflection. In this framework, errors in the boundary region are tolerable. Therefore, the required thickness of the ABCs can be reduced. Considering that the field in the boundary region is recalculated in each iteration, there is no need to store these data, which can further reduce memory usage.

Implementation

Fig 2. The implementation of the virtual boundary condition.

The detailed implementation of the ASM is integrated with the original calculation of the MBS. In the MBS, the only propagation operation is based on a three-dimensional convolution, which consists of a forward fast Fourier transforms (FFT), multiplication with the frequency domain Green’s function, and invers FFT. With the VBC, the 3D FFT is decomposed into a 2D FFT on the x-and y-axes and a 1D FFT on the z-axis to conduct the ASM. The ASM is used to propagate the field based on the field at the reference plane. It is noted that the operations between the 2D FFT, including the ASM propagation, 1D FFT, and multiplication, can be decoupled in the x- and y-axes. Therefore, the data is extracted and operated in the depth direction, as shown in Fig. 2, which avoids dealing with large and bulky data. The memory required by the decoupled VBC can be compressed to a negligible level compared with that of traditional ABCs.

Examples validation

Three examples, including the propagation of a single plane wave, the diffraction of the extreme ultraviolet lithography (EUVL) mask, and the scattering of biological cells, are provided to demonstrate the performance of the proposed method. The propagation of the plane wave, a simple and straightforward situation, is used to validate the performance by the amplitude fluctuation of the field result. With ultra-low memory usage, the proposed method performs better than other contemporary methods.

For other rigorous field solvers

The viability of the proposed method is based on three assumptions. First, the VBC is limited to frequency domain solvers due to the adoption of the ASM. Second, there is a pseudo propagation during the solving process. Third, the calculation must be iterative, and the next iteration is based on the updated solution.

Besides the MBS method, there are many popular frequency-domain solvers, such as the finite-difference frequency-domain and finite element methods. These methods are formulated as Ax = b and solved by iterative algorithms for large-scale simulations. Usually, A is a large sparse matrix with a small bandwidth, which leads to extremely slow pseudo propagation during the iterations. Taking the finite-difference frequency-domain method as an example, the pseudo propagation speed is one cell per iteration [3], which means that only one layer of the ABC is needed with the VBC. Unfortunately, the application of the VBC is limited by the fact that most modern algorithms for linear equations do not use the updated solution for the next iteration. Therefore, the VBC has the potential to be applied to other methods with competent stationary iterative algorithms.

Reference

[1] P. He, J. Liu, H. Gu, H. Jiang, and S. Liu, "Modified Born series with virtual absorbing boundary enabling large-scale electromagnetic simulation," Commun. Phys. 7, 383 (2024). https://doi.org/10.1038/s42005-024-01882-5

[2] G. Osnabrugge, S. Leedumrongwatthanakun, and I. M. Vellekoop, "A convergent Born series for solving the inhomogeneous Helmholtz equation in arbitrarily large media." J. Comput. Phys. 322, 113-124 (2016). https://doi.org/10.1016/j.jcp.2016.06.034

[3] J. W. Demmel, Applied Numerical Linear Algebra (Society for Industrial and Applied Mathematics, 1997). https://doi.org/10.1137/1.9781611971446