An Efficient Surface-Integral-Equation Based Nyström Method With an Over-Determined Testing Scheme for Broadband Grating Scattering Modeling

Abstract

The design complexity of photonic crystals and periodic gratings has been continuously increasing, driven by exploration of their unique physical phenomena and widespread applications. However, existing approaches for scattering modeling of periodic structures potentially encounter challenges when adapting to complex configurations, especially in the context of accurate near-field analysis and frequency responses near resonance. Meanwhile, they often exhibit difficulties in computational efficiency considering broadband simulations. Therefore, the development of an efficient and general scattering modeling approach to overcome these limitations has emerged as a crucial task. In this paper, an efficient surface integration equation (SIE)-based method is developed to model the scattering properties of arbitrary-shaped 2D gratings with 1D periodicity. The SIE is solved with a Nyström approach, which incorporates a local correction scheme and a Gaussian-Legendre quadrature rule. The evaluation of periodic Green's functions is achieved by combining an advanced imaginary wavenumber extraction technique with an integral transformation approach, which significantly increase the broadband simulation efficiency. Additionally, an over-determined matrix equation is constructed by testing the SIE with redundant observation points to mitigate potential internal resonance phenomena. The proposed approach is assessed through various numerical examples involving scatterers of different shapes and arrangements to demonstrate its accuracy and efficiency. The transmissivity spectra and surface field results, considering both normal and grazing incidence, are computed and compared against traditional approaches. The method proposed is found to be superior in accuracy and efficiency, especially when complicated evanescent modes are excited, and for broadband simulations.