Impact of mode completeness on the accuracy of the coupling theory of quasinormal modes: A strict numerical demonstration

The coupling theory of quasinormal modes (QNMs) for a coupled system of generally lossy and dispersive optical nanoresonators has been established in a rigorous manner based on the first principle of Maxwell's equations [Phys. Rev. B 102, 045430 (2020)], and can achieve superior computational efficiency and physical intuitiveness compared with full-wave numerical methods if a small set of modes can achieve a high accuracy. The QNMs suffer from an exponential divergence of far field and can form a complete basis inside but not outside the resonator. In the QNM coupling theory (QCT), it is required that the QNMs of each resonator form a complete basis in expanding the scattered field both inside and outside the resonator, which can be achieved by using regularized QNMs (RQNMs). However, a strict numerical demonstration of the impact of the mode completeness of RQNMs on the accuracy of QCT by using a virtually complete basis of RQNMs is still absent. In this paper, we will provide such a numerical demonstration along with an improvement of the QCT and some theoretical demonstrations on a rigorous incorporation of RQNMs into the QCT. The RQNMs are obtained by introducing an equivalent surface current (ESC) encircling the resonator (called ESC-RQNMs) or the perfectly matched layer (PML) surrounding the computational domain (called PML-RQNMs). The numerical example is selected as two one-dimensional resonators of slabs in the extreme coupling case of direct contact, for which a virtually complete basis of RQNMs can be solved either analytically (for ESC-RQNMs) or numerically (for PML-RQNMs). The results show that by using a virtually complete basis of RQNMs, the QCT can achieve a high accuracy in predicting both the source-free eigenmodes and the source-excited scattered field of the coupled system, which is not true if using the incomplete basis of not-regularized QNMs (i.e., physical QNMs).