Full analysis on coupling strengths between split ring resonators for double negative microwave tight-binding models

Previous studies have shown that split-ring resonators (SRRs) can be utilized to achieve finely tuned nearest-neighbor coupling strengths in various one-dimensional hopping models. In our study, we present a systematic investigation of resonator coupling, providing a comprehensive quantitative description of the interaction between SRRs and complementary split-ring resonators (CSRRs) for any orientation combination. Our method includes an estimation of the coupling strength through a linear combination of periodic functions based on two orientation angles, with a sinusoidal expansion of up to the 3rd order, allowing for efficient and streamlined microwave structure design. Through our approach, we offer a satisfactory explanation of the band structure of SRR chains using a microwave-hopping model, which facilitates the exploration of exotic photonic band structures based on tight-binding theory.