Concentric circular gratings are classic optical elements that are critical to engineer light-matter interactions in the form of bullseye cavities, annulus cavities, and circular Bragg resonators. However, due to the absence of translational symmetry, to calculating their band structure along the radial direction has remained challenging. In this paper, we introduce a physically intuitive generalized radial Bloch condition for cylindrical systems, mathematically prove it, and use it to build the eigenvalue equation for circular gratings. Energy bands of the circular grating with a Hankel wave period are computed using the transfer matrix method. The approach has been validated against experimental data. And reflectance spectra of annulus cavities and field distribution of resonance modes are studied. It is shown that these properties are highly related to the energy bands along the radial direction. This work provides a direct, intuitive, and computationally efficient method to analyze and design complex circular grating devices, resolving a persistent bottleneck in the modeling of these critical photonic components.
Kangling Xiong (Fri,) studied this question.