The band structure of helical waveguide arrays in topological photonics: a tutorial

The helical waveguide array is a well accepted model for fabricating the photonic topological insulator, since it introduces a gauge field to break the equivalent time-reversal symmetry of the system. However, there is a threshold, from both physical and mathematical point of view, for acquiring the corresponding band structure, especially for the beginners and even grown-up researchers. As far as we know, a tutorial on how to solve this problem is still absent from the scientific literature. To break the technique fence and provide a friendly documentation, we sort out this tutorial in detail based on our almost ten-year work experience in this area. Utilizing the tight-binding method with the nearest-neighbor coupling considered, we provide two methods to numerically calculate the band structure: the Floquet operator method and the Fourier expansion method. The main formulae and key codes are displayed. We believe this tutorial is a useful introduction in topological photonics and may be a shortcut for scientists who would like to venture into research areas related with topological physical objects.