Self-collimation in the square lattice photonic crystals composed of multi-circular ring scatterers

In this paper, fractal supercells are constructed according to the Thue-Morse theory and arranged in a square lattice to form fractal photonic crystals (PCs), which are called the first-order (1st) and second-order (2nd) approximate of Thue-Morse structure. The multi-circular ring scatterers (MCRSs) are innovatively applied in them and the effect of the number (H) and width (W) variation of the rings on the equifrequency contours (EFCs) of PCs is investigated. To calculate the energy band diagrams and EFC mappings of the first-order (1st) and second-order (2nd) approximate of Thue-Morse structure, an improved plane wave expansion (PWE) method is also introduced. Then, the possibility of achieving polarization-insensitive self-collimation (PISC) in the approximate of Thue-Morse structure and conventional square lattice (CSL) PCs are explored. This phenomenon was verified by simulations at normalized frequencies of 0.128, 0.068, and 0.27 (omega a/2 pi c), respectively, and the phenomenon of vertical incidence self-collimation of light (VISCOL) was proposed on this basis. Finally, a reflection on the value of the multi-ring nested type of scatterers is presented. Replacing the circular rings in the PCs with square rings, it is found that the EFCs for TE polarization change less while the EFCs for TM polarization have larger variations, which leads to a reduction in the VISCOL frequency range and provides a possibility for beam splitter design.