Universal appearance tendency of modes at the finite periodic PT-symmetric structure

We investigate the emergence of specific modes in finite periodic (FP) parity-time (PT) symmetric photonic structures using the transfer matrix (T-matrix) formalism. Three distinct resonant states are identified-the coherent perfect absorber-laser (M_CL), the unidirectional invisibility (M_UI), and the anti-reflection (M_AR) modes. We show that, unlike in a single-period PT bilayer-where the M_CL mode arises only once-FP structures support recurring M_CL modes as the number of periods increases. M_AR modes appear between successive M_CL modes. The required imaginary part of the refractive index (n^'') for the emergence of each mode is determined by the ratio l/N, where l is the mode order and N the number of periods. For small l/N, n^'' increases linearly, whereas for large l/N, n^'' approaches the PT critical point. Furthermore, the phase diagram shows that M_CL modes exist solely in the PT-broken phase, while M_AR modes remain confined to the PT-symmetric phase. These findings are validated using photonic nanobeam simulations, demonstrating the systematic realization of the three modes in FP PT-symmetric photonic systems.