Universal appearance tendency of modes at the finite periodic ????????-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}) mode, the unidirectional invisibility (M_{UI}) mode, and the anti-reflection (M_{AR}) mode. Through detailed analysis of transmission, reflection, scattering behavior, dispersion relations, and phase diagrams, 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.