Universal appearance tendency of modes at the finite periodic <i>P</i> <i>T</i> -symmetric structure

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