  We theoretically and numerically investigate the scattering behavior of a
periodic parity-time (PT)-symmetric waveguide network composed of a finite
number of unit cells. Specifically, we put forward rigorous and formally exact
expressions for wave propagation, bi-directional reflectionless, and coherent
perfect absorption and lasing (CPAL) occuring in a finite periodic optical
waveguide network. Through the use of the generalized parametric space derived
from observation of PT-symmetric transfer matrix, Lorentz reciprocity theorem
and non-imaginary Bloch phase, we observe that when the unit cell is operated
at the PT broken phase or exceptional point, the system can always have
propagating modes, independent of the number and transmission phase of the unit
cell. On the other hand, when the unit cell is operated at the exact
PT-symmetric phase, the formation of propagating waves would depend on the
transmission phase of the unit cell. More interestingly, we find that even
though the unit cell is not operated at the exceptional point, reflectionless
with bi-directionality as well as unity transmittance can be achieved by
choosing appropriate number of unite cells and specific PT phases. We also find
two approaches to implement CPAL. One is to exploit odd number of the unit cell
operated at the CPAL point. Another way is to manipulate specific broken phase
with an appropriate number of the unit cells, while making transmission phase
to be null. We believe this work may offer a theoretical underpinnings for
studying extraordinary wave phenomena of PT-symmetric photonics and may open
avenues for manipulation of light.
