In (f), the input field is reflectionless from left side and the (absolute) amplitude of transmitted field is same as incident one, while the corresponding transmission phase is 0.31. As expected, the field in (e) has parity symmetry outside PT-symmetric system, but not inside. With comsol Multiphysics (a finite element solver), we plot the corresponding strength of electric field in (e), (f), and (g) for PSR, ATR, and CPAL (close), respectively. Comparison with the results provided by parametric spaces (b) and (c), the spectra provides a specific information related to system configuration. We also mark PSR, ATR, and CPAL, by red star, light blue cross, and black star, respectively. The corresponding transmittance and reflectance from right and left illumination are shown in (d). In principle, transmission phases can be arbitrary. In (c), we demonstrate a clear relation between transmission phase and these extraordinary scattering phenomena. When ω = 1.35 ω 0, the system response is close to CPAL and belongs to broken phase. When ω = 1.245 ω 0, it is ATR, associated with exceptional point (the emergence of degenerate scattering eigenvectors). When ω = 1.21 ω 0, the PT has PSR situation. In the beginning ω = 1.2 ω 0, the PT belongs to symmetry phase regime.

(b) By adjusting operating frequency from ω =, we investigate its parametric variation of PT-symmetric transfer matrix in (b) 2D ( α and β) and (c) 3D ( α, β, and ϕ). Each composition has its permittivity, Eq. ( 4), satisfied by Kramers-Kronig relation and causality. (a) Schematic configuration of a PT-symmetric system made of two composition of gain-gap-loss. PSR stands for parity symmetry of reflection coefficients for right and left incidences marked by a black dashed line in (c), (e), and (f). Based on this 2D parametric diagram, we plot transmittance T, reflectances for left and right incidences, R L and R R, respectively, in (d)–(f). CPAL stands for coherent perfect absorber laser, corresponding to green line, located at the allowed-forbidden boundary. Between these two phases, there is an exceptional point regime associated with degeneracy of eigenvalues, marked by blue and red dashed lines, corresponding to anisotropic transmission resonance (ATR). The gray region denotes PT-symmetry phases, while the bright yellow denotes broken symmetry phase. We thus plot a two-dimensional (2D) parametric diagram with α and β in which allowed parameters are illustrated by gray and bright yellow colors in (c). In our context, we find extraordinary PT scattering phenomena are related to parameters α and β. Allowed parametric space is depicted by yellow. (b) Three-dimensional parametric space in terms of α, β, and ϕ for general PT-symmetric transfer matrix. (a) Schematic configuration of one-dimensional scattering system interacted with two independent input waves a and d and accompanied with output waves b and c.

Star reflection symmetry