Generalized parametric space, parity symmetry of reflection, and systematic design approach for parity-time-symmetric photonic systems

Based on the reciprocity theorem, we put forward a generalized parametric space for arbitrary transfer matrix with parity-time (PT) symmetry. Through this parametric space, one can extract complete information involving PT-symmetric phases, reflectances, transmittances, and known extraordinary scattering phenomena. We demonstrate a PT-symmetric heterostructure with coherent perfect absorption-lasing, anisotropic transmission resonance, and parity symmetry of reflection coefficients at the frequencies of interest. In addition, with the parametric space and the analytical formula, the corresponding complex dielectric permittivities for a simple PT-symmetric system made of a gain, an air gap, and a loss media in deeply subwavelength is derived to achieve various exotic PT-symmetric functionality. This work could offer an alternative route to design versatile optical and photonic PT-symmetric devices.

Figure 1

(a) Schematic configuration of one-dimensional scattering system interacted with two independent input waves $a$ and $d$ and accompanied with output waves $b$ and $c$. (b) Three-dimensional parametric space in terms of $\alpha$, $\beta$, and $\varphi$ for general PT-symmetric transfer matrix. Allowed parametric space is depicted by yellow. In our context, we find extraordinary PT scattering phenomena are related to parameters $\alpha$ and $\beta$. We thus plot a two-dimensional (2D) parametric diagram with $\alpha$ and $\beta$ in which allowed parameters are illustrated by gray and bright yellow colors in (c). The gray region denotes PT-symmetry phases, while the bright yellow denotes broken symmetry phase. 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). CPAL stands for coherent perfect absorber laser, corresponding to green line, located at the allowed-forbidden boundary. 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). PSR stands for parity symmetry of reflection coefficients for right and left incidences marked by a black dashed line in (c), (e), and (f).

Figure 2

(a) Schematic configuration of a PT-symmetric system made of two composition of gain-gap-loss. Each composition has its permittivity, Eq. (4), satisfied by Kramers-Kronig relation and causality. (b) By adjusting operating frequency from $\omega =[1.2{\omega }_0,1.5{\omega }_0]$, we investigate its parametric variation of PT-symmetric transfer matrix in (b) 2D ($\alpha$ and $\beta$) and (c) 3D ($\alpha$, $\beta$, and $\varphi$). In the beginning $\omega =1.2{\omega }_0$, the PT belongs to symmetry phase regime. When $\omega =1.21{\omega }_0$, the PT has PSR situation. When $\omega =1.245{\omega }_0$, it is ATR, associated with exceptional point (the emergence of degenerate scattering eigenvectors). When $\omega =1.35{\omega }_0$, the system response is close to CPAL and belongs to broken phase. In (c), we demonstrate a clear relation between transmission phase and these extraordinary scattering phenomena. In principle, transmission phases can be arbitrary. The corresponding transmittance and reflectance from right and left illumination are shown in (d). We also mark PSR, ATR, and CPAL, by red star, light blue cross, and black star, respectively. Comparison with the results provided by parametric spaces (b) and (c), the spectra provides a specific information related to system configuration. 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. As expected, the field in (e) has parity symmetry outside PT-symmetric system, but not inside. 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 incident field from right side, there has a reflected wave, indicated in bottom side of (f). In (g), this system performs CPAL (close). Enhanced transmitted and reflected waves from opposite illuminations are shown. In ideal CPAL, the amplitudes of transmitted and reflected waves are infinite, corresponding to amplifying.

Figure 3

(a) Schematic configuration of a PT-symmetric system made of a ultrathin gain (denoted as $G$) and a ultrathin loss (denoted as $L$) separated by a gap (denoted as $P$). We choose two sets of parametrization to manipulate PT phenomena marked by red square and blue cross in (b). Under $k_0{l}_2$ given, we can find the corresponding lossy permittivity of Eq. (5) by tuning $k_0{l}_1$ from 0.1–0.5 in (c) and (d). Orange triangles denote the results by numerical calculation, while blue circles denote the results from analytical expression, Eq. (5). Numerical and analytical results match very well.