Unambiguous scattering matrix for non-Hermitian systems

$\mathcal{PT}$ symmetry is a unique platform for light manipulation and versatile use in unidirectional invisibility, lasing, sensing, etc. Broken and unbroken $\mathcal{PT}$-symmetric states in non-Hermitian open systems are described by scattering matrices. A multilayer structure, as a simplest example of the open system, has no certain definition of the scattering matrix, since the output ports can be permuted. The uncertainty in definition of the exceptional points bordering $\mathcal{PT}$-symmetric and $\mathcal{PT}$-symmetry-broken states poses an important problem, because the exceptional points are indispensable in applications such as sensing and mode discrimination. Here we derive the proper scattering matrix from the unambiguous relation between the $\mathcal{PT}$-symmetric Hamiltonian and scattering matrix. We reveal that the exceptional points of the scattering matrix with permuted output ports are not related to the $\mathcal{PT}$ symmetry breaking. Nevertheless, they can be employed for finding a lasing onset as demonstrated in our time-domain calculations and scattering-matrix pole analysis. Our results are important for various applications of the non-Hermitian systems including encircling exceptional points, coherent perfect absorption, $\mathcal{PT}$-symmetric plasmonics, etc.

Figure 1

$\mathcal{PT}$-symmetric multilayer system with alternating loss and gain slabs. We employ parameters $d=50$ nm and ${\varepsilon }_a=1$ throughout the paper.

Figure 2

(a) Curves of the loss and gain balance (difference of the energy in loss and gain layers) for different ${\varepsilon }^{″}$ in a bilayer. (b) Energy density across the bilayer for points 1–4 in (a). $|a_R|$ and $\varphi$ are the absolute value and argument of the complex amplitude $a_R$ at fixed $a_L=1$. The real part of permittivity is ${\varepsilon }^{\prime }=2$.

Figure 3

(a) Transmittance $T$ and reflectances $R_L$ and $R_R$ of the loss-gain bilayer with ${\varepsilon }^{\prime }=4$ and $d=50$ nm as a function of ${\varepsilon }^{″}$. The results of calculation using the transfer-matrix method and Maxwell-Bloch equations are shown by the solid lines and symbols, respectively. Vertical dotted lines mark positions of the exceptional points calculated using the scattering matrices ${\widehat{S}}_1$ (${\mathrm{EP}}^{\left(1\right)}$) and ${\widehat{S}}_2$ (${\mathrm{EP}}^{\left(2\right)}$). (b) Temporal profiles of transmitted intensity for ${\varepsilon }^{″}$ just below and above the lasing threshold near the ${\mathrm{EP}}^{\left(1\right)}$. The inset shows the curve for ${\varepsilon }^{″}=13.8$ on the longer scale.

Figure 4

The positions of the poles in the complex-frequency plane. The cases of the lossless bilayer (black dots) and the bilayer with ${\varepsilon }^{″}=6.8245$ (blue dot) are shown.

Figure 5

The same as in Fig. 3, but for $d=74.308$ nm (near the pole shown in Fig. 4).