$\mathcal{P}\mathcal{T}$-Symmetry Breaking and Laser-Absorber Modes in Optical Scattering Systems

Using a scattering matrix formalism, we derive the general scattering properties of optical structures that are symmetric under a combination of parity and time reversal ($\mathcal{P}\mathcal{T}$). We demonstrate the existence of a transition between $\mathcal{P}\mathcal{T}$-symmetric scattering eigenstates, which are norm preserving, and symmetry-broken pairs of eigenstates exhibiting net amplification and loss. The system proposed by Longhi [Phys. Rev. A 82, 031801 (2010).], which can act simultaneously as a laser and coherent perfect absorber, occurs at discrete points in the broken-symmetry phase, when a pole and zero of the $S$ matrix coincide.

Figure 1

Semilog plot of $S$-matrix eigenvalue intensities ${log}_{10}|s_{\pm }{|}^2$ versus frequency $\omega$ (solid curves), for a 1D $\mathcal{P}\mathcal{T}$-symmetric system of length $L$ with balanced refractive index $n=3\pm 0.005i$ in each half (inset). The $\mathcal{P}\mathcal{T}$ symmetry is spontaneously broken at ${\omega }_c\approx 1418.21/L$. Dashed curves show the minimum scattered intensity for equal-intensity input beams of variable relative phase; in the broken-symmetry regime, net amplification always occurs.

Figure 2

Phase diagram of $S$-matrix eigenvalues for 1D $\mathcal{P}\mathcal{T}$-symmetric systems. The structure in (a) is the same as in Fig. 1. The structure in (b) has a real-index region of length $L/10$ in the center, with all other parameters the same as in (a). It shows reentrant behavior as $\omega$ is varied. White areas show the $\mathcal{P}\mathcal{T}$-symmetric phase, and grey areas show the $\mathcal{P}\mathcal{T}$-broken phase; circles in (a) show the CPA-laser solutions. In (a), the dashed line shows the approximate phase boundary given by Eq. (11); the points $A$, $B$, and $C$ on this boundary are referenced in Fig. 3. For fixed $\omega$, stronger $\mathcal{T}$ breaking always favors broken symmetry.

Figure 3

(a) Trajectories of $S$-matrix poles and zeros for the two-layer system of Fig. 1. Filled circles and squares show the zeros and poles for $\tau =|\mathrm{Im}[n]|=0$; solid curves show their trajectories as $\tau$ is increased. Filled stars show CPA-laser solutions where zeros and poles meet on the real axis. For the middle set of trajectories, this happens at $\omega L=1468.7$, $\tau =4.982\times {10}^{-3}$. The transition occurs at the “anticrossings, ” indicated by the representative points $A$, $B$, and $C$, which match the values of $\{\mathrm{Re}[\omega ],\tau \}$ marked in Fig. 2. (b) Log of the transmittance for $\tau =0.005$ and $\tau =0$, showing the doubled free spectral range in the CPA laser.

Figure 4

Semilog plot of $S$-matrix eigenvalue intensities ${log}_{10}|s{|}^2$ versus frequency $\omega$, for a 2D $\mathcal{P}\mathcal{T}$-symmetric disk of radius $R$ and refractive indices $1.5\pm 0.1i$. The $S$ matrix is truncated to angular momenta $|m|\le 20$. Most of the 41 eigenvalues are unimodular at these frequencies; those with higher average $m$, near resonances of the passive disk, show greater symmetry breaking. The colors distinguish several different pairs of $\mathcal{P}\mathcal{T}$-broken eigenvalues.