High-order spectral singularity

Exceptional point and spectral singularity are two types of singularity that are unique to non-Hermitian systems. Here, we report the high-order spectral singularity as a high-order pole of the scattering matrix for a non-Hermitian scattering system, and the high-order spectral singularity is a unification of the exceptional point and spectral singularity. At the high-order spectral singularity, the scattering coefficients have high-order divergence and the scattering system stimulates high-order lasing. The wave emission intensity is polynomially enhanced, and the order of the growth in the polynomial intensity linearly scales with the order of the spectral singularity. Furthermore, the coherent input controls and alters the order of the spectral singularity. Our findings provide profound insights into the fundamentals and applications of high-order spectral singularities.

Figure 1

Schematic of a discrete system. The port sites are shown in cyan. The scattering center sites are shown in red. Inset: The scattering system is formally decomposed into the ports ending with gain ${\omega }_c-\kappa e^{-ik}$, which are coupled through the connection Hamiltonian $h_c\left(k\right)$ as shaded in orange.

Figure 2

(a) $N$-port scattering system at the $N\mathrm{th}$-order SS. (b) Scaling behaviors near $k_s=\pi /2$ and (c) intensity growth in the outputs of the ports $q\in [1,5]$ for the input in the first port $p=1$ in the situation where the unidirectional coupling equals the uniform coupling $\mu =\kappa =1$.

Figure 3

The profile of time evolution for the injected Gaussian wave packet. (a)–(e) correspond to the $|{\phi }_{q1}(j,t)|$ with $q\in [1,5]$. The colored lines from lightness to darkness represent the simulations at $t=50,100,150$ and the black line represents analytical results. In the simulation, the system has five ports and each port has 401 sites, and the site number $j$ ranges from 0 to 400. The Gaussian wave packets have $\sigma =20$ and $k_c=\pi /2$.