Spectral singularities with directional sensitivity

We propose a class of spectral singularities that are sensitive to the direction of excitation and arise in nonlinear systems with broken parity symmetry. These spectral singularities are sensitive to the direction of the incident beam and result in diverging transmission and reflection for the left (right) incident, while the transmission and reflection of the right (left) side of the system remain finite. For the pedagogical reason, first we review the scattering formalism in nonlinear systems using an abstract $\delta$-function model. Then, using a parity symmetry broken nonlinear system consisting of two $\delta$ functions, one linear and the other nonlinear, we prove the existence of our proposed spectral singularities. Finally, we use an experimentally feasible realistic model based on coupled disk resonators to demonstrate the spectral singularity with directional sensitivity (SSDS). Our proposed SSDS might have applications in the design of nonlinear sensors and might provide a solution for the hole-burning problem in pumped laser ring resonators.

Figure 1

Left spectral singularity with directional singularity. (a) Transmission and reflection curves vs wave vector $k$ associated with a system composed of two $\delta$ functions, one linear with permittivity $n_1=2-3.4641i$ placed at $x=-2.7207$ (in units of $k^{-1}$) and the other with $n_2=3-3.4641i$ and $\chi =3$ placed at $x=2.7207$ (in units of $k^{-1}$). Left transmission and left reflection are diverging at $k\approx 0.288675$, while transmission and reflection for the right incident beam remain finite. (b) The phase associated with the $t_i,r_i, i=l,r$. At the SSDS, a $\pi$ shift occurs for the left reflection (red $\blacksquare$) and transmission (black $•$), while the right reflection and transmission curves are smooth.

Figure 2

Right spectral singularity with directional singularity. (a),(b) Similar to the one presented in Fig. 1, with $\delta$ functions placed at $x\approx \pm 2.32$ with $n_1=2-2.95712i, n_2=3-2.95712i$, and $\chi =3$. Here the SSDS occurs for right transmission and reflection.

Figure 3

(a) Transmission and reflection coefficients of the coupled disk resonators depicted in the upper inset. The resonance frequency of the disks at the transmission line is ${\omega }_c=0$, the resonance frequency of the linear (blue) disk with gain is ${\omega }_{\_}=0.503461i$, while the linear part of the resonance frequency of the nonlinear disk is ${\omega }_{+}=2{\omega }_{\_}$ and its Kerr coefficient is $\chi =1$. Right transmission and right reflection are diverging at $k^{★}=0.754559$ (in units of couplings), indicating the existence of a right SSDS mode. (b) The corresponding phases. We observe the $\pi$ phase shift at the spectral singularity (green $⧫$ and blue $▴$).