Hierarchical Construction of Higher-Order Exceptional Points

The realization of higher-order exceptional points (HOEPs) can lead to orders of magnitude enhancement in light-matter interactions beyond the current fundamental limits. Unfortunately, implementing HOEPs in the existing schemes is a rather difficult task, due to the complexity and sensitivity to fabrication imperfections. Here we introduce a hierarchical approach for engineering photonic structures having HOEPs that are easier to build and more resilient to experimental uncertainties. We demonstrate our technique by an example that involves parity-time symmetric optical microring resonators with chiral coupling among the internal optical modes of each resonator. Interestingly, we find that the uniform coupling profile is not required to achieve HOEPs in this system—a feature that implies the emergence of HOEPs from disorder and provides resilience against some fabrication errors. Our results are confirmed by using full-wave simulations based on Maxwell’s equation in realistic optical material systems.

Figure 1

By introducing a unidirectional coupling between two different Hamiltonians $H_a$ and $H_b$, each of which exhibits an EP of order $N$, the combined system in general has $M$ EPs, each of which is of order $<2N$. As described in the text, under certain conditions, $M=1$, the system has only one EP with order $2N$.

Figure 2

A schematic of two identical copies of a PT symmetric $J_x$ array. Each lattice has an EP of order $N$. By introducing disordered unidirectional coupling $k$’s, the combined array can exhibit an EP of order $2N$.

Figure 3

(a) A PT symmetric system consisting of two coupled microring resonators having equal loss ($-i\gamma$) and gain ($i\gamma$) implements an EP of order 2. By introducing an evanescent coupling between any of the rings (the lossy ring here) and a waveguide with an end mirror, the system realizes the Hamiltonian ${\mathcal{H}}_4$ of Eq. (6). The nanoparticle close to the gain ring is used as a small perturbation to confirm that the combined system indeed has an EP of order 4 in the simulations. (b) Equivalent real space implementation to the system in (a), detailing the gain-loss distribution and the coupling profile between various CW and CCW components.

Figure 4

(a) Real and (b) imaginary splitting of the eigenfrequency of the system in Fig. 3 as a result of a perturbation $|\epsilon |$ (see the Supplemental Material [22]) introduced by a nanoparticle. Dots represent the results obtained from full-wave FEM simulations for the ring resonator parameters given in the Supplemental Material [22], while the solid black line is a plot of the analytical formula of Eq. (7). (c) Log scale plot of the real frequency shift $\mathrm{Re}[{\omega }_1-{\omega }_0]$ as a function of $|\epsilon |$. The slope of the linear fit is $\sim 0.24$, confirming that the system is operating in the vicinity of an EP of order 4.