Synthesizing exceptional points with three resonators

In non-Hermitian coupled-resonator networks, the eigenvectors of degenerate eigenmodes may become parallel due to the singularity at so-called exceptional points (EP). To exploit the extraordinary parametric sensitivity at EPs, an important problem is, given an arbitrary set of resonators, how to generate a desired EP by properly coupling them together. This paper provides the solution for the case of three resonators. We show that all physically admissible EPs can be realized with either weakly coupled linear networks or strongly coupled circular networks, and the latter type of EPs has not been reported in the literature. Each admissible EP eigenvalue is associated with two and only two resonator-network realizations, and the formulas for calculating the required coupling constants are provided. The characteristics of these EPs are illustrated by the sharp change of transmission spectra near them, which verifies the enhanced sensitivity induced by the singularity of EPs. The obtained results lay a basis for designing EP structures towards broader applications.

Figure 1

The clockwise ($a_k^{\mathrm{cw}}$) and counterclockwise ($a_k^{\mathrm{ccw}}$) chiral modes in a waveguide-coupled-resonator network. The cw (ccw) modes interact only with ccw (cw) modes with opposite chirality in their neighboring resonators.

Figure 2

Schematic diagram for possible topologies of three-resonator networks with input and output fields coupled to the middle resonator: (a) one resonator is isolated from the other two coupled resonators; (b) three linearly coupled resonators; and (c) three circularly coupled resonators. The fields travels unidirectionally in (a) and (b), while in (c) both chiral modes coexist.

Figure 3

The distribution of degenerate eigenvalues and their associated twin EPs in a given three-resonator network (${\gamma }_1\le {\gamma }_2\le {\gamma }_3$). Each point in the brown (red, blue) horizontal line segments represents an ${\mathrm{EP}}_2$ network (with eigenvalue being real numbers) in the weak-coupling regime, which is realized by linear networks with ${\kappa }_{13}=0$ (${\kappa }_{12}=0,{\kappa }_{23}=0$), respectively. The two vertical lines represent the twin EPs (with eigenvalues being imaginary numbers) in the strong-coupling regime, which are realized by circular networks. The two yellow points at the origin represent the twin critical-coupling ${\mathrm{EP}}_3$ realizations.

Figure 4

The twin weak-coupling EP realizations [(a, b) for the first and (c, d) for the second] with resonators ${\gamma }_1/\mathrm{\Delta }=-1.9640,{\gamma }_2/\mathrm{\Delta }=0.6547$, and ${\gamma }_3/\mathrm{\Delta }=1.3093$. The plots (a) and (c) are the coupling strengths for each pair of twin EPs, while (b) and (d) are the power distribution in the corresponding EP eigenmodes.

Figure 5

The twin strong-coupling EP realizations [(a, b) for the first and (c, d) for the second] with resonators ${\gamma }_1/\mathrm{\Delta }=-1.9640,{\gamma }_2/\mathrm{\Delta }=0.6547$, and ${\gamma }_3/\mathrm{\Delta }=1.3093$. The plots (a) and (c) are the coupling strengths for each admissible EP, while (b) and (d) are the power distributions in the corresponding EP eigenmodes.

Figure 6

The eigenmodes (left) and the transmission spectra (right) near the twin weak-coupling ${\mathrm{EP}}_2$ [corresponding to (a, c) and (d, f)] designed at $\sigma /\mathrm{\Delta }=0.1$. The transmission spectra at the ${\mathrm{EP}}_2$ are the black solid curves, while the red dotted (blue dash-dotted) curves are the transmission spectra when ${\gamma }_3$ is perturbed by $\epsilon /\mathrm{\Delta }=-{10}^{-5}$ ($\epsilon /\mathrm{\Delta }={10}^{-5}$). The loss (gain) rates of the resonators are chosen as ${\gamma }_1/\mathrm{\Delta }=-1.3093,{\gamma }_2/\mathrm{\Delta }=-0.6547$, and ${\gamma }_3/\mathrm{\Delta }=1.9640$.

Figure 7

The eigenmodes (left) and the transmission spectra (right) near the twin strong-coupling ${\mathrm{EP}}_2$ [corresponding to (a)–(c) and (d)–(f)] designed at $\sigma /\mathrm{\Delta }=0.1i$. The transmission spectra at the EPs are shown by black solid curves, while the red dotted (blue dash-dotted) curves are the spectra when ${\gamma }_3$ is perturbed by $\epsilon /\mathrm{\Delta }=-{10}^{-5}$ ($\epsilon /\mathrm{\Delta }={10}^{-5}$). The loss (gain) rates of the resonators are chosen as ${\gamma }_1/\mathrm{\Delta }=-1.3093,{\gamma }_2/\mathrm{\Delta }=-0.6547$, and ${\gamma }_3/\mathrm{\Delta }=1.9640$.

Figure 8

The eigenmodes (left) and the transmission spectra (right) near the twin critical-coupling ${\mathrm{EP}}_3$ [corresponding to (a)–(c) and (d)–(f), respectively] designed at $\sigma =0$. The black solid curves in the transmission spectra are those at the EP, while the red dotted (blue dash-dotted) curves are the spectra with perturbation $\epsilon /\mathrm{\Delta }=-{10}^{-8}$ ($\epsilon /\mathrm{\Delta }={10}^{-8}$) on ${\gamma }_3/\mathrm{\Delta }$. The loss (gain) rates of the resonators are chosen as ${\gamma }_1/\mathrm{\Delta }=-1.3093,{\gamma }_2/\mathrm{\Delta }=-0.6547$, and ${\gamma }_3/\mathrm{\Delta }=1.9640$.

Figure 9

The eigenmodes (left) and the transmission spectra (right) near the twin EPs in critical-coupling regime, in which one is an ${\mathrm{EP}}_3$ [(a)–(c)] and the other is an ${\mathrm{EP}}_2$ [(d)–(f)]. The black solid curves in the transmission spectra are those at the EP, while the red dotted (blue dash-dotted) curves are the spectra with perturbation $\epsilon /\mathrm{\Delta }=-{10}^{-8}$ ($\epsilon /\mathrm{\Delta }={10}^{-8}$) on ${\gamma }_3$. The loss (gain) rates of the resonators are chosen as ${\gamma }_1/\mathrm{\Delta }=-1.7321,{\gamma }_2/\mathrm{\Delta }=0$, and ${\gamma }_3/\mathrm{\Delta }=1.7321$.

Figure 10

The existence of ${\mathrm{EP}}_2$ in circular networks, where the cross of the cubic curve and the horizontal line implies that Eq. (B8) always has two positive roots smaller than $x_1$ and $x_2$. They correspond to ${\kappa }_{31}^2$ associated with the twin EP realizations.