Local chirality at exceptional points in optical whispering-gallery microcavities

We study the field chirality at the exceptional points (EPs) in an optical whispering gallery mode (WGM) microcavity, which is commonly believed to be globally perfect. We have discovered a locally imperfect or locally perfect chirality of eigenmodes at the EPs in a WGM microcavity perturbed by two strong nanoscatterers. We find that the generally local and imperfect chirality at the EPs tends to be globally perfect with the decrease of the scattering effect induced by the nanoscatterers, and the chirality also becomes locally perfect with the decrease of the relative azimuthal angle between the two strong nanoscatterers. With a first-principles-based model considering a dynamic multiple-scattering process of the azimuthally propagating modes (APMs), all the above counterintuitive phenomena of imperfect or perfect chirality can be respectively explained by a strong or weak frequency dependence of the APM scattering coefficients at an effective scatterer composed of the two nanoscatterers. In this paper, we provide increased understanding of the general properties of chirality at EPs which will benefit potential applications enabled by the chirality features of non-Hermitian systems at EPs.

Figure 1

(a) Schematic of a $z$-invariant cylindrical microcavity with two nanoholes. $r$ and ϕ denote the radial and azimuthal coordinates, respectively. The blue region represents the effective scatterer. $a_{\mathrm{ccw}\left(\mathrm{cw}\right)}$ and $b_{\mathrm{ccw}\left(\mathrm{cw}\right)}$ denote the complex-amplitude coefficients of the azimuthally propagating modes (APMs) outside and inside the effective scatterer, respectively. (b1) and (b2) Definition of the effective reflection ($R_{\mathrm{cw}}$ or $R_{\mathrm{ccw}}$) and transmission ($T$) coefficients of the APM at the effective scatterer with an incident normalized counterclockwise (CCW) or clockwise (CW) APM, respectively. (c1) and (c2) Like (b1) and (b2) but for a complementary effective scatterer. (d1) and (d2) Fundamental reflection (${\rho }_j$) and transmission (${\tau }_j$) coefficients of the APM at the nanohole $j=1$ and 2, respectively. The incident APM is denoted by a solid red arrow, while the solid and dashed black arrows in (b1) to (d2) indicate the transmitted and reflected APMs, respectively.

Figure 2

(a) Local chiralities ${\alpha }_{\mathrm{out}\left(\mathrm{in}\right)}$ at the exceptional points (EPs) plotted as functions of size $r_2$ of nanohole 2. The circles and squares show the finite-element method (FEM) results of ${\alpha }_{\mathrm{out}}$ and ${\alpha }_{\mathrm{in}}$, respectively. The solid and dashed curves show the model prediction of ${\alpha }_{\mathrm{out}}$ and ${\alpha }_{\mathrm{in}}$, respectively. The blue and red curves show the results of the pair of nearly degenerate resonant modes. (b) Electric-field intensity distribution at the EPs obtained with the FEM shown in (a). The five columns are for different $r_2$, and the two rows are for the pairs of nearly degenerate resonant modes.

Figure 3

(a) Chiralities ${\alpha }_{\mathrm{out}\left(\mathrm{in}\right)}$ at exceptional points (EPs) plotted as functions of β. The hollow (solid) circles and the squares denote the full-wave finite-element method (FEM) results (model predictions) of ${\alpha }_{\mathrm{out}}$ and ${\alpha }_{\mathrm{in}}$, respectively. Blue and red correspond to two resonant modes. (b) Electric-field intensity distribution at the EPs for different β values corresponding to the FEM results shown in (a). The different columns are for different β, and the two rows are for the pair of nearly degenerate resonant modes.

Figure 4

(a) Real and imaginary parts of $k_0{n}_{\mathrm{eff}}{R}_0$ plotted as functions of the size $r_2$ of nanohole 2, which are obtained with the model at the exceptional points (EPs) already shown in Fig. 2. (b) $|\partial v/\partial k_0|$ (circles) and $|\partial w/\partial k_0|$ (triangles) at the EPs for the different β values already shown in Fig. 3. The solid and dashed curves show the results of $\beta R_0\left|n_{\mathrm{eff},0}\right|$ and $(2\pi -\beta )R_0\left|n_{\mathrm{eff},0}\right|$, respectively.