Far-Field Correlations Verifying Non-Hermitian Degeneracy of Optical Modes

An experimental verification of an exceptional point (EP) in a stand-alone chaotic microcavity is a tough issue because as deformation parameters are fixed the traditional frequency analysis methods cannot be applied any more. Through numerical investigations with an asymmetric Reuleaux triangle microcavity (ARTM), we find that the eigenvalue difference of paired modes can approach near-zero regardless of nonorthogonality of the modes. In this case, for a definite verification of EPs in experiments, wave function coalescence should be confirmed. For this, we suggest the method of exploiting correlation of far-field patterns (FFPs), which is directly related to spatial mode patterns. In an ARTM, we demonstrate that the FFP correlation of paired modes can be used to confirm wave function coalescence when an eigenvalue difference approaches near zero.

Figure 1

(a) Wave function of mode ($kR=121.8254-i7.2056\times {10}^{-4}$) localized on the period-5 classical orbit. (b) FFP of the mode. The inset is FFP in the polar coordinate. The emission robes $\{f_2,f_4\}$ ($\{f_1,f3\}$) correspond to the CW (CCW) direction. (c) The chiralities versus $\mathrm{Re}(kR)$ of fifteen pairs of modes localized on the period-5 classical orbit.

Figure 2

Relations among (a) $|\mathrm{\Delta }kR|$, (b) the overlap $S$, and (c) the far-field intensity correlation ${\sigma }_F$ for modes obtained in $3.299\le n\le 3.3$. (a) Six dips of $|\mathrm{\Delta }kR|$ of selected five modes (A, B, C, D, E) in Fig. 1 are, respectively, marked by symbols (red circle, green triangle, black diamond, blue square, and inverted orange triangle). Corresponding peak values of $S$ and ${\sigma }_F$ are marked by the same symbols as well.

Figure 3

(a), (e) Avoided resonance crossings of $nkR$ as a function of $n$ around IV and VI in Fig. 2. (b), (f) Wave function intensity in the log scale; (c), (g) Husimi functions; and (d), (h) FFPs of modes at IV and VI. In (a) and (e), $nkR\mathrm{s}$ are shifted as the amount of $\kappa$ ($\in C$; mean $nkR$ of the paired modes at IV and VI; dashed vertical line). In (b), (c), (d), (f), (g), and (h), the upper (lower) panels correspond to the mode having a higher (lower) $\mathrm{Im}(nkR)$ crossing the dashed vertical line in (a) and (e).

Figure 4

(a) Wave numbers $kR\in \mathbb{C}$ of modes in ARTM. The inset enlarges one of the nearly degenerated paired modes. (b) The chirality $\alpha$ as a function of the wave overlap $S$ in Eq. (7). (c) Histogram of modes having a large $S$ ($>0.8$; dashed line) that reveals CW ($\alpha <0$)-dominant chirality.

Figure 5

The distribution of the wave overlap $S$ versus (a) $|\mathrm{\Delta }kR|=|k_a-k_b|R$ and (b) the far-field correlation ${\sigma }_F$, where $(a,b)$ represents the paired-two modes [see, e.g., the inset in Fig. 4]. $S$ has no strong correlation with $|\mathrm{\Delta }kR|$, while it shows a intense concentration along a nontrivial curve with ${\sigma }_F$. In particular, the pairs exhibiting $S>0.8$ (dashed line) are well aligned on the curve $S=2{\sigma }_F-1$ deduced in Eq. (10).