Fermi resonance in optical microcavities

Fermi resonance is a phenomenon of quantum mechanical superposition, which most often occurs between normal and overtone modes in molecular systems that are nearly coincident in energy. We find that scarred resonances in deformed dielectric microcavities are the very phenomenon of Fermi resonance, that is, a pair of quasinormal modes interact with each other due to coupling and a pair of resonances are generated through an avoided resonance crossing. Then the quantum number difference of a pair of quasinormal modes, which is a consequence of quantum mechanical superposition, equals periodic orbits, whereby the resonances are localized on the periodic orbits. We derive the relation between the quantum number difference and the periodic orbits and confirm it in an elliptic, a rectangular, and a stadium-shaped dielectric microcavity.

Figure 1

Eigenvalues in an elliptic microcavity depending on the eccentricity: (a) the real eigenvalues and (b) the imaginary eigenvalues. Curves 1, 2, 3, and 4 exhibit resonance trapping for bow-tie SLRs and their pairs through ARCs.

Figure 2

Intensity plots and Husimi functions of resonances in an elliptic microcavity depending on $\epsilon$. (a) and (b) Bouncing-ball-type quasinormal modes before an ARC at $\epsilon =0.76$. (c) and (d) Bow-tie SLR and its pair, respectively, at $\epsilon =0.793$. (e) and (f) Recovered bouncing-ball-type resonances after the ARC at $\epsilon =0.81$. (g) and (h) Husimi functions of (c) and (d), respectively. In the Husimi functions, thick solid elliptic lines are resonant tori corresponding to the (4,1) periodic orbit. Here $\chi$ is the incident angle, $S$ is the arc length from the $x$ axis, and $S_{max}$ is the total boundary length.

Figure 3

Intensity plots of superposed resonances: (a) the addition of two bouncing-ball-type quasinormal modes $[(11,5)+(7,6)]/\sqrt{2}$ whose shape is a bow tie and (b) the subtraction $[(11,5)-(7,6)]/\sqrt{2}$.

Figure 4

Eigenvalues in a rectangular microcavity depending on the deformation parameter $\epsilon$: (a) the real eigenvalues and (b) the imaginary eigenvalues.

Figure 5

Pairs of SLRs in a rectangular dielectric microcavity at (a) and (b) points A and B, respectively, around $\epsilon =0.046$; (c) and (d) points C and D, respectively, around $\epsilon =0.051$; and (e) and (f) points E and F, respectively, around $\epsilon =0.119$ in Fig. 4. The lines are the trajectories of the nonisolated periodic orbits where the SLRs are localized.

Figure 6

Eigenvalues and the resonances in a stadium-shaped microcavity depending on $\epsilon$. (a) and (b) Real and imaginary eigenvalues of the (2,36) and (3,32) quasinormal modes. (c) and (d) The (2,36) and (3,32) quasinormal modes at $\epsilon =0.0$, respectively. (e) and (f) The diamond-shaped scarred resonance and its pair of rectangular-shaped resonances at $\epsilon =0.01$, respectively. (g) and (h) Superposed states of $[(2,36)-(3,32)]/\sqrt{2}$ and $[(2,36)+(3,32)]/\sqrt{2}$, respectively. The insets in (a) and (b) are the eigenvalues in the region of small deformation.