Corrected perturbation theory for transverse-electric whispering-gallery modes in deformed microdisks

The perturbation theory by Ge et al. [Phys. Rev. A 87, 023833 (2013)] for transverse-electric polarized modes in weakly deformed microdisks omits terms related to the variation of the normal derivative of the magnetic field along the boundary. Here, we show that these terms are necessary to accurately describe microdisks with a strongly winding boundary. In particular, it is demonstrated that the corrected perturbation theory allows one to describe the counterintuitive phenomenon of $Q$-factor enhancement due to weak boundary deformation. We discuss in detail the microflower cavity and the limaçon cavity. Good agreement of the corrected perturbation theory with full numerical results is observed.

Figure 1

Boundary of the microflower cavity (left) with $\kappa =10$ and $\varepsilon =0.1$ and the limaçon cavity (right), including the regular (dash-dotted light gray) and shifted (red or dark gray) boundary, with $\varepsilon =0.4$. The unperturbed circular cavity is shown as the dashed curve. The radial coordinate is here dimensionless as $R=1$ is chosen.

Figure 2

Microflower cavity with $n=2.63$, $\kappa =10$, $\varepsilon =0.13$: intensity distributions of the even-parity (left) and odd-parity mode (right) with the mode numbers $l=1$ and $m=5$ calculated by the BEM; the deformation strength $\varepsilon$ is rather large because it is taken at the maximum $Q$ factor according to the BEM results in Fig. 4.

Figure 3

Microflower cavity with $n=2.63$, $\kappa =10$: real (top) and imaginary parts (bottom) of the dimensionless frequency in dependence on the dimensionless deformation strength. Solid (dashed) curves show the results for the even-parity (odd-parity) mode with the mode numbers $l=1$ and $m=5$. The triangles (circle and diamond) correspond to the uncorrected (corrected) perturbation theory. Symbols without curves correspond to the BEM.

Figure 4

Microflower cavity with $n=2.63$, $\kappa =10$, $l=1$, $m=5$: $Q$ factor versus dimensionless deformation strength; the dotted vertical line marks the maximum $Q$ factor according to the BEM.

Figure 5

Microflower cavity with $n=2.63$, $\kappa =10$, $\varepsilon =0.03$: intensity distributions of the even-parity (left) and odd-parity mode (right) with the mode numbers $l=1$ and $m=5$ for the BEM (top) and corrected PT (bottom).

Figure 6

Microflower cavity with $n=2.63$, $\varepsilon =0.03$, and $l=1$: dimensionless deviation of the two PT's real (top) and imaginary (bottom, on a logarithmic scale) parts of the frequency from the BEM reference $\mathrm{\Delta }x=x_{\text{PT}}-x_{\text{BEM}}$ versus the azimuthal mode number $m=\kappa /2$.

Figure 7

Limaçon cavity with $n=2.63$, $l=1$, and $m=4$: dimensionless deviation of the two PTs' real parts of the frequency from the BEM reference in dependence on the dimensionless deformation strength; the lower panel shows the same data on a logarithmic $y$ axis; the dips appear where the PT results cross those of the BEM.

Figure 8

Limaçon cavity with $n=2.63$, $l=1$, and $m=4$: dimensionless deviation of the two PTs' imaginary parts of the frequency from the BEM reference in dependence on the dimensionless deformation strength; the lower panel shows the same data on a logarithmic $y$ axis.

Figure 9

Limaçon cavity with $n=2.63$, $l=1$, and $m=4$: dimensionless deviation of the TM PT and the corrected TE PT real (top) and imaginary parts (bottom) of the frequency from the BEM reference versus the dimensionless deformation strength in semilogarithmic plots; the dip appears where the PT results cross those of the BEM.