Perturbation theory of nearly spherical dielectric optical resonators

Dielectric spheres of various sizes may sustain electromagnetic whispering-gallery modes resonating at optical frequencies with very narrow linewidths. Arbitrary small deviations from the spherical shape typically shift and broaden such resonances. Our goal is to determine these shifted and broadened resonances. A boundary-condition perturbation theory for the acoustic vibrations of nearly circular membranes was developed by Rayleigh more than a century ago. We extend this theory to describe the electromagnetic excitations of nearly spherical dielectric cavities. This approach permits us to avoid dealing with decaying quasinormal modes. We explicitly find the frequencies and the linewidths of the optical resonances for arbitrarily deformed nearly spherical dielectric cavities, as power series expansions by a small parameter, up to and including second-order terms. We thoroughly discuss the physical conditions for the applicability of perturbation theory.

Figure 1

(a) Illustration of the dielectric spherical and nearly spherical resonators of equations $r-a=0$ and $r-a=a[1+h(\theta ,\varphi \left)\right]$, respectively, where $a$ is the radius of the sphere. (b) Geometry of a section of the spherical (dark blue) and of the nearly spherical (light blue) resonators. Here ${\widehat{\mathbf{e}}}_r$ is the radial unit vector and $\mathbf{n}(\theta ,\varphi )$ is the vector normal to the deformed surface (41). From (48) it follows that $cos\gamma ={\widehat{\mathbf{e}}}_r·\mathbf{n}/\left|\mathbf{n}\right|$.

Figure 2

Spectrum of the TE and TM resonances of a dielectric sphere of radius $a$ and refractive index $n_1=1.5$, surrounded by vacuum with $n_2=1$. The values of $x_{ln}=k_{ln}a$ for $1\le n\le 10$ and $1\le l\le 10$ are shown as blue bands. The vertical position of the center of each band is equal to $Re\left(k_{ln}a\right)$ and the thickness is equal to $Im\left(k_{ln}a\right)$. For the first radial mode $n=1$ (lighter blue bands) the imaginary part of $k_{nl}a$ quickly decreases as $l$ increases from left to right. Each resonance characterized by the pair of azimuthal and radial numbers $l$ and $n$ is $2l+1$ times degenerate.

Figure 3

(a) Comparison between numerical calculations (orange open circles) and perturbation theory predictions (blue closed circles) for the TE complex-valued resonances $x=ka$ of a dielectric approximate oblate spheroid with $\delta =0.01$ and refractive index $n_1=2$. Here $k=k_{l_0{n}_0}$, where $l_0=10$ and $n_0=1$ is the first radial number. The numbers $10,9,...,0$ nearthe resonances mark the values of $0\le \left|m\right|\le l_0$. (b) Relative error $\delta x/x$ between numerical and perturbative calculations, calculated according to (250). The dashed orange line gives the average relative error.

Figure 4

Same as Fig. 3 but for TM waves.

Figure 5

Same as Fig. 3 but for a bigger value for the magnitude of the deformation $\delta =0.05$.

Figure 6

Same as Fig. 5 but for TM waves.