Leaky modes of solid dielectric spheres

In the absence of external excitation, light trapped within a dielectric medium generally decays by leaking out, and also by getting absorbed within the medium. We analyze the leaky modes of solid dielectric spheres by examining solutions of Maxwell's equations for simple homogeneous, isotropic, linearly dispersive media that admit complex-valued oscillation frequencies. We show that, under appropriate circumstances, these leaky modes constitute a complete set into which an initial electromagnetic field distribution inside a dielectric sphere can be expanded. We provide the outline of a completeness proof, and also present results of numerical calculations that illustrate the close relationship between the leaky modes and the resonances of solid dielectric spherical cavities.

Figure 1

Locations in the $\omega$ plane of the poles and zeros of $\varepsilon \left(\omega \right)$, whose square root contributes to the refractive index $n\left(\omega \right)$ in accordance with Eq. (2). A similar set of poles and zeros, albeit at different locations in the $\omega$ plane, represents $\mu \left(\omega \right)$. The dashed lines connecting pairs of adjacent poles and zeros constitute branch cuts for the function $n\left(\omega \right)$. In accordance with the Cauchy-Goursat theorem [18], the integral of a meromorphic function, such as $f\left(\omega \right)$, over a circle of radius $R_c$ is $2\pi i$ times the sum of the residues of the function at the poles of $f\left(\omega \right)$ that reside within the circle.

Figure 2

Contours in the complex $\omega$ plane representing regions where $\mathrm{Re}\left[F\left(\omega \right)\right]=0$ (solid red lines) and $\mathrm{Im}\left[F\left(\omega \right)\right]=0$ (dashed blue lines). The real and imaginary axes are normalized by the reference frequency ${\omega }_{\mathrm{ref}}=1.216\times {10}^{15}$ rad/s. Where a solid red and a dashed blue curve cross, $F\left(\omega \right)$ vanishes; these crossing points (some of them marked with small circles) correspond to TE leaky-mode frequencies ${\omega }_q={\omega }_q^{\prime }+i{\omega }_q^{\prime \prime }$ of the spherical cavity at the chosen value of $\ell =10$. The spherical particle has radius $R=1.55\mu \mathrm{m}$, permeability $\mu \left(\omega \right)=1.0$, and refractive index $n\left(\omega \right)=\sqrt{\varepsilon \left(\omega \right)}$ governed by a single Lorentz oscillator. The fourth quadrant pole and zero of $n\left(\omega \right)$ are, respectively, at ${\mathrm{\Omega }}_{3e}\cong \left(2.0-0.01i\right){\omega }_{\mathrm{ref}}$ and ${\mathrm{\Omega }}_{1e}\cong \left(5.385-0.01i\right){\omega }_{\mathrm{ref}}$.

Figure 3

Logarithmic plot of the ratio of the $E$ field inside the sphere to the incident $E$ field, for a dielectric sphere of radius $R=1.55\mu \mathrm{m}$ at $\ell =10$. The horizontal axis represents the normalized excitation frequency $\omega /{\omega }_{\mathrm{ref}}$. The refractive index $n\left(\omega \right)=\sqrt{\varepsilon \left(\omega \right)}$ of the spherical particle is governed by a single Lorentz oscillator. The fourth quadrant pole and zero of $n\left(\omega \right)$ are at ${\mathrm{\Omega }}_{3e}\cong \left(2.0-0.01i\right){\omega }_{\mathrm{ref}}$ and ${\mathrm{\Omega }}_{1e}\cong \left(5.385-0.01i\right){\omega }_{\mathrm{ref}}$, respectively. The EM field hardly penetrates the dielectric sphere in the frequency interval between the pole and zero of $n\left(\omega \right)$. Outside this “forbidden” interval, the $E$-field amplitude ratio exhibits sharp peaks at certain frequencies, which is indicative of resonant behavior.

Figure 4

The $E$-field amplitude inside and outside a solid dielectric sphere of radius $R=1.55\mu \mathrm{m}$, plotted versus the normalized radial coordinate $r/R$ for three $\ell =10$ TE modes. The real part of the field is shown in (a), while its imaginary part appears in (b). The solid black, dashed red, and dash-dotted blue curves correspond, respectively, to $\omega /{\omega }_{\mathrm{ref}}=0.76253+0.00128i$, $0.938779+0.00199i$, and $1.08039+0.00275i$ (${\omega }_{\mathrm{ref}}=1.216\times {10}^{15}$ rad/s). The particle, whose permeability is $\mu \left(\omega \right)=1.0$, has a refractive index $n\left(\omega \right)=\sqrt{\varepsilon \left(\omega \right)}$ governed by a single Lorentz oscillator (${\omega }_r=2{\omega }_{\mathrm{ref}}$, ${\omega }_p=5{\omega }_{\mathrm{ref}}$, and $\gamma =0.02{\omega }_{\mathrm{ref}}$).

Figure 5

Comparison of the $E$ field inside a spherical cavity with its expansion as a superposition of leaky modes ($R=1.55\mu \mathrm{m}$, $\ell =10$ TE mode, $\omega =1.8{\omega }_{\mathrm{ref}}$). The real and imaginary parts of $E_{\mathrm{inside}}\left(r\right)$ are shown as solid lines—black and red (gray), respectively. The superposed symbols (i.e., small solid circles) represent the result of leaky-mode expansion, ${\sum }_q{E}_{\mathrm{leaky}}\left(r\right)$, composed of 100 terms.

Figure 6

Computed $Q$ factor versus the resonance frequency for a solid dielectric sphere ($R=77.5\mu \mathrm{m}$, $\mu =1$, $n=1.5$) in the vicinity of ${\omega }_{\mathrm{ref}}=1.216\times {10}^{15}\mathrm{rad}/\mathrm{s}$. The leaky frequencies ${\omega }_q={\omega }_q^{\prime }+i{\omega }_q^{\prime \prime }$ are solutions of $F\left(\omega \right)=0$, which have been found numerically. The ratio $|{\omega }_q^{\prime }/{\omega }_q^{\prime \prime }|$ is used as a measure of the cavity $Q$ factor at the excitation frequency $\omega ={\omega }_q^{\prime }$. Shown are computed $Q$ factors for both TE and TM modes (solid blue squares for TE, open red circles for TM) at several resonance frequencies of the dielectric sphere corresponding to $\ell =340$.

Figure 7

Plots of the amplitude ratio of the $E$ field inside the dielectric sphere $\left(R=77.5\mu \mathrm{m},\mu =1,n=1.5\right)$ to the incident $E$ field for the $\ell =340$ spherical harmonic. The horizontal axis represents the excitation frequency $\omega$ normalized by ${\omega }_{\mathrm{ref}}=1.216\times {10}^{15}\mathrm{rad}/\mathrm{s}$. (a) TE mode. (b) TM mode. Note that the cutoff frequency for both modes is $\omega \cong 0.78{\omega }_{\mathrm{ref}}$, below which no resonances are excited. Above the cutoff, in between adjacent resonances, the field amplitude inside the cavity drops to exceedingly small values. The occurrence of extremely large resonance peaks in these plots is due to the assumed value of the refractive index $n$ being purely real. (c) Close-up view of the resonance lines of the glass ball for the $\ell =340$ spherical harmonic, showing the TM resonances (dashed red lines) being slightly shifted away from the TE resonances (solid black lines). (d) Magnified view of an individual TE resonance line centered at ${\omega }_R=1.00207{\omega }_{\mathrm{ref}}$.

Figure 8

Excitation frequency dependence of the ratio of the $E$ field inside a glass sphere to the incident $E$ field for $\ell =10$ (solid black), $\ell =20$ (dashed blue), and $\ell =25$ (dash-dotted red) TE spherical harmonics.

Figure 9

Similar to Fig. 6, except that the refractive index $n=n^{\prime }+i{n}^{\prime \prime }$ of the dielectric sphere is now allowed to have a small nonzero imaginary part, $n^{\prime \prime }$, representing absorption within the material.