Application of a trace formula to the spectra of flat three-dimensional dielectric resonators

The length spectra of flat three-dimensional dielectric resonators of circular shape were determined from a microwave experiment. They were compared to a semiclassical trace formula obtained within a two-dimensional model based on the effective index of refraction approximation and a good agreement was found. It was necessary to take into account the dispersion of the effective index of refraction for the two-dimensional approximation. Furthermore, small deviations between the experimental length spectrum and the trace formula prediction were attributed to the systematic error of the effective index of refraction approximation. In summary, the methods developed in this article enable the application of the trace formula for two-dimensional dielectric resonators also to realistic, flat three-dimensional dielectric microcavities and -lasers, allowing for the interpretation of their spectra in terms of classical periodic orbits.

Figure 1

(a) Photograph of the experimental setup with disk B. (b) Sketch of the setup (not to scale). Three metal pillars are used to support a Teflon disk with radius $R$ and thickness $b$. A special foam (see text) is used to isolate the disk from the pillars. Two vertical wire antennas protruding from coaxial rf cables are used to couple microwave power into and out of the disk. A metal plate with variable distance $D$ to the disk (not depicted in the photo) can be added to determine the polarization of the resonances.

Figure 2

Measured frequency spectrum of disk B. The polarization and quantum numbers are indicated by TM or TE $(m,n_r)$. In the right part, only the relevant TM modes are indicated. The free spectral range ($\mathrm{FSR}$) is the spacing between resonances with the same polarization and radial quantum number.

Figure 3

Frequency spectrum without metal plate (solid line) and with metal plate [see Fig. 1b] at distances $D=14$ mm (dashed), 10 mm (dotted), and 8 mm (dot-dashed) for disk B. With decreasing distance $D$ the TE modes are shifted to higher frequencies, while the TM modes are shifted to lower frequencies.

Figure 4

Ray traveling through an infinite dielectric slab with thickness $b$ and index of refraction $n$. The wave vector $\vec{k}$ is decomposed into its components perpendicular ($k_z{\vec{e}}_z$) and parallel (${\vec{k}}_{\parallel }$) to the plane of the disk, where $k_{\parallel }=n_{\mathrm{eff}}k$. The angle of incidence on the top and bottom surface is $\theta$.

Figure 5

Effective index of refraction for the TM modes with $\zeta =0$ of disk A (solid line) and disk B (dashed line).

Figure 6

Length spectrum for the TM modes of disk A. The solid line in the top part is the experimental length spectrum and the dashed line the FT of the semiclassical trace formula. The curves in the bottom part are the contributions of the individual POs to the semiclassical trace formula. The solid arrows indicate the lengths of the POs denoted by $(q,\eta )$ and the dashed arrows indicate the peak positions computed from Eq. (17). The circumference $2\pi R$ of the circle is also indicated.

Figure 7

Critical angle ${\alpha }_{\mathrm{crit}}=\arcsin (1/n_{\mathrm{eff}})$ for TIR with respect to the frequency for the TM modes of disk A. The angles of incidence of the $(q,\eta )$ orbits are indicated by the horizontal lines, and the critical frequencies $f_{\mathrm{crit}}$ at which they become confined by TIR by the vertical lines. The $(3,1)$ and $(4,1)$ orbits are not confined for the frequency range considered here.

Figure 8

Length spectrum for the TM modes of disk B. The solid line in the top part is the experimental length spectrum and the dashed line the FT of the semiclassical trace formula. The curves in the bottom part are the contributions of the individual POs to the semiclassical trace formula. The solid arrows indicate the lengths of the POs denoted by $(q,\eta )$ and the dashed arrows indicate the peak positions computed from Eq. (17). The circumference $2\pi R$ of the circle is also indicated.

Figure 9

Length spectrum for the TM modes of disk A. The solid line is the experimental length spectrum, the dashed line the FT of the semiclassical, and the dotted line the FT of the exact trace formula. The dashed arrows indicate the lengths ${\ell }_{\mathrm{peak}}$. A magnification around the peak corresponding to the $(6,1)$ orbit is shown in the inset. Here, $\Delta {\ell }_{\max }$ denotes the difference between the positions of the maxima of the experimental length spectrum and those of the FT of the exact trace formula.

Figure 10

Length spectrum for the TM modes of disk B. The solid line is the experimental length spectrum, the dashed line the FT of the semiclassical, and the dotted line the FT of the exact trace formula. The dashed arrows indicate the lengths ${\ell }_{\mathrm{peak}}$. The solid and dashed lines are the same as in the top part of Fig. 8.

Figure 11

(a) Difference between the calculated and the measured resonance frequencies with respect to the measured frequency for the TM modes with $n_r=1$ of disk A. The modes with $n_r>1$ show a similar behavior (adapted from Ref. [34]). (b) Measured (dots) and calculated (solid line) free spectral ranges for TM modes with $n_r=1$. The calculated FSR is plotted as a curve instead of data points to guide the eye. The inset shows a magnification in the frequency range of 10–18 GHz.

Figure 12

Geometry and coordinate system for the calculation of the reflection coefficient $r\left(D\right)$.