Trace formula for chaotic dielectric resonators tested with microwave experiments

We measured the resonance spectra of two stadium-shaped dielectric microwave resonators and tested a semiclassical trace formula for chaotic dielectric resonators proposed by Bogomolny et al. [Phys. Rev. E 78, 056202 (2008)]. We found good qualitative agreement between the experimental data and the predictions of the trace formula. Deviations could be attributed to missing resonances in the measured spectra in accordance with previous experiments [Phys. Rev. E 81, 066215 (2010)]. The investigation of the numerical length spectrum showed good qualitative and reasonable quantitative agreement with the trace formula. It demonstrated, however, the need for higher-order corrections of the trace formula. The application of a curvature correction to the Fresnel reflection coefficients entering the trace formula yielded better agreement, but deviations remained, indicating the necessity of further investigations.

Figure 1

(a) Geometry of the two Teflon plates S1 (left) and S2 (right). The solid lines indicate the boundaries of the plates and the dashed lines the common border of the semicircular and the rectangular parts of the stadia, where $R$ is the radius of the semicircles and $L$ the length of the rectangular part. 1b Schematic side view (section) of the experimental setup (not to scale). The Teflon plate with thickness $d$ is put between two copper plates. Two antennas are led through small holes in the top copper plate next to the sidewalls of the Teflon plate. Reprinted from Ref. [21].

Figure 2

The measured frequency spectrum of the stadium S2. The inset in the top part shows the shape of the stadium with the positions of the antennas indicated by crosses. The arrows in the bottom panel indicate a subset of resonances that are approximately equidistant, i.e., they have a constant free spectral range $f_{\mathrm{FSR}}$.

Figure 3

Measured free spectral range (FSR) of the subset of equidistant resonances (marked by arrows in the bottom panel of Fig. 2) of resonator S2 with respect to the frequency. The horizontal line is the mean value of the measured FSR, and the gray bar indicates the standard deviation.

Figure 4

Length spectrum for the stadium S1. The top graph shows the experimental length spectrum and the bottom graph the FT of the trace formula. Note the different scales of the top and bottom graphs. The vertical line in the top graph indicates the path length $l_1$ defined in Eq. (3), the gray bar its error, and the arrow the circumference $U$ of the stadium. The vertical lines in the bottom graph indicate the lengths and the amplitudes $A_{p,r}$ of the POs [Eq. (13)] used for the calculation of the trace formula. Some POs are shown as insets.

Figure 5

Length spectrum for the stadium S2. The top graph shows the experimental length spectrum, and the bottom graph the FT of the trace formula. Note the different scales of the top and bottom graphs. The vertical line in the top graph indicates the path length $l_2$, the gray bar its error, and the arrow the circumference $U$ of the stadium. The vertical lines in the bottom graph indicate the lengths and the amplitudes $A_{p,r}$ of the POs. Some POs are shown as insets.

Figure 6

Real vs imaginary parts of the eigenvalues for stadium S1 in frequency units. The $\times$ marks are the experimental data, and the dots the numerically calculated data.

Figure 7

Length spectrum for the stadium S1 evaluated in the frequency range $f=1.0$–$13.2$ GHz. Shown are the length spectrum of the numerically calculated eigenvalues (dashed line), the experimental one (solid line), the FT of the trace formula with the ordinary Fresnel reflection coefficients [Eq. (8)] inserted (dotted line), and that obtained with the Fresnel reflection coefficients for curved interfaces, Eq. (16) (dash-dotted line). The arrow indicates the circumference $U$.

Figure 8

Modulus of the Fresnel reflection coefficient with respect to the angle of incidence $\theta$ for $n=1.425$. Shown is the modulus of the reflection coefficient including a curvature correction, $|r_{\mathrm{curv}}|$, according to Eq. (16) for $kR=3.1$ corresponding to $f=1$ GHz with $R=0.15$ m (solid line), for $kR=6.3$ (2 GHz, dashed line), for $kR=15.7$ (5 GHz, dotted line), and for $kR=40.9$ (13 GHz, dash-dotted line). The gray solid line shows the modulus of the ordinary Fresnel coefficient, $\left|r\right|$, defined in Eq. (8).