Experimental test of a trace formula for two-dimensional dielectric resonators

Resonance spectra of two-dimensional dielectric microwave resonators of circular and square shapes have been measured. The deduced length spectra of periodic orbits were analyzed and a trace formula for dielectric resonators recently proposed by Bogomolny et al. [Phys. Rev. E 78, 056202 (2008)] was tested. The observed deviations between the experimental length spectra and the predictions of the trace formula are attributed to a large number of missing resonances in the measured spectra. We show that by taking into account the systematics of observed and missing resonances the experimental length spectra are fully understood. In particular, a connection between the most long-lived resonances and certain periodic orbits is established experimentally.

Figure 1

Schematic picture of the experimental setup (not to scale). (a) Side view: the dielectric plate is placed between two copper plates. Two dipole antennas entering the resonator through small holes in the top plate are placed next to the sidewalls of the dielectric plate. The dipole antennas are attached to a vectorial network analyzer via rf cables. (b) Top view: the solid lines denote the contour of the copper plates; the dashed lines denote that of the dielectric plates (circle with radius $R$ and squares with side length $a$). The crosses indicate the positions of the antennas.

Figure 2

Frequency spectrum of the Teflon circle. The modulus of the transmission amplitude $S_{21}$ is plotted with respect to the frequency $f$. The classical trajectories associated with two resonances with $n_r=1$ and $n_r=2$ are shown as insets, and $\alpha$ denotes the angle of incidence with respect to the surface normal of these trajectories. The bars above the graph denote the frequency regimes in which modes with certain radial quantum numbers $n_r$ are observed.

Figure 3

The integrated resonance density $N\left(f\right)$ for the Teflon circle. The solid line is deduced from the measured frequency spectrum in Fig. 2; the dashed line is deduced from Weyl’s law. All in all, 716 resonances were identified in the spectrum up to 20.5 GHz, which is only about 10% of the total number of resonances.

Figure 4

Length spectrum for the Teflon circle. The full line results from the measured spectrum, the dashed line results from a complete calculated spectrum, and the dotted line shows the semiclassical expression $\left|{\tilde{\rho }}_{\text{scl}}\left(\ell \right)\right|$. The arrows indicate the lengths of the depicted POs and of the circumference $2\pi R$ of the circle. The semiclassical expression and the calculated length spectrum agree well except for the square orbit. The experimental length spectrum has smaller amplitudes than the calculated one. However, the deviations are smaller for the higher-order polygon orbits.

Figure 5

Length spectrum for a single resonance family. Only modes with radial quantum number $n_r=1$ were considered and only up to a certain frequency $f_{\text{max}}$ (solid line: 20.5 GHz; dashed line: 15 GHz; dotted line: 10 GHz). The arrows denote the lengths of some POs and of the circumference $2\pi R$ of the circle. Apparently, the position of the maximum of $\left|\tilde{\rho }\left(\ell \right)\right|$ only depends on the frequency $f_{\text{max}}$ and is not related to any PO.

Figure 6

Length spectra for different resonance families. (a) Only one radial quantum number taken into account each (solid line: $n_r=1$; dashed line: $n_r=2$; dotted line: $n_r=3$). (b) Several families combined. The solid line is the length spectrum for the families with radial quantum numbers $n_r=1$ and 2 combined, the dashed line is for $n_r=1$, 2, and 3 combined, and the dotted line is for all resonances (identical with the solid line in Fig. 4). The peaks of the length spectra in (b) originate from the interference between the different resonance families, and the main contributions of the different families are focused on different orbits.

Figure 7

Frequency spectrum of the Teflon square. The series of equidistant resonances atop the slowly oscillating background are superscarred states localized on a family of POs including the diamond orbit (shown as inset). The arrows indicate the computed resonance frequencies of the superscarred states, and the corresponding quantum numbers $m=35$ and $m=80$ are indicated for two examples, respectively.

Figure 8

Length spectrum of the Teflon square. The top graph shows the experimental length spectrum (solid line) and the length spectrum for a single family of superscars (dashed line), and the bottom graph the semiclassical expression. Note the different scales of top and bottom graphs. The arrows denote the lengths of the POs in the square (dashed arrows for orbits not contained by TIR), and the indices $\left(n_x,n_y\right)$ denote the type of orbit. Two examples of POs are shown as insets [Fabry-Perot orbit (left) and diamond orbit (right)]. The experimental length spectrum features only the diamond orbit and its repetitions, which are the only POs confined by TIR.

Figure 9

(a) Ratio between the peak amplitudes of the experimental and the superscar length spectra. The diamonds are the relative amplitudes for the indicated POs, and the full line is an exponential fit of Eq. (16) whose slope corresponds to a resonance width of $\Gamma =56.1\text{MHz}$. (b) Experimental length spectrum (top graph) and superscar length spectrum multiplied with the exponential fit (bottom graph). The two curves show excellent agreement.

Figure 10

Frequency spectrum of the alumina square. In contrast to the cases of the Teflon circle and square, the structure of the spectrum seems irregular, and it cannot be divided into one or more families of equidistant resonances. The lower graph shows the part of the spectrum from 2 to 3 GHz. The arrows indicate the resonance frequencies computed according to a generalized superscar model, and the quantum numbers $\left(m_x,m_y\right)$ and the corresponding angles of incidence $\tilde{\chi }$ are given for each. The computed resonance frequencies agree well with the measured spectrum in most cases.

Figure 11

Length spectrum of the alumina square. The upright graph in the top panel is the experimental length spectrum, the inverse graph is based on the generalized superscar model, and the bottom panel shows the semiclassical expression. Note the different scales of top and bottom panels. The lengths of the POs are indicated by arrows, with solid arrows for those POs confined by TIR and dashed arrows for those not confined. The indices $\left(n_x,n_y\right)$ characterizing the confined POs are also given. There are several peaks corresponding to POs in the experimental length spectrum, but most of them hardly stand out above the noise level, and their peak amplitudes are much lower than semiclassically predicted. The inverse graph in the top panel is the length spectrum for a set of modes computed according to the generalized superscar model. Only modes with $\tilde{\chi }\ge 28°$ were taken into account in this set, and no possible degeneracies. This length spectrum reproduces the experimental one well except for the cases of the (1,1) and (2,2) orbits.

Figure 12

The ratio between the peak amplitudes of the experimental and semiclassical length spectra with respect to the angle of incidence of the corresponding PO. The indices $\left(n_x,n_y\right)$ of the POs are indicated. The vertical line signifies the critical angle for TIR. The relative amplitudes for POs close to the critical angle are smaller compared to those of other POs.