Dielectric square resonator investigated with microwave experiments

We present a detailed experimental study of the symmetry properties and the momentum space representation of the field distributions of a dielectric square resonator as well as the comparison with a semiclassical model. The experiments have been performed with a flat ceramic microwave resonator. Both the resonance spectra and the field distributions were measured. The momentum space representations of the latter evidenced that the resonant states are each related to a specific classical torus, leading to the regular structure of the spectrum. Furthermore, they allow for a precise determination of the refractive index. Measurements with different arrangements of the emitting and the receiving antennas were performed and their influence on the symmetry properties of the field distributions was investigated in detail, showing that resonances with specific symmetries can be selected purposefully. In addition, the length spectrum deduced from the measured resonance spectra and the trace formula for the dielectric square resonator are discussed in the framework of the semiclassical model.

Figure 1

Experimental setup. (a) Photograph of the experimental setup. (b) Sketch (not to scale) of the setup (reprinted from Ref. [21]). (c) Drawings of the two antenna configurations used, i.e., the antenna was either placed at the side wall of (left) or below (right) the ceramic plate.

Figure 2

Measured frequency spectra for different antenna combinations. The resonances identified as ${\mathrm{TM}}_0$ modes are indicated by the arrows. The insets indicate the positions of the antennas at the side walls, respectively, below the cavity. (a) The excitation antenna was placed $a/4$ above the lower left corner while the receiving antenna was placed $a/4$ below the upper right corner. (b) The excitation antenna was positioned at the upper left and the receiving antenna at the lower right corner. (c) The excitation antenna was put beneath the center of the cavity and the receiving antenna at the upper left corner. (d) The excitation antenna was placed at the midpoint of the left and the receiving antenna at the midpoint of the right cavity edge.

Figure 3

Measured wave function (a) and momentum distribution (b) of a ${\mathrm{TM}}_0$ resonance at $6.835$ GHz. The modulus of ${\Psi }_{\mathrm{expt}}(x,y)$ and ${\tilde{\Psi }}_{\mathrm{expt}}(k_x,k_y)$ is shown in false colors, respectively. The straight white lines in panel (b) indicate the eight major momentum components, the white circle indicates their modulus $k_{\parallel }$, and ${\alpha }_{\mathrm{inc}}$ is the angle of incidence of the corresponding family of classical trajectories. Adapted from Ref. [21].

Figure 4

(a) Measured effective refractive index. The black data points were obtained from the experimental momentum distributions (see text) and correspond to the three waveguide modes ${\mathrm{TM}}_0$, ${\mathrm{TM}}_1$, and ${\mathrm{TM}}_2$. The white lines were calculated from the theoretical expression for $n_{\mathrm{eff}}$, Eq. (4), with values for $n_1$ deduced from the experimental data. The gray lines indicate the one sigma error interval of $n_{\mathrm{eff}}$. (b) Refractive index $n_1$ of the alumina deduced from the measured effective refractive index for the ${\mathrm{TM}}_0$ modes (black points). The white line indicates the mean value of $n_1$ in the range of 5–20 GHz and the gray bar the standard deviation. Note that some data points below 5 GHz are outside of the displayed range.

Figure 5

(a) Measured WF of a ${\mathrm{TM}}_0$ resonance at $9.146$ GHz. The excitation antenna was placed at the midpoint of the left side. See Fig. 3 for the color scale. (b) Corresponding filtered WF. (c) Corresponding momentum distribution. The outer white circle indicates $k{n}_{\mathrm{eff}}\left({\mathrm{TM}}_0\right)$ and the inner one $k{n}_{\mathrm{eff}}\left({\mathrm{TM}}_1\right)$. See Fig. 3 for the color scale. (d) Filtered momentum distribution. The dashed white circle indicates the filter radius $k_{\mathrm{filt}}$.

Figure 6

Measured WFs (left subpanels) and corresponding model WFs (right subpanels). See Fig. 3 for the color scale. The corresponding measured resonance frequencies (left subpanels) and quantum numbers (right subpanels) are indicated. The WFs presented in (a)–(c) have $(--)$ symmetry and were excited by the fixed antenna placed $a/4$ above the lower left corner, the WFs in (d)–(f) have $(-+)$ symmetry and were excited by the fixed antenna placed at the upper left corner, and the WFs in (g)–(i) have $(++)$ symmetry and were excited by the fixed antenna placed in the middle of the resonator.

Figure 7

(a) Symmetry ratios $|C_{s_1{s}_2}{|}^2$ of resonant states unambiguously assigned to $(m_x,m_y,s_1{s}_2)$ modes versus the frequency. The data set comprises modes belonging to the $A_{1,2}$ and $B_{1,2}$ representations as well as to the $E$ representation with $(-+)$ symmetry, the latter being excited by an antenna at the corner of the resonator. (b) Symmetry ratios of the ${\mathrm{TM}}_0$ modes belonging to the $E$ representation where the excitation antenna was placed at the midpoint of the left edge [cf. Fig. 2]. The different symmetry ratios are marked by $+$ for $|C_{y+}{|}^2$, $\circ$ for $|C_{x-}{|}^2$, $\times$ for $|C_{-+}{|}^2$, and $\square$ for $|C_{+-}{|}^2$.

Figure 8

Overview of the identified measured resonances. The angle of incidence ${\alpha }_{\mathrm{inc}}$ of the corresponding set of classical trajectories is plotted versus the measured resonance frequency. The different symbols indicate the symmetry class, marked by $\square$ for $(++)$, $\times$ for $(--)$, and $\circ$ for $(-+)$. The quantum numbers $(m_x,m_y)$ of some resonances are indicated. The horizontal dotted line indicates the series of resonances with transverse quantum number $p=0.5$, and the vertical dotted line indicates the series with longitudinal quantum number $m=35$. The solid line indicates the critical angle for TIR, ${\alpha }_{\mathrm{crit}}$.

Figure 9

Difference between measured and calculated resonance frequencies versus the measured frequency. The gray scale indicates the overlap between the experimental and model WFs. The solid line is a linear fit to $\Delta f$.

Figure 10

Examples of POs in the square billiard. (a) $(1,0)$ orbit, which is also called Fabry-Pérot orbit, (b) $(1,1)$ or diamond orbit, (c) $(2,1)$ orbit, and (d) $(3,2)$ orbit. The angles of incidence of an orbit on the edges perpendicular to the $x$ ($y$) axis are ${\chi }_x$ (${\chi }_y$).

Figure 11

Length spectrum $|\widehat{\rho }\left(\ell \right)|$ and FT of the trace formula for the dielectric square resonator in the length regimes (a) $\ell =0.5$–$3.5$ m and (b) $\ell =3.5$–$6.5$ m. Each panel is divided into two parts. The solid line in the lower parts is the experimental length spectrum. In the upper parts, the gray line is the FT of the trace formula, Eq. (28), and the solid line is the length spectrum deduced from the set of modes that are confined by TIR. Note the different scales of the top and bottom parts. The arrows indicate the lengths ${\ell }_{\mathrm{peak}}\approx {\ell }_{\mathrm{po}}$ according to Eq. (32) of the POs labeled by their indices $(n_x,n_y)$. The dotted arrows denote those POs not confined by TIR. Only the POs that are clearly visible in the experimental length spectrum are indicated in the bottom parts.

Figure 12

Difference $\Delta {\ell }_{\mathrm{peak}}$ between the peak positions of the experimental and the semiclassical length spectrum versus the peak position ${\ell }_{\mathrm{peak}}^{\mathrm{scl}}$. The solid line is a linear fit.

Figure 13

Relative peak amplitudes ${\mathcal{A}}_{\mathrm{expt}}/{\mathcal{A}}_{\mathrm{scl}}$ for the POs observed in the experimental length spectrum versus their angle of incidence ${\chi }_{\mathrm{po}}$. The indices of the POs are indicated as $(n_x,n_y)$.