Figure 1

(a) The physical dimensions of the quarter bow-tie chaotic microwave resonator are shown along with the position of the single coupling port. Two metallic perturbations are systematically scanned and rotated throughout the entire volume of the cavity to generate the cavity ensemble. (b) The details of the coupling port (antenna) and cavity height $h$ are shown in cross section. (c) The implementation of the radiation case is shown, in which commercial microwave absorber is used to line the inner walls of the cavity to minimize reflections.Reuse & Permissions

Figure 2

(a) Polar contour density plot for the real and imaginary components of the normalized cavity $s$ $[s=\mid s\mid \exp \left(i{\varphi }_s\right)]$ for loss case 0 in the frequency range of $6–9.6\mathrm{GHz}$. The angular slices with the symbols (triangles, circles, squares) indicate the regions where the PDF of $\mid s\mid$ is calculated and shown in (b). Observe that the PDFs of the three regions are essentially identical. (c) The PDF of the phase ${\varphi }_s$ of the normalized scattering coefficient $s$ for two annuli defined by $0⩽\mid s\mid ⩽0.3$ (stars) and $0.3<\mid s\mid ⩽0.6$ (hexagons). Observe that these phase PDFs are nearly uniform in distribution. The uniform distribution is shown by the solid line $[P\left(\varphi \right)=1∕2\pi ]$. This is consistent with the prediction that the $\mid s\mid$ is statistically independent of the phase ${\varphi }_s$ of $s$.Reuse & Permissions

Figure 3

(a) PDF for the un-normalized loss case 0 cavity ${\mid S\mid }^2$ in the frequency range of $6–11.85\mathrm{GHz}$ for two different coupling antenna diameters $2a=0.635\mathrm{mm}$ (circles) and $2a=1.27\mathrm{mm}$ (solid stars). (b) PDF for the normalized cavity ${\mid s\mid }^2$ in the frequency range of $6–11.85\mathrm{GHz}$ for two different coupling antenna diameters $2a=0.635\mathrm{mm}$ (circles) and $2a=1.27\mathrm{mm}$ (solid stars). Note that the disparities seen in (a) on account of the different coupling geometries disappear after normalization. (c) PDF for the un-normalized cavity phase $\left({\varphi }_S\right)$ for loss case 0 in the frequency range of $6–11.85\mathrm{GHz}$ for two different coupling antenna diameters $2a=0.635\mathrm{mm}$ (circles) and $2a=1.27\mathrm{mm}$ (solid stars). (d) PDF for the normalized cavity phase $\left({\varphi }_s\right)$ in the frequency range of $6–11.85\mathrm{GHz}$ for two different coupling antenna diameters $2a=0.635\mathrm{mm}$ (circles) and $2a=1.27\mathrm{mm}$ (solid stars). The normalized phase PDFs for the stars and circles in (d) are nearly uniformly distributed [the gray line in Fig. 4d shows a perfectly uniform distribution, $P\left(\varphi \right)=1∕2\pi$].Reuse & Permissions

Figure 4

PDFs for the (a) real and (b) imaginary parts of the normalized cavity impedance $z$ for a wave chaotic microwave cavity between 6.5 and $7.8\mathrm{GHz}$ with $h=7.87\mathrm{mm}$ and $2a=1.27\mathrm{mm}$, for three values of loss in the cavity (open stars: 0, circles: 1, squares: 3 strips of absorber). Also shown are single parameter, simultaneous fits for both PDFs (solid curves), where the loss parameter ${\tilde{k}}^2∕Q$ is obtained from the variance of the data in (a) and (b).Reuse & Permissions

Figure 5

PDF for the normalized power reflection coefficient $r={\mid s\mid }^2$ on a natural log scale for loss case 0, 1, 3 (stars, circles, and squares, respectively) in the frequency range of $6.5–7.8\mathrm{GHz}$. These are from the same data as used to obtain Fig. 4. Also shown is the prediction of the model in Ref. 17 (solid lines) for $P\left(r\right)$ using the values of ${\tilde{k}}^2∕Q$ obtained from the variances of the distributions in Fig. 4.Reuse & Permissions

Figure 6

Dependence of the average of the cavity power reflection coefficient $\overline{{\mid s^{\prime }\mid }^2}$ on the magnitude of the radiation scattering coefficient $\mid s_{\mathit{rad}}^{\prime }\mid$, for different loss cases (loss case 0: stars; loss case 1: circles; loss case 3: squares). The data are shown for the frequency range of $6.5–7.8\mathrm{GHz}$. Also shown are the numerical simulations from the RMT based upon the ${\tilde{k}}^2∕Q$ values 0.81, 2.4 and 6.5 for loss cases 0, 1, and 3, respectively (solid lines).Reuse & Permissions

Figure 7

(Color online) Polar plot for the cavity scattering coefficient $S=\mathrm{Re}\left(S\right)+i\mathrm{Im}\left(S\right)$ is shown for a frequency range of $6–12\mathrm{GHz}$ for loss case 0 and with a coupling port of diameter $2a=1.27\mathrm{mm}$. The blue trace represents one single rendition of the cavity for a selected position and orientation of the perturbers. Each circular loop represents an isolated resonance. The black trace is the ensemble average ${⟨S⟩}_{100}$ over 100 different locations and orientations of the perturbers within the cavity. The meandering nature of the black trace shows that remnants of the cavity resonances are still present because of the finite number of ensemble averages. The red trace shows the radiation scattering coefficient for the same port. This trace is smooth because the radiation scattering coefficient approximates the cavity boundaries being extended to infinity.Reuse & Permissions

Figure 8

(a) The experimental PDF for the loss case 0 cavity power reflection coefficient $\left({\mid S\mid }^2\right)$ (stars) over a frequency range of $6–7.5\mathrm{GHz}$. Also shown is the numerical estimate $P\left({\mid \tilde{S}\mid }^2\right)$ (solid trace) determined from RMT and the experimentally measured radiation impedance of the port $\left(Z_{\mathit{rad}}\right)$. (b) The experimental PDF for the loss case 0 cavity scattering phase $\left({\varphi }_s\right)$ (stars) over a frequency range of $6–7.5\mathrm{GHz}$. Also shown is the numerical estimate $\mathrm{P}\left({\varphi }_{\tilde{S}}\right)$ (solid trace) determined from RMT and the experimentally measured radiation impedance of the port. We observe good agreement between the measured data and the numerically estimated PDFs. The inset shows the fluctuation in $\mid {⟨S⟩}_{100}\mid$ (stars) over the frequency range of $6–7.5\mathrm{GHz}$, while the solid trace shows the magnitude of the experimentally measured radiation scattering coefficient $\left(\mid S_{\mathit{rad}}\mid \right)$.Reuse & Permissions