Distribution of Reflection Coefficients in Absorbing Chaotic Microwave Cavities

The distribution of reflection coefficients $P(R)$ for chaotic microwave cavities with time-reversal symmetry is investigated in different absorption and antenna coupling regimes. For all regimes the agreement between experimental distributions and random-matrix theory predictions is very good, provided both the antenna coupling $T_a$ and the wall absorption strength $T_w$ are taken into account in an appropriate way. These parameters are determined by independent experimental quantities.

Figure 1

Sketch of the microwave billiards. (left panel) Half Sinai billiard ($a_1=43\mathrm{c}\mathrm{m}$, $b_1=23.7\mathrm{c}\mathrm{m}$, height $h_1=7.8\mathrm{m}\mathrm{m}$). The half disk ($d=12\mathrm{c}\mathrm{m}$) was moved along $a_1$. For the second set of measurements one wall was coated with an absorber. (right panel) Disordered billiard ($a_2=34\mathrm{c}\mathrm{m}$, $b_2=24\mathrm{c}\mathrm{m}$, height $h_2=8\mathrm{m}\mathrm{m}$) [16]. The billiards are drawn at scale.

Figure 2

Typical absorption spectra for different absorption regimes: (a) Weak absorption and weak coupling—disordered billiard, (b) intermediate absorption and coupling—half Sinai billiard, and (c) strong absorption and almost perfect coupling—half Sinai billiard with absorber.

Figure 3

Average reflection $⟨R⟩$ as a function of the frequency $\nu$ for half Sinai (triangles), half Sinai with absorber (squares), and disordered (diamonds) cavities. The shaded boxes mark the cases subjected to a detailed analysis. The frequency average is $\Delta \nu =450\mathrm{M}\mathrm{H}\mathrm{z}$.

Figure 4

Distribution of the reflection coefficient $R$ for the half Sinai billiard in the strong absorption limit (13.5–15.5 GHz). The solid line corresponds to Eq. (2) and the dotted one to the numerical simulation.

Figure 5

Distribution of the reflection coefficients, $P(R)$, for weak coupling strength and small absorption (3.5–4.0 GHz), obtained from the Lorentzian fit corrected spectrum (for details see text). For comparison the distribution obtained from the uncorrected spectrum is shown in light grey. The inset enlarges the region of $R$ between 0.95 and 1.005. The dotted line is obtained from the random-matrix simulation.

Figure 6

Distribution of the reflection coefficients $P(R)$ for the intermediate absorption regime (half Sinai billiard). The data were collected at different frequency windows marked in Fig. 3. The dashed lines stand for the random-matrix simulations. In (a) the experimental distribution was obtained from the Lorentzian fit corrected spectrum.