Direct Processes in Chaotic Microwave Cavities in the Presence of Absorption

We quantify the presence of direct processes in the $S$ matrix of chaotic microwave cavities with absorption in the one-channel case. To this end the full distribution $P_S(S)$ of the $S$ matrix, i.e., $S=\sqrt{R}{e}^{i\theta }$, is studied in cavities with time-reversal symmetry for different antenna coupling strengths $T_a$ or direct processes. The experimental results are compared with random-matrix calculations and with numerical simulations including absorption. The theoretical result is a generalization of the Poisson kernel. The experimental and the numerical distributions are in excellent agreement with theoretical predictions for all cases.

Figure 1

Sketch of the model for scattering with direct processes and absorption. In (a) $S_0$ describes the scattering of a billiard with absorption $\gamma$ but perfect coupling to the lead ($T_a=1$). In (b) we associate the $S$ matrix $S_a$ given by Eq. (5) to the barrier that describes the nonideal coupling. The resulting $S$ matrix of the system with absorption $\gamma$ and coupling $T_a$ can be written in terms of $S_0$ and $S_a$.

Figure 2

The phase $\theta$ of the $S$ matrix for different coupling and absorption regimes: (a) $\gamma =0.56$, $T_a=0.116$ (weak absorption, weak coupling), (b) $\gamma =2.42$, $T_a=0.754$, (c) $\gamma =8.40$, $T_a=0.989$, and (d) $\gamma =48.00$, $T_a=0.998$ (strong absorption, nearly perfect coupling). On the right-hand side, the Argand diagrams show $\mathrm{Im}(S)$ versus $\mathrm{Re}(S)$.

Figure 3

Distribution of the $S$ matrix for the same ranges of coupling and absorption of Fig. 2. The upper, central, and lower rows correspond to experiment, numerics, and theory, respectively. Note the change of scales for $R$ and $\theta$.

Figure 4

Experimental distribution $P_{\theta ,T}(\theta )$ (histogram) for the same regimes of coupling and absorption as given in Fig. 2. Additionally, the numerical (crosses) and the theoretical (solid line) is plotted as well. The dotted curves correspond to the Poisson kernel without absorption [see Eq. (1)].