Experimental examination of the effect of short ray trajectories in two-port wave-chaotic scattering systems

Predicting the statistics of realistic wave-chaotic scattering systems requires, in addition to random matrix theory, introduction of system-specific information. This paper investigates experimentally one aspect of system-specific behavior, namely, the effects of short ray trajectories in wave-chaotic systems open to outside scattering channels. In particular, we consider ray trajectories of limited length that enter a scattering region through a channel (port) and subsequently exit through a channel (port). We show that a suitably averaged value of the impedance can be computed from these trajectories and that this can improve the ability to describe the statistical properties of the scattering systems. We illustrate and test these points through experiments on a realistic two-port microwave scattering billiard.

Figure 1

(Color online) Plot of corrections to the impedance due to (a) one-port with one wall (wall B), (b) one-port with two walls (walls C and D), (c) two-port with one wall (wall B), and (d) two-port with two walls (walls B and C) short trajectories, versus frequency from 6 to 10 GHz inside the 1/4-bow-tie cavity. Shown are the theoretical predictions ($\text{Re}\left[z_{cor}^{\left(L_M\right)}\right]$ and $\text{Im}\left[z_{cor}^{\left(L_M\right)}\right]$) in blue (dashed) curves and experimental data ($\text{Re}\left[z_{cor}\right]$ and $\text{Im}\left[z_{cor}\right]$) in red (solid) curves. The insets show the 1/4-bow-tie billiard with thicker lines representing walls covered by microwave absorbers, the ports are shown as circular dots, and a few short trajectories are represented with lines as illustrations.

Figure 2

(Color online) Plot of the loss parameter $\alpha$ and the attenuation parameter $\kappa$ versus frequency inside the empty 1/4-bow-tie cavity.

Figure 3

(Color online) Plot of the absolute values of the impedance correction in the length domain $\epsilon \left(L\right)$ in the cases of (a) one-port with one wall (wall B), (b) one-port with two walls (walls C and D), (c) two-port with one wall (wall B), and (d) two-port with two walls (walls B and C) short trajectories, versus length from 0 to 120 cm inside the 1/4-bow-tie cavity. Shown are the measured data in red curves with circular symbols and the theory in blue dashed curves with triangular symbols. The insets show the 1/4-bow-tie billiard with labeled trajectories that show the sources of error (E1, E2, and E3).

Figure 4

(Color online) Plot of the impedance due to the trajectories of four side-walls, versus frequency from 6 to 8 GHz inside the empty 1/4-bow-tie cavity. Shown are (a) the real part of $Z_{11}$, (b) the imaginary part of $Z_{11}$, (c) the real part of $Z_{12}$ and (d) the imaginary part of $Z_{12}$. The three curves are the measured impedance $Z$ (red thinner), the theoretical impedance $Z^{\left(L_M\right)}$ (blue dashed), as well as the radiation impedance $Z_R$ (black).

Figure 5

(Color online) Plot of the smoothed impedance versus frequency from 6 to 8 GHz. Shown are, (a) for $Z_{11}$ and (b) for $Z_{12}$, the real (three upper curves) and the imaginary part (three lower curves) of the smoothed impedance for the theory ($Z^{\left(L_M\right)}$ with $L_M=200\text{cm}$, blue dashed) and the experiment (red solid), as well as the measured (un-smoothed) radiation impedance of the port ($Z_R$, black thick).

Figure 6

(Color online) Plot of the RMS errors for (a) the one-port experiment and (b) the two-port experiment. Shown are ${\zeta }_{\mu }$ for the mean of the real part (red circles) and the imaginary part (blue triangles) of the normalized impedance difference and ${\zeta }_{\sigma }$ for the difference of standard deviations (green rectangles), versus short-trajectory corrections with the maximum length from 0 to 200 cm. Data are taken for a single realization of the bow-tie cavity.

Figure 7

(Color online) Plot of the average impedance versus frequency from 6 to 8 GHz. Shown are, (a) for $Z_{11}$ and (b) for $Z_{12}$, the real (three upper curves) and the imaginary part (three lower curves) of the average impedance for the theory ($Z_{avg}^{\left(L_M\right)}$ with $L_M=200\text{cm}$, blue dashed) and the configuration average experiment ($⟨Z⟩$ red solid), as well as the measured radiation impedance of the ports ($Z_R$, black thick). Plot (c) shows the real part of the measured impedance $Z_{12}$ (black thin) in a single realization comparing with $Z_{avg,12}^{\left(L_M\right)}$ and $⟨Z_{12}⟩$. Inset: The wave-chaotic two-dimensional cavity with perturbers and two ports.

Figure 8

(Color online) Plot of probability distributions of (a) the real part and (b) the imaginary part of the normalized impedance eigenvalues in the frequency range $6.8\text{GHz}\sim 7.0\text{GHz}$, and (c) the real part and (d) the imaginary part of the normalized impedance in the frequency range $11.0\text{GHz}\sim 11.2\text{GHz}$. The blue (squares) curve is $i{\xi }^{\left(L_M\right)}$ that is normalized with the short-trajectory-corrected impedance, the green (circles) curve is $i{\xi }_0$ that is normalized with the radiation impedance only, and the black curve shows the PDF generated from numerical RMT.

Figure 9

(Color online) Plot of (a) an example probability distribution of the phase ${\phi }_s$ of eigenvalues of the normalized scattering matrix $s^{\left(L_M\right)}$ from $0\sim 2\pi$, taken over the 11–11.5 GHz frequency window inside the 1/4-bow-tie cavity of 100 realizations with perturbers. The blue (squares) curve is from data normalized with $L_M=200\text{cm}$, the green (circles) curve is from data normalized with the radiation impedance only, and the black line shows the perfectly uniform distribution for comparison. Plot of (b) the average RMS error of distributions of ${\phi }_s$, where the normalized scattering matrix $s^{\left(L_M\right)}$ is calculated from impedance normalized with the radiation impedance only (green circles), with the measured ensemble averaged impedance (pink stars), or with theoretical impedance with $L_M=50\text{cm}$ (red triangles), 100 cm (orange invert triangles), up to 200 cm (blue squares), versus frequency window size from 10 MHz to 1.0 GHz.