Resonance Widths in Open Microwave Cavities Studied by Harmonic Inversion

From the measurement of a reflection spectrum of an open microwave cavity, the poles of the scattering matrix in the complex plane have been determined. The resonances have been extracted by means of the harmonic inversion method. By this, it became possible to resolve the resonances in a regime where the linewidths exceed the mean level spacing up to a factor of 10, a value inaccessible in experiments up to now. The obtained experimental distributions of linewidths were found to be in perfect agreement with predictions from random matrix theory when wall absorption and fluctuations caused by couplings to additional channels are considered.

Figure 1

Modulus square $|\widehat{S}(t){|}^2$ of the Fourier transform of the part of the spectrum $S(\nu )$ ranging from 14.5 to 16.5 GHz (see inset). The dashed horizontal line indicates the noise level of the Fourier transform. The dotted vertical line shows the cutoff time applied for the harmonic inversion.

Figure 2

Real and imaginary part of the spectrum in the regime of overlapping resonances. Additionally, the reconstructed spectrum using the resonances identified by the harmonic inversion is shown in the dashed line. The vertical lines indicate the positions of the resonances and the horizontal lines the corresponding widths at half maximum. On top of each figure, the difference between the measured spectrum and the calculated one is shown. The mean level spacing $\Delta$ is marked by a horizontal bar.

Figure 3

Normalized widths $\Gamma /\Delta$ for 14.7 to 15.7 GHz with $T_t=7.00$ and $T_a=0.989$. The horizontal dashed line corresponds to the width coming from the total absorption [${\Gamma }_t=T_t/(4\pi )$], whereas the dotted line (only visible in the inset) corresponds to the constant part of the absorption [${\Gamma }_{\mathrm{const}}=T_{\mathrm{const}}/(4\pi )$]. The inset is an enlargement of the small width region.

Figure 4

Normalized width distribution in the frequency range 4 to 5 GHz with $T_a=0.116$ and $T_t=0.75$. The black histogram was obtained from the harmonic inversion, the light one from a Lorentzian fitting procedure of the isolated resonances. The dashed curve corresponds to the theoretical prediction from Eq. (6) shifted by a constant offset ${\Gamma }_t=T_t/(4\pi )$ to account for absorption in the walls. The solid line in addition takes into account the fluctuations using $T_{\mathrm{fluc}}=0.6$ and $N_c=10$ (see the text for details). The inset shows a comparison between a random matrix simulation and the theoretical prediction using the same values for $T_a$, $T_t$, $T_{\mathrm{fluc}}$, and $N_c$ as for the experimental data.

Figure 5

Same as Fig. 4, but for the frequency range 14.7 to 15.7 GHz with $T_a=0.989$ and $T_t=7.0$. Additionally, the asymptotics of the theoretical expectation for perfect coupling is plotted as a dotted curve (see text). For the simulations and the theoretical curve, $T_{\mathrm{fluc}}=6.4$ and $N_c=20$ have been used.