## 3 Experimental Procedure  
The experiments were performed using standard coaxial cables with different lengths and identical characteristic impedance \(Z_{c}\), combined with metallic T-shaped connectors. The validity of our results is subject to the requirement that the cross section of the waveguide is negligible compared to their length and to the propagation wavelength. Indeed, it is well established that within such assumption and for frequencies below the cut-off frequency (\(f_{c}\)), only transverse electromagnetic modes (TEM) can propagate. The threshold frequency \(f_{c}\) below which the propagation becomes monomode, is given by [76]  
\[f_{c}=190.85/(\varepsilon_{r}(d_{out}+d_{in}))\ (GHz), \tag{12}\]  
where \(d_{out}=3\pm 0.1\) mm is the dielectric diameter (i.e., the interior diameter of the outer conductor), \(d_{in}=0.8\pm 0.02\) mm is the diameter of the inner conductor and \(\varepsilon_{r}\) is the dielectric permittivity (\(\varepsilon_{r}=2.3\)) of the polyethylene between the two conductors, which corresponds to a nominal propagation speed of \(0.66c\). For \(d_{out}=3\) mm and \(d_{in}=0.8\) mm, one obtains \(f_{c}=33.11\) GHz which is sufficiently above the domain of frequencies considered in this work.  
The characteristic impedance \(Z_{c}\) of the cables is given by [76]  
\[Z_{c}=138\log(d_{out}/d_{in})/\sqrt{\varepsilon_{r}}. \tag{13}\]  
From the above values of \(d_{out}\) and \(d_{in}\), one can find the characteristic impedance \(Z_{c}=50\pm 3\,\Omega\). The electromagnetic impedance at the free ends of the resonators of lengths \(d_{2}\) (Figure 1a) and \(d_{3}\) (Figure 1b), may be adapted in order to realize the boundary condition \(E=0\) (short-circuit) or \(H=0\) (open-circuit).  
The experimental measurements (including the amplitude and the phase of the transmission coefficients) were measured in the frequency range 0-200 MHz by means of a broadband vector network analyzer (VNA) Agilent PNA-X N5242A. The VNA allows accurate measurements of the four S parameters with an error less than 0.2%. The uncertainty on the measurements of the amplitude and phase are around 0.5 mV and 0.01\({}^{\circ}\) respectively.  
The attenuation inside the coaxial cables was simulated by introducing a relative dielectric permittivity \(\varepsilon\) (\(\varepsilon=\varepsilon^{\prime}-j\varepsilon^{\prime\prime}\)) where \(\varepsilon^{\prime}\) and \(\varepsilon^{\prime\prime}\) are the real and imaginary parts of the dielectric permittivity respectively. The attenuation coefficient \(\alpha^{\prime\prime}\) can be expressed as \(\alpha^{\prime\prime}=\varepsilon^{\prime\prime}\omega/c\). From the data supplied by the manufacturer of the coaxial cables in the frequency range of 0 to 200 MHz, one can deduce a fitted expression of \(\alpha^{\prime\prime}\) such as \(\ln(\alpha^{\prime\prime})=\gamma+\delta\ln(\omega)\), where \(\gamma\) and \(\delta\) are two constants. This fitting procedure enables to deduce a useful expression for \(\varepsilon^{\prime\prime}\) versus the frequency, namely \(\varepsilon^{\prime\prime}=(\frac{f}{f_{0}})^{-0.5}\) where the frequency \(f\) is expressed in Hz and \(f_{0}=9200\) Hz. Further details and a photograph of the experimental setup can be found elsewhere [77; 78].