## 2 Materials and Methods
### Impedance in ESICL  
The impedance \(Z_{0}\) of an ESICL depends on the dimensions of the cross section (\(A\), \(B\), \(t\) and \(w\)). The following approximation for the impedance can be found in Reference [34]:  
\[Z_{0}=\frac{\eta_{0}}{4\sqrt{\epsilon_{r}}}\left[\frac{1}{\frac{w/B}{B/t-1}+ \frac{2}{\pi}\ln\left(\frac{1}{1-t/B}+\coth\frac{\pi A}{2B}\right)}\right](\Omega) \tag{5}\]  
In the ESICL, the heights of the inner and outer conductor are fixed by the height of the substrate layers. In this work, all substrate layers will be Rogers 4003C substrates of height \(h=0.813\) mm plus electrodeposited copper foils of 17 \(\upmu\)m. After taking into account the thickness of the soldering paste \(t=0.866\) mm and \(B=2.618\) mm. From Equation (5), it can be derived that \(Z_{0}\) does not change significantly with the width of the outer conductor \(A\). Figure 4 shows the variation of \(Z_{0}\) with \(A\) using Equation (5), and fixing the other dimensions of the cross section. It can be observed that for \(A\geq 6\) mm, the impedance is almost constant. So \(A\) is fixed to 6 mm, and once \(A\), \(B\) and \(t\) are fixed, \(Z_{0}\) depends only on the width of the inner conductor \(w\). Figure 5 shows the variation of \(Z_{0}\) with \(w\), while \(A\), \(B\) and \(t\) are fixed. This impedance was calculated with the analytic formula of Equation (5), and also with a commercial electromagnetic solver (computer simulation technology (CST) Studio). Both results are not in good agreement, so the most accurate results of the full-wave commercial simulator were considered. It must be noted that in order to implement the stop band filter we need an impedance variation along the propagation direction as shown in Figure 3. That is, we must be able to change the impedance in a range from approximately 32 to 50 \(\Omega\). As shown in Figure 5, by changing \(w\) from 1 to 5.5 mm we can change \(Z_{0}\) in a range that goes from 17 to 66 \(\Omega\), which is enough for implementing the stop band filter. Using the relationship between \(Z_{0}\) and \(w\) obtained with CST and shown in Figure 5, we can now derive the variation of the inner conductor width \(w(z)\) needed to achieve the variation of the impedance \(Z_{0}(z)\) of Figure 3 that provides the desired coupling coefficient \(K(z)\), and therefore, the desired stop band response. Figure 6 shows this variation \(w(z)\) for three different cases. In the first place, the value of \(w(z)\) derived using the analytic formula, which has proven to be inaccurate, is plotted with a solid red line. Next, the accurate value of \(w(z)\) computed with the impedance obtained with CST is plotted  
Figure 4: Variation of the impedance (\(Z_{0}\)) of the ESICL as a function of the outer conductor width. \(B=2.618\) mm, \(w=2.2823315\) mm, \(t=0.866\) mm. The analytic formula of Equation (5) was used. in solid black line. And finally, a sinusoidal approximation to the accurate value of \(w(z)\) is plotted in green round marks. This approximation is calculated using the following expression:  
\[w=A_{w}\cos\left(\frac{2\pi z}{Z_{p}}+P\right)+B_{w} \tag{6}\]  
After 450 iterations, an optimization process with the Nelder Mead simplex algorithm [35] determined the optimum values of the parameters \(A_{w}\), \(B_{w}\) and \(P\) for an optimum match of this sinusoidal approximation with the accurate value of \(w(z)\). It gave \(A_{w}=0.8305\), \(B_{w}=2.61103\) and \(P=-3.0817\). This approximation, which matches almost perfectly with the accurate value, was preferred to the accurate one, because it eases the modeling and simulation of the ESICL continuous profile filter in the electromagnetic simulator. Figure 5: Variation of the impedance (\(Z_{0}\)) of the ESICL as a function of the inner conductor width. \(B=2.618\) mm, \(A=6\) mm, \(t=0.866\) mm. Comparison between analytic formula of Equation (5) and simulation with a computer simulation technology (CST) electromagnetic (EM) solver.