## 2 Materials and Methods
### Coplanar Interleaved Dielectric Materials  
The coupling between microstrip lines has also been widely studied [48; 49; 50; 51], and the coupling coefficient between two microstrip lines, as shown in Figure 3a, is given by:  
\[\text{k}=\frac{1}{2}\frac{\Gamma_{e}-\Gamma_{o}}{1-0.5\ \Gamma_{g}(\Gamma_{e}+ \Gamma_{o})} \tag{5}\]  
where the \(\Gamma_{e}\) and \(\Gamma_{o}\) are the even and odd mode reflection coefficients, respectively, and the \(\Gamma_{g}\) is the reflection coefficient at the receiver element. The coupled impedance is given by:  
\[Z_{0e}=\frac{60\pi}{\sqrt{\varepsilon_{r}}}\frac{1}{\Gamma_{e}/(\varepsilon_{ 0}\varepsilon_{r})} \tag{6}\]  
\[Z_{0o}=\frac{60\pi}{\sqrt{\varepsilon_{r}}}\frac{1}{\Gamma_{o}/(\varepsilon_{ 0}\varepsilon_{r})} \tag{7}\]  
where \(\varepsilon_{r}\) is the relative permittivity of the dielectric material, and \(C_{e}\) and \(C_{o}\) are the even and odd capacitance, respectively. The even mode is related to a magnetic wall between the microstrip lines, and the odd mode has an electric wall. The coupling is dependent on the wave propagation constant and the type of loading. For the proposed arrangement, the frequencies, and therefore the wavelength, differ, which will reduce the coupling between the lines. Notice that the mutual capacitances are also dependent on the relative dielectric material [52].  
Figure 3: (**a**) The models for analyzing the coupling between two microstrips depending on the dielectric material and line type, for a uniform dielectric material (**b**) double straight line (uniform-DSL), (**c**) one straight line and one gap line (uniform-OSLOGL), (**d**) double gapped lines (uniform-DGL), (**e**) (interleaved-DSL), (**f**) (interleaved-OSLOGL) and (**g**) (interleaved-DGL).  
We applied EM simulations for an analysis of two microstrip lines to determine the coupling between them in different scenarios, following the models shown in Figure 3. For this analysis, we modeled two microstrips with similar lengths and strip widths. The analysis consisted of checking the effects of the gap size and dielectric material. The first case used a uniform dielectric material for both lines, consisting of two straight lines (uniform-DSL), one straight line, one gapped line (uniform-OSLOGL), or two gapped lines (uniform-DGL). The second case used interleaved dielectric materials, for which one line had a dielectric material with lower relative permittivity than the other. In this case, the line structure was that same as before, with two straight lines (interleaved-DSL), one straight line and one gapped line (interleaved-OSLOGL), or two gapped lines (interleaved-DGL). The models for this analysis are shown in Figure 3b-g. The separation between the microstrips was 20 mm. The materials used for the simulations were \(\varepsilon_{1}\) Rogers RT/duroid 6006 and \(\varepsilon_{2}\) Rogers RT/duroid 6010.2 LM, with a relative permittivity of 6.45 and 10.7, respectively. The S21 parameters and coupling parameters are shown in Figure 4. These results indicate that the best decoupling was achieved by using the double gapped and interleaved dielectric material, as compared to the double straight lines, which showed higher coupling than the other configurations.