## Appendix A
### Scenario 2  
\[ \begin{array}{rl}\mathbf{E}_{\mathrm{tr}}&=\mathrm{T}_{1}\mathbf{E}_{\mathrm{ inc}}+\mathrm{T}_{1}\Big{(}\mathrm{e}^{-\mathrm{j2 dk}_{0}}\mathrm{R}_{2} \mathrm{R}_{1}\Big{)}\mathbf{E}_{\mathrm{inc}}+\cdots+\mathrm{T}_{1}\Big{(} \mathrm{e}^{-\mathrm{j2 dk}_{0}}\mathrm{R}_{2}\mathrm{R}_{1}\Big{)}^{\mathrm{n }}\mathbf{E}_{\mathrm{inc}}\\ &\qquad=\mathrm{T}_{1}\Big{(}1+\Big{(}\mathrm{e}^{-\mathrm{j2 dk}_{0}}\mathrm{ R}_{2}\mathrm{R}_{1}\Big{)}^{1}+\cdots+\Big{(}\mathrm{e}^{-\mathrm{j2 dk}_{0}} \mathrm{R}_{2}\mathrm{R}_{1}\Big{)}^{\mathrm{n}}\Big{)}\mathbf{E}_{\mathrm{inc }}\end{array} . \tag{10}\]  
Equation (10) is a geometric series with a common ratio \(\mathrm{q}=\mathrm{e}^{-\mathrm{2 dk}_{0}}\mathrm{R}_{2}\mathrm{R}_{1}\). Since \(\big{|}\mathrm{q}\big{|}<1\), (10) converges to the following expression for \(\mathrm{n}\rightarrow\infty\) :  
\[\mathbf{E}_{\mathrm{tr}}=\mathrm{T}_{1}\frac{1}{1-\mathrm{e}^{-\mathrm{j2 dk}_{ 0}}\mathrm{R}_{2}\mathrm{R}_{1}}\mathbf{E}_{\mathrm{inc}}. \tag{11}\]  
with \(\mathrm{R}_{2}=-1\), (11) becomes:  
\[\mathbf{E}_{\mathrm{tr}}=\mathrm{T}_{1}\ \frac{1}{1+\mathrm{e}^{-\mathrm{j2 dk}_{0} }\mathrm{R}_{1}}\mathbf{E}_{\mathrm{inc}}. \tag{12}\]  
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