## 3 Theory  
The longitudinal component of magnetic field associated with a TE waveguide mode is described by the following equation:  
\[H_{zi}=ib_{i}^{\pm}\cos(k_{x}x)\cos(k_{y}y)\exp(\pm iK_{i}z) \tag{1}\]  
where \(+\) and \(-\) correspond to the waves propagating along waveguide axis \(z\) in positive and negative directions, respectively, subscript \(i\) = 0, 1, 2, 3 labels vacuum, substrate, PyC film and vacuum, respectively, \(x\), \(y\) are cross sectional coordinates of the rectangular waveguide, \(k_{x}=\frac{\pi n}{a},k_{y}=\frac{\pi m}{b}\), where \(a\), \(b\) are the lateral waveguide dimensions, \(n\), \(m\) are integers, \(K_{i}=\sqrt{k^{2}\varepsilon_{i}-\left(\frac{\pi n}{a}\right)^{2}-\left(\frac {\pi m}{b}\right)^{2}}\) is the longitudinal wavenumber in the \(i\)-th medium, \(b_{i}\) is mode amplitudes in \(i\)-th medium to be found from boundary conditions. In particular, if the waveguide supports \(\mathrm{TE}_{1,0}\) mode, \(n\) = 1, \(m\) = 0 and \(K_{i}=\sqrt{k^{2}\varepsilon_{i}-\left(\frac{\pi}{a}\right)^{2}}\). Let us consider the \"quartz + PyC\" geometry when the EM wave is incident on the quartz substrate. Boundary conditions in this geometry have the following form [7]:  
\[\left\{ \begin{array}{c}b_{0}^{+}+b_{0}^{-}=b_{1}^{+}+b_{1}^{-}\\ K_{0}(b_{0}^{+}-b_{0}^{-})=K_{1}(b_{1}^{+}-b_{1}^{-})\\ b_{1}^{+}e^{iK_{1}l_{1}}+b_{1}^{-}e^{-iK_{1}l_{1}}=b_{2}^{+}+b_{2}^{-}\\ K_{1}\big{(}b_{1}^{+}e^{iK_{1}l_{1}}-b_{1}^{-}e^{-iK_{1}l_{1}}\big{)}=K_{2}(b_ {2}^{+}-b_{2}^{-})\\ b_{2}^{+}e^{iK_{2}l_{2}}+b_{2}^{-}e^{-iK_{2}l_{2}}=b_{3}^{+}\\ K_{2}\big{(}b_{2}^{+}e^{iK_{2}l_{2}}-b_{2}^{-}e^{-iK_{2}l_{2}}\big{)}=K_{0}b_ {3}^{+}\end{array} \right. \tag{2}\]  
Here \(l_{1}\) and \(l_{2}\) are the thicknesses of substrate and PyC film, respectively. By solving Equation (2) under assumptions \(K_{2}l_{2}<<1\), \(\mathbf{Im}\{\varepsilon_{2}\}>>1\) and \(\left|K_{2}\right|^{2}>>K_{1}K_{0}\) we arrive at the following equations for the complex transmission ( \(t=b_{3}^{+}/b_{0}^{+}\) ) and reflection ( \(r=b_{0}^{-}/b_{0}^{+}\) ) coefficients in the \"quartz + PyC\" geometry:  
Figure 2: The waveguide measurements setup in the “quartz + PyC” geometry. \[t_{qp} =T_{0}\frac{e^{i\phi}}{1+\delta-\sqrt{R_{0}}\left(\sqrt{R_{0}}+ \delta\right)e^{2i\phi}} \tag{3}\] \[r_{qp} =\frac{\sqrt{R_{0}}\left(1+\delta\right)-\left(\sqrt{R_{0}}+ \delta\right)e^{2i\phi}}{1+\delta-\sqrt{R_{0}}\left(\sqrt{R_{0}}+\delta\right) e^{2i\phi}}\]  
where \(\phi=K_{1}l_{1}\) is the phase accrued by the wave after one pass through the substrate, \(R_{0}=\left|K_{1}-K_{0}\right|^{2}/\left|K_{1}+K_{0}\right|^{2}\) and \(T_{0}=4K_{1}K_{0}/\left|K_{1}+K_{0}\right|^{2}\) are reflectance and transmittance of the vacuum-substrate interface, respectively, \(\delta=l_{2}/l_{\sigma}\) where \(l_{\sigma}=c^{2}(K_{0}-K_{1})/4\pi\omega\sigma\) is the characteristic length that describe EM properties of the two-layer structure. In order to clarify the meaning of \(l_{\sigma}\) it is instructive to consider the interaction of the EM with a free standing thin conductive film. In such a case the amplitude reflection/transmission coefficients can be obtained from Equation (2) at \(K_{0}=K_{1}\) and \(l_{1}=0\). By substituting \(T_{0}=1\) and \(R_{0}=\phi=0\) into Equation (3) we arrive at the following equations for the transmittance and reflectance of the conductive film with thickness \(l_{2}\):  
\[T =\frac{l_{\sigma}^{2}}{\left(l_{2}+l_{\sigma}\right)^{2}}, \tag{4}\] \[R =\frac{l^{2}}{\left(l_{2}+l_{\sigma}\right)^{2}},\] \[A =1-T-R=\frac{2l_{2}l_{\sigma}}{\left(l_{2}+l_{\sigma}\right)^{2}}\]  
One can observe from Equation (4) that at \(l_{2}=l_{\sigma}\) absorption losses in the film are maximum and \(A=0.5\). In the \"PyC + quartz\" geometry, the amplitude transmission coefficient (and, hence, transmittance) is expectedly the same as that in the \"quartz + PyC\" geometry.