# Multilayer Bolometric Structures for Efficient Wideband Communication Signal Reception
## 2 Modelling of Electromagnetic Wave Propagation in Periodic Heterostructures  
It is well known that the energy levels of a quantum system split when moving from a potential well to a pair of connected identical wells. In Dirac comb-type systems with many potential wells, typically relatively broad energy zones of the total energy levels appear to exist. The quantum mechanical analogy tells us that it is possible to obtain relatively wide absorption bands instead of narrow resonance peaks by replacing one set of \"bolometer + dielectric resonator\" layers with several such sets [49; 50; 51; 52; 53; 54; 55]. Such a structure with allowed and forbidden spectral intervals for electromagnetic wave propagation can be considered as a photon crystal. Previous studies have shown that it is challenging for the radio frequency range to consider a pair of bolometric and dielectric layers as a potential pit and a high barrier. It is all the more important to verify the assumption made from general considerations by an accurate calculation for electromagnetic fields and to compare the results of the analysis with experimental data. Our theoretical investigations of the propagation of the electromagnetic wave through the \"semiconductor-dielectric\" heterostructure embedded in the rectangular waveguide (we plan to consider a sequence of regions with different transparency) are similar to those applied in [41]. We assume that only the lowest \(H_{10}\) mode propagates in the waveguide:the electric field vector has only a transverse component, while the magnetic field vector is characterized by both transverse and longitudinal components [44]. The analysis of the electromagnetic field propagation should be carried out using the Helmholtz equation for the electric field strength \(E_{\omega}\)[43; 44]:  
\[\frac{\partial^{2}E_{\omega}}{\partial z^{2}}+k_{\perp}^{2}E_{\omega}=0. \tag{1}\]  
Here, the \(z\)-axis is directed along the waveguide, \(\omega\) is the angular frequency of the radiation, and  
\[k_{\perp}^{2}=\frac{\omega^{2}}{c^{2}}\bigg{(}\varepsilon(z)-\frac{\omega_{c }^{2}}{\omega^{2}\varepsilon(z)}\bigg{)} \tag{2}\]  
is the transverse wave number, which depends on the permittivity profile along the z-axis \(\varepsilon(z)\); \(c\) is the speed of light in vacuum, and \(\omega_{c}\) is the cut-off frequency in the waveguide. For the empty waveguide, it is given by the expression \(\omega_{c}=\pi c/l\), where \(l\) is the internal transverse size of the waveguide. This cut-off frequency represents the minimum value of the wave frequency that can be transmitted in the given waveguide. In this paper, we consider different configurations of the sequence of dielectric and conductive layers, including the two-layer and four-layer periodic structures. Specifically, the permittivity profiles of the studied heterostructures are written in the following forms (the electromagnetic wave is incident from the right):  
\[\varepsilon(z)=\left\{ \begin{array}{c}\varepsilon_{air},z\leq 0,\\ \varepsilon_{p},0<z\leq d_{1},\\ \varepsilon_{d},d_{1}<z\leq a+d_{1},\\ \varepsilon_{air},a+d_{1}<L_{max}\end{array} \right. \tag{3}\]  
\[\varepsilon(z)=\left\{ \begin{array}{c}\varepsilon_{air},\quad\quad z\leq 0,\\ \varepsilon_{p},\quad 0<z\leq d_{1},\\ \varepsilon_{d},\quad d_{1}\leq z\leq a+d_{1},\\ \varepsilon_{p},\quad\quad a+d_{1}<z\leq a+d_{1}+d_{2},\\ \varepsilon_{d},a+d_{1}+d_{2}<z\leq 2a+d_{1}+d_{2},\\ \varepsilon_{air},\quad\quad 2a+d_{1}+d_{2}<L_{max}\\ \cdot\end{array} \right. \tag{4}\]  
\[\varepsilon(z)=\left\{ \begin{array}{c}\varepsilon_{air},\quad\quad z\leq 0, \\ \varepsilon_{p},\quad 0<z\leq d_{2},\\ \varepsilon_{d},\quad d_{2}\leq z\leq a+d_{2},\\ \varepsilon_{p},\quad a+d_{2}<z\leq a+d_{1}+d_{2},\\ \varepsilon_{d},a+d_{1}+d_{2}<z\leq 2a+d_{1}+d_{2},\\ \varepsilon_{air},\quad\quad 2a+d_{1}+d_{2}<L_{max}\\ \cdot\end{array} \right.