## 2 Material Parameters of the Structure  
Let us consider a symmetric microresonator structure formed of two BMs and a resonator cavity separating them, designed to be filled with a controlled active structure (As can be seen Figure 1). We assume that the period of the BMs consists of two layers of isotropic dielectrics SiO\({}_{2}\) and ZrO\({}_{2}\) with permittivities \(\varepsilon_{1}\) and \(\varepsilon_{2}\), and with the same optical thicknesses  
Figure 1: Symmetric MCR structure formed by two BMs and a resonator cavity designed to be filled with a controllable active structure. \(L_{0}=L_{1}\sqrt{\varepsilon_{1}}=L_{2}\sqrt{\varepsilon_{2}}=\lambda_{0}/4\). Each of the Bragg mirrors consists of \(a\) periods \((\text{SiO}_{2}/\text{ZrO}_{2})\). The cavity has a length \(L_{3}=\lambda_{0}\) and permittivity \(\varepsilon_{3}=1\). We also assume that the permittivity of the coating medium and substrate \(\varepsilon_{a}=\varepsilon_{b}=1\). For a microresonator with a free cavity, the transfer matrix that relates the wave field at the input and output of the structure has the form:  
\[G=(N_{1}N_{2})^{a}N_{3}(N_{2}N_{1})^{a}, \tag{1}\]  
where the transfer matrices of individual layers has the form:  
\[N_{j}=\left( \begin{array}{cc}\cos k_{j}L_{j}&i\sqrt{\varepsilon_{j}}\sin k_{ j}L_{j}\\ (i/\sqrt{\varepsilon_{j}})\sin k_{j}L_{j}&\cos k_{j}L_{j}\end{array} \right), \tag{2}\]  
where \(j=1-3\), \(k_{j}=k_{0}\sqrt{\varepsilon_{j}}\) are propagation constants in the respective layers, \(k_{0}=\omega/c\) and \(\omega\) and \(c\) are the frequency and speed of the wave in a vacuum, respectively. The transmission and reflection amplitude coefficients for the entire MCR structure are determined in terms of the matrix elements of the transfer matrix:  
\[r=\frac{G_{11}+G_{12}-G_{21}-G_{22}}{G_{11}+G_{12}+G_{21}+G_{22}},\quad t= \frac{2}{G_{11}+G_{12}+G_{21}+G_{22}}. \tag{3}\]  
The energy transmission and reflection coefficients in the case of a symmetrical environment (we assume that the structure is in a vacuum) have the form \(R=|r|^{2}\), \(T=|t|^{2}\). When the absorption in the layers is taken into account, the fraction of the energy absorbed by the structure is determined by the quantity \(A=1-R-T\). The negativity of absorption should indicate the presence of amplification in the structure. To reveal the spectral features of an MCR that arises when a cavity is filled with an effective medium, we first consider the distribution of the wave field in a structure with an unfilled cavity. Figure 2 shows the distribution of the dielectric permittivity and the squared modulus of the electric field strength of the TM wave (solid and dashed lines) along the structure, and (bottom) shows the transmission spectrum for the considered resonator structure with the number of periods in BMs \(a=5\). In the numerical analysis, the following values of the structure parameters are used: \(\varepsilon_{1}=2.1\) and \(\varepsilon_{2}=4.16\), the optical thickness of the layers \(L_{0}=25\) um, the real thicknesses of the layers in the BMs \(L_{1}=17.2\) um and \(L_{2}=12.2\) um, period is \(L_{1}+L_{2}=29.5\) um, the cavity has a length \(L_{3}=100\) um, and permittivity \(\varepsilon_{3}=1\), the operating frequency \(f_{0}=3\) THz, \(\lambda_{0}=c/f_{0}=100\) um. It can be seen that, for the chosen parameters, the amplitude of the electric field is minimal at the center of the cavity and reaches a maximum at its lateral boundaries. Figure 2: The distribution over the structure of the modulus square electric field strength of the wave normalized to the amplitude \(E_{max}\) (solid line) and the profile of the refractive index (dashed line) (**a**) and the transmission spectrum of the MCR structure (**b**) with the thickness of the resonator \(L_{3}=\lambda_{0}/\sqrt{\varepsilon_{3}}\) filled with air at the parameters of the layers \(\varepsilon_{1}=2.1\) (SiO\({}_{2}\)), \(\varepsilon_{2}=4.16\) (ZrO\({}_{2}\)), \(\varepsilon_{3}=1\) (air) and \(a=5\).