## 3 Spectra of a Microresonator with a Filled Cavity  
Next, we will consider the transmission and reflection spectra for an MCR with a resonator cavity filled with a graphene-containing periodic structure. Initially, we will consider the \"graphene-dielectric\" structure as such a filling, the transfer matrix of which has the form \(N_{3}=(N_{g}N_{d})^{b}N_{g}\), where the number of periods \(b\) hereinafter is taken equal to five. The matrices of individual layers have the form (2), and the transfer matrix of a MCR with a filled cavity, which connects the wave field at the input and output of the structure, has the form (1). The material of the dielectric layers is silicon dioxide SiO\({}_{2}\) with a permittivity \(\varepsilon_{d}=5.07\), the thickness of the dielectric layers is \(\ell_{d}=\lambda_{0}/4\sqrt{\varepsilon_{d}}=11\)\(\mu\)m, and the thickness of the graphene monolayer is \(\ell_{g}\) = 0.335 nm. The two following figures show the frequency dependencies of the coefficients \(T\) and \(R\) for a microcavity structure with a different number of periods in BMs (Figure 4, \(a=5,7\) left and right, respectively) and a cavity filled with an dielectric-graphene structure. It can be seen from the given dependencies that, when the cavity is filled with the indicated structure, a single peak of the defect mode observed in the center of the PBG of an empty MCR splits into two narrow and closely spaced transmission peaks of almost identical amplitude. This is due to the fact that when the cavity is filled with a structure with quantum wells (graphene monolayers), a regime of strong exciton-photon interaction between the cavity defect mode and quantum well excitons is realized. This leads to the splitting of the exciton and photon levels into two polariton states, which leads to a significant transformation of the transmission and reflection spectra [35]. Since the imaginary part of the graphene conductivity (hence, the real part of its permittivity) increases in the PBG region, these peaks also shift towards a higher frequency (for \(E_{F}\) = 0, the position of the two peaks is determined by the frequencies \(f_{-}=2.75\) THz and \(f_{+}=3.25\) THz, while for \(E_{F}\) = 300 meV, \(f_{-}=2.8\) THz and \(f_{+}=3.31\) THz). Thus, due to the presence of graphene in the cavity, the effective permittivity of the entire insert changes with an increase in the Fermi energy and the frequencies of the two polariton states also change. The energy state of graphene monolayers significantly affects the character of the transmission and reflection spectra not only in the PBG region, but also over the entire spectrum width. Thus, at a zero value of the Fermi energy, the coefficients \(T\) and \(R\) do not exceed unity. In the PBG region, the transmission peaks occurring at \(E_{F}=0\), are almost completely suppressed with increasing \(E_{F}\). With an increase in the Fermi energy in the region of a negative real part of the conductivity (\(f\leq 0.6\) THz), an amplification is observed and the coefficients \(T\) and \(R\) become significantly greater than unity. In this region, the most efficient control of the spectra \(T(f)\) and \(R(f)\) is also observed with the help of an electric field. Figure 5 shows the transmission and reflection spectra for an MCR with a number of periods in the BMs \(a=7\) and a resonator cavity filled with a \"semiconductor/graphene\" structure with a transfer matrix \((N_{g}N_{s})^{b}N_{g}\). The material of the semiconductor layers is n-InSb (its parameters are given in the description of Figure 3). The spectra are plotted for the Fermi energies \(E_{F}=0\), 100, 200, 300 meV (a-d) in the absence of a magnetic field (\(H_{0}=0\)). The semiconductor used as barrier layers is a material with free charge carriers and with a higher permittivity value than SiO\({}_{2}\), partially modifying the spectra compared to those presented in Figure 4b.