Next, we take into account the effect of the FBG mirrors on the mode functions. We assume that the two FBG mirrors are identical, having the same complex reflection and transmission coefficients R and T , respectively, for the guided modes in a broad bandwidth around the atomic transition frequency ω$_{0}$ . In general, we have | R | 2 + | T | 2 ≤ 1, where the equality (inequality) occurs for lossless (lossy) gratings. Without loss of essential physics, we assume that the gratings are lossless, that is, | R | $^{2}$+ | T | 2 = 1. Let the mirrors be located at the positions z = ± L/ 2 along the fiber, where L is the distance between the mirrors. The guided modes are modified by the presence of the mirrors. The mode functions of the cavity-modified guided modes are obtained, as usual in the Fabry-Perot theory, by summing the geometric series resulting from the multiple reflections by the mirrors [31–33]. Inside the cavity, the mode functions of the cavity-modified guided modes are given by ˜ e ( ω,f,l ) = e ( ω,f,l ) T 1 − R $^{2}$e2 iβL + e ( ω, − f,l $_{)}$T Re iβ ( L − 2 fz ) 1 − R $^{2}$e2 iβL , (5) and, hence, the electric positive-frequency component of the field in the cavity-modified guided modes is E (+) cavgyd = i ∑ µ √ ¯ hωβ ′ 4 πϵ$_{0}$ a$_{µ}$ ˜ e ( µ $^{)}$e i ( fβz + lϕ ) − $^{iωt}$. (6) We assume that the FBG mirrors do not reflect the radiation modes. This approximation is reasonable when the distance L between the FBG mirrors is large as compared to the fiber radius a and to the wavelength λ$_{0}$ = 2 π/k$_{0}$ , with k$_{0}$ = ω$_{0}$/c being the wave number of the atomic transition. In the framework of this approximation, the mode functions of the radiation modes are not affected by the presence of the FBG mirrors. We drive the FBG cavity by a classical probe light field propagating along the fiber in a guided mode µ$_{p}$ = ( ω$_{p}$, f$_{p}$, l$_{p}$ ). Let P$_{in}$ be the incident power. The transmitted power is given by P$_{out}$ = P$_{in}$ (1 − | R | $^{2}$)2 (1 − | R | $^{2}$)2 + 4 | R $^{2}$sin$^{2}$Θ( ω$_{p}$ ) . (7) Here Θ( ω ) = β ( ω ) L + φ$_{R}$ is the phase shift caused by a single cavity crossing and a single reflection, with φ$_{R}$ being the phase of the reflection coefficient R , that is, R = | R | e $^{iφ$_{R}$}$. We assume that the probe field frequency ω$_{p}$ is close to a resonant cavity frequency ω$_{c}$ , which is determined by the equation Θ( ω$_{c}$ ) = mπ , with m being an integer number. To the first order in ω$_{p}$ − ω$_{c}$ , we have the expansion Θ( ω$_{p}$ ) = mπ + ( L/v$_{g}$ )( ω$_{p}$ − ω$_{c}$ ). Hence, we find P$_{out}$ = P$_{in}$ (1 − | R | $^{2}$)2 (1 − | R | $^{2}$)2 + 4 | R | $^{2}$( L/v$_{g}$ ) $^{2}$( ω$_{p}$ − ω$_{c}$ ) $_{2}$. (8) 5