# Ultrafast Optical Signal Processing with Bragg Structures
## 3 Optical Signal Processing in Resonantly Absorbing Bragg Reflector (RABR)  
### Theoretical Considerations  
RABR is produced by adding narrow stripes, doped by two-level atoms resonantly interacting with the transmitted electromagnetic fields, to a BG structure, as schematically shown in Figure 1[9; 10; 11]. Light-matter interaction in RABR, built of infinitely thin atomic layers, is modeled by the Maxwell-Bloch equations for electric-field components \(E^{+}\) and \(E^{-}\) of the forward- and backward-propagating waves, and population W[12]:  
\[\frac{\partial\Sigma^{\pm}}{\partial\tau}\pm\frac{\partial\Sigma^{\pm}}{ \partial\zeta}=i\eta\Sigma^{\pm}+P, \tag{2}\]  
\[\frac{\partial P}{\partial\tau}=-i\delta P+(\Sigma^{+}+\Sigma^{-})W \tag{3}\]  
\[\frac{\partial W}{\partial\tau}=-\text{Re}\big{[}(\Sigma^{+}+\Sigma^{-})P^{*} \big{]} \tag{4}\]  
where \(\Sigma^{\pm}\equiv(2\tau_{c}\mu/\hbar)E^{\pm}\), \(\tau_{c}\equiv(2\hbar n_{0}/\mu_{0}c^{2}\omega_{c}\mu\rho)^{1/2}\) is the cooperative time (ranging from pico- to femto-seconds )which is determined by the presence of the two-level atoms; \(n_{0}\) is the average refraction index; \(\omega_{c}\) is the resonant frequency of the two-level atom; \(\mu\) is the magnitude of the dipole matrix element \(\rho\) is the density of the dopant atoms; \(\eta\equiv(n_{1}\omega\tau_{c})/4\) is the dimensionless coupling constant; \(P\) and \(W\) are the material polarization and population inversion density, respectively; \(\delta\equiv(\omega-\omega_{c})/\tau_{c}\) is the dimensionless detuning; \(\tau\equiv t/\tau_{c}\) and \(\zeta\equiv(n_{0}/c\tau_{c})z\) are the dimensionless time and spatial coordinates, respectively; and \(\omega\) is the frequency of the incident light. Here, \(\tau_{c}\) is equal to 300 fs[13]. Solutions of the Maxwell-Bloch equations, some being available in an analytical form[10], produce a vast family of stable gap solitons, of both a standing and moving nature. Compared to the self-induced-transparency (SIT) in uniform media, the solitons generated in RABR may have an arbitrary pulse area, while in uniform media solitons, are only created by pulses with an area exactly equal to \(2\pi\)[44; 45]. Stable dark solitons can also be excited in RABR[11]. Thus, the theoretical analysis demonstrates that light signals with any velocity can indeed exist in the RABR. An optical pulse with a hyperbolic-secant shape, generated by the input with a small area, undergoes complete Bragg reflection in the RABR. With an increase in the input intensity, the SIT solitons can be excited, making it possible for light pulses to propagate without loss. If the intensity is still higher, the splitting of the pulse occurs. Most remarkably, with a suitable incident pulse, an oscillating gap soliton trapped in the RABR as a standing wave can be created. Figure 2a shows the evolution of the optical energy, which is represent by the population distribution \(W\)[12; 13], in this case. Furthermore, we have found that multiple gap solitons can be simultaneously created as standing modes, thus predicting a possibility of efficient storage of the optical energy in the RABR structure (Figure 2b). Figure 1: A scheme of RABR with black stripes representing thin layers of two-level atoms. White and gray bands represent a periodically structured nonabsorbing medium (After Reference[11]). For an input pulse with a suitable area, the numerical simulations have shown that the pulse can evolve into a standing gap soliton that can be stored as a stable self-localized state. The minimum length of the sample, necessary for the realization of the storage of the standing light pulse, is 1200 BG periods [12; 13]. The self-trapping dynamics of the laser pulse may be considered as the motion of a quasi-particle in an effective potential representing the nonlinear interaction between the input pulse and the two-level atomic medium [46]. As a result, the pulse decelerates and eventually comes to a halt inside the structure, under the action of the trapping potential.