# Ultrafast Optical Signal Processing with Bragg Structures
## 3 Optical Signal Processing in Resonantly Absorbing Bragg Reflector (RABR)
### Theoretical Considerations
The propagation of light in the DRBR is modeled by the Maxwell-Bloch equations, which take into account the Raman and Kerr nonlinearities [7]:  
\[\pm\frac{\partial\Sigma_{g}^{\pm}}{\partial_{\tau}^{\pm}}+\frac{\partial\Sigma _{g}^{\pm}}{\partial\tau}=-\frac{G_{\rm F}}{2}\Sigma_{p}^{\pm}(\left|\Sigma_{ g}^{+}\right|^{2}+\left|\Sigma_{s}^{-}\right|^{2})+P+i\Gamma_{p}\Sigma_{p}^{\pm} \left[\left|\Sigma_{p}^{\mp}\right|^{2}+\left|\Sigma_{p}^{\mp}\right|^{2}+(2-f_ {R})(\left|\Sigma_{s}^{+}\right|^{2}+\left|\Sigma_{s}^{-}\right|^{2})\right] \tag{5}\]  
\[\pm\frac{\partial\Sigma_{g}^{\pm}}{\partial_{\tau}^{\pm}}+\frac{\partial\Sigma _{g}^{\pm}}{\partial\tau}=-\frac{G_{\rm F}}{2}\Sigma_{s}^{\pm}(\left|\Sigma_{ p}^{+}\right|^{2}+\left|\Sigma_{p}^{-}\right|^{2})+iK\Sigma_{s}^{\mp}+i \Delta\Sigma_{s}^{\pm}+i\tau_{s}\Sigma_{s}^{\mp}\left[\left|\Sigma_{s}^{\pm} \right|^{2}+\left|\Sigma_{s}^{\mp}\right|^{2}+(2-f_{R})(\left|\Sigma_{p}^{+} \right|^{2}+\left|\Sigma_{p}^{-}\right|^{2})\right] \tag{6}\]  
\[\frac{\partial P}{\partial\tau}=(\Sigma_{p}^{+}+\Sigma_{p}^{+})W, \tag{7}\]  
\[\frac{\partial W}{\partial\tau}=-\text{Re}\Big{[}(\Sigma_{p}^{+}+\Sigma_{p}^{ +})P^{*}\Big{]}, \tag{8}\]  
where \(\Sigma_{p,s}^{\pm}\equiv(2\tau_{c}\mu/\hbar)E_{p,s}^{\pm}\) represents the forward- and backward-propagating pump and Stokes waves; \(K\equiv\kappa c\tau_{c}/n_{0}\) and \(\Delta\equiv\delta c\tau_{c}/n_{0}\) are the dimensionless coupling constants and detuning, respectively; \(f_{R}\) is the fraction of the nonlinearity arising from molecular vibrations with a typical value of 0.18 [45]; and \(G_{p,s}\equiv(\varepsilon_{0}c^{2}\hbar^{2}/8\mu^{2}\tau_{c})g_{p,s}\) and \(\Gamma_{p,s}\equiv\omega_{p,s}\mu_{2}\varepsilon_{0}c\hbar^{2}/8\mu^{2}\tau_{c}\) are the dimensionless nonlinearity and gain coefficients for the pump and Stokes waves, respectively. When the time of the interaction of the standing Raman active medium and soliton is large enough, the intensity of the zero-velocity Stokes pulse starts to increase at the expense of the pump field. Following the efficient power exchange between the pump and Stokes pulses, the energy of the Stokes pulse can leak out from both edges of the DRBR. By comparing the output energy of the Stokes pulses with the energy of the standing pump pulse, the efficiency of the Raman shift can be estimated (Figure 4). Remarkably, it may exceed 85%, i.e., higher than the efficiency in the bulk medium [47]. In practice, such an efficient conversion can be achieved by using a periodically arranged multi-quantum-well structure, [48]. The required length and power are only a few millimeters and 100 uJ, respectively. RABR can also be used to reshape and compress ultrashort laser pulses. From the Maxwell-Bloch Equation (1), one can show that the RABR may compress the sech-shaped input, with a pulse width ranging from \(2\tau_{c}\) to \(5\tau_{c}\) (\(\tau_{c}\) is the cooperative time), into a \(2\pi\) SIT soliton [49]. Furthermore, input pulses with a multiple-peak shape can be re-shaped to produce a single-peak output. Figure 5 shows the transformation of the three-peak pulse into a single-peak pulse in the RABR structure. In fact, SIT in bulk media can also be applied to compressed optical pulses. However, in that case, the input pulse with single peak is split into multiple ones, rather than morphed into a single-peak SIT soliton, as in RABR. The difference is explained by the fact that reshaping in the RABR originates from  
\[\frac{\partial P}{\partial\tau}=-i\delta P+(\Sigma^{+}+\Sigma^{-})W \tag{3}\]0 The evolution of the energy density of (**a**) pump and (**b**) stokes pulses in the DRBR under the action of the SRS [7]. multiple reflections in the Bragg structure, while no reflections occur in the the bulk SIT medium. Other interesting phenomena, such as the negatively refracted light in the RABR, have also been discovered [50]. The validity of the numerical predictions was also checked by the comparison with a more realistic finite-atomic-width approach [51].