# Ultrafast Optical Signal Processing with Bragg Structures
## 4 Optical Signal Processing in Chirped Bragg Structures  
Chirped fiber BGs with a gradually varying local BG period, have been widely used to provide a strong positive/negative dispersion in a fiber system to stretch and compress the optical pulse [26]. Recently, the nonlinear propagation of the optical pulse in a silicon chirp Bragg structure was investigated [14; 15]. The simulation result shows that the optical pulse can be buffered and released in the silicon chirp Bragg structure. Generally, for the purpose of creating very slow BG solitons, two conditions should be met. First, the grating-induced dispersion must be balanced by the nonlinearity [53]. Second, since the optical field in the 1D Bragg structure is represented by forward-traveling and backward-traveling waves, an initial configuration should have nearly-equal powers in the two waves [54]. The former condition can be achieved by using a high-power input, and therefore, sufficient nonlinearity [12]. The latter condition is much harder to meet with the single incident pulse [55], as it does not initially contain any backward component. Figure 5: The input (solid line) and output (dash line) from the RABR, when a three-peak pulse is used as the input [40]. In other studies [14; 15], concatenated BGs were used to generate slow ultrashort pulses, or even standing ones. Such concatenated BGs are built by linking a linearly chirped grating with a uniform one. The main advantage of this setting is that the initial conditions for pulses at the input edge of the uniform BG segment may be manipulated by means of the preceding chirped BG, which provides a possibility of preparing the right mix of forward and backward fields. As in Bragg superstructures, light propagation in chirped gratings can be described by the standard coupled-mode theory [53; 56; 57; 58; 59]. For slowly varying envelopes of forward and backward waves, \(E_{f}\) and \(E_{b}\), the coupled-mode equations are written as [14; 56]:  
\[\pm i\frac{\partial E_{f,b}}{\partial z}+\frac{i}{v_{g}}\frac{\partial E_{f,b} }{\partial t}+\delta(z)E_{f,b}+\kappa(z)E_{b,f}+\gamma\big{(}\big{|}E_{f,b} \big{|}^{2}+2\big{|}E_{b,f}\big{|}^{2}\big{)}E_{f,b}=0 \tag{9}\]  
where \(t\) is the time; \(v_{g}=c/n_{0}\) is the group velocity in the material of which the BG is fabricated; and \(\gamma=n_{2}\omega/c\) is the nonlinearity strength, where \(\omega\) is the frequency of the carrier wave and \(n_{2}\) is the Kerr coefficient. Further, \(z\) is the propagation distance, \(L\) is the total length of the grating, \(n_{0}\) is the average refractive index with modulation depth \(\Delta n\), \(\Lambda_{0}\) is the BG period at the input edge, and \(C\) is the chirp. The wavenumber-detuning parameter \(\delta\) (Figure 6), and the coupling between the forward and backward field, \(\kappa=\pi\Delta n/\Lambda_{0}\), are functions of the propagation distance. The pulse-propagation properties are first investigated with a different input-pulse peak intensity (Figure 6). The analysis leads to the following conclusions. 1. At low intensities, e.g., 0.65 GW/cm\({}^{2}\), the pulse is almost totally reflected by the Bragg structure due to the presence of the photonic bandgap (Figure 6 bd\({}_{1}\)). Since the pulse's dispersion is not compensated by the weak nonlinearity, pulse stretching is observed. 2. At higher intensities, an unstable standing light pulse trapped at the interface is generated. In particular, at \(I_{P}\) = 2.26 GW/cm\({}^{2}\), the pulse will be reflected after stopping for a short time, as seen in panel (d\({}_{2}\)) of Figure 6b. Slightly increasing \(I_{P}\) to 2.29 GW/cm\({}^{2}\), in the range of 2.26 to 2.29 GW/cm\({}^{2}\), gives rise to a stopping time of \(\sim\)1.3 ns for the pulse that is eventually reflected, as shown in panel (d\({}_{3}\)). 3. A slow moving Bragg soliton can be observed, as shown in panel (d\({}_{4}\)), for \(I_{P}\) = 2.30 GW/cm\({}^{2}\).