# Ultrafast Optical Signal Processing with Bragg Structures
## 5 Ultrafast Pulse Modulations Based on Cholesteric Liquid Crystal (CLCs) Bragg Gratings  
For modulating the optical signal in a compact photonic device, the following properties of the device materials are required: (i) Fast response to process a high speed optical signal; and (ii) Strong dispersion and nonlinearity to achieve a shorter dispersion and nonlinear length to decrease the required propagation length. Many works on the pulse modulation have been completed in fiber Bragg grating, and for which, the typical nonlinear coefficient is on the order of \(10^{-16}\) cm\({}^{2}\)/W; in conjunction with a peak power of kilowatts [26], the resulting nonlinear effective length is on the millimeter scale [60]. In this section, we show that naturally occurring Bragg grating in a highly nonlinear CLCS can enable the same ultrafast (femtoseconds-picoseconds) pulse modulation operations in sub-mm interaction lengths. After decades of studies, liquid crystals have emerged as highly versatile and nonlinear optical materials, due to their organic molecular constituents and unique liquid crystalline properties [61]. In liquid crystals, the underlying mechanisms generally fall into two classes, as depicted in Figure 11 for the nematic phase (which include the chiral nematic phase often called cholesteric liquid crystals (CLCs)): (i) Macroscopic crystalline responses including the director axis, reorientation by light  
Figure 10: A typical example of the predicted femtosecond diode effect. The pulses with a width of 100 fs are injected from the left (**a**) and right (**b**) hand sides, respectively. The incident peak intensity of the pulse is \(I=0.35\) GW/cm\({}^{2}\). (**a**) The femtosecond soliton can propagate from A to B; (**b**) the soliton bounces back when it is injected from B [16]. Figure 9: The simulation result of the trapping of three pulses in the BG structure. Here (\(S\); \(d_{w}\); \(\epsilon\)) = (0.132 cm; 50 \(\upmu\)m; 0.04). The peak intensities of the three pulses are 3.04, 2.78, and 2.07 GW/cm\({}^{2}\). Such pulses are launched into the gratings at \(t=0\), \(t=3\), and \(t=6\) ns, respectively [15]. fields or a light/DC field induced photorefractive effect, index/birefringence change by laser induced order parameter and/or temperature changes, and flow reorientation; these bulk effects generally respond in the millisecond to a sub-microsecond regime [62]; and (ii) Individual molecule's nonlinear polarization associated with single- and multiple photonic transitions within the molecular energy levels; these respond on the sub-picoseconds to a femtoseconds scale [63; 64; 65; 66; 67]. CLCs are formed by introducing a chiral agent in standard nematic liquid crystals used in ubiquitous liquid crystal display devices. In CLC, the birefringent molecules self-assemble with the director (crystalline) axis arranged in a spiral, with a pitch on the order of the optical wavelength stretching from the UV to the infrared regime, as seen in Figure 12. As a result, CLC's possess not only the advantageous features of liquid crystals such as fabrication ease and low cost, and a very wide spectral dynamic range of operation [68; 69], but also the 1-D photonic crystals' unique ability to enhance the nonlinear ultrafast all-optical responses of the CLC constituent molecules [mostly due to the nematic constituent], in addition to the dispersion effect at the photonic band-edges. A typical magnitude of the non-resonant nonlinear index coefficient \(n_{2}\) is in the order of \(10^{-14}\)-\(10^{-13}\) cm\({}^{2}\)/W, but owing to the photonic crystal band-edge enhancement, the magnitude of the effective \(n_{2}\) can be as large as \(10^{-12}\)-\(10^{-11}\) cm\({}^{2}\)/W. In comparison to other materials [70] used for ultrafast pulsed laser modulation applications, these \(n_{2}\) values are orders of magnitude larger, and thus, one can envision a tremendous miniaturization possibility.