# Ultrafast Optical Signal Processing with Bragg Structures
## 5 Ultrafast Pulse Modulations Based on Cholesteric Liquid Crystal (CLCs) Bragg Gratings
Figure 11: Observed optical nonlinearity in terms of the nonlinear index coefficients of nematic liquid crystals (including chiral nematics) for several mechanisms and the characteristic relaxation time constants. Depending on the electronic resonances of the nematic constituent used in synthesizing a CLC and the wavelength of the laser under study, the sign of n\({}_{2}\) can be positive [18] or negative [17]--a common feature found in the electronic nonlinearities of most materials [70]. As shown in Figure 12b for the nematic compound used in [18], the refractive index change induced by the laser causes the bandgap to shift towards the shorter wavelength region (i.e., blue-shift), and consequently gives rise to an increasing intensity dependence since the laser wavelength is located at the long wavelength edge of the CLC (Figure 12b). In Reference [17], the CLC sample with self- defocusing nonlinear properties is used to compress the femtosecond pulse (Figure 13). The nonlinear coefficient of the CLC used in that study was measured to be about \(-10^{-11}\) cm\({}^{2}\)/W using a similar intensity dependent transmission measurement. This is four orders of magnitude higher than silica [12]; as a result, the required the thickness of the CLC sample is merely 6 microns in order to compress a 100 fs laser pulse to about 50 fs [17]. If the constituent nematic molecules possess a lower optical nonlinearity, thicker CLC cells of several 100's microns are required [18], which are nevertheless thin/short compared to other materials used for ultrafast laser pulse modulations. Figure 12: (**a**) Observed transmission spectrum of linearly polarized light through the CLC cell used in the experiment; (**b**) Observed dependence of the transmission of a left-handed circularly polarized laser pulse (\(\lambda\) = 815 nm) on the peak intensity [17]. Figure 13: Observed direct pulse compression effect on an initial transform-limited 100 fs pulse (black line) to a 48 fs output pulse (red line); the inset figure corresponds to the simulation results using the measured experimental parameters [17]. The pulse propagation can be simulated by the nonlinear coupld-mode theory presented in Part 4. In [17; 18], it is shown that the simulations for the pulse width and the spectra are in good agreement with the experimental results, as seen in Figures 13 and 14. The compression ability for the pulse with different widths is determined by the propagation length, since the differences of dispersion for different widths lead to a difference in the dispersion length. Another work reported in Reference [18] shows that by using a CLC sample with a 500 \(\upmu\)m cell gap, the sub-picosecond pulse with 800 fs can be compressed to 286 fs. The corresponding spectra broadening can also be observed (Figure 15). This result clearly indicates that by cascading the CLC sample with a different cell gap, the pulse can be compressed in a wide temporal range. Figure 14: Observed spectral broadening effect due to the pulse compression for an input pulse peak intensity of 1.04 \(\text{GW}/\text{cm}^{2}\); the inset depict simulation results for the spectrum (solid lines) and spectral phase (dashed lines) [17]. Figure 15: (**a**) Observed direct pulse compression effect on an initial transform-limited 847 fs pulse (open circles) to a 286 fs output pulse (open squares); the inset figure corresponds to the simulation results using the measured experimental parameters; (**b**) Observed and simulated spectra for the input and output (compressed) pulses [18]. Another possible pulse-modulation device is based on two tandem CLC cells that are tailor-made so that their respective blue/red photonic band-gap edges match the operating laser wavelength, and to utilize the opposite linear dispersions (i.e., without involving the optical nonlinearity) from these band edges to impart an opposite chirping effect on these pulses.