# Quantification of Optical Chirality in Cellulose Nanocrystal Films Prepared by Shear-Coating
## 3 Results and Discussion
### Polarization Properties of CNC Chiral Films  
A complete description of the polarization and depolarization properties of a linear optical system is given by the Stokes\(-\)Mueller formalism [18]. In this approach, light beams are assumed to travel along the \(z\)-axis and are represented by column vectors \(\mathbf{S}\) = [\(l\), \(Q\), \(U\), \(V\)]T (T means transpose), where \(l\) represents the total irradiance, \(Q\) > 0 (<0) accounts the tendency for linear polarization along the \(x\)-axis (\(y\)-axis), \(U\) > 0 (< 0) is the tendency for linear polarization at +45\({}^{\circ}\) (-45\({}^{\circ}\)) in the \(xy\)-plane, and \(V\) > 0 (<0) the right (left) circular character of polarization. After interacting with a sample, the Stokes vector of the resulting beam is given by \(\mathbf{S^{\prime}}\) = \(\mathbf{MS}\), where \(\mathbf{M}\) = \(\{M_{ij}\}\) (\(i\),\(j\) = 1... 4) is the 4 \(\times\) 4 Mueller matrix representing the sample. In this work, normalized Mueller matrices (\(m_{ij}\) = \(M_{ij}\)/\(M_{11}\)) and Stokes vectors (\(S_{0}\) = 1) are used. For measurements in the reflection mode at the oblique incidence, the \(xy\) axes were replaced with those parallel (p) and perpendicular (s) to the plane of incidence. Figure 3 shows Mueller matrices measured in the reflection mode as a contour map representation. The angle of incidence dependence is on the vertical axis and the wavelength dependence is on the horizontal axis. This contour map looks similar to that reported for the beetle _Cotinis mutabilis_[21] and displays the same symmetries among elements, as reported for multidomain free-standing CNC chiral films [19]. In general, the symmetries expected for a chiral system are fulfilled [23], as follows: \(m_{12}=m_{21}\), \(m_{13}=-m_{31}\), \(m_{14}=m_{41}\), \(m_{23}=-m_{32}\), \(m_{24}=m_{42}\), and \(m_{34}=-m_{43}\). As for unpolarized incident light \(\mathbf{S}\) = [0,0,0,1]T, the reflected beam is given as \(\mathbf{S}\) = [1, \(m_{21}\), \(m_{31}\), \(m_{41}\)]T. The negative values in the \(m_{41}\) element for small angles of incidence account for selective Bragg reflection with left-handed polarization. Homogeneous blue color was perceived on large areas of the CNC film, as shown in Figure 2, so it was expected to find selective reflection at short wavelengths of the visible range. Indeed, for an angle of incidence 20\({}^{\circ}\), \(m_{41}\) had a minimum at about 400 nm and the band of selective reflection extended to the visible range. The spectra of the normalized Mueller matrices are shown in Figure A1 in Appendix A. The minimum in \(m_{41}\) shifted to shorter wavelengths as the angle of incidence increased, which is indicative of iridescence, a property distinctive of self-assembled CNC films. Figure 3: Contour map of the reflection Mueller matrix of a CNC glass-supported film. Notice the reflection band with negative values in the \(m_{41}\) element. The experimental Mueller matrix in transmission mode at normal incidence shows selective Bragg reflection in a narrow range, as can be noticed in Figure A2 in Appendix A. Data below a wavelength 300 nm were omitted because of the strong absorption of the glass substrate. To highlight the most important features, Figure 4 shows the experimental data (continuous curves) in the range of 300-600 nm. Thus, for unpolarized incident light the transmitted beam is right-handed polarized (\(m_{41}>0\)) in a band centered at a wavelength of about 415 nm. The latter value agrees with that of 400 nm found for the reflection at \(\theta=20^{\circ}\), according to Equation (1), taking \(n_{\text{av}}=1.54\) for cellulose [24]. The mean pitch is about 270 nm. Data represented with dashed curves in Figure 4 correspond to the fit described below in Section 3.4. To quantify the polarization properties, the capability of the system to depolarize incident polarized light is tested first.