# Quantification of Optical Chirality in Cellulose Nanocrystal Films Prepared by Shear-Coating
## 3 Results and Discussion
### Effective Structural Chirality Parameter Determined from Tellegen Constitutive Equations  
The structural chirality can also be assessed using the Tellegen constitutive equations, which has recently been applied to the scarab beetle _Cetonia aurata_[28]. For a reciprocal medium, the electromagnetic coupling is described by the chirality tensor \(\kappa\), and Tellegen constitutive equations are given by the following  
\[\begin{array}{l}\mathbf{D}=\varepsilon_{0}\varepsilon\mathbf{E}+ic^{-1} \kappa\mathbf{H}\\ \mathbf{B}=-ic^{-1}\kappa^{\mathbf{T}}\mathbf{E}+\mu_{0}\mathbf{\mu}\mathbf{H},\end{array} \tag{3}\]  
where \(\mathbf{D}\), \(\mathbf{E}\), \(\mathbf{B}\), and \(\mathbf{H}\) are the electric displacement field, electric field, magnetic flux density, and magnetic field, respectively. The dielectric tensor \(\varepsilon=\text{diag}({n_{x}}^{2},\,{n_{y}}^{2},\,{n_{z}}^{2})\) in the principal axis frame, the complex chirality tensor \(\kappa=\kappa\mathbf{I}\) (\(\kappa=\kappa^{\prime}+i\kappa^{\prime\prime}\)), and the permeability tensor \(\mu\), represent the materials properties (for non-magnetic materials \(\mu=\mathbf{I}\)); fundamental constants of vacuum are the speed of light \(c\), permittivity \(\varepsilon_{0}\), and permeability \(\mu_{0}\). The refractive indices were modeled with the Cauchy expression, as follows  
\[n_{j}=A_{j}+\frac{B_{j}}{\lambda^{2}}+\frac{C_{j}}{\lambda^{4}}, \tag{4}\]  
where \(A_{j}\), \(B_{j}\), and \(C_{j}\) are fitting parameters and \(\lambda\) is given in units of \(\mu\)m.  
The chirality parameter was modeled with gaussian functions,  
\[\begin{array}{l}\kappa^{\prime\prime}=\sum\limits_{j=1}^{2}A_{j}\bigg{[} \exp\biggl{[}-\left(\frac{E-E_{0j}}{\Gamma_{j}}\right)^{2}\biggr{]}-\exp \biggl{[}-\left(\frac{E+E_{0j}}{\Gamma_{j}}\right)^{2}\biggr{]}\bigg{]}\\ \kappa^{\prime}=\kappa_{0}+\text{KK}(\kappa_{2}),\end{array} \tag{5}\]  
where \(A_{j}\), \(E_{0j}\), and \(\Gamma_{j}\) are the amplitude, central energy, and broadening, respectively; \(\kappa_{0}\) is an offset; and KK refer to a Kramers-Kronig transformation. The thickness of the film is \(d=\) 18.1 \(\pm\) 0.4 \(\mu\)m, as determined from a cross-sectional SEM image (Figure 15 in Appendix A). The values of the fitting parameters in Equations (4) and (5) are shown in Tables 1 and 2, respectively. From the fitting, \(\varphi=-180.2^{\circ}\) is obtained, which is consistent with the sample position during the measurement. As can be seen in Figure 4, Tellegen constitutive equations give a very good description of the experimental Mueller matrix. Furthermore, Figure 6a shows that circular polarization properties and the chirality parameter fulfill the relationship \(CB+\text{{l}cd}=4\pi d\kappa/\lambda\). The retardation obtained from the analytical inversion agrees very well with that determined by the relationship \(\delta=2\pi d\Delta n/\lambda\) (Figure 6b), where \(\Delta n=n_{y}-n_{x}\) is the film birefringence.  
Figure 6: Elementary polarization properties determined from the analytical inversion of experimental Mueller matrix data in Figure 4. (**a**) Circular dichroism (\(CD\)) and circular birefringence (\(CB\)). (**b**) Linear retardance (\(\delta\)) and linear diattenuation (\(p\)). The dashed curves correspond to properties determined using the Tellegen constitutive equations.