# Optical Chirality Determined from Mueller Matrices
## 4 Results
### Structural Optical Activity in Cuticle from Cetonia Aurata
#### 4.3.3 Method III  
The Mueller matrix from sample CA1 was used in nonlinear fitting whereby the index difference \(\Delta N_{x^{\prime}y^{\prime}}=\Delta n_{x^{\prime}y^{\prime}}+\mathrm{i}\Delta k _{x^{\prime}y^{\prime}}\) and the chirality \(\kappa=\kappa_{\mathrm{Re}}+\mathrm{i}\kappa_{\mathrm{Im}}\) in Equation (12) were fitted in a single layer model with film thickness 20 \(\upmu\)m and with dispersion models for the optical functions. Two Gaussian resonances were used for \(\kappa\) according to  
\[\kappa_{\mathrm{Im}}(E)=\sum_{j=1}^{2}\left[A_{j}e^{-\left(\frac{E-E_{0j}}{ \Gamma_{j}}\right)^{2}}-A_{j}e^{-\left(\frac{E+E_{0j}}{\Gamma_{j}}\right)^{2}}\right] \tag{16a}\] \[\kappa_{\mathrm{Re}}(E)=\kappa_{0}+\mathrm{KK}(\kappa_{\mathrm{Im}}), \tag{16b}\]  
where \(A_{j}\), \(E_{0j}\), and \(\Gamma_{j}\) are the amplitude, resonance energy, and broadening of resonance \(j\), respectively. \(\kappa_{0}\) is a constant, and KK stands for a Kramers-Kronig transform. In Equation (16), the spectral dependences are expressed versus the photon energy \(E=hc/\lambda\) (in units of eV), where \(h\) is Planck's constant. Details about modeling the linear properties are found elsewhere [9]; however, in brief, linear birefringence \(\Delta n_{x^{\prime}y^{\prime}}\) is fitted with a Cauchy expression similar to that in Equation (15) but with only three terms, and a small linear dichroism \(\Delta k_{x^{\prime}y^{\prime}}\) is fitted with an exponential term, a so-called Urbach tail [9].  
The sample optical axes are not aligned to the reference axes \(x\) and \(y\) of the instrument, and a Euler rotation angle \(\phi\) is included in the fit, and a value \(\phi=-52.4^{\circ}\) was found. This is close to \(\phi=-45^{\circ}\) explaining why \(L^{\prime}_{\mathrm{B}}\) and \(L^{\prime}_{\mathrm{D}}\) are larger compared to \(L_{\mathrm{B}}\) and \(L_{\mathrm{D}}\).  
In Figure 9, the chirality parameter \(\kappa\) as obtained by regression and with best-fit parameters as given in Table 3 is shown. The asymmetry seen in \(m_{41}\) and \(m_{14}\) and in \(C_{\mathrm{D}}\) in Figure 6 is clearly revealed in \(\kappa_{\mathrm{Im}}\) in Figure 9 and is reasonably modeled with the two overlapping Gaussian resonances. The physical origin of the double resonance might be a chiral structure with a gradient in pitch throughout the sample or with two different pitches at different depths. An attempt to model the Mueller-matrix data from specimen CA2 using regression analysis failed as the large depolarization could not be reproduced.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Resonance** & \(A_{j}\) **(-)** & \(E_{0j}\) **(eV)** & \(\Gamma_{j}\) **(eV)** \\ \hline
1 & \(0.00042\pm 0.00002\) & \(2.138\pm 0.003\) & \(0.197\pm 0.004\) \\
2 & \(0.00081\pm 0.00002\) & \(2.173\pm 0.001\) & \(0.066\pm 0.002\) \\ \hline \hline \end{tabular}
\end{table}
Table 3: Gaussian parameters and corresponding confidence limits for \(\kappa\) for the cuticle of _C. aurata_ (sample CA1).  
Figure 8: Birefringence and dichroism of the elytron from _C. aurata_ (sample CA2).