# Optical Chirality Determined from Mueller Matrices
## 5 Discussion  
The results obtained directly from a Mueller matrix (method I) may be considered as primary data. The only filtering is noise reduction depending on the data acquisition time chosen during the measurements. Values of \(C_{\text{B}}\) and \(C_{\text{D}}\) or the derived properties \(\alpha\) and \(\epsilon\), respectively, may, therefore, be affected by instrument noise and systematic errors. This also holds for the differential decomposition approach (method II); however, as evaluation is done on the logarithm of a Mueller matrix, noise and systematic errors will propagate and, thereby, will be weighted differently compared to the original Mueller matrix. In method III, a nonlinear regression is performed on the Mueller matrix elements using dispersion functions for the spectral variation in the materials properties. The values of these functions, which constitute the results of the analysis, are noise free, which means strong filtering. Depending on the function chosen, this may imply that small features in the data are missed. Systematic errors from the system as well as from model mismatch may, in addition, introduce false information in the results. The example on a glucose solution should be considered as a demonstration on a simple case and not to promote the use of MMSE instead of polarimetry. Polarimetry is a very mature technique and has a resolution of the order of \(\pm 0.01^{\circ}\) or better in the polarization angle rotation \(\alpha\) as found on vendor sites. Low-cost instruments with a resolution of \(\pm 0.003^{\circ}\) were presented [33]. In our results, we have an estimated resolution of \(\pm 0.02^{\circ}\) determined from the noise level in \(\alpha\) in an MMSE measurement with a 10 s acquisition time. However, no efforts have been made to determine the attainable accuracy. We used a medium precision balance and an ordinary measuring glass to prepare the glucose solution. For the rather high concentrations used, a concentration adjustment of \([\alpha]_{589}^{20}\)[34] should be made. Furthermore, the sodium D-line used is a doublet, and it makes some difference if \(\lambda=589.0\) or \(589.6\) nm is used. However, the specific rotations of \(52.5\), \(52.5\), and \(52.0\) obtained with methods I, II, and II, respectively, are surprisingly close to the data-base value \([\alpha]_{589}^{20}=+52.7\) [\(\deg\text{dm}^{-1}\,\text{cm}^{3}\,\text{g}^{-1}\)] [34]. The spectral variation of \(\kappa\) is complex as it may arise from various chemical groups in the glucose molecule as discussed by [PERSON] et al. [35]. [PERSON] used a Drude-type dispersion model to account for resonances in the ultraviolet spectral region. We found it sufficient to use a Cauchy model but with four terms to account for the complexity due to overlapping CB features from different subgroups. [PERSON] detected small CB features below 300 nm [36], which, however, could not be seen in our data. Figure 15: Model function with two Gaussian resonances for the chirality parameter \(\kappa\) of the CNC film. If a sample only has CB and/or CD, it is not necessary to measure a complete Mueller matrix. Figure 16 shows an example of a measurement on the same D-glucose solution as in Figure 1 using a rotating analyzer ellipsometer with a fixed compensator. This type of instrument only records three rows of a Mueller matrix. From \(m_{23}\) and \(m_{32}\), we can determine the specific rotation in a similar way as in Section 3.1, and we find, at \(\lambda=589\,\mathrm{nm}\), that \([a]_{589}^{20}=+52.5\,[\mathrm{deg\,dm^{-1}\,cm^{3}\,g^{-1}}]\). It is also possible to use symmetry conditions to expand a non-depolarizing Mueller matrix with three rows to a full Mueller matrix [6]. In Method II, the natural logarithm of an experimental Mueller matrix is calculated. This corresponds to a transformation of experimental data, from measured transmission properties in terms of Mueller matrix elements to fundamental optical properties (\(C_{\text{B}}\), \(C_{\text{D}}\) etc.)