# Optical Chirality Determined from Mueller Matrices
## 5 Discussion
of the sample, but without distortion except for filtering if used. An alternative method to determine these properties is analytic inversion of a Muller matrix as suggested by [PERSON] and [PERSON] [13]. This method also requires a homogeneous sample; however, in addition, the sample is non-depolarizing with a so-called Mueller-Jones matrix. We only discussed natural and structural optical activity manifested as occurrences of CB and/or CD. In the presence of a quasistatic magnetic field, we may also observe similar effects, referred to as the Faraday effect, when the magnetic field is along the optical path. More generally, it is called the optical Hall effect and has been reviewed by [PERSON] et al. [37]. However, the birefringence and dichroic effects then arise from the breaking of symmetry in the permittivity tensor and are not contained in the chirality tensor. The use of MMSE for studies of optical activity is a valuable complement to methods for the direct determination of spectral CB and CD properties. However, the recording of a normalized Mueller matrix is more complex as its 15 parameters are determined instead of a single parameter like a rotation angle in ORD or ellipticity in CD spectroscopy. In MMSE, compromises are made to assure precision in all elements as compared to the high resolution and accuracy obtainable in ORD and CD spectroscopy. An ORD instrument is referred to as a polarimeter and is based on measuring changes in the direction of polarization, whereas, in CD-spectroscopy, a dichrometer measures changes in ellipticity. Modulation techniques are, thereby, employed, including photoelastic modulators and lock-in amplifiers to enhance the resolution and signal to noise ratio [38; 39]. However, the fundamental properties measured, i.e., polarization changes, are the same in MMSE, ORD, and CD with the difference that MMSE in addition allows depolarization to be monitored. The lower resolution of an ellipsometer compared to a polarimeter or a dichrometer with available instruments is, therefore, not a principle limitation but purely technical. Accordingly, in ORD and CD spectroscopy, it is implied that neither linear effects nor depolarization are present in the studied specimen. Consequently, incorrect values of unpredictable error magnitude for CD and CB will be reported if these effects cannot be  
Figure 16: Partial Mueller matrix measured on a D-glucose solution using a rotating analyzer/fixed compensator spectroscopic ellipsometer. ignored. Therefore, MMSE can be seen as a generalization of the established methods to study chirality. In cases when Mueller-matrices are used, a major advantage is that access to both \(C_{\text{B}}\) and \(C_{\text{D}}\) makes it possible to check the Kramers-Kronig consistency. Another advantage is that samples with depolarization can be addressed. With differential decomposition (method II), the depolarization properties are separated from the bianisotropic properties and can also be analyzed. In electromagnetic modeling (method III), it is possible to address small to medium depolarization by smearing, i.e., by assigning a distribution of parameters during modeling. With methods I and II, as well as with ORD and CD spectroscopy, average sample properties are obtained, whereby it is assumed that samples are homogeneous along the optical path. As discussed above, linear effects may distort the results in method I, whereas isotropic effects do not interfere as long as the absorption is sufficiently low to obtain enough light throughput. However, if a sample is structured, e.g., has a layered structure, methods I and II fail to deliver correct results, but method III may still be applicable. In reflection mode, MMSE is routinely used with the electromagnetic modeling of structured samples, and layer thicknesses and permittivity tensors can be determined [40]. Such procedures are also applied in transmission mode although less common.