# Optical Chirality Determined from Mueller Matrices
###### Abstract  
Optical chirality, in terms of circular birefringence and circular dichroism, is described by its electromagnetic and magnetoelectric material tensors, and the corresponding optical activity contributes to the Mueller matrix. Here, spectroscopic ellipsometry in the spectral range 210-1690 nm is used to address chiral phenomena by measuring Mueller matrices in transmission. Three approaches to determine chirality parameters are discussed. In the first approach, applicable in the absence of linear polarization effects, circular birefringence and circular dichroism are evaluated directly from elements of a Mueller matrix. In the second method, differential decomposition is employed, which allows for the unique separation of chirality parameters from linear anisotropic parameters as well as from depolarization provided that the sample is homogeneous along the optical path. Finally, electromagnetic modeling using the Tellegen constitutive relations is presented. The last method also allows structural effects to be included. The three methods to quantify optical chirality are demonstrated for selected materials, including sugar solutions, \(\alpha\)-quartz, liquid crystals, beetle cuticle, and films of cellulose nanocrystals. 10 June 2021  
or ORD are not reached with commercially available Mueller-matrix ellipsometers. Still, the Mueller-matrix methodology is an important complement. To facilitate a comparison among different approaches, the use of transmission Mueller matrices to determine the chirality in samples is briefly reviewed and examples of data analysis from liquid and solid homogeneous as well as structured samples are discussed. Methodology based on optical activity is used as a standard technique in many fields of science. Optical rotatory dispersion (ORD) is used in polarimetry to determine the concentrations of sugar-based solutions and also finds frequent use to determine chiral purity, e.g., for the separation of enantiomers in chemistry during drug development. Essentially, the magnitude and direction of rotation of polarized light is measured, often at a single wavelength, and related to molecular properties. The rotation can be relatively large--on the order of degrees--and is easy to record. ORD can also be used for the structural analysis of biomolecules; however, in most cases, CD spectroscopy is used as it is more sensitive, has better spectral resolution, and has less overlap of spectral features [1]. An example of structural studies is analysis of the amount of \(\alpha\)-helices, \(\beta\)-sheets, and random coils in proteins. In CD spectroscopy, the change in ellipticity is measured, which, in most cases, is specified in units of degrees. It can be very small and on the order of millidegrees. Both CB and CD are included in a Mueller matrix and can be extracted directly in the absence of linear birefringence and linear dichroism. If linear effects are also present, a differential decomposition can be used to separate linear and circular effects as proposed by [PERSON] [2] and further detailed by [PERSON] [3]. Later, [PERSON] included the separation of depolarization [4]. Even though the methodology was proposed long ago, it has been ignored in interpretation of Mueller matrices. More recently, [PERSON] and [PERSON] determined CD and CB from spectroscopic Mueller matrices measured on nickel sulfate hexahydrate crystals [5]. Giant optical activity was observed in thin films of enantiopure squaraines with opposite handedness [6]. Optical activity originating from structural effects in beetle cucticles was determined together with LB and LD as well as depolarization [7]. Chiral nanocrystalline cellulose free-standing films exhibit both linear and circular anisotropy and can be resolved by the differential decomposition of Mueller matrices [8]. Optical activity is contained in the electro-magnetic and magneto-electric tensors in the Tellegen constitutive relations and can be analyzed with Kramers-Kronig consistent dispersion models using Mueller matrices from beetle cucticles [9].