# Optical Chirality Determined from Mueller Matrices
## 3 Theory
### Method III: Modeling Using the Tellegen Constitutive Relations  
An effective medium approach is used to determine effective chiral properties of a sample. The method is based on the Tellegen constitutive relations  
\[\mathbf{D}=\varepsilon_{0}\boldsymbol{\varepsilon}\mathbf{E}+c^{-1}\boldsymbol{ \xi}\mathbf{H} \tag{11a}\] \[\mathbf{B}=c^{-1}\boldsymbol{\xi}\mathbf{E}+\mu_{0}\boldsymbol{\mu}\mathbf{H}, \tag{11b}\]  
where the electric displacement field \(\mathbf{D}\), the electric field \(\mathbf{E}\), the magnetic field \(\mathbf{H}\), and the magnetic flux density \(\mathbf{B}\) are interrelated by the materials tensor permittivity \(\boldsymbol{\varepsilon}\), permeability \(\boldsymbol{\mu}\), and the magnetoelectric tensors \(\boldsymbol{\zeta}\) and \(\boldsymbol{\zeta}\). The vacuum permittivity and permeability are denoted as \(\varepsilon_{0}\) and \(\mu_{0}\), respectively, and \(c\) is the vacuum speed of light. Notice that \(\boldsymbol{\varepsilon}\) and \(\boldsymbol{\mu}\) are defined in the \(xyz\) coordinate system. The tensor \(\boldsymbol{\zeta}\) can be expanded as \(\boldsymbol{\xi}=\boldsymbol{\chi}+\mathbf{i}\mathbf{\kappa}\) where \(\boldsymbol{\chi}\) and \(\boldsymbol{\kappa}\) are the nonreciprocity tensor and chirality tensor, respectively. The materials studied here are reciprocal implying \(\boldsymbol{\chi}=0\), which leads to \(\boldsymbol{\zeta}=-\boldsymbol{\zeta}^{T}=\mathbf{i}\mathbf{\kappa}\) where T indicates transposition [16]. In a transmission measurement at normal incident in a Cartesian \(xyz\) coordinate system, aligned with the instrument framework and with \(z\) being the direction of wave propagation, we do not probe the \(z\)-components of the material tensors. Furthermore, in the presence of biaxial anisotropy, the optical axes \(x^{\prime}\) and \(y^{\prime}\) of a sample may differ an angle \(\phi\) (in-plane Euler angle) from the axes \(x\) and \(y\). The samples studied here are either isotropic or have their helical axis along the \(z\)-direction, and the general form of the chirality tensor used is \(\boldsymbol{\kappa}=\text{diag}(\kappa_{x},\kappa_{y},\kappa_{z})\). Furthermore, \(\kappa_{x}=\kappa_{y}\) and, as \(\kappa_{z}\) can be set arbitrarily, we can use \(\kappa_{x}=\kappa_{y}=\kappa_{z}=\kappa\) and simplify to \(\boldsymbol{\kappa}=\kappa\mathbf{I}\) where \(\kappa=\kappa_{\text{Re}}+\mathrm{i}\kappa_{\text{Im}}\) is a complex-valued scalar representation of chirality and \(\mathbf{I}\) is the identity matrix. The permittivity tensor is assumed to be diagonal \(\boldsymbol{\varepsilon}_{x^{\prime}y^{\prime}}=\text{diag}(\varepsilon_{x^{ \prime}},\varepsilon_{y^{\prime}},\varepsilon_{z})\). Its elements are related to the refractive index by \(N_{j}=n_{j}+\mathrm{i}k_{j}=\sqrt{\varepsilon_{j}}\) (\(j=x^{\prime},y^{\prime},z\)). In transmission ellipsometry at normal incidence, the absolute phase is not accessible, and there is no sensitivity to the absolute values of \(n_{x^{\prime}}\) and \(n_{y^{\prime}}\). However, the difference \(\Delta n_{x^{\prime}y^{\prime}}=n_{x^{\prime}}-n_{y^{\prime}}\) can be evaluated. With normalized Mueller matrices as used here, we also lose sensitivity to isotropic absorption; however, \(\Delta k_{x^{\prime}y^{\prime}}=k_{x^{\prime}}-k_{y^{\prime}}\) can be determined. As \(\varepsilon_{z}\) can be set arbitrarily, we use \(\boldsymbol{\epsilon}=\mathbf{A}_{\text{E}}\text{diag}(N_{x^{\prime}}^{2},( N_{x^{\prime}}-\Delta n_{x^{\prime}y^{\prime}}-\mathrm{i}\Delta k_{x^{ \prime}y^{\prime}}),N_{x^{\prime}}^{2})\mathbf{A}_{\text{E}}^{-1}\), where \(\mathbf{A}_{\text{E}}\) is the Euler rotation matrix with only the in-plane rotation angle \(\phi\
eq 0\). Equation (11) can now be reformulated as  
\[\mathbf{D}=\varepsilon_{0}\mathbf{A}_{\text{E}}\text{diag}(N_{x^{\prime}}^{2 },(N_{x}-\Delta n+\mathrm{i}\Delta k)^{2},N_{x}^{2})\mathbf{A}_{\text{E}}^{-1} \mathbf{E}+c^{-1}\mathrm{i}\kappa\mathbf{H} \tag{12a}\] \[\mathbf{B}=-c^{-1}\mathrm{i}\kappa\mathbf{E}+\mu_{0}\mathbf{H}, \tag{12b}\]  
where \(\mu=\mathbf{I}\) is assumed.