Anisotropic materials can be represented by a 9 element permittivity tensor \( \varepsilon _{ij} \) such that the electric and displacement fields are related via the relation.
$$ D_{i}=\varepsilon_{ij} E_{j} $$
where summation over j is implied on the right hand side. The full anisotropy tensor can be written as
$$\boldsymbol{\varepsilon} = \begin{bmatrix} \varepsilon_{11} & \varepsilon_{12} & \varepsilon_{13} \\ \varepsilon_{21} & \varepsilon_{22} & \varepsilon_{23} \\ \varepsilon_{31} & \varepsilon_{32} & \varepsilon_{33} \end{bmatrix} $$
The input of anisotropic materials is simple when the permittivity tensor is diagonal
$$\boldsymbol{\varepsilon} = \begin{bmatrix} \varepsilon_{x} & 0& 0 \\ 0 & \varepsilon_{y} & 0 \\ 0 & 0 & \varepsilon_{z} \end{bmatrix}$$
Users may find the Liquid crystal simulation video helpful.
Diagonal anisotropic materials
To define an anisotropic material, set the Anisotropy field in the Material database to Diagonal and specify the material model parameters for each diagonal component. When viewing the material data with the material explorer, use the 'axis' property to select the diagonal component to visualize.
General anisotropic materials
If you have a more general form of anisotropy, you must first diagonalize the permittivity matrix (use the ` eig` script command) and find both the eigenvalues and the unitary transformation that makes the permittivity diagonal. Then you can set your permittivity in the general form:
$$\varepsilon _D=U\varepsilon U^\dagger$$
where \(U\) is a unitary matrix, \(U^† = U^{-1}\) is the complex conjugate transpose of \(U\) and \(\varepsilon _D\) is diagonal. The diagonal values of \(\varepsilon_D\) should be entered into the materials database then a matrix transform grid attribute needs to be added using \(U\). Thus you are required to define \(U\) the matrix transformation associated grid attribute and \(\varepsilon_D\) the principle diagonal material anisotropy.
For more information, see the following sub-topics.
Grid attribute tips and introduction , LC rotation grid attribute, Permittivity rotation grid attribute , Matrix transformation grid attribute
For examples of using a diagonalized permittivity matrix, see:
MOKE, Faraday effect
Simple anisotropic indices
Anisotropic index values can be set using the material tab in a structure, if a dielectric material is used. To specify an anisotropic refractive index, use a semicolon to separate the diagonal \(xx\), \(yy\), \(zz\) indices. Eg. 1;1.5;1.
Example diagonal anisotropic simulation
$$\varepsilon = \begin{bmatrix} \varepsilon_{x} & 0& 0 \\ 0 & \varepsilon_{y} & 0 \\ 0 & 0 & \varepsilon_{z} \end{bmatrix}$$
The diagonally anisotropic material in the example file has a permittivity of n_xx, n_yy, n_zz = [2, 2, 1.001]. The anisotropic nature of the material is easily visible by comparing the field profiles of the Ex and Ez fields. Notice the reflection and refraction visible in the Ex fields that is not present in the Ez fields.
