Creating anisotropic optical materials in FDTD and MODE

FDTD MODE

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 in the following 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. $U=V^†$ where $V$ is the eigenvector matrix of $\varepsilon$ returned by eig.

The diagonal values of $\varepsilon_D$ should be entered into the materials database then a matrix transformation grid attribute needs to be added using $U$. Thus you are required to define $U$, the matrix transformation grid attribute and $\varepsilon_D$ the diagonal material anisotropy.

Please note that convention used above for matrix diagonalization may be different from the convention used elsewhere, such as the script command eig, where the equation is typically written in the form $\varepsilon_D=V^\dagger\varepsilon V$ where $V$ is the eigenvector matrix of $\varepsilon$. In other words, $U$ above is the complex conjugate transpose of the eigenvector matrix $V$ of $\varepsilon$ returned by eig: $U=V^†$.

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.