Birefringence

Crystal symmetries and Birefringence

KostaCLOUD supports materials with the following Crystal structures. By default it is assumed that materials are amorphous and do not have any particular symmetry groups. But if one is to use a crystal within the program here are the following symmetry groups for permittivity. As a quick reminder, here is the relationship between permittivity and refractive index. Typically μ is close to 1 at optical frequencies.

$$n = \sqrt{\epsilon_r \mu_r}$$

Triclinic Crystals have the following form:

$$\epsilon = \begin{bmatrix} a & b & c\\ b & d & e\\ c & e & f \end{bmatrix}$$

Monoclinic Crystals have the following form:

$$\epsilon = \begin{bmatrix} a & 0 & b\\ 0 & c & 0\\ d & 0 & e \end{bmatrix}$$

Orthorhombic Crystals have the following form:

$$\epsilon = \begin{bmatrix} a & 0 & 0\\ 0 & b & 0\\ 0 & 0 & c \end{bmatrix}$$

Tetragonal Crystals have the following form:

$$\epsilon = \begin{bmatrix} a & 0 & 0\\ 0 & b & 0\\ 0 & 0 & b \end{bmatrix}$$

Trigonal Crystals have the following form:

$$\epsilon = \begin{bmatrix} a & 0 & 0\\ 0 & a & 0\\ 0 & 0 & b \end{bmatrix}$$

Hexagonal Crystals have the following form:

$$\epsilon = \begin{bmatrix} a & 0 & 0\\ 0 & a & 0\\ 0 & 0 & b \end{bmatrix}$$

Cubic Crystals have the following form:

$$\epsilon = \begin{bmatrix} a & 0 & 0\\ 0 & a & 0\\ 0 & 0 & a \end{bmatrix}$$

Crystal Axis rotations

Additionally any crystal axis can be rotated relative to the part's mechanical axis, which is the default crystal axis. We can do this by applying a rotation matrix to our permittivity. Where we define our rotation matrix similarly to how we define it for geometric rotations. (ZYX rotation order)

$$\epsilon' = \begin{bmatrix} a_{xx} & a_{xy} & a_{xz}\\ a_{yx} & a_{yy} & a_{yz}\\ a_{zx} & a_{zy} & a_{zz} \end{bmatrix} \epsilon \\ \quad \\ \begin{bmatrix} a_{xx} & a_{xy} & a_{xz}\\ a_{yx} & a_{yy} & a_{yz}\\ a_{zx} & a_{zy} & a_{zz} \end{bmatrix} = \begin{bmatrix} \cos\theta\cos\phi & \cos\theta\sin\phi\sin\psi-\sin\theta\sin\phi & \cos\theta\sin\phi\cos\psi+\sin\theta\sin\psi\\ \sin\theta\cos\phi& \sin\theta\sin\phi\sin\psi+\cos\theta\cos\psi& \sin\theta\sin\phi\cos\psi-\cos\theta\sin\psi\\ -\sin\phi & \cos\phi\sin\psi & \cos\phi\cos\psi \end{bmatrix}$$