Birefringence is the optical property of a material having a refractive index that depends on the polarization and propagation direction of light. It is one of the oldest methods to determine orientations of crystals in thin sections of rocks.

## Import Olivine Data

In order to illustrate the effect of birefringence lets consider a olivine data set.

## The refractive index tensor

The refractive index of a material describes the dependence of the speed of light with respect to the propagation direction and the polarization direction. In a linear world this relation ship is modeled by a symmetric rank 2 tensor - the so called refractive index tensor, which is usually given by it principle values: n_alpha, n_beta and n_gamma. In orthorhombic minerals such as olivine the principal values are parallel to the crystallographic axes. Care has to be applied when associating the principle values with the correct axes.

For Forsterite the principle refractive values are

with the largest refractive index n_gamma being aligned with the a-axis, the intermediate index n_beta with the c-axis and the smallest refractive index n_alpha with the b-axis. Hence, the refractive index tensor for Forsterite takes the form

For Fayalite the principle refractive values

are aligned to the crystallographic axes in an analogous way. Which leads to the refractive index tensor

The refractive index of composite materials like Olivine can now be modeled as the weighted sum of the of the refractive index tensors of Forsterite and Fayalite. Lets assume that the relative Forsterite content (atomic percentage) is given my

Then is refractive index tensor becomes

## Birefringence

The birefringence describes the difference `n`

in diffraction index between the fastest polarization direction `pMax`

and the slowest polarization direction `pMin`

for a given propagation direction `vprop`

.

If the polarization direction is omitted the results are spherical functions which can be easily visualized.

## The Optical Axis

The optical axes are all directions where the birefringence is zero

## Spectral Transmission

If white light with a certain polarization is transmitted though a crystal with isotropic refractive index the light changes wavelength and hence appears colored. The resulting color depending on the propagation direction, the polarization direction and the thickness can be computed by

Effectively, the rgb value depend only on the angle tau between the polarization direction and the slowest polarization direction `pMin`

. Instead of the polarization direction this angle may be specified directly

If the angle tau is fixed and the propagation direction is omitted as input MTEX returns the rgb values as a spherical function. Lets plot these functions for different values of tau.

Usually, the polarization direction is chosen at angle phi = 90 degree of the analyzer. The following plots demonstrate how to change this angle

## Spectral Transmission at Thin Sections

All the above computations have been performed in crystal coordinates. However, in practical applications the direction of the polarizer as well as the propagation direction are given in terms of specimen coordinates.

As usual we have two options: Either we transform the refractive index tensor into specimen coordinates or we transform the polarization direction and the propagation directions into crystal coordinates. Lets start with the first option:

and compare it with option two:

## Spectral Transmission as a color key

The above computations can be automated by defining a spectral transmission color key.

As usual we me visualize the color key as a colorization of the orientation space, e.g., by plotting it in sigma sections:

## Circular Polarizer

In order to simulate we a circular polarizer we simply set the polarizer direction to empty, i.e.