# Optical Activity of Quartz

• Paul Colby
In summary: It can take only one of two forms,##\epsilon_{ijl}=\left(\begin{array}{ccc}\epsilon_{a}&0&0\cr 0&\epsilon_{a}&0\cr 0 & 0 & \epsilon_{b}\end{array}\right)####\epsilon_{ijl}=\left(\begin{array}{ccc}\mu_{a}&0&0\cr 0&\mu_{a}&0\cr 0 & 0 & \mu_{b}\end{array}\right)##The first form is not allowed, because the permittivity tensor has to have the

#### Paul Colby

Gold Member
Hi,

I've been looking at the optics of ##\alpha##-quartz which comes in two parities, left and right. Quartz is optically active which means that the plane of a linearly polarized beam propagating along the optic axis is rotated by an angle proportional to the distance traveled. I would like to express this in terms of tensor constitutive parameters, ##\epsilon_{nm}## and ##\mu_{nm}##. Here is where this runs aground. Crystal symmetry limits the form of the permittivity tensor to the form,

##\epsilon_{nm} = \left(\begin{array}{ccc}\epsilon_{a}&0&0\cr 0&\epsilon_{a}&0\cr 0 & 0 & \epsilon_{b}\end{array}\right)##​

The very same symmetry arguments would require,

##\mu_{nm} = \left(\begin{array}{ccc}\mu_{a}&0&0\cr 0&\mu_{a}&0\cr 0 & 0 & \mu_{b}\end{array}\right)##​

Okay, for a beam propagating along the optic (##z##-axis) no optical activity can be generated from constitutive relations of this form? What gives?

You have to take spatial dispersion into account. I.e., the electric polarisation does not only depend on the field at the same point in space but also on the field in nearby points. In Fourier space, this means that epsilon on the wavevector k. To obtain optical activity, the first term in a Taylor expansion of epsilon in powers of k has to be taken into account, i.e.,
## \epsilon_{ij}(k)=\epsilon^0_{ij}+\epsilon^{(1)}_{ijl} k_l+\ldots##. Magnetic effects can be taken into account with the quadratic terms in k, so no need for a separate tensor ##\mu##.
However, if you really want so, you can alternatively introduce an additional tensor which describes the dependence of ##P## on ##B## to get optical activity.

Details can be found e.g. in Landau Lifshetz, Electrodynamics of continua.

vanhees71 and Paul Colby
Thanks, I even have that book. Time to use it. Ah, ##\epsilon_{ijl}## clearly has different symmetry constrains.

vanhees71

## What is optical activity in quartz?

Optical activity in quartz refers to the ability of quartz crystals to rotate the plane of polarized light as it passes through the crystal. This phenomenon is caused by the asymmetric arrangement of atoms within the crystal lattice, which creates a difference in the refractive index for left- and right-handed circularly polarized light.

## How does optical activity in quartz occur?

Optical activity in quartz occurs due to a phenomenon called optical rotation, where the orientation of polarized light is rotated as it passes through the crystal. This is caused by the difference in refractive index for the two circular polarizations, which results in a phase shift between them.

## What factors affect the degree of optical activity in quartz?

The degree of optical activity in quartz is affected by several factors, including the thickness of the crystal, the wavelength of light, and the temperature of the crystal. Thicker crystals and longer wavelengths of light result in a greater degree of optical rotation, while higher temperatures can decrease the level of optical activity.

## What are some applications of optical activity in quartz?

Optical activity in quartz has numerous applications in the fields of optics, geology, and materials science. It is used in optical devices such as polarizing filters and waveplates, as well as in studying the properties of minerals and rocks. It is also used in the production of optically active materials, such as certain types of glass and polymers.

## Can the optical activity of quartz be measured?

Yes, the optical activity of quartz can be measured using various techniques, such as polarimetry and circular dichroism. These methods involve measuring the rotation of polarized light as it passes through a quartz crystal and using mathematical calculations to determine the degree of optical activity. These measurements are important in understanding the properties of quartz and its potential applications.