The Indicatrix

Steven Dutch, Professor Emeritus, Natural and Applied Sciences, Universityof Wisconsin - Green Bay

This is an overview of the indicatrix concept and some fundamental terms. The uniaxial and biaxial indicatrices are discussed in more detail on other pages.

Vibration Directions

Light is made up of oscillating electric and magnetic fields. We could describe optical phenomena equally well in terms of either the electric or magnetic field components, but not equally simply. We generally deal with the electric field component since it's much easier to picture. Electrons move in the direction of the electric field. In contrast, electrons move perpendicular to magnetic fields, making the description way more complicated.

In this crystal, an electric field oscillating in the direction shown in red will interact with the electrons in the atoms a certain way.

Any light with its electric field vector oscillating in that direction will interact with the material the same.

Thus we meet the counterintuitive fundamental principle of optical crystallography: vibration direction, not ray path, is paramount.

Not only do an infinite number of rays interact with the material the same way, but light traveling in a given direction, but polarized differently, will interact differently with the material.

If we construct vectors whose direction corresponds to the vibration direction, and whose length corresponds to the refractive index of the material for light with that vibration direction, their tips will define an imaginary surface called the indicatrix.
The indicatrix is a shape called an ellipsoid. Every cross section of an ellipsoid is an ellipse. The longest direction of the ellipsoid is its major axis. The shortest direction, perpendicular to the major axis, is called the minor axis. Perpendicular to the major and minor axes is the intermediate axis. These three axes are called the principal axes.

If light has the same refractive index for all vibration directions in the material, the material is said to be isotropic. Two types of material are isotropic: isometric materials like halite, fluorite, or garnet, and noncrystalline materials like glass. Since the refractive index is the same for all vibration directions, the indicatrix is a sphere.

Also, since all light in the material travels at the same speed (a consequence of having a single refractive index) there is no retardation. The birefringence is zero, the retardation is zero, and the material remains continuously dark when you rotate the stage.

Obviously, if the cross-section of the indicatrix is a circle, all light vibrating in that plane will have the same refractive index and in that plane only, the material will be isotropic. The direction perpendicular to that plane is called the optic axis. Every ellipsoid has at least one circular cross-section. Therefore, every indicatrix has at least one optic axis, and every material has at least one direction where it appears isotropic.

Uniaxial Minerals

If we rotate an ellipse around its major or minor axis, we sweep out an ellipse with a circular cross section. Such an ellipsoid is called an ellipsoid of revolution. If we rotate around the minor axis we get a shape that is flattened along the rotation axis (left); if we rotate around the major axis the ellipsoid is elongated along the rotation axis.

A material with an indicatrix of this shape is called uniaxial. There is only one optic axis: down the axis of revolution. Uniaxial materials obviously must have one principal symmetry axis and be tetragonal, hexagonal, or trigonal. Quartz, calcite, corundum, beryl, rutile, tourmaline and nepheline are important uniaxial minerals.

It's useful to know not just whether a mineral is uniaxial, but which of the two types of indicatrix it has. If the refractive index along the symmetry axis is smaller, the mineral has the indicatrix at left and is said to be optically negative. If the refractive index along the symmetry axis is greaterer, the mineral has the indicatrix at right and is said to be optically positive.

There's an easy memory trick to keep the two straight. The negative indicatrix is flat like a minus sign. The positive indicatrix is tall like the upright stroke in a plus sign.

Biaxial Minerals

An ellipse with three unequal principal axes has two circular cross sections. The plane containing the major and minor axis cuts the ellipsoid in an ellipse with the maximum and minimum possible radii. Somewhere in between is a radius equal to the intermediate axis. The two circular sections have radius equal to the intermediate axis and intersect along the intermediate axis. They are shown in blue and purple at left.
The direction perpendicular to each circular section is the optic axis. For a generic ellipsoid with three unequal axes, there are two optic axes. Such materials are called biaxial.

Biaxial minerals are orthorhombic, monoclinic or triclinic and include the pyroxenes, amphiboles, micas, feldspars, and olivine.

Biaxial minerals can also be optically positive or negative. If the intermediate axis is closer to the major axis in length, the circular sections will make small angles with the major axis and the optic axes will be close to the minor axis. The indicatrix will look much like a uniaxial negative indicatrix, and the mineral is optically negative (left). If the intermediate axis is close to the minor axis in length, the circular sections will make small angles with the minor axis and the optic axes will be close to the major axis. The indicatrix will look much like a uniaxial positive indicatrix, and the mineral is optically positive (right). If the optic axes are perpendicular to each other, the mineral is on the dividing line between positive and negative.

Determining Vibration Directions

How can you tell what the vibration directions will be for a randomly-oriented ray of light? Fortunately, there's an easy rule.

Given a random ray direction (red):
  • Find the two planes containing the ray path and each optic axis (gray and yellow).
  • Bisect the angles where the planes intersect.
  • The bisectors are the vibration directions (green).

This is the Biot-Fresnel Rule (the same Biot after whom biotite is named).

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Created 22 February 2005, Last Update