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The index ellipsoid is a diagram of an ellipsoid that depicts the orientation and relative magnitude of refractive indices in a crystal.## Uniaxial indicatrix

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The equation for the ellipsoid is constructed using the electric displacement vector, D, and the dielectric constants. Defining the field energy, W, as:

- $8pi\; W=\; D^2\_1/epsilon\_1\; +D^2\_2/epsilon\_2\; +\; D^2\_3/epsilon\_3$

and the reduced displacement as:

- $R\_i=\; D\_i/sqrt\{8/pi\; W\}$

then the index ellipsoid is defined by the equation,

- $R\_1^2/epsilon\_1\; +R\_2^2/epsilon\_2\; +R\_3^2/epsilon\_3\; =\; 1$ .

The semiaxes of this ellipsoid are dielectric constants of the crystal.

This ellipsoid can be used to determine the polarization of an incoming wave, with wave vector, s, by taking the intersection of the plane $R\; cdot\; s\; =0$ with the index ellipsoid. The axes of the resulting ellipse are the resulting polarization directions.

An important special case of the index ellipsoid occurs when the ellipsoid is an ellipsoid of revolution, e.g. constructed by rotating an ellipse around either the minor or major axis. In this case, there is only one optical axis; the axis of rotation. In such a case, the material is uniaxial and has only one principal symmetry axis.

- Index ellipsoid calculator on Tal Carmon Research Homepage
- Tensors and Ellipsoids
- Uniaxial Indicatrix

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Last updated on Wednesday June 04, 2008 at 13:37:58 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Wednesday June 04, 2008 at 13:37:58 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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