Optical indicatrix

In optics, an index ellipsoid is a diagram of an ellipsoid that depicts the orientation and relative magnitude of refractive indices in a crystal .

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

and the reduced displacement as

then the index ellipsoid is defined by the equation

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 ${\displaystyle {\vec {s}}}$ by taking the intersection of the plane ${\displaystyle {\vec {R}}\cdot {\vec {s}}=0}$ with the index ellipsoid. The axes of the resulting ellipse are the resulting polarization directions.

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