Index ellipsoid

In optics, an index ellipsoid is a diagram of an ellipsoid that depicts the orientation and relative magnitude of refractive indices in a crystal .[1]

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

${\displaystyle 8\pi W={\frac {D_{1}^{2}}{\varepsilon _{1}}}+{\frac {D_{2}^{2}}{\varepsilon _{2}}}+{\frac {D_{3}^{2}}{\varepsilon _{3}}}.}$

and the reduced displacement as

${\displaystyle R_{i}={\frac {D_{i}}{\sqrt {8\pi W}}},}$

then the index ellipsoid is defined by the equation

${\displaystyle {\frac {R_{1}^{2}}{\varepsilon _{1}}}+{\frac {R_{2}^{2}}{\varepsilon _{2}}}+{\frac {R_{3}^{2}}{\varepsilon _{3}}}=1.}$

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 ${\displaystyle {\vec {s}}}$ 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.