Index ellipsoid (optical indicatrix)

The index ellipsoid, or optical indicatrix, is a geometric construction that gives the refractive indices and polarization directions of the two waves that can propagate in any direction through an anisotropic crystal. In the crystal's principal axes it is the surface

$$\frac{x^2}{n_x^2} + \frac{y^2}{n_y^2} + \frac{z^2}{n_z^2} = 1,$$

whose semi-axes are the three principal indices. For propagation along a unit vector $\hat{k}$, the plane through the center perpendicular to $\hat{k}$ cuts the ellipsoid in an ellipse; the lengths of its semi-axes are the two indices, and their directions are the directions of the electric displacement $\mathbf{D}$ of the two eigenpolarizations. In lithium niobate at 1550 nm ($n_o = 2.211$, $n_e = 2.138$), a wave traveling at 45° to the optic axis has indices 2.211 and 2.174.

Where it comes from

The ellipsoid is the surface of constant electric energy density written in terms of $\mathbf{D}$: with the relative impermeability tensor $\eta = \varepsilon_r^{-1}$, the condition $\eta_{ij}x_i x_j = 1$ is the same equation with $\eta$ in place of $1/n^2$. In principal axes $\eta$ is diagonal, $\eta_{ii} = 1/n_i^2$. Writing the ellipsoid in terms of $\eta$ is what makes it convenient for perturbations: the electro-optic effect and the photoelastic effect are expressed as small changes $\Delta\eta_{ij}$, which deform and rotate the ellipsoid, after which the new principal axes and indices are found by diagonalizing it.

Uniaxial crystals

When two principal indices are equal, $n_x = n_y = n_o$ and $n_z = n_e$, the ellipsoid is a spheroid about the $z$ axis, which is the optic axis. Every cross-section contains a semi-axis of length $n_o$, so one wave, the ordinary wave, always has index $n_o$. The other semi-axis lies in the plane of $\hat{k}$ and the optic axis and gives the extraordinary index at angle $\theta$ to the axis:

$$\frac{1}{n_e(\theta)^2} = \frac{\cos^2\theta}{n_o^2} + \frac{\sin^2\theta}{n_e^2}.$$

For lithium niobate at 1550 nm this gives 2.192 at 30°, 2.174 at 45° and 2.156 at 60°, moving from $n_o$ along the axis to $n_e$ perpendicular to it. Quartz, calcite, lithium niobate and YVO₄ are uniaxial; the birefringence entry tabulates their indices. Because $\mathbf{D}$, not $\mathbf{E}$, lies along the semi-axis, the energy of the extraordinary wave flows at a small angle to $\hat{k}$, the walk-off angle.

Biaxial crystals

When all three indices differ, as in KTP, LBO and mica, the ellipsoid has two circular cross-sections, both of radius $n_y$ when the axes are labeled so that $n_x < n_y < n_z$. The normals to these circles are the two optic axes, along which the two waves have the same index. They lie in the $xz$ plane at an angle $V$ to the $z$ axis given by

$$\tan^2 V = \frac{n_x^{-2} - n_y^{-2}}{n_y^{-2} - n_z^{-2}}.$$

For other directions the two indices must be found from the elliptical section, and neither wave is ordinary in general. Phase-matching calculations in biaxial crystals are therefore usually restricted to the principal planes, where one wave has a fixed index.

Using it in practice

The construction answers the routine questions of polarization optics: which polarization a waveplate cut along a given direction retards, what index an extraordinary beam sees at a chosen phase-matching angle, and how a field applied to a Pockels cell rotates the principal axes. In lithium niobate modulators, for example, a field along $z$ changes the lengths of the semi-axes without rotating them, and z-cut and x-cut devices apply it with light polarized along $z$ to use the largest coefficient, $r_{33}$ (about 31 pm/V).

Pitfalls

The ellipsoid gives indices for directions of the wave vector; the ray (Poynting) direction differs, and a separate ray ellipsoid with semi-axes $1/n_i$ applies to rays. Principal indices depend on wavelength and temperature, and in some monoclinic and triclinic crystals the principal axes themselves rotate with wavelength. Published tables differ in whether they label axes by $n_x < n_y < n_z$ or by crystallographic axes, so the axis labels must be checked before using the biaxial formulas.

Common questions

How do you find the refractive index for an arbitrary direction?

Take the cross-section of the index ellipsoid perpendicular to the propagation direction. Its two semi-axes are the two refractive indices, and their directions are the polarizations ($\mathbf{D}$) of the corresponding waves.

What is the difference between a uniaxial and a biaxial crystal?

A uniaxial crystal has two equal principal indices and one optic axis; its indicatrix is a spheroid. A biaxial crystal has three different principal indices and two optic axes; its indicatrix is a general ellipsoid.

References: A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, 1984); M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999); B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 3rd ed. (Wiley, 2019).