Consider a plane wave with wave vector k propagating through a medium with a dielectric tensor

.

In a coordinate system with its axes along the principal axes, the equation D = ε₀n²E_┴ for a component of D along one of the principal axes becomes
ε_iE_i = ε₀n²(E_i – (1/k²)k_i(k∙E)),
since E_┴ = E - e_k(e_k∙E) = E – (1/k²)k(k∙E)).
This can be rewritten as ε_iΣ_j(δ_ijE_j)  = ε₀n²(Σ_j(δ_ijE_j) – (1/k²)k_iΣ_j(k_jE_j)), or

with .

In matrix form this may be written as ME = 0, with

.

Here we have used n_i² = ε_i/ε₀.

If the ε_i are known, and we choose a particular direction of propagation e_k for the wave front, then we can solve for E.

ME = 0 only if the characteristic equation is satisfied and det(M) = 0.  The characteristic equation appears cubic in n², but when written in terms of a polynomial in n² the coefficients of the n⁶ terms vanish, leaving a quadratic equation in n².  This equation has two positive roots, n₁² and n₂².

For a given direction of propagation e_k, there are in general two values for the refractive index, n₁ and n₂.  We can solve for corresponding components of E up to a multiplicative constant.  Let the solutions E₁ and E₂ correspond to n₁ and n₂, respectively.

We may resolve E into components parallel and perpendicular to e_k.

If n₁² is not equal to n₂², then D₁ and D₂ are perpendicular to each other.

If n₁² is equal to n₂², then the directions of D₁ and D₂ are arbitrary as long as they are perpendicular to e_k and it is convenient to choose them perpendicular to each other.

If the parallel components of both E₁ and E₂ are not zero, then E₁·E₂ =  E_1|| ·E_2|| ≠ 0.  (Parallel refers to the direction of k.)    The parallel components of E₁ and E₂ are only zero if the wave propagates along one of the principal axes of the crystal and the matrix M is diagonal.

Example:

Let e_k be the z-direction, i.e. the wave front propagates along the z-direction. Then k = k_z and

.

The characteristic equation det(M) = 0, or (n²-n_x²) (n²-n_y²) n_z² = 0 implies that n² = n_x², or n² = n_y², or n_z² = 0.  But n_z² = 0 is not a physical solution, since ε_z > 0.  We therefore have two solutions, n₁² = n_x² and n₂² = n_y².

The direction of E₁ = D₁/( ε₀n₁²) corresponding to n₁ is the x-direction and the direction of E₂ = D₂/( ε₀n₂²) corresponding to n₂ is the y-direction.  E is parallel to D.

When D is not polarized along the x- or y-direction, then E is not parallel to D, as required.  Such a wave does not propagate through the material.  A wave propagating along a principal axis must be linearly polarized along one of the two remaining principal axes of the dielectric tensor.  An arbitrarily polarized wave must be treated a linear superposition of two waves each linearly polarized along one of the two remaining principal axes, and these two waves propagate independently with different phase velocities.

The phase velocity of the wave polarized along the x-direction is c/n₁, and the phase velocity of the wave polarized along the x-direction is c/n₂.