Chapter V: Propagation of Light Through Crystals

Source: Gerrard and Burch, Introduction to Matrix Methods in Optics (1975), Chapter V.

This chapter contains no numbered illustrative problems. It applies matrix notation to vector products, the dielectric tensor, plane waves in uniaxial crystals, and Huygens wavelets. The compact derivation below records the chapter’s central reusable result.

Plane-wave eigenproblem

For a nonmagnetic anisotropic dielectric, insert \(\mathbf E=\mathbf E_0e^{i(\mathbf k\cdot\mathbf r-\omega t)}\) into Maxwell’s equations. Eliminating \(\mathbf H\) gives

\[\mathbf k\times(\mathbf k\times\mathbf E_0) +\frac{\omega^2}{c^2}\boldsymbol\epsilon_r\mathbf E_0=0.\]

Writing the cross product as the antisymmetric matrix \(K(\mathbf k)\), the allowed waves satisfy

\[\left[K(\mathbf k)^2+ \frac{\omega^2}{c^2}\boldsymbol\epsilon_r\right]\mathbf E_0=0, \qquad \det\left[K^2+ rac{\omega^2}{c^2}\boldsymbol\epsilon_r\right]=0.\]

For a uniaxial crystal with optic axis along \(z\), \(\boldsymbol\epsilon_r=\operatorname{diag}(n_o^2,n_o^2,n_e^2)\). The determinant separates into the ordinary sphere and extraordinary ellipsoid. The ordinary wave has \(n=n_o\); the extraordinary effective index obeys

\[\boxed{\frac{1}{n_e(\theta)^2} =\frac{\cos^2\theta}{n_o^2} +\frac{\sin^2\theta}{n_e^2}},\]

with \(\theta\) measured from the optic axis under this convention.

Check

Propagation along the optic axis makes the two indices equal to \(n_o\), so there is no double refraction. Perpendicular propagation recovers the principal extraordinary value \(n_e\).