6
1 Introduction
crystal exhibits two optical axes (which will not be parallel to the principal axes of
the crystal) and is termed biaxial.
1.1.3 Plane Wave Propagation Through an Anisotropic
Medium
In this section we will consider the propagation of electromagnetic waves through
anisotropic dielectric media. We will treat the orthogonal electric and magnetic field
components of electromagnetic waves as plane waves of the form
E = E 0 e
−i(k·r−ωt)
,
H = H 0 e
−i(k·r−ωt)
,
(1.15)
describing the wave at position r with a wavevector k = k u oriented along the unit
vector u.
1.1.3.1 Dispersion Relation in an Anisotropic Medium
Maxwell’s equations in a dielectric medium can be expressed as
∇ × E = −μ 0
∂H
∂t
,
(1.16)
∇ × H = − ·
∂E
∂t
.
(1.17)
By considering the electric and magnetic fields as plane waves of the form given in
Eq. 1.15 and using the relation D = E, Eqs. 1.16 and 1.17 reduce to
k × E = ωμ 0 H,
(1.18)
k × H = −ωD.
(1.19)
These equations define the geometry of the vectors describing an electromagnetic
wave in a dielectric medium, shown schematically in Fig. 1.2: H is oriented perpedicular to both k and E, whilst D is perpendicular to both k and H. Defining the
direction of energy flow via the Poynting vector S = E × H sets it perpendicular to
both E and H. Therefore the vectors E, D, k and S all lie in the plane perpedicular
to H; however, due to Eq. 1.5 E and D (and therefore k and S) are not necessarily
parallel to each other. Rearranging Eq. 1.18 for H and substituting into Eq. 1.19, we
obtain
k × (k × E) + ω
2
μ 0 E = 0.
(1.20)