The Maxwell equations in the presence of a medium are given in SI units (m kg s A) as: 

∇⋅ D = 0                 ∇× E = - ∂/∂t B
∇⋅ B = 0                 ∇× H =   ∂/∂t D

Furthermore there are the phenomenological relations between the fields

D = ε ε₀ E                  B = μ μ₀ H

In vacuum ε=μ=1, in a medium a microscopic theory is necessary to determine ε and μ. The propagation of an electromagnetic wave through the medium is determined by the wave equation, which can be derived from the above equations by considering :

∇ × (∇ × E) = -∇ × (∂/∂t B)

The left-hand side simplifies with the help of ∇D = 0 to

∇ × (∇ × E) = ∇ × (∇ E) - (∇²) E  = -∇²E

The right-hand side can be rewritten as

-∇ × ∂/∂tB = ∂/∂t ∇ × μ μ₀H = μ μ₀ ∂²/∂t²D = ε ε₀ μ μ₀ ∂²∂t²E

yielding

∇² E = 1/c² ∂²/∂t²E

with the speed of light in the medium given by c = 1/sqrt( ε ε₀ μ μ₀).

The refractive index n is then defined as the ratio between the speed of light in vacuum (ε=μ=1) and the speed of light in the medium:

n = c₀ / c = sqrt(ε μ).

Solutions of the wave equation are plane waves

E = E₀ exp(i k r - ωt)

which have to fulfill the dispersion relation

ω = c k = (c₀/n) k

where k = || k ||.
 
 

Boundary conditions for the k vector

Let us consider a wave travelling through medium 1 and hitting a plane interface with medium 2. The most general situation will be that part of the incident wave is reflected and part of it refracted into medium 2. The solutions of the wave equation in both media are subject to boundary conditions at the interface. First of all, the frequencies of the three wave must be equal on either side of the interface:

ω = ω'

k' = n k

k'_|| = k_||

k'_z² = k'² - k'_||² = (n k)² - k_||² = (n k)² - (k² - k_z²) = (n²-1) k² - k_z²