Question

Obtain expressions for reflection and transmission amplitude coefficients when
electric vector associated with a plane monochromatic e. m. wave is in the plane of
incidence.

Answer 1

We note that any plane wave may be represented as a superposition of two orthogonal linearly polarized waves.

Ignoring the rapidly varying parts of the light wave and keeping only the complex amplitudes:

$E_i+E_r=E_t (1)$

where E_i is the incident amplitude, E_r the reflected amplitude, E_t is the transmitted amplitude

For the electric field the continuity relation becomes (θ_i= θ_r):

$-E_i×\cos θ_i + E_r×\cos θ_i = -E_t×\cos θ_t (2)$

We can write for the magnetic field

$-B_i×\cos θ_i + B_r×\cos θ_i = -B_t×\cos θ_t (3)$

The link between E and B is given by formula

$B=\frac {nE}{c} (4)$

We put (4) in (3)

$-n_i×(E_r- E_i)\cos θ_i = -n_t×E_t×\cos θ_t (5)$

We put (1) in (5)

$-n_i×(E_r- E_i)\cos θ_i = -n_t×(E_i+E_r)×\cos θ_t (5)$

Solving (5) for $\frac {E_r}{ E_i }$ yields reflection and transmission coefficients for perpendicularly polarized light (Fresnel Equations)

$n=\frac {n_i×\cos θ_i -n_t×\cos θ_t}{n_i×\cos θ_i + n_t×\cos θ_t}$

$t=\frac {2n_i×\cos θ_i}{n_i×\cos θ_i + n_t×\cos θ_t}$