Fresnel Equations

10.1 Derivation at a Dielectric Interface

When light strikes a planar interface between media with refractive indices $n_1$ and $n_2$The Amplitudes of the reflected and transmitted waves depend on the polarisation.

For an incident wave with amplitude $E_i$The reflection and transmission coefficients are:

s-polarisation (perpendicular to the plane of incidence):

$r_s = \frac{n_1\cos\theta_i - n_2\cos\theta_t}{n_1\cos\theta_i + n_2\cos\theta_t}, \quad t_s = \frac{2n_1\cos\theta_i}{n_1\cos\theta_i + n_2\cos\theta_t}$

p-polarisation (parallel to the plane of incidence):

$r_p = \frac{n_2\cos\theta_i - n_1\cos\theta_t}{n_2\cos\theta_i + n_1\cos\theta_t}, \quad t_p = \frac{2n_1\cos\theta_i}{n_2\cos\theta_i + n_1\cos\theta_t}$

Reflectance and transmittance (energy fractions):

$R = |r|^2, \quad T = \frac{n_2\cos\theta_t}{n_1\cos\theta_i}|t|^2$

With $R + T = 1$ (energy conservation).

10.2 Brewster”s Angle

At the Brewster angle $\theta_B$The reflected beam for p-polarised light has zero amplitude: $r_p = 0$:

$\tan\theta_B = \frac{n_2}{n_1}$

Proof. Setting $r_p = 0$: $n_2\cos\theta_i = n_1\cos\theta_t$. Using Snell’s law $n_1\sin\theta_i = n_2\sin\theta_t$:

$\frac{\cos\theta_i}{\sin\theta_i} = \frac{\cos\theta_t}{\sin\theta_t}$

$\cot\theta_i = \cot\theta_t \implies \theta_i + \theta_t = 90^\circ$

So $\tan\theta_i = \tan\theta_B = n_2/n_1$. $\blacksquare$

At Brewster’s angle, the reflected and refracted beams are perpendicular. This is why polarising Filters work at specific angles for reflected glare.

10.3 Total Internal Reflection and the Evanescent Wave

When $n_1 \gt n_2$ and $\theta_i \gt \theta_c = \arcsin(n_2/n_1)$, $\sin\theta_t \gt 1$So $\cos\theta_t = i\sqrt{\sin^2\theta_t - 1}$ becomes imaginary.

The transmitted field becomes an evanescent wave:

$E_t \propto e^{-\kappa x}\, e^{i(k_z z - \omega t)}$

Where $\kappa = k_0\sqrt{n_1^2\sin^2\theta_i - n_2^2}$ and $k_z = k_0 n_1\sin\theta_i$.

The field decays exponentially with penetration depth $\delta = 1/\kappa$ but propagates along the Interface. No energy is transported into the second medium: $R = 1$.

Frustrated total internal reflection. If a third medium is brought within a few wavelengths of The interface, energy can tunnel across the gap (analogous to quantum tunnelling).