Fresnel Equations: Reflection and Refraction at Boundaries

When an electromagnetic wave meets the boundary between two materials, part of it reflects and part refracts into the second material. The Fresnel equations tell us how much of the electric field is reflected and transmitted for each polarisation.

These equations explain why a lake surface shines at grazing angles, why anti-reflection coatings work, and how fibre optics trap light. They are the quantitative form of reflection and refraction for waves.

Key Concepts

At a boundary, the incident wave makes angle θ_i with the normal, the reflected wave makes θ_r = θ_i, and the transmitted wave makes θ_t given by Snell law. Light has two polarisation cases: s-polarisation, with E perpendicular to the plane of incidence, and p-polarisation, with E parallel to that plane. Fresnel coefficients r and t are ratios of electric field amplitudes. Reflectance R = |r|^2 is the fraction of power reflected, and transmittance T is the fraction transmitted, with R + T = 1 when there is no absorption.

Formulas

Snell law: n1 sin θ_i = n2 sin θ_t, with refractive indices n1, n2 dimensionless. For s-polarisation: r_s = (n1 cos θ_i - n2 cos θ_t) / (n1 cos θ_i + n2 cos θ_t). For p-polarisation: r_p = (n2 cos θ_i - n1 cos θ_t) / (n2 cos θ_i + n1 cos θ_t). At normal incidence both reduce in magnitude to r = (n1 - n2)/(n1 + n2) and R = ((n1 - n2)/(n1 + n2))^2. These forms assume smooth, non-absorbing, non-magnetic dielectrics.

Worked Example

Light in air (n1 = 1.00) hits glass (n2 = 1.50) at normal incidence. Find reflectance. Step 1: r = (1.00 - 1.50)/(1.00 + 1.50) = (-0.50)/(2.50) = -0.20. The negative sign means a phase change on reflection. Step 2: R = r^2 = 0.04. Step 3: So 4 percent of the power is reflected and about 96 percent enters the glass, ignoring absorption. This is why each uncoated glass surface gives a noticeable reflection and why camera lenses use coatings to reduce it.

Common Misconceptions

• Reflection is not total at every glass surface. At normal incidence on ordinary glass only about 4 percent of power reflects.
• p-polarisation can vanish at Brewster angle, but s-polarisation does not. That is why polarised sunglasses work on reflected light.
• Fresnel coefficients are amplitude ratios, not directly power fractions. Square them to get reflectance.

Real-World Applications

• Anti-reflection coatings on spectacles, solar panels and camera lenses.
• Fibre-optic communication relies on total internal reflection, the limit of refraction at large angles.
• Remote sensing and glossy-surface rendering use Fresnel reflection to model water, glass and metals.

The Fresnel equations split an incident wave into reflected and refracted parts according to angle, refractive indices and polarisation. They turn the familiar mirror shine of water and glass into precise numbers.