Reflection and the Fresnel Equations

The Fresnel equations give the fraction of light reflected and transmitted at an interface as a function of polarization and angle of incidence.

Definition

The set of relations, derived from electromagnetic boundary conditions, that determine the amplitudes and phases of the reflected and transmitted waves at an interface for each polarization and angle of incidence.

Scope

This topic covers what happens to light at the boundary between two media. It includes the derivation of the amplitude reflection and transmission coefficients from the boundary conditions on the electromagnetic fields, the separate behaviour of the two polarization components parallel and perpendicular to the plane of incidence, the reflectance and transmittance in intensity, Brewster's angle at which the reflected wave is fully polarized, total internal reflection and the accompanying evanescent wave and phase shift, and the special case of normal incidence. It connects polarization to the everyday phenomena of reflection.

Core questions

• What fraction of light is reflected at an interface and how does it depend on angle?
• Why do the two polarization components reflect differently?
• What is Brewster's angle and why is the reflected light there fully polarized?
• What happens at and beyond the angle of total internal reflection?

Key concepts

• Fresnel equations
• reflection coefficient
• transmission coefficient
• reflectance and transmittance
• plane of incidence
• Brewster's angle
• total internal reflection
• evanescent wave

Key theories

Fresnel reflection and transmission coefficients

Applying the continuity of the tangential electric and magnetic fields at an interface yields the Fresnel equations, giving the amplitude reflection and transmission coefficients for the parallel and perpendicular polarizations as functions of the angle of incidence and the refractive indices.

Brewster's angle and polarization on reflection

At Brewster's angle the parallel-polarized reflection vanishes, so reflected light is completely perpendicular-polarized; this follows directly from the Fresnel equations and explains the polarization of glare.

Clinical relevance

Fresnel reflection governs the light lost at the surfaces of lenses, fibres, and tissue interfaces in medical optical systems, motivating anti-reflection coatings and index matching, while total internal reflection underlies fibre-optic endoscopes and certain biosensors.

History

Fresnel derived his reflection and transmission formulae in the early 1820s from his transverse-wave theory of light, accounting for the polarization changes Malus and Brewster had observed on reflection. Maxwell's electromagnetic theory later provided the rigorous derivation from the boundary conditions on the fields.

Key figures

• Augustin-Jean Fresnel
• David Brewster
• James Clerk Maxwell

Seminal works

• hecht2017
• bornwolf1999

Frequently asked questions

Why is glare from water or roads polarized?

Light reflecting near Brewster's angle is strongly suppressed for the polarization in the plane of incidence, leaving the reflected glare predominantly polarized parallel to the surface, which polarizing filters can then block.

Does total internal reflection lose any light into the second medium?

No energy is transmitted on average, but the field does not stop abruptly at the boundary; an evanescent wave penetrates a fraction of a wavelength into the second medium before decaying, which is exploited in sensing and coupling applications.