Fresnel equations

The Fresnel equations give reflection and transmission amplitudes for both transverse-electric and transverse-magnetic polarization of an electromagnetic wave at a planar interface. They follow by applying interface conditions for electromagnetic fields to incident, reflected and refracted plane electromagnetic waves.

The specular reflection of a wave at a planar dielectric interface with amplitudes described by the Fresnel equations. Below the critical angle, the reflected fraction is generally nonzero even when transmission is allowed; above the critical threshold, ideal lossless total internal reflection has unit reflected power. The reflected electric-field sign depends on polarization and the chosen basis conventions.

With one fixed tangential electric-field direction, normal-incidence reflection from impedance $Z_1$ to $Z_2$ gives $r_E=(Z_2-Z_1)/(Z_2+Z_1)$. Using one fixed tangential magnetic-field direction reverses the reflected wave's sign, giving $r_H=-r_E$. Reversing the chosen reflected basis can produce the same change. The reflectance $|r{|}^2$ is invariant, but coherent amplitudes require a consistent convention throughout. Two internal reflections at identical opposite interfaces give the same round-trip product in either convention.

For a ray meeting an interface at nonzero incidence angle, the plane spanned by the incoming wavevector and the surface normal. transverse electric polarization has electric field perpendicular to this plane, while transverse magnetic polarization has electric field in it. At exact normal incidence, the surface normal and ray alone do not select a unique plane.

For reflection or refraction at a flat isotropic interface, p-polarization has its electric field in the incidence plane and perpendicular to its wavevector. At a lossless nonmagnetic dielectric interface, its reflected amplitude vanishes at Brewster's angle. This does not mean that its field is generally parallel to the interface surface.

For reflection or refraction at a flat isotropic interface, s-polarization has its electric field perpendicular to the plane containing the incident wavevector and interface normal. At an isotropic dielectric boundary it is an eigenpolarization of the Fresnel reflection problem. The label is relative to the specified interface, not an absolute direction in space.

For real wave impedances at normal incidence, the electric-field transmission coefficient $t$ gives transmitted power fraction $T=(Z_1/Z_2)|t{|}^2$. Reflection gives $R=|r{|}^2$. Enforcing both tangential electric and magnetic boundary conditions gives $r=(Z_2-Z_1)/(Z_2+Z_1)$ and $t=2Z_2/(Z_1+Z_2)$, which satisfy $R+T=1$.

For an interface between two lossless, isotropic, nonmagnetic media with unequal refractive index values $n_1,n_2$, Brewster's angle is the incidence angle at which the reflected amplitude in transverse magnetic polarization vanishes. The Fresnel equations and Snell's law give $\tan {\theta }_B=n_2/n_1$; the reflected and transmitted rays are perpendicular there. The usual formula assumes equal magnetic permeabilities.

For sunlight reflected from a horizontal flat dielectric interface, incidence angle is measured from the vertical while solar elevation is measured from the horizontal. The Brewster condition $\tan {\theta }_B=n_2/n_1$ therefore gives $h_B=\pi /2-{\theta }_B={\tan }^{-1}(n_1/n_2)$. Reflected light has its electric field perpendicular to the vertical incidence plane, so its surviving linear polarization is horizontal. This ideal result assumes real refractive indices, negligible absorption, nonmagnetic media and a smooth interface.

At a transparent nonmagnetic dielectric interface, the Fresnel reflection coefficients differ for s and p linear polarization. At Brewster incidence the reflected p amplitude vanishes and reflected unpolarized light becomes s polarized. For a horizontal reflecting surface this is horizontal glare. Away from Brewster incidence it is generally partial polarization. Refraction and internal reflection in droplets also select the two components differently, explaining polarization of rainbow light without a biological mechanism.

A vacuum ray incident normally on a dielectric moving parallel to its surface becomes oblique in the material rest frame, with $\sin i=|\beta |$. For a transparent nonmagnetic dielectric, the rest-frame Brewster angle satisfies $\tan i_B=n$, giving $|{\beta }_B|=n/\sqrt{1+n^2}$. The reflected electric field is perpendicular to the plane containing the ray and the velocity. A real refractive index $n\ne 1$ is assumed; dispersion makes the condition implicit through the rest-frame frequency, and absorption generally removes exact extinction.

In the sharp planar, nonmagnetic ionosphere model with frequency above the plasma frequency, the refractive index is $n=\sqrt{1-{\omega }_p^2/{\omega }^2}<1$. From air, the Brewster angle satisfies $\tan {\theta }_B=n$. At that angle the reflected p-polarized wave vanishes, while s polarization retains a nonzero reflected component. A ground-to-ground single reflection from height $h$ spans $D=2h\tan {\theta }_B=2hn$. This is below the critical angle for total internal reflection for total internal reflection, which reflects both polarizations in this lossless isotropic model. The distinction diagnoses ordinary Brewster reflection rather than total internal reflection.

When the permeabilities agree, the signed reflected-to-incident electric-field ratio for polarization normal to the plane of incidence isFor incidence from lower to higher refractive index it never vanishes.

For equal permeabilities on the two sides of an interface, tangential-field continuity and Snell law for electromagnetic waves givefor consistently signed electric amplitudes normal to the plane of incidence.