The Fresnel equations describe the behavior of electromagnetic waves at the interface between two different media, specifically the reflection and transmission coefficients for both parallel (p) and perpendicular (s) polarizations. These equations are named after the French physicist Augustin-Jean Fresnel, who derived them in the early 19th century.

The Fresnel equations can be expressed in terms of the angle of incidence (θ1), angle of refraction (θ2), and the refractive indices (n1 and n2) of the two media. They calculate the reflection coefficients (rp and rs) and transmission coefficients (tp and ts) for the parallel and perpendicular polarizations, respectively:

Reflection coefficients:

rs = (n1*cos(θ1) – n2*cos(θ2)) / (n1*cos(θ1) + n2*cos(θ2))

- rp = (n2
*cos(θ1) – n1*cos(θ2)) / (n2*cos(θ1) + n1*cos(θ2))

Transmission coefficients:

ts = (2*n1*cos(θ1)) / (n1*cos(θ1) + n2*cos(θ2))

- tp = (2
*n1*cos(θ1)) / (n2*cos(θ1) + n1*cos(θ2))

The coefficients represent the amplitude ratios of the reflected and transmitted waves to the incident wave. To obtain the power reflection and transmission coefficients (which represent the fractions of power reflected and transmitted), the amplitude coefficients should be squared:

Reflectance (power reflection coefficient) for s and p polarizations:

Rs = |rs|^2

Rp = |rp|^2

Transmittance (power transmission coefficient) for s and p polarizations:

Ts = |ts|^2

Tp = |tp|^2

The Fresnel equations are fundamental to understanding how electromagnetic waves interact with different media and are essential in various applications, such as designing anti-reflective coatings, optical devices, and studying the behavior of waves in different environments.