The equations assume the interface between the media is flat and that the media are homogeneous and

There are two sets of Fresnel coefficients for two different linear polarization components of the incident wave. Since any polarization state can be resolved into a combination of two orthogonal linear polarizations, this is sufficient for any problem. Likewise, unpolarized (or "randomly polarized") light has an equal amount of power in each of two linear polarizations.

The s polarization refers to polarization of a wave's electric field normal to the plane of incidence (the z direction in the derivation below); then the magnetic field is in the plane of incidence. The p polarization refers to polarization of the electric field in the plane of incidence (the xy plane in the derivation below); then the magnetic field is normal to the plane of incidence.

Although the reflectivity and transmission are dependent on polarization, at normal incidence (θ = 0) there is no distinction between them so all polarization states are governed by a single set of Fresnel coefficients (and another special case is mentioned below in which that is true).

Power (intensity) reflection and transmission coefficients i = θ r , {\displaystyle \theta _{\mathrm {i} }=\theta _{\mathrm {r} },}

and Snell's law:

${\displaystyle n_{1}\sin \theta _{$

The relationship between these angles is given by the law of reflection:

and Snell's law:

${\$

The behavior of light striking the interface is solved by considering the electric and magnetic fields that constitute an electromagnetic wave, and the laws of electromagnetism, as shown below. The ratio of waves' electric field (or magnetic field) amplitudes are obtained, but in practice one is more often interested in formulae which determine power coefficients, since power (or irradiance) is what can be directly measured at optical frequencies. The power of a wave is generally proportional to the square of the electric (or magnetic) field amplitude.

We call the fraction of the incident power that is reflected from the interface the reflectance (or "reflectivity", or "power reflection coefficient") R, and the fraction that is refracted into the second medium is called the transmittance (or "transmissivity", or "power transmission coefficient") T . Note that these are what would be measured right at each side of an interface and do not account for attenuation of a wave in an absorbing medium following transmission or reflection.^[2]

The reflectance for s-polarized light is

${\displaystyle R_{\mathrm{s}}={|\frac{}{}}^{}}$

We call the fraction of the incident power that is reflected from the interface the reflectance (or "reflectivity", or "power reflection coefficient") R, and the fraction that is refracted into the second medium is called the transmittance (or "transmissivity", or "power transmission coefficient") T . Note that these are what would be measured right at each side of an interface and do not account for attenuation of a wave in an absorbing medium following transmission or reflection.^[2]

The reflectance for s-polarized light is

while the reflectance for p-polarized light is

${\displaystyle R_{\mathrm {p} }=\left|{\frac {Z_{2}\cos \theta _{\mathrm {t} }-Z_{1}\cos \theta _{\mathrm {i} }}{Z_{2}\cos \theta _{\mathrm {t} }+Z_{1}\cos \theta _{\mathrm {i} }}}\right|^{2},}$

where Z₁ and Z₂ are the wave impedances of media 1 and 2, respectively.

We assume that the media are non-magnetic (i.e., μ₁ = μ₂ = μ₀), which is typically a good approximation at optical frequencies (and for transparent media at other frequencies).^[3] Then the wave impedances are determined solely by the refractive indices n₁ and n₂:

${\displaystyle Z_{i}={\frac {Z_{0}}{n_{i}}}\,,}$

where Z₀ is the impedance of free space and i = 1, 2. Making this substitution, we obtain equations using the refractive indices:

${\displaystyle T_{\mathrm {s} }=1-R_{\mathrm {s} }}$