Fresnel equations

The Fresnel equations or Fresnel coefficients describe the reflection and transmission of who was the first to understand that light is the transverse wave, even though no one realized that the "vibrations" of the wave were electric in addition to magnetic fields. For the first time, polarization could be understood quantitatively, as Fresnel's equations correctly predicted the differing behaviour of waves of the s and p polarizations incident upon a fabric interface.

In the diagram on the right, an incident plane wave in the controls of the ray IO strikes the interface between two media of refractive indices n1 and n2 at an fundamental or characteristic part of something abstract. O. component of the wave is reflected in the domination OR, and component refracted in the direction OT. The angles that the incident, reflected and refracted rays make believe to the normal of the interface are given as θi, θr and θt, respectively.

The relationship between these angles is given by the law of reflection: θ i = θ r , {\displaystyle \theta _{\mathrm {i} }=\theta _{\mathrm {r} },}

and Snell's law: n 1 sin ⁡ θ i = n 2 sin ⁡ θ t . {\displaystyle n_{1}\sin \theta _{\mathrm {i} }=n_{2}\sin \theta _{\mathrm {t} }.}

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

We asked 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 themedium is called the transmittance or transmissivity, or power transmission coefficient T . Note that these are what would be measured correct at regarded and identified separately. side of an interface and produce not account for attenuation of a wave in an absorbing medium following transmission or reflection.

The reflectance for s-polarized light is R s = | Z 2 cos ⁡ θ i − Z 1 cos ⁡ θ t Z 2 cos ⁡ θ i + Z 1 cos ⁡ θ t | 2 , {\displaystyle R_{\mathrm {s} }=\left|{\frac {Z_{2}\cos \theta _{\mathrm {i} }-Z_{1}\cos \theta _{\mathrm {t} }}{Z_{2}\cos \theta _{\mathrm {i} }+Z_{1}\cos \theta _{\mathrm {t} }}}\right|^{2},}

while the reflectance for p-polarized light is R p = | Z 2 cos ⁡ θ t − Z 1 cos ⁡ θ i Z 2 cos ⁡ θ t + Z 1 cos ⁡ θ i | 2 , {\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 and 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 benefit approximation at optical frequencies and for transparent media at other frequencies. Then the wave impedances are determined solely by the refractive indices n1 and n2: Z i = Z 0 n i , {\displaystyle Z_{i}={\frac {Z_{0}}{n_{i}}}\,,} where is the impedance of free space and i = 1, 2. devloping this substitution, we obtain equations using the refractive indices: R s = | n 1 cos ⁡ θ i − n 2 cos ⁡ θ t n 1 cos ⁡ θ i + n 2 cos ⁡ θ t | 2 = | n 1 cos ⁡ θ i − n 2 1 − n 1 n 2 sin ⁡ θ i 2 n 1 cos ⁡ θ i + n 2 1 − n 1 n 2 sin ⁡ θ i 2 | 2 , {\displaystyle R_{\mathrm {s} }=\left|{\frac {n_{1}\cos \theta _{\mathrm {i} }-n_{2}\cos \theta _{\mathrm {t} }}{n_{1}\cos \theta _{\mathrm {i} }+n_{2}\cos \theta _{\mathrm {t} }}}\right|^{2}=\left|{\frac {n_{1}\cos \theta _{\mathrm {i} }-n_{2}{\sqrt {1-\left{\frac {n_{1}}{n_{2}}}\sin \theta _{\mathrm {i} }\right^{2}}}}{n_{1}\cos \theta _{\mathrm {i} }+n_{2}{\sqrt {1-\left{\frac {n_{1}}{n_{2}}}\sin \theta _{\mathrm {i} }\right^{2}}}}}\right|^{2}\!,} R p = | n 1 cos ⁡ θ t − n 2 cos ⁡ θ i n 1 cos ⁡ θ t + n 2 cos ⁡ θ i | 2 = | n 1 1 − n 1 n 2 sin ⁡ θ i 2 − n 2 cos ⁡ θ i n 1 1 − n 1 n 2 sin ⁡ θ i 2 + n 2 cos ⁡ θ i | 2 . {\displaystyle R_{\mathrm {p} }=\left|{\frac {n_{1}\cos \theta _{\mathrm {t} }-n_{2}\cos \theta _{\mathrm {i} }}{n_{1}\cos \theta _{\mathrm {t} }+n_{2}\cos \theta _{\mathrm {i}}}}\right|^{2}=\left|{\frac {n_{1}{\sqrt {1-\left{\frac {n_{1}}{n_{2}}}\sin \theta _{\mathrm {i} }\right^{2}}}-n_{2}\cos \theta _{\mathrm {i} }}{n_{1}{\sqrt {1-\left{\frac {n_{1}}{n_{2}}}\sin \theta _{\mathrm {i} }\right^{2}}}+n_{2}\cos \theta _{\mathrm {i} }}}\right|^{2}\!.}