The Fresnel–Arago laws are three laws which summarise some of the more important properties of interference between light of different states of polarization. Augustin-Jean Fresnel and François Arago, both discovered the laws, which bear their name.

Statement

The laws are as follows:

Formulation and discussion

Consider the interference of two waves given by the form where the boldface indicates that the relevant quantity is a vector. The intensity of light goes as the electric field absolute square (in fact,, where the angled brackets denote a time average), and so we just add the fields before squaring them. Extensive algebra yields an interference term in the intensity of the resultant wave, namely: where the initial fields are involved in a complex dot product ; the cosine argument is a phase difference \delta arising from a combined path length and initial phase-angle difference is: Now it can be seen that if is perpendicular to (as in the case of the first Fresnel–Arago law), I_{12}=0 and there is no interference. On the other hand, if is parallel to (as in the case of the second Fresnel–Arago law), the interference term produces a variation in the light intensity corresponding to \cos\delta. Finally, if natural light is decomposed into orthogonal linear polarizations (as in the third Fresnel–Arago law), these states are incoherent, meaning that the phase difference \delta will be fluctuating so quickly and randomly that after time-averaging we have, so again I_{12}=0 and there is no interference (even if is rotated so that it is parallel to ).