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.
The laws are as follows:[1] 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 [2] 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
arising from a combined path length and initial phase-angle difference is: Now it can be seen that if
(as in the case of the second Fresnel–Arago law), the interference term produces a variation in the light intensity corresponding to
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
will be fluctuating so quickly and randomly that after time-averaging we have
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