Fresnel–Arago laws

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:

One may understand this more clearly when considering two waves, given by the form ${\displaystyle \mathbf {E_{1}} (\mathbf {r} ,t)=\mathbf {E} _{01}\cos(\mathbf {k_{1}\cdot r} -\omega t+\epsilon _{1})}$ and ${\displaystyle \mathbf {E_{2}} (\mathbf {r} ,t)=\mathbf {E} _{02}\cos(\mathbf {k_{2}\cdot r} -\omega t+\epsilon _{2})}$, where the boldface indicates that the relevant quantity is a vector, interfering. We know that the intensity of light goes as the electric field squared (in fact, ${\displaystyle I=\epsilon v\langle \mathbf {E} ^{2}\rangle _{T}}$, 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: ${\displaystyle I_{12}=\epsilon v\mathbf {E_{01}\cdot E_{02}} \cos \delta }$, where ${\displaystyle \delta =(\mathbf {k_{1}\cdot r-k_{2}\cdot r} +\epsilon _{1}-\epsilon _{2})}$ represents the phase difference arising from a combined path length and initial phase-angle difference.

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