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.

Statement
The laws are as follows:


 * 1) Two orthogonal, coherent linearly polarized waves cannot interfere.
 * 2) Two parallel coherent linearly polarized waves will interfere in the same way as natural light.
 * 3) The two constituent orthogonal linearly polarized states of natural light cannot interfere to form a readily observable interference pattern, even if rotated into alignment (because they are incoherent).

Formulation and discussion
Consider the interference of two waves given by the form
 * $$\mathbf{E_1}(\mathbf{r},t)=\mathbf{E}_{01}\cos(\mathbf{k_1\cdot r}-\omega t + \epsilon_1)$$
 * $$\mathbf{E_2}(\mathbf{r},t)=\mathbf{E}_{02}\cos(\mathbf{k_2\cdot r}-\omega t + \epsilon_2),$$

where the boldface indicates that the relevant quantity is a vector. The intensity of light goes as the electric field absolute square (in fact, $$I=\epsilon v \langle \mathbf{\|E\|}^2 \rangle_T$$, 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:
 * $$I_{12}=\epsilon v \mathbf{E_{01}\cdot E_{02}}\cos\delta,$$

where the initial fields are involved in a complex dot product $$\mathbf{E_{01} \cdot E_{02}}$$; the cosine argument is a phase difference $$\delta$$ arising from a combined path length and initial phase-angle difference is:
 * $$\delta=\mathbf{k_1\cdot r - k_2 \cdot r}+\epsilon_1-\epsilon_2$$

Now it can be seen that if $$\mathbf{E_{01}}$$ is perpendicular to $$\mathbf{E_{02}}$$ (as in the case of the first Fresnel–Arago law), $$I_{12}=0$$ and there is no interference. On the other hand, if $$\mathbf{E_{01}}$$ is parallel to $$\mathbf{E_{02}}$$ (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 $$\langle\cos\delta\rangle_T=0$$, so again $$I_{12}=0$$ and there is no interference (even if $$\mathbf{E_{01}}$$ is rotated so that it is parallel to $$\mathbf{E_{02}}$$).