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:^[1]

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).

One may understand this more clearly when considering two waves, given by the form $\mathbf{E_1}(\mathbf{r},t)=\mathbf{E}_{01}\cos(\mathbf{k_1\cdot r}-\omega t + \epsilon_1)$ and $\mathbf{E_2}(\mathbf{r},t)=\mathbf{E}_{02}\cos(\mathbf{k_2\cdot r}-\omega t + \epsilon_2)$, 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, $I=\epsilon v \langle \mathbf{E}^2 \rangle_T$, where the angled brackets denote a time average), and so we just add the fields before squaring them. Extensive algebra ^[2] 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 $\delta=(\mathbf{k_1\cdot r - k_2 \cdot r}+\epsilon_1-\epsilon_2)$ represents the phase difference arising from a combined path length and initial phase-angle difference.

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 $\mathbf{E_{02}}$).

References

1. World of Physics; http://scienceworld.wolfram.com/physics/Fresnel-AragoLaws.html
2. Optics, Hecht, 4th edition, pp. 386-7