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:

# Two orthogonal, coherent linearly polarized waves cannot interfere.
# Two parallel coherent linearly polarized waves will interfere in the same way as natural light.
# 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 $\mathbf(\mathbf,t)=\mathbf_\cos(\mathbf-\omega t + \epsilon_1)$ and $\mathbf(\mathbf,t)=\mathbf_\cos(\mathbf-\omega t + \epsilon_2)$, 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, $I=\epsilon v \langle \mathbf^2 \rangle_T$, where the angled brackets denote a time average), and so we just add the fields before squaring them. Extensive algebra Optics, Hecht, 4th edition, pp. 386-7 yields an interference term in the intensity of the resultant wave, namely:
$I_=\epsilon v \mathbf\cos\delta$, where $\delta=(\mathbf+\epsilon_1-\epsilon_2)$ represents the phase difference arising from a combined path length and initial phase-angle difference.

Now it can be seen that if $\mathbf$ is perpendicular to $\mathbf$ (as in the case of the first Fresnel–Arago law), $I_=0$ and there is no interference. On the other hand, if $\mathbf$ is parallel to $\mathbf$ (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 $I_=0$ and there is no interference (even if $\mathbf$ is rotated so that it is parallel to $\mathbf$).

References

Interference
Polarization (waves)

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