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:

# 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).

Formulation and discussion

Consider the interference of two waves given by the form
:$\mathbf(\mathbf,t)=\mathbf_\cos(\mathbf-\omega t + \epsilon_1)$
:$\mathbf(\mathbf,t)=\mathbf_\cos(\mathbf-\omega t + \epsilon_2),$
where the boldface indicates that the relevant quantity is a vector. The intensity of light goes as the electric field absolute square (in fact, $I=\epsilon v \langle \mathbf^2 \rangle_T$, 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 the initial fields are involved in a complex dot product $\mathbf$; the cosine argument is a phase difference $\delta$ arising from a combined path length and initial phase-angle difference is:
:$\delta=\mathbf+\epsilon_1-\epsilon_2$

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 $\mathbf$).

See also

*Unpolarized light

References

Interference
Polarization (waves)

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