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 E 1 ( r , t ) = E 01 cos ( k 1 ⋅ r − ω t + ϵ 1 ) and E 2 ( r , t ) = E 02 cos ( k 2 ⋅ r − ω t + ϵ 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 = ϵ v ⟨ E 2 ⟩ 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 = ϵ v E 01 ⋅ E 02 cos δ , where δ = ( k 1 ⋅ r − k 2 ⋅ r + ϵ 1 − ϵ 2 ) represents the phase difference arising from a combined path length and initial phase-angle difference.
Now it can be seen that if E 01 is perpendicular to 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 E 01 is parallel to 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 δ . 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 δ will be fluctuating so quickly and randomly that after time-averaging we have ⟨ cos δ ⟩ T = 0 , so again I 12 = 0 and there is no interference (even if E 01 is rotated so that it is parallel to E 02 ).
