Английская Википедия: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.
- 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
- <math>\mathbf{E_1}(\mathbf{r},t)=\mathbf{E}_{01}\cos(\mathbf{k_1\cdot r}-\omega t + \epsilon_1)</math>
- <math>\mathbf{E_2}(\mathbf{r},t)=\mathbf{E}_{02}\cos(\mathbf{k_2\cdot r}-\omega t + \epsilon_2),</math>
where the boldface indicates that the relevant quantity is a vector. The intensity of light goes as the electric field absolute square (in fact, <math>I=\epsilon v \langle \mathbf{\|E\|}^2 \rangle_T</math>, 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:
- <math>I_{12}=\epsilon v \mathbf{E_{01}\cdot E_{02}}\cos\delta,</math>
where the initial fields are involved in a complex dot product <math>\mathbf{E_{01} \cdot E_{02}}</math>; the cosine argument is a phase difference <math>\delta</math> arising from a combined path length and initial phase-angle difference is:
- <math>\delta=\mathbf{k_1\cdot r - k_2 \cdot r}+\epsilon_1-\epsilon_2</math>
Now it can be seen that if <math>\mathbf{E_{01}}</math> is perpendicular to <math>\mathbf{E_{02}}</math> (as in the case of the first Fresnel–Arago law), <math>I_{12}=0</math> and there is no interference. On the other hand, if <math>\mathbf{E_{01}}</math> is parallel to <math>\mathbf{E_{02}}</math> (as in the case of the second Fresnel–Arago law), the interference term produces a variation in the light intensity corresponding to <math>\cos\delta</math>. 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 <math>\delta</math> will be fluctuating so quickly and randomly that after time-averaging we have <math>\langle\cos\delta\rangle_T=0</math>, so again <math>I_{12}=0</math> and there is no interference (even if <math>\mathbf{E_{01}}</math> is rotated so that it is parallel to <math>\mathbf{E_{02}}</math>).
See also
References
- ↑ World of Physics; http://scienceworld.wolfram.com/physics/Fresnel-AragoLaws.html
- ↑ Optics, Hecht, 4th edition, pp. 386-7
