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The '''Fresnel–Arago laws''' are three laws which summarise some of the more important properties of [[Interference (wave propagation)|interference]] between light of different states of [[polarization (waves)|polarization]]. [[Augustin-Jean Fresnel]] and [[François Arago]], both discovered the laws, which bear their name. | |||
The laws are as follows:<ref>World of Physics; http://scienceworld.wolfram.com/physics/Fresnel-AragoLaws.html</ref> | |||
1. Two [[orthogonal]], [[Coherence (physics)|coherent]] linearly [[polarized]] waves cannot interfere. <br> | |||
2. Two parallel coherent linearly polarized waves will interfere in the same way as [[Sunlight|natural light]].<br> | |||
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 <math>\mathbf{E_1}(\mathbf{r},t)=\mathbf{E}_{01}\cos(\mathbf{k_1\cdot r}-\omega t + \epsilon_1)</math> and <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 [[Euclidean vector|vector]], interfering. We know that the [[intensity (physics)|intensity]] of light goes as the electric field squared (in fact, <math>I=\epsilon v < \mathbf{E}^2 >_T)</math>, where the angled brackets denote a time average), and so we just add the fields before squaring them. Extensive algebra <ref>Which can be found in, e.g Optics, Hecht, 4th edition, pp. 386-7</ref> yields a result for the intensity of the resultant wave, namely that: | |||
<math>I_{12}=\mathbf{E_{01}\cdot E_{02}}\cos\delta</math> where <math>\delta</math> represents the interference term; the [[phase difference]] arising from a combined [[path length]] and initial phase-angle difference; <math>\delta=(\mathbf{k_1\cdot r - k_2 \cdot r}+\epsilon_1-\epsilon_2)</math>. | |||
At this point, substituting various correct values of <math>\delta</math> yields laws 1 & 3 above. | |||
With classical wave formulation for law 1 at absorption the polarisation axis are <math>Y = sin\alpha sin(\omega t)</math> and <math>X = cos\alpha sin(\omega t + \phi)</math> which combine to <math>I = \sqrt{Y^2+X^2}</math>. Here <math>\alpha </math> is the angle between the incoming polarization and the (orthogonal) axis of the two polarisers and <math>\phi </math> is the phase shift between both waves. According quantum mechanics the probability of absorption is <math>\int_0^{2\pi} \! I^2\,d\omega t = ... = \pi</math>, so independent of <math>\phi</math>. | |||
== References == | |||
<references /> | |||
<!-- It may look as if the above is plagarised directly from the reference. However, the almost exact same text appears in Optics, 4th Edition, by Eugene Hecht (a standard textbook of undergraduate optics) and can be found on page 392, near the bottom. I do not see how to state the laws differently. --> | |||
{{DEFAULTSORT:Fresnel-Arago laws}} | |||
[[Category:Interference]] | |||
[[Category:Polarization (waves)]] | |||
{{optics-stub}} | |||
Revision as of 12:06, 21 December 2013
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 and , 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, , where the angled brackets denote a time average), and so we just add the fields before squaring them. Extensive algebra [2] yields a result for the intensity of the resultant wave, namely that: where represents the interference term; the phase difference arising from a combined path length and initial phase-angle difference; .
At this point, substituting various correct values of yields laws 1 & 3 above.
With classical wave formulation for law 1 at absorption the polarisation axis are and which combine to . Here is the angle between the incoming polarization and the (orthogonal) axis of the two polarisers and is the phase shift between both waves. According quantum mechanics the probability of absorption is , so independent of .
References
- ↑ World of Physics; http://scienceworld.wolfram.com/physics/Fresnel-AragoLaws.html
- ↑ Which can be found in, e.g Optics, Hecht, 4th edition, pp. 386-7