Fresnel–Arago Laws
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The Fresnel–Arago laws are three laws which summarise some of the more important properties of
interference Interference is the act of interfering, invading, or poaching. Interference may also refer to: Communications * Interference (communication), anything which alters, modifies, or disrupts a message * Adjacent-channel interference, caused by extra ...
between light of different states of
polarization Polarization or polarisation may refer to: Mathematics *Polarization of an Abelian variety, in the mathematics of complex manifolds *Polarization of an algebraic form, a technique for expressing a homogeneous polynomial in a simpler fashion by ...
.
Augustin-Jean Fresnel Augustin-Jean Fresnel (10 May 1788 – 14 July 1827) was a French civil engineer and physicist whose research in optics led to the almost unanimous acceptance of the wave theory of light, excluding any remnant of Isaac Newton, Newton's c ...
and
François Arago Dominique François Jean Arago (), known simply as François Arago (; Catalan: , ; 26 February 17862 October 1853), was a French mathematician, physicist, astronomer, freemason, supporter of the Carbonari revolutionaries and politician. Early l ...
, both discovered the laws, which bear their name.


Statement

The laws are as follows: # Two
orthogonal In mathematics, orthogonality (mathematics), orthogonality is the generalization of the geometric notion of ''perpendicularity''. Although many authors use the two terms ''perpendicular'' and ''orthogonal'' interchangeably, the term ''perpendic ...
,
coherent Coherence is, in general, a state or situation in which all the parts or ideas fit together well so that they form a united whole. More specifically, coherence, coherency, or coherent may refer to the following: Physics * Coherence (physics ...
linearly polarized waves cannot interfere. # Two parallel coherent linearly polarized waves will interfere in the same way as
natural light Natural Light, formerly Anheuser-Busch Natural Light, nicknamed Natty, is an American reduced-calorie light lager brewed by Anheuser-Busch. Its ingredients are listed as water, barley malt, cereal grains, yeast, and hops. One serving contains ...
. # 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 Vector most often refers to: * Euclidean vector, a quantity with a magnitude and a direction * Disease vector, an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematics a ...
. The
intensity Intensity may refer to: In colloquial use * Strength (disambiguation) *Amplitude * Level (disambiguation) * Magnitude (disambiguation) In physical sciences Physics *Intensity (physics), power per unit area (W/m2) *Field strength of electric, m ...
of light goes as the electric field
absolute square In mathematics, a square is the result of multiplying a number by itself. The verb "to square" is used to denote this operation. Squaring is the same as raising to the power  2, and is denoted by a superscript 2; for instance, the square o ...
(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 In physics and mathematics, the phase (symbol φ or ϕ) of a wave or other periodic function F of some real variable t (such as time) is an angle-like quantity representing the fraction of the cycle covered up to t. It is expressed in such a s ...
\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 Unpolarized light is light with a random, time-varying polarization. Natural light, like most other common sources of visible light, is produced independently by a large number of atoms or molecules whose emissions are uncorrelated. Unpolarized ...


References

Interference Polarization (waves) {{optics-stub