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[[Image:Refraction - Huygens-Fresnel principle.svg|thumb|300px|right|Wave [[refraction]] in the manner of Huygens]]
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[[Image:Refraction on an aperture - Huygens-Fresnel principle.svg|thumb|300px|right|Wave diffraction in the manner of Huygens and Fresnel]]
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The '''Huygens–Fresnel principle''' (named after [[Netherlands|Dutch]] [[physicist]] [[Christiaan Huygens]] and [[France|French]] physicist [[Augustin-Jean Fresnel]]) is a method of analysis applied to problems of [[wave propagation]] both in the [[Far-field diffraction pattern|far-field limit]] and in near-field [[diffraction]].
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[[File:Huygens Fresnel Principle.gif|thumb|[[Diffraction]] of a plane wave at a slit whose width is several times the wavelength]]
 
[[File:Wavelength=slitwidth.gif|thumb|[[Diffraction]] of a plane wave when the slit width equals the wavelength]]
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In 1678, Huygens<ref>Chr. Huygens, ''Traitė de la Lumiere'' (completed in 1678, published in Leyden in 1690)</ref> proposed that every point which a luminous disturbance reaches becomes a source of a spherical wave; the sum of these secondary waves determines the form of the wave at any subsequent time. He assumed that the secondary waves travelled only in the "forward" direction and it is not explained in the theory why this is the case. He was able to provide a qualitative explanation of linear and spherical wave propagation, and to derive the laws of reflection and refraction using this principle, but could not explain the deviations from rectilinear propagation that occur when light encounters edges, apertures and screens, commonly known as [[diffraction]] effects.<ref name = "Insight into Optics">OS Heavens and RW Ditchburn, ''Insight into Optics'', 1987, Wiley & Sons, Chichester ISBN 0-471-92769-4</ref>
 
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In 1816, Fresnel<ref>A. Fresnel, ''Ann Chim et Phys'', (2), 1 (1816), Oeuvres, Vol. 1, 89, 129</ref> showed that Huygens' principle, together with his own principle of [[Interference (wave propagation)|interference]] could explain both the rectilinear propagation of light and also [[diffraction]] effects. To obtain agreement with experimental results, he had to include additional arbitrary assumptions about the phase and amplitude of the secondary waves, and also an obliquity factor. These assumptions have no obvious physical foundation but led to predictions that agreed with many experimental observations, including the [[Arago spot]].
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[[Siméon Denis Poisson|Poisson]] was a member of the French Academy, which reviewed Fresnel's work.<ref name = "Born and Wolf">[[Max Born]] and Emil Wolf, ''Principles of Optics'', 1999, Cambridge University Press ISBN 978-0-521-64222-4</ref> He used Fresnel's theory to predict that a bright spot will appear in the center of the shadow of a small disc and deduced from this that the theory was incorrect. However, Arago, another member of the committee, performed the experiment and showed that the prediction was correct. (Lisle had actually observed this fifty years earlier.<ref name = "Insight into Optics"/>) This was one of the investigations that led to the victory of the wave theory of light over the then predominant corpuscular theory.
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The Huygens–Fresnel principle provides a good basis for understanding and predicting the wave propagation of light. However, this article<ref>[http://www.mathpages.com/home/kmath242/kmath242.htm Huygens' Principle]</ref> provides an interesting discussion of the limitations of the principle and also of different scientists' views as to whether it is an accurate representation of reality or whether "Huygens' principle actually does give the right answer but for the wrong reasons".
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[[Kirchhoff's diffraction formula]] provides a rigorous mathematical foundation for diffraction, based on the wave equation. The arbitrary assumptions made by Fresnel to arrive at the Huygens–Fresnel equation emerge automatically from the mathematics in this derivation.<ref>MV Klein & TE Furtak, ''Optics'', 1986, John Wiley & Sons, New York</ref>
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A simple example of the operation of the principle can be seen when two rooms are connected by an open doorway and a sound is produced in a remote corner of one of them. A person in the other room will hear the sound as if it originated at the doorway. As far as the second room is concerned, the vibrating air in the doorway is the source of the sound.
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==Mathematical expression of the principle==
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[[Image:Huygens-Fresnel BW.svg|thumb|300px|right|Geometric arrangement for Fresnel's calculation]] Consider the case of a point source located at a point '''P'''<sub>0</sub>, vibrating at a [[frequency]] ''f''. The disturbance may be described by a complex variable ''U''<sub>0</sub> known as the complex amplitude. It produces a spherical wave with [[wavelength]] λ, [[wavenumber]] ''k'' = 2π/λ. The complex amplitude of the primary wave at the point '''Q''' located at a distance ''r''<sub>0</sub> from '''P'''<sub>0</sub> is given by
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:<math>U(r_0) = \frac {U_0 e^{ikr_0}}{r_0} </math>
</ul>
 
since the [[amplitude|magnitude]] decreases in inverse proportion to the distance travelled, and the phase changes as ''k'' times the distance travelled.
 
Using Huygens' theory and the [[superposition principle|principle of superposition]] of waves, the complex amplitude at a further point '''P''' is found by summing the contributions from each point on the sphere of radius ''r''<sub>0</sub>. In order to get agreement with experimental results, Fresnel found that the individual contributions from the secondary waves on the sphere had to be multiplied by a constant, −''i''/λ, and by an additional inclination factor, ''K''(χ). The first assumption means that the secondary waves oscillate at a quarter of a cycle out of phase with respect to the primary wave, and that the magnitude of the secondary waves are in a ratio of 1:λ to the primary wave. He also assumed that ''K''(χ) had a maximum value when χ = 0, and was equal to zero when χ = π/2. The complex amplitude at '''P''' is then given by:
 
:<math> U(P) = -\frac{i}{\lambda} U(r_0) \int_{S} \frac {e^{iks}}{s} K(\chi)\,dS </math>
 
where ''S'' describes the surface of the sphere, and ''s'' is the distance between '''Q''' and '''P'''.  
Fresnel used a zone construction method to find approximate values of ''K'' for the different zones,<ref name = "Born and Wolf"/> which enabled him to make predictions that were in agreement with experimental results.
 
The various assumptions made by Fresnel emerge automatically in [[Kirchhoff's diffraction formula]],<ref name = "Born and Wolf"/> to which the Huygens–Fresnel principle can be considered to be an approximation. Kirchhoff gave the following expression for ''K''(χ):
 
:<math>~K(\chi )= \frac{1}{2}(1+\cos \chi)</math>
 
''K'' has a maximum value at χ = 0 as in the Huygens–Fresnel principle; however, ''K'' is not equal to zero at χ = π/2.
 
==Huygens' principle and quantum electrodynamics==
 
Huygens' principle can be seen as a consequence of the [[isotropy]] of space—all directions in space are equal. Any disturbance created in a sufficiently small region of isotropic space (or in an isotropic medium) propagates from that region in all radial directions. The waves created by this disturbance, in turn, create disturbances in other regions, and so on. The [[superposition]] of all the waves results in the observed pattern of wave propagation.
 
Isotropy of space is fundamental to [[quantum electrodynamics]] (QED) where the [[wave function]] of any object propagates along all available unobstructed paths. When integrated along all possible paths, with a [[Phase (waves)|phase]] factor proportional to the path length, the interference of the wave-functions correctly predicts observable phenomena.
 
== See also ==
 
{{Commons|Huygens' principle}}
* [[Kirchhoff's diffraction formula]]
* [[Green's function]]
* [[Green's theorem]]
* [[Green's identities]]
* [[Near-field diffraction pattern]]
* [[Double-slit experiment]]
* [[Knife-edge effect]]
* [[Fermat's principle]]
* [[Fourier optics]]
* [[Wave field synthesis]]
 
==References==
 
{{reflist}}
 
==Further reading==
 
Stratton, Julius Adams: ''Electromagnetic Theory'', McGraw-Hill, 1941. (Reissued by Wiley – IEEE Press, ISBN 978-0-470-13153-4).
 
{{DEFAULTSORT:Huygens-Fresnel Principle}}
[[Category:Wave mechanics]]
[[Category:Dutch Golden Age]]
[[Category:Diffraction]]

Revision as of 20:08, 10 February 2014

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