Editing Huygens–Fresnel principle (section)

==Mathematical expression of the principle==
[[File: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]] {{math|''k'' {{=}} 2''&pi;''/''&lambda;''}}. Within a constant of proportionality, 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:

:<math>U(r_0) \propto \frac {U_0 e^{ikr_0}}{r_0}. </math>

Note that [[amplitude|magnitude]] decreases in inverse proportion to the distance traveled, and the phase changes as ''k'' times the distance traveled.

Using Huygens's theory and the [[superposition principle|principle of superposition]] of waves, the complex amplitude at a further point '''P''' is found by summing the contribution from each point on the sphere of radius ''r''<sub>0</sub>. In order to get an 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, where χ is the angle between the normal of the primary wavefront and the normal of the secondary wavefront. The complex amplitude at '''P''', due to the contribution of secondary waves, is then given by:<ref name = "Introduction to Fourier Optics">{{cite book|author=J. Goodman|year=2005|title=Introduction to Fourier Optics|edition=3rd|publisher=Roberts & Co Publishers|isbn=978-0-9747077-2-3|url= https://books.google.com/books?id=ow5xs_Rtt9AC}}</ref>

:<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 [[Kirchhoff integral theorem|integral theorem of Kirchhoff]] includes the basic idea of Huygens–Fresnel principle. Kirchhoff showed that in many cases, the theorem can be approximated to a simpler form that is equivalent to the formation of Fresnel's formulation.<ref name="Born and Wolf"/>

For an aperture illumination consisting of a single expanding spherical wave, if the radius of the curvature of the wave is sufficiently large, Kirchhoff gave the following expression for ''K''(χ):<ref name="Born and Wolf"/>
:<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, but at χ = π.

Above derivation of ''K''(χ) assumed that the diffracting aperture is illuminated by a single spherical wave with a sufficiently large radius of curvature. However, the principle holds for more general illuminations.<ref name="Introduction to Fourier Optics"/> An arbitrary illumination can be decomposed into a collection of point sources, and the linearity of the wave equation can be invoked to apply the principle to each point source individually. ''K''(χ) can be generally expressed as:<ref name="Introduction to Fourier Optics"/>

:<math>~K(\chi )= \cos \chi</math>

In this case, ''K'' satisfies the conditions stated above (maximum value at χ = 0 and zero at χ = π/2).