Editing Arago spot (section)

=== Finite source size and spatial coherence ===

The main reason why the Arago spot is hard to observe in circular shadows from conventional light sources is that such light sources are bad approximations of point sources. If the wave source has a finite size ''S'' then the Arago spot will have an extent that is given by ''Sb''/''g'', as if the circular object acted like a lens.<ref name="sommerfeld"/> At the same time the intensity of the Arago spot is reduced with respect to the intensity of the undisturbed wave front. Defining the relative intensity <math>I_\text{rel}</math>as the intensity divided by the intensity of the undisturbed wavefront, the relative intensity for an extended circular source of diameter w can be expressed exactly using the following equation:<ref name=":0">{{Cite journal|last1=Reisinger|first1=T|last2=Leufke|first2=P M|last3=Gleiter|first3=H| last4=Hahn|first4=H| date=2017-03-14|title=On the relative intensity of Poisson's spot|journal=New Journal of Physics|volume=19| issue=3| pages=033022| doi=10.1088/1367-2630/aa5e7f|bibcode=2017NJPh...19c3022R|issn=1367-2630|doi-access=free}}</ref>
<math display="block">I_\text{rel}(w) = J_0^2\left(\frac{w R \pi}{g \lambda}\right) + J_1^2\left(\frac{w R \pi}{g \lambda}\right)</math>
where <math>J_0</math>and <math>J_1</math>are the Bessel functions of the first kind. <math>R</math> is the radius of the disc casting the shadow, <math>\lambda</math> the wavelength and <math>g</math> the distance between source and disc. For large sources the following asymptotic approximation applies:<ref name=":0" />
<math display="block">I_\text{rel}(w) \approx \frac{2 g \lambda }{\pi^2 w R}</math>