Editing Arago spot (section)

=== Calculation of diffraction images ===

To calculate the full diffraction image that is visible on the screen one has to consider the surface integral of the previous section. One cannot exploit circular symmetry anymore, since the line between the source and an arbitrary point on the screen does not pass through the center of the circular object. With the aperture function <math>g(r,\theta)</math> which is 1 for transparent parts of the object plane and 0 otherwise (i.e. It is 0 if the direct line between source and the point on the screen passes through the blocking circular object.) the integral that needs to be solved is given by:
<math display="block">U(P_1) \propto \int_0^{2\pi} \int_0^\infty g(r,\theta) e^{\frac{\mathbf{i} \pi \rho^2}{\lambda} \left( \frac{1}{g} + \frac{1}{b} \right)} \rho \, d\rho \, d\theta.</math>

Numerical calculation of the integral using the [[trapezoidal rule]] or [[Simpson's rule]] is not efficient and becomes numerically unstable especially for configurations with large [[Fresnel number]]. However, it is possible to solve the radial part of the integral so that only the integration over the azimuth angle remains to be done numerically.<ref name="dauger"/> For a particular angle one must solve the line integral for the ray with origin at the intersection point of the line P<sub>0</sub>P<sub>1</sub> with the circular object plane. The contribution for a particular ray with azimuth angle <math>\theta_1</math> and passing a transparent part of the object plane from <math>r = s</math> to <math>r = t</math> is:
<math display="block">R(\theta_1) \propto e^{\frac{\pi}{2} \mathbf{i} s^2} - e^{\frac{\pi}{2} \mathbf{i} t^2}.</math>

So for each angle one has to compute the intersection point(''s'') of the ray with the circular object and then sum the contributions <math>I(\theta_1)</math> for a certain number of angles between 0 and <math>2\pi</math>. Results of such a calculation are shown in the following images.

[[File:poissonspot simulation d4mm.jpg|200px]]
[[File:poissonspot simulation d2mm.jpg|200px]]
[[File:poissonspot simulation d1mm.jpg|200px]]

The images are simulations of the Arago spot in the shadow of discs of diameter 4&nbsp;mm, 2&nbsp;mm, and 1&nbsp;mm, imaged 1&nbsp;m behind each disc. The disks are illuminated by light of wavelength of 633&nbsp;nm, diverging from a point 1&nbsp;m in front of each disc. Each image is 16&nbsp;mm wide.

The Arago spot can also be visualized using lines of average energy flow calculated numerically by averaging the [[Poynting vector]] of the electromagnetic field.<ref name=Gondran-2010/><ref>{{Cite book |last=Born |first=Max |title=Principles of optics: electromagnetic theory of propagation, interference and diffraction of light |last2=Wolf |first2=Emil |date=1993 |publisher=Pergamon Press |isbn=978-0-08-026481-3 |edition=6. ed., reprinted (with corrections) |location=Oxford}}</ref>{{rp|575}}
[[File:Arago spot.jpg|thumb|left|700px|Numerical simulation of the intensity of monochromatic light of wavelength λ&nbsp;=&nbsp;0.5&nbsp;μm behind a circular obstacle of radius {{nowrap|1=R = 5 μm = 10λ}}.<ref name=Gondran-2010>{{Cite journal |last=Gondran |first=Michel |last2=Gondran |first2=Alexandre |date=2010-06-01 |title=Energy flow lines and the spot of Poisson–Arago |url=https://pubs.aip.org/aapt/ajp/article-abstract/78/6/598/1042061/Energy-flow-lines-and-the-spot-of-Poisson-Arago?redirectedFrom=fulltext |journal=American Journal of Physics |volume=78 |issue=6 |pages=598–602 |doi=10.1119/1.3291215 |issn=0002-9505|arxiv=0909.2302 }}</ref>]]

{{clear}}