Arago spot

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In optics, an Arago spot, Fresnel bright spot, or Poisson spot is a bright point that appears at the center of a circular object's shadow due to Fresnel diffraction.[1][2][3][4] This spot played an important role in the discovery of the wave nature of light and is a common way to demonstrate that light behaves as a wave (for example, in undergraduate physics laboratory exercises).

The basic experimental setup requires a "point source," such as an illuminated pinhole, or a diverging laser beam. The dimensions of the setup must comply with the requirements for Fresnel diffraction. Namely, the Fresnel number must satisfy

F = \frac{d^2}{\ell \lambda} \gtrsim 1


d is the diameter of the circular object
is the distance between the object and the screen
λ the wavelength of the source


These conditions together explain why the bright spot is not encountered in everyday life. However, with the laser sources available today, it is undemanding to perform an Arago spot experiment.[5]


When light shines on the circular obstacle, Huygens' principle says that every point in the plane of the obstacle acts as a new point source of light. The light coming from points on the circumference of the obstacle, and going to the center of the shadow, travels exactly the same distance; so all the light passing close by the object arrives at the screen in phase and constructively interferes. This results in a bright spot at the shadow's center, where geometrical optics and particle theories of light predict that there should be no light at all.


At the beginning of the 19th century, the idea that light does not simply propagate along straight lines gained traction. Thomas Young published his double-slit experiment in 1807.[6] The original Arago spot experiment was carried out a decade later, and was the deciding experiment on the question of whether light is a particle or a wave. It is thus an example of an experimentum crucis.

At that time, many favored Isaac Newton's corpuscular theory of light, among them the theoretician Siméon Denis Poisson.[7] In 1818 the French Academy of Sciences launched a competition to explain the properties of light, where Poisson was one of the members of the judging committee. The civil engineer Augustin-Jean Fresnel entered this competition by submitting a new wave theory of light.[8]

Poisson studied Fresnel's theory in detail, and, being a supporter of the particle-theory of light, looked for a way to prove it wrong. Poisson thought he had found a flaw when he argued that a consequence of Fresnel's theory was that there would exist an on-axis bright spot in the shadow of a circular obstacle, where there should be complete darkness according to the particle-theory of light. Since the Arago spot is not easily observed in every-day situations, Poisson interpreted it as an absurd result and that it should disprove Fresnel's theory.

However, the head of the committee, Dominique-François-Jean Arago—who incidentally later became Prime Minister of France—decided to perform the experiment in more detail. He molded a 2 mm metallic disk to a glass plate with wax.[9] He succeeded in observing the predicted spot, which convinced most scientists of the wave-nature of light, and gave Fresnel the win.

Arago later noted that the phenomenon (which was later to be known as Poisson's Spot or the Spot of Arago) had already been observed by Delisle[10] and Maraldi[11] a century earlier. It only turned out much later (in one of Einstein's Annus Mirabilis papers) that light can be equally described as a particle (wave–particle duality of light).




U(P_1) =  \frac{A e^{\mathbf{i} k r_0}}{r_0}  \int_S \frac{e^{\mathbf{i} k r_1}}{r_1} K(\chi) \, dS,

where the inclination factor K(\chi) which ensures that the secondary wavelets do not propagate backwards is given by

K(\chi) = \frac{\mathbf{i}}{2 \lambda} (1 + \cos(\chi))


A is the amplitude of the source wave
S is the unobstructed surface.



U(P_1) = - \frac{\mathbf{i}}{\lambda}  \frac{A e^{\mathbf{i} k (g+b)}}{g b}  2\pi \int_a^\infty  e^{\mathbf{i} k \frac{1}{2} \left(\frac{1}{g} + \frac{1}{b}\right)r^2} r \, dr.

This integral can be solved numerically (see below). If g is large and b is small so that the angle \chi is not negligible one can write the integral for the on-axis case (P1 is at the center of the shadow) as (see [14]):

U(P_1) = \frac{A e^{\mathbf{i} k g}}{g} \frac{b}{\sqrt{b^2+a^2}}  e^{\mathbf{i} k \sqrt{b^2+a^2}}.

The source intensity, which is the square of the field amplitude, is I_0=\left|\frac{A e^{\mathbf{i} k g}}{g}\right|^2 and the intensity at the screen I = \left | U(P_1) \right |^2. The on-axis intensity as a function of the distance b is hence given by:

I = \frac{b^2}{b^2+a^2} I_0.


Calculation of diffraction images[edit]


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.

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.[15] For a particular angle one must solve the line integral for the ray with origin at the intersection point of the line P0P1 with the circular object plane. The contribution for a particular ray with azimuth angle \theta_1 and passing a transparent part of the object plane from r=s to r=t is:

R(\theta_1) \propto e^{\pi \mathbf{i} s^2/2} - e^{\pi \mathbf{i} t^2/2}.

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

Poissonspot simulation d4mm.jpg Poissonspot simulation d2mm.jpg Poissonspot simulation d1mm.jpg


Experimental aspects[edit]

Intensity and size[edit]



U(P_1,r) \propto  J_0^2 \left(\frac{\pi r d}{\lambda b}\right)


d is the diameter of circular object
λ is the wavelength
b is the distance between circular object and screen.


Poissonspot simulation d4mm lateral.jpg Poissonspot simulation d2mm lateral.jpg Poissonspot simulation d1mm lateral.jpg


Finite source size and spatial coherence[edit]

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 S×b/g, as if the circular object acted like a lens.[12] At the same time the intensity of the Arago spot is reduced with respect to the intensity of the undisturbed wave front.

Deviation from circularity[edit]


The circular object's surface roughness[edit]



The simulation includes a regular sinusoidal corrugation of the circular shape of amplitude 10 µm, 50 µm and 100 µm, respectively. Note, that the 100 µm edge corrugation almost completely removes the central bright spot.

This effect can be best understood using the Fresnel zone concept. The circular object blocks a certain number of Fresnel zones. The Fresnel zone that begins with the edge of the circular object is the only one that contributes to the Arago spot. All the Fresnel zones that are further out destructively interfere with each other and thus cancel. Random edge corrugation whose amplitude is of the same order as the width of that adjacent Fresnel zone reduces the Arago spot intensity. Contributions from the parts of the edge whose radius has been increased by the corrugation to about the width of the adjacent Fresnel zone now destructively interfere with those contributions from the parts which have not been affected by the corrugation.

The adjacent Fresnel zone is approximately given by:[18]

\Delta r \approx \sqrt{r^2 + \lambda \frac{g b}{g+b}} - r.


Arago spot with matter waves[edit]

Recently, the Arago spot experiment was demonstrated with a supersonic expansion beam of deuterium molecules (an example of neutral matter waves).[18] Material particles behaving like waves is known from quantum mechanics. The wave-nature of particles actually dates back to de Broglie's hypothesis[19] as well as Davisson and Germer's experiments.[20] An Arago spot of electrons, which also constitute matter waves, can be observed in transmission electron microscopes when examining circular structures of a certain size.


Other applications[edit]

Beside the demonstration of wave-behavior the Arago spot also has a few other applications. One of the ideas is to use the Arago spot as a straight line reference in alignment systems (see Feier et al.). Another is to probe aberrations in laser beams by using the spot's sensitivity to beam aberrations.[16]


  5. ^
  9. ^ Fresnel 1868, p. 369
  10. ^ Delisle, J.-N. (1715), 'Reflexions' in Mémoires de l’Académie Royale des Sciences, p. 166 
  11. ^ Maraldi, G.F. (1723), 'Diverses expèriences d'optique' in Mémoires de l’Académie Royale des Sciences, Imprimerie impériale, p. 111 
  12. ^ a b Sommerfeld, Arnold (1978), Vorlesungen über Theoretische Physik: Optik 4 (3rd ed.), Deutsch (Harri), ISBN 3-87144-377-8 
  14. ^ Sommerfeld 1978, p. 186
  15. ^ Dauger, D.E. (November 1996), "Simulation and Study of Fresnel Diffraction for Arbitrary Two-Dimensional Apertures", Comput. Phys. (AIOP) 10 (6): 591–604, Bibcode:1996ComPh..10..591D, doi:10.1063/1.168584 
  16. ^ a b Harvey, James E.; Forgham, James L. (1984), "The spot of Arago: New relevance for an old phenomenon", American Journal of Physics (AAPT) 52 (3): 243–247, Bibcode:1984AmJPh..52..243H, doi:10.1119/1.13681 
  17. ^ Coulson, John; Becknell, G. G. (1922), "Reciprocal Diffraction Relations between Circular and Elliptical Plates", Phys. Rev. (American Physical Society) 20 (6): 594–600, Bibcode:1922PhRv...20..594C, doi:10.1103/PhysRev.20.594 
  18. ^ a b c Reisinger, Thomas; Patel, A. Amil; Reingruber, Herbert; Fladischer, Katrin; Ernst, Wolfgang E.; Bracco, Gianangelo; Smith, Henry I.; Holst, Bodil (2009), "Poisson's spot with molecules", Phys. Rev. A (American Physical Society) 79 (5): 053823, Bibcode:2009PhRvA..79e3823R, doi:10.1103/PhysRevA.79.053823 
  19. ^ de Broglie, Louis (1923), "Waves and Quanta", Nature 112 (2815): 540, Bibcode:1923Natur.112..540D, doi:10.1038/112540a0 
  20. ^ Davisson, C.; Germer, L. (1927), "Diffraction of Electrons by a Crystal of Nickel", Nature 119: 558, Bibcode:1927Natur.119..558D, doi:10.1038/119558a0