Arago spot

The basic experimental setup requires a point source, such as an illuminated pinhole or a diverging laser beam.

[8] In astronomy, the Arago spot can also be observed in the strongly defocussed image of a star in a Newtonian telescope.

There, the star provides an almost ideal point source at infinity, and the secondary mirror of the telescope constitutes the circular obstacle.

At the beginning of the 19th century, the idea that light does not simply propagate along straight lines gained traction.

At that time, many favored Isaac Newton's corpuscular theory of light, among them the theoretician Siméon Denis Poisson.

[10] 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.

Although Arago's experimental result was overwhelming evidence in favor of the wave theory, a century later, in conjunction with the birth of quantum mechanics (and first suggested in one of Albert Einstein's Annus Mirabilis papers), it became understood that light (as well as all forms of matter and energy) must be described as both a particle and a wave (wave–particle duality).

Before the advent of quantum theory in the late 1920s, only the wave nature of light could explain phenomena such as diffraction and interference.

Today it is known that a diffraction pattern appears through the mosaic-like buildup of bright spots caused by single photons, as predicted by Dirac's quantum theory.

With increasing light intensity the bright dots in the mosaic diffraction pattern just assemble faster.

In contrast, the wave theory predicts the formation of an extended continuous pattern whose overall brightness increases with light intensity.

At the heart of Fresnel's wave theory is the Huygens–Fresnel principle, which states that every unobstructed point of a wavefront becomes the source of a secondary spherical wavelet and that the amplitude of the optical field E at a point on the screen is given by the superposition of all those secondary wavelets taking into account their relative phases.

In order to derive the intensity behind the circular obstacle using this integral one assumes that the experimental parameters fulfill the requirements of the near-field diffraction regime (the size of the circular obstacle is large compared to the wavelength and small compared to the distances g = P0C and b = CP1).

Going to polar coordinates[dubious – discuss] then yields the integral for a circular object of radius a (see for example Born and Wolf[13]):

is not negligible[dubious – discuss] one can write the integral for the on-axis case (P1 is at the center of the shadow) as (see Sommerfeld[19]):

To calculate the full diffraction image that is visible on the screen one has to consider the surface integral of the previous section.

It is 0 if the direct line between source and the point on the screen passes through the blocking circular object.)

For an ideal point source, the intensity of the Arago spot equals that of the undisturbed wave front.

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.

Therefore, the image of the extended source only becomes washed out due to the convolution with the point-spread function, but it does not decrease in overall intensity.

This means that a small amount of surface roughness of the circular object can completely cancel out the bright spot.

Note, that the 100 μm edge corrugation almost completely removes the central bright spot.

If the variance in the radius of the obstacle are much smaller than the width of Fresnel zone near the edge, the contributions form radial segments are approximately in phase and interfere constructively.

However, if random edge corrugation have amplitude comparable to or greater than the width of that adjacent Fresnel zone, the contributions from radial segments are no longer in phase and cancel each other reducing the Arago spot intensity.

In the above simulations with the 4 mm diameter disc the adjacent Fresnel zone has a width of about 77 μm.

In 2009, the Arago spot experiment was demonstrated with a supersonic expansion beam of deuterium molecules (an example of neutral matter waves).

The wave nature of particles actually dates back to de Broglie's hypothesis[25] as well as Davisson and Germer's experiments.

The observation of an Arago spot with large molecules, thus proving their wave-nature, is a topic of current research.

[21] Finally, the aragoscope has been proposed as a method for dramatically improving the diffraction-limited resolution of space-based telescopes.