Arago spot

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. Arago later noted that the phenomenon (later known as "Poisson's spot" or the "spot of Arago") had already been observed by Delisle and Maraldi a century earlier. 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). However the particle associated with electromagnetic waves, the photon, has nothing in common with the particles imagined in the corpuscular theory that had been dominant before the rise of the wave theory and Arago's powerful demonstration. 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. == Theory ==

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. == Experimental aspects ==

Intensity and size , contradicting the prediction of geometric optics. For an ideal point source, the intensity of the Arago spot equals that of the undisturbed wave front. Only the width of the Arago spot intensity peak depends on the distances between source, circular object and screen, as well as the source's wavelength and the diameter of the circular object. This means that one can compensate for a reduction in the source's wavelength by increasing the distance between the circular object and screen or reducing the circular object's diameter. The lateral intensity distribution on the screen has in fact the shape of a squared zeroth Bessel function of the first kind when close to the optical axis and using a plane wave source (point source at infinity): 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) where J_0and J_1are the Bessel functions of the first kind. R is the radius of the disc casting the shadow, \lambda the wavelength and g the distance between source and disc. For large sources the following asymptotic approximation applies: I_\text{rel}(w) \approx \frac{2 g \lambda }{\pi^2 w R} Deviation from circularity If the cross-section of the circular object deviates slightly from its circular shape (but it still has a sharp edge on a smaller scale) the shape of the point-source Arago spot changes. In particular, if the object has an ellipsoidal cross-section the Arago spot has the shape of an evolute. Note that this is only the case if the source is close to an ideal point source. From an extended source the Arago spot is only affected marginally, since one can interpret the Arago spot as a point-spread function. 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. Surface roughness of circular object The Arago spot is very sensitive to small-scale deviations from the ideal circular cross-section. This means that a small amount of surface roughness of the circular object can completely cancel out the bright spot. This is shown in the following three diagrams which are simulations of the Arago spot from a 4 mm diameter disc (): 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 field transmitted by a radial segment that stems from a point on the obstacle edge provides a contribution whose phase is tight to the position of the edge point relative to Fresnel zones. 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. The adjacent Fresnel zone is approximately given by: \Delta r \approx \sqrt{r^2 + \lambda \frac{g b}{g + b}} - r. The edge corrugation should not be much more than 10% of this width to see a close to ideal Arago spot. In the above simulations with the 4 mm diameter disc the adjacent Fresnel zone has a width of about 77 μm. == Arago spot with matter waves ==

In 2009, the Arago spot experiment was demonstrated with a supersonic expansion beam of deuterium molecules (an example of neutral matter waves). Material particles behaving like waves is known from quantum mechanics. The wave nature of particles actually dates back to de Broglie's hypothesis as well as Davisson and Germer's experiments. 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. The observation of an Arago spot with large molecules, thus proving their wave-nature, is a topic of current research. == Other applications==

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. Another is to probe aberrations in laser beams by using the spot's sensitivity to beam aberrations. == See also ==