User:Treisinger/PoissonSpot



In optics, a Poisson spot is a bright point which appears due to Fresnel diffraction at the center of the wave shadow of a circular object. It is also sometimes called an Arago spot or a Fresnel bright spot. The experiment played an important role in the discovery of the wave nature of light (see history section below) and is a common way to demonstrate that light behaves as a wave for example in undergraduate physics laboratory exercises. The basic experimental setup is shown in the figure on the right. The wave source must be at least smaller in diameter than the circular object casting the shadow and the dimensions of the setup must comply with the requirements for Fresnel diffraction. Namely, we must have that the Fresnel number


 * $$F = \frac{d^{2}}{l \lambda} \gtrsim 1$$

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

Finally, the edge of the circular object must be sufficiently smooth. These conditions together explain why the bright spot is not encountered in every-day life. However, with the abundance of laser sources available to day it is easy to perform a Poisson spot experiment (see for example here). In astronomy, the Poisson spot can be also easily 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.

The presence of Poisson's spot can be easily understood. When light shines on a 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.

History
The original Poisson-spot experiment was carried out in the beginning of the 19th century and played an important role in the history of science. Then it turned out to be the deciding experiment of whether light is a particle or a wave. It is thus a great example of a so-called experimentum crucis. 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).

At the beginning of the 19th century it became more and more evident that light does not simply propagate along straight lines (Thomas Young published his double-slit experiment in 1807 ). However, many still favored Isaac Newton's corpuscular theory of light, among them the great theoretician Siméon-Denis Poisson. In 1818 the French Academy launched therefore 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. Poisson studied Fresnel's theory in detail and of course looked for a way to prove it wrong being a supporter of the particle-theory of light. Poisson thought that 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. We mentioned before that the Poisson spot is not easily observed in every-day situations, so it was only natural for him to interpret it as an absurd result and that it should disprove Fresnel's theory.

However, the head of the committee, Dominique-Francois-Jean Arago, and 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. To everyone's surprise he succeeded in observing the predicted spot, which convinced most scientists of the wave-nature of light. In the end Fresnel won the competition, much to Poisson's chagrin. 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 and Maraldi a century earlier.

Theory behind the Poisson spot


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. This means that the field at a point P1 on the screen is given by a surface integral:



U(P_1) = \frac{A e^{\mathbf{i} k r_0}}{r_0} \int \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)) $$

and


 * A is the amplitude of the source wave
 * k = $$\frac{2\pi}{\lambda}$$ is the wavenumber
 * S is the unobstracted surface

The first term outside of the integral represents the oscillations from the source wave at a distance r0. Similarly, the term inside the integral represents the oscillations from the secondary wavelets at distances r1.

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 then yields the the integral for a circular object of radius a (see for example ):



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} (\frac{1}{g} + \frac{1}{b})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 ):



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:


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

This shows that the on-axis intensity at the center of the shadow tends to the source intensity, as if the circular object was not present at all. Furthermore, this means that the Poisson spot is present even just a few obstacle diameters behind the disc.

Calculation of Poisson spot 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 $$g(r,\theta)$$ 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 aperture we are trying to calculate the intensity for passes through the blocking circular object.) the integral we need solve is given by:



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. 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.



The images show simulated Poisson spots in the shadow of a disc of varying diameter (4mm, 2mm, 1mm - left to right) at a distance of 1 m from the disc. The point source has a wavelength of 633 nm (e.g. He-Ne Laser) and is located 1 m from the disc. The image width corresponds to 16 mm.

Experimental aspects
Observation of the Poisson spot with a conventional light source can be challenging. This section summarizes how the various experimental parameters affect the visibility of the Poisson spot.

Intensity and size of Poisson spot
For an ideal point source the intensity of the Poisson spot equals that of the undisturbed wave front. Only the width of the Poisson 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 is clear from the simulation images above. This means that one can compensate for a reduction in the source's wavelength by increasing the distance l between 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) :



U(P_1,r) \propto J_0^2(\frac{\pi r d}{\lambda b}) $$ where
 * r ... distance of the point $$P_1$$ on the screen from the optical axis
 * d ... diameter of circular object
 * $$\lambda$$ ... wavelength
 * b ... distance between circular object and screen

The following images show the radial intensity distribution of the simulated poisson spot images above:



The red lines in these three graphs correspond to the simulated images above, and the green lines were computed by applying the corresponding parameters to the squared bessel function given above.

Finite source size / spatial coherence
The main reason why the Poisson 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 Poisson spot will have an extend that is given by S*b/g, as if the circular object acted like a lens. At the same time the intensity of the Poisson spot is reduced with respect to the intensity of the undisturbed wave front.

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 Poisson spot changes. In particular, if the object has an ellipsoidal cross-section the Poisson 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 Poisson spot is only affected marginally, since one can interpret the Poisson spot as a point-spread function. And so 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 over all intensity.

The circular object's surface roughness
The Poisson 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 Poisson spot from a 4-mm-diameter disc (g=b=1m):



The simulation includes a regular sinusoidal corrugation of the circular shape of amplitude 10µm, 50µm and 100µm, 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 Poisson spot. All the Fresnel zones that are further out destructively [interference (light)|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 Poisson 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 :

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

The edge corrugation should not be much more than 10% of this width to see a close to ideal Poisson spot. In the above simulations with the 4-mm-diameter disc the adjacent Fresnel zone has a width of about 77µm.

Poisson spot with matter waves
Recently, the Poisson spot experiment has been demonstrated with a supersonic expansion beam of deuterium molecules, so-called neutral matter waves. Material particles behave like waves as we know from quantum mechanics. The wave-nature of particles actually dates back to de Broglie's hypothesis as well as Davisson and Germer's experiments. A Poisson 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 a Poisson spot with large molecules, and thus proving their wave-nature is a topic of current research.

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