## Poisson's Spot (aka Spot of Arago)

Occulter/aperture size is \(\sf D = 60\)mm.

Wavelength is \(\sf \lambda = 600\)nm.

Downstream distance starts at \(\sf z = 25 D = 1.5\)m and tends toward infinity.

The formula for the downstream electromagnetic field associated with a plane wave after passing through a circular opening measured as a function of radius \(\sf \rho\) from the beam/shadow's center is:

\(\sf
E_{hole}(\rho) = \frac{2 \pi}{i \lambda z} e^{\frac{i \pi \rho^2 }{ \lambda z}}
\int_0^{D/2} J_0(2\pi \rho r / \lambda z) e^{\frac{i \pi r^2 }{ \lambda z}} r dr
\)

By Babinet's principle, the downstream electric field associated with a circular occulter
is just one minus the above electric field:

\(\sf
E_{occulter}(\rho) = 1 - E_{hole}(\rho)
\)

The intensity (aka brightness) of the image is the magnitude-squared
of the complex electromagnetic field. The movies below show the brightness.

In the 2D animations shown on the left below, each pixel in each frame is a realization of
a Poisson random variable having a mean value proportional to the current intensity at that location.

These values represent the number of discrete photons arriving during an "exposure".

The movie starts out slow but has a dramatic ending---hit the "play" button.

Matlab codes:

Occulter shadow (Poisson4.m)

Light beam from hole (antiPoisson4.m)

Individual movies:

Occulter shadow in 2D

Occulter shadow radial intensity profile

Light beam from hole in 2D

Light beam from hole radial intensity profile

Here's a closeup showing Poisson's spot at the center...

###
Click here
for information about our High-Contrast Imaging research group.