Arago Spot - Theory

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

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

where the inclination factor 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

is the wavenumber

S is the unobstructed 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 integral for a circular object of radius a (see for example Born and Wolf):

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

This integral can be solved numerically (see below). If g is large and b is small so that the angle 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 and the intensity at the screen . The on-axis intensity as a function of the distance b is hence given by:

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 Arago spot is present even just a few obstacle diameters behind the disc.

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