Diffraction patterns using the Rayleigh-Sommerfeld integrals

Using the Rayleigh-Sommerfeld diffraction theory it is possible to calculate the diffracted field at an arbitrary aperture. The 2-D integral integral is transformed into a 1-D integral (see figure at left). When a parallel wave front is diffracted by an aperture the diffracted waves obtained using the Rayleigh-Sommerfeld diffraction integrals take the general form [7] [8]:

$$U_{\text{RS}}^{\text{I}}(P) = C z \int_{\phi'_{\text{min}}}^{\p... ... \left[\frac{e^{i k s}}{s} \right]_{s_{\text{max}}}^{s_{\text{max}}} d \phi'$$

and

$$U_{\text{RS}}^{\text{II}}(P) = C \int_{\phi'_{\text{min}}}^{\ph... ...xt{max}}} \left[ e^{i k s} \right]_{s_{\text{min}}}^{s_{\text{max}}} d \phi' .$$

These integrals do not have mostly closed-form solution but can be evaluated by numerical integration.

The figure at left shows the model used for the simulations for 2 shells of circular apertures. The lattice constant is $D$ and each aperture has radius $a$.

A C++ program using the GNU Scientific Library (GSL 1.3) was written to calculate the intensity and phase of the scalar field above the array.

From upper left to lower right: $z = \lambda$, $z = 15 \lambda$, $z = 18 \lambda$, $z = 30 \lambda$, $z = 45 \lambda$, $z = 60 \lambda$

Relative field intensity of the simulation for images of size 20$\times$20 . Lattice constant: $D =$   3176 . Radius of the aperture $a =$   250 and wavelength $\lambda =$   488 . The images have been calculated using the RS diffraction integral of the first kind, using an array of circular apertures with 5 hexagonal shells and a total of 91 apertures. For each image is indicated the distance to the diffraction array $z = z(\lambda)$.

Below are shown the relative intensities calculated for 3 axis parallel to $z$ with coordinates
$P_{0} = (0, 0, z)$, $P_{1} = ( D / 2, D \sqrt{3} / 6, z)$ and $P_{2} = ( D / 2, 0, z)$.