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Diffraction patterns using the Rayleigh-Sommerfeld integrals

\includegraphics[width=0.8\textwidth]{FIGURES/DBHEXP2.eps}

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]:

$\displaystyle 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

$\displaystyle 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.

\includegraphics[width=0.8\textwidth]{IMAGES/RSDP_CIRCHEXPS5L488ZXXL.eps}
From upper left to lower right: $ \scriptstyle z = \lambda$, $ \scriptstyle z = 15 \lambda$, $ \scriptstyle z = 18 \lambda$, $ \scriptstyle z = 30 \lambda$, $ \scriptstyle z = 45 \lambda$, $ \scriptstyle 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
$ \scriptstyle P_{0} = (0, 0, z)$, $ \scriptstyle P_{1} = (
D / 2, D \sqrt{3} / 6, z)$ and $ \scriptstyle P_{2} = ( D /
2, 0, z)$.

\includegraphics[width=.80\textwidth]{FIGURES/CIRCHEXPS5L488PZ60L_P0P1P2x.eps}


next up previous
Next: Conclusion Up: Self-imaging observed on colloidal Previous: Field intensity measured above
Othmar Marti
Experimentelle Physik
Universiät Ulm