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I tried to implement difraction as explained here. I used the formula on page 4, right before the simplification to the Rayleigh-Sommerfeld diffraction formula,

$$u(x_0)=\frac{1}{4\pi}\int_A u(x)\frac{\partial g_{x_0}}{\partial\nu}\Bigg|_x dS(x)$$

with

$$\frac{\partial g_{x_0}}{\partial\nu}\Bigg|_x=-2(ik-\frac{1}{\|x\|-\|x_0\|})G(x-x_0)\cos(\theta(x, x_0))$$

because they say

Since we are interested in the regime where the observation point $x_0$ is far from the aperture

and I wanted to play around with various distances, so I think the formula should be just fine?

The only thing which I don't understand, from reading through the lectures, it seems like we are just modelling monochromatic electromagnetic waves, so we should also get all kinds of real world situations where diffraction is not visible.

When I setup a monochromatic point source behind some large aperture, I get a sharp shadow from the aperture on the screen. If I move the point source around (along the $XY$-plane), the shadow would also move around on the $XY$-plane. But I'm neither able to get this behaviour out of my simulation, nor do I see how this would work, as I compute a large number of spherical wavefronts travelling in the z-direction from each point on the aperture, and bundling their energy along their travelling direction, which is the $\cos\theta(x, x_0)$-term.

So here is how I picture the current process:

And here is the other real life situation which should also have an explanation in terms of electromagnetic waves:

Is the correct way of thinking about this that a sharp shadow only results from spherical wavefronts cancelling each other precisely behind the egde of the shadow (and my simulation not providing enough resolution for this phenomenon)?

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  • $\begingroup$ Please avoid external links that could break in the future. Typeset the necessary equations in your equations with Mathjax (very close to LaTeX syntax). $\endgroup$
    – Miyase
    Jul 17, 2022 at 9:25
  • $\begingroup$ Thanks for your comment, are you talking about my sketches or about the lecture notes? $\endgroup$
    – fweth
    Jul 17, 2022 at 9:26
  • $\begingroup$ Your last image corresponds to a very large aperture (in units of wavelengths). Try increasing wavenumber in your simulation, and you should see something sensible (I tried changing u_WaveNumber from 350 to 1000 and it worked, tried even higher, and signal-to-noise ratio rapidly dropped to unusability). Ideally you should try 10000 or more to see the pattern (after having fixed the simulation SNR). $\endgroup$
    – Ruslan
    Jul 17, 2022 at 9:57
  • $\begingroup$ Thanks a lot, I guess that's the answer then? So a sharp shadow is actually the result of interference of a large number of high frequency waves? Feel free to post as an answer if you like! $\endgroup$
    – fweth
    Jul 17, 2022 at 10:00

1 Answer 1

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As usual in wave optics and quantum mechanics, the observations known from ray optics and classical mechanics, i.e. sharp shadows, are the result of high wavenumbers combined with macroscopic obstacles/apertures (i.e. large compared to wavelength).

If you increase wavenumber in your simulation by one or two orders of magnitude, you should be able to get a high frequency oscillation in the lit part of the screen and rapid decay of illuminance in the shaded part.

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