A Fresnel zone plate, in higher resolution

This second simulation of a Fresnel zone plate uses a simulation grid with half the mesh size than the previous simulation • A Fresnel zone plate and with a slightly smaller wavelength, therefore allowing to see more details in the diffraction pattern.
Fresnel zone plates are devices with interesting optical properties, similar to lenses, but that use diffraction instead of refraction to get the desired effect. In this simulation, the lens approximately focuses an incoming linear wave to a point. Unlike for usual lenses the index of refraction of the zone plate is constant, equal to 1, but instead the transparency of the plate varies like [1 + cos(k r^2)]/2, where r is the distance to the center of the plate. It is also possible to replace the smoothly varying transparency by a transparency switching between two fixed values.
The position of the focal point depends on the parameter k and on the wavelength of the incoming wave, see en.wikipedia.org/wiki/Zone_plate for more details. The transparency of the plate is controlled here by adding a dissipation term inside the plate, which varies according to the above function of the distance to the center.
The simulation has two parts, showing respectively the amplitude of the wave, and the average energy from the beginning of the simulation, on a logarithmic scale:
Wave amplitude: 0:00
Average wave energy (log scale): 1:01
Render time: 1 hour 26 minutes
Color scheme: Twilight by Bastian Bechtold
github.com/bastibe/twilight
Music: "Old Vienna" by Endless Love@Endless Love
See also images.math.cnrs.fr/Des-ondes... for more explanations (in French) on a few previous simulations of wave equations.
The simulation solves the wave equation by discretization. The algorithm is adapted from the paper hplgit.github.io/fdm-book/doc...
C code: github.com/nilsberglund-orlea...
www.idpoisson.fr/berglund/sof...
Many thanks to my colleague Marco Mancini for helping me to accelerate my code!

[@KitagumaIgen]

Nice! How would the focusing change if the innermost zone was perfectly reflective and blocked the wave completely? Would it be easier to distinguish the contributions from the peripheral zones? How does the focusing change with focal length? Questions, questions...

[@NilsBerglund]

Good questions, thanks!

[@samuelsegal3542]

I absolutely love these videos! Really helps me visualize optics

[@NilsBerglund]

Awesome! Thank you!

[@aubergine1115]

cool video

[@vk2zay]

What does it look like when the source wavefronts aren't parallel to the axis? I imagine they will be focused in the same image plane, but offset radially.

[@NilsBerglund]

Good question, I suppose I can try simulating it. It should also be possible to determine what happens from the Huygens-Fresnel principle.

[@mltonsorangestapler]

what you're describing is essentially an interference spectrum at the focal length. One side or the other is perfectly bright and the other is perfectly dark AKA perfectly constructive and destructive interference, respectively, with lobes of bright and dark interference patterns in nodes along the spectrum.

[@dzmitryv.krukau4327]

Does it looks like "a flaw" (wake) of liquid/optical hydraulics combined with multiple slot (filter) [lenz] "illusion" ?

[@mltonsorangestapler]

Can you do the far-field too? I'm assuming this is the near-field

[@TiagoTiagoT]

In the average wave energy section of the video, it looks like the scale is missing the darker red/wine color range, it seems to stop the light orange/peach color at the top end...

[@NilsBerglund]

I should probably have shown a larger range. I set the range by hand, the reason being that since I don't know the range beforehand, the color is determined by computing tanh(s*h), where h is the wave height, and s is a slope parameter controlling the contrast. So h can be arbitrarily large, there will just be a saturation effect.

[@TiagoTiagoT]

@@NilsBerglund Perhaps you could first render the video as grayscale frames in an image format that allows for 32bits float colors like OpenEXR, and then before encoding the final video file, run thru the frames to get the range of values, and add the scale overlay and the color gradient mapping?

[@Kram1032]

very cool. I'd like to see a much longer run of the average energy one so it can converge everywhere. Would make for a nice poster too I'd imagine.
Maybe you can somehow use the speed of wave propagation to adjust from when the averaging ought to start, delaying that in places further to the right, so you get a quicker convergence to the steady state average

[@NilsBerglund]

Thanks, good idea!

[@WHYNKO]

Ideally it looks like only the first few do the work of concentration of the beams...

[@NilsBerglund]

It may be the case that for smaller wavelengths, more sections would contribute to the concentration. A similar effect is observed for Fresnel lenses.

[@WHYNKO]

@@NilsBerglund i was in fact thinking this is how Fresnel lenses work... But what was the wavelength used in this simulation?

[@WHYNKO]

@@NilsBerglund really nice work tho 👍🙂

[@NilsBerglund]

You can estimate it from the first part of the video. The width of the zone plate contains about 2.5 wavelengths.

[@basisTermium]

Woah, the zone plate has graded index?
beautiful follow up!

[@NilsBerglund]

Thanks! The index of refraction is constant in a Fresnel zone plate, but the "transparency" depends on the distance to the center of the plate.

[@basisTermium]

​@@NilsBerglund Interesting, for the transparent gradient, is there a specific reason why?

[@NilsBerglund]

It happens to create a diffraction pattern that focuses the wave. A similar effect is achieved with alternating transparent and opaque sections. There are more details in the page the description links to.