Numerical aperture of a lens
Channel: Nils Berglund
Date: 21/11/2023

This video is the first of a new series based on a collaboration with the Huygens Optics channel https://www.youtube.com/@HuygensOptics , illustrating the concept of numerical aperture (NA).
A video with more detailed explanations will be released by Huygens Optics later this week, but here is a short explanation. The first half of the video shows a lens with two boundaries made of circular arcs. Two sources separated by a distance of a few wavelengths emit waves synchronously. The distance between sources and lens and the index of refraction inside the lens have been chosen in such a way that an image should form at an about equal distance to the right of the lens. One can indeed see the waves concentrate in two points that mirror the positions of the sources. This situation will produce an image with a good resolution, and is said to have a large NA.
The second half of the video shows a situation in which only the central part of the lens has been kept. This results in less light going through the lens, and fewer diffraction orders contributing to the final image. Hence the image will be of poorer quality: the sources cannot be resolved. This situation is said to have a small NA. 
Forthcoming videos in this series will show various set-ups, with extra lenses and/or absorbing walls. 
The video has four parts, showing two different numerical apertures and two different color schemes:
Large NA, wave height: 0:00
Large NA, averaged energy: 1:41
Small NA, wave height: 3:24
Small NA, averaged energy: 5:05
In the first and third parts, the color hue shows the wave height. In the second and fourth parts, it shows the energy, averaged from the beginning of the simulation. There are absorbing boundary conditions on the outer boundaries of the simulation. 

Render time: Part 1 - 32 minutes 14 seconds
Part 2 - 29 minutes 26 seconds
Color scheme: Parts 1 and 3 -  Twilight by Bastian Bechtold
https://github.com/bastibe/twilight
Parts 2 and 4 - Magma by Nathaniel J. Smith and Stefan van der Walt
https://github.com/BIDS/colormap

Music: "Images of Tomorrow" by the Unicorn Heads@UnicornHeads

See also https://images.math.cnrs.fr/Des-ondes-dans-mon-billard-partie-I.html 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 https://hplgit.github.io/fdm-book/doc/pub/wave/pdf/wave-4print.pdf 
C code: https://github.com/nilsberglund-orleans/YouTube-simulations
https://www.idpoisson.fr/berglund/software.html
Many thanks to Marco Mancini and Julian Kauth for helping me to accelerate my code!

#wave #wave_equation #interferenceoflight #numerical_aperture