Nils Berglund | The double peak disappears when the source filmed with a gradient index lens is slower @NilsBerglund | Uploaded April 2024 | Updated October 2024, 58 seconds ago.
The recent video youtu.be/idB6qn802lM showed a source of light moving at about 33% of the speed of light being filmed by a gradient index lens. It showed a somewhat strange double peak, or "echo" being formed by the lens, making the source appear doubled. The question arose in the comments whether this was due to the very fast speed of the source. This simulation shows a similar situation, with a source moving at a slower speed, about 17% of the speed of light. It turns out that there is no double peak here, confirming that in the other simulation it must have been an effect of the higher speed.
The refractive index n(r) of the gradient index lens used here depends on the distance r to the horizontal symmetry axis of the lens in a quadratic way, like n(r) = n0 - a*r², with n0 = 1.25 and a = 0.4375, making the refractive index smaller than 1 at the outer boundary of the lens, where r = 1. In order to visualize the refractive index as well, the luminosity of the background depends on the refractive index.
The videos are inspired by Huygens Optics' recent short youtube.com/shorts/VGd3Ajnp6e0 showing the principle of a gradient index lens.
Lenses focus incoming rays of light by delaying them more near the center of the lens than at its periphery. This is often done with a material of constant index of refraction, by making the lens thicker near the center, as shown for instance in the simulation youtu.be/rrJJBh9ubUE . However, one can also build lenses of constant thickness, by making the index of refraction of their material depend on the location in the lens. In this simulation, the index decreases like the square of the distance to the center (that is, it is of the form n0 - a*r², where r is the distance to the axis of symmetry). This results in the incoming waves being focused at two points in the (estimated) focal plane, marked by a vertical line. The plot to the right shows a time-averaged value of the field along that plane.
This video has two parts, showing the same evolution with two different color gradients:
Wave height: 0:00
Averaged wave energy: 2:16
In the first part, the color hue depends on the height of the wave. In the second part, it depends on the energy of the wave, averaged over a sliding time window.
There are absorbing boundary conditions on the borders of the simulated rectangle. The display at the right shows the signal along the focal plane, which is indicated by a vertical line.
Render time: 1 hour 6 minutes
Compression: crf 23
Color scheme: Part 1 - Viridis, by Nathaniel J. Smith, Stefan van der Walt and Eric Firing
Part 2 - Plasma by Nathaniel J. Smith and Stefan van der Walt
github.com/BIDS/colormap
Music: "Lazy Boy Blues" 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 hplgit.github.io/fdm-book/doc/pub/wave/pdf/wave-4print.pdf
C code: 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 #lens #gradient_index
The recent video youtu.be/idB6qn802lM showed a source of light moving at about 33% of the speed of light being filmed by a gradient index lens. It showed a somewhat strange double peak, or "echo" being formed by the lens, making the source appear doubled. The question arose in the comments whether this was due to the very fast speed of the source. This simulation shows a similar situation, with a source moving at a slower speed, about 17% of the speed of light. It turns out that there is no double peak here, confirming that in the other simulation it must have been an effect of the higher speed.
The refractive index n(r) of the gradient index lens used here depends on the distance r to the horizontal symmetry axis of the lens in a quadratic way, like n(r) = n0 - a*r², with n0 = 1.25 and a = 0.4375, making the refractive index smaller than 1 at the outer boundary of the lens, where r = 1. In order to visualize the refractive index as well, the luminosity of the background depends on the refractive index.
The videos are inspired by Huygens Optics' recent short youtube.com/shorts/VGd3Ajnp6e0 showing the principle of a gradient index lens.
Lenses focus incoming rays of light by delaying them more near the center of the lens than at its periphery. This is often done with a material of constant index of refraction, by making the lens thicker near the center, as shown for instance in the simulation youtu.be/rrJJBh9ubUE . However, one can also build lenses of constant thickness, by making the index of refraction of their material depend on the location in the lens. In this simulation, the index decreases like the square of the distance to the center (that is, it is of the form n0 - a*r², where r is the distance to the axis of symmetry). This results in the incoming waves being focused at two points in the (estimated) focal plane, marked by a vertical line. The plot to the right shows a time-averaged value of the field along that plane.
This video has two parts, showing the same evolution with two different color gradients:
Wave height: 0:00
Averaged wave energy: 2:16
In the first part, the color hue depends on the height of the wave. In the second part, it depends on the energy of the wave, averaged over a sliding time window.
There are absorbing boundary conditions on the borders of the simulated rectangle. The display at the right shows the signal along the focal plane, which is indicated by a vertical line.
Render time: 1 hour 6 minutes
Compression: crf 23
Color scheme: Part 1 - Viridis, by Nathaniel J. Smith, Stefan van der Walt and Eric Firing
Part 2 - Plasma by Nathaniel J. Smith and Stefan van der Walt
github.com/BIDS/colormap
Music: "Lazy Boy Blues" 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 hplgit.github.io/fdm-book/doc/pub/wave/pdf/wave-4print.pdf
C code: 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 #lens #gradient_index
