Nils Berglund | 3D representation of a gradient index lens with quadratic refractive index @NilsBerglund | Uploaded April 2024 | Updated October 2024, 2 minutes ago.
This is a 3D render of the video youtu.be/idqg-zchKGE , showing a gradient index lens exposed to two light sources. The z-coordinate is used to visualize the value of the wave or its energy, as well as the refractive index. The refractive index n(r) 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.
The lens is indeed able to focus the light on two spots located in the focal plane. In practice, one seldom has light sources located so close to the lens (except perhaps in macro photography), but this set-up was better suited to the simulation method used here.
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 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: 1:15
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 from the beginning of the simulation. In both cases, the z-coordinate shows the sum of the refractive index and the value of the wave height or energy. There are absorbing boundary conditions on the borders of the simulated rectangle.
Render time: 1 hour 19 minutes
Compression: crf 23
Color scheme: Part 1 - Twilight by Bastian Bechtold
github.com/bastibe/twilight
Part 2 - Plasma by Nathaniel J. Smith and Stefan van der Walt
github.com/BIDS/colormap
Music: "Free Spirit" by Sarah, the Illstrumentalist@Sarah2ill
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
This is a 3D render of the video youtu.be/idqg-zchKGE , showing a gradient index lens exposed to two light sources. The z-coordinate is used to visualize the value of the wave or its energy, as well as the refractive index. The refractive index n(r) 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.
The lens is indeed able to focus the light on two spots located in the focal plane. In practice, one seldom has light sources located so close to the lens (except perhaps in macro photography), but this set-up was better suited to the simulation method used here.
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 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: 1:15
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 from the beginning of the simulation. In both cases, the z-coordinate shows the sum of the refractive index and the value of the wave height or energy. There are absorbing boundary conditions on the borders of the simulated rectangle.
Render time: 1 hour 19 minutes
Compression: crf 23
Color scheme: Part 1 - Twilight by Bastian Bechtold
github.com/bastibe/twilight
Part 2 - Plasma by Nathaniel J. Smith and Stefan van der Walt
github.com/BIDS/colormap
Music: "Free Spirit" by Sarah, the Illstrumentalist@Sarah2ill
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
