A chaotic rainbow: where Cézanne could have met Poincaré


The design of this show-experiment was motivated by the challenge to propose a broad audience scientific setup, aimed at visually illustrating the richness and complexity of chaotic motion. The full realization of this setup has benefited from the financial support of the government of the Region Franche-Comté, in the framework of the 2015 International Year of Light. The scientific concept of the experiment is directly related to my first academic publication about a wavelength chaos setup operating with a tunable DBR semiconductor laser subject to a nonlinear delayed feedback. This experiment is an attempt to adapt the tiny few tenth of nm wavelength fluctuations of the original experiment, to the full visible spectrum one (ca. 300 nm, from the blue to the red). It makes use of a supercontinuum (SC) light source which white light is spectrally sliced by an acousto-optic tunable filter (AOTF), thus providing the wavelength tunable light source in the visible range. An appropriate birefringent filter (exhibiting maximum two free spectral range -FSR- within the full visible spectrum) thus nonlinearly converts the wavelength deviation into an intensity (according to the modulation transfer function of a two-wave interference). An electronic board is then slowing down the measured intensity fluctuations (3 Hz cut-off), so that these motion can be captured by human eye bandwidth. The board also performs a long delay (0.512 s) through an electronic delay line. The output signal is adequately amplified and biased with an appropriate offset before driving a voltage controlled oscillator, which RF output, after amplification, drives the AOTF in order to determine the next selected colorized slice of the SC white light source... which is itself again nonlinearly converted into intensity fluctuations, and so on...

A movie of the setup can be dowloaded here. It shows how a properly chosen time delay τD can reveal in a 2D space (xt xt-τD) the deterministic structure of the chaotic trajectory evolving in an infinite dimensional phase space of a delay dynamical system. The horizontally rotating points in the movie are suddenly revealing the nonlinear transformation connecting two 'time delay'-shifted color fluctuations x(t), for the correct value of the time shift. These points are directly picked from an experimental time trace, where we have selected the values x(t) at which the time derivative vanishes, (dx)/(dt)=0. The video also shows the live chaotic motion through the observation of the diffracted zero order at the AOTF output: this optical output corresponds to the SC light beam input of the AOTF, minus the selected color. A dark line is thus chaotically moving within a rainbow.

This broad audience dissemination action also led to a publication in Physical Review A, where interesting nonlinear delay dynamics issues have been reported in details.

Aknowledgements for the design of this setup are going to many contributors: Pierre Roux (IUT level internship, mechanical design) and Emmanuel Dordor (mechanical design), Joseph Nunes (electronic PCB), Ludovic Gauthier-Manuel (thickness reduction and polishing of the birefringent quarz slab), Anaïs Boisson and Etienne Genier (Bachelor degree internship in 2015), Quentin Ansel (Bachelor degree internship in 2013), Luca Furfaro, Remo Giust, Jean-Marc Merolla, Lucas Illing, Maxime Jacquot, Yanne Chembo, John Dudley (for helpful discussions), Roger Bourquin (for providing the X-cut quarz slab), Benjamin Beausire Félicien Berthet, Jennifer Nguyen, Jean-Christophe Savoie, Pierre Roblot, Laurent Tournier, Vincent Armbruster (for their contributions during an internship I proposed for the design of a prototype within the Superior Technical School in Photonics Victor Bérard in Morez, Jura).


Because of the dramatic events occured on the 13th of November 2015, two meanings of the word Chaos are sadly, unintentionally, temporally and linguistically colliding. It appeared thus necessary to make a small comment about such a sad and unintended coincidence. The Chaos I am used to deal with scientifically is of course at huge parsec distance from what is sometimes said by journalists about the social and moral earthquakes generated by e.g. the barbarism of terrorists. There is a need here to underline, from a moral, social, and civilization viewpoint, how antagonist Science is from extremism: This is the main aim of this footnote.

Science is by practice and construction, open minded and attracted by the potential to meet, understand and learn the unknown: This only can lead to the goal of an improved society, with less pain, more life, more happiness, through the technological implementation of scientific discoveries and novel knowledge. On the total contrary, terrorism, fanaticism, and extremism is based on confinement, absolute refusal to learn anything different from a forced, blind, and imposed way of thinking, and it is always leading to death and destruction, but never to any Human or social progress.

Chaos needs to be commented here as well, to emphasize on the actual signification probably thought on one side by Cézanne from a painting artist perspective, and also on the other side from a scientific perspective, by the mathematicians Li and Yorke. The latter researchers indeed proposed "Chaos" as a word  referring to complex motions exhibited by nonlinear dynamical systems, such as the ones first explored by Poincaré in the late XIXth century (three body problems), and re-discovered numerically by Edward Lorenz in the 1960s (simplified model for the dynamics of the atmosphere). Chaos in Science refers to richness, diversity, complexity, and harmonious coexistence of different kinds of behavior,... thus these notions are completely orthogonal to any disaster caused by extremisms.