Presentation at Chaos 2018: The Chaos theory is a tool used to study old problems from a new point of view, based on two main concepts: deterministic nonlinear systems with few degree of freedom and sensitiveness to initial conditions. If we consider the geometric optics of an incident ray of light in a spherical drop, we can see these concepts of chaotic systems present in this system. The nonlinearity is present in the abrupt change of the refraction index at the border of the drop, with the trajectory of the light ray changing following just two laws of reflection and refraction in some points.

1. Rainbows, Billiards and Chaos
Alberto Tufaile, Adriana P. B. Tufaile
Soft Matter Laboratory
Escola de Artes, Ciências e Humanidades
Universidade de São Paulo, Brazil

2. Rainbow and Glory
• One of the most beautiful atmospheric phenomena observed by many
people is the rainbow, as can be seen in Fig. 1(a). The rainbow involves the
formation of a perfect circular arc with the presence of beautiful colors [1].
Other beautiful phenomena is the Glory effect [2], which presents
concentric colored halos around the shadow of the observer, as it is shown
in Fig. 1(b). These kinds of phenomena inspire myths, songs and physicists
to study their properties. According to Nussenzveig, some of the most
powerful tools of mathematical physics were created to solve the questions
raised by the observation of rainbows and with closely related problems,
such as the Glory effect.

3. A drop as
an open
billiard
• The Chaos theory is a tool used to study old problems from a new point of view,
based on two main concepts: deterministic nonlinear systems with few degree of
freedom and sensitiveness to initial conditions. If we consider the geometric optics
of an incident ray of light in a spherical drop of Fig. 2(a), we can see these concepts
of chaotic systems present in this system. The nonlinearity is present in the abrupt
change of the refraction index at the border of the drop, with the trajectory of the
light ray changing following just two laws of reflection and refraction in some
points.
• The signature of chaotic systems is observed in Fig. 2(b), when we try to follow
some trajectories of some light rays in a single drop: the light rays split and bounce
back and forth following the equations (1), making the long term prediction of the
trajectories of these rays very difficult, as the same as it is observed in systems
known as open billiards.

5. White Rainbow

7. Catastrophes

8. Rays and Waves

9. Cusped wavefront

10. The Spiral and the Rainbow
From Spheres to Cylinders
2D & 3D

11. 3D

12. 3D

13. Multiple reflections

14. Horseshoe & reflections
Cusp

15. 2D

16. 2D

18. Supernumerary
Bows
Interference between
the two arms of the
cusped wavefronts
producing supernumerary
bows

19. Interference in the cusped wavefront

20. Rainbow angle

22. Look at the shadow of your plane!

24. Surface
waves

25. Surface waves in a Reuleaux Bubble

26. Chaos, Acoustic and Quantum
Systems, some concepts:

27. Emergence of Chaos

28. Periodic, Chaotic, Random

29. Chaos in a regular acoustic
resonator

30. Quartz blocks
Effect of ray splitting
in the fluctuation
statistics of fused
quartz blocks. The
curve labeled “8
GOE” corresponds to
a superposition of
eight independent
Gaussian Orthogonal
Ensemble, each with
fluctuation
properties of the
Gaussian distribution
family.

31. Conclusions
• Quantum and classical systems present an
interesting frontier: semi-classical systems.
The concepts of particles/rays and waves is
not enough to understand this region, and the
use of diffracted rays is interesting to improve
our comprehension of physical systems
presenting this type of duality, since from
rainbows to quantum chaos, because ray
theory breaks down when diffraction is
present.

32. • Besides diffraction effects, we observed that the
mechanism of ray splitting is one of the main
features of chaotic behavior, because the
trajectories diverge in an abrupt fashion, not only
exponentially. This ray splitting introduces
another degree of divergence in each case, and
whole system has to be examined all at once,
using the different points of view of classical and
quantum systems, probably because ray splitting
tends to destroy invariant tori and stable islands
in the phase space, increasing the ergodic
component of the dynamics.

33. • In our studies, we have found the interesting
case of “spiral rainbow”, based in the
experiment of a laser scattering in a glass
cylinder. In this spiral rainbow, we have
observed something similar to the horseshoe
map for the multiples reflections in the
cylinder.

34. Quartz Blocks
• The mechanism of ray splitting is also present in
the elastomechanics of the fused quartz blocks,
involving mode conversion as a symmetry-
breaking mechanism that acts to mix transverse
and longitudinal wave motion. The resonances in
these quartz blocks are mixtures of transverse
and longitudinal motion. Although numerical
simulation shows that the classical rectangular
three-dimensional ray-splitting billiard is not
chaotic, the spectral fluctuations statistics of the
measured eigenfrequencies follow superposed
GOE spectra.

35. Acknowledgments
• This work was partially supported by Conselho
Nacional de Desenvolvimento Científico e
Tecnológico (CNPq), Instituto Nacional de
Ciência e Tecnologia de Fluidos Complexos
(INCT-FCx), and by Fundação de Amparo à
Pesquisa do Estado de São Paulo (FAPESP)
FAPES/CNPq#573560/2008-0.