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A Study of Evolving Optical Caustics
Formed by Evaporating Water Droplets
Samantha Scibelli
Laser Teaching Center
Stony Brook University
Optical caustics are concentrated envelopes of light rays reflected or refracted by a curved
surface of an object, or the projection of that envelope of rays on another surface.
There are many familiar examples: rainbows, bright lines in water droplets and on the bottoms of
swimming pools, and the sharp light curves formed by a wine glass or coffee mug.
These seemingly random effects share an underlying mathematical structure, which was revealed
when Berry [1] applied catastrophe theory to optical caustics.
Catastrophe theory, developed by the French mathematician René Thom in the 1970's, is the
study of how gradually changing control variables can lead to sudden qualitative changes in the
solution of a differential equation. Such phenomena are analogous to the
popular idiom "the
straw that broke the camel's back."
In this project, which was inspired by the work of Nye and Lock et al. [2,3,4], I
observed and mathematically modeled the evolution of caustics created by passing a light beam
through the irregular edges of evaporating water droplets.
Each droplet was contained within a very shallow (~0.1 mm thick) well 6 mm in diameter on a
vertical microscope slide. The light source was a horizontal laser beam (λ = 633 nm) that
was expanded to ~ 6 mm diameter with two lenses.
The patterns were observed on a screen at a distance of 40 cm from the drop and photographed
with a Nikon digital SLR camera. I observed the evolution of six drops in all.
The patterns slowly evolved with time (over about an hour) revealing a sequence of distinctive
caustic features. The time it took for each caustic to appear and its general appearance
depended on the placement of the beam and the size of the drop. However, the evolution of the
caustics for each of the six trials was similar in that first a fold appeared, then an elliptic
umbilic formed, the two features merged to form a parabolic singularity, and finally the process
reversed itself until the droplet had completely evaporated. The overall process is part of the
unfolding of the parabolic umbilic in the catastrophe theory.
What's striking is that even though each drop is unique its evolution takes place through
the same sequence of steps.
I am in the process of deriving the caustic curve equations for each of the seven elementary
catastrophes in order to compare their mathematical shapes to my experimental observations.
These equations originate from each catastrophe's generating function, which is the combination of
the unfolding terms (containing the control variables) and the catastrophe's characteristic germ.
I arrived at the equation for the caustic curve by setting the second partial derivative of the
generating function equal to zero and solving for the control variables.
The graph of this equation allowed for a visual comparison between the theoretical elementary
catastrophe and the observed feature.
I have derived and plotted with MATHEMATICA the caustic curves for the first
three catastrophes: the fold, the cusp, and the swallowtail. Further study includes analysis of
the caustic curves for the remaining four catastrophes.
By analyzing the curves a better understanding of how caustics evolved in this experiment as well
as the structure of light's overall natural focusing power can be achieved.
This work was supported by the Laser Teaching Center and the Simons Foundation. I also wish to
thank Melia Bonomo and John Noé for helpful discussions.
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