A Study of Evolving Optical Caustics Formed by Evaporating Water Droplets
Samantha Scibelli
Laser Teaching Center, Department of Physics and Astronomy
Stony Brook University
Introduction
Before coming to the Laser Teaching Center this summer I was interested in conducting a research project that involved gravitational lensing. Gravitational lensing is the deflection of a light source from sources more distant than the lens and depends on the gravitational potential. In essence, it is the process where light rays are defected by a massive density of matter. It can often be described as analogous to the deflection of light by lenses in optics. My mission was to connect gravitational lensing to a project I could accomplish in the physics lab.
Online research first led me to geometric lensing and geometric optics. I read about how 2D cones are good examples of what the intrinsic curvature of space can be. I went through an exercise where I imagined a 2D universe on the surface of the cone at a point. To the creatures living on this world, it appears flat. The light rays cannot tell they are not traveling in a flat plane. The 2D cone can be slit along the side at a line (think about this as the light ray) and placed flat. This shows that the object can be seen at each edge of the flat cone. So, two light rays can appear to be originating from two objects, one on each side of the singularity of the cone. This doubling effect is geometric lensing.
Geometric optics is used describe various properties of imaging; including optical aberrations. These occur when light from one point of an object does not converge into (or does not diverge from) a single point after transmission through the system. Aberration leads to blurring of the image produced by an image-forming optical system. I learned that makers of optical instruments need to correct optical systems to compensate for aberration.
Geometric optics and optical aberrations led me to investigate caustics. The word caustic actually means burning. Broadly, it is the concentration of light. At first caustics intrigued me because I remember hearing a talk by a graduate student at Rensselaer Polytechnic Institute who was studying dark matter caustics. I began to read numerous papers that described various applications of caustics. For example, caustics were used to measure distances and to focus light into a clear portrait like images.
In addition to reading journal articles about caustics, I read papers about topological defects formed from Nematic liquid crystals. These defects are believed to form when symmetry is broken in a phase transition. Gravitational lensing actually plays into this phenomenon. Objects such as cosmic strings change their topology and, therefore, introduce phenomena like gravitational lensing.
After reading numerous papers I decided on studying caustics formed from water droplets. I conducted a project where I studied the evolution of caustics formed from an evaporating water droplets viewed in the far field.
Background
OPTICAL CAUSTICS
What are caustics exactly? Caustics are 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. Examples of caustics include rainbows, bright lines in water droplets, the sharp light curves off of a wine glass or coffee mug, and sheets or lines of light in a swimming pool (Fig. 1).
Fig. 1 – The image to the left is a photograph of
caustics reflecting off of a soda
bottle. The image to the right was taken from the Optics Picture of the Day website that shows
rapidly moving caustics at the bottom of a stream and the distortion of a trout caught by
caustics.
I was interested in studying caustic patterns because the same organization patterns of caustics explain the twinkling of stars and underlie the phenomenon of gravitational lensing, which makes visible to us the most distant objects in the universe.
An optical caustic can more precisely be referred to as the locus of the points at which the intensity of the deflected rays becomes infinite, as the envelope of the deflected rays, and as the locus of the principal centers of curvature of the deflected wavefronts (Lock et al., 1990). There is an interest in optical caustics because their structure and formation can be calculated mathematically. For example, it can be visually seen that when rays from a distant point source of light pass through an irregular drop on a dirty glass, they come to focus not at a point but in a system of caustics (Nye 1978). The evolution of caustics from evaporating water droplets provides a good test case where a comparison between the observed features and the mathematical model of the caustics analyzed can be preformed (Nye 1999).
CATASTROPHE THEORY
A mathematical model known as catastrophe theory is used to analyze the caustic shapes of evaporating water droplets. Catastrophe theory, developed by the French mathematician Rene Thom in the 1970's, is the study of how changing control variables leads to qualitative changes in the solutions of an equation. It has been applied to different phenomena, such as the stability of ships and bridge collapses, however optical caustics are one of the best visual illustrations of how catastrophe theory can be applied (Nye 1999). Berry (1976) first applied the theory to caustics seen by evaporating water droplets. Later, Nye (1978) preformed an experiment of his own that involved viewing evaporating water droplets in the near field.
Thom’s theorem, which is at the heart of the classification of structurally stable caustics, states that there are only a strictly limited number of different types of shapes, or catastrophes, for any given number of dimensions in control space or codimension (Table 1 from Nye 1999). There are seven ‘elementary catastrophes’ in control space less than or equal to four (Berry 1980). In this experiment the equations that generate each elementary catastrophe were studied.
Experimental Methods
FAR FIELD VIEW
When laser light passes through a water droplet the perturbations produce a far field caustic, known as the parabolic umbilic in the catastrophe theory classification (Lock et al., 1990). A Laser beam (λ = 633nm) was expanded to about 6mm in diameter with two lenses, with the combined focal 7 length of 16cm, and illuminated a vertically placed microscope slide (Fig. 2).
Fig. 2 – Diagram of the far field set-up used for this
experiment.
MICROSCOPE SLIDE SETUP
A microscope slide was placed vertically for the far field viewing method. A Pipette was used to place a water droplet close to 6mm in diameter onto dirty slide with 6mm pre-made circular hole outlines about 0.1mm thick for each trial (Fig. 3). The slide was illuminated on the side opposite the water droplet with the horizontal broadened laser beam and at a region about 40cm away caustics could be seen on a viewing screen.
Fig. 3 – The diagram on the left shows a drop of water is in a circular
traced
hole on a vertical
glass
slide. The image to the right is of the microscope slide that is used to
place the
water
droplets in. The
drops were placed in the hole circled in red
Data/Results
In total I have preformed six trials. In this report I only present three. Images from the remaining trails can be found in the caustic pictures link on my webpage. Each of the trials progressed in a similar fashion. Below are pictures of the caustic evolution for the first three trials. Under each of the pictures there are tables that correspond with the pictures above. The tables includes the time, initial observations, and picture number for each transition a caustic had for each trial.
The first trial was the only trial where the entire process took around 30 minutes. The evolution for the remaining trails took around 60 minutes. The setup from Lock et al., describes the process taking around 60 minutes. There could be many factors that play into the time it take for the evolution of these water droplets and the varying appearances droplet evolves.
Each transition sequence for each trail was similar in that a fold caustic was present first. Many times diffraction stars, or elliptic umbilic foci, would appear. Then, a prominent elliptic umbilic would appear and beginning to slowly merge in the fold, which slowly appeared as a cusp point, until it was completely indistinguishable as two different features. Then, the process would reverse to just a fold until it evaporated completely and no caustic was seen. Note, when the elliptic umbilic focus lies exactly on the fold we have the parabolic singularity itself (Fig. 4). This process is part of the unfolding of the parabolic umbilic.
Fig. 4 – Image of parabolic singularity. Photo taken from
the
first trial conducted.
Analysis
Each evolution observed shows the parabolic umbilic catastrophe unfolding as the water droplet evaporates. Catastrophes can be thought of as functions that describe sudden jumps or changes in the number of equlibra of a system. The parabolic umbilic is type of catastrophe that can occur for four control factors and two behavior axes. The parabolic umbilic is an intermediate between the hyperbolic umbilic and elliptic umbilic. Fig. 5 shows that unfolding process taken from Nye 1978. The parabolic umbilic catastrophe is given by the unfolding, F(x, y, a, b, c, d) = x4 + xy2 + ax2 + by2 + cx + dy of its germ, f(x,y) = x4 + xy2. Note that not every step can be noticed visually because they are blurred out by diffraction.
Fig. 5 – Unfoldings of the parabolic umbilic (D5) from
Nye
1978, adapted from Thom 1972.
Properties of a particular caustic can be deduced by considering critical points of gradient mappings. Caustics occur when two rays meet, that is when two stationary phase points join. These critical points on a map move and annihilate each other, or are created or dispersed from each other, in a state space as the control parameters are varied. By changing these parameters by an external method, equilibrium is displaced.
Think about a circle with a line through it. The line intersects the circle at two points. As the line slowly moves to the top, or bottom, of the circle the two points “smash” into a tangent point. This is a catastrophe. More specifically this is a fold catastrophe. The higher order catastrophes when higher orders of equilibrium are broken. When the critical points move and “smash” into each other from the state parameters of each catastrophe then the control parameters are varied and the shape evolves. You can think about this: larger quantitative changes in a system will result in qualitative changes that can be observed.
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. Mathematically caustics are studied by analyzing phase functions that generate a particular caustic. The caustic curve 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.
The generating function for a caustic is created into the standard form of the coordinates obtained from the germ and unfolding terms from Table 1. The formation of the ray is expressed as the derivative of the generating function. The formation of the caustic is expressed as the second derivative of the generating function. Sometimes overlapping caustics are formed and expressed as the third derivative of the generating function. The formation of the caustic curve is obtained by setting the second derivative (aka the formation of caustic) equal to zero.
I have derived and plotted with MATHEMATICA the caustic curves for the first three catastrophes: the fold, the cusp, and the swallowtail (Fig 6, 7, and 8). Further study includes analysis of the caustic curves for the remaining four catastrophes.
Discussion
The evolution of the caustics formed from the evaporating water droplets do follow a similar pattern described by Lock (1990), Nye (1979), and Berry (1980). Within these similar patterns, however, the times as well as the appearance of the caustic vary from transformation to transformation.
I believe that reason for the increased transformation time for the first trial was because of the placement of the laser beam on the microscope slide. In the first trial the laser beam was not in the center of the circular hole, the way the slide was placed made the beam hit the bottom of drop. The reason I assume this is effect took place is because the area responsible for the caustic is actually at the bottom of the droplet.
Fig. 9 – The left image shows the beam from the laser hitting the bottom of the
circular hole where the water droplet is placed. The right image shows the laser beam in the center
of the circular hole.
The size of the droplet affects how the caustic develops and what it looks like at the beginning of its transition. This is seen numerous times in the literature. In the first trail the droplet was larger than the others. The second trial droplet was the smallest of the three and the third trail’s droplet was in-between in size. In each trial after the third I made sure the water drops filled in the 6mm hole completely. What I saw was that each droplet’s irregular nature made it so that no transition was ever identical for each trial. They were, however, very similar. I plan to study these variables more in depth in future trials.
The caustic patterns do not rotate rigidly but continuously reorganize themselves in such a way that the bright convex fold remains at the top of the field. The umbilic points on the surface of a drop are stable features; they are not destroyed by small perturbations. The caustic can pass from one class to another and can also undergo reaction by an encounter with another umbilic point (Nye 1979).
The dynamic nature of the caustic formed from the evaporating water droplet does not affect the internal structure of the caustic analyzed. The structural changes I noticed are consistent to the ones notices by Lock, Nye, and Berry. What’s important is that each irregularly shaped water droplet follows a similar pattern of evolution. The time it takes for each transition to occur and the general appearance of the caustic depend on the placement of the beam and the size of the drop. Nonetheless, each caustic evolution for each trial follows the unfolding process of the parabolic umbilic.
By analyzing the curves that form the caustic catastrophes a better understanding of how they evolved in the experiment can be achieved. Caustics curves have been analyzed for the first three catastrophes that are in the first, second and third control dimensions. Future analysis of the catastrophes in the fourth control dimension will be studied. Overall, a fundamental understanding of nature’s focusing power can be achieved by studying caustics.
Future Studies
Further analysis of caustic behavior of water droplets evaporating in the far field will be conducted. Also, I wish to obtain a clearer understanding of the caustic curves for higher dimension catastrophes, such as the elliptic umbilic, hyperbolic umbilic, and parabolic umbilic. Investigation of caustics of higher order singularities (ex: symbolic umbilic E6) would also be interesting.
Fig. 10 – Images of separate water drop caustics that formed in the far field as the
size of the water droplet changed rapidly.
I also hope to be able to study caustics formed from evaporating water droplets in the near field by using a microscope. By studing the near field, clear ellipitc umbilic foci can be observed.
Fig. 11 – The diagram on the left is of the microscope slide setup for the near
field. The image on the right is of an irregularly shaped drop of water as it evaporates in the near
filed and elliptic umbilic foci appear (Nye 1978).
Acknowledgements
I would like to thank The Laser Teaching Center (LTC), Dr. Noé, Melia Bonomo, Professor Metcalf, Marty Cohen, Sam Goldwasser, Dave Battin, and the summer students working in the LTC this summer for the resources and guidance needed to conduct this project.