Microdroplets: Chaos in a Spray Can

Microdroplets are of interest in atmospheric science, combustion science and biology. From the optics point of view, they belong to the family of dielectric spheroids.

If you had a crystal ball, it's unlikely that you'd be able to see the future with it. However, if you could see the future, you may well find tiny crystal balls in it.

Oscillating droplet

Microspheres could show up as wavelength-selective devices in future micro-optical applications. They are already a widely used tool in quantum optics labs, but their technological potential is not fully exploited yet.
Shown here is a water droplet falling vertically while undergoing natural shape oscillations.

What makes droplets and other spheroids interesting ?

It's the shape-dependence of their optical properties. The properties that I am talking about include:
  1. Wavelength-selectivity, i.e. filter action
  2. Ability to trap light for long times
  3. Preferential coupling to adjacent optical components
  4. Near-field and internal field profile
The questions that can be investigated range from engineering issues via applied mathematics to chaos physics.

We have studied both solid and liquid forms of spheroids. For engineering applications, one can make microspheres from fused silica (glass). A big advantage of silica is that individual resonances can be spectrally resolved with greater easy than in liquids. For basic experimental studies of shape-dependent effects, on the other hand, it may be better to use a material whose shape can be varied more easily. That is one of the advantages of liquid droplets. They can have a wide variety of shapes, and one can moreover make those shapes reproducibly. What we learn from the study of deformed droplets can then potentially be transfered to solid spheroids.

Meanwhile, it has been demonstrated that controllable shape variations in silica spheroids can be achieved. The work done by Scott Lacey and Hailin Wang in this direction has given rise to a collaboration centering on weakly deformed, non-axial spheroids that extends the work described on this page in a new direction.

Some links for further information

The technological relevance of such cavities, based for example on silica, is shown in K.Vahala's group at Caltech.

To learn about the scientific significance of microspheres, check out the work of Steve Arnold and the page at the Laboratoire Kastler-Brossel.

Microspheres appeared, e.g., in a special issue of Science Magazine: "Manipulating coherence"
See the review by Mabuchi and Doherty on page 1372.

New ways of exploiting the strong internal field enhancement within microdroplets are still being discovered: Plasma formation dynamics within a water microdroplet on femtosecond time scales (Courvoisier et alia, Optics Letters February 2003). Some of the optics underlying these short-pulse microcavity experiments is discussed in my recent book chapter with R.K. Chang.

Learning Resonator design from nature

Microcavities are useful because they can act as optical resonators. Resonances are a wave phenomenon which relies on interference effects. When one designs optical devices, it is often useful to start from a simplified treatment which ignores such wave effects. That's the ray picture - obtained from the exact wave equations by making a short-wavelength approximation.

The fascination of marbles and beads comes mostly from the ray-based tricks that light can play in such simple objects. Nature itself presents spectacular effects such as the rainbow as light is caught by the bead-like droplets in a cloud. A wide variety of other phenomena occur in our atmosphere (see also the hexagon page).

In the laboratory, droplets can be produced by squeezing a jet of liquid through a nozzle. Of particular interest to us are ways to make individual droplets of well-defined shape. For that purpose, one uses a vibrating type of nozzle that chops the stream in a periodic way, and then images the falling droplets stroboscopically to catch them reproducibly at a predetermined stage of their shape oscillation.

The ray-wave connection

The rainbow is a good example for using rays as a starting point and then gaining physical understanding by adding wave corrections. See the general discussion of quantum chaos for more on this approach of interpreting wave effects as the flesh on the skeleton made up by the classical rays.

Why bother with rays ?

The use of ray approximations opens up a set of powerful theoretical methods, such as catastrophe theory and chaos analysis.

To illustrate why this can be extremely useful for our understanding, let's look at liquid microdroplets more closely.

Here is a wave calculation for the intensity distribution in a droplet shaped like a mathematical ellipse. It shows enhanced light emission near the points of highest curvature, as one would expect intuitively because of the large "bending" of the waveguiding walls there.

The ray-wave connection can be visualized in this surface of section. The black lines are generated by following a couple of ray trajectories launched within the cavity, and at each successive encounter with the boundary recording the polar angle theta along the boundary (see inset). A two-dimensional plot is generated because we also record the angle of incidence, chi, with respect to the outward normal.

The rays inside the ellipse generate continuous curves as shown above. This tells us that there is no chaos in the classical ray dynamics. In color, I have superimposed the wave intensity of the cavity mode shown above. As can be seen from the red band, the wave intensity "condenses" onto the classical structure in this " phase space" spanned by theta and sin(chi).

Here is a wave calculation for the intensity distribution in a droplet shaped like a quadrupole. As can be seen, the emission directionality of this cavity is very different from that of the ellipse, even though the two shapes are quite hard to tell apart by eye. In particular, the points of globally highest curvature are located at the same positions in this and the previous example.

What this implies is that the emission properties can sometimes be very counter-intuitive: light in this case does not emanate tangential to the highest-curvature points, as expected from a naive bending-loss argument.

The good thing about the ray approach is that it can provide an explanation for such effetcs. As you can see here, the surface of section for the quadrupole shows structure that is not present in the ellipse. The waves see this phase-space structure and form different patterns as well. This is shown in the color representation of the wave field. One thing to notice here is that wave intensity is "pushed away" from the points of highest curvature (which in this plot lie at theta=0 and 180°), as a result of some elliptic islands into which the intensity cannot penetrate.

Chaotic whispering-gallery modes

We have called the phenomenon just described dynamic eclipsing. It is a fingerprint of the partially chaotic phase space structure as it exists, e.g., in the quadrupole. The cavity resonances giving rise to this effect are chaotic whispering-gallery modes. See the Hitchhiker's guide to dielectric cavities for more on whispering-gallery modes.

Microdroplets were one of the first systems in which this nonintuitive physics was observed experimentally. The details are to be found in
"Observation of emission from chaotic lasing modes in deformed microsperes: displacement by the stable orbit modes ",
S.S.Chang, J.U.Nöckel, R.K.Chang and A.D.Stone, JOSA B 17, 1828 (2000).

This page © Copyright Jens Uwe Nöckel, 06/2003

Last modified: Sat Jun 22 09:00:44 PDT 2013