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.
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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.
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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.
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What makes droplets and other spheroids interesting ?
It's the shape-dependence of their
optical
properties. The properties that I am talking about include:
- Wavelength-selectivity, i.e. filter action
- Ability to trap light for long times
- Preferential coupling to adjacent optical components
- 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 also appear in a recent special issue of Science
Magazine: "Manipulating coherence"
See
the review by Mabuchi and Doherty on page 1372.
Exciting new basic research is also being done with
microdroplets:
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.
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.
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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.
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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.
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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).
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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 th 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.
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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. |
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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.
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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).
PDF.
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