A hitchhiker's guide to dielectric cavities*
Light's growing weightYou don't need a great deal of imagination to foresee an increasing significance of lightwave technology in data processing and telecommunications. Here are some arguments in favor of light:
Miniaturization of electronic circuits leads to increased resistances and hence larger dissipation. Photons don't suffer from losses in the same degree because their interaction is much weaker than that of electrons. The speed of light in a dielectric is much larger than the propagation speed of electronic signals in metals. The bandwidths available for signal transmission are a few hundred kHz on copper cables, versus roughly a THz in a typical glass fiber - even now it is feasible to carry half a million telephone conversations over a single glass fiber. Photons are the method of choice for massively parallel data processing and storage.
More could be said, but this has already been done by others: See this online article on microphotonics in general, and an article describing my own field of work in the photonics industry since May 2000.
At the heart of these developments is the availability of small but efficient lasers which deliver the required intense and coherent light.
Microlaser designAll of us (physicists) have probably been "exposed" to the He-Ne laser in some graduate student lab. But of course the most ubiquitous lasers are by now the semiconductor diode lasers. Both of these incarnations rely on the parallel-mirror configuration to provide the feedback that makes laser action possible. This type of resonator is also known from the Fabry-Perot interferometer.
As a rather logical continuation of the same principle, one has progressed
photonic crystals which employ the Bragg principle in more than one
spatial direction and can in principle be used to make extremely small
photonic cavities. The price one pays is that one needs many periods of
the artificial crystal lattice in order to obtain high reflectivities, so
total size of the structure ends up being much larger than the cavity
itself. Higher and higher reflectivities are required, on the other hand,
if one wants to make a laser out of such a microcavity. The simple
reason is that a small cavity can host only a small amount of amplifying
material, and therefore it becomes more difficult for amplification to win
over the losses in a microcavity laser.
This is achieved with the "whispering-gallery" resonator which is at the heart of the lowest-threshold lasers made so far. This low threshold becomes possible as a consequence of the small size that can be achieved with these resonators. They are essentially circular disks in which the light circulates around close to the dielectric interface. Such modes are especially low in losses.
Semiconductors are far from being the only application of the whispering-gallery mechanism. The first laser resonators in the submillimeter size regime were made of liquid droplets containing a lasing organic dye. The highest-quality optical microresonators have been achieved using fused-silica spheres (i.e., glass). Although these materials have a refractive index closer to unity than a semiconductor, they still support whispering-gallery modes. In that context, they are often called morphology-dependent resonances (MDRs).
Both the semiconductor and the droplet realizations of the whispering gallery are illustrated on the cover of
Stable and unstable resonatorsOther mirror arrangements provide different advantages. In particular, there has been a considerable body of work employing concave or convex mirrors. E.g., concave mirrors separated by less than their radii of curvature added together, make a stable resonator in which light rays undergo focussing while being multiply reflected between the mirrors. Light can then be coupled out by making one of the mirrors slightly transparent.
When the output couplig is small, the theoretical treatment of such a laser can often be performed by neglecting the leakage and hence assuming the existence of some orthogonal set of modal eigenfunctions.
If one wants to avoid the use of partially transparent mirrors (which need
to have very low losses for high-power applications), one alternative
design is the unstable resonator containing defocussing
elements [see the
exhaustive textbook by A.E.Siegman,
(University Science Books, Mill Valley, CA (1986)].
Such unstable lasers differ from stable resonators in their mode structure: A set of well-defined bound modes is not available for the expansion of the laser field, because they all couple to the outside. Therefore, it has been necessary to use quasibound states in the calculations.
Lasers are fundamentally open systems, so a description in terms of quasibound states seems only natural. These states are, however, not as familiar a tool as the usual square-integrable eigenfunctions one knows from bound systems. Their properties are still a topic of current research.
Chaotic resonatorsAs an extention of the unstable-resonator idea, one can think of two concave mirrors in a defocussing setup combined with some lateral (sideways) guiding of the light between the mirrors. A naive reasoning could be this:
We want lasing from light spilling out near one of the mirrors, but we don't want the escape angle with the optical axis to be too large, hoping thereby to improve the spatial mode pattern (focussing). So we put additional mirrors along the open sides joining the mirrors.
Now combine this idea with the use of dielectric interfaces as (partially transparent) mirrors, and one is lead quite directly to consider the so-called stadium resonator (or a generalization thereof).
Here is an illustration of the stadium shape and of how it scatters
an incident ray:
It is taken from J.H.Jensen, J.Opt.Soc.Am.A 10 (1993).
To arrive at the idea of using a chaotic resonator cavity, one can either
start from the unstable-resonator concept as described above,
from the whispering-gallery design. We came from the latter direction. The
argument leading to an oval dielectric resonator is simply that a circular
whispering-gallery cavity does not have a preferred emission direction,
owing to its rotational symmetry. In addition, one wishes to have a
parameter with which the resonance lifetimes of the cavity can be
controlled. This is achieved by deforming its shape.
The Cavity QED group of T. Mossberg at the University of Oregon provides some info on why the ensuing internal focussing is desirable: Cavity quantum electrodynamic effects allow to modify the rate of spontaneous emission of the laser active material. To that end, one has to go to small mode volumes. But as is pointed out in this web seminar, the cavity volume isn't necessarily what counts. With a focused ray pattern as in the confocal resonator, the light field is especially strong in only certain portions of the resonator, notably the focal point in the center. And that is where the desired strong coupling between the light and the active medium occurs.
Bowtie laserNow we put all of the above together, but for the price of one...
The microcylinder laser shown here is not circular, but not a stadium shape, either. The stadium has fully chaotic ray dynamics, the circle has no chaos at all. This oval shape has a mixed phase space. As a by-product of the transition to chaos which takes place with increasing deformation, a bowtie-shaped ray path is born that does not exist below a certain eccentricity. This pattern combines internal and external focussing, and its lifetime is long enough for lasing because the rays hit the surface close to the critical angle for total internal reflection.
This is the world's most powerful microlaser to
The driven pendulum is not autonomous. A standard autonomous system showing chaos is the double pendulum, which can be admired in this JAVA simulation.
In Optics, there is a slight confusion of terminology about the concept of chaos, because it is traditionally found when people want to describe the statistical properties of a photon source. "Chaotic light" in that context has a much shallower meaning - it just means "random" thermal distribution of photons as it is found in blackbody radiation.
Chaos in the deterministic sense already has a place in optics as well, but again we have to make a distinction to our work. In multimode lasing one can look at the temporal and/or spatial evolution of the laser emission and finds that the signal can become very irregular. By mapping this behavior onto an artificial (usually many-dimensional) space, e.g. by a so-called time-delay embedding, one then sometimes finds that the system follows a trajectory on a "chaotic attractor". That's a type of structure one finds in dissipative nonlinear classical systems. This is what people have studied in nonlinear optics for a long time now - an example of a chaotic attractor for a laser can be seen here.
There are many lists of chaos-science links, one example being the one at NIST.
The optical resonator in the ray picture is a realization of what mathematicians call a billiard. See this short article for an entertaining introduction to billiards. Only non-chaotic billiards are shown there: the circle and the ellipse (note that this math definition of a billiard doesn't conform with what we know from the local pub). But generic oval billiards display chaotic dynamics. To take the step into the world of chaotic billiards, follow this link to the polygonal and stadium billiard (among others).
If you have any further questions about chaos, you may well find an
this informative FAQ site maintained by
Meiss. Further information, including a host of graphics and
is also available from the chaos group at the
One of the pioneers of quantum chaos, Martin C. Gutzwiller, has written a beautiful introduction to this field in Scientific American. See in particular the third figure describing the central place of quantum chaos in our our understanding of quantum mechanics. An important lesson here is: Playing around with the simple standard systems, such as harmonic oscillators, we barely scratch the surface of what the classical-quantum transition really entails. If we want to go beyond pedestrian descriptions of this transition, classically chaotic systems are where the action is! This also holds for much-discussed fundamental topics such as "decoherence", see the example of periodically "kicked" Cesium atom. As a by-product, quantum chaos has brought together an arsenal of powerful techniques. My first chance to study these was a graduate course at Yale taught by Prof. Gutzwiller in 1993/94; he also accompanied my thesis work on chaotic optical cavities through discussions and as a reader at dissertation time.
As it turns out, many of the intrinsic emission properties of dielectric optical resonators have a classical origin. The significance of this for quantum chaos is that comparison between ray model and numerical solutions of the wave equations uncover corrections to the ray model. Alternatively, one can also discover such wave corrections by comparing the ray predictions to an actual experiment. We follow both approaches.
Such wave corrections become especially interesting when the underlying
classical dynamics is partially chaotic, as is the case in the asymmetric
dielectric resonators. In that setting, two major new effects arise:
In dielectric cavities, the effect of such phenomena on resonance lifetimes and emission directionality, and of course on resonance frequencies, can be studied. Emission directionality is in itself a completely new question to investigate from the viewpoint of quantum chaos: when decay occurs, e.g.,in nuclear physics or chemistry, any anisotropy of the individual process is averaged out in the observation of an ensemble - but microlasers can be looked at individually, and from various directions. If they are bounded only by a dielectric interface, the emission pattern is determined by the phase-space structure. This is an important focus of my work: the short-wavelength asymptotics of systems that are chaotic and open.
What this means is illustrated in a slightly different example on picture page . There, we studied the relation between resonance lifetimes and dynamical tunneling (since it involves tunneling into a chaotic portion of phase space, it is also called "chaos-assisted tunneling").
Eric Heller's group provides some information on dynamical tunneling in a different system.
A list of Quantum Chaos groups is found here.
Related information is found on the following
© Jens Uwe Nöckel 2001-Aug-21