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Microlasers and ray chaos

A hitchhiker's guide to dielectric cavities*

Contents of this page:

Light's growing weight

You 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 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.

A more specific example of how microphotonics can make an impact is described in this PDF-article describing my field of work in the photonics industry from May 2000 until August 2001. The material system discussed there Indium Phosphide, a semiconductor compound. Other material systems for microphotonics can be found among polymers, glasses, porous media - to name a few.

At the heart of these developments is the availability of small but efficient lasers which deliver the required intense and coherent light.

If you have any doubts that the laser is one of the twentieth century's most important achievements in science and technology, please read about the impact and history of laser light at this new website.

For an amusing but also informative glimpse of laser physics, see the Britney Spears Guide to Semiconductor Physics.

Wikipedia is a good source of information and links on laser physics.

Microlaser design

All 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.

Trapping light with interference

One common way of making especially good parallel mirrors is to use Bragg reflection at multiple layers of dielectric films. See, e.g., the Wikipedia entry on "Vertical-Cavity Surface Emitting Lasers". The Bragg principle is based on the destructive interference between waves in successive layers of a stack of dielectric layers.

As a rather logical continuation of the same principle, one has progressed to 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 that the 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.

Whispering-gallery resonators - trapping without interference

In solid-state laser materials, it is often possible to realize the mirrors simply by exploiting total internal reflection at the interface between the high-index solid and the surrounding medium (e.g., air). In contrast to the Bragg principle, this confinement mechanism for light is to lowest order frequency-independent and can therefore be called a classical effect - it can be described without explicit use of the wave nature of light, by using Fermat's principle.
This is good because it means that a device based on this confinement mechanism will in principle be able to work over a very broad range of wavelengths - in stark contrast to photonic crystals. Nevertheless, one can use total internal reflection to make three-dimensionally confined resonators with high frequency selectivity (or "finesse"), provided one can force wavefronts inside the cavity to interfere with themselves.

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.

Whispering-gallery waves:
To illustrate the whispering-gallery effect, the movie shows a cross sectional view of a curved interface (black circle) between glass and air, with a circulating wave radiating in all directions.
The color represents the electric field, and in the first animation the field inside the resonator is only slightly higher than outside. This is not a good resonator because it is very "lossy".
In the second movie, the wavelength is about 4 times shorter than above. In this case, the field outside the resonator is much weaker than inside it, meaning that we are confining the light much better. In both animations, the wave fronts look slanted, especially on the outside. Comparing the two clips, you will notice, however, that the wave fronts right at the circular interface are perfectly radial in the bottom image. This is what makes the two scenarios different: the straight wave fronts at the interface correspond to grazing propagation along the curved boundary.
There is still a wave emanating from the cavity at the bottom, but its amplitude relative to that at the interface is now much smaller. Observe also the central region of the dielectric circle, which is essentially field-free. The intensity is highly concentrated near the surface.

Even in the more strongly confined case, shown here, the wave penetrates slightly into the surrounding medium. In reflection off a straight dielectric interface, this penetration is known to go along with the Goos-Hänchen effect, a lateral displacement of the scattered beam. A calculation of the analogous effect in reflection off a curved interface can be done starting from the circular geometry. Since the Goos-Hänchen effect can be incorporated into a ray model, it improves semiclassical calculations for non-circular cavities.

A detailed introduction to the Goos-Hänchen effect and our relevant work is presented on a separate page.

To find out more about the spiral patterns shown in these movies, read about wavefronts in open systems.

plot
plot

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

cover
Optical Processes in Microcavities, edited by R.K.Chang and A.J.Campillo (World Scientific Publishers, 1996).
The lasing droplets are seen on the left side, and a "thumbtack" microlaser with its rotationally symmetric calculated emission pattern appears in the main panel.

Remark:
This book contains 11 chapters on important experimental and theoretical aspects of dielectric microcavities. Chapter 11 represents the status of our work as of summer 1995:
"Chaotic Light: a theory of asymmetric cavity resonators",
J.U.Nöckel and A.D.Stone
PDF - (warning: large files)

Don't be square!

The question that arises naturally in lasing microdroplets is: how strongly can a dielectric resonator be deformed before whispering-gallery modes cease to exist, or become degraded by leakage? The intuitive answer is, "the rounder, the better". However, even shapes with sharp corners can sustain modes that have every right to be called whispering-gallery phenomena. In fact, these types of whispering-gallery modes cannot be understood purely on the basis of ray optics. This is discussed in our work on hexagonal nanoporous microlasers.

Intriguingly, hexagonal zinc oxide nanocrystals have recently become the smallest resonators sustaining whispering-gallery type modes ever observed.

Being round is not a prerequisite for whispering-gallery action. So there is a huge space of possible shapes (practically from circle to square) that could possibly be considered as whispering-gallery type resonators. If we had a choice, what should the ideal shape be? This clearly depends on the application context, but in any case it would be desirable to have some design rules. In the following, we begin to discuss some design issues, and point out how our work in particular aims to provide the design rules just mentioned, based on approximate methods such as the ray picture.

Stable and unstable resonators

Other 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 coupling 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, Lasers (University Science Books, Mill Valley, CA (1986)].
E.g., two concave mirrors separated by more than their added radii of curvature cause rays to diverge out from the optical axis after several reflections. Outcoupling occurs when the light spills over the edge of one of the mirrors (which hence need not be partially transparent themselves).

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.

Remark:
Important work on such "quasi-normal modes" has also been carried out by Kenneth Young, Pui-Tang Leung and co-workers. The central problem from the point of view of laser physics is this:
In order to define photons in the first place, we expect to have at our disposal a set of normal modes for which we then write the creation and annihilation operators. But metastable states are not eigenstates of a Hermitian differential operator, because they represent energy escaping to infinity. Therefore, familiar precedures involving expansions in normal modes run into problems.

Nevertheless, their use makes a lot of sense when discussing the emission properties of individual metastable states, such as their frequency shifts as a result of a perturbation in the resonator's shape or dielectric constant.

Or - just to mention a really far-out example: metastable states find application in the study of gravitational waves emitted from a black hole [P.T.Leung et al., Phys.Rev.Lett. 78, 2894 (1997)]

Chaotic resonators

As 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:
billard

It is taken from J.H.Jensen, J.Opt.Soc.Am.A 10 (1993).

Remark on previous work:
Jensen seems to have been the first to attack the ray-wave duality for a stadium-shaped dielectric resonator, in particular taking into account the inevitable ray-splitting into reflected and transmitted portions that occurs at the sharp dielectric interface of the chaotic resonator (thanks to R.K. Chang and A. Poon for pointing out the reference). However, he did not consider the long-lived resonances that such a cavity could support, which are a prerequisite for lasing. Instead, Jensen's paper gives a quasiclassical analysis of the rainbow-peaks for this structure. For more on rainbows, see this Atmospheric Optics web site. Ray splitting has received renewed interest in recent years (in my own ray optics simulations, it is taken into account as well - it becomes essential in high-index materials).

We are not the only ones to consider chaotic dielectric resonators. However, we were the first (to my knowledge) to seriously apply chaos analysis to the emission properties of quasibound states in dielectric resonators, see
"Q spoiling and directionality in deformed ring cavities",
J.U.Nöckel, A.D.Stone and R.K.Chang, Optics Letters 19, 1693 (1994).
PDF.
This is a theory paper in which we address the consequences of emerging ray chaos for the lifetimes and emission directionality of deformed dielectric resonators.

The first experiment in which the correspondence between emission anisotropy and chaotic structure in the classical ray dynamics was successfully applied to dielectric microlasers is
"Ray chaos and Q-spoiling in lasing droplets",
A.Mekis, J.U.Nöckel, G.Chen, A.D.Stone and R.K.Chang, Phys.Rev.Lett. 75, 2682 (1995).
PDF.
In this paper, we studied lasing microdroplets with a nonspherical shape, which leads to a strongly anisotropic light output along the droplet surface. The total-intensity profile was imaged and compared with a ray model, yielding an explanation for the observed features.

To arrive at the idea of using a chaotic resonator cavity, one can either start from the unstable-resonator concept as described above, or  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.

Confocal resonators

Inbetween stable and unstable resonators, there is another useful mirror configuration, called confocal. It has the advantage of creating a focussing effect inside the resonator, which in turn amounts to producing a smaller effective mode volume for the laser. Instead of the whole volume between the mirrors, it is possible to utilize only a smaller volume around the coinciding focal points of the mirrors. The ray pattern that forms in a confocal arrangement of two concave mirrors can sometimes take on the shape of a bowtie (depending on the shape of the mirrors). This well-known configuration is found in etalons but also in lasers. The simplest confocal cavity would consist of two circle segments with a common focus. A less trivial example is the case of two confocal paraboloids, i.e., surfaces of revolution generated by opposing parabolas that share their focal point:
dome   plot
The righthand picture shows two bowtie rays going through the focus. There are many other ray paths that never go through the focus, but they form caustics which are reminiscent of this basic shape. For a study if this type of (three-dimensional) mirror configuration, see my work with Izo Abram's group at CNET, "Mode structure and ray dynamics of a parabolic dome microcavity". This is the manuscript:
PDF.

Microresonators such as this can find application in quantum electrodynamics because they allow to modify the rate of spontaneous emission of atoms or quantum dots interacting with the electromagnetic field. To that end, one has to go to small mode volumes. But 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 laser

Now we put all of the above together, but for the price of one...

plot

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 date.
To understand why this very desirable intensity distribution arises in the smooth oval shape we chose here, but not in the circle or the stadium, one has to use methods of classical nonlinear dynamics. This is explained in our article,
" High power directional emission from lasers with chaotic resonators ",
C.Gmachl, F.Capasso, E.E.Narimanov, J.U.Nöckel A.D.Stone, J.Faist, D.Sivco and A.Cho, Science 280, 1556 (1998)
PDF, cond-mat/9806183.

Remark:
In this paper, the oval-resonator concept is combined with a very innovative laser material that turns out to be particularly compatible with a disk-shaped resonator geometry: the quantum cascade laser.

This active material consists of a semiconductor heterostructure in which an electrical current leads to the emission of photons. But in contrast to more conventional quantum-well diode lasers, the optical transitions responsible for the creation of the photons take place exclusively within the nanostructured conduction band (between quantum well subbands). Electron-hole recombination across the valence band (the usual mechanism) is not involved here, leading to various advantages.

F.Capasso and J.Faist are among the winners of the 1998 Rank Prize for the invention of the quantum cascade laser.

The basic ideas of our work are illustrated on picture pages starting with a galery of magazine covers and continuing with a special type of shape called the Robnik billiard (also known as the dipole shape or limacon billiard).

How to learn more:

What is chaos ? And what in the world  is quantum chaos ?

Chaos is not just chaos

We are talking here about deterministic chaos. The term refers to the fact that even simple classical systems governed by simple equations such as Newton's laws can exhibit highly irregular motion that defies long-term predictions. One example for such a simple physical system is the double pendulum; as the following animation shows, the two degrees of freedom represented by the two angles θ and ψ are coupled, and this leads to a non-periodic, unpredictable-looking combined motion:
plot

In Optics, there is a slight confusion of terminology about the concept of chaos, because it is traditionally found (in quantum optics) 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.

There are many lists of chaos-science links; see for example the Wikipedia artticle on this subject. For more on the the relation between our work and the more traditional nonlinear optics, see below.

Chaos in billiards

In the classical ray picture for our microresonators, the fact that boundaries are penetrable does not (to lowest order in the wavelength) affect the shape of the trajectories, and hence our internal ray dynamics is that of a non-dissipative, closed system.

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 answer at this informative FAQ site maintained by Jim Meiss. Further information, including a host of graphics and animations, is also available from the chaos group at the University of Maryland.

Quantum Chaos

Quantum chaos sounds like a contradiction in terms because linear wave equations such as the Schrödinger equation do not exhibit the sensitivity to initial conditions that gives rise to chaos. Nonetheless, classical mechanics is just a limiting case of quantum mechanics, just as ray optics is the limit of wave optics for short wavelengths. So one should expect "signatures of chaos" in the wave solutions. To find and understand these, semiclassical methods are indispensable.

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:
dynamical localization and dynamical tunneling.

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 a picture page describing the annular billiard. 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").

Is quantum chaos just a mathematical-conceptual game without relevance for experiments? Our work has been among the first to propose actual applications of quantum chaos phenomena, and to my knowledge the two patents I co-authored were the very first to rely on such phenomena.

Nonlinear dynamics

Chaos, belonging to the field of nonlinear dynamics, is known to laser physicists in another guise as well: pattern formation, in particular vortices and vortex lattices, due to the nonlinearity of the lasing medium, has been studied much longer than our type of chaotic phenomena which rely on the boundary effects. Of course, there can be a cross-over from one regime to the other, e.g. from nonlinear vortices to linear vortices which in a circular resonator are encountered as whispering-gallery modes.

What I'm discussing above is chaos in the linear wave equation. This phenomenon often dominates the physics, especially near the lasing threshold. At higher powers the nonlinearity of the medium itself becomes more important.

This is something we had earlier addressed in an invited conference contribution , and also commented on in a book chapter titled "2-d Microcavities: Theory and Experiments" .

*Note:

Last significant revision: 09/09/04.
This page represents a compilation of information relevant to our work on microlaser resonators. Naturally, it cannot claim to be complete in any way. However, I felt it appropriate to provide some context because the questions we are discussing are at the interface between two fields of study that traditionally haven't had much overlap: micro-optics and quantum chaos.

These fields have more in common than meets the eye.
But that by no means implies that one community cannot learn from the other...

Since this is a NET DOCUMENT, I am trying to refer mostly to other documents that are available online, instead of citing things printed on dead trees. But if you have something you'd like me to include, feel free to let me know.

Related information is found on the following web pages:

This page © Copyright Jens Uwe Nöckel, 2002-2004

Last modified: Sat Jun 22 09:20:46 PDT 2013