Kathy

Barbara Ortscheid "translates" the following article:

When light hits an object, it bounces off at
the same angle that it struck the object. Throw a ball against
a wall. If you throw it straight to the wall, it will bounce
straight back to you. Throw it at an angle to the right and it
will bounce off to the right. Throw it at a wide angle and it
will bounce away at a wide angle. Notice the change in the
pattern of angles.This principle is the basis for the games of
pool and billiards.  [It is also what happens to light (lasers) when
it is constrained within a specific geometric form]

Now change the shape of your table to a circle. The same
basic science principle, but the bounce pattern changes. You
might want to play around with this with paper, pencil, angle
measure. With a change of shape, the patterns change.

Next flatten the circle on two side so that it is short one way and
long the other. When the ball (light) begins bouncing around inside
this shape, it creates a bow-tie pattern, a much more complicate
pattern. Then take a hammer (imaginary) and bang and dent the edge
of the circular table  inside and outside. The edges are no longer
straight or curved. When the ball (light) bounces around inside this
shape, it creates a pattern of mind-boggling complexity (chaos).

It is in the changing of the edges of the figure that creates different
patterns, and the more complex the figure the more complex the
patterns. [Thus, geometry affects the expression of laser light]

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LASER PHYSICS:
Enhanced: Geometrical Shaping of Microlaser Emission Patterns

Erich Gornik

Optoelectronics--the conversion of electronic signals into light and back
again--is part of our everyday lives. Though unnoticed when we pick up the
phone, listen to a compact disk player [HN1], or touch the television
remote control, optoelectronics is based on highly refined technology, a
well-balanced combination of transistors, semiconductor lasers, and
detectors [HN2]. Although enormous effort is still going into the
improvement of these "standard" devices in major laboratories and
university institutions, significant breakthroughs have been achieved in
the area of laser sources. The field is experiencing an explosion of fresh
and unconventional ideas; the control of threshold currents and emission
patterns make many new applications possible. The research article by
Gmachl et al. (1) on page 1556 in this issue represents a milestone in this
respect. It not only provides a concept for lasing at extremely low
currents but also for directionality control; it provides an example in
which an active optical system has entered a regime [HN3] where the
boundary to chaos can be controlled by geometrical design.

Micrometer-size lasers are of enormous interest as they promise to satisfy
the demands for ever smaller and more power-efficient systems.
Nevertheless, these devices still face several difficulties. One concept
that offers an extremely low current threshold is the "whispering-gallery"
disc laser (2); unfortunately this design still has problems with low power
output and uncontrollable directionality [see (A) of figure]. Gmachl et al.
(1) have developed a new type of semiconductor microlaser that is a
dramatic improvement over earlier related lasers by using better resonator
optics [HN4], chaos theory [HN5], and semiconductor quantum-engineering
[HN6]. The authors demonstrate a power increase by several orders of
magnitude (from ~10 mW to ~10 mW) and output directionality of miniature
cylinder lasers by fabricating them in a geometry that is smoothly deformed
from circular symmetry. The lasers are, in fact, quadrupolar; that is, they
have a circular cross section that has been elongated in one direction and
squeezed in the perpendicular direction. At small deformations this results
in chaotic whispering-gallery resonances, which are explained below. At
larger deformations the lasers operate on bow-tie-shaped modes that are
completely new to these little resonators and are highly advantageous. The
nature of these resonances becomes quite clear from the lower part of the
figure. In contrast to the circular lasers and those with very small
deformations, these resonances use only parts of the cylinder laser's
perimeter as resonator mirrors. This is responsible for the strongly
directional light output. The reflectivity of the boundary is very high,
but not quite unity, allowing the laser to have a low threshold and reach a
high output power.

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<Picture: Figure 1>
A la mode. (A) Schematic diagram of a whispering-gallery resonance in a
circular symmetric cylinder laser; the light remains trapped within the
laser resonator by continuous specular reflection from the boundary that
essentially circles along the cylinder perimeter. Light only weakly leaks
out from the resonator. (B) Schematic diagram of a bow-tie resonance in a
smoothly deformed cylinder laser of the kind described by Gmachl et al.
(1); the light bounces back and forth across the resonator emitting strong
light (by refraction) into narrow angles.

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The authors applied this technique to mid-infrared lasers, an important
class of devices because of their many applications, such as pollution
monitoring, medical diagnostics, and combustion or process control.
Nevertheless, the demonstrated concept is universal to resonators made of
high-refractive index material and as such is applicable to a much broader
range of materials, in particular to semiconductor lasers of all different
wavelength ranges.

The gain medium is an electrically pumped semiconductor cylinder laser.
When the cross section is circular, laser action takes place on so-called
whispering-gallery resonances [HN7]. This old and well-known concept comes
from medieval churches, where sound was known to travel along the curved
inner surfaces of arches and domes. Whispering-gallery lasers are some of
the tiniest monolithic lasers. The resonator operates by confining the
light through total internal reflection [HN8] within a spherical or
cylindrical dielectric medium. In such a case the long-lived
(low-threshold) resonances can be described by light rays that reflect
repeatedly from the boundary with the same angle of incidence, which is
greater than the angle for refracting out of the medium; hence, the light
circulates around the boundary, perpendicular to the symmetry axis, trapped
indefinitely (top part of figure). Light leaks out very weakly and equally
in all directions by a process that may be described as quantum-mechanical
tunneling of photons. In principle this circular symmetric design allows
one to make a very compact resonator. However, the very long lifetime and
particularly the isotropic emission from the optical resonances of such
symmetric dielectric resonators make them unsuitable as laser resonators
for technological applications because such lasers produce very low power
and require additional components to direct the emitted light.

Some time ago Nöckel, Stone, and Chang (3) suggested that by substantially
deforming the resonator from perfect circularity, resonators with highly
directional emission could be achieved. The interesting point is that the
motion of rays in such asymmetric resonant cavities [HN9] is not simple,
but instead extremely complex. In fact, if one puts aside for the moment
the possibility that such rays can escape from the resonator eventually and
applies the familiar law of specular reflection from a boundary, the
problem of ray motion within such a resonator becomes identical to a
classic problem in nonlinear dynamics: that of the motion of an elastic
billiard ball [HN10] on an oddly shaped billiard table. For most table
shapes the motion is chaotic, which means that the trajectories of two
balls (or rays) with almost identical initial conditions would diverge from
one another exponentially. One can think of this as arising from the
slightly different "kicks" each ray is given at the boundary, the effect of
which is amplified by the nonlinear dependence of the angle of incidence on
the previous angle of incidence. This is quite remarkable since it opens
the door for chaotic studies with light rays.

This analogy has initiated experimental searches for chaotic effects in
deformed quantum cascade lasers [HN11]. These lasers (4) are nearly ideal
two-dimensional optical billiards because light propagates in the plane of
the semiconductor layers polarized perpendicularly to the layers.

Although the existence of chaotic billiard motion means that the trajectory
of any one ray is in practice impossible to predict, it does not imply that
the emission of light from such a resonance is unpredictable. The
electromagnetic resonance is equivalent to an ensemble of such rays, and it
is possible to predict where the rays are most likely to escape and in what
directions. The escape process as described by Nöckel et al. (3) involves a
diffusive "spiraling in" of the angle of incidence of the trapped rays
until this angle falls below the critical angle for total internal
reflection and escapes by refraction. This has a strong tendency to happen
at or near the points of highest curvature on the boundary, leading to
highly directional emission in the far-field. This approach was used to
explain the highly anisotropic emission of laser light from deformed liquid
droplets [HN12] containing a lasing dye (5), a long-observed but poorly
understood effect. A quantitative theory of the emission directionality for
these materials with a relatively small index of refraction has been
developed (6).

In contrast, the new lasers are based on the stable ray motion that
survives when the resonator is substantially deformed: a "bow-tie"
resonance (lower part of figure) develops as the remaining stable mode.
However, the chaotic behavior in the rest of phase space [HN13]
(surrounding the bow-tie) plays an important role as it suppresses
competing lasing resonances, which are still present from lower
deformations. This is an important point: the chaotic resonances basically
feed the stable orbit, and the gain is only limited by diffusion processes
within the pump medium. In addition, this lasing principle opens new
possibilities for injection control in lasers. By changing the lateral
distribution of the injection current through special contact geometry the
interplay between the chaotic region and the stable mode can be controlled
and dynamic processes within the nonlinear medium can be studied.

A new parameter for laser design is thus introduced with this work:
deformation of the resonator. In conventional lasers, the output power
depends on the resonator length, whereas here the power increases
exponentially with deformation. It is remarkable that the transition from
whispering-gallery modes to bow-tie modes appears at certain deformations
where simultaneously the spectral properties are also improved.

These new laser-resonators provide a system for fundamental studies of mode
behavior [HN14] at the boundary to chaos, thus creating a playground for
mesoscopic physics in optical systems. The theoretical concepts do not
directly relate to optical physics but to the field of quantum chaos
[HN15]. This field, which has been an active branch of theoretical physics
for about two decades, seeks to understand the consequences of chaotic
classical motion for the associated quantum dynamics. The same concepts
apply to the wave equation for light in the short wavelength limit where
ray optics apply: light rays exhibit chaotic motion. Quantum or "wave"
chaos theory allows one to classify and understand the possible solutions
of the wave equation, which are hard to find with numerical methods because
of unsymmetrical boundary conditions.
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References
1.C. Gmachl et al., Science 280, 1556 (1998). 2.S. McCall et al., Appl.
Phys. Lett. 60, 289 (1991). 3.J. U. Nöckel, A. D. Stone, R. K. Chang, Opt.
Lett. 19, 1693 (1994). 4.F. Capasso et al., Solid State Commun. 102, 231
(1997). 5.A. Mekis et al., Phys. Rev. Lett. 75, 2682 (1995) [APS]. 6.J. U.
Nöckel and A. D. Stone, Nature 385, 45 (1997).

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The author [HN16] is at the Institut für Festkörperelektronik, Technische
Universität Wien, A-1040 Wien, Austria. E-mail:
gornik@macmisz.fke.tuwien.ac.at

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