Lasers, chaos and bow-ties
Physics World 11 (#9), 23-24 (September 1998) -- Physics in Action
From Malvin C. Teich in the Department of Electrical and Computer Engineering, Boston University, US
Since the invention of the laser nearly 40 years ago, there has been an inexorable trend towards creating devices that are smaller, more efficient and tunable over a wider range of wavelengths. This has been achieved through continuous improvements in the three principal building blocks of the laser an active medium that provides amplification, a pump that serves as the source of energy and a resonator that provides feedback. In the latest advance, researchers from Lucent Technologies and Yale University in the US, and the Max Planck Institute for the physics of complex systems in Dresden, Germany, have tackled the limitations imposed by laser resonators in a fresh and effective way (C Gmachl et al. 1998 Science 280 1556). They have obtained strong beams of laser light from microdisk lasers, which were previously only able to produce weak laser emission.
Bow-tie models |
Laser technology has come a long way since the 1960s. The very earliest lasers, such as ruby and heliumneon devices, were formidable contraptions. They relied on dilute active media with discrete energy levels such as gases or dopant ions scattered in a solid, external pumping mechanisms such as bulky coiled flashlamps or auxiliary gases excited by radio-frequency coils, and cumbersome external resonators consisting of highly flat and reflective mirrors.
The advent of the semiconductor laser in 1962 developed almost simultaneously at the MIT Lincoln Laboratory, General Electric and IBM changed all of that. The active medium, which consisted of a semiconductor pn junction, was in this case a dense three-dimensional solid. The junction could be made from various materials, so that the band-gap energy, and hence the emission wavelength, could be tuned by altering the composition.
The pumping process was simple: a current flowing through the junction created electronhole pairs that recombined to emit photons. Resonator mirrors were readily provided by the surface reflection associated with the high refractive index of the semiconductor material. Size plummeted, efficiency soared and the accessible wavelength range increased dramatically. It is not an exaggeration to view the invention of the semiconductor injection laser as the start of a new era of quantum electronics one that was ultimately to broker a fair partnership between photonics and electronics.
Many important developments in semiconductor lasers followed. Double heterostructures replaced simple pn junctions, lowering the laser threshold and increasing efficiency. Compounds of three or four semiconductors, coupled with band-gap engineering, provided an extensive palette of wavelengths. Gratings could be integrated directly with the active medium, creating distributed-feedback (DFB) lasers to complement the existing lumped-resonator lasers. The very dimensionality of the active medium was reduced; two dimensions gave birth to the quantum-well laser, while a single dimension yielded the quantum-wire laser. Even zero-dimensional devices have come to the fore in the form of quantum-dot lasers comprising just a few thousand atoms. And the vertical-cavity configuration allowed tiny lasers to be built on a single chip, providing integrated two-dimensional laser arrays.
As important as these developments were, remarkable achievements have continued to be made. In the early 1990s devices were created that did not require electronhole recombination for stimulated emission. Instead light is generated from electron transitions in a cascade of coupled quantum wells. These quantum cascade lasers (QCLs) were constructed using an exquisitely refined form of band-gap engineering made possible by molecular-beam-epitaxy technology (for a recent update see F Capasso et al. 1997 Solid State Communications 102 231). QCLs provided strong and tunable laser action in the middle infrared region of the spectrum at room temperature, promising new and important applications.
In the new work, Claire Gmachl of Lucent and her co-workers adapted the quantum cascade laser to a tiny oblong configuration (50 X 70 µm) with a very low threshold current and excellent directionality. They achieved this by turning their attention to the two-dimensional circular resonators used in microdisk semiconductor lasers. These resonators support "whispering-gallery" modes, so-named because of the ease with which an acoustic whisper can bounce along the convex surface of a church dome or gallery. Such modes rely on total internal reflection and skim around the inside rim of the resonator with an angle of incidence that is always greater than the critical angle, preventing them from refracting out of the device. Although these lasers are among the smallest in the world, light only emerges from them via evanescence (photon tunnelling), which makes the emitted light weak and cylindrically symmetric.
What Gmachl and colleagues have done is to flatten the resonator circle to create a stadium-shaped structure. This establishes preferred location angles around the perimeter of the stadium at which strong and highly directional beams are emitted. The power emitted in these beams is up to a thousand times greater than that emitted by the circular lasers.
The team has also discovered a new type of "bow-tie" mode that emerges if the structure is flattened sufficiently. This mode is so-named because of the bow-tie shape of the four-bounce, round-trip ray path within the resonator, much the same as that followed by the rays in a classical confocal resonator (see figure). These modes are sustainable because the resonator is not circular, and so provides a range of mirror curvatures. Although the effect was demonstrated using a quantum-cascade active medium, it should also be observed in other laser media with sufficiently high refractive indices.
Why should these preferred angles emerge? We are used to thinking about laser resonators in terms of solutions to the Helmholtz equation, or rays traced according to the laws of geometrical optics. It is surprising, then, that in this case the behaviour of the resonator can be explained in terms of nonlinear dynamics. The nonlinearity resides in the dependence of a given angle of incidence on its precursor incidence and location angles. When the flattening of the resonator is minimal, phase-space studies of ray-trajectories (in which the angle of incidence is plotted against location angle) in the stadium-shaped resonator show that chaotic versions of the whispering-gallery modes emerge. For more pronounced flattening, regions of stable and regular ray motion give rise to the new bow-tie laser modes. These arise from localized reflections at four particular locations on the perimeter of the resonator those that provide the curvature of a classical confocal resonator and that have adequate reflectance to exceed the laser threshold.
Moreover, because the Helmholtz equation describing optical fields is closely related to the Schrödinger equation, there is a formal relationship between their short-wavelength counterparts ray optics for the Helmholtz equation and Newtonian mechanics for the Schrödinger equation. A non-separable Helmholtz equation is associated with full or partially chaotic ray dynamics, just as a non-separable Schrödinger equation is associated with chaotic classical dynamics.
Resonator physics therefore turns out to be rich, beautiful and useful. Nonlinear dynamics and optical physics have previously intersected via the laser's active medium and pump, since nonlinearity in the laser-rate equations can give rise to chaotic time dynamics of the emitted intensity (the "green problem"). The resonator connection discussed here provides another testbed for investigating the intersection of laser physics and nonlinear dynamics.
As laser active media and pumping mechanisms have evolved in the past few decades, so too have resonator structures. The original plane-parallel mirrors that comprised one-dimensional FabryPérot resonators were soon replaced by spherical mirrors to make alignment easier. Unstable resonators and two-dimensional ring-laser structures were developed for particular applications. Distributed-feedback configurations provided an alternative to lumped resonators. And now the circular cross-section of the microdisk semiconductor laser has given way to an improved stadium-shaped version.