Optics Group -

Physics and Mathematical Physics, Univ. of Adelaide

nonlinear dynamics of lasers

People: Kerry Corbett, Tim Hill, Laurence Stamatescu, Murray Hamilton

1. Overview

We got into this game because in studying nonlinear optical processes and noise we were working with things that happen very fast, on a timescale of 10-15s. Basically, electrons in an atom or molecule oscillate up and down once every 10-15s or so, and generate light waves with the same period. In a nonlinear optical process the light might have a somewhat different period but not very different - it's still damn fast! When things happen that fast you can't see directly the effect of external noise on dynamical changes. There is no detector fast enough to "see" the electric field in a light wave oscillating up and down. You can only measure the intensity averaged over maybe 10-11s, and that corresponds to a pretty fast detector too.

In order to get around this limitation of not being able to follow the dynamics directly we decided to switch to a system with slower dynamics. The laser! A laser has fast dynamics as well as slow. The fast is just the oscillation of electrons at optical frequencies that gives the laser light. The slow dynamics is a movement of stored energy back and forth between the laser medium and the electromagnetic field (the light) confined between the mirrors of the laser. If you just look at the total intensity of the laser output beam you can pretty much ignore the optical frequency oscillation and just concentrate on the slower dynamics.

What timescale these slow dynamics have depends very much on the type of laser and the details of its design. In some lasers, such as the common HeNe laser, the atomic physics of the gaseous medium actually washes out the slow dynamics pretty much completely. In other lasers such as CO2 or neodymium, the dynamics is fairly slow - it has a timescale of around about 1 microsecond. In diode lasers it's about 1ns. That is slow!

If you change the conditions of the laser suddenly, it will oscillate about its new operating point for a bit. The oscillations will die down over about 10-6s as the laser settles down to its new operating point. It's called a relaxation oscillation. So what's so interesting about that? Well if you keep perturbing the laser at the right frequency, you can keep the oscillations going. Furthermore they are nonlinear which means that they start to alter themselves as they get stronger. This leads to all the usual nonlinear phenomena that you may have heard of: bifurcations, limit cycles, chaos ...

...and then you can start to study what happens when the nonlinear relaxation oscillator is perturbed by noise that is deliberately brought into the system.

So why do that? The hope is that such work can tell us something about how general systems interact with their environments. General systems are typically nonlinear, and typically complex. But the environment is, almost by definition really, much bigger and more complex. So all of the myriad little interactions that the environment has with the system cannot be tracked individually - statistical methods must be used. What this boils down to is that the influence of the environment is pretty much random. Noise is random too, so you can mimic, or model, the effect of the enviroment by perturbing the system with noise.

This talk of system and enviroment is pretty abstract. What's a concrete example? The common interpretation of the word "environment" is that which is all around us and enables us to exist as a species, or a society, or a civilisation ... What the environment actually comprises depends a bit on which of these you're talking about! All a bit vague really! But to look for an example away from lasers and physics, consider a lizard basking on a rock in the sun. As an organism it's a complex nonlinear system. The environment is this case is the immediate surroundings of the lizard; trees, rocks, air and so on. The sun is part of it's enviroment; so is the cloud which temporarily blocks the sun; the wind is also. The sum total of all of these influences on the lizard is really rather random, though there is a certain degree of predictability - the sun will set after all. This system, the lizard, is vastly more complex than those that we are tackling, but, Hey!, you've got to begin somwhere.

So what are we up to at the moment? Well it turned out that the lasers we had for this study were a bit more complex that we'd bargained on! They were multimode, meaning that they put out light which had several components, each with a slightly different wavelength. Each of these components is called a mode, a longitudinal mode to be specific. (There are also transverse modes. )

This then led us to consider

and the detailed dynamics of
  • diode lasers, (Section 4) taking into account their multimode nature and some of the complications of semiconductor physics.

In this work we have produced results which indicate that the effect of noisy excitation of an oscillator array is different from single frequency exitation. The relative contributions of the individual oscillators in the array to the collective modes of the array seems to be different in each case.

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queries and comments to mwh@physics.adelaide.edu.au