One of our lasers for studying laser dynamics is a small Nd:YAG laser. The active species is the Nd3+ ion which is embedded in an Yttrium Aluminium Garnet host crystal. This is the most common of hosts for neodymium lasers. The lasing wavelength is 1,064nm in the infra-red. The laser is short, with an optical path length of just 2.5cm. The picture shows how the components are laid out.

The laser is pumped (that is energy is supplied to it) by another laser, a diode laser. This has a wavelength of 808nm, also in the infra-red. We use an acousto-optic modulator to modulate the intensity of the pump light, and thus, if we get the modulation frequency right we can excite relaxation oscillations. The oscillations can also be excited to a degree by ambient noise - that is random vibrations coming from the floor, random airpressure fluctuations and so on. When we look at the laser power with a photodiode and take the output of the photodiode (photocurrent) to a spectrum analyser, we can see peaks in the spectrum that correspond to the oscillations excited by the noise or by the modulation.

The laser operates on several longitudinal modes. The picture to the right shows what the optical spectrum of a laser operating with 5 modes would look like if we used a very high resolution spectrometer. The frequency of each mode corresponds to a wavelength of about 1,064nm (~1015 Hz). The spacing between each peak is quite small compared to the absolute frequency; in the case of our laser about 6GHz compared to 1015 Hz.

Each longitudinal mode is a (relaxation) oscillator in its own right, but is coupled to all of the others through having to share the same gain medium. That is they have to compete for the energy that is stored in the Nd3+ ions. When one mode gets energy through stimulated emission, the others suffer to a greater or lesser extent. Thus we have a set of coupled oscillators which forms an array. As with other types of coupled array, the whole array behaves collectively.

The interaction of these gives rise to some interesting effects, whereby the laser can operate in an antiphased regime. This means that individual longitudinal modes have fluctuations that, at certain frequencies, cancel the fluctuations of the other longitudinal modes. At other frequencies, typically just one, the fluctuations all add up in phase. This results in relatively simple total behaviour: the fluctuation spectrum looks like that of just a single oscillator rather than an array. That is it would show just one peak. If we look at the fluctuation spectrum of just one of the longitudinal modes, which can be separated out of the laser beam with a Fabry-Perot interferometer, we see a multipeak spectrum that is characteristic of the array.

This effect in lasers has been known for some years through the work of several Russian workers and of Otsuka et al. who studied this primarily through the power spectrum of the fluctuations of the total intensity and the modal intensity. However the antiphasing phenomenon in nonlinear oscillator arrays was first predicted in Josephson Junctions, arrays of which are used in voltage standards. With lasers in a limit-cycle (pulsed) or a chaotic regime of operation the antiphase dynamics can be observed directly just by looking at the (simultaneous) time series data for each of the laser modes. However antiphasing is also present in the relaxation oscillations as the lasers settles down to a steady-state operating point (i.e. in laser jargon, a continuous wave operation). This is a little harder to observe and quantify.

Our contribution to this field has been to use transfer function techniques to actually dissect the array and directly determine whether the fluctuations of any two give modes are interfering constructively or destructively. In some earlier work the transfer function magnitude was measured but not its phase. It is the phase which enables the antiphasing to be directly confirmed. To find out more about our measurements of how individual oscillators contribute to the array, follow this link.

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