The basic laser construction is illustrated in Figure 1.

Figure 1. A schematic illustration of the diode pumped Nd:YAG laser. The optical beams are shown as thick lines, black for the pump wavelength and grey for the Nd:YAG laser wavelength. The lenses used to collimate and refocus the pump laser light are not shown, neither are the optical isolators between the pump and Nd:YAG lasers, and between the Nd:YAG laser and the Fabry-Perot.

The laser medium that we used was a Nd:YAG rod cut with one end perpendicular to the rod axis and the other at Brewster's angle. A high reflection coating on the perpendicularly cut end formed one end mirror of the laser cavity and the output coupler was a partially transmitting mirror (T=1%) mounted on a piezoelectric tube for fine cavity length control. The Brewster cut face served to ensure that all modes were linearly polarised in the same plane. The Nd laser was end pumped by two 40mW diode lasers through the coated end of the rod (T=65% at 808nm). The optical path length of the cavity was 24.3mm corresponding to a longitudinal mode separation of 6.16GHz. Modulation of the pump was achieved with an acousto-optic modulator which avoided the intensity noise of the pump laser that can occur if its injection current is modulated. We also had optical isolators, comprising a polariser and Fresnel rhomb, between the pump laser and the Nd laser, and before the Fabry-Perot to minimise the destabilising effects of optical feedback.

2.2 Early experiments

Initial experiments consisted of modulating the pump laser intensity at or about the frequency of the laser relaxation oscillation, thus exciting this oscillation and if the modulation were strong enough, exciting chaotic behaviour in the total laser intensity, as recorded with an A/D converter. When the pump laser intensity was modulated at or close to the relaxation oscillation frequency, or one of its sub-harmonics or harmonics, pulsed (limit cycle) and chaotic output power from the Nd:YAG laser could be achieved depending on the strength and frequency of the modulation. One of the early experiments with this intial setup was to record chaotic intensity data and then apply singular value decomposition to recreate a model of the chaotic attractor. This showed that the attractor could be embedded in a three dimensional space, i.e. its dimension was less than three. Subsequently the output of the laser was checked with a scanning Fabry-Perot interferometer and shown to be multimode, with between 2 and 7 longitudinal modes running simultaneously. This implies that number of degrees of freedom was in fact rather larger than the embedded attractor would suggest and that somehow several of these were being "frozen out" of the dynamics.

2.3 Transfer function measurements

In order to analyse the multimode behaviour we measured transfer functions for the total intensity and the intensities individual laser modes. Operationally, this means that we modulated the pump input over a range of frequencies and measured the amount of the modulation frequency in the intensity in question. The strength of the modulation was small so that the laser remained close to steady state. The output of the Nd laser was split into two beams so that the total intensity and intensity of one mode could be measured simultaneously. With a pump level of around 1.3 times threshold we observed between four and six laser modes, depending on the exact cavity length. The data shown below is for five mode operation. A Fabry-Perot interferometer was used in one output beam to select individual laser modes. The intensities, total and modal, were measured with low noise photodiode detectors and the outputs of these were directed to a Hewlett Packard HP35670A signal analyser. For transfer function measurements the analyser supplies the modulating signal, scanned repetitively in frequency.

A transfer function can be expressed either as a ratio of polynomials (pole-zero representation) or as a partial fraction (pole-residue representation) (Blackman 1977):

The poles [lambda]_j are the eigenvalues of A, the zeros are z_i and the residues are r_j . Interesting insights into the problem can be gained from both the pole-zero representation and the pole-residue representation. The latter gives a particularly vivid picture of the operation of the antiphase dynamics as the residues contain information about the contribution of each oscillator to the collective modes.

In our experiment a parallel theoretical analysis is made. Although the residues are determined by fitting a general transfer function, in the form of a ratio of polynomials, to the data, we can also calculate the residues on the basis of a suitable theoretical model. It turns out that the comparison of the measured and calculated functions is quite a sensitive test of the model.

We start with the rate equations due to Tang, Statz and deMars (1963), modified slightly to include the small pump rate modulation w = msin[omega]t. These equations are appropriate to a Nd:YAG laser with a high finesse Fabry-Perot cavity and take into account the spatial holeburning. It is assumed that the inversion density, averaged over a few optical wavelengths, does not vary along the laser medium in a direction parallel to the axis of the laser mode. Also implicit in these equations is the assumption that the laser medium fills the cavity. These assumptions are not strictly valid for our laser, but in the present absence of a better model the TSdM equations are still useful to illustrate how our measurements of transfer functions can discriminate against a model.

For comparing the measured transfer functions to the theory, several parameters needed to be determined. We measured the cavity lifetime to be [tau]_c =3.56ns and took the response time of the medium to be [tau]_f =230us (Koechner 1992). For the gains [gamma]_i we assumed a Gaussian distribution of width 135GHz, normalized to one at the peak and with an adjustable offset of the comb of cavity modes from the center of the gain profile.

The transfer functions are complex valued and contain phase and magnitude information about the response of the modes or total intensity. The residues are extracted by fitting general transfer functions in the pole-zero representation to the data and converting the fitted functions to the pole residue representation. Table 1 below shows the residues extracted from the data for operation with pump power 1.3 times threshold and five modes lasing. For comparison we show the residues in table 2 calculated on the basis of the Tang Statz deMars equations. The frequencies shown are those of the poles.

Table 1. Measured residues for five mode operation. The columns correspond to the low frequency collective modes and the rows to the laser cavity modes.

Table 2.. Theoretical residues for five mode operation. The columns correspond to the low frequency collective modes and the rows to the laser cavity modes.

Work to resolve the discrepancies by taking into account these factors is currently in progress. We conclude from this study that the transfer function approach gives much more detailed information and is a more sensitive means of studying the dynamics of the modes than excitation with broadband noise.

to section 2

to section 1

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