Physics and Mathematical Physics, Univ. of Adelaide
The diode whose output wehave obtained our preliminary results is a
GaAs diode with output around 830nm. Its output beam passes through
an optical isolator and part is directed to a grating spectrometer and
part to a high speed photodiode (BW = 6GHz). With the former we can
study the laser spectrum and with the latter, the spectrum of the
intensity fluctuations of the laser. An example of the data obtained
with the spectrometer is shown below. The data obtained for the
spectrum of the intensity fluctuations are in broad agreement with the
multimode model results shown in figure 4.2 below.
For small amplitude modulation the rate equations can be approximated
by a linear damped oscillator exhibiting a resonance at a
characteristic frequency fRO. This frequency
corresponds to the frequency at which the system oscillates as it
approaches its steady state. For larger amplitude modulation we find
that this frequency is tuned to a lower frequency as the modulation
amplitude is increased. The behaviour of this resonant frequency is
intimately linked with the bifurcations that the system undergoes.
Since the noise inherent in the system acts to continually perturb it
from its stable state then the transient frequencies will become
amplified in the power spectra. One consequence of this is that these
frequencies are readily detectable in experiments and can be used as a
means to probe the dynamics of the system; in particular the various
bifurcations which the system may undergo.
The single mode rate equations are well known in the literature and
can be solved, with and without additive Langevin noise terms , and
driving terms. We find that in the numerical solution of these
equations period doubling bifurcations and chaos are predicted, but
are not observed in the experiments, neither ours nor those of other
workers. This model does correctly predict, at least qualitatively,
the pulling behaviour of the relaxation oscillation peak in the
intensity noise power spectrum, to the driving frequency or its
subharmonic as the driving strength is increased.
We in fact use a more sophisticated model than the single mode model.
Two main modifications are made; the model allows for multiple
longitudinal modes and the dependence of the gain on the carrier
density is taken into account more realistically. In a semiconductor
laser photons are generated by the recombination of a conduction band
electron and a valence band hole. We consider a two band model of the
semiconductor laser active region with parabolic bands. Population
inversion is achieved by pumping electrons into the conduction band
and holes into the valence band. The gain depends on the distribution
of carriers in the bands and so depends strongly both on the
temperature and the carrier density in the active region. The peak
gain occurs at different wavelengths for different carrrier densities
with important consequences for the dynamics of the laser under
modulation.
The multimode rate equations used are from Tarucha and Otsuka (1981).
In these equations the gain is approximated by a quadratic function of
fixed shape. However in our work, both the peak gain and its
position in
wavelength are made functions of the carrier density. In order to consider
the effects of the laser noise we add Langevin noise terms
FN(t) and FPj(t) to the multimode rate
equations. In the simulations, generally 25 modes were used.
However, in most cases it is only the central modes, near the peak of
the gain curve that have any significant amplitude.
Figure 4.2 below shows results for the multimodel model
illustrating a period doubling bifurcation. The behaviour is
qualitatively similar to the results obtained for the single mode
model where the relaxation oscillation peak is tuned (or pulled) to
half of the modulation frequency before the onset of the period
doubling bifurcation. We find however that for realisitic ranges of
parameters that the first period doubling bifurcation is the only one
predicted by the multimode model which is in better agreement with
experiment (in contrast to the period doubling cascades to chaos
predicted by the single mode model).
We can also calculate the laser spectrum (as opposed to the spectrum
of intensity noise) which can be compared with Fig 4.1. A typical
result is shown below in figure 4.3. A prediction of the model is
that as the modulation frequency or one of its harmonics is tuned
through the relaxation oscillation frequency the system becomes
multimode with the spectrum shifting towards the shorter wavelength
modes. The reason for the occurence of this phenomena can be partly
understood by considering the effect that the phase change,
experienced by the carrier density as the system passes through a
resonance, affects the gain and hence the photon density of the modes
themselves. This is consistent with what is observed in
experiments.
Another difference between the two models is the occurence of a
Hopf bifurcation in the multimode model. This phenomena results in the
appearance of extra peaks in the power spectra of intensity
fluctuations and generally occurs for modulation frequencies near the
relaxation oscillation. No experimental observations of this
phenomena have been made to date in these experiments and it is quite
possibly an artificial feature. Further work needs to be done on this
matter.
References
S Tarucha and K Otsuka (1981), IEEE J. Quantum Electron. QE17, 810
People: Kerry Corbett, Murray Hamilton
4 Semiconductor laser dynamics
A semiconductor laser is inherently noisy because of spontaneous
emission and the discreet nature of the carrier recombination process.
A consequence of noise is that it continually perturbs the system from
its stable behaviour and hence the transient dynamics of the system is
also important. Measuring the power spectrum of a noise driven system
is, in fact, one way to probe the transient behaviour of a stable
system. For example, in a semiconductor laser the power spectrum of
intensity fluctuations features a peak at the relaxation oscillation
frequency of the laser. It has been shown also in previous studies
that the transient behaviour of a noise driven system, as a parameter
of the system is varied, can act as a signature for a bifurcation. In
particular this has been shown for period doubling in a modulated
semiconductor laser
4.1 Experimental configuration and results
Fig 4.1 . Data showing laser spectra obtained with a
grating spectrometer, with the exit slit replaced by a photodiode
array, for several modulation frequencies showing the complex
multimode behaviour of the laser when modulated at the relaxation
oscillation frequency. When the modulation frequency is close to the
relaxation oscillation frequency the laser becomes distinctly
multimode. Although the laser appears to be single-mode at other
modulation frequencies, there is still much more power in the
"side-modes" than is the case for no modulation. The data also shows
the shift in peak wavelength as the relaxation oscillation is
excited.
4.2 Modelling
4.2.1 Single mode laser model
4.2.2 Multimode laser model
Figure 4.2 Consecutive power spectra of the total photon
density exhibiting a period doubling as the modulation index is
increased from m=0.4 to m = 0.75. The injection current is
Ib=1.5. The vertical bars on the horizontal axis indicate
the modulation frequency (2GHz) and its second harmonic. The low peak
to the left of the peak at the modulation frequency is the (incipient)
relaxation oscillation. As the modulation index is increased this
peak is pulled toward the subharmonic frequency of the modulation.
Figure 4.3. Calculated powers of the modes as a function
of modulation frequency for modulation index m=0.4 and injection
current Ib=1.5. The modes are labelled 5 - 12, the
numbering being arbitrary.