EUROPHYSICS LETTERS
Europhys. Lett., 40 (6), pp. 619-624 (1997)
15 December 1997
Statistical properties of the dynamics of
semiconductor lasers with optical feedback
G. Huyet, S. Hegarty, M. Giudici( ), B. de Bruyn and J. G. McInerney
Physics Department, National University of Ireland, University College - Cork, Ireland
(received 7 August 1997; accepted in nal form 27 October 1997)
PACS. 42.65Sf { Dynamics of nonlinear optical systems; optical instabilities, optical chaos,
and optical spatio-temporal dynamics.
PACS. 42.55Px { Semiconductor lasers; laser diodes.
Abstract. { We measure the time-resolved probability distribution of the light emitted by
a semiconductor laser with optical feedback. We show and analyse, for the rst time, highfrequency uctuations of the laser intensity in the coherence collapse regime. Our results suggest
that the noise induces the power drop-outs and the laser recovers in a deterministic fashion. We
also give a new mechanism for a transition between erratic and synchronised behaviour in the
laser dynamics.
Delayed dynamical systems have been studied in many elds during the last several years.
Delay has been used particularly to control unstable solutions of these systems, but often has
the paradoxical eect of inducing instabilities. A semiconductor laser with optical feedback,
i.e. using an external reector to return a portion of the laser output back to the main
resonator, is one example of such a delayed dynamical system, being rst considered in order
to narrow the laser linewidth. For moderate feedback levels, close to the laser threshold, the
laser is generally stable with a narrow spectral linewidth. For intermediate injection currents,
the laser becomes unstable and displays power drop-outs at low frequency (see g. 1), this state
being usually referred to as Low-Frequency Fluctuations (LFF). At high injection current, the
laser loses its coherence properties and the laser intensity shows strong uctuations, a condition
termed coherence collapse. Since their discovery [1], several authors have investigated whether
the LFF are a consequence of stochastic or deterministic behaviour [2]-[5]. This answer to this
question is of interest both from a fundamental viewpoint and for practical considerations in
eliminating or controlling these uctuations.
Attempts to model this dynamics include an iteration map where the drop-outs were
initiated by noise [3]. However, they remained even if the noise was turned o during the
remainder of the numerical simulation. From these results, the authors concluded that the
underlying dynamics might be chaotic. Numerical simulation of the Lang-Kobayashi (LK)
( ) Institut Non Lineaire de Nice, Universite de Nice Sophia Antipolis, CNRS UMR 129, 1365 route
des Lucioles, 06560 Valbonne.
c Les Editions de Physique
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Fig. 1. { Typical time series of power drop-outs observed on a 500 MHz oscilloscope.
equations [6] also showed similar power drop-outs [7], [8]. The numerical experiments exhibited
strong irregular picosecond light pulses at a rate associated with the relaxation oscillation
frequency, typically a few GHz [8]. Experimental results showed the existence of high-frequency
uctuations [9], but these uctuations clearly diered from the numerical simulations [10].
Among the many studies of the LFF regime [11]-[14], there is little experimental information
available about the high-frequency behaviour of the laser intensity. This is mainly due to the
diculty of digitising non-periodic signals at frequencies as high as the 5 GHz relaxation
oscillations of semiconductor lasers. One option is to use a single-shot streak camera which
can acquire a short (ns) time series with picosecond resolution [9]. However, the time spanned
by the data acquired would need to be at least on the order of the recovery time of the
laser intensity, typically tens of ns, to give information about the temporal dynamics. These
experimental requirements, demanding information at two very dierent time scales, make
a detailed analysis of the high-frequency behaviour dicult. An alternative approach is to
measure statistical quantities of the system from many single-shot measurements.
In this letter, we measure the probabilility distribution of the laser intensity and nd it
strongly asymmetric about the most probable intensity. This demonstrates that the laser
intensity contains fast uctuations several times larger than its average value. However,
these dier from the numerical simulation of the LK equations [10]. We then analyse the
time-resolved probability distribution and characterise the temporal behaviour of the laser
BS
M
CSA
LD
ISO
SD
ESA
APD
OSC
Fig. 2. { Experimental arrangement. LD is the laser diode (Hitachi HLP-1400), M the high-reectivity
mirror, ISO the optical isolator, BS the beam splitter, SD the Schottky diode (New Focus model 1435),
APD the avalanche photodiode, ESA the electronic spectrum analyser, CSA the communications signal
analyser (Tek CSA 803) and OSC the 500 MHz oscilloscope (Tek 640A).
g. huyet et al.: statistical properties of the dynamics etc.
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Fig. 3. { Power spectra of the laser intensity for two dierent values of the injection current. a) i =
53 mA corresponding to the LFF regime and b) i = 90 mA, to the coherence collapse regime.
intensity at high frequency. Our results indicate that the power drop-outs are deterministic
objects that close to threshold are induced by noise. At higher injection currents they are
induced by a fast high-intensity transient following the preceding drop-out and are synchronised
with this transient. This behaviour, which is similar to that encountered in many excitable
systems [15], [16], has already been identied as such in [5].
The experiment was carried out with a commercial Hitachi HLP-1400 AlGaAs diode laser.
The light emitted by the front facet was focussed onto a high-reectivity mirror and re-injected
into the laser (see g. 2) forming an external cavity of round trip length 44 cm. A neutral
density lter placed in the external cavity allowed control of the feedback level. All results
were obtained for a xed level of feedback involving a threshold reduction of 10% (the solitary
laser threshold was 53 mA and the laser threshold with feedback was 47 mA). The light from
the back facet was collimated through an optical isolator and coupled to a 25 GHz bandwidth
InGaAs Schottky diode (New Focus model 1435), to analyse the fast dynamics with a 21
GHz spectrum analyser (Tektronix 492 P). Part of the collimated beam was also used for
low-frequency detection with a 500 MHz oscilloscope. Figure 3 contains power spectra of
the laser intensity for three dierent values of the injection current. Close to threshold, the
spectrum consisted of narrow peaks at the round trip frequency and the intensity observed
on the 500 MHz oscilloscope only shows small uctuations at the round trip frequency. In
g. 3 a), at 53 mA injection current, which was close to the solitary laser threshold, the laser
operated in the LFF regime. The peaks in the power spectrum at the round trip frequency
were broadened and the spectrum contained a broadband low-frequency background. As the
current was increased to 90 mA, g. 3 b), the laser entered the coherence collapse regime. This
last regime, in which the laser intensity showed strong uctuations, corresponded to injection
currents typically larger than 75 mA. Higher frequencies, up to 14 GHz, appeared in the
spectrum and the time series showed many closely spaced power drop-outs.
To examine the statistical features of these high-frequency components, we connected the
fast photodiode to a 40 GHz sampling head (Tektronix SD-30) inserted in a digital signal
analyser (Tektronix CSA 803). By using the internal trigger of the signal analyser, which was
not synchronised with the light emitted by the laser, we randomly sampled the waveform at a
large number of points. We then used these data to calculate the probability distribution of the
laser intensity with a frequency bandwidth of 0{23 GHz. The system bandwidth was tested
with 115 fs pulses from a mode-locked Titanium Sapphire laser and found to have a rise
time of 15 ps. These pulses were also used to verify the validity of our statistical measurement
method.
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Fig. 4. { Probability distribution of the laser intensity for injection current values of 70, 80, 90 and
99 mA. Note that the x-axis is renormalised to the average intensity.
The xed optical feedback level, giving a laser threshold reduction of 10%, was chosen
to extend the LFF regime over the widest range of injection current. With this feedback
level the LFF regime occurs near the solitary laser threshold with an average time between
consecutive drop-outs of 40 ns. This time decreases as the injection current increases until
the laser reaches the coherence collapse regime. In g. 4, we show the probability distribution
of the laser intensity for dierent values of the injection current. The curves, which show a
maximum near the average intensity, indicate the presence of uctuations from zero intensity
to two or three times the intensity average value. (Note that the x-axis was normalised
with respect to the average intensity.) These curves also present a strong asymmetry about
their maxima, with an exponential decay on the high-intensity side. We then ltered the
photodiode signal by passing it through a 1 GHz amplier and contrasted this low-pass{ltered
probability distribution with the unltered distribution. The ltered distribution no longer
has an exponential tail at high intensity and we therefore conclude that the high-intensity
values are associated with high-frequency uctuations. The ltered distribution has a long
tail at low intensity, which is a signature of low-frequency power drop-outs. We also note
from g. 4 that the distributions obtained for dierent injection currents fall into the same
curve after renormalisation of the intensity to the average intensity. This indicates that the
amplitude of the uctuations increases linearly with the average intensity. These uctuations
may correspond to either transient pulses during the recovery of laser intensity after the power
drop-out or to continuous fast uctuations around the laser mean intensity.
In order to investigate this point further, we constructed the time-resolved probability
distribution of the laser intensity. This was achieved by triggering the signal analyser on
power drop-outs detected with the low-bandwidth avalanche photodiode. For technical reasons,
the signal analyser takes its earliest possible acquisition 36 ns after the trigger event, so
that we were required to insert a bre delay-line in front of the fast photodiode. Due to
losses in coupling to the bre, it was then necessary to include a high-bandwidth amplier
(0.05{18 GHz) after the Schottky diode, with resultant loss of the DC information. Figure 5
shows a time-resolved probability distribution. On this screen dump, taken at 60 mA injection
current, we see the drop-out occurring at 44 ns. This corresponds to the length of our delay
line minus the length of cabling of the low-frequency detector. Following the drop-out shown
in g. 5, the laser recovered in a series of steps each of which triggered damped relaxation
oscillations. Although the signal following the drop-out was clear for several round trip times,
later uctuations became blurred with a large amplitude. This indicated the appearance
of non-periodic uctuations in the signal corresponding to a strongly modulated transient
regime. After this regime, the laser relaxed to a quiescent state with low uctuations. After
g. huyet et al.: statistical properties of the dynamics etc.
623
Fig. 5. { Time resolved density plot of the probability distribution of the laser intensity for an injection
current of 60 mA. On the top picture, where the span is 100 ns, one can see the power drop-out and the
high-intensity pulse transient following it for about 30 ns. The lower picture zooms on the drop-out
region, showing recovery at the round trip frequency, with one relaxation oscillation between successive
pulses.
the drop-out, the relaxation oscillations were not necessarily synchronised with any external
cavity frequency. However, they may be synchronised when the signal is blurred since the
microwave spectra contain only broad peaks at external cavity frequencies.
For lower injection currents close to the solitary laser threshold, where the power dropouts occur erratically and infrequently, the system recovered with external cavity round trip
pulses followed by strong uctuations that damp over a period of 30 ns. We note that the
relaxation oscillations were not visible since their frequency was lower than the external round
trip frequency at low injection currents. It has been shown that noise induces the power
drop-outs [5] and these results indicate that the laser recovers in a deterministic fashion.
This behaviour, called excitability, has been observed in many dierent systems [15], [16]. For
higher currents, where we observed the relaxation oscillations in the rst ns of the recovery, the
transient was damped but the next drop-out was activated before the laser reaches its steady
state. This indicates that the strongly modulated transient synchronises the power drop-outs.
For yet higher injection currents, corresponding to coherence collapse, this transient was no
longer damped and the high-frequency uctuations remained all the time. This regime may
correspond to chaotic behaviour of the laser intensity.
In conclusion, our detailed analysis of the high-frequency dynamics of a semiconductor
laser with feedback provides new data not compatible with the existing theoretical models. To
explain the discrepancy, we note that two possibly signicant eects have often been omitted
from the standard description of the LFF [6]-[8]. The rst is that the equations consider only
one external cavity round trip. This approximation, which is good for low feedback levels,
may not be valid for the LFFr egime where the feedback level is not so low. The second is
that the equations consider only one solitary laser mode. It has been shown [17] that the laser
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emits multiple solitary laser modes in the LFF regime even though it operates in a single mode
without feedback.
***
We would like to acknowledge B. Kearns, S. Balle, J. R. Tredicce, P. O'Brien,
A. V. Uskov and R. Gillen for useful advice and assistance, and the Commission of the
European Union for funding Marie Curie research fellowships for two of us (GH and MG).
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