Phase synchronization in mutually coupled chaotic diode lasers
Y. Aviad,
1
I. Reidler,
1
W. Kinzel,
2
I. Kanter,
1
and M. Rosenbluh
1
1
Jack and Pearl Resnick Institute for Advanced Technology, Department of Physics, Bar-Ilan University, Ramat-Gan 52900, Israel
2
Institut fur Theoretische Physik, Universitat Wurzburg, Am Hubland 97074 Wurzburg, Germany
Received 20 May 2008; published 19 August 2008
Semiconductor lasers with optical feedback have chaotically pulsating output behavior. When two similar
chaotic lasers are optically coupled, they can become synchronized in their optical fluctuations. Here we show
that the synchronization is not only in the amplitude and in the timing of the pulses but that the short pulses are
also phase coherent with each other. This is true even when the lasers are separated by distances much larger
than their coherence length.
DOI: 10.1103/PhysRevE.78.025204 PACS numbers: 05.45.Xt, 05.45.Ac, 05.45.Vx, 42.65.Sf
The synchronization of the optical phase of two mutually
coupled chaotic diode lasers is experimentally examined un-
der isochronal and achronal conditions. We show that the
emergence of full or partial correlations in the chaotic laser
intensities is accompanied by similar optical phase correla-
tions. The chaotic lasers are thus coherent with each other
either instantaneously isochronal or with a time delay be-
tween them equal to an integer multiple of the optical propa-
gation time between the lasers achronal.
Two similar mutually coupled semiconductor lasers, each
with self-feedback in addition to their mutual coupling, have
been shown recently 1–4 to exhibit zero-lag isochronal
synchronization in their chaotic intensity output. The chaoti-
cally fluctuating intensity pattern of each laser has been ex-
perimentally examined with greater than 100 ps resolution
5 and has been shown to consist of short random intensity
spikes, emitted by each laser. The spike trains emitted by the
lasers are nearly identical with zero time delay between
them, in spite of the fact that the two lasers can be physically
separated by an arbitrarily large distance. Such systems are
of great interest in the general fundamental study of coupled
dynamics as well as in such diverse fields as neural networks
6, cryptography 7–9 and secure optical communications
10–13.
If the two similar coupled lasers lack self-feedback, a con-
figuration known as face-to-face, chaotic fluctuations are
again observed. Synchronization between the lasers is ob-
served in this configuration as well, but in this case the syn-
chronization is achronal, that is delayed by the propagation
time of the coupling light between the lasers. The achronal
synchronization mode can be of the leader-laggard type, in
which one of the lasers always precedes the other in time, or
a mode where the leader position is taken randomly by each
of the lasers with each laser taking an equal share of leader
and laggard positions 14,15. The simplest configuration of
course is two lasers which are unidirectionally coupled in
which case the receiving laser is injection locked; the receiv-
ing laser copies the time-dependent intensity of the transmit-
ting laser. If the transmitting laser happens to be chaotic due
to self-feedback, for example the receiving laser will copy
the intensity fluctuations with a delay corresponding to the
light propagation time.
A natural question arises as to whether the two intensity
wise, isochronally or achronally synchronized sources are
also phase synchronized? On the one hand, one would natu-
rally expect that the intensity spike emitted by a laser at a
specific time and with a specific amplitude is determined by
the precise time varying phase and phase history in the laser
cavity. Thus if two lasers emit synchronized intensity spikes
their instantaneous phases would also be synchronized and
they should be coherent with each other. On the other hand,
the phase in a semiconductor lasers varies greatly on very
short time scales and it is possible that the long intensity
spike emitted by the laser represents some time average of
this rapidly varying phase. This is especially the case where
the optical distance between a pair of mutually coupled la-
sers is much larger than the solitary laser coherence length.
In this case one might expect that only some average phase
of the two synchronized lasers is required to be the same and
the two lasers would be instantaneously phase incoherent.
The phase coherence, or lack of it, is even less intuitive in
a face-to-face, achronal or anticipated synchronization con-
figuration. For such configurations the intensity correlation
between the laser pulse trains is not perfect, and only a par-
tial overlap is observed in the time shifted correlation of the
intensities of the lasers, while for isochronal synchronization,
the unshifted intensity correlation based on numerical calcu-
lations as well as experiment is near perfect. Thus for achro-
nal synchronization with only a partial intensity correlation,
which occurs for long time shifts, it is not obvious whether
the partial time shifted intensity correlation necessarily im-
plies a partial time shifted phase correlation. Similar ques-
tions of phase synchronization have been addressed in a solid
state laser array system 16.
In this paper we address this question by directly measur-
ing the phase coherence for two isochronally synchronized
diode lasers, as well as for two lasers in a face-to-face con-
figuration. We show that intensity correlation between two
lasers, whether achronally or isochronally synchronized, also
implies a corresponding phase correlation and coherence be-
tween the two laser outputs. The intensity correlation is mea-
sured by, t1:
t =
i
I
A
i
- I
A
i
I
B
i+t
- I
B
i+t
i
I
A
i
- I
A
i
2
i
I
B
i+t
- I
B
i+t
2
, 1
where I
A
and I
B
are the time-dependent intensities of lasers A
and B, respectively, and
i
stands for an average over a
given window size. When isochronal synchronization is es-
PHYSICAL REVIEW E 78, 025204R2008
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