\documentstyle[spie,epsfig,amsmath]{article}

%.......................................... my definitions
\newcommand{\bcen}{\begin{center}}
\newcommand{\ecen}{\end{center}}
\newcommand{\bfig}[1]{\begin{figure}[#1]}
\newcommand{\efig}{\end{figure}}
\def\nsi{ns$^{-1}$}
\def\e{\times 10}
\def\nl{\\ \hline \rule[-1ex]{0pt}{3.5ex}}
\def\rt{\vec r_\perp}
\def\r{\vec r}
\def\si{\sum_{n=0}^\infty}
%.........................................

\title{Dynamical Behavior of Two Distant Mutually-Coupled Semiconductor Lasers}

\author{Josep Mulet\supit{$\ast$}\supit{a}, Claudio R. Mirasso\supit{b},
Tilmann Heil\supit{c} and Ingo Fischer\supit{c}
\skiplinehalf
\supit{a}Instituto Mediterraneo de Estudios Avanzados, CSIC-UIB,
E-07071\\ Palma de Mallorca, Spain.\\
\supit{b}Departament de
F\'{\i}sica, Universitat de les Illes Balears, E-07071\\ Palma de Mallorca,
Spain. \\
\supit{c}Institute of Applied Physics, Darmstadt University of Technology,
Schlo{\ss}gartenstr. 7,\\ D--64289 Darmstadt, Germany.}

\authorinfo{\supit{$\ast$}Correspondence: Email:
mulet@imedea.uib.es; URL: http://www.imedea.uib.es/PhysDept;
Telephone: +34 971 172536; Fax: +34 971 173426}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%>>>> uncomment following for page numbers
% \pagestyle{plain}
%>>>> uncomment following to start page numbering at 301 \setcounter{page}{301}
\begin{document}
\maketitle

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{abstract}
In this paper we present numerical and experimental investigations
on the synchronization of the instabilities originated by the
mutual coupling of two semiconductor lasers in face to face
configuration. We have restricted ourselves to the analysis of two
lasers with identical parameters and operating at the same
frequency. Numerical simulations are based on standard rate
equations for each semiconductor laser whereas the mutual
injection is modeled by including delayed optical fields.
Experiments are performed using almost identical  Fabry Perot
lasers coupled through the TE component. As soon as the coupling
strength is increased we observe fluctuations in the power
dynamics that appears synchronized except for a small time lag.
This asymmetric operation of the perfectly symmetric system allows
to differentiate between {\it leader} and {\it laggard} lasers.
Synchronization properties are studied making use of the
synchronization plots and cross-correlation measurements.
Extensive investigations of the dependence of the time traces and
correlation degree on the coupling strength and current level
demonstrate good agreement between numerical and experimental
observations.
\end{abstract}
\keywords{Semiconductor laser, Nonlinear oscillator, Chaos
synchronization, Symmetry breaking.}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{INTRODUCTION}
\label{sect:intro}

Coupled nonlinear oscillators have been extensively studied in the
last years due to their rich variety of possible behaviors and
applications. Periodic and chaotic oscillations have been reported
in chemical oscillators, population dynamics, physiological
interactions, coupled neurons, mechanical oscillators and lasers,
etc. \cite{Coffman86,Schafer98,Ernst95,Roy94}. Thus, the
understanding of the dynamics of nonlinear-coupled oscillators is
essential for a wide range of scientific investigations. Many of
the real systems include a significant time delay between
subsystems when they are coupled. However, and for the sake of
simplicity, this delay is usually neglected when modeling the
system dynamics in terms of, in general, ordinary or partial
non-linear differential equations. This delay could yield
absolutely unexpected behaviors, mainly due to the additional
degrees of freedom introduced into the system. Mathematically, a
delay in a differential equation yields an infinite dimensional
system. While experimental studies on the impact of the delay on
the dynamics of nonlinear coupled oscillators has been only
carried out recently \cite{prl}, theoretical investigations have
already demonstrated bistability between synchronized and
incoherent states \cite{Yeung99} and stochastic resonance
\cite{Kim99}.

The dynamics of two mutually coupled spatially separated
semiconductor lasers (SCL) has been recently addressed. The two
lasers were coupled via the optical field with a delay determined
by the propagation time between subsystems, which was much larger
than the internal roundtrip time. The main results reported
\cite{prl} include a spontaneous symmetry breaking when the two
lasers are identical and a well defined leader-laggard dynamics.
These studies complement some previous ones where two SCL were
weakly coupled \cite{hohl97,hohl99}. For the latter, localized
synchronization of periodic oscillations were reported. There the
limit of small time delays was considered, being much smaller than
the relaxation oscillation period of the solitary laser. 

Experiments including spatially separated SCLs are easy to
implement and have several advantages.  First, the nonlinear
dynamical behavior of these lasers is widely understood. Second,
the parameters of SCLs are well known and can be controlled
accurately. Third, SCLs have an immense potential for practical
applications, in particular for future telecommunications
technologies: the proposal of novel communication systems using
chaotic carriers based on chaos synchronization of distanced
lasers \cite{VanWiggeren98,Fischer00} has further boosted the
interest in coupling and synchronization phenomena in SCLs.

In this paper, we present theoretical and experimental
investigations of the dynamical behavior of two delayed-coupled
SCLs. We show that the large delay and the strong coupling give
rise to very interesting dynamical phenomena: we give theoretical
and experimental evidence for fast coupling-induced instabilities,
and sub-nanosecond time scale chaos synchronization for a wide
range of coupling strength and levels of the injection currents.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{THE MODEL}
\label{sect:themodel}

The system of two mutually coupled semiconductor lasers, as those
depicted in Fig. \ref{fig:f1} as LD$_{1,2}$, can be described in
terms of the slowly varying electrical fields amplitudes,
$E_{1,2}$, together with terms accounting for the delayed
injection from one laser to the other one. We also need equations
describing the carrier numbers in each SCLs.

\bfig{h}
\bcen
\begin{tabular}{c}
\psfig{file=Figs/f1spie.eps,width=9cm,clip=true}\vspace{-0.5cm}
\end{tabular}
\ecen \caption{Face to face configuration of two device-identical
semiconductor lasers placed by a distance $L_e=c\tau$. $r$ and
$r'$ are the field reflectivities of each laser facet.}
\label{fig:f1} \efig

Our starting point, as in many other previous works, is the
single-mode rate equations that are modified to take into account
external perturbations, in our case the mutual coupling. We find
\cite{pra-model} that in the case of identical lasers (symmetric
case) the perturbation of the solitary equations appears as a
series expansion of orders $x \equiv \rho r'$. We have defined
$r'$ as the facet reflectivity and $\rho^2$ is the power
attenuation of the light traveling from the right facet of LD$_1$
to the left facet of LD$_2$ or viceversa. This attenuation is
mainly due to the optics inserted in between the two lasers in the
experimental setup [See Fig. \ref{fig:setup}]. The most important
contribution in the expansion is $o(x)$ that arises from the
injection of a delayed field from one laser to its counterpart.
This order in the expansion involves terms like $E_{2,1}(t-\tau)$,
being $\tau=L_e/c$ the flying time between the two lasers. The
next order $o(x^2)$ describes the feedback of the light emitted
from one laser after a roundtrip time $2\tau$, that has been
reflected at the facet of the other laser. Since $x\ll 1$, it is
justified to neglect the feedback effects when comparing with the
mutual injection terms.

The solitary lasers are assumed to be operating around the optical
frequencies $\omega_{1,2}$. From these frequencies we define the
symmetric reference frame $\omega_0=(\omega_1+\omega_2)/2$ and the
detuning by $\Delta=(\omega_2-\omega_1)/2$. With this definition
the laser 2 is detuned at higher frequency than laser 1 if
$\Delta>0$. For simplicity, we have assumed identical intrinsic
parameters for both lasers although small dissimilarities can be
present in the experiment. At the order $o(x)$, the coupled system
is governed by the following equations
%
\begin{eqnarray}
\dot{E}_{1,2}(t)&=&\mp i\Delta E_{1,2}(t)
+\frac{1}{2}(1+i\alpha)[G_{1,2}-\gamma]E_{1,2}(t)
+\kappa e^{-i\omega_0\tau} E_{2,1}(t-\tau)
+\sqrt{2\beta N_{1,2}}\,\xi_{1,2}(t), \label{eq:eqne}\\
%
\dot{N}_{1,2}&=&p_{1,2}J_{th}^{sol}-\gamma_e N_{1,2}-G_{1,2}
|E_{1,2}|^2, \label{eq:eqnn}\\
%
G_{1,2}&=&\frac{g(N_{1,2}-N_t)}{1+\varepsilon
|E_{1,2}|^2}. \label{eq:eqng}
\end{eqnarray}
%
In this paper, we restrict the analysis to the situation in which
the free running lasers emit at the same frequency, i.e.
$\Delta=0$. The lasers are pumped at the same injection levels
$p_{1,2}=p\equiv I/I_{th}^{sol}$ with $I_{th}^{sol} \approx 60$
mA. Therefore, the equations are perfectly symmetric under the
interchange of the two lasers except for noise terms. The
parameters used in our numerical simulations are adopted to the
actual experimental conditions. The linewidth enhancement factor
is $\alpha=3.5$, the cavity losses $\gamma=240$ \nsi, the
differential gain $g=3.2\times 10^{-6}$ \nsi. The coupling
strength is given by $\kappa=\rho (1-r^{\prime 2})/(\tau_{in} r')$
being $\tau_{in}=2 n_g L/c$ the internal round-trip time and $n_g$
the group refractive index. We have estimated that the maximum
coupling rate achieved in the experiment corresponds to
$\kappa\sim 30$ \nsi. The term $e^{-i\omega_0\tau}$ provides the
change of the optical phase in a trip time $\tau$. The coupling
time is $\tau=4$ ns that provides a separation of $L_e=120$ cm,
the carrier decay rate $\gamma_e=1.66$ \nsi, the carrier value at
transparency $N_t=1.5\times 10^{8}$, the gain saturation parameter
$\varepsilon=10^{-7}$, the spontaneous emission rate
$\beta=10^{-6}$ \nsi. The spontaneous emission processes are
modeled by Langevin noise sources that consist in white Gaussian
numbers $\xi_{1,2}(t)$ with zero mean $\langle \xi_i(t)\rangle =
0$ and $\delta$-correlated in time, i.e. $\langle \xi_i^* (t)
\xi_j (t^\prime) \rangle = 2\delta_{i,j} \delta(t-t^\prime)$ for
$i$=1, 2.

Equations (\ref{eq:eqne})-(\ref{eq:eqng}) have similarities to those
describing a system of two solid state lasers laterally coupled by 
evanescent fields \cite{terry99,ashwin-e} when neglecting the time delay.  
Also some similarities with a vertical-cavity surface emitting laser (VCSEL)
with $\pi/2$-rotated optical feedback \cite{li} can be found when
comparing the two orthogonal polarizations of the VCSEL with the
fields from each SCL. Nevertheless, we point out that the delay
present in our case arises from the longitudinal configuration
system.


%-----------------------------------------------------------------------------
\subsection{Steady States}
\label{sect:ss}

We start our analysis by looking at the steady state solutions of
Eqns. (\ref{eq:eqne})-(\ref{eq:eqng}).  These solutions can be
obtained imposing the frequency locking conditions,
$\varphi_1(t)=\Omega t$ and $\varphi_2(t)=\Omega t + \phi$ with
$\phi$ a relative phase difference between the two electrical
fields $ E_{1,2}(t)=\sqrt{P_{1,2}(t)} e^{i\varphi_{1,2}(t)}$,
$\dot{P}_1=\dot{P}_2=0$ and $\dot{N}_1=\dot{N}_2=0$. Neglecting
Langevin noise sources and introducing these conditions into Eqns.
(\ref{eq:eqne})-(\ref{eq:eqng}), we arrive to a set of non-linear
equations that reads
%
\begin{eqnarray} \eta+\delta&=&-a C
\sin{\left(\eta+\arctan\alpha+\varphi_0-\phi\right)},
\label{eq:st1}\\
%
\eta-\delta&=&-\frac{1}{a} C
\sin{\left(\eta+\arctan\alpha+\varphi_0+\phi\right)},
\label{eq:st2}\\
%
0&=&J_k-\gamma_e N_{k}-G_{k} P_k \;\;\;\;\mathrm{for} \;\;\;\;\
k=1,2. \label{eq:st3}
\end{eqnarray}
%
At this point, we have defined the following quantities: the
compound system mode frequency $\eta=\Omega\tau$,  the power ratio
$a^{2}=P_2/P_1$, the normalized detuning $\delta=\Delta\cdot\tau$,
the injection phase $\varphi_0=\omega_0\tau \mod 2\pi$ and the
effective coupling strength $C=\kappa\tau\sqrt{1+\alpha^2}$. A
solution of the system of Eqns. (\ref{eq:st1})-(\ref{eq:st3})
determines a vector of six unknowns, ($P_1$,$P_2$, $N_1$, $N_2$,
$\eta$, $\phi$), that identify a frequency-locked (FL) solution of
the coupled system.

Given the two optical phases $\varphi_1(t)$ and $\varphi_2(t)$, we
define in-phase cw solution (IP) \cite{mandel00} if the condition
$\varphi_2(t)=\varphi_1(t)$ is achieved whereas the
weaker condition $|\varphi_2(t)-\varphi_1(t)|\rightarrow \phi \mod 2\pi$
applies for phase locking (PL) \cite{terry99}. At resonance
($\Delta=0$) and under equal injection levels ($p_1=p_2$) for both
lasers, the high degree of symmetry in the system allows us to
differentiate between IP solutions with $\phi=0$ and PL solutions
with $\phi\ne0$.  The IP cw solutions verify the following
equation
%
\begin{equation}
\eta=-C\sin(\eta+\arctan\alpha+\varphi_0). \label{eq:lk}
\end{equation}
%
This equation is equivalent to the external cavity mode condition
obtained within the Lang-Kobayashi model \cite{lk} for a feedback
problem with roundtrip $\tau$.  It is remarkable that the IP cw
solutions are perfectly symmetric under the interchange of the two
lasers, i.e. the symmetry ($P_1$, $P_2$, $N_1$, $N_2$, $\eta$,
0)$\rightarrow$ ($P_2$, $P_1$, $N_2$, $N_1$, $\eta$, 0) is
satisfied. The IP cw solutions are located in an ellipse in the
$\eta-N$ phase space and they are depicted by solid circles in
Fig. \ref{fig:steady}. The number of IP cw solutions is proportional
to the effective coupling strength $C$. The shape of the ellipse
changes with the value of $\alpha$ modifying the stability
properties of the fixed points. In analogy to the feedback
case we refer the upper branch of the ellipse as antimode-IP cw
solutions and the lower-branch as mode-IP cw solutions. All these
solutions describe a symmetric operation of the system where both
lasers evolve with the same value of the variables.


\bfig{h}
\bcen
\begin{tabular}{c}
\psfig{file=Figs/f2spie.eps,width=10cm,clip=true}\vspace{-0.5cm}
\end{tabular}
\ecen \caption{Steady states representation for one laser in the
carrier vs. mode frequency subspace for (a) p=1.00 and (b) p=1.01.
The parameters used correspond to the described in the text except
for $\kappa=5$ ns$^{-1}$ in order to simplify the figure. IP (PL)
cw solutions are represented by solid (empty) circles. The inset in
panel (b) represents the relative phase $\phi$ for the different
PL cw solutions.} \label{fig:steady} \efig

Even though the high degree of symmetry of the problem, PL
solutions can exist. In particular, these solutions might be of
interest since, as we will show in the next section \ref{sect:nr},
under dynamical conditions the lasers do not operate in a
symmetrical way. PL solutions are asymmetric under the interchange
of the two lasers. However, it is easy to demonstrate that the
transformation ($P_1$, $P_2$, $N_1$, $N_2$, $\eta$,
$\phi$)$\rightarrow$ ($P_2$, $P_1$, $N_2$, $N_1$, $\eta$, -$\phi$)
provides a new solution only if $\Delta=0$. Therefore,  pairs of
PL solutions with the same $\eta$ (or the same frequency) appear
recovering the initial symmetry of the problem. PL solutions
calculated from Eqns. (\ref{eq:st1})-(\ref{eq:st3}) are depicted
in Fig. \ref{fig:steady} by empty circles. We plot in Fig.
\ref{fig:steady}(a) the steady states at the solitary threshold
(p=1.00) and in  Fig. \ref{fig:steady}(b) slightly above threshold
(p=1.01). First, we can observe that the number of PL solutions
depends on the injection level $p$ approaching to the number of IP cw
solutions for large current levels. It can be seen that the mode
frequency $\eta_s$ of each of these solutions remain constant for
high injection levels. Furthermore, from Fig. \ref{fig:steady}(a)
it can be easily seen that the a pair of PL solutions appears
encircling a mode-IP cw solution for negative $\eta$ and around an
antimode-IP cw solutions for positive $\eta$ although with a small
frequency mismatch. This asymmetry between positive and negative
$\eta$, that is remarkable in Fig. \ref{fig:steady}(b), might be
attributed to the role of the alpha factor. Furthermore, the
variation of the relative phase for the different PL solutions can
be seen in the inset of Fig. \ref{fig:steady}(b). The relative
phase, $\phi$, is approximately zero at the edges of the ellipse
and approaching to $\pi/2$ for modes close to the solitary
frequency mode.

Although an extensive stability analysis of the steady state
solutions has not yet been performed, from numerical simulations
we find that a large number of these states are unstable for large
to moderate values of the linewidth enhancement factor. We observe
that the trajectory of the system moves close these steady state
solutions although for small time intervals before being repelled
by a fix-point towards its neighboring.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{NUMERICAL RESULTS}
\label{sect:nr} In this section we restrict the discussion of the
instabilities that arises under weak to moderate coupling
conditions (only 5\% of the emitted light is injected) and long
coupling times, providing a time delay of $\tau \simeq 4$ ns.
We find that under these conditions the external flying time,
being much larger than any other typical time scale of the laser,
plays an important role in the determination of the dynamics of
the global system. On the other hand, the coupling itself provides
the synchronization mechanisms of the before mentioned
instabilities.


%...............................................................................
\subsection{Coupling induced instabilities and its synchronization}
\label{sect:ciinr}

In the Fig. \ref{fig:kdepen-num}(a) we represent the typical time
traces  of the optical power emitted by the two lasers ($LD_{1}$
upper panel,  $LD_{2}$ lower panel) under strong coupling
conditions $\kappa=30$ ns$^{-1}$.  In these numerical simulations
the lasers are biased at their solitary threshold. Therefore, in
the absence of coupling the lasers emit fully uncorrelated
spontaneous emission but as soon as the coupling is increased the
signals start to display a behavior that consists in irregular
pulses with small correlation. When increasing further the
coupling strength, we obtain similar fast pulses but now
accompanied with sudden power dropouts followed by a gradual
built-up of the optical power. This low frequency dynamics, much
slower than any other time scale of the system, displays a good
correlation between the two time series [See Fig.
\ref{fig:kdepen-num}(a)]. We find that power dropouts appear for a
wide range of coupling rates and injection currents close to the
solitary laser threshold. The appearance of power dropouts is
manifested in the power spectrum\cite{prl} by a broad peak at low
frequencies ($\sim$10 MHz). When the coupling strength is strong
enough, synchronization is achieved and the power spectra of both
lasers almost coincide. At higher frequencies we observe a
sequence of peaks located at $n/(2\tau)$ for $n$=1, 2, $\cdots$.
The frequency separation between two consecutive peaks is
$1/(2\tau)$ that corresponds to one roundtrip time in the external
cavity defined by the free-space between the two lasers. We note
that this structure in the power spectrum resembles that of a
feedback problem with roundtrip time $2\tau$, although no feedback
reflections at $2\tau$ are considered within our model.

\bfig{h}
\bcen
\begin{tabular}{c}
\psfig{file=Figs/f3spieb.eps,width=10cm,clip=true}\vspace{-0.25cm}
\psfig{file=Figs/f3spie-return.eps,width=6.5cm,clip=true}\vspace{-0.25cm}
\end{tabular}
\ecen
\caption{(a) Typical time traces when both lasers operate at their solitary threshold
current and coupled by a rate $\kappa=30$ \nsi, (b) synchronization plot between
power dropouts when one series is shifted by a time $\tau$.}
\label{fig:kdepen-num}
\efig

When zooming into Fig. \ref{fig:kdepen-num}(a), we can see that
power dropouts do not occur simultaneously but with a small time
lag $\tau_0$ [See Fig. \ref{fig:zoom}(a)]. By analyzing a large
number of power dropouts under different coupling conditions we
have found that the time lag almost corresponds to the flying time
$\tau$ present in our equations. It is remarkable that although
all the power dropouts appear with this small time lag, the laser
that drops first changes from time to time. The laser that drops
first is indicated in Fig. \ref{fig:kdepen-num}(a) with vertical
arrows. We have observed that this striking phenomenon disappears
for large enough detunings where always the laser detuned at
higher frequency drops first. In addition and by looking at slower
time scales, we observe a fast pulsing behavior that appears well
correlated only if one time series is shifted by a time $\tau_0$
with respect the other [Fig. \ref{fig:zoom}(b)]. This kind of
evolution, where the two lasers play asymmetric roles, has been
referred as {\it leader-laggard} operation\cite{prl}. 
A similar phenomenon was previously observed in a system of two
coupled chaotic oscillators that display lag synchronization
although the equations do not contain any delayed term
\cite{piko}. At a first sight, it is hard to decide which signal
has to be shifted due to the high degree of symmetry in this
system. To overcome this problem, we present a method based on
cross-correlation measurements.
In absence of spontaneous emission and forcing both lasers to
start from identical initial conditions, we have observed that the
amplitude dynamics is perfect synchronized with no time lag and
the system evolves in a phase synchronized state, i.e. $\phi=0$.
However, as soon as a small perturbation is introduced the 
system switches to a state of time-retarded synchronization.

\bfig{h}
\bcen
\begin{tabular}{c}
\psfig{file=Figs/f4spieb.eps,width=10cm,clip=true}\vspace{-0.5cm}
\end{tabular}
\ecen \caption{Zooms of the Fig. \ref{fig:kdepen-num}(a) showing
(a) non-shifted time traces during a dropout (b) time shifted
series between two dropouts.} \label{fig:zoom} \efig

We find {\it leader-laggard} intensity dynamics that evolves
synchronized except for a small time lag. These interesting
findings motivate us to investigate the mechanisms that originates
this asymmetric role between the two subsystems. For these
purposes,  we introduce the cross-correlation\cite{piko} function
$S(\Delta t)$ between two variables $x_1(t)$ and $x_2(t)$ (with
mean values being subtracted) defined by
%
\begin{equation}
\label{eq:corr} S(\Delta t)=\frac{\langle x_1(t)\, x_2(t+\Delta
t)\rangle}{\left[\langle x_1^2(t)\rangle \, \langle
x_2^2(t)\rangle\right]^{1/2}}.
\end{equation}
%
We look for the time shift $\tau_0$ where one get the maximum
correlation $S(\tau_0)$ referred as correlation degree. Complete
synchronization (CS) is achieved when the two signals have
identical evolution at the same time $\tau_0=0$, i.e.
$x_1(t)\approx x_2(t)$.  A generalized concept of synchronization
consists of retarded synchronization\cite{parlitz} (RS), 
where the two signals evolve
with a time shift, i.e. $x_1(t)\approx x_2(t+\tau_0)$. Let us
assume that $\tau_0>0$, this can be interpreted as the subsystem 1
leads the dynamics (leader) whereas the subsystem 2 (laggard)
follows the evolution of its counterpart.   When analyzing the
function $S(\Delta t)$, one find a nearly symmetric function
around $\Delta t=0$ with dominant peaks at $\Delta t=\pm n\tau$
for $n=1,2,\cdots$  with decaying correlation as the index $n$
increases being $S(\Delta t=0)\approx 0$. We observe that the
correlation degree increases from zero very rapidly when the
coupling strength is enhanced until it reaches a critical value.
Above this critical value the correlation degree does not
significantly increase displaying a clear plateau with a
relatively high maximum correlation around 90\%.

We can study synchronization by plotting the output power of the
Laser 2 versus the output power of the Laser 1.  However, by
plotting the two signals without any time shift one only get a
cloud of points without any tendency. Since our amplitude
dynamics is lagged by a time $\tau$, we plot in the Fig.
\ref{fig:kdepen-num}(b) the synchronization diagram when a 
signal is shifted by this time with respect to the other.
For the figure we only consider
the fast dynamics between power dropouts while neglecting a 
short time after the drops when the lasers interchange the {\it leader
laggard} roles.
A narrow cloud of points, that is
arranged around a 45 degree straight line, can be seen. The
dispersion of the points with respect to the linear tendency is
linked with the maximum correlation degree achieved.

%...............................................................................
\subsection{Injection current dependence}
\label{sect:idnr}
%
It is also interesting to characterize how robust is the
synchronization scheme against variations of the accessing
parameters of the system. Two external parameters are easily
controllable:  the coupling strength and the current injection
levels of the lasers. In Fig. \ref{fig:f5}(a), we plot the time
traces and the synchronization diagram of the two lasers with an
injection current $p=1.5$. We observe a rather chaotic output
power that displays irregular pulsations around the solitary laser
output. The power dropouts are not anymore distinguishable under
these conditions. Power spectrum displays a broad band of
frequencies as an indicator of the chaoticity of the signal.
In addition, we also observe retarded synchronization.
In Fig. \ref{fig:f5}(b), we represent the
synchronization plot, that again is arranged along a 45 degrees
straight line. The maximum degree of correlation is achieved by
shifting one time signal by a time $\tau$.  The correlation degree
for this case is still considerably high, $S(\tau_0)\sim 0.5$, but
smaller when comparing with Fig. \ref{fig:kdepen-num}(b) when both
lasers operate at the solitary threshold. A possible
interpretation for this phenomenon is that we have increased the
chaoticity of the system by increasing the injection current while
maintaining the same coupling rate, being this too weak to achieve
good synchronization.

In conclusion, we found found that the synchronization scheme is
rather robust under variations of both the coupling strength and
the injection level achieving relatively high values of the
correlation degree only if one is able to compensate the time lag
by shifting one time series.

 \bfig{h} \bcen
\begin{tabular}{c}
\psfig{file=Figs/f5spie.eps,width=10cm,clip=true}\vspace{-0.25cm}
\psfig{file=Figs/f5spie-return.eps,width=6.5cm,clip=true}\vspace{-0.25cm}
\end{tabular}
\ecen \caption{(a) Typical time traces when both lasers operate at
high current injection levels $p$=1.5 and coupled by a rate
$\kappa=30$ \nsi, (b) synchronization plot when one power series
is shifted by a time $\tau$.} \label{fig:f5} \efig

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{EXPERIMENTS}
\label{sect:exp}

\subsection{Experimental scheme}
\label{sect:scheme}
\bfig{h}
\bcen
\begin{tabular}{c}
\psfig{file=Figs/f6spie.eps,width=6.cm,clip=true,angle=-90}\vspace{-0.5cm}
\end{tabular}
\ecen
\caption{Scheme of the experimental setup.}
\label{fig:setup}
\efig
A scheme of our experimental setup is depicted
in Fig. \ref{fig:setup}. The system consists of two uncoated
Hitachi HLP1400 Fabry-P\'erot sc lasers. We have selected two
device-identical lasers produced from the same wafer in order to
achieve the highest possible degree of symmetry in the system
although small dissimilarities might be present in the experiment.
The temperature of each laser is stabilized to better than
0.01\,K, and selected such that the frequencies of the two lasers
match with an accuracy better than 1\,GHz. The lasers are
coherently coupled via the dominant TE component of the optical
field. The coupling time $\tau$ is determined by the propagation
of the light between the lasers. The neutral density filter (NDF)
controls the coupling strength, and the polarizer (Pol.)
guarantees the coherent coupling between the lasers via the TE
mode.

We detect the intensity dynamics of both lasers simultaneously via
their rear-facet-emission using two photoreceivers with DC to
$6$\,GHz bandwidth. The signal of the photoreceivers is analyzed
by a fast oscilloscope of $3$\,GHz analog bandwidth, and by an
electrical spectrum analyzer (ESA). Thus, our detection equipment
allows us to record the intensity time series of both lasers
simultaneously on sub-nanosecond time scale. In addition, we
monitor the optical spectra of the two lasers with a grating
spectrometer (OSA) with $0.1$\,nm resolution, and detect the time
averaged output power with two p-i-n photodiodes. In the
experiments, we inject a well controlled amount of the emitted
light of either laser into the other one, approximately $5\%$ of
the respective output power, leading to a threshold reduction of
$6\%$ in both lasers.


%...............................................................................
\subsection{Experimental results}
In Fig. \ref{fig:f7}(a) we show the experimental time traces of
the optical power when the lasers operate at their solitary
threshold and under 5\% mutual injection conditions. The
solitary laser frequencies have been controlled to match within
the actual experimental accuracy. In the experiment, the coupling
induced instabilities consists in fast pulsations of the optical
power combined with power dropouts that occur at very much slower
time scales. We observe that power dropouts occur well correlated
in correspondence with the numerical results. The fast pulsing
dynamics that appears between two consecutive power dropouts can
be measured using a fast detector. It has been found\cite{prl}
that also the fast dynamics is synchronized with good correlation
only if one time series is shifted by a time $\tau$. Therefore, we
find a leader-laggard dynamics in the experiment as result of a
spontaneous symmetry breaking that takes place when the lasers
operate at resonance. A different situation occurs under detuned
operation of the lasers, where the sign of the detuning determines
the direction of the symmetry breaking. We observe that always the
laser detuned to higher frequencies takes the lead in agreement
with the numerical results. The leading laser satisfy the same
conditions for the cross-correlation function Eq. (\ref{eq:corr})
than those introduced in Sec. \ref{sect:ciinr}. Therefore, the
overall dynamics is driven by the leader laser and the laggard
laser only follows the chaotic motion of its counterpart.
Furthermore, when the synchronization is achieved the power and
optical spectra of leader and laggard lasers are experimentally
indistinguishable.
%%
We quantify the synchronization properties in terms of the
synchronization plot shown in Fig. \ref{fig:f7}(b),  as discussed for
Fig. \ref{fig:kdepen-num}(b),  and the
synchronization degree $S(\tau_0)$. On one hand, we observe a
clear linear tendency for the $P_1(t)$ vs. $P_2(t-\tau)$
representation indicating that the amplitude dynamics during two
consecutive power dropouts evolves well correlated. The dispersion
of the plot is rather small obtaining relatively large correlation
degree between the two series, $S(\tau_0)\approx 0.85$, being only
a little smaller than the numerical value. In the experiment we
have obtained relatively high values of the correlation degree in
a wide range of coupling strengths in correspondence with
numerical results.

\label{sect:ciiexp}
\bfig{h}
\bcen
\begin{tabular}{c}
\psfig{file=Figs/f7spie.eps,width=10cm,clip=true}\vspace{-0.25cm}
\psfig{file=Figs/f7spie-return.eps,width=6.5cm,clip=true}\vspace{-0.25cm}
%\psfig{file=Figs/f7spie-b.eps,width=6.5cm,clip=true}\vspace{-0.25cm}
\end{tabular}
\ecen \caption{(a) Typical time traces when both lasers operate at
their solitary threshold ($p=1$) current under 5\% mutual injection conditions,
(b) synchronization plot between power dropouts
when one series is shifted by a time $\tau$.} \label{fig:f7}
\efig


On the other hand, it is also interesting to study the dependence
of the synchronization properties on the injection current. Fig.
\ref{fig:f8}(a) depicts the evolution of the optical power when
the lasers are injected at higher current levels, $p=1.16$. The
time traces are similar to those obtained numerically in Fig.
\ref{fig:f5}(a). The dynamics of the optical power consists in
irregular pulsations around the solitary power level. This
irregular behavior of the laser outputs does not prevent some
degree of correlation between the two signals. The correlation
degree, like in the numerical results, is moderate being
$S(\tau_0)\approx 0.55$. Fig. \ref{fig:f8}(b) depicts the
synchronization plot under this last situation. We can still
observe some linear tendency but being more spread than Fig.
\ref{fig:f7}(b).

\bfig{h} \bcen
\begin{tabular}{c}
\psfig{file=Figs/f8spie.eps,width=10cm,clip=true}\vspace{-0.25cm}
\psfig{file=Figs/f8spie-return.eps,width=6.5cm,clip=true}\vspace{-0.25cm}
\end{tabular}
\ecen \caption{(a) Typical time traces when both lasers operate at
current $p$=1.16 and 5\% mutual injection conditions, (b) synchronization plot
when one power series is shifted by a time $\tau$.} \label{fig:f8}
\efig

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Conclusions}
\label{sect:conclusions}

In this paper we have numerically and experimentally investigated
the amplitude dynamics of a system of two mutually delay-coupled
semiconductor lasers. We have taken
identical lasers operating at the same solitary frequencies.
We have demonstrated that the intensity dynamics appears synchronized but with a
time lag that coincides with the propagation time
between the two lasers. At long time scales, the intensity
dynamics displays synchronized power dropouts followed by a
gradual recovering process. Looking at short time scales, the
power dynamics is dominated by a fast pulsing behavior that also
appears synchronized with a time lag. From cross-correlation measurements we
have found relatively high values of the correlation degree for
currents near to the solitary threshold and moderate to strong
coupling conditions.  The discussion has been accompanied by
synchronization plots that depict a clear linear tendency only
when one time series is shifted with respect to the other. The
asymmetric role of the lasers have been also determined from the
cross-correlation function allowing to define {\it leader} and
{\it laggard} lasers.

In general, we have obtained good agreement between numerical and
experimental results: occurrence of correlated amplitude dynamics
and its dependence on coupling strength and injection current
parameters. Nevertheless, there are still some interesting
questions about the role of the detuning on the determination of
the leader and laggard dynamics. On the other hand, further
investigation on stability analysis should provide an explanation
of the preference of the system to operate in a retard synchronized
instead of a perfect synchronous state.


%%-----------------------------------------------------------
\acknowledgments This work has been funded through the Acci\'on
Integrada HA98-29. J.M. and C.R.M. also acknowledge financial
support from Comisi\'on Interministerial de Ciencia y
Tecnolog\'{\i}a under project TIC98-0418-C05-02.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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