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\title{\bf
Frequency Dynamics of Gain-Switched Injection-Locked
Semiconductor Lasers
\author{
J. Dellunde, M.C. Torrent$^{\ast}$, J.M. Sancho and M. San
 Miguel$^{\ast\ast}$\\
\normalsize
\it Departament d'Estructura i Constituents de la Mat\`{e}ria,
Facultat de F\'{\i}sica, \\
\normalsize
\it Universitat de Barcelona, Diagonal 647, E-08028
Barcelona,
Spain. \\
\normalsize
$^{\ast}$ \it Departament de F\'{\i}sica i Enginyeria Nuclear, EUETIT, \\
\normalsize
\it Universitat Polit\`{e}cnica de Catalunya, Colom 1,
E-08222 Terrassa,
Spain. \\
\normalsize
$^{\ast\ast}$ \it Instituto Mediterr\'{a}neo de Estudios Avanzados IMEDEA
(CSIC-UIB), \\
\normalsize
\it
Campus Universitat de les Illes Balears, \\
\normalsize
\it E-07071 Palma de Mallorca, Spain \\
}}
\date{}
\begin{document}
\maketitle
\begin{abstract}
The frequency dynamics of gain-switched single-mode semiconductor lasers subject
to optical injection is investigated. The requirements for low time jitter
and reduced frequency chirp operation are 
studied as a function of the frequency mismatch between the master and slave lasers.
Suppression of the power overshoot, typical during gain-switched operation,
can be achieved for selected frequency detunings.
\end{abstract}
\section{Introduction}

Gain-switched pulses from single-mode semiconductor lasers are used in a
variety of applications. In optical communication systems
working at high bit rates, narrowband spectra under
fast current modulation and  low time jitter at the detection end are
required \cite{agrawal2}.
Several mechanisms have been designed to provide such characteristics,
including mode locking to an external cavity \cite{langleysm}, optoelectronic
feedback \cite{dellundet}, self-seeding \cite{schellu} and optical injection \cite{torrentb,scholl}.
Optical narrowband injection, usually from a similar master laser,
seems to be the most promising technique for controlled,
low jitter, single frequency pulse generation from laser diodes.
While good performance of the injection-locked lasers at moderate
injection levels was early demonstrated \cite{hui2,lin,olssont},
only more recently the possibility of injecting a considerable amount of
light into the active region of the laser structure has allowed to obtain
extremely high repetition rates in the pulsed operation, with narrow spectra
and wide eye openings \cite{andrekson,mohrdiek1,mohrdiek2,wang}.

The frequency dynamics during the turn-on of a solitary semiconductor laser was
considered in \cite{ballea,marga}. In that case,
the evolution of carrier  and photon 
populations is decoupled from the phase evolution.
The instantaneous frequency follows the  carrier dynamics through
the linewidth enhancement factor $\alpha$, responsible for both 
the linewidth broadening which appears in steady-state operation and
the
frequency chirp observed during current modulation of the laser diode.
The dynamics of an optically injected semiconductor laser is far
more complex than that of a free-running laser, because the relevant variables
intensity, phase and carrier number are nonlinearly coupled. Instabilities can occur 
 even when the master and slave frequencies are
perfectly matched, due to the depressed carrier number induced by
optical injection. The DC biased slave laser exhibits several behaviors
depending on the injection level and the frequency mismatch between the
two lasers, including steady-states \cite{lang}, periodic self-oscillations \cite{ye},
chaotic output \cite{lodi,sacherb,simpson}, low-frequency fluctuations \cite{dedushenko},
four-wave mixing and frequency conversion \cite{liu}, among others. 
The steady-state, in which the slave laser is frequency locked to the
master laser, has received particular attention \cite{lang,mogensen,petitbon}.
Injection-locking phenomena in semiconductor lasers are currently under 
active research \cite{huali,espana,tbsimpson} due to their potential 
applications.

Our aim is to provide a framework where 
the frequency dynamics of the optically injected laser can be
elucidated, in the strong dynamical conditions following the
laser switch-on.
Attention is focused on the ability of the slave laser
to lock at the master frequency. The paper is organized as follows. 
In section 2, a transient potential for the field components is introduced which 
is useful to understand the frequency dynamics during the first stages of 
the light intensity build-up. The relaxation of the slave output to a 
possible steady-state and the conditions for which the power overshoot 
and the frequency chirp can be efficiently suppressed are studied. 
The timing information is analyzed in section 3, where strong 
links between the turn-on time statistics and the frequency 
dynamics appear. Finally, some general conclusions are drawn in section 4. 

\section{Frequency dynamics}

Under optical injection, some amount of the output power
launched from a master laser enters the 
cavity of the slave laser at the laser facet with reflectivity $R_2$.
When a single-mode semiconductor laser is considered,
the field rate equations for the amplitude of the optical field
and the carrier number in the active region of the slave are \cite{dellundetm}
\begin{eqnarray}
\label{injection}
\frac{dE}{dt}&=&\frac{1+i \alpha}{2}\left[\frac{g(N(t)-N_0)}{1+s |E(t)|^2}-\gamma\right]
E(t) + k_m E_m e^{i \Delta\omega t}+ \\
&+&\sqrt{2 \beta N(t)} \xi(t) \nonumber \\
\frac{dN}{dt}&=& C-\gamma_e N(t)-\frac{g(N(t)-N_0)}{1+s |E(t)|^2} |E(t)|^2
\end{eqnarray}
The optical field $E(t)$ is normalized in such a way that $|E(t)|^2$ gives
the photon number inside the laser cavity. Typical numerical values assigned to the
parameters are listed in Table I. 
The random spontaneous emission
processes are modeled by means of a complex Gaussian white noise term $\xi(t)$ of
zero mean and correlation $<\xi(t)\xi^*(t')>=2\delta(t-t')$.
We allow for a frequency mismatch between the master laser, with angular frequency
$\omega_m$, and the solitary slave laser, with angular frequency $\omega_s$.
This frequency mismatch is considered in the field rate equations through the
frequency detuning $\Delta\omega=\omega_m-\omega_s$.
$k_m$ is the parameter controlling the injection strength.
The injected field $E_m$ is affected by
power losses arising from filtering, mode matchings and other effects different
to the losses introduced by the laser facet and are considered
through the coupling losses parameter $\eta_{ext}$,
which is always lower than one. The injection
strength $k_m$ is then given as
\begin{equation}
\label{strengthinj}
k_m=\frac{1}{\tau_L} \frac{\sqrt{1-R_2}}{\sqrt{R_2}} \eta_{ext}
\end{equation}
where $\tau_L$ is the slave cavity roundtrip time. Inputs from a far above threshold
biased master laser are considered, so that $E_m$ can be reasonably taken to
be constant. Injected fields with a finite linewidth generally lead to
reduced efficiencies \cite{dellundetsm}.
When the phase or amplitude dynamics of the optical field is under study, equation (\ref{injection})
can be split in two equations.
The optical field is written as 
$E(t)=R(t) e^{i [\Delta\omega t+u(t)]}$ in (\ref{injection}) to give
\begin{eqnarray}
\frac{dR}{dt}&=&\frac{1}{2}\left[\frac{g(N-N_0)}{1+s R^2}-\gamma\right] R
+k_m E_m \cos(u)+
\frac{2\beta N}{R}+\sqrt{\beta N} \xi_R(t)\\
\frac{du}{dt}&=&\frac{\alpha}{2}\left[\frac{g(N-N_0)}{1+s R^2}-\gamma\right]-\Delta\omega-\frac{k_m E_m}{R}\sin(u) 
+\frac{\sqrt{\beta N}}{R} \xi_{u}(t)
\end{eqnarray}
$u(t)$ is understood as the residual phase during locked operation, because
the frequency component $\Delta\omega$ has been substracted from the
field evolution. Amplitude and field noises have zero mean and a time
correlation given by $<\xi_i(t)\xi_j(t')>=2\delta_{ij}\delta(t-t')$.

Strong injection is considered, and the particular injection level used
in this work is $I_L=(E_m/R_{so})^2=0.04$ ($R_{so}^2$ is the light intensity
of the solitary slave in the on state).
The final state depends on the driving current  $C_{on}$, the
master field amplitude $E_m$ and the frequency detuning $\Delta\omega$.
Steady-states exist inside a locking range given by detunings
$|\Delta\omega|<\Delta\omega_L$, where
\begin{equation}
\label{lockingrange}
\Delta\omega_L=\frac{k_mE_m}{R_s}\sqrt{1+\alpha^2}
\end{equation}
In these steady-states, the laser oscillates at the same frequency
of the master laser, and the slave laser is said to be {\em locked}.
The steady-state values $R_s$, $u_s$ and $N_s$ are obtained by setting to zero
the corresponding time derivatives.
At low injection currents only one steady-state solution exists, which is identified as
the {\em off} solution. Three solutions exist in general, two {\em on} solutions and one
off solution. Only one of the two on states is linearly stable.
Small perturbations to the steady-state values $R_s$, $N_s$ and
$u_s$ are considered in order to investigate the linear stability of
the solutions. The stability matrix for the solitary laser gives three eigenvalues. 
One of them, corresponding to the phase perturbations, 
 is zero, and the other two are conjugate complex numbers. 
For the slave laser, two eigenvalues of the stability matrix, $\lambda_1=-\Gamma_s+i\Omega_s$
and $\lambda_2=-\Gamma_s-i\Omega_s$, are conjugate complex numbers. The third
eigenvalue, $\lambda_3$, is a pure real number. $\Gamma_s$ is the damping
constant of the relaxation oscillations, with angular frequency $\Omega_s$.
When $\Gamma_s$ is positive and $\lambda_3$ is negative, the steady-state
is stable under small perturbations. The ensemble of frequency detunings 
inside the locking range which lead to stable steady-states is called
{\em stable locking range}. The inclusion of gain compression effects in the
stability conditions reveals that gain saturation contributes
to an increase of the stable locking range, effectively suppressing
relaxation oscillations.
Additional damping and, as a consequence,  further stability 
can be obtained with a careful selection of the frequency detuning.
The criteria for detuning selection are illustrated in figure 1.
The locking range extends beyond 60 GHz. Negative
detunings beyond $-35$ GHz show negative real parts of the three eigenvalues,
leading to stable steady-states. The eigenvalues of the solitary laser are included as solid lines.
Detunings exceeding $-40$ GHz lead to damping factors significantly higher
than
those of the solitary laser.

In the present section, the interest is concentrated on the relaxation to a steady-state
following a gain-switch.
The laser is initially below threshold with an injection current $C_b$. The laser is suddenly
switched on by means of an increase of the driving current
to $C_{on}$. In our numerical simulations, $C_{on}$ is 2.1 times above its threshold value, $C_{th}$. 
Different turn-on situations are analyzed based on the previously described detuning
ranges.
The nonlinear equations describing the laser switch-on are numerically
simulated by means
of a first-order Euler algorithm, with an integration time step of $0.01$ ps.
The bias current is taken to be $C_b=0.9 C_{th}$.
The results are presented in figure 2. The intensity, phase and carrier number time
evolutions at detunings of $-30$ and -$60$ GHz are included, and compared to the
same evolutions without
optical injection.
Intensity and carrier number are normalized according to the steady-state values of the
free-running slave laser.
The solitary laser exhibits large overshoots
following the switch-on, with a  peak power five times above the steady-state  output power. The field
phase is rather erratic and displays 
a fast dynamics up to the final steady-state.
The injected laser with a frequency detuning of $-30$ GHz shows periodic oscillations because
the chosen detuning lies just outside the stable locking range. The reader will observe,
however, that the laser is initially locked before the switch-on time.
The  off solution is stable for low injection currents. The laser
gets unlocked as time goes on because the light intensity grows up, and the locking range
is reduced accordingly (see equation (\ref{lockingrange})).
With a detuning of $-60$ GHz, the final state is stable and with a strong damping
of any small fluctuation. Strong damping during gain-switched operation is evident
as well in figure 2c, where the power overshoot is almost completely suppressed.
The frequency dynamics is rather simple in this case. Both initial and final states
have well-defined locked phases $u_s$, different in general to each other.
The phase dynamics undergoes a smooth transient between the two states, pointing to the
fact that the laser is effectively locked.
Only a residual instantaneous frequency shift
remains as a result of the smooth transient.
We stress the fact that a careful examination
of the steady-states, available by means of standard analytical calculations,
is able to predict the more complicated nonlinear dynamics of the laser and
the suppression of relaxation oscillations, which involve the
numerical simulation of modified stochastic rate equations.
 The possible suppression of the relaxation oscillations is also shown. 

The locking process during the laser turn-on can be understood by means
of a potential for the components of the optical field, valid during the
transient. This potential requires the change to a new reference frame
$E(t)=G(t) e^{i \phi(t)}$ 
\begin{equation}
\label{omega}
\dot{\phi}=\frac{\alpha}{2} g (N(t)-N_{th})
\end{equation}
where
$ N_{th}= N_{0}+ \gamma/g $
and  $G(t)=G_1(t)+i G_2(t)$.
During the transient and before reaching a significant output power,
light saturation is not operative and then one can take $s=0$ in  the 
field rate equations.
The nonlinear terms in the carrier evolution equation
can be neglected as well. Under these reasonable assumptions, the carrier
dynamics can be independently solved to give
\begin{equation}
\label{carrierevolution}
N(t)=\frac{C_{on}}{\gamma_e}+\frac{1}{\gamma_e} (C_b-C_{on}) e^{-\gamma_e t}
\end{equation}
Then the time evolution of the
field components $G_1(t)$ and $G_2(t)$, can be expressed in terms of
 the potential
\begin{eqnarray}
\label{potencom}
V(G_1,G_2,t)&=&-\frac{1}{4} g (N(t)-N_{th}) (G_1^2+G_2^2)+ \\ \nonumber
&+&k_mE_m\left(G_1 \cos[\Phi(t)]+
G_2 \sin[\Phi(t)]\right)
\end{eqnarray}
with $\dot{\Phi}=\Delta\omega-\dot{\phi}$
and the time evolutions of the field components turn out to be
\begin{equation}
\frac{dG_i}{dt}=-\frac{\partial V(G_1,G_2,t)}{\partial G_i}
\end{equation}
where $i=1,2$.
Let's closely examine this last expression. Without optical
injection,
the extremal values of the potential correspond to a state with a vanishing
intensity. Below threshold there is a potential minimum, showing that the laser is
stably turned-off because of a negative net gain. The stability is
lost as time goes on and the carrier number approaches its threshold value,
as identified with an increasingly small potential curvature. The transition
occurs at the threshold time, when a maximum in the potential appears
and the zero-intensity state becomes unstable. At this stage, any fluctuation
in the photon population originated from spontaneous emission is able to
trigger the laser switch-on. In this case, $\phi(t)$ is nothing else
but the instantaneous emission frequency of the solitary laser \cite{freins}.
It is evident from equation (\ref{omega}) that the
linewidth enhancement factor $\alpha$ linearly couples the frequency
to the carrier number.

The previously presented picture completely changes
in the presence of optical injection.
Stable states with a non-vanishing intensity can appear even below the threshold of the solitary laser.
The time at which the instability occurs is shifted with respect to the
threshold time of the solitary laser. The most remarkable result is,
however, the
rotation with time of the potential extrema in the complex $(G_1,G_2)$ plane, with an angular frequency
given by $\dot{\Phi}=\Delta\omega-\dot{\phi}$ (see equation (\ref{potencom})). The injected field is able to
compensate the $\alpha$-driven frequency shift appreciated in the
solitary laser and  lock the slave laser emission to the master frequency.
In this way, efficient frequency locking even under strong dynamic
conditions is predicted. Nevertheless, we point to the fact that noise
and phase matchings to the steady-states can suppress this perfect
locking in some extent.

In what follows, a more systematic study of the locked operation during the whole transient
is presented, following the method in \cite{ballea} in order to get
a statistical description of the frequency dynamics.
In single-mode lasers, in spite of their single frequency
behavior, a broader spectrum in comparison with the steady state operation can
be measured. This broadening is caused by the carrier-induced frequency shift
through the linewidth enhancement factor $\alpha$.
In the gain-switching process the definition of the field spectrum requires a
given time window. The
interest is focused on the transient build-up of intensity, so that the final steady state
should be avoided when evaluating the spectrum. As a compromise,
 a time $t_f$ after the first optical pulse of the solitary laser
is considered for the time window of the FFT.
The transient averaged field spectrum is then defined as
\begin{equation}
S(w)=\frac{1}{C_s}\int_0^{t_f}<E(t)E^*(0) e^{-i \omega t}> dt
\end{equation}
where $C_s$ is a normalization constant giving
\begin{equation}
\int_{-\infty}^{+\infty}|S(w)|^2 dw = 1
\end{equation}
The normalized field spectrum $S(w)$ allows the calculations of relevant parameters,
as the mean frequency
\begin{equation}
<\nu>=\int_{-\infty}^{+\infty}\frac{\omega}{2 \pi} |S(w)|^2 dw
\end{equation}
and the mean spectrum width
\begin{equation}
<\Delta\nu>^2=\int_{-\infty}^{+\infty}\left(\frac{\omega}{2 \pi}-<\nu>\right)^2 |S(w)|^2 dw
\end{equation}
$<\Delta\nu>$ is the dynamic equivalent of the linewidth in steady state operation. It is usually
called
{\em mean frequency chirp} or simply {\em chirp}. In our numerical simulations the time window
of the FFT is set to $t_f=280$ ps
and the averages are taken over $10^4$ turn-on events.
We say that the laser is dynamically locked
when the slave mean frequency is the same as the master frequency and the chirp
is significantly lower than the solitary laser chirp. Both criteria have to be met to
elucidate the transient frequency locking. The range of detunings giving rise to
transient locking define the {\em dynamical locking range}. Similar definitions
of the dynamical locking range were reported in previous works \cite{mohrdiek2} for current modulated
laser diodes. Some results are presented in figure 3. Frequency locking is identified
as a straight line in the mean frequency versus detuning curve. In the chirp versus detuning
curve a plateau is observed, with a chirp as low as $10$ GHz (the solitary laser chirp value is
around 40 GHz). In the present example, the dynamical locking range goes approximately
from $-70$ GHz to $30$ GHz. As in steady-state operation, this locking range is asymmetric
and favorable to negative detunings. The low carrier population,  induced by a high intensity
level, leads to a preferred emission at reduced  frequencies.
The reader will appreciate that the dynamical locking range is broader than
the stable locking range (about $25$ GHz). This remarkable fact is a result of the whole
turn-on process, which include stages of low intensity and thus allows
wide locking regimes (see equation (\ref{lockingrange})).

\section{Jitter and dynamical locking} 

In this section, the turn-on time statistics is analyzed in terms of the
results obtained in section 2 on the frequency dynamics.
In the solitary laser,
the turn-on is triggered by spontaneous emission random events, which
lead to an important dispersion of turn-on times and to a broadening of
the corresponding probability density function (PDF). The variance in the
PDF of turn-on times can be
considerably reduced when the turn-on process is triggered by a deterministic
seed instead of the random noise. 
This reduction of jitter 
in the presence
of a weak optical fiels was analytically studied in \cite{torrentb}. The 
results
were reported as a function of the frequency detuning between the master and
the slave lasers. For large detunings, both the mean turn-on time and the
time jitter turned out
to be the same as for the solitary laser, while small values of the
detuning parameter led to reduced values of that quantities.
The lower values corresponded to a master laser completely tuned to the slave
emission frequency, and symmetrical detunings around the zero value gave
the same results for the PDF of turn-on times. 
Experimental verification of this
behavior as a function of the frequency detuning has been 
reported in agreement with the theoretical results \cite{seo,seo2}.
Additional experimental observations were a linear increase in the
SMSR and pulse broadening, specially noticeable at high injection
levels.

Under strong injection conditions the
analytical approximations in \cite{torrentb} lose their validity.
An alternative method was developed in \cite{dellundets}
for lasers with a simpler dynamics (class-A), but the
more complex dynamics present in semiconductor lasers  prevents
its direct implementation.
Nonlinear effects arise in the turn-on of the optically injected slave laser
when large frequency detunings are forced. A typical example is depicted in
figure 4a, where a relatively high peak power  prepulse appears before
the final optical pulse has developed. A detuning of -70 GHz has been used
just outside the negative side of the locking range, and the nonlinear
effects are enhanced accordingly. 
As we mentioned in the previous section, if a detuning value inside 
the stable locking range is selected, a smooth transient occurs and 
prepulses are avoided.

The formation of these prepulses can be
a source of large variances in the PDF of turn-on times if the reference
intensity is chosen to lie near the prepulse intensity peak. This effect
is illustrated in figure 4b, where the PDF is shown to be double-peaked.
On the contrary, choices of the frequency detuning inside the stable locking
range give rise to a lower time jitter than for the solitary laser.
The long tail in the PDF of turn-on times typical of semiconductor
 lasers without optical injection,
consequence of a turn-on triggered by noise, completely disappear when a master laser
is used to seed the slave.
Strong optical injection forces the laser to reach the reference intensity
before the carrier number has crossed its threshold value, so that
the performance of the slave laser is similar to that of a 
semiconductor optical amplifier.

Nonlinear effects can also be appreciated at lower values of the
injection level ($I_L=0.001$).
The results concerning the mean turn-on time and time jitter are in figure 5
as a function of the frequency detuning, at different reference intensities. The
modified rate equations have been numerically solved and averaged over
$10^4$ turn-on events.
The frequency detuning that gives a minimum in the mean turn-on time curve
is generally shifted to negative detunings.
Considerable reductions in the time jitter values, up to a 80 \% those of
the solitary laser, can be achieved by selecting positive detunings.

The choice of several reference intensities gives significantly different
turn-on time PDFs. This is a known effect also present in class-A lasers
subject to strong light injection \cite{dellundets} and is a consequence
of the non-exponential growth of the light intensity during the laser turn-on. Peaks in the
time jitter appear at selected detunings. In these turn-on
events the light intensity crosses its reference value with
an almost vanishing time slope, leading to a higher dispersion of turn-on
times driven by noise-induced fluctuations.

\section{Conclusions}

The frequency dynamics of optically injected single-mode
semiconductor lasers has been studied during the laser turn-on.
A transient potential for the components of the optical field
has been introduced which qualitatively describe the transient locking
process. 
A simple
linear stability analysis provides relevant information to characterize
the relaxation of the gain-switched laser to a steady-state.  Transient locked operation
is appreciated inside a dynamical locking range of detunings where the mean frequency
of the slave laser is that of the master laser.  The mean frequency
chirp can be as low as a 75\% its numerical value in
the free-running slave laser.
This dynamical locking range is broader than the stable locking range.
In addition, relaxation oscillations in the laser turn-on can be selectively suppressed .

The optically injected laser operates as a low time jitter source. 
 A 80\% reduction in the numerical value of jitter can be achieved
if carefully selected values of frequency detuning and reference level 
are used.
In order to obtain the above-mentioned performance, the master laser must be
biased above threshold in CW operation.

These conclusions are of practical importance when injection-locked
lasers are used as light sources in intensity modulated communication systems,
where the
 direct detection requires a low time jitter.

\section{Acknowledgments}

The work was supported by the program Human Capital \& Mobility,
Refs. CHRX-CT93-0331.
Work also supported
by Comisi\'on Interministerial de Ciencia y Tecnolog\'\i a, Project
PB93-0769-C02-01 (JD,MCT and JMS) and PB94-1167 (MSM).

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\end{thebibliography}

\newpage
{\bf Table I}: Semiconductor laser parameters
\begin{table}[htb]
\begin{center}
\begin{tabular}{ccc}  \hline
Parameter & Meaning & Value  \\ \hline
g & differential gain & $5.6\,\, 10^4$ $s^{-1}$  \\
$\gamma$  & losses   & $4\,\,  10^{11}$ $s^{-1}$ \\
$\gamma_e$ & inverse carrier lifetime & $5\,\,10^8 s^{-1}$ \\
$N_0$ & transparency value for the carrier number &
$6.8\,\,10^7$ \\
$\beta$& spontaneous emission rate& $1.1\,\,10^4 s^{-1}$ \\
$\alpha$ & linewidth enhancement factor & $5$ \\
$
s$ &  gain compression factor     & $0.7\,\,10^{-6}$\\
$\tau_L$ & cavity roundtrip time & $7\,\,10^{-12}$ $s$ \\
$R_2$ & facet reflectivity& $0.32$ \\
$\eta_{ext}$ & coupling losses &$1$ \\
\hline
\end{tabular}
\end{center}
\end{table}

\newpage
{\large \bf Figure captions}
\vspace{0.5cm}


{\bf Figure 1}: Eigenvalues of the stability matrix. Imaginary (a) and real (b) parts.
The solid lines correspond to the eigenvalues of the solitary laser.

{\bf Figure 2}: Time traces of the normalized intensity,
normalized carrier number and field phase after the laser switch-on. (a) Without
light injection. (b) With light injection and a detuning of -30 GHz. (c)
With light injection and a detuning of -60 GHz.

{\bf Figure 3}: mean frequency (stars) and mean frequency chirp (circles) as a function
of the frequency detuning.

{\bf Figure 4}: (a) Time trace of the normalized intensity after the gain-switching
using a detuning of -70 GHz, outside the locking range. (b) PDF of turn-on times
for the solitary laser (long dashed line), and optically injected lasers with detunings
of -60 GHz (short dashed line) and -70 GHz (solid line).

{\bf Figure 5}: Mean turn-on time (above) and time jitter (below) at two reference
intensities ($I_L= 0.001$). The percentages refer to the steady-state
intensity of the solitary laser.  Separate points at zero detuning correspond
to the values of the solitary slave laser. 





\end{document}