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\begin{document}
\title{Transient Dynamics of a Single Mode Semiconductor Laser Subjected to
Both \\ Optical Feedback and External Light Injection}

\author{Jaume Dellunde$^1$, Claudio R. Mirasso$^2$, M. C. Torrent$^3$,
\\ J. M.  Sancho$^1$ and
Emilio Hern\'{a}ndez-Garc\'{\i}a$^2$}

\date{~}
\maketitle
{}~~~~~
\vspace{-3cm}

{\large
\noindent
1- Departament d'Estructura i Constituents de la
Mat\`eria,
Universitat de Barcelona, Diagonal 647, E-08028, Barcelona,
Spain.

\noindent
2- Departament de F\'\i sica,
Universitat de les Illes Balears,
E-07071 Palma de Mallorca, Spain.

\noindent
3- Departament de F\'{\i}sica i Enginyeria Nuclear,
EUETIT, Universitat
Polit\`{e}cnica de Catalunya, Colom 1, E-08222, Terrassa, Spain.}
\vspace{0.5cm}

\begin{center}
{\Large \bf {Abstract}}
\end{center}

\noindent
Transient dynamics of a single-mode semiconductor laser subjected to both
weak optical feedback and light injection
is studied.
Even at detuning values for which
injection locking is
not effective, turn-on jitter and frequency jitter
can be still strongly reduced
by either adjusting the length of the external
cavity or the detuning.
\vspace{0.5cm}

\pagebreak

Turn-on delay jitter (TOJ) is of considerable importance for practical
applications
of semiconductor lasers. It causes a degradation of the temporal resolution
and it acts as a limiting factor in the performance of high-bit rate optical
communication systems. Jitter properties have been extensively studied both
experimentally and numerically in the last years [1]-[3].
%\cite{blanes}-\cite{CPM}.

Optical feedback is an important effect to take into account when considering
the performance of a laser diode in an optical communication system. The
effect of feedback on transient dynamics of single-mode semiconductor lasers
has been recently studied for both long [4]
%\cite{Alan}
and short [5]
%\cite{CE}
external cavity lengths ($L_{ext}$). For $L_{ext}$ of the order of
centimeters it has been pointed out that TOJ increases with feedback
strength
[4].
%\cite{Alan}
For short external
cavities, up to a few millimeters, the mean turn-on time (MTOT)
periodically oscillates when changing
the position of the external reflector.
The associated TOJ is quite
insensitive to the external cavity length when it is very short,
but shows large
peaks at particular (and periodically spaced)
positions of the external mirror [5]
%\cite{CE}
for external cavities in the range
of millimeters.

Injection locked semiconductor lasers
have been studied under CW [6,7]
%\cite{mogensen}
and transient dynamics [8,9]
%\cite{leiden},\cite{surette}
regimes
both experimentally and analytically. A small frequency mismatch
between the external source of light (master laser (ML)) and the laser
(slave laser (SL))
makes the SL to operate in a single-mode regime, yielding also
a reduction of jitter and of the frequency chirp [9].
%cite{mohrdiek}
Measurements of MTOT
have been proposed to detect weak external signals [10].
%\cite{SCMJM}.

In this letter we study the
transient dynamics of a single-mode semiconductor
laser subjected to both weak optical feedback
and light injection. In this situation
the response of the SL can be separated in three regimes. If the frequency
mismatch is small (locking condition), the ML improves SL
speed and consequently a reduction of the TOJ and
MTOT time will be observed with increasing strength of the
injected light.
If the frequency mismatch is very large, light injection is inefficient and
only the effects of feedback are noticeable.
There exists, however, a region in which light injection is inefficient in the
absence of feedback but becomes relevant in cooperation with it. An oscillatory
dependence of the MTOT and the TOJ on
the relative phase between the injected
and the feedback fields appears, different from the
one of the laser without external light injection.
It will be shown that for this region, and
for any position of the
external reflector (in the range of $L_{ext}$ considered here) there exists
an optimum frequency mismatch that gives a large reduction of the TOJ,
up to 50 \% the one of the free laser. Alternatively, a similar effect is
obtained for fixed detuning at optimal external cavity lengths.
Moreover, as
will be described below, the TOJ and the switch-on to switch-on frequency
jitter (FJ), defined
as the fluctuations of the frequency chirp,
is such that a decrease in the TOJ yields
a decrease of the FJ. This last consequence is very important
for pulse propagation in optical fibers [11].
%\cite{CLA}.

We describe the transient response of the SL in terms of the
noise-driven single-mode rate equations for the electric field and
carrier number inside the cavity
\begin{eqnarray}
\frac{dE}{dt} & = & (1+j\alpha)(G-\gamma)\frac{E}{2}+k_mE_m e^{j t
\Delta\omega}+
\kappa e^{-j\omega_s \tau} E(t-\tau)
\nonumber \\ & + & \sqrt{2\beta N} \xi(t) \\
\frac{dN}{dt} & = & \frac{I(t)}{e}-\gamma_e N-G P
\end{eqnarray}
where $G=g(N-N_0)/(1+s P)$, $P=|E|^{2}$ is the light intensity. We use
parameter values typical of Multi-Quantum Well lasers: $g=5.6 \times
10^{-8}$
ps$^{-1}$ is the gain parameter, $\alpha=5.5$ is the linewidth
 enhancement
 factor, $\gamma=0.4$ ps$^{-1}$ is  the inverse photon lifetime, $\gamma_e=5
\times 10^{-4}$  ps$^{-1}$
is the inverse
carrier lifetime, $N_0=6.8 \times 10^{7}$ is the  carrier
number at
transparency, $\beta=1.1 \times 10^{-8}$ ps$^{-1}$ is the spontaneous
emission rate, $e=1.602 \times 10^{-19}$ C is the electronic
charge and
the gain saturation coeficient $s=5 \times 10^{-7}$. The SL is
coupled to the external cavity, of round trip time $\tau$,
through the coupling parameter
$\kappa= \frac{\sqrt{R_{ext}} (1-r)}{\tau_L\sqrt{r}}=0.025$
ps$^{-1}$. $R_{ext} \approx  0.016$ is the power external
reflectivity, $r = 0.32$ is the power laser-facet reflectivity and $\tau_L=6$
ps is the laser round trip time. $k_m$ is the coupling parameter of the
injected
field, of amplitude $E_m$ such that $k_m E_m=3$ ps$^{-1}$ (this corresponds
to a power of $\sim$ -35 dB with respect to the solitary laser). The
random spontaneous emission process is modeled by a complex
Gaussian white noise term $\xi(t)$ of zero mean and correlation
$<\xi(t)\xi^*(t')>=2\delta(t-t')$.
The frequency mismatch between
the ML
and the SL is $\Delta\omega=\omega_m-\omega_s$.

We consider the case of repetitive gain switching, i.e. the injection
current of the
SL is suddenly changed from a bias current ($I_b$) below the threshold
current ($I_t$)
to a value well
above threshold $(I=3.5 \, I_t)$.
We concentrate our discussion on two different ranges of external cavity
lengths:
$L_{ext} \sim 16 \, \mu m$ and $L_{ext} \sim 750  \, \mu m$.
These cavity lengths and feedback
reflectivities can be met in diode modules where unwanted reflection coming
from lenses, fiber faces, and other package components may affect the laser
behavior. These values of $L_{ext}$
allow us to perform some analytical calculations which capture
in a general way some of the basic issues of the problem. In addition,
numerical simulations are reported.

For $\kappa\tau<<1$
we can
expand $\kappa E(t-\tau) \approx \kappa E(t) - \kappa \tau \, dE(t)/dt$ in
eq.(1).
Switch-on events are best analyzed in the reference frame of the
ML emitting frequency, i. e.  $E(t)={\cal E}(t)
e^{j(t \Delta\omega+\phi)}$.
We have
\begin{eqnarray}
\frac{d{\cal E}}{dt}&=&\frac{1}{2}[(1+j\alpha)(G-\gamma)
+2\kappa e^{-j\omega_s\tau}-2j\Delta\omega] R e^{j\phi}{\cal E}(t) \nonumber\\
 &+&k_m (R E_m)+\sqrt{2(\beta R^2) N} \xi(t)
\end{eqnarray}
where $R^2 =  \left [1+(\kappa\tau)^2 + 2 \kappa \tau \cos{\omega_s \tau}
\right ]^{-1}$ and
$\phi = \arctan{\left [ \kappa \tau \sin{\omega_s \tau}/
(1 + \kappa \tau \cos{\omega_s \tau)}\right ]}$.

The MTOT and TOJ of a semiconductor laser subjected
to optical injection were studied in [10]
%\cite{SCMJM}
in the context of detection
of weak optical signals. There, low values of the detuning parameter
$\Delta\omega$ produce both MTOT and TOJ reduction, as plotted
in figure 1 of ref. [10].
%\cite{SCMJM}.
The minimum in that figure is placed
at the angular frequency $\omega_m=\omega_s$,
meaning that the maximum reduction
in the MTOT and TOJ appears when the two lasers
are in resonance at the time the amplification becomes first possible,
i.e. when the SL reaches the threshold (the frequency there is
$\omega_s$) [10].
%\cite{SCMJM}.
This idea can be extended in the presence of feedback. In this case,
the threshold value for the carrier number from (3) is (during the switch-on of
the SL, $P$ is very small and gain saturation effects can be
neglected in eqs.(1)-(2)):
\begin{equation}
N_t=N_0+\frac{\gamma}{g}-\frac{2\kappa\cos(\omega_s\tau-\phi)+2\Delta\omega
\sin \phi }{g(\cos \phi -\alpha\sin \phi )}
\end{equation}
By replacing $N$ by $N_t$ into (3) we obtain at the moment the laser reaches
threshold:
\begin{eqnarray}
\frac{d{\cal E}}{dt} & = & -j\frac{R}{\cos \phi -\alpha\sin \phi }
[\Delta\omega+\kappa\sin(\omega_s\tau+\arctan \alpha )]{\cal E}(t) \nonumber \\
& + &
k_m(R E_m)+\sqrt{2(\beta R^2) N_t} \xi(t)~~~~~,
\end{eqnarray}
i.e., feedback into the SL
results in a shift of its frequency that leads to an effective
frequency mismatch ($\Delta \nu =\Delta\omega / 2\pi$)
\begin{equation}
\label{12}
\Delta\nu_{ef}=\frac{R}{\cos \phi -\alpha\sin \phi
}[\Delta\nu+\frac{\kappa}{2\pi}\sqrt{1+\alpha^2}\sin(\omega_s\tau+
\arctan \alpha )]
\end{equation}
According to the ideas of Ref.[10], we get dynamic resonance when
$\Delta\nu_{ef}=0$, i.e., from (6), for a detuning
\begin{equation}
\label{10}
\Delta\nu_{min}=-\frac{\kappa}{2\pi}\sqrt{1+\alpha^2}\sin(\omega_s\tau+
\arctan\alpha)
\end{equation}
which gives the minimum of the MTOT and TOJ.
This expression coincides with the frequency shift introduced purely
by optical feedback under CW operation [12].

As we mentioned above, feedback and light injection compete for some range
of detunings. The minimum detuning value for competition
can be estimated
from eq.(28)
of ref. [10],
%\cita{SCMJM}
assuming that this expression yields the maximum frequency mismatch
for which light
injection reduces the MTOT, in our case
\begin{equation}
\Delta\nu_0 \sim \frac{1}{\pi} \sqrt{(\ln2) \, g \frac{(I-I_t)}{e}(1+\alpha^2)}
\end{equation}
For $I=3.5 \, I_t$, $\Delta\nu_0 \sim 107$ GHz.
The range of competition is limited by the maximum shift that can be introduced
by feedback. It is calculated from eq.(6)
by taking the maximum
value for the sine function: $\Delta \nu_{int}=\frac{\kappa}{2 \pi}
\sqrt{1+\alpha^2}\approx 22$ GHz, for our parameter values.
This means that the range of frequencies of the SL that can be tuned
with this method
is of about 44 GHz.
It should be mentioned, however, that even when the resonance condition
$\Delta \nu_{ef}=0$ is not reached, large reductions of TOJ can be obtained
as will be shown below.
By changing the values of $I$ and $\kappa$ the range of competition
can be varied.

We have checked the validity of eq. (7) through  numerical
simulation of the stochastic differential equations (1)-(2)
averaging over $10^4$ turn-on events.
The turn-on time
is defined as the time the optical intensity takes to reach
a 50 \% of the value
for the free-running SL in the on-state,
and the time jitter as its standard deviation.
We plot in figure 1 the value of the $\Delta \nu_{min}$, the detuning value for
which the larger MTOT and TOJ reduction is obtained, as evaluated
from (7), and the corresponding numerical results for the two
ranges of $L_{ext}$.
The agreement is good for the whole range of
values of the feedback delay time.

In fig. 2 we present the results of TOJ vs $L_{ext}$ obtained by
solving numerically (1) and (2) for the detuning parameter $\Delta \nu=88$ GHz.
For $L_{ext} \sim 16 \, \mu m$ a flat region can be seen
with a TOJ around  4.25 ps, which is close to the value of the free laser.
However, a large reduction of TOJ, up to 50 \% can be obtained at particular
locations of the external reflector. We have checked that the MTOT also
presents minima at the same points. For $L_{ext} \sim 750 \, \mu m$ the same
qualitative
features can be seen, showing that even for much larger external cavities
large reductions of TOJ are also obtained.
As is evident in the figure, TOJ becomes highly dependent of the
distance of the external reflector, for the range of $\Delta \nu$ for which
there is a competition between feedback and light injection.
It should be noticed that with our $\kappa$ value we can not reach the
perfect resonance condition.

The reduction of TOJ yields a reduction of the FJ
for both ranges $L_{ext}$.
This effect can be seen in fig. 3 where we plot the FJ vs. the TOJ for
both $L_{ext}$.
When TOJ $\sim$ 4.25-4.75 ps, FJ $\sim$ 4-5 GHz,
while
when TOJ $\sim$ 2.5 ps,  FJ $\sim$ 1.6 GHz, i.e. when the maximum reduction of
TOJ is reached FJ is about 65 \% less than its value in absence of light
injection,
an important feature for
optical pulses propagating in fibers [11].

In summary,
light injection from a master laser can be used to obtain jitter reduction
in light pulse generation, even for large frequency mismatches, of the
order of
100 GHz, by using external feedback from short external cavities.
On the other hand, if a given laser
is subjected to an unwanted weak feedback, the use of a  master laser with a
frequency mismatch as large as 100 GHz, easily available,
can be used to improve turn-on jitter and also frequency jitter.
Using this method, reductions of time jitter and frequency jitter, up
to 50 \% and 65 \%, respectively, their values of the free SL,
can be obtained.
\vspace{0.25cm}

\noindent
{\Large{\bf Acknowledgments}}

J.D., M.C.T. and J.M.S. acknowledge the
Comisi\'on Interministerial de Ciencia y Tecnolog\'\i a, Project 
PB93-0769-C02-01 
and Fundaci\'o Catalana per a la Recerca-Centre de Supercomputaci\'o de
Catalunya.
The work of C.R.M. and E.H.G. was supported
by the
Comisi\'on Interministerial de Ciencia y Tecnolog\'\i a, Project
TIC93-0744.
\pagebreak

\noindent
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Optical Systems", ed. by R.Vilaseca and R.Corbalan, Springer-Verlag,
(1991) p. 259, and references therein.

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\bibitem{}
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\end{thebibliography}
\pagebreak

\noindent
{\Large \bf {Figure Captions}}
\vspace{0.75 cm}

\noindent Fig. 1. $\Delta \nu_{min}$ vs. $\tau$ for $\tau \sim 0.1$ ps,
$L_{ext} \sim 16 \, \mu m$ (stars and upper scale) and
$\tau \sim 5$ ps, $L_{ext} \sim 750 \, \mu m$
(squares and lower scale). Continuous line corresponds to eq.(7).
\vspace{0.5cm}

\noindent Fig. 2. TOJ vs. Cavity Length. $L_{ext} \sim 16 \, \mu m$
(stars and upper scale) and
$L_{ext} \sim 750 \, \mu m$
(squares and lower scale).

\noindent Fig. 3. FJ vs. TOJ. Stars correpond to $L_{ext} \sim 16 \, \mu m$.
Squares correspond to $L_{ext} \sim 750 \, \mu m$


\end{document}