\documentstyle[12pt]{article}
\def\baselinestretch{1.2}
\textheight 9.in
\topmargin 0cm
\textwidth 6.2in
\oddsidemargin .25in
\evensidemargin 0in
\parskip .25in
\pagestyle{myheadings}

\begin{document}

\title{Analytical Calculations of Switch-on Time and Timing Jitter in
Diode Lasers Subjected to
Optical Feedback and External Light Injection}

\author{Jaume Dellunde$^1$, M. C. Torrent$^2$, Claudio R. Mirasso$^3$, \\ 
Emilio Hern\'{a}ndez-Garc\'{\i}a$^3$ and J. M. Sancho$^1$\\
\ \\
$^1$Departament d'Estructura i Constituents de la Mat\`eria,\\
Universitat de Barcelona, Diagonal 647, \\
E-08028, Barcelona, Spain.\\
$^2$Departament de F\'{\i}sica i Enginyeria Nuclear, EUETIT, \\
Universitat Polit\`{e}cnica de Catalunya, Colom 1, \\
E-08222, Terrassa, Spain.\\
$^3$Departament de F\'\i sica, Universitat de les Illes Balears,\\
E-07071 Palma de Mallorca, Spain.
}
\date{~}
\maketitle

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

Analytical expressions for the switch-on time and timing jitter are
obtained for a single-mode semiconductor laser subjected to both optical
feedback and external light injection. The expressions are validated by
numerical simulation of the noise-driven rate equations. It is shown
that even for a large frequency mismatch, when the external light
injection is inefficient to improve the response of the laser, an
important reduction of the timing jitter can be
obtained when optical feedback is present. 
\vfill
Emilio Hern\'andez-Garc\'\i a's e-mail: {\tt dfsehg4@ps.uib.es}
\pagebreak

Single-mode gain-switched semiconductor lasers are one of the most
important light sources in optical communication systems. Several mechanisms
affect the performance of these lasers: spontaneous emission noise
produces an uncertainty in the switch-on time (SOT),  timing jitter
(TJ), which causes a degradation of the temporal resolution and acts as
a limiting factor in the performance of the system when working at Gb/s
rates \cite{spano}; the change of the injection current from a given value
to a much larger one (gain-switching), leads inevitably to a frequency 
%down-
chirp due to the change of the carrier density \cite{salvador1}.
This frequency chirp makes pulses to be dispersed when propagating in
optical fibers \cite{agra_fib}. 

It is well known
that the light coming from a secondary laser (usually called master
laser (ML)) injected into the first laser (slave laser (SL)) can improve
the response of the SL by reducing the jitter and the frequency chirp
under nearly locking conditions \cite{stefan}. However, there is another
important effect that usually appears in, for example, laser diode
modules. This effect is the optical feedback and is commonly due to
unwanted reflections coming from lenses, fiber facets and other package
components. Transient dynamics of diode lasers subjected to optical feedback 
were considered
in both large \cite{alan} and short \cite{ec} external cavity lengths
($L_{ext}$). For $L_{ext}$ of the order of centimeters it has been
pointed out that TJ increases with feedback strength \cite{alan}. For
short external cavities, up to a few millimeters, the SOT periodically
oscillates when changing the position of the external reflector. The
associated TJ 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 \cite{ec} for external cavities
in the range of millimeters. When both optical feedback and light
injection are present the response of the SL can be separated into three
regimes. If the frequency mismatch is small (locking condition), the ML
improves SL response and consequently a reduction of the TJ, the SOT and
frequency chirp 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. We have
recently shown numerically \cite{CLEO} that TJ can be strongly reduced
in this region even for a large frequency mismatch between the ML and
the SL (of the order of 100 GHz). 

In this letter we develop an analytical approach which allows us to
calculate the SOT and TJ when both optical feedback and external light
injection are present. It will be shown that the analytical expressions
so obtained are in very good agreement with numerical simulation of the
noise driven rate equations for the SL, for external cavity lengths up
to some hundreds of microns. 

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 \cite{agra_laser} 
\begin{eqnarray}
\label{1}
\frac{dE}{dt'} & = & (1+i\alpha)(G-\gamma)\frac{E}{2}+k_mE_m e^{i t'
\Delta\omega}+
%\nonumber \\
 \kappa e^{-i\omega_s \tau} E(t'-\tau)+\sqrt{2\beta N} \xi(t') \\
\label{2}
\frac{dN}{dt'} & = & \frac{I(t')}{e}-\gamma_e N-G |E|^2
\end{eqnarray}
where $G=g(N-N_0)/(1+s |E|^{2})$. $g$ is the differential gain, $\gamma$
is the inverse of the photon lifetime, $\gamma_e$ is the inverse of the
carrier lifetime, $N_0$ is the carrier number at transparency, $s$ is
the inverse saturation intensity, $\beta$ is the spontaneous emission rate,
$e$ is the electronic charge, and $I(t)$ is the injection current.
% in $mA$ units.
The SL is coupled to the external cavity,
of round trip time $\tau$ (length $L_{ext}=c\tau/2$), through the 
coupling parameter $\kappa$.
$k_m$ is the coupling parameter of the injected field, of constant
complex amplitude 
$E_m$. The frequency mismatch between the ML and the SL is
$\Delta\omega=\omega_m-\omega_s$. The random spontaneous emission
process is modeled by a complex Gaussian white noise term $\xi(t)$ of
zero mean and correlation $<\xi(t_1)\xi^*(t_2)>=2\delta(t_1-t_2)$. 

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 $(I=3.5 \,
I_t)$. The calculation of switch-on  times after the laser gain
switching involves only the first stages of evolution when the light
intensity is small, so that gain saturation can be neglected in (2). We 
concentrate our discussion on the range of short external cavity 
lengths, up to some hundreds of microns, as expected to be found in diode
modules.  For these range of $L_{ext}$ we can expand $\kappa E(t'-\tau)
\approx 
\kappa E(t') - \kappa \tau \, \dot{E}(t')$ for $\kappa\tau<<1$ \,\,\, in
Eq.(1). Defining the dimensionless parameters
$A^2 =  \left [1+(\kappa\tau)^2 + 2 \kappa \tau \cos{\omega_s \tau}
\right ]$ and $\phi = -\arctan{\left [ \kappa \tau \sin{\omega_s \tau}/
(1 + \kappa \tau \cos{\omega_s \tau)}\right ]}$,
the equation for the electric field reads
\begin{equation}
\label{2bis}
\frac{dE}{dt'}=\frac{1}{A e^{i\phi}}
\left\{\frac{(1+i\alpha)}{2}[g(N(t')-N_0)-\gamma]+
\kappa e^{-i\omega_s \tau}\right\} E(t') 
\\
+k_mE_m e^{i t' \Delta\omega}+ \sqrt{2\beta N} \xi(t')
\end{equation}
The threshold value for the
carrier number $N_{th}$ at the onset of
the amplification process can be evaluated from (\ref{2bis}) as ($Re$
denotes the real part) 
\begin{equation}
\label{9bis}
Re\left\{\frac{1}{A e^{i\phi}}
\left[\frac{(1+i\alpha)}{2}[g(N_{th}-N_0)-\gamma]+
\kappa e^{-i\omega_s \tau}\right]\right\}=0
\end{equation}
so that
\begin{equation}
\label{10}
N_{th}=N_0+\frac{\gamma}{g}-\frac{2[\kappa\cos(\omega_s\tau)
+\kappa^2\tau]}{g[1+\kappa\tau(\cos(\omega_s\tau)-\alpha
\sin(\omega_s\tau))]}~~~. 
\end{equation}
The last term in the r.h.s. of (\ref{10}) accounts for the change in
threshold due to feedback. We consider that
noticeable laser emission cannot occur before the carrier number has
crossed this threshold value. Then we can neglect the last term in Eq.(3)
and obtain  
%\begin{equation}
%\label{3}
$N(t')=N_{on}+[N(0)-N_{on}] e^{-\gamma_e t'}$, 
%\end{equation}
where $N_{on}=I/\gamma_e$ and $N(0)=I_b/\gamma_e$. The time
at which $N$ crosses $N_{th}$ is 
%\begin{equation}
%\label{3bis}
%\bar{t}=\frac{1}{\gamma_e}\ln\frac{N_{on}-N(0)}{N_{on}-N_{th}}.
$\bar{t}=\gamma_e^{-1}\ln[(N_{on}-N(0))/(N_{on}-N_{th})]$.
%\end{equation}
For times larger than $\bar t$ we solve eqs.(\ref{2}),(\ref{2bis}) with 
initial conditions
$N(\bar{t})=N_{th}$ and $E(\bar{t})=0$. Following the analytical method
described in \cite{pere}
, we decompose the expression
for the electric field in the variable $t=t'-\bar{t}$ as
%\begin{equation}
$E(t)=h(t) e^{Q(t)}$, 
%\end{equation}
with
\begin{equation}
Q(t)=\frac{1}{A e^{i \phi}}\int_0^t\left[\frac{1+ i\alpha}{2}
[g(N(t')-N_0)-\gamma]+\kappa e^{-i \omega_s \tau}\right] dt'
\end{equation}
The Gaussian stochastic process $h(t)$ retains all the information about
the spontaneous emission noise and reads 
\begin{equation}
\label{5}
h(t)=\frac{1}{A e^{i \phi}}\int_0^t[k_mE_m 
e^{i t' \Delta\omega}+\sqrt{2\beta N(t')} \xi(t')] e^{-Q(t')} dt'
\end{equation}
The statistical properties of $h(t)$ can be calculated from this equation.
For the mean value we have
\begin{eqnarray}
\label{16}
<h(t)>&=&\frac{k_mE_m}{A e^{i \phi}} \int_0^t dt'
\exp[-\gamma_e\bar{y}_1 t'+(1-e^{-\gamma_e t'}) \bar{y}_0]  \nonumber \\
&=&\frac{k_mE_m}{\gamma_e A e^{i \phi}}
\frac{e^{\bar{y}_0}}{\bar{y}_0^{\bar{y}_1}}
[\gamma(\bar{y}_1,\bar{y}_0)-\gamma(\bar{y}_1,\bar{y}_0e^{-\gamma_e t})]
\end{eqnarray}
where $\gamma(a,z)$ denotes the incomplete $\gamma$ function of $a$ up
to $z$ \cite{abra}, the relevant dimensionless parameters are
defined by  
\begin{eqnarray}
\label{8}
\bar{y}_0&=&\frac{g(1+ i \alpha)}{2 \gamma_e A e^{i \phi}}
(N_{on}-N_{th}) \\ \label{9}
\bar{y}_1&=&\frac{1}{A e^{i \phi}} \left[\frac{g(1+ i \alpha)}{2
\gamma_e}(N_{on}-N_0-\frac{\gamma}{g})
+\frac{\kappa e^{-i \omega_s \tau}}{\gamma_e}\right]
-i\frac{\Delta \omega}{\gamma_e}\ \ , 
\end{eqnarray}
and $y_0=Re(\bar{y}_0)=Re(\bar{y}_1)$.
For typical semiconductor laser parameters, $y_0 >> 1$. In this case, we
can neglect the second incomplete gamma function in front of the first
one in eq.(\ref{16}), so that  $h(t)$ becomes a time-independent random
variable $h$, and we can  replace $\gamma(\bar{y}_1,\bar{y}_0)$ by the
Gamma function $\Gamma(\bar{y}_1)$. An expansion for large values of the
argument $\bar{y}_1$ gives the final expression for the mean value of $h$ 
\begin{equation}
\label{6}
<h> = \frac{k_mE_m }{\gamma_e A e^{i \phi}}
e^{\bar{y}_0-\bar{y}_1}\left(\frac{\bar{y}_1}{\bar{y}_0}\right)^{\bar{y}_1}
\sqrt{\frac{2\pi}{\bar{y}_1}}
\end{equation}
The variance of $h(t)$ can be calculated in a similar way. This variance is
given by
\begin{equation}
\sigma^2_h(t)=<|h(t)|^2>-|<h(t)>|^2=\frac{4\beta}{A^2}\int_0^t dt' N(t')
e^{-2 Re Q(t')}
\end{equation}
Under the same approximations used in the calculation of 
$\left<h(t)\right>$, we get the 
final expression
\begin{equation}
\label{7}
\sigma^2_h=\frac{4 \beta}{g[1+\kappa\tau(\cos(\omega_s\tau)-\alpha 
\sin(\omega_s\tau))]}+\frac{4 \beta N_{th}}{A^2 \gamma_e}
\sqrt{\frac{2\pi}{y_0}}
\end{equation}
The turn-on time $T$ is defined as the time the optical intensity takes
to reach a reference intensity $S_r$. It is a random
variable depending on the spontaneous emission events triggering
the switch-on. 
From its definition we have $S_r=|h|^2 e^{2 Re \{Q(T-\bar{t})\}}$, 
which yields $T$ as a function 
of the random variable $h$. Then the statistical properties of $T$
can be obtained from those of $h$ (Eqs. (\ref{6},\ref{7})). For the 
times involved $\gamma_e(T-\bar{t}) << 1$, so that 
%\begin{equation}
$T=\bar{t}+\gamma_e^{-1}[ 
{2}{y_0}^{-1}\ln\left({S_r}/{|h|^2}\right) ]^{1/2}$. 
%\end{equation}
We calculate the statistical properties of $T$ through
the generating function $W(\rho)$. Following \cite{salvador} and using
(\ref{6}), (\ref{7}), we get  
\begin{eqnarray}
W(\rho)=<e^{-\rho (T-\bar{t})}>&\approx&\exp\left[-\frac{\rho}{\gamma_e}
y_1-\delta\right]\Gamma\left(1+\frac{\rho}{\gamma_e y_0 y_1}\right) \nonumber 
\times \\
&\times&M\left[\left(1+\frac{\rho}{\gamma_e y_0 y_1}\right)+1,1,\delta\right]
\end{eqnarray}
where $M(a,b,z)$ is the confluent hypergeometric function \cite{abra} and
\begin{equation}
y_1=\sqrt{\frac{2}{y_0} \ln\left(\frac{S_r}{\sigma^2_h}\right)}
\end{equation}
From this generating function we get
\begin{eqnarray}
\label{133}
<T>&=&\bar{t}+\left[-\frac{d}{d\rho}\ln W(\rho)\right]_{\rho=0} =
 \nonumber \\ &=& \bar{t}+ \frac{y_1}{\gamma_e}
-\frac{1}{\gamma_e y_0 y_1}[E_1(\delta)+\ln(\delta)]
\end{eqnarray}
where $\delta=|<h>|^2/\sigma^2_h$ is the natural scaling parameter which
appears in the calculations and is a measure of the relative strength of
the injected field. $E_1(z)$ is the integral exponential of order 1 and
$\psi(z)$ is the digamma function \cite{abra}. TJ is
defined as the variance of $T$. From the generating function:  
\begin{eqnarray}
\label{13bis}
\sigma^2_t&=&\left[\frac{d^2}{d\rho^2}\ln W(\rho)\right]_{\rho=0} = \nonumber \\
&=&\frac{1}{(\gamma_e y_0 y_1)^2}\left[\psi'(1)-[E_1(\delta)-\psi(1)
+\ln(\delta)]^2
+2\sum_{n=2}^{\infty}\frac{(-\delta)^n}{n!n}\sum_{i=1}^{n-1}\frac{1}{i}\right]
\end{eqnarray}

We have checked the validity of the expressions (\ref{133}),(\ref{13bis}) by
numerically solving (\ref{1}),(\ref{2}) 
%.
%The stochastic rate equations have been solved by means of a first-order
%Euler algorithm with a time step of 0.02 ps 
and performing averages over
$10^4$ turn-on events. The reference value $S_r$ has been chosen as a 
50 \% of the optical intensity value for the free-running SL in the on-state.
%Random numbers were generated by using a very fast time consuming 
%algorithm \cite{raul}. 
The results are plotted in figures (1) and (2) for an external field $k_m
E_m =$ 3 ps$^{-1}$ and external cavity lengths of around 100 $\mu m$.
$\kappa= 
%\frac{\sqrt{R_{ext}} (1-r)}{\tau_L\sqrt{r}}=
0.025$ 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.
The laser parameters were taken as: $g=5.6 \times
10^{-8}$ ps$^{-1}$, $\gamma=0.4$ ps$^{-1}$, $\gamma_e=5 \times 10^{-4}$ 
ps$^{-1}$ and $\beta=1.1 \times 10^{-8}$ ps$^{-1}$. 

In figure 1 we plot the dependence of SOT and TJ on the detuning between
the ML and the SL for two values (near 100 $\mu$m) of $L_{ext}$, 
showing the presence of a value of the detuning which
optimizes the speed of the laser. The main effect of changing the 
external cavity length \cite{CLEO} is that the curves, and then this 
optimum detuning,  
oscillate between the two curves presented in the figure. 
As a reference, the curve without feedback is also included.

In figure 2 the SOT and TJ are plotted vs. external cavity length,
around $L_{ext} \sim 100$ $\mu$m, for a detuning of 88 GHz. The detuned
external field can efficiently produce a reduction of SOT and TJ at
selected positions of the external reflector. The analytical calculations
are able to predict this effect.

In both figures, a good agreement is found between theory and numerical
simulation when calculating the SOT. The TJ behavior is correctly
described in the range of external cavity lengths and detunings that give a
maximum reduction of its value, and qualitatively agrees in the whole range. 
In diode modules it is expected that the external cavity length providing the 
optical feedback is fixed. In this case a fine tuning between the ML and the SL 
could yield a reduction of the TJ up to $\sim$ 40\% its value in the absence of
the external light, even for a frequency mismatch between both lasers
as large as $\sim$ 100 GHz. 

In summary, we have developed an analytical approach to calculate
switch-on time and timing jitter in a single-mode semiconductor laser
subjected to both optical feedback and external light injection. This 
approach, developed for external cavity lengths up to some hundreds of
microns, yields expressions that are in very good agreement with numerical
simulations of the stochastic rate equations. The main result is that
in a region of large frequency mismatch (of the order of 100 GHz) 
between the master and
slave laser, where external light injection alone is inefficient to 
improve the slave laser behavior, timing jitter can still be reduced if
optical feedback is present. 

\vskip 0.5cm
\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, Human Capital and Mobility Program of the European 
Union, Ref. CHRX-CT93-0331, 
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 

\begin{thebibliography}{99}

\bibitem{spano}P. Spano, A. Mecozzi, A. Sapia and A. D'Ottavi, 
%``Noise and Transient Dynamics in Semiconductor Lasers", 
in {\sl Third 
International Workshop on Non-Linear Dynamics and Quantum 
Phenomena in Optical Systems}, ed. by R.Vilaseca and R.Corbalan,
Springer-Verlag, pp. 259-292, 1991, and references therein.

%\noindent
\bibitem{salvador1} S. Balle, N. B. Abraham, P. Colet 
and M. San Miguel, IEEE J. Quantum Electron., {\bf 29}, 33, (1993).

%\noindent
\bibitem{agra_fib} G. P. Agrawal, {\it Nonlinear Fiber Optics}. San Diego:  
Academic Press, (1989).

%\noindent
\bibitem{stefan} S. Mohrdiek, H. Burkhard and H. Walter,
J. Lightwave Tech. {\bf 12}, 418, (1994).

%\noindent
\bibitem{alan}
L. N. Langley and K. A. Shore,
IEEE J. Lightwave Technology {\bf 11}, 434, (1993).

%\noindent
\bibitem{ec} E. Hern\'andez-Garc\'{\i}a, C. R. Mirasso, K. A. Shore
and M. San Miguel,
IEEE J. Quantum Electron. {\bf 30}, 241,  (1994);
C. R. Mirasso and E. Hern\'{a}ndez-Garc\'{\i}a,
IEEE J. Quantum Electron. {\bf 30}, 2281 (1994).

%\noindent
\bibitem{CLEO} C. R. Mirasso, E. Hern\'andez-Garc\'{\i}a, J. Dellunde,
J. M. Sancho and M. C. Torrent, {\sl Technical Digest of Conference on Laser 
and Electro-Optics/Europe}, p. 159, (1994); J. Dellunde, C.R. Mirasso,
M.C. Torrent, J.M. Sancho, and E. Hern\'andez-Garc\'\i a (to appear in
Opt. and Quantum Electron.).

%\noindent
\bibitem{agra_laser} G. P. Agrawal and N. K. Dutta, {\sl Long Wavelength 
Semiconductor Lasers}, Van Nostrand Reinhold Company, New York, 
(1986).

%\noindent
\bibitem{pere} S. Balle, P. Colet and M. San Miguel, 
%``Statistics for the Transient Response of Single-Mode Semiconductor Laser
%Gain Switching",
Phys. Rev. A {\bf 43}, 498, (1991).

%\noindent
\bibitem{abra}
{\it Handbook of Mathematical Functions}, edited by M. Abramowitz and I.A.
 Stegun
 (Dover, New York, 1972).

%\noindent
\bibitem{salvador} M. C. Torrent, S. Balle, M. San Miguel and J. M. Sancho,
%``Detection of a Weak External Signal Via the Switch-on Time Statistics
%of a Semiconductor Laser", 
 Phys. Rev. A {\bf 47}, 3390, (1993).
\end{thebibliography}

\newpage
\begin{center}
{\bf \Large Figure Captions}
\end{center}
\vspace{0.5cm}

\noindent
{\bf Fig.1}: Mean switch-on time and time jitter vs.
frequency mismatch (points are simulation results and lines are from theory).
We plot the values without feedback (solid line,
circles) and compare with the cases of feedback with
external 
%round-trip times of 0.6666 ps 
cavity length of 99.99 $\mu$m (dotted line,
stars) and  
%0.6689 ps 
100.34 $\mu$m (dashed line, triangles).
\vspace{0.5cm}

\noindent
{\bf Fig.2}: Mean switch-on time and timing jitter vs.
external cavity length. We plot the simulation points
(stars) and the theory (solid line) for a detuning
of 88 GHz.
% and compare with the simulation points without light injection (triangles).

%\input figures

\end{document}