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\title{EFFECT OF PHASE--CONJUGATE OPTICAL FEEDBACK 
ON TURN--ON JITTER IN LASER DIODES}

\author{J. Revuelta, L. Pesquera}

\address{Instituto de F\'\i sica de Cantabria, CSIC--UC, and
Departamento de F\'\i sica Moderna, 
Universidad de Cantabria, E-39005 Santander, Spain.}

\author{E. Hern\'andez-Garc\'\i a and Claudio R. Mirasso$^*$ }

\address{Departamento de F\'\i sica, Universitat de les Illes Balears, 
E-07071 Palma de Mallorca, Spain}


\maketitle
\begin{abstract}
In this work the turn--on delay jitter of single--mode semiconductor
lasers subjected to  optical feedback from a phase--conjugate mirror 
in short external cavities is studied.
We develop a theory, validated by numerical simulations, to obtain
the turn--on time distribution. It is shown that the turn--on time 
statistics is very sensitive to the linewidth enhancement factor.
\end{abstract}

It is well known that the performance of  a laser diode (LD) is 
extremely sensitive to external optical feedback.\cite{ref1,ref2}
% A general classification of the effect of feedback on 
%laser diode (LD) properties under CW operation has been reported.\cite{ref2}
The switch--on dynamics of LD subjected to optical 
feedback has been recently considered \cite{ref5}, finding that
the turn--on delay jitter is extremely sensitive to the reflector 
location. 
This is due to the phase shift acquired in the 
external cavity, that can be 
eliminated if the feedback occurs from a phase--conjugate mirror (PCM),
 phase--conjugate feedback (PCF), since the phase of the returned light 
is reversed during reflection. 
Therefore there are significant differences between the behavior of LD 
with PCF and conventional optical feedback (COF). 
Analysis of the stability \cite{ref7,ref8,ref9} 
of LD in the presence of PCF have been reported.
Concerning the noise characteristics, it has been shown 
\cite{ref9,ref11} that 
for weak values of PCF the intensity and frequency noise are 
reduced at low frequencies.
The objective of this Letter is to investigate the effect of PCF on the 
turn--on time statistics of LD. A theory, validated by numerical 
simulations, is developed to obtain the turn--on time probability 
density $P(T)$. Our results show that, unlike the case with COF,\cite{ref5} 
the turn--on time is not very sensitive to small 
variations in the position of the PCM on the optical wavelength scales.
This property, due to the lack of the external roundtrip phase change, 
is the main advantage of PCF compared to COF.
However, the effective laser threshold is dependent upon the phase that 
changes due to chirping. Therefore $P(T)$
becomes sensitive to the value of the linewidth enhancement factor 
$\alpha$. Morever, when $\alpha \ne 0$ the stability range of a LD
subjected to PCF is reduced.\cite{ref7,ref9} This is the main disadvantage
of PCF compared with COF.

Here we consider a degenerate four--wave mixing 
PCM in a fast--responding nonlinear medium\cite{ref9}
 with a time response smaller than 1 ps. 
Our analysis is based on noise--driven rate equations for the optical 
field $E=E_1+i E_2=\sqrt I \exp (i\phi)$ ($E_i$ being the field components)
and carrier number $N$. 
In the presence of PCF, these equations can be 
written as (assuming single--mode operation) 
\cite{ref7,ref8,ref9}
\begin{equation}
\dot E  =  q(t) E(t)
 + \kappa E^*(t-\tau )+\sqrt{2\beta N(t)} \xi (t), \label{eq1}
\end{equation}
\begin{equation}
\dot N =C(t) -N(t)\gamma_e - G(t)I(t) , \label{eq2}
\end{equation}
where $q=q_r+ iq_i= (1-i\alpha) (G-\gamma)/2$, $G=g(N-N_0)/\sqrt{1+ sI}$ is 
the optical gain, $g=5.6 \times 10^4$ s$^{-1}$ is the gain rate per 
carrier, $\gamma =4 \times 10^{11}$ s$^{-1}$ and 
$\gamma_e =5 \times 10^8$ s$^{-1}$ are the inverse photon and carrier
lifetimes, $N_0 =6.8 \times 10^7$ is the carrier number at transparency,
$s=1.2 \times 10^{-6}$ is the nonlinear--gain parameter, and $\beta= 
1.1 \times 10^4$ s$^{-1}$ and $\xi(t)$ are the spontaneous emission rate
and noise, respectively.
The feedback parameters are the feedback rate 
\cite{ref2} $\kappa$ and the external cavity roundtrip delay $\tau$.

We consider a situation of repetitive gain--switching from stationary 
initial conditions determined by the bias level.
The injection current $C(t)$ is switched from a value $C_b= 0.9\, 
C_{th}$ below the threshold 
current, $C_{th}=3.76 \times 10^{16}$ s$^{-1}$,
 to a value $C_{on}=3.5\, C_{th}$ above it.
The turn--on time $T$ is defined as the time at which $\mid E\mid^2 =I_r$, 
where $I_r=3.04 \times 10^5$ corresponds to 13\% of the steady--state 
intensity in the absence of feedback.
In the initial stage of evolution the intensity is small, so the term 
$GI$ in (\ref{eq2}) and the non--linear gain saturation are 
negligible. Within this approximation $G(t)$ can be obtained 
\cite{ref5} by solving 
(\ref{eq2}). For short external cavities such that $\kappa \tau$ is small 
Eq. (1) can be approximated at early times by
\begin{eqnarray}
\dot E  =&& \bigl( q-\kappa^2 \tau \exp ({-2 \tau q_r  }) \bigr) E (t) 
+\kappa \exp ({-\tau q^* }) E^* (t) \nonumber\\
&&+\sqrt {2\beta N} \bigl( \xi (t)- \kappa \tau \exp ({-\tau q^* }) 
\xi^* (t) \bigr) . \label{eq3}
\end{eqnarray}
From the corresponding Eq. for $I$ the carrier number needed for 
the beginning of amplification (that is the effective 
threshold)  is given to first order in $\tau$ by
\begin{eqnarray}
N_{th}^{eff} =N_{th} -{{2\kappa}\over g} \bigl[ \cos 2\phi +\kappa \tau 
\sin 2\phi (\alpha \cos 2\phi -\sin 2\phi) \bigr], \nonumber
\end{eqnarray}
where $N_{th}=N_0+\gamma/g$. Then $N_{th}^{eff}$ is phase--dependent.
If the reinjected field is in phase with the field in the cavity the 
threshold is reduced. The reverse effect occurs in the phase opposition 
case. When $\alpha \ne 0$ the instantaneous frequency of the LD changes
with time\cite{ref1} (chirping). 
This leads to phase oscillations that are followed
by $N_{th}^{eff}$ (see Fig. 1). When the 
linewidth enhancement factor $\alpha$ is small (quantum--well lasers) 
the phase and $N_{th}^{eff}$ are nearly constant in time. In the 
limiting case $\alpha=0$ the  field components $E_1$ and  $E_2$ 
decouple, having  $E_i$ a threshold given by
the minimum and maximum values of $N_{th}^{eff}$, 
$N_{thi}=N_{th}+ (-1)^i 2\kappa/g$. 

Our approach consists in calculating the turn--on time distribution 
$P(T)$ from $P(T)=-\int_0^{I_r} \dot f(I,T) \, dI$, where $\dot f(I,T)$
is the time derivative of the intensity distribution. $f(I,T)$
is easily obtained from the field distribution, which is Gaussian of
moments, $x_i= \langle E_i^2 \rangle$ and 
$x_{12}= \langle E_1 E_2 \rangle$. The evaluation of the field moments 
from (\ref{eq3}) is in general cumbersome. We give here some explicit 
expressions in some cases with $\alpha=0$ and the results in other 
cases are displayed in the figures.
When $\alpha=0$, $x_{12}=0$. If in addition $\kappa$ 
is small the two field components contribute to the laser switch--on
and we get
\begin{equation}
x_i (t) = 2\beta_i \sqrt {{2\pi} \over {a_i}} N_{thi} \exp \Bigl( 
{{a_i} \over 2} (t-t_{thi})^2 \Bigr), \label{eq5}
\end{equation}
where $a_i = g(C_{on} -C_{th})/ (1-(-1)^i \kappa \tau)$, $\beta_i = 
\beta /(1-(-1)^i \kappa \tau)^2$ and $t_{thi}$ is the time needed 
for the carrier number to reach the effective threshold $N_{thi}$.
When $\kappa$ increases only the component with the minimum threshold 
contributes to the laser switch--on. Then $x_1$ is still given by 
(\ref{eq5}), but $x_2$ is negligible. In this case the following 
expression is obtained for the turn--on distribution,
\begin{eqnarray}
P(T) =&& \sqrt{{I_r}\over {2\pi b_1}} a_1 (T-t_{th1}) \exp \biggl(-{a_1 
\over 4} (T-t_{th1})^2 \biggr) \nonumber\\
&&\times \exp \biggl[ -{{I_r}\over {2b_1}} 
\exp \biggl( -{a_1 \over 2} (T-t_{th1})^2 \biggr) \biggr], \label{eq6}
\end{eqnarray}
where $b_1 = 2\beta_1 \sqrt {{2\pi} / {a_1}} N_{th1} $.

For values of $\alpha \ne 0$ typical of bulk LD $P(T)$ changes from a 
single--maximum distribution to a multimodal one by increasing $\kappa$
(see Fig. 1). 
The maxima and minima of the turn-on time distribution correspond to the 
effective threshold oscillations due to the 
chirping. Our theory reproduces well $P(T)$ for short external cavities. 

We have obtained the mean turn--on time $\langle T\rangle$ and its 
standard deviation (jitter) $\sigma_T$ when $P(T)$ has a 
single maximum, i. e. when  $\kappa$ and/or  $\alpha$ are small.
The gross magnitude of $\langle T\rangle$  in Fig. 2 is mainly due to
the time spent by the carriers to reach threshold (see Fig. 1).
We consider first the case with small $\kappa$ for different values 
of $\alpha$. Since $T$ is defined in terms of the intensity, all results
are symmetric with respect to $\alpha$ changing sign.
 When $\alpha $ increases
$\langle T\rangle$ is found to increase, whereas the jitter decreases
(see Fig. 2). As discussed above when $\alpha $ is small 
the two field components $E_i$ have thresholds
given by the maximum and minimum values of $N_{th}^{eff}$.
The turn--on time is smaller than the one in the absence of feedback due to 
the component with the minimum threshold. However, the jitter increases
with the PCF because of the different turn--on times for photons with 
different phases. 
When $\alpha$ increases the effective threshold oscillates in time with 
the phase, and these differences are washed
out. When the frequency of these oscillations is large, 
values of $\langle T\rangle$ and $\sigma_T$ close to those without PCF are 
recovered.
The same behavior is obtained when the external cavity length increases, 
due to the decrease of the number of the feedback photons that 
contribute to the switch--on. When $\alpha=0$ the jitter decrease with 
$\tau$ corresponds to a reduction of the difference between the field 
components in (\ref{eq5}). 
We note that in contrast with the results for conventional feedback 
\cite{ref5} 
a non--oscillatory behavior with $\tau$ is observed, due to the lack of 
the external roundtrip phase change.

We now consider the low chirping case ($\alpha $ small).
In the limiting case $\alpha=0$ the 
theory simplifies and $P(T)$ can be obtained from (\ref{eq5}) 
for low $\kappa$ and it is given by (\ref{eq6}) for large $\kappa$.
The turn-on time decreases with $\kappa$
since the minimum value of $N_{th}^{eff}$ decreases. 
Fig. 3 shows that for  $\alpha=0$ and small $\tau$ the jitter increases with 
$\kappa$ until a nearly constant value is reached.
When $\kappa$ increases the range of phases of the spontaneously emitted
photons that 
contribute to the switch--on is reduced. Then the intensity noise and 
the jitter increase until a constant value is reached when only photons 
 with the minimum threshold contribute.
It is shown in Fig. 3 that the one--component theory (\ref{eq6}) is valid when 
$\kappa$ is greater than 0.04 ps$^{-1}$.
As discussed above $\sigma_T$ is reduced when $\alpha$ increases  
for small $\kappa$. However, for large $\kappa$ the behavior is the 
opposite, since the photons have the same phase and
the phase evolution due to $\alpha$ increases the effective 
threshold and $\sigma_T$. The differences due to 
$\alpha$ are smaller for large enough $\kappa$, since the phase tends to 
be fixed by the PCF.
The behavior of the jitter with respect to the external cavity roundtrip 
delay is also different for small and large 
$\kappa$ (see Figs. 2 and 3).
When the feedback is weak $\sigma_T$ decreases with $\tau$.
For strong feedback the photon phase is fixed and
only one component contributes to the jitter. The photon rate
amplification given by $a_1$ decreases with 
$\tau$, resulting in a greater value of $\sigma_T$.
When $\tau$ is greater than 10 ps the theory is not valid and $\sigma_T$ 
decreases with $\tau$ to the value without PCF, since the contribution 
of feedback photons can be neglected.


The research is supported by EEC CHRX--CT94--0594 project and by
CICYT (Spain) project TIC93--0744.

$^*$Present address, Instituto de Estructura de la Materia, CSIC, Serrano 123, 
E-28006 Madrid, Spain.


\begin{references}

\bibitem{ref1} G. P. Agrawal and N. K. Dutta, {\it Long--Wavelenth 
Semiconductor Lasers} (Van Nostrand Reinhold, New York, 1986).

\bibitem{ref2} K. Petermann, {\it Laser Diode Modulation and Noise}
(Kluwer Academics, Dordrecht, The Netherlands, 1988).


\bibitem{ref5} E. Hern\'andez--Garc\'\i a, C. R. Mirasso, K. A. Shore, and M. 
San Miguel, IEEE J. Quantum Electron. {\bf 30}, 241 (1994).

\bibitem{ref7} G. P. Agrawal and J. T. Klaus, Opt. Lett. {\bf 16}, 1325 
(1991).

\bibitem{ref8} G. H. M. van Tartwijk, H. J. C. van der Linden, and D. 
Lenstra, Opt. Lett. {\bf 17}, 1590 (1992).

\bibitem{ref9} G. P. Agrawal and G. R. Gray, Phys. Rev. A {\bf 46}, 5890 
(1992).


\bibitem{ref11} L. N. Langley and K. A. Shore, Opt. Lett. {\bf 18}, 1432 
(1993).

\end{references}

\begin{figure}
\caption{ Carrier number (a) dotted line). Effective threshold in a) 
and turn-on time distribution in b) correspond to numerical simulations
for $\alpha =0$ (dashed lines) and $\alpha=5.5$ (solid lines).
Dash--dotted lines in b) correspond to the theory.
Feedback parameter values: $\kappa = 0.1$ ps$^{-1}$ and $\tau =1$ ps.
\label{fig1}}
\end{figure}

\begin{figure}
\caption{Turn-on time $<T>$ and its jitter $\sigma_T$ vs. $\alpha$ 
for $\kappa = 0.02$ ps$^{-1}$, $\tau =1$ ps 
(stars) and 5 ps (diamonds)
 from numerical simulations (symbols) and theory (solid lines).
 The arrows show the values without feedback.
\label{fig2}}
\end{figure}

\begin{figure}
\caption{Turn--on jitter vs. $\kappa$ ($\tau$= 1 ps) and 
$\tau$ ($\kappa =0.1$ ps$^{-1}$) for two values of $\alpha$: 0 (stars) 
and 1 (diamonds). Solid lines correspond to the theory and dashed line
to Eq. (5).
\label{fig3}}
\end{figure}

\end{document}