%\documentstyle[12pt,epsf]{article} % comment out for twocolumn \documentstyle[12pt]{article} % comment out for twocolumn \textheight 21cm % comment out for twocolumn \textwidth 14.5cm % comment out for twocolumn \oddsidemargin 0.96cm % margins on all sides of 3.5 cm, \evensidemargin 0.96cm % correcting 1 inch default on left-hand \topmargin -0.31cm % side, and 1.5 inch default on top. \columnsep 0.5in \raggedbottom \newcommand{\BE}{\begin{equation}} \newcommand{\EE}{\end{equation}} % construct "greater than or of the order of" \gto % construct ".... and less than or of the order of" \lto \newcommand {\gto } {\,\vcenter{\hbox{$\buildrel\textstyle>\over\sim$}}\,} \newcommand {\lto } {\,\vcenter{\hbox{$\buildrel\textstyle<\over\sim$}}\,} % \begin{document} \baselineskip 24pt % comment out for twocolumn \setcounter{page}{0} \thispagestyle{empty} \ \\ \vfill \ \begin{center} {\Large \bf Current Modulation and Transient Dynamics of Single-Mode Semiconductor Lasers under different Feedback Conditions} \bigskip \vfill Claudio R. Mirasso$^1$, Emilio Hern\'{a}ndez-Garc\'{\i}a$^1$, Jaume Dellunde$^2$, \\ M.C. Torrent$^3$, and J.M. Sancho$^2$ \bigskip \end{center} \vfill \noindent $^1$ Departament de F\'\i sica, Universitat de les Illes Balears, E-07071 Palma de Mallorca, Spain. \noindent $^2$ Departament d'Estructura i Constituents de la Mat\`eria, Universitat de Barcelona, Diagonal 647, E-08028, Barcelona, Spain. \noindent $^3$ Departament de F\'{\i}sica i Enginyeria Nuclear, EUETIT, Universitat Polit\`{e}cnica de Catalunya, Colom 1, E-08222, Terrassa, Spain. \vfill \newpage \vfill \begin{center} {\Large \bf Current Modulation and Transient Dynamics of Single-Mode Semiconductor Lasers under different Feedback Conditions} \bigskip Claudio R. Mirasso$^1$, Emilio Hern\'{a}ndez-Garc\'{\i}a$^1$, Jaume Dellunde$^2$, \\ M.C. Torrent$^3$, and J.M. Sancho$^2$ \bigskip \end{center} \noindent $^1$ Departament de F\'\i sica, Universitat de les Illes Balears, E-07071 Palma de Mallorca, Spain. \noindent $^2$ Departament d'Estructura i Constituents de la Mat\`eria, Universitat de Barcelona, Diagonal 647, E-08028, Barcelona, Spain. \noindent $^3$ Departament de F\'{\i}sica i Enginyeria Nuclear, EUETIT, Universitat Polit\`{e}cnica de Catalunya, Colom 1, E-08222, Terrassa, Spain. \vfill \begin{center} {\Large \bf Abstract} \end{center} We study numerically the response of a single mode semiconductor laser under modulation of the injection current in the presence of optical feedback, for different external cavity lengths and feedback reflectivities. For an external reflectivity larger than $R_{ext} \approx 1.6 \times 10^{-4}$ feedback degrades considerably the statistical properties of optical pulses. When the external round trip time is equal to the period of the modulation of the laser the optical pulses differ widely from each other and then they are not adequate for use in optical communication systems. In almost no case we obtain better pulses as compared with the ones in the absence of feedback. As happens in the solitary laser case, pattern effects, due to random sequences of inputs ``0" and ``1" bits, play an important role. \newpage \section*{1. Introduction} Statistical properties of optical pulses like turn-on time (TOT), pulse width (PW), maximum output power (MOP), etc. and their standard deviations (STD), are of considerable importance for practical applications of semiconductor lasers when working at GHz rates. Some of these quantities, specially the turn-on time jitter (TOJ), defined as the standard deviation of the turn-on time, have been extensively studied both experimentally and numerically \cite{SpanoPT}-\cite{shen}. Most recently the statistical properties of optical pulses on bias level and modulation frequency of the injection current have been considered \cite{Claudio} showing that biasing below threshold can be advantageous to reduce the pulse fluctuations in a situation of signal transmission at high speed (Gbit/sec). The connection of pattern effects with the randomness of the turn-on time have also been evidentiated \cite{shen}\cite{Pere}. Optical feedback is an important and well-known effect to take into account when considering the performance of a laser diode in an optical communication system \cite{Petermann,lenstra}. Most of the works including this effect deal with CW operation of the laser. It has been shown that a small amount of feedback is useful for linewidth reduction, but feedback intensities likely to occur in optical communication systems degrade its performance through the occurrence of the ``coherence collapse'' \cite{lenstra2,wang} giving rise to linewidths of several GHz and a chaotic intensity signal \cite{elsasser}. Recent experimental results using feedback from short external cavities show that the relaxation-oscillation side-bands can be suppressed and that the linewidth can be strongly reduced in the CW regime when using an appropriate value of the external feedback \cite{simonsen}. A general classification of the effect of feedback on spectral and dynamic properties of semiconductor lasers has been reported \cite{Petermann,14jap}. However, few studies are available of external cavity lasers under direct-modulation conditions or in the transient regime following a gain-switching \cite{clarke}-\cite{1}. Only recently the degradation in the performance for a digital intensity modulated direct detection system caused by optical feedback has been considered in some detail \cite{clarke}-\cite{ce}. In this context the turn-on dynamics and associated jitter properties of semiconductor lasers affected by optical feedback have been examined. The characterization of optical pulses of semiconductor lasers in the presence of optical feedback seems a worthwhile task given its practical implications in optical communication systems. In this paper we perform an extensive numerical analysis of the stochastic rate equations to characterize output optical pulses. For a fixed value of the modulation frequency of the laser ($\sim$ 4 GHz) we consider three different bias currents ($C_b$), $C_b=0.98 \, C_{th}$, $C_b= C_{th}$, and $C_b=1.1 \, C_{th}$, where $C_{th}=6$ mA is the threshold current of the laser. We also vary the position of the external reflector between 3 mm and 45 mm. This range of distances is intermediate between longer \cite{clarke}-\cite{3} and shorter \cite{ce} values considered in previous studies, and it is expected to be found in integrated optical devices, where unwanted reflections coming from lenses, pigtails, amplifiers, etc. are present. We characterize the optical pulses through the mean TOT, the mean PW, the MOP and their respective STD. The outline of the paper is as follows. In section 2 we describe the rate equations and laser parameters. In section 3 we present and discuss the results, and finally in section 4 some general conclusions are summarized. \section*{2. Rate equations and dynamical model} We consider a situation in which a single-mode semiconductor laser is weakly coupled to an external cavity of lenght $L_{ext}$ (round-trip time $\tau=2 L_{ext}/c)$ . The delay-differential equations appropriate to describe this configuration are the Lang-Kobayashi equations \cite{LK} which describe the coupled time evolution of the complex amplitude of the electric field $E$ of the laser (in the slowly varying envelope approximation) and the carrier number $N$ inside the laser cavity. Supplemented with Langevin noise terms \cite{Petermann} these equations can be written as: \begin{eqnarray} \label{field} \dot E(t) &=& {1 + i \alpha \over 2} \left[ G - \gamma \right] E(t) + K(\tau) E(t-\tau)+\sqrt{2\beta N(t)} \xi (t) \; , \\ \label{N} \dot N &=& {C(t) \over e} - \gamma_e N(t) - G |E(t)|^2 \; , \\ \label{G} G &=& \frac{g (N(t) - N_0)}{\sqrt{1 + s |E(t)|^2}} \end{eqnarray} The equation for the complex field $E$ is written in the frame of reference in which the electric field is constant except for phase diffusion when the laser is ``on" and in CW operation in the absence of feedback. The meaning and values used for numerical calculations of the parameters in Eqs.(\ref{field}-\ref{G}) are given in Table 1. Spontaneous emission noise is modeled through the last term in (\ref{field}). The complex random process $\xi(t)$ is taken to be Gaussian of zero mean and correlations given by $\left< \xi(t) \xi^*(t') \right> = 2 \delta (t-t')$. $K(\tau)$ in (\ref{field}) is given by \BE \label{kappa} K(\tau)=\kappa e^{-i \omega_0 \tau}\ , \EE where the feedback coupling parameter $\kappa= \frac{\sqrt{R_{ext}} (1-r)}{\tau_L\sqrt{r}}$. $R_{ext} \approx 1.6 \times 10^{-2}, \, 4 \times 10^{-3}$, and $1.6 \times 10^{-4}$ will be the values taken for the power external reflectivity, $r = 0.32$ is the power laser-facet reflectivity and $\tau_L = 6$ ps is the laser round trip time. We consider here values of $\kappa$ appropriate for weak external feedback situations while we will leave the delay feedback time $\tau$ measuring the length of the external cavity as a free parameter of our study. $C(t)$ is the injection current. We take it to follow for a bit ``1'' a square-wave form of maximum value $C_{on}=21.5$ mA during a time $t_{on}=90$ ps and a minimum value $C_b=5.88, \, 6 \, \rm{or} \, 6.6$ mA during $t_{off}=150$ ps ($T=t_{on}+t_{off}$ is the period of the modulation). For a ``0'' bit $C(t)=C_b$ during the whole period $T$ (IM/DD RZ scheme). The values of $C_{on}$ and $t_{on}$ are taken as in ref. \cite{Claudio}. Concerning the bias current, it is well known that a value not far from the threshold value is appropriate to avoid pattern effects and pulse superposition at the output \cite{shen}-\cite{Pere}. We consider two types of modulations: Periodic Modulation (PM, a periodic sequence of all ``1" bits) and Pseudorandom Word Modulation (PRWM, a random sequence of ``1" and ``0" bits). \section*{3. Numerical Results} The intention of the present paper is to characterize the pulse statistics during the transient dynamics of the external cavity laser in a relatively simple form. We will put special emphasis not only on the statistical properties of the optical pulses but also on their shape, since this aspect is very important for pulse propagation in optical fibers. As will be shown latter the pulse shape can be strongly distorted for some positions of the external reflector yielding a very poor on-off ratio which would not be able to propagate along the fiber. Numerical simulations are performed using a stochastic Euler algorithm, with a time step of 0.01 $ps$ and a Gaussian random number generator \cite{raul} to simulate spontaneous emission white noise. The values of the external cavity length are varied by steps of 0.6 $mm$. For fixed $L_{ext}$ we perform averages over 5000 pulses in the PM case and 10000 pulses in the PRWM case ($\sim$ 5000 ``1" bits in the sequence). We considered three different reference levels to define the switch-on time: 0.5, 0.75 and 1 times the output power under CW operation of the laser ($P_{CW} \approx 5.1$ mW). The reference level is important since for some positions of the external reflector and reflectivity values the degradation of the pulse shape can make some pulses not to cross all the reference levels considered here. For the range of external cavity lengths $L_{ext}$ considered here ($3 {\rm mm} \le L_{ext} \le 45 {\rm mm}$, i.e. $20 {\rm ps} \le \tau \le 300 {\rm p}s$) an interesting value of $\tau$ seems, {\sl a priori}, to be the one which coincides with the modulation period of the injection current, $T=240\ ps$. It is expected that when $\tau \approx T$ the reinjected field interacts with the internal electrical field in the moment when the next optical pulse is being emitted (at the turn-on time). Due to spontaneous emission the turn-on time fluctuates around a mean value, i.e. sometimes the reinjected light will strongly affect the pulse and sometime it will not, so that large fluctuations of the statistical properties of the optical pulses are expected when $\tau$ is close to $T$. We begin with the PM case. In fig. 1 and 2 we plot the turn-on time, the mean pulse width and the maximum output power, and their respective STD, for bias current $C_b=C_{th}$ and $C_b=1.1 \, C_{th}$ respectively, and $R_{ext} \approx 1.6 \times 10^{-2}, \, 4 \times 10^{-3}$ and $1.6 \times 10^{-4}$, as a function of the round trip delay time from 20 to 300 ps for two reference levels: $P_{ref} = 0.5 \, P_{CW}$ and $P_{ref} = P_{CW}$. We have observed for the other value of the bias current mentioned above the same qualitative, and a very similar quantitative, behaviour as the one for $C_b=C_{th}$. As expected, the larger the reflection of the external mirror the stronger its effect on the statistical properties of the optical pulses. For all the values of the reflection coefficient considered here the mean TOT and the MOP decrease as compared with the case without feedback (represented by symbols in the figures), while the mean PW increases, except for $\tau$'s around 180 ps. For these $\tau$'s the maximum output power is very low and the fluctuations in the width are as large as their mean values, implying that some pulses have problems to reach the reference levels. In figs. 1 and 2 a) and f) it can be clearly seen that the STD of the turn-on time and the maximum output power presents peaks at $\tau \approx 80$ and 240 ps. The second $\tau$ coincides with the period of modulation of the injection current. As described above the fluctuations in the turn-on time make the reinjected pulses reenter the cavity at a time close to $T$ but not exactly its value. As soon as the reinjected field interacts with the carriers, which for $\tau \sim T \approx 240\ ps$ are around its maximum value, they recombine and the laser emits a new optical pulse, yielding different turn-on times (and also different maximum output powers) for different pulses and a large time jitter (and standard deviation of $P_{max}$). This large jitter can be also seen in fig. 3 b) where we plot the evolution of the output power for $\tau=T$ for ten different pulses. Interference effects of the reinjected field with the internal field already present in the cavity should in principle be taken into account to complete this argument, but the internal field is very small at the time the carriers reach threshold. Other features of Figs. (1) and (2) can be understood by thinking about optical feedback as a kind of external light injection. The effect of light injection during the switch-on of a single-mode semiconductor laser has been recently studied \cite{carme}. Light injection at the appropriate frequency accelerates the switch-on process and results in a reduction of the mean turn-on time. The largest interaction between the injected and the internal field occurs at times when the net gain is positive but the optical pulse has not yet been emitted. For our pulses, these times range approximately from 15 ps to 50 ps after the beginning of an ``1'' bit in the current. The laser results in being very sensitive to light injection in this critical range of times. The large reduction in the switch-on time observed around 180 ps corresponds to reinjection of light from the maximum of the previous optical pulse at these critical times. For external round-trip times smaller than 90 ps no interaction with the maximum of the previous pulse is expected. At such small reinjection times there is an interaction of a pulse with itself that results in the formation of an adjacent secondary small optical pulse. It can be seen in fig. 3 c) where we plot ten typical pulses for $\tau \sim 80$ ps. For this value of $\tau$ the secondary pulse can reach the critical range of the following pulse and lead to large fluctuations of the switch-on time and the maximum output power. This last effect is not appreciated for low values of the reflectivity, for which the intensity of the secondary pulse is very small. In fig. 3 a) we plot for comparison the output pulses obtained with the same laser parameters but without feedback. As stated above, for $170\, {\rm ps} < \tau < T=240$ ps the turn-on time and the maximum output power are reduced since the reinjected field makes the laser to emit before the carriers reach their natural maximum value without feedback. The smallest fluctuations of the optical pulses are obtained for $100 \, \rm ps < \tau < 150 \, \rm ps$. For $\tau \approx 100$ ps reasonably good values for the mean TOT, mean PW and MOP are obtained. In fig. 4 a) we show the pulses obtained with $R_{ext} \approx 1.6 \times 10^{-2}$, $C_b=C_{th}$, and $\tau=110$ ps. They present a different shape as compared with the ones of Fig. 3 a). Smaller fluctuations of the optical pulses are observed and these pulses could be good enough for propagation in fibers. In Fig. 4 b) it can be seen that for $R_{ext} \approx 1.6 \times 10^{-4}$ the pulses are very similar to the ones without feedback. This indicates that this value of $R_{ext}$ is very close to the minimum value above which the feedback begins to affect the dynamical evolution of the laser. In the case of PRWM Figs. 5 and 6 are different as compared with Figs. 1 and 2. This was expected especially for $C_b=1.1 \, C_{th}$ since pattern effects, due to the random sequence of ``0" and ``1" bits, change the pulse statistics as compared with the PM case \cite{Claudio}. The peaks in the jitter and STD of the MOP observed under PM disappeared for $C_b=C_{th}$. In this case, and for $R_{ext} \approx 1.6 \times 10^{-2}$, at the $\tau$ at which the mean TOT and the MOP reach a minimum, a maximum in their STD appear, while when the mean PW reaches a maximum its STD also reaches a maximum. In the case $C_b=1.1 \, C_{th}$ all the curves seem to be smoothed when comparing with the PM case. In Fig. 7 a) we plot 10 optical pulses for $\tau=170$ ps, $C_b=C_{th}$ and $R_{ext} \approx 1.6 \times 10^{-2}$. Two different kinds of pulses are identified. The widest correspond to a ``1" bit preceded by another ``1" bit, while the narrowest correspond to a bit ``1" preceded by a bit ``0". This behaviour evidentiate again that pattern effects play a very important role in the laser output. These pattern effects also make some pulses inappropriate for optical fiber transmission. Finally in fig.7 b) we plot, as an example, ten output pulses for $C_b=1.1 \, C_{th}$ and $\tau=240$ ps. Still large fluctuations, as compared with the ones of the PM case, appear. \section*{4. Summary and Conclusions} We have studied numerically the response of a single mode semiconductor laser in the presence of feedback for external cavities in the range of 3 to 45 mm. These distances and the level of feedback used in this paper are expected to be present in optical integrated devices, where unwanted reflections coming from lenses, pigtails, amplifiers, etc. are present. The most relevant conclusion that can be extracted from our results is that for a level of external reflection larger than $R_{ext} \approx 1.6 \times 10^{-4}$ feedback effects are very important even when the external round trip time is not in resonance with the period of the modulation. When $\tau$ is in resonance with $T$ the situation is even more delicate since pulses obtained in this condition are far from being the pulses needed for propagation in optical fibers. For some special values of $\tau$ the fluctuations of the statistical properties of the pulses are reduced but they become wider and less intense. In almost no case we obtain better pulses as compared with the ones in the absence of feedback. As happens in the solitary laser, pattern effects due to the random sequences of input ``0" and ``1" bits play an important role in the shape of the output pulses. \section*{Acknowledgments} 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. J.D., M.C.T. and J.M.S. acknowledge the Comisi\'on Interministerial de Ciencia y Tecnolog\'\i a, Project PB90-0030 and Fundaci\'o Catalana per a la Recerca-Centre de Supercomputaci\'o de Catalunya. \newpage \begin{thebibliography}{99} \bibitem{SpanoPT} P. Spano, A. Mecozzi, A. Sapia and A. D'Ottavi, ``Noise and Transient Dynamics in Semiconductor Lasers", in ``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. \bibitem{Botcher} E. H. B\"ottcher, K. Ketterer and D. Bimberg, "Turn-on Delay Time Fluctuations in Gain-Switched AlGaAs/GaAs Multiple-Quantum-Well Lasers", J. Appl. Phys. {\bf 63}, 2469-2471 (1988). \bibitem{weber} A. Weber, W. Ronghan, E. B\"ottcher, M. Schell, D. Bimberg, "Measurement and Simulation of the Turn-on Delay Time Jitter in Gain-Switched Semiconductor Lasers", IEEE J. Quantum Electron., {\bf QE-28}, 441-445 (1992). \bibitem{shen} T. Shen, "Timing Jitter in Semiconductor Lasers Under Pseudorandom Word Modulation", J. Lightwave Tech. {\bf LT-7}, 1394-1399 (1989). \bibitem{Claudio} C. R. Mirasso, P. Colet and M. San Miguel, "Dependence of Timing Jitter on Bias Level for Single-Mode Semiconductor Lasers under High-Speed Operation", IEEE J. Quantum Electron. {\bf QE-29}, 23-32 (1993). \bibitem{Sapia} A. Sapia, P. Spano, C.R. Mirasso, P. Colet, and M. San Miguel, "Pattern Effects in Time Jitter of Semiconductor Lasers", Applied Phys. Lett. {\bf 61}, 1748-1750 (1991). \bibitem{Pere} P. Colet, C. R. Mirasso and M. San Miguel, ``Memory Diagram of Single-Mode Semiconductor Lasers", IEEE J. Quantum Electron., vol. 29, pp. 1624-1630 (1993); C. R. Mirasso, A. Valle, L. Pesquera and P. Colet, ``Simple Method for Estimating the Memory Diagram in Single mode Semiconductor Lasers", IEE Proc.-Optoelectronics, {\bf 141}, 109-113 (1994). \bibitem{Petermann} K. Petermann, {\sl Laser Diode Modulation and Noise}, (Kluwer Academic Publishers, Dordrecht, 1988). \bibitem{lenstra} D. Lenstra, "Feedback Noise in Single-Mode Semiconductor Laser", SPIE Proc. {\bf 1376}, 245-258 (1991). \bibitem{lenstra2} D. Lenstra, B.H. Verbeek, and A.J. den Boef, "Coherence Collapse in Single-Mode Semiconductor Lasers due to Optical Feedback", IEEE J. Quantum Electronics {\bf QE-21}, 674-679 (1985). \bibitem{wang} J. Wang and K. Petermann, "Noise Analysis of Semiconductor Lasers within the Coherence Collapse Regime", IEEE J. Quantum Electronics {\bf QE-27}, 3-9 (1991); see also correction in IEEE J. Quantum Electronics {\bf QE-27}, 2365 (1991). \bibitem{elsasser} J. Sacher, W. Els\"asser, and E.O. G\"obel, "Nonlinear dynamics of semiconductor laser emission under variable feedback conditions", IEEE J. Quantum Electronics {\bf QE-27}, 373-379 (1991). \bibitem{simonsen} H. Simonsen, ``Frequency Noise Reduction of Visible InGaAlP Laser Diodes by Different Optical Feedback Methods", J. Quantum Electron. vol. 29, pp. 877-884 (1993). \bibitem{14jap} R.W. Tkach and A.R. Chiraplyvy, "Regimes of feedback effects in $1.5\ \mu m$ distributed feedback lasers", IEEE J. Lightwave Tech. {\bf LT-4}, 1655-1661 (1986). \bibitem{clarke} B.R. Clarke, "The Effect of Reflections on the System Performance of Intensity Modulated Laser Diodes", IEEE J. Lightwave Tech. {\bf LT-9}, 741-749 (1991). \bibitem{wu} H. Wu, H. Chang, "Turn-on Jitter in Semiconductor Lasers with Moderate Reflecting Feedback", IEEE Photonics Tech. Lett. {\bf 4}, 339-342 (1992). \bibitem{2} L.N.Langley and K.A.Shore,' The effect of external optical feedback on the turn-on delay statistics of laser diodes under pseudorandom modulation ' IEEE Photonics Tech. Letts. {\bf 4}, 1207-1209 (1992). \bibitem{3}L.N.Langley and K.A.Shore,'The effect of external optical Feedback on timing jitter in modulated laser diodes' IEEE J. Lightwave Technology {\bf LT-11}, 434-441 (1993). \bibitem{ce} E. Hern\'{a}ndez-Garc\'{\i}a, C. R. Mirasso, K. A. Shore and M. San Miguel, "Turn-on jitter of external cavity semiconductor lasers", IEEE, J. Quantum Electron. {\bf 30}, 241-248, (1994); C. R. Mirasso and E. Hern\'{a}ndez-Garc\'{\i}a, "Effects of Current Modulation on Timing Jitter of Single-Mode Semiconductor Lasers in Short External Cavities", IEEE J. Quantum Electron., to be published (October 1994). \bibitem{1} E. Hern\'andez-Garc\'\i a, N.B. Abraham, M. San Miguel and F. de Pasquale, "Frequency selection and transient dynamics in single-mode lasers with optical feedback", J. Applied Phys. {\bf 72}, 1225-1236 (1992). \bibitem{LK} R. Lang and K. Kobayashi, "External optical feedback effects on semiconductor injection laser properties", IEEE J. Quantum Electron. {\bf QE-16}, 347-355 (1980). \bibitem{raul} R. Toral and A. Chakrabarti, ``Generation of Gaussian Distributed Random Numbers by Using a Numerical Inversion Method", Computer Phys. Comm. {\bf 74}, pp. 327-334, (1993). \bibitem{carme} 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}, 3390-3395, (1993). \end{thebibliography} \newpage \section*{Figure captions} \noindent Fig. 1- a) Mean turn-on time , b) Standard deviation of the turn-on time, c) Mean pulse width, d) Standard deviation of the pulse width, e) Maximum output power and f) Standard deviation of the maximum output power as a function of the external cavity round trip time. PM and $C_b=C_{th}$. Solid line corresponds to $R_{ext} \approx 1.6 \times 10^{-2}$ and $P_{ref}=0.5 \, P_{CW}$; Dotted line corresponds to $R_{ext} \approx 1.6 \times 10^{-2}$ and $P_{ref}=P_{CW}$; Dashed line corresponds to $R_{ext} \approx 4 \times 10^{-3}$ and $P_{ref}=0.5 \, P_{CW}$; Long Dashes corresponds to $R_{ext} \approx 4 \times 10^{-3}$ and $P_{ref}=P_{CW}$; Dash-Dot line corresponds to $R_{ext} \approx 1.6 \times 10^{-4}$ and $P_{ref}=0.5 \, P_{CW}$ and Dash-Three Dots corresponds to $R_{ext} \approx 1.6 \times 10^{-4}$ and $P_{ref}=P_{CW}$; Star, Triangle and Diamond correspond to to the results in the absence of feedback for $P_{ref}=0.5 \, P_{CW}$, $P_{ref}=0.75 \, P_{CW}$, and $P_{ref}=P_{CW}$, respectively. \noindent Fig. 2- The same as fig. 1 but for $C_b=1.1 \, C_{th}$. \noindent Fig.3- Ten typical optical pulses as a function of time for $C_b=C_{th}$ and PM. $R_{ext} \approx 1.6 \times 10^{-2}$. a) without feedback; b) $\tau=240$ ps and c) $\tau=80$ ps. \noindent Fig. 4- Ten typical optical pulses as a function of time for $C_b=1.1 \, C_{th}$, PM and $\tau=110$ ps. a) $R_{ext} \approx 1.6 \times 10^{-2}$ b) .$R_{ext} \approx 1.6 \times 10^{-4}$. \noindent Fig. 5- The same as fig.1 but for Pseudorandom Word Modulation. \noindent Fig. 6- The same as fig. 2 but for Pseudorandom Word Modulation. \noindent Fig. 7- Ten typical optical pulses as a function of time for PRWM and $R_{ext} \approx 1.6 \times 10^{-2}$. a) $C_b=C_{th}$ and $\tau=110$ ps. b) $C_b=1.1 \, C_{th}$ and $\tau=240$ ps. \newpage \begin{table} \caption{Meanings and values of the parameters in Eqs. (1) and (2) } \bigskip \begin{tabular} {cccc} Parameter&Meaning&Value&Units\\ $e$&Electronic charge&$1.6022\ 10^{-19}$&$C$\\ $g$&Gain parameter&$5.6\ 10^{-8}$&$ps^{-1}$\\ $\gamma$&Inverse photon lifetime&$0.4$&$ps^{-1}$\\ $\gamma_e$&Inverse carrier lifetime&$5\ 10^{-4}$&$ps^{-1}$\\ $\alpha$&Linewidth enhancement factor&$5.5$&adimensional\\ $\beta$&Spontaneous emission rate&$1.1\ 10^{-8}$&$ps^{-1}$\\ $\omega_0$&Optical angular frequency&$1.216\ 10^{3}$&$ps^{-1}$\\ $N_0$&Carrier number at transparency&$6.8\ 10^7$&adimensional\\ $C_{th}$&Threshold current&$6$&mA\\ $C_{on}$&Injection current after gain switching&$21.5$&mA\\ $\tau$&Feedback delay time&variable&$ps$. \end{tabular} \end{table} %\input figures \end{document}