%********0*********0*********0*********0*********0*********0*********0***75 % % NOTE: % If you have the IEEEtran.sty file (obtainable from % the web: http://www.ieee.org % This tex file, can produce real IEEEJQE output % To do that uncomment the following 17 lines: %\documentstyle[12pt,twoside,draft]{IEEEtran} %\begin{document} %\title{Optical Feedback on Self-Pulsating Semiconductor Lasers} %\author{G. H. M.~van Tartwijk and M.~San Miguel\thanks{G. H. M. van %Tartwijk is with the Departament de F\'{\i}sica, %Universitat de les Illes Balears, E-07071 Palma de Mallorca, Spain. %From November 1995 he will be with the Institute of Optics, the %University of Rochester, Rochester, NY 14627 .}\thanks{M. San Miguel %is with the Instituto Mediterraneo de Estudios Avanzados, IMEDEA %(CSIC-UIB) and with the Departament de %F\'{\i}sica, Universitat de les Illes Balears, %E-07071 Palma de Mallorca, Spain .}\thanks{E-mail: %guido@hp1.uib.es; dfsmsm0@ps.uib.es .}} %\markboth{IEEE Journal of Quantum Electronics, Vol. XX, %No. Y, Month 1999}{van Tartwijk and San Miguel: Optical Feedback on %Self-Pulsating Semiconductor Lasers} %\maketitle % % ###### until here ############### % Then: comment the following 28 lines \documentstyle[12pt]{article} \setlength{\topmargin}{-1.00cm} \addtolength{\textheight}{4.00cm} \addtolength{\evensidemargin}{-3.00cm} \addtolength{\oddsidemargin}{-0.50cm} \addtolength{\textwidth}{3.0cm} \textheight 43\baselineskip \setlength{\evensidemargin}{0.00cm} \setlength{\oddsidemargin}{0.00cm} \setlength{\textwidth}{16.00cm} \renewcommand{\topfraction}{0.995} \renewcommand{\bottomfraction}{0.995} \renewcommand{\textfraction}{0.001} \begin{document} \title{Optical Feedback on Self-Pulsating Semiconductor Lasers} \author{G. H. M.~van Tartwijk$^{1,2}$ and M.~San Miguel$^{1,3}$\\ $^1$Departament de F\'{\i}sica,\\ Universitat de les Illes Balears,\\ E-07071 Palma de Mallorca, Spain.\\ $^2$Present address: The Institute of Optics,\\ The University of Rochester,\\ Rochester, NY 14627\\ $^3$also at: Instituto Mediterraneo de Estudios Avanzados,\\ IMEDEA (CSIC-UIB) } \pagestyle{myheadings} \markboth{submitted to IEEE J. of Quantum Electron. 1995} {van Tartwijk and San Miguel} \maketitle % Until here !!!! % Now, there are two more things to do: % 1) Goto the first line of the introduction and uncomment the line % starting with PARstart, and comment out the next line % % 2) uncomment the biographies at the end of the text % Good Luck % \begin{abstract} Weak optical feedback effects on the statistical properties of self-pulsations in narrow-stripe semiconductor lasers are analyzed using Lang-Kobayashi-type equations. The self-pulsation features are compared with the characteristics of excited relaxation oscillations. We determine the operating regime in which the randomizing effect of spontaneous emission noise destroys pulse coherence. In this regime, only phase-insensitive effects of optical feedback are possible and optimum jitter reduction is achieved with delay times of the order of an integer odd multiple of the free running pulsation period. In the high pump operating regime interpulse coherence is retained and the optical feedback phase is shown to be instrumental for pulse jitter control. Our results show that for cavity lengths up to $10$ cm variations on the order of half an optical wavelength induce jitter variations of one order of magnitude. \end{abstract} \section{Introduction} %\PARstart{S}{elf-pulsating} semiconductor lasers (SPSL) have been Self-pulsating semiconductor lasers (SPSL) have been studied since the first diode lasers became available in the late 1960s \cite{Basov68}. The first semiconductor lasers, although designed to operate in CW mode, showed self-induced pulsations of the light intensity because of (a combination of) two reasons: (i) the laser resonance is internally excited through the nonlinear interaction of various longitudinal laser modes, thus causing mode beating; (ii) defects in the active material serve as saturable absorbing areas, thus causing absorptive Q-switching processes. In the case of self-pulsations caused by saturable-absorbing effects, the pulsation frequency depends on the pump current in qualitatively the same way as the relaxation oscillation (RO) frequency \cite{AGDUT}. Indeed, it was shown that the RO is the underlying mechanism for self-pulsations in these devices \cite{AGDUT,Yamada93}. Saturable absorption effects, causing self-pulsations in stripe-geometry lasers have been investigated since the early 1980s \cite{Paoli79,Ueno85}. Saturable absorption is also responsible for self-pulsations in double-section laser diodes \cite{Avrutin93}. A similar mechanism of dispersive Q-switching has been recently invoked to describe self-pulsations in multisection Distributed Feedback Lasers \cite{Bandelow93,BandelowPREP}. Shortly after the advent of optical data storage systems it was found that conventional, i.e., CW semiconductor lasers show a very high sensitivity to unwanted weak reflections from the optical system, often reaching a state denoted as ``coherence collapse" \cite{Lenstra85,Mork90,vanTartwijk95}. It was soon recognized that the the phenomenon of selfsustained pulsations could be used to reduce optical feedback noise \cite{Matsui83}, and as a consequence, SPSLs have been commercially used in CD devices. The reason for this low sensitivity is the assumption that each optical pulse builds up from spontaneous emission noise, so that feedback can not affect the phase noise (an important cause for coherence collapse) \cite{Avrutin93,Simler92}. SPSLs have been also studied for a number of other applications which include all-optical synchronization, i.e., as optical clocks in all-optical communication systems \cite{Phelan92}, tunable electrical signal generators \cite{Simler92}, etc. These other applications of SPSLs require stable and repeatable pulses with minimum jitter and stringent control of phase noise characteristics. Thus, there appear to exist conflicting requirements on SPSLs for different applications: on the one hand the pulse to pulse incoherence and associated insensitivity to optical feedback is considered to be beneficial for optical data storage systems, while it is highly detrimental in optoelectronic sampling applications. This fact calls for detailed studies of the statistical properties of optical pulses in SPSLs which permit to discriminate among different operating regimes appropriate for different applications. In this paper we report one such study for the self-pulsating behavior of a narrow-stripe SL subjected to (weak) external optical feedback (EOF). Control of the pulse frequency and jitter of SPSLs has been considered since long ago. Already in 1970, it was experimentally shown that various types of feedback (optoelectronic, electrical, and optical) can favorably change the pulse rates and widths \cite{Paoli70}. A model explaining these optical feedback-induced changes in pulse frequency and pulse width was given by Lau {\em et al.} \cite{Lau80}. External optical injection \cite{Simler92} as well as electrical modulation \cite{Egan95} and optoelectronic feedback \cite{Lau84} have been shown to have an improving effect on the pulsing characteristics, such as jitter. The studies on EOF of Paoli and Ripper \cite{Paoli70} dealt with an external cavity of $\sim 75$ cm ($5$ ns delay). They showed experimentally that, provided the feedback is strong enough, the pulse rate always locks on a higher harmonic of the external cavity resonance. They also observed that the pulse rate is always pulled to a higher frequency than the feedback-free value. These studies were made long before the discovery of the importance of the linewidth enhancement factor in semiconductor lasers \cite{Henry82}, and their explanation \cite{Lau80} was given in terms of photon number equations, disregarding phase effects. Such effects have also not been considered in later studies of EOF. These include a report on chaotic, period-2, and period-3 pulsations in external cavities with lengths around $10$ cm. \cite{Kuznetsov86} and the effects of strong feedback from high-Q external (ring) cavities with lengths between $1$ and $3$ m \cite{OGorman88}. More recently \cite{Sartorius95}, it has been experimentally shown that the phase of reflected light is crucial to control self-pulsations in Distributed Feedback Lasers with phase tuning sections. However, no theoretical study of phase effects in SPSLs with EOF seems available. In this paper we investigate the role of the phase of the optical field in the pulsation characteristics and we study shorter external cavities (up to $9$ cm) focusing on pulse-to-pulse jitter. In summary, the aims of this paper are to (i) provide an analysis of the steady states and their stability of the free running laser, thereby identifying self-pulsations as sustained ROs and identifying regimes in which pulses are built from noise and regimes with pulse to pulse coherence; (ii) investigate, through statistical characterizations of the pulses, the possibilities of reducing the jitter of the self-pulsations using weak EOF, and to (iii) elucidate the role of the phase of the optical field in this system. In section II we explain our model for a narrow-stripe SL, based on recent work by Yamada \cite{Yamada93}, which is closely related to the model of two-section SPSL as used by Avrutin \cite{Avrutin93}. The steady state (CW) solutions and their stability are analytically investigated, and we find a window for self-pulsations in the LI-curve. In section III we test our analytical results with simulations, and discuss the relation between the self-pulsation frequency and the instabilities, as well as discuss the role of spontaneous emission noise. The latter invites for an estimate where phase-effects in feedback could be observable, and hence the greatest jitter reduction could be achieved. In section IV we discuss EOF in general and introduce our model, which is a Lang-Kobayashi-type \cite{LangKob} extension of our model for the free running laser. In section V we report on the pulsation statistics as a function of feedback strength and global delay time. In section VI we address the phase sensitivity of the pulsations. There we show the dramatic dependence of the pulsation statistics on minute changes in the delay time, corresponding to a fraction of an optical period. \section{Free running laser model} Common SPSLs are nowadays the narrow-stripe geometry lasers, such as V-channel substrate inner-stripe structures and ridge-waveguide self-aligned structures, although multi-section lasers are also often employed. Our model is somewhat focused on the narrow-stripe geometry, see Fig.\ \ref{fig1_sp}, but with some small adaptations is also applicable to a two-section laser \cite{Avrutin93}. Recently, Yamada \cite{Yamada93} put forward a theoretical description of self-sustained pulsation phenomena in narrow-stripe semiconductor lasers. More precisely, he studied a ridge-waveguide inner-stripe geometry. These lasers are weakly index-guided \cite{AGDUT}, due to a small built-in refractive index step. Self-pulsation arises in these structures due to combination of pumped amplifying (A) material and unpumped (blocked) saturable absorbing (SA) material. Yamada's model consists of three rate-equations, one for the total number of photons in the device, and one for each of the carrier densities in the active and saturable absorbing regions. Taking into account one lateral mode (x-direction), and allowing only the parameters of the functional form of this mode to change in time (time-dependent lateral confinement), the basic mechanism leading to the self-pulsations was argued to be similar to the phenomenon of sustained relaxation oscillations. Current dependencies of the output power and the pulse repetition frequency were shown, as well as the stripe width dependence of the pulsing. It was found that a time-dependent treatment of the optical field distribution across the stripe width is unnecessary after the first few integration steps, when a suitable pair of confinement factors has been found. Yuri {\em et al.}\ \cite{Yuri95} reported that a full two-dimensional analysis of the field, i.e., in the longitudinal and lateral direction, give qualitatively similar results that can be compared quantitatively with experimental results. Since we want to study the effect of EOF, and more precisely, the role of the phase of the field, we start with Yamada's model but include the phase of the optical field. Doing that, we implicitly assume single longitudinal mode operation. In Yamada's model, only the number of photons is considered, and no assumption regarding the longitudinal modes has to be made. The single-mode approximation has been shown to be very helpful in finding the underlying physical mechanisms in semiconductor laser dynamics, especially under optical feedback conditions \cite{vanTartwijk95}. Indeed, we will see that this approximation does not give qualitatively unrealistic results. Furthermore, we simplify the ``self-consistent" treatment of the lateral field distribution, by assuming time-independent confinement factors (as is justified by Yamada's results). For the free-running laser the single-mode approximation is irrelevant, as it will turn out that the phase is a slave variable. Our results for the free-running laser are independent of the phase. The single longitudinal mode of the optical field under consideration is written as ${\cal E}(t) = E(t)\exp{(i\omega_0 t)}$, where $E(t)$ is the slowly varying complex amplitude of the field with frequency $\omega_0$. The equations for the complex field amplitude $E(t)=\sqrt{P(t)}\exp{(i\varphi(t))}$, the carrier number in the active (pumped) region $N_1$, and the carrier number in the absorbing (unpumped) region $N_2$ read: \begin{eqnarray} \frac{dE}{dt} &=& \frac{1}{2} \left [ (1+i\alpha_1)\Gamma_1\xi_1(N_1-N_{t1}) +(1+i\alpha_2)\times \right . \nonumber \\ &&\left . \times \Gamma_2\xi_2(N_2-N_{t2}) - \Gamma_0 \right ] E + F_E(t), \label{FRrateE_uns}\\ \frac{dN_1}{dt} &=& \frac{J}{e} - \frac{N_1}{T_1(N_1)} + \frac{N_2}{T_{21}} - \Gamma_1\xi_1(N_1-N_{t1})P, \label{FRrateN1_uns}\\ \frac{dN_2}{dt} &=& - \frac{N_2}{T_2(N_2)} + \frac{N_1}{T_{12}} - \Gamma_2\xi_2(N_2-N_{t2})P. \label{FRrateN2_uns} \end{eqnarray} The parameters in Eqs.\ (\ref{FRrateE_uns},\ref{FRrateN2_uns}) are specified in Table I. The subscript $i=1$ denotes the pumped (A) region, while $i=2$ denotes the saturable absorbing (SA) region. The complex optical gain is linearized around the transparency numbers $N_{ti}$, with differential gain coefficients $\xi_i$, and total optical confinement factors $\Gamma_i$. $T_i(N_i)$ are the carrier lifetimes in the two regions, $T_{ij}$ are the diffusion times of carriers from region $i$ to region $j$. The ratio between the volume $V_1$ of region A and the effective \cite{Yamada93} volume $V_2$ of region SA, is equal to the ratio of the two diffusion times: \begin{equation} f_1 \equiv \frac{V_1}{V_2} = \frac{T_{12}}{T_{21}}. \label{f1} \end{equation} The carrier diffusion between the two regions seems crucial for self-pulsations to arise. When it is left out of the problem, the introduction of nonlinear gain (gain saturation) can also induce SP, as is shown in the case of two-section SLs \cite{Avrutin93}, and in studies on solid-state and gas lasers with saturable absorbers \cite{Lugiato78,Mandel80,Arimondo85}. In the case of the narrow-stripe laser, however, lateral diffusion of carriers appears to be the important effect to consider. Note that we explicitly take into account the carrier dependence of the carrier decay rates $T_{i}^{-1}(N_i)$. In conventional SLs, one often approximates the carrier lifetime to be constant, since the carrier number is clamped above threshold in CW operation. We will see that due to the saturable absorption the carrier numbers do not show this clamping behavior. Furthermore, SP cause such large variations in the carriers that a constant carrier lifetime becomes a too crude approximation. The carrier lifetimes are given by \cite{AGDUT}: \begin{equation} T_{i}^{-1}(N_i) = A_{nr,i} + B_i N_i + C_i N_{i}^{2}, \label{TiABC} \end{equation} where the parameters are explained in Table I. In Eq.\ (\ref{FRrateE_uns}), $F_E(t)$ models the spontaneous emission noise and is a Langevin force describing a (Gaussian) white noise process. Its strength is given by \begin{equation} \langle F_{E}(t)F_{E}^{\ast}(t^{\prime}) \rangle = C_{sp}B_1 N_{1}^{2} \delta (t-t^{\prime}) , \label{BNsq} \end{equation} where $C_{sp}$ is the fraction of the spontaneously emitted photons that end up in the single longitudinal mode under consideration. In principle, Langevin noise terms should also be added to the the carrier equations (\ref{FRrateN1_uns},\ref{FRrateN2_uns}), but these can be neglected \cite{Henry82}. In Table I we list the values of the parameters that we will use in our investigation. To facilitate the analytical work, we make all variables and parameters dimensionless, using the following obvious rescaling: \begin{equation} \begin{array}{lll} \hat{t}=\Gamma_0t, & \hat{E}=E/\sqrt{N_{t1}}, &\hat{N}_i=\frac{N_i-N_{ti}}{N_{ti}},\\ \hat{T}_{ij}=\Gamma_0 T_{ij},&g_i=\Gamma_i\xi_iN_{ti}/\Gamma_0, &\hat{J}=J/e\Gamma_0N_{t1},\\ \hat{A}_{nr,i}=A_{nr,i}/\Gamma_0,&\hat{B}_i=B_iN_{ti}/\Gamma_0, &\hat{C}_i=C_iN^{2}_{ti}/\Gamma_0. \end{array} \label{rescal} \end{equation} Furthermore we define ``dressed" carrier decay rates (inverse lifetimes) by including the diffusion times according to: \begin{equation} \hat{\gamma}_{i}(\hat{N}_i) = \frac{1}{\hat{T}_{ij}} + \hat{A}_{nr,i} + \hat{B}_i(\hat{N}_i +1) + \hat{C}_i(\hat{N}_i + 1)^2. \label{gamsti} \end{equation} Our model equations for the free running laser read in dimensionless form: \begin{eqnarray} \frac{d\hat{E}}{d\hat{t}} &=& \frac{1}{2} \left [ (1+i\alpha_1)g_1\hat{N}_1 + (1+i\alpha_2)g_2\hat{N}_2 - 1 \right ] \hat{E} \nonumber \\ &+& F_E(\hat{t}), \label{FRrateE}\\ \frac{d\hat{N}_1}{d\hat{t}} &=& \hat{J} - (\hat{N}_1+1)\hat{\gamma}_1(\hat{N}_1) + \frac{f_1(\hat{N}_2+1)}{N_{12}\hat{T}_{12}} \nonumber \\ &-& g_1\hat{N}_1 \hat{P},\label{FRrateN1}\\ \frac{d\hat{N}_2}{d\hat{t}} &=& - (\hat{N}_2+1)\hat{\gamma}_2(\hat{N}_2) + \frac{N_{12}(\hat{N}_1+1)}{\hat{T}_{12}}\nonumber \\ &-& g_2N_{12}\hat{N}_2 \hat{P}, \label{FRrateN2} \end{eqnarray} where: \begin{equation} N_{12} \equiv N_{t1}/N_{t2}, \label{defN12} \end{equation} and $F_E(\hat{t})$ has two-point correlation: \begin{equation} \langle F_E(\hat{t_1}) F^{\ast}_E(\hat{t_2}) \rangle = C_{sp}\hat{B}_{1}(\hat{N}_1+1)^2 \delta (\hat{t_1}-\hat{t_2}). \label{rescaled_noise} \end{equation} In the remainder of this paper we omit the hats. We note that for the free running laser described by Eqs.\ (\ref{FRrateE}-\ref{FRrateN2}) the dynamics of the phase of the optical field decouples from the equations for photon- and carrier- number. Therefore, the single mode approximation is irrelevant for a photon number description and we will not deal with the phase of the optical field until the introduction of optical feedback in section IV. As a first step, we find the steady states of the free running laser, and their stability. We therefore neglect the spontaneous emission noise. Steady states are defined by monochromatic fields and constant carrier numbers: $P(t)=P_s$, $\varphi (t) = \Delta \omega_s t$, and $N_i(t) = N_{is}$. The frequency shift $\Delta\omega_s$ with respect to the optical frequency $\omega_0$ is for all steady states given by: \begin{equation} \Delta\omega_s = \frac{1}{2}(\alpha_1g_1N_{1s} + \alpha_2g_2N_{2s}). \label{DWs_fp} \end{equation} Note that the steady-state behavior of the phase is dictated by the carrier numbers, i.e.\ the phase is slaved. The trivial, i.e., non-lasing solutions are found to be given by: \begin{equation} N_{1s} + 1 = \frac{T_{12}}{N_{12}}(N_{2s}+1)\gamma_{2}(N_{2s}), \label{N1ssubthr} \end{equation} where $N_{2s}$ is any of the physical meaningful (positive carrier number $N_{is} > -1$) real zeros of a $9^{th}$-order polynomial in $N_{2s}$ (see Appendix for details). In Fig.\ \ref{fig2_sp}(a) we show the non-lasing solution $N_{1s}$ as a function of the pump current. Just as in the conventional laser, the carrier number builds up from zero, i.e., $N_{1s}=-1$. Fig.\ \ref{fig2_sp}(b) shows the carrier dependence of the carrier lifetime for this solution, and it is clear that taking a constant value for $T_i$ is not justified. More interesting are of course the lasing solutions. Since now $P_s \neq 0$ the gain must equal all losses and they are given by: \begin{eqnarray} N_{1s} &=& \frac{1}{g_1}-\frac{g_2}{g_1}N_{2s}, \label{GeqLsupthr}\\ P_s &=& \frac{N_{12}(N_{1s}+1) - \gamma_2(N_{2s})T_{12}(N_{2s}+1)} {g_2 N_{12} T_{12} N_{2s}}, \label{Pnz} \end{eqnarray} where $N_{2s}$ is any of the physical meaningful ($N_{is} > -1$) real zeros of a quartic polynomial in $N_{2s}$ (see Appendix for details). In Fig.\ \ref{fig2_sp}(c) we show the lasing solutions for $N_{1s}$ as a function of pump current. These show a dramatic different dependence on current than the conventional lasing solution: in the pump window between $48$ and $58$ mA, there are two lasing solutions possible. For currents larger than $58$ mA, only one solutions survives, which gradually saturates towards $N_{1s} \sim 1.5 N_{t1}$. Note that this is completely different from the clamping behavior in conventional SL, confirmed by the carrier lifetime, see Fig.\ \ref{fig2_sp}(d), which shows changes of almost $10 \%$ on the interval between $58$ and $150$ mA. Summarizing the steady state solutions, we show in Fig.\ \ref{fig3_sp} the light-current (LI) curve of the free running laser. To fully appreciate this curve, we determine the stability of the steady state solutions. Around each steady state solution $(P_s,N_{1s},N_{2s})$ a linear stability analysis gives the following characteristic equation: \begin{eqnarray} z^3 &+& [2(\lambda_1+\lambda_2) - (g_1N_{1s}+g_2N_{2s}-1)] z^2 \nonumber \\ &+& \left[ 4\lambda_1\lambda_2 - \frac{f_1}{T_{12}^2} -2(\lambda_1+\lambda_2)(g_1N_{1s}+g_2N_{2s}-1) \right ]z \nonumber \\ &+& \left [g_1^2N_{1s}P_s + g_2^2N_{12}N_{2s} \right ] z \nonumber \\ &+& (g_1N_{1s}+g_2N_{2s}-1) \left (\frac{f_1}{T_{12}^2} -4\lambda_1\lambda_2\right ) \nonumber \\ &+& g_1N_{1s}\left( 2\lambda_2g_1P_s + \frac{N_{12}g_2P_s}{T_{12}}\right) \nonumber \\ &+& g_2N_{12}N_{2s}\left(2\lambda_1g_2P_s+ \frac{f_1g_1P_s}{N_{12}T_{12}}\right) = 0 , \label{chareq} \end{eqnarray} where \begin{eqnarray} \lambda_1 &=& \frac{1}{2}\left\{ g_1P_s + \gamma_1(N_{1s})\right. \nonumber\\ &+& \left.(N_{1s}+1)[B_1+2C_1(N_{1s}+1)] \right\}, \label{lambda1}\\ \lambda_2 &=& \frac{1}{2}\left \{g_2N_{12}P_s +\gamma_2(N_{2s})\right. \nonumber\\ &+& \left.(N_{2s}+1) [B_2+2C_2(N_{2s}+1)]\right\}, \label{lambda2} \end{eqnarray} are ``relaxation oscillation"-like damping rates. The zeros $\{z\}$ of Eq.\ (\ref{chareq}) are the complex growth rates of the perturbations around the solution under consideration. The solution is stable only if all zeros have negative real part. The nonlasing solution (Fig.\ \ref{fig2_sp}(a) and the line at $P=0$ in Fig.\ \ref{fig3_sp}) has three real characteristic roots, all of which are negative for pump current less than $58$ mA, while one of the roots is positive above $58$ mA. The nonlasing solution is therefore unstable above $58$ mA indicated by the dashed line. The lasing solutions (Fig.\ \ref{fig2_sp}(c) and the curve in Fig.\ \ref{fig3_sp}) can be divided into two branches. The upper branch exists from $48$ mA onwards, while the middle branch only exists in the triple solution region $48 - 58$ mA. The latter is always unstable, and its characteristic zeros are all real. The upper branch shows a more delicate behavior. Starting at high pump, e.g.\ at $125$ mA, the lasing solution is stable: one characteristic zero is real and negative, while the other two form a complex conjugate pair with negative real part. The latter indicates that perturbations in the lasing solutions will damp out through an oscillation. In conventional semiconductor lasers, a similar phenomenon is known as damped relaxation oscillation. In those systems it is well known that external influences such as optical feedback can excite these relaxation oscillations and make them sustained or induce a route to chaos \cite{Mork90}. The oscillation here is similar to the conventional RO, except for the fact that it involves a third party: the photon number, and both carrier populations. The complex conjugate pair of characteristic roots of the upper branch solution is shown in Fig.\ \ref{fig4_sp}. Here, we see that by reducing the current below $118$ mA, the real part of the complex conjugate pair becomes positive, and the perturbations will show oscillatory growth. One should realize that this result is only valid in a short time after the perturbation, since we have carried out a linear stability analysis. Indeed, we will see in the next section that for currents only a little smaller than $118$ mA, we observe a growing oscillation evolving into self-pulsations. It is commonly stated that this type of self-pulsations is analogous to sustained relaxation oscillations. Decreasing the pump further, the frequency of the oscillation (imaginary parts of the characteristic roots) decreases until zero, while the growth rate increases, shortly after which the solution itself ceases to exist. From this analysis we expect self-pulsations to be restricted to the pump window $58 - 118$ mA, where the pulse frequency decreases with decreasing pump, while the pulse strength (modulation depth) increases. The triple solution region thus involves only one stable solution, the non-lasing solution. A definition of the threshold current is the current at which the non-lasing solution looses stability, in this case $58$ mA. One can also define threshold current as the lowest current at which lasing is observed. For conventional lasers, both definitions give nonconflicting results, but in the case of the self-pulsating laser neither definitions seem to make much sense. The situation is best described by the statement that at $48$ mA there occurs a sub-critical Hopf-bifurcation at zero frequency of the nonlasing solution. The expressions derived here for the steady states and their stability are specific for the narrow-stripe semiconductor laser, in the sense that we include lateral carrier diffusion and carrier-dependent carrier-lifetimes. From a more general point of view, the steady-states and their stability show the same characteristics as those of a $CO_2$ laser with saturable absorber (see e.g.\ Arimondo {\em et al.} \cite{Arimondo85} and De Tomasi {\em et al.} \cite{deTomasi89}). A thorough analysis of the bifurcation phenomena in a similar laser system was carried out by Erneux and Mandel \cite{Mandel80,Erneux81}. In the next section we study the dynamical behavior of the free running laser with special attention on the self-pulsation characteristics, and relate them to the phenomena found here, such as the relation between the self-pulsing frequency and the frequency of the instability. \section{Self pulsation characteristics for the free running laser} We have numerically solved the model equations (\ref{FRrateE},\ref{FRrateN2}) for parameters as listed in Table I for different pump values. A Runge-Kutta algorithm has been used, with a time step of $0.1$ ps to safeguard numerical accuracy. Spontaneous emission white noise is simulated using a Gaussian random number generator \cite{Toral93}. Self-pulsations are obtained for pump values between $58$ and $118$ mA. Pulsations are characterized by their (average) period and jitter. The pulse period is the average time $$ between subsequent pulses, where the time of a pulse is the moment at which the power surpasses from below the reference value $P_{ref} \equiv 2.5 \times P_s$, where $P_s$ is the steady state power of the (unstable) lasing solution. We characterize the pulse-to-pulse jitter by the standard deviation: \begin{equation} \sigma = \sqrt{-^2}, \label{jitter} \end{equation} which is a suitable measure for the half width at half maximum of the distribution function of pulse periods. Statistical averages are performed over at least $10^{4}$ pulses and we estimate the statistical error in the jitter $\sigma$ to be less than $0.1$ ps. First, we investigate the relation between the pulse frequencies (PF), i.e., inverse pulse periods, and the imaginary parts of the complex conjugate pair of roots of the upper branch solution, see Fig.\ \ref{fig4_sp}(b). In Fig.\ \ref{fig5_sp} we show the both quantities, where the PF is measured in simulations with and without spontaneous emission noise. Close to $118$ mA, the pulse frequency bends towards the imaginary part, but for pump values not very much smaller, they differ already substantially. Although the frequency of the (linear) instability is a poor quantitative measure for the real dynamical effect, the instability is the cause of the pulsing, as it is observed in the transients after starting on the unstable lasing solution. The effect of noise on the PF, as shown in Fig.\ \ref{fig5_sp}, is only observable for pump currents smaller than $100$ mA. Also, the jitter shows a sudden change around this value, as shown in Fig.\ \ref{fig6_sp}(b). There, it is shown that pump values between $70$ and $110$ mA yield pulse periods between $0.25$ and $0.50$ ns, with jitter $\approx 10$ ps. The role of noise becomes clearer as we take a look at the floor value of the photon number in between pulses. Without noise, this quantity is basically zero at $60$ mA and increases to about $10^5$ at $117.5$ mA. The peak photon number of $\sim 10^{7}$ remains roughly the same over the pump interval. The pulse width, $14$ps at $60$ mA, reaches its minimum of $12$ ps at $80$ mA, and then increases to $18$ ps at $117.5$ mA. In Fig.\ \ref{fig7_sp} we show the effect of noise: for pump values larger than $100$ mA, no significant difference is observed in the floor value with and without noise. Note in Fig.\ \ref{fig6_sp}(b) that jitter does not vanish for those pump values, but is indeed much smaller than for currents smaller than $100$. We may conclude from these results that the pulses build up ``from spontaneous emission noise" only for currents below $100$ mA, while above this current, the pulses retain correlation. This will be of importance when studying the phase effect in the situation with feedback. \section{Optical feedback model and general considerations} In the remainder of this paper we look at the effects of EOF on the self-pulsation characteristics, i.e, change of the PF and jitter. In our analysis of the free running SP laser the optical phase was only mentioned in the first part, since it decouples from photon- and carrier- number dynamics. As soon as external optical modulation, such as injection or feedback are employed, the dynamics of the photon number is coupled to the phase of the optical field. In our analysis of the steady states and their stability we reported some significant difference between the self-pulsating laser and conventional lasers. In both systems, a typical oscillation plays an important role. This relaxation oscillation is shown to be the driving force of the self-pulsations, while in conventional lasers it has been shown to easily become excited when subjecting the system to EOF. Taking the analogy further, reduction of jitter in self-pulsating lasers can be compared with narrowing the relaxation oscillation side bands in a conventional laser with optical feedback. Our model consists of the SP-laser and an external cavity, formed by the laser and an external mirror. For ordinary semiconductor lasers, external cavity lengths have been studied in a wide range: from the short cavity limit ($\mu$m) to the long cavity limit ($\sim 10$ m). Roughly speaking, short cavities are useful for narrowing the laser linewidth and tuning the operating frequency, whereas long-cavities tend to destabilize the system \cite{QSO95}. One should realize that these qualitative remarks are in principle only valid for the CW laser, and may only serve as a guide in our investigation of feedback effects in the self-pulsating case. In this paper we will study relatively short (a few cm) cavities with time delays of the order of the pulse period, since we focus to investigate feedback effects on the pulse-to-pulse jitter. The theoretical description of the laser subjected to optical feedback from a cavity of arbitrary length is chosen to be a simple extension of Eqs.\ (\ref{FRrateE}-\ref{FRrateN2}) using a Lang-Kobayashi-type term for the feedback \cite{QSO95}. Only the field equation needs modification and reads: \begin{eqnarray} \frac{dE}{dt} &=& \frac{1}{2} \left [ (1+i\alpha_1)g_1N_1 + (1+i\alpha_2)g_2N_2 - 1 \right ] E \nonumber\\ &+& \gamma \exp{(-i\omega_{0}\tau)}E(t-\tau) + F_E(t). \label{FBrateE} \end{eqnarray} Here, $\tau$ is the external cavity round-trip time, straightforwardly related to the external cavity length $L_{ext}$ as $\tau = 2L_{ext}/c$, where $c$ is the speed of light, and $\gamma$ is the feedback rate. This feedback rate is a measure for the strength of the feedback: the ratio between the power reflected from the external mirror and that from the laser facet facing the external cavity is given by $(\gamma\tau_{in})^2$ where $\tau_{in}$ is the round-trip time of the field inside the laser cavity. For a typical laser of length $300$ $\mu$m, a $1\%$ power reflection gives a feedback rate of $\approx 15$ (ns)$^{-1}$. The effect of the feedback depends of course not only on the feedback rate but also on the delay time. As can be seen in Eq.\ (\ref{FBrateE}), the delay time has two simultaneous effects: one on the slowly varying amplitude $E(t-\tau)$, which is a relatively global effect, and the second in the feedback phase $\omega_{0}\tau$, which is a more delicate effect. When operating with an external cavity of a few cm, i.e., a feedback time of $\approx 0.1$ ns, a change in the cavity length on the order of half an optical wavelength ($400$ $\mu$m) hardly changes the global role of the delay time, but results in a completely different value of the feedback phase, since it will have changed over $2\pi$. In fact, for not too short cavities the two effects of the delay time can be regarded independently: the feedback parameters are then $\gamma$, $\tau$, and $\omega_0\tau$. When the external cavity is shortened the global and local effect of the delay time can not be distinguished anymore \cite{Hern94}, and only the two independent feedback parameters ($\gamma$ and $\tau$) remain. When the self-pulsation characteristics depend on $\omega_{0}\tau$, the phase of the field apparently remains a meaningful quantity between pulses; when the value of the feedback phase is irrelevant for the self-pulsations we operate in the regime of phase-uncorrelated pulses. Since in general very few analytical results can be obtained in systems with delayed feedback, we will base our analysis on numerical simulations. In our simulations we use the same integrating scheme as in the free running case and explore some part of the vast parameter space around two characteristic pump currents. The first is $80$ mA, where the analysis of the previous section indicates that the pulses are incoherent, while the second is at $100$ mA, where the pulses can keep some of their correlation in spite of the spontaneous emission noise. Also, we analyze the pulsing behavior on the edge of the pulsating window, at $117.5$ mA. \section{The effect of feedback rate and delay time on self-pulsations} In this section we discuss the global effects of weak EOF on the self-pulsation characteristics. These effects are associated with the feedback-rate $\gamma$ and delay time $\tau$. Dependence on the feedback phase $\omega_0\tau$ will be discussed in the next section. We will keep the feedback phase fixed at $\omega_0\tau = 3\pi/2$, which is always possible for not too small delay times. In the $80$ mA case, the free running laser has an average pulse period of $=0.387$ ns with $\sigma = 12$ ps jitter. Choosing a feedback rate of $10$ ns$^{-1}$, corresponding to a feedback power ratio of $\approx 0.5 \%$, we determine the self-pulsation characteristics for delay times up to $0.6$ ns. Fig.\ \ref{fig8_sp}(a) shows the observed average pulse periods. Note the sudden jump in pulse period around $\tau = 0.45$ ns. This jump in pulse period must be attributed to a switch of the locked pulse frequency to a neighboring compound cavity mode, i.e., the resulting resonance frequency of the laser mode (with relaxation oscillations) and one of the external cavity resonances. It should be noted that the self pulsing frequency here does not lock simply to a higher harmonic of the external cavity resonance frequency. This is due to the fact that we are considering relatively short cavities \cite{OGorman88}. The tuning range of the pulse period decreases with increasing delay times. Fig.\ \ref{fig8_sp}(b) shows the corresponding jitter. Keeping in mind that the free running value of the jitter is $12$ ps, we see that feedback can induce substantial jitter reduction, provided the delay time is not close to the free running pulse period: at $\tau = 0.54 _{FR} = 0.21$ ns we find a jitter reduction of $12$ dB and at $\tau = 1.47 _{FR} = 0.57$ ns a jitter reduction of $10$ dB is found. In contrast when $\tau = _{FR}$ the jitter is hardly changed at all. These results suggest that the maximum jitter reduction is found when the delay times is an odd multiple of $_{FR}/2$, and that it is particularly ineffective to choose a delay time which is equal to the self-pulsating period. For increasing delay times, the effect of the feedback on the jitter slowly diminishes. These notions are confirmed only for $$ in the situation at $100$ mA, as shown in Fig.\ \ref{fig9_sp}(a). At this pump current, the free running laser has an average pulse period of $0.242$ ns with $\sigma = 7$ ps jitter. Again taking a feedback rate of $10$ ns$^{-1}$, we perform the same diagnostics as in the $80$ mA case. Fig.\ \ref{fig9_sp}(a) shows qualitatively the same behavior as Fig.\ \ref{fig8_sp}(a): it also shows a jump to a neighboring compound cavity mode, roughly when the delay time equals the free running pulse period. The jitter, Fig.\ \ref{fig9_sp}(b), shows different behavior: at $\tau=0.3$ ns, equal to $1.25 $ we see a sharp decrease in jitter. A similar sharp change in the jitter, but in the opposite direction, is observed at $0.5 $. From the free running laser analysis, we found that phase effects may become important starting at $100$ mA. The sudden changes in the jitter values in Fig.\ \ref{fig9_sp}(b) may be caused by this phase effect, which will be further investigated in the next section. The optimum jitter reduction achieved in this set of simulations is $11.9$ dB. To investigate the effect of the feedback rate, we performed similar simulations, again at the two pump values $80$ and $100$ mA. In Figs.\ \ref{fig10_sp} and \ref{fig11_sp} we plot the dependence of $$ and $\sigma$ on the feedback rate. For both cases we used a feedback phase of $\omega_0\tau = 3\pi/2$. At $80$ mA we chose the delay time to be $\tau = 0.57 \mbox{ ns} = 1.47 _{FR}$, while at $100$ mA the delay time used is $\tau = 0.3\mbox{ns} = 1.25 _{FR}$. For very weak feedback, the pulse period and the jitter differ only slightly from their solitary values. As the feedback becomes stronger, the optimum jitter reduction is arrived at, for both pump currents, around $\gamma = 10$ (ns)$^{-1}$. These optimum conditions were explored in Figs.\ \ref{fig8_sp} and \ref{fig9_sp}. The effect of the feedback rate is qualitatively similar to the case of the conventional laser: weak reflections (less than $1\%$) can have a beneficial effect, while already reflections of a few $\%$ can be fully detrimental. The increase in jitter, i.e.\ in the variance of the pulse period, manifests itself by a multi-peaked, for very high feedback-rates simply very broad, pulse period distribution. This seems to agree with the period doubling route to chaos as reported in \cite{Kuznetsov86}. \section{Phase effects} When numerically solving the equations with feedback, a clear dependence of the feedback phase is observed when spontaneous emission noise is switched-off by setting $C_{sp}=0$. This will come as no surprise because the phase in included in the equations in a nonlinear way. The real issue is of course whether these deterministic effects will survive realistic amounts of spontaneous emission noise. A first example of a feedback phase effect is shown in Fig.\ \ref{fig12_sp}. Plotted is the time between subsequent pulses versus the pulse number, i.e., $t_n - t_{n-1}$ versus $n$. The pump current is $117.5$ mA, very close at the limit of self-pulsations, but still showing pulses with a peak to valley ratio of more than $20$ dB. In the nine ``time traces" we show the evolution of the time between subsequent pulses as we switch on the feedback at $n = 75$, and the noise at $n = 389$ (dotted line). Each trace corresponds to the same delay time of $0.21$ ns and feedback rate of $5.0$ ns$^{-1}$, but with different feedback phases, which are indicated in the figures. First of all, we note that feedback introduces ``jitter" in the absence of noise in all cases, most notably at $\omega_0 \tau = 0$ and $\omega_0 \tau = \pi/3$. This behavior is not stochastic but fully deterministic, and could be considered as a quasi-periodic modulation of the pulses. The feedback phase of $3\pi/2$ appears to suit the system best, since the ``noisy" behavior is minimal. After switching the noise on, the behavior does not change much. If we reduce the pump current, and look at the same traces, we observe that noise very efficiently ``washes" away the direct deterministic effects of the feedback phase. In those cases we have to rely on pulse statistics to observe any feedback phase effect. Returning to the two cases of the previous section, $80$ and $100$ mA, we choose the optimum parameter sets (resulting in lowest jitter) and vary $\omega_0 \tau $ over $2\pi$. In the $80$ mA case we take the feedback rate to be $\gamma=10$ ns$^{-1}$. The delay time is chosen to be $\tau = 0.57$ ns, corresponding to an external cavity length of $8.5$ cm. Changing the feedback phase over $2\pi$ corresponds to relative changes in the external cavity length of $10^{-5}$. The results are shown in Fig.\ \ref{fig13_sp}. Spontaneous emission is strong enough to wash out the deterministic effect completely. We conclude that for this pump value there is no pulse coherence and pulses build up from noise. Only global feedback effects ($\gamma$ and $\tau$) are relevant. In the $100$ mA case we also take the feedback rate to be $\gamma=10$ ns$^{-1}$. The delay time is chosen to be $\tau = 0.3$ ns, corresponding to an external cavity length of $4.5$ cm. Changing the feedback phase over $2\pi$ corresponds to relative changes in the external cavity length of $2\times 10^{-5}$. Fig. \ref{fig14_sp} shows the phase effect at $100$ mA, and it is clearly observable: by merely changing the feedback phase the jitter changes $9$ dB. For pump currents of $100$ mA and larger there is interpulse coherence and feedback phase effects can be instrumental for pulse control. \section{Conclusions} The phenomena of self-pulsations in narrow-stripe semiconductor lasers are theoretically and numerically analyzed using a semiconductor rate-equation model. Around threshold the system suffers from a sub-critical Hopf bifurcation, and the self-pulsation features are compared with the characteristics of sustained relaxation oscillations. The randomizing effect of spontaneous emission noise on the pulse-pulse correlation is found to be limited to the lower pump current regime. This notion is substantiated by applying weak external optical feedback in a Lang-Kobayashi type extension of the model. Assuming single-mode (longitudinal, lateral, and transverse) operation, the effects of feedback are numerically investigated. In a real experiment, single mode operation can be ensured by feedback from a grating, which is a well-known technique to control the emission frequency of a semiconductor laser. In the noisy regime, feedback with delay time on the order of an odd multiple of half of the free running laser pulsation period is shown to be most effective for jitter reduction. In the deterministic regime, where noise is not capable of totally destroying the pulse-pulse correlation, phase-effects, i.e., changes in the delay time corresponding to a fraction of an optical period, have dramatic effects on the jitter. \section*{Acknowledgement} Financial support from the European Union HCM project CHRX-CT94-0594 and from Direcci{\'o}n General de Investigaci{\'o}n Cient\'{\i}fica y T{\'e}cnica (Spain) Project PB94-1167 is acknowledged. We thank J.~O'Gorman for bringing the problem of optical feedback on self-pulsations to our attention. We acknowledge suggestions and fruitful discussions with A. Egan and J.~O'Gorman. \section*{Appendix} \subsection{Zero Power solutions} The zero power solutions, i.e., $P_s = 0$, are the real zeros of the following $9^{th}$ order polynomial in $N_{2s}$, while $N_{1s}$ depends on $N_{2s}$ through (\ref{N1ssubthr}): \begin{equation} \sum_{i=0}^{9} d_i (N_{2s}+1)^{i} = 0, \label{ninthorderpoly} \end{equation} where the coefficients $d_i$ are given by \begin{eqnarray} d_0 &=& - J ,\nonumber\\ % d_1 &=& \left(A_{nr,2} + \frac{1}{T_{21}}\right) \left(A_{nr,1} + \frac{1}{T_{12}}\right) \frac{T_{12}}{N_{12}} \nonumber \\ &-&\frac{f_1}{T_{12}N_{12}}, \nonumber\\ % d_2 &=& \left(A_{nr,2} + \frac{1}{T_{21}}\right)^2 B_1 \left (\frac{T_{12}}{N_{12}}\right)^2\nonumber \\ &+& B_2 \frac{T_{12}}{N_{12}}\left(A_{nr,1} + \frac{1}{T_{12}}\right), \nonumber\\ d_3 &=& \left(A_{nr,2} + \frac{1}{T_{21}}\right)^3 C_1 \left (\frac{T_{12}}{N_{12}}\right)^3\nonumber\\ &+& 2B_2\left(A_{nr,2} + \frac{1}{T_{21}}\right) B_1 \left (\frac{T_{12}}{N_{12}}\right)^2\nonumber\\ &+& C_2 \frac{T_{12}}{N_{12}}\left(A_{nr,1} + \frac{1}{T_{12}}\right), \nonumber\\ d_4 &=& 3B_2\left(A_{nr,2} + \frac{1}{T_{21}}\right)^2 C_1 \left (\frac{T_{12}}{N_{12}}\right)^3\nonumber\\ &+& \left[2C_2\left(A_{nr,2} + \frac{1}{T_{21}}\right) + B_2^2 \right] B_1 \left (\frac{T_{12}}{N_{12}}\right)^2, \nonumber\\ d_5 &=& 3\left(A_{nr,2} + \frac{1}{T_{21}}\right) \left[C_2 \left( A_{nr,2} + \frac{1}{T_{21}} \right) + B_2^2 \right] \times \nonumber\\ & &\times C_1 \left (\frac{T_{12}}{N_{12}}\right)^3 + 2 C_2 B_2 B_1 \left (\frac{T_{12}}{N_{12}}\right)^2, \nonumber\\ d_6 &=& \left[6C_2\left(A_{nr,2} + \frac{1}{T_{21}}\right)+B_2^2\right] C_1 \left (\frac{T_{12}}{N_{12}}\right)^3\nonumber\\ &+& 2 C_2^2 B_1\left (\frac{T_{12}}{N_{12}}\right)^2, \nonumber\\ d_7 &=& 3 C_2 \left[ C_2 \left( A_{nr,2} + \frac{1}{T_{21}} \right) + B_2^2 \right] C_1 \left( \frac{T_{12}}{N_{12}} \right)^3, \nonumber\\ d_8 &=& 3 B_2 C_2^2 C_1 \left( \frac{T_{12}}{N_{12}} \right)^3, \nonumber\\ d_9 &=& C_2^3 C_1 \left( \frac{T_{12}}{N_{12}} \right)^3. \label{d0_9} \end{eqnarray} \subsection{Nonzero Power solutions} The lasing solutions are defined by the real zeros of the following quartic equation in $N_{2s}$, while $N_{1s}$ and $P_s$ simply depend on $N_{2s}$ according to Eqs.\ (\ref{GeqLsupthr},\ref{Pnz}): \begin{equation} \sum_{i=0}^{4} c_i (N_{2s}+1)^{i} = 0, \label{4thorderpoly} \end{equation} where the coefficients are given by: \begin{eqnarray} c_0 &=& -J - \frac{1}{T_{12}}\left(\frac{1+g_2}{g_2}\right) \left(\frac{1+g_1+g_2}{g_1}\right ) \nonumber\\ &+& \gamma_1\left(\frac{1+g_2}{g_1}\right) \left(\frac{1+g_1+g_2}{g_1}\right ), \nonumber\\ c_1 &=&J - \gamma_1\left (\frac{1 + g_2}{g_1}\right ) \left(\frac{1+g_1+2g_2}{g_1}\right ) \nonumber\\ &-& \frac{g_2}{g_1}\left ( \frac{1+g_1+g_2}{g_1}\right ) \left[B_1+2C_1\left(\frac{1+g_1+g_2}{g_1}\right)\right] \nonumber \\ &-& \frac{f_1}{T_{12}N_{12}} + \frac{1}{N_{12}}\left[ \left(\frac{1+g_2}{g_2}\right)\times \right.\nonumber\\ &&\times \left(\frac{g_2N_{12}}{g_1T_{12}}+\frac{1}{T_{21}}+ A_{nr,2}\right) \nonumber\\ &+&\left. \frac{N_{12}}{T_{12}} \left (\frac{1 + g_1+ g_2}{g_1}\right ) \right], \nonumber\\ c_2 &=& C_1 \left( \frac{g_2}{g_1}\right )^2 \left ( \frac{1+g_1+g_2}{g_1}\right ) + \gamma_1\left(\frac{1+g_2}{g_1}\right) \frac{g_2}{g_1} \nonumber \\ &+& \frac{f_1}{T_{12}N_{12}} - \frac{1}{N_{12}}\left[A_{nr,2} - B_2\left(1+\frac{1}{g_2}\right) \right .\nonumber\\ &+& \left. \frac{1}{T_{12}}\left(1 + \frac{g_2N_{12}}{g_1}\right)\right], \nonumber\\ c_3 &=&-C_1\left(\frac{g_2}{g_1}\right)^2 \left( \frac{1+g_1+2g_2}{g_1} \right)\nonumber\\ &-& \left(\frac{g_2}{g_1}\right)^2 \left[B_1 + 2C_1\left ( \frac{1 + g_1+ g_2}{g_1}\right )\right] \nonumber \\ &-& \frac{1}{N_{12}}\left[B_2-C_2\left(\frac{1+g_2}{g_2}\right) \right], \nonumber\\ c_4 &=& C_1\left( \frac{g_2}{g_1}\right)^3 - \frac{C_2}{N_{12}}. \label{c0_4} \end{eqnarray} \begin{thebibliography}{99x} \bibitem{Basov68} N. 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Quantum Electron.}, vol.\ {\bf 30}, pp.\ 241-248, 1994. \end{thebibliography} %\begin{biography}{Guido van Tartwijk} was born in The Netherlands %in 1967. In 1990 he received the M.Sc.\ degree in theoretical physics %from Eindhoven University of Technology, Eindhoven, The Netherlands, %studying magneto-tunneling in double-barrier structures. %In 1994 he received the Ph.D.\ degree in physics from the Vrije %Universiteit, Amsterdam, The Netherlands. His thesis was on %semiconductor laser dynamics with optical injection and feedback. %In 1995 he was at the Universitat de les Illes Balears, Spain, in %the framework of a Human Capital and Mobility project of the %European Union. Some of the results of this project are %described in the present paper. In 1995 he was awarded a %stipend from the Netherlands Organization for Scientific Research %(NWO), which enabled him to start a post-doctoral stay at %the Institute of Optics, University of Rochester, Rochester, NY, %in November 1995. His research interests are semiconductor laser %(spatio-temporal) dynamics, phase-conjugation, delayed optical %feedback, and nonlinear optics. Dr.\ van Tartwijk is a member of the %Optical Society of America. %\end{biography} % %\begin{biography}{}For a photograph and biography of %Maxi San Miguel, see {\em IEEE J. Quantum Electron.}, vol.\ {\bf 29}, %pp.\ 1624-1630, 1993. %\end{biography} %\clearpage \begin{table}[t] \renewcommand{\arraystretch}{0.75} \caption{Narrow Stripe Laser Parameters. The labels $1,2$ refer to the active region and saturable absorbing regions, respectively.} \begin{center} {\tt \begin{tabular}{|llll|} \hline $L$&$300$&$\mu m$&Laser Length\\ \hline $w$&$3$ &$\mu m$ &Stripe Width\\ \hline $d$&$0.08$ &$\mu m$ &Layer Thickness\\ \hline $V_1$ &$72$ &$(\mu m)^3$ &Volume\\ \hline $V_2$ &$45.62$ &$(\mu m)^3$ &Volume\\ \hline $T_{12}$ &$2.647$ &$ns$ &Diffusion time\\ \hline $n_g$ &$3.6$ &$-$ &Group Index\\ \hline $\Gamma_0$ &$0.367$ &$(ps)^{-1}$ &Cavity Decay Rate\\ \hline $\xi_1$ &$4.28\times 10^4$ &$s^{-1}$ &Differential Gain \\ \hline $\xi_2$ &$2.70\times 10^4$ &$s^{-1}$ &Differential Gain \\ \hline $\Gamma_1$ &$0.1834$ &$-$ &Total Confinement\\ \hline $\Gamma_2$ &$0.1748$ &$-$ &Total Confinement\\ \hline $N_{t1}$ &$1.008\times 10^8$ &$-$ &Transparency Number \\ \hline $N_{t2}$ &$7.3\times 10^7$ &$-$ &Transparency Number \\ \hline $\alpha_1$ &$5$ &$-$ &$\alpha$-factor\\ \hline $\alpha_2$ &$5$ &$-$ &$\alpha$-factor\\ \hline $A_{nr,i}$ &$0.1$ &$(ns)^{-1}$ &Nonradiative\\ & & &Recombination\\ & & &Rate\\ \hline $B_1N_{t1}$ &$0.42$ &$(ns)^{-1}$ &Bimolecular\\ & & &Recombination\\ & & &Rate at $N_{t1}$\\ \hline $B_2N_{t2}$ &$0.48$ &$(ns)^{-1}$ &Bimolecular\\ & & &Recombination\\ &&&Rate at $N_{t2}$\\ \hline $C_1N_{t1}^2$ &$0.1372$ &$(ns)^{-1}$ &Auger\\ & & &Recombination\\ & & &Rate at $N_{t1}$\\ \hline $C_2N_{t2}^2$ &$0.1792$ &$(ns)^{-1}$ &Auger\\ & & &Recombination\\ & & &Rate at $N_{t2}$\\ \hline $C_{sp}$ &$10^{-5}$ &$-$ &Spontaneous\\ & & &Emission Ratio\\ \hline $\lambda_0$ &$800$ &$nm$ &Wavelength\\ \hline \end{tabular} } \end{center} \end{table} \clearpage \begin{figure}[f] \caption{Schematic of the narrow-stripe semiconductor laser. Two current blocking (CB) layers only allow the pump current through a stripe of width $w$ of the central layer. This central layer is thus divided into an active region (A) and two saturable absorbing (SA) regions.} \label{fig1_sp} \end{figure} \begin{figure}[f] \caption[]{The fixed point values of the carrier number in the active region as a function of the pump current. The top figure (a) shows $N_{1s}$ for the nonlasing solution which becomes unstable at $58$ mA. The bottom figure (b) shows the lasing solution which only is stable for currents larger than $118$ mA. Solid lines indicate linearly stable solutions and dashed lines indicate linearly unstable solutions, as described after Eq.\ (\ref{chareq}).} \label{fig2_sp} \end{figure} \begin{figure}[f] \caption[]{The light-current characteristic of the narrow-stripe laser with parameters as listed in Table I. Dashed lines indicate instability. Note that for currents smaller than the threshold current, at which the zero solution becomes unstable, already lasing solutions are possible.} \label{fig3_sp} \end{figure} \begin{figure}[f] \caption[]{The real (a) and imaginary (b) parts of the complex conjugate pair of roots of the characteristic equation responsible for the unstable behavior of the upper branch of lasing solutions. Coming from high pump, at $118$ mA the relaxation oscillation are not damped any more. They remain ``undamped" until the solution itself vanishes around $48$ mA.} \label{fig4_sp} \end{figure} \begin{figure}[f] \caption[]{The pulse frequency with and without noise compared to the imaginary part of the relevant characteristic exponent.} \label{fig5_sp} \end{figure} \begin{figure}[f] \caption[]{The pulse period $$ (a) and the jitter $\sigma$ (b) for the free running self pulsating laser. From $118$ mA onwards, no pulses are observed.} \label{fig6_sp} \end{figure} \begin{figure}[f] \caption[]{The floor value of the photon number reached in between pulses, with and without noise. The effect of noise is negligible for pump values larger than $\approx 100$ mA.} \label{fig7_sp} \end{figure} \begin{figure}[f] \caption[]{The effect of the feedback time on the pulse period (top) and the jitter (bottom) at pump current of $80$ mA, feedback rate $\gamma = 10$ ns$^{-1}$, and feedback phase $\omega_{0}\tau = 3\pi/2$. The free running laser has an average pulse period $ = 0.387$ ns with jitter $12$ ps. } \label{fig8_sp} \end{figure} \begin{figure}[f] \caption[]{The effect of the feedback time on the pulse period (top) and the jitter (bottom) at pump current of $100$ mA, feedback rate $\gamma = 10$ ns$^{-1}$, and feedback phase $\omega_{0}\tau = 3\pi/2$. The free running laser has an average pulse period $ = 0.242$ ns with jitter $7$ ps. } \label{fig9_sp} \end{figure} \begin{figure}[f] \caption[]{The effect of the feedback rate on the pulse period (top) and the jitter (bottom) at pump current of $80$ mA, delay time $\tau = 0.57$ ns, and feedback phase $\omega_{0}\tau = 3\pi/2$. The free running laser has an average pulse period $ = 0.387$ ns with jitter $12$ ps. } \label{fig10_sp} \end{figure} \begin{figure}[f] \caption[]{The effect of the feedback rate on the pulse period (top) and the jitter (bottom) at pump current of $100$ mA, delay time $\tau = 0.3$ ns, and feedback phase $\omega_{0}\tau = 3\pi/2$. The free running laser has an average pulse period $ = 0.242$ ns with jitter $7$ ps.} \label{fig11_sp} \end{figure} \begin{figure}[h] \caption[]{The effect of the feedback phase on the the time between two consecutive pulses, in the case of a pump current of $117.5$ mA, a feedback rate of $\gamma = 5$ (ns)$^{-1}$, and a delay time of $\tau = 0.21$ ns. Feedback is switched on at $n =75$ while noise is switched on at $n = 389$ (dotted line). } \label{fig12_sp} \end{figure} \begin{figure}[h] \caption[]{The effect of the feedback phase on the pulse period (top) and the jitter (bottom) at pump current of $80$ mA, feedback rate $\gamma = 10$ ns$^{-1}$, and delay time $\tau = 0.57$ ns. The free running laser has an average pulse period $ = 0.387$ ns with jitter $12$ ps. } \label{fig13_sp} \end{figure} \begin{figure}[h] \caption[]{The effect of the feedback phase on the pulse period (top) and the jitter (bottom) at pump current of $100$ mA, feedback rate $\gamma = 10$ ns$^{-1}$, and $\tau = 0.3$ ns. The free running laser has an average pulse period $ = 0.242$ ns with jitter $7$ ps.} \label{fig14_sp} \end{figure} %\clearpage %\begin{biography}{Guido van Tartwijk} was born in The Netherlands %in 1967. In 1990 he received the M.Sc.\ degree in theoretical physics %from Eindhoven University of Technology, Eindhoven, The Netherlands, %studying magneto-tunneling in double-barrier structures. %In 1994 he received the Ph.D.\ degree in physics from the Vrije %Universiteit, Amsterdam, The Netherlands. His thesis was on %semiconductor laser dynamics with optical injection and feedback. %In 1995 he was at the Universitat de les Illes Balears, Spain, in %the framework of a Human Capital and Mobility project of the %European Union. Some of the results of this project are %described in the present paper. In 1995 he was awarded a %stipend from the Netherlands Organization for Scientific Research %(NWO), which enabled him to start a post-doctoral stay at %the Institute of Optics, University of Rochester, Rochester, NY, %in November 1995. His research interests are semiconductor laser %(spatio-temporal) dynamics, phase-conjugation, delayed optical %feedback, and nonlinear optics. Dr.\ van Tartwijk is a member of the %Optical Society of America. %\end{biography} %\begin{biography}{}For a photograph and biography of %Maxi San Miguel, see {\em IEEE J. Quantum Electron.}, vol.\ {\bf 29}, %pp.\ 1624-1630, 1993. %\end{biography} \end{document}