\documentstyle[12pt]{article} %\pagestyle{myheadings} \leftmargin 1.5625in \topmargin 0cm \textwidth 6.in \textheight 8.2in \begin{document} \title{Modulation response of Quantum Well lasers with carrier transport effects under weak optical feedback} \author{M. Homar$^1$, C. R. Mirasso$^2$, I. Esquivias$^3$, and M. San Miguel$^{1,4}$} \date{~} \maketitle \vspace{-1cm} \baselineskip 24pt \noindent {1- Departament de F\'\i sica, Universitat de les Illes Balears, E-07071 Palma de Mallorca, Spain.} {\noindent 2- Department of Physics and Astronomy, Vrije Universiteit, De Boelelaan 1081, 1081 HV, Amsterdam, The Netherlands} {\noindent 3- Departamento de Tecnolog\'{\i}a Fot\'{o}nica, Universidad Polit\'{e}cnica de Madrid, E-28040 Madrid, Spain.} {\noindent 4- Instituto Mediterraneo de Estudios Avanzados, IMEDEA (CSIC-UIB), E-07071 Palma de Mallorca, Spain.} \begin{center} \section*{Abstract} \end{center} \vspace{0.5cm} We theoretically investigate the influence of optical feedback on the modulation response of quantum-well lasers. Controlled weak optical feedback is shown to be useful in overcoming the high frequency limitations imposed by carrier-transport effects. With a fine tuning of the external cavity length the maximum attainable modulation bandwidth can be increased. \pagebreak Modulation bandwidths ($f_{3dB}$) higher than in bulk devices have been recently demonstrated in Quantum-Well (QW) lasers, achieving 40 GHz \cite{1} and 25 GHz \cite{2} for InGaAs/GaAs and InGaAsP/InP material systems, respectively. However, while the maximum intrinsic modulation bandwidth $f^{max}_{3dB}$ is limited in bulk lasers by damping, carrier transport introduces additional limitations in QW lasers \cite{4,5}. For bulk lasers $f^{max}_{3dB}$ is expressed in terms of the $K$-factor as $f^{max}_{3dB} \approx 8.9/K$ \cite{3}, with $K = 4\pi^2 (\tau_p + \frac{\epsilon}{\nu_g g_n})$ (see Table I for parameter definitions). Carrier transport in QW lasers, including carrier diffusion across the confinement layer, carrier capture into the QW (both processes characterized by a single effective capture time $\tau_{cap}$), and carrier escape out of the QW (with effective escape time $\tau_{esc}$) influences the small signal modulation response (MR) in two ways: {\it i)} it introduces a parasitic-like low frequency roll-off, with cut-off frequency $(2\pi \tau_{cap})^{-1}$ and {\it ii)} it decreases the effective differential gain and increases the apparent K-factor when the ratio $\tau_{cap}/\tau_{esc}$ is comparable to or higher than one. The consequence is that $f^{max}_{3dB}$ in QW lasers, due to transport effects, can be reduced in comparison with the attainable values for laser structures where these effects do not come into play. The performance of a laser diode can be greatly modified by optical feedback \cite{peter},\cite{lenstra}. In particular, the MR of bulk lasers is known to be significantly affected by optical feedback \cite{LK}. It is also known that small amounts of feedback are useful for linewidth reduction, but large feedback intensities can degrade the laser performance through the occurrence of the "coherence collapse" which gives rise to several GHz linewidth \cite{peter},\cite{lenstra}. Pattern effects and jitter of large-signal intensity modulated lasers are also drastically changed by weak optical feedback in extremely short external cavities as has been shown analitically and numerically \cite{alan}, and experimentally \cite{jaume}. In this work we explore the effects of weak optical feedback from short external cavities of length $L_E \approx 1.5$ mm (which correspond to an external round-trip time $\tau_E \simeq 10$ ps) in the small signal modulation properties of QW lasers, and we find that it can be instrumental in modifying and compensating the detrimental carrier transport effects. Our analysis is based on standard single-mode rate equations for QW lasers \cite{4}, which incorporate the dynamics of the unconfined carriers in the confinement and barrier regions (core). These equations are supplemented by weak feedback terms describing single reflection for an external reflector in the way described by Lang and Kobayashi for bulk lasers \cite{LK}. Optical feedback is characterized by $\tau_E$ and the coupling parameter $\kappa= \frac{\sqrt{R_{ext}} (1-r)}{\tau_L \sqrt{r}}$, with $R_{ext}$ the power external reflectivity, $r$ the laser-facet reflectivity and $\tau_L$ the laser cavity round-trip time. The equations for the photon number $S$ in the cavity, the optical phase $\phi$, the carrier number in the QWs $N_{\omega}$, and the carrier number in the core $N_b$ are: \begin{eqnarray} \frac{dS(t)}{dt} & = & \left( \frac{\Gamma \nu_g g_n (N_{\omega}-N_t) }{V_{QW}(1+\epsilon S(t))}- \frac{1}{\tau_p}\right) \, S(t) + \frac{\beta}{\tau_{eff}} \, N_{\omega} \nonumber \\ &+& 2 \, \kappa \sqrt{S(t-\tau_E) S(t)} \cos\left[\omega_s \tau_E + \phi(t) - \phi(t-\tau_E)\right] \\ \frac{d\phi(t)}{dt} & = & \omega_t - \omega_m + \frac{\alpha}{2} \frac{\Gamma \nu_g g_n (N_{\omega}-N_t)}{V_{QW}(1+\epsilon S(t))} \nonumber \\ &-&\kappa \sqrt{\frac{S(t-\tau_E)}{S(t)}} \sin\left[\omega_s \tau_E + \phi(t) - \phi(t-\tau_E)\right] \\ \frac{dN_{\omega}}{dt} & = & \frac{N_b}{\tau_{cap}} - \frac{N_w}{\tau_{esc}} - \frac{N_w}{\tau_{eff}} - \frac{\Gamma \nu_g g_n (N_{\omega}-N_t)}{V_{QW}(1+\epsilon S(t))} \, S(t) \\ \frac{dN_{b}}{dt} & = & \frac{I}{e} - \frac{N_b}{\tau_{cap}} + \frac{N_{\omega}}{\tau_{esc}}~~~~, \end{eqnarray} \noindent where $I$ denotes the bias current, $e$ the electronic charge, $\omega_s$ and $\omega_m$ the stationary optical frequencies with and without feedback, respectively, and $\omega_t$ the frequency at transparency. Other parameters are defined in table I. We find the stationary solution of (1)-(4) which describes the feedback- modified laser emission, and calculate the MR by a small-signal analysis in the vicinity of such solution. We illustrate the effects of feedback on the MR using the experimentally determined parameters for a p-doped InGaAs/GaAs QW laser [10], given in table I. These lasers have demonstrated 30 GHz modulation bandwidth and the K-factor yields $f^{max}_{3dB} = 63$ GHz. Transport effects are not important for this laser structure ($\tau_{cap} \approx 2$ ps and $\tau_{esc} > 1$ ns \cite{11}), but we study the feedback influence by assuming different $\tau_{cap}$ and $\tau_{esc}$. In the calculations we consider a fixed $\kappa=0.05$ ps$^{-1}$, which correspond to $R_{ext} \approx 0.05$, and use $\tau_E$ as the parameter controlling feedback. Monitoring $\tau_E$ fixes the phase of the reinjected field. The variations in threshold current ($I_{th} \simeq 17.6$ mA) and emission wavelength ($\lambda = 1.095$ $\mu$m) due to feedback are $<10\%$ and $<< 1\%$, respectively. Figure 1 shows typical modifications of the MR due to optical feedback. We plot the MR at $I=3.5 I_{th}$ for the solitary laser ($\kappa = 0$, curve e) and the laser with feedback (curves a-d). It is clear that both, the apparent resonance frequency and the damping, are extremely sensitive to $\tau_E$. By an appropriate choice of $\tau_E$, it is possible to increase $f_{3dB}$ up to $20 - 30 \%$ with respect to the solitary laser. The dependence of $f_{3dB}$ on $\tau_E$ is shown in figure 2, considering the cases $\tau_{cap} << \tau_{esc}$, $\tau_{cap} \sim \tau_{esc}$ and $\tau_{cap} \rightarrow 0$, $\tau_{esc} \rightarrow \infty$, the latter being the limit for a bulk laser. The modulation bandwidth oscillates periodically with the external round-trip time $\tau_E$ with a period of $0.005$ ps. Consequently, the high-frequency performance can be improved if $\tau_E$ is carefully adjusted with a precision $<< \Delta \tau_E$, i.e. $L_E$ should be controlled within half optical wavelength. Such fine control can be achieved either by using an external reflector controlled with high-precision positioners, or by integrating the laser diode together with a reflector and controlling the optical length, similarly to the phase control in multisection lasers. Our results show that the optimum $\tau_E$ presents a weak dependence on the injection level and therefore a close-loop system could look automatically for the best feedback conditions. The physics behind the feedback effects on the MB is related to the coupling between the intensity and the frequency modulation, as it has been discussed by Peterman \cite{peter} and therefore it is aplicable to both bulk and quantum well lasers. Simulations on the feedback influence on a bulk laser ($\tau_{cap} \rightarrow 0$, $\tau_{esc} \rightarrow \infty$ and standard bulk laser parameters \cite{agra}) yield variations of the MR qualitatively similar to those shown in figs. 1 and 2. We emphasize the possible improvements in QW lasers due to their high speed potentialities and to their intrinsic limitations caused by transport effects. On the other hand, figures 1 and 2 show that a weak unintentional feedback level can reduce drastically the high-frequency performance without apparently affecting other device characteristics such as emission wavelength or threshold current and this should be taken into account when operating the laser in real conditions or when measuring its characteristics. Figure 3 illustrates the possible improvements on $f_{3dB}$ for the QW laser whose parameters are shown in table I and with different regimes of $\tau_{cap}$ and $\tau_{esc}$. We plot the dependence of $f_{3dB}$ on the injection current considering the optimum feedback $\tau_{esc}= 10.003$ $ps$ and different regimes of $\tau_{esc}$ and $\tau_{cap}$. In fig. 3a) we consider $\tau_{cap}=2$ ps, i.e., the cut-off frequency due to $\tau_{cap}$ is much higher than the $K$-factor limit. For $\tau_{esc}=2$ ns, $f^{max}_{3dB}$ for the solitary laser is $\simeq 56$ GHz (curve A), corresponding to $\approx 8.9/K$, while the response with optimized feedback can achieve up to $\simeq 65$ GHz (curve B). On the other hand, for $\tau_{esc}=2$ ps, $f^{max}_{3dB}$ for the solitary laser is reduced due to the high value of $\tau_{cap}/\tau_{esc}$ down to 30 GHz (curve C). The use of optimized feedback improves $f^{max}_{3dB}$ up to $\simeq 50$ GHz (curve D) almost reaching the value for the laser with negligible transport effects. In fig. 3b) we consider the case of lasers with $\tau_{cap}=5$ ps in which $f^{max}_{3dB}$ is limited by $\tau_{cap}$. For the solitary laser with $\tau_{esc}=2$ ns, $f_{3dB}$ has a maximum of $\simeq 35$ GHz but saturates to $(2 \pi \tau_{cap})^{-1} \simeq 30$ GHz (curve A). By using an optimum feedback, $f_{3dB}$ is enhanced up to $\simeq 55$ GHz (curve B). The improvement is more significant at a certain current range. Finally, both detrimental transport effects are considered for the solitary laser in curve C ($\tau_{esc} = \tau_{cap})$. The response is again improved by controlled feedback (curve D). However, for large input currents $f_{3dB}$ saturates very fast to the value without feedback. In summary, controlled weak optical feedback gives the possibility of overcoming the limitations in the maximum intrinsic modulation frequency imposed by carrier transport effects in QW lasers. In cases in which $\tau_{cap}$ alone does not limit $f^{max}_{3dB}$ but $\tau_{cap} \simeq \tau_{esc}$, it is possible to choose the external cavity length in such a way that optical feedback improves the $f_{3dB}$ to those of situations in which carrier transport effects can be neglected. Moreover, if in this situation the external mirror is carefully adjusted the relaxation oscillation frequency could be very small with a still large $f_{3dB}$. In cases in which $f^{max}_{3dB}$ is limited by $\tau_{cap}$ there is an optimum range of injection currents for which the maximum attainable frequency can be increased by weak optical feedback. \vspace{0.5cm} \section*{Acknowledgment} This work was partially supported by the Comisi\'on Interministerial de Ciencia y Tecnolog\'\i a, Project TIC93/0744 and Project HCM-CHRX-CT94-0594 of the European Union. \pagebreak \begin{thebibliography}{99} \bibitem{1}S. Weisser, E. C. Larkins, K. Czotscher, W. Benz, J. Daleiden, I. Esquivias, J. Fleissner, J. D. Ralston, B. Romero, R. E. Sah, A. Schenfelder and J. Reosenzweig, %``Damping-limited modulation bandwidths up to 40 GHz in undoped Short-Cavity $In_0.35Ga_0.65As/GaAs$ Multiple Quantum Well Lasers", to be publushed in IEEE Photon. Technol. Lett. \bibitem{2} P. A. Morton, R. A. Logan, T. Tanbun-Ek, P. F. Sciortino Jr., A. M. Sergent, R. K. Montgomery and B. T. Lee, %``25 GHz bandwith 1.55 $\mu$m GaInAsP $p$-doped strained multiquantum well lasers", Electron. Lett {\bf 28}, pp. 2156-2157, 1992. \bibitem{4} R. Nagarajan, M. Ishikawa, T. Fukushima, R. S. Gills and J. E. Bowers, %``High speed quantum-well lasers and carrier transport effects", IEEE J. Quantum Electron. {\bf QE-28}, pp. 1990-2008, 1992. \bibitem{5} S. C. Kan, D. Vassilovski, T. C. Wu and K. Y. Lau, %``On the effects of carrier diffusion and quantum capture in high speed modulation of quantum well lasers", Appl. Phys. Lett. {\bf 61}, pp. 752-754, 1992. \bibitem{3} R. Olshansky, P. Hill, V. Lanziera and W. Powazinik, %``Frequency response of a 1.3 $\mu$m InGaAsP high speed semiconductor lasers", IEEE J. Quantum Electron. {\bf QE-23}, pp. 1410-1418, 1987. \bibitem{peter} K. Petermann, {Laser Diode Modulation and Noise}, (Kluwer Academic Publishers, Dordrecht, 1988). \bibitem{lenstra} D. Lenstra, %``Feedback Noise in Single-Mode Semiconductor Lasers", SPIE Proc. 1376, pp. 245-258, 1991. \bibitem{LK} R. Lang and K. Kobayashi, %``External optical feedback effects on semiconductor injection laser properties", IEEE J. Quantum Electron. {\bf QE-16}, pp. 347-355, 1980. \bibitem{alan} E. Hern\' andez 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 QE-30}, pp. 241-248, 1994. \bibitem{jaume} J. Dellunde and A. Sapia, %``Gain-Switching of Laser Diodes Coupled to Optical Fibers", IEEE Phot. Tech. Lett. {\bf 7}, pp. 1258-1260, 1995. \bibitem{10} J. D. Ralston, S. Weisser, I. Esquivias, E. C. Larkins, J. Rosenzweig, P. J. Tasker, and J. Fleissner, %``Control of Differential Gain, Non-Linear Gain and Damping Factor for High Speed Application of GaAs-Based MQW Laser", IEEE J. Quantum Electron. {\bf QE-29}, pp. 1648-1659, 1993. \bibitem{11} S. Weisser, I. Esquivias, P. J. Tasker, J. D. Ralston, and J. Rosenzweig, %``Impedance, Modulation Response, and Equivalent Circuit of Ultra-High-Speed In$_{0.35}$Ga$_{0.65}$As/GaAs MQW Lasers with p-doping", IEEE Photon. Technol. Lett. {\bf 6}, pp. 782-785, 1994. \bibitem{agra}G. P. Agrawal and N. K. Dutta, {\em Long-Wavelength Semiconductor Lasers}, Van Nostrand Reinhold, New York, 1986. \end{thebibliography} \pagebreak \begin{table}[h] \begin{center} \section*{Table I} {\bf Meaning and values of the different parameters appearing in the model} \vspace{1cm} \begin{tabular}{|c|c|c|c|} \hline Parameter & Meaning & Value & Units \\ \hline \hline $\Gamma$ & Optical confinement factor & $0.088$ &adim.\\ $V_{QW}$ & Volume of Quantum Wells & $2.05 \times 10^{-11}$ & cm$^3$ \\ $\nu_g$ & Group velocity & $7.7 \times 10^{9}$ & cm/s \\ $g_n$ & Differential gain & $2.5 \times 10^{-15}$ & cm$^{2}$ \\ $N_t$ & Carrier number at transparency & $8.2 \times 10^6$ & adim. \\ $\epsilon$ & Nonlinear gain saturation & $2 \times 10^{-7}$ & adim. \\ $\beta $ & Spontaneous emission factor & 1$\times 10^{-5}$ & adim. \\ $\alpha$ & Linewidth enhancement factor & 1.4 & adim. \\ $\tau_{p}$ & Photon lifetime & 1.2 & ps \\ $\tau_{eff}$ & Carrier lifetime in the QWs & 0.17 & ns \\ \hline \end{tabular} \end{center} \end{table} \pagebreak \section*{Figure Captions} \noindent Figure 1. Modulation response ($f_{3db}$) vs. modulation frequency for $\kappa=0.05$ ps$^{-1}$ and different $\tau_E$ (A-D) and $\kappa=0$ (E), with $\tau_{cap}=2$ ps and $\tau_{esc}=2$ ns. \vspace{0.5cm} \noindent Figure 2. Modulation response ($f_{3db}$) vs. the external round-trip time $\tau_E$, for $\tau_E \sim 10$ ps, $\kappa=0.05$ ps$^{-1}$ and injection current 3.5 $I_{th}$. Solid line corresponds to a bulk laser, dashed line corresponds to a QW laser with $\tau_{cap}=5$ ps and $\tau_{esc}=2$ ns and three-dot-dash line corresponds to a QW laser with $\tau_{cap}=\tau_{esc}=5$ ps. The horizontal lines correspond to the value of the $f_{3db}$ without feedback. \vspace{0.5cm} \noindent Figure 3. Modulation bandwidth vs. $(I-I_{th})^{1/2}$ for $\tau_{cap}=2$ (panel a) and 5 ps (panel b). Solid lines $\kappa=0$ (A: $\tau_{esc}=2$ ns, and C: $\tau_{esc}=\tau_{cap}$); dashed lines $\kappa=0.05$ ps$^{-1}$ and $\tau_E=10.003$ ps (B: $\tau_{esc}=2$ ns, and D: $\tau_{esc}=\tau_{cap}$). \end{document}