3o'9' - The University of Adelaide Department of Physics and Mathematical Physics Modelling and Experiments on the Behaviour of Iniection Current Modulated Multimode Semiconductor Lasers By Kerry Corbett Thesis submitted for the degree of Doctor of Philosophy kr March, 1999 Abstract This thesis investigates the behaviour of semiconductor lasers under large amplitude sinu- soidal current modulation. As these systems have applications in optical fibre communication systems, an understanding of their behaviour is highly desirable. Semiconductor lasers are also nonlinear systems and can exhibit complicated dynamics depending upon their operat- ing conditions. For this reason, they are also widely studied as a system in which to test, experimentally, the theory of nonlinear dynamical systems. This thesis focuses directly on the behaviour of Fabry-Perot (FP) lasers under direct modulation of the injection current. It encompasses both modelling of these devices using the laser rate equations and experi- mental measurements. Experimentally, the dynamics of the laser is probed via measurement of the power spectra of the output intensity and of the time average longitudinal mode spec- tra. These measurements are performed over a wide range of modulation frequencies and modulation amplitudes. A main component of the research involves numerical modelling using the laser rate equations. In most previous investigations the single mode rate equations were used. How- ever, a FP laser operates with multiple longitudinal modes. Therefore, in this thesis vr'e use the multimode rate equations to investigate the bifurcation scenarios occurring with change in the modulation frequency and amplitude. We also investigate the effect on the dynamics of the shift in the gain peak with an increase in carrier density, resulting from the popula- tion of higher energy electronic states (referred to as band-filling)[1]. We show that both these effects contribute significantly to the predicted bifurcation scenarios leading to large deviation from the single mode predictions. In particular, we show that inclusion of multiple longitudinal modes acts to reduce the extent of bifurcations occurring under current modu- lation. Such predictions are more consistent with the bifurcations (or lack thereof) observed experimentally in FP semiconductor lasers[2]. Previous studies[l] on the effects of multiple longitudinal modes and band-filling on the behaviour of the time average longitudinal mode spectra variation with modulation frequency showed that, when the modulation frequency is coincident wiih the relaxation oscillation frequency, multimode operation ensues. Moreover, band-filling effects lead to a spectral shift of the longitudinal mode spectra towards the shorter wavelengthmodes[1,36]. We demonstrate, experimentally and numerically, that this behaviour also occurs when a harmonic of the modulation frequency is coincident with the relaxation osciliation frequency. In previous studies it was shown that a system's sensitivity to noise near a bifurcation manifests itself as structure in the por,¡/er spectra of the system observables. Such structures are called noise precursors. Investigations into the dynamics of period doubling bifurcations in a current modulated semiconductor laser have shown that in these systems the precursor to the bifurcation consists of the relaxation oscillation frequency being pulled to half the modulation frequency before the onset of the bifurcationl3,4]. This has been observed experimentally in a (single mode) distributed feed-back semiconductor laser and explained theoretically using the single mode rate equations[4]. We show that this is also predicted by the multimode rate equations and show experimental resuits for a multimode FP laser. Other bifurcations predicted by the multimode rate equations include the saddle- node bifurcations and Hopf bifurcations. We show that at a saddle-node bifurcation point, the multimode rate equations predict an abrupt mode hop to an adjacent longitudinal mode, which is usually accompanied by hysteresis. Hopf bifurcation on the other hand leads to the presence of another frequency, other than the driving frequency and its harmonics in the power spectrum of the photon density. To our knowledge neither of these two bifurcations have been previously discussed in reference to the multimode semiconductor laser rate equa- tions. ll 1V Acknowledg*ents I would foremost like to thank my supervisor, Murray Hamilton, for his help and patience throughout the period of my candidature. I would also like to thank Laurence Stamatescu whose help and encouragement, especially during the earlier years of my PhD, was invaluable. Thanks also go to Shu and Damien for their thorough proof reading of parts of the manuscript and to Blair Middlemiss for his technical support. Finally, I wish to thank the other members of the optics group, who have made it impossible for me to regard my time here with anything other than pleasure. Last, but not least, I wish to thank my mother and father and Uncle Larry and Fong without whom I would never have started let alone completed my PhD. I would especially like to thank my mother and father for their support throughout my entire tertiary education. V V1 Contents 1 Introduction ¿) 2 Semiconductor Laser Theory I 2.1 Introduction I 2.2 Semiconductor Laser Structures 10 2.2.1 Introduction 10 2.2.2 Semiconductor Laser Materials 11 2.2.3 Heterostructures 12 2.2.4 Laser Modes 13 2.3 Relevant Concepts from Solid State Physics 15 2.3.1 Introduction 15 2.3.2 Semiconductor Band Structure t7 2.3.3 Quantum Description of the Semiconductor 23 2.3.4 Fermi-Dirac Distribution and Density of States 30 2.4 Semiclassical Laser Theory 33 2.4.1 Introduction 33 2.4.2 Electromagnetic Field Equations 34 2.4.3 Free Carrier Theory 37 2.4.4 The Rate Equation Approximation 41. 2.4.5 Free Carrier Gain 44 2.4.6 The Multimode Rate Equations 47 2.4.7 Laser Noise 52 2.5 Other Considerations 53 2.5.1 Coulomb Interactions 53 2.5.2 Linewidth Enhancement Factor (a) 54 2.5.3 Device Parasitics 55 2.5.4 Carrier Diflusron bb 2.5.5 Nonlinear Gain 56 2.5.6 Summary Ðt 1 2 CONTENTS 3 Semiconductor Laser Diagnostics 59 3.1 Experimental Arrangement 59 3.2 Steady State Operation 60 at t). t) Rate Equation Parameter Values 64 3.3.1 Carrier Lifetime 64 3.3.2 Gain Spectra 6( 3.3.3 Spontaneous Emission Parameter 72 3.3.4 Summary of Results 72 4 Bifurcation Scenarios tÐ 4.I Overview 76 4.L.1 Introduction to Nonlinear Dynamics 76 4.L.2 Nonlinear Dynamics of Lasers ll 4.7.3 Nonlinear Dynamics of Semiconductor Lasers 79 4.2 Attractors 83 4.3 Bifurcation Diagrams 86 4.3.7 Comparison of Single Mode and Multimode Predictions 86 4.3.2 Effects of Band-filling . 90 4.4 Modal Behaviour 97 4.4.7 NumericalResults 97 4.4.2 Experimental Results 101 +.5 Spontaneous Emission Factor 103 4.6 Bifurcations 108 4.6.7 Period Doubling Bifurcations 108 4.6.2 Hysteresis 110 4.6.3 Hopf Bifurcations TT2 4.7 Global Behavrour 115 4.7.l NumericalResults 115 4.7.2 Experimental Results . I17 4.8 Quantitative Comparisons I2I 4.9 Concluding Remarks t24 5 Approach to Bifurcations L27 5.1 Introduction I27 5.2 Nonlinear Differential Equations t28 5.2.7 Linear Stability Analysis 128 5.2.2 Stochastic Differential Equations t32 5.3 Bifurcations and Transients 734 CO¡\ITE^rTS ,) 5.3.1 Nonlinear Oscillators 734 5.3.2 Overview 135 5.3.3 Period Doubling Bifurcations 138 5.3.4 Hopf Bifurcations 740 5.3.5 Saddle-node Bifurcations 143 5.3.6 Discussion 143 5.4 Period Doubling Bifurcations r45 5.4.I Single Mode System 745 5.4.2 Multimode System 749 5.5 Saddle-node Bifurcations 156 5.5.1 Single Mode System 160 5.5.2 Multimode System 160 5.6 Hopf Bifurcations 165 5.6.1 NumericalResults 165 5.6.2 Experimental Results r67 Ð.1 Global Behaviour 169 5.7.1 Single Mode System 170 5.7.2 Multimode Equations 175 5.8 Summary 178 6 Conclusion 185 A Material Properties of AlGaAs 189 B Field Quantisation 191 8.1 Classical Langrangian Field Theory 191 8.2 Field Quantisation t92 C Laser Diode Drive Circuit 195 D Steady State Solutions and Linearised Rate Equations 199 D.1 Single Mode Rate Equations 199 D.2 Multimode Equations 206 E Numerical Integration 2Lt F Variational Equation 2L3 4 CO¡\ITEf\rTS Chapter 1 Introduction Large amplitude modulation of the injection current of a semiconductor laser at microwave frequencies has applications in optical fibre communication systems[5,6,7,8]. For this application, ideally we require the output intensity to exactly follow the evolution of the input injection current and therefore a linear device is desirable. A semiconductor laser is, however, like most physical systems, inherently nonlinear. Its behaviour under direct modulation of the injection current is analogous to that of a nonlinear oscillatorf9, 10]. It exhibits a damped relaxation oscillation whose physical origin lies in the different decay rates of the electron and hole populations and the photon density within the semiconductor medium[6]. For applications in optical communications, large amplitude modulation of the injection current is required in order to achieve significant signal to noise ratio. Under these conditions the nonlinearities become important ieading to distortion of the input signal. Therefore, knowledge and understanding of this behaviour is essential for the implementation of these devices in practical applications. Unlike a linear oscillator, the resonant frequency of a nonlinear oscillator is not constant but depends on the amplitude and frequency of the input modulation[11]. Further- more, if strongly driven, this resonance can become unstabie, through bifurcation, leading to an undamped oscillation[12]. These types of instabilities are a very important aspect of dissipative nonlinear dynamical systems as they are responsible for the vast array of be- haviours exhibited by these systems, including strange attractors and chaotic motion, as opposed to the fixed point and periodic motions of a linear system[13].