Microscopic Spatio-Temporal Dynamics of Semiconductor Quantum Well Lasers and Amplifiers Von der Fakult¨at Mathematik und Physik der Universit¨at Stuttgart zur Erlangung der Wu¨rde eines Doktors der Naturwissenschaften (Dr. rer. nat.) genehmigte Abhandlung Vorgelegt von Klaus B¨ohringer aus Waiblingen Hauptberichter: Prof. Dr. O. Hess Mitberichter: Prof. Dr. G. Mahler Tag der mu¨ndlichen Pru¨fung: 4. April 2007 Institut fu¨r Technische Physik Deutsches Zentrum fu¨r Luft- und Raumfahrt (DLR) Pfaffenwaldring 38–40 D-70569 Stuttgart 2007 Summary This work discusses light-matter interaction and optical nonlinearities in semiconductor nanostructuresand presents a detailednumericalanalysis of the spatio-temporaldynam- ics in novel high-power diode lasers. We derive a microscopic, spatially resolved model that combines a density matrix approach to the optoelectronic properties of quantum well gain media with the macroscopic Maxwell equations for the electromagnetic field dynamics. We present Maxwell semiconductor Bloch equations in full time-domain that cover many-particle interactions, a diversity of time scales and gain saturation mech- anisms, and inclose the fast-oscillating carrier wave and a sub-wavelength spatial res- olution. Microscopically calculated scattering rates are incorporated into our spatially resolved model. Our work focuses on ultrafast carrier effects, a quantitative understanding of optical nonlinearities, the engineering of the mode structure in microcavities, and their im- pact on the laser emission characteristics. Optical dephasing and carrier and energy redistribution due to the screened Coulomb interaction and scattering with phonons are explored in detail. We study the technologically important structure of a broad area edge-emitting laser within the framework of the paraxial wave approximation. The excitation of multiple transverse modes and the occurrence of unstable optical filaments are quantitatively analysed. We show how transverse instabilities originate from spatial hole burning, gain- and index-guiding and from self-focussing. We investigate the dependence of emis- sion dynamics on characteristics of the gain material (e.g. the amplitude-phase coupling factor), the stripe width, pumping and carrier diffusion. Depending on the width of the laser, several dynamic emission regimes can be distinguished. We also project the spatio-temporal dynamics onto the laser modes. We analyse VCSEL devices with a periodically structured defect as an example of a photonic band edge band gap laser. In particular, we explore the utilisation of photonic crystal structures: gain enhancement for band edge modes due to the more efficient interaction of photons with the gain medium and increased localisation over the active layers, and the reduction of optical losses. We numerically confirm that photonic crystal effects can be obtained for finite crystal structures, and demonstrate that they lead to a significant improvement in laser performance, e.g. reduced lasing thresholds. Optically pumped VECSEL are a device concept designed to increase the power out- put of surface-emitters in combination with near-diffraction-limited beam quality. We explore the complex interplay between the intracavity optical fields and the quantum well gain material in VECSEL structures. Our simulations reveal the dynamical balance between carrier generation due to pumping into high energy states, momentum relax- ation of carriers, and stimulated recombination from states near the band edge. We show that the longitudinal multi-mode behaviour is composed of several external cavity modes. We also consider the interaction of high-intensity femtosecond and picosecond pulses with semiconductor structures. We identify the microscopic origin of the fast nonlin- earities, and consider the physical effects behind the various saturation mechanisms. We also obtain the nonlinear gain coefficients and recovery rates. It is demonstrated that group velocity dispersion, dynamical gain saturation and fast self-phase modula- tion are the main causes for changes and asymmetries in the amplified pulse shape and spectrum. We show that the time constants of the intraband scattering processes are critical to gain recovery. Our results are essential for the interpretation and the quan- titative understanding of nonlinear pulse reshaping in semiconductor optical amplifiers and absorbers. Theaccurateandspectrallybroadmodellingofsemiconductorgainandcomplexstruc- tured laser cavities that is presented in this work extends the scientific discussion of semiconductorlasersystems. Builtuponefficientnumericalalgorithmsandtheincreased availability of inexpensive high-performance computing resources, our microscopic time- domain approach is also well suitable for the engineering and design optimisation of modern nanostructured high-power diode lasers. CONTENTS v Contents Summary iii Contents v 1 Introduction, Overview and Outlook 1 2 Microscopic Description of the Gain Dynamics 11 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2 Parabolic Band Structure Approximation and Confinement Functions . . 13 2.3 Density Matrix Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.4 Coherent Coupling to the Optical Field . . . . . . . . . . . . . . . . . . . 20 2.5 Coupling to a Full Time-Domain Scheme . . . . . . . . . . . . . . . . . . 23 2.6 Phenomenological Terms and Additional Many-Body Hamiltonians . . . 26 2.7 Many-Body Interactions—Hartree-Fock Terms . . . . . . . . . . . . . . . 30 2.8 Many-Body Interactions—Correlation Terms . . . . . . . . . . . . . . . . 32 2.9 Quantum Dot Lasers and Multi-Level Bloch Equations . . . . . . . . . . 40 2.10 Fitting the Dielectric Function . . . . . . . . . . . . . . . . . . . . . . . . 42 2.11 Spectral Summation and Coupling to the Maxwell Equations . . . . . . . 44 2.12 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3 Light Field Dynamics in Laser Cavities 55 3.1 Introduction—Macroscopic Maxwell Equations . . . . . . . . . . . . . . . 55 3.2 Paraxial Approximation—Transverse Wave Equation . . . . . . . . . . . 58 3.3 Full Time-Domain Maxwell Equations . . . . . . . . . . . . . . . . . . . 64 3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 4 Semiconductor Laser Fundamentals 69 4.1 Calculation of Scattering Rates . . . . . . . . . . . . . . . . . . . . . . . 69 4.1.1 Microscopic Scattering Rates in Semiconductor Quantum Wells . 69 4.1.2 Extension to the Multi-Subband Case . . . . . . . . . . . . . . . . 74 4.2 Calculations of Quantum Well Laser Gain . . . . . . . . . . . . . . . . . 78 4.3 Relaxation Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 4.4 Quantisation of Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 4.5 Thermodynamics of Semiconductor Lasers . . . . . . . . . . . . . . . . . 97 4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 vi CONTENTS 5 Transverse Multi-Mode Laser Dynamics 101 5.1 Introduction—Effects of Spatial Degrees of Freedom . . . . . . . . . . . . 101 5.2 Transverse Instabilities in Broad Area Lasers . . . . . . . . . . . . . . . . 103 5.3 Simulations—Different Dynamic Emission Regimes . . . . . . . . . . . . 105 5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 6 Longitudinal Multi-Mode Laser Dynamics 117 6.1 VCSEL with Resonant Periodic Gain and Refractive Index Structures . . 117 6.2 Optically Pumped VECSEL . . . . . . . . . . . . . . . . . . . . . . . . . 124 6.3 Small Signal Gain Calculations . . . . . . . . . . . . . . . . . . . . . . . 129 6.4 Nonlinear Pulse Propagation . . . . . . . . . . . . . . . . . . . . . . . . . 132 6.5 Ultrafast Gain Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 6.6 Chirped Pulse Amplification . . . . . . . . . . . . . . . . . . . . . . . . . 147 6.7 Ultrashort Coherent Optical Pulse Interactions . . . . . . . . . . . . . . . 150 6.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 A Discretisation Schemes and Numerical Implementations 159 A.1 Discretisation Scheme of the Transverse Wave Equation Model . . . . . . 160 A.2 Discretisation Scheme of the Longitudinal Full Time-Domain Model . . . 162 A.3 Numerical Analysis of the Scattering Integrals . . . . . . . . . . . . . . . 167 A.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 B Dynamical Treatment of the Scattering Contributions 173 C Zusammenfassung 177 C.1 Kurzzusammenfassung . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 C.2 Einleitung und Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . 178 C.3 U¨berblick . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 C.4 Ausblick . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 Bibliography 189 1 Introduction, Overview and Outlook Introduction and Motivation The concept of light amplification by the process of stimulated emission of radiation (i.e. LASER operation [1]) in semiconductor gain materials was first demonstrated in the year 1962. Following a theoretical suggestion by Basov, three laboratories independently reported lasing in direct band gap compound semiconductor structures, with pulsed op- eration at cryogenic temperatures [2–4]. Stimulated emission of radiation was realised in thesesemiconductordiodelasersbytheradiativerecombinationofelectron-holepairsin- jected across a pn-junction. The population inversion necessary for lasing was created in thedepletionregionofaGaAs(P)homojunction. Theintroductionofheterostructuresin 1970 was a pioneering contribution to the development of efficient optoelectronic devices and a breakthrough towards industrial applications. Heterostructures are composed of multiple layers of compound semiconductors with different compositions. In particular, double heterostructures consisting of an intrinsic active GaAs layer, a thin film for light generation and amplification, sandwiched between two doped AlGaAs cladding layers enabled continuous wave operation at room temperature [3]. The improved performance characteristics (e.g. high efficiency, low heat dissipation) and reduced lasing thresholds in these structures were achieved by enhanced carrier and optical confinement [2–4]. Sophisticated crystal growth techniques (e.g. molecular beam epitaxy, metal-organic chemical vapor deposition, liquid-phase epitaxy, self-organised growth methods) [5], and processingandetchingtechnologies,allowthemanufactureofhighquality,complexsemi- conductor nanostructures (such as graded index layers, superlattices, quantum wells and quantum dots, or photonic crystals). These technologies permit better control over the electronic properties of the gain material by employing the concepts of size quantisa- tion and reduced dimensionality, and engineering of the optical mode structure and the density of states in functional photonic materials. At the same time, theory and simula- tion of semiconductor laser dynamics and structures has developed into a topic of more than only an academic interest. Due to its predictive character, computational mod- elling constitutes an invaluable tool for the engineering of novel active structures and improved device designs. Driven by the increasing demands for higher power output, dif- ferent operating wavelengths, and high-speed modulation performance, the development of novel laser and amplifier structures and the introduction of new gain material systems and lasing concepts have been pushed forward. This is accompanied with the succes- sive miniaturisation of coherent light sources and amplifiers (with the aim of on-chip realisation). This intense research in the world of semiconductor lasers stimulated the transition from a research structure to a mass product with applications in our everyday 2 INTRODUCTION, OVERVIEW AND OUTLOOK life. Semiconductor diode lasers feature some key advantages compared to other coher- ent light sources: compact size, reliability, very high conversion efficiency of pump energy into coherent light, tunability and tailoring of the optoelectronic properties, direct signal modulation up to tens of GHz, and inexpensive production costs. This has created a huge market for diode lasers, with worldwide sales revenue for semiconductor lasers es- timated at $3.2 billion in 2005. The main industrial applications of diode lasers include optical storage (CD- and DVD-based technologies), optical fibre communications, high- powerapplications(suchassolid-statelaserpumping)andmedicaltherapeutics. Besides their technological relevance, the complexity of physical interactions in diode lasers, the nonlinear coupling of various subsystems, and the diversity of time and length scales also make semiconductor lasers ideal laboratories for investigating nonlinear [6,7] and quan- tum optical phenomena [8], as well as ultrafast interaction processes in semiconductors, and concepts of nonlinear dynamics and synergetics [9,10]. New and exciting developments are aimed to control and manipulate light-matter interaction,lightemissionandpropagation,andtoengineertheoptoelectronicproperties of semiconductor gain media. In the following we give an overview on recent research activities that are related to this work. Laser and amplifier structures: Vertical cavity surface-emitting lasers (VCSEL) [11, 12] are designed as highly reflective Bragg mirrors with an enclosed defect cavity and additional transverse confinement (using oxide apertures). In particular, VCSEL show strong coupling between active gain material and the light field. Microcavities [13,14] promise high quality factors, small mode volumes and low lasing thresholds. They are fundamental tools to study and tailor the emission characteristics of a light source, and allow the modification of light-matter interactions, e.g. spontaneous emission, as the local optical density of states is defined by the dielectric structure. Semiconductor optical amplifiers (SOA) are attractive as repeaters and functional devices in all-optical networks. Resonant periodic gain structures [11,15] offer an enhancement of the effective modal gain. Vertical external cavity surface-emitters (VECSEL) [16,17] are a design scheme for increasing laser output powers with high quality output beams, and give easy access to nonlinear optical effects, such as dynamical gain saturation and pulse shaping or frequency conversion. Photonic crystals and photonic band gap materials facilitate light propagation with very low group velocities [18,19]. They can function as frequency-selectivemirrorsoropticalmicrocavities, andofferaguidingmechanismbased on multiple Bragg interference and not on total internal reflection [20]. This guiding in the low index core (air) is used in photonic crystal fibres. Semiconductor gain material systems: New semiconductor compound materials (bi- nary, ternary and quaternary alloys) can be designed for a wide range of optical emission wavelengths spanning from the infrared to the ultraviolet. Most recent research focuses on II-VI compounds, wide band gap group-III nitride-based semiconductors [21–23], and dilute nitrides, e.g. GaInNAs [24], for advanced telecommunications device applica- tions. Quantum confined active nanostructures [25] with reduced dimensionality, such as quantum wells [26], nanowires or quantum dashes [27,28], and quantum dots [29], fundamentally change the electronic band structure and optoelectronic properties by 3 applying the concept of size quantisation, resulting in more efficient carrier inversions. The application of tensile or compressive strain allows further band and gain engineer- ing. In addition, organic semiconductor lasers [30] may become viable candidates for visible solid-state semiconductor lasers and provide green light in display and illumi- nation technologies. The scope for simple fabrication and shaping of these conjugated polymer films is attractive [31]. Most lasers of this type utilise a corrugated structure which acts as a resonator based on the concept of distributed feedback. Laser and gain concepts: Optical pumping schemes provide an alternative to electrical pumping, and offer the possibility of defining the spatial distribution of the population inversion over large areas. The generation of ultrashort optical pulses by passive mode- lockingusingsaturableabsorberelements[32,33]isbaseduponthesystematicutilisation of nonlinear effects and the complex dynamic interplay between gain and light field. The proposed gain material concept and structure of quantum cascade lasers depends on in- tersubband transitions (i.e. intraband polarisations). This intersubband nature (involv- ingonlyasingletypeofcarriers)givesrisetoseveralkeyadvantagescomparedtodevices based on stimulated electron-hole recombinations [34]. For example, tunability on basis of the size quantisation concept in multiple quantum well heterostructures (i.e. no re- striction given by the energy band gaps), and higher efficiency. Achieving optical gain for recombination processes in indirect band gap semiconductor materials is challenging, which motivates the investigation of other physical interactions and gain mechanisms. Recently, light amplification and lasing in silicon has been demonstrated [35], where amplification is achieved by stimulated Raman scattering. The numerical simulation of laser dynamics plays an important role in the develop- ment of novel structures, concepts, and improved designs of active semiconductor laser devices. It is also valuable in analysing the underlying physical limitations and optical and electronic properties of the various subsystems. Because of the complex nature of the problem and the nonlinear character of the governing coupled equations, an analyt- ical treatment is difficult. With only little restriction on the geometrical and physical setup, time-domain methods offer a flexible and expandable tool appropriate to tackle the coupled dynamics of intracavity optical fields and active gain material. The only restriction in the implementation of such methods is set by the required computing re- sources. The basis for semiconductor laser models are the fundamental laser equations, derived by Haken and Lamb, Jr. [36]: The basic set of dynamical variables is given by the optical field, the induced polarisation of the active material, and the carrier inver- sion. An overview of commonly applied time-domain models to simulate semiconductor lasers within the semiclassical framework (i.e. the combination of classical electromag- netic fields with the quantum electronic properties of the active material) is presented in [3,37]. We summarise the various approaches to modelling semiconductor gain dy- namics in Figure 1.1, and the dynamics of the optical fields in Figure 1.2. Therein, we also define the scope and merits of our novel models compared to other methods. Indistinctiontootherlaserstructures, semiconductordiodelasersarecharacterisedby some unique features that have to be reflected in realistic computer models [2,4,37]: Due to the high carrier density, many-body interactions are important, particularly in gain 4 INTRODUCTION, OVERVIEW AND OUTLOOK descriptiontwo-level Bloch equationsphenomenological gain adiabatic elimination of P macroscopic SBE, Debye and DrudeDE(Lorentzianlineshape) plusdynamical polarisationmulti-level Bloch equationsmacroscopic SBE with the (small signal) gain and induced refractive index spectraeffective SBE: fitting the dielectric function (susceptibility), real-valued polarisations: b. (2.37)+(2.38) c. (2.41)-(2.44) a. -polarisation as derived from density matrix theory,plus additional phenomenological terms; three formulations:(multi-subband) SBE in relaxation rate approximation ( ) interactions Hartree-Fockand correlation contributions from many-bodyband-resolved semiconductor Bloch equations (SBE) with quantum kinetic theory unrealistic coupling of amplitude and phase dynamics atomic-like material models unrealistic modelling of gain spectra and dephasing separation of (decay) time scales effective SBE models full time-domain frequency-/time-domain requirementsmuch reduced numerical complexity and computationalbut:(homogeneousand inhomogeneous broadening);physical motivation of coupling between the different statesno spectral hole burning, no nonlinear gain saturation, loss of simplified treatment of correlation contributions/scattering band-resolved models variables, separation of scales: parametric space dependenceTaylor expansion of the Wigner representation of microscopic Figure 1.1: Overview on commonly applied approaches for describing the semicon- ductor quantum well gain dynamics as the source term of an optical wave equation. SymbolsandappliedapproximationsarediscussedinChapter2. Oursimulationsem- ploy the band-resolved semiconductor Bloch equations in relaxation rate approximation and effective SBE models (as highlighted in the drawing).