| PREFACE
The book is divided into nine chapters. Except for the first two introductory chapters, each chapter is independent and restricted to a particular subject to be studied. To the best of the author s knowledge, the most appropriate theories have been chosen to model the specific topic of VCSELs. In Chapters 3 and 4, theoretical models have been developed to analyze the modal profile and polarization, respectively, of VCSELs. The most popular structure of VCSELs is a cylindrical symmetric cavity, which is assumed in the derivation of the models. In addition, this configuration of VCSELs allows investigation of the modal profile and polarization separately such that the complexity of theoretical models can be reduced. In Chapter 3, different methods of solving the wave equation for the modal profile of VCSELs are discussed in detail. The reader can choose the most appropriate model with the required speed and accuracy to analyze the problems. In Chapter 4, two- and four-level models are described to study the polarization properties of the fundamental transverse mode. These simplified models can evaluate the dominant factors that control the polarization properties of VCSELs. It must be noted that the investigation of VCSELs using cold cavity approximations is not realistic. This is so because most of the measurable data, such as threshold current, lasing wavelength, slope efficiency, and output power, all depend on the operating temperature of lasers. Furthermore, the optical behavior of VCSELs is affected by thermal lensing (i.e., self-focusing of transverse modes into the core region of the active layer). Therefore, the thermal properties of VCSELs are investigated in Chapter 5. The method of effective temperature using a simple rate equation model is presented. Effective thermal conductivity and heat generation rate are also derived. The objective in defining effective temperature is to simplify the study by using a rate equation model so that the computational efficiency can be improved. However, this approach will not provide detailed information on heat distribution. Detailed heat distribution inside the laser cavity is studied by solving the heat equation numerically. In this case, the influence of thermal lensing on the optical field profile can be evaluated. Spatial hole burning of carrier concentration also has significant influence on the modal profile of VCSELs. Therefore, Chapter 6 describes the use of a simple rate equation to evaluate the distribution of carrier concentration inside the active region. In this case, self-consistent calculation of optical gain and carrier concentration (i.e., self-consistent calculation of the Poisson and Schrödinger equations) is ignored to simplify the calculation. Different methods for approximating the nonuniform distribution of carrier concentration are also discussed. On the other hand, nonuniform distributions of electric potential and current are required as the input parameters to calculate the heat distribution inside the laser cavity. They have to be solved numerically using the Poisson and continuity equations simultaneously with appropriate boundary conditions. The electric potential across the active layer and the corresponding carrier concentration can be linked together by a simple diode equation. This is so because the simplified relation between optical gain and carrier concentration has been assumed. The self-consistent calculation of optical field, heat, and electrical characteristics of VCSELs is also described in Chapter 6. The dynamic response of VCSELs is analyzed in Chapter 7. Preliminary investigation of the dynamic response of VCSELs using a simple rate equation model is described. Hence, the time variation of carrier concentration and photon density inside the active layer can be calculated. Furthermore, detailed analysis of optical fields can be considered using the beam propagation method such that the influence of optical confinement on the dynamic response of VCSELs can be evaluated. However, detailed investigation of the transient response of heat and electrical properties is avoided in the self-consistent calculation. This is because the time variation of heat and voltage, which are related to heat and the Poisson equations, is much slower than that of photon density and carrier concentration. This assumption significantly reduces the computation time of the model without sacrificing the accuracy of the calculation. The influence of various transportation mechanisms inside the quantum well (QW) active region on the dynamic response of VCSELs is also discussed in this chapter. The methods used to evaluate the spontaneous emission and linewidth of VCSELs are described in Chapter 8. Simple models have been developed to study these parameters quantitatively through the investigation of the spontaneous emission factor and linewidth enhancement factor. On the other hand, the magnitude of the spontaneous emission factor and linewidth enhancement factor is evaluated using rate equation model by empirically fitting the measurable data. Hence, design criteria to optimize the spontaneous emission of VCSELs are obtained. Other nonlinear features of VCSELs such as self-sustained pulsation, bistability, dual-wavelength operation, and wavelength tunability are studied in Chapter 9 using rate equation models. The advantage of using simple rate equation models is that the parameters that describe the nonlinear behavior of VCSELs can be easily extracted through some measurable data such as injection current and lasing power. In conclusion, this book presents the most effective way to implement laser models of VCSELs, which the reader can easily understand. However, the readers are assumed to have the usual undergraduate background knowledge of electromagnetic theory and solid-state physics as well as basic computational skills. Materials of this research monograph concentrate on the evaluation of modeling techniques to analyze VCSELs under various operating conditions. As each chapter of this book is mostly independent of the other chapters, readers can selectively study any chapter for their own interest. Although this book is of most interest to the design engineer of VCSELs, it also provides valuable information to CAD tool designers in other fields of semiconductor lasers. Siu Fung Yu Singapore |
Design and fabrication of vertical cavity surface emitting lasers (VCSELs) requires an iterative process, which is extremely expensive and time-consuming. The use of computer-aided design (CAD) tools can help shorten the design cycle and speed up the development process. Laser models, which are found in the literature, can be used to implement CAD tools for the analysis and design of VCSELs. However, some comprehensive models, which perform sophisticated functions, are difficult to implement and show low computational efficiency. Other simplified models exhibit high computing speed but deliver inadequate descriptions of the observed effects. As a result, inconsistent conclusions may be obtained because different assumptions are applied. This book attempts to provide a guideline for the derivation of models based on appropriate assumptions for a particular problem so that the phenomena observed by the experiment can be easily explained. In fact, the objective throughout this book is to search for the simplest and most direct treatment for modeling VCSELs. The author believes that the laser models covered in this book can help the readers customize their CAD tools to fit into their applications. In addition, the readers should have no difficulty in implementing their own laser models.
TABLE OF CONTENTS Chapter 6 - Electrical Characteristics of Vertical Cavity Surface Emitting Lasers
| CHAPTER 6 Electrical Characteristics of Vertical Cavity Surface Emitting Lasers Methods of modeling spatial hole burning using rate equations of carrier concentration are discussed. The influence of transverse modes and injection current on the formation of spatial hole burning is investigated. Detailed calculation of current transport mechanism at the heterointerfaces of the multilayered mirrors is also explained. The approach to model the quasi-3D distribution of electrical potential and current density using Poisson s equation and resistance network, respectively, is described. A comprehensive self-consistent calculation of VCSELs including the quasi-3D distribution of electric potential, current flow, and temperature, as well as the transverse distribution of transverse modes and carrier concentration inside the active layer, is also presented. 6.1 INTRODUCTION Single mode operation, low threshold current and high wallplug efficiency can be achieved in vertical cavity surface emitting lasers (VCSELs) if the distribution of carrier concentration inside the active region is controlled carefully. It is noted that the performance of VCSELs with gain-guided [1,2] and index-guided waveguides [3 5] deteriorates significantly in the presence of spatial hole burning (SHB). SHB is excited by stimulated recombination of carrier concentration inside the active layer, and the shape of SHB follows the profile of transverse fields. For VCSELs operating at high power, SHB reduces the modal gain of the fundamental mode so that higher-order transverse mode can be excited. This is due to the reduction of overlap between the transverse distribution of optical gain and the fundamental mode. As a result, the wallplug efficiency of VCSELs is reduced and the corresponding threshold current is increased. The self-focusing effect is also observed in VCSELs due to SHB (i.e., carrier-induced index change) inside the active layer. The self-focusing effect narrows the beamwidth of the transverse modes so that the stability operation of the fundamental mode is further reduced. Current spreading is the diffusion of carrier concentration toward the cladding region of the active layer. Current spreading reduces the wallplug efficiency and increases the threshold current so that VCSELs with a strong confinement of injection current inside the core region of the active layer are desired [6]. It is expected that VCSELs with buried heterostructure have better wallplug efficiency than that with ion-implanted [7] or oxide-confined structures [8,9]. Current spreading is critically dependent on the detailed confinement structure of injection current such as the injection current path and the resistance of heterointerfaces. In VCSELs with buried heterostructure [3,4], the injection current is strongly confined inside the core region and the diffusion of carriers into the cladding region of the active layer is minimized. In VCSELs with double-oxide-confinement structure of injection current, the diffusion of carriers into the cladding region of the active layer can be minimized, provided the separation between the two oxide layers is optimized [8,9]. Current crowding, which has arisen from the accumulation of carriers at a particular location of the active layer, has significant influence on the uniformity of the injection current profile. The performance of VCSELs can deteriorate by current crowding through the excitation of SHB as well as thermal lensing effect (i.e., due to the excessive heat sources) [1,7,8]. It must be noted that thermal lensing increases the strength of self-focusing. Current crowding usually occurs in VCSELs with a strong current confinement structure of carrier concentration such as those with buried heterostructure or oxide-confined structure. Most of the time, the crowded current supplies higher carrier density at the periphery of the core region of active layer. As mentioned above, SHB, current spreading, and current crowding significantly influence the performance of VCSELs. In addition, the excessive injection of carriers generates heat inside the laser cavity. In order to obtain an accurate analysis of the performance of VCSELs, the electrical, thermal, and optical mechanisms have to be calculated simultaneously through their linkage of optical gain and refractive index inside the active layer. In this chapter, the models using a rate equation of carrier concentration, with the consideration of injection current, transverse modes, and carrier transport effect, are developed to analyze SHB. In addition, the influence of SHB on threshold current and transverse-mode profiles of VCSELs is investigated. The approach to calculate the resistivity of heterointerfaces is explained, and the methods employed to minimize the resistance of heterointerfaces using graded interface and modulation doped are discussed. The dependence of current spreading and current crowding on the geometry of VCSELs is also investigated. This can be done by solving the quasi-3D distribution of electrical potential using Poisson s equation and the appropriated boundary conditions. In addition, a simple resistance network is developed to solve the quasi-3D current flow. It is noted that the SHB, current flow, and temperature distribution are mutually linked together through the optical gain and the refractive index distribution inside the active layer. A self-consistent calculation is suggested to simultaneously solve the electrical, thermal, and optical behaviors of VCSELs, including calculation of SHB, current spreading, and crowding, as well as self-focusing. |
