| 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 4A.3 - Adiabatic Elimination of Spin Dynamics
4A.3 Adiabatic Elimination of Spin Dynamics To identify the role of carrier dynamics in the polarization behavior of the laser, the variable n will be eliminated adiabatically from (4.74) with the approximated expression valid under the conditions [46]  In (4.74), the adiabatically eliminated n becomes  where ∂n/∂t → 0 is assumed due to the slowly varying components on the RHS of (4.74). It is convenient to separate the optical intensity and polarization by introducing the so-called Stokes vector  where S is the photon intensity and the angles ψ and Φ represent the polarization state. The angle 0 ≤ Φ≤ π characterizes the polarization; it is the angle between the long axis of the polarization ellipse and the x-axis. The angle – π/4 ≤ Ψ ≤ π/4 characterizes the ellipticity of the light; ψ = 0 corresponds to the linearly polarized light, whereas ψ = ±π/4 corresponds to circularly polarized light. Using the angles (2Ψ, 2Φ) as spherical coordinates, the polarization state can be conveniently depicted as a single point on the Poincaré sphere [43]. Now, substituting (4A.17) into (4.72) and (4.73) to replace the variable n gives the following version of rate equation with spin eliminated:   In (4A.18a) and (4A.18b), N disappears as any variation of N from its equilibrium will lead to equal gain or loss for all Stokes parameters and thus cannot affect the field polarization. It is noted that τd/τJ > 100 is commonly found in VCSELs, and S can be assumed to be less dependent on the angles (2Ψ, 2Φ), so that S in (4A.1l8a) and (4A.18b) can be considered as a constant. If the loss anisotropy and birefringence are separated from (4A.18a) and (4A.18b), a more general spin-eliminated version can be simplified to  where Φr represents the orientation of the loss anisotropy and birefringence with respect to the crystal axis. The other parameters are defined as ωlin= 2γbf , γlin = 2γa , γnon = SτJ/(τdτp). The parameters ωlin and γlin are the linear birefringence and linear anisotropy, respectively. To remove the various sine and cosine functions in (4A.19), Φ and ψ can be expanded to first order as Φ , ψ ≪ 1. The steady-state angles thus found are  Equation (4A.20a) is asymmetric in ωlin; large ellipticity is most likely for negative ωlin , specifically for the case of dominant linear birefringence (ωlin >> γlin, γnon). It is also found that Φs≪ ψs. For ψs, Φs ≪ 1 the linearized polarization rate equations, including noise, are  where the Langevin noise sources fΦ and fψ have been introduced into linearized polarization rate equations. The simplicity of these results is due to the fact that, after spin elimination, the polarization dynamics (Φ,ψ) are separated almost completely from other dynamics such as intensity S and its corresponding phase, average inversion N. The only coupling is via the intensity dependence of γnon, and this coupling disappears when the intensity is reasonably constant (i.e., under the condition that the fluctuations are limited or at frequencies very different from those of the polarization dynamics). |
= (s0, s1, s2, s3) as
