| 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 4.2.4 - Modeling of Polarization Modes in VCSELs
4.2.4 Modeling of Polarization Modes in VCSELs In this section, a rate equation model is deduced to analyze the polarization properties of VCSELs [25]. Using this model, the influence of in-plane gain anisotropy and gain saturation on the polarization properties of VCSELs is studied. The rate  Figure 4.5 In-plane gain anisotropy for the different orientation growth of five epitaxial layers are plotted as a function of an in-plane angle θ (after Ref. 24). equation model is also applied to investigate the influence of birefringence (i.e., refractive index anisotropy) on the modal gain of the two orthogonal polarizations [26]. It can be shown that a relatively small amount of refractive index anisotropy (~0.005%) is adequate to induce in-plane gain anisotropy to discriminate the polarization state of VCSELs. The polarization properties of VCSELs with symmetry cylindrical waveguide structure are analyzed by studying the modal gain of the polarized transverse modes. Suppose that the transverse modes are weakly guided inside the core region of the active layer so that the LP mode approximation can be applied. Hence, polarized electric field distribution, Ψh,lm and Ψv,lmm, can be expressed as  where Φlm (r) is the radial distribution of the transverse mode of order lm, which has an expression of (3.26) with the corresponding eigenvalue equation given by (3.28). The corresponding azimuthal variation of the polarized electric field is described by the functions sin(lθ) and cos(lθ) for the h and v modes, respectively. These functions ensure the orthogonal orientation of the two polarized fields. Hence, the modal gain gη,lm of Ψη,lm can be obtained by a standard perturbation analysis, expressed as [25]  where the asterisk superscript (*) represents a complex conjugate, η = v, h and g (r,θ) is the in-plane gain distribution of the active layer. Hence, the corresponding photon rate equations can be obtained by substituting (4.37) into (4.13) as deduced in Section 4.2.1, with the consideration of higher-order transverse modes and spontaneous emission. It can be shown that the corresponding photon rate equation can be written as  where a spontaneous emission term is deliberately introduced into the photon rate equation and  where τn is the carrier lifetime, Dn is the carrier diffusion constant, d is the thickness of the active layer, and J is the injection current density. Hence, the transverse polarized fields at or above threshold can be obtained numerically by simultaneously solving (4.38) and (4.39) using the finite difference method. If fundamental transverse mode (i.e., LP01) with two orthogonal polarizations is considered only in the analysis of the polarization properties of VCSELs, the rate equation models can be simplified to [27]  where gN,h (= 3.0 x 10–16 cm–2) and gN,v (= 2.85 x 10–16 cm–2) are the differential gain coefficients of the h and v modes, respectively and Nt (= 2.0 x 1018 cm–3) is the carrier concentration at transparency. In the derivation of (4.40), it is assumed that the carrier concentration N is uniformly distributed along the active layer and the QW optical gain of is linearly approximated with the injected carrier concentration (i.e., for more appropriate calculation, the logarithmic gain–carrier density relation should be used). In addition, it is assumed that In the preceding calculations, the oscillation frequencies of the two orthogonal polarizations degenerate as both polarizations are of the same fundamental transverse-mode profile. However, it is seldom found experimentally that the two orthogonal polarizations have identical oscillation frequencies [28]. This is because the two polarizations experience birefringence (i.e., anisotropy in refractive index), which induces a split in oscillation frequency. In addition, birefringence may alternate the radial distribution of the two orthogonal polarizations and introduces in-plane gain anisotropy. In order to analyze the influence of birefringence on the polarization properties of VCSELs, the photon rate equations derived in Section 4.2.1 have to be modified. Consider that the total field E includes the two polarized states of the fundamental transverse mode (i.e., LP0l mode), which can be written as  and Ψη (η = v, h) satisfies the Holmholtz equation as [29]  where βeff, η is the corresponding effective propagation coefficient in the transverse direction. Substituting (4.41) into (4.5) and using (4.42), it can be shown that  The term given below  represents the effective optical gain experienced by the polarized fields, and the term 2βη (βeff,η − βη) represents the detuning from the waveguide mode. Hence, the rate equation model can be simplified to  and the corresponding carrier rate equation is similar to (4.40c). These two photon rate equations have ignored the influence of self- and cross-gain saturation, and the calculation of gv and gh is different from that given in (4.40) as birefringence has been taken into calculation. Figure 4.6 shows the calculation of the normalized difference between the h and v modes (i.e., with transverse distribution of LP01), that is, δΨ = Ψh(r) / < Ψh > − Ψv(r) / < Ψv >. It is assumed that the corresponding background permittivity of the two orthogonal polarizations are εb,h (r < W) = 12.2514, εb,h (r > W) = 11.56, εb,v (r < W) = 12.25, and εb,v (r > W) = 11.56. The core radius W of the symmetry cylindrical waveguide of VCSELs is 3 µm. It is noted that the refractive index of the h mode in the core region is larger than that of the v mode, so the h mode has better transverse confinement than the υ mode. Therefore, it is expected that the h mode has higher optical gain than the v mode and the corresponding gain difference is found to be ~ 0.04 cm¯1, which is sufficient to maintain a single h-mode operation above threshold [30]. Furthermore, it is expected that the birefringence separates the oscillation frequency of the two orthogonal polarizations and the lasing mode, the h mode has higher oscillation frequency. This is because εb,h > εb,v implies βeff,h > βeff,v so that the lasing mode has higher oscillation frequency than the nonlasing mode. Hence, polarization mode with higher oscillation frequency will be excited by birefringence only if VCSELs whave uniform in-plane gain distribution. In the following section, the origin of birefringence in VCSELs is analyzed more rigorously.  Figure 4.6 The normalized intensity difference δΨ versus radial coordinate (after Ref. 30). |
N
is the average carrier concentration over the active layer. The photon lifetime τp and spontaneous emission factor βs are assumed to be dependent on the profile of the transverse modes. Furthermore, in order to determine the modal properties of the polarized modes for the laser operating above threshold, it is necessary to calculate the spatial distribution of carrier concentration inside the active layer using the carrier rate equation [25]
, the self- and cross-saturation coefficients for both polarization modes are assumed identical (ζsv = ζsh = 1.0 X 10–17 cm3 and ζcvh = ζchv = 2.0 x 10–17 cm3). It is noted that the differential gain coefficients for the h and v modes are different and the gN,h is assumed larger than gN,v. In addition, the value of the cross-saturation coefficients is assumed to be larger than that of the self-saturation coefficients. These assumptions are based on the experimental observation of bistable operation of two orthogonal polarizations. In fact, to realize this bistable operation, the condition of optical gain anisotropy with ζsv × ζsh < ζcvh× ζchv is required [27]. Hence, it can be shown that the v mode is well suppressed by more than 10 dB and only the h mode is supported for the laser biased about twice the threshold (i.e., equivalent to gain difference of 5%). However, further increase of the lasing mode intensity reduces the modal gain of the h mode through the gain saturation and mode switching is observed. Assuming that gN,v = gN,h (= 3 X 10–16 cm–2) is applied to the model and the other parameters remained unchanged, the numerical calculation has demonstrated that both orthogonal polarizations can be excited but are dependent on the initial conditions. If no randomly generated spontaneous emission noise is used on βs, both polarizations can be excited simultaneously above threshold.
