Analysis of Modes in a Vertical Cavity Surface Emitting Laser with Multilayer Bragg Reflectors

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A numerical method is introduced which is suitable for mode analysis in an optical resonator with complicated refractive-index variations such as vertical cavity surface emitting lasers (VCSELs). In this method, the optical field of a laser mode is expressed as a linear combination of component fields with their coefficents to be determined. After a hypothetical boundary is set surrounding the region to be analyzed, the component fields are obtained by numerically integrating the wave equation in the inside region using the conditions on part of the boundary as the initial values of the integration. The total field, which is a linear combination of these fields, satisfies the equation and the selected part of the boundary conditions regardless of the coefficients. The conditions imposed on the total field on the rest of the boundary lead to a matrix eigenvalue problem, from which the optical frequency and the coefficients are obtained. The matrix expresses only boundary conditions and, therefore, its size is much smaller than that of a matrix expressing bulk conditions, as appears in the finite element method or the finite difference method. At the same time, this method has the advantage of adaptability for graded-index problems in contrast to conventional boundary formalisms such as the boundary element method and the mode matching method, because in the present method the component fields (or base functions) are calculated for individual index distributions while in these methods an inflexible set/of base functions is used. As an example of the application of the method, mode properties in gain-guided VCSELs are analyzed using this method based on a two-dimensional model. This is the first model that takes into account the effects of standing-wave formation in the resonator and of the incident angle-and polarization-dependence of reflectivity. The ability to treat these effects makes the present method suitable for VCSELs equipped with a thin active layer and with multi-layer reflectors. Basic properties including polarization, threshold gain, oscillation wavelengths, and deflection of far-field patterns have been predicted for various cavity sizes and for various gradients in gain distributions. The major results of the analysis are: TE modes have lower thresholds than TM modes; the laser beam can be steered by tailoring the gain distribution as with edge-emitting lasers.
社団法人電子情報通信学会の論文
1994-09-25

Analysis of Modes in a Vertical Cavity Surface Emitting Laser with Multilayer Bragg Reflectors

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