Proceedings Article | 2 January 1998
KEYWORDS: Optical semiconductors, Semiconductors, Optical components, Vertical cavity surface emitting lasers, Quantum wells, Chemical elements, Beam propagation method, Modeling, Gaussian beams, Solid state lasers
With the increasing use of the new generation of semiconductor (s/c) optical devices that support weakly diffracting fields it is seen that the conventional analysis, widely used in the context of (longitudinally) uniform structures, is not readily applicable. Prominent examples of weakly non-uniform devices are: i) High Power Semiconductor Optical Sources using adiabatically tapered structures, [1], [2]; ii) VCSELs with Quantum Well layers that are pumped over a small area but have no explicit lateral guiding, [3], Field analysis using eigenfunction expansions seems a suitable approach and this paper presents results of calculations on tapered optical sources that are obtained by using the Hermite-Gauss (HG) functions in a field expansion scheme with the Collocation method for numerical solution.Entirely numerical procedures, e.g., Finite Elements, Beam Propagation Method, or Cellular Automata [4] have been developed for analysing optical fields in non-uniform structures. However, there are advantages in developing quasi-analytic methods since they tend to provide a closer link with the physical model and may lead to very efficient computation schemes. Such features can be invaluable in the modelling of, particularly, active devices where, in general, a self consistent solution is obtained only after several iterations and hence an efficient computation technique is desired. Although the Gaussian Beam method [5] has been applied often in the past to analyse, for example, gas and solid state lasers, and is a quasi-analytic approach, it appears to be less amenable for use with laterally inhomogeneous structures (typical of s/c devices), [6]. Thus the more systematic eigenfunction expansion procedure is described in this paper since it is far better suited to analyse the s/c optical devices of present interest Specifically, considering a two-dimensional (x,z) structure that supports fields propagating predominantly in the z-direction, the total field solution is written in the general form, [7] F(x,z) = X Am(z)\|/m(x) (1)m where yra(x) belong to a complete set of orthonormal basis functions, and Am(z) are the z-dependent expansion coefficients. Whereas any complete function set may be used in eqn. (1), e.g., [8], [9], [10] in the present formulation the set of HG functions has been considered to be the most apt since the complete set of HG functions is discrete, and both the HG functions and the actual, physically sustainable total field of any waveguide decay to zero along the transverse (x) axis at infinity (this is so even in the presence of radiation modes), [11], [12]. Substitution of eqn. (1) into the wave equation (along with the boundary conditions) yields a set of coupledjlordinary) differential equations in the expansion coefficients, Am(z). It has been argued that this set of equations may be numerically solved very efficiently by the Collocation method, [13], and hence this procedure has been used for the present calculations. Test results obtained from the HG-Collocation method compare very favourably with those from analytic and/or purely numerical schemes for both, purely real and complex refractive index media (with small imaginary part). The proper analysis of active devices requires calculations which self-consistently solve for the optical field (intensity) and the carrier (inversion population) density distribution. Several methods exist for solving the non-linear diffusion equation for the carrier density, [14]. However, in the present analysis the HG-Collocation scheme is used and is shown to be a very effective method for solving for the carrier distribution. The talk will present a brief description of the theoretical development of the model and will provide a comparison of several experimental and theoretical results not only to prove the suitability of the HG-Collocation method, but also to confirm the substantial advantages in using the proposed scheme, not least of which is the fast and efficient numerical model that has been achieved.
