3D finite element simulation of optical modes in VCSELs

Vertical-cavity surface-emitting lasers (VCSELs) are light sources with unique properties and potential applications of great interest. 1 Solution of Maxwell's equations for realistic 3D VCSELs is a challenging task. 2 Since the VCSEL resonator is realized by distributed Bragg reflectors (DBR), the geometry inherits a pronounced multiscale structure. The devices are relatively large (thousands of cubic wavelengths) and contain subwavelength DBR layers, very thin active zones and structured apertures. Further, the infinite exterior adjacent to the VCSEL containing also a layered structure has to be modeled to obtain realistic predictions of radiation loss and lasing threshold. A variety of methods has been used to compute optical VCSEL modes. These include FEM, finite difference time-domain methods (FDTD), modal expansion and approximative methods. [2][3][4][5] In most standard approaches the optical problem is restricted to purely 1D or to cylindrically symmetric structures. Nevertheless, many realistic 3D devices cannot be restricted in this way. This is the reason why reliable full 3D simulations become so important. Accuracy limitations of state-of-the-art 3D solvers, including also FEM solvers, have recently been discussed. 2,5 The finite element method is very well suited for simulation of nano-optical systems and devices. 6,7 Its main features are the capability of exact geometric modeling due to usage of unstructured meshes and high accuracy at low computational cost. The finite element method offers great flexibility to approximate the solution: different mesh sizes h and polynomial ansatz functions of varying degree p can be combined to obtain high convergence rates. As a result, very demanding problems can be solved on standard workstations. 8 We demonstrate that a FEM eigenmode solver with higher-order finite elements, adaptive meshing and a rigorous implementation of transparent boundary conditions is a powerful method for 3D VCSEL mode computation.

The main physical effects in a VCSEL are associated to time scales ranging over several orders of magnitude. Since the frequency of the optical modes is much higher than those of all other effects, a time-harmonic ansatz for the electric field E(x, y, z) is well-justified:

where ω denotes the frequency. Using this ansatz in Maxwell’s equations, the following second order equation for the electric field can be derived:

In this equation no exterior current or charge density sources are present. Physically, the light field of a VCSEL is created by coupling of the electron system in the active layer to the eigenmodes of the structure. In Maxwell’s equations this usually enters via the complex permittivity tensor ε (in all relevant optical materials the magnetic permeability µ is a constant). The resonance problem then consists of finding pairs (E, ω), such that Maxwell’s equations (2) on the given geometry are fulfilled. Furthermore, the so called radiation condition has to be satisfied which requires that the resonance modes are purely outward radiating. For solving equations (2) we use the FEM package JCMsuite developed by ZIB and JCMwave.

The aim of this paper is to demonstrate that very low numerical errors can be reached in full 3D VCSEL simulation. In order to quantify the error of a numerical solution we compare it to a reference solution. Because for the full 3D VCSEL problem no accurate quantitative results are available as independent reference, we use a cylindrically symmetric VCSEL setup in this convergence study. A very accurate solution to this problem can be obtained using a 2D solver in cylindrical coordinates. 7 Restriction of 3D resonance mode computations to a 2D cross section due to the cylindrical symmetry leads to substantial savings in computational time and memory requirements. The 3D, rotationally symmetric solution can also be compared to results from the literature. 3 With the reference solution at hand we perform simulation of the same physical setup using a full 3D FEM model. Numerical accuracy of the 3D results from this model is then obtained from comparison to the reference solution.

In a further sub-section, we demonstrate 3D simulation of the thermal lensing effect in a VCSEL.

As physical model we choose a VCSEL setup as described by Bienstman et al. 3 The laser consists of two DBR mirrors with alternating AlGaAs-GaAs layers. The InGaAs quantum well layer (gain material) is embedded in the central GaAs cavity region. An AlOx aperture is placed in the upper region of the lowest AlGaAs layer of the top mirror. The whole structure is situated on a GaAs substrate. Figure 1a shows a 2D cross section through the cylindrically symmetric setup. Note that the layered structure extends infinitely in radial direction and modal confinement is reached through the finite AlOx aperture and the finite active region.

We use a 2D FEM solver in cylindrical coordinates to obtain the near field solution E and the complex eigenfrequenc

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