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The VCSEL Design Tool in Ansys Lumerical Multiphysics™ software is a solution for simulating vertical cavity surface emitting laser (VCSEL) devices. It includes a new VCSEL Solver that can solve self-consistent equations governing the laser cavity, transport of charge carriers into the cavity, quantum effects in the active layer, and thermal effects to accurately predict key laser performance metrics.
Note: The current version of the VCSEL Design Tool applies to a cylindrically symmetric structure. The simulation domain is 2D, which enables an exact simulation of cylindrical 3D cavities by exploiting the symmetry.
For cold cavity simulations of cylindrical cavities, the VCSEL Design Tool is much faster than the time domain 3D FDTD, while retaining the exactness of the solution. For opto-electro-thermally coupled simulations of cylindrical cavities, the VCSEL Design Tool is much more accurate, complete, and faster than the traditional FDTD, even when using FDTD gain material plugins. The VCSEL Design Tool naturally outputs important figures of merit, such as the resonance frequency, mode profile, beam profile, carrier density, temperature profiles, and LIV curve.
VCSEL Devices
A VCSEL device typically consists of the components shown in the diagram below. On the left you can see a cross-sectional view of a cylindrical VCSEL device, and on the right you can see its corresponding 3D view.
The VCSEL Design Tool can couple all necessary physics in the structure and efficiently simulate its behaviour to extract important figures of merit. Due to the complexity of physics involved, the VCSEL Design Tool leverages coupling from other solvers in the Ansys Lumerical Multiphysics™ product, including CHARGE, MQW, and HEAT solvers.
In the following sections, we present the optics concepts of VCSELs and how to couple electrical, thermal and optical physics.
Cavity and Photon Rate Equations
The VCSEL solver is based on the cavity and photon rate equations. These equations are derived from Maxwell’s equations, in the limit where the optical frequency of the laser is much faster compared to the modulating frequency [1]. Using these equations, accurate field distributions and photon numbers inside the laser cavity can be efficiently calculated, and key figures of merit for VCSEL devices can be extracted. Compared to methods such as the Finite Difference Time Domain (FDTD) method, where a very small time-step is required to resolve the high-frequency optical signal, resulting in long simulation times, simulations using the VCSEL Solver will have a larger time-step, and will therefore complete much quicker. Also, due to its ability to couple with other key physics, as described below, the VCSEL Solver can produce more accurate predictions of laser performance compared to FDTD Solver.
Cold Cavity Solution
In the case of simulations without including electrical and thermal effects, known as “cold cavity” simulations, only the cavity equation is solved. The coupling to electrical and thermal components is described in this article in the section “Coupling to CHARGE, HEAT, and MQW”.
In cold cavity simulations, the cavity equations can be solved either in a source-free configuration, similar to an Eigenmode simulation with FEEM Solver, or in a configuration with a single dipole source, similar to 3D electromagnetic light wave simulation with FDTD Solver.
In the source-free configuration, the cavity equation is a linear eigenvalue problem. This equation is solved in the frequency domain for every modulation time step, and has the following form:
$$
\nabla \times \mu_r^{-1}(\mathbf{r}) \cdot [\nabla \times \mathbf{E}_k(\mathbf{r}, t)] = \frac{\omega_k^2(t)}{c_0^2} \epsilon_r(\mathbf{r}, t, \omega_0) \cdot \mathbf{E}_k(\mathbf{r}, t)
$$
where \(\mathbf{r}\) is the spatial coordinate, \(c_0\) is the speed of light in vacuum, \(\mu_r\) and \(\varepsilon_r\) are the relative permeability and permittivity of the material, respectively, \(k\) is the index of the optical mode, \(\omega_k\) is the complex frequency of mode \(k\), defined as \(\omega_k(t) = \omega_k^\prime(t) + i\omega_k^{\prime\prime}(t)\), \(\omega_0\) is the estimated center frequency of the laser, and \(t\) is the time coordinate of the modulation signal.
In the configuration with a single dipole source, the cavity equation is as follow:
$$
\nabla \times \mu_r^{-1}\left(\mathbf{r}\right) \cdot \left[\nabla \times \mathbf{E}\left(\mathbf{r}\right)\right] - \frac{\omega^2}{c_0^2} \varepsilon_r\left(\mathbf{r}, \omega\right) \cdot \mathbf{E}\left(\mathbf{r}\right) = \mathbf{F}(\mathbf{r}_0, \omega)
$$
where \(\mathbf{F}(\mathbf{r}_0, \omega)\) represents a single frequency dipole source at position \(\mathbf{r}_0\), and all other variables are as described earlier.
Note: The current version of the VCSEL Design Tool solves only for steady state solutions, and time-dependence of the modulation signal is not considered.
Symmetrical Solution and Modes
VCSEL assumes a cylindrical symmetry of the optical cavity, allowing for further simplification of the cavity equation.
Specifically, the electric fields are expanded into Fourier bases as follows, given the vectorial definition of \(\mathbf{E} = E_\rho \hat{\rho} + E_\phi \hat{\phi} + E_z \hat{z}\) and \(\mathbf{E}_T = E_\rho \hat{\rho} + E_z \hat{z}\):
$$
\mathbf{E}_s\left(\rho, \phi, z\right) = \mathbf{E}_{T,0}\left(\rho, z\right) + \sum_{m=1} \left[\mathbf{E}_{T,m}\left(\rho, z\right) \cos(m\phi) + E_{\phi,m}\left(\rho, z\right) \sin(m\phi) \hat{\phi}\right]
$$
$$
\mathbf{E}_a\left(\rho, \phi, z\right) = E_{\phi,0}\left(\rho, z\right) \hat{\phi} + \sum_{m=1} \left[\mathbf{E}_{T,-m}\left(\rho, z\right) \sin(m\phi) + E_{\phi,-m}\left(\rho, z\right) \cos(m\phi) \hat{\phi}\right]
$$
$$
\mathbf{E}\left(\rho, \phi, z\right) = \mathbf{E}_s\left(\rho, \phi, z\right) + \mathbf{E}_a\left(\rho, \phi, z\right)
$$
In the solver, only a finite number of Fourier components, \(m\), are included in the solutions, and can be set using the “Fourier Component” option in the VCSEL solver options. Due to the cylindrical symmetry, the fields displayed in the results are \(\mathbf{E}_\rho\), \(\mathbf{E}_\phi\), and \(\mathbf{E}_z\) evaluated at \(\phi = 0\) only, separately for each \(m\).
The Fourier component index, \(m\), and the eigen frequency index \(k\), fully describe a mode in the device.
Note: The word “Mode Number,” appearing as a parameter name in the visualizer, as well as the word “Number of trial modes,” appearing as an option in the VCSEL Solver object, both refer to the index \(k\).
Coupling to CHARGE, HEAT, and MQW
In the coupled solution, the photon rate equation, as shown below, is solved alongside the cavity equation described above.
The photon rate equation is as follows:
$$
\frac{d}{dt} S_k\left(t\right) = -2\omega_k^{\prime\prime} S_k\left(t\right) + R_k^{sp}(t)
$$
where \(S_k\) represents the number of photons in the laser cavity for mode \(k\), and \(R_k^{sp}(t)\) represents the spontaneous recombination rate in the laser cavity for mode \(k\), and all other variables are as described earlier.
To self-consistently couple the optical equations to the electro-thermal simulations and quantum-mechanical simulations, the VCSEL Design Tool utilizes the CHARGE, HEAT and MQW solvers within Ansys Lumerical Multiphysics™.
The diagram below illustrates the variables exchanged between various components of the fully coupled simulation. Further information on the physics of the CHARGE and HEAT solvers can be found in the CHARGE Solver Introduction article and HEAT Solver introduction, respectively. Further information on the physics of the MQW solver can be found in the MQW Solver Introduction article.
For information on achieving convergence between these physics’ components, see Troubleshooting Convergence Errors in VCSEL.
Reference
- M.Stereiff, “Opto-Electro-Thermal VCSEL Device Simulation”, Ph.D. Disseration, Swiss Federal Institute of Technology in Zurich, Zurich, Switzerland, 2004.
See Also
VCSEL Design Tool - User Manual, Troubleshooting Convergence Errors in VCSEL, Workflows for VCSEL Devices