Tutorial for computing the edge emission of a multilayer stack using a 2D simulation

[oskooi]

Edge-emitting semiconductor lasers are generally based on a multilayer stack where the wafer has been cleaved to expose an edge facet. It would be useful to provide a tutorial demonstration of the computation of the extraction efficiency and radiation pattern of such a structure. This would be based on a 2D cell in $xy$ and Brillouin-zone integration in $z$ (i.e. the out-of-plane direction). A schematic of the simulation setup is shown below. The active region forms the cavity and consists of an extended (spatially incoherent) source such as a quantum well.

There are three things to note regarding this calculation:

1. A demonstration of Brillouin-zone integration for the case of a 1D cell is provided in Tutorial/Dipole Emission of a Light Emitting Diode as a Multilayer Stack. However, in this example, the radiation pattern is computed using a near-to-far field transformation (rather than directly from the DFT flux).
2. The emission from the active region must be computed by combining the response from a collection of dipoles for which the emission is non zero. The dipole emission is expected to decrease with increasing distance from the edge facet.
3. The radiation pattern $P(\theta, \phi)$ is computed for $\theta$ in [-π/2, π/2] (where $\theta$ = -π/2 is the $-y$ direction and $\theta$ = π/2 is $+y$) and $\phi$ in [0, π/2] (where $\phi$ = 0 is the $xy$ plane and $\phi$ = π/2 is the $yz$ plane).

[oskooi]

Here is the initial script for computing the extraction efficiency of a dipole placed in the active region of the cleaved multilayer stack. A schematic of the simulation layout obtained using Simulation.plot2D is shown below. In this example, the active region is a lossless dielectric with refractive index of 2.3 surrounded by cladding layers of index 1.5. The quantum well is positioned in the middle of the active region. The Brillouin-zone integration involves sweeping $k_z$ in the range [0, ω] using 11 equally spaced values. Note: there is no need to compute negative values of $k_z$ since their flux is equivalent to positive $k_z$. The flux emitted by the dipole in the active region is obtained using the LDoS feature (rather than an enclosing box of DFT monitors). Meep supports a 2D cell with out-of-plane wavevector. The kz_2d parameter of the Simulation constructor was set to "complex" (default).

A plot of the extraction efficiency vs. dipole position for a dipole with $E_x$ polariztaion demonstrates that the extraction efficiency approaches a constant value of ~28% with increasing distance of the dipole from the edge of the cleaved facet. A constant nonzero value is expected because the active region contains no loss.