This cavity supports two modes at 1430nm and 1491nm. Here, a quantum dot (QD) is placed in the setup and the structure is analyzed.
Examination of the mode profile below shows that there is a nice position that has fairly high intensity and same polarization along the x axis, where the QD can be placed. The simulation will be pumped at 1430nm and use the resonant mode as means to enhance the excitation of the QD and optical pumping, and then lase at 1491nm. To pump the system optically, a high numerical aperture beam, NA of 0.6 is used. The beam is placed on top of the QD to maximally pump the QD, and also break the symmetry. If the beam is placed on top of the center of the cavity, it will have opposite symmetry to the two modes, which will resultantly not excite the modes at all.
The QD is meant to be really small. To represent the QD, rather than trying to resolve it on FDTD mesh, we have represented it at a single position of FDTD mesh. In order to do so, the material that represents the QD needs to be carefully placed with respect to the Yee cell (please refer to the images below for the placement and the Yee cell). In this case, QD needs to be placed at a particular location, which is the midpoint of the mesh cell, and excite the polarization in the x direction. Also care needs to be taken with the placement of the time monitor.
Initial simulations showed that the lasing threshold could not be reached due to the low Q factor of the cavity. To increase the gain, we artificially increased the size of the QD to cover 3x3x3 = 27 grid cells of FDTD mesh, as can be seen below. In principle, the same result can be achieved by increasing the electron density in the QD, one of the parameters in the material model, but this creates so much gain in a single cell that the simulation becomes numerically unstable for reasonable values of dt (reduction of dt causes simulation memory requirement to increase, which is not desired). Another option for dealing with the issue would be to re-engineer the cavity to achieve much higher quality (Q) factor, but in this case the QD has been artificially increased in size. This 3D simulation was run for 150p. The simulation is pumped with the maximum field amplitude of 1e7V/m.
Open and run the simulation file ppc_cavity_QD.fsp. Having run the simulation, run the script file ppc_cavity_QD_analysis.lsf to generate the images below. Because of the maximum field amplitude of 1e7V/m, population inversion is achieved in a relatively short time. As can be seen in the images below, steady state is reached but the level populations are oscillating. Level population of N2 is substantially larger than level population of N1 on average, which means substantial gain, however, because of oscillating, QD is actually absorbing radiation when N1 is larger than N2.
The image below shows the field profile inside the cavity at the lasing wavelength, we see that the cavity mode is excited and the bright spot near the edge of the mode where the QD is positioned and where the amplification is happening.
Four-Level Two-Electron Material model