FDTD material database contains a Four-Level Two-Electron Material model as one of the built-in material plugins. This simple example demonstrates the basic concept of this model and how one might use it in a simulation.
As can be seen in the image above, a gain structure is placed in the simulation region. The gain structure's material utilizes the "Four-Level Two-Electron" material model available at the Material Database (see figure below). Two sources, one acting as the pump and the other acting as the probe are added to the simulation. The pump is at 750 nm, corresponding to the energy difference between the level 3 and level 0 (E3-E0). The maximum amplitude of the pump is 2e6 V/m. The probe is at 1500 nm, corresponding to the energy difference between the level 2 and level 1 (E2-E1), maximum amplitude of the probe is 1e4 V/m, an amplitude much less than that of the pump. The simulation will be pumped with a 4 ps pulse, centered at 8 ps. The simulation will be probed with an ultra short pulse, at 30 ps.
Open the simulation file pump_probe.fsp. Run the simulation and then run the script file pump_probe_analysis.lsf. The plots below will be generated. From the time signal plot we see the pump and the pulse and that their respective shapes and positions reflect the source settings. The signals in the time signal plot are scaled to show on the same graph. The pump peak intensity is 4x104 times larger than probe peak intensity.
From the level population plot, we see what is happening in the material as the pump enters. Electron is brought from the ground state to the third state, which rapidly decays to the second state. Levels 0 and 3 are not perturbed much. But we end up bringing up the level 2 and down the 1st level. After around 10 ps, the desired population inversion is reached. After the pump pulse dies away, level population decays back to its original state with certain time constants. Probe pulse has completely negligible effect on the level population due to its low amplitude, but here it is almost like a linear system, so at 30 ps, there is a gain.
The gain at around 1500nm where we have this inverted level population can be seen from the transmission plot. At other wavelengths, we have 100% transmission.
- Shih-Hui Chang and Allen Taflove, "Finite-difference time-domain model of lasing action in a four-level two-electron atomic system", Optics Express, Vol. 12 Issue 16, pp.3827-3833 (2004)
- Taflove, Computational Electromagnetics: The Finite-Difference Time-Domain Method. Boston: Artech House, (2005).