Most of the notes describing the licensing as well as the numerical algorithm behind the mqwgain script command apply to mqwindex command as well, with the difference that mqwindex returns complex index in addition to gain and spontaneous emission. The complex index returned by mqwindex is given as \(dn+i*\kappa\), where \(dn\) is the change in refractive index due to MQW absorption or gain, and \(\kappa\) is the imaginary part of complex index.
Using mqwindex avoids the need to solve the Kramers-Kronig relations to obtain the real part of index from gain or absorption. Solving this integral is numerically challenging due to the need to integrate over a wide frequency range of gain values, much wider than both the range of interest and the range where mqwgain gives reasonably accurate results.
The complex index output of mqwindex can be easily exported to one of our optical solvers, such as FDTD, FDE, and FEEM and calculate transmission through a structure that includes an MQW region, for example.
The example below benchmarks the accuracy of mqwindex command against reference .
The main file which should be run from the script environment of any of our products is [[mqwindex_50kVpercm_sqw_Ahn1988_index_test.lsf]]. This file implements and solves mqwindex command for a device based on an AlGaAs/GaAs/AlGaAs single quantum well under 50 kV/cm electric field. Other script files used by the main file and that should be placed in the same folder with the main file are: [[mqw_material_build_functions.lsf]] and [[mqw_material_database.lsf]], which create custom material properties based on the reference rather than using our material database in order to make the comparison fair, and [[plot_wavefunction_banddiagram.lsf]], which implements a function to plot wave functions on top of band diagram.
Upon running the main script file following results will be plotted. The band diagram accompanied by corresponding conduction and valence band wave functions looks like:
The relative change of the real part of index (refractive index) compared to the reference for two different charge densities can be seen in the following figure:
The corresponding imaginary part of index is shown in the figure below:
Similarly to mqwgain, the user can plot various auxiliary outputs directly using our visualizer, such as non-parabolic valence bands:
Additionaly, gain and spontaneous emission are returned by mqwindex, and their value will be the same as the results of mqwgain for the same structure.
It is important to note that it is necessary to set a sufficiently large portion of the Brillouin zone in mqwindex calculation in order to have a stable converged results for the real part of index. Typically this portion is closer to 30%, compared to 10-20% needed for the mqwgain calculation and, as with mqwgain, it will be temperature dependent (higher temperature require larger portion in order to include all relevant energy levels).
 D. Ahn, S. L. Chuang, and Y.C. Chang, Valenceband mixing effects on the gain and the refractive index change of quantumwell lasers, J. Appl. Phys. 64, 4056 (1988).