Dispersion of effective refractive indices of mid-infrared quantum cascade lasers

We point out the difference between the two mostly used methods for calculating the effective refractive index of mid-infrared quantum cascade lasers, namely by solving the Maxwell's wave equation and by analyzing the frequency spacing of the longitudinal modes of the Fabry-Perot cavity. The effective refractive indices obtained by these methods are shown to be different, as one refers to the phase effective refractive index while the other refers to the group effective refractive index, respectively. Dispersion relationships for these two effective refractive indices are deduced for mid-infrared quantum cascade lasers, which show an increase in group refractive index and a decrease in phase refractive index with wavelength. Experiments are conducted to obtain the group effective refractive indices of quantum cascade lasers emitting at various wavelengths, and good agreement has been achieved between the experimental data and theory.