Numerical aspects of QCL frequency comb modeling

Abstract

Quantum cascade lasers (QCLs) are attracting a great deal of interest due to their wide range of applications in science and industry. The QCL represents a compact and integrable source of coherent light in the infrared and the terahertz regime. To expand the bandwidth of the QCL in order to enlarge the number of detectable analytes, the cascade design can be slightly changed along the growth direction, leading to a different emission frequency of the segments. This structure, namely the heterogeneous QCL, has already been shown in various publications. In order to deepen the understanding of these devices and, among other things, to analyze the generation conditions of optical frequency combs with heterogeneous QCLs, the conventional numerical model of a QCL developed by N. Opačak et al. was numerically analyzed and extended in this work.Starting with the first-order wave-equation, the description was expanded to include losses, group-velocity dispersion effects, and gain. In case of the losses, the exponential integrator was utilized. This method implements the losses analytically, which is in deep contrast to the previously used first-order approximation. The group velocity dispersion (GVD) implementation was stabilized. This is essential since in case of the simulation of heterogeneous QCLs, the polarization elimination, which was utilized in the master equation model developed by N. Opačak et al., cannot be applied. If this simplification is used, the gain induces losses above and below the center frequency stabilizing the overall simulation. To overcome the inherent stability problem, a new method was developed which retains the advantage of parallel computing.This method is based on the Laplace transformation of the governing wave-equation. The implementation becomes stable by conducting a Taylor expansion and including higher order terms. The presented method can also be used for a stable implementation of higher-order effects, such as third or fourth-order, and enables their stable joint usage.Instead of using the master equation formalism with its simplified gain implementation, a slightly simplified version of the Maxwell-Bloch equations was implemented. The equation for the upper state population was carried out by a first-order explicit scheme. This scheme cannot be applied to the polarization equation since it induces numerical losses above and below the center frequency. To overcome this numerical problem, the bilinear Z-transformation was applied to the Laplace transformed polarization equation, which eliminates the unwanted losses. The implementation of the gain was expanded towards the simulation of heterogeneous QCL by a multiple usage of the polarization equation. The gain peaks can be readily shifted in the frequency domain to simulate different cascade designs.Apart from the heterogeneous QCL, the emission-spectrum can be enriched by nonlinear e effects. To obtain noticeable broadening e effects, pulses with high intensities are required. In case of QCL, the generation of pulses is challenging due to the fast gain properties of the medium. A powerful technique to overcome this limitation is presented by using a QCL emitting a frequency-modulated optical frequency comb and by compensating its chirp by a compressor. To generate short pulses, the compressor must fully compensate for the chirp induced by the QCL. Different shaping methods of the QCL were analyzed to align these two components, and their impact was simulated based on the master equation for fast gain media. Among the parameters, three important shaping mechanisms were found: laser bias, cavity design, and the reflectivity of the facets. The spatially varying laser bias was simulated by using a more section device, and the cavity design was tested by applying a tapered structure to the device. The reflectivity of the facets influences the extent of their influence. All three parameters combined reveal a versatile set of tools to tailor the locking state of the QCL, enabling the generation of ultra-short pulses.