Abstract:
The ability to mode lock a laser to produce short, powerful pulses has useful applications in many areas, including telecommunication systems. In this project, we consider a system with gain & absorption, as well as a spatially extended nonlinear fibre element. In the first instance, the dynamics of the pulse through the fibre are calculated with the split-step method of solving the Nonlinear Schrödinger Equation, while the other components (saturable absorber, gain medium, and spectral filter) are modeled by discrete maps. We find that output of the simulated laser system has regions of stability and instability, much like that of an actual laser. The goal is to replace the integration step for the fibre with a simple map that captures the relevant behavior and provides us with greater insight into the pulse dynamics. To this end, we study the dynamics of the pulse within the fibre itself with questions such as: How does the pulse change as it moves through the fibre? How does the length of the fibre affect the distortion of the pulse? We create discrete input-output maps that capture this behavior with polynomial surfaces in the case of a normal dispersion fibre, and linearly interpolated surfaces in the case of an anomalous dispersion fibre. After successfully reconstructing the simulated behavior of the pulse energy with our maps, we use the numerical continuation tools of bifurcation theory to find the fixed points and dynamical behavior of the system.
