RESEARCH PROJECTS

Mode-locked lasers have become a thriving research topic in the math community in last decade due to their wide range of applications in GPS navigation, satellite communication, medicine, etc. These lasers generate periodically stationary pulses by balancing gain and loss, dispersion and nonlinearity. I have developed a gradient based optimization method in conjunction with a PDE solver to discover these pulses. I perform the linear stability analysis of the pulses obtained using optimization by numerically computing the spectrum of the monodromy operator. I have proven the existence and uniqueness of the monodromy operator using the theory of semigroups of linear operators. I then use concepts of functional analysis together with the theory of multiplication operators to derive a formula for the essential spectrum of the monodromy operator for a stretched pulse laser. I also perform parameter continuation studies to observe bifurcations of periodically stationary pulses in the stretched pulse laser. 

My work presents the first stability analysis of periodically stationary pulses in a realistic lumped model of an experimental short pulse laser. Moreover, I introduce an analytical formula for the essential spectrum of the monodromy operator through rigorous proofs. This formula combined with the computational method for stability analysis will eventually allow the modellers to determine regions in the parameter space that support stable periodically stationary pulses, and ultimately to quantify the effects that optical noise and other perturbations have on the system performance.

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