Laser pulse shaping via extremum seeking

Extremum seeking, a non-model based optimization scheme, is employed to design laser pulse shapes that maximize the amount of stored energy extracted from the amplifier gain medium for a fixed input energy and inversion density. For this pulse shaping problem, a double-pass laser amplifier whose dynamics are fully coupled and composed of two nonlinear, first-order hyperbolic partial differential equations, with time delays in the boundary conditions, and a nonlinear ordinary differential equation, is considered. These complex dynamics make the optimization problem difficult, if not impossible, to solve analytically and make the application of non-model based optimization techniques necessary. Hence, the laser pulse shaping problem is formulated as a finite-time optimal control problem, which is solved by first, parameterizing the input pulse and pumping rate over the system's finite time interval and then, utilizing extremum seeking to maximize the associated cost function. The advantage of the approach is that the model information is not required for optimization. The extremum seeking methodology reveals that a rather non-obvious laser pumping rate waveform increases the laser gain by inducing a resonant response in the laser's nonlinear dynamics. Numerical simulations illustrate the effectiveness of the approach proposed in the paper.