Laser Pulse Shaping via Iterative Learning Control and Infinite-dimensional Extremum Seeking

We investigate pulse shaping and optimization for a laser amplifier. Due to the complex character of the nonlinear PDE dynamics involved in the laser model, it is of interest to consider non-model based methods for pulse shaping. We determine input pulse shapes for an unknown laser dynamics model using iterative learning control (ILC) and infinite-dimensional extremum seeking (ES), which is a real-time optimization strategy. We utilize ILC to obtain the input pulse shape that generates a desired output pulse shape and ES to find the input pulse that maximizes the energy amplifier gain. Both single-pass and double-pass laser models are investigated. The effectiveness of these approaches is illustrated via numerical simulations. INTRODUCTION Control of the shape of laser pulses is a key problem in photolithography light sources, where fundamental physics and control theory come together. The goals are to provide a stable output characteristic (pulse-to-pulse consistency and robustness to manufacturing tolerances, thermal effects, and mild optical damage), extract maximum stored energy from the amplifier gain medium with minimum input signal, and minimize the sensitivity of the total (integrated over a pulse period) output energy to total input energy. Some of these objectives may be competing, which calls not only for feedback design techniques but also for real-time optimization techniques. In [1], the growth of a radiation pulse in a laser amplifier was described by nonlinear, time-dependent photon transport equations, which account for the effect of the radiation on the medium ∗Correspondence author. as well as vice versa. In [2], a one-dimensional model including Poisson’s equation to consider the space-charge for a dischargeexcited ArF excimer laser has been developed. Due to the complex character of the nonlinear partial differential equation (PDE) dynamics involved, it is of interest to consider non-model based methods for pulse shaping. The operation of light sources for lithography is inherently repetitive. Exploiting the periodic nature of the process and its corresponding control actions can be done effectively by incorporating iterative learning control (ILC) methodologies into the light source design. ILC is a control methodology that improves system performance over repeated trials and has become a wellestablished field with many successful theoretical results and applications that range from robotics to hard disk drives [3–5]. However, there are only a few works [6, 7], which apply ILC to distributed parameter systems governed by PDEs. In this paper, we adopt the proportional-differential-type (PD-type) ILC scheme for pulse shaping of the laser amplifier system with the output pulse shape given, where the spatio-temporal evolution of irradiance and excimer population in a laser amplifier, is governed by a coupled nonlinear first-order hyperbolic PDE and a nonlinear ordinary differential equation (ODE) (extended from the classical model [1]). Numerous optimization problems exist in the design and real-time operation of light sources for photolithography. One of them, which is pursued in this paper, is pulse shape optimization. Extremum seeking (ES) [8] is employed in this work to design the finite-time optimal input signal to maximize the amount of stored energy extracted from the amplifier gain medium with a fixed input energy level. ES is a non-model based real-time optimization approach for dynamic problems where only limited