Thermocapillary Shaping is a novel method for fabrication of optical phase masks. By controlling the heat gradients on a thin liquid film’s surface, we drive surface deformations via the thermocapillary effect. The underlying physics of this method is described by a highly nonlinear partial differential equation with Dirichlet boundary conditions (pinning).
We have previously demonstrated the viability of the method by subjecting a thin film to a heat flux from below, as prescribed by the steady state solution of our thermocapillary equation over an infinite domain.1 However, the result, while consistent with theory, still exhibited non-negligible inaccuracies and required a long convergence time.
In this work, we present the development of a 2D dynamical numerical solver for simulated closed loop control of the Thermocapillary Shaping method in finite domains. We implemented closed-loop control via a PID controller that regulates the heat flux actuation at the base of the liquid chamber based on an array of surface topography measurements.
We show convergence rates two orders of magnitude faster than the previous open-loop approach, while achieving surface accuracies below 10 nm – as required for high-precision optical elements. Furthermore, unlike the open loop approach, our algorithm converges even with imperfect initial conditions, and volumetric errors.
