Shape-based optimal estimation and design of curve evolution processes with application to plasma etching

Abstract

This paper considers the problem of determining a finite number of discrete parameters appearing in a nonlinear partial differential equation describing a curve evolution process. The method is applied to the plasma etching of thin films for semiconductor manufacturing. Results are obtained within the mathematical framework of level set methods. Here, the evolution of the curve under study is captured through the evolution of a level set function. The underlying physics of the process are completely contained in a scalar function called the speed function. The degree of difficulty of treating the evolution equation depends on the functional dependencies of the speed function. This paper presents optimal estimation and design techniques based on analytical gradient computations for a class of position and orientation dependent speed functions. The technique is demonstrated on a plasma etching model taken from the literature. Only simulation results are presented here, but the model under study has been shown to reproduce experimental data with reasonable accuracy. In the estimation problem, parameters in the model are fit to best match the feature shape measured in experiments. In the optimal design problem, parameter values are selected to most closely attain a desired feature shape.

Optimal Control of the Propagation of a Graph in Inhomogeneous Media

“…It is known for such an initial value problem that when formulated as finding Φ(x, t) satisfying Φ t = a|∇Φ|, Φ(x, 0) = Φ 0 (x) and Γ(t) is the zero level set of Φ(·, t) that there is a unique viscosity solution. Such a problem arises in etching out a surface Γ(t) with a prescribed etch rate which may depend on the medium, [12,3,2]. It is natural to control the final surface at time T say using the etch rate.…”

Numerical analysis of an inverse problem for the eikonal equation

“…In Section 4 we finally present a series of numerical tests in which we consider the more general functional (1.6) and apply our discretization strategy to varying geometries and numbers of degrees of freedom for a. Let us finish this introduction by referring to [5,9], where optimal control problems for time dependent Hamilton-Jacobi equations were considered.…”