Optimal Control and Identification for Optical Lithography

The creation of a fine line using a positive optical photoresist involves several essentially irreversible steps. The cleaning, spin coating, prebake, exposure, post-exposure bake, and development/rinse steps must all be processed within specified tolerances to insure that each subsequent step may proceed. In the past, control of the process was based on whether a specified critical criterion was within tolerance. On-line optimal control of the development phase of the optical lithography process has been found to be feasible. This paper describes the use of both the measurements available during development and the process models associated with exposure and development to obtain optimal estimates of linewidth. Since process states cannot be completely measured, and there is disturbance and measurement noise during the process, an extended Kalman filter is used to identify the process states. A recursive least squares algorithm is used to update uncertain model parameters based upon the process measurements. Using the optimally estimated process states for the development stage, an optimal control algorithm, derived from calculus of variations and Pontryagin´s minimum principle, is implemented. The feasibilty of using Kalman filtering and on-line parameter identification for the development step of optical lithography is examined.