Alternating phase-shifting mask design for low aberration sensitivity

Abstract

The aberration present in the lenses of exposure systems can cause placement errors to the images produced by alternating phase-shifting masks (PSMs). In reality, when the aberration signature varies from one lens to another, the magnitude of placement error also varies. It remains a question of how the alternating PSM should be designed, so that the image placement error, on average, can be minimized. To achieve this goal, we are interested in optimizing the phase width of an alternating PSM with a fixed critical dimension (CD). The constraint of the optimization is the mean of root mean square (rms) aberrations for a set of interest of exposure systems. To begin the analysis, the image placement error is expressed as a function of illumination, mask spectrum, and wave aberration. A Monte Carlo technique is then applied to produce random samples of wave aberration and image placement error. This analysis shows that a global minimum of mean image placement error is likely to occur at phase widths between 0.2[λ/numerical aperture (NA)] and 0.4(λ/NA). This is further confirmed by analytically considering the expected value of the square of the image placement error. The methodology of finding the optimal phase width is applicable to the design of all alternating PSMs. © 2005 Society of Photo-Optical Instrumentation Engineers.