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[/numerical aperture (NA)] and 0.4(/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.
Theories are developed to optimize the mask structure of alternating phase-shifting masks (PSMs) to minimize the average image placement error towards aberration under coherent imaging. The constraint of the optimization is a given mean value of RMS aberration, which corresponds to infinitely many sets of random Zernike coefficients. To begin the analysis, the image placement error is expressed as a function of the mask spectrum and the wave aberration. Monte Carlo analysis on the Zernike coefficients is then performed, which assures us that a global minimum of average image placement error is likely to occur at low phase widths. This result is confirmed by analytically considering the expected value of the square of the image placement error. By Golden Section Search, the optimal phase width is found to be 0.3707 (λ/NA) at 0.07 λ RMS aberration. This result is applicable to the design of all alternating PSMs.