Aerial image through focus in the presence of aberrations and electromagnetic edge effects modeled by adding phase at pattern edges is expanded by a quadratic equation with respect to focus. The quadratic equation is expressed by four coefficients that are adequately independent of both mask layout and the variations in the optical setting in projection printing, thus saving the computation cost of the quadratic fit for each individual layout edge position in a new mask pattern or variation from a nominal optical setting. The error of this method is less than 1% for any typical integrated circuit features. This accuracy holds when the defocus is less than one Rayleigh unit (, where λ is a wavelength and NA is the numerical aperture) and the root mean square of the existing aberration is less than , which encompasses current lithography practice. More importantly, the method is a foundation for future first-cut accurate algebraic imaging models that have sufficient speed for assessing the desired or undesired changes in the through-focus images of millions of features as the optical system conditions change. These optical system changes occur naturally across the image field, and aberration levels are even programmed in tuning modern tooling to compensate for electromagnetic mask edge effects.
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