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As is well known, the mask in projection printing is a thin (about 1,am thick) object, composed of numerous variable domains. The various domains contain different complex refractive indices. These refractive indices form a discontinuous "mask function" ,appearing as a coefficient in the Maxwell-Material (MM) equation1 that describes the transmission of the electric field through the 3D mask domain. The right-hand side of this equation involves the gradient of the dot-product of the electric field with the gradient of the log of the refractive index. Within each feature, the refractive index is a constant. Therefore its gradient vanishes, and the resulting local MM equation is the wave equation. However, at a feature boundary (chrome-quartz, phase-i— phase-2, etc.) the complex refractive index is discontinuous, resulting in a highly singular delta-function-like coefficient. The trailing components of the field display the standard Maxwell discontinuity conditions, resulting in two very singular terms, i.e. a derivative of a delta function and a product of two delta functions. Such a source term, with such a remarkable oscillation, will be the origin of extreme instability in the numerical solution of the equations, no matter which direct solution algorithm is employed. Accordingly, the derivation of a preconditioner, namely a function which contains most of the unpleasantness of the equation times a function to be found, is absolutely essential to obtain accurate solutions in a reasonable time. Through an elaborate uniformly asymptotic scheme and scale analysis, we have derived such a preconditioner. The solution of the direct MM equation was compared with the solution using the preconditioner, and it gave us the confidence that we were on the right track. It actually means that one wishes to find the mask's Green function which, when convolved with the electric field incident on the mask, yields the exiting field which then enters the imaging optical system. It is the purpose of this paper to demonstrate the results obtained thus far, as well as to illustrate its implications on defocus linewidth control.
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