4.4 Numerical Implementation

In this section we describe the numerical implementation of Abbe's method of aerial image simulation. From the above discussion it can be seen that basically two Fourier transforms are required:

• Forward Transform. The Fourier coefficients T_nm of the mask transmission function t(x, y) have to be calculated. As in our model t(x, y) consists of piecewise constant regions with ideal sharp transitions, a simple sampling is not adequate because t(x, y) is definitely no low-pass function. Hence, we developed a semi-analytical method to avoid errors arising from aliasing.
• Backward Transform. The Fourier coefficients T_nm are weighted with the pupil function and have then to be transformed back to the spatial domain to obtain the image field. As the projection lens acts as a low-pass filter due to the finite extent of the pupil function the image has only spectral components up to a certain order. Therefore sampling can be employed for the backward transform without introducing any aliasing.

Consequently, the calculation of the aerial image is in a certain sense ``exact'', i.e., no additional discretization errors are introduced by the numerical implementation. All approximations made throughout can be physically explained and therefore constitute a sound physical and mathematical basis of the aerial image simulator.