On the accuracy of different Fourier transforms of VLSI designs

We evaluate the accuracy of the simulated aerial image computed using different types of Fourier transform: the Continuous Fourier transform (CFT), continuous Fourier series (CFS) and discrete Fourier transform (DFT), the transform currently used in practise. Lithography simulation necessitates efficient algorithms to accurately estimate the Fourier transform. The rectilinear mask structure can be used to compute the CFS efficiently, in most cases much faster than the DFT. We present a rigorous analysis and comparison of FFT-simulated aerial image and CFS-simulated aerial image. We also present the conditions under which the most accurate simulated aerial image for each type of Fourier transform can be obtained. We show that there are two main sources of inaccuracy in the computation of aerial image using the DFT. First, we establish that aliasing from the inherent discontinuity of rectilinear polygons is the main source of inaccuracy. We show the conditions under which this results in an over-estimate or underestimate of the Critical Dimension (CD). We thus describe how to perform aliasing-minimizing sampling. This adapts the sampling to the polygon pattern of the mask, by ensuring each discontinuity is exactly at the middle of two consecutive samples. Second, we show that sampling at twice the Nyquist rate of the filtered signal can in practise circumvent the inaccuracy due to the approximation of the integral by a discrete sum.