Analytic form for the power spectral density in one, two, and three dimensions

Analytical expressions for the power spectral density (PSD) are often useful in stochastic lithography simulation and the metrology of roughness. Using a common stretched exponential correlation function with three parameters (standard deviation, correlation length, and roughness exponent), the PSD can be computed as the Fourier transform of the autocorrelation function. For the special cases of roughness exponent equal to 0.5 and 1, the PSD can be computed analytically for one, two, and three dimensions. In this paper, the analytical results of these calculations are given. The resulting equations can be used when modeling rough lines, surfaces, or volumes.