A functional-integral representation of the statistics of rough surfaces is developed. The assumption of locality, defined in the text, produces a probability distribution for the shape of the surface that has the form of an exponential of a power series in surface height, slope, curvature, and higher surface derivatives. Each term in this series is shown to have a straightforward interpretation with respect to the surface statistics. For example, the lowest-order quadratic terms yield the mean-squared height and the correlation length of the surface. To the lowest order and within the assumption of locality the two-point correlation function is shown to be essentially a K0 modified Bessel function away from the origin. At the origin it is shown to have a finite value equal to the rms height variation of the surface. The power spectrum corresponding to this result for the correlation function is in good agreement with measured power spectra. In order to account for the atomic structure of real surfaces the high-spatial-frequency behavior of the statistics must be controlled. Here we use a cutoff at high spatial frequencies. We also show that fractal behavior occurs naturally in this formalism, owing to the anomalous scaling of the correlation functions when higher-order terms in the power series are included in the calculation.
© 1991 Optical Society of AmericaFull Article | PDF Article
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