It is shown by numerical simulations for a random, one-dimensional surface defined by the equationx 3=ζ(x 1), where the surface profile function ζ(x 1) is a stationary, stochastic, Gaussian process, that the transverse correlation lengtha of the surface roughness is a good measure of the mean distance 〈d〉 between consecutive peaks and valleys on the surface. In the case that the surface height correlation function 〈ζ(x 1)ζ(x 1′)〉/〈ζ2(x 1)〉=W (|x 1−x 1′|) has the Lorentzian formW(|x 1|)=a 2/(x 1 2 +a 2) we find that 〈d〉=0.9080a; when it has the Gaussian formW(|x 1|)=exp(−x 1 2 /a 2), we find that 〈d〉=1.2837a; and when it has the nonmonotonic formW(|x 1|)=sin(πx 1/a)/(πx 1/a), we find that 〈d〉=1.2883a. These results suggest that 〈d〉 is larger, the faster the surface structure factorg(|Q|) [the Fourier transform ofW(|x 1|)] decays to zero with increasing |Q|. We also obtain the functionP(itx 1), which is defined in such a way that, ifx 1=0 is a zero of ζ′(x 1),P(x 1)dx 1 is the probability that the nearest zero of ζ′(x 1) for positivex 1 lies betweenx 1 andx 1+dx 1.
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H. Davies,Proc. Inst. Elect. Eng. (London)101(IV):209 (1954).
J. Crowell and R. H. Ritchie,J. Opt. Soc. Am. 60:794 (1970).
E. Kretschraann, T. L. Ferrell, and J. C. Ashley,Phys. Rev. Lett. 42:1312 (1979).
A. A. Maradudin, E. R. Méndez, and T. Michel, inScattering from Volumes and Surfaces, J. C. Dainty and M. Nieto-Vesperinas, eds. (North-Holland, Amsterdam, 1989).
M. deBilly, G. Quentin, and E. Baron,J. Appl. Phys.,61:2140 (1987).
J. C. Dainty, M.-J. Kim, and A. J. Sant, Measurements of enhanced backscattering of light from one and two dimensional random rough surfaces, Notes for the Workshop “Recent Progress in Surface and Volume Scattering,” Madrid, September 14–16, 1988, and for the International Working Group Meeting on “Wave Propagation in Random Media,” Tallinn, September 19–23, 1988.
R. Kubo,J. Phys. Soc. Japan 17:1100 (1962).
M. Schlesinger, inSittatistical Mechanics and Statistical Methods in Theory and Application, U. Landman, ed. (Plenum Press, New York, 1977), p. 507.
E. I. Thorsos,J. Acoust. Soc. Am. 83:78 (1988).
A. Stuart and J. Keith Ord,Kendall's Advanced Theory of Statistics (Charles Griffin, London, 1987), pp. 308–309.
A. A. Maradudin, E. R. Méndez, and T. Michel,J. Opt. Soc. Am. B (in press).
S. O. Rice, inSelected Papers on Noise and Stochastic Processes, Nelson Wax, ed. (Dover, New York, 1965), p. 133.
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Maradudin, A.A., Michel, T. The transverse correlation length for randomly rough surfaces. J Stat Phys 58, 485–501 (1990). https://doi.org/10.1007/BF01112758
- Transverse correlation length
- rough surfaces