Abstract

We present in this paper, approximate analytical expressions for the intensity of light scattered by a rough surface, whose elevation ξ(x,y) in the z-direction is a zero mean stationary Gaussian random variable. With (x,y) and (x,y) being two points on the surface, we have ξ(x,y)=0 with a correlation, ξ(x,y)ξ(x,y)=σ2g(r), where r=[(xx)2+(yy)2]1/2 is the distance between these two points. We consider g(r)=exp[(r/l)β] with 1β2, showing that g(0)=1 and g(r)0 for rl. The intensity expression is sought to be expressed as f(vxy)={1+(c/2y)[vx2+vy2]}y, where vx and vy are the wave vectors of scattering, as defined by the Beckmann notation. In the paper, we present expressions for c and y, in terms of σ, l, and β. The closed form expressions are verified to be true, for the cases β=1 and β=2, for which exact expressions are known. For other cases, i.e., β1, 2 we present approximate expressions for the scattered intensity, in the range, vxy=(vx2+vy2)1/26.0 and show that the relation for f(vxy), given above, expresses the scattered intensity quite accurately, thus providing a simple computational methods in situations of practical importance.

© 2013 Optical Society of America

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