Intensity profile of light scattered from a rough surface.

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=[(x-x')2+(y-y')2]1/2 is the distance between these two points. We consider g(r)=exp[-(r/l)ß] with 1≤ß≤2, showing that g(0)=1 and g(r)→0 for r≫l. 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 ß. The closed form expressions are verified to be true, for the cases ß=1 and ß=2, for which exact expressions are known. For other cases, i.e., ß≠1, 2 we present approximate expressions for the scattered intensity, in the range, vxy=(vx2+vy2)1/2≤6.0 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.