Intensity profile of light scattered from a rough surface

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 x0; y0 being two points on the surface, we have hξ x; y i 0 with a correlation, hξ x; y ξ x0; y0 i σ2g r , where r x − x0 2 y − y0 2 1∕2 is the distance between these two points. We consider g r exp − r∕l β with 1 ≤ β ≤ 2, showing that g 0 1 and g r → 0 for r ≫ l. The intensity expression is sought to be expressed as f vxy f1 c∕2y v2x v2y g−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., β ≠ 1, 2 we present approximate expressions for the scattered intensity, in the range, vxy v2x v2y, 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.