Abstract

We present in this paper, approximate analytical expressions for the intensity of light scattered by a rough surface, whose elevation $\xi (x,y)$ in the $z$-direction is a zero mean stationary Gaussian random variable. With $(x,y)$ and $(x^{\prime },y^{\prime })$ being two points on the surface, we have $〈\xi (x,y)〉=0$ with a correlation, $〈\xi (x,y)\xi (x^{\prime },y^{\prime })〉={\sigma }^{2}g(r)$, where $r={[{(x-x^{\prime })}^2+{(y-y^{\prime })}^2]}^{1/2}$ is the distance between these two points. We consider $g(r)=\exp [-{(r/l)}^{\beta }]$ with $1\le \beta \le 2$, showing that $g(0)=1$ and $g(r)\to 0$ for $r\gg l$. The intensity expression is sought to be expressed as $f(v_{xy})={\{1+(c/2y)[v_x^2+v_y^2]\}}^{-y}$, where $v_x$ and $v_y$ 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 $\sigma$, $l$, and $\beta$. The closed form expressions are verified to be true, for the cases $\beta =1$ and $\beta =2$, for which exact expressions are known. For other cases, i.e., $\beta \ne 1$, 2 we present approximate expressions for the scattered intensity, in the range, $v_{xy}={(v_x^2+v_y^2)}^{1/2}\le 6.0$ and show that the relation for $f(v_{xy})$, 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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