随机表面散射光场的格林函数法与基尔霍夫近似的比较

Green's function method of light scattering from random surfaces compares with Kirchhoff's approximation

从简谐光波满足的亥姆霍兹方程出发 ,将由格林定理得到的介质分界面上的积分方程转化为以表面上的光波及其导数为未知量的线性方程组 ,并对其进行数值求解 ,实现了光场的数值计算 .同时 ,由透射光场的格林函数积分得出了基尔霍夫近似下光场的表达式 .通过类比推导夫琅和费面上散斑场自相关函数的方法 ,提出了产生随机表面及其导数的傅里叶变换方法 .在此基础上 ,对采用基尔霍夫近似进行自仿射分形随机表面的散射光场数值计算的精确程度进行了研究 .发现在随机表面粗糙度比较小时 ,基尔霍夫近似的精度比较高 ;在粗糙度相同的情况下 ,表面的分形维数越小 ,基尔霍夫近似的精度越高 .

Starting from the Helmholtz Equation, we obtain the integral equations of the light field at the medium interfaces by use of Green's theorem. Then the integral equations are discretized into a linear equation set, from which the values of the light field and its derivatives at the interface can be numerically solved. We also obtain the expression for the transmissive light waves from the Green's-function integral in the case of Kirchhoff's approximation. By an analogy to the derivation process of the autocorrelation functions of speckles in Frauhofer plane, we propose the method for the generation of random self-affine fractal surfaces and Fourier transformation method for the numerical derivative of random surfaces. Then we study the accuracy of Kirchhoff's approximation in the scattering of light field from the random self-affine fractal surface. We find that the accuracy of Kirchhoff's approximation is relatively high when the root-mean-square roughness w is small. for random surfaces with the same value of w but smaller values of roughness exponent α, the Kirchhoff's approximation gives higher accuracy in the calculation of scattered light fields. We believe that the results of this paper would be of significance in understanding the validity range of the Kirchhoff's approximation when it is applied to light scattering from self-affine random surfaces.