Surface roughness dependence of the first-order probability density function for integrated speckle

The surface roughness dependence of the first-order probability density function (PDF) of an integrated speckle pattern, produced in the condition of circular detection aperture and gaussian scattering spot, was theoretically investigated. The mutual intensity J_A(x₁, y₁; x₂, y₂) containing two roughness parameters, dispersion of surface height ø² lateral correlation length x_c, was calculated. The exact first-order probability density function was analytically derived and numerically calculated by means of the Karhunen-Loeve expansion and fast Fourier transform (FFT). As a diffuse object became smooth, the first order probability density function was changed from negative exponential to sharp peak gaussian centered around mean intensity.