The surface roughness effect on the first order probability density function of an integrated speckle pattern, produced in the condition of circular detection aperture and Gaussian scattering spot was theoretically investigated.
The mutual intensity $J_A$($X_1$, $Y_1$ ; $X_2$, $Y_2$) containing two roughness parameters, dispersion of surface height $<\phi^2>$ and 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.
The ranges of the roughness parameters in which the probability density function change significantly was estimated under the experimental condition, λ= 633nm (the frequency of He-Ne laser), z = 0.5m (distance between rough surface and detector) and $r_m$ = 10 μm (the pixel size of CCD).
The shape of the probability density function depended mostly on $<\phi^2>$. $<\phi^2>$ ranged from 10 $rad^2$ to 18 $rad^2$ for $r_0$ = 0.3mm (common value of a He-Ne laser beam spot) and from 14 $rad^2$ to 22 $rad^2$ for $r_0$ = 1.5mm. These ranges corresponds to those of surface roughness encountered when lens or mirror is polished by emery powder #100-#1000.
This leads to the conclusion that the ranges of the major changes may be adjusted with suitable choices of the parameters, $r_0$ z and λ, and that the surface roughness can be determined with precision if the ranges can be adjusted so as to cover the roughness of the surface we are interested in.