Theoretical And Experimental Study On Random Surfaces And Their Scattered Optical Fields

Random surfaces and their scattered light measurements are of great importance in many scientific and technological fields such as material growth, precision machining and medicine diagnosis etc. The study of optical fields scattered from random surfaces is the main content of statistical optics. From these optical fields the properties of the surfaces can also be explored. Based on the theory of the scattered optical field, the light intensity distribution scattered by a random surface is defined as the scattered light intensity profile. The relation between the scattered light intensity profile and the statistical properties of the random surface, or the extraction of surface parameters from the measured scattered light intensity profile, is the ultimate purpose of the study about random surfaces and their scattered optical fields. In recent years, many researchers have made great efforts in both studying the scattered light measurements of the random surfaces and retrieving the surface parameters from different scattered light intensity profiles. However, most of the experimental setups reported in the literature are designed for one or two parameters of a surface from a single scattered light intensity profile, or some methods are used in different experimental setups for the measurement for all parameters of a surface. The report about the simultaneity extraction of all parameters of a surface from one single scattered light intensity profile is quite rare.In this paper, a new method is proposed for the simultaneous extraction of the three parameters of the self-affine fractal surfaces from a single scattered light intensity profile. This new method uses Levenberg-Marquardt algorithm to fit the theoretical equation of scattered light intensity profile in the integral form to the experimental profile data, and the three parameters of the self-affine fractal surface are extracted by Visual C++ programmes. The surface model of the self-affine fractal is demonstrated to be more adequate for the comprehensive description of a wide category of random surfaces at present. Besides the traditional parameters of root-mean-square roughness w and correlation lengthξ, a new parameter of the roughness exponentαis introduced in this model to characterize the short-range fractal properties of the surfaces. In the measurement of the scattered light intensity profiles of transmission-mode random surfaces, an experimental setup is constructed combining a Boxcar of gated integration to reduce the signal noise with computer-controlled intensity detection technique to acquire the data automatically. In order to describe accurately the statistical properties of random surfaces, various kinds of surface samples preparations are made in the laboratory. The surface topographies are scanned by the atomic force microscopy (AFM). The main contents and results of this thesis can be summarized as follows.1. Based on Kirchhoff approximation theory, we obtain the mathematical expression for the scattered optical fields of random surfaces in Fresnel and Fraunhofer diffraction regions. And the approximately relational formula between statistical parameters of the random surface and the scattered light intensity profile is presented. We analyze the characteristics of light scattered from different random surfaces, such as Gaussian correlation random surfaces and self-affine fractal random surfaces, and we use the method to retrieve statistical parameters of different random surfaces. In the experimental measurement, root-mean-square roughness w is determined on the relationship between the pulse light intensity of the scattered light intensity profile and the parallel component of light wave vector. The lateral correlation lengthξcan be obtained from the change of the full width at half-maximum (FWHM) of the scattered light intensity profile. And the roughness exponentαcan also be calculated by the relationship between the scattered light intensity profile and the scattered structure factor.2. According to mathematical models, the various kinds of surface samples preparations are made in the laboratory. We measure these kinds of random surfaces such as Gaussian correlation random surfaces, self-affine fractal random surfaces and weak scatter random surfaces by AFM. The one-dimensional, two-dimensional and three-dimensional topographies of the surfaces are scanned by AFM on the basic image formation principle. Based on Fourier transforms of AFM images, we study statistical properties of surface topographies. Otherwise, we also calculate the statistical parameters and the autocorrelation function of random surfaces based on AFM data.3. The experimental system is set up to measure the light scattered from the reflection-mode random surface. In this system, the He-Ne laser irradiates the random surface after through the filter and the lens. The light scattered by the random surface forms the scattered optical field on the Fraunhofer diffraction region. Light intensities in the scattered optical fields are detected by Panasonic WV-BP310 charge-coupled device (CCD). And then the simulated signals are converted into the digital data by DH-VT110 image collecting card. The scattered light intensity profile is acquired via the computer programme.4. In the experiment to measure the light scattered from the reflection-mode random surface, we adopt the roughness backside of Si(110) wafer as the sample of self-affine fractal random surface. CCD detects the light intensities scattered from the sample surface at the incidence angles of 45°, 50°, 55°, 60°, 65°, 70°, 75°, 80°, 83°and 85°. The scattered light intensity profiles at different incident angles are calculated by the symmetric decline function in mathematics. The roughness exponent of the random surface is extracted from the scattered light intensity profile, based on the relationship between FWHM of the scattered light intensity profile and the light wave vector. In addition, we also measured the roughness exponent by AFM. It is obvious that the outcome derive from the former is corresponding with the latter.5. The transmission-mode random surface is used as the self-affine surface sample. According to the Fourier-Bessel transform, the mathematical express is taken to describe the relationship between the scattered light intensity profile and the parameters of random surface. The result shows that this relational formula is a double exponential integral function including the Bessel function. Therefore, it is difficulty to extract the surface parameters from the experimental measured data by the general fitting method. In this paper, the nonlinear least-square Levenberg-Marquardt algorithm is used to find the minimum of the sum-squared error between the experimental data of scattered intensities and the theoretical function. After some times of iterative, the best-fitted values of surface parameters w ,ξandαare obtained.6. In the measurement of the scattered intensities from the transmission-mode random surface, an experimental setup is constructed combining a Boxcar of gated integration to reduce the signal noise with computer-controlled intensity detection technique to acquire the data automatically. Three transmission random surfaces are used as the practical samples for the light scattering measurement. A photomultiplier tube (PMT) used as the light intensity detector, the signal from the detector PMT is sent to the Boxcar and is averaged for the inherent electric noise to be suppressed. The theoretical equation of the scattering intensity profile is fit to the experimental profile data by Levenberg-Marquardt algorithm, with the three parameters of the random surface extracted simultaneously. The obtained parameter values of w ,ξandαfor each samples are compared with those obtained by the atomic force microscopy measurement, and the results are in good conformance.