Precision measurement and characterisation of spherical and aspheric surfaces
This thesis describes an investigation into the measurement and the characterisation of spherical and aspheric surfaces. The reliability of the nonlinear least-squares sphere fitting algorithm has been investigated. The study is focused on spherical surfaces, superimposed with random surface irregularities, within small surface segment angles. A new method based on Box’s estimation has been developed to calculate the bias of estimated parameters. The method is significantly faster and more convenient than a computer simulation process. By combining this estimate with the calculation of the uncertainty, a comprehensive understanding of the nonlinear least-squares sphere fitting algorithm has been achieved. Aspheric surface fitting algorithms are also of interest. Two methods have been developed to fit aspheric surfaces. The advantage of these two methods is that they both estimate the complete surface parameters which can then be compared with the design parameters. The first one is an indirect method. The calculation of surface parameters is based on the estimation of vertex radius, and errors in this estimate will influence all other parameters. To overcome this disadvantage, a direct method has been proposed. The method uses the nonlinear least-squares technique in which all parameters can be estimated simultaneously. The fitting algorithm has been tested on both computer simulated surfaces and measured surfaces. Issues regarding applying this method to measured surfaces have also been discussed. In addition to these theoretical studies, experimental work is presented. The dominant systematic errors have been studied in a white light confocal scanning system, which suggests that non-contact optical scanning system can be used for precision surface measurements.