Uncertainty Evaluation For Interferometric Testing Of Absolute Surface Figure Error

Modern high-end optical systems,especially the lithographic objective lens system,have taken interferometrical surface figure measurement of optics to the extreme.The surface figure specification requires to nanometer rms even sub-nanometer rms,which is a huge challenge for surface figure testing.Traditional test techniques can not work.Thus the test result can not be the guidance to optical fabrication.As GUM says,a complete statement of the result of a measurement includes an approximation or estimate of the value of the measurand,as well as information about the uncertainty of measurement.Study on the uncertainty evaluation of surface figure testing is a hard problem.But it is the right key to improve measurement uncertainty.Absolute test,i.e.,test procedures that enable separation of systematic errors(mainly the reference errors)from the relative test datas,can break through the limitation of reference standards.So it is possible to reach the lowest uncertainty.However,the measurement uncertainty of the absolute test itself should be accurately evaluated.Aiming at the requirements of uncertainty evaluation in absolute test,this dissertation focuses on the research of a generally universal uncertainty evaluation approach for absolute test and contains the following sections:1.The domestic and overseas development status about the uncertainty evalucation is investigated.Both the top-down approach and bottom-up approach are compared with each other,aims to find the applicability of each method using in surface figure measurement.Various absolut test techniques are compared and analyzed,including the three-flat test and its variants,two-sphere test,random-ball test,shift-rotation test and reverse optimization.For each test technique,the four main sources of uncertainty are analyzed: measurement instrument,people,method and procedure,and environment.2.The following parts are systematically studied,forward modeling for absolute test,inverse model solution for the model,and error propagation and uncertainty evaluation.A universal linear regress model is used to express the whole absolute test process.Three main model solvers are introduced: least squares estimator(LSE),maximum likelihood estimator(MLE),and iterative methods.Uncertainty evaluation methods based on normal distribution hypothesis and Monte Carlo error propagation are presented.3.A generalized iterative method for absolute measurement of optical flats has been discussed.Both simulation results and experimental results demonstrated that the generalized method can correctly reconstruct absolute figures with pixel-level spatial resolution;are easy to understand and implement;and computationally efficient.Also it saves much computational costs and memory space requirements.For three-flat test,several sources of uncertainty are analyzed systematically.4.A quasi-MCM is proposed to estimate the uncertainty of absolute test,which can handle the uncertainty problems with small samples,without any assumption of the probability distribution.Quasi-MCM estimates the surface figure at every pixel,which makes the absolute test result and its uncertainty both matrices.Thus the evaluation is more complete compared with other methods using single value like rms.Taking the three-flat test as an example,both GUM and Quasi-MCM are used to evaluate the measurement uncertainty of the three-flat test result.To prove the reliability of the uncertainty evaluation,several independent cross-check experiments are designed.5.Both error allocations and accuracy guarantees for three-flat test are analyzed.As to the sag problem introduced by gravity for a horizontally mounted optical flat,a reverse optimization method based on the FEM model and the real difference rotation test result is proposed.Experimental results demonstrated that this reverse optimization method can effectively reconstruct the sagging information due to gravity.6.Through sensitivity analysis of reconstruction error and rotation angle,it is noticed that some surface error terms cannot be reconstructed by the one-angle rotation model.By adding measurements with additional rotation angles,the error terms that cannot be estimated can be compensated and the measurement uncertainty is improved as well.Both simulation and experimental results indicate that the proposed iterative optimization method is effective for solving the three-flat problem with pixel-level spatial resolution and the measuring precision of two separate measurements for multiple rotations is 0.24 nm rms,while the result is 0.35 nm rms for one-angle rotation.