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The effect of curvature change of a mirror surface on the removal function of ion beam figuring was explored.A new higher precision figuring method for a highly steep off-axis asphere was proposed by using a correction matrix to calculate the removal function in different dwell points.The coordinate transformation for the off-axis asphere was used to decrease the effect of steep change on the removal function,the material removal rate of ion beam figuring asphere was analyzed based on Sigmund sputtering theory and a mathematics model of the removal function of the ion beam figuring asphere was established to calculate the variation of the material removal rate in different well points.The FWHMs(full wave and half maximum)of removal functions on different dwell points werecalculated based on projection theory.Then the correction matrix for removal function was obtained based on the dwell point matrix,by which the information of removal function on every dwell points was calculated.Furthermore,the dwell time could be computed by Computer Controlled Optical Surfacing(CCOS)theory.At last,the experiments were performed by combing with a simulation.A silicon off-axis sphere mirror with a size of 900 mm×680 mm and an off-axis magnitude of 350 mm was figured.Obtained the RMS value of profile accuracy for the off-axis asphere mirror changes from32.041 nm to 11.566 nm and the rate of convergence is up to 2.77.The experimental result demonstrates that the correction matrix for removal function of ion beam figuring satisfies the requirement of optical fabrication.

Circular Subaperture Stitching Interferometry(CSSI) and Annular Subaperture Stitching Interferometry(ASSI) were expounded to realize the test of large aspheric surfaces without the null optics.The theories and principle of the technique were analysised,and the synthetical optimization stitching model and an effective stitching algorithm were established based on homogeneous coordinate transformation and simultaneous least-squares fitting.The softwares of CSSI and ASSI were devised,respectively,then the experiment was carried out for a Φ350 mm hyperboloid by CSSI,ASSI and a null test,respectively.Experiments show that the differences of PV and RMS errors between null test and CSSI are 0.031λ and 0.004λ,and those between null test and ASSI are 0.028λ and 0.006λ,respectively,while their synthesized surface maps are consistent to the entire surface map from the null test.The technique proposed in this paper provides another quantitive measurement for testing large aperture aspheres besides the null-compensation method.