The Heisenberg uncertainty principle indicates the position and momentum for one particle cannot be determined at the same time. In signal processing, it points out that the frequency domain resolution and time domain resolution of one signal cannot be too small at the same time. As a general form of Fourier transform, the linear canonical transform (LCT) has made great contributions in signal processing. In this paper, an uncertainty principle for n-dimensional LCT, based on which some generalizations for LCT are obtained.
In this paper, a novel in-situ surface measuring technique for optical elements with aspheric surface is presented. It is a
contact type probe, and can be used for measuring ground surfaces. The theory of this technique develops from
coordinate measuring machine (CMM), and the measurement accuracy of this technique is depended on the accuracy of
computer numerical controlled (CNC). By installing a special equipment with high accuracy measuring head in main
spindle of CNC machine, and moving the probe along the path which is described by a mathematical aspheric expression
precisely, we could get relative errors of sag height of any position in this path. With this technique, the repeat
positioning error caused by traditional off-line measurement will be avoided. The author also has finished a special
software with VC++ 6.0. With this software, the form error of ground work piece could be corrected rapidly. This
software can calculate and handle the arrangement automatically with all parameters which are required to input in
operation interface. In the correction stage, the software can analyze and process error data and generate a new NC
program with corrected data for next grinding stage. After 2 or 3 times measuring and correction, the surface shape error
of the aspheric optical element will be less than 1μm. The finished work piece has a very good surface finish and can be
polished with high quality.
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