A parametric smoothing model is developed to quantitatively describe the smoothing action of polishing tools that use visco-elastic materials. These materials flow to conform to the aspheric shape of the workpieces, yet behave as a rigid solid for short duration caused by tool motion over surface irregularities. The smoothing effect naturally corrects the mid-to-high frequency errors on the workpiece while a large polishing lap still removes large scale errors effectively in a short time. Quantifying the smoothing effect allows improvements in efficiency for finishing large precision optics. We define normalized smoothing factor SF which can be described with two parameters. A series of experiments using a conventional pitch tool and the rigid conformal (RC) lap was performed and compared to verify the parametric smoothing model. The linear trend of the SF function was clearly verified. Also, the limiting minimum ripple magnitude PVmin from the smoothing actions and SF function slope change due to the total compressive stiffness of the whole tool were measured. These data were successfully fit using the parametric smoothing model.
A mathematical approach for the third order solution for a general zoom lens design is proposed. The design starts with a first-order layout. Lens elements with the proper refracting power are placed at the proper distances to meet the physical constraints of the intended lens system. For the third-order design stage, a matrix notation called "Aberration Polynomial," which clarifies the linearity of the transformation from a normal thin group configuration to a general thin group configuration by pupil shift and conjugate shift theory is implemented.The purpose of the method is correcting low-order aberrations during the preliminary design of zoom lenses. The goal is to mathematically reduce to zero the four aberration coefficients of the third-order (spherical aberration, coma, astigmatism, and distortion) rather than searching for a minimum by commercial design software. Once this theory is proven and accepted, it becomes possible to determine how many groups are needed for a particular optical system. The method of aberration polynomials establishes the number of groups needed to correct a given number of aberrations at a given number of zoom positions. Furthermore, it provides the shape or bending of the elements, from where it will be possible to continue to optimize with standard methods..
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