‘Efficiency’ is significant in several ways. Only the starting values for the ranges of summation are shown explicitly in this work options for the upper limits are discussed separately. Where c and k are the axial curvature and conic constant and A 4, A 6, A 8,… are monomial degrees of freedom. One of the traditional specifications for rotationally symmetric surfaces of this type involves a well-known set of coefficients: In cylindrical polar coordinates, for example, such a surface can be expressed explicitly as z= f( ρ,θ). Regarding ‘generality’, it is sufficient for many purposes to allow the exclusion of hyper-hemispheres and the like so that surfaces can then be specified by a single-valued sag function. We are primarily concerned here with particular aspects of efficiency. Among the criteria for the effectiveness of any process for characterizing shape are ‘generality’, ‘efficiency’, and ‘robustness’. Sitting at the foundation of all this is the shared requirement for effective conventions to characterize surface shapes. Unique challenges are faced by at least four different industries: those that provide optical design software as well as the hardware systems for generating (grinding, diamond turning, etc.), polishing/finishing, and measuring optical surfaces. Although symmetry typically allows sections of spheres to be fabricated and tested more simply, the successful exploitation of non-spherical optics of growing complexity offers no end of difficulties.
Aspheric and freeform optics have come to play an increasingly central role across this domain.
These applications vary from commercial high-volume products such as miniature mobile phone cameras to scientific instruments such as giant monolithic or segmented telescopes and industrial tools such as ultra-high performance lithographic projection systems. Production limitations are constantly being tackled to advance the many applications of precision optics.
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