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Scientific Papers of the Bureau of Standards. 'A Mexican Physicist Solved a 2,000-Year Old Problem That Will Lead to Cheaper, Sharper Lenses'.

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'General formula for bi-aspheric singlet lens design free of spherical aberration'. 'Goodbye Aberration: Physicist Solves 2,000-Year-Old Optical Problem'. For example, in a design consisting of a single lens with spherical surfaces and a given object distance o, image distance i, and refractive index n, one can minimize spherical aberration by adjusting the radii of curvature R 1 See also For simple designs one can sometimes analytically calculate parameters that minimize spherical aberration. Lens systems with aberration correction are usually designed by numerical ray tracing. In lens systems, aberrations can be minimized using combinations of convex and concave lenses, or by using aspheric lenses or aplanatic lenses. Longitudinal sections through a focused beam with negative (top row), zero (middle row), and positive spherical aberration (bottom row). The effect is proportional to the fourth power of the diameter and inversely proportional to the third power of the focal length, so it is much more pronounced at short focal ratios, i.e., 'fast' lenses. 'Negative' spherical aberration means peripheral rays are not bent enough.

'Positive' spherical aberration means peripheral rays are bent too much. In many cases, it is cheaper to use multiple spherical elements to compensate for spherical aberration than it is to use a single aspheric lens. This is an important effect, because spherical shapes are much easier to produce than aspherical ones. It is often considered to be an imperfection of telescopes and other instruments which makes their focusing less than ideal due to the spherical shape of lenses and mirrors.

A typical value of refractive index for crown glass is 1.5 (see list), which indicates that only about 43% of the area (67% of diameter) of a spherical lens is useful.

3 Estimation of the aberrated spot diameterДЄ spherical lens has an aplanatic point (i.e., no spherical aberration) only at a radius that equals the radius of the sphere divided by the index of refraction of the lens material.

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