Crystal ball Claude Alexander, was often billed as "Alexander the Crystal Seer". A crystal ball is essentially a bi-convex spherical lens with a uniform radius of curvature, although without its edges and center material truncated as in a conventional lens construction. Thus the principles of optics may be applied to analyze its imaging characteristics. As a lens, a transparent sphere of any material with refractive index greater than one bends rays of light to a focal point. An image is formed with significant coma, field curvature, and other optical aberrations inherent to spherical lenses. The refractive index of typical materials used for
Luneburg lens one side, and formula_14 only on the other: The parameter formula_19 is a constant for any given ray, but differs between rays passing at different distances from the centre of the lens. For rays passing through the centre, it is zero. In some special cases, such as for Maxwell's fish-eye, this first order equation can be further integrated to give a formula for formula_14 as a function or formula_13. In general it provides the relative rates of change of formula_14 and formula_13, which may be integrated numerically to follow the path of the ray through the lens. Luneburg lens A