Thin lens positive if the surface is concave, and negative if it is convex. This is an arbitrary sign convention; some authors choose different signs for the radii, which changes the equation for the focal length. Certain rays follow simple rules when passing through a thin lens, in the paraxial ray approximation: By tracing these rays, the relationship between the object distance "s" and the image distance "s′" can be shown to be which is known as the thin lens equation. In scalar wave optics a lens is a part which shifts the phase of the wave-front. Mathematically this can be understood
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