Handbook of Optical Design, 3rd Edition

Appendix 2: Mathematical Representation of Optical Surfaces

A2.1  SPHERICAL AND ASPHERICAL SURFACES

An optical surface may have many shapes (Herzberger and Hoadley 1946; Malacara 1992; Mertz 1979, 1981; Shannon 1980; Schulz 1988), but the most common is spherical, whose sagitta for a radius of curvature r and a semidiameter S = x² + y² may be written as

$Z=r-\sqrt{r^2-S^2}.$

(A2.1)

However, this representation fails for flat surfaces. A better form is

$Z=\frac{c{S}^2}{1+\sqrt{1-c^2{S}^2}},$

(A2.2)

where, as usual, c = 1/r and S² = x² + y².

A conic surface is characterized by its eccentricity e. If we define a conic constant K = –e², then the expression for a conic of revolution may be written as

$Z=\frac{1}{K+1}[r-\sqrt{r^2- (K+1)S^2}],$

(A2.3)

which works for all conics except the paraboloid. ...