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* = *x*^{2} + *y*^{2} 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 *S*^{2} = *x*^{2} + *y*^{2}.

A conic surface is characterized by its eccentricity *e*. If we define a conic constant *K* = –*e*^{2}, 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. ...

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