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Handbook of Optical Design, 3rd Edition by Zacarías Malacara-Hernández, Daniel Malacara-Hernández

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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=rr2S2.

(A2.1)

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

Z=cS21+1c2S2,

(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=1K+1[rr2 (K+1)S2],

(A2.3)

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

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