Zopt 2Rc x y2

By expanding the 1 + sc2x/R terms, the representation of the shape of a segment becomes

2 3 s2c3

2R 2R2 8R3 3 5

Let us consider the corresponding cylindrical coordinates (p,9, z) where p = r/a is the normalized radius, r =(x2 + y2)1/2 and a is the segment semi-aperture. We set 9 = 0 in the (x, z) plane. The local shape of a paraboloid segment can be expanded in the form zopt = S Onmpn cosm9 with 0 < p < 1, (7.32)

where the optics coefficients anm are determined from identifications. Also using c = (1 + u2)-l/2 and s = u(1 + u2)-l/2 leads to the sags s2c n2

33 a3sc3

SR2"

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