Page 129 - Modern Optical Engineering The Design of Optical Systems

112 Chapter Six

where z is the longitudinal coordinate (abscissa) of a point on the surface
which is a distance s from the z axis and
2
2
s y x 2
For the purposes of computing the third-order contributions, we can
assume that the aspheric surface is represented by a power series in s 2

1
4
3
2
1
z
C e s (
C e K ) s . . . (6.2p)
8
2
6
in which the terms in s and higher may be neglected. For aspheric
surfaces given in the form of Eq. 6.2o, the equivalent curvature C e and
equivalent fourth-order deformation constant K may be determined
from
(6.2q)
C e c 2A 2
A 2 2 2
K A 4 (4A 2 6cA 2 3c ) (6.2r)
4
where c, A 2 , and A 4 are the curvature and second- and fourth-order
deformation terms, respectively, of Eq. 6.2o. Note that if A 2 is zero,
3
C e c and K A 4 ; see Chap. 18 for conics, where A 4 /8R .
The aspheric surface contributions are determined by first com-
puting the contributions for the equivalent spherical surface C e
using Eqs. 6.2g through 6.2m. Then the contributions due to the
equivalent fourth-order deformation constant K are computed by the
following equations and added to those of the equivalent spherical
surface to obtain the total third-order aberration contribution of the
aspheric surface.

4K (n′ n)
W (6.2s)
Inv
4
TSC a Wy h (6.2t)
3
CC a Wy y p h (6.2u)
2
2
TAC a Wy y p h (6.2v)
TPC a 0 (6.2w)
3
DC a Wyy p h (6.2x)
TAchC a 0 (6.2y)
TchC a 0 (6.2z)