Page 130 - Modern Optical Engineering The Design of Optical Systems

Third-Order Aberration Theory and Calculation 113

It is worth noting that if the aspheric surface is located at the aper-
ture stop (or at a pupil), then y p 0, and the only third-order aberra-
tion that is affected by the aspheric term is spherical aberration. The
Schmidt camera makes use of this by placing its aspheric corrector
plate at the stop so that only the spherical aberration of the spherical
mirror is affected by the plate. Conversely, if an aspheric is expected to
affect coma, astigmatism, or distortion, it must be located a significant
distance from the stop.

6.4 Third-Order Aberrations:Thin Lenses;
Stop Shift Equations
When the elements of an optical system are relatively thin, it is fre-
quently convenient to assume that their thickness is zero. As we have
previously noted, this assumption results in simplified approximate
expressions for element focal lengths, which are nonetheless quite
useful for rough preliminary calculations. This approximation can be
applied to third-order aberration calculations; the results form a very
useful tool for preliminary analytical optical system design. The follow-
ing equations may be derived by application of the surface contri-
bution equations of the preceding section to a lens element of zero
thickness.
The thin-lens third-order aberrations are found by tracing an axial
and a principal ray through the system of thin lenses, in the manner
outlined in Chap. 4. The equations used are

u′ u y (6.3a)
(6.3b)
y 2 y 1 du′ 1
where u and u′ are the ray slopes before and after refraction by the
element, is the element power (reciprocal focal length), y is the
height at which the ray strikes the element, and d is the spacing
between adjacent elements.
From Sec. 3.5 we also recall that the power of a thin element is
given by

1/f
(n 1) (c 1 c 2 ) (6.3c)
(n 1) c

where c c 1 c 2 and c 1 and c 2 are the curvatures (reciprocal radii) of
the first and second surfaces of the element.