Page 134 - Modern Optical Engineering The Design of Optical Systems

Third-Order Aberration Theory and Calculation 117

values are equal to some desired set of values, as will be evident in
Chap. 16. For aspheric surfaced lenses, the contributions from the
asphericity are calculated (by Eqs. 6.2r through 6.2y) and added to the
contributions calculated for spherical surfaced lenses.
Equations 6.3f to 6.3l are called stop shift equations. They may also
be applied to the surface contributions (from Eq. 6.2) to determine the
third-order aberrations for a new, or changed, stop position by setting

∗
(y p y p )
Q
y
∗
where y p is the ray height of the “new” principal ray (i.e., after the
stop is shifted) and y p and y are as indicated in Sec. 6.3. Note that Q is
∗
an invariant; thus the values for y p , y p , and y may be taken at any
convenient surface. When the equations are used this way the
unstarred terms (SC, CC, etc.) refer to the aberrations with the stop in
∗
∗
the original position, while the starred terms (SC , CC , etc.) refer to
the aberrations with the stop in the new position. Another conse-
quence of the invariant nature of this definition of Q is the fact that
the stop shift may be applied to either the individual surface contri-
butions or to the contribution sums of the entire system or any portion
thereof.
The implications of the stop shift equations (Eqs. 6.3f through 6.3l)
are worthy of note. If all the third-order aberrations are corrected for
a given stop position, then moving the stop will not change them.
Similarly, if there is no spherical, the coma is not affected by a stop
shift. This is the case with the paraboloid mirror which, because it has
no spherical aberration, has the same amount of coma regardless of
where the stop is placed. But because it has coma, the astigmatism is
a function of the stop position.

6.5 Sample Calculations
Since it is highly unlikely that a reader interested in the material of
this chapter will want to carry out any of the aberration calculations
“by hand,” we will abjure our usual set of exercises in favor of a
demonstration of computer software calculation. Our subject will be a
quite ordinary Cooke triplet anastigmat at a focal length of 101 mm,
a speed of f/3.5, and a total field of 23.8 . The reader is invited to
duplicate (either by hand or by computer) the calculations to validate
his computations.
A fairly complete raytrace analysis of the lens is shown in Fig. 6.2,
which is available as a “one click” feature in the software program that
I use (OSLO from Lambda Research Corp.).