Page 123 - Modern Optical Engineering The Design of Optical Systems

106 Chapter Six

Third-order aberrations Fifth-order aberrations
Exponents Name Exponents Name
y 3 spherical y 5 5th-order spherical
y 2 h coma y 4 h linear coma
yh 2 astigmatism yh 4 5th-order astigmatism
yh 2 Petzval yh 4 5th-order Petzval
h 3 distortion h 5 5th-order distortion
y 3 h 2 oblique spherical
y 2 h 3 elliptical coma

An examination of the “C” terms in Eqs. 5.1 and 5.2 will indicate the
complexity of the fifth-order aberrations, since unlike the third-order
terms, for several of the aberrations there is more than one coefficient,
and the shape of the aberration blur can vary significantly because of
this.
Happily, it turns out that the third-order aberration surface contri-
butions can be calculated from the raytrace data of two paraxial rays,
the axial ray and the principal, or chief ray. The fifth-order aberrations
can also be calculated from the same data, but the fifth-order contribu-
tions from a surface are not determined solely by the ray data at the
surface in question, but also from the ray data or aberration contribu-
tions from the other surfaces of the system. (See the Buchdahl refe-
rence.) An illustration of this effect can be found in the discussion of the
design of telescope objectives in Chap.16. Most full-service optical soft-
ware programs can calculate both the third- and fifth-order aberration
contributions.
The chief value of the contribution equations is that they not only
allow the calculation of the amount of the aberrations, but they calcu-
late the contribution of each individual surface to the final aberration,
which is simply the sum of all the surface contributions. The third-
order contributions not only indicate where the third-order aberrations
originate, but they are a fair indicator of the source of the higher order
aberrations. This relationship is not a simple one, but if a system is
cursed with a fifth-order spherical aberration for example, it is quite
likely that the problem originates mostly at those surfaces which have
outstandingly large third-order spherical contributions.
Another value of the third-order contribution equations lies in the
fact that while it is relatively easy to change the third-order aberra-
tions by changing a constructional parameter such as a curvature, a
spacing, or an index, the higher order aberrations are relatively stable
and difficult to change. Thus if one changes a parameter and finds that
a certain amount of change in the third-order aberration is produced, it
is likely that a very similar amount of change will be found in the calcu-
lation of the aberration by an exact trigonometric raytracing.