Page 116 - Modern Optical Engineering The Design of Optical Systems

The Primary Aberrations 99

correspond to the two points. Ordinarily, the reference ray for OPD is
either the optical axis or the principal ray (for an oblique bundle).
Thus the OPD for a given ray is usually the area under the ray intercept
plot between the center point and the ray.
Mathematically speaking, then, the OPD is the integral of the H–tan U
plot and the defocus is the slope or first derivative. The coma is
related to the curvature or second derivative of the plot, as a glance at
Fig. 5.24d will show.
It should be apparent that a more general ray intercept plot for a given
object point can be considered as a power series expansion of the form
2
3
4
5
H′ h a bx cx dx ex fx . . . (5.9)
where h is the paraxial image height, a is the distortion, and x is the
aperture variable (e.g., tan U′). Then the art of interpreting a ray
intercept plot becomes analogous to decomposing the plot into its various
2
4
terms. For example, cx and ex represent third- and fifth-order coma,
while dx and fx are the third- and fifth-order spherical. The bx term
3
5
is due to a defocusing from the paraxial focus and could be due to cur-
vature of field. Note that the constants a, b, c, etc., will be different for
points of differing distances from the axis. For the primary aberra-
tions, the constants will vary according to the table of Fig. 5.16, and in
general per Eqs. 5.1 and 5.2.

5.9 The Relationships between Longitudinal
Aberration,Transverse Aberration,
Wave-Front Aberration (OPD),
and Angular Aberration
The various ways of assigning a numerical value to an aberration are
very simply related. Given the value of one measure of an aberration,
the corresponding values for the other measures of that aberration can
readily be found. Figure 5.25 illustrates the case of spherical aberra-
tion, which can be specified in four different ways:

1. as a longitudinal aberration
2. as a transverse aberration
3. as a wave-front aberration and
4. as an angular aberration

Defocusing, spherical, astigmatism, Petzval curvature, and axial
chromatic aberrations can all be expressed in any of the four mea-
sures. Since coma, distortion, and lateral chromatic aberrations are