Page 92 - Modern Optical Engineering The Design of Optical Systems

The Primary Aberrations 75

lens increases, the position of the ray intersection with the optical axis
moves farther and farther from the paraxial focus. The distance from
the paraxial focus to the axial intersection of the ray is called longitu-
dinal spherical aberration. Transverse, or lateral, spherical aberration is
the name given to the aberration when it is measured in the “vertical”
direction. Thus, in Fig. 5.2 AB is the longitudinal, and AC the trans-
verse spherical aberration of ray R.
Since the magnitude of the aberration obviously depends on the
height of the ray, it is convenient to specify the particular ray with
which a certain amount of aberration is associated. For example, mar-
ginal spherical aberration refers to the aberration of the ray through
the edge or margin of the lens aperture. It is often written as LA m or TA m .
Spherical aberration is determined by tracing a paraxial ray and a
trigonometric ray from the same axial object point and determining
their final intercept distances l′ and L′. In Fig. 5.2, l′ is distance OA
and L′ (for ray R) is distance OB. The longitudinal spherical aberra-
tion of the image point is abbreviated LA′ and
LA′ L′ l′ (5.3)

Transverse spherical aberration is related to LA′ by the expression

TA′ LA′ tan U′ (L′ l′) tan U′ (5.4)
R R R
where U′ R is the angle the ray R makes with the axis. Using this sign
convention, spherical aberration with a negative sign is called under-
corrected spherical, since it is usually associated with simple uncor-
rected positive elements. Similarly, positive spherical is called
overcorrected and is generally associated with diverging elements.
The spherical aberration of a system is usually represented graphi-
cally. Longitudinal spherical is plotted against the ray height at the
lens, as shown in Fig. 5.3a, and transverse spherical is plotted against
the final slope of the ray, as shown in Fig. 5.3b. Figure 5.3b is called a
ray intercept curve. It is conventional to plot the ray through the top of
the lens on the right in a ray intercept plot, regardless of the sign
convention used for ray slope angles.
For a given aperture and focal length, the amount of spherical aber-
ration in a simple lens is a function of object position and the shape, or
bending, of the lens. For example, a thin glass lens with its object at
infinity has a minimum amount of spherical at a nearly plano-convex
shape, with the convex surface toward the object. A meniscus shape,
either convex-concave or concave-convex has much more spherical
aberration. If the object and image are of equal size (each being two
focal lengths from the lens), then the element shape which gives the
minimum spherical is equiconvex. Usually, a uniform distribution of