Page 102 - Modern Optical Engineering The Design of Optical Systems

The Primary Aberrations 85

reveals that for a given index and thickness, there is an infinite num-
ber of combinations of R 1 and R 2 which will produce a given focal
length. Thus a lens of some desired power may take on any number of
different shapes or “bendings.” The aberrations of the lens are changed
markedly as the shape is changed; this effect is the basic tool of optical
design.
As an illustrative example, we will consider the aberrations of a thin
positive lens made of borosilicate crown glass with a focal length of
100 mm and a clear aperture of 10 mm (a speed of f/10) which is to
image an infinitely distant object over a field of view of 17°. A typical
borosilicate crown is 517:642, which has an index of 1.517 for the
helium d line ( 5876 Å), an index of 1.51432 for C light ( 6563 Å),
and an index of 1.52238 for F light ( 4861 Å).
(The aberration data presented in the following paragraphs were
calculated by means of the thin-lens third-order aberration equations
of Chap. 6.)
If we first assume that the stop or limiting aperture is in coincidence
with the lens, we find that several aberrations do not vary as the lens
shape is varied. Axial chromatic aberration is constant at a value of
1.55 mm (undercorrected); thus the blue focus (F light) is 1.55 mm
nearer the lens than the red focus (C light). The astigmatism and field
curvature are also constant. At the edge of the field (30 mm from the
axis) the sagittal focus is 7.5 mm closer to the lens than the paraxial
focus, and the tangential focus is 16.5 mm inside the paraxial focus.
Two aberrations, distortion and lateral color, are zero when the stop is
at the lens.
Spherical aberration and coma, however, vary greatly as the lens
shape is changed. Figure 5.12 shows the amount of these two aberra-
tions plotted against the curvature of the first surface of the lens.
Notice that coma varies linearly with lens shape, taking a large positive
value when the lens is a meniscus with both surfaces concave toward
the object. As the lens is bent through plano-convex, convex-plano, and
convex meniscus shapes, the amount of coma becomes more negative,
assuming a zero value near the convex-plano form.
The spherical aberration of this lens is always undercorrected; its
plot has the shape of a parabola with a vertical axis. Notice that the
spherical aberration reaches a minimum (or more accurately, a maximum)
value at approximately the same shape for which the coma is zero.
This, then, is the shape that one would select if the lens were to be
used as a telescope objective to cover a rather small field of view. Note
that if both object and image are “real” (i.e., not virtual), the spherical
aberration of a positive lens is always negative (undercorrected).
Let us now select a particular shape for the lens, say, C 1 0.02
and investigate the effect of placing the stop away from the lens, as