Astronomy Principles and Practice Fourth Edition.pdf

Refractors 251
Figure 16.13. A ray diagram illustrating the effect of chromatic aberration resulting from the use of a single
positive lens.
which can be obtained. The image in this plane of sharpest focus is not a point image. It is known as
the circle of least confusion and its size can be determined from the physical properties of the lens.
The spread of the image along the optic axis is known as longitudinal chromatic aberration and the
spread of the image in the plane of sharpest focus is known as lateral chromatic aberration.
It will be remembered that it was first supposed that chromatic aberration could not be removed
from lens systems but, eventually, Dolland proposed a method for achromatism. It is possible by
using in combination with a positive lens, a negative lens of different material—and hence refractive
power—to cancel the dispersion without complete cancellation of the refractive power. This is easily
demonstrated as follows.
If two lenses of focal lengths F 1 and F 2 are fixed together by optical cement, the focal length of
the combination, F,isgivenby
1
F = 1 + 1 .
F 1 F 2
Hence,
F = F 1 F 2
. (16.18)
F 1 + F 2
By expressing the individual focal lengths in terms of the shape of each lens and their refracting
powers, equation (16.18) can be rewritten as
K 1 K 2
F =
K 2 (n 1 − 1) + K 1 (n 2 − 1) . (16.19)
Now the aim of the combination is to provide a system whose focal length is identical for the blue
and red ends of the spectrum. If the refractive indices of the materials of the lenses are denoted by n 1B ,
n 2B , for blue light and n 1R , n 2R , for red light, the achromatic condition is achieved when
K 1 K 2
K 2 (n 1B − 1) + K 1 (n 2B − 1) = K 1 K 2
K 2 (n 1R − 1) + K 1 (n 2R − 1)
i.e.
K 1
=− n 1B − n 1R
. (16.20)
K 2 n 2B − n 2R
Since n 1B > n 1R and n 2B > n 2R by the general properties of optical materials, K 1 /K 2 must be
negative. Hence, achromatization can only be achieved by combining a positive and a negative lens.