Chapter 6
Practical Optical Systems
The Telescope
The telescope in its most basic
form is a special case of two thin lenses
in air. In this form it is a
simple astronomical refracting telescope.
Basically, a telescope is used
to increase the angle that a distant
object subtends at the eye.
If the eye is relaxed for distant viewing, the
telescope simply produces an angular magnification
in which an incident
(approximately) parallel beam from a point in
a distant source, making an angle
say
with respect to the optic axis, emerges as a
parallel beam making a larger
angle '
with respect to the axis. When the
object is viewed with the
instrument, the image formed on the retina
of the eye will be larger by an
amount proportional to
'/.
the angular
magnification, M. This telescopic
system is characterized by a zero in the
lower left hand corner of the matrix,
[A]_{O1,O2}, which we obtain if d = f_{1} +
f_{2}.
Then, from
{A]_{O1,O2} above

(70) 
or M =
'/.
=
 f_{1}/f_{2}.
Thus the angular magnification is
the negative ratio of the focal length of the first lens,
the objective lens,
to the focal lengths of the second lens, the eyepiece lens.
As illustrated
in Fig. 26 the image as viewed through the telescope
is inverted, and the
image formed by the objective lens is in the second
focal plane of that lens
which is also the first focal plane of the eyepiece lens.
Figure 26.
For a terrestrial telescope one may insert an erector lens in
between the objective and eyepiece lenses such that the image
formed by the objective acts as an object for the erector lens
which in turn forms an inverted image at the first focal point
of the eyepiece lens. The matrix of this system
of three lenses will have a zero in the lower left
hand corner and the angular magnification is positive.
Exercise 5
Find the matrix of the system of the three lenses
of Fig. 27 and show
that they form a telescopic system with positive
angular magnification.
Solve for the case in which f_{1} = 20 cm,
f_{i} =
2 cm and f_{2} = 5 cm.
(Answer M = + 4)
Figure 27.
Exercise 6
A Galilean telescope has an objective lens with
f = 20 cm and the eyepiece
lens with f = 5 cm. The lenses are separated
by 15 cm . Calculate the
matrix for this system and show that it is a
telescopic system with M = +4.
The Ocular
In most telescopes it is common practice to have a system of at least
two lenses called an ocular to perform the function
of the eyepiece lens. The first lens or field lens improves
the field of view and in combination with the second lens or
eye lens can be chosen to reduce some of the aberrations
inherent in a single lens. The most common
oculars are the Ramsden and Huygen types.
Exercise 7
A Ramsden eyepiece consists of two equal focal
length lenses, f_{1}. If the
separation is equal to f_{1}, chromatic aberration
is minimized. In practice the lenses are usually moved closer
together (Jenkins and White, p. 182).
Calculate the position of the focal points and
the focal lengths of the
ocular in the case in which d = (3/4) f_{1}.
The ocular discussed above as well as just a
single thin lens can be
used as a magnifying glass. When an object is
examined an image is formed at
the retina in the eye and this retinal image may
be increased in size using a
magnifier. For continued viewing, it is usual
to assume that an object 25 cm
in front of the eye can be comfortably viewed.
For this case then, an object
of height y subtends an angle
_{o} at the eye
with _{o} =
y/25, as in Fig. 28.
Now if the object is placed at the focal point of
an ocular, the emergent rays
from the ocular may be focused by a relaxed eye as
also shown in Fig. 28. The
rays from y
now make an angle of
_{i} =
y/f where f is
the focal length of the
ocular. Thus the change in the size of the retinal
image is given approximately by
_{i}/_{o}
= (y/f)/(y/25) = 25/f.
Figure 28.
Figure 29.
The ocular may also be used as in Fig. 29 with the
object just inside the
focal point of the ocular for an eye accommodated
to the 25 cm viewing
distance. In this case the magnifying power
(which is usually quoted on the
eyepiece) is (1 + 25/f). This is an approximate
formula which assumes that the
image distance from the second principal point is 25 cm and then
Exercise 8
Calculate the magnifying power for the Ramsden
eyepiece of the above exercise
7 for both a relaxed eye and an accommodated eye if
f_{1} = 2 cm.