Page 310 - Modern Optical Engineering The Design of Optical Systems

Optical System Layout 289

for the eyelens is a “virtual” object, no inversion occurs, and the final
image presented to the eye is erect and unreversed. Since there is no
real image formed in a Galilean telescope, there is no location where
cross hairs or a reticle may be inserted.
Assuming the components of the telescope to be thin lenses, we can
derive several important relationships which apply to all telescopes
and afocal systems and which are of great utility. First, it is readily
apparent that the length (D) of a simple telescope is equal to the sum
of the focal lengths of the objective and eyelens.

D f f (13.1)
o e
Note that in the Galilean telescope, the spacing is the difference
between the absolute values of the focal lengths since f e is negative.
The magnification, or magnifying power, of the telescope is the ratio
between u e , the angle subtended by the image, and u o , the angle sub-
tended by the object. The size (h) of the internal image formed by the
objective will be

h u f (13.2)
o o
and the angle subtended by this image from the first principal point of
the eyelens will be
h
u (13.3)
e
f
e
Combining Eqs. 13.2 and 13.3, we get the magnification
u f
MP e o (13.4)
u f
o e
and
f D/(1 MP)
e
f MPD/(1 MP)
o
The sign convention here is that a positive magnification indicates an
erect image. Thus, if objective and eyelens both have positive focal
lengths, MP is negative and the telescope is inverting. The Galilean
scope with objective and eyelens of opposite sign produces a positive
MP and an erect image.
Note that u o can represent the real angular field of view of the tele-
scope and u e the apparent angular field of view, and that Eq. 13.4
defines the relationship between the real and apparent fields for small
angles. For large angles, the tangents of the half-field angles should be
substituted in this expression.