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 $\alpha$ say with respect to the optic axis, emerges as a parallel beam making a larger angle $\alpha'$ ' 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 alpha '/alpha. 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 = f1 + f2. Then, from {A]O1,O2 above
$$\bmatrix{y\cr \alpha}=
\bmatrix{ -f_1/f_2& -(f_1+f_2) \cr 0 & -f_2/f_1}_{O_1 O_2}
\bmatrix{y'\cr \alpha'}_{O_2}\eqno(70)$$ (70)

or M = alpha '/alpha. = - f1/f2. 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 f1 = 20 cm, fi = 2 cm and f2 = 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, f1. If the separation is equal to f1, 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) f1.

Magnifiers

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 $\theta_o$o at the eye with $\theta_oo = 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 $\theta_ii = 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 $\theta_ii/$\theta_io = (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

$$M={\theta_i\over \theta_o}={y/s\over y/25}
= {25\over s}=25\left({1\over f}-{1\over s}\right)
={25 \over f}+1 \ {\rm (since}\ \ s'=25\ {\rm cm)}$$

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 f1 = 2 cm.