Eyepieces
 

In the previous discussion on the magnification of the telescope, it has been assumed for simplicity that the eyepiece, or ocular, is in the form of a single positive lens. It is obvious that one of the drawbacks of using a single lens for an eyepiece is that the viewed image would suffer from chromatic aberration. In practice, an eyepiece is generally in the form of a combination of lenses and there are several types which are frequently used; and perhaps the most traditional are the Huygens eyepiece and the Ramsden eyepiece. These eyepieces are constructed of two simple separated lenses made of the same material. The two-lens system serves two purposes: it allows a field of increased size to be viewed and also allows compensation for chromatic aberration. The ability of such a system to overcome chromatic aberration is demonstrated here.
If two lenses of focal length F₁ and F₂ are separated by a distance, x, the focal length of the combination, F, is given by

Hence,
(1)
By expressing the individual focal lengths in terms of the shape of each lens and their refractive powers and by letting the lenses be made of the same type of glass so that n₁ = n₂ = n,

Figure 1. The Huygens eyepiece.
equation (1) can be re-written as
(2)
The focal length of the combination, F, is independent of wavelength, i.e. free from chromatic aberration, if dF/dλ is zero. For a given pair of lenses, K₁ and K₂ are constant so that this condition
can only be obtained if dF/dn = 0. By differentiating equation (2) and putting dF/dn = 0, it follows that
(3)
Thus, for an achromatic system involving two positive lenses made of the same material, their separation must be equal to half the sum of their focal lengths.
The optical arrangement of the Huygens eyepiece is depicted in figure 1. The first lens of the system is known as the field lens and the second is known as the eye lens. The ratio of the focal length of the field lens to that of the eye lens is usually in the range 1·5:3·0. It can be seen from figure 1 that the field lens lies inside the focal plane of the telescope and that it forms a real image between the two lenses of the eyepiece, at the focus of the eye lens. The purpose of the field lens, as its name implies, is to increase the field of view. When the eyepiece is in the correct position, the light rays
emerge from the eyepiece in a parallel bundle and the viewed image appears to be at infinity. If it is desired to view the field against a cross-wire or graticule, it should be located between the lenses at the focus of the eye lens. Thus, the observed images and graticule will be in focus simultaneously. Although the combined lens system corrects for chromatic aberration in the observed images, the image of the graticule is obtained by using only the eye lens and, consequently, this image suffers from coloured effects. The Huygens eyepiece also suffers from spherical aberration, astigmatism and pincushion distortion. These aberrations are very noticeable if the eyepiece is used with a telescope of focal ratio smaller than about f/12. The exit pupil corresponds to the image of the collector which is produced by the lens combination and is close to the eye lens. The small eye relief sometimes makes this type of eyepiece difficult to use.
In the Ramsden eyepiece, the two lenses have the same focal length and to compensate for chromatic aberration , their separation, therefore, should be equal to the value of this focal length. If this were to be done rigorously, the eye lens would only produce a virtual image at infinity when the field lens is placed in the focal plane of the telescope. Any graticule would also be required to be placed on the field lens and dust on the same lens would be in focus with the images under inspection.
At the expense of introducing a small amount of chromatic aberration, the two lenses of the Ramsden eyepiece are brought together so that their separation is just less than their focal lengths. The arrangement of the two lenses is depicted in figure 2. It can be seen that the field lens is situated

Figure 2. The Ramsden eyepiece.
outside the focal plane of the telescope and it is an easy matter to arrange to have a graticule in this same plane. The Ramsden eyepiece is superior to the Huygens in that the aberrations are much smaller and it is more convenient to use because of its greater eye relief.
By using a doublet instead of a single lens for the eye lens, it is possible to utilize the desirable qualities of the Ramsden eyepiece and, at the same time, remove the chromatic aberration. This type of ocular is known as a Kellner eyepiece.
In the case of the single element eyepiece, it is a simple matter to measure its focal length and the expected magnification, when used with a particular telescope, can be obtained from knowledge of the focal length of the telescope. It is usually more difficult to make a direct measurement of the effective focal length of a compound eyepiece and the magnification that it produces is best measured when it is used on the telescope. A simple way to do this is to point the telescope at the daytime blue sky and measure the diameter of the parallel beam (exit pupil) emerging from the eyepiece.
By joining simple ray diagrams, illustrating the effect of each of the optical components, it is easy to demonstrate that an astronomical telescope, using one of the previously described eyepieces, provides an inverted image. Amore complicated eyepiece giving an upright field (a terrestrial eyepiece) is of no real advantage to the astronomer. In fact, its extra lenses lose more light.