cm (one-fourth meter) from the eye without the magnifier (Fig 7-2). Assuming small angles, we therefore divide the focal length of the lens by 4. Thus, a +10 D lens is considered a 2.5× magnifier and is held one-tenth of a meter, its focal length, from the object of interest. Note that rays coming from the object leave the magnifier with zero vergence, so that the user of the magnifier can gaze through it from any distance she prefers.
Figure 7-1 A plus lens used to view an object positioned in the focal plane of the lens. O = object; f = focal length. (Redrawn
from Basic and Clinical Science Course Section 2: Optics, Refraction, and Contact Lenses. San Francisco: American Academy of Ophthalmology; 1986–1987:73. Fig 43.)
Figure 7-2 The reference viewing distance of 25 cm, used in the definition of magnification of simple magnifiers. (Redrawn
from Basic and Clinical Science Course Section 2: Optics, Refraction, and Contact Lenses. San Francisco: American Academy of Ophthalmology; 1986–1987:73. Fig 44.)
Telescopes
A telescope is an optical system designed to increase the angle subtended at the eye by distant objects. It is called afocal because pencils of light entering with zero vergence come out with zero vergence. The first lens, the objective, forms an image of the distant object. The second lens, the eyepiece or ocular, is then used to view the image formed by the objective. With small-angle approximations, a telescope’s angular magnification (or “minification,” if you look through the
telescope turned the other way around) is the longer focal length of the objective divided by the shorter focal length of the ocular, with a minus sign to enable us to figure out whether the final image is upright or inverted:
where
fobj = focal length of objective feye = focal length of eyepiece Peye = power of eyepiece Pobj = power of objective
Galilean Telescope
In a Galilean telescope, the objective is a plus lens, and the eyepiece is a minus lens. Bundles of rays with approximately zero vergence, emanating from various points on a distant object, pass through a low-power plus objective lens. Before the rays are able to arrive to focus at the secondary focal point of the first lens, they meet a higher-power minus lens, the eyepiece, which has been placed so that its primary focal point coincides with the secondary focal point of the first lens. Thus, the rays leave the second lens with no vergence, and the telescope is therefore said to be afocal (Fig 7-3). The image is magnified and upright.
Figure 7-3 Galilean telescope. (Redrawn from Guyton D, West C, Miller J, Wisnicki H. Ophthalmic Optics and Clinical Refraction. London: Prism Press;1999:39.)
The distance separating the eyepiece from the objective, which is the length of the telescope, equals the difference between the focal lengths of the objective and the eyepiece.
Astronomical Telescope
In an astronomical, or Keplerian, telescope, both the objective and the eyepiece are plus lenses. Light without vergence enters a low-power objective lens, just as in the Galilean telescope. Whereas in the Galilean telescope a minus lens is placed in the path before the light reaches its secondary focal point, in the astronomical telescope a stronger, convex lens is placed beyond the secondary focal point of the first convex lens, with its primary focal point coinciding with the secondary focal point of the first lens. The distance separating the eyepiece from the objective is the sum of the focal lengths of the objective and the eyepiece. Once again, we have an afocal system; rays that enter with zero vergence exit with zero vergence. A large low-power objective lens collects relatively large-angle pencils of light from a distant object; in particular, it collects more of the light coming from that object than would enter the viewer’s smaller entrance pupil without the telescope (Fig 7-4).
Figure 7-4 Astronomical telescope. (Redrawn from Guyton D, West C, Miller J, Wisnicki H. Ophthalmic Optics and Clinical Refraction. London: Prism Press; 1999:39.)
The image seen through the astronomical telescope is inverted as well as magnified. If we wish to render the image upright, we must invert it again by passing the light path through at least one more lens or through a set of internally reflecting prisms such as those inside typical binoculars. Because they “fold” the light path, the prisms used in binoculars also enable the binoculars to be more compact and to have the right and left objective lenses spread farther apart than the viewer’s eyes, enhancing the perception of depth. Note that the minus sign in the formula for the telescope’s angular magnification yields, as it should, a positive power for the Galilean telescope’s upright image and a negative power for the astronomical telescope’s inverted image.
Accommodation Through a Telescope
If you look at an object one-third of a meter away, your otherwise emmetropic eye must accommodate 3 diopters. If you look at the same object one-third of a meter away through an afocal telescope, you have to accommodate much more: the accommodation required through the telescope is the usual amount, 3 D, multiplied by the square of the magnification of the telescope. For instance, the accommodation required through a 2× telescope would be 3 × (2)2 = 12 D, which is too difficult for an adult to achieve. Therefore, in order to enable us to see near objects through a telescope, we need to alter the telescope, in the manner we next describe, to form a loupe.