Figure 1-41 Illustration of Knapp’s law. If the refractive power of eyes is the same but the axial length varies (a, b, c), a correcting lens placed at the anterior focal point of each eye (Feye) will produce an identical retinal image size regardless
of the axial length. In this example, the power of the correcting lens will change depending on the axial length of the eye. However, the retinal image size will remain constant. (Illustration b y C. H. Wooley.)
For example, if eyes have identical refractive power and differ only in axial length, then placing a lens at the anterior focal point of each eye will produce retinal images identical in size. However, it is rare that the difference between eyes is purely axial. In addition, the anterior focal point of the eye is approximately 17 mm in front of the cornea (see Chapter 2). Although it is possible to wear glasses so the spectacle lens is 17 mm in front of the eye, most people prefer to wear them at a corneal vertex distance of 10–15 mm. Because the clinician is rarely certain that any ametropia is purely axial, Knapp’s law has limited clinical application.
Manual lensmeters make use of the same principle, although for an entirely different reason. When applied to lensmeters, Knapp’s law is called the Badal principle. One type of optometer used for performing objective refraction is based on a variation of Knapp’s law wherein the posterior focal plane of the correcting lens coincides with the anterior nodal point of the eye. The effect is the same. Retinal image size remains constant. In this application, the law is called the optometer principle. Optical engineers use a variation of Knapp’s law called telecentricity to improve the performance of telescopes and microscopes. Regardless of the name, the principle remains the same.
Afocal Systems
Consider an optical system consisting of 2 thin lenses in air (Fig 1-42). The lens powers are +2 D and –5 D, respectively. Where is Fp for this system? The posterior focal point is where incoming parallel rays focus. However, as ray tracing demonstrates, rays entering the system parallel to the optical axis emerge parallel to the axis. This system has no focal points; in other words, it is an afocal system.
Figure 1-42 The Galilean telescope, an afocal system. The lenses are separated by the difference in focal lengths. F is simultaneously the posterior focal point of the plus lens and the anterior focal point of the minus lens. (Illustration developed
b y Kevin M. Miller, MD, and rendered b y C. H. Wooley.)
If an object is 2 m in front of the first lens, where is the image and what is the transverse magnification? Vergence calculations show that the image is virtual, that it is 44 cm to the left of the second lens (14 cm to the left of the first lens), and that the transverse magnification is 0.4×. If an object is 4 m in front of the first lens, vergence calculations show that the image is virtual, that it is 76 cm to the left of the second lens, and that the transverse magnification is exactly 0.4. In afocal systems, the transverse magnification is the same for every object regardless of location.
Where are the principal planes for this system? Actually, it has no principal planes. Remember, the principal planes are the unique conjugates with a transverse magnification of 1. In this system, the transverse magnification is always 0.4 and never 1. If the transverse magnification were equal to 1, it would be 1 for every pair of conjugates. Consequently, there would be no unique set of planes that could be designated principal planes. In general, afocal systems do not have cardinal points.
Afocal systems are used clinically as telescopes or low vision aids. The 2 basic types of refracting telescopes are the Galilean telescope (named for, but not invented by, Galileo) and the Keplerian, or astronomical telescope (invented by Johannes Kepler). The Galilean telescope consists of 2 lenses. The first lens, the objective lens, is always positive and usually has a low power, whereas the second lens, the eyepiece, or ocular, is always negative and usually has a high power. The lenses are separated by the difference in their focal lengths. The afocal system depicted in Figure 1-42 is a Galilean telescope. The Galilean telescope is also used in some slit-lamp biomicroscopes.
The Keplerian telescope also consists of 2 lenses, a low-power objective and a high-power ocular, but both are positive and separated by the sum of their focal lengths. The image is inverted. For comparison, construct a Keplerian telescope using +2 D and +5 D trial lenses.
For each telescope,
where
Peye = power of the eyepiece or ocular Pobj = power of the objective lens
fobj = focal length of the objective lens
feye = focal length of the eyepiece (negative for concave lenses)
For afocal telescopes like the Galilean and the Keplerian telescopes, the focal point of the objective lens and the focal point of the ocular lens are in the same position.
Each form of telescope has advantages and disadvantages. The advantage of a Galilean telescope is that it produces an upright image and is shorter than a Keplerian telescope. These features make the Galilean telescope popular as a spectacle-mounted visual aid or in surgical loupes.
Conversely, the Keplerian telescope uses light more efficiently, making faint objects easier to see (Fig 1-43). In the Keplerian design, all the light from an object point collected by the objective lens ultimately enters the eye. In the Galilean design, some of the light collected by the objective is lost. Because astronomical observation is largely a matter of making faint stars visible, all astronomical telescopes are of the Keplerian design. The inverted image is not a problem for astronomers, but inverting prisms are placed inside the telescope. Common binoculars and handheld visual aids are usually of the Keplerian design.
Figure 1-43 Comparison of Galilean and Keplerian telescopes. In the Galilean telescope (A), some of the light collected by the objective is lost. In the Keplerian telescope (B), all the light collected enters the eye. (Illustration developed b y Kevin M. Miller,
MD, and rendered b y C. H. Wooley.)