where α is the angle subtended by the object at the eye entrance pupil, and α′ is the angle subtended by the image at the exit pupil of the optical system. These angles are given by:
where h and h′ are object and image heights, respectively, q is distance from the eye entrance pupil to the object, and b is the distance from the exit pupil to the image. Replacing the right-hand sides of equations 3a and 3b for α and α′, respectively, into equation 2 gives:
Here, MT is the transverse magnification h′/h. The transverse magnification of a single surface with reduced object and image vergences L and L′ is given by L/L′. For an optical system consisting of k surfaces, this relationship can be cascaded to:
For a single spectacle lens with the subscripts “1” and “2” referring to the front and back surfaces, respectively, we have:
Substituting the right-hand side of equation 5a for M T into equation 4 gives:
Starting with an object at distance l 1 from the first surface of an ophthalmic lens in a medium of refractive index n 0 (not necessarily air) and using first surface power F 1 and second surface power F 2, lens central thickness t, and lens refractive index n 1, raytracing through an ophthalmic lens using refraction and transfer equations gives the reduced vergences as:
Distance q is given by:
where d is the distance from the second surface of the lens to the eye entrance pupil.
Distance b depends on the position of the stop.
Stop at Eye Entrance Pupil
If the stop is the eye entrance pupil (Fig. 1), the exit pupil of the optical system coincides with the eye entrance pupil. Accordingly, the distance b is given by:
Substituting the right-hand side expressions of equations 11 and 12 for q and b, respectively, into equation 6 gives:
Substituting the right-hand sides of equations 8, 9, and 10 for L′1, L 2, and L′2, respectively, into equation 13 gives:
As spectacle magnification is often expressed in terms of one surface power, usually the first surface power F 1, and the back vertex power F′v, we obtain an equation for F 2 as:
Substituting the right-hand side of equation 15 for F 2 into equation 14 gives:
Stop at First Surface of Lens
If the stop is at the first surface of the lens (Fig. 2), the exit pupil of the optical system is the image of the stop in the back surface of the lens. Accordingly, the distance b is given by:
where l′2 is the distance from the lens second surface to the image and l′p is the distance from the second surface of the lens to the exit pupil. l′2 is given by:
and l′p is given by:
Substituting the right-hand sides of equations 18 and 19 for l′2 and l′p, respectively, into equation 17 gives:
We proceed as for the case of the stop at the eye entrance pupil. Substituting the right-hand sides of equations 11 and 20 for q and b, respectively, into equation 6 gives:
Substituting the right-hand sides of equations 8 and 9 for L′1 and L 2, respectively, into equation 21 gives:
Finally, substituting the right-hand side of equation 15 for F 2 into equation 22 gives:
RESULTS AND DISCUSSION
The main equations that describe spectacle magnification for the cases of the stop at the eye entrance pupil and at the lens are equations 16 and 23, respectively. As appropriate, equation 13 or 14 can be used instead of equation 16, and equation 21 or 22 can be used instead of equation 23.
Equations 16 and 23 simplify under certain conditions. For example, if the object is in air and at infinity, setting n 0 = 1 and L 1 = 0 into equation 16 gives the well-known equation:
Treatments of spectacle magnification appear in many texts. Ogle’s equation 28 is equivalent to equation 16 when the object is in air.4 Jalie1 and Bennett2,3 gave the standard equation 24 and then described changes to it when the object is at a finite distance.
For example, Jalie1 gave the equation:
With appropriate substitutions for L′2 and F 2 from equations 10 and 15, respectively, this equation is equivalent to equation 16 when the object is in air. Bennett derived a proximity factor N to convert distance spectacle magnification as given by equation 24 into near spectacle magnification:
This can be derived from equation 16 for the spectacle magnification by assuming a thin lens (t = 0) in air (n 0 = 1), using the binomial expansion, and ignoring powers in d higher than the second.
Spectacle magnification will now be determined for different conditions. Table 1 provides a summary. The eye entrance pupil is taken as 3.047 mm inside the eye to match that of the Gullstrand number 1 eye.8,9
Comparing Ophthalmic Lenses with Stop at the Eye Entrance Pupil and at the Lens
Fig. 3 shows spectacle magnifications of a range of ophthalmic lens powers for the two stop positions and for object distances of infinity and −0.4 m from the anterior surfaces of lenses. The distance between lenses and the eye entrance pupil is taken as 17.05 mm. When the stop corresponds to the eye entrance pupil, SM for distance vision varies from 0.86, at −10 D power, to 1.32 at +10 D power. Changing the object distance to near has minor effects only, at least for typical reading distances. This would be expected from the value of the proximity factor, N, given by equation 26, which is always close to unity because d is small. When the stop is at the lens, because of minimal deviation of the pupil ray at the lens, spectacle magnification varies little across the range of powers; for distance, it is in the range 1.01 to 1.02, and for near vision, it is in the range 1.05 to 1.09.
These results show that placing artificial pupils next to ophthalmic lenses gives spectacle magnifications that are little affected by lens power. This is in contrast to the situation where there is no artificial stop but only the pupil of the eye when, as shown above, the spectacle magnification can differ substantially from unity.
For vision in water without any spectacle aid or artificial stop, into equation 16, we can set n 0 = 1.333, F 1 = 0, and t = 0 to give:
Here F′v is the back surface power F 2 of the water environment, where the radius of curvature corresponds to that of the anterior cornea r c, and it is considered that there is an infinitesimally thin air film between the water and the cornea. Setting L 1 to zero and F′v = F 2 simplifies equation 27 to:
where F 2 is given by:
For the Gullstrand number 1 eye, r c = 7.7 mm to give F 2 = −43.247 D. Using this value, and setting d = 3.047 mm (as there is a zero vertex distance) gives SM = 1.1778. Note that the distance d is very important here because of the large value of F′v. If we ignored d, we would have SM = 1.333.
Air-Filled Plano Goggles Surrounded by Water and with the Natural Pupil
In equation 16 we set n 0 = 1.333, the refractive index of the water. n 1 and t are the refractive index and thickness of the faceplate of the goggles, respectively. Assuming that the goggles are flat at the front and have zero back vertex power (F 1 = F′v = 0), we have:
Setting L 1 to zero reduces this to SM = 1.333. This magnification is well known, but probably it is not realized that spectacle magnification is increased by only about 1.333/1.18 = 1.13 times compared with the water environment when no goggles are worn. However, the most important difference between the situations is, of course, that, in the goggle-wearing case, the retinal images can be brought to an accurate focus because of the absence of the high negative power of the water-air interface as given in equation 28 for the goggle-free case.
Plano Water-Filled Goggles in Air and with the Natural Pupil
This case is of interest in relation to studies in which it is attempted to simulate underwater vision through the wearing of water-filled goggles, the test targets being in air. It is then necessary to compensate for refractive effects at the goggle surface. If we ignore the small effect of the material of the faceplate of the goggles, we effectively have a “correction” of water whose thickness t extends all the way between the anterior surface of the faceplate and the eye. The back vertex power of this “correction” again corresponds to the power of the water-air interface at the assumed thin film of air in front of the cornea (equation 28). Thus, n 0 and n 1 are the indices of air and water, respectively, and d is the distance of the eye entrance pupil from the second surface of the “water correction,” that is, the anterior cornea. In equation 16, setting n 0 = 1, n 1 =1.333, and F 1 = 0, we have:
For a distance object, setting L 1 = 0 gives:
which for d = 3.047 mm and F′v = −43.247 D gives SM = 0.8836.
Water-Filled Goggles, Object in Air, with the Stop at the Front of the Goggle
For a simulation of the effects of pupil size on vision in water, we placed water-filled goggles on the eye with artificial pupils in front of the goggles. This simulation is shown in Fig. 4. As before, we essentially have a thick “water correction” and, in equation 23, setting n 0 = 1, n 1 = 1.333, and F 1 = 0, we have:
Setting L 1 = 0 gives:
where F′v is again given by equation 28. Spectacle magnification in this case is highly dependent on water thickness t in the goggles because of the high F′v. In Fig. 5, we show spectacle magnification for water-filled goggles as a function of water thickness. For typical depths of water within the goggles (≈15 mm), the magnification is quite high (≈1.5), substantially higher than might be expected on the basis of Snell’s law at a plane surface.
Using equations 27a and 31a, the ratio of spectacle magnification of a distant object observed in water through the natural pupil to that with the water-filled goggles with an artificial pupil in the plane of the anterior faceplate is given by:
This is the factor by which visual acuity with the simulation should be multiplied to estimate the true visual acuity in water. This is also shown in Fig. 5. It has been applied in a study of the vision in the water environment.10
We have developed equations showing the spectacle magnification when the limiting stop is either the entrance pupil of the eye or an artificial pupil in front of a lens. The reference was the retinal image for an uncorrected eye in air. For the artificial pupil in front of lenses, unlike the usual situation, spectacle magnification is hardly affected by lens power. In water, spectacle magnification is highly sensitive to the distance between the cornea and eye entrance pupil. In water, retinal images are approximately 18% bigger than in air. Wearing air-filled goggles in water increases retinal image size by about 13% compared with that when they are not worn.
School of Optometry and Institute of Health
and Biomedical Innovation
Queensland University of Technology
60 Musk Avenue
Kelvin Grove, Queensland 4059
The first author is grateful to Mo Jalie for bringing equation 4 to his attention.
Received July 12, 2013; accepted August 28, 2013.
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Keywords:© 2014 American Academy of Optometry
aperture; entrance pupil; spectacle magnification; water