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3.10.An angle of 45° corresponds to how many prism diopters (Δ)?

a.45.0Δ

b.22.5Δ

c.90.0Δ

d.100.0Δ

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3.11.When bifocal lenses are prescribed for a patient with myopia,

a.the practitioner should leave the choice of the segment type to the optician

b.a round-top segment is preferred because of its thin upper edge, which causes less prismatic effect

c.a flat-top segment is preferred because it lessens image jump

d.the 1.piece shape is indicated for adds greater than +2.00 D

e.a split bifocal should be used because patients with myopia do not accept bifocal lenses easily

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Appendix 3.1

Common Guidelines for Prescribing Cylinders for Spectacle Correction

[The material in this appendix is modified from Guyton DL. Prescribing cylinders: the problem of distortion. Surv Ophthalmol. 1977;22(3):177–188. Copyright © 1977, Survey of Ophthalmology]

Commonly taught guidelines for spectacle correction are the following:

1.Children accept the full astigmatic correction.

2.If an adult cannot tolerate the full astigmatic correction, rotate the cylinder axis toward 90° or 180° and/or reduce the cylinder power to decrease distortion. When reducing the cylinder power, keep the spherical equivalent constant by making the appropriate adjustment of sphere.

3.With older patients, beware of changing the cylinder axis.

The problem: distortion

Why have such guidelines been developed? Why can some patients not tolerate the full astigmatic correction in the first place? One text on clinical refraction states that full correction of a high astigmatic error may initially result in considerable blurring of vision. Another teaching is that with the full astigmatic correction the image is too sharp—the patient is not used to seeing so clearly. Statements such as these are not only misleading; they are incorrect.

The reason for intolerance of astigmatic spectacle corrections is distortion caused by meridional

before the right eye, and a +1.00 diopter cylinder, axis 45° before the left eye. If the patient looks at a vertical rod 3 m away, the retinal images of the rod will be tilted toward each other at the top (declination error) by approximately 0.4° each, a barely perceptible amount under monocular conditions. But under binocular conditions, the vertical rod will theoretically appear tilted toward the patient (inclination) by approximately 35°! Such large errors in stereoscopic spatial localization are clearly intolerable, seemingly out of proportion to the amount of monocular distortion that produces them. Oblique distortion in 1 or both eyes causes more distressing binocular symptoms than does vertical or horizontal distortion, and movement accentuates the symptoms.

Fortunately, errors in stereoscopic spatial localization are usually compensated for in most patients by experiential factors: perspective clues, the known size and shape of familiar objects, the knowledge of what is level and what is perpendicular, etc. The possibility of permanent adaptation to binocular distortion is discussed later, but the fact remains that some patients cannot or will not tolerate binocular spatial distortion, and herein lies our problem.

We have no effective means of treating binocular spatial distortion except by altering or eliminating the monocular distortion that produces it. Complete understanding of the causes and management of monocular distortion is difficult, involving several areas of physiological optics that the average practitioner would prefer to avoid: blur theory, spectacle lens design, obliquely crossed cylinders, and theory of the Jackson cross cylinder. However, with a few key facts from each of these areas, we can quickly gain a working understanding of astigmatic spectacle corrections.

Sources of monocular distortion

As illustrated in Figure 3-41 monocular distortion is caused by meridional magnification. We can identify 2 basic sources of meridional magnification, 1 involving the design of the spectacle lens and the other involving the location of the spectacle lens with respect to the entrance pupil of the eye.

Shape factor of the spectacle lens All spectacle lenses having curved front surfaces produce a magnification inherent to the lens itself. The more convex the front surface and the thicker the lens, the greater this shape factor magnification. If the front surface of the lens is spherical, the shape factor magnification will be the same in all meridians, producing only an overall size change in the retinal image. However, if the front surface of the lens is cylindrical or toric, the shape factor magnification will vary from 1 meridian to another, producing distortion of the retinal image. Again, this occurs only with lenses in which the cylinder ground is on the front surface, the so-called plus cylinder form or anterior toric spectacle lens. Lenses in which the cylinder ground is on the back surface (minus cylinder lenses, posterior toric lenses, iseikonic lenses) do not produce differential meridional magnification due to the shape factor, because the front surface power is the same in all meridians. The lens clock may be used to check the front and back curves.

Meridional magnification arising from the shape factor of plus cylinder spectacle lenses is rarely more than 1% to 2%. Many patients can perceive this difference, however, and for this reason and others, since the mid-1960s, minus cylinder spectacle lenses have become the preferred form for routine dispensing.

Distance of the spectacle lens from the entrance pupil More important than the shape factor of the

spectacle lens in producing distortion is the location of the spectacle lens relative to the entrance pupil of the eye. We consider both the conventional method and the general method of analyzing the magnification produced.

To investigate the general nature of meridional magnification, we must consider what happens in the case of blurred retinal images. The size of a blurred retinal image is defined as the distance between the centers of those blur circles which represent the extremities of the object.1 Each blur circle is formed by a bundle of rays limited by the entrance pupil of the eye; the chief ray of the bundle passes toward the center of the entrance pupil and forms the center of the blur circle on the retina. Therefore, the chief rays from the extremities of the object determine the size of the retinal image, and because the chief rays pass toward the center of the entrance pupil, the angle subtended by the object at the entrance pupil is proportional to the size of the retinal image.2

Illustrating the location effect of cylindrical lenses

Figure 3-43 illustrates the general case of distortion of the retinal image produced by a cylindrical lens placed before a nonastigmatic eye. In Figure 3-43 rays ef and gh represent the chief rays from the vertical extremities of the object. These chief rays cross at the center of the entrance pupil and continue on to form the vertical extremities of the retinal image.3 No distortion of the retinal image is present in Figure 3-43A.

Figure 3-43 The relationship between the position of a cylindrical lens relative to the entrance pupil of a nonastigmatic eye and the distortion of the retinal image produced. A, No distortion in the absence of the cylindrical lens. B, Distortion produced by the cylindrical lens located in the usual spectacle plane. C, No distortion when the cylindrical lens is placed hypothetically in the entrance pupil of the eye.

In Figure 3-43B a cylindrical lens is placed before the eye in the usual spectacle plane with its axis xy oriented in the 45° meridian. Chief rays ijk and lmn pass through the cylindrical lens at points away from the axis xy and are therefore bent by the lens. Chief ray ijk undergoes a small prismatic deviation down and to the patient’s left, while chief ray lmn undergoes a small prismatic deviation up and to the patient’s right. The chief rays continue on through the center of the entrance pupil to the retina, but the ray segments jk and mn lie in a tilted plane because of the prismatic deviations that occurred in opposite horizontal directions at the cylindrical lens. (Note that ray segments i and I do not lie in the same common plane as the segments jk and mn, and neither do they follow the same paths as ray segments e and g in Figure 3-43A.) Because ray segments jk and mn lie in a tilted plane, the vertical arrow in the retinal image of Figure 3-43B is tilted. In fact, the entire retinal image in Figure 3-43B is distorted. In the case of this retinal image as a whole, we may speak of the distortion as arising from meridional magnification caused by the cylindrical lens—a meridional magnification in the direction of the double arrows (at corners), perpendicular to axis xy of the cylindrical lens.

In Figure 3-43C the cylindrical lens has been moved toward the eye until it is located hypothetically within the entrance pupil. Chief rays op and qr pass to the center of the entrance pupil, where they now pass undeviated through the cylindrical lens because they strike the lens along its axis. Chief rays op and qr continue on to the retina and may be seen to be identical with rays ef and gh of Figure 3-43A causing no distortion of the retinal image.

Summary of the general case

To summarize the lesson learned from Figure 3-43 astigmatic refracting surfaces located away from the entrance pupil of the eye cause meridional magnification and distortion of the retinal image. The direction of meridional magnification is determined by the axis orientation of the astigmatic refracting surface, and the amount of meridional magnification increases not only with the power of the astigmatic refracting surface but also with increased distance of the astigmatic refracting surface from the entrance pupil of the eye.4

Distortion with uncorrected and inappropriately corrected astigmatism

We can now predict what sort of meridional magnification and resultant distortion to expect in the case of uncorrected astigmatism or inappropriately corrected astigmatism. In uncorrected astigmatism, the astigmatic refracting surfaces are those of the cornea or lens. Because these surfaces are located near the entrance pupil, meridional magnification and distortion will be minimal. The meridional magnification produced by uncorrected corneal astigmatism is approximately 0.3% per diopter of astigmatism, which is a rather small amount.

The situation with inappropriately corrected astigmatism is much the same as the situation with properly corrected astigmatism. Whenever an astigmatic spectacle lens is placed before an eye, whether it is the correct lens or not, significant meridional magnification is likely to occur because of the relatively remote location of the spectacle lens from the entrance pupil. Even though the eye may be astigmatic, the effect of astigmatic surfaces located near the entrance pupil is so small that the

direction and amount of meridional magnification are determined, not by the astigmatism of the eye itself, but primarily by the axis and power of the astigmatic spectacle lens—whatever the axis and power may be, correct or not.

Minimizing monocular distortion

Specifying minus cylinder (posterior toric) spectacle lenses The small amount of meridional magnification

caused by plus cylinder (anterior toric) spectacle lenses is avoided simply by specifying minus cylinder lenses in the prescriptions. In practice, this distinction is rarely necessary because minus cylinder lenses have become the preferred form for routine dispensing. Most dispensers will choose the minus cylinder form automatically and reserve the plus cylinder form only for duplication of an old pair of plus cylinder lenses.

Minimizing vertex distance As previously discussed, meridional magnification decreases as the correcting spectacle lens is placed closer and closer to the eye (see Fig 3-42). There is often little room for manipulation of vertex distance with common styles of frames, but when distortion is anticipated, we can at least avoid the so-called fashionable glasses that sit at the end of the patient’s nose.

With contact lenses, the vertex distance is reduced to zero and distortion is practically eliminated. Contact lenses should always be considered an alternative to spectacle correction if the patient remains unsatisfied with other attempts to reduce distortion. In fact, contact lenses may be the only means available (other than iseikonic corrections) to reduce distortion while maintaining clear imagery, given that further attempts to reduce distortion by manipulating the spectacle lens correction, as discussed in the next section, involve a certain sacrifice in the sharpness of the retinal image.

Altering the astigmatic correction: rotating the cylinder axis Clinical experience suggests that new

astigmatic spectacle corrections in adults are better tolerated if the axis of the cylinder is at 90° or 180° rather than in an oblique meridian. In fact, it has long been taught that oblique axes should be rotated toward 90° or 180°, if visual acuity does not suffer too much, to avoid problems from oblique distortion. This makes sense because the direction of meridional magnification is determined principally by the axis orientation of the correcting cylinder, whether or not the axis is correct, and vertical or horizontal aniseikonia is known to be more tolerable than oblique aniseikonia.

There have been recurrent arguments in the literature regarding why spectacle cylinder axes should not be rotated away from the correct position, but these arguments are based primarily on the misconception that the axis and power of the residual astigmatism determine the direction and amount of distortion. Rather, it is the axis and power of the spectacle lens itself, correct or not, that principally determine the direction and amount of distortion. With this concept understood, and on the basis of clinical experience, we may thus state that the direction of distortion may be made more tolerable, if necessary, by rotating the cylinder axis toward 90° or 180°.

There is another situation in which the cylinder axis should sometimes be rotated away from the correct position. An older patient may have adapted to an incorrect axis position in his or her previous spectacles, and may not tolerate the change in direction of distortion produced by rotating the cylinder axis to the correct position. The nature of such adaptation is discussed elsewhere, but there is no question that it occurs and can cause problems. In this case, the cylinder axis should be rotated toward the position of the old cylinder axis, even if the old cylinder axis is oblique. This maneuver does not reduce distortion but does change the direction of distortion back toward the

position of adaptation.

Altering the astigmatic correction: reducing the cylinder power The other method that is commonly used to

lessen distortion is reduction of the power of the correcting cylinder. This method makes sense, for we have seen that the amount of meridional magnification is largely determined by the power of the correcting cylinder—the less the cylinder power, the less the meridional magnification.

Altering the astigmatic correction: residual astigmatism Whenever the cylinder power is reduced from its

correct value, or the cylinder axis is rotated away from its correct position, residual astigmatism appears. The residual astigmatism does not produce distortion, but it does produce blur of the retinal image, limiting the amount that the cylinder power may be reduced or the amount that the cylinder axis may be rotated away from its correct position. If we must minimize distortion, we must be careful at the same time not to create excessive blur from residual astigmatism.

It is the amount of residual astigmatism that produces the blur, not the axis5 of the residual astigmatism. It is easy to judge the amount of residual astigmatism when reducing the cylinder power of a spectacle correction, for the residual astigmatism is simply equal to the amount the cylinder power is reduced. It is more difficult, however, to judge the amount of residual astigmatism induced by rotating the cylinder axis away from the correct position. The resulting residual astigmatism may always be calculated using the rules for combination of obliquely crossed cylinders, but this calculation requires considerable mental gymnastics. Residual astigmatism may be minimized with the Jackson cross-cylinder test for cylinder power. The spherical equivalent concept will guide the adjustment of the sphere. The guidelines for prescribing cylinders for spectacle correction are summarized below.

Revised guidelines for prescribing cylinders for spectacle correction

1.In children, give the full astigmatic correction.

2.In adults, try the full astigmatic correction first. Give warning and encouragement. If problems are anticipated, try a walking-around trial with trial frames before prescribing.

3.To minimize distortion, use minus cylinder lenses and minimize vertex distances.

4.Spatial distortion from astigmatic spectacle corrections is a binocular phenomenon. Occlude 1 eye to verify that this is indeed the cause of the patient’s complaints.

5.If necessary, reduce distortion still further by rotating the cylinder axis toward 180° or 90° (or toward the old axis) and/or by reducing the cylinder power. Balance the resulting blur with the remaining distortion, using careful adjustment of cylinder power and sphere. Residual astigmatism at any position of the cylinder axis may be minimized with the Jackson crosscylinder test for cylinder power. Adjust the sphere using the spherical equivalent concept as a guide, but rely on a final subjective check to obtain best visual acuity.

6.If distortion cannot be reduced sufficiently by altering the astigmatic spectacle correction, consider contact lenses (which cause no appreciable distortion) or iseikonic corrections.

Special cases occasionally arise. If a patient’s sense of spatial distortion seems out of proportion to his or her astigmatic correction, consider the possibility of spherical aniseikonia as the cause of his symptoms, and prescribe accordingly.

If patients with moderate to high astigmatism have no complaints about distance spectacle

correction but have difficulty reading at near, remember that changes in the astigmatic axes of the eyes (from cyclorotations) and changes in the effectivity of astigmatic corrections may cause problems at near. Such patients may require separate reading glasses.

Finally, patients who desire spectacles only for part-time wear may not be able to adapt to distortion during short periods of wear. In such cases, the astigmatic correction should be altered to reduce distortion according to the principles outlined above.

The revised set of guidelines for prescribing cylinders is now complete. With a rational basis for these guidelines, we should be able to place our confidence in them and prescribe cylinders more from knowledge and less from empirical rumors.

For a complete list of references, see the original article: Guyton DL. Prescribing cylinders: The problem of distortion. Surv Ophthalmol. 1977;22(3):177–188.

Appendix Notes

1.In the case of astigmatism, the blur patches on the retina may be ellipses or lines instead of blur circles, and the size of the blurred retinal image may appear somewhat altered by the effect of the shape of the blur patches on the outline of the image. This effect, however, only affects the outline of the retinal image and does not cause the type of monocular distortion (tilting of lines, etc.) that concerns us here.

2.The size of the retinal image is usually computed as proportional to the angle subtended by the object at the first nodal point of the eye (which is approximately 4 mm posterior to the center of the entrance pupil), but this is true only for sharp retinal images. The nodal point cannot be used to compute the size of blurred retinal images.

3.Chief rays such as those in Figure 3-43 although they initially pass toward the center of the entrance pupil, are actually refracted by the cornea, pass through the center of the real pupil, and are further refracted by the crystalline lens before continuing on to the retina. However, each pair of chief rays, such as ef and gh or jk and mn, because of the axial symmetry of the ocular media, remain in the same plane as one another other as they pass through the eye’s optics. Therefore, although the retinal images in Figure 3-43 are not exactly the proper size because of the simplified representation of the chief rays, the presence or absence of distortion is accurately represented.

4.An alternate way to analyze the effect of a cylindrical lens is to consider the lens an integral part of the eye’s optical system and calculate the position of the entrance pupil for the system as a whole in each principal meridian. The pairs of chief rays in the 2 principal meridians would then cross the optical axis at different points, producing the same differential meridional magnification as obtained with the present analysis.

5.The axis of residual astigmatism is of no consequence in the consideration of blur, except perhaps as it may affect the reading of letters that have predominantly vertical strokes. If the axis of the residual astigmatism is vertical or horizontal, and if the patient is able to clear the vertical strokes of the letters by accommodating, his or her reading ability may be somewhat better than would otherwise be predicted.