Of the five monochromatic lens aberrations, three – radial (or oblique) astigmatism, power error or curvature of field, and distortion – significantly degrade vision through off-axis areas of a spectacle lens. The magnification effects of lenses and the difference between the magnifications of the two lenses will also influence a patient’s vision and must be considered. The other two monochromatic aberrations, spherical aberration and coma, are the result of variations in power across an aperture. These two aberrations are generally negligible in moderate power single vision lenses because the pupil of the eye is relatively small, blocking the aberrated rays. Coma may be of importance in the design of progressive addition lenses.¹³

When a patient turns his or her eye to look at a point object that is not along the optic axis of a spectacle lens, the image of the point as formed by the lens may no longer be a point. Instead, the point may be imaged as two lines perpendicular to each other with one line closer than the other to the lens (Fig. 5). This aberration of the image is radial astigmatism. Radial astigmatism occurs because the refractive power through the tangential (or radial) meridian of the lens varies more rapidly away from the optic axis than the power through the sagittal (or circumferential) meridian perpendicular to it. This causes a difference in focal power between the two meridians at oblique angles of gaze. The effects of radial astigmatism on image formation are similar to those of astigmatism in general. If one of the line images falls on the far-point sphere, the patient sees the image of the point as a line. If the two foci straddle the far-point sphere, no sharp image is formed. Instead, the image is round and blurry (a blur circle). At other positions of the line images relative to the far-point sphere, the image formed is blurry and elliptical in shape. The separation of the two line foci formed by astigmatism is termed the Interval of Sturm, and the out-of-focus circular image formed between the two line foci is known as the circle of least confusion.¹⁴

If an off-axis extended object is viewed through a lens that has radial astigmatism, each point on the object is imaged as two line foci at different distances from the lens. For the example of Figure 6, each point on the vertical arms of the cross-object is imaged as a short horizontal line at the focus closer to the lens. Each point on the horizontal arms is also imaged as a large number of short horizontal lines, but the line images overlap along the length of the arms. If this image were to be formed at the far-point sphere of a patient’s eye, the patient would see the horizontal arms of the cross in sharp focus, while the vertical arms would appear blurry. At the focus farther from the lens, the situation is reversed with the vertical cross arms in sharp focus and the horizontal arms blurry. At the circle of least confusion, all points on the cross are imaged as out-of-focus circles, and both arms of the cross appear equally blurry. The appearance of the image differs at different distances from the lens.

A standard terminology describes the foci created by radial astigmatism. Suppose a patient were to look laterally (sideways) away from the center of a lens at a cross target. If the patient’s field of view is considered to be a circle on a vertical plane, with the cross pattern at the lateral edge of this circle, then the vertical arms of the cross are tangent to this circle. The focus of the vertical arms by the lens is termed the tangential focus, and errors in their focus position relative to the far-point sphere are tangential errors. Errors in focus that affect the cross arms perpendicular to the tangential detail (i.e., the horizontal cross-arms) are termed sagittal errors, and the focus of these lines is called the sagittal focus. If the patient were to look upward at a cross at the top of the field of view, the horizontal cross-arms would be tangential to the field-of-view circle, the focus of the horizontal cross-arms would be the tangential focus, and the image of the vertical cross-arms would form the sagittal focus. Tangential and sagittal focus errors increase as the distance from the center of a lens increases.

The amount of radial astigmatism present at a given point on a lens is defined as the dioptric difference between the two line foci, as measured from some reference surface. Changing the base curve of a lens alters the amount of radial astigmatism present, and there is usually an optimal base curve that results in zero radial astigmatism for a lens of given power.

In the absence of radial astigmatism, the two focal lines formed by a lens collapse onto a single curved image surface known as Petzval’s surface. Curvature of field represents the error between a flat plane and Petzval’s surface. However, the ideal focusing plane of the eye, the far-point sphere, is also curved, and will not typically coincide with Petzval’s surface. This means that for most prescriptions some residual focusing error will remain, even when radial astigmatism has been eliminated. The error between Petzval’s surface and the far-point sphere is the power error of the lens for a given viewing angle. In the presence of radial astigmatism, power error represents the average error of the two focal lines formed by the lens from the far-point sphere. Consequently, power error is analogous to the spherical equivalent of the errors from the desired spectacle prescription (Fig. 7).

Because the tangential and sagittal astigmatic errors vary as a function of base curve, the average of these errors also varies. Therefore, power error, like radial astigmatism, changes with the base curve chosen for a lens, and there is usually an optimal base curve that will minimize or eliminate power error. This condition is satisfied when the tangential and sagittal errors are equal in magnitude but opposite in sign, resulting in an average error of zero. Consequently, with the exception of extremely high minus-power lenses in which both the sagittal and tangential errors can be completely eliminated at the same time, the base curve that eliminates power error is generally not equal to the base curve that eliminates radial astigmatism.¹⁵ When radial astigmatism is eliminated, there will be some residual power error, and when power error is eliminated, some uncorrected radial astigmatism will remain (Fig. 8). Choosing which aberration to eliminate completely is left to the discretion of the lens designer. In some cases, a lens design criterion may be chosen that represents a compromise, such as eliminating the tangential error, instead of eliminating either power error or radial astigmatism.

Distortion is a lens aberration caused by a variation in magnification from the center to the edge of a lens (Fig. 9). It has little effect on resolution of an image, but it does affect the image shape. When an extended object is imaged by a high minus-power lens that has decreasing magnification toward the lens periphery, the images of portions of the object farther from the lens center are magnified less than those closer to the center, resulting in a “barrel” distortion of the image. A plus-power spectacle lens, which has increasing magnification to the periphery, exhibits “pincushion” distortion. The effects of distortion are most noticeable in high plus-power (aphakic) spectacle lenses. Patients who wear these lenses often report that door frames and other rectangular or square objects appear bent. In addition, the combination of magnification and distortion in thick plus-power lenses results in a swimming sensation as the patient turns his or her head. These effects can be reduced in high plus-power lenses by using flatter aspheric designs.

The various wavelengths that constitute visible light generally travel at different velocities through transparent media, which means that the refractive index of a lens material also varies as a function of wavelength or color. Short wavelength blue light, for instance, travels more slowly—and undergoes more refraction—through a lens material than longer wavelength red light, resulting in the phenomenon known as chromatic dispersion (Fig. 10), as is often seen through prisms. The degradation of image quality through an optical system that occurs as a result of dispersion is referred to as chromatic aberration, and two forms of chromatic aberration are typically recognized.

Longitudinal chromatic aberration or axial chromatic aberration occurs along the optical axis of an optical system (Fig. 11A). Light rays of different wavelength passing through the lens focus at different distances from the lens because the index of refraction is different for each wavelength. In effect, the lens has a different power for every wavelength of light. Longitudinal chromatic aberration does not seem to have much effect on visual acuity under normal circumstances. This may in part be due to the depth of field of the eye. In addition, the red and blue wavelengths at the edges of the visible spectrum tend to be the most out-of-focus, but the eye is relatively insensitive to these wavelengths.¹⁶

Transverse chromatic aberration or lateral chromatic aberration, on the other hand, can have significant effects on off-axis vision. As a patient looks laterally to view a point object through a lens, the different wavelengths of light from the object are refracted at different angles, “smearing” the image into its component colors (Fig. 11B). Patients may describe this as blur or color fringes around objects. For an extended object such as a cross, the effects of transverse chromatic aberration will differ for different meridians. For example, when a patient looks laterally to view a cross object, each of the wavelengths of light from any point on the vertical arms of the cross is refracted horizontally at a slightly different angle, smearing the image of the vertical arms horizontally and resulting in color fringes that surround the image. The image of the horizontal cross-arms is relatively unaffected because the dispersion of light into its component colors occurs along the arms of the cross and is not noticeable except at the ends of the cross. Therefore, it can be argued that the effects of transverse chromatic aberration are more important for the tangential focus than for the sagittal focus. Transverse chromatic aberration is related to spectacle lens prismatic effects and is analogous to the dispersive effects visible when looking through a prism.

The prismatic effects of a lens vary with lens power and with the distance from the lens optical center that a light ray passes through the lens. The relationship is termed Prentice’s rule:¹⁷

where P is the amount of prism created, measured in prism diopters, h is the distance from the lens optical center, measured in centimeters, and F is the lens power in diopters. For example, the prism at a point 15 mm away from the optical center of a 4.00 D lens is 1.5 × 4.00, or 6 prism diopters. (The prism direction depends upon where the light ray passes through the lens and upon the sign of the lens power.) Prismatic effects are larger for higher-power lenses and larger angles of view, so transverse chromatic aberration, which is related to the prismatic effects of a lens, will be worse for these same conditions. High-index lens materials have more dispersion than CR-39 plastic or crown glass, so transverse chromatic aberration will also be more noticeable for high-index materials. Clinically, patients report color fringes or blur around objects only when looking away from the center of high-power, high-index spectacle lenses, and it is unusual for patients to report these problems unless their lens powers are greater than approximately ±5.00 D. The lens designer cannot vary the dispersion of a given lens material, so in a sense, nothing can be done about the visual effects caused by transverse chromatic aberration. However, proper lens design to minimize or eliminate other aberrations and proper fitting techniques in the optical dispensary can minimize total off-axis blur and improve off-axis visual acuity.

Figure 13 illustrates the positioning of a frame and lens on the face. Most lenses are worn with the bottom tipped slightly toward the eye through an angle of about 6 to 10 degrees. This is the pantoscopic angle, and the frame is said to have pantoscopic tilt. A plane connecting the top and bottom of the orbit (the eyebrows and cheekbone) has some slight tilt from a vertical plane, so a slight amount of pantoscopic tilt is cosmetically attractive and maximizes field of view through a lens. Most frames, especially plastic frames, fit the nose such that the center of the lens is a few millimeters below the pupil when the patient is looking straight ahead. Fortunately, the combination of downward tip and lowered fitting position tends to keep the optic axis of the lens close to the center of rotation of the eye (within a few millimeters), and the lens back surface is automatically approximately on the reference sphere. In short, Figure 13 is Figure 12 tipped downward with some flexibility added.

The fit of a frame can be evaluated by drawing an imaginary line perpendicular to the eyewire at its vertical center and projecting backward into the eyeball, as shown by the solid line in Figure 13. The edge of a ruler placed against the frame and cheek area aids in visualization of the line. This line indicates the location of the optic axis of the lens. It should seem to pass approximately through the center of rotation. This point is behind the canthus, as shown in the figure.

Because the greatest use of lenses is for straight-ahead and downward viewing, the normal cosmetically attractive pantoscopic angle is also functionally desirable. For most prescriptions, departures from the ideal alignment shown in Figures 12 and 13 do not cause serious errors. However, when fitting high-power lenses (lenses for extremely high hyperopes and myopes, aphakic lenses), aspheric lenses, atoric lenses, and high-index lenses of relatively high power, pantoscopic tilt can create some problems. One, pantoscopic tilt alters the effective power through which the patient is looking when looking straight-ahead.¹⁹ As an example, adding pantoscopic tilt to a minus power lens increases the effective minus power of the lens and also adds plus-cylinder power at an axis of 90 degrees or minus-cylinder power axis 180 degrees. (Some undercorrected myopes may report this effect, stating that their vision is better when they tilt their glasses.) The power change is proportional to the lens power, so a patient’s vision can be blurred by the effects of pantoscopic tilt when lens power is high. Formulas that describe the effective power change caused by the tilt of a spherical power lens were described by Martin,¹⁹ and more general formulas that calculate the power change for any lens power are also available.^20,21 It is possible to prescribe a lens power that compensates a patient’s spectacles for the effects of pantoscopic tilt, but this is not commonly done, at least for single vision lenses. Two, pantoscopic tilt alters the off-axis optical performance of a lens. When the pantoscopic tilt of a frame is such that the optic axis of a lens does not pass through the center of rotation of the eye, the fitting geometry used by the lens designer is no longer correct, and off-axis monochromatic aberrations will increase.²² Aspheric lenses are more sensitive to this effect than are standard spherical lens designs.²³ High-index lenses have off-axis optical quality problems because of transverse chromatic aberration, and increases in monochromatic aberrations will only make the off-axis optical quality of these lenses worse.