An ophthalmic lens composed of a spherical surface and an opposing toric surface is called a toric lens. The toric surface has two principal meridians; the difference in refracting powers of the two principal meridians represents the astigmatism of the surface (see Figures 11 and 12). For practical purposes, the astigmatism of the toric surface of a toric lens can be assumed to be constant over the entire lens surface. The grinding and polishing of a toric surface is accomplished with opposing lens and tool having oscillatory and translational movements with respect to each other but without rotation of either.

Both spherical and toric surfaces are easily made; ophthalmic lenses with these surfaces are used almost exclusively.

The monochromatic blurring aberrations, curvature of the field, and marginal astigmatism, in both spherical and toric lenses, are reduced by appropriate bending of the lens, ie, the relative curvatures of the two surfaces of the lens, known as coflexure. Corrected curve spherical and toric lenses which permit the widest and clearest field of vision depend on specific coflexures for specific powers of the lenses to minimize the blurring aberrations. The appropriate coflexure also reduces distortion but does not eliminate it.

The correction of aberrations of spherical lenses by coflexure is quite effective for minus power lenses as strong as -20.00 D and for plus power lenses up to +7.50 D. For lens powers above +7.50 D, aberrations cannot be reduced satisfactorily by coflexure. Excessive lateral power and lateral astigmatism can be corrected only by gradually changing the curvature of one of the surfaces of the spectacle lens, usually the anterior surface. By a continuous and regular reduction in curvature from the apex of the front surface to its periphery, both lateral power excess and lateral astigmatism are reduced. Such a surface is shaped like the end of an oval or egg, ie, ellipsoidal. Strong plus power lenses such as cataract lenses which have ellipsoidal front surfaces combined with spherical or toric back surfaces are far superior to spherical and toric lenses of the same back vertex power: the field of vision for given size lenses is somewhat larger in the aspheric lenses due to the more uniform magnification, and vision is considerably clearer throughout the field.

The grinding technique used for spherical lenses cannot be applied to aspheric lenses. Although it rotates about its own axis during the grinding and polishing, the spherical lens moves with respect to the axis of the grinding tool so that the lens and tool axes do not coincide and are continually changing their angular relationship with respect to each other. In the grinding procedure, broad areas of the rotating grinding tool and the rotating lens are in contact and precise surfaces are obtained. In the grinding of aspheric surfaces other than toric surfaces, only limited areas of the surface of the grinding tool can be in contact with the lens surface; it is difficult to produce and maintain the desired surface contour of an aspheric lens throughout the grinding and polishing procedures. Various techniques are utilized to overcome the difficulties in the production of aspheric surfaces. All aspheric lenses other than toric lenses are high in cost as compared with spherical lenses, and it is the cost which has limited the use of aspheric ophthalmic lenses.

THE CORNEA

The anterior surface of the human cornea closely resembles an ellipsoid, ie, highly curved centrally and flattening toward the periphery (Fig 1). The central area of the anterior surface of the cornea may be free of astigmatism, resembling a spherical surface, while the remainder of the cornea becomes increasingly flat and astigmatic peripherally. The central, practically spherical, area subserves central vision; such a cornea is said to be free of astigmatism.

Alternatively, the central area may resemble a toric surface; such a cornea is said to be astigmatic, regardless of the peripheral portions of the cornea.

An idealized version of a principal section of the anterior corneal surface is shown in Figure 2, wherein a spherical surface of a specific radius of curvature matches or “osculates” the central area of the ellipsoidal cornea. The two surfaces appear to be in contact over a relatively large central area.

The simplest instrument for studying the corneal contour is a keratoscope, such as the Placido's disc (Fig 3), which consists of a circular target with alternate black and white annuli and a handle. Usually a +5.00-D lens is placed behind an aperture in the center of the disc to enable the examiner to see the image of the target produced by specular reflection from the smooth tear film overlying the cornea. The target, either illuminated indirectly or self-luminous, has an image formed of it by the cornea which is observed through the lens of the instrument. Figure 4 illustrates the appearance of the central corneal image of the target when there is no central astigmatism; the image consists of a series of successively larger circular white rings centered about the corneal apex. Figure 5 demonstrates the appearance of the image of the target as produced by the astigmatic periphery of the cornea. When the central cornea is astigmatic the images of the target rings are elliptic, with the long axis of each of the ellipses along the flatter meridian of the cornea.

Hydrophilic soft contact lenses are now made by two distinct methods. The first method is the same as that used for methylmethacrylate hard contact lenses and involves the generating and polishing of the lens in its hardened nonhydrated state. At the completion of the process the lens in its hardened state is immersed in water which it absorbs until it reaches its fully hydrated state; therefore, it is necessary to take into account the expansion of the lens due to hydration when considering what surfaces are to be generated and polished. The second method involves the casting of the soft lens material in its liquid state into molds where it is allowed to polymerize.

Soft contact lenses which have generated and polished ellipsoidal back surfaces are now being produced. The fitting procedure takes into account the apical radius of curvature of the cornea and its ellipsoidal shape. The apical radius of curvature of the ellipsoidal back surface of the contact lens bears a fairly definite relationship to that of the cornea. Once the appropriate corneal surface of the lens is determined, the power of the front surface of the lens required to correct the refractive error of the eye is known. Hence for a given back surface, the amount of lens power is controlled by the amount of curvature of the front surface. Such aspheric soft contact lenses are used overseas, but they are not yet available in the United States.

Softens, Bausch & Lomb's trademark for its hydrophilic soft lens, utilizes a fixed specific front surface curvature for a given series of lenses and varies the power of the lens by variations in the apical curvature of paraboloidal back surfaces. For a series of eyes of a given corneal shape and apical curvature and having a range of refractive errors, the range of powers of the lens results from changes in the apical radius of curvature of the paraboloidal back surface. Hence the fit relationship between the lens and the cornea must vary according to the amount of power of the lens. Consequently some lenses fit well while others fit poorly, depending on the refractive error of the eye within the given range of powers for the given series of lenses. This method of fitting soft contact lenses is the consequence of the method of manufacturing of Softens, in which the paraboloidal back surface of the lens is obtained by spinning the liquid polymer while it rests in a fixed curvature concave mold. Different paraboloidal back surfaces are obtained from different rates of spin, variations in polymer, and changes in temperature of the polymer; for a given series of lenses, the front surface of fixed curvature is produced by the mold.

In the lower powers of the range of subnormal vision lenses, only one surface need be aspheric; in the higher power lenses both surfaces must be aspheric if aberrations are to be adequately reduced. Figure 6 shows an example of a subnormal vision lens in which both surfaces are aspheric¹. Ellipsoids, paraboloids, and hyperboloids are good aspheric optical surfaces for subnormal vision lenses; all such surfaces are classified geometrically as conoids or conicoids.

The proper use of a conoid aspheric spectacle lens for subnormal vision necessitates that the spectacle frame be so adjusted that the lens is centered directly in front of and close to the eye, with its optical axis coinciding with the line of sight of the eye when the eye is directed forward for reading or close work. Pantoscopic tilt of the spectacle frame and lens should be minimal or absent to permit the reading material, which must be perpendicular to the optical axis of the lens, to be brought into the anterior focal plane of the lens (assuming no accommodation of the eye). Generally the reading material is maintained in the focal plane and moved laterally to bring new material successively into the field of view. Should the head be rotated during reading, the plane of the reading material becomes tilted with respect to the optical axis of the lens and it is, therefore, impossible for the reading material to remain in focus across the field of view. At the same time the distance from the lens to the reading material tends to become increased with head turning, resulting in blurred vision.

A most useful type of aspheric cataract lens is the conoid cataract lens; this series of lenses, which covers the range of prescriptions usually found in aphakia, has a series of ellipsoids serving as the positive power front surfaces^2,3. As the average power or spherical equivalent power of the lens increases, the curvature of the apex of the front surface increases, while the rate at which the lens flattens peripherally also increases. The average power of the back surface remains at -3.00 D.

Conoid cataract lenses are made of glass. Other types of aspheric cataract lenses are made of plastic, both in the lenticular form and the full-lens form. Plastic aspheric cataract lenses have the advantage of less weight than glass aspheric cataract lenses, but they have the disadvantage of being slightly thicker in the full-lens form since the index of refraction of ophthalmic plastic is less than that of ophthalmic crown glass. Ophthalmic plastic is much softer than ophthalmic crown glass and is more easily scratched; therefore, special care must be taken in handling and cleaning plastic lenses.

Aspheric indirect ophthalmoscopy lenses must be designed to serve two functions:¹ to gather light emitted from the indirect ophthalmoscope and focus it to a very small precise image approximately in the plane of the pupil of the eye, ie, serving the function of a condensing lens (Fig 8), and² to form a three-dimensional, flat, aerial image of the fundus of the eye between the indirect ophthalmoscopy lens and the ophthalmoscope (Fig 9). In order to perform these two functions effectively, one of the two surfaces must be aspheric in order to eliminate aberrations; the other may be plane or spherical. In a properly designed indirect ophthalmoscopy lens there must be a specific balance between the aspheric and the spherical surface, and the rate of change in curvature from the center to the edge of the aspheric surface also must be specific. In conoid indirect ophthalmoscopy lenses the elimination of aberrations is accomplished by using the appropriate conoid surface in combination with the required balancing spherical surface. As the powers and the diameters of the lenses vary, so must the apical curvatures and the rate of change of flattening of the conoid surfaces as well as the curvatures of the balancing spherical surfaces.

In use, the indirect ophthalmoscopy lens must be held at a distance in front of the entrance pupil of the eye approximately equal to the secondary focal distance of the lens. The aerial image of the fundus will then be at approximately the same distance on the opposite side of the lens in an emmetropic eye, closer to the lens in myopia, and farther from the eye in hyperopia (Fig 10).

An obvious disadvantage of bifocals and trifocals is the restricted range of distances for clear vision through each portion of the lens. For example, if the reading addition is +2.50 D and there is no accommodation (full presbyopia) the distance of clear reading vision is 16 inches, with a depth of focus of approximately 2 inches on either side of the 16-inch distance. At a reading distance of less than 14 inches, vision is blurred; at distances from 18 inches to approximately 5 feet, vision is blurred, whether through the bifocal segment or through the distance vision portion of the lens. The intermediate segment of the trifocal provides a second focal distance approximately twice that of the reading segment, and there is a greater depth of focus through it; but there are still intermediate distances between the focal distances of the two segments at which vision is blurred despite the intermediate segment.

In a sense, a progressive lens can be thought of as an infinite number of multifocal segments extending horizontally across the entire lens, with one above the other; refractive powers are arranged so that there is a progressive increase in refractive power of each of the segments from an upper lens portion to a lower lens portion. In such a lens there is no discontinuity in central vision through any portion of the lens. Central vision along a vertical line of symmetry of the lens in front of the pupil is clear for distance vision through the upper part of the lens; as gaze through the lens is directed increasingly downward, vision is clear for increasingly closer objects. Although spherical refractive power of the lens increases progressively down the lens along the vertical line of symmetry, the amount of change in power along the line across any area of the lens approximately the size of the pupil is insufficient to cause blurring of vision.

On either side of the vertical line of symmetry at any given level of the lens, there is astigmatism at oblique axes; the amount of astigmatism increases with lateral distance from the line of symmetry and at a rate of increase proportional to the rate of increase in refractive power along the line of symmetry at the given level.

A properly designed progressive lens has the advantage of a single continuous field of view, with minimal distortion and an unlimited range of clear viewing distances from far to the usual near work distance through areas of the lens in the vicinity of the vertical line of symmetry. It has the disadvantage of astigmatically blurred vision in lateral gaze through the lens, the amount of blurring increasing with increasingly lateral gaze. Just as the wearer of a bifocal or a trifocal must learn to avoid direct vision along the segment line or lines which intrude on the most important areas of a lens (areas normally used in single vision lenses) and to disregard the diplopia and blurring produced by joining segments in a lens, a wearer of a progressive lens must learn to disregard the blurring effects of astigmatism in lateral parts of the lens. Should a progressive lens have an abrupt or localized increase in the rate of change in curvature along the vertical line of symmetry, there will be a corresponding horizontal zone of the lens in which there is severe oblique astigmatism and severe distortion laterally. Such a progressive lens may appear to have or may actually have a discontinuity in the field seen through the lens in lateral gaze at the level of the severe localized distortion and astigmatism. Any gain in the extent of clear lateral vision in the distance vision area of the lens will be accompanied by a reduction in the size of the clear field at the reading level. A properly designed progressive lens must take into account the rates of change in power down the entire vertical line of symmetry if astigmatism laterally is not to be excessive at any level of the lens.

Consider the geometry of a toric surface. If a circular arc is rotated about an axis in the plane of the arc, which axis is not a diameter of the circular arc, a toric surface will be generated. If the axis lies between the circular arc and its center of curvature, the surface generated is termed a spindle torus and here the meridian curve has a greater radius of curvature than the equatorial curve (Fig 13). In the spindle torus, the normals from the meridian curve, which are directed toward its center of curvature, intersect the axis of revolution at progressively shorter distances, with increasing distances of the normals from the equatorial curve. Each normal from the meridian curve to the axis of revolution represents a sagittal or transmeridian radius of curvature, and, as shown in Figure 13, the transmeridian radius of curvature progressively decreases while the radius of curvature of the meridian curve is constant. When the difference in the two principal curvatures at the equator is small, the difference in meridian and transmeridian curvatures remains practically constant for a relatively large distance from the equator. When the difference in the two principal curvatures at the equator is large, the difference in meridian and transmeridian curvatures becomes significantly different from that at the equator within a relatively short distance from the equator.

If the center of curvature of the meridian curve lies between the axis of revolution and the meridian curve, the surface generated is a barrel torus and here the transmeridian curvature decreases with distance from the equator (Fig 14). Just as described above for the spindle torus, the rate at which the difference between the meridian and transmeridian curvatures becomes significant with distance from the equator in the barrel torus is proportional to the difference in the two principal curvatures at the equator.

Toric spectacle lenses can completely neutralize the effect of ocular astigmatism if it is of a regular type in which there are two distinct principal refractive powers of the eye for central vision. As shown above, toric surfaces are not of constant curvature but vary along a meridian. It is important that the design of toric lenses take into account the geometry of toric surfaces if aberrations of central vision through the periphery of the lenses are to be minimized by bending of the lenses.

1. Volk D: Conoid lenses in legal blindness. Am J Ophthalmol 56: 195, 1963

2. Volk D: Conoid cataract lenses for the correction of aphakia. Am J Ophthalmol 51:615, 1961

3. Volk D: Correction of aphakia with conoid aspherical cataract spectacle lenses. Eye Ear Nose Throat Digest 25:25, 1963

4. Sudarsky RD, Volk D: Aspherical objective lenses as an aid in indirect ophthalmoscopy: A preliminary report. Am J Ophthalmol 47:572, 1959