Spectacle Lens Powers
GREGORY L. STEPHENS and JOHN K. DAVIS
Table Of Contents
MYOPIA, HYPEROPIA, AND ASTIGMATISM|
FOCAL POINTS AND FOCAL LENGTHS
BACK VERTEX POWER
THE LENS CLOCK
THE BASE CURVE
HOW LENSES ARE SUPPLIED
|The primary goal of a refractive examination is to discover the lens power
that will result in a sharp image of a distant object being formed
on a patient's retina when the patient is not accommodating. There
may be other requirements, such as a need for clear vision when reading
or when performing occupational tasks, but in this chapter we are
concerned only with distant vision. With the correcting lens in position, parallel
rays from a distant object entering the spectacle lens are
either converged or diverged to compensate for the patient's refractive
error. The patient then is able to see distant objects clearly
and can accommodate or use multifocal lenses to see objects at other
The word parallel is used in this chapter to describe a bundle of light rays coming from a single point on a distant object, at the point where the eye's line of sight intercepts the object. If an eye with a 7-mm diameter pupil looks at a road sign 100 m away, the bundle of rays that enter the eye from any single point on the sign have diverged only of a degree, so we call them parallel. In a refracting room, the rays from a single point on a letter on the visual acuity chart are not quite parallel. At 20 feet (6 m), the light rays have a divergent power of -0.16 diopters (D) as they enter the refractor. At 10 feet (3 m), the light rays diverge -0.33 D. Some doctors adjust for this factor when they determine the spectacle prescription.
|MYOPIA, HYPEROPIA, AND ASTIGMATISM|
|When an uncorrected myope looks at a distant object with accommodation
relaxed, parallel light rays entering the eye form an image in front of
the retina (Fig. 1). For the image to fall on the retina, the light entering the eye must
therefore be made divergent. There are two ways to accomplish this objective. The
object may be moved closer to the eye so that the object
itself becomes a source of divergent light. The object position at which
light from the object forms a focused image on the retina is the patient's far point (punctum remotum). Alternatively, if the object is to remain far away, at
optical infinity (approximately 6 m), a divergent (minus power) lens
may be placed in front of the eye to correct the refractive error. When
the lens has the proper power, parallel light from the object is
diverged so it appears to originate from the patient's far point. In
the terminology of geometric optics, the lens forms a virtual image
of the distant object at the patient's far point. This image becomes
the object for the eye, which converges the light to form the image
on the retina.|
An analogous situation occurs for the uncorrected hyperope (Fig. 2). When the patient views a distant object without accommodating, the image falls behind the retina. Placing an image on the retina requires convergent light, a situation that cannot be achieved by moving the object, but only with a lens in front of the cornea or with accommodation. A convergent (plus power) lens placed in front of the eye converges the parallel light rays. This convergent light incident on the cornea is further converged by the cornea and lens to form a retinal image. Unlike the situation of the myope, the far point in the hyperope is a point behind the retina. A convex lens or accommodation converges the light from the distant object toward the far point so that the final image is formed by the eye on the retina. Young hyperopes who can bring the image onto the retina by accommodating may not seek a refractive correction until they get older and begin to lose their accommodative ability or until prolonged near work causes fatigue because of the prolonged accommodation required.
When a patient with astigmatism views a distant object point without accommodating, the point is imaged as two lines oriented 90 degrees apart (Fig. 3). In essence, the astigmatic eye has two different far points. (The astigmatism may be combined with either hyperopia or myopia.) Correction of the patient's refractive error now requires a lens that forms two different images, one at each of the far points. Such a lens has two different powers and is termed a spherocylinder, or toric, lens to distinguish it from a spherical, or sphere, lens, which corrects only myopia or hyperopia. With the proper spherocylinder lens in position, the distant point object can be focused as a single point image on the retina.
|The power of a spectacle lens is obtained from a combination of the powers
of its front and back surfaces. Each surface of a lens may be either
convex, concave, or flat (plano or zero power), resulting in six general
lens shapes (Fig. 4). Spectacle lenses used to correct refractive errors are almost always
meniscus lenses, with both lens surfaces concave toward the eye. Plano-concave
and biconcave shapes are used only for high minus power lenses. Biconvex
and planoconvex shapes are not used for spectacle lenses
because of their poor off-axis optical quality when mounted in a spectacle
frame. However, with proper design and the use of aspheric surfaces, these
lenses are useful as, for example, magnifying lenses (low-vision
aids) and ophthalmoscopy lenses.|
|The use of a sign convention for measurement of distances simplifies the calculation of lens power. One commonly used convention is to specify distances relative to the direction of travel of light. In this chapter, light is always considered to be traveling from left to right. If a distance is measured in the same direction as the direction of travel of light, then the distance is positive. If it is measured in the opposite direction of light, the distance is negative.|
|FOCAL POINTS AND FOCAL LENGTHS|
|As illustrated in Figures 1 and 2, the lens that corrects a refractive error images parallel light rays
from a distant object. The point at which a lens brings parallel light
to a focus is the secondary focal point of the lens, and the distance
from the lens to the focal point is the focal length. The term secondary focal point is useful when discussing complex optical systems and instrument lenses. For
clinicians, there is nothing secondary about the focal point for
incident parallel light. It is the only focal point of interest. Some writers have called it the back focal point, and the distance from the back vertex or back surface of a lens to its
back focal point is the back vertex focal length, usually expressed by the symbol fV. The reciprocal of this distance in meters is the back vertex power of the lens, D, which is measured in units of diopters. Back vertex power
is the only power used by clinicians for spectacle lenses.|
The position of the back focal point determines the sign of the lens power. In Figure 1, the back focus of the correcting lens is to the left of the lens back surface. The back vertex focal length, measured from the lens back surface to the back focal point, is opposite the direction of travel of the incident parallel light, so both the back vertex focal length and the lens power are negative. That is why this type of lens is termed a negative or minus lens. The back focus of the correcting lens in Figure 2 is to the right of the lens, the lens power is positive, and the lens is a plus lens. A spherocylinder lens has two powers, and each power may be either plus or minus. If the two powers properly correct the patient's refractive error, there is a single sharp point image on the retina for each point on the object.
|When a refraction is performed on a patient, the refraction is correct
only for the position of the lens that is placed in front of the eye. The
position of the lens is specified by its vertex distance, the distance
from the back surface of the lens to the apex of the patient's
cornea. The patient's far point is located at the back focal point
of the lens, which is located at a distance from the back surface
of the correcting lens equal to the reciprocal of the lens power. When
a spectacle lens is prescribed, the lens must be positioned in the patient's
spectacle frame at the same vertex distance as for the refracting
lens if the back focal point of the prescribed lens is to fall
at the patient's far point. The effectiveness or effectivity of
the prescribed lens and the refraction will then be the same.|
If the spectacle lens prescribed for the patient is not positioned at the same vertex distance as that of the refracting lens, then the spectacle lens no longer corrects the patient's refractive error properly. The error in position of the back focal point relative to the patient's far point is equal to the error in vertex distance. A lens that has effectively the same power as that of the lens used in the refraction must be prescribed; that is, the prescribed lens must have a different power so that its back focal point once again coincides with the patient's far point. The new effective power or “effectivity” can be calculated as follows:
effective power = 1(fV + d)
where fV is the back vertex focal length of the refracting lens and d is the difference in vertex distance of the two lenses, with both distances measured in meters. If the spectacle lens is fitted forward of the position of the refractor lens (at a longer vertex distance than that of the refractor lens), then d is positive. If the spectacle lens is closer to the eye than the refractor lens, d is negative. Figure 5 provides an example. A refraction has been performed on a patient, and a power of -10.00 D is found with the refractor at a vertex distance of 10 mm. This lens has a back vertex focal length of -10 cm, so the patient's far point is 10 cm in front of the lens back surface. If the patient will be wearing spectacles that position the lens at a vertex distance of 15 mm, the distance, d, is + 5 mm, and the spectacle lens power needed is 1/(-0.10 + 0.005), or -10.53 D (-10.50 D when rounded to the nearest 0.25 D).
A simple rule of thumb can be used to determine whether a vertex distance change is sufficient to require a change in the power of the lens ordered for a patient. The change in effectivity, or the change in lens power needed, is approximately equal to:
effectivity change = d × D2
where d is measured in meters and D is the lens power in diopters. Table 1 shows the results of this calculation for a displacement value (d) of 5 mm. Effective power changes are not significant for lower-power lenses. It is only for higher-power lenses and larger changes in lens position that the clinician must take effectivity changes into account.
|As previously mentioned, the power of a spectacle lens is determined by
the powers of its front and back surfaces. The power of a lens surface
is given by:|
D = (n2- n1)r
where D is the surface power in diopters, n1 is the index of refraction of the medium containing the incident light, n2 is the index of the medium containing the refracted light, and r is the radius of curvature of the surface, specified as the distance from the surface to its center of curvature, measured in meters. A sign convention similar to that used for focal lengths must be used for radii, and an example is shown in Figure 6. With light traveling from left to right, the distances ÜobA1C1 and ÜobA2C2 are both positive. The index of the lens is always greater than 1, so the front surface power (D1) is positive and the back surface power (D2) is negative, the typical surface power relationship for most spectacle lenses.
|BACK VERTEX POWER|
|The power of a spectacle lens is not just the sum of its surface powers (D1 + D2). The clinician is interested in the power of a lens measured relative
to the position of the lens back surface, which requires that the effectivity
of the lens front surface be calculated at the lens back surface
before the surface powers are added. We can treat the front surface
as a thin plus-power lens displaced forward from the back surface by
the thickness between the surfaces. The back vertex power (D) of the lens, the power of the lens calculated at the lens back surface, then
D = D11 - tn D1 + D2
where D1 and D2 are the lens surface powers, t is the center thickness of the lens in meters, and n is the index of refraction of the lens material. The t/n term is analogous to the displacement d in the effectivity formula (equation 1). The displacement in an optical material is the distance divided by its index of refraction. (This is why the bottom of a clear lake appears to be closer than it actually is.) Equation 4 provides the power of a lens referred to the lens back surface, and the reciprocal of this power, measured in meters, is the back vertex focal length, the distance from the lens back surface to the back focal point.
Back vertex power has a number of advantages over other methods used to specify lens powers. Most importantly, back vertex power uses a reference point, the back surface of the lens, whose position can be easily specified using the vertex distance. Second, because back vertex power is specified relative to the position of the back surface of the lens, lenses of any shape (e.g., biconcave, meniscus) can be used to correct a given patient's refractive error, as long as they all have the same back vertex power and are positioned at the same vertex distance (Fig. 7). The back focal point for each lens falls at the same distance from the eye and coincides with the patient's far point, correcting the refractive error. Thus, a patient's refractive error can be measured with refractor lenses or trial lenses that are biconcave, biconvex, plano-convex, or planoconcave in shape, whereas spectacle lenses are prescribed that are meniscus in shape, with no worries about the effect of the shape of the lens on the adequacy of the refractive correction.
In some cases, it is convenient to ignore the effect of thickness on lens power (i.e., to consider the lens to be infinitely thin). Lens power then becomes the sum of the surface powers, and this power is termed the thin lens power or approximate power of a lens. Approximate or thin lens power is used in many of the examples that follow.
MEASURING BACK VERTEX POWER
The instrument used for measuring the back vertex power of a spectacle lens is the focimeter or Lensometer (Reichert Ophthalmic Instruments, a Division of Leica Microsystems, Buffalo, NY). Some focimeters are manually operated, and the operator focuses the instrument to obtain a sharp target image. Others are fully automatic and print out the measured prescription.
|THE LENS CLOCK|
|Lens surface powers are measured using a lens clock or lens measure (Fig. 8). When placed against a curved surface, the middle of the three lens clock
pegs moves up or down, measuring the sagittal depth of the curve
delineated by the two outer pegs. The radius of curvature of the surface
is directly related to the sagittal depth. Surface power then is calculated
using equation 3. Equation 3 requires that a value for the index
of refraction of the lens be chosen, and the value used for all lens
clocks is 1.53. Although no lens materials currently available have
an index of 1.53, lens clocks are useful in checking and identifying
lenses because in the optical industry, the 1.53 values are considered
|THE BASE CURVE|
|The clinician rarely specifies lens surface powers when writing a spectacle
prescription because optical laboratory personnel generally choose
the surface powers based on the recommendations of lens designers. However, in
certain cases, such as when a patient is having trouble adapting
to a prescription change or when magnification effects are of concern, the
practitioner may have to order a specific surface power or
curve as part of the prescription. By convention, the surface power ordered
is termed the base curve. The base curve is the reference curve that is the base or basis from
which all other surface powers are calculated, and for a single-vision
lens it usually is defined as the power of the lens front surface. This
base curve is a 1.53 curve, meaning that the base curve will be the
value read by the lens clock, not the true surface power. Once the base
curve of a lens and the lens power are specified, both of the lens
surface powers are completely defined. There can be only one way for the
optical laboratory to make the lens.|
The base curve of a multifocal lens is defined as the distance portion power or curve on the side of the lens containing the multifocal. Almost all multifocals are on the lens front surface, so the base curve is usually the distance portion power on the lens front surface.
|The surface powers of a spectacle lens are not generally specified in the
optical industry as the true or exact surface power. Instead, the surface
power is changed or compensated for the effects of two other variables. First, as
previously mentioned, lens manufacturers and optical
laboratories specify the surface powers of all lenses using equation 3, with
the index set to 1.53. This practice originated in the early
days of spectacle manufacturing, when the index of the glass then in use
was approximately 1.53. As time went by and newer materials of many
different indices came into use, it was still sensible to keep the same
tools for grinding surface powers.|
When using 1.53 curves to specify the power of a lens surface, laboratories and lens manufacturers are, in essence, specifying the surface by its radius of curvature. As an example, a lens made of polycarbonate plastic of index 1.586 with a 1.53 base curve of + 5.00 D has a front surface radius, calculated from equation 3 with n2 set equal to 1.53 and n1 set equal to 1, of + 0.106 m. The true front surface power, determined by substituting the true index of refraction into the same formula, along with the calculated lens radius of + 0.106 m, is + 5.53 D.
The practice of specifying exact 1.53 curves for the front curves of spectacle lenses means that manufacturers and optical laboratories can use one set of surfacing tools, standardized to an index of 1.53, to grind the surface curves for all lenses, regardless of lens index. Some manufactures use true power toric tools for the concave side of finished lenses so that cylinder values are precise.
A second compensation of lens surface powers was introduced because the thickness of a lens influences its back vertex power. Before computer use became common in optical laboratories, the 1.53 value of a front base curve was chosen so that a laboratory technician could select a lens blank and know what 1.53 curve to grind on the lens back surface, without a need for difficult calculations. For example, a surfacing chart provided by a lens manufacturer to an optical laboratory might state that a + 4.00 D lens should be made with a + 10.00 D base curve and a -6.00 D back curve. Such a simple calculation of lens surface powers should not work because the calculation seems to ignore both the effect of thickness on power and the use of 1.53 surface curves. However, the lens manufacturer adjusts, or compensates, the base curve of the lens blank from its labeled value so that the simple calculation works accurately. A typical lens thickness is assumed when making the compensation. The actual power of a lens front surface is usually less than its labeled value to compensate for the thickness of the lens.
Today, accurate 1.53 front curves are supplied by the lens manufacturer, and computers are used in the optical laboratory to calculate the thickness and back curves. However, even with computers, unavoidable small errors in lens power occur because the equipment used to grind power onto the lens back surface uses steps of 0.10 or 0.12 D.
|Spherocylinder, or toric, lenses, used to correct astigmatism, have two
different powers, one for each principal meridian. These lenses have
one spherical surface and one toroidal surface. A toroidal surface has
different radii of curvature or different powers in different meridians, with
the meridians of maximum and minimum power corresponding to the
principal meridians of the lens. The shape of a toroidal surface can
be thought of geometrically as the surface obtained by rotating an arc
of a circle about an axis (Fig. 9). Depending on the radius of the arc, the figure has somewhat the shape
of an automobile tire, a doughnut, or a barrel.|
The cross-sectional shapes of a spherocylinder lens in its principal meridians are different (Fig. 10). Light is brought to a separate focus by each principal meridian.
PRESCRIPTION NOTATION AND DESIGN OF SPHEROCYLINDER LENSES
When astigmatic corrections were first used, the corrective lenses were fabricated by cementing a second nonspherical lens onto one of spherical power. These cemented-on lenses were “slices” of a cylinder (Fig. 11). If used alone, a cylinder lens focuses light in only one meridian because the power is in only one principal meridian (the power meridian), the other having 0 or plano power (the axis meridian). The power meridian and cylinder axis are 90 degrees to each other. If a patient needed a + 3.00-D correction in the vertical direction and a + 5.00-D correction in the horizontal, a + 2.00-D cylinder with its axis at 90 degrees could be cemented onto the flat back surface of a + 3.00-D spherical lens, as shown in the upper drawing of Figure 12. Thus, the finished lens had a power of + 3.00 D in the vertical and + 5.00 D in the horizontal direction. It literally was a + 3.00-D sphere combined with a + 2.00-D cylinder. This was the origin of the custom of writing a prescription for astigmatic corrections as a sphere with a cylinder, in this case plus cylinder notation (e.g., + 3.00 sphere + 2.00 cylinder axis 090, abbreviated + 3.00 + 2.00 × 090). The same lens power could have been achieved with a + 5.00-D sphere and a -2.00-D cylinder with its axis horizontal at 0 degrees (see Fig. 12, lower drawing), or as more commonly stated, 180 degrees (+ 5.00 -2.00 × 180). Astigmatic corrections were originally prescribed and fabricated with both plus and minus cylinders. The diagrams of Figure 12 illustrate the concept of what came to be known as a spherocylinder lens.
The next logical step was to grind the spherical and cylindrical powers on one piece of glass instead of cementing two components together. At first, one surface had the entire sphere power and the other was a plano cylinder. As optical manufacturing technology and production methods improved and commonly used lenses became more steeply curved, toric surfaces (with both meridians curved, rather than one plano meridian and one curved meridian) became necessary and available. Currently, nearly all lenses that correct astigmatism have one spherical surface and one toric surface. The toric surface is the back surface for essentially all lenses, so this lens type is termed a minus cylinder design. A patient's lenses are made in minus cylinder design regardless of whether the lens prescription is written in plus-cylinder or minus-cylinder notation.
Figure 13 illustrates, using optical crosses, the surface powers used to make a minus cylinder design lens of power + 3.00 + 2.00 × 090, or + 5.00 -2.00 × 180. The lens has a base curve of + 7.75 D. Many possible combinations of surface powers exist that provide the proper lens power. The surface powers chosen by a lens designer are usually those that provide a lens with the sharpest image across the patient's field of view.
|HOW LENSES ARE SUPPLIED|
|A finished or edged spectacle lens that goes into a patient's frame
is made from a lens blank. Optical laboratories buy blanks from lens
manufacturers and then perform other operations such as surfacing (grinding
the proper power onto a surface of a lens), edging (cutting the
lens to the proper shape), tinting, and treating for impact resistance. The
following three types of lens blanks are in use.|
Glass lenses are formed by grinding and polishing a molded pressing or rough blank (Fig. 14A). A pressing is a rough, unpolished piece of glass having a meniscus shape, with the surface curves approximating those to be eventually ground on it. Its diameter is that intended for the finished lens. Pressings are usually formed immediately after a gob of molten glass is drawn from the tank where the raw ingredients have been combined. Each gob is pressed between the curved surfaces of a metal mold and allowed to cool. Alternatively, the glass can be drawn from the tank and rolled out into sheets that are cut into lens-sized squares, then shipped. In the molding room of a lens manufacturer, these squares can be reheated and formed into round meniscus-shaped pressings. Optical laboratories rarely use pressings.
A semifinished blank is a lens blank that has been finished on one surface (see Fig. 14B). Glass semifinished blanks are made from a pressing. The pressing is “blocked” or attached to a holding device so that the front surface can be ground and polished to the final power. If the lens is to be a multifocal, a segment or segments must be constructed on the lens front surface. The concave back surface is then given a short grinding and polishing to “clear” the surface. This practice improves the appearance, which is a competitive factor, and also facilitates inspection of the surface and the interior of the glass itself. After inspection, semifinished blanks are packaged and shipped from the lens manufacturer to local optical laboratories through a variety of distribution channels.
The need for a fixed orientation of the multifocal segment prevents rotation of a multifocal lens blank to obtain the proper cylinder axis. Therefore, multifocal lenses are almost universally supplied as semifinished blanks, and the back surface is surfaced by the optical laboratory to obtain the proper lens power and to position the optical center correctly relative to the multifocal segment. Some finished bifocal spheres are available. Single-vision semifinished blanks usually are used only for stronger-than-average lens powers that are not routinely available as uncut lenses.
Plastic semifinished blanks, which are usually multifocal lenses, are made by pouring (CR-39, PPG Industries, Pittsburgh, PA) or injecting (polycarbonate) the plastic into glass or metal molds, then cooling it at a controlled rate. Polycarbonate blanks are then coated for abrasion resistance.
Any one semifinished blank may be used for a variety of prescriptions, depending on what ocular (back) surface power is ground at the local laboratory. A wide range of choices are possible, depending on the thickness of the blank: the thicker the blank, the greater the latitude in curve selection. Prescription accuracy at the optical center and the control of off-axis lens aberrations restrict the choice. One of the objectives of lens design is to design a “set” of these semifinished blanks. This set or series of 10 to 20 lens blanks of different front surface powers and center thicknesses, used in conjunction with a chart of instructions, can provide for hundreds of well-designed but different prescription items. An index of the quality of a series of blanks is their number. Available series number from 4 to approximately 20.
FINISHED UNCUT LENSES
Finished lenses or finished “uncut” lenses have had both surfaces ground to the final optical quality (see Figs. 14C and D). The steps are the same as those described previously, except that thinner pressings or molds can be used because the lens power will not be further modified. The savings in lens material and the lower labor costs of mass production make this a less expensive route than semifinished blanks for popular single-vision prescription items. However, even in this procedure there are economic reasons for using a limited number of surface powers and molds. The uncut single-vision lenses provided by a lens manufacturer should be similar to those made by an optical laboratory from semifinished blanks of the same manufacturer. This provides consistent patient response.
The finished uncut lens must be edged or shaped to fit into a patient's frame. “Layout” for edging includes rotation of the lens to orient the cylinder axis properly and decentration of the lens optical center to position it properly for the patient's interpupillary distance (PD). Glass lenses are further processed after edging to increase impact resistance, whereas plastic lenses may be tinted or coated for abrasion resistance or to reduce reflections. The lenses are then inserted into the frame, and the accuracy of the prescription is verified.