High-Powered Lenses and Thickness

By Darryl Meister

Introduction

This course will present the fundamental principles of lens geometry, lens thickness, lens

design, and fitting for high-powered lenses, including a review of high-index lens materials,

surface geometry, and the optical implications of high-powered lenses. This is a technical,

intermediate level course intended for dispensing opticians, laboratory technicians, and

paraoptometric personnel. An understanding of both basic mathematics and basic optics is

required.

Refractive Index

Waves of light travel at a constant velocity of approximately 300,000 km/s in free space. In

other transparent media, including lens materials, waves of light will travel at a slower rate.

The velocity of light in other media will vary as a function of the refractive index for that

material. The refractive index of a transparent medium is essentially a measure of the

"optical resistance" of the material to light. It is defined as the ratio of the velocity of light

in air compared to the velocity of light in the material:

Refractive Index = Velocity in Air ÷ Velocity in Material

The refractive index of a material is often abbreviated 'n.' Except for air, which has a

refractive index of 1, the refractive index of most substances is greater than 1 (n > 1).

Radius of Curvature

Before we can fully understand the thickness of a lens, we need to understand the geometry

of a lens surface. A typical spherical lens surface is simply a section of a sphere. The

curvature of a spherical lens surface is dictated by its radius of curvature, which controls

how "steep" or "flat" the surface is. The larger the radius of curvature, the flatter the

curvature of the surface. Conversely, the shorter the radius of curvature, the steeper the

curvature of the surface. For instance, Earth, which has a relatively large radius of

curvature, appears quite flat to us as we walk across it. However, a boulder—which has has

a much shorter radius of curvature—would appear quite steep to us as we walked across its

surface.

The radius of curvature of a surface is inversely proportional to its surface power. This

means that, as the radius of curvature increases, the surface power decreases. Conversely,

as the radius of curvature decreases, the surface power increases.

This simply means that steeper surfaces produce stronger surface powers than flatter

surfaces. For instance, the radius of a 6.00 diopter surface is twice as long as the radius of a

12.00 diopter surface. The relationship between the power of a surface, in diopters, and its

radius of curvature, in meters, is given by:

Surface Power = (Refractive Index - 1) ÷ Radius

For example, consider a lens surface with an 83.2 mm (0.0832 m) radius of curvature,

ground onto hard resin, which has a refractive index of 1.499. The surface power is equal to

(1.499 - 1) ÷ 0.0832 = 6.00 D.

Surface Height

The height (or depth) of a convex or concave lens surface is referred to as the sagitta, or

simply sag, of that surface. Geometrically speaking, the sagitta (Latin for "arrow") of a

curve is the perpendicular distance from the vertex of the curve to some plane cutting

through the curve. Consequently, the sagitta of a surface is also known as the vertex depth.

The distance (or chord) from one side of the curve to the other across the plane that "cuts"

through the curve is associated with the diameter of the lens. For a spherical surface, the

sagitta of a lens surface is completely defined by the radius of curvature of the surface at a

given diameter (that is, the size of the lens blank). Further, the sagitta of a convex curve is

equivalent to the sagitta of a concave curve when the radii of curvature of both surfaces are

equal in magnitude.

Most eyecare professionals are undoubtedly aware of the fact that the sagitta of a lens

surface will vary with both the radius of curvature of the lens surface and its diameter. As

the radius of curvature of the surface shortens, the sagitta increases for a given diameter.

Since the power of the surface is inversely proportional to its radius of curvature, this

means that the height of the surface increases as the power of the surface increases. Further,

as the diameter—or size—of the lens blank increases, the height of the surface must also

increase.

The equation for calculating the exact sagitta (height) of a surface is derived in the

Appendix. However, we will use an approximation of this equation, which is simpler to

work with—yet still yields answers that are reasonably accurate in most cases. The

approximate sagitta of a lens surface, in millimeters, is given by the approximate sagitta

formula:

Sagitta = (½ Diameter)

2

2000 · Radius

Now that we have examined the relationship between the sagitta of a lens surface, its

diameter, and its radius of curvature—which, in turn, is related to its surface power—we

can develop a single mathematical relationship between them all. We will combine the

approximate formula for the sagitta of a lens surface with the formula for surface power,

which will obviate the need to calculate the radius of curvature directly. The approximate

sagitta of a lens surface, in millimeters, is therefore also given by:

Sagitta = (½ Diameter)

2 · Surface Power

2000 · (Index - 1)

Note that this equation will yield a negative (-) number for the sagitta of a concave

surface—which has a negative (-) surface power. This just serves as an indication that the

sagittal value describes a concave surface. From this equation, we can derive several useful

rules regarding the height of a lens surface:

Surface power: As the magnitude of the surface power increases, the sagitta also

increases; steeper surfaces have greater surface heights than flatter surfaces.

Refractive index: For a given surface power, as the refractive index the lens

material decreases, the sagitta increases.

Lens diameter: As the size of the lens increases, the sagitta increases; a change in

diameter affects the sagitta more rapidly than a comparable change in surface

power.

The effect of refractive index should not be overlooked. A lens material with a higher

refractive index will bend light more for the same amount of curvature. Therefore, a lens

material with a higher refractive index requires less curvature to produce the same surface

power. In turn, less curvature (i.e., longer radius of curvature) results in a smaller sagitta for

a given lens diameter. This means that high-index materials will produce shallower lens

surfaces for the amount of surface power.

For example, consider the previous lens surface with a 6.00 D curve, ground into hard resin

(index = 1.499), at a diameter of 60 mm. The approximate sagitta is equal to (½ Diameter)2

· Surface Power / [2000 · (Index - 1)] = (½ 60)2 · 6.00 / [2000 · (1.499 - 1)] = 5400 / 998 =

5.4 mm. In this example, the actual sagitta is 5.6 mm, which means that the approximation

is quite accurate—within 5% of the actual value. However, as the diameter increases or as

the radius of curvature decreases, this approximation will begin to lose accuracy.

Lens Geometry

A lens is little more than two surfaces in a lens material separated by a finite amount of

center or edge thickness. Note that "center" in these cases refers to the optical center of the

lens, which is the point of zero prismatic effect. The power of each surface contributes to

the total focal power of the lens. In fact, the total focal power of the lens, which is the

capacity of a lens to add either convergence or divergence to incident waves of light, is

simply the net effect of its two surfaces. The focal power of a lens, in diopters, is therefore

given very nearly by:

Focal Power = Front Surface Power + Back Surface Power

Modern ophthalmic lenses are generally meniscus—or "crescent-shaped"—in form. This

means that they typically have a convex front surface (i.e., positive power) and a concave

back surface (i.e., negative power). Because focal power is the net effect of both surfaces,

you can make three general statements concerning the relationship between the front and

back surfaces and the thickness of the lens:

Plus lenses: When the positive (convex) front surface is stronger than the negative

(concave) back surface, the lens is generally a plus lens. The sagitta (or height) of

the curve will be higher with stronger surface powers, which means that the convex

front surface will be the highest surface. Consequently, by necessity, a plus lens will

generally be thicker at the center and thinner at the edge.

Minus lenses: When the negative (concave) back surface is stronger than the

positive (convex) front surface, the net power of the two surfaces is negative and the

lens is generally a minus lens. This means that the concave back surface will be the

highest surface. Consequently, by necessity, a minus lens will generally be thicker

at the edge and thinner at the center.

Plano lenses: When the positive (convex) front surface is equal to the negative

(concave) back surface, the net power of the two surfaces is essentially zero and the

lens is usually a plano lens. Consequently, by necessity, a plano lens will generally

have roughly the same thickness at both the center and the edge.

Since the thinnest point on a plus lens occurs at the edge, the edge thickness of a plus lens

represents its minimum thickness. Conversely, the thickest point on a plus lens occurs at the

center, which represents its maximum thickness lens. The center thickness is given by:

Center Thickness = Edge Thickness + (Front Sagitta + Back Sagitta)

Recall that, consistent with our earlier sign convention, concave surfaces have negative (-)

sagittal values. Since the thinnest point on a minus lens occurs at the center, the center

thickness of a minus lens represents its minimum thickness. Conversely, the thinnest point

on a minus lens occurs at the edge, which represents its maximum thickness lens. The edge

thickness is given by:

Edge Thickness = Center Thickness - (Front Sagitta + Back Sagitta)

Note that the highest surface of a minus lens is concave, which has a negative (-) value by

convention, so the sum of the front and back sagittal values is also negative. However,

since this negative sum is actually subtracted from the center thickness in this equation, the

edge thickness value is still positive.

The previous two equations are used to find the maximum thickness of the lens based upon

both the minimum thickness and the sum of the front and back sagittal values. By

considering only the magnitude, or absolute value, of the sum of the front and back sagittal

values by ignoring the sign of the answer, we can express these two equations using a

single formula:

Maximum Thickness = Minimum Thickness + |Front Sagitta + Back Sagitta|

(The absolute value of a number, denoted by the || symbols, ignores the minus sign for

negative numbers.)

Keep in mind that minus lenses are never made to a "zero" center thickness, and plus lenses

are very seldom surfaced to a "zero"—or knife-edged—edge thickness. There is always

some minimum substance or thickness to the lens. Typical minimum thickness guidelines

for traditional ophthalmic lenses range from a minimum edge thickness of 1 mm to a

minimum center thickness of 2 mm.

The manufacturer's minimum thickness guidelines ensure that the lens will have enough

thickness to provide sufficient impact resistance for eye protection and sufficient flexural

stability while processing. Most minus lenses will be either surfaced to or supplied in

finished form with centers between 1.0 mm and 2.2 mm, depending upon the power of the

lens, type of lens material, and lens design. Certain lens materials, such as polycarbonate,

have a relatively high tensile strength and satisfy the FDA's impact-resistance requirement

even with thinner centers. Additionally, in some cases, the manufacturer may apply an

impact absorbing primer coating in order to allow for thinner centers. However, lenses

intended for rimless or safety frames may be supplied with a greater center thickness.

Calculating Lens Thickness

Recall that the maximum thickness represents the center thickness of plus lenses and the

edge thickness of minus lenses. Now that we have a mathematical description of the

maximum thickness of a lens, we can substitute our approximate sagitta formula for the

front and back sagittal values. The maximum thickness of a lens, in millimeters, is given

by:

Maximum Thickness = Minimum Thickness + |Front Sagitta + Back Sagitta|

Now, we can substitute our approximate sagitta formula for the front and back sagittal

terms. This allows us to express the equation in terms of surface powers, which gives us the

following expression for the quantity |Front Sagitta + Back Sagitta|:

(½ Diameter)

2 · Front Power

+ (½ Diameter)

2 · Back Power

2000 · (Index - 1) 2000 · (Index - 1)

Which, after rearranging, gives us the expression:

|Front Power + Back Power| · (½ Diameter)

2

2000 · (Index - 1)

And, since the sum of the front surface power and back surface power is equal to the total

focal power of the lens, we finally arrive at one simple equation for the maximum thickness

of both plus and minus lenses:

Maximum Thickness = Minimum Thickness + (½ Diameter)

2 · |Focal Power|

2000 · (Index - 1)

This provides us with a relatively simple equation for calculating the approximate

maximum thickness of a lens. Note that the absolute value of the focal power is used in this

equation, which means that the negative (-) sign for minus lens powers should be ignored.

Of course, the actual thickness of the lens will vary slightly, since this equation is based on

the approximate sagitta of each surface. When accuracy is critical, the exact sagitta formula

should be used to calculate thickness (see Appendix).

Essentially, we are treating the meniscus lens as a simpler, flat lens—with one flat or plano

surface and one concave or convex surface whose power represents the total focal power of

the lens. This treats a plus lens as a simple plano-convex with the front surface equal to the

power of the lens and a minus lens as a simple plano-concave lens. This approach has the

added advantage that neither the front nor back surface needs to be known, since only the

power of the lens is used in the calculation.

As with the approximate sagitta formula, this equation for lens thickness demonstrates

several important relationships between the thickness of a lens, the focal power of the lens,

the refractive index of the lens material, and the diameter of the lens blank:

Focal power: As the magnitude (plus or minus) of the focal power increases, the

maximum lens thickness also increases—and vice versa.

Refractive index: As the refractive index of the lens material decreases, the

maximum lens thickness increases—and vice versa.

Lens diameter: As the size of the lens blank increases, the maximum lens thickness

increases (rapidly)—and vice versa.

We can solve this approximate equation for various values of the diameter and the

refractive index in advance. A table of such constants, which we will call thickness factors,

can be prepared and kept readily available. These thickness factors represent millimeters of

thickness per diopter of power for a given lens size and refractive index. To approximate

the maximum thickness for a given refractive index and diameter, we can simply multiply

the appropriate thickness factor by the power of the lens, and then add any minimum

thickness:

Maximum Thickness = Minimum Thicknes + Thickness Factor × |Power|

Thickness Factors by Blank Diameter

Lens

Material

40

mm

45

mm

50

mm

55

mm

60

mm

65

mm

70

mm

75

mm

80

mm

Hard Resin 0.40 0.51 0.63 0.76 0.90 1.06 1.23 1.41 1.60

Crown Glass 0.38 0.48 0.60 0.72 0.86 1.01 1.17 1.34 1.53

Spectralite 0.37 0.47 0.58 0.70 0.83 0.98 1.13 1.30 1.48

Polycarbonate 0.34 0.43 0.53 0.65 0.77 0.90 1.05 1.20 1.37

1.60 Index 0.33 0.42 0.52 0.63 0.75 0.88 1.02 1.17 1.33

1.66 Index 0.30 0.38 0.47 0.57 0.68 0.80 0.93 1.07 1.21

1.70 Index 0.29 0.36 0.45 0.54 0.64 0.75 0.88 1.00 1.14

For example, consider a -4.00 D hard resin lens edged to a diameter of 60 mm with a center

thickness of 2 mm. Hard resin has a thickness factor of 0.90 mm per diopter at a diameter

of 60 mm. This gives us a maximum (edge) thickness of 0.90 × 4.00 = 3.6 mm. After

adding the minimum center thickness, we have a total edge thickness of 3.6 + 2.0 = 5.6

mm.

Minimum Blank Size

In order to determine the thickness of a lens we need to know at least the focal power of the

lens and the diameter of the lens blank. Of course, the power of the lens is a given, since it

is specified by the prescription. And, up to this point, we have assumed a lens diameter

(size). However, if the diameter is unknown, a few more computations may be necessary,

especially for minus lenses.

It is important to note that the center thickness of a finished lens is fixed with respect to the

initial diameter of the lens blank. Once cast by the lens manufacturer in the case of factory-

finished (or "stock") lenses, or processed to the desired power by the surfacing laboratory in

the case of semi-finished (or "surfaced") lens blanks, the center thickness of the lens is

permanently established. This has several important consequences for lens thickness:

Minus lenses—whose maximum thickness occurs at the edge—can be made

thinner by edging them to a reduced diameter. Consequently, the initial diameter of

a minus lens does not affect the final edge thickness of the finished (edged) lens

shape.

Plus lenses—whose maximum thickness occurs at the center—cannot be made

thinner by edging them to a reduced diameter. Consequently, when ordering

finished uncut or stock plus lenses, the smallest blank size should be utilized. When

ordering uncut lenses from a laboratory, providing them complete details about the

job will allow them to determine the smallest blank size necessary.

The smallest circle, centered at the geometric center (GC) of a frame, that completely

encircles the aperture (or opening) of the frame is known as the effective diameter (ED) of

the frame. The effective diameter of the frame is equal to twice its effective radius (ER),

which is the distance from the geometric center of the frame to the farthest point along the

perimeter—or eyewire—of the frame. The smallest possible lens diameter that will cut-out

for a given Rx job—including frame style and fitting measurements—is known as the

minimum blank size. In the case of single vision lenses, this represents the size of the lens

blank required to completely cover the aperture (opening) of the frame after the lens has

been decentered to the wearer's interpupillary distance (PD).

Decentration is the horizontal displacement of the optical center of the lens from the

geometric center of the frame in order to position the optical center in front of the wearer's

line of sight. If the lens requires no decentration, the minimum blank size of the lens is

equal to the effective diameter of the frame. Otherwise, the minimum blank size is equal to

twice the new effective radius of the decentered lens, which is the distance from the newly

positioned optical center of the lens to the farthest point along the perimeter (eyewire) of

the frame.

Determining the exact minimum blank size is difficult to do, since it requires calculating

the maximum distance from the optical center of the lens to the perimeter of the frame.

However, if we assume the "worst case scenario"—that is, if we assume that the

decentration occurs along the meridian of the frame containing the effective diameter—we

can arrive at a simple rule-of-thumb formula for determining the minimum blank size of a

decentered lens. This rule-of-thumb formula is known as the minimum blank size

formula:

Minimum Blank Size = Effective Diameter + 2 × Decentration

Because this formula assumes the "worst case," the minimum blank size of a single vision

lens will never be larger than the blank size predicted by this equation. Consequently,

using this formula will always err on the side of caution. Additionally, certain prescription

combinations—such as plus prescriptions with minus cylinder power—may necessitate

even smaller minimum blank sizes at certain cylinder axis angles, though this consideration

is beyond the scope of this article. If this formula is used to determine the minimum blank

size for edging purposes, an additional allowance of 1 to 2 mm should made in order to

account for the bevel.

Frame Shape and Blank Size

As described earlier, the minimum blank size for a given job will vary with the decentration

required. Since thickness increases rapidly with the minimum blank size, XXX. Therefore,

when lens thickness is a consideration, it is important to select a frame that minimizes

decentration as much as possible. If the decentration is not already known, it can be

calculated from the eyesize (or "A" measurement), bridge size (or "DBL"), and the wearer's

interpupillary distance (PD) using:

Decentration = (Eyesize + Bridge Size - PD) ÷ 2

For minus lenses

For plus lenses, While the maximum thickness , the thickness of a plus lens XXX.

COnseqeutnly, it .

The shape of the frame will also impact the minimum blank size needed for a given job.

Two frames with identical A (eyesize) and B (depth) measurements can have very different

effective diameters. Frames with exotic shapes—such as "harlequin" or "aviator" styles—

often require larger minimum blank sizes than round or oval frame styles. The meridian of

the effective diameter (or twice the distance from the center of the frame to the farthest

point along its perimeter) with exotic frame shapes often occurs, which XXX.

Cylinder Power and Thickness

When a prescription calls for cylinder power, the curvature and power of the lens varies

from meridian to meridian. For a lens made in minus cylinder form, which is usually the

case, this also means that the edge thickness of the uncut lens varies from meridian to

meridian. When the cylinder power of a prescription is significant, the contribution of

cylinder power to edge thickness should be considered, particularly for minus lenses.

The meridian associated with the axis of the prescription contains only the sphere power,

and there is zero cylinder power along this meridian. In the absence of prism, the thinnest

points along the edge of an uncut lens will occur at this meridian. Away from this meridian,

the contribution of the cylinder power to the total power through any meridian begins to

increase—in a sinusoidal fashion. At 90° from the axis of the prescription, the total power

contains the maximum cylinder power, so that the total power is equal to the sum of the

sphere and cylinder power. The thickest points along the edge of an uncut lens will occur at

this meridian.

Consequently, the edge thickness of a minus lens through any particular meridian is due to

both the sphere power and the contribution of the cylinder power through that particular

meridian. For a typical frame shape at least, the edge thickness of the lens generally reaches

its maximum along the horizontal (180°) meridian, since the distance from the optical

center to the perimeter of the frame (or the effective radius) is greatest near this meridian

after decentration. Therefore, we are typically interested in finding the contribution of

cylinder power along the horizontal meridian. Mathematically, the total power along the

horizontal meridian can be calculated using the sine-squared rule:

Horizontal Power = Sphere Power + Cylinder Power × sin2 Axis

For example, consider a prescription of -1.00 DS -1.50 DC × 045. The total power along

the horizontal meridian is equal to -1.00 + (-1.50) × sin2 45 = -1.75 D. When axis of the

prescription is close to 180, the power through the horizontal meridian is roughly equal to

the sphere power. When the axis is close to 90, the power through the horizontal meridian

is roughly equal to the sum of the sphere and cylinder power. A table of results appears

below.

Contribution of Cylinder Power Along the Horizontal (180°) Meridian

Axis 000 015 030 045 060 075 090 105 120 135 150 165 180

Cyl 0% 7% 25% 50% 75% 93% 100% 93% 75% 50% 25% 7% 0%

Lens Form and Thickness

Although it is not apparent from the use of our approximate lens thickness formula, the

maximum thickness of a lens—for a given prescription—actually varies with the form of a

lens. Since the sagitta (height) of a lens surface increases more rapidly than its surface

power, the maximum thickness of the lens for a given focal power increases as the surfaces

become steeper. Consequently, flatter lens forms—with shallower curves—are slightly

thinner than steeper lens forms. Since the lenses are thinner, they also have less mass—

making them lighter in weight as well.

In addition to lens thickness, varying the lens form will also produce significant differences

in the plate height, or overall bulge, between lenses of the same power. Essentially, plate

height is the height of a lens as measured from a flat plane. Plus lenses with flatter plate

heights do not fall out of frames as easily, which is especially important with large or exotic

frame shapes. In addition, flatter plate heights are also more cosmetically pleasing than

steeper, bulbous ones—particularly in plus powers.

A reduction in plate height will also provide a significant reduction in the magnification

associated with plus lenses. Since a flatter plate height brings the back surface closer to the

eye, the minification associated with minus lenses is also reduced slightly. This gives the

wearer's eyes a more natural appearance through the lenses.

We can evaluate the maximum thickness, plate height, and weight for a range of lens forms

to demonstrate the effects of lens form upon cosmesis for a given prescription. The table,

below, represents a range of +4.00 D lenses in hard resin plastic, edged to a 70-mm

diameter and a 1-mm minimum edge thickness.

+4.00 D Lenses

Base Curve Center Plate Weight

10.00 D Base 6.9 mm 15.3 mm 21.7 g

8.00 D Base 6.3 mm 11.7 mm 19.5 g

6.00 D Base 6.0 mm 8.7 mm 18.3 g

4.00 D Base 5.9 mm 6.0 mm 17.7 g

Note how the lenses become gradually thinner, flatter, and lighter in weight as the base

curve is reduced—or flattened.

Minimizing Lens Thickness

Use of small frame sizes

Decentation.

Refractive index. Use of high-index lens materials

Aspheric lens designs Use of aspheric lens designs

Use of high-index lens materials

Use of edge treatments and modifications

Because the maximum edge thickness of a minus lens is more obvious to the wearer (and

others) than the maximum center thickness of a plus lens, myopes (nearsighted wearers) are

often more concerned with lens thickness. Therefore, it is especially helpful to have a

strategy in place for estimating and minimizing edge thickness for minus lenses. Using the

principles presented earlier, there are four basic steps for estimating the approximate edge

thickness of a minus lens:

1. Determine the minimum blank size of the job using the effective diameter and

decentration measurements.

2. Determine the power of the lens through the horizontal (180°) meridian using either

a table or formula.

3. Determine the thickness of the lens for the chosen refractive index using either a

table of thickness factors or the approximate sag formula.

4. Determine the total thickness of the lens by adding the minimum center thickness to

the calculated edge thickness.

RULE OF THUMB THICKNESS REDUCTION BY INDEX

High-Index Materials

When dispensing lenses of moderate to high power, "high-index" lens materials are often

utilized to minimize lens thickness and weight. There are three properties of lens materials

that are especially relevant in the context of high-powered lenses:

Refractive index: Which is associated with the thickness of the lens.

Abbe value: Which is associated with the optics of the lens.

Density: Which is associated with the weight of the lens.

When the refractive index of a lens material is greater than the standard tooling index of

1.530, the material is often referred to as a high-index material. In some cases, however,

lens materials may be further categorized as follows:

Standard-index: Index < 1.53

Mid-index: 1.53 ≤ Index < 1.60

High-index: 1.60 ≤ Index < 1.66

Ultra-index: Index ≥ 1.66

The refractive index of a lens material is actually due to the refractivity of the individual

elements that constitute the molecules of the material. The "atomic refractivity" of different

elements varies significantly. The primary elements that form the molecules of polymer

plastics include carbon, hydrogen, and oxygen, which have relatively low refractivity.

Some elements, however, have a high refractivity, including sulfur and the various metals.

It therefore becomes possible to increase the overall refractive index of a lens material by

adding such elements to the molecules that make up the material. Sulfur and various

aromatic structures are commonly used to increase the refractive index of polymer plastics.

Common "high-index" plastic materials are often made from various forms of

polyurethane, which is a cross-linked thermosetting resin that generally has a good

combination of tensile strength and surface hardness. MR-7, a common high-index plastic

material with a refractive index of 1.66 to 1.67, has 30% sulfur content. MR-174, a newer

high-index plastic material with a refractive index of 1.73 to 1.74, has 60% sulfur content.

Unfortunately, there are limits on the amount of sulfur that can be added to the material.

Increasing the refractive index of a polymer plastic beyond 1.80 will most likely require the

addition of metal elements to the molecules of the material.

Several "mid-index" materials are also made from acrylic, which may be either

thermoplastic or thermosetting (cross-linked) and may also be combined with urethane.

Mid-index materials, which frequently have a refractive index between 1.53 and 1.56,

typically have a higher Abbe value than high-index materials. Another extremely popular

lens material is polycarbonate, which is a thermoplastic resin that demonstrates an

extremely high tensile strength for excellent impact impact resistance. Polycarbonate has a

refractive index of 1.59 (1.586) and a very low density (1.20).

Crown glass is an amorphous (non-crystalline) material made primarily of quartz sand

(silicon), soda, and lime. High-index glass materials are generally produced by adding

various metal oxides with a higher atomic refractivity to the composition of the glass,

including lead (used in early flint glass bifocal segments), titanium, and lanthanum.

However, the addition of these metals typically increases the density of the material.

Ophthalmic glass is currently available in refractive indices up to 1.90.

The higher the refractive index of a lens material, the slower the light will travel through it.

In reality, the refractive index of any material varies slightly as a function of the

wavelength (color). This means that various colors of light will each actually have a

slightly different refractive index in the same lens material! This phenomenon is

responsible for chromatic dispersion, or the breaking up of white light into its component

colors by prisms and lenses. Blue light, which has a higher refractive index than red light, is

therefore refracted—or bent—more than red light as it passes through a lens or prism.

The degree to which a given lens material will disperse light is described by a measure of

its refractive efficiency or, more commonly, its Abbe value (after Ernst Abbe). It is also

referred to as constringence. Lenses with a high Abbe value will disperse light less than

lenses with low Abbe values. In general, high-index materials produce lower Abbe values

than conventional plastic and crown glass lens materials, which makes these materials more

likely to produce symptoms of chromatic aberration. Furthermore, the higher the refractive

index of the material, the lower the Abbe value is likey to be.

The weight of a lens material is determined by its density, which is the mass of the material

per unit volume (generally measured in grams per cubic centimeter, or g/cc). The lower the

density, the lighter the material will be for a given volume. (For spectacle lenses, the

volume depends on the size and thickness of the lens.) Another term frequently used is

specific gravity, which is the ratio of the mass of a material or liquid compared to the mass

of an equal volume of water (at 4°C). When measured in g/cc, density is synonymous with

specific gravity since a gram is equal to one cubic centimeter of water. The properties of

some common lens materials are listed below.

Lens Material Properties

Lens Material Index Abbe Density

Hard Resin 1.499 58 1.32

Spectralite 1.537 47 1.21

Ormex 1.558 37 1.23

Polycarbonate 1.586 30 1.20

MR6 Plastic 1.597 36 1.34

MR7 Plastic 1.658 32 1.35

MR174 Plastic 1.732 33 1.47

Crown Glass 1.523 59 2.54

1.6 Glass 1.601 40 2.62

1.7 Glass 1.701 30 2.93

1.8 Glass 1.805 25 3.37

Aspheric Lens Designs

The use of special, non-spherical surfaces, referred to as aspheric surfaces, allows lens

designers to flatten a lens form in order to improve cosmesis, without sacrificing opical

performance. Flattening the form of a lens with a spherical base curve from traditional "best

form" recommendations will result in significant optical errors—or lens aberrations—in

the periphery of the lens, including oblique astigmatism. However, the lens aberrations

produced by using flattened lens forms are neutralized by the surface astigmatism of the

aspheric design.

While aspheric lenses do not necessarily provide better vision than traditional lenses in

lower powers, they do provide equivalent vision in a flatter, thinner, and lighter lens.

However, in high plus prescriptions—above roughly +8.00 D—aspheric lens designs can

provide considerably better vision than spherical base curves in any form.

Aspheric lenses allow lens designers to produce lenses that are considerably flatter, thinner,

and lighter in weight than conventional (i.e., "best form") lenses. Aspheric surfaces produce

thinner lenses for two reasons:

Aspheric lenses generally use flatter front curves, which reduce the center thickness

in plus lenses and the edge thickness in minus lenses.

The geometry of an aspheric surface also provides additional thickness reduction.

Some aspheric lenses are even designed solely for cosmesis, and actually use more

asphericity than what is optically required. This produces a thinner lens at the

expense of reduced optical performance.

Ideally, aspheric lenses should be optimized for each individual focal power. In practice,

however, small ranges of powers are grouped upon common aspheric base curves—just

like with best form lenses. Nevertheless, asphericity gives lens designers the freedom to

optimize just about any base (front) curve for the chosen focal power—or range of powers.

(Generally, flatter base curves are chosen for cosmesis.)

Aspheric base curves free lens designers from the constraints of conventional (best form)

lenses—which use simple spherical base curves. Lenses can be made flatter, thinner, and

lighter, while maintaining the same excellent optical performance. Moreover, aspheric

lenses are essential in high plus prescriptions, since traditional spherical lens designs cannot

eliminate lens aberrations in this prescription range.

The table, below, represents a comparison of lens designs for a +4.00 D prescription in hard

resin plastic, edged to a 70-mm diameter and a 1-mm minimum edge thickness. Note that

the best form lens design provides good peripheral optics (that is, very little oblique

astigmatism off-center), while the flattened lens design (that is, made using a flatter base

curve) provides a thinner, lighter, and flatter profile with poor optics. Finally, the aspheric

lens design provides both good optics and the thinnest, lightest, and flattest lens profile.

Comparison of Lens Designs for +4.00 D

Best

Form Flattened Aspheric

Front Curve 10.00 D 6.00 D 6.00 D

Center

Thickness 6.9 mm 5.9 mm 5.0 mm

Weight 21.7 g 17.7 g 14.6 g

Plate Height 15.3 mm 6.0 mm 5.1 mm

Obl.

Astigmatism 0.07 D 0.98 D 0.07 D

High-Powered Lenses

As the power of a spectacle lens increases, the optical and mechanical issues associated

with lens power also increase. For prescription powers stronger than ±4.00 diopters, the fit

and design of the lens become especialy critical due to the sensitivity of these higher

powers to changes in lens design or position. This means that great care must be taken

when fitting and dispensing high-powered lenses in order to ensure maximum optical

performance and visual comfort for the wearer. The following factors should be considered

when fitting and dispensing high-powered lenses lenses:

Magnification and field of view. The magnification produced by plus lenses

results in a relatively small field of view through the spectacle lens. This

magnification also results in the unappealing "bug-eye" effect visible to others.

Keeping the vertex distance of the lenses as short as possible and using a flatter,

aspheric lens design will minimize magnification and its related effects.

Lens reflections. The thick edges of high minus lenses produce internal reflections,

known as power rings, which are visible to others and exacerbate the apparent

thickness of the lens. An anti-reflection coating will eliminate these annoying

reflections and also improve the overall appearance of the lens.

Optical aberrations. High-powered lenses are subject to greater optical aberrations

in the periphery of the lens than low-powered lenses, including oblique astigmatism

and chromatic aberration. Therefore, proper base curve selection in critical in order

to ensure that the wearer enjoys a wide field of clear vision. Lens materials with

high Abbe values are also highly recommended.

Vertex distance. As the vertex distance—or distance from the back surface of the

lens to the cornea of the eye—changes, the power of the lens as perceived by the

wearer effectively changes as well. Increasing the vertex distance, for instance,

increases the effective power of a plus lens and decreases the effective power of a

minus lens. In some cases the refractionist may note a refracted vertex distance,

which is the vertex distance of the trial lenses used during the examination. If the

fitted vertex distance of the actual frame differs from the refracted vertex distance,

the ordered powers of the lens should be adjusted accordingly. OptiCampus has

available online a vertex distance compensation tool that will perform the necessary

calculations.

Lens tilt. Any excessive lens tilt, including pantoscopic tilt (that is, lens tilt toward

the cheek) and face-form tilt (that is, "wrap"), will also induce a form of oblique

astigmatism due to lens tilt. The oblique astigmatism induced by lens tilt can be

minimized by ensuring that the optical axis of the lens passes through the center of

rotation (C) of the eye. We can accomplish this by manipulating the relationship

between the pantoscopic tilt and the height (H) of the wearer's pupil center above

the optical center (OC) of the lens according to Martin's rule of tilt, which states

that you should Ensure 1 mm of Optical Center Drop (H) for Every 2° of

Pantoscopic Tilt. You can also compensate for the effect by changing the ordered

powers of the lens. OptiCampus has available online a lens tilt compensation tool

that will perform the necessary calculations.

Because of differences in magnification and thickness, plus-powered lenses and minus-

powered lenses each present unique optical and mechanical challenges. Special lens designs

have been devised to address the optical and mechanical requirements specific to high plus

lenses and to high minus lenses.

High Plus Lens Designs

Many dispensers have come to rely on high-index lens materials as the only option for

minimizing lens thickness. For low to moderate plus powers, center thickness and weight

can generally be controlled satisfactorily using high-index materials, aspheric lens designs,

and sensible frame styles. However, for higher plus powers, particularly those for post-

operative cataract patients with no crystalline lenses, more radical lens designs may be

necessary to control thickness and weight adequately. Indeed, for extremely high-powered

plus lenses, it may not be possible to fabricate a lens blank beyond 50 to 60 mm in diameter

using conventional designs.

Moreover, while high-index lens materials can afford some reduction in thickness, the

effects of chromatic aberration may become intolerable to the wearer for these

prescriptions. However, significant thickness reduction can be obtained by using alternative

plus lens designs. Common lens designs for high-powered plus lenses, beyond +8.00 or

+9.00 D—which are often referred to as cataract lenses, include:

Full field design: This is the most common single vision lens design, and is

generally what you would receive if you do not specify a "specialty" design. The

desired power is provided across the entire diameter of a full field design. You can

reduce the thickness of full field designs by using an aspheric lens design, though

this is less effective in extremely high plus powers. Above +8.00 D, traditional

spherical lens designs will not adequately eliminate lens aberrations in the

periphery; only aspheric full field lens designs should be used in this prescription

range.

Aspheric lenticular design: This is a specialty lens design that can provide

significant thickness reductions. The desired power is provided by a convex

"bowl"—or aperture—with a lenticular design. This bowl is often around 40 mm

in diameter, and protrudes from a much flatter carrier curve. The curvature of the

carrier is such that when combined with the range of back curves typically used for

the recommended prescription range, the carrier becomes nearly plano in power.

This near-plano carrier provides a very slim profile compared to the full field

design, since it retains the center thickness of a 40-mm diameter lens, regardless of

the actual blank size. However, the vision beyond the bowl is extremely blurred and

a scotoma exists at the junction between the bowl and the carrier. The bowl is also

very conspicuous to others.

Continuous surface and zonal aspheric designs: This is a category of lens designs

that provides the apparent full field of view of a full field design (without a visible

lenticular region), yet provides much of the weight and thickness reduction of a

lenticular design. A continuous surface design is similar to an aspheric lenticular

with the central optical bowl "blended" into the peripheral carrier using asphericity.

This results in a thin, lightweight design with the optical performance of an aspheric

lenticular. With a zonal aspheric design, asphericity is exaggerated across the

surface in order to thin the lens profile. However, this extreme asphericity (on the

order of 4.00 diopters or more of surface astigmatism) results in blurred vision in

the periphery.

Another optical issue characteristic of high-powered plus lenses is the annular blind area, or

scotoma, that is produced in the wearer's periphery. Because of the high magnification

produced by plus lenses, the field of view through the lens is relatively narrow.

Consequently, there is a blind area surrounding the periphery of the lens between the field

of view through the lens and the object field immediately visible outside the lens edge (or

bowl, in the case of lenticulars). This also results in a somewhat startling "jack-in-the-box"

effect for the wearer as objects pass through the scotoma and suddenly reappear on the

other side it.

Since the advent of intraocular lens implants, which are artificial crystalline lens implants,

high-powered "cataract" lens designs have become relatively scarce. However, several

cataract lens design options are still available through a handful of manufacturers.

High Minus Lens Designs

As with plus powers, for low to moderate minus powers, edge thickness and weight can

generally be controlled satisfactorily using high-index materials, aspheric lens designs, and

sensible frame styles. Again, for higher minus powers, more radical lens designs may be

necessary to control thickness and weight adequately. For instance, it may not even be

possible to close the temples completely on a frame with a pair of -10.00 D lenses because

of the excessive edge thickness. The wearer should be made aware of other options in these

cases, particularly if he or she has expressed concern over the thickness of previous

eyewear.

Further, as with high plus prescriptions, the effects of chromatic aberration must also be

considered. However, significant thickness reduction can again be obtained by using

alternative minus lens designs. Common lens designs for high-powered minus lenses

include:

Full field design: As with plus lenses, this is the most common single vision lens

design, and is generally what you would receive if you do not specify a "specialty"

design. The desired power is provided across the entire diameter of a full field

design. You can reduce the thickness of full field designs by using an aspheric lens

design, using a high-index material, rolling the edges, and so on. Unlike the optical

performance of plus lenses, lens aberrations in the periphery of most full field lens

designs are negligible for high minus prescriptions.

Myodisc design: This is a specialty lens design that can provide significant

thickness reductions. The desired power is provided within a concave "bowl"—or

aperture—with a myodisc design. The bowl is ground into a carrier, which is very

nearly plano in power. This near-plano carrier provides a very slim edge profile

compared to the full field design, and remains at a relatively constant edge thickness

regardless of blank size. However, the vision beyond the bowl is virtually unusable,

and an image jump occurs at the junction between the bowl and the carrier because

of the difference in prism. Further, the bowl is obvious to others. Generally, any

cylinder power is surfaced upon the front (cylinder power in the actual bowl would

make it elliptical in shape).

Minus lenticular design: This is a specialty lens design very similar in nature to

the myodisc design. As with a myodisc, the desired power is provided within the

bowl of a minus lenticular design. However, the bowl is ground into a convex

carrier, instead of a plano carrier, which provides an even slimmer edge profile than

the myodisc. The minus lenticular design also suffers from the same disadvantages

as the myodisc design, including an even more pronounced image jump between the

concave bowl and the convex carrier.

Aspheric lenticulars and other high plus lens designs employ special surfaces that must be

fabricated using specialized manufacturing equipment and molds. However, high minus

lens designs can be fabricated using traditional surfacing equipment, though this of course

involves a great deal of skill. The image, below, is a photo of a minus lenticular lens. Note

the difference in size between the letters that appear through the carrier region versus those

that appear through the central bowl region.

To overcome the cosmetic issues surrounding the bowl of these specialty designs, one

manufacturer (Younger Optical) introduced a blended lenticular design, referred to as a

Blended Myodisc. The edge of the bowl in this design has been smoothed and blended into

the carrier, making the demarcation between the carrier and the bowl virtually invisible.

The blended surface of these designs is also molded by the manufacturer, instead of being

surfaced by the laboratory, which simplifies the processing greatly. For extremely high

minus powers, the Blended Myodisc design should certainly be considered.

Lens Aberrations

We've just discussed the obvious mechanical and cosmetic advantages of flatter lens forms

(with their flatter plate heights). However, the principal impetus behind lens form selection

is optical performance. Base curves are typically chosen to provide a wide field of clear

vision. It turns out that the form of a lens will have a significant impact on the clarity of

peripheral vision experienced by the wearer. Although vision through the center of a lens

will be relatively sharp no matter what the form, vision through the periphery of a lens will

vary greatly as a function of lens form.

Peripheral vision generally requires the wearer to look away from the optical center of the

lens. As a result, the wearer's line of sight makes an angle to optical axis of the lens, which

is the imaginary line passing through the optical center. Consequently, we often refer to the

peripheral performance of a spectacle lens as its off-axis or off-center performance. During

peripheral and dynamic vision, the line of sight makes an angle to the optical axis of up to

30° or more as the wearer observes objects in the visual field.

The focal power formula, P = F + B, adequately describes the behavior of the lens near its

optical center, within an area referred to as the paraxial region, since incident rays of light

make very small angles to its optical axis. These small angles result in a well-behaved

refraction of the incident light rays, allowing us to simplify Snell's law of refraction using a

mathematical simplification known as a first-order approximation. Light rays refracted

through the paraxial region will form a sharp point focus at the desired focal point of the

lens and ultimately upon the retina of the eye.

However, away from the paraxial region, the incident rays of light make larger and larger

angles to the optical axis, and the first-order approximation no longer accurately describes

the refraction of light rays. Incident rays of light are no longer brought to a single point

focus at the desired focal point of the lens, as described by our simple focal power formula.

This error in focus is referred to as a lens aberration.

Lens aberrations act as errors in power from the desired prescription, and can degrade the

image quality produced by the lens as the wearer gazes away from—or obliquely to—its

optical axis. There are six different lens aberrations that can affect the quality of peripheral

vision through a spectacle lens:

Oblique Astigmatism

Power Error

Spherical Aberration

Coma

Distortion

Chromatic Aberration

The first five lens aberrations are referred to as the monochromatic aberrations, since

they occur independently of color. They are also referred to as the Seidel aberrations,

since Ludwig Von Seidel first derived equations for assessing these aberrations using a

third-order approximation (which is more accurate than the first-order approximation).

We will concentrate mainly on oblique astigmatism and power error, which are the two

primary lens aberrations that must be reduced or eliminated when designing ophthalmic

lenses.

The sixth lens aberration, chromatic aberration, is a consequence of the dispersive

properties of the actual lens material, and is not a function of lens design.

You can also think of a lens aberration as the failure of a lens, which has otherwise been

made correctly, to produce a sharp focus at the desired focal point of the lens as the eye

rotates behind it in order to view objects in the periphery. The focal power of the lens is

prescribed to produce a focus at the far-point of the eye. The far-point (FP) of the eye is

conjugate to the retina, meaning that rays of light from a lens that come to a focus at the

far-point will also be brought to a focus at the retina once refracted by the eye. Hence, the

far-point represents the ideal focal plane of the spectacle lens.

As the eye rotates vertically and horizontally behind the lens, the far-point moves with the

eye at a fixed distance from its center of rotation (C). This movement describes an

imaginary spherical surface, known as the far-point sphere, which represents the ideal

locus of focal points for the lens as the eye rotates to look through it. Lens aberrations result

when light refracted by a lens fails to come to a focus at the far-point sphere.

The higher the refractive index of a lens material, the slower the light will travel through it.

In reality, the refractive index of any material varies slightly as a function of the

wavelength. This means that various colors of light will each actually have a slightly

different refractive index in the same lens material! This phenomenon is responsible for

chromatic dispersion, or the breaking up of white light into its component colors by

prisms and lenses. Blue light, which has a higher refractive index than red light, is

refracted—or bent—more than red. Chromatic dispersion is a result of the fact that colors

of light with shorter wavelengths, like blue, travel more slowly through most transparent

materials than colors with longer wavelengths, like red. Therefore, blue light generally has

a higher refractive index than red light. However, this is usually only a concern for certain

high-index lens materials, since the differences in refractive index between colors is more

dramatic.

Appendix: Exact Sagitta Formula

The mathematics described above make frequent use of the approximate formula for the

sagitta (depth) of a lens surface. In reality, the sagitta of a curve actually increases slightly

faster than its surface power, particularly at larger diameters. When accuracy is critical,

such as for surfacing calculations, the exact sagitta formula should be applied to each

individual surface. The derivation of the exact sagitta formula appears below.

Note that this equation can be expressed (using the power ½ to represent square-roots) as:

s = r - [r2 - (½ d)

2]

1/2

Using the Binomial theorem, and eliminating the "higher order" terms whose contributions

are relatively small compared to the overall answer, gives us:

s = (½ d)

2

2 · r

Since the radius (r) is equal to 1000 · (n - 1) / F, where (F) is the surface power and (n) is

the refractive index of the material, we can substitute this expression into our equation to

arrive at the approximate sagitta formula:

s = (½ d)

2 · F

2000 · (n - 1)