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Introduction

Focal PowerLens Form

BaseCurve Selection

Mechanical Factors of Lens

Form

Optical Factors of Lens Form

Tangential and Sagittal Errors

Oblique Astigmatism

Power Error

Other Lens AberrationsBest Form Lens Design

Vis ion and Lens Design

Asphericity

Aspheric Lens Design

Atoric Lens Design

Return to CE Courses

Ophthalm ic Lens Des ign

By Darry l Meis ter

In t roduc t ion

This course will present the fundamental principles of ophthalmic lens design, including a r

of lens aberrations, corrected curve theory, and asphericity. This is a technical, intermedi

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

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

Focal Pow er

The ability of a lens to refract and focus lightby either converging or diverging itis refer

as its foca l power or re f rac t ive power . The focal power of a lens is simply equal to t

net effect of its front and back surfaces. When a refractionist writes a prescription for an

ophthalmic lens, he/she is specifying the focal power of the lens. The focal power of a lens

diopters, is given by:*

Focal Pow er = Front Surface Power + Back Surface Pow er

Or:

P = F + B

Where (P) is focal power in diopters, (F) is the front surface power in diopters, and (B) is th

back surface power in diopters. Both focal power and surface power are measured in units

called diopters (abbreviated 'D').

For example, consider a lens with a 6.00 D front curve and a -4.00 D back curve . The fo

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power (P) is equal to F + B = 6.00 + (-4.00) = +2.00 D.

* Th is is an approximation to k eep our math s imp le ; the fo rmu la become

a b i t m ore complex w hen lens th ick ness is considered .

Lens Form

The relationship between the front and back surface curves of a lens is referred to as the l

fo rm (or lens prof i le ). A lens with a given focal power c an be produced by many differ

lens forms, as long as the sumof the front and back surface powers remains constantor

least nearly so (neglecting thickness). A given lens power c an be produced with an almos

endless variety of lens forms, as long as the sum of the front and back surface powers rem

equal to the desired focal power.

Historically, spectacle lenses fall into either one of two general categories of lens form:

q Bent lenses: Modern lens form are generally bent, or meniscuswhich means

"crescent-shaped." Bent lenses use convexfront curves and concaveback curves.

q Flat lenses: The earliest lens forms were flat. For pluslenses, flat lenses use either

convexcurves for both the front and back (i.e., bi-convex lenses), or one convexcu

and one plano(plano) curve (i.e., plano-convex lenses). For minuslenses, flat len

use either concavecurves for both the front and back (i.e., bi-concave lenses), or o

concavecurve and one plano(plano) curve (i.e., plano-concave lenses).



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Base Curve Select ion

The form of a given lens is determined by "base curve selection." The base curve of a

is the surface curve that serves as the basis or starting point from which the remaining cu

will be calculated. For semi-f in ished lens blanks, the base curve will be the factory-fini

curve , which is generally located on the frontof the blank. The surfacing laboratory is

ultimately responsible for choosing the appropriate base curve for a given prescription (o

power) before surfacing the lens. For f in ished lens blanks, which have already been fabr

to the desired power, the curves are chosen beforehand by the manufacturer.

Manufacturers typically produce a series of semi-finished lens blanks, each with its own ba

curve . This "base curve series" is a system of lens blanks that increases incrementally i

surface power (e.g., +0.50 D, +2.00 D, +4.00 D, and so on). Each base curve in the serie

used for producing a small range of prescriptions, as specified by the manufacturer.

Consequently, the more base curves available in the series, the broader the prescription ra

of the product. Manufacturers make base curve selection charts available that provide the

recommended prescription ranges for each base curve in the series.

A Typica l Base Curve Select ion Chart

Pow er Range Base Curve

+8.00 D to +4.75 D 10.00 D

+2.25 D to +4.50 D 8.00 D

+2.00 D to -2.00 D 6.00 D

-2.25 D to -4.00 D 4.00 D

-4.25 D to -7.00 D 2.50 D



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-7.25 D to -12.00 D 0.50 D

The base curve of a lens may affect certain aspects of v is ion , such as distortion and

magnification, and wearers may notice perceptual differences between lenses with differen

base curves. Consequently, some practitioners may specify "match base curves" on a new

prescription. Some feel that these perceptual differences should be minimized by employin

same base curves when the wearer obtains new eyewear. This would conceivably make it

easier for particularly sensitive wearers to "adapt" to their new eyewear.

However , changes in the spectacle prescription will also create unavoidable perceptual

differences. Moreover, the wearer will generally adjust to these perceptual differences with

week or so. If the same base curve is continually used as the wearer's prescription chang

which might necessitate a change in the manufacturer's recommended base curve , the

peripheral optical performance of the lens may suffer as a consequence. When duplicating

lenses of the same lens material, design, and power, matching base curves should not pos

problemand is a recommended practice. Otherwise, unless the wearer has shown a prev

sensitivity to base curve changes, you should use the manufacturer's recommended base

curve when changing the prescription, or when using different lens materials and/or desig

There are some exceptions to this rule, though they are rare. Some wearers with particular

long eyelashes may have been given steeper base curves at some point in order to preven

their lashes from rubbing against the back lens surface when their ver tex d is tanceor

distance between the lens and the eyeis small, though this practice is very uncommon.

Additionally, some wearers with a significant difference in prescription between the right an

eyes may suffer from anise ikonia , or unequal retinal image sizes, and require unusual b

curve combinations in order to minimize the magnification disparity produced by the diffe

in lens powers. In these situations, a discussion with the prescriber may be in order before

changing base curves.

Since the power of a lens c an be produced by an almost infinite range of lens forms, why

choose one base curve over another? There are two principal factors that influence the

selection of base curves (and their resulting lens forms):

q Mechan ica l fac t o rs

q Opt ica l fac to rs

Mechanica l Fact ors o f Lens Form



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The maximum thickness of a lens, for a given prescription, varies with the form of a lens. F

lens forms are slightly thinnerthan steeper lens forms, and vice versa. Since the lenses ar

thinner, they also have less massmaking them lighter in weight as well. In addition to len

thickness, varying the lens form will also produce significant differences in the plate he ig

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

lens as measured from a flat plane.

Pluslenses with f la t te r plate heights do not fall out of frames as easily, which is especial

important with large or exotic frame shapes. In addition, f la t te r plate heights are also mo

cosmetically pleasing than steeper, bulbous onesparticularly in plus powers.

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

associated with pluslenses. Since a f la t te r plate height brings the back surface closer to

eye, the minificationassociated with minuslenses is also reduced slightly. This gives the

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

We c an evaluate the maximum thickness, plate height, and weight for a range of lens form

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

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

mm minimum edge thickness.

+4.00 D Lenses

Base Curve Cent er Plat e 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



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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, f la t te r , and lighter in weight as the base

curve is reducedor flattened. The table, below, represents a range of -4.00 D lenses in

resin, edged to a 70-mm diameter and a 2-mm minimum center thickness.

-4.00 D Lense s

Base Curve Edge Plat e Weight

6.00 D Base 8.7 mm 16.4 mm 25.4 g

4.00 D Base 7.8 mm 12.8 mm 24.0 g

2.00 D Base 7.3 mm 9.7 mm 23.2 g

0.00 D Base 7.0 mm 7.0 mm 22.8 g

Again, the lenses become gradually thinner, f la t te r , and lighter in weight as the base cur

reduced. In summary, f la t te r lens forms provide the following mechanical and cosmetic

benefits:

q Flat ter ( less "bu lge")

q Thinner center t h ick ness (p lus) or edge th ick ness (minus)

q L igh te r in w e igh t

q Less magni f icat ion (or min i f icat ion)

q Bet te r f rame re ten t ion ( in p lus powers )

Optica l Fact ors o f Lens Form

We've just discussed the obvious mechanical and cosmetic advantages of f la t te r lens fo

(with their f la t te r plate heights). However , the principal impetus behind lens form select

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

turns out that the form of a lens will have a significant impact on the clarity of peripheral v i

experienced by the wearer. Although v is ion through the centerof a lens will be relatively

no matter what the form, v is ion through the peripheryof a lens will vary greatly as a funct

lens form.

Peripheral v is ion generally requires the wearer to look away from the optical centerof the

As a result, the wearer's line of sightmakes an angle to opt ica l ax is of the lens, which is

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

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

and dynamic v is ion , the line of sight makes an angle to the optical axis of up to 30 or mo



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the wearer observes objects in the visual field.

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

optical center, within an area referred to as the parax ia l reg ion , since incident rays of li

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

of the incident light rays, allowing us to simplify Snell's law of refractionusing a mathemati

simplification known as a f i rs t -order approx imat ion . Light rays refracted through the

paraxial region will form a sharp pointfocus at the desired focal point of the lens and ultima

upon the retinaof 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

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

desired focal point of the lens, as described by our simple focal power formula. This error i

focus is referred to as a lens aberrat ion .

Lens aberrations act as errors in power from the desired prescription, and c an degrade th

image quality produced by the lens as the wearer gazes away fromor obliquely toits op

axis. There are six different lens aberrations that c an affect the quality of peripheral v is io

through a spectacle lens:

q Obl ique Ast igmat ism

q Power Error

q Spher ica l Aberrat ion

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q Coma

q Distor t ion

q Chromat ic Aberrat ion

The first five lens aberrations are referred to as the monochromat ic aber ra t ions, sin

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

Ludwig Von Seidel first derived equations for assessing these aberrations using a th ird-o

approx imat ion (which is more accurate than the first-order approximation). We will

concentrate mainly on oblique astigmatismand power error, which are the two primarylen

aberrations that must be reduced or eliminated when designing ophthalmic lenses.

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

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

You c an also think of a lens aberrat ion as the failure of a lens, which has otherwise be

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

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

produce a focus at the far-pointof the eye. The far-point (FP) of the eye is conjugateto t

retina, meaning that rays of light from a lens that come to a focus at the far-point will also b

brought to a focus at the retina once refracted by the eye. Hence, the far-point represents t

ideal focal plane of the spectacle lens.

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

a fixed distance from its cen te r o f ro ta t ion (C). This movement describes an imaginary

spherical surface, known as the far-po in t sphere, which represents the ideal locus of fo

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.

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Tangent ia l and Sagi t ta l Errors

In our discussion of lens aberrations, we will often refer to the tangentialand sagittal errors

the desired power. The tangen t ia l plane of the lens represents the meridian of the lens t

radiates out from the optical center; these planes are analogous to the spokes of a bicycle

wheel. The sag i t ta l plane of the lens represents the meridian of the lens that is perpendic

to the tangential plane (i.e., at a 90 angle to it) at any point; these planes circumscribe the

optical center.

When rays of light from an object strike the lens obliquely, the principal refraction of these

occurs through the tangential and sagittal meridians, much like the principal refraction of a

sphero-cylindricallens occurs through its principal power meridians. The tangent ia l err



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the error from the desired focus through the tangentialmeridian of the lens as a result of le

aberrations. The sag i t ta l e r ro r is the error from the desired focus through the sagittal

meridian of the lens.

Obl ique Ast igmat ism

Obl ique ast igmat ism is an aberration that results when rays of light from an object in t

periphery strike the lens obliquely, and are refracted differently by the tangential and sagitt

meridians of the lens. When a lens suffers from oblique astigmatism, the tangential meridia

of the lens refracts incident light more than the sagittal meridian (S) perpendicular to it.

Consequently, incident light from an off-axis object point is brought to a focus at two diffe

locations (i.e., the tangential focus and the sagittal focus). The image of the object point is

longer focused to a single point, but rather separated into two focal linesinstead.



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Note that two focal lines are produced from each single object point by the tangential and

sagittal meridians of the lens, instead of a single point focus. The dioptric difference betwe

these two focal lines is known as the astigmatic errorof the lens. Oblique astigmatism is si

to the normal ocular astigmatism for which refractionists prescribe cylinder power. Howev

oblique astigmatism only occurs when the wearer looks through the lens at an angle (henc

"oblique") or through the peripheral regions of the lens. Since it is an astigmatic focusing e

this error is similar in effect to unwanted cylinder powerin a prescription.

When light is incident upon a surface at an angle, it produces such an astigmatic focus. Sin

light is refracted by two surfaces as it passes through a lens, the total oblique astigmatism

produced by the lens depends upon the net astigmatism produced at each surface.

Consequently, oblique astigmatism is dependent upon the formof the lensthat is, the

relationship between the front and back curves. Certain lens forms will produce more obliq

astigmatism than others.

In terms of our tangential and sagittal errors, the astigmatic error is given by:

Ast igm at ic Error = Tangent ia l Error - Sagi t ta l Error

In addition to the oblique astigmatism that occurs while viewing off-axis objects in the

periphery, you c an introduce oblique astigmatism by simply tiltinga lens, since this also p

the line of sight at a significant angle to the optical axis of the lens. This is sometimes refer

as "astigmatism due to lens tilt." The oblique astigmatism induced by lens tilt c an be minim

by ensuring that the optical axis of the lens passes through the center of rotation (C) of the

We c an accomplish this by manipulating the relationship between the pantoscop ic t i l t



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which is lens tilt toward the cheeksand the height (H) of the wearer's pupil center above

optical center (OC) of the lens according to the following rule-of-thumb:

Ensure 1 m m of Opt ica l Center Drop (H) for Every 2 o f Pantoscopic T i

Power Error

In the absence of oblique astigmatism, a spectacle lens brings light to a focus across a cu

image plane referred to as the Petzval sur fac e . Curvature of t he f ie ld is an aberra

that results from the difference in focus between a flatfocal plane and the curvedcollection

actual focal points on the Petzval surface. This aberration is a concern for optical devices t

require a flat image plane, such as cameras. However , recall that the ideal image plane

eye, the far-point sphere, is also curved. Unfortunately, the Petzval surface is generally f la

than the far-point sphere. Power error is an aberration that results from the difference in

focus between the Petzval surface (PS) and the far-point sphere (FPS) of the eye.



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Power error is a result of the fact that the focal plane of the lens for off-axis object points

departs from the far-point sphere of the eye, even when the lens is free from oblique

astigmatism. In the presence of power error, light from an object point may be refracted to

single point focus by the tangential (T) and sagittal (S) meridians, but this point focus does

lie on the far-point sphere (FPS). The dioptric difference between the actual focal point of t

lens and its desired focal point is the power errorof the lens. Power error is a spherical-like

focusing error, and is similar in effect to unwanted sphere powerin a prescription. This is in

contrast to the astigmatic errorproduced by oblique astigmatism.

In the presence of oblique astigmatism, there is no singlefocal point but rather twofocal lin

In this case, the power error is equal to the averagedioptric difference between the two

astigmatic focal lines and the desired focal point of the lens, just as the spher ica l



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equiva lent is equal to the average power of a sphero-cylindrical prescription.

In terms of our tangential and sagittal errors, the power error is given by:

Pow er Error = (Tangent ial Error + Sagit t al Error) 2

For example, consider a +4.00 D lens that produces a power of +5.00 D through the tange

meridian and a power of +4.50 D through the sagittal meridian at some distance from the o

center. This represents a tangential error of 5.00 - 4.00 = +1.00 D and a sagittal error of 4.5

4.00 = +0.50 D. The astigmatic erroris equal to 1.00 - 0.50 = 0.50 D, while the power error

equal to (1.00 + 0.50) / 2 = +0.75 D.

Other Lens Aberrat ions

Spher ica l aberrat ion and c oma occur because the focal power of a lens effectively

increases away from its optical axis. As a result, rays of light refracted by the peripheral reg

of the lens are focused closer to the lens than light rays refracted through the central, para

region. Spherical aberrationaffects rays of light from objects situated near the optical axis,

comaaffects rays of light from objects away from the optical axis. The small pupi l apertu

the eye minimizes these aberrations by restricting the region of the lens that admits rays of

into the eye at any one time. Conseqently, these two aberrations are generally not a conce

ophthalmic lens designers.

Distor t ion does not affect the focal quality of an image, but rather its size and shapeor

geometric reproduction. Unlike oblique astigmatism and power error, distortion does not

produce a focal error that produces blur. Just as the focal powerof a lens effectively increa



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away from its optical axis, so does the magnification(or minification) produced by the lens

excess magnification (or minification) causes objects to appear curved or misshapen

particularly in higher powers. Since it cannot be eliminated using conventional base curve

ranges, distortion is usually not a consideration for ophthalmic lens design.

A lens free from distortion exhibits or thoscopy (or no distortion). The excess minification

the periphery of minuslenses generally produces barre l distortion, while the excess

magnification of pluslenses generally produces pin-cushion distortion.

Best Form Lens Design

The peripheral v is ion through a lens that suffers from these lens aberrations is blurred ,

the wearer experiences a limited field of clear v is ion . For conventional lenses, which utiliz

base curves with spherical surfaces instead of aspheric surfaces, base curve selection is

of the primary tools used to reduce these aberrations.

Recall that amount of oblique astigmatism produced by a lens depends on the relationship

between the front and back surfaces (or lens form). When the curves of lens are chosen in

attempt to produce a lens form with a minimum of lens aberrations, the resulting lens is oft

referred to as a cor rec ted curve lens design. (This term refers to the fact that the base

curvehas been chosen in order to correct certain aberrations.) Since properly designed

corrected curve lenses will have the least amount of the most detrimental aberrations, the

also called best form lens designs.

In 1804, when W. Wollaston experimented with different lens forms to improve peripheral

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v is ion . He created a series of lens forms that neutralized the astigmatism produced at the

surface of the lens using the astigmatism produced at the back surface. Essentially, the for

was chosen such that the astigmatism produced at the two surfaces canceled each other o

He referred to his lens experimental series as "periscopic," meaning "look around." Howe

his lens forms turned out to be too steep for practical production as spectacle lenses. In 18

F. Ostwalt developed another system of lenses free from oblique astigmatism, which were

f la t te r than Wollaston's.

In 1904, M. Tscherning demonstrated mathematically that there were in fact tworecomme

or "best form" spherical best curves for each lens power: a steeperseries and a flatterser

Wollaston'slenses had been based upon the steepersolutions to Tscherning's formula, w

Ostwalt'shad been based upon the flattersolutions. Tschern ing 's e l l ipse is the locus

points that plot out the two recommended front curves for each lens focal power. The f la t t

Ostwalt branch of the ellipse serves as the basis for modern best form lenses.

Tshcerning's ellipse indicates the optimum spherical base (or front) curve to use for each

power in order to reduce or eliminate lens aberrations. For instance, the recommended bes

form base curve for a lens with +2.00 D of focal power is roughly 8.00 D. After examining

Tscherning's ellipse, we c an draw a few conclusions:

q Tscherning's ellipse recommends relatively steepbase (front) curves for many

prescriptions. Unfortunately, the "best form" base curves necessary to provide good

peripheral v is ion also produce relatively steep, thick lens forms. Flatter lens forms, w

thinner and lighter, generally produce significant lens aberrations, including large



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astigmatic and spherical power errors. Consequently, best form lens designs, while

optically superiorto f la t te r lens forms, are cosmetically inferiorto them.

q The focal power limit of the ellipse is around +7.50 D; above this power, lens aberratio

cannot be eliminated without using special, aspheric lens designs.

q According to Tscherning's ellipse, each individual lens power should be made using a

separate base curve . Early best form lenses utilized this approach, which required a

massive and costly inventory of lens blanks. As stated earlier, manufacturers now gro

small ranges of prescriptions together upon common lens blanks (that is, base curves

minimize inventory requirements and keep costs down.

This results in somewhat of an optical compromise, but the errors are usually negligible.

Nonetheless, the more base curves a given base curve series has, the more precisely the

aberrations c an be minimized.

It is generally notpossible to eliminate alllens aberrations completely. A lens that is entirel

of oblique astigmatism will generally have a small amount of residual power error remainin

and vice versa. It is up to the lens designers of each individual manufacturer to determine w

of the two aberrations (or combination thereof) they plan to reduce or eliminate.

Some manufacturers may choose to eliminate oblique astigmatism completely, some may

choose to eliminate power completely, and others may choose to eliminate or minimize a

combination of the two aberrations. Each of these separate approaches utilizes a slightly

different lens form and base curve for each focal power. Consequently, different manufac

may have slightly different base curve recommendations for their lenses.

http://www.opticampus.com/cecourse.php?url=lens_design/#aspherichttp://www.opticampus.com/cecourse.php?url=lens_design/#aspheric


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The recommended "best form" base curve for a given prescription c an be approximated

Vogel 's formulas . For plus(+) prescription powers, Vogel's formula is:

Base Curve = Sphere Rx + 6.00

For minus(-) prescription powers, Vogel's formula is:

Base Curve = Sphere Rx / 2 + 6.00

When the prescription also calls for cylinder, use the spher ica l equiva lentor Sphere

Cylinder / 2. For example, consider a prescription of -4.50 DS -1.00 DC 180. The spheric

equivalent would be -4.50 + (-1.00) / 2 = -5.00. And, since this is a minus prescription, the

recommended base curve would be -5.00 / 2 + 6.00 = 3.50.

Vis ion and Lens Design

Lens aberrations manifest themselves as departures from the desired prescription. For ins

the lens aberrations produced by "flattening" a lens form (i.e., using a base curve that is

f la t te r than recommended) increase the spherical focal power perceived by the wearer in

periphery of the lenses and induce unwanted cylinder power (astigmatism). The result is a

change in the effective power of the prescription away from the optical axis(or optical cent

the lens, leaving a "residual" refractive error.

The errors from the desired prescription produced by these lens aberrations result in blur

v is ion in the periphery and a restricted field of clear v is ion . Consequently, an imprudent

flattened lens design, while thinner and lighter in weight than a "best form" lens design,

produces inferior peripheral v is ion . The best form lens design, on the other hand, offers a



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wide, clear field of v is ion .

The prescription errors caused by lens aberrations will increase with the following factors:

q Distance f rom the opt ica l ax is /cent er : The farther the wearer looks into the

peripery of a lens, the greater the potential for lens aberrationsand the more rapidly

those aberrations will increase.

q Departure f rom best form des ign: The farther the lens form departs from the

recommended "best form," the greater the potential for lens aberrations.

q Strength of the prescr ip t ion: The stronger (plus or minus) the focal power of th

lens, the greater the potential for lens aberrations.

To summarize, the goal of best form lens design is to determine the most "optically approp

base curve for a given focal power (or range of focal powers). This means selecting a bas

curve that will produce a lens form free from the lens aberrations that c an blur v is ion th

the periphery of the lens. This process is referred to as lens des ign or opt imizat ion . W

the doctor prescribes a certain prescription, he/she is really specifying the focal power "on

The focal power "off-axis ," however , is ultimately controlled by the design of the finishe

lens.

Aspher ic i ty

Best form lenses, although optically superior to f la t te r lens forms, are somewhat steep, t

and heavy for many focal powersat least compared to the f la t te r lens forms. Does this

that providing good optics precludes the ability to provide good cosmesis? Fortunately, len



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designers have another tool at their disposal when designing lenses: asphericity.

Put simply, an aspher ic surface is a surface that departs from being perfectly spherical.

Aspheric base curves are surfaces that vary gradually in surface power from the center tow

the edge, in a radial fashion (meaning the asphericity is the same in every meridian of the

like the spokes of a bicycle wheel).

Unlike a sphericalsurface, which has the samecurvature in any direction across the entire

surface, a typical aspher ic surface becomes progressively flatter (or, in some cases, ste

awayfrom the center of the lensi.e., the tangen t ia l meridian of the lens. However , th

aspheric surface changes very little aroundthe circumference of the lens, which is the sag

meridian of the lens perpendicular to the tangential meridian.

This difference in surface curvature (and power) produces sur face as t igmat ism , whic

means that the surface literally produces cy l inder power away from its center. Furtherm

this surface astigmatismis used to counteract and neutralize the oblique astigmatismprod

by looking through the lens off-axis . Essentially, the difference in surfacepower on an

aspheric surface cancels out the difference in off-axis focalpower produced through the

by oblique astigmatism. An aspheric surface departs more and more from a spherical surfa

away from its center, just as oblique astigmatism would normally increase more and more

looking away from the center.

To produce a three-dimensional aspheric surface, an aspheric curve is rotated about an a

of symmet ry . As a result, aspheric surfaces are "rotationally-symmetrical." The central r

of an aspheric surface will be nearly spherical. Away from this central region, the amount o



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surface astigmatism smoothly increases towards the periphery of the lens.

In most cases the difference in surface heightor sag i t tabetween an aspheric curve

conventional spherical curve results in a thinner profile for aspheric lenses. This means th

lens using an aspheric surface will generally be thinner than a lens using a spherical surfac

the same power. Moreover, it is possible to exaggerate the asphericity of a surface in orde

maximize cosmesis, though this will generally result in excessive levels of unwanted

astigmatism. Some cataract lenses, such as the Welsh 4-Drop, employed such an approac

Many aspheric surfaces are made by rotating special curves, called con ic sections, aroun

their axis of symmetry. Conic sections include the parabola, the hyperbola, the oblate ellip

and the prolate ellipse. More generaland sophisticatedaspheric surfaces are described

polynomialequations of the form:

Z = A2X2 + A4X

4 + A6X6 .. .

where Z is the height of the surface at a distance X from its center and the coefficients A2,

A6 ... control the shape of the aspheric surface. This type of surface offers more flexibility to

lens designer than a simple conicoid surface. For instance, the designer may choose to

minimize power errors out to a certain point and then exaggerate asphericity to improve

cosmesis beyond that point.

Aspher ic Lens Des ign

Asphericityallows lens designers to flatten a lens form in order to improve cosmesis, witho

sacrificing opical performance. The lens aberrations produced by using flattened lens form



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simply eliminated using the surface astigmatism of the aspheric design. While aspheric len

do notprovide betterv is ion than best form lenses, they do provide equivalentv is ion in a

f la t te r , thinner, and lighter lens.

Aspheric lenses were originally employed to provide acceptable v is ion in high-plus, pos t

ca ta rac t lenses that exceeded the +7.50 D limit of Tscherning's ellipse. Today, aspheric

surfaces are mainly used to allow lens designers to produce f la t te r , thinner lenses with th

superior optical performance of the steeper corrected curve , or best form, lenses.

Aspheric lenses allow lens designers to produce lenses that are considerably f la t te r , thin

and lighter in weight than conventional best form lenses. It is interesting to note that asphe

surfaces produce thinner lenses for two reasons:

q Aspheric lenses generally use f la t te r front curves, which reduce the centerthickness

pluslenses and the edgethickness in minuslenses.

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

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

than what is optically required. This produces a thinner lens at the expense of reduced

optical performance.

As with the base curve of a best form lens, the amount or degree of asphericity will depen

upon the focal power of the lens. Additionally, the surface (that is, front or back) upon whic

asphericity has been applied will also make a difference:

q Plus lenses. If asphericity is applied to the frontsurface of a plus lens, the surface w

become flatteraway from the center. If it is applied to the backsurface, the surface w

become steeperaway from the center.

q Minus lenses. If asphericity is applied to the frontsurface of a minus lens, the surfa

will become steeperaway from the center. If it is applied to the backsurface, the surfa

will become flatteraway from the center.

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

however , small ranges of powers are grouped upon common aspheric base curvesjus

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

just about any base (front) curve for the chosen focal poweror range of powers. (Gener

f la t te r base curves are chosen for cosmesis.)



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This is unlike best form lens design, which requires a specific front curve that conforms to

Tscherning's ellipse in order to provide good v is ion for a given focal power. Consequently

before the appropriate aspheric design c an be determined, the lens designer must first de

upon the base curve value of the lens blank as well as its intended focal poweror range

focal powers.

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

lenses, which use simple spherical base curves. Lenses c an be made f la t te r , thinner, a

lighter, while maintaining the same excellent optical performance. In summary, aspheric le

q

Utilize a non-spherical surface with surface astigmatism to neutralize the oblique

astigmatism produced by off-center refraction

q Provide both the visualadvantages of best formlenses and the cosmeticadvantages

flatlenses

q Do notnecessarily provide betteroptical performance than best form lenses, but simp

provide comparable performance without the restrictions imposed by best form base

curve selection

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

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

best formlens design provides good peripheral optics (that is, very little oblique astigmatism

center), while the flattenedlens design (that is, made using a f la t te r base curve) provid

thinner, lighter, and f la t te r profile with poor optics. Finally, the asphericlens design provi

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



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Compariso n of Lens Designs for +4.00 D

Be st For m Fl at t ene d As ph er ic

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

Of course, in some cases it may be desirable to use base or front curves that are actually

steeperthan "best form" curves. For instance, "wrap" sunwear frames often necessitate hig

curved and steeply tilted lenses. Typically, non-standard base curves with a nominal front

curve of roughly 8.00 D are chosen for lenses glazed into these frames, resulting in

compromised optical performance for many prescriptions. In this case, asphericity c an be

applied to regain the optical performance, normally afforded by f la t te r "best form" lenses

using a steeper front curve .

Ator ic Lens Des ign

Tschernings ellipse demonstrates that each prescription power requires a unique front cu

or lens design in order to achieve optimal optics. A single, rotationally-symmetrical lens sur

cannot completely eliminate the power errors produced simultaneously by both principal

meridians of sphero-cylindrical lenses with cylinder power. Consequently, standard best fo

and aspheric lens designs with prescribed cylinder power represent an optical compromise

Lenses must therefore be designed to optimize peripheral optical performance for either th

sphere power meridian, the cylinder power meridian, or some power in between the two.

It is possible, however , to eliminate the power errors associated with both the sphere pow

and cylinder power meridians of sphero-cylindrical lenses using a non-rotationally-symmet

surface in which the asphericity varies from meridian to meridian. Ator ic surfaces have tw

planes of symmetry corresponding to the principal meridians of the lens, each with a uniqu

amount of asphericity to minimize the peripheral optical aberrations that would otherwise o

through that power meridian. Unlike the change in curvature away from the center of an

aspheric lens surface, which remains the same through every radial meridian of the lens, t

change in curvature away from the center of an atoric lens surface varies from meridian to

meridian.



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Since any atoricity is most often applied to the toric surface of the lens, atoricc an be thou

as a departure from a standard toric surface, just as asphericrefers to a departure from a

spherical surface. When the prescription calls for significant cylinder power, the field of clea

v is ion is often restricted through one or more planes of the lens, since a traditional base

curve or aspheric lens design c an only correct the optical aberrations associated with on

power meridian. Atoric lens designs, on the ohter hand, provide unrestricted fields of clear

v is ion , regardless of the power meridian of the lens, since the optical aberrations associa

with each power meridian are corrected individually.

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