Progressive Addition Lenses Ang - Essilor AcademyTHE PROGRESSIVE ADDITION LENS CONCEPT Basic design...

Progressive AdditionLenses

OPHTHALMICOPTICS FILES

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SUMMARY

INTRODUCTION o oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo 3

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THE PROGRESSIVE ADDITION LENS CONCEPT

Basic design differences between Progressive, Single Vision, Bifocaland Trifocal Lenses oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo 4-7

Advantages of progressive addition lenses1) Continuous field of clear vision oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo 82) Comfortable intermediate vision ooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo 83) Continuous support to the eye’s accommodation ooooooooooooooooooooooooooooooooooooooooooooo 94) Continuous perception of space oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo 9

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PHYSIOLOGICAL CONSIDERATIONS IN PROGRESSIVE LENS DESIGN

Foveal vision1) Accommodation, body and head postures and vertical eye movements ooooooo 102) Horizontal eye and head movements ooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo 113) Visual acuity ooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo 11

Extra-foveal vision1) Space and form perception ooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo 122) Perception of movement oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo 12

Binocular vision1) Corresponding retinal points ooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo 132) Similar images oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo 13

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DESIGNING PROGRESSIVE ADDITION LENSES

Modern conception of ophthalmic lens design1) The ophthalmic lens as an optical system ooooooooooooooooooooooooooooooooooooooooooooooooo 14-152) “Optimization” software and “Merit Function” ooooooooooooooooooooooooooooooooooooooooooooooooo 16

Designing progressive addition lenses1) Specific optical requirements of a PAL oooooooooooooooooooooooooooooooooooooooooooooooooooooo 17-192) Clinical studies and prototypes ooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo 19

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DESCRIPTION AND CONTROL OF PROGRESSIVE LENS DESIGNS

Optical description of a progressive lens1) Power profile oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo 202) Contour plot ooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo 203) Grid plot ooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo 214) Three dimensional plot oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo 21

Progressive lens design control1) In lens development oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo 222) In lens production oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo 22

SUPPLEMENT:MATHEMATICAL DESCRIPTION OF PROGRESSIVE SURFACES

Local mathematical description of surfaces oooooooooooooooooooooooooooooooooooooooooooooooooooooooo 23

Mathematical characterization of surfaces in a circular domain ooooooooooooooooooooo 23

Mathematical modelization of surfaces with B. splinespolynomial functions ooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo 24-25

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ENHANCING THE CHARACTERISTICSOF FINISHED PROGRESSIVE LENSES

“Equithin” progressive lenses ooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo 26

“Precalibration” ooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo 27

Pre-decentered uncut finished progressive lenses oooooooooooooooooooooooooooooooooooooooooooooooo 27C

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THE EVOLUTION OF PROGRESSIVE ADDITION LENSES

The evolution of progressive addition lenses ooooooooooooooooooooooooooooooooooooooooooooooooo 28-31

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INTRODUCTION

INTR

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UCT

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Since their introduction by Essilor in1959, progressive addition lenses (PALs)have gained worldwide acceptance as themost performant ophthalmic lenses forthe correction of presbyopia becausethey provide comfortable vision at alldistances.

They successfully and advantageouslyreplace single vision and bifocal lenses,presbyopic corrections that do not offersuch an advantage.

Experts project that, worldwide, about22 percent of all presbyopes will befitted with progressive lenses by 1994

and that this segment of the presbyopiacorrection market will continue to growby about 10 percent a year.

Since more and more progressivepractitioners systematically use PALswith most of their presbyopic patients,many presbyopes already enjoy thebenefits of progressive lenses, and manymore will do so in the future.

This volume of The Essilor OphthalmicOptics Files series reviews the basicphysiological and technical conceptsbehind progressive addition lenses.

PROGRESSIVE ADDITION LENSES

Fig. 1: Progressive addition lens.

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THEPROGRESSIVE ADDITION

LENS CONCEPT

Basic design differencesbetween Progressive, SingleVision, Bifocal and TrifocalLenses:

A single vision reading lens consists of asingle sphere of appropriate radius providingcorrection for near vision only (see Fig. 2a).Distance vision through the lens is blurred andthere is no specific correction for intermediatevision.

Bifocal, trifocal, and progressive lens designscombine areas of correction for both distanceand near vision in a single lens and link them indifferent ways:

– In a bifocal lens, a distance vision sphere isplaced above a near vision sphere and linked bya single “step” creating a visible segment line.

A

Fig. 2: Basic design principles of single vision, bifocal, trifocal andprogressive lenses.a) Single vision.b) Bifocal.

Fig. 3: Comparative fields of clear vision with single vision, bifocal,trifocal and progressive lenses (2.00D add).a) Single vision (2.00D add).b) Bifocal (2.00D add).

Fig. 2

ooooooooooooooooooooooooooo a ooooooooooooooooooooooooooo

ooooooooooooooooooooooooooo b ooooooooooooooooooooooooooo

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LENS CONCEPT

Fig. 3



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LENS CONCEPT

– In a trifocal lens, a third sphere is addedbetween the distance and near vision spheres toproduce an intermediate vision power. This givesrise to two segment lines on the lens surface.

– In a progressive lens, an uninterrupted seriesof horizontal curves links distance vision,intermediate vision, and near vision with novisible separation. Lens power increasessmoothly from the distance vision area at thetop of the lens, through an intermediate visionarea in the middle, to the near vision area at thebottom of the lens.

Fig. 2 (contd.)

ooooooooooooooooooooooooooo c ooooooooooooooooooooooooooo

ooooooooooooooooooooooooooo d ooooooooooooooooooooooooooo

Fig. 2 (contd.):c) Trifocal.d) Progressive.

Fig. 3 (contd.):c) Trifocal (2.00D add).d) Progressive (2.00D add).

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LENS CONCEPT

Fig. 3 (contd.)

ooooooooooooooooooooooooooo c ooooooooooooooooooooooooooo

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D I S T A N C E

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Acc. = 1.50 d (maxi)

Acc. = 0.00 d

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LENS CONCEPT

Advantages of ProgressiveAddition Lenses:

Progressive addition lenses offera continuous field of clear visionfrom distance to near. Singlevision reading lenses offer a field

of clear vision limited to the near area only, whilethe abrupt change of power in a bifocal createscompletely divided fields for distance and nearvision (Fig. 3).

Progressive addition lenses arethe only lenses providing clearand comfortable intermediatevision whatever the addition as

the progression of power gives rise to an areaspecifically designed for intermediate distancecorrection.

Only in the early stage of presbyopia, can singlevision and bifocal wearers enjoy clearintermediate vision, as they can stillaccommodate and adjust their head position(Fig. 4).

For high additions, progressive lenses continueto offer clear vision at intermediate distancewhile bifocal and single vision lenses do not.For the latter, the ageing eye can no longeraccommodate to compensate for the lack ofintermediate vision power.

Despite their clear intermediate field of vision,trifocal lenses are not ideal, since wearers mustcope with the image jumps at the two segmentlines.

2) Comfortableintermediatevision:

1) Continuousfield of clearvision:

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LENS CONCEPT

In a single vision reading lens,the eye’s accommodation issupported for near visiononly. In a bifocal lens, theeye’s accommodation

experiences abrupt changes when the gaze shiftsfrom distance to near vision across the segmentline. Only, for each point of the progressive lensmeridian does the power exactly correspond tothe eye’s focusing distance.

Progressive lenses offer globalperception of space: the powerchanges continuously andgradually in all directions. Single

vision reading lenses do not allow real spatialperception, since they provide only a near visioncorrection. The two portions of bifocal lensessplit and alter spatial relationships. Vertical andhorizontal lines appear broken and image jumphampers the wearers’ vision.

Many clinical studies conducted over the last20 years have demonstrated the superiority ofprogressive addition lenses over bifocal andsingle vision reading lenses for correctingpresbyopia. Studies such as by Drs Borish (1), (2),Cho (3), Davidson (4), Krefman (5) in the U.S.A., andby Dr Gresset (6) in Canada have documented asuccess rate higher than 95 percent withprogressive lenses while a survey by Dr Shultz (7)

had shown that 11 percent of bifocal wearersnever adapt to their lenses.

References:(1) - Borish I.M., Hitzeman S.A., and Brookman K.E.; Double-masked Studyof Progressive Lenses, Journal of the American Optometric Association,Vol. 51, No 10, October 1980.(2) - Borish I.M., Hitzeman S.A.; Comparison of the Acceptance of ProgressiveMultifocals With Blended Bifocals, Journal of the American OptometricAssociation, Vol. 54, No 5, May 1983.(3) - Cho M.H., Barnette C.B., Aiken B., Shipp M.; A Clinical Study of PatientAceeptance and Satisfaction of Varilux Plus and Varilux Infinity Lenses,Journal of the American Optometric Association, Vol. 62, No 6, June 1991.(4) - Gatoura J., Study Cites Success Of Spectacle Lens Presbribed ForPresbyopia, Ophthalmology Times, Vol. 9, No 12, 1984.(5) - Krefman R.A., Comparison of Three Progressive Addition Lens Designs:A Clinical Trial, Southern Journal of Optometry, Summer 1991.(6) - Gresset J.; Subjective Evaluation of a New Multi-Design ProgressiveLens, Journal of the American Optometric Association, Vol. 62, No 9,September 1991.(7) Shultz L.B.; Adaptation to Bifocals, American Journal of Optometry andArchives of the American Academy of Optometry, Vol. 50, No 3, March1973.

4) Continuousperceptionof space:

3) Continuoussupportto the eye’saccommodation:

Fig. 4



ooooooooooooooooooooooooooo c oooooooooooooooooooooooooooFig. 4: Intermediate vision.a) Using the distance vision portion of a bifocal.b) Using the near vision segment of a bifocal.c) Using the intermediate portion of a progressive.

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PHYSIOLOGICAL CONSIDERATIONSIN PROGRESSIVE

LENS DESIGN

A progressive lens is designed not only torestore a presbyope’s ability to see clearly atall distances but also to optimally respect allphysiological visual functions, in particular:

Foveal vision:

The retina’s foveal area permits sharp vision ofdetails at any distance within a very small fieldwhich follows the eye’s rotation (usually within a30° angle). To this end, the lens areas used forfoveal vision must provide for perfect retinalimages.

The wearer’s naturalbody and head positionsdetermine the verticalrotation of the eye fornear and distance vision,

and therefore, the optimal length of the lens’power progression. Furthermore, thecoordination of the body, head, and eyemovements, in relation to the objects’ location inthe vision field, defines the power value neededat each point of the progression.

1) Accommodation,body and headpostures and verticaleye movements:

Fig. 5: Progression of power in relation to viewing distance, headposture and eye movements.

Fig. 5

0.000.00

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LENS DESIGN

Likewise, the naturalcoordination of horizontal eyeand head movementsdetermines the field of gaze in

physiological conditions and defines the width ofthe lens’ zone used for foveal vision (usually lessthan 15°).

To maximize wearer’s visual acuity inthe lens’ central area, the unwanted,induced cylinder of the progressive

lens must be kept to a minimum and be pushedto the peripheral parts of the lens.

3) Visualacuity:

2) Horizontaleye and headmovements:

25°

Fig. 6

Fig. 6: Horizontal eye/head movement coordination and width offield.

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LENS DESIGN

Extra-foveal vision:

Extra-foveal vision refers to the visual perceptionprovided by the periphery of the retina. In extra-foveal vision, wearers do not see objects sharplybut locate them in space, perceive their formsand detect their movements.

Space and form perception isprovided by the retina’s periphery,and is directly influenced by thedistribution of

prism on the progressive lens surface. Dependingon the orientation and magnitude of theseprismatic effects, the power progressionintroduces slight deformations of horizontal andvertical lines thus altering the wearer’s visualcomfort.

1) Spaceand formperception:

Movement is perceived by thewhole of the retina which isalmost homogeneously sensitive

to motion. Here also, the variation of prismaticeffects plays a role in the wearer’s comfort,where it must be slow and smooth across thewhole lens to ensure comfortable dynamicvision.

2) Perceptionof movement:

Fig. 7: Perception of form and movement through a progressivelens.

Fig. 7

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LENS DESIGN

Binocular vision:

Binocular vision refers to the simultaneousperception of the two eyes. For optimal fusion,the images produced by the right and left lensesmust be formed on corresponding retinal pointsand display similar optical properties.

The eyes naturally convergewhen the wearer’s gaze islowered for near vision. The

power progression must be positioned in thelens in order to follow the eyes’ path ofconvergence downwards in the nasal direction.For ease of motor fusion, in all directions of gaze,both right and left lenses must offerapproximately equal vertical prism on each sideof the power progression path.

1) Correspondingretinal points:

To ensure sensorial fusion, the retinalimages formed in both eyes must besimilar in all directions of gaze. For

that purpose, the power and astigmatismencountered on corresponding points of rightand left lenses must be approximately equal.

Progressive lens designers work towardsrespecting these physiological functions. Thefollowing section, illustrates the methods used toaccomplish this.

2) Similarimages:

Fig. 8: Binocular vision with progressive lenses.

Fig. 8

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DESIGNINGPROGRESSIVE ADDITION

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Modern conception ofophthalmic lens design:

An ophthalmic lens is anoptical system designed toform images of objects onthe far point sphere of the

eye. This sphere is the optical conjugate of theretina of the unaccommodated eye in rotation.The image of an object point formed on thissphere is usually a blurred spot instead of asharp point due to the aberrations of the lens. Tomeasure the quality of the image of any objectpoint, the lens designer “sends” a set of selectedlight rays which enter the eye’s pupil afterrefraction through the lens and calculates theirintersections with the eye’s far point sphere. Theimage quality is determined by the diameter ofthe blur spot formed on this sphere. The lensdesigners strive to improve the quality of thisimage by controlling the lens’ optical aberrationsin the best possible way.

1) The ophthalmiclens as an opticalsystem:

Fig. 9: Image formation on the far-point sphere.

Fig. 10: Lens + Eye optical model.

Fig. 11: Image calculation.

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Fig. 9

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LENSES

MERITFUNCTION

optimization software

contour plotof the optimized lens

contour plotof the initial lens to be optimized

Fig. 11

STOP (w')Q' w'

center of rotation of the eyechief ray

Fig. 10

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LENSES

Lens designers cannotcreate optimized opticalsystems in a single step.Instead, most designers

employ an iterative process using anoptimization computer software. In this process,the designer defines an initial optical system anda “Merit Function” used to rate the opticalsystem’s overall performance. After rating theinitial optical system, the optimization softwarerecomputes the parameters of an upgradedsystem. This process is repeated until a finaloptimized optical system is found.

The Merit Function evaluates numerous pointsof the lens. For each point, a target value anda specific weight are assigned to each opticalcharacteristic: power, astigmatism, prismaticcomponents and their gradients. The MeritFunction calculated at each point is the weightedsum of the quadratic differences between the setoptical characteristics Tj and the actualcharacteristics Aj of the system.

2) “Optimization”software and“Merit Function”:

The overall performance of the lens is thenevaluated by the weighted sum of the foundMerit Function values according to the followingformula:

with Pi weight for the point iWj weight for the optical characteristic jTj target for the optical characteristic jAj current value of the optical

characteristic j

The concept of Merit Function is a classicalmethod used for managing large numbers ofpartially conflicting constraints. Merit Functionapplied to ophthalmic lenses links physiologicalrequirements and lens calculations.

Fig. 12: Optimization software.

0

3

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image spot

object point

beam of incident rays

spot diagram of the image

profile of the wave frontwhich makes the image

refraction of the beam through a single surface(Snell - Descartes law)

Q'

Fig. 12

Merit Function = Pi . W j .(T j − Aj )2

j=1

j=n

∑i=1

i=m

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LENSES

Designing progressiveaddition lenses:

The optical characteristicsof a progressive lens aredefined by the visualphysiology and postural

behavior of the wearers, as determined byclinical experiments. They can be divided intotwo categories:

1) characteristics that must respect strictlydetermined values,

2) characteristics that should be kept belowgiven thresholds.

1) Specific opticalrequirementsof a PAL:

Fig. 13: Near and intermediate vision.

Fig. 13

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LENSES

a) Power progression requirements:

The primary function of a progressive lensconsists in restoring near and intermediate visionwhile maintaining clear distance vision. Lensdesigners must respect distance and near visionpowers but avail of more freedom in defining theprogression, especially:

– Vertical location of the near vision area:physiological considerations, such as strain ofextra-ocular muscles or limited range ofbinocular fusion with downward gaze, favor ahigh position of the near vision area in the lens.Unfortunately, a short progression usually resultsin rapidly varying peripheral aberrations. Lensdesigners redress this conflict. A goodcompromise consists in locating the usable nearvision at a downward gaze position of about25 degrees.

– Profile of power progression: a suitable powerprogression, along the meridional line, enablesthe wearer to explore the object field withouttiresome vertical head movements. This isachieved by associating the shape of the powerprogression to the orientation of the verticalhoropter linked to the natural tilting of readingmaterial.

– Horizontal (lateral) location of the near visionarea (meridian): once the power profile has beendefined, its lateral positioning on the lens mustbe adapted to the natural convergence of theeyes and the value of the addition. Since, themore advanced the presbyopia, the closer thereading distance, the meridional line must beshifted nasally as the addition increases.

Fig. 14: Power progression.

0 2.5

distance vision

intermediate vision

near vision

diopters

Fig. 14

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LENSES

b) Visual perception requirements:

To ensure optimal performance in foveal vision,image aberrations must be kept at the lowestpossible levels on the lens surface, in particularalong the meridional line and in its vicinity.

In the central lens area great care must be takenbalancing vertical prism between right and lefteyes to perfectly respect retinal image fusion inbinocular vision. This is achieved by anasymmetrical design of the progressive lenssurface coupled with proper positioning of themeridional line.

In the lens periphery, used in extra-foveal vision,aberrations cannot be totally eliminated. In thisregion, image quality constraints are lessdemanding whilst the control of prismatic effectsis of utmost importance. Motion perception is akey factor when considering the lens periphery,where the gradient of variation of the residualaberrations is more important than theirabsolute value.

All of the above optical requirements introducedin the Merit Function are then integrated into thelens design optimization software.

Following the lens optimization andcalculation process, the designteam – composed of physiologistsand engineers – propose several

tentatively optimized designs. Numerous lensprototypes of each design are then producedand tested through rigorous clinical trials.Comparative lens evaluations are made after in-depth analysis of clinical evidence and patients’comments, leading to a final selection of lensdesign.

Since there is usually no exact relationshipbetween lens design calculations and wearersatisfaction, information gathered from clinicalexperiments is also used to improve the MeritFunction. This Merit Function represents theaccumulated theoretical and clinical expertiseand savoir faire of the design team.

2) Clinicalstudies andprototypes:

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DESCRIPTION AND CONTROLOF PROGRESSIVE

LENS DESIGNS

Optical description of aprogressive lens:

Lens designers use different methods tographically represent the optical characteristicsof progressive lenses, in particular:

The curve represents the powerprogression of the lens along itsmeridional line from distance to nearvision. This power progression is a

result of a continuous shortening of the radius ofcurvature of the front surface.

This is a two dimensional map ofthe lens representing either thedistribution of power (Fig. 16) or of

astigmatism (Fig. 17). The map shows lines ofequal dioptric value (iso-power or iso-astigmatism). Between two consecutive lines, thepower or astigmatism varies by a constant value,0.50 D, in these examples.

2) Contourplot:

1) Powerprofile:(Fig. 15)

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-1 1 2 3 4

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Fig. 15: Power profile of a PAL (2.00D add).

Fig. 16: Power contour-plot of a PAL (+ 2.00DV with a 2.50D add).

Fig. 17: Astigmatism contour-plot of a PAL (2.50D add).

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LENS DESIGNS

Fig. 18: Grid-plot of a PAL (2.50D add).

0

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Beta in degrees

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Fig. 21

Fig. 19: 3D power plot of a PAL (2.50D add).

Fig. 20: 3D astigmatism plot of PAL (2.50D add).

Fig. 21: 3D power gradients plot of PAL(2.50D add).

-20-30-40-50-60-70 -10 605040302010 70

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Fig. 18

A three dimensionalrepresentation whichplots vertically the value

of a given optical characteristic at each point ofthe lens in relation to a reference plane. It maybe used to show the distribution of power(Fig. 19), astigmatism (Fig. 20), prismatic effects,gradients of power variation (Fig. 21), etc...These three-dimensional plots are moredemonstrative of lens characteristics thancontour-plots.

Plot interpretation:

Though useful in the lens design process, allthese plots are mere representations of PALcharacteristics and do not really correlate withwearers’ acceptance. As such, plots cannot beused to make significant PAL design comparisonsor predict patients’ visual comfort. The onlytrustworthy way for assessing or comparinglenses consists in conducting well monitoredwearer tests.

4) Three dimensionalplot:

The grid highlights the distributionof the prismatic effects of the lensby showing how they alter a

regular rectangular grid.

3) Grid plot:(Fig. 18)

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LENS DESIGNS

Progressive lens designcontrol:

Progressive lens design control is a very criticalbut little known activity of lens designers andmanufacturers.

In the PAL designing process,checking the conformity of afinished lens to a wearer’s needs,

requires the re-creation of the true conditionsunder which the eye will use the lens. This isdone in one of two ways:

1) A direct method which uses a specialfocimeter simulating lens wearing conditions.

2) An indirect method in which the progressivelens characteristics are measured with thewearer’s eye being simulated by calculations.

1) In lensdevelopment:

Three-dimensional mechanical measurement ordeflectometric methods (analysis of deviation oflight rays produced by the surface) may be usedto measure the geometry of a progressive lens’surface.

During mass production, theconformity of a lens to its technicalspecifications and its

reproducibility are checked either by means oftraditional focimetry, direct geometric orinterferometric measurement of the progressivelens surface.

2) In lensproduction:

Fig. 22: Coordinate Measuring Machine.

Fig. 22

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SUPPLEMENTMATHEMATICAL DESCRIPTIONOF PROGRESSIVE SURFACES

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Local mathematicaldescription of surfaces:

Any surface defined by a z=f(x,y) equation canbe mathematically expressed in a 3D coordinatereference system Oxyz - xOy being thetangential plane to the surface at point O - by aquadratic equation plus terms of higher degrees.This quadric surface is osculatory with thesurface at point O (ie. its curvatures are identicalto those of the real surface) and is defined bythe equation:

z = rx2 + 2sxy + ty2

where r,s,t are local derivatives of the surface:

r=d2z/dx2, s=d2z/dxdy, t=d2z/dy2.

This quadric surface defines the local axis andmain curvatures of the surface at O.Furthermore, since any surface can beassimilated locally to a toric surface,characterized by its orthogonal main curvaturesC1 and C2 and by its axis derived from thefollowing equations:

where p = dz/dx, q = dz/dy,

Axis = Arctg (m) with m solution of the quadraticequation:

Mathematicalcharacterization of surfacesin a circular domain:

Any portion of a complex surface can be definedby using the reference system known as Zernikepolynomials. This system is used tomathematically express the surface by a sumof a series of specific polynomials. The first tenZernike polynomials give rise to remarkablemathematical and physical applications: the 5thgives access to the mean curvature of thesurface, the 4th and 6th to its cylinder and axis,the 7th and 10th to its slope of curvaturevariation. Zernike polynomials are also usedin the determination of the local power,astigmatism, coma and spherical aberration ofthe lens by means of wavefront analysis. The lenssurface is mathematically expressed by:

Pi: Zernike polynomialswith Zi: Coefficients

y, z: reduced variables

Geometricalor optical meaning Zernike polynomials Coefficients

Piston Z0

Tilt in y Z1

Tilt in z Z2

Asti ±45° Z3

Defocus Z4

Asti 0,90° Z5

Coma tr y Z6

Coma y Z7

Coma z Z8

Coma tr z Z9

Expansion of a surface into the first 10 Zernikepolynomials.

z3 − 3.z.y2

−2.z + 3.z.y2 + 3.z3

−2.y + 3.y.z2 + 3.y3

3.y.z2 − y3

z2 − y2

−1+ 2.y2 + 2.z2

2.y.z zy 1

A B

C1.C2 = r.t − s2

(1+ p2 + q2)2

C1 + C2

2 =

t .(1+ p2) + r.(1+ q2) − 2.p.q.s

2.(1+ p2 + q2)3

2

t .p.q − s.(1+ q2)[ ].m2 + t .(1+ p2) − r.(1+ q2)[ ].m+s.(1+ p2) − r.p.q = 0

(Meancurvature)

(Total curvature)

f y,z( ) = Zi .Pii=0

i=9

∑

24


Mathematical modelizationof surfaces with B. splinespolynomial functions:

Any bi-regular surface can be represented by aset of numerous ordinates evenly distributed onthe surface according to a regular reference grid.The local characteristics of the surface at an x, ycoordinate point, z=f (x,y), p, q, r, s, t, arededuced from the values of the discreteordinates in the vicinity of this point by theirlinear combination on a squared matrix. Thesecharacteristics are calculated according to thefollowing formulae:

Fig. A: Local description of a surface.

Fig. B: Graphic representation of the 8th Zernike polynomial.

Fig. C: Modelization of a surface with B. splines functions.

C

z = λ i, j .ai, ji, j∑

p = dfdx

= wi, jx .

i, j∑ (ai +1, j − ai, j )

q = dfdy

= wi, jy . (ai, j +1 − ai, j )

i, j∑

r = d2fdx2 = wi, j

xx . (ai +2, j − 2.ai +1, j + ai, j )i, j∑

s = d2fdxdy

= wi, jxy . (ai +1, j +1 − ai +1, j + ai, j − ai, j +1 )

i, j∑

t = d2fdy2 = wi, j

yy . (ai, j +2 − 2.ai, j +1 + ai, j )i, j∑

with λ i, j ,wi, jx ,wi, j

y ,wi, jxx ,wi, j

xy ,wi, jyy , being

tabulated coefficients.


x

y

z

P

C1

C2

O

normal vector

principal sectionswhose main curvatures arc C1 and C2

tangent plane in O

Fig. A

Y X

Z

XiYj

ai,j

mesh

Fig. C

- 1.5

- 2.5

- 3.5

- 4.5

- 0.5

0.5

1.5

Z

1

0

- 1

- 2

- 3

- 4

- 5

010

2030

-10

-20

-30

X mm / 10

40

-40

-10

-20

-30

010

20

Y mm / 1

0

3040

-40

Fig. B

25

SUP

PLE

MEN

T- M

ATH

EMAT

ICA

LD

ESC

RIP

TIO

NO

FP

RO

GR

ESSI

VESU

RFA

CES

26

ENHANCING THE CHARACTERISTICSOF FINISHED PROGRESSIVE

LENSES

V

ENH

AN

CIN

GTH

ECH

AR

ACT

ERIS

TICS

OF

FIN

ISH

EDPR

OG

RES

SIVE

LEN

SES

“Equithin” ProgressiveLenses:

As a result of the increase of curvature of theprogressive surface in the near vision portion, aPAL is naturally thin at the bottom and thick atthe top (Fig. 23a). To produce thinner lenses,lens surfacers generally use an “equithin”technique which consists of tilting the back sideof the lens to equalize its thickness at the topand bottom (Fig. 23b). This “equithin” processinduces a base-down prism; its value - expressedin prismatic diopters or cm/m - is generally 2/3the value of the addition and can be measuredat the optical center of the lens. For example, ina 3.00 D add “equithin” progressive lens, a2 diopter base down prism would be read.“Equithin” prisms of identical value must beprovided on both right and left lenses to avoidthe introduction of any vertical prism imbalance.

The effect of the “equithin” prism is a slightupward shift of the whole vision field which hasbeen clinically proven to have no significanteffect on wearers’ visual comfort. Since it offersdramatically thinner, lighter, more comfortablelenses, the use of “equithin” is highlyrecommended for progressive lenses of anydistance and add powers.

Fig. 23: “Equithin” progressive lens.

Fig. 23

A

V

a b

27

ENHANCING THE CHARACTERISTICSOF FINISHED PROGRESSIVE

LENSES

“Precalibration”

The most effective way of reducing the centerthickness of plus lenses is to produce them in“precalibrated” form. This consists in surfacingthe lens as thin as possible based on the chosenframe and the patient’s Rx. Wearer’s PDs, fittingheights, frame shape and size are transmitted tothe lens surfacer who calculates the optimalcenter thickness of the finished lens. Althoughnot specific to progressive lenses, the resultsobtained with “precalibration” are morespectacular for this type of lens which alreadybenefits from the “equithin” process.

Pre-decentered uncutfinished progressive lenses

In the markets where round uncut lenses aredistributed (Europe), pre-decentration of PALs isa method used by manufacturers to producethinner plus-power lenses. To thin the lenses, thelens diameter is reduced, and so as not to losetemporal capacity, the progressive surface isnasally decentered. Pre-decentered finishedlenses are produced, eg. in a 65/70mmdiameter, meaning that the lens has a 65mmgeometric diameter but a 70mm effectivediameter.

Lens pre-decentration is also used for semi-finished lenses for the purpose of increasingblank effective diameter.

Fig. 24

Fig. 24: Pre-decentered uncut progressive lenses.

C B

CIRCLES

LINK

CIRCLES

Fig. 25

ELLIPSES

CIRCLE

PARABOLAS

HYPERBOLAS

Fig. 26

28

THE EVOLUTION OFPROGRESSIVE ADDITION

LENSES

The very first progressive lens - introduced byEssilor under the name of Varilux 1 (1959) - hada basic design linking two large and sphericaldistance and near vision zones. In designing thelens, more attention was paid to distance andnear vision rather than peripheral vision. The second generation of progressive lenses wasintroduced as Varilux 2 the “physiological”progressive addition lens (1972). While providinglarge distance, intermediate and near fields of

vision, V2 also took into account the importanceof extra-foveal and dynamic vision thanks to thenew concept of “horizontal optical modulation”.Binocular vision was optimized as a result of anasymmetric design.The overall design of the lensis represented by a succession of conic sections(Fig. 26).

Fig. 25: The “First” progressive addition lens.

Fig. 26: The “Physiological” progressive addition lens.

VI

THE

EVO

LUTI

ON

OF

PR

OG

RES

SIVE

AD

DIT

ION

LEN

SES

VI

29


LENSES

During the decade following the introduction ofVarilux 2, other manufacturers developedalternative progressive lens designs focusing onspecific optical characteristics. Someemphazised large near and distance vision zones,while concentrating unwanted astigmatism in thelens periphery (American Optical Ultravue,Rodenstock Progressiv R, Silor SuperNoLine,Sola VIP). Others took the opposite approach,reducing the amount of unwanted astigmatism inthe periphery by spreading it more widely in thelens (American Optical Truvision Omni). Stillothers placed special emphasis on the conceptof lens asymmetry and comfortable binocularvision (Zeiss Gradal HS).

A further step in the enhancement of progressivelens performance was the introduction of theMulti-Design concept with the Varilux Multi-Design / Varilux Infinity lens (1988). This lensused distinct designs to match the wearer’schanging needs as presbyopia advances. Multi-design concept aims at optimizing visual comfortfor each stage of presbyopia and makingchanges of addition easier for the wearer. The“Multi-Design” concept is well illustrated with thechange of power progression profile by addition(Fig. 27).

power increase

0.75 1.50 2.50 3.50

Distance vision

Near vision

Intermediate vision

Height

Fig. 27

Fig. 27: The “Multi-Design” progressive addition lens.

30


LENSES

The latest generation of progressive lenses,introduced under the name of Varilux Comfort(1993), offers wearers more “natural vision” thanany previous progressive lens.

Varilux Comfort’s near visionarea is located high in the lensso that the wearer can reach iteasily and naturally on

downward gaze (Fig. 28). To explore the near andintermediate vision fields, fewer head and eyemovements are required and the wearer enjoysmore comfortable posture (Fig. 29).

Comfortableposturein near vision:

These advantages are a result of Varilux Comfort’sspecific power progression profile (Fig. 30) : for a2.00 D add, 85% of the addition is reached12mm below the distance fitting cross, comparedto a minimum of 14 to 15mm for a classicalprogressive.

Fig. 28: Comparative head and eye posture: Varilux Comfort,Classical Progressive and “non presbyope”.

Fig. 29: Comparative head and eye movements (vertical plane):Varilux Comfort, Classical Progressive and “non presbyope”.

25°

35°30°

30°

45°

15°

a) "non presbyope" b) "classical progressive" c) Varilux Comfort

Fig. 28

10°

10°10°

30°

5°

5°

15°

35°45°

20°

a) "non presbyope" b) "classical progressive" c) Varilux Comfort

Fig. 29

31


LENSES

Varilux Comfort offers the wearerlarger fields of clear vision as wellas additional comfort inperipheral and dynamic vision.This is due to the softness of the

lens periphery, which greatly reduces horizontalhead movements necessary to explore the fullwidth of the field (Fig. 31). Furthermore, there isa dramatic reduction of all “swim effects” togreatly improve wearer comfort in dynamicvision.

True comfortin peripheraland dynamicvision:

Moreover, Varilux Comfortoffers perfectly balancedbinocular vision thanks to

its asymmetry, and also integrates the multi-design concept of previous Varilux generationlenses which have been retained and improved.

Binocular comfortand multi-design:

+4

-8

-14

-20

085% 100% add

DISTANCE

NEAR

( add 2.00 )

INTERMEDIATE

Fig. 30

Fig. 30: Power profile of Varilux Comfort add 2.00.

Fig. 31: Comparative head and eye movements (horizontal plane):Classical Progressive and Varilux Comfort.

12° 13° 6° 19°Fig. 31

32

CONCLUSION

Latest advances in progressive lens technology have furtherimproved patient satisfaction.

Progressive lenses will continue to develop ensuring successto eye-care practitioners and visual comfort to an ever-increasing number of their presbyopic patients.

ESSI

LOR

INTE

RN

ATI

ON

AL

- R.C

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RIS

B 7

12

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18

.

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