Ophthal. Phvsiol. Opt. Vol, 17. No, 1. pp, 81-87. 1997Copyright C 1996 The College of Optometrists, Published by Elsevier Science Ltd

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TECHNICAL NOTE

Measurement of mechanical warpage in CR-39lenses

Jose Alonso\ Asuncion Perar, Juan Carlos Sanz' andEusebio Bernabeu^

^ optics Department, Universidad Complutense de Madrid, School of Optics, Arcos del Jalon,s/n, 28037, Madrid; and ^Optics Department, Universidad Complutense de Madrid, Facultyof Physics, Ciudad Universitaria, s/n, 28040, Madrid, Spain

Summary

The warpage of CR-39 ophthalmic spherical lenses of a variety of powers and compressedin the radial direction is measured as a function of the applied tangential force. The lenses arecut with circular shape and then mounted on a frame specially designed to measure thetangential force by means of a calibrated coil spring. The measurements are performed witha high resolution micrometer at different points on the lens surfaces. The warpage of the twosurfaces of each lens is calculated and then a quantitative value of surface power variationis given. From these data, vertex power and oblique astigmatism variation are shown. Depar-ture of the surfaces from sphere as a function of the applied tangential force is analysed.Copyright <& 1996 The College of Optometrists. Published by Elsevier Science Ltd.

Introduction

It is well knowti that a variety of factors can induce warpageon the surfaces of ophthalmic lenses. The magtiitude ofthewarpage depends both on the nature and magnitude ofthe warping factor and on the material the lens is madewith. The best known warping factors are lens softeningduring thermal hardening for glass lenses, and mechanicalwarping due to stress, aging of the lens material andsoftening of cut lenses due to exposure to high temperaturesin the frame-mounting process for plastic lenses. Forexample. CR-39 shrinks as it ages and the consequentreduction in volume produces an inerease in the curvatureradius of the surfaces of a lens made with this material.This effect has been analysed by Kislin et al. (1972).

Smith and Wientzen (1978) have studied the visual effectsderived from a cylindrical one diopter warpage of the spec-tacle lens surface. This particular value is the maximumtolerance specified in the prescription lens standard ANSIZ 80-1-72. Dowaliby et al. (1980) measured, by means ofa lens gauge, the bitoricity of hard resin lenses induced bythe pressure exerted by the frame. They measured thelenses of a sample of 275 completed spectacles chosen at

Received: 26 October 19^5Revised form: 1 May 1996

random. Metal frames produced deformation exceeding theANSI standard in 1% ofthe lenses. Other frame materialsyielded smaller deformations. After these works, an impor-tant question remains unanswered: this is the quantitativerelation between the force exerted by the frame and theinduced warpage. In this paper we address this problem.

The pressure exerted by the rim of the frame on a mountedlens produces in general a quite complicated stress field.The geometry of this field depends on the shape of the rimas well as on the lens thickness over the lens surface. Theresulting warpage as a function ofthe applied tangential forcecould be deduced by solving the compatibility equations forthe strain tensor along with the equilibrium equations for thestress tensor and the boundary eonditions (Kuske andRobertson. 1974). In general there will not be analyticalsolutions to this set of equations for arbitrary rim shapesand lens types. For this reason, we have adopted a pheno-menological approach to this problem. As a first step, wehave restricted our work to the study of spherical hard resinlenses and circular shaped rims.

As we show below, warpage greater than the maximumtolerance specified in the prescription lens standard ANSI Z80-1-72 is likely to appear in practical situations, especiallywhen metallic frames are used. Although we give somepreliminary results on the vertex power and obliqueastigmatism variations, a detailed analysis ofthe change in

81

82 Ophthal. Physiol. Opt. 1997 17: No 1

optical performance for the measured warpage is postponedto be presented in a future work. Another question thatarises when warpage is strong, is the departure of thestressed surfaces from their design geometry. In effect, notonly the curvature radius changes, but other higher orderparameters describing the surface geometry. In the presentstudy, surfaces of revolution have been scanned by measuringthe vertical coordinate of a collection of surface points. Thedata is then fitted to spherical surfaces, and the residues ofthe fitting give us the information of initial asphericity andthe way it changes under increasing compression.

Experimental

The force is applied to the lens by means of the specialframe shown in Figure I. The frame is a channelled circularaluminium ring which holds the lens in a similar way as atraditional spectacle frame does. The inner diameter of therim is 55 mm, and the depth of the channel 1 mm. Thepressure is exerted in a controlled way by means of acalibrated spring coil. The screw shown in the figure com-pressed the spring, so that this exerts a tangential forcewhich tends to close the frame. At any given position of thescrew this force is inferred by means of Hooke's law fromthe spring elongation, which in turn is measured with avernier caliper accurate to 0.1 mm. The maximum tangen-tial force we can get with this technique is determined bythe (negative) elongation at which adjacent turns of thespring touch themselves. The employed spring allowed usa maximum force of about 150 N.

Screw

Figure 1. Scheme of the circular frame used to apply acontrolled amount of tangential force.

The calibration ofthe spring was performed by measuringits length variation with an accuracy of 0.01 mm under thecompression of six different reference weights. The elasticconstant t( of the spring is then obtained as the slope of theleast square linear fitting, with the result k = ( — 3.626 +0.012) X 10"- mN"'. The correlation of the fitting was0.9991. In this case, as well as in all the results of thepresent work, we characterize the error of the parameter bymeans of the square root of its estimated variance (Martin,1971).

The lens surfaces have been scanned along a lens dia-meter by means of a dial indicator micrometer, accurate to0.001 mm. The symmetry ofthe problem allows us to reducethe otherwise two-dimensional scan to a one-dimensionalone. Each surface has been measured at 21 radial points,with intervals of 2 mm between them, so that the measuringarea stretches to 40 mm. We obtain for each surface a setof points c,, ( = 1 , 2 , . . ., 21, which correspond to theradial coordinates x, = - 20 + 2(/ — 1). We then fit thedata to the equation

where Zo and XQ are the centre of the fitting sphere, R itscurvature radius and r ^ 1.00 mm is the radius of the steelball at the tip ofthe micrometer. The residues ofthe fitting,5^,, are given by

-(X- (2)

where the minus and plus signs correspond to the front andback surfaces, respectively. These residues give us infor-mation about the departures of the actual surface geometryfrom the sphere.

We have measured the warpage for six levels of appliedtangential force at the two surfaces of nine CR-39 lensesfrom three difFerent manufacturers. The characteristics ofthe lenses employed are shown in Table 1.

An important question that arises at this point is whetherthe forces applied in our frame are comparable with theforces that a typical spectacle frame can exert on a mountedlens. To answer this question, we mounted two identicalCR-39 lenses with vertex power -3 .0 D in our aluminiumring and in an ordinary metallic frame. The lens cor-responding to the latter was cut with a diameter 0.2 mmlarger than necessary to fit in the spectacle frame. We thenprogressively pushed the two frames to their limits, charac-terized by the maximum (negative) elongation ofthe springin our ring and the breaking of the screw in the spectacleframe. To compare the generated stress fields, we used a

Table 1.

Lens

e^ (mm)

Vertex power ^P /̂I

1

- 4 . 91,95

and central

2

-4.61,80

width (e,-

3

-1.91.80

) of tbe

4

- 1 .2,

nine

115

measured

5

-0 .632.00

lenses

6

0.862.60

23

7

.6,85

35

8

.9

.6046

9

.4

.25

Measurement of mechanical warpage in CR-39 lenses: J. Alonso et al. 83

Figure 2. Encoded grey levels of the isoclinic and isochromatic patternsat maximum tangential force corresponding with the frame of Figure 1(a), and an actual metallic spectacle frame (b).

plane polariscope with monochromatic light. Figure 2 showsthe corresponding false-colour patterns for our frame (a)and the spectacle's frame (b) at maximum tangential force.In these patterns, isochromatic fringes appear as darkfringes near the rim at the left side ofthe circular frame andat the bottom and top of the spectacle frame. The fringesappears in both patterns approximately at the same distancefrom the rim, at points with very similar lens thickness, sowe can conclude that the stress fields are quite similar.

Results

We will suppose in this section that the surfaces of thestrained lens maintain their spherical shape at all theachievable force levels. This is a good approximation forspherical lenses with centred revolution surfaces. Underthis assumption, we only need one parameter to describethe effect of tangential force on the surface geometry, forexample, the curvature radius R. The functional relations/?i(F) and ,/?;(F), where the subscripts I and 2 refer tothe front and back surfaces respectively, is all we needto fully characterize the optical properties of the lens asa function of the applied force (the central thickness vari-ations turned out to be smaller than 0.01 mm, so we canneglect them).

In Figure 3 we present experimental measurements ofcurvature radius vs applied tangential force. Based on thehigh value ofthe linear correlation parameter r,tF (̂ ^̂ \eastof which was 0.98), we propose a linear model for thisdependence in the range of achievable applied forces

To prevent overwhelming the paper with figures, weonly show the complete experimental data for the front andback surface of the three most representative lenses fromthe set of nine. The squares in Figure I correspond withlens 1, the most powerful negative lens. The circles cor-respond to lens 5, the less powerful of all, and the rhombus

150

140 -

11025 IOO 75 50 25

25 50 75 IOO

Tangential force (N)125 150

Figure 3. Dependence of curvature radius with appliedtangential force. The filled symbols correspond with frontsurfaces, R,, whereas hollow symbols correspond withback surfaces, Rj- Tbe x-axis for R2 is inverted. Squares,lens 1; circles, lens 5; rhombus, lens 9.

to lens 9, the most powerful positive lens. For the threelenses the filled symbols correspond with the front surface,whereas the hollow symbols correspond with the backsurface. The plots in Figure 3 include the fitting lines andthe confidence intervals. The results of the fitting for allthe lenses are summarized in Figure 4. Some conclusionsare derived from inspection of the data presented in thisfigure.

84 Ophthal. Physiol. Opt. 1997 17: No 1

250

200 -

o. - 100

50 -

-0,03 -

Z -0,06 -

-0,12 -

-0,15

J

- j

J

r - 1

min (4)

4 5 6Number of lens

Figure 4. Parameters of tbe linear fitting fl(F) = ^F + R(0).Bars at the left of each lens number correspond with frontsurfaces. Bars at the right correspond with back surfaces.

1. For negative lenses, the coefficients of curvature radiusvariation ,̂ are smaller (in absolute value) in the backsurface than in the front surface. The opposite holds forpositive lenses.

2. The coefficients ^, increase with the curvature radius.The flatter the surface, the larger the variation of itscurvature radius. This could be the reason lens 7 doesnot obey the former rule.

3. For lenses of similar vertex power in absolute value, thecoefficients ,̂ are smaller in negative lenses, withthe exception of lens 7, which is aspheric.

There are two reasons for the positive lenses to he lesssensitive to compression. First, the central thickness islarger than in negative lenses of similar absolute power.Second, the bases for positive lenses are steeper.

Once we have the model for the curvature radius variationas a function of tangential force, we can proceed to thecalculation of any opticai property ofthe lens. Also, we cangive an answer to the question raised in the introduction ofthis paper: the estimation ofthe force required to producea one diopter warpage in any of the two lens surfaces. Letus call F,,̂ this limiting force. It is straightforward todeduce from Equation (3) that F,,,,, is given by

The values of the limiting forces for the complete set oflenses is shown in Figure 6. According to the previousdiscussion, for negative lenses the limiting tangential forceis larger in the back surface. The opposite holds for positivelenses. Lenses 1 and 2 require a force in the range 300—400 N to become warped by an amount of 1 Diopter in anyof their surfaces. Low power lenses (3 to 6) require a forcein the range 160-320 N. Finally negative tenses of higherpower (7 to 9) require higher tangential forces, in the range600-1100 N.

These predicted force levels exceed the maximum forcewe can apply with our frame. Besides, according to theisochromatic patterns obtained with a standard spectacleframe, it seems unlikely that even with the sturdiestspectacle frame forces larger than 300 N could be applied.According to this, our results on limiting tangential forcesare obtained by extrapolating the model represented byEquation (3) beyond the 150 N range for which it has beenconstructed. We want to state clearly that the predictions ofthe linear model for F > 150 N should be understood to becorrect as long as the linear model holds for F> 150.Nevertheless, due to the smallness ofthe lens deformations,we are confident that the differences of the curvature radiusactual variation from the predictions of our linear model aregoing to be small even for the highest values of F,,,^.

From Equation (3) and the results shown in Figure 5, wecan deduce that the warpage generated in lenses 1 to 6 bya tangential force of --150 N is ^0.5 D. except on lens 3,which turned out to be the most easily deformed lens of all.For this - 2 Diopters lens, the refractive power variation at

1200

1000 -

800 -

600 -

400 -

200 -

3 4 5 6 7

Number of lens

Figure 5. Limiting force at which the warpage equals onediopter. Bars at the left of each lens number correspondwith front surfaces. Bars at the right correspond with backsurfaces.

Measurement of mechanical warpage in CR-39 lenses: J. Alonso et al. 85

each surface was --0.8 D. This example demonstrates thatit is possible, although not usual, to produce the maximumwarpage tolerated by the ANSI standard by applying pres-sure with an actual metallic spectacle frame.

We have computed the vertex power and obliqued astig-matism at different tangential force levels. The results areshown on Figure 6 for lenses I, 5 and 9. The filled circlesand squares represent respectively the vertex power andoblique astigmatism derived directly from the experimentalmeasurement of the curvature radius. The continuous linesrepresent the same magnitudes computed from model (3) witherror intervals. The oblique astigmatism has been derived forfar objects from the classical expression (Jalie, 1992)

OA -y'P

n(n-\)X {in + 2)-P; + P.

-- 1)

2) ] - .(n - \) + tiP;(5)

where P^ ^ P, -I- P;, Lj is the inverse of the distance fromthe rotation centre of the eye to the pole of the lens backsurface and _v is the height at which the ray crosses the lensfor an angle of 40 degrees. The variations encountered areapproximately 0.14, 0.02 and 0.05 Diopters for lenses 1,5 and 9, respectively. It is clear that the oblique astigmatismvariations are negligible for low powered and positivelenses inside the applied force range. Figure 7 shows thevertex power and oblique astigmatism variations for alllenses at the corresponding limiting forces. The powervariations of lens 3 is ^—0.2 D, large enough to be notice-able for anyone requiring the correcting power of this lens.For lenses 7 to 9, power variation is ofthe same order, hutthe limiting force required to reach this variation is unlikelyto happen in practice.

With respect to oblique astigmatism, the problem trans-lates to high and medium powered negative lenses, forwhich variation greater than 0.1 D can be expected. This

•0,60

-0,65 -

-0,70 -

-0,75 -

-0,80'0

400

-4.80 -

-4,85

-4.90 -

50 IOO 150

300 200

200

100

- -0,08

250

0

0,4100 200

Tangential force (N)300 400

Figure 6. Vertex power (circles) and oblique astigmatism (squares) as afunct ion of applied force, (a) Lens 1; (b) lens 2; (c) lens 3. The x-axisis inverted for tbe oblique astigmatism in plots (a) and (c).

86 Ophthal. Physiol. Opt. 1997 17: No 1

- 0 a

1 2 1 4 5 6 7 8 9

Number of lens

Figure 7. Vertex power variation at limiting force (bars attbe left of eacb lens number), and oblique astigmatismvariation at limiting force (bars at the right).

oblique astigmatism variation can produce lack of comfortin the wearing of a newly adapted and compressed CR-39lens.

Deviations from spherical shape

We have assumed that the surfaces ofthe lenses are sphericaland they maintain this geometry as the radial compressionis increased. Although this is a good assumption to studythe overall behaviour of the lens under compression, ingeneral we can expect that there will be small deviationsfrom sphericity in the unperturbed surface that will grow asthe tangential force increases. The measurement methodemployed allows us to find out these deviations and theirdependence on the applied force.

The lack of sphericity at the tens surfaces is directlyrelated with the magnitude and structure of the residues.Figure 8 shows the fitting residues for the two surfaces oflens number 5 and for all the applied tangential forces. Thediagrams corresponding to the other lenses are quite similar.

For the unstrained lens, the magnitude of the residuesrarely exceed ±5 /tin. Besides, they are distributed as aripple whose period is -- 15 mm in the back surfaces and-10 mm in the front surfaces. As the compression in-creases and the lens becomes warped, the magnitude oftheripple increases in both surfaces. The period and the positionof the nodes of the ripple keep constant on the back surfacewhereas they change on the front surface for the firstvalues of tangential force. On this surface, the period oftheripple without compression is smaller, but as tangentialforce is increased, the residues distribution changes in sucha way that period and nodes match those on the backsurface. From the third value of applied force, the residues

20

1,0

J 0

-10 -

20

10

n3 0

-10

15 20

Figure 8. Residues of tbe nonlinear fitting for lens 5. Thedifferent force levels are encoded in grey levels, (a) Frontsurface; (b) back surface.

distribution present the same pattern on the front and hacksurfaces.

This behaviour is quite similar in all lenses except 7,which is aspheric. Although we have not measured theouter regions of this lens, at which the aspheric surfacediffers more appreciably from the spherical shape, theresidues in the central region, which is mainly spherical,present a ripple with the smallest period of all the measuredlenses (about 10 mm on the eye surface and 5 mm on thefront surface). The matching ofthe ripple characteristics onboth surfaces as the tangential force increases does notoccur in the aspheric lens.

Figure 9 shows the dependence of A(Xi^, ZQ, R) as afunction of the applied force for all the measured lenses. Inthe case of the front surface (Figure 9a), the behaviour ofA is completely different for negative (filled symbols) andpositive lenses (hollow symbols). The former clearly becomeaspheric as the compression increases. The effect is strongerfor the powered lenses I and 2. By the contrary, the sumof residues for the front surfaces of positive lenses is almostindependent of the applied force.

In the case ofthe back surface (Figure 9(b)), the behaviouris similar in positive and negative lenses. The aspheric lensnumber 7 begins with the highest value of A in the unstrainedstate and it sufFers small increases of this initial value withincreasing force. It should be recalled that the thickness

Measurement of mechanical warpage in CR-39 lenses: J. Alonso et al. 87

45

30 -

(b)

^~ >^

,0

t '• - - o -

„ • • • • - • •

- • o • ~

1

^ /

o

*

"^ 0

D

25 50 7,"; IOO 125 150

Tangential force (N)

25 .•iO 75 100Tangential force (N)

125

Figure 9. Sum of the residues squared (A) for all tbe lenses.Filled symbols correspond with negative lenses. Hollowsymbols witb positive lenses, (a} Front surface; (b) backsurface. Key for filled symbols; circle, lens 1; square, lens2; rbombus, lens 3; triangle, lens 4; star, lens 5. Key forbollow symbols: circle, lens 9; square, lens 8; rbombus,lens 7; triangle, lens 6.

variation in the central region is smaller in positive lenses(and especially in the aspheric) than in negative lenses. Thisexplains the smaller deviations from the initial values of A.In the aspheric lens, this effect is even more noticeable,because the two surfaces are comparatively flatter and thecentral thickness is smaller. The central spherical region ofthis lens can be deformed more easily without importantchanges in A, Unfortunately, we have not yet data on theouter, aspheric regions of this lens, hut we suspect thatthe delicate geometry of aspheric and progressive lensescan be seriously affected by mechanical warping.

Conclusions

We have characterized the warpage of compressed lenseswith revolution symmetry surfaces as a function of appliedtangential force. In our measurements, we have appliedforces similar in magnitude to those achievable with actualmetallic spectacle frames. According to the experimentaldata, a linear model for the curvature radius variation as afunction of applied tangential force is proposed. We havefound an example in which actual compression levels caninduce warpages similar to the maximum tolerance allowedin the prescription standard ANSI Z 80-1-72. The modelcan be used to compute the optical properties of the lensunder compression, and we have found important differencesin behaviour between positive and negative lenses. Finally,increasing asphericities of the warped surfaces have beendemonstrated, and they are produced by the magnificationof the initial defects on the unperturbed surface.

Acknowledgements

We thank Dr. C. Iilueca for his suggestions. We thank toothe company INCOSA-Framo (INDO group) for its col-laboration. This work has been partially supported by thePlan Regional de Investigacion de la Comunidad de Madrid,I + D 0119/94.

References

Dowaliby. M., Nichols, E, and Bailey, M. E, (1980).Comparative study involving bitoric efFect of hard resinlenses mounted in optyl frames, zylonite frames, and metalframes. Am. J. Optom. Pliysiot. Opt. 57(2), 109-112,

Kislin. B, etat. (1972), Warpage of CR-39 Pta.stic Len.ses.USAF School of Aerospace Medicine, Brooks Air ForceBase, Texas, USA, SAM-TR 72-4.

Kuske, A, and Robertson, G. (1974). Photoetastic StressAnatysis. John Wiley & Sons, London, UK.

Jalie, M, (1992). The Principtes of Ophthatmic Lenses. 4thedn. The Association of British Dispensing Opticians,London. UK,

Martin. B. R. (1971). Statistics for Physicists. AcademicPress, London. UK,

Smith. F. D, and Wientzen, R. V, (1978), Prediction of visualeffects from the warpage of spectacle lenses. J. Optom.Arch. Am. Acad. Optom.. 616—631.