ORIGINAL PAPER

Effect of spectacle and contact lenses on theeffective corneal refractive zone

Clin Exp Optom 2009; 92: 2: 99–103 DOI:10.1111/j.1444-0938.2008.00346.x

William F Harris BSc(Eng) BOptom PhDHonsBSc FRSSAfDepartment of Optometry, University ofJohannesburg, Johannesburg, SouthAfricaE-mail: wharris@uj.ac.za

The effective corneal refractive zone is that portion of the cornea traversed by the lightthat enters the pupil of the eye from object points at a specified angle from the line ofsight. It is of relevance in corneal surgery and for understanding the effect of cornealopacities and lesions on vision. Gaussian optics is used in this paper to obtain explicitequations for the geometry of the effective corneal refractive zone for a simplified eye,when spectacle and contact lenses are worn. The theory shows that lenses of positivepower increase the diameter of the effective corneal refractive zone and lenses of nega-tive power decrease the diameter. For axial object points the diameter of the effectivecorneal refractive zone increases by about 0.015 mm per dioptre increase in the power ofthe spectacle or contact lens. For object points at 30 degrees from the longitudinal axis,the increase is about twice as much in the case of contact lenses and more than four timesas much in the case of spectacle lenses.

Submitted: 5 August 2008Revised: 16 October 2008Accepted for publication: 23 October2008

Key words: ablation zone, contact lenses, effective corneal refractive zone, equivalent naked eye, spectacle lenses

The effective corneal refractive zone isthat portion of the cornea that may bedirectly involved in vision for object pointsat a particular angle to the line of sight. Itis of relevance in connection with the abla-tion zone in refractive surgery amongother things.1,2 Light from an object pointthat enters the cornea in the effectivecorneal refractive zone passes through thepupil. Light that enters outside the zoneis blocked by the iris. Brown and Freed-man1,2 used geometrical optics to deter-mine the geometry of the effective cor-neal refractive zone in the case of a sim-plified eye (a centred circular pupil anda spherical cornea with a single refract-ing surface). The resulting equations aremessy and can be solved only numerically

and with relatively sophisticated software.Ultimately the problem lies in the trigono-metric functions that arise in the analysisbecause of Snell’s equation for refraction.

In Gaussian optics, the small-angle as-sumption overcomes the problem andleads to neat and explicit formulae for thegeometry of the effective corneal zone.3

Although accuracy falls off with increasingangle that object points make with the lineof sight, the formulae have the advantageof directly showing the dependence of thegeometry of the zone on clinically impor-tant parameters: anterior chamber depth,corneal power and pupil diameter.

So far, the analyses are for the nakedeye. The purpose of this paper is to showhow the Gaussian approach can be readily

extended to obtain explicit formulae forthe effect of spectacle and contact lenseson the effective corneal refractive zone.

We begin by reviewing the results forthe case of a naked eye. Equations arethen set up for an eye with a thin lens infront of it. They lead to the conclusionthat as far as the effective corneal refrac-tive zone is concerned, an eye with a lensin front of it behaves as an equivalentnaked eye. Tables show the numericaleffect of spectacle and contact lenses.

EFFECTIVE CORNEAL REFRACTIVEZONE FOR THE NAKED EYE

Consider a simplified eye with a single-surface spherical cornea of dioptric power

C L I N I C A L A N D E X P E R I M E N T A L

OPTOMETRY

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Journal compilation © 2008 Optometrists Association Australia 99

F and a circular pupil of diameter d(Figure 1). The distance za between thecorneal plane K and the iridial plane Irepresents the anterior chamber depth.The index of refraction outside the eye is1 and in the anterior chamber na. Thereduced depth of the anterior chamber isza. The cornea and pupil are centred on acommon longitudinal axis Z.

Consider distant object points at somesmall angle aO to Z. Rays 1 and 2 inFigure 1A are from a point below Z; theyarrive at the iridial plane at the bottomand top margins, respectively, of thepupil. Similarly rays 3 and 4 are from apoint above Z. Rays from the first point

between rays 1 and 2 enter the pupil. Allother rays from the point are blocked bythe iris. The same holds for rays 3 and4 from the other point. Gaussian opticstreats refraction as taking place in a trans-verse tangent plane, in this case K. Theportion of the corneal plane that is effec-tive in transmitting rays from these pointsis shown as a thick line segment of lengthdKo in the corneal plane in Figure 1A. Incross-section, it consists of two overlap-ping portions, one associated with eachof the two points. When viewed from thefront along Z, the effective corneal refrac-tive zone is circular with diameter dKo asshown in B.

For a sufficiently large angle aO the situ-ation is as in Figures 1C and 1D. Beyonda critical value of aO the overlapping por-tions of the cornea in cross-section in Abecome separated. The view from thefront (D) shows the annular effectivecorneal refractive zone. Its outer diameteris dKo and its inner diameter is dKi.

From Gaussian optics one finds that thecircular effective corneal refractive zonehas diameter (Equation 7 of Harris3):

dd a

FKo

a O

a-= + 2

1ζ

ζ(1)

for aO � aOcrit where aOcrit is the criticalangle given by (Equation 6 of Harris3):

ad

Ocrita

=2ζ

(2)

For aO > aOcrit the annular refractivezone has outer diameter given by Equa-tion 1 and inner diameter given by (Equa-tion 8 of Harris3):

dd a

FKi

a O

a

--

= + 21

ζζ

(3)

EFFECTIVE CORNEAL REFRACTIVEZONE WITH A LENS IN FRONT OFTHE EYE

We now consider the same eye but with athin spectacle lens L of power FL located adistance z from K in front of it (Figure 2).The rays are deflected by the lens but oth-erwise the situation is the same as that inFigure 1.

Consider ray 2 in Figure 2C. It intersectslens L, corneal plane K and iridial plane Iat yL, yK and yI, respectively. It has inclina-tions aO before the lens, aL between thelens and the corneal plane, and aI in theanterior chamber. All transverse positionsand inclinations are relative to the longi-tudinal axis Z.

GENERAL EQUATIONS

At the spectacle lens the ray undergoes adeflection given by:

a a y FL O L L- -= (4)

dKo

dKi

dKo

1

23

4

aOI

Z

K

za

1

2

3

4

aO

y I

K

yK

I

dZa I

C

A B

D

Figure 1. Rays from distant object points entering a simpli-fied eye at corneal plane K and arriving at iridial plane I. Thepupil has diameter d and the anterior chamber depth is za. Theobject points are at angle aO to longitudinal axis Z. A and Bare for objects at a relatively small angle aO and C and D forobjects at a relatively large angle. B is a front view of theeffective corneal refractive zone corresponding to the rela-tively small angle aO in A. It is circular and has diameter dKo.D shows the annular effective corneal refractive zone corre-sponding to the larger angle aO in C. The zone has outerdiameter dKo and inner diameter dKi.

Effective corneal refractive zone Harris

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100 Journal compilation © 2008 Optometrists Association Australia

In effect, this is what is commonly calledPrentice’s equation4 but, as Bennett5 haspointed out, the credit should probablygo to Imbert.6 Between the lens and thecornea we have:

ay y

zL

K L-= (5)

The deflection at the corneal plane isgiven by a generalised version7 of Pren-tice’s equation:

n a a y Fa I L K- -= (6)

Finally, across the anterior chamber:

ay y

zI

I K

a

-= (7)

Substituting from Equation 5 into Equa-tion 4 and solving for yL, we obtain:

yy za

zFL

K O

L

--

=1

(8)

Then substituting from Equations 5, 7and 8 into Equation 6 and solving for yK wefind that:

yy

zFa

zFF F zFF

K

Ia

LO

a

LL L

--

--

-=

+( )

ζ

ζ1

11

(9)

where za = za/na is the reduced depth ofthe anterior chamber.

If we define:

′ =ζ ζa

a

L-1 zF(10)

and

′ = +F F F zFFL L- (11)

Equation 9 becomes:

yy a

FK

I a O

a

--

= ′′ ′

ζζ1

(12)

From Equation 11, one sees that F ′ iswhat is commonly called the equivalentpower of the system consisting of spectaclelens and cornea.

Setting FL = 0 is equivalent to the situa-tion with no lens in front of the eye.Equation 12 then reduces to the sameequation obtained previously (Equation 3of Harris3). In general, Equation 12 is thesame as the equation obtained for thenaked eye except that ′ζa, given by Equa-tion 10, now replaces za and F ′, given byEquation 11, replaces F. It follows that allof the equations in the previous paper3

for the naked eye now apply for an eyelooking through a thin lens, provided thesame replacements are made.

In particular, the critical angle becomes(Equation 2, above):

ad

Ocrita

=′2ζ

(13)

For object points located such thataO � aOcrit the effective corneal refractivezone is circular with diameter (Equa-tion 1, above):

dd a

FKo

a O

a-= + ′

′ ′2

1ζ

ζ(14)

For aO > aOcrit the effective zone isannular with outer diameter given byEquation 14 and inner diameter given by(Equation 3, above):

dd a

FKi

a O

a

--

= + ′′ ′

21

ζζ (15)

Subtracting Equation 15 from Equa-tion 14 and halving the result gives thewidth of the annulus in the case of anannular zone:

wd

F=

′ ′1 - aζ (16)

which is also equal to the diameter of theeffective refractive zone for object pointson the longitudinal axis.

EQUIVALENT NAKED EYE

It follows from the equations above that asfar as the effective corneal refractive zone

1

2

3

4 LC

A

1

23

4

B

D

dKoZ

K

IaO

Z

I

K

dKo

dKi

y I

a I

za

aLyL

aO

z yKd

Figure 2. The eye is the same as in Figure 1 except that thereis a thin lens of power FL at distance z from the corneal planeK

Effective corneal refractive zone Harris

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Journal compilation © 2008 Optometrists Association Australia 101

is concerned, an eye with a thin spectaclelens in front of it behaves as a naked eyewith an anterior chamber of reducedwidth ′ζa and cornea of power F ′ given byEquations 10 and 11, respectively. Wemay refer to such an eye as the equivalentnaked eye. F ′ is the corneal power ofthe equivalent naked eye and ′ζa is thereduced anterior chamber depth of theequivalent naked eye.

Some values for anterior chamberdepth and corneal power for an equiva-lent naked eye are presented in Table 1.Each dioptre increase in power of thespectacle lens effectively stretches theanterior chamber of the equivalent nakedeye by about 0.05 mm and increases thepower of its cornea by about 0.5 D.

EFFECT OF SPECTACLE LENSES ONTHE CORNEAL REFRACTIVE ZONE

Table 1 also shows the diameter of theeffective corneal refractive zone for objectpoints on the longitudinal axis and at30 degrees to the axis. It was calculatedusing Gaussian optics (Equation 14). Forpurposes of comparison, we also give(in brackets) the diameter obtained bynumerical solution of the equations ob-tained via geometrical optics including theexact form of Snell’s equation. (The latternumber depends on the form of the spec-tacle lens; here we have assumed that the

lens is such that it obeys Prentice’s orImbert’s equation exactly.) The criticalangular position is given in Table 1, asis the width of the annulus for effectiverefractive zones that are annular. One seesfrom Table 1 that the diameter of theeffective corneal refractive zone increasesfor spectacles lenses of positive power anddecreases for spectacle lenses of negativepower. The rate is at about 0.016 mm perdioptre for objects on the longitudinalaxis. The rate increases for objects off thelongitudinal axis becoming more thanfour times as much (about 0.073 mm perdioptre) for objects at 30 degrees from thelongitudinal axis.

The bracketed numbers in Table 1show that, for the values of the param-eters indicated, the Gaussian approach isconservative in the sense that it overesti-mates the diameter of the effective zone.However, this is not always the case. Foraxial objects and a thin spectacle lens ofpower -15 D, for example, the Gaussianapproach gives a slight underestimate.The overestimate is greater for objectsat larger angles from the longitudinalaxis.

EFFECT OF A CONTACT LENS ONTHE CORNEAL REFRACTIVE ZONE

For a thin contact lens z = 0, Equations 10and 11 reduce to:

′ =ζ ζa a (17)

and

′ = +F F FL (18)

as would be expected. Here FL is the powerof the contact lens or the combined powerof the contact and tear lens if the latteris significant. F ′ is the power of the wholethin lens system from air in front of thecontact lens through to aqueous immedi-ately after the cornea. In effect a thincontact lens merely adds its power to theeye’s corneal power. It has no effect on thecritical angle aOcrit.

Table 2 gives the diameter of the effec-tive corneal refractive zone for contactlenses of several powers. Again the num-bers obtained via geometric optics aregiven for comparison. The diameter ofthe zone increases at about 0.015 mm perdioptre for objects on the longitudinalaxis. The rate is nearly twice as much(about 0.029 mm per dioptre) for ob-jects at 30 degrees to the longitudinalaxis.

CONCLUSION

Gaussian optics applied to a simplified eyewearing a spectacle or contact lens hasresulted in an explicit expression (Equa-tion 14) for the diameter of the effective

FL (D) ′za (mm) F � (D) dKo (mm) aOcrit (deg) w (mm)

aO = 0° aO = 30°-10 3.57 38.2 4.45 (4.44) 7.57 (6.98) 42.9 3.90-2 3.91 42.0 4.56 (4.52) 8.05 (7.36) 39.2 3.980 4 43 4.59 (4.56) 8.18 (7.47) 38.3 4.002 4.10 44.0 4.62 (4.58) 8.33 (7.59) 37.4 4.02

10 4.55 47.8 4.78 (4.72) 9.03 (8.22) 33.7 4.14

Table 1. The effect of a thin spectacle lens of power FL 12 mm in front of the eye withcorneal power 43 D, anterior chamber depth 4 mm, and pupil diameter 4 mm. F � is thecorneal power and ′za the anterior chamber depth of the equivalent naked eye. dKo isthe diameter of the effective corneal refractive zone calculated via Gaussian optics withthe number calculated via geometrical optics in brackets. aOcrit is the critical angularposition of the object points and w is the width of the annulus for an annular zone. Theindex of refraction of the anterior chamber has been taken as na = 1.3375.

FL (D) dKo (mm)

aO = 0° aO = 30°-10 4.44 (4.42) 7.91 (7.26)-2 4.56 (4.53) 8.13 (7.43)0 4.59 (5.56) 8.18 (7.47)2 4.62 (4.58) 8.24 (7.52)

10 4.75 (4.70) 8.48 (7.70)

Table 2. The effect of a thin contact lensof power FL on an eye with corneal power43 D, anterior chamber depth 4 mm, andpupil diameter 4 mm. dKo is the diameterof the effective corneal refractive zonecalculated via Gaussian optics with thenumber calculated via geometrical opticsin brackets.

Effective corneal refractive zone Harris

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102 Journal compilation © 2008 Optometrists Association Australia

corneal refractive zone for distant objectpoints. For object points that are at an-gular positions further from the longitudi-nal axis than the critical angle given byEquation 13, the effective corneal refrac-tive zone is annular; Equation 14 gives theouter diameter and Equation 15 the innerdiameter of the annulus.

The size of the effective corneal refrac-tive zone is increased by spectacle andcontact lenses of positive power and de-creased by lenses of negative power. Inthe case of object points on the longitu-dinal axis the rate is approximately thesame for contact and spectacle lenses,about 0.015 mm per dioptre. The raterises more rapidly for spectacle lensesfor objects off the longitudinal axis. Forobjects at 30 degrees, for example, therate is about double for contact lensesand more than four times for spectaclelenses.

ACKNOWLEDGEMENTS

I thank C Novis and SM Brown for com-ments on the manuscript.

REFERENCES

1. Brown SM, Freedman KA. Effective cornealrefractive diameter as a function of theobject tangent angle in visual space. J Cata-ract Refract Surg 2005; 31: 2356–2362.

2. Freedman KA, Brown SM, Mathews SM,Young RSL. Pupil size and the ablation zonein laser refractive surgery; considerationsbased on geometric optics. J Cataract RefractSurg 2003; 29: 1924–1931.

3. Harris WF. Effective corneal refractive zonein terms of Gaussian optics. J Cataract RefractSurg 2008; 34: 2030–2035.

4. Prentice CF. A metric system of numberingand measuring prisms. Arch Ophthalmol 1890;19:128–135.

5. Bennett AG. Prismatic effects of sphericaland astigmatic lenses: Imbert’s pioneeranalysis of 1886. Ophthalmic Physiol Opt 1990;10: 397–398.

6. Imbert A. Calcul de l’effet prismatique desverres décentrés. Annal Oculist 1886; 95: 146–153.

7. Harris WF. Ray vector fields, prismatic effectand thick astigmatic optical systems. OptomVis Sci 1996; 73: 418–423.

Corresponding author:Professor WF HarrisDepartment of OptometryUniversity of JohannesburgPO Box 524Auckland Park 2006SOUTH AFRICAE-mail: wharris@uj.ac.za

Effective corneal refractive zone Harris

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