JOURNAL OF THE OPTICAL SOCIETY OF AMERICA

Corneal thickness measured by interferometry

Daniel G. Green, Bartley R. Frueh,* and Jerrold M. Shapiro

Department of Ophthalmology, University of Michigan, Ann Arbor, Michigan 48104(Received 19 August 1974)

An optical method for measuring the thickness of transparent structures has been developed, and hasbeen used to measure, in vivo, the thickness of the human cornea. The thickness is measured byplacing the anterior surface of the cornea at the focus of a beam of coherent laser light and thenmeasuring the spacing between the interference fringes generated by the reflected light. The thicknessis then calculated from the fringe spacing. The method has been used to measure corneal thicknessin frog and human corneas. These measurements have been correlated with histologic and pachometermeasurements of corneal thickness. A significant capability ofthickness of optically opaque corneas.

Index Headings: Interferometry; Laser; Vision.

This paper presents a method for measuring the thick-ness of transparent or semitransparent biological struc-tures. Although in principle the technique is applicableto a more-general class of problem, the results reportedhere relate to the cornea and precorneal tear film.

Our measurements are based on the principle that ifa coherent source of light is directed at a thin layer oftransparent material that has an index of refraction dif-ferent from the media on either side of it, reflectionwill occur at the front and back surfaces of the layer.The wave fronts reflected from the two surfaces willinterfere, forming a pattern of alternating dark and lightfringes. The angular separation between successiveinterference fringes will vary inversely with the thick-ness of the layer.

The precise relationship between fringe frequency andthickness when a converging beam of monochromaticcoherent light is focused onto the cornea, which is con-sidered as a thin convex object with concentric surfaces,is derived in the following section. With this relation-ship and the relevant geometric quantities, we then usethe experimentally measured spacing between interfer-ence fringes from the living human cornea to determineits thickness.

Light incident on the eye encounters a number of trans-parent structures before it reaches the retina. At eachinterface between two structures of different indices ofrefraction, some of the light is reflected; eventually aportion of the reflected light exits from the eye. If theeye is illuminated with coherent light, the beams reflect-ed from different interfaces will interfere. We havechosen a particular geometry of illumination so that wecan use the spacing of the interference fringes to calcu-late the thickness of the cornea. The analysis is begunby treating the cornea as a plane transparent plate (Fig.2), and then, as a concentric spherical shell (Fig. 3).

When a transparent plate with two parallel surfaces isilluminated by a point source, reflections from the frontand back surfaces of the plate produce virtual images ofthe source. These two images act as secondary sourcesthat produce waves which augment and cancel, producingfringes. The apparent separation of the virtual point

this technique is to measure the

sources can be calculated from the fringe spacing. Then,given the index of refraction of the plate, the plate thick-ness can be derived.

To derive the relationship between the spacing of thefringes and that of the secondary sources, consider thesituation shown in Fig. 1. Two point sources lie alongthe x axis of a Cartesian coordinate system, with onesource at the origin and the other a distance a behind it.The observation screen is a plane perpendicular to thexy plane and intersecting it in line L. The ratio of the,irradiances falling on the observation screen from thepoint sources at (-a, 0) and the origin, respectively, isdenoted by p 2 . Because the virtual point sources arederived from the same source, they are spatially andtemporally coherent with one another, and their phasedifference is arbitrarily assumed to be zero. Scalardiffraction theory can be used to calculate the field onthe screen due to the two sources because both the point-source separation and the distance to the screen arelarge compared to the wavelength.

In our experimental setup, the screen was placed tooclose to the sources to allow the Fraunhofer approxima-tion' to be used. However, the Fresnel approximation" 2

is valid. Although the analysis is carried out on a plane,it is easily extended to three dimensions. The locus of

y

( x,y )

x0

Line L

FIG. 1. Interference-fringe formation.

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THEORY

VOLUME 65, NUMBER 2 FEBRUARY 1975

GREEN, FRUEH, AND SHAPIRO

D

H

J

FIG. 2. Geometric relationships for case of plane transparentplate.

each interference fringe is a hyperboloid of revolutionabout the x axis.

The field at the point (x,y) due to the two sources is

it(x, y ) = 1eiro/j Xro +peihl/j X,

where X is the wavelength of the light, To is the distancefrom the origin to point (x,y), and r is the distance from(-a, 0) to (x,y). Because a is very small compared tor7, we make the usual binomial approximation to r1 ,factor out terms in To, and take the square of the fieldamplitude to get the intensity at point (x, y) on line L,

I(x, y) = Io[1 + p2 + 2p cos(kax/ro) .

From Fig. 1, we see that x/ro equals cosS, where 0 isthe angle between 70 and the x axis. The irradiancealong line L varies sinusoidally with cosO, and with aperiod equal to the change of 0 necessary to vary kax cos0 by 21T rad. The distance between successive max-ima on the screen, 6, can be shown to be given by

6 = Xro/da sinG, (1)

where d is the distance from the sources to the screenalong a line normal to L.

The quantity a sinG may be regarded as the projectionof a onto a line through the origin that is perpendicularto P1. Thus the second source, at (-a, 0), could be lo-cated anywhere on T, in the vicinity of the origin, with-out changing the fringe spacing.

To determine the relation between the separation ofthe apparent point sources, a, and the thickness of atransparent plate, t, consider Fig. 2. The virtual pointsource due to reflection from the air-plate surface islocated along line CH. The virtual point source, due to

reflection at the second surface, is located along lineJE. The quantity a sine of Eq. (1) is the distance CN ofFig. 2. By the geometric relationships shown in Fig. 2,we find that

CN/CM = sinzCMN= sin2a

and

CF/CM l = sin .Using the relationship

CF = FD tank = t tangs,

we can solve for CN. We find that

CN= ttank (sin2a)/sinu .By the law of refraction

n' sinca = n sink .

Using the above, we obtain

t sin2o!CN= (n/n' )2 _ sin2a1 /2

and by substitution into Eq. 1 we find t,

2 [(n/nz')2 - sin'al]1 /2

t =Xr~f d sin2 a (2)

wheref is the frequency of the fringes, i. e., f= 1/6.

Equation (2) is based upon a simple model for thecornea, a transparent plate. The central region of thecornea is more-accurately modeled as a hollow trans-parent sphere. We will show that if a small area of thesurface of a thin-walled hollow transparent sphere isilluminated, then the derived equation for thickness re-duces to Eq. (2).

The hollow sphere shown in Fig. 3 is bounded byspheres of radii R and R - t at its outer and inner sur-faces, respectively. A ray incident at A and with angleof incidence a is refracted at the outer surface, reflect-ed at B from the inner surface, and refracted again atC, where it exits at an angle of a + 4 with respect toline OA, where 0 is the central angle between points Aand C. Because t is much less than R, P is a smallangle, and triangles BCE and BOE are approximatelyright triangles. With this approximation, we get

*) si P BE ttanP 332 2 R t ()t

(a) (b)

A'

D

FIG. 3. Geometric relationships for (a) concentric sphericalshell and (b) detail of rays reflected from outer shell.

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CORNEAL THICKNESS MEASURED BY INTERFEROMETRY

P

FIG. 4. Experimental apparatus.

The quantity a sinO of Eq. (1) is called ap in Fig. 3(b).Under the small-angle assumptions,

ap= R ¢ cosa,

and, when we substitute for 0 from Eq. (3) and for tankfrom Snell's law of refraction, we get, after simplifica-tion,

t sin2 a RaP=T(nl/n')2 - sin 2 all/ 2 R - t

Equation (4) for the spherical surface reduces to theequation for CN for the plane surface when R is muchgreater than t. For more accuracy, ap from Eq. (4)may be substituted for a cos0 in Eq. (1), and the resultsolved for t to get

t= 1 (5)1/t2 + 1/R'

where t2 is calculated from Eq. (2). For the humancornea, R/(R - t) is about 1. 07, so Eq. (5) should beused for the best accuracy.

METHOD

The experimental apparatus for measuring cornealand tear-film thickness is shown schematically in Fig.4. The output from a 1 mW He-Ne gas laser was madedivergent with a negative lens, L1. A photographic-ob-jective lens, L2 , focuses the light on the eye betweenthe anterior surface of the precorneal tear film and theposterior surface of the cornea. The source is correct-ly focused when the reflected beam has a circular crosssection. The reflected and incident beams' axes weremade approximately perpendicular by positioning theanimal's body or by having the subject fixate his eyesat an appropriate position. The cross section of thereflected beam was registered on film by placing a cam-era back (no lens) with a focal plane shutter, P1, in thebeam.

The corneal thickness was calculated by use of a mod-ified version of Eq. (2). Most of the measurementswere made at the center of the reflected beam, wherero= d. Then the first term in Eq. (2) becomes

Xr2gf /d= Adf .

This, in turn, can be written

df = X(d/D)fD,

where D is the diameter of the beam on the film. Thequantity d/D is simply the aperture ratio of the cone ofilluminating light, and is measured before the subject's

eye is placed in the beam. The frequency, f, of thecorneal lines was measured directly from either thePolaroid positive or the film negative with a measuringmicroscope. The diameter, D, of the reflected beamwas measured directly with a millimeter scale. Thequantity t2 is computed according to

t=- X(d/D)fD (n2- sin2a)l/22- ~sin2a

where n is taken to be 4 The corneal thickness is thencomputed from Eq. (5), after measuring the cornealcurvature with an ophthalmometer.

If the distance of the film from the eye were measured,df could be used in place of (d/D)fD, and the diameterof the reflected beam could be ignored.

RESULTS

Our initial experiments measured the thickness of aglass slide, which could be checked with a micrometer.As this proved feasible, a living bullfrog (Rana cates-beiana) was anesthetized with 200 mg/Kg urethane andmounted in a restraining device. The light from thegas laser was brought to focus on the cornea, and theirradiance distribution in the reflected beam was regis-tered on Polaroid film.

Figure 5(a) shows a typical example of the patternsobtained from the eye of the bullfrog. Parallel, low-contrast stripes are oriented normal to the plane formedby axial rays of the incident and reflected bundle.Fringes with the same orientation would be producedby reflections from the cornea, a precorneal tear film,or from structures internal to the cornea itself.

To test which structure produced these fringes, theaqueous was removed from the anterior chamber byparacentesis and replaced with air. This increased theratio of the indices of refraction of the cornea and themedia posterior to it, increasing the reflectance of theposterior surface of the cornea. Figure 5 shows photo-graphs of the interference pattern taken immediatelybefore (a) and after (b) the injection of air into the ante-rior chamber of the bullfrog eye. The increase ofcontrast without a change of fringe spacing shows thata reflection from one of the surfaces bordering the an-terior chamber is participating in forming the interfer-ence pattern. Two observations eliminate the possibil-

FIG. 5. Interference fringes generated by light reflected from(a) normal bullfrog cornea and (b) after injection of air into an-terior chamber of the eye.

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FIG. 6. Patterns obtained from normal human cornea. Insertshows fringes from central portion of pattern at 2. 4 x magnifi-cation.

ity that the pattern comes from the ocular lens: First,the same interference pattern is observed when thebeam strikes the iris as when it passes through the lensand is incident on deeper structures. Second, from thefringe spacing, we calculate that the two reflecting sur-faces are separated by about 100 gim. This separationagrees with the thickness of the frog cornea, as deter-mined by histological sections of this same eye. Theseexperiments therefore indicate that the posterior andanterior surfaces of the frog's cornea are the reflectingsurfaces that give rise to the two virtual point sourcesthat produce the interference pattern.

A normal human eye was similarly examined (Fig. 6).The relatively high-contrast, high-frequency verticalfringes are generated by the cornea and indicate a thick-ness of 0. 55 mm. Pachometer1 3 readings made with anunmodified Haag-Streit pachometer on the same area ofthe cornea indicate a thickness of 0. 58 mm.

A human cornea thinned from a lye burn was macro-scopically too opaque to measure with a pachometer.However, interference fringes were generated (Fig. 7)and indicate a thickness of 0. 28 mm, corroborating theclinical impression.

FIG. 7 Fringes from patient with clinically opaque cornea ow-ing to a lye burn. Insert shows central pattern at 1. 6 x magnifi-cation.

FIG. 8. Fringes from tear film on human eye.

To determine whether it might be possible to measurethe thickness of the precorneal tear film, 2% methylcellulose was placed thickly on an artificial eye and itsthickness examined with the interferometer. Goodfringes were produced initially, but as the fluid ran off,leaving only a thin layer, the fringe spacing became,predictably, wider. This convinced us that it shouldbe possible to detect fringes from the tear film. Uponcareful examination of our photographs from the humaneye, we thought we could usually detect 1 to 2 broadfringes (Fig. 8). However, the low contrast makes itdifficult to be certain of the presence of the fringes, andthe small number of fringes makes accurate thicknessmeasurements difficult. However, 2 or fewer fringes(shown in Fig. 8) would indicate a tear-film thicknessof 10 jim or less, which is in general agreement withthe 3-10 gim found by others. 3 6

DISCUSSION

Blix, in 1880, was the first to show that the thicknessof the cornea of the living human eye was about 0. 5 mm.The method he used was to position a microscope sothat it was focused first on the anterior surface of thecornea and then to move it normal to the surface untilthe posterior surface was brought into focus. The dis-tance between the posterior and anterior positions offocus is the apparent thickness of the cornea. Knowingthe angular relationship between the axis of the micro-scope and the normal to the corneal surfaces, Blix cal-culated the true thickness by use of a formula similarto Eq. (4). Since 1880, several other methods havebeen introduced7 -1 1 and refined. 12i14 These have allbeen visual methods that use direct observation of re-flections from the anterior air-epithelial interface andposterior endothelial-aqueous interface. Excellent re-sults have been obtained, with accuracies of ± 10 jimreported for such determinations.

We now add still another technique for measuringcorneal thickness, that of measuring the spatial frequen-cy of the interference fringes produced by the reflectionof coherent light from the cornea. Corneal thicknesscan be derived from the measured separation betweensuccessive fringes.

Thomas Young first demonstrated the interference oflight in his famous double-pinhole experiments. Al-though Fizeaul5 showed that interference could be used

GREEN, FRUEH, AND SHAPIRO Vol. 65

CORNEAL THICKNESS MEASURED BY INTERFEROMETRY

to measure the thickness of plane objects over 100 yearsago, it was not practical to measure optical-path-length difference simply by reflecting a light from thefront and back of a surface of an object until the laserwas developed. Now, as every coherent-optical-systemdesigner well knows, it is nearly too easy to obtain in-terference fringes from reflected images. Yet, as faras we are aware, reflected laser light has not previous-ly been used to measure the distance between curvedconcentric surfaces. It seems possible that we are notthe first to notice the fine pattern of interference fringesproduced by reflecting coherent light from the cornea.Schweitzer16 published a "reflectogram" of the normalhuman cornea obtained with a He-Ne laser that illumi-nated a 3-mm-diameter area of the cornea. The "re-flectogram" in his paper does not show interferencefringes, but in the text Schweitzer comments that somepictures showed "a very subtle pattern of lines. " Thisseems like an apt description of the fringes that we haveused to measure cornea thickness.

Interferometry may in some instances offer advantagesover more-traditional methods. The equipment requiredis, in principle, simply a laser and a camera. It ispotentially a very rapid and accurate method for deter-mining thickness. In addition, because the thicknessesof microscopic areas of the cornea are being sampled,we have found it possible to obtain results where othertechniques fail. The ability of coherent light to traversea moderately opaque optical medium has been used pre-viously to measure subjective visual acuity in patientswith clinically opaque media. 17-20 Similarly, even whenthe cornea is not sufficiently clear to allow direct visualmeasurement, the coherent light reflected from the eyemay contain interference fringes of sufficient regularityto allow the thickness to be determined (see Fig. 7, forexample). This is one real advantage of this technique

over its predecessors.

ACKNOWLEDGMENT

We thank Dr. P. Bolich for his assistance. Thiswork was supported by a grant from the National EyeInstitute, U. S. Public Health Service (EY00379).

*Present address: Department of Ophthalmology, UniversityMedical Center, Columbia, Missouri 65201.

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