cone prism: principles of optical design and linear measurement of the applanation diameter or area...

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Cone prism: principles of optical design andlinear measurement of the applanation diameteror area of the cornea

Jianguo Ma

The cone prism introduced is mainly used in applanation tonometers to act as an applanation prism forlinear measurement of the applanation diameter or the applanation area of the cornea. Its principles ofoptical design and linear measurement are expounded in detail. The measuring body, which comprisesthe cone prism and other optical and electronic parts, is also briefly introduced. © 1999 Optical Societyof America

OCIS codes: 170.3890, 170.4470, 230.5480.

1. Introduction

The measuring technology of the applanation diam-eter or area of the cornea is mainly used in applana-tion tonometry. It is also a technique crucial to anapplanation tonometer. Designs of every kind of ap-planation tonometer center on how to measure theapplanation diameter or area of the cornea accu-rately. Here are several typical applanation tonom-eters. In the early stages, what is called theMaklakov tonometer is actually a cylinder with agiven weight. One of two end surfaces of the cylin-der acts as the applanation surface in contact withthe cornea and is daubed evenly with a layer of ster-ilized argentam colloidal glycerin or other solutions.When the applanation surface flattens the cornea,the glycerin solution on it is removed by the cornea,and then its flattening stamp is printed on a paper.Thereby the diameter of the flattening stamp ~i.e., theapplanation diameter of the cornea! is measured by apecial transparent ruler. This method has noween abandoned mainly because of errors. Theoldmann tonometer1 is regarded as the most accu-

ate tonometer at present, and its applanation prisms like a cone but consists of two semicone prisms.

hen it flattens the cornea, with the help of corneauorescein dyeing and the light of an ultraviolet

The author is with the Anhui College of Mechanical and Elec-trical Engineering, Wuhu, Anhui, 241000, China.

Received 23 November 1998; revised manuscript received 7 July1998.

0003-6935y99y102086-06$15.00y0© 1999 Optical Society of America

2086 APPLIED OPTICS y Vol. 38, No. 10 y 1 April 1999

lamp, the flattening stamp is divided into two semi-circles, the upper and the lower, which are separatedand tangential to each other horizontally. At thattime, the applanation diameter of the cornea is just3.06 mm. The major weakness of this method isthat it needs more auxiliary devices and certain skillsin operation. Perkins2 had made efforts to make theGoldmann tonometer portable, but the optical prin-ciple of measuring the applanation diameter of 3.06mm is the same as that of the Goldmann tonometer.Consequently something either cumbersome in itsuse or complicated in its structure still exists. In hispatented applanation tonometer3 Dreager describedthe triangular applanation prism whose applanationsurface is square. In terms of total internal reflec-tion of light in the prism, the applanation area, ratherthan the diameter, of the cornea can be calibrated asa function of the light returned from the triangularprism when the prism flattens the cornea.

The cone prism introduced in this paper is roughlyof the same principle as that of the triangular prismabove. However, owing to the circular symmetry ofthe cone prism and consequently its possession ofsome particular optical characteristics, it can serve tocarry out direct measurement of both the applanationdiameter, as the Goldmann tonometer does, and theapplanation area, as the applanation tonometer does.The principles of optical design and measurement arepresented below.

2. Optical Principle of the Cone Prism

Figures 1 and 2 show the cone prism at a great scale.In Fig. 1, D2 and D1 are the diameters of the top andthe bottom surface, respectively, of the cone prism

ci

and R0 5 ~D2 1 D1!y4. The bottom surface of theone prism is called the applanation surface becauset is in contact with the cornea. u1 and u2 are inci-

dence angles of light rays at the inclined flank andthe bottom surface, respectively. E0 is the intensityof incident light. A beam of parallel rays enters thecone prism and is totally reflected at the inclinedflank and delivered to the bottom surface. At thisplace, it is again totally reflected and arrives over theopposite inclined flank on the top of the cone prism.If, however, in accordance with Fig. 3, the prism is incontact with the eye, it follows that on the flattenedportion of the cornea, there is no reflection or only aweak one because an important part of the light en-ters the eye. In view of the above, the taper of thecone prism is designed. As an estimate, suppose therefractive indexes of air, glass, and cornea are 1, 1.5,1.37, respectively; by use of the formula of the total-reflection critical angle, we know that u1 should begreater than 41.8° and that 41.8° , u2 , 66.5°, where

Fig. 1. Schematic configuration of the prism and optical paths.

Fig. 2. Illustration of the relationship among R1, R2, and r.

41.8° and 66.5° are total-reflection critical angles ofglass versus air and cornea, respectively. The taperof the cone prism equals u1 and u2 5 180° 2 2u1.

It is known that the triangular prism used in theapplanation tonometer3 to measure the applanationarea of the cornea is based on the fact that the lightintensity distribution on the applanation surface ishomogeneous, namely, the area density of luminousflux is a constant. Consequently the luminous fluxon an arbitrary area of the applanation surface isproportional to the area. However, it is obvious thatthe cone prism does not possess the characteristicbecause of its circular symmetry. From the follow-ing analysis we know that the luminous flux on anarbitrary circular area of the applanation surface is alinear function of the radius, rather than the area, ofthe circular area. This is why the cone prism can beused to measure directly the applanation diameter ofthe cornea.

The distribution function of the luminous flux onthe applanation surface w~r! is deduced here.

From analyzing Fig. 1, we know that the luminousflux dw on the areal element ds 5 2prdr of the ap-planation surface is dw 5 dw1 1 dw2, where

dw1 5 2E02pR1dR1 ~D1y2 # R1 # R0!, (1)

dw2 5 E02pR2dR2 ~R0 # R2 # D2y2!. (2)

To obtain the function w~r!, the relationships amongR1, R2, and r should be found first.

From DABC and DBCD in Fig. 2, the following canbe concluded:

R1 5 r0 1 ~r0 2 r!sin a; (3)

thus

dR1 5 2sin a dr. (4)

Fig. 3. Cone prism in contact with the eye and the applanationdiameter of the cornea d.

1 April 1999 y Vol. 38, No. 10 y APPLIED OPTICS 2087

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2

Likewise, we can conclude from DAB9C9 and DB9C9D9in Fig. 2 that

R2 5 r0 1 ~r0 1 r!sin a; (39)

thus

dR2 5 sin a dr. (49)

Replace Eqs. ~1! and ~2! with Eqs. ~3! and ~4! and~3!9 and ~4!9, respectively; then

dw1 5 2pE0@r0 1 ~r0 2 r!sin a#sin a dr,

dw2 5 2pE0@r0 1 ~r0 1 r!sin a#sin a dr;

thus

dw 5 dw1 1 dw2 5 4pr0 E0 sin a~1 1 sin a!dr, (5)

here r0 5 D1y2, a 5 90° 2 u2, and K 5 4pr0E0 sina~1 1 sin a!. So formulas ~5! are simplified to

dw 5 K dr. (6)

By integrating two sides of formula ~6!, i.e.,

*w0

w

dw 5 K *0

r

dr,

we obtain the function w 5 w~r!:

w~r! 5 Kr 1 w0, (7)

where r is an arbitrary radius of the bottom surface ofhe cone prism. Formula ~7! shows that the distri-ution function of luminous flux w~r! on the applana-ion surface is a linear function of radius r.

In view of formula ~7!, the applanation diameter ofhe cornea can be linearly measured. The measur-ng principle is given in Section 3.

3. Measuring Principle of the Applanation Diameter ofthe Cornea

As stated above, when the cone prism is removedfrom the eye, all the light entering the prism is totallyreturned. If, however, in accordance with Fig. 3, theprism is flattening the eye, only a diminished part oflight corresponding to the applanation diametercomes out of the cone prism. This diminished part oflight can serve to calibrate the applanation diameterof the cornea.

Suppose F is the total luminous flux entering theone prism and w9 is the luminous flux returned fromhe prism. It is obvious that w9 5 F when the conerism is not in contact with the eye. Also supposehat the reflection and refraction coefficients of therism versus the cornea are respectively Rn and Rtnd Rn 1 Rt 5 1, when medium absorption is notonsidered. According to Fig. 3, the returned lumi-ous flux w9 is

w9 5 F 2 w 1 Rnw


or

w9 5 F 2 Rtw. (8)

Then the diminished quantity of luminous flux re-turned from the prism is Dw 5 F 2 w9, i.e.,

Dw 5 Rtw. (9)

Replace w with formula ~7!; then the relationship be-tween the applanation radius of the cornea r and Dws obtained:

r 5Dw

KRt2

w0

K. (10)

Of course, the applanation diameter d is

d 5 2g 52

KRtDw 2

2w0

K, (109)

where w0 is the luminous flux near the circle center onthe applanation surface, which is a constant.

Obviously, on the basis of formula ~10!9 or ~10!, theiminished quantity of luminous flux Dw can serve asmeasure for the size of the applanation diameter of

he cornea after being properly calibrated.

4. Principles of Light Intensity Modulation and LinearMeasurement of the Applanation Area of the Cornea

What is mentioned above is the way in which the coneprism is used to measure directly the applanationdiameter rather then the applanation area of the cor-nea. In order to make the cone prism serve to mea-sure directly the applanation area of the cornea, asthe way the triangular prism in the Applanationtonometer3 works, the light intensity distribution onits applanation surface should be homogeneous, asstated above. However, we find from formula ~7!hat the light intensity distribution E~r! is inverselyroportional to the radius r, i.e., E~r! 5 dwyds 5

ky2pr @or E~r! ; 1yr#, where ds 5 2prdr. If, how-ever, the incident light intensity E0 is properly mod-ulated spatially, E~r! will be a constant or, say, thehomogeneous intensity distribution on the applana-tion surface of the cone prism will be brought out.

For the reason given above, a circular modulatingflake of light intensity is inserted into the opticalpath, as shown in Fig. 4. In Fig. 4, E0 and E09~R!represent the light intensity in the front of and be-hind the modulating flake V, respectively. Then themodulating function of the modulating flake can bedefined as V~R! 5 E09~R!yE0. It is not difficult tofind that when the modulation function V~R! followsthe function curve shown in Fig. 5, the light intensitydistribution E~r! on the applanation surface of thecone prism will be a constant. The optical principleis presented below.

In accordance with Fig. 5, the modulation functionsand the boundary conditions are assumed to be

V~R! 5 0 ~0 # R , D1y2!, (11)

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~

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w

sa

o

HV~R1! 5 a1 R1 1 b1

V~R0! 5 0 ~D1y2 < R1 < R0!V~D1y2! 5 1

, (12)

HV~R2! 5 a2 R2 1 b2

V~R0! 5 0 ~R0 < R2 < D2y2!V~D2y2! 5 1

, (13)

respectively, where a1, b1, a2, and b2 are all constantso be defined. Formula ~11! means to obstruct theight beam that arrives on the bottom surface directlyrom the top surface of the cone prism.

From formula ~12!, V~R1! is obtained:

V~R1! 54

D2 2 D1~R0 2 R1! ~D1y2 < R1 < R0!; (14)

then

E09~R1! 5 E0 V~R1! 54E0

D2 2 D1~R0 2 R1!. (15)

Likewise, from formula ~13!, V~R2! is obtained:

V~R2! 54

D2 2 D1~R2 2 R0! ~R0 < R2 < D2y2!;

(16)

Fig. 4. Modulating flake of light intensity V is inserted into theptical path.

Fig. 5. Functional curve that the modulation function V~R! willfollow.

then

E09~R2! 5 E0 V~R2! 54E0

D2 2 D1~R2 2 R0!. (17)

By use of formulas ~15! and ~17!, with reference tothe introduction in Section 2, the light intensity dis-tribution on the applanation surface of the cone prismis concluded. As above, the luminous flux dw on theareal element ds ~ds 5 2prdr! is

dw 5 dw1 1 dw2

5 2p@2E09~R1!R1dR1 1 E09~R2!R2dR2#. (18)

When formula ~18! is replaced with formulas ~15!,17!, ~3!, ~4!, ~3!9, and ~4!9 and by use of the equality R0

5 r0~1 1 sin a!, formula ~18! can be simplified to

dw 5 K0 2prdr 5 K0ds,

where K0 5 @~4E0D1!y~D2 2 D1!#sin2 a~1 1 sin a! is aconstant; therefore

E 5dw

ds5 K0. (19)

Formula ~19! shows that the light intensity distri-bution on the applanation surface of the cone prism ishomogeneous; so the cone prism can work in the sameway as the triangular prism does, i.e., to measuredirectly the applanation area rather then the appla-nation diameter of the cornea. Formula ~19! alsomeans that a directly proportional relationship be-tween the flux w and the area s can consequently bebtained, i.e.,

w 5 K0 s, (20)

where s is a arbitrary area of the bottom surface ofthe cone prism. When formula ~20! is comparedwith formula ~7!, it is easy to see that two formulasare essentially different.

On the basis of formula ~20!, with reference to theintroduction in Section 3, the principle of linear mea-surement of the applanation area of the cornea iseasily comprehended.

When the cone prism is in contact with the eye ~seeFig. 3! and the applanation area is s, at the momentonly a diminished part of light corresponding to theapplanation area comes out of the prism. As statedin Section 3, the diminished quantity of the luminousflux returned from the prism is Dw 5 Rtw. Replace w

ith formula ~20!; then

Dw 5 K0 Rt S (21)

or

S 51

K0 RtDw. (219)

Formula ~219! shows that Dw can serve as a mea-ure for the size of the applanation area of the corneafter being properly calibrated.


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5. Configuration of the Measuring Body with the ConePrism

The configuration of the measuring body is similarto that of the tonometer advocated by Zeimer andWilensky,4 as shown in Figs. 6~a! and 6~b!, theormer without the modulating flake serving to

easure directly the applanation diameter of theornea, and the latter with the modulating flakeerving to measure the applanation area. In ac-ordance with Fig. 6~a! or Fig. 6~b!, the light re-urned from the cone prism is reflected by aemireflection mirror toward a photodiode. Thehotodiode then delivers a signal corresponding to

Fig. 6. Schematic configurations of two types of the measuringbody that comprises the cone prism.


he applanation diameter or area to a microproces-or. Consequently the size of the applanation di-meter or area is calculated by the microprocessor.Compared with the measuring body in Dreager’s

pplanation tonometer,3 which consists of the tri-angular prism, as shown in Fig. 7~a! or 7~b!, it can

e seen that the structure of the measuring bodyomprising the cone prism is superior to thatomprising the triangular prism in aspects of theptical parts’ integrality and coaxiality. This su-eriority makes the configuration of the measuringody comprising the cone prism simple and reliablend consequently means that a corresponding tech-ique can be easily turned into products, i.e., ap-lanation tonometers.

6. Conclusion

To sum up, the cone prism introduced in this papercan act as an applanation prism in applanationtonometry. By use of total internal reflection, onecan calibrate the applanation diameter or area ofthe cornea as a function of the diminished quantityof light returned from the cone prism. On the ba-sis of two kinds of different optical principles, twotypes of the measuring body can comprise the coneprism and other optical and electronic parts, which

Fig. 7. Schematic configurations of two types of the measuringbody that comprises the triangular prism.

2. E. S. Perkins, “Hand-held applanation tonometer,” Br. Ophthal-
will meet the need of linear measurement of eitherthe applanation diameter or the applanation area ofthe cornea.
References1. R. A. Moses, “The Goldmann applanation tonometer,” Am. J.

Ophthalmol. 46, 865–869 ~1958!.

mol. 49, 591–593 ~1965!.3. J. Dreager, “Applanation tonometer,” U.S. patent 5,203,331A

~20 April 1993!.4. R. C. Zeimer and J. T. Wilensky, “An instrument for self-

measurement of intraocular pressure,” IEEE Trans. Biomed.Eng. 29, 178–183 ~1982!.


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