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The refractive structure and optical properties of the isolated crystalline lens of the cat
Vision Res. Vol. 30, No. 5, pp. 723-738, 1990 Printedin GreatBritain.All rightsreserved
0042-6989/9053.00+ 0.00 CopyrightQ 1990PcrgamonPras pk
THE REFRACTIVE STRUCTURE AND OPTICAL PROPERTIES OF THE ISOLATED CRYSTALLINE LENS OF THE CAT W. S. JAGGER National Vision Research Institute, 386 Cardigan Street, Carlton. Victoria 3053, Australia (Received 22 May 1989; in revisedform 21 September 1989) Abn&act-Direct measurements of the shape and internal refractive index distribution of the isolated cat lens were used to construct individual optical models of the lenses from the left eyes of five cats. The right eyes were used in a companion study of the optics of the cat eye. The lens refractive index spatial distribution and dispersion were measured with a spekally designed Pulfricb areal refractometer. Agreement of calculated ray paths tbrougb these models with the obacrved paths of laser beam fans incident parallel to, and at an oblique angle to the kns axis indicates that the models, which contain no freely adjustabk parameters, are good physical and optical representations of the isolated (accommodated) crystalline knees. Crystalline lens Optics
Refractometer
Lens model
Gradient nfkactive index
INTRODUCTION
The internal refractive structure of the crystalline lens is the key to understanding the optics of the eye, yet has been difhcult to access directly (Hughes, 1986). Present knowledge about lens refractive index gradients and isoindicial surfaces is based on the measurement of the index of small samples taken from a lens (Matthiessen, 1877), interferometry on thin sections (Kleberger, Pliihn & Simonsohn, 1%8), schlieren methods applied to a lens section (Nakao, Fujimoto, Nagata & Iwata 1968; Nakao, Otto, Nagata & Iwata, 1969), and inverse ray tracing to infer the refractive structure (Campbell & Hughes, 1981; Campbell, 1984, Axelrod, Lemer & Sands, 1988; Chan, Ennis, Pierscionek & Smith, 1988). Further approaches measured lens mass concentration or protein distribution (Philipson, 1969, Fagerholm, Philipson & Lindstriim, 1981; Bando, Nakajima, Nakagawa & Hiraoka, 1976; and BOW, Fijdisch & Hockwin, 1987). Measuring the refractive index of local samples from a lens is a tedious procedure, and is subject to location uncertainty of the samples and possible mixing and dilution of the lens substance. Interferometry on thin sections is subject to readily apparent geometrical distortion. Schlieren methods require considerable mathematical processing, as the method yields
Dispersion
Cat
the index gradient rather than the index directly. In addition, an undistorted lens section is required. Inverse ray tracing has the advantage of being performed on an intact lens, but is subject to the criticism that the structure might not be unique and is inferential rather than direct. The distribution of index within the lens has been described by the workers above as following a parabolic increase as the lens centre is approached. The distribution of dry mass or protein within the lens can also be used to indirectly demonstrate the presence of a parabolic increase of index (however, Fagerholm et al. found a plateau of dry mass and hence index in the centre of the human lens). Pulfrich refractometry (Van Heel, 1959) applied to an area was discussed by Marshall, Mellerio and Pahner (1973), who experienced dilflculties when they attempted to use it on the pigeon lens. The present paper introduces an improved Pulfrich area1 refractometer, which, applied to the cut surface of a lens, yields isoindicial curves at desired values of refractive index. If the lens is symmetric for rotation about its axis, and the cut plane includes the axis, the shape and spacing of these curves determine the refractive index throughout the lens. Measurement of the lens shape and the internal refractive structure of the lens allowed the constmction of individual optical models with the minimum number of parameters required to
723
W. S. JACKXR
724
specify their physical structure. The parabolic rule of index increase and the geometrical similarity of isoindicial surfaces found to apply to the 87% of the lens volume for which measurements were made was assumed to apply to the remainder of the lens. Five individua1 models were made rather than only one average model to avoid obscuring correlation of parameters within individual lenses. Any developmental mechanisms that balance one parameter against another to achieve required optical performance might be obscured in an average model. The course of fans of parallel meridional laser beams (at Odeg and at an oblique angle to the lens axis) through each lens was compared with the calcutated paths of rays through its model. The good agreement between models and measurements indicates that the lens models are good physical and optical representations of the isolated cat lens. The isolated cat lens has been shown to be rather rigid (Fisher, 1971),and it was thought at first (Vakkur & Bishop, 1963) that no shape change occurs upon accommodation. The isolated cat lens, however, difTers slightly in shape from the lens in the ~~~~a~ state (Suzuki, 1973; O’Neill & Brodkey, 1970), exhibits elasticity (Ejiri, Thompson C O’Neill, 1969) and changes its optical properties with deformation (Sunderland 8r O’Neill, 1976). The isolated lens probably corres~~s to the accommodated state in viva. METHODS 3;he
isolated fens
Five lenses were obtained from the kft eyes of five adult cats anaesthetised with halothane. (The right eye of each animal was used in a companion study of the optics of the cat eye.) The globe was cut posterior to the ora serrata, and the anterior segment piaeed on cotton wool saturated with saline solution, to avoid pressure damage to the lens. The vitreous was gently removed~ and the cornea was cut away, leaving some of the zonule fibres attached to the kns. Subaeq~t handling of the tens was only by means of these fibres. The leas was susPended in a small rectangular glass walled e fflkd with physioIogicai saline solution, and photograPhed with an adjacent scale from a dire&m along its axis, and perpendicuiu to its axis (Fig. 1). Photography was carried out with a 35mm camera with a 1SOmm focal length telephoto objective on Kodak 2415 8lm and
prints were made on dimensionally stable waterproof paper. The long focal length objective allowed photographs to be made from a distance sufficient to reduce the effect of perspective and allow the true shape of the surfaces to be recorded. Checks for distortion in the camera and enlarger optics were carried out by photographing a machine ruled grid and a precision ruled scale with magnifications similar to those used for photographing a lens, making prints and examining them. At the edge of the print, distortion was less than -i-0.3% (pincushion type; Kingslake, 1978). Because distortion can be expected to depend on the third power of the distance from the print center, avoiding the print outer regions reduced any error due to distortion to less than other measurement errors. The external surfaces of the lens were fit to conic sections by digitizing a print of the lens and employing a curve-fitting computer program. ~~~tio~
of laser beams by the isolated lens
A set of narrow parallel beams spaced 1 mm vertically generating meridional fans from a helium-neon laser of wavelength 633 nm were passed in the physiologi~l direction through the lens parallel to its axis, and at an oblique angle to the axis. Precise parallelism and spacing of the beams in a fan were ensured by placing the laser axis at 90deg to the lens axis, and spacing each beam by moving a refkcting prism on a micrometer-driven precision translator along the laser axis. Vergence of beams in the fan measured over a 5 m path was less than 0.05 D. A small drop of milk was added to the saline to make the beam paths visible from the side by means of scattering, and the paths of a fan of beams were photographed by multiple exposure. All work proceeded at room temperature, about 20°C. The PtJfrich area1 refractometer For refractometry, the lens was embedded in dental impression material with its axis horizontal on a freezing microtome stage. When frozen (within about 5 mm), sections were removed to the level of its axis. Because the sections were removed from the solidly frozen lens, the lens itself was unaffected by geometrical distortion that might apply to thin sections. After refmctometry rn~~ents were completed, the residual iens half was removed and a plaster cast made of the space it had occupied. Measurements on this cast allowed confirmation that the cut lay within 0.2 mm and 2 deg of the lens axis.
Fig. 1. An isolated lens (No. 4) seen from the side suspended in saline by zonule fibres before refractometry. The anterior surface faces left. Scale: 5 mm.
725
Fig. 3. Series of Ptdfrich anal refractometer photasraphs of the cut surface of a cat kns (No. 4) bWted in a plane in&ding the axis, with anterior pole facin8 left. The outer boundary d each phMqr& is the is&&i&d curve for the i&x and camera elavation angle (from luh to right): 1.396 (48.2 da& 1.4fO (U.Oda& 1.430 (41.8dq& and 1.450 (38Sdcg). The photographs were taken in hi@ of SWmn wavelength. l&b psrotosrsdr is for&d by an amount given by equation (5) along the axial dinxtion (the hosir.Wal dire&on in thcpt pho~ogaphs). The centml dark area is a region of bad optical contact between the lens material above n = 1.46 and the prism.
726
727
Fig. 2. Diagram (elevation; not to scak) of the Fulfrich areal mfmctom&r. The&=rilbtJngleprirm ofindexN,Pndsidelmgthsisseenfromtheside,withthclear,~inaptneiad~ibuir, on the bottom face of the prism with ib anterior pole facing Mt. The camera, with artnna rpcshue at 0, is mountad on an arm pivoting vertically ahout the prism vertex V. The camera akvatkmaagkisa. A general ray kaves a point P of index n on the cut kns surface. is refracted at the vertkal prirm fw, and proceeds to the camera at the devation angk 4.
Pulfrich arcal refractometry was carried out by placing a glass right-angle prism of index N, = 1.5820 at 550 nm with one side horizontal on the froxcn cut kns surface, and the other vertical and parallel to the lens equator (Fig. 2). The dispersion curve of the prism glass had been measured by the method of minimum deviation for eight mercury lines between 405 and 691 nm (see legend to Fig. 4). The lens, which had become white and opaque upon freezing, was then allowed to thaw, becoming crystal clear again. Because the lens half was supported by the embedding material, distortion of the lens shape due to freezing and thawing was small. Figure 7 shows that the initial dimensions (vertical solid lines) differ little from the final dimensions after thawing (vertically arrayed points). The prism surface made optical contact with the cut surface of the lens. The cut surface was illuminated with light from a halogen incandescent lamp passed through an interference filter of wavelength 550 nm and bandwidth at half-maximum of 30nm. This wavelength is close to the peak of the cat’s green cone photopic system (Loop, Millican & Thomas, 1987). As the cut surface was observed through the vertical prism face the region of index higher than that defining the critical angle for internal reflection at the lensprism interface appeared light because of total internal r&&ion within the prism, and the
region of lower index appeared dark. This will occur for inflnitcsimahy small areas of lens, and depends only on the index value. The refractometer can therefore operate properly on gradient-index kns substance, with the lightdark boundary d&ning the shape of the isoindicial curve. By varying the vertical angle (elevation) of observation, isoindicial curves for various vale of n within the lens could be observed and photographed (Fig. 3). In practice some 20 isoindicial curves were photographed per lens. The instrument’s operation is governed by the following refractometer equations (l-5), which apply to Fig. 2. The ray leaving a point P of kns index II at the critical angle for internal r&cction follows the relation: NE-b/u = n ~sin(90°)- PJand thus; a = b-N&t.
(1)
From Snell’s Law at the vertical prism face: sin 4 = NC-c/a;
(2)
c==Jp=F).
(3)
and clearly
These equations ddlnc theangle 4 for observing
the isoindicial curves of refractive in&x II at the wint c _~.~.P. To relate them to the stationarvI 1DriSm
W. S. JAGOER
728
additionally minim& this error. The precision and accuracy of the machine-divided graduated sector used to position the camera were better than 1 min arc, and the scale reading was to within 0.1 deg, resulting in no more than 0.0007 error in index. Photographs of isoindicial curves are foreshortened in the direction of view by the factor: b/(c ‘COSf#J);
cAMERAasv~T(owuuaE
a
Fig. 4. Plot of lens isoindicial curve index VCAIU(L autwa elevation angle a for the Pulfrish arcal refractometer. These six curws mprcwt ~0Mions to tht refrnctometer squatiorm for each of six wavdc#hs. From the I&, these wavek@lS,WiththCprirm&lSlIidXitlpacontho#,~: 700 rtm (l.S72S),650nm (l.S753), 600 mu (1.578Q).550 mn (1.5820), SQonm (1.5875) ad 45onm (1.5947).
vertex Vabout which the camera elevation angle u is measured:
(5)
and were corrected by digitising the curve, multiplying the appropriate coordinate by the necessary factor to correct the foreshortening, and plotting the corrected curve. Foreshortening can be minimised over the range of index values occurring in a lens by choosing NI to allow the broad curve of foreshortening to fall with its minimum in the midrange of measured index values. For the cat lens, the prism index NE= 1.5820 (550 nm) is close to this optimal value. The minimum foreshortening factor of 2.81 falls at about n = 1.39. Dispersion of the lens substance
Dispersion (variation of refractive index with wavelength) of the lens material was molwtd tan$=(L+sina -c)/(L.cosa -s); (4) with a series of interference filters of bandwldtb whereListhediatancefromthepriamvertex V 30 run. For a series of niue elevation settings, the equatorial width of the isoindicial curve was tothecameraaperture,andsisthelengthofthe measured for each of six wavelengths batween prism side. These equations were solved munuitally for a as a function of n or vice versa for 450 and 7OOnm. A plot of nine cures of this given values of L, s, IV,, and b (the disQusa5of width (ordinate) vs wavelength ( ) was the centre of the lens to the vertical prism face). made, with elevation angle as paraanettr, and One degree of camera elevation angle u corre- a non-linear interpolation was made between sponds to about 0.0065 units of n. The angle a curves to find the elevation setting for each and n are approximately linearly related for the wavelength that wouId yield the isoindicial range of refractive index of interest (Fig. 4). The curve width measured for 550 nm. Points on this arm supporting the camera pivoted verucally horizontal line of constant curve width clearly about the prism vertex V, and 45 deg camera refer to the same lens substance. The refractelevation was de&red by sighting through the ometer equations were then solved for the lens camera along the hypotenuse of the prism. index using these elevation settings and the Desired camera elevation angles were set on a known dispersion curve of the p&m glass. These points yield the dispersion curves of Fig. 5. 200mm radius precision graduated sector. The lens cut surface has a finite size, and to minim& the error caused by viewing different Calculations on model lenses ‘&e paths of meridional fans of incident rays parts from di&rent aagles, the camera entrance aperture was placed 1010mm distant (a 450 mm through a model lens were calculated by means focal length telephoto camera objective was of the ray-tracing program Drishti (Snds, used to yield adequate image sixe). This reduces 1984).This powerful program is able to treat the the error across a lens 8 mm in axial dimen- aspheric surfaces and refractive index gradients sion to 0.001 index units. The finite size of the found in animal eyes. In addition to calculating camera aperture will cause an isoindicial bound- cardinal properties, it is able to trace ray paths ary unsharpness error, e&mated to be about through an inhomoganeous model lens by 0.001 units for the 6 mm entrance aperture used. solving the difIerential quations descr&ng the The film was developed to high contrast to paths using a Runge-Kutta-Merson algorithm
729
Optics of the isolated cat lens
[
'.44-
Y F
1.42 -
distort the lens slightly by their tension, changing the surface s&e and decreasing the axial thickness. Viewed along the axis, the isolated lens is very nearly circular (diameters differ less than 1% from each other) and axial symmetry with coincidence of optical and physical axes was assumed. The external shape of the lens viewed from the side (Fig. 1) was fit to the anterior and posterior conic curves of polar radius RO and conic constant CC (Table 1). The standard error of estimate of the sag on the aperture averaged over the ten surfaces was 0.019 mm, indicating very good fits. Additional higher-order parameters were therefore judged unnecessary. RO and CC are defined by:
Y L E
1.40-
1.38 -
1
1
I
I
RO = A :/A,
1
450500ssoamaio
700'
ULIVELEMGTH tnll ,
Fig. 5. Dispersion of cat lens substance in the range 45&700 mn. Beaulse the full set of CmVeswas not mcaswed for any one of the five knses, this complete set was measured on an additional kns. Agmemcnt between data from diiTermtknm?swas~.EadlcmVeKmsistsofstraiglltlines connecting six points, calcula+d at each of the abscissa w-avekngths.
and
CC = (AJAX)* - 1;
where A, is the semi-axis perpendicular to the lens axis, and A, is the semi-axis parallel to the axis. A circle has a conic constant equal to xero, that of an oblate ellipse is greater than zero, and that of a prolate ellipse lies between - 1 and 0. Axial thicknesses measured from lens photographs are also shown in Table 1. Puljiiich area1refractometry
(not by using a series of homogeneous shells). The position of best focus can be found by minimising the r.m.s. radius (R.r.m.s.) of a pencil or fan of rays. The R.r.m.s. is relevant to the image-forming ability of a lens, because it is a measure of ray concentration, while the minimum total width of a pencil or fan may reflect only the course of highly aberrated rays. Optical conventions used are those of modern optical engineering practice (Smith, 1966; Ringslake, 1978). Figures are oriented so that light travelling in the physiological direction proceeds from left to right, in the positive direction. Modem convention differs from optometric usage regarding longitudinal spherical aberration. A double convex glass lens with spherical surfaces will focus marginal rays closer to the lens than rays near the axis. Modem convection designates this as undercorrected longitudinal spherical aberration, of negative sign. To avoid confusion, the state of correction will also be given. RESULT!3
External shape of the isolated lens The isolated cat lens is free of xonular fibre tension. In the intact eye, the zonular fibres
A series of Pulfrich area1 refractometry photographs is shown in Fig. 3, taken with the apparatus illustrated in Fig. 2. The outer boundary corresponds to the isoindicial curve. The dark area in the centre lies above n = 1.46, and is apparently caused by inadequate optical contact of material above this index value to the glass prism. This feature was seen in all five lenses, accounting for about onequarter of the cross-sectional area and about 13% of the total lens volume. Data from this type of photograph, with each isoindicial curve individually corrected for foreshortening and superimposed, are shown in the form of a contour map of refractive index in Fig. 6. The irregularity apparent in the curves is probably due to a combination of intrinsic irregularity and measurement error. Although this map shows only nine curves for clarity, typically twenty photographs per lens were taken. The full set of data allowed the refractive index along and perpendicular to the axis to be determined (Fig. 7). Dispersion of lens material over the range 450-700 nm is given in Fig. 5. As the lens centre is approached, and the index rises, the curves become steeper. The Abbe number vd, an inverse measure of dispersion defined as: vd= [n(588 nm)- l]/[n(486 nm) - n(656 nm)]
730
W. S. JAGGER
Individual lens models Each of the five lenses was modelled individually, using the same principles struction. parameters
The minimum
was used to adequately
physical structure
Fig.6.Isobdic&-ofnsc&onthrou&atypk&ut kns (No. 4) includingthe axis with antctior pole facingleft. Thccllrves have bcm cJxmctd for fofcshortening.This diagram shows for clarity only nine curves of a total of about twaty that were nconled ftx each lens. The mkp of index is from 1.380 to 1.460 spaced 0.01 in index. SC& 5 mm.
was evaluated by interpolation b&wecn data points of each curve by mcatl~ of a secondorder least-squares curve fit. Values of the Abbe number range between 50 at n = 1.38, decreasing to about 40 at n 3: 1.46.
of model con-
number, nine, of model specify the
of each lens at
550 nm, based on the external shape and the internal form of the isoindicial curves and their spacing. Each lens model was assumed to have rotational symmetry about its physical and optical axis, so that each isoindicial curve defines an isoindicial surface when rotated about the axis. The following discussion is concerned with plane sections containing the axis. The external shape was fully specified by the polar radii and conic constants of the anterior and posterior surfaces, and the total thickness, as discussed above. The capsule is so thin as to be optically ineffective, and was not treated separately. The internal structure was mod&d by a set of isoindicial curves spaced following a parabolic rule of the form given by Matthiesscn in 1877: n = NC,,- (N,, - N,,,)WW;
(6)
where r is the distance from the centre to the element of index n, and R is the distance from the core to the cortex. As discussed in the introduction, the parabolic rule has been found to apply to a variety of lenses. The model parabolic rule, shown as solid curves in Fig. 7, describes the measured index well (points in Fig. 7) from the cortex to the highest measured isoindicial curve. The small
Fig. 7. Isoindibal curve width in the udal and squrtorial @erpendiculpr to the axis) directions for a typiud lens (No. 4) The solid mrva duaibe the parabolicvaribon of index used in UICmodel of this lass. The vertical lines are the lens boundaries.
Optics of the isolated cat lens
rounding at the edges of the measured curves is probably due to a small fluid redistribution across the lens edge; drying cannot occur because there is no contact with air. Values of index used in the models are based on the solid curves, which compensate for the rounding. Measurements in this laboratory also show that the parabolic rule applies to cats younger than about four months, in which no central area of bad optical contact occurs, allowing measurement up to the peak index (which lies below 1.46). Therefore, on the weight of the overwhelming evidence from work on other lenses, and the agreement with the present index measurements, it was assumed that this rule of increase holds for the fully mature animal to its peak index value. The values of the model core and cortex indices were found from curves of the type shown in Fig. 7. Each model isoindicial curve consists of an anterior and a posterior part. Each part is half of an ellipse, with major axes coincident and perpendicular to the axis. The minor axis of each ellipse was chosen to fit the observed isoindicial curve shapes. Because the major axes of both anterior and posterior ellipses coincide, discontinuities in the curves or their derivatives at the point of joining the anterior and posterior mode1 curves are avoided. These curves therefore meet smoothly at the equatorial plane, as the measured isoindicial curves indicate. As the refractive index increases, the shape of the isoindicial curves was found to remain very similar geometrically. In the absence of a trend away from geometrical similarity, the remaining isoindicial surfaces in the unmeasured central 13% of the lens volume were assigned the same geometrical shape as those of the remainder of the lens. The axial location of the index peak varied from lens to lens, and its position was incorporated in each model. An axial section of a cat lens model is shown in Fig. 8. The smooth
Fig. 8 Section of a typical cat lens model (No. 4) including the axis showing the internal refractive struch~rc. The isoindicial curves for the anterior (left) part arc sets of half-&pm geometrically similar to each other, and likewise for the poF&ior part (right). Their indices follow the parabolic rule of equation (6) as the ccntre of the lens is approached. and the curves shown arc those for n = 1.3900-1.4900. spaced 0.01 units apart. Ticks on the axes arc spaced 1 mm apart.
mode1 curves are free of the irregularity seen in the measured curves (Fig. 6). The set of nine independent parameters needed to describe the physical structure of each lens is shown in Table 1, together with their average values. Optical properties of the lens
Figure 9a is a photograph showing the paths of a meridional fan of laser beams incident parallel to the kns axis. The distance from the lens posterior vertex at which each beam crosses
Table 1. Model parameters for each of the five isolated cat lens models. The average value of each parameter is shown in the last column. Lens thicknew is measured along the axis. Index peak location (IP) is measured from the anterior surface. MA ind&tes the major axis of the outer isoindicial curve. Refractive indices are for 550 MI. Radii and d&anccs are in mm LAxls Anterior RO Anterior CC Thickness Posterior RO Posterior CC IP MA NX.NX VR W-F
1
2
3
4
5
AWraEC
6.63 0.37 8.09 -7.44 0.73 4.14 10.74 1.3870 1.4683
6.86 0.28 8.34 - 7.85 0.74 4.15 11.04 1.3890 1.4774
6.88 0.31 8.58 -7.55 0.54 4.18 11.34 1.3880 1.4818
7.42 0.31 8.73 -8.16 0.45 5.01 12.12 1.3857 1.4902
7.36 0.47 8.41 -8.19 0.81 4.52 11.51 1.3890 1.4890
7.03 0.35 8.43 -7.84 0.65 4.40 11.35 1.3877 1.4813
732
W. S. JAGGW~
(b)
J
1
I
1
1
6
8
10
12
14
RAY cslosww
l.ocATKm
mm
Fig. IO.(a) L.ongitudinal spherical aberration of an M&d cat lens (No. 4). Cid show the dktance (abscissa) from the Iens post&or vertex at which irkdent j7eraM laser beams of varkxu he&&s (ordittate) croastheaxisafternfractionby~Isnr.Tbecolidcurwcisth!longit~~sphericniPbarption~ for the model of this lens. Meamued points for + 1 and - 1 mm ray heights an not shown because the shallowanslckrtw#athenfiscradbepmtlBdthePrisco~largeuncertsintyintheaxis~ location. (b) Lot@tud&A absmrion along the cekf ray for an obliquely incident (54 dog) ttmidioti fin of parallel rays (lens No. 4). Absoissa: positkm of ray i&me&on with the chief ray mausumd along the chief ray from the pIane taqpnt to the po6Wior lens pole. Ordbate: distance of the krcidcet ray from the chief ray in the hxident paraM fan. So&l curve: calculated aberration for the model of lenu No. 4. Points: aberration measured from a fan of laser beams incident upon lens No. 4 as in Fig. 9h.
the axis after refraction is shown in Fig. 1Oa vs incident beam height. All five lenses displayed positive longitudinal spherical aberration (overcorrection; that is, marginal rays focussed farther from the lens than did rays close to the axis) ranging from + 2.9 to +4.4 mm for incident ray height 5 mm. This is clearly seen in Figs 9a and lOa. Figure 9b shows a similar meridional fan, but incident at 45 deg to the lens axis. It is clear that the lens still produces a tight focus. The fan width in Fig. 9b at the position of best focus is 0.55 mm. Calculated optical prqerties of the models A meridional fan of rays equivalent to the 633 nm wavelength laser beams shown in Fig. 9a was traced through each model lens using the ray-tracing program Drishti. To perform this ray tracing, the core and cortical indkes measured at 55Onm had first to be corrected downward to their values at 633 nm. Dispersion measurements (Fig. 5) show that cat lens substance measurccd at 550 nm to have an index of 1.450 has an index at 633 nm about 0.005 units
lower. For lens substance of index 1.380 at 550 inn, this difkence is about 0.002. The surrounding fluid index was 1.333at 633 nm. Small additional adjustments were then made to the values of the core and cortical lens indices, within their ranges of measummen t error. Final index values for 550 nm used in the mode& listed in Table 1, result in the solid curves of Figs 7 and 10, showing good agreement with the measured points of refractive indq longitudinal spherical aberration, and oblique km&tudinal aberration measured by means of laser beams. The calculated positions of best focus are indicated on Figs 9a and b. The width of the oblique fan calculated for the model of the lens of Fig. 9b is 0.49 mm at best focus, close to the measured value of 0.55 mm. Similar good agreement between measurements and calculations on each model was seen in all lenses. The calculated optical properties of eaoh individual model are given in Table 2. These quantities were also calculated for the “average” lens posserkng the average parameter vahaes shown in Table 1.
Fig. 9.(a) The upper photograph shows an isolated lens (No. 3) suspended in a saline bath by zonuk fibres. A meridional fan of parallel laser beams incident from the left spaced I .OO mm apart is refracted by the lens and crosses the axis to the right of the lens. Note the curved beam paths within the kns, a result of the index gradient. The beams of outer zones cross the axis farther from the lens than do the inner rays, showing overcorrected (positive) longitudinal spherical aberration. The left arrow indicates the position of paraxial focus, and the right arrow indicates the position of best focus (minimum R.r.m.s.), calculated for the model of this lens. (b) The lower photograph shows the same kns with beams of the meridional fan incident at 45 deg to the lens axis. The position of best focus calculated for the model of this lens lies between the arrow points.
733
Optics of the isolated cat kns
735
Table 2. Optical propertiescalculated at 633 nm wavekngth for each of the isolated cat kns models and for the lens composed of average parhtkr v&es. Anterior principal point position (PP) is measumd from the anteriorkns pole; the posterior principal point position is measuredfrom the posterior pole. Nodal points coincide with principal points. Paraxialfocal kngths (FL) are measuredfrom the principal points. The paraxkl bath focal dktance (RFD) is measuredfrom the posterior kns pole. Longitudinal sphetical aberration(LSA) is measured from the pamxial focus for ray height 5 mm. Units (except for power) are mm 1
Lens
2
3
4
5
Av. Lens
Anterior
PP FL Posterior
3.86 - 24.46
3.96 -23.43
4.10 - 22.94
4.16 - 23.58
3.99 -22.88
4.01 - 23.39
-3.97 24.46 20.49 2.9 54.50
-4.09 23.43 19.35 4.4 56.88
-4.19 22.94 18.75 3.5 58.11
-4.23 23.58 19.3s ;::3
-4.69 22.88 18.79 4.0 58.27
-4.11 23.39 19.28 4.0 56.99
FL BFD LSA Power, D
The sensitivity of the optical properties of a model lens to changes in the model parameters is shown in Table 3. The effect is shown of a small change in the size of each parameter, approximately of the sire of the measurement uncertainty of that parameter, upon lens focal length, longitudinal spherical aberration, and the minimum R.r.m.s. and position of best focus for an oblique fan. DISCUSSION
Physical and optical nature of the lens The surface of a cat lens is easily deformed, yet regains its form after small deformations when pressure is released (Ejiri et al., 1969). If the capsule is removed, and the lens held between the fingers, the outer jellylike material is easily rubbed away, leaving increasingly solid material as the centre is approached, until finally only a hard and gummy core material remains. It is therefore not surprising that the
curves of constant refractive index show the form seen in Fig. 6 and the increase of index as the centre is approached shown in Fig. 7. The method of Pulfrich areal refractometry has the considerable advantage of yielding the shape of the isoindicial curves for 87% of the lens volume and their relative spacing directly. The slight disadvantage of having to freexe the lens before sectioning could cause a dimensional change of up to 1 or 2%, which is probably reversible upon thawing, when the actual measurement is made. Redistribution of water during the quick freezing process is also possible, but is probably inconsequential, given the success of the models based on the method to predict the optical behaviour of the intact lens. Freytag (1910), quoted by Philipson (1969), found a cortical index for the cat lens of 1.38-1.40 and a central value of 1.46. These cortical values agree with the range of model values used here (1.3857-l .3890). Freytag’s central value of 1.46 corresponds to the highest
Table 3. Effectof small changes in the nine independentmodel pammemmupon the calculated optical properties of a typical model kns (No. 3). Chant& the outer isoindicial curve major axis al%ctsall isoindicial curves proportionally,becausethey remaingeometricellysimi~.Theoptical~rhownue~lfocrl~ (FL), longitudinal spherical aberrationof a ray of height 5 mm (LSA), the minimum R.r.m.r. of an obliquely incident meridional fan of width 10mm huzidentat 45 deg and the coordinates of the position of best focus of this obliqtk fan. The initial FL and LSA are 22.94 mm and 3.50 mm, mspectively. The initial oblique R.r.m.s. is 0.218 mm. The initial position of oblique best focus is x = 8.928 mm from the posterior lens pok. and y - 13.306mm above the axis. Units are mm Delta parameter
Anterior RO Anterior CC Thickness Posterior RO Posterior CC IP MA J”N,
+O.lOmm +0.05 +O.Olmm +O.lOmm +0.05 +O.lOmm +0.10 mm +0.001 +0.001
FL
LSA
R.r.m.s.
x
+0.050
+0.281 -0.358 -0.040 -0.228 -0.170 +0.114 -0.319 -0.194 +0.041
+0.006
+0.092 -0.067 +o.otM -0.019 -0.034 +0.013 +o.oss +0.028 +0.091
+o.ooo +0.016 -0.028 0.000 +0.17a +0.275 +0.097 -0.182
+0.002 -0.001 -0.004 -0.006 -0.004 -0.005 -0.906 +0.003
Y
+0.081
-0.059 +0.012 - 0.028 -0.040 +0.029 +0.052 +0.025 +0.091
736
W. S. JAGOER
directly measured index before poor optical contact between lens substance and refractometer prism would appear (Fig. 3), limiting his maximum measured value to less than the assumed model central value of about 1.48. The series of nine dispersion curves of lens material shown in Fig. 5 seems to show no discontinuity, and probably results from a smoothly decreasing water fraction in the lens as the centre is approached. The value of the Abbe number decreases from 50 at the periphery to about 40 in the near-central region of index 1.46. Palmer and Sivak (198 1) published dispersion data for various animals and man, and the shapes of their curves are generally similar to those found here for the cat. However, to calculate values of V (almost identical in definition to the Abbe number) in their Table 1 they apparently used individual point values, which reflect scatter, rather than points on the smooth curves drawn through their points, which are a better representation of their data. Dispersion curves of transparent homogeneous media such as aqueous solutions, glasses, and organic solvents do not demonstrate fine structure in the visible region, but rather show smooth curves, as is to be expected on physical grounds, and the same is to be expected for lens substance. Their procedure has a drastic effect on the values of V, vitiating its ability to characterize dispcrsive behaviour, and their Table 1 is to be interpreted with care. If points on their smooth curves are used to calculate V, both rabbit and pigeon show a decrease in V as the centre is approached, rather than an increase, and the unrealistically high value for the human peripheral V value decreases. The optical behaviour shown in Fig. 9a indicates overcorrection for longitudinal spherical aberration on the axis. This behaviour, clearly evident in all five lenses, was noted by Vakkur and Bishop (1963), but was not seen by Sivak and Kreuzer (1983), who found spherical aberration of both signs in the cat lens. Their Fig. 7 expresses longitudinal spherical aberration in terms of back focal power. If the range of longitudinal spherical aberration found in the present study, + 2.9 to +4.4 mm, is expressed also as back focal power, it is +5.8 to +9.OD. This amount of longitudinal spherical aberration is very small on their scale. It may be lost in the variation shown by their points within the 1 mm zone. Small uncertainty in beam position due to irregular lens structure or measurement error will strongly affect the calculated position
of beam axis crossing for beams within the 1 mm zone because of their shallow angle with the axis. This may account for the relatively large irregular spherical abberation these authors found for the cat lens within this zone. In considering spherical aberration of the isolated lens, it should be pointed out that in uivo, rays impinging upon the lens from a distant point object are not parallel, but of course are convergent, having been refracted by the cornea. Lens axial spherical aberration in unaberrated convergent light decreases as the degree of convergence increases. Its value decreases from that for parallel light as the incident light’s virtual focus approaches the lens. It becomes zero for a virtual focus about 6 mm from the lens posterior surface, and becomes negative as the focus moves still closer. The spherical aberration of the cornea-lens combination may be more or less than that of the lens alone, depending on the ability of the cornea to compensate for the lens aberration. The purpose of using parallel rays in this study is not to equate directly the spherical aberration found with that contributed by the lens in the intact eye, but rather to serve as a check on the accuracy of the models developed. Nature of the lens models The strength of the type of lens model presented here is that it is constructed from the external shape of the lens and from data from the Pulfrich area1 refractometer. The gap in measurements near the core of the lens corresponds to only about 13% of the lens volume. The assumed structure in this region is firmly based on the overwhelming evidence from work on other lenses confirming the parabolic rule of index increase and the good fit of a parabolic curve and the uniform trend of geometrical similarity of isoindicial surfaces in the remaining 87% of the lens. There are no freely adjustable parameters in the models. Only minor adjustment in the values of the cortex and core indices, within their ranges of experimental error, was necessary to make all five individual models predict the paraxial focal length and longitudinal spherical aberration (and hence compkteiy specify the axial imagery), the axial position of best focus, and the longitudinal aberration of a meridional fan incident at an oblique angle to the axis. They are therefore good physical and optical representations of the isolated lenses. Inverse ray tracing methods suffer from the criticism that they may result in a lens model
Optics of the isolated cat lens
that is not unique, and does not correspond to the physical structure of the l&s. It is not tr&! in general that a lens model that predicts axial spherical aberration of the lens is unique, unless it is also shown that the lens internal and external structure is spherically symmetric. In the absence of this symmetry, it must be shown that a model can predict off-axis optical properties. Sensitivity of a model lens to parameter changes
The optical properties of the models used for each of the five lenses studied are not drastically affected by changes in their measured parameters of a size approximating their measurement uncertainty. Table 3 shows the effect of such small changes in each of the nine lens model parameters upon the axial and oblique optical properties of a typical model, No. 3. The anterior and posterior polar radius changes have relatively weak effects on the optics of the lens; the anterior radius change affects the focal length more than does the posterior change largely because of its greater distance from the focus. The conic constants do not afIect the paraxial focal length, as these constants have no influence on surface curvature in the infinitely small paraxial region. They do, however, affect the surface curvature off axis, and therefore influence the longitudinal spherical aberration, Increasing the conic constant will increase the curvature of the per-ipheral lens surface (for a given polar radius), giving marginal rays a shorter focus and decreasing the longitudinal spherical aberration. Positive axial displacement of the peak location of the refractive index distribution increases the focal length and the longitudinal spherical aberration. Elongating the isoindical curves in the equatorial direction increases the focal length, while decreasing the longitudinal spherical aberration. The cortical and core index values were adjusted slightly for final fine tuning. of each model. While increasing the cortex index increases the paraxial focal length and decreases the longitudinal spherical aberration, increasing the core index has the opposite effect. However, each index affects the paraxial focal length and the longitudinal spherical aberration to differing degrees. The increase in cortex index decreases the longitudinal spherical aberration about twice as much as it increases the focal length, while the increase in core index decreases the focal length nearly five times the
737
amount it increases the longitudinal spherical ab&&atibn*
Advantage of a lens with a refractive index gradient Why has such an apparently complex structure as a lens with a gradient index evolved? The answer lies largely in aberration control in the whole eye, and the ability to accommodate, and will be treated in the companion paper on the optics of the cat eye. However, one important advantage of a gradient index is that it allows construction of a high power lens from material of low index. This can be intuitively understood by the observation that a non-axial beam traversing a gradient-index lens is continuously refracted within the lens and follows a curved path (Fig. 9). We can construct a homogeneous model lens of the same shape as a normal lens with a constant index (the “total index” of Matthiessen, 1880) such that the paraxial back focal distance is the same as that of the actual lens. The index required is 1.559, much higher than the core index in the inhomogeneous lens. A rule given by Matthiessen on the basis of lens structure with a parabolic index gradient predicts that the differena between the total index and the core index is qua1 to the difference between the core index and the cortical index. For model No. 3, these two quantities are in fact of similar size, 0.081 and 0.085, respectively. For the homogeneous lens, however, the axial longitudinal spherical aberration is - 10.1 mm, instead of +3.5 mm for the gradient index lens. The axial position of meridional fan best focus lies 9.8 mm inside that of the actual lens, where its axial R.r.m.s. is 0.55 mm instead of 0.15 mm for the inhomogeneous lens model. At 45 deg off axis, the position of meridional fan best focus is 3.7 mm inside that of the inhomogeneous lens, and its focussed R.r.m.s. at 45 deg is 0.39 mm, compared to 0.22 mm for the gradient index model lens. In short, the optical properties are grossly different from those of the gradient index lens, and the entire eye, including cornea and retina, would require redesign to use such a lens. A further problem is the biological production of a lens with the high index of 1.559, which is much higher than that of the core of fish lenses, about 1.51 (Matthiessen, 1880), which probably represents the upper limit to a biologically possible lens index. These cores are very rigid, and accommodation by shape change would be out of the question for such a lens.
W. S. JAOOER
738
Acknowledgements-This work was supported by the Australian National Health and Medical Research Council project grant No. 870792 and the Viktor and Ema Hasselblad Foundation. The National Vision Research Institute is an affiliate of the University of Melbourne.
REFERENCES Axelrod, D., Lemer, D. & Sands, P. J. (1988). Refractive index within the lens of a goldfish eye determined from the paths of thin laser beams. Vision Research, 28, 57-65. Bando, M., Nakajima, A., Nakagawa, M. St Hiraoka, T. (1976). Measurement of protein distribution in human lens by microspectrophotometry. Experimentcrl Eye Research, 22, 389-392. Bouts, J., F&lirh, H. J. & Hockwin, 0. (1987). Age-related changes in water and crystallin content of the fetal and adult human lens, demonstrated by a microsectioning technique. Ophtkalmic Research, 19, 235-239. Campbell, M. C. W. (1984). Measurement of refractive index in an intact crystalline kns. Vision Research, 24, 409415. Campbell, M. C. W. & Hughes, A. (1981). An analytic, gradient index schematic lens and eye for the rat which predicts aberrations for finite pupils. Virion Research, 21, 1129-1148. Chan, D. Y. C., Ennis, J. P., Pierscionek, B. K. & Smith, G. (1988). Determination and mode&g of the 3-D gradient refractive indias in crystalline lenses. Appried Optics, 27, 926-931. Ejiri, M., Thompson, H. E. & O’Neill, W. D. (1969). Dynamic visco-elastic properties of the kns. Vision Research, 9, 233-244. Fagerhohn, P. P., Philipson, B. T. & Lindstr6m. B. (1981). Normal human lens--the distribution of protein. Experimentol Eye Research, 33, 615420. Fisher, R. F. (1971). The elastic constants of the human lens. Journal of Physiology, 212, 147-180. Freytag, G. (1910). Die Brechungsindkes der Lime und der flilssigen Augenmedien bei der Katz und beim Kaninchen. Archiv fiir vergkichenckr Ophtltabnoh?gie, 1, 61. Hughes, A. (1986). The schematic eye comes of age. In Pettigrew, J. D., Sanderson, K. J. & Levick, W. R. (Eds.), Visual neuroscience. Cambridge: Cambridge University Press. Kingslake, R. (1978). L.ms &sign fwrdoment&. Orlando: Academic Press. Kieberger, E., Pldhn, H.-J. & Simonsohn, G. (1968). Zur
Brechungsindexverteiiung in der Augenlinse. Abecht von GrPefPs Archiv fiir kliniscke und experimenteile Ophthalmologic, 176, 155-159. Loop, M. S., Millican, C. L. & Thomas, S. R. (1987). Photopic spectral sensitivity of the cat. Journal of Physiology, 382, 537-553. Marshall, J., Mellerio, J. & Palmer, D. A. (1973). A schematic eye for the pigeon. Virion Research, 13, 2449-2453. Matthiessen, L. (1877). Grrsldriss &r Dioptrik geschichtete Linsensysteme. Leipzig: Teubner. Matthiessen, L. (1880). Untersuchungen ilber den Apianatismus und die Periscopie der Krystalllinsen in den Augen der F&he. P&gem Archiv fur die gesammte Physiologic, 21, 287-307. Nakao, S., Fugimoto S., Nagata R. & Iwata K. (1968). Model of refractive index distribution in the rabbit crystalline lens. Journal of the Optical Society of America, 58, 1125-1130. Nakao, S., Ono T., Nagata R. & Iwata, K. (1969). The distribution of refractive indii in the human crystalline lens. Japanese Journal of Clinical Ophthalmoiogy, 23, 903-906. O’Neiii, W. D. & Brodkey J. S. (1970). A nonlinear analysis of the mechanics of accommodation. Vision Research, IO, 375-391. Palmer, D. A. 8c Sivak, J. (1981). Crystalline lens diiion. Journal of the Optical Society of America, 71, 780-782. Philipson, B. (1969). Distribution of protein within the normal rat kns. Investigative OphtMmology, 8,258270. Sands, P. (1984). Drishti, a computer program for analyaing symmetrical optical systems incorporating inhomogeneous media. Tech&al Report 8, Canberra: CSIUO Division of Computing Research. Sivak, J. G. & Kreuxer, R. 0. (1983). Spherical aberration of the crystalline lens. Vision Researek, 23, 59-m. Smith, W. J. (1966). Modern opticalengineering. New York: McGraw-Hill. Sunderland, H. R. & O’Neill, W. D. (1976). Fun&o4 dependence of optical parameters on cimumfemntial forces in the cat lens. Vision Reseurch, 16, 1151-1158. Suzuki, H. (1973). Observations on the intraocular changes associated with accommodation: An exparimental study using radiographic technique. fiprrineental Eye Research, 17, 119-128. Vakkur, G. J. & Bishop, P. 0. (lW3). The schematic eye in the cat. Vision Research, 3, 357-381. Van Heel, A. C. S. (1959). Op&l instruments. In Kronig, R. (Ed.) Textbook of physics (pp. 795-868). New York: Pergamon Press.
0042-6989/9053.00+ 0.00 CopyrightQ 1990PcrgamonPras pk
THE REFRACTIVE STRUCTURE AND OPTICAL PROPERTIES OF THE ISOLATED CRYSTALLINE LENS OF THE CAT W. S. JAGGER National Vision Research Institute, 386 Cardigan Street, Carlton. Victoria 3053, Australia (Received 22 May 1989; in revisedform 21 September 1989) Abn&act-Direct measurements of the shape and internal refractive index distribution of the isolated cat lens were used to construct individual optical models of the lenses from the left eyes of five cats. The right eyes were used in a companion study of the optics of the cat eye. The lens refractive index spatial distribution and dispersion were measured with a spekally designed Pulfricb areal refractometer. Agreement of calculated ray paths tbrougb these models with the obacrved paths of laser beam fans incident parallel to, and at an oblique angle to the kns axis indicates that the models, which contain no freely adjustabk parameters, are good physical and optical representations of the isolated (accommodated) crystalline knees. Crystalline lens Optics
Refractometer
Lens model
Gradient nfkactive index
INTRODUCTION
The internal refractive structure of the crystalline lens is the key to understanding the optics of the eye, yet has been difhcult to access directly (Hughes, 1986). Present knowledge about lens refractive index gradients and isoindicial surfaces is based on the measurement of the index of small samples taken from a lens (Matthiessen, 1877), interferometry on thin sections (Kleberger, Pliihn & Simonsohn, 1%8), schlieren methods applied to a lens section (Nakao, Fujimoto, Nagata & Iwata 1968; Nakao, Otto, Nagata & Iwata, 1969), and inverse ray tracing to infer the refractive structure (Campbell & Hughes, 1981; Campbell, 1984, Axelrod, Lemer & Sands, 1988; Chan, Ennis, Pierscionek & Smith, 1988). Further approaches measured lens mass concentration or protein distribution (Philipson, 1969, Fagerholm, Philipson & Lindstriim, 1981; Bando, Nakajima, Nakagawa & Hiraoka, 1976; and BOW, Fijdisch & Hockwin, 1987). Measuring the refractive index of local samples from a lens is a tedious procedure, and is subject to location uncertainty of the samples and possible mixing and dilution of the lens substance. Interferometry on thin sections is subject to readily apparent geometrical distortion. Schlieren methods require considerable mathematical processing, as the method yields
Dispersion
Cat
the index gradient rather than the index directly. In addition, an undistorted lens section is required. Inverse ray tracing has the advantage of being performed on an intact lens, but is subject to the criticism that the structure might not be unique and is inferential rather than direct. The distribution of index within the lens has been described by the workers above as following a parabolic increase as the lens centre is approached. The distribution of dry mass or protein within the lens can also be used to indirectly demonstrate the presence of a parabolic increase of index (however, Fagerholm et al. found a plateau of dry mass and hence index in the centre of the human lens). Pulfrich refractometry (Van Heel, 1959) applied to an area was discussed by Marshall, Mellerio and Pahner (1973), who experienced dilflculties when they attempted to use it on the pigeon lens. The present paper introduces an improved Pulfrich area1 refractometer, which, applied to the cut surface of a lens, yields isoindicial curves at desired values of refractive index. If the lens is symmetric for rotation about its axis, and the cut plane includes the axis, the shape and spacing of these curves determine the refractive index throughout the lens. Measurement of the lens shape and the internal refractive structure of the lens allowed the constmction of individual optical models with the minimum number of parameters required to
723
W. S. JACKXR
724
specify their physical structure. The parabolic rule of index increase and the geometrical similarity of isoindicial surfaces found to apply to the 87% of the lens volume for which measurements were made was assumed to apply to the remainder of the lens. Five individua1 models were made rather than only one average model to avoid obscuring correlation of parameters within individual lenses. Any developmental mechanisms that balance one parameter against another to achieve required optical performance might be obscured in an average model. The course of fans of parallel meridional laser beams (at Odeg and at an oblique angle to the lens axis) through each lens was compared with the calcutated paths of rays through its model. The good agreement between models and measurements indicates that the lens models are good physical and optical representations of the isolated cat lens. The isolated cat lens has been shown to be rather rigid (Fisher, 1971),and it was thought at first (Vakkur & Bishop, 1963) that no shape change occurs upon accommodation. The isolated cat lens, however, difTers slightly in shape from the lens in the ~~~~a~ state (Suzuki, 1973; O’Neill & Brodkey, 1970), exhibits elasticity (Ejiri, Thompson C O’Neill, 1969) and changes its optical properties with deformation (Sunderland 8r O’Neill, 1976). The isolated lens probably corres~~s to the accommodated state in viva. METHODS 3;he
isolated fens
Five lenses were obtained from the kft eyes of five adult cats anaesthetised with halothane. (The right eye of each animal was used in a companion study of the optics of the cat eye.) The globe was cut posterior to the ora serrata, and the anterior segment piaeed on cotton wool saturated with saline solution, to avoid pressure damage to the lens. The vitreous was gently removed~ and the cornea was cut away, leaving some of the zonule fibres attached to the kns. Subaeq~t handling of the tens was only by means of these fibres. The leas was susPended in a small rectangular glass walled e fflkd with physioIogicai saline solution, and photograPhed with an adjacent scale from a dire&m along its axis, and perpendicuiu to its axis (Fig. 1). Photography was carried out with a 35mm camera with a 1SOmm focal length telephoto objective on Kodak 2415 8lm and
prints were made on dimensionally stable waterproof paper. The long focal length objective allowed photographs to be made from a distance sufficient to reduce the effect of perspective and allow the true shape of the surfaces to be recorded. Checks for distortion in the camera and enlarger optics were carried out by photographing a machine ruled grid and a precision ruled scale with magnifications similar to those used for photographing a lens, making prints and examining them. At the edge of the print, distortion was less than -i-0.3% (pincushion type; Kingslake, 1978). Because distortion can be expected to depend on the third power of the distance from the print center, avoiding the print outer regions reduced any error due to distortion to less than other measurement errors. The external surfaces of the lens were fit to conic sections by digitizing a print of the lens and employing a curve-fitting computer program. ~~~tio~
of laser beams by the isolated lens
A set of narrow parallel beams spaced 1 mm vertically generating meridional fans from a helium-neon laser of wavelength 633 nm were passed in the physiologi~l direction through the lens parallel to its axis, and at an oblique angle to the axis. Precise parallelism and spacing of the beams in a fan were ensured by placing the laser axis at 90deg to the lens axis, and spacing each beam by moving a refkcting prism on a micrometer-driven precision translator along the laser axis. Vergence of beams in the fan measured over a 5 m path was less than 0.05 D. A small drop of milk was added to the saline to make the beam paths visible from the side by means of scattering, and the paths of a fan of beams were photographed by multiple exposure. All work proceeded at room temperature, about 20°C. The PtJfrich area1 refractometer For refractometry, the lens was embedded in dental impression material with its axis horizontal on a freezing microtome stage. When frozen (within about 5 mm), sections were removed to the level of its axis. Because the sections were removed from the solidly frozen lens, the lens itself was unaffected by geometrical distortion that might apply to thin sections. After refmctometry rn~~ents were completed, the residual iens half was removed and a plaster cast made of the space it had occupied. Measurements on this cast allowed confirmation that the cut lay within 0.2 mm and 2 deg of the lens axis.
Fig. 1. An isolated lens (No. 4) seen from the side suspended in saline by zonule fibres before refractometry. The anterior surface faces left. Scale: 5 mm.
725
Fig. 3. Series of Ptdfrich anal refractometer photasraphs of the cut surface of a cat kns (No. 4) bWted in a plane in&ding the axis, with anterior pole facin8 left. The outer boundary d each phMqr& is the is&&i&d curve for the i&x and camera elavation angle (from luh to right): 1.396 (48.2 da& 1.4fO (U.Oda& 1.430 (41.8dq& and 1.450 (38Sdcg). The photographs were taken in hi@ of SWmn wavelength. l&b psrotosrsdr is for&d by an amount given by equation (5) along the axial dinxtion (the hosir.Wal dire&on in thcpt pho~ogaphs). The centml dark area is a region of bad optical contact between the lens material above n = 1.46 and the prism.
726
727
Fig. 2. Diagram (elevation; not to scak) of the Fulfrich areal mfmctom&r. The&=rilbtJngleprirm ofindexN,Pndsidelmgthsisseenfromtheside,withthclear,~inaptneiad~ibuir, on the bottom face of the prism with ib anterior pole facing Mt. The camera, with artnna rpcshue at 0, is mountad on an arm pivoting vertically ahout the prism vertex V. The camera akvatkmaagkisa. A general ray kaves a point P of index n on the cut kns surface. is refracted at the vertkal prirm fw, and proceeds to the camera at the devation angk 4.
Pulfrich arcal refractometry was carried out by placing a glass right-angle prism of index N, = 1.5820 at 550 nm with one side horizontal on the froxcn cut kns surface, and the other vertical and parallel to the lens equator (Fig. 2). The dispersion curve of the prism glass had been measured by the method of minimum deviation for eight mercury lines between 405 and 691 nm (see legend to Fig. 4). The lens, which had become white and opaque upon freezing, was then allowed to thaw, becoming crystal clear again. Because the lens half was supported by the embedding material, distortion of the lens shape due to freezing and thawing was small. Figure 7 shows that the initial dimensions (vertical solid lines) differ little from the final dimensions after thawing (vertically arrayed points). The prism surface made optical contact with the cut surface of the lens. The cut surface was illuminated with light from a halogen incandescent lamp passed through an interference filter of wavelength 550 nm and bandwidth at half-maximum of 30nm. This wavelength is close to the peak of the cat’s green cone photopic system (Loop, Millican & Thomas, 1987). As the cut surface was observed through the vertical prism face the region of index higher than that defining the critical angle for internal reflection at the lensprism interface appeared light because of total internal r&&ion within the prism, and the
region of lower index appeared dark. This will occur for inflnitcsimahy small areas of lens, and depends only on the index value. The refractometer can therefore operate properly on gradient-index kns substance, with the lightdark boundary d&ning the shape of the isoindicial curve. By varying the vertical angle (elevation) of observation, isoindicial curves for various vale of n within the lens could be observed and photographed (Fig. 3). In practice some 20 isoindicial curves were photographed per lens. The instrument’s operation is governed by the following refractometer equations (l-5), which apply to Fig. 2. The ray leaving a point P of kns index II at the critical angle for internal r&cction follows the relation: NE-b/u = n ~sin(90°)- PJand thus; a = b-N&t.
(1)
From Snell’s Law at the vertical prism face: sin 4 = NC-c/a;
(2)
c==Jp=F).
(3)
and clearly
These equations ddlnc theangle 4 for observing
the isoindicial curves of refractive in&x II at the wint c _~.~.P. To relate them to the stationarvI 1DriSm
W. S. JAGOER
728
additionally minim& this error. The precision and accuracy of the machine-divided graduated sector used to position the camera were better than 1 min arc, and the scale reading was to within 0.1 deg, resulting in no more than 0.0007 error in index. Photographs of isoindicial curves are foreshortened in the direction of view by the factor: b/(c ‘COSf#J);
cAMERAasv~T(owuuaE
a
Fig. 4. Plot of lens isoindicial curve index VCAIU(L autwa elevation angle a for the Pulfrish arcal refractometer. These six curws mprcwt ~0Mions to tht refrnctometer squatiorm for each of six wavdc#hs. From the I&, these wavek@lS,WiththCprirm&lSlIidXitlpacontho#,~: 700 rtm (l.S72S),650nm (l.S753), 600 mu (1.578Q).550 mn (1.5820), SQonm (1.5875) ad 45onm (1.5947).
vertex Vabout which the camera elevation angle u is measured:
(5)
and were corrected by digitising the curve, multiplying the appropriate coordinate by the necessary factor to correct the foreshortening, and plotting the corrected curve. Foreshortening can be minimised over the range of index values occurring in a lens by choosing NI to allow the broad curve of foreshortening to fall with its minimum in the midrange of measured index values. For the cat lens, the prism index NE= 1.5820 (550 nm) is close to this optimal value. The minimum foreshortening factor of 2.81 falls at about n = 1.39. Dispersion of the lens substance
Dispersion (variation of refractive index with wavelength) of the lens material was molwtd tan$=(L+sina -c)/(L.cosa -s); (4) with a series of interference filters of bandwldtb whereListhediatancefromthepriamvertex V 30 run. For a series of niue elevation settings, the equatorial width of the isoindicial curve was tothecameraaperture,andsisthelengthofthe measured for each of six wavelengths batween prism side. These equations were solved munuitally for a as a function of n or vice versa for 450 and 7OOnm. A plot of nine cures of this given values of L, s, IV,, and b (the disQusa5of width (ordinate) vs wavelength ( ) was the centre of the lens to the vertical prism face). made, with elevation angle as paraanettr, and One degree of camera elevation angle u corre- a non-linear interpolation was made between sponds to about 0.0065 units of n. The angle a curves to find the elevation setting for each and n are approximately linearly related for the wavelength that wouId yield the isoindicial range of refractive index of interest (Fig. 4). The curve width measured for 550 nm. Points on this arm supporting the camera pivoted verucally horizontal line of constant curve width clearly about the prism vertex V, and 45 deg camera refer to the same lens substance. The refractelevation was de&red by sighting through the ometer equations were then solved for the lens camera along the hypotenuse of the prism. index using these elevation settings and the Desired camera elevation angles were set on a known dispersion curve of the p&m glass. These points yield the dispersion curves of Fig. 5. 200mm radius precision graduated sector. The lens cut surface has a finite size, and to minim& the error caused by viewing different Calculations on model lenses ‘&e paths of meridional fans of incident rays parts from di&rent aagles, the camera entrance aperture was placed 1010mm distant (a 450 mm through a model lens were calculated by means focal length telephoto camera objective was of the ray-tracing program Drishti (Snds, used to yield adequate image sixe). This reduces 1984).This powerful program is able to treat the the error across a lens 8 mm in axial dimen- aspheric surfaces and refractive index gradients sion to 0.001 index units. The finite size of the found in animal eyes. In addition to calculating camera aperture will cause an isoindicial bound- cardinal properties, it is able to trace ray paths ary unsharpness error, e&mated to be about through an inhomoganeous model lens by 0.001 units for the 6 mm entrance aperture used. solving the difIerential quations descr&ng the The film was developed to high contrast to paths using a Runge-Kutta-Merson algorithm
729
Optics of the isolated cat lens
[
'.44-
Y F
1.42 -
distort the lens slightly by their tension, changing the surface s&e and decreasing the axial thickness. Viewed along the axis, the isolated lens is very nearly circular (diameters differ less than 1% from each other) and axial symmetry with coincidence of optical and physical axes was assumed. The external shape of the lens viewed from the side (Fig. 1) was fit to the anterior and posterior conic curves of polar radius RO and conic constant CC (Table 1). The standard error of estimate of the sag on the aperture averaged over the ten surfaces was 0.019 mm, indicating very good fits. Additional higher-order parameters were therefore judged unnecessary. RO and CC are defined by:
Y L E
1.40-
1.38 -
1
1
I
I
RO = A :/A,
1
450500ssoamaio
700'
ULIVELEMGTH tnll ,
Fig. 5. Dispersion of cat lens substance in the range 45&700 mn. Beaulse the full set of CmVeswas not mcaswed for any one of the five knses, this complete set was measured on an additional kns. Agmemcnt between data from diiTermtknm?swas~.EadlcmVeKmsistsofstraiglltlines connecting six points, calcula+d at each of the abscissa w-avekngths.
and
CC = (AJAX)* - 1;
where A, is the semi-axis perpendicular to the lens axis, and A, is the semi-axis parallel to the axis. A circle has a conic constant equal to xero, that of an oblate ellipse is greater than zero, and that of a prolate ellipse lies between - 1 and 0. Axial thicknesses measured from lens photographs are also shown in Table 1. Puljiiich area1refractometry
(not by using a series of homogeneous shells). The position of best focus can be found by minimising the r.m.s. radius (R.r.m.s.) of a pencil or fan of rays. The R.r.m.s. is relevant to the image-forming ability of a lens, because it is a measure of ray concentration, while the minimum total width of a pencil or fan may reflect only the course of highly aberrated rays. Optical conventions used are those of modern optical engineering practice (Smith, 1966; Ringslake, 1978). Figures are oriented so that light travelling in the physiological direction proceeds from left to right, in the positive direction. Modem convention differs from optometric usage regarding longitudinal spherical aberration. A double convex glass lens with spherical surfaces will focus marginal rays closer to the lens than rays near the axis. Modem convection designates this as undercorrected longitudinal spherical aberration, of negative sign. To avoid confusion, the state of correction will also be given. RESULT!3
External shape of the isolated lens The isolated cat lens is free of xonular fibre tension. In the intact eye, the zonular fibres
A series of Pulfrich area1 refractometry photographs is shown in Fig. 3, taken with the apparatus illustrated in Fig. 2. The outer boundary corresponds to the isoindicial curve. The dark area in the centre lies above n = 1.46, and is apparently caused by inadequate optical contact of material above this index value to the glass prism. This feature was seen in all five lenses, accounting for about onequarter of the cross-sectional area and about 13% of the total lens volume. Data from this type of photograph, with each isoindicial curve individually corrected for foreshortening and superimposed, are shown in the form of a contour map of refractive index in Fig. 6. The irregularity apparent in the curves is probably due to a combination of intrinsic irregularity and measurement error. Although this map shows only nine curves for clarity, typically twenty photographs per lens were taken. The full set of data allowed the refractive index along and perpendicular to the axis to be determined (Fig. 7). Dispersion of lens material over the range 450-700 nm is given in Fig. 5. As the lens centre is approached, and the index rises, the curves become steeper. The Abbe number vd, an inverse measure of dispersion defined as: vd= [n(588 nm)- l]/[n(486 nm) - n(656 nm)]
730
W. S. JAGGER
Individual lens models Each of the five lenses was modelled individually, using the same principles struction. parameters
The minimum
was used to adequately
physical structure
Fig.6.Isobdic&-ofnsc&onthrou&atypk&ut kns (No. 4) includingthe axis with antctior pole facingleft. Thccllrves have bcm cJxmctd for fofcshortening.This diagram shows for clarity only nine curves of a total of about twaty that were nconled ftx each lens. The mkp of index is from 1.380 to 1.460 spaced 0.01 in index. SC& 5 mm.
was evaluated by interpolation b&wecn data points of each curve by mcatl~ of a secondorder least-squares curve fit. Values of the Abbe number range between 50 at n = 1.38, decreasing to about 40 at n 3: 1.46.
of model con-
number, nine, of model specify the
of each lens at
550 nm, based on the external shape and the internal form of the isoindicial curves and their spacing. Each lens model was assumed to have rotational symmetry about its physical and optical axis, so that each isoindicial curve defines an isoindicial surface when rotated about the axis. The following discussion is concerned with plane sections containing the axis. The external shape was fully specified by the polar radii and conic constants of the anterior and posterior surfaces, and the total thickness, as discussed above. The capsule is so thin as to be optically ineffective, and was not treated separately. The internal structure was mod&d by a set of isoindicial curves spaced following a parabolic rule of the form given by Matthiesscn in 1877: n = NC,,- (N,, - N,,,)WW;
(6)
where r is the distance from the centre to the element of index n, and R is the distance from the core to the cortex. As discussed in the introduction, the parabolic rule has been found to apply to a variety of lenses. The model parabolic rule, shown as solid curves in Fig. 7, describes the measured index well (points in Fig. 7) from the cortex to the highest measured isoindicial curve. The small
Fig. 7. Isoindibal curve width in the udal and squrtorial @erpendiculpr to the axis) directions for a typiud lens (No. 4) The solid mrva duaibe the parabolicvaribon of index used in UICmodel of this lass. The vertical lines are the lens boundaries.
Optics of the isolated cat lens
rounding at the edges of the measured curves is probably due to a small fluid redistribution across the lens edge; drying cannot occur because there is no contact with air. Values of index used in the models are based on the solid curves, which compensate for the rounding. Measurements in this laboratory also show that the parabolic rule applies to cats younger than about four months, in which no central area of bad optical contact occurs, allowing measurement up to the peak index (which lies below 1.46). Therefore, on the weight of the overwhelming evidence from work on other lenses, and the agreement with the present index measurements, it was assumed that this rule of increase holds for the fully mature animal to its peak index value. The values of the model core and cortex indices were found from curves of the type shown in Fig. 7. Each model isoindicial curve consists of an anterior and a posterior part. Each part is half of an ellipse, with major axes coincident and perpendicular to the axis. The minor axis of each ellipse was chosen to fit the observed isoindicial curve shapes. Because the major axes of both anterior and posterior ellipses coincide, discontinuities in the curves or their derivatives at the point of joining the anterior and posterior mode1 curves are avoided. These curves therefore meet smoothly at the equatorial plane, as the measured isoindicial curves indicate. As the refractive index increases, the shape of the isoindicial curves was found to remain very similar geometrically. In the absence of a trend away from geometrical similarity, the remaining isoindicial surfaces in the unmeasured central 13% of the lens volume were assigned the same geometrical shape as those of the remainder of the lens. The axial location of the index peak varied from lens to lens, and its position was incorporated in each model. An axial section of a cat lens model is shown in Fig. 8. The smooth
Fig. 8 Section of a typical cat lens model (No. 4) including the axis showing the internal refractive struch~rc. The isoindicial curves for the anterior (left) part arc sets of half-&pm geometrically similar to each other, and likewise for the poF&ior part (right). Their indices follow the parabolic rule of equation (6) as the ccntre of the lens is approached. and the curves shown arc those for n = 1.3900-1.4900. spaced 0.01 units apart. Ticks on the axes arc spaced 1 mm apart.
mode1 curves are free of the irregularity seen in the measured curves (Fig. 6). The set of nine independent parameters needed to describe the physical structure of each lens is shown in Table 1, together with their average values. Optical properties of the lens
Figure 9a is a photograph showing the paths of a meridional fan of laser beams incident parallel to the kns axis. The distance from the lens posterior vertex at which each beam crosses
Table 1. Model parameters for each of the five isolated cat lens models. The average value of each parameter is shown in the last column. Lens thicknew is measured along the axis. Index peak location (IP) is measured from the anterior surface. MA ind&tes the major axis of the outer isoindicial curve. Refractive indices are for 550 MI. Radii and d&anccs are in mm LAxls Anterior RO Anterior CC Thickness Posterior RO Posterior CC IP MA NX.NX VR W-F
1
2
3
4
5
AWraEC
6.63 0.37 8.09 -7.44 0.73 4.14 10.74 1.3870 1.4683
6.86 0.28 8.34 - 7.85 0.74 4.15 11.04 1.3890 1.4774
6.88 0.31 8.58 -7.55 0.54 4.18 11.34 1.3880 1.4818
7.42 0.31 8.73 -8.16 0.45 5.01 12.12 1.3857 1.4902
7.36 0.47 8.41 -8.19 0.81 4.52 11.51 1.3890 1.4890
7.03 0.35 8.43 -7.84 0.65 4.40 11.35 1.3877 1.4813
732
W. S. JAGGW~
(b)
J
1
I
1
1
6
8
10
12
14
RAY cslosww
l.ocATKm
mm
Fig. IO.(a) L.ongitudinal spherical aberration of an M&d cat lens (No. 4). Cid show the dktance (abscissa) from the Iens post&or vertex at which irkdent j7eraM laser beams of varkxu he&&s (ordittate) croastheaxisafternfractionby~Isnr.Tbecolidcurwcisth!longit~~sphericniPbarption~ for the model of this lens. Meamued points for + 1 and - 1 mm ray heights an not shown because the shallowanslckrtw#athenfiscradbepmtlBdthePrisco~largeuncertsintyintheaxis~ location. (b) Lot@tud&A absmrion along the cekf ray for an obliquely incident (54 dog) ttmidioti fin of parallel rays (lens No. 4). Absoissa: positkm of ray i&me&on with the chief ray mausumd along the chief ray from the pIane taqpnt to the po6Wior lens pole. Ordbate: distance of the krcidcet ray from the chief ray in the hxident paraM fan. So&l curve: calculated aberration for the model of lenu No. 4. Points: aberration measured from a fan of laser beams incident upon lens No. 4 as in Fig. 9h.
the axis after refraction is shown in Fig. 1Oa vs incident beam height. All five lenses displayed positive longitudinal spherical aberration (overcorrection; that is, marginal rays focussed farther from the lens than did rays close to the axis) ranging from + 2.9 to +4.4 mm for incident ray height 5 mm. This is clearly seen in Figs 9a and lOa. Figure 9b shows a similar meridional fan, but incident at 45 deg to the lens axis. It is clear that the lens still produces a tight focus. The fan width in Fig. 9b at the position of best focus is 0.55 mm. Calculated optical prqerties of the models A meridional fan of rays equivalent to the 633 nm wavelength laser beams shown in Fig. 9a was traced through each model lens using the ray-tracing program Drishti. To perform this ray tracing, the core and cortical indkes measured at 55Onm had first to be corrected downward to their values at 633 nm. Dispersion measurements (Fig. 5) show that cat lens substance measurccd at 550 nm to have an index of 1.450 has an index at 633 nm about 0.005 units
lower. For lens substance of index 1.380 at 550 inn, this difkence is about 0.002. The surrounding fluid index was 1.333at 633 nm. Small additional adjustments were then made to the values of the core and cortical lens indices, within their ranges of measummen t error. Final index values for 550 nm used in the mode& listed in Table 1, result in the solid curves of Figs 7 and 10, showing good agreement with the measured points of refractive indq longitudinal spherical aberration, and oblique km&tudinal aberration measured by means of laser beams. The calculated positions of best focus are indicated on Figs 9a and b. The width of the oblique fan calculated for the model of the lens of Fig. 9b is 0.49 mm at best focus, close to the measured value of 0.55 mm. Similar good agreement between measurements and calculations on each model was seen in all lenses. The calculated optical properties of eaoh individual model are given in Table 2. These quantities were also calculated for the “average” lens posserkng the average parameter vahaes shown in Table 1.
Fig. 9.(a) The upper photograph shows an isolated lens (No. 3) suspended in a saline bath by zonuk fibres. A meridional fan of parallel laser beams incident from the left spaced I .OO mm apart is refracted by the lens and crosses the axis to the right of the lens. Note the curved beam paths within the kns, a result of the index gradient. The beams of outer zones cross the axis farther from the lens than do the inner rays, showing overcorrected (positive) longitudinal spherical aberration. The left arrow indicates the position of paraxial focus, and the right arrow indicates the position of best focus (minimum R.r.m.s.), calculated for the model of this lens. (b) The lower photograph shows the same kns with beams of the meridional fan incident at 45 deg to the lens axis. The position of best focus calculated for the model of this lens lies between the arrow points.
733
Optics of the isolated cat kns
735
Table 2. Optical propertiescalculated at 633 nm wavekngth for each of the isolated cat kns models and for the lens composed of average parhtkr v&es. Anterior principal point position (PP) is measumd from the anteriorkns pole; the posterior principal point position is measuredfrom the posterior pole. Nodal points coincide with principal points. Paraxialfocal kngths (FL) are measuredfrom the principal points. The paraxkl bath focal dktance (RFD) is measuredfrom the posterior kns pole. Longitudinal sphetical aberration(LSA) is measured from the pamxial focus for ray height 5 mm. Units (except for power) are mm 1
Lens
2
3
4
5
Av. Lens
Anterior
PP FL Posterior
3.86 - 24.46
3.96 -23.43
4.10 - 22.94
4.16 - 23.58
3.99 -22.88
4.01 - 23.39
-3.97 24.46 20.49 2.9 54.50
-4.09 23.43 19.35 4.4 56.88
-4.19 22.94 18.75 3.5 58.11
-4.23 23.58 19.3s ;::3
-4.69 22.88 18.79 4.0 58.27
-4.11 23.39 19.28 4.0 56.99
FL BFD LSA Power, D
The sensitivity of the optical properties of a model lens to changes in the model parameters is shown in Table 3. The effect is shown of a small change in the size of each parameter, approximately of the sire of the measurement uncertainty of that parameter, upon lens focal length, longitudinal spherical aberration, and the minimum R.r.m.s. and position of best focus for an oblique fan. DISCUSSION
Physical and optical nature of the lens The surface of a cat lens is easily deformed, yet regains its form after small deformations when pressure is released (Ejiri et al., 1969). If the capsule is removed, and the lens held between the fingers, the outer jellylike material is easily rubbed away, leaving increasingly solid material as the centre is approached, until finally only a hard and gummy core material remains. It is therefore not surprising that the
curves of constant refractive index show the form seen in Fig. 6 and the increase of index as the centre is approached shown in Fig. 7. The method of Pulfrich areal refractometry has the considerable advantage of yielding the shape of the isoindicial curves for 87% of the lens volume and their relative spacing directly. The slight disadvantage of having to freexe the lens before sectioning could cause a dimensional change of up to 1 or 2%, which is probably reversible upon thawing, when the actual measurement is made. Redistribution of water during the quick freezing process is also possible, but is probably inconsequential, given the success of the models based on the method to predict the optical behaviour of the intact lens. Freytag (1910), quoted by Philipson (1969), found a cortical index for the cat lens of 1.38-1.40 and a central value of 1.46. These cortical values agree with the range of model values used here (1.3857-l .3890). Freytag’s central value of 1.46 corresponds to the highest
Table 3. Effectof small changes in the nine independentmodel pammemmupon the calculated optical properties of a typical model kns (No. 3). Chant& the outer isoindicial curve major axis al%ctsall isoindicial curves proportionally,becausethey remaingeometricellysimi~.Theoptical~rhownue~lfocrl~ (FL), longitudinal spherical aberrationof a ray of height 5 mm (LSA), the minimum R.r.m.r. of an obliquely incident meridional fan of width 10mm huzidentat 45 deg and the coordinates of the position of best focus of this obliqtk fan. The initial FL and LSA are 22.94 mm and 3.50 mm, mspectively. The initial oblique R.r.m.s. is 0.218 mm. The initial position of oblique best focus is x = 8.928 mm from the posterior lens pok. and y - 13.306mm above the axis. Units are mm Delta parameter
Anterior RO Anterior CC Thickness Posterior RO Posterior CC IP MA J”N,
+O.lOmm +0.05 +O.Olmm +O.lOmm +0.05 +O.lOmm +0.10 mm +0.001 +0.001
FL
LSA
R.r.m.s.
x
+0.050
+0.281 -0.358 -0.040 -0.228 -0.170 +0.114 -0.319 -0.194 +0.041
+0.006
+0.092 -0.067 +o.otM -0.019 -0.034 +0.013 +o.oss +0.028 +0.091
+o.ooo +0.016 -0.028 0.000 +0.17a +0.275 +0.097 -0.182
+0.002 -0.001 -0.004 -0.006 -0.004 -0.005 -0.906 +0.003
Y
+0.081
-0.059 +0.012 - 0.028 -0.040 +0.029 +0.052 +0.025 +0.091
736
W. S. JAGOER
directly measured index before poor optical contact between lens substance and refractometer prism would appear (Fig. 3), limiting his maximum measured value to less than the assumed model central value of about 1.48. The series of nine dispersion curves of lens material shown in Fig. 5 seems to show no discontinuity, and probably results from a smoothly decreasing water fraction in the lens as the centre is approached. The value of the Abbe number decreases from 50 at the periphery to about 40 in the near-central region of index 1.46. Palmer and Sivak (198 1) published dispersion data for various animals and man, and the shapes of their curves are generally similar to those found here for the cat. However, to calculate values of V (almost identical in definition to the Abbe number) in their Table 1 they apparently used individual point values, which reflect scatter, rather than points on the smooth curves drawn through their points, which are a better representation of their data. Dispersion curves of transparent homogeneous media such as aqueous solutions, glasses, and organic solvents do not demonstrate fine structure in the visible region, but rather show smooth curves, as is to be expected on physical grounds, and the same is to be expected for lens substance. Their procedure has a drastic effect on the values of V, vitiating its ability to characterize dispcrsive behaviour, and their Table 1 is to be interpreted with care. If points on their smooth curves are used to calculate V, both rabbit and pigeon show a decrease in V as the centre is approached, rather than an increase, and the unrealistically high value for the human peripheral V value decreases. The optical behaviour shown in Fig. 9a indicates overcorrection for longitudinal spherical aberration on the axis. This behaviour, clearly evident in all five lenses, was noted by Vakkur and Bishop (1963), but was not seen by Sivak and Kreuzer (1983), who found spherical aberration of both signs in the cat lens. Their Fig. 7 expresses longitudinal spherical aberration in terms of back focal power. If the range of longitudinal spherical aberration found in the present study, + 2.9 to +4.4 mm, is expressed also as back focal power, it is +5.8 to +9.OD. This amount of longitudinal spherical aberration is very small on their scale. It may be lost in the variation shown by their points within the 1 mm zone. Small uncertainty in beam position due to irregular lens structure or measurement error will strongly affect the calculated position
of beam axis crossing for beams within the 1 mm zone because of their shallow angle with the axis. This may account for the relatively large irregular spherical abberation these authors found for the cat lens within this zone. In considering spherical aberration of the isolated lens, it should be pointed out that in uivo, rays impinging upon the lens from a distant point object are not parallel, but of course are convergent, having been refracted by the cornea. Lens axial spherical aberration in unaberrated convergent light decreases as the degree of convergence increases. Its value decreases from that for parallel light as the incident light’s virtual focus approaches the lens. It becomes zero for a virtual focus about 6 mm from the lens posterior surface, and becomes negative as the focus moves still closer. The spherical aberration of the cornea-lens combination may be more or less than that of the lens alone, depending on the ability of the cornea to compensate for the lens aberration. The purpose of using parallel rays in this study is not to equate directly the spherical aberration found with that contributed by the lens in the intact eye, but rather to serve as a check on the accuracy of the models developed. Nature of the lens models The strength of the type of lens model presented here is that it is constructed from the external shape of the lens and from data from the Pulfrich area1 refractometer. The gap in measurements near the core of the lens corresponds to only about 13% of the lens volume. The assumed structure in this region is firmly based on the overwhelming evidence from work on other lenses confirming the parabolic rule of index increase and the good fit of a parabolic curve and the uniform trend of geometrical similarity of isoindicial surfaces in the remaining 87% of the lens. There are no freely adjustable parameters in the models. Only minor adjustment in the values of the cortex and core indices, within their ranges of experimental error, was necessary to make all five individual models predict the paraxial focal length and longitudinal spherical aberration (and hence compkteiy specify the axial imagery), the axial position of best focus, and the longitudinal aberration of a meridional fan incident at an oblique angle to the axis. They are therefore good physical and optical representations of the isolated lenses. Inverse ray tracing methods suffer from the criticism that they may result in a lens model
Optics of the isolated cat lens
that is not unique, and does not correspond to the physical structure of the l&s. It is not tr&! in general that a lens model that predicts axial spherical aberration of the lens is unique, unless it is also shown that the lens internal and external structure is spherically symmetric. In the absence of this symmetry, it must be shown that a model can predict off-axis optical properties. Sensitivity of a model lens to parameter changes
The optical properties of the models used for each of the five lenses studied are not drastically affected by changes in their measured parameters of a size approximating their measurement uncertainty. Table 3 shows the effect of such small changes in each of the nine lens model parameters upon the axial and oblique optical properties of a typical model, No. 3. The anterior and posterior polar radius changes have relatively weak effects on the optics of the lens; the anterior radius change affects the focal length more than does the posterior change largely because of its greater distance from the focus. The conic constants do not afIect the paraxial focal length, as these constants have no influence on surface curvature in the infinitely small paraxial region. They do, however, affect the surface curvature off axis, and therefore influence the longitudinal spherical aberration, Increasing the conic constant will increase the curvature of the per-ipheral lens surface (for a given polar radius), giving marginal rays a shorter focus and decreasing the longitudinal spherical aberration. Positive axial displacement of the peak location of the refractive index distribution increases the focal length and the longitudinal spherical aberration. Elongating the isoindical curves in the equatorial direction increases the focal length, while decreasing the longitudinal spherical aberration. The cortical and core index values were adjusted slightly for final fine tuning. of each model. While increasing the cortex index increases the paraxial focal length and decreases the longitudinal spherical aberration, increasing the core index has the opposite effect. However, each index affects the paraxial focal length and the longitudinal spherical aberration to differing degrees. The increase in cortex index decreases the longitudinal spherical aberration about twice as much as it increases the focal length, while the increase in core index decreases the focal length nearly five times the
737
amount it increases the longitudinal spherical ab&&atibn*
Advantage of a lens with a refractive index gradient Why has such an apparently complex structure as a lens with a gradient index evolved? The answer lies largely in aberration control in the whole eye, and the ability to accommodate, and will be treated in the companion paper on the optics of the cat eye. However, one important advantage of a gradient index is that it allows construction of a high power lens from material of low index. This can be intuitively understood by the observation that a non-axial beam traversing a gradient-index lens is continuously refracted within the lens and follows a curved path (Fig. 9). We can construct a homogeneous model lens of the same shape as a normal lens with a constant index (the “total index” of Matthiessen, 1880) such that the paraxial back focal distance is the same as that of the actual lens. The index required is 1.559, much higher than the core index in the inhomogeneous lens. A rule given by Matthiessen on the basis of lens structure with a parabolic index gradient predicts that the differena between the total index and the core index is qua1 to the difference between the core index and the cortical index. For model No. 3, these two quantities are in fact of similar size, 0.081 and 0.085, respectively. For the homogeneous lens, however, the axial longitudinal spherical aberration is - 10.1 mm, instead of +3.5 mm for the gradient index lens. The axial position of meridional fan best focus lies 9.8 mm inside that of the actual lens, where its axial R.r.m.s. is 0.55 mm instead of 0.15 mm for the inhomogeneous lens model. At 45 deg off axis, the position of meridional fan best focus is 3.7 mm inside that of the inhomogeneous lens, and its focussed R.r.m.s. at 45 deg is 0.39 mm, compared to 0.22 mm for the gradient index model lens. In short, the optical properties are grossly different from those of the gradient index lens, and the entire eye, including cornea and retina, would require redesign to use such a lens. A further problem is the biological production of a lens with the high index of 1.559, which is much higher than that of the core of fish lenses, about 1.51 (Matthiessen, 1880), which probably represents the upper limit to a biologically possible lens index. These cores are very rigid, and accommodation by shape change would be out of the question for such a lens.
W. S. JAOOER
738
Acknowledgements-This work was supported by the Australian National Health and Medical Research Council project grant No. 870792 and the Viktor and Ema Hasselblad Foundation. The National Vision Research Institute is an affiliate of the University of Melbourne.
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