Refractive index distribution and spherical aberration in the crystalline lens of the African cichlid fish haplochromis burtoni

Pergamon

0042-6989(93)EOO34-S

Vision Res. Vol. 34, No. 14, pp. 181H822, 1994 Copyright 0 1994 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0042-6989/94$7.00 + 0.00

Refractive Index Distribution and Spherical Aberration in the Crystalline Lens of the African Cichlid Fish H~pl~ch~~rni~ burtoni RONALD H. H. KRtjGER,*t RUSSELL D. FERNALDg

MELANIE

C. W, CAMPBELL,$

REJEAN MUNGER,$

Received I8 December f992; in revised~orm 3 November 1993

Refractive index distribution in the teleost crystalline lens was measured with a nondestructive method in freshly excised lenses of the African teleost fish Haplochromis burtoni. Independently, spherical aberration was measured in a parallel set of lenses. The measured refractive index profiles show a continual decrease of refractive index from the center to the surface of the lens. The H. burtoni lens is of high optical quality and slightly overcorrected for spherical aberration. Details of the small residual spherical aberration were accurately predicted by ray-tracing model calculations based on the measured refractive index profile. The refractive index profile and the spherical aberration both show more complex characteristics than suggested by earlier measurements and lens models. Fish Lens

Physiological ~a~i~cbrorn~~burt~ni

optics

Refractive index gradient

INTRODUCTION The most difficult task in the analysis of the optics of vertebrate eyes is to create a good model of the crystalline lens. Most vertebrate lenses have a gradient of refractive index which increases from the surface toward the center of the lens. This gradient increases the total refractive power of the lens and reduces aberrations, most notably spherical aberration (e.g. Maxwell, 1854; Matthiessen, 1882, 1893). An eye model with good predictive power must incorporate the refractive index gradient within the lens in detail since even small differences in the shape of the refractive index profile in the lens can lead to significant differences in the optical quality of the lens (e.g. Campbell & Sands, 1984; Campbell & Harrison, 1990; Jagger, 1992). Exact knowledge of the three-dimensional distribution of refractive indices within the lens is therefore a precondition for successful modelling of the performance of the eye. The simple geometry of the spherical lens of a variety of fish species has led many workers to look for analytical or approximative solutions for the distribution of -_. *Institute of Neuroscience, University of Oregon, Eugene, OR 97403, U.S.A. tTo whom all ~rres~ndence should be addressed at: Anatomisches Institut. E~rhard-Karls-Universit~t Tiibingen, Osterbergstr. 3, 72074 Tiibingen, Germany. $School of Optometry, University of Waterloo, Waterloo, Ontario, Canada N2L 3G 1. §Neuroscience Program and Department of Psychology, Stanford University, Stanford, CA 94305, U.S.A.

Spherical aberration

Eye model

refractive indices in the fish lens (e.g. Maxwell, 1854; Matthiessen, 1882, 1893; Fletcher, Murphy & Young, 1954; Jagger, 1992). Those models assume that the fish lens is almost perfectly corrected for spherical aberration since it is known, from measurements of spherical aberration in a variety of species, that many fish lenses come close to this ideal (Sroczyfiski, 1975a, b, 1977; Sivak & Kreuzer, 1983). However, fish lenses with poor correction of spherical aberration have also been reported (Sivak & Kreuzer, 1983). It is therefore necessary to correlate measurements of the refractive index gradient and spherical aberration of the lens in any particular species (Campbell & Hughes, 1979). We present here results from high resolution measurements of the refractive index profile and of the spherical aberration of the lens of Haplochromis burtoni, an African cichlid fish from Lake Tanganyika. Ray-tracing model calculations based on the measured refractive index profile were used to test the predictive power of the lens model for ~apZochromis burtoni. H. burtoni has a highly developed visual system (Fer-

nald, 1977, 1984, 1987). It has been established in earlier experiments that its crystalline lens is of high optical quality (Fernald & Wright, 1983, 1985). Measurements on freeze-cut sections (KrSger & Fernald, 1994) revealed that naso-temporal and axial radii of II. burtoni lenses differed by 0.5% average lens radius with a measurement accuracy (standard deviation, n = 113) of 0.8% lens radius. That means that the lens is spherical within the resolution of the method.

I815

1816

RONALD

H. H. KRijGER

Fernald and Wright (1983) estimated the distribution of refractive indices within the lens by successively peeling off layers from the surface of the lens and measuring the optical properties of the remaining part of the lens. Their results suggested that the lens has a homogeneous core of approximately constant refractive index covered with layers of decreasing refractive indices toward the lens surface. However, those results were inconsistent with ray-tracing models that showed that the high optical quality of the lens could not be achieved with a lens core of constant refractive index (Campbell & Sands, 1984). We measured the refractive indices in crystalline lenses of H. hurtoni using a destruction free method introduced by Campbell (1984). The method is based on analysing the deflection of thin laser beams shone through the lens. That method appears to be more accurate than the method used by Axelrod, Lerner and Sands (1988) which relies on entrance and exit positions of laser beams incident on the lens to calculate the refractive index profile. In the present work, Campbell’s original method was slightly modified to increase resolution and accuracy. Spherical aberration was measured in a second set of H. burtoni lenses of similar sizes. Our results show that the refractive index profile of the lens of H. burtoni has a continual decrease of refractive indices from the center to the surface with a complex curve shape. The lens is well corrected for spherical aberration and the small residual spherical aberration found in the lenses of H. burtoni is correctly predicted by the lens model. MATERIALS AND METHODS The animals used in this study were bred from a laboratory population that was drawn from a large and diverse founder population (Fernald & Hirata, 1977). Inbreeding has been largely avoided since then. The animals were kept in white light under conditions as described in Fernald (1989). The gene pool of the laboratory population was refreshed on a regular basis. All animals used in this study were males and were about 10 months old. Although we did not expect to find sex related differences in the lenses, we used the larger males because they had lenses of adequate size. The animals were sacrificed by cerebral section before the lenses were removed. Since the refractive properties of the lens change with increasing size in at least some fish species (e.g. Sroczynski, 1975b, 1979), lenses of similar sizes were chosen for this study. The optical performance the immersion medium

of a lens is strongly afected

index profile in one set of lenses and calculated spherical aberration with a ray-tracing lens model. In a parallel set of lenses, spherical aberration was measured directly. Comparison of the results of model calculations with direct measurements of spherical aberration is an independent test for the accuracy of the two methods (Fig. 1). Rejiactive index measurements

Since the lens capsule is very thin its contribution to the refractive properties of the lens is negligible (Campbell, 1984) and was not measured. Refractive index of lens material just inside the capsule (surface index) was measured with an Abbe refractometer in two freshly excised lenses. Small incisions were made in the lens capsule so that some material of the almost liquid outer layers of the lens cortex was squeezed out by intralenticular pressure. Due to the steep rise in refractive indices close to the surface of the lens, the error distribution of measurements of the surface index is strongly biased toward higher values and can be expected to show a sharp drop-off at the lowest, real value. We therefore used the lowest measured value as the surface index of the lens. For measurements of refractive index distribution, the lens was excised and immersed in an isotonic (290-305 mosm), neutralized (pH 7.0-7.5) solution of NaCl and polyvinylpyrrolidone (PVP) at room temperature as described in Campbell (1984). Refractive index of the solution matched the index of the outermost layers of the lens cortex. The lens rested on a pad of black plasticine that had a small, circular pit to keep the lens from rolling off. The plasticine was held by a turnable disc of black PVC (Fig. 2). The lenses almost floated in the immersion solution and the H. burtoni lens is very rigid so that compression of the lens due to its own weight was negligible. Lens orientation was random since the freshly excised H. burtoni lens is so clear and symmetric that we found it impossible to position the SPW

0 +

s LENS

LENS CALC%!Z%JNS

I

RAY-TRACING SALINE+ n=l.361 n = 1.334

by

Measurements of refractive index distributions were made in a medium that had the same refractive index as the lens surface while spherical aberration was measured in isotonic saline which refractive index is close to the index of the aqueous and vitreous humors. Both types of measurements are therefore independent and different in principle, although identical methods of data gathering can be used. In this study, we measured the refractive

(‘I tri

,

0 +

I

I

SALINE

t?c tiLi-I: n = 1.334

INDEXPIMplLE FIGURE

1. Refractive

index

measurements in combination with were used to produce results which could be compared with direct measurements of spherical aberration. The complex method of refractive index measurements and the predictive power of the resulting lens model couid thus be tested against a simple and independent measurement of the refractive properties of the lens.

ray-tracing model calculations

REFRACTIVE

INDEX DISTRIBUTION

Side view a.

Video camera

LENS

1817

after the lens had been rotated by 90” so that beam paths were determined along a different axis. Deterioration of the lens and/or asymmetry in the refractive index profile would cause differences in the results from both measurements.

’

Measurements of spherical aberration

/ HeNe laser \

Lens ’ f=50mm * -

EFz$gr ,

Top view

Lens &ath J

-I@ I

/

J

Metric grid

FIGURE 2. Experimental set-up. Laser beams were shown through a meridional plane of the fish lens that was immersed in solutions that matched the refractive index either of the lens surface (refractive index measurements) or of the aqueous/vitreous (measurements of spherical aberration). Laser beam deflection by the lens was monitored with a video camera and analyzed by the computer. The lasexjfocussing lens assembly was moved by a computer controlled stepper to scan the laser beam through the lens. The lens hoider could be turned to do measurements in the same plane of the iens from different angles. To obtain a scaling factor, the camera was moved over a metric grid that was mounted in the same vertical position as the plane of measurements.

reliably in the same orientation. Thin, parallel laser beams (HeNe, 1 = 633 nm) were focussed with a 50mm focal length lens to reduce beam width in the measurement area (Axelrod ef al., 1988) to about IOOpm. The focussed beams were shone through the lens such that they were refracted in a meridional plane of the lens. Beam paths were determined from video images of that plane with an image capturing and processing system. Beams were digitized for at least 70 and usually about 100 beam entrance positions on each lens. In areas were exit angles were hard to determine, the number of measured beams was increased until we felt confident that we had satisfactorily covered the critical region. After all data had been collected from the first lens (1 to 14hr), the second lens from the same fish was excised and treated in the same way. All data were transformed to a metric scale by digitizing a metric grid and calculating a scaling matrix for each lens. Lens diameter and the geometrical center of the lens were determined by two beams that just grazed the lens on each side. Those beams were digitized at the end of each experiment. The differences between the entrance angles and the exit angles were recorded for different beam entrance positions, i.e. as a function D (y) of the separation between the entrance beam and the optical axis. The validity and accuracy of our results critically depends on whether the lens stayed fresh for the duration of the measurements and whether the precondition of radial symmetry in the distribution of refractive indices is met by the H. burtoni lens. To address those questions, we measured the refractive index profile twice in the same lens. The second measurement was done Ienses

IN TELEOST CRYSTALLINE

As described above, lenses were quickly removed from animals after sacrifice. For measurements of spherical aberration, the lenses were placed in isotonic NaCl solution. Refractive index of the saline solution was 1.334 (633 nm) which is close to the refractive indices of aqueous and vitreous humors (1.336, 633 nm). Back vertex distances (BVDs), i.e. the distances between the posterior vertex of the lens and the intersection point of laser beams with the optical axis after refraction by the lens, were measured for 70-100 beams in a meridional plane of the lens. Lens diameter and the geometrical center of the lens were determined from grazing beams as in the refractive index measurements. A metric grid was digitized for each lens to transform the data to a metric scale. BVDs could be measured for maximum beam entrance positions of up to approximately 95% lens radius. Resolution at the edges of the lenses was limited by the thickness of the laser beams and by the poor optical quality of the very periphery of the lens. Exit beams became very distorted and weak for beam entrance positions between 95 and 100% lens radius. Data gathering and analysis were done with computer programs specifically developed for those purposes. Ray-tracing model calculations were done with a program developed by Kriiger (1989). The algorithms used for cubic splines and linear interpolation were taken from Press, Flannery, Teukolsky and Vetterling (1989). All computations were done with 10 bytes (19-20 digits) accuracy. Data anatysis

The following formulas are used for the calculation of refractive index profiles from the deflection of beams (Chu, 1977; Campbell, 1984): r(a)=@

exp(-h(cw)j

n [r (a)] = ns exp(h (cx)> h(a)=

(1)

(2)

1;

with IZ= refractive index at radial position r, ns = refractive index of the lens surface, R = lens radius, y = beam entrance position, CI= integration constant. In order to apply the method, the discrete, measured deflection data have to be transformed into a continuous function so that numerical integration of equation (3) becomes possible. Campbell (1984) fitted a two-term polynomial to the data for the rat lens and for the rock bass lens (Campbell & Harrison, 1990): D(y)=Ay

-By”

where m is a high order, odd integer.

(4)

RONALD H. H. KROCER e/ r/i

1818

Other investigators used cubic splines for interpolation between data points in the deflection function (Chan, Ennis, Pierscionek & Smith, 1988; Pierscionek, Chan, Ennis & Smith, 1988; Pierscionek, 1989; Munger, Cambell & Kroger, 1992a) because it is then possible to exactly follow the deflection data. About 20 data points

are sufficient to determine a smooth refractive index profile with near maximum accuracy (Munger et c/l.. 1992a). In our data sets, however, cubic splines were prone to extreme oscillations in the neighborhood of data points where D (y, )<< D (y2)and J‘, z ,t‘?. Due to small spacings between beams and to experimental

3.0

2.0

1.0

0.0

-1.0

-2.0

1.51 ti %A7 $ &A3 ki a 1.39

FIGURE 3. Analysis of beam deflection data for refractive index measurements in a representative data set. (A) Deflection function: in order to numerically solve the integral in equation (3), other investigators used cubic spline interpolation between data points in the deflection function. In the majority of our data sets, however, cubic splines (smooth line connecting data points) showed fluctuations so that we used linear interpolation between data points instead. (B) Enlarged view of thedeflection function with linear interpolation between data points: the geometrical center of the lens was initially determined by two beams that just grazed the lens on each side. For the integration, the position of the optical axis was determined more accurately by the intercept of the connecting line between the two least deflected beams with the beam entrance position axis. A beam in that position would not have been deflected at all. All data were recentered to the optical axis. (C) Refractive index profiles of both halves of the lens: for each half of the lens, a refractive index profile was calculated by numerically solving equation (3) for 100 values of the integration constant IX.Since r has to be separately calculated from c1for each half of the lens (equation (I)], the n (r) data points are of unequal spacings along the r-axis between the two halves of the lens (only every 5th dafa point is shown). Differences between the two profiles are due to uncertainty in the position of the optical axis and to experimental error in the measurements of beam delIaction angles. (D) Averaged refractive index profile: a cubic spline was generated for the profile of each lens half. By interpolating between points, 100 equally spaced n (r) points were calculated for each lens half and averaged between the two halves of the lens. Spacings between n (r) points were chosen as a parabolic function to increase resolution in the peripheral, steep part of the profile (only every other data point is shown). In the averaged profile, differences in the two profiles due to uncertainty in the position of the optical axis have been removed. The arrows point to slight depressions in the profile which were found in most of our data sets.

REFRACTIVE

INDEX DISTRIBUTION

uncertainty in beam entrance positions, such data constellations occurred quite frequently in our data [Fig. 3(A)], especially in regions of the lens where the refractive index profile shows irregularities. Substantial oscillations in cubic splines of the deflection function were observed in more the 50% of our data sets. We therefore used linear inte~olation instead of cubic splines to conserve the advantage of exactly following all data points while avoiding artefacts due to oscillations in the cubic splines. Strictly speaking, the spherical H. burtoni lens has no single optical axis in vitro. In the context of beam path measurements we use the term “optical axis” to describe the axis in the measurement plane in which a laser beam for a given angle of incidence would not be deflected at all by the lens. That axis was determined by interpolating between the two least deflected beams in each data set [Fig. 3(B)]. Exact determination of the optical axis is necessary since otherwise a non-integrable singularity would occur for a = 0 [see equation (3)]. The difference between the optical axis and the geometrical center of the lens was very small (average difference = 0.012% lens radius) and well within the experimental uncertainty. Measurements were taken across the entire lens, i.e. deflection angles were obtained for beam entrance positions ranging across the full lens. Differences between the two halves of a lens [Fig. 3(C)] induced by uncertainty in the position of the optical axis can be removed by averaging the profiles from the halves (Campbell, 1982). To calculate this average refractive index profile, we normalized the refractive index profiles of both halves to unit lens radius. After normalization, cubic splines were generated for both profiles with 100 n(r) points each. By interpolating between calculated points with the cubic splines, we calculated refractive index values for equally spaced r-positions and averaged n (+r) with n (-r) for 100 r-positions. The spacings between the r-positions were chosen as a parabolic function so that many data points were calculated for the periphery of the lens where the refractive index gradient was steepest [Fig. 3(D)].

IN TELEOST CRYSTALLINE

1819

LENS

ipl

1.38

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 NORMALIZED DISTANCE FROM LENS CENTER

FIGURE 4. The refractive index profile of a H. burfoni lens was measured twice. The lens was rotated by 90” between the experiments. The close agreement between the results from both measurements indicates that the lens did not change its refractive properties for the duration of two measurements (approx. 2: hr) and that the lens has radial symmetry in the distribution of refractive indices, an important precondition for the applicability of the method.

index profile of the H. burtoni lens. Figure 5 shows the representative refractive index profile of H. burtoni in comparison to profiles proposed for carp and cod by Matthiessen (1880), goldfish (Axelrod et al., 1988) rock bass (Campbell & Harrison, 1990), and for a hypothetical fish lens with very little spherical aberration (Jagger, 1992). A two-dimensional model of the H. burtoni lens based on the representative refractive index profile was used in a ray-tracing simulation. The lens was represented in the model as a structure with 30,000 concentric layers. A cubic spline was generated for the 100 data points of the representative refractive index profile. By interpolating between data points to simulate a smooth gradient, a constant refractive index was calculated for each layer. The path taken by a ray of light from one layer to the next was calculated using Snell’s law. To increase the

RESULTS

The lowest surface index measured in 5 samples from two lenses was 1.3622 at 590 nm. This corresponds to an index of about 1.3610 at 633 nm (Kroger, 1992). Refractive index distribution was measured in 17 lenses from 10 fish. The average lens radius was 1.24 f 0.06 mm (standard deviation). The refractive index of the lens immersion solution was 1.3610 + 0.0015 (633 nm, maximum deviation). In two lenses measured again after the lens had been rotated by 90”, the results from the second measurements were very similar to the data from the first runs (Fig. 4). This indicates that the lenses stayed fresh for at least twice the time necessary to gather a complete set of data and that the lens of H. burtoni indeed has a radially symmetric distribution of refractive indices. The normalized refractive index profiles from all lenses were averaged to obtain a representative refractive

, 0.0

I

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

NORMALIZED DISTANCE FROM LENS CENTER FIGURE 5. The average refractive index profile for 17 lenses from 10 fish. The two halves of each lens had been averaged before all profiles were averaged. Error bars are standard deviations. The arrows point to shght depressions in the individ~l profiles [see also Fig. 3(R)] which were preserved after averaging a11profiles. In the inset, refractive index distributions suggested for other fish are shown in comparison to the H. burroni profile (heavy line) (cod, carp: Matthiessen, 1880; goldfish: Axelrod et al., 1988; rock bass: Campbell 8c Harrison, 1990). The dashed line is the refractive index profile of a hypothetical fish lens suggested by Jagger (1992). Small differences in the ways of calculating the refractive index due to the different wavelengths used by the various authors have been neglected since dispersion of ocular media is small between 590 and 633 nm (e.g. Sivak & Mandelman, 1982).

1820

RONALD OPTICAL

1 so 1.45

AXIS

t

0.0

H. H

A

0.1

0.2 0.3 NORMALIZED

0.4 0.6 0.6 BEAM ENTRANCE

0.7 0.6 POSITION

0.9

1.0

FIGURE 6. Comparison of direct measurements of spherical aberration in lenses in 0.9% saline (10 lenses, 5 fish) with predictions from ray-tracing model calculations based on the H. burtoni refractive index profile shown in Fig. 5. The two halves of all lenses have been combined on one side of the optical axis. For each lens, BVD(.r) data points were calculated for beam entrance positions separated by O.OlR (R = lens radius) by using linear interpolation between measured data points. Thereafter corresponding BVD(y) data points were averaged to obtain the spherical aberration shown in the figure. Large error bars are the standard deviations of the means of interpolated BVD(y) data points separated by 0.1 R. Small error bars are the standard deviations in the means of BVD(y) if all data sets are normalized to the same average BVD. These error bars indicate the uncertainty in curve shape. The predicted curve was generated by a ray-tracing model calculation with the average refractive index profile shown in Fig. 5.

smoothness of model calculations, the spacings between layers were chosen as a parabolic function so that many thin layers were placed in the periphery of the lens where the refractive index gradient is steepest. The ray-tracing program was tested by modelling a lens which analytically has been shown to be free of spherical aberration (Luneburg, 1944). Increasing the number of layers from 1000 to 30,000 has previously been shown not to influence the predictions of the model except for increasing the smoothness of the results (Kriiger, 1989). Back vertex distances equivalent to direct measurements of BVDs in saline were calculated with a ray-tracing model that used the measured refractive index profiles and had the refractive index of the surrounding medium set to 1.334 (633 nm). Spherical aberration was measured in 10 lenses from 5 fish with an average lens radius of 1.19 + 0.07 mm. The measured spherical aberration is compared with predictions from the lens model in Fig. 6. Calculations are based on the representative refractive index profiles shown in Fig. 5. Paraxial focal length of the H. burtoni lens was calculated from the BVDs of beams with entrance positions between -0.25 and +0.25 R (R = lens radius) to make the results comparable with data from Fernald and Wright (1983, see also Fernald & Wright, 1984). For the red light of the HeNe laser (A = 633 nm), we found f = 2.238 f 0.043 R.

DISCUSSION

The shape of the refractive index profile in H. burtoni lens is more complex than suggested

the by

KRijGER

CI ul.

previous studies in this species (Fernald & Wright, 1983; Campbell & Sands, 1984) and the profiles proposed for other fish species (Matthiessen, 1880; Axelrod ct al.. 1988; Campbell & Harrison, 1990). As suggested by Campbell and Sands (1984) based on model calculations and by Matthiessen (1880) Axelrod et ul. ( 1988). and Campbell and Harrison (1990) based on measurements in other fish species, the refractive index profile of the H. burtoni lens also shows a continual decrease of refractive indices from the center to the surface of the lens and the index profile is parabolic in first order approximation. However, details of curve shape of the spherical aberration can not be correctly predicted if the refractive index profile of H. burtoni is approximated by a simple function. Even if a polynomial with all even terms up to the 20th degree is fitted to the measured refractive indices and is used in the model calculations, the predictions of lens performance do not show the detailed agreement with directly measured spherical aberration that was achieved by using a cubic spline to interpolate between points in the representative refractive index profile (Figs 6 and 7). Model calculations that are based on simple approximations of the refractive index profile should therefore not be expected to accurately predict the image quality of a crystalline lens. The H. burtoni lens is of high optical quahty and slightly overcorrected for spherical aberration in the periphery. Shortening of the focal length near the optical axis (Fig. 6) makes a negligible contribution to image quality. The refractive index profile of H. burtoni results in much better correction of spherical aberration than predicted by the lens models for other species (Fig. 8). Steplike increases in BVD (Fig. 6) correspond to barely visible depressions in the index profile [Figs. 3(D) and 51. Steps are much more obvious in the spherical aberration than in the refractive index profile, indicating the sensitivity of image quality to the exact profile. Depressions were visible in 82% of the refractive index profiles at about 60% lens radius and in 65% of the profiles at OPTICAL

I

1.50

AXIS

a 1.26 19th DKiREE

P 1.20 1.15 1.101 0.0

. 0.1

_

A2 0.2 0.4 nolwLtl%DBcIw~~

0.6

.

.

OS

0.7

. 88

0.9

1.0

FIGURE 7. Comparison of the predictions of different approximations of the refractive index profile of the H. burtoni lens. The excellent predictive power of the model using a cubic spline intcrpolation between n(r) data points has been shown in Fig. 6. The complexity of the refractive index profile is demonstrated by the failure to produce similar results when polynomials with all even numbered terms up to 10th and 20th degree are fitted to the refractive index profile and used in the model calculation. Simple mathematical description of the profile appears to be impossible.

REFRACTIVE

INDEX DISTRIBUTION

_ _ ) OPTICAL AXIS

0.0

0.1

0.2 0.3 0.4 0.6 0.6 0.7 0.8 0.9 NORMALIZED BEAM ENTRANCE POSlllON

1.0

FIGURE 8. Spherical aberrations as in Figs 6 and 7 from ray-tracing calculations with refractive index profiles suggested for various fish species and with a hypothetical profile suggested for a generalized iish lens by Jagger (1992). The corresponding refractive index profiles are shown in Fig. 5. The refractive index profile of H. burfoni measured in this study results in better correction of spherical aberration than profiles previously proposed for fish lenses. The H. burtoni lens has a shorter focal length and it is slightly less well corrected for spherical aberration than the generalized fish lens proposed by Jagger (1992). The cod and carp profiles have quadratic decreases in refractive index with distance from the lens center. In the rock bass and goldfish profiles, the squared refractive index decreases with distance from the lens center as a polynomial with even numbered terms of up to 8th degree which is similar to the refractive index profile proposed by Jagger ( 1992).

about 85% lens radius. These depressions correspond to steps in the spherical aberration for beam entrance positions of about 70% and 90% lens radius since the beams come closer to the center of the lens on their curved paths through the lens. In the visual system of H. burtoni, calculation of focal length as a function of aperture size would be much more difficult than the standard Seidel approximation that the best focal plane falls half way between the paraxial and marginal plane (Welford, 1989). Although the aberration of the H. b~rtun~ lens is small (Fig. 6) it is certainly not simply 3rd order. Very faint, spherical zones of backscattered light were visible as rings in freshly excised lenses and were also noted by Fernald and Wright (1983) in the same species. They were in approximately the same radial positions as the steps in the spherical aberration and the depressions in the index profile. It is not known whether the backscattering of light was due to opacity or to reflection at a discontinuity, i.e. a steplike increase or decrease in refractive index. The depressions found in the refractive index profiles may actually be small steps that have been smoothed by the techniques of measurement and data analysis. The observations of backscattered light, depressions in the refractive index profile, and steps in the spherical,aberration at corresponding positions all point to anomalies in the fine structure of the refractive index profile and hence in the spherical aberration of the H. burtoni lens. Although the results from each method alone may not be sufficient evidence for such a fine structure, the combination of all results shows that there are indeed anomalies in the refractive index distribution of the H, b~rt~ni lens.

IN TELEOST CRYSTALLINE

LENS

1821

It is unclear whether the rings observed in excised lenses are also present in vivo. Slit lamp photography failed to show such rings in living animals, but that may be due to insufficient resolution. The backscattered light seen in excised lenses was extremely faint in most fresh lenses. The rings become increasingly more pronounced within a few hours and are correlated with measurable changes in the refractive index profiles. These postmortem changes have been investigated in a separate study (Munger, Campbell, Kriiger & Burns, 1992b). The exact orientations of the lenses to the laser beams were not known in this study since, except for the ligament which was used as a handle to manipulate the lenses, there are no landmarks on the H. burt~~~ lens that are visible in immersed lenses. It has been shown in the rabbit that lens sutures can have significant effects on measurements of spherical aberration (Kuszak, Sirak & Weerheim, 1991). Some of the variability in our data may therefore have resulted from varying angles between the planes of the laser beam scans and lens suture planes. However, the variation in BVDs near the optical axis is correctly predicted from the spherically symmetric refractive index profile measured in a large number of lenses with varying orientations to the laser beams used for measurements. It is therefore unlikely that the observed drop in BVDs for beam entrance positions close to the optical axis was caused by lens sutures. Fernald and Wright (1983) found a paraxial focal length of 2.252 _t 0.046 R at 494 nm by focussing a microscope on the focal points of excised lenses from the same stock of animals as used in this study. From the data of Fernald and Wright (1985) we estimate the longitudinal chromatic aberration between 633 and 494 nm to about 1.5% focal length. If our value of the focal length is adjusted to 494 nm, it is 2.196 + 0.043 R, which is slightly lower than the previous results. However, the difference is not statistically significant and while in our method all beams make an equal contribution, beams very close to the optical axis contribute little to the appearance of the focal point in a microscope. Due to the steep drop-off in BVDs for beam entrance positions close to the optical axis (Fig. 6), focal lengths calculated from BVDs decrease more rapidly with decreasing aperture than measurements of focal lengths with a microscope. Excellent correction of spherical aberration, and focal lengths similar to the focal length of the H. burtoni lens were found by Sroczyriski in the lenses of rainbow trout (1975a, f = 2.244-2.387 R), pike (1975b, f = 2.370-2.440 R), and roach (1977, f = 2.169-2.294 R). The shape of the refractive index profiles of those species may therefore also be similar to the profile of H. burtuni. In comparison to the hypothetical lens model proposed by Jagger (1992), the H. burtoni lens has a considerably shorter focal length and it is slightly less well corrected for spherical aberration (Fig. 8). Refractive index of the H. burtoni lens is higher at the center and lower at the surface than assumed by Jagger (1992) (Fig. 5). The correct prediction of details of the spherical

1822

RONALD

H. H. KRiiGER

aberration by the lens model based on the measured refractive index profiles is an indication for the high resolution and accuracy that can be achieved in refractive index measurements in crystalline lenses with the new video technique and by using interpolation between data points in the deflection function. High resolution and accuracy is possible even in lenses as small as the H. burtoni lens. This method can be applied to all lenses of spherical or ellipsoidal symmetry (Campbell, 1984). Detailed measurements of refractive index profiles can be the basis for lens models that are much more realistic than previously possible. In this study, we have shown that the refractive index profile of the lens of H. burtoni has a continual decrease of refractive indices from the center to the surface with a complex curve shape. The lens is of high optical quality with little residual spherical aberration. Details in the curve shape of spherical aberration measured in the lenses of H. burtoni are correctly predicted by the lens model. 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, 5145. Campbell, M. C. W. (1982). Gradient refractive index optics and image quality in the rat eye. Dissertation: Australian National University, Canberra. Campbell, M. C. W. (1984). Measurement of refractive index in an intact crystalline lens. Vision Research, 24, 409415. Campbell, M. C. W. & Harrison, E. M. (1990). Refractive-index distribution within the crystalline lens of the rock bass and its variation with age. Optical Society of America Annual Meeting Technical Digesl, p. 149. Campbell, M. C. W. & Hughes, A. (1981). An analytic, gradient index schematic lens and eye for the rat which predicts aberration for finite pupils. Vision Research, 21, 1129-l 148. Campbell, M. C. W. & Sands, P. J. (1984). Optical quality during crystalline lens growth. Nature, London, 312, 219-292. Chan, D. Y. C., Ennis, J. P., Pierscionek, B. & Smith, G. (1988). Determination and modeling of the 3-D gradient refractive indices in crystalline lenses. Applied Optics, 27, 92693 1. Chu, P. L. (1977). Nondestructive measurement of index profile of an optical-tibre preform. Electronics Letters, 13, 736738. Fernald, R. D. (1977). Quantitative behavioural observations of Haplochromis burtoni under semi-natural conditions. Animal Behaviour, 25, 643653.

Fernald, R. D. (1984). Vision and behavior in an African cichlid fish. American Scientist, 72, 58-65.

Femald, R. D. (1987). Aquatic adaptations in fish eyes. In Atema, J., Fay, R. R., Popper, A. N. & Tavolga, W. N. (Eds), Sensory biology of aquatic animals Berlin: Springer. Femald, R. D. (1989). Fish vision. In Finlay, B. L. & Sengelaub, D. R. (Eds), Development of the vertebrate retina (pp. 247-266). New York: Plenum Press. Femald, R. D. 8~ Hirata, N. R. (1977). Field study of Haplochromis burtoni, habitats and co-habitants. Environmemal Biology of Fish, 2,

e/ ul

Jagger. W. S. (1992). The optics of the spherical fish lens. )‘r.crorr Research, 32, 1271-1284.

Kroger, R. H. H. (1989). Dioptrik, Funktion der Pupille und Akkommodation bei Zahnwalen. Dissertation: Eberhard-Karls-Universitat. Tubingen. Kroger, R. H. H. (1992). Methods to estimate dispersion in vertebrate ocular media. Journal of the Optical Society. of‘ Americ,a A, 9. 14861490.

Kriiger, R. H. H. & Fernald, F. D. (1994). Regulation of eye growth in the African cichlid fish. Haplochromis burtoni. Vi.sion Research, 34, 1807-1814.

Kuszak, J. R., Sivak, J. G. & Weerheim, J. A. (1991). Lens optical quality is a direct function of lens sutural architecture. Inuesfigatire Ophthalmology and Visual Science, 32, 2 119-2 129. Luneburg, R. K. (1944). Mathematical theory of optics. Providence, R. 1.: Brown University. Matthiessen, L. (1880). Untersuchungen iiber den Aplanatismus und die Periscopie der Kristalllinsen in den Augen der Fische. PjIr7ger.s Archiv, 21, 287.-307.

Matthiessen, L. (1882). Ueber die Beziehungen, welche zwischen dem Brechungsindex des Kerncentrums der Krystalllinse und den Dimensionen des Auges bestehen. P’tigers Archiu, 27, 510-523. Matthiessen, L. (1893). X. Beitrlge zur Dioptrik der Kristall-Linse. Zeitschrift fur vergleichende Augenheilkunde,

7. 102-146.

Maxwell, J. C. (1854). Solutions of problems. Cambridge and Dublin Mathematical Journal, 8, 1888195. Reprinted in Niven, W. D. (Ed.), The scienrific papers of James Clerk MaxweN (Vol. 1). (1890). Cambridge: Cambridge University Press. Munger, R., Campbell, M. C. W. & Kriiger, R. H. H. (1992a). Gradient index profile calculation using cubic splines. Optical Society of America 1992 Technical Digest Series, 3. 59 62.

Munger. R., Campbell, M. C. W., Kroger, R. H. H. & Burns, C. M. (1992b). Refractive index profiles of crystalline lenses with a visible region of opacity. Investigative Ophthalmology and Visual Science. 33, I 169. Pierscionek, B. K., Chan, D. Y. C., Ennis, J. P. & Smith, G. (1988). Non-destructive three-dimensional gradient index models for crystalline lenses: 1. Theory and experiment. American Journal of Optometry and Physiological Optics, 65, 481-491.

Pierscionek, B. K. (1989). Growth and ageing effects on the refractive index in the equatorial plane of the bovine lens. Vision Research, 29, 175991766.

Press, W. H., Flannery, B. P., Teukolsky, S. A. & Vetterling, W. T. (1989). Numerical recipes in Pascal. Cambridge: Cambridge University Press. Sivak, J. G. & Mandelman, T (1982). Chromatic dispersion of the ocular media. Vision Research, 22, 997-1003. Sivak, J. G. & Kreuzer, R. 0. (1983). Spherical aberration of the crystalline lens. Vision Research, 23, 59970. Sroczynski, S. (1975a). Die sphiirische Aberration der Augenlinse der Regenbogenforelle (Salmo gairdneri Rich.). Zoologische Jahrbiicher (Physiologie), 79, 204-2 12. Sroczynski, S. (1975b). Die spharische Aberration der Augenlinse des Hechts (Esos lucius L.). Zoologische Jahrbri’cher (Physiologic), 79. 547-558.

Sroczynski, S. (1977). Spherical aberration of crystalline lens in the roach, Rut&s rutilus L. Journal of Comparative Phvsiology A. 121, 1355144.

Sroczynski, S. (1979). Das optische System des Auges des Flussbarsches. Zoologische Jahrbiicher (Physioiogie), 83, 224252. Welford, W. T. (1989). Aberrations of optical syslems. Bristol: Adam Hilger.

2999308.

Fernald, during Femaldl in the

R. D. & Wright, S. E. (1983). Maintenance of optical quality crystalline lens growth. Nature, London, 301, 618620. R.-D. & Wright, S. E. (1985). Growth of the visual system African cichlid fish, Haplochromis burtoni; optics. Vision

Research, 25, 155-161.

Fletcher, A., Murphy, T. & Young, A. (1954). Solutions of two optical problems. Proceedings of the Royal Society of London A, 223, 216-225.

Acknowledgements-We

gratefully acknowledge the help of Catherine M. Burns and Sabine Kriiger in the experimental work. We also wish to thank Anthony P. Cullen for performing the slit lamp photography. RHHK was supported by DFG grant Kr1078/1-1. This study was also supported in part by NIH grant EY05051 to RDF and by NSERC Canada. URF, and Operating Grant to MCWC.