Introduction

The optics of the eye lens varies according to species requirements and the variation is attributed to differences in lens shape and in the distribution of refractive index. The latter parameter has been the more difficult to determine chiefly because the lenticular index is not uniform. Various methods for its measurement have been tried. Studies which required slicing of the tissue1, 2, 3, 4, 5, 6, 7, 8, 9, 10 could not avoid some smearing of cellular contents across the sliced layer. More sophisticated procedures, using fibre optic sensing with minimal tissue violation, have been introduced.11, 12, 13 Noninvasive ray tracing methods14, 15, 16, 17, 18, 19, 20, 21, 22 originally derived for the measurement of refractive index distributions in optic fibre preforms23 necessitated assumptions about the lens shape and index contours. Complex modelling techniques have also been used,24 and more recently magnetic resonance imaging has been promoted as a method of index measurement.25, 26

The refractive index of the lens varies with temperature and the wavelength of light but the effects of such variations on the refractive index profile have not been studied. This paper presents the first study of the refractive index distribution in the porcine lens for two different wavelengths using the method of ray tracing to see whether there is a wavelength dependent difference in the magnitude and shape of the refractive index gradient. The paper also examines the ray-tracing methods to see what effect the value of the refractive index at the lens edge has on the refractive index profile.

Materials and methods

A total of 17 porcine lenses, obtained from the local abattoir, were used in these experiments. All were used within 12 h of death. Lenses were weighed immediately after removal from the eye and reweighed after experimentation. The weight range of the lenses was 0.321–0.509 g (pre-experimental weight). The difference in pre- and postexperiment weights was less than 5% in all but two lenses in which the differences were 9 and 12%. These larger differences were not the result of any alteration to the lens (eg swelling) during experimentation but due to incidents which occurred after experimentation. (In one case some of the gel medium adhered to the surface of the lens capsule; as much of this as possible was gently wiped off. In the other case, the capsule peeled off the lens on removal from the gel. Although the capsule and lens were weighed together, the sticky lens surface adhered to the spatula and some outer cell layers were lost).

During experimentation the lenses were set in a 0.5% agarose gel made up in a solvent isotonic with the aqueous, as described in previous work.16 The lens and gel were contained in a specially designed L-shaped cell which allowed measurements to be taken in the sagittal and the equatorial planes.16

The optical system has been described previously.16 Briefly, the cell containing the immobilized lens was placed in the path of parallel laser rays. The single beam source was split into 10 parallel rays in accordance with the method of Pierscionek.27 The passage of rays through the lens was photographed using high-resolution Canon Powershot Pro70 digital camera with a total image sensor of 1 680 000 pixels. The images were transferred into a computer and analysed using the Ulead PhotoImage v.4.0 software. Ray path parameters: positions of ray entry into and exit out of the lens, and angles the emergent rays made with the optic axis were measured from the images. The maximum error in the measurement was ±0.13 mm and ±0.2°. Measurements were taken for the two light sources: Uniphase He–Ne class IIIb (633 nm: red) and Coherent DPSS 532 Class IIIb (532 nm: green). All lenses were subjected to ray tracing using the 633 nm source; 10 of the lenses were subjected to additional ray tracing with the 532 nm source. In all cases, measurements were made in the equatorial and the sagittal planes.

The mathematical analysis, which was applied to the measured data, was based on the method of measuring gradient index profiles in optic fibre preforms23 as amended and described by Chan et al.28 The height of the incident ray above the optic axis (y) and the angle ψ(y), made by the emergent ray and the optic axis, are related to the refractive index profile n(r) by the expression

where: ns is the refractive index of the surrounding medium, n(r) the refractive index profile across the lens, ρ the distance of boundary from the lens centre (ie equatorial radius), and rm the function of the displacement y and can be obtained from the equation

However, when the refractive index of the surrounding medium is not matched to that of the lens edge, it is necessary to obtain an estimate of the magnitude of the refractive index mismatch before the refractive index profile can be determined.

Following Chan et al,28 equation (1) can be transformed by introducing the variable ξ and a function g(ξ) defined by

and

to give the following expression:

where is the ratio between the refractive index at the edge of the lens (n(ρ)) and that of the surrounding (ns). The term between {⋯} in equation (5) vanishes when the system is matched and after inversion it reduces to

The effect of the refraction across the lens boundary due to index mismatch can be removed by transforming the experimentally determined refraction data [y,ψ(y)] into an equivalent set of index-matched data28 where

and

The index matched data derived from equations (7) and (8) are used to calculate g(ξ) (Equation (6)) which is subsequently used in Equation (4) to calculate the refractive index distribution n(r).

Measurement of edge index value

The refractive index values at the extreme edge of the lens were measured by magnifying each equatorial image and applying Snell's law:

where: ns is the refractive index of the surrounding gel, ne is the refractive index at the lens edge, θi is the angle at which the ray is incident on the interface, θe is the angle at which the ray leaves the interface between lens surface and the surrounding medium. These measurements were taken at entry and exit points of rays in the upper or lower third of lenses. Here, refraction was sufficiently large to enable accurate measurement of refractive index at the edge of the lens to be made. Rays close to the centre did not bend sufficiently to make such calculations feasible. In all, 13 measurements were taken. The mean of these was 1.354 with a standard deviation of ±0.002. Edge index values were only measured for 633 nm light; the 532 nm source did not provide sufficient contrast to obtain accurate values. The edge index values for 532 nm were calculated from the values measured for the 633nm source (as described in results).

Results

Refractive index profile

Figure 1 shows ray passage through the equatorial plane of a porcine lens taken for 633nm (Figure 1a) and 532 nm (Figure 1b) light. While the sagittal plane is the one relevant to the path of light in vision, the calculation of refractive index, using the Eikonal equation, requires the symmetry provided by the equatorial plane. In previous publications, the equatorial index profile was replotted for the sagittal plane by a simple transposition requiring re-scaling of the equatorial radius to the sagittal width.16, 28 The assumption underlying this transposition was that the contours of refractive index are concentric. This was supported by the findings of subsequent studies, in bovine lenses, in which the refractive index was measured directly along the optic axis12 and the profile obtained corresponded in shape and magnitude to that found in the equatorial plane.16, 18

Figure 1
figure 1

Ray trace photographs taken through the equatorial planes of a porcine lens for (a) 633 nm and (b) 532 nm light.

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The parameters necessary for the derivation of refractive index are the displacement of the incident ray from the optic axis, (y), and the angle at which each emergent ray meets the optic axis (ψ(y)). Figure 2(a) and (b) shows (ψ(y)) plotted against (y) in the equatorial plane for 633 nm and 532 nm light, respectively. Data for all lenses are combined because, in all cases, the shape in the equatorial plane is the same and the lenses were from the same age group and were of the same size. The plots were transformed into an equivalent set of index matched data (as described in the Materials and methods and using edge index values discussed below), from which the refractive index, as a function of distance from the lens centre, was calculated.

Figure 2
figure 2

The angle formed by the emergent ray and the optic axis (ψ(y)) in degrees) plotted against the displacement of the incident ray from the optic axis (y in mm) for light of wavelength (a) 633 nm and (b) 532 nm. Measurements were made in the equatorial plane.

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Figures 3(a) and (b) show the refractive index plotted against distance from the centre of the lens for 633 nm and 532 nm light respectively. There is a small wavelength dependent difference between the profiles and this difference increases very slightly from periphery to centre. The index at the edge of the lens is 1.354 for 633 nm (from measurement) and 1.358 for 532 nm (from calculation described below). The index values in the central part of the lens plateau at 1.396 for 633 nm and 1.404 for 532 nm. Polynomials of second order fitted closely to both index profiles (R2=0.9760 (633 nm); R2=0.9972 (532 nm), P≪0.005 (both profiles)) and hence parabolic fits were considered to provide good approximations.

Figure 3
figure 3

Refractive index (n(r)) plotted against radial distance from the centre of the lens (r in mm) for (a) 633 nm light and (b) 532 nm light.

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Edge index value

The results for 633 nm source ranged between 1.351 and 1.358, giving an average of 1.354±0.002. The edge index values for the 532 nm source were more difficult to obtain because the greater scatter of the shorter wavelength light reduced the contrast of the rays. However, it was possible to calculate the index value at the edge of the lens, for 532 nm light, from protein concentrations which were obtained from the 633 nm index value by the Gladstone-Dale formula:29

where n is the refractive index at the edge of the lens, ns the refractive index of the solution (in this case extracellular fluid), Δ the specific refractive increment (the increase in refractive index for every 1% increase in protein concentration (ml/g)), and C is the protein concentration (g/ml).

The refractive index of the surrounding gel was, to a close approximation, the same as water.30 The refractive index of water at 20°C for light of 633 nm=1.3325.31

Taking n=1.354, ns=1.333, and Δ=0.186 ml/g (see the Appendix) gives a protein concentration at the lens edge of 0.11 g/ml. Putting this value of protein concentration back into the Gladstone–Dale formula and using ns=1.33731 and Δ=0.194 ml/g (see the Appendix) gives an edge refractive index of 1.358 for 532 nm light.

The refractive index of the surrounding gel was not matched to the value of the index at the lens edge, requiring further calculations involving the edge index values (as described in Materials and methods). In order to see what effect the edge index value (and hence the mismatch between edge and surround indices) may have on the refractive index distribution across the lens, the edge index values for both 633 nm and 532 nm were each used with the ray trace parameters measured with 633 nm light, to calculate the respective index profiles. The distributions are shown in Figure 4 and show that the magnitude of the edge index makes little difference to the shape and magnitude of the final distribution.

Figure 4
figure 4

Refractive index (n(r)) plotted against radial distance from the centre of the lens (r in mm) for 633 nm using both the 633 nm (filled diamond) and the 532 nm (open square) edge index values.

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Discussion

This paper, presents results of ray-trace measurements and calculations of the refractive index profiles in the equatorial planes of porcine lenses. Measurements were made for two different wavelengths, within the visible spectrum, to investigate whether any wavelength dependent variations in refractive index could be detected.

The shape of the index. profile in the porcine lens is similar to the shapes reported in other animal lenses.7, 10, 12, 14, 16, 17, 18, 21, 22 The results of this study show that there is some, albeit small, wavelength dependent difference in the magnitude of refractive index: the value is slightly higher when measured with 532 nm light compared to the measurements made with 633 nm light. This is consistent with known wavelength dependent variations in refractive index: the refractive index increases as wavelength decreases. Both wavelengths were within the visible wavelength range and only about 100 nm apart and so large differences in refractive index were not expected.

Measurement of the complete refractive index gradient of the porcine lens has not been reported previously, although refractive index, for four wavelengths, has been measured, using Pulfrich refractometry, in outer and inner sections of four porcine lenses, by Sivak and Mandelman.32 It was found that from 650 to 440 nm, the refractive index in the inner region increased from 1.4218 to 1.4346.32 These values are higher than those reported in this study. This may be because of age differences between the lenses used by Sivak and Mandelman and the lenses used in this study. Refractive index has been shown to increase with age in bovine lenses.18 However, as no physical dimensions were reported by Sivak and Mandelman,32 it is not possible to make any comparisons based on age.

In terms of profile shape, the results of this study support findings from measurements of porcine lens protein concentration profiles as determined by Raman microspectroscopy.33 This latter study showed that the profile of protein concentration in the porcine lens has a shape which could be approximated to parabolic. de Korte et al33 report that the protein content rises from around 0.3 g/cm3 in the peripheral sections of the porcine lens to around 0.7 g/cm3 in the centre. Using these values as protein concentrations, 1.333 for the refractive index of water and 0.19 for the specific refractive increment, the Gladstone–Dale formula gives approximate refractive index values of 1.390 and 1.466 for the periphery and centre respectively. These values are higher than those found in this study and in previous work.32 Discrepancies may be attributed to usage of different techniques and to the units of the protein content used in the presentation of results. The protein contents, reported by de Korte et al33 are based on mass percentages taken as the complement of water mass, the latter calculated from the ratio of the intensities of the Raman peaks.34 These values, therefore, give content on a relative mass/mass basis and do not take into account the partial specific volume of proteins (around 0.72 ml/g). The values derived from the Gladstone–Dale formula give concentrations as mass of solute (g) dissolved in a given volume of solvent (1 ml). Furthermore, the measurements made by de Korte et al33 were made on 1 mm thick lens slices and so the result is an average value over the sample. Any method which involves slicing tissue, no matter how accurate, will cause some smearing of the contents of adjacent layers.

This study shows that there is a very small change in dispersion across the lens, increasing into the centre. This may be indicative of the variations in the proportions of the three protein classes but in the absence of data about the protein distributions in the porcine lens, no further specific conclusions can be made. Structure/function links between the proteins of the lens and the refractive index have been studied in bovine and human lenses.35, 36, 37 No direct link between the distribution of any particular protein class and the refractive index profile has, as yet, been found. Nevertheless, some general observations, chiefly from interspecies comparisons about the structure/function relationship in the lens, are worthy of mention. The interaction between the different protein classes has an effect on water content and hence on the refractive index. Bettelheim and Finkel38 showed that the more complex the combination of crystallin proteins, the higher the amount of water that can be absorbed. γ-crystallins (the smallest class of proteins) tend not to interact with the other protein groups and it has been noted that in species where γ-crystallin content is relatively high, for example in rat lenses, the lenses have a high protein density and therefore high refractive index. γ-crystallin is also the protein with the highest refractive increment of all the crystallins.39 It is possible that the porcine lens has a similar pattern of protein distribution to the bovine lens, because of the similarity in shape of their respective refractive index profiles. In such case, the nuclear region of the porcine lens, should have the highest level of γ-crystallins. However, this alone cannot explain the slightly higher dispersion in the centre of the lens compared to the periphery, because the refractive increment of γ-crystallins has the lowest variation with wavelength (Appendix). The explanation is likely to lie in the structural organisation and interaction of the crystallins in the intact lens. An in depth understanding of the contribution of proteins to the optical properties will require complex models which can accurately predict and simulate the natural environment of the crystallins.

Conclusion

The complete refractive index distributions in the porcine lens have been measured for the first time for two wavelengths: 532 and 633 nm, using a ray tracing method. The two wavelengths used for the measurement were about 100 nm apart, and wavelength-related differences in the profile, were apparent. The index distributions are, like those in other animal lenses, approximately parabolic.