Age dependence of the average refractive index of the isolated human crystalline lens

Abstract

We measured the average group refractive index (RI) of 120 isolated lenses from 120 human donors (age: 0.03 to 61 years). The average group RI was calculated from a measurement of the optical thickness of the lens using optical coherence tomography and the apparent window shift of the test chamber caused by the lens. The estimated measurement uncertainty was ±0.004. The group RI at 880 nm was converted to phase RI at 589 nm using the dispersion equation of water and protein. From 2 to 61 years, the mean value of the RI was 1.415 ± 0.002 (group index at 880 nm) and 1.406 ± 0.002 (phase index at 589 nm) independent of age (p = 0.774). Two lenses from donors of age 0.33 and 3 months had significantly lower RI (group index: 1.405 and 1.403; phase index: 1.396 and 1.394). From age 2 to 61, the average lens RI is constant with age within the measurement uncertainty (±0.004).

1. Introduction

The human lens has a non-uniform refractive index distribution (gradient refractive index, GRIN) that contributes to the lens power and aberrations. The refractive index distribution changes with age due to the continuous growth of the lens [1–6]. For instance, there is evidence that the compaction of lens fibers in the lens center as new lens fibers are generated results in the progressive formation and enlargement of a central plateau in the refractive index distribution [4,7]. Age-related changes in refractive index distribution combined with changes in lens shape contribute to the age dependence of ocular power and aberrations [8–11]. Therefore, understanding how the gradient profile changes with age is crucial for comprehending the age-related changes in the optics of the lens and of the eye, in particular during refractive development in childhood or later in adult life in the age range leading to presbyopia. The lens refractive index gradient is directly related to the gradient in protein concentration [2,12,13]. Quantifying the average index and its age dependence can also provide information regarding age-related changes in the protein concentration.

Several techniques have been used to measure the refractive index distribution of intact human lenses. Measurements on intact in vitro human lenses have been acquired using MRI [3,14,15], X-ray interferometry [6], and inverse methods based on Optical Coherence Tomography (OCT) [5,8,16]. MRI has also been used to acquire measurements in vivo in human subjects [1]. In these studies, the axial variation of the refractive index in the anterior and posterior regions of the lens is generally represented as a power function of the form:

$$n(z)=n_N-(n_N-n_C){\left(\frac{z}{t}\right)}^p$$ (1)

where z is the axial coordinate, n_N is the refractive index of the lens center (nucleus), n_C is the refractive index at the lens surface (outer cortex), t is the anterior or posterior lens thickness, and p is the power coefficient. The power coefficient determines the shape of the profile. A value of p = 2 corresponds to a parabolic profile. As the power coefficient increases, the axial profile steepens in the periphery and flattens in the lens center, leading to a more pronounced central refractive index plateau.

In vivo MRI studies found that the power coefficient increases with age, with average values of 4.2 at age 19-29 and 6.7 at age 60-70 [1]. On the other hand, a study using OCT to reconstruct the GRIN produced a constant power coefficient in the axial direction, with a value of p = 2.39 on average [8]. Similarly, measurements using X-ray interferometry found no statistically significant age dependence of the power coefficient along the optical axis, with an average of 4.7, and a relatively large variability across individuals with values ranging from about 3.2 to 6.6 [6]. The cortex and nucleus indices have been found to vary little with age, with values that vary significantly between studies. The values obtained using MRI in vivo are n_N= 1.410 and n = 1.379 on average for young (19-29) and older (60-70) years [1,17]. OCT measurements produce values that range from 1.356 to 1.388 for the lens surface, and 1.396 to 1.434 for the nucleus [5]. X-ray interferometry finds a surface index around 1.35 and a nuclear index ranging from 1.420 to 1.441 with a slight age dependence, with an average value of around 1.433 from age 16 to 58 and 1.429 from age 60 to 91 [6]. The variability of the gradient parameters between studies reflects the challenges in measuring the refractive index distribution with high accuracy and precision.

The validity of the refractive index profiles produced by different studies can be assessed by measuring the average refractive index of the lens. The average refractive index is the average value of the axial refractive index profile. For the axial profile of Eq. (1), the expression of the average index is [18]:

$$n_{avg}=\frac{1}{t}{\int }_0^{t}n(z)d_z=n_N-\frac{n_N-n_c}{p+1}$$ (2)

The average refractive index is also the index value that must be used to calculate the geometrical thickness of the lens, t, from measurements of lens optical thickness, OT, acquired using optical biometers (n_avg= OT/t) and for distortion correction of lens OCT images [19]. Note that the average index is different from the “equivalent” index of the lens. The equivalent index is the index of the homogenous lens that has the same shape and dioptric power as the real lens. The equivalent index is calculated from measurements of the lens anterior and posterior radii of curvature, lens thickness, and lens power [20,21].

Measurements of the average index and its changes with age can help assess the accuracy of gradient index profiles from prior studies. For instance, the values of the in vivo MRI studies [1] predict an average index of 1.404 at age 19-29 years (n_C= 1.379, n_N= 1.410, p = 4.9) and 1.406 at 60-70 years (n_C= 1.379, n_N= 1.410, p = 6.7). On the other hand, a study using OCT to reconstruct the GRIN [8] produced an average index of 1.394, independent of age, while an earlier study [5] produced an average RI of 1.417 ± 0.011. With n_C= 1.350 and p = 4.7 on average, the study using X-ray interferometry [6] produces an average index of 1.418 from age 20 to 58 where n_N= 1.433 and 1.415 from age 60 to 90 where n_N= 1.429. The validity of these reconstructed GRIN profiles can be assessed by evaluating if the average RI that they produce is in agreement with direct measurements of the average index.

The average value of the axial refractive index profile of intact in vitro lenses was previously measured using OCT on 33 human lenses aged 6 to 82 years and found to be 1.408 ± 0.005 [18]. There was a trend for a decrease with age, but the sample size was insufficient to make a definite conclusion regarding age dependence. On the other hand, an analysis compiling refractive index gradient measurements from different studies predicts that the average refractive index increases with age, consistent with the progressive formation of a refractive index plateau in the nucleus [22].

These conflicting findings regarding the gradient parameters and the average index illustrate the need for additional measurements over a broader age range. The current study aims to measure the average refractive index of intact isolated human lenses as a function of age using the OCT-based method of Uhlhorn et al. [18].

2. Methods

2.1. Donor information

Isolated crystalline lenses or whole globes were acquired from the Ramayamma International Eye Bank (RIEB), LVPEI, Hyderabad. The eye bank procured the tissues from surrounding regional hospitals. All lens tissues were obtained and used in accordance with the Declaration of Helsinki for experimental research purposes. The excised globes were carefully preserved by placing them in a tightly sealed chamber, enveloped in saline-soaked gauze. Additionally, the lenses were individually stored in sealed glass vials, ensuring that the vitreous humor remained intact. For whole globes, the lenses were extracted within 3 hours of receiving the globe by sectioning the cornea, removing the iris, and cutting the ciliary zonules and vitreous. The tissues used in the isolated lens experiment were prepared according to the procedure outlined by Heilman et al. [9].

After excluding the lenses with capsular or cortical damage or insufficient image quality, a total of 137 intact lenses from 34 bilateral and 103 unilateral eyes were available for the study. For the 34 lenses from bilateral eyes, the lens from the left or right eye was randomly selected for inclusion in the statistical analyses, leaving a final dataset of 120 lenses from 120 donors, ranging in age from 0.03 (10 days) to 61 years. The sample included two infant lenses (0.03, 0.25 years) belonging to the fast growth period of the lens, which ends before the first postnatal year. Since there were only two samples within this age range, we excluded these two eyes from the statistical analysis of age-dependency. The final statistical analysis includes 118 lenses ranging from 2 to 61 years. For these 118 eyes (118 donors), the mean ± standard deviation of the age was 25.96 ± 12.68 years and the median (interquartile) post-mortem time was 38 (27, 55) hours. The tissues included intact globes (n = 5) and lenses extracted from globes (n = 113) with the vitreous. Ninety-five donors were males, and 23 were females.

2.2. OCT system

The lenses were imaged with the OCT module of a custom-built combined OCT-laser Ray Tracing aberrometer instrument (LRT-OCT). A detailed description of the LRT-OCT system can be found in Ruggeri et al. [23]. The OCT module uses a commercial 880 nm SD-OCT engine (Bioptigen ENVISU R4400, Bioptigen Inc., NC) coupled with a custom-built beam delivery unit. The SD-OCT system has an acquisition speed of 32,000 A-lines/s. The beam delivery system produces a telecentric scan with a focused spot diameter of 53 µm. For the present study, 3D images of the lens were acquired using a raster scan with 2048 A-lines per B-scan covering a 15 × 15 mm zone. Each A-line included 2048 pixels over a depth of 15.18 mm, corresponding to a pixel resolution of 7.4 µm/pixel in air (approx. 5.5 microns in tissue) in the axial direction. The lens was placed on a custom-built holder in a tissue chamber filled with Balanced Salt Solution (BSS; Alcon Laboratories, Inc. Fort Worth, TX, USA). OCT imaging was used in real-time to precisely align the center of the lens with the beam delivery unit [23].

2.3. Calculation of the refractive index

The average refractive index was determined by using the same method as Uhlhorn et al. [18]. The method relies on measurements of the optical thickness of the lens and of the distortion of the window of the tissue cell caused by the lens. Since the lens has a higher refractive index than the surrounding fluid, the window boundary appears to be shifted posteriorly in the central region of the image containing the lens relative to the peripheral regions where the rays propagate only through the fluid. The average index can be calculated from the optical thickness of the lens and the apparent shift of the window boundary.

The optical thickness and window shift for each lens were measured manually using ImageJ by three independent observers using a double-masked approach. We used a manual approach instead of automated segmentation to maximize reproducibility and minimize the uncertainty in identifying the boundary positions. A standard operating procedure was developed to ensure consistency of measurements across lenses and operators. The operator zoomed into the region of interest until the individual pixels were visible. The operator then moved the mouse to the boundary of interest and recorded the boundary’s pixel coordinates. Pixel coordinates were measured for the vertex of the anterior lens surface (AL) and posterior lens surface (PL), unshifted cell window (W1), and shifted cell window (W2). The coordinates of the lens and shifted boundary were measured along the same A-line passing through the lens center. The coordinates of the unshifted boundary were measured at the image periphery along A-lines passing left (W1-L) and right (W1-R) of the lens equator. Measurements were taken on both sides of the image to account for slight tilts of the cell. (Fig. 1) The average refractive index was calculated from the pixel coordinates using the following equation:

$$n_{avg(880)(group)}=\frac{n_{(group)(BSS)}}{1-\frac{W_2-W_1}{P_L-A_L}}$$ (3)

where n_(group)(BSS)= 1.343 is the group refractive index of BSS at 880 nm, and W₁ is the average of the measurements acquired on the left and right sides of the image (W1-L and W1-R). The group refractive index of BSS at 880 nm was measured by filling the tissue cell with seven different volumes of BSS and measuring the resulting window shift. The pixel coordinate of the window was plotted versus the pixel coordinate of the fluid level (air-BSS interface), and a linear regression was performed. The group refractive index of BSS was calculated from the linear regression slope (n_(group)(BSS) = 1 – slope).

Fig. 1.

OCT image showing the anterior lens (AL), posterior lens (PL), unshifted cell window right (W1, left and right) and shifted lens window (W2). W1 was calculated by taking the average of W1(L) and W1(R)

Since the OCT source is a broadband source, the average index is the group refractive index at 880 nm. To enable comparison with previous studies, the group refractive index at 880 nm was then converted to the phase refractive index at 589 nm. A detailed description of the conversion based on published data on the refractive increment and its dispersion is included as an Appendix [24,25]. The relation between the lens phase index at 589 nm and the lens group index at 880 nm is:

$$n_{avg(589)(phase)}=n_{avg(880)(group)}-0.009$$ (4)

2.4. Measurement uncertainty

The measurement uncertainty was calculated from Eq. (3), By taking the partial derivatives relative to the measured parameters.

$$\Delta n_{avg(880)(group)}=\left|\frac{\mathrm{\partial }{n}_{avg(880)(group)}}{\mathrm{\partial }Shift}\right|\Delta Shift+\left|\frac{\mathrm{\partial }{n}_{avg(880)(group)}}{\mathrm{\partial }OT}\right|\Delta OT$$ (5)

where Shift = W₂-W₁ is the window shift, OT = P_L-A_L is the optical thickness of the lens and ΔShift and ΔOT are the uncertainties in the window shift and optical thickness, respectively. Combining this formula with Eq. (3), gives the following expression of the measurement uncertainty:

$$\Delta n_{avg(880)(group)}=n_{(group)(BSS)}\left(\frac{1}{OT}\Delta Shift+\frac{Shift}{O{T}^2}\Delta OT\right)$$ (6)

Given the characteristics of the lens and image boundaries in the OCT images, we estimated that the uncertainty in the position of the boundaries was ±1 pixel for the window and ±5 pixels for the lens surfaces. Since the shift and lens optical thickness are the differences between the two measurements, the resulting uncertainty in the group refractive index was calculated from the measured values of the window shift and lens optical thickness using Eq. (6) assuming ΔShift=±2 pixels and ΔOT=±10 pixels. In addition, the measurement uncertainty was estimated by comparing the measurements of boundary positions and refractive index obtained by the three independent observers who processed the images using a double-masked approach.

2.5. Lens diameter and thickness-to-diameter ratio

The diameter of each lens was measured manually by one operator using Image J, following the same general approach as for the thickness measurement. The operator manually selected the left-most and right-most pixels in the equatorial plane of the lens. The distance between these two pixels was taken as the lens diameter. The thickness to diameter ratio was then calculated for each lens to ensure that the lenses were in their normal hydration state [26].

2.6. Data analysis

Statistical analyses were performed using SPSS Statistics 28 (Armonk, NY, USA). Univariate and multivariate analyses were performed to quantify the effect of gender, post-mortem time and age on lens average refractive index, lens thickness, lens diameter, and diameter to thickness ratio. A p-value of <0.05 was considered statistically significant. We also quantified the inter-operator variability of the window shift and optical thickness.

3. Results

The measurement uncertainty calculated using Eq. (6) was ±0.004, with a contribution of ±0.0034 from the uncertainty on the shift and ±0.0009 from the uncertainty on the optical thickness. The analysis shows that the uncertainty on the shift is the largest contributor to the overall measurement uncertainty.

For the window shift, the interoperator variability for all 120 eyes ranged from −2 to +1.5 pixels with a maximum average difference of 0.1 ± 0.7 pixels between operators. For the optical thickness of the lens, the interoperator variability ranged from −18 to +35 pixels with a maximum average difference of 1.6 ± 6 pixels between operators for the lens optical thickness. These differences produced an interoperator variability ranging from −0.0038 to +0.0034 for the group refractive index with a maximum average difference of 0.0002 ± 0.0014. Overall, these results confirm our prediction that measurement uncertainty of the average group refractive index is within ±0.004.

Dataset 1 ^{(20.9KB, xlsx)} [27] provides the age, post-mortem time, average group and phase refractive index, diameter, thickness and thickness to diameter ratio (Fig. 2) and average group and phase refractive index (Fig. 3) for all 120 lenses. For the two infant lenses included in the study (ages: 0.03 and 0.25 years), the average group refractive index was significantly lower (1.405, 1.403) compared to the remainder of the study population. For the 118 eyes ranging from 2 to 61 years, the general linear model was used to test the contribution of gender, age and post-mortem time to each of the dependent variables: average refractive index, lens thickness, diameter, and the thickness/diameter ratio. As initial analyses suggested a significant curvilinear component for lens thickness against age, a second-order coefficient for age was included in the general linear model for lens thickness. The resultant parameter estimates from the general linear models are given in Table 1. For all dependent variables, gender does not show a statistically significant contribution (p > 0.1). Average refractive index was statistically significantly associated with post-mortem time (p = 0.029), however the coefficient of -1.8 × 10⁻⁵ indicated that over the range of post-mortem time within the data set, the effect amounted to a change only in the third decimal place and thus, insignificant in practice. The linear component of age was found to have a significant contribution to lens thickness, diameter and thickness/diameter ratio (p < 0.01) but not to average refractive index (p = 0.77). A significant second order component of age on thickness (p < 0.01) indicated that thickness tended to decrease in earlier ages then increase in later ages.

Fig. 2.

Correlation between age and lens thickness (T), lens diameter (D), and the ratio of lens thickness to diameter. Data points for infants (stars), individuals between age 2 and 20 years (circles) and above age 20 years (squares) are shown. The regression analysis includes only subjects older than 20 years (n = 80). The regression equations for the overall subjects with respect to the age since vesicle closure (+0.63) are 0.0111x + 3.8125 for thickness, 0.0165x + 8.4164 for diameter and 0.0004x + 0.4543 for thickness/diameter ratio.

Fig. 3.

Distribution of the calculated group refractive index at 880 nm (p = 0.40) and phase refractive index at 589 nm (p = 0.40) with respect to age. The two infants (stars) were not included in the regression analysis. Additionally, regression analysis was conducted separately for two age groups: aged 2 to 20 (n = 44) and those older than 20 (n = 74). There was no statistical significance for age-dependence (p-values were 0.91 and 0.29).

Table 1. Parameter estimates for the general linear model analysis.

Dependent variable	Independent variable	Parameter Value	p-value
Average index	Age	-4.0 10−6	0.77
	Post-mortem time	-1.8 10−5	0.03 a
	Gender [Male]	-0.001	0.13
Lens thickness	Age	-0.027	<0.01 a
	Age squared	4.8 10−4	<0.01 a
	Post-mortem time	0.002	0.12
	Gender [Male]	-0.002	0.98
Lens diameter	Age	0.531	<0.01 a
	Post-mortem time	-0.001	0.39
	Gender [Male]	0.105	0.23
Thickness/Diameter ratio	Age	-0.039	<0.001 a
	Post-mortem time	0.000	0.10
	Gender [Male]	-0.010	0.36

^a indicates statistically significance

The mean value of the average group refractive index in this age range was 1.415 ± 0.002, corresponding to a mean value of the average phase refractive index of 1.406 ± 0.002 (Fig. 3). The distribution of the average group refractive index shows that the values for eyes above age two are normally distributed with a range that corresponds to the estimated uncertainty of ±0.004 (Fig. 4).

Fig. 4.

Histogram depicting the distribution of the calculated group refractive indices.

4. Discussion

This study presents measurements of the average refractive index along the optical axis for human lenses using OCT imaging. These measurements resulted in a mean value of n = 1.406 ± 0.002 for the phase index for the age range 2 to 61 years with no age dependence. Interestingly, we found that two infant lenses (0.03 and 0.25 years) exhibit an average refractive index significantly lower (1.393 and 1.395) than the remainder of the study population which is older than age 2. This finding suggests a possible correlation with the rapid growth phase of the lens, as evidenced by the dependence of lens weight or dimensions on age [2]. A larger number of samples is required to confirm this finding.

The mean thickness to diameter ratio was 0.48 ± 0.05 (close to 0.5), suggesting that lenses were in their normal hydration state [26]. The age dependence of lens thickness and diameter (Fig. 2) is comparable to results from previous studies that employed other methodologies to quantify the age dependence of lens shape on a subset of the same in vitro lenses. For instance, for adult lenses (age 20 and above), the increases in lens thickness (0.0111 mm/year) and lens diameter (0.0165 mm/year) with age are comparable to the values found by Mohamed et al. (0.0119 mm/year and 0.0119 mm/year) [28] and Martinez-Enriquez et al. (0.0146 mm/year and 0.0229 mm/year) [29].

Uhlhorn et.al reported a statistically significant decrease in the average group RI with age for a smaller sample size of 40 human lenses [18]. However, the data was more variable and as stated in the author’s conclusion, the sample size was not sufficient to accurately determine the age-dependence. In the current study, we repeated the same methodology with a larger sample size of 120 lenses and found that there is no significant change with age. The finding that the average refractive index does not change with age after age 2 is consistent with previous studies that used MRI, OCT or X-ray interferometry to measure the age-dependence of the gradient profiles [6,8,14]. Our average value of 1.406 ± 0.002 is in good agreement with the average index obtained using in vivo MRI, which is 1.405 for individuals aged 19 to 29 years and 1.406 for those aged 60 to 70 years [1] and values estimated form protein concentrations (see below). On the other hand, the values are different from those calculated from the axial index profiles measured using OCT-based gradient reconstruction (1.394) [8] or X-ray interferometry (1.415 to 1.418) [6], suggesting that these methods may have underestimated (OCT) or overestimated (X-ray) some of reconstructed gradient parameters. The differences appear to be due primarily to a difference in the value of the power coefficient obtained by the OCT-based reconstruction and in the nuclear index obtained using X-ray interferometry. The value of the power coefficient from the OCT-based reconstruction (p = 2.3) is much lower than the values obtained using in vivo MRI (p = 4.9 to 6.7) or X-ray interferometry (p = 4.7). The value of the nuclear index obtained using X-ray interferometry (1.429 to 1.433) is much higher than the values obtained using OCT-based reconstruction (1.409 to 1.414 based on the regression analysis) or MRI (1.410). According to Eq. (2), a lower power coefficient results in a lower average RI and a higher nuclear index results in a higher average RI.

At first sight, the constancy of the average index appears to be at odds with the observation that there is a progressive formation of a plateau of refractive index with age [7,15]. A plateau corresponds to a wider region of higher refractive index in the center of the lens. The formation of a plateau should therefore be associated with an increase in the average refractive index since a wider region of the lens has a higher refractive index, assuming that the nuclear and surface indices are approximately constant. In fact, plateau formation is associated with only a slight change in average refractive index. For instance, if the nuclear and cortical indices are 1.410 and 1.379 and the power coefficient increases from 4.9 to 6.7 with age [1], then the average index changes only from 1.405 to 1.406. This increase of 0.001 is below the sensitivity of our measurements. Similarly, if the nuclear and cortical indices are 1.410 and 1.379 and the power coefficient is 4.9 then the average index is 1.405. A measurement uncertainty of ±0.002 (i.e., 1.403 to 1.407) or ±0.004 (1.401 to 1.409) corresponds to a power coefficient ranging from 3.4 to 9.3 or 2.4 to 30, respectively. This uncertainty in the power coefficient is more than the expected changes with the formation of a plateau.

The above analyses of the GRIN profiles from previous studies rely on the power model of Eq. (1), which is commonly used to model the axial gradient profile. Equation (1) is either used to fit experimental profiles (MRI, X-ray interferometry) or to extract gradient parameters through an optimization process (OCT). The power model provides a good overall fit to the experimental data but it does not capture fluctuations or discontinuities that are observed in the refractive index profiles [1,6,7]. The average RI calculated using Eq. (1) is therefore an approximation assuming a smooth profile. However, the fluctuations of the index along the profile and deviations from the fits are small, approximately within ±0.002 [1,6,7]. The magnitude of the fitting error is therefore too small to explain the difference of approximately 0.01 between our measurements and the values obtained from the previous data [6,8].

The refractive index is directly related to the protein concentration. Assuming a refractive index increment of 0.198 and a refractive index of 1.336 for the aqueous phase, the average index of 1.406 corresponds to a protein concentration of 35.6%, consistent with prior studies [2,30]. The uncertainty of ±0.002 or ±0.004 in the average index corresponds to an uncertainty of ±2% (34.3% to 36.4%) or ±4% (33.3% to 37.4%) in the protein concentration. The constancy of the average index therefore suggests that the average of the protein concentration along the sagittal axis of the lens remains constant with age after age 2, within ±2% (34.3% to 36.4%) to ±4% (33.3% to 37.4%). For the two youngest lenses, the index corresponds to a concentration of 29% and 30%. The constancy of the average of the sagittal protein concentration profile indicates that the formation of a plateau in the lens center is compensated by a concurrent increase in the thickness of the cortex. Alternatively, the nuclear index could decrease. The X-ray, MRI and OCT studies suggest that there may be a slight decrease in the nuclear index, however, the age dependence that was found is too small compensate for the increase expected from the formation of a plateau.

5. Conclusions

In conclusion, the lens's average refractive index remains constant in humans after two years of age, within the measurement uncertainty of ±0.004. Initial data from two infant lenses indicate a rapid increase in refractive index within the first two years post-birth, aligning with the lens's rapid growth phase. However, further data is essential to validate this observation.

Appendix: conversion from group index at 880 nm to phase index at 589 nm

The dispersion equation for the refractive index of the lens can be written:

$${\text{n}}_{\text{lens}}(\mathrm{\lambda })={\text{n}}_{\text{aq}}(\mathrm{\lambda })+\mathrm{\alpha }(\mathrm{\lambda })\times \text{c}$$ (A1)

where n_aq(λ) is the refractive index of the aqueous phase, α(λ) is the refractive index increment and c is the concentration of protein. The refractive index increment is α(λ) = n_p(λ) – n_aq(λ) where n_p(λ) is the lens protein refractive index.

The phase refractive index of the lens at 589 nm is:

$${\text{n}}_{\text{lens}}(589)={\text{n}}_{\text{aq}}(589)+\mathrm{\alpha }(589)\times \text{c}$$ (A2)

The group refractive index of the lens at 880 nm is:

$${\text{n}}_{\text{g}\mathrm{\_}\text{lens}}(880)={\text{n}}_{\text{g}\mathrm{\_}\text{aq}}(880)+{\mathrm{\alpha }}_{\text{g}}(880)\times \text{c}$$ (A3)

where the group refractive index is related to the phase refractive index by:

$${\text{n}}_{\text{g}}(\mathrm{\lambda })=\text{n}(\mathrm{\lambda })-\mathrm{\lambda }\frac{\text{dn}}{\mathrm{d}\mathrm{\lambda }}$$ (A4)

Combining equations (A2) and (A3) gives the relation between the phase refractive index at 589 nm and the group refractive index at 880 nm:

$${\text{n}}_{\text{lens}}(589)={\text{n}}_{\text{g}\mathrm{\_}\text{lens}}(880)+[{\text{n}}_{\text{aq}}(589)-{\text{n}}_{\text{g}\mathrm{\_}\text{aq}}(880)]+[\mathrm{\alpha }(589)-{\mathrm{\alpha }}_{\text{g}}(880)]\times \text{c}$$ (A5)

We assume that the difference between the phase index and group index of the aqueous phase of the lens is the same as for BSS. We measured the phase index of BSS at 589 nm using an Abbe refractometer and the group index of BSS using the method described in the text. We found n_BSS(589) = 1.335 and n_g(BSS)(880) = 1.343.

We then used the relation of Perlmann and Longsworth [24] to model the dispersion of the refractive increment:

$$\mathrm{\alpha }(\mathrm{\lambda })=\mathrm{\alpha }(578)\times \left(0.940+\frac{20000}{\mathrm{\lambda }{(\text{nm})}^2}\right)$$ (A6)

Note that the dispersion was estimated from measurement in the range from about 450 to 650 nm (see their Fig. 2). We assume that the dependence extends to 880 nm. Combining Eq. (A6) and Eq. (A4) gives the expression of the group refractive increment:

$${\mathrm{\alpha }}_{\text{g}}(\mathrm{\lambda })=\mathrm{\alpha }(\mathrm{\lambda })-\mathrm{\lambda }\times \frac{\mathrm{d}\mathrm{\alpha }}{\mathrm{d}\mathrm{\lambda }}=\mathrm{\alpha }(578)\times \left(0.940+\frac{60000}{\mathrm{\lambda }{(\text{nm})}^2}\right)$$ (A7)

Combining Eq. (A6) and Eq. (A7) gives the difference between the phase refractive increment at 589 nm and the group refractive increment at 880 nm:

$$\mathrm{\alpha }(589)-{\mathrm{\alpha }}_{\text{g}}(880)=\mathrm{\alpha }(578)\times \left(\frac{20000}{{589}^2}-\frac{60000}{{880}^2}\right)$$ (A8)

We assume that the refractive increment of the lens at 578 nm is α(578) = 0.198 [24,30]. With this value, we find α(589) - α_g(880) = -0.00393.

With the measured values of the refractive indices of BSS and the estimated values of the refractive index increment of the lens, we obtain the following relation between the phase refractive index at 589 nm and the group refractive index at 880 nm:

$${\text{n}}_{\text{lens}}(589)={\text{n}}_{\text{g}\mathrm{\_}\text{lens}}(880)-0.008-0.00393\times \text{c}$$ (A9)

Assuming a protein concentration of 35% (c = 0.35) gives:

$${\text{n}}_{\text{lens}}(589)={\text{n}}_{\text{g}\mathrm{\_}\text{lens}}(880)-0.009$$ (A10)

Funding

National Eye Institute10.13039/100000053 (P30EY14801, R01EY021834); Hyderabad Eye Research Foundation10.13039/501100005809; Beauty of Sight Foundation10.13039/100016850; Henri and Flore Lesieur Foundation10.13039/100015590.

Disclosures

There are no financial interests to disclose.

Data availability

Data underlying the results presented in this paper are available in Dataset 1 ^{(20.9KB, xlsx)} , Ref. [27].