Determination and modeling of the 3-D gradient refractive indices in crystalline lenses

The effect of fibre cell remodelling on the power and optical quality of the lens

Vertebrate eye lenses are uniquely adapted to form a refractive index gradient (GRIN) for improved acuity, and to grow slowly in size despite constant cell proliferation. The mechanisms behind these adaptations remain poorly understood. We hypothesize that cell compaction contributes to both. To test this notion, we examined the relationship between lens size and shape, refractive characteristics and the cross-sectional areas of constituent fibre cells in mice of different ages. We developed a block-face imaging method to visualize cellular cross sections and found that the cross-sectional areas of fibre cells rose and then decreased over time, with the most significant reduction occurring in denucleating cells in the adult lens cortex, followed by cells in the embryonic nucleus. These findings help reconcile differences between the predictions of lens growth models and empirical data. Biomechanical simulations suggested that compressive forces generated from continuous deposition of fibre cells could contribute to cellular compaction. However, optical measurements revealed that the GRIN did not mirror the pattern of cellular compaction, implying that compaction alone cannot account for GRIN formation and that additional mechanisms are likely to be involved.

Tunable multilayered lens made of PDMS with a biconical surface profile design and manufacture

A polymer that has been used for the development of optical components and has had a significant impact is polydimethylsiloxane (PDMS) due to its remarkable mechanical and optical properties and easy handling. We present a practical and straightforward technique for designing and manufacturing a tunable graded index, graphical input (GRIN)-type lenses, and tunable lenses with a homogeneous refractive index made of PDMS. Implementing a biconical surface profile in a tunable plane-convex lens is proposed for elaborating both a homogeneous refractive index lens and a multilayered GRIN-type lens with a constant increased variation of 0.014 on its refractive index. Likewise, we introduce a mechanical mounting system that aims to modify their curvatures and therefore their focal lengths through mechanical stimuli applied on the lenses. Simulations of the optomechanical behavior and optical characterization of the lenses are also presented.

Experimental ray-tracing with point diffraction interferometry and its application in focal length measurement

A novel method that could experimentally trace the ray propagation of an optical system based on point diffraction interferometry (PDI) is presented. The ray represented by the line connecting two point light sources (PLS) intersects with two parallel photographic planes, which are separated at a distance. The intersections of the ray defined by the PLSs with the planes occur where the optical path difference is a maximum and can be determined from interferograms on the photographic planes. The ray is determined by connecting the two intersections. Using fibers as the PLSs and CCD arrays as the photographic planes, we demonstrate its principle and its application in focal length measurement through experiments.

The eye lens: age-related trends and individual variations in refractive index and shape parameters

The eye lens grows throughout life by cell accrual on its surface and can change shape to adjust the focussing power of the eye. Varying concentrations of proteins in successive cell layers create a refractive index gradient. The continued growth of the lens and age-related changes in proteins render it less able to alter shape with loss of capacity by the end of the sixth decade of life. Growth and protein ageing alter the refractive index but as accurate measurement of this parameter is difficult, the nature of such alterations remains uncertain. The most accurate method to date for measuring refractive index in intact lenses has been developed at the SPring-8 synchrotron. The technique, based on Talbot interferometry, has an X-ray source and was used to measure refractive index in sixty-six human lenses, aged from 16 to 91 years. Height and width were measured for forty-five lenses. Refractive index contours show decentration in some older lenses but individual variations mask age-related trends. Refractive index profiles along the optic axis have relatively flat central sections with distinct micro-fluctuations and a steep gradient in the cortex but do not exhibit an age-related trend. The refractive index profiles in the equatorial aspect show statistical significance with age, particularly for lenses below the age of sixty that had capacity to alter shape in vivo. The maximum refractive index in the lens centre decreases slightly with age with considerable scatter in the data and there are age-related variations in sagittal thickness and equatorial height.

Simplifying numerical ray tracing for characterization of optical systems

Ray tracing, a computational method for tracing the trajectories of rays of light through matter, is often used to characterize mechanical or biological visual systems with aberrations that are larger than the effect of diffraction inherent in the system. For example, ray tracing may be used to calculate geometric point spread functions (PSFs), which describe the image of a point source after it passes through an optical system. Calculating a geometric PSF is useful because it gives an estimate of the detail and quality of the image formed by a given optical system. However, when using ray tracing to calculate a PSF, the accuracy of the estimated PSF directly depends on the number of discrete rays used in the calculation; higher accuracies may require more computational power. Furthermore, adding optical components to a modeled system will increase its complexity and require critical modifications so that the model will describe the system correctly, sometimes necessitating a completely new model. Here, we address these challenges by developing a method that represents rays of light as a continuous function that depends on the light's initial direction. By utilizing Chebyshev approximations (via the chebfun toolbox in MATLAB) for the implementation of this method, we greatly simplified the calculations for the location and direction of the rays. This method provides high precision and fast calculation speeds that allow the characterization of any symmetrical optical system (with a centered point source) in an analytical-like manner. Next, we demonstrate our methods by showing how they can easily calculate PSFs for complicated optical systems that contain multiple refractive and/or reflective interfaces.

Adjustable internal structure for reconstructing gradient index profile of crystalline lens

Employing advanced technologies in studying the crystalline lens of the eye has improved our understanding of the refractive index gradient of the lens. Reconstructing and studying such a complex structure requires models with adaptable internal geometry that can be altered to simulate geometrical and optical changes of the lens with aging. In this Letter, we introduce an optically well-defined, geometrical structure for modeling the gradient refractive index profile of the crystalline lens with the advantage of an adjustable internal structure that is not available with existing models. The refractive index profile assigned to this rotationally symmetric geometry is calculated numerically, yet it is shown that this does not limit the model. The study provides a basis for developing lens models with sophisticated external and internal structures without the need for analytical solutions to calculate refractive index profiles.

Geometry-invariant gradient refractive index lens: analytical ray tracing

A new class of gradient refractive index (GRIN) lens is introduced and analyzed. The interior iso-indicial contours mimic the external shape of the lens, which leads to an invariant geometry of the GRIN structure. The lens model employs a conventional surface representation using a coincoid of revolution with a higher-order aspheric term. This model has a unique feature, namely, it allows analytical paraxial ray tracing. The height and the angle of an arbitrary incident ray can be found inside the lens in a closed-form expression, which is used to calculate the main optical characteristics of the lens, including the optical power and third-order monochromatic aberration coefficients. Moreover, due to strong coupling of the external surface shape to the GRIN structure, the proposed GRIN lens is well suited for studying accommodation mechanism in the eye. To show the power of the model, several examples are given emphasizing the usefulness of the analytical solution. The presented geometry-invariant GRIN lens can be used for modeling and reconstructing the crystalline lens of the human eye and other types of eyes featuring a GRIN lens.

The gradient index lens of the eye: an opto-biological synchrony

The refractive power of a lens is determined largely by its surface curvatures and the refractive index of its medium. These properties can also be used to control the sharpness of focus and hence the image quality. One of the most effective ways of doing this is with a gradient index. Eye lenses of all species, thus far, measured, are gradient index (GRIN) structures. The index gradation is one that increases from the periphery of the lens to its centre but the steepness of the gradient and the magnitudes of the refractive index vary so that the optics of the lens accords with visual demands. The structural proteins, the crystallins, which create the index gradient, also vary from species to species, in type and relative distribution across the tissue. The crystallin classes do not contribute equally to the refractive index, and this may be related to their structure and amino acid content. This article compares GRIN forms in eye lenses of varying species, the relevance of these forms to visual requirements, and the relationship between refractive index and the structural proteins. Consideration is given to the dynamics of a living lens, potential variations in the GRIN form with physiological changes and the possible link between discontinuities in the gradient and growth. Finally, the property of birefringence and the characteristic polarisation patterns seen in highly ordered crystals that have also been observed in specially prepared eye lenses are described and discussed.

Accuracy of the reconstruction of the crystalline lens gradient index with optimization methods from ray tracing and Optical Coherence Tomography data

The accuracy of the reconstruction of the Gradient Refractive Index (GRIN) of the crystalline lens from optimization methods was evaluated. Different input data, including direction cosines of deflected rays, ray impacts obtained from ray tracing and optical path differences from Optical Coherence Tomography (OCT) were studied. Three different GRIN models were analyzed. The experimental errors of the different experimental input data were estimated from comparisons of measurements and simulations using artificial lenses of known geometries. The experimental errors in the surfaces shape measurement and the influence of the number of rays were also considered. OCT-based input data produced the most accurate GRIN reconstructions. We found that optimization methods (combining global and local search algorithms) allow GRIN reconstructions with acceptable accuracies for moderate noise level.

Optical properties of in situ eye lenses measured with X-ray Talbot interferometry: a novel measure of growth processes

The lens, a major optical component of the eye, has a gradient refractive index, which is required to provide sufficient refractive power and image quality. The refractive index variations across the lens are dependent on the distributions and concentrations of the varying protein classes. In this study, we present the first measurements of the refractive index in the in situ eye lens from five species using a specially constructed X-ray Talbot grating interferometer. The measurements have been conducted in two planes: the one containing the optic axis (the sagittal plane) and the plane orthogonal to this (the equatorial plane). The results show previously undetected discontinuities and fluctuations in the refractive index profile that vary in different species. These may be linked to growth processes and may be the first optical evidence of discrete developmental stages.

Three-dimensional reconstruction of the crystalline lens gradient index distribution from OCT imaging

We present an optimization method to retrieve the gradient index (GRIN) distribution of the in-vitro crystalline lens from optical path difference data extracted from OCT images. Three-dimensional OCT images of the crystalline lens are obtained in two orientations (with the anterior surface up and posterior surface up), allowing to obtain the lens geometry. The GRIN reconstruction method is based on a genetic algorithm that searches for the parameters of a 4-variable GRIN model that best fits the distorted posterior surface of the lens. Computer simulations showed that, for noise of 5 μm in the surface elevations, the GRIN is recovered with an accuracy of 0.003 and 0.010 in the refractive indices of the nucleus and surface of the lens, respectively. The method was applied to retrieve three-dimensionally the GRIN of a porcine crystalline lens in vitro. We found a refractive index ranging from 1.362 in the surface to 1.443 in the nucleus of the lens, an axial exponential decay of the GRIN profile of 2.62 and a meridional exponential decay ranging from 3.56 to 5.18. The effect of GRIN on the aberrations of the lens also studied. The estimated spherical aberration of the measured porcine lens was 2.87 μm assuming a homogenous equivalent refractive index, and the presence of GRIN shifted the spherical aberration toward negative values (-0.97 μm), for a 6-mm pupil.

Wide-field schematic eye models with gradient-index lens

We propose a wide-field schematic eye model, which provides a more realistic description of the optical system of the eye in relation to its anatomical structure. The wide-field model incorporates a gradient-index (GRIN) lens, which enables it to fulfill properties of two well-known schematic eye models, namely, Navarro's model for off-axis aberrations and Thibos's chromatic on-axis model (the Indiana eye). These two models are based on extensive experimental data, which makes the derived wide-field eye model also consistent with that data. A mathematical method to construct a GRIN lens with its iso-indicial contours following the optical surfaces of given asphericity is presented. The efficiency of the method is demonstrated with three variants related to different age groups. The role of the GRIN structure in relation to the lens paradox is analyzed. The wide-field model with a GRIN lens can be used as a starting design for the eye inverse problem, i.e., reconstructing the optical structure of the eye from off-axis wavefront measurements. Anatomically more accurate age-dependent optical models of the eye could ultimately help an optical designer to improve wide-field retinal imaging.

Tomographic method for measurement of the gradient refractive index of the crystalline lens. II. The rotationally symmetrical lens

In the first part of this paper we presented a tomographic method to reconstruct the refractive index profile of spherically symmetrical lenses. Here we perform the generalization to lenses that are rotationally symmetrical around the optical axis, as is the ideal human lens. Analysis of the accuracy and versatility of this method is carried out by performing numerical simulations in which different magnitudes of experimental errors and two extreme case scenarios for the likely shape of the refractive index distribution of the human lens are considered. Finally, experimental results for a porcine lens are shown. Conceptually simple and computationally swift, this method could prove to be a valuable tool for the accurate retrieval of the gradient index of a broad spectrum of rotationally symmetrical crystalline lenses.

Refractive index distribution and optical properties of the isolated human lens measured using magnetic resonance imaging (MRI)

Using a non-invasive MRI technique for measuring the refractive index distribution through the crystalline lens, refractive index maps were obtained through 20 intact isolated human lenses (7-82years). Focal length measurements, obtained by simulated light ray propagation through each index map were found to be in agreement with direct measurements performed on a scanning laser apparatus. With increasing age, the refractive index profiles became flatter in the central region, accompanied by steepening of the profile in the periphery. This appears to be an important mechanism underlying the observed changes in power and longitudinal aberration of the human lens.

Tomographic method for measurement of the gradient refractive index of the crystalline lens. I. The spherical fish lens

We present an iterative tomographic algorithm to reconstruct refractive-index profiles for meridional planes of the lens of the spherical fish eye from measurements of deflection angles of refracted rays. Numerical simulations show that the algorithm allows accuracy up to the fourth decimal place, provided that the refractive index can be regarded as an analytical function of the radial coordinate and the experimental errors are neglected. An experimental demonstration is given by applying the algorithm to retrieve the refractive-index profile of a spherical fish lens. The method is conceptually simple and does not require matching of the index of the surrounding medium to that of the surface of the lens, and the related iterative algorithm rapidly converges.

Refractive index distribution in the porcine eye lens for 532 nm and 633 nm light

AIMS

To measure the refractive index distribution in porcine eye lenses for two wavelengths from the visible spectrum: 532 and 633 nm, in order to determine whether there are any discernible wavelength dependent differences in the shape of the profile and in the magnitude of refractive index.

METHODS

Rays were traced through 17 porcine lenses of the same age group and of similar size. Ray trace parameters were used to calculate the refractive index distributions for 633 nm light in all 17 lenses and for 532 nm light in 10 lenses. The effect of the refractive index at the edge of the lens, on the rest of the profile, was considered because the mismatch between refractive index at the lens edge and the refractive index of the surrounding gel necessitated a further step in calculations.

RESULTS

The shape of the refractive index distributions is parabolic. There is a small wavelength dependent difference in the magnitude of the refractive index across the profile and this increases very slightly into the centre of the lens. The value of the refractive index at the edge of the lens does not appreciably affect the index profile.

CONCLUSIONS

The wavelength dependent differences in refractive index between light of 633 and 532 nm are small but discernible.

Wave aberrations of the isolated crystalline lens

A method to measure wave aberrations in the isolated crystalline lens is demonstrated. The method employs a laser scanning technique in which the trajectories of narrow refracted laser beams are measured for an array of sample positions incident on the lens. The local slope of the emerging wavefront is calculated for each sample position, and a least squares procedure is used to fit a Zernike polynomial function to define the wave aberration. Measurements of the aberrations of an isolated porcine lens and macaque lens undergoing changes in accommodative state with mechanical stretching are shown. Many aberrations were present, but negative spherical aberration dominated. In the macaque lens, many aberrations underwent systematic changes with accommodation, most notably the 4th order spherical aberration, which became more negative, and the 6th order spherical aberration, which progressed from negative to positive.

Analysis of Inhomogeneous Optical Systems by the Use of Ray Tracing. II. Three-dimensional Systems with Symmetry

We describe a new approach to the index reconstruction of three-dimensional optical systems with rotational symmetry, which is based on sampling ray paths that lie in the sagittal plane. Since the observed rays are distorted by the optical system itself, they cannot be used directly for index reconstruction. We present an iterative procedure to compute the true ray paths and then to find the index distribution. The utility of the method is verified on the model problem.

Analysis of inhomogeneous optical systems by the use of ray tracing. I. Planar systems

We describe a novel approach to refractive-index reconstruction in two-dimensional systems with no special symmetry, based on observation of traces of rays that travel through the optical system. The mathematical model of ray-tracing analysis is presented in detail, and both the analytical and numerical solutions are given. Methods of data processing in the presence of experimental errors are developed and applied to model problems.

Refractive index contours in the human lens

The refractive index values along the equatorial and sagittal planes of lenses of varying ages were measured using a reflectometric fibre optic sensor. In younger lenses (from the third decade) and in one older lens, the index profiles from the two planes did not concur when plotted on a normalized scale refuting, in these lenses, the assumption of concentric, isoindicial contours which follow the shape of the lens. Agreement between the normalized profiles did occur with all other lenses investigated (aged 47 and older).

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

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.

Growth related changes to functional parameters in the bovine lens

Dimensions, volumes and protein contents were measured for bovine lenses with wet weights ranging from 0.17-3.07 g (2 months gestation to 19 years post-natal). All increase in a non-linear fashion. The lens becomes flatter with age due to a more rapid increase in the equatorial plane, but the ratio of anterior to posterior sagittal distances remains constant (1.19). The radius of curvature increases from 4.9 to 15 for the anterior surface and from 4.4 to 13 for the posterior. Protein content increases more rapidly than volume resulting in an increased average protein concentration from around 18% in the early prenatal lens to nearly 50% in the 19 year old. Total protein content (TPC) was found to be related to wet weight (We) according to the equation, TPC = 0.3We1.33. It is suggested that TPC is a better parameter for describing growth than wet weight or age. The refractive index, in the equatorial plane, increases towards the centre, from 1.38 at the edge of the lens. The maximum index, in the centre, increases with lens size up to 1.474 in the largest lens studied. This corresponds to a protein concentration of 70%. In all lenses, refractive index and protein concentration gradients were superimposable when plotted from the outside towards the centre. The optical performance of the lenses was assessed by measuring the back focal length which increases gradually from 24 to 51.5 mm over the 0.17 to 3.07 g size range. This was attributed to the increased radii of curvature.

The refractive structure and optical properties of the isolated crystalline lens of the cat

Direct measurements of the shape and internal refractive index distribution of the isolated cat lens were used to construct individual optical models of the lenses from the left eyes of five cats. The right eyes were used in a companion study of the optics of the cat eye. The lens refractive index spatial distribution and dispersion were measured with a specially designed Pulfrich areal refractometer. Agreement of calculated ray paths through these models with the observed paths of laser beam fans incident parallel to, and at an oblique angle to the lens axis indicates that the models, which contain no freely adjustable parameters, are good physical and optical representations of the isolated (accommodated) crystalline lenses.

Growth and ageing effects on the refractive index in the equatorial plane of the bovine lens

The refractive index profile in the equatorial plane of bovine lenses from over a wide age range is presented. The form of the profile is parabolic and the shape, already apparent in lenses from early prenatal age, is maintained throughout the span investigated. With age the magnitude of the refractive index increases at all points. An empirical formula which relates the value of the refractive index to any point along a radial distance from the centre is derived. This is applicable to lenses of all sizes in the range studied.