Contribution of the gradient refractive index and shape to the crystalline lens spherical aberration and astigmatism

The optical properties of the crystalline lens are determined by its shape and refractive index distribution. However, to date, those properties have not been measured together in the same lens, and therefore their relative contributions to optical aberrations are not fully understood. The shape, the optical path difference, and the focal length of ten porcine lenses (age around 6 months) were measured in vitro using Optical Coherence Tomography and laser ray tracing. The 3D Gradient Refractive Index distribution (GRIN) was reconstructed by means of an optimization method based on genetic algorithms. The optimization method searched for the parameters of a 4-variable GRIN model that best fits the distorted posterior surface of the lens in 18 different meridians. Spherical aberration and astigmatism of the lenses were estimated using computational ray tracing, with the reconstructed GRIN lens and an equivalent homogeneous refractive index. For all lenses the posterior radius of curvature was systematically steeper than the anterior one, and the conic constant of both the anterior and posterior positive surfaces was positive. In average, the measured focal length increased with increasing pupil diameter, consistent with a crystalline lens negative spherical aberration. The refractive index of nucleus and surface was reconstructed to an average value of 1.427 and 1.364, respectively, for 633 nm. The results of the GRIN reconstruction showed a wide distribution of the index in all lens samples. The GRIN shifted spherical aberration towards negative values when compared to a homogeneous index. A negative spherical aberration with GRIN was found in 8 of the 10 lenses. The presence of GRIN also produced a decrease in the total amount of lens astigmatism in most lenses, while the axis of astigmatism was only little influenced by the presence of GRIN. To our knowledge, this study is the first systematic experimental study of the relative contribution of geometry and GRIN to the aberrations in a mammal lens.     

The optics of the spherical fish lens.

The optical design of the fish eye is particularly simple because immersion renders the cornea optically ineffective and the lens is nearly spherical in shape. Measurements have shown that an approximately parabolic gradient of refractive index exists within the lens. If full internal and external spherical symmetry of the lens applies, the geometrical-optical behaviour of the lens is then a function only of the refractive index of the surrounding medium, that of the lens core and cortex, and of the form of the index gradient. The theoretical optical performance of models of the spherical fish lens is calculated by means of the ray-tracing program Drishti as a basis for understanding the optical design of real fish and aquatic eyes. Models based on the gradients proposed by earlier workers are shown to be unable to predict reported spherical aberration and image quality. A model of the fish lens with a polynomial gradient is proposed that yields spherical aberration, image quality and chromatic aberration similar to that reported for the fish.     

Changes in spherical aberration after lens refilling with a silicone oil

PURPOSE: It may be possible to restore accommodation to presbyopic human eyes by refilling the lens capsular bag with a soft polymer. In the present study, optical changes were measured that occurred in a pig eye model after the refilling of the capsular bag. METHODS: The optical power and spherical aberration in 10 isolated pig lenses were measured under different conditions. They were first determined by using a scanning laser ray-tracing technique over an effective pupil size of 6 mm. Second, the contours of the anterior and posterior lens surface were photographed, and a mathematical ray-tracing was performed by using a polynomial fit through the digitized surface contours, to determine optical power and spherical aberration. Third, the lenses were refilled with a silicone oil until their preoperative lens thickness was reached, and scanning laser ray-tracing, contour photography, and mathematical ray-tracing were repeated. Comparisons between the measurements were made to determine how the change from a gradient refractive index to a homogeneous refractive index influenced spherical aberration. The influence of the change in lens contour on spherical aberration after lens refilling was also studied. RESULTS: The natural lenses had a higher lens power than the refilled lenses (49.9 +/- 1.5 D vs. 36.8 +/- 1.5 D; P < 0.001). Moreover, there was a change in sign from negative spherical aberration before lens refilling (-3.6 D) to positive spherical aberration after lens refilling (7.9 D; P < 0.001). The comparison between scanning laser ray-tracing of the natural lens and mathematical ray-tracing of the photographed surface contours of the natural lens to determine the effect of refractive index substitution (i.e., replacement of a gradient refractive index by a homogeneous refractive index) showed a significant change in spherical aberration from -3.6 +/- 2.0 to 11.0 +/- 2.1 D (P < 0.001). The change in lens contour did not result in a significant change in spherical aberration (P = 0.08) before and after lens refilling with an equal refractive index. CONCLUSIONS: The lower lens power of refilled pig lenses compared to natural lenses was due to the lower refractive index of the refill material. Refilling pig lenses with the silicone oil used in this study resulted in an increase in spherical aberration. This increase was mainly caused by the change from a gradient refractive index to a homogeneous refractive index. The change in lens curvature after lens refilling did not result in an increase in spherical aberration. The influence of lens refilling on spherical aberration of human lenses must be determined in similar experiments in human eyes.     

A Wide-angle Gradient Index Optical Model of the Crystalline Lens and Eye of the Rainbow Trout

Trout lens external shape and internal refractive index gradient structure were measured and used to construct an optical lens model that predicts by ray tracing the average longitudinal spherical and chromatic aberration, focal length and image quality. The nearly spherical shape of the lens was measured from photographs, and the internal refractive gradient structure was measured directly with a special Pulfrich areal refractometer. Longitudinal spherical aberration and back focal length were measured using a simplified Hartmann test using laser beams and a Schlieren test which additionally made refractive index gradient fine structure visible and detected scattering, axial symmetry and structural irregularity. Axial focus shift caused by longitudinal chromatic aberration was measured using a star test. The model lens was then incorporated into a model trout eye based on vertical and horizontal eye frozen sections. Calculated model function yields insight into the relation between eye and lens structure and optical behaviour. Semi-random secondary structural features act as perturbations on the basic model, and will result in point image fine structure. Copyright © 1996 Elsevier Science Ltd.     

Age-related changes in refractive index distribution and power of the human lens as measured by magnetic resonance micro-imaging in vitro

We report a new technique for non-invasively mapping the refractive index distribution through the eye lens using magnetic resonance micro-imaging. The technique is applied to map the refractive index distribution throughout the sagittal plane of 18 human eye lenses ranging in age from 14 to 82 years in vitro. The results are compared with standard models for the human eye lens. They confirm that the refractive index distribution, when plotted as a function of normalised lens radius, is a function of lens age and differs both between the equatorial and axial directions and between the anterior and posterior halves of the optical axis. The refractive index of the lens nucleus exhibits a significant reduction with age amounting to 3.4+/-0.6 x 10(-4) years(-1). The contribution of the gradient index (GRIN) to the lens power decreases by 0.286+/-0.067 D/year, accounting almost entirely for the estimated overall change in lens power with age for these lenses, which were probably in their most accommodated state. The results provide experimental verification of hypothesised changes in the GRIN that have previously been invoked as contributing to presbyopia and support the hypothesis that changes in the GRIN are sufficient to offset effects of increasing curvature of human lenses with age in their unaccommodated state.     

Numerical study of the influence of the shell structure of the crystalline lens on the refractive properties of the human eye

In this paper, the numerical model of the refractive properties of the human eye is given. A special account is given of the laminated structure of the crystalline lens. The crystalline lens is presented in the form of hundreds of shells with rotational symmetry, and with the refractive index constant within each shell. The shells have the form of two semi-ellipsoids joined along the equator. The refractive index increases from the cortical shell to the inner one, according to the exponential dependency. The cornea, approximated by two ellipsoidal surfaces, is added in front of the crystalline lens. A ray-tracing procedure is applied to study the refractive properties of such a system: refractive power, spherical aberration and energy distribution. The optical properties of the given model are analysed by changing some parameters such as refractive index profile and number of shells. Calculations show that the gradient of the refractive index inside the crystalline lens results in the generation of many focal planes in such an optical system.     

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.     

Wavefront aberrations following laser in situ keratomileusis and refractive lens exchange for hypermetropia

PURPOSE: To compare the magnitude of aberrations in eyes after elective hypermetropic laser in situ keratomileusis (LASIK) and refractive lens exchange (clear lens replacement). METHODS: Forty-nine patients (92 eyes) had hypermetropic LASIK and 28 (48 eyes) had refractive lens exchange; 23 hypermetropic subjects (41 eyes) were the control group. LASIK was performed with the Nidek EC-5000 excimer laser; ablation zones 5.5 to 6.0-mm in diameter with transition zones 7.5 to 8-mm in diameter. For refractive lens exchange, all but four IOLs were made of foldable acrylic. Aberrations and corneal topography were measured with the Nidek OPD-Scan model ARK-10000 more than 12 months after surgery. The higher-order root-mean-square (HORMS) wave aberrations for combined third to sixth Zernike aberration orders and the Zernike spherical aberration coefficient C(0)(4) at both 4.2-mm and 6.0-mm pupil sizes were calculated. RESULTS: For the LASIK group, surgical refractive change correlated significantly with total, corneal, and internal HORMS and spherical aberrations (except with internal spherical aberration for a 4.2-mm diameter pupil). For the refractive lens exchange group, there were no significant correlations of surgical refractive change with any of these factors. Similarly, there were no significant correlations of refraction with any of these factors for the control group. For a 3-diopter change in refraction with 6-mm pupils, LASIK doubled the total HORMS aberrations. LASIK changed the sign of spherical aberration from positive to negative by increasing the negative asphericity of the anterior cornea. Taking age differences between groups into account, refractive lens exchange increased the total HORMS aberrations by 40% compared with that of the control group, but this was not statistically significant. However, refractive lens exchange significantly increased total spherical aberration. CONCLUSION: Refractive lens exchange was a better refractive procedure than LASIK for minimizing total higher order optical aberrations that accompany hypermetropic refractive surgery.     

A new intraocular lens design to reduce spherical aberration of pseudophakic eyes

PURPOSE: The aim of this study was to design and evaluate in the laboratory a new intraocular lens (IOL) intended to provide superior ocular optical quality by reducing spherical aberration. METHODS: Corneal topography measurements were performed on 71 cataract patients using an Orbscan I. The measured corneal surface shapes were used to determine the wavefront aberration of each cornea. A model cornea was then designed to reproduce the measured average spherical aberration. This model cornea was used to design IOLs having a fixed amount of negative spherical aberration that partially compensates for the average positive spherical aberration of the cornea. Theoretical and physical eye models were used to assess the expected improvement in optical quality of an eye implanted with this lens. RESULTS: Measurements of optical quality provided evidence that if this modified prolate IOL was centered within 0.4 mm and tilted less than 7 degrees, it would exceed the optical performance of a conventional spherical IOL. This improvement occurred without an apparent loss in depth of focus. CONCLUSION: A new IOL with a prolate anterior surface, designed to partially compensate for the average spherical aberration of the cornea, is intended to improve the ocular optical quality of pseudophakic patients.     

An analytic,gradient index schematic lens and eye for the rat which predicts aberrations for finite pupils

Schematic eyes with a homogeneous equivalent lens are inadequate for non-paraxial optics and available versions have been invalidly fitted to non-paraxial properties.A new model eye is here analytically derived from the refractive index profile of the crystalline lens and anatomical measurements of the rat eye. It predicts spherical abberration, coma, paraxial properties and the variation of refractive state with pupil size in accord with experimental measurements.The cornea counteracts the spherical aberration and linear coma of the lens so that overall aberration is reduced and the eye is of good optical quality.The nodal point is invariant with object eccentricity in a manner advantageous to a species whose visual axis is displaced from the optic axis. The potential of the model lies in its extension to a study of such off-axis optics.     

The spherical aberration of aphakic eyes corrected with intra-ocular lenses

The spherical aberration of eyes corrected with intra-ocular lenses is investigated using a model eye with realistic levels of corneal asphericity. The results indicate that the aberration is intermediate between that of paraxial schematic eyes and real eyes. By using standard optical aberration theory, it is shown that for a plano-convex lens with the curved surface facing the cornea, the aberration is similar to that of normal phakic eyes and therefore probably too low to be of any clinical significance. However, for other lens orientations or designs, the level of aberration is usually higher and may lead to a refractive error varying with pupil size and a loss of acuity with large pupil diameters.     

Embryonic lens of the human eye as an optical structure

The spherical aberration of the excised embryonic (18 to 22 weeks) human lens was determined by photographing the refractive effects of the lens on fine parallel laser beams. All lenses showed little evidence of focal variation with laser position although a slight tendency toward positive (undercorrected) spherical aberration was noted. One lens from the eye of a newborn showed slight negative (overcorrected) aberration. It is assumed that the continued growth of the lens, with central compression of old tissue, produces a refractive index distribution which contributes to lens optical quality, even in the uterine environment. Frozen sections of embryo eyes of the same period (18 to 22 weeks) indicate that a major part of the change from a spherical to an elliptical lens shape takes place between the fourth and fifth month of development.     

Relationship between refractive error and monochromatic aberrations of the eye.

PURPOSE: To examine the relationship between ametropia and optical aberrations in a population of 200 normal human eyes with refractive errors spanning the range from +5.00 to -10.00 D. METHODS: Using a reduced-eye model of ametropia, we tested the hypothesis that the optical system of the eye is uncorrelated with the degree of ametropia. These predictions were evaluated experimentally with a Shack-Hartmann aberrometer that measured the monochromatic aberrations across the central 6 mm of the dilated pupil in well-corrected, cyclopleged eyes. RESULTS: Optical theory predicted, and control experiments on a model eye verified, that Shack-Hartmann measurements of spherical aberration will vary with axial elongation of the eye even if the dioptric components of the eye are fixed. Contrary to these predictions, spherical aberration was not significantly different from emmetropic eyes. Root mean square of third-order aberrations, fourth-order aberrations, and total higher aberrations (third to 10th) in myopic and hyperopic eyes were also uncorrelated with refractive error. Astigmatic eyes tended to have larger total higher-order aberrations than nonastigmatic eyes. CONCLUSIONS: We conclude that a reduced-eye model of myopia assuming fixed optical parameters and variable axial length is not tenable.     

On-eye evaluation of optical performance of rigid and soft contact lenses.

A Shack-Hartmann aberrometer was used to assess the optical performances of eyes corrected with rigid or soft contact lenses compared with spectacles. Metrics of optical quality derived from the measured wave aberrations were consistent with the subjective rating of visual clarity by subjects. Optical aberration analysis illustrated the differences in aberration structures of eyes wearing different optical corrections. For our subjects, correction with a rigid gas-permeable lens yielded significantly better optical quality than did the soft contact lens or spectacle lens. This was due to a reduction in the eye's asymmetric (odd-order) aberrations and a reduction in the amount of the eye's positive spherical aberration. These observations can be explained by theoretical calculations of the aberrations of the eye plus lens optical system. We conclude that aberrometry provides a better understanding of the optical effects of contact lenses in situ and could be useful for optimizing future designs of contact lenses.     

基于儿童正视眼的眼光学模型构建

目的根据正视儿童眼部参数,构建正视儿童的眼光学模型。方法实验研究。基于“安阳儿童眼病研究”数据,包括角膜曲率半径、角膜厚度、前房深度、晶状体厚度、屈光度和眼轴长度,取右眼数据,应用ZEMAX光学设计软件构建一个符合我国儿童眼球特点的正视眼光学模型。正态性检验采用单样本K-S分析。结果共纳入正视儿童332名,年龄（7.1±0.4）岁,等效球镜度（SE）为（0.11±0.24）D。构建的正视儿童眼光学模型的光学参数为：角膜前表面曲率半径7.78 mm,非球面系数-0.18;后表面曲率半径6.4 mm,非球面系数-0.60;厚度0.54 mm,折射率1.376。前房深度3.00 mm,房水折射率1.336。晶状体前表面曲率半径12.4 mm,非球面系数-0.94;后表面曲率半径-8.1 mm,非球面系数0.96;厚度3.55 mm,折射率为梯度渐变折射率。玻璃体厚度15.94 mm,折射率1.336。视网膜曲率半径-12.3 mm,眼轴长度23.03 mm,总屈光力62.55 D。结论本研究构建了一个符合儿童正视眼特点的眼光学模型,该模型眼的总屈光力为62.55 D,眼轴长度23.03 mm,该模型可作为儿童眼正视化和近视研究的参考工具。     

Optimizing higher-order aberrations with intraocular lens technology

PURPOSE OF REVIEW: Aspheric intraocular lens technology has been implemented during the past 5 years, and more and more intraocular lenses with different amounts of asphericity are becoming available. Despite the efficacy in the correction of spherical aberration and the good results on implanted eyes, the theoretical advantages of aspheric intraocular lenses are still controversial. RECENT FINDINGS: All investigations showed the ability of the aspheric intraocular lenses to correct the positive spherical aberration of the cornea, with variable impact on the total eye wavefront but with constant advantages in the optical quality of the eyes as measured by the Modulation Transfer Function, the Point Spread Function, and the contrast sensitivity of implanted patients. Theoretical studies on model eyes underlined some possible limitations of aspheric intraocular lenses, especially sensitivity to decentration. In addition, the actual optical quality in implanted eyes is also affected by light scattering, a parameter missed by simple aberration analysis. SUMMARY: Aspheric intraocular lenses effectively reduce spherical aberration in implanted eyes, with improvement in optical quality over the parent spherical intraocular lens. The advantages for implanted eyes could be limited by decentration, by small pupil diameter, and by reduced media transparency.     

Adjusting a light dispersion model to fit measurements from vertebrate ocular media as well as ray-tracing in fish lenses

Color dispersion, i.e., the dependency of refractive index of any transparent material on the wavelength of light, has important consequences for the function of optical instruments and animal eyes. Using a multi-objective goal attainment optimization algorithm, a dispersion model was successfully fitted to measured refractive indices of various ocular media and the longitudinal chromatic aberration determined by laser-scanning in the crystalline lens of the African cichlid fish, Astatotilapia burtoni. The model describes the effects of color dispersion in fish lenses and may be applicable to the eyes of other vertebrates as well.     

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.     

Optical ray tracing for the calculation of optimized corneal ablation profiles in refractive treatment planning

PURPOSE: Optical calculations have shown that wavefront-based ablation profiles as well as intraocular lens (IOL) implantations can cause residual aberrations or even induce significant additional aberrations due to the poor registry between the eye's optical components. These effects can be exacerbated in eyes that require higher corrections. Individualized eye models can provide accurate ablation profiles for these cases. The aim of this report is to analyze the relevance of individualized eye models for refractive treatment planning from a theoretical point of view. METHODS: A method for the customization of eye models based on various types of measurement data of a specific patient is presented and the calculation of optimal ablation profiles and IOL shapes by means of ray tracing through customized model eyes are discussed. Topography data with an original centration on the corneal apex were aligned on the pupil center for the creation of eye models. RESULTS: An ideal ablation profile or a customized IOL can be calculated based on the obtained individualized eye models. Calculations have shown that in theory ray-tracing optimized ablation profiles do not leave any residual aberrations in the eye whereas wavefront-guided corrections were found to have the potential to increase specific types of aberrations by a factor of two. This is due to the negligence of the multi-lens structure of the eye. CONCLUSIONS: Optical ray tracing algorithms allow the highest degree of customization. The systematic induction of higher order aberrations by means of wavefront-guided treatments or standard IOLs has to be overcome by such a method. However, these theoretical observations must be compared to the corresponding measurement accuracies and precisions and they must be supported by future clinical trials.