Adaptive model of the gradient index of the human lens. I. Formulation and model of aging ex vivo lenses

A simple, parametric adaptive model of the refractive index distribution of the ex vivo crystalline lens is presented. It assumes conicoid (or nonrevolution quadric in 3D) iso-indical surfaces, concentric with the external surfaces of the lens. The model uses a minimum number of internal structural parameters, while the shape of the iso-indical surfaces adapts automatically to the external geometry. In this way, it is able to adapt and fit individual distributions as well as adapt to the changes of the lens shape and structure with age and accommodation. The model is fit to experimental data for individual eyes spanning ages 7 to 82 years, where for each eye the crystalline lens dimensions and iso-indical index data are known. The analysis demonstrates that only one age-dependent structural parameter is needed to replicate the internal iso-indical index structure, given age-dependent models for the external surfaces. An age-dependent-parameter global model is derived and is shown to predict age-dependent changes in the ex vivo lens power and longitudinal spherical aberration with age.     

Single dispersive gradient-index profile for the aging human lens

We provide a single gradient-index (GRIN) profile for the crystalline lens in an updated age-dependent emmetropic-eye model. The parameters defining the GRIN profile include their variation with age and the dispersion of the refractive index in order to account for the increase in the positive-wave spherical aberration, for the constant chromatic difference in the refraction of the human eye, as well as for the decrease in the retinal-image quality with aging. In accounting for these ocular properties, the results show that first, the value of the dispersion parameters are invariant with age. Second, those parameters defining the distribution of the lens index cause the lens-center-index value to decrease slightly, and its position along the lens axis changes with age. Furthermore, these findings are in agreement with the lens paradox.     

Aging of the human lens: changes in lens shape upon accommodation and with accommodative loss.

Accommodation in the human eye occurs through controlled changes in crystalline lens shape, thickness, and refractive surface placement relative to the cornea. The changes in lens curvatures, whether surface or internal, have been characterized as a function of accommodation and subject age by use of quantitative analysis of Scheimpflug slit-lamp photographic images. Radii of curvature of the major lens refractive surfaces--the external and nuclear boundaries--decrease linearly with increasing accommodation in all eyes that are capable of accommodation. The rates at which they change with accommodation are age dependent, decreasing steadily toward zero with increased age. For the curves visible in each lens half, arising from boundaries between adjacent zones of discontinuity, radius of curvature and location are linearly related over the entire accommodative range for a given lens and over the age range for the population. The slope of this relationship changes with both accommodation and age, decreasing linearly in both cases. The relationship between these geometric changes and the loss of accommodative amplitude is discussed.     

Nested shell optical model of the lens of the human eye

A nested shell model of the human lens is developed based on the known anatomical construction of the lens, on the known way in which the lens grows throughout its life, on the measured characteristics of the lens surfaces as a function of the age of the lens, on the measured changes in the shape of the lens during accommodation, and on measured material characteristics of the lens materials, such as density and index of refraction throughout. The observed changes in central surface curvature and thickness force the shell thicknesses to vary in a predictable way and in turn force the shell surface asphericity to take certain values. Thus, in addition to giving the shape of each shell, the model predicts the change expected in the asphericity of the lens surfaces as the lens ages and adds cortical cell layers. Two examples are given, one for a 25-year-old lens and one for a 40-year-old lens, to show how the cortical layers change their shapes throughout the cortex and over time as the lens ages. The performance of the model of this paper is compared to that of two other nested shell models, one where the layers have constant thickness and one where the lens posterior is fixed within the eye over time, to show the superior performance of this model with respect to maintaining a constant refractive error for the eye as the lens ages and grows.     

眼球系统中关键屈光元件的光学模型研究

角膜和晶状体是人眼光学系统中两个关键的屈光元件,为了理解眼内各介质和整个眼球的屈光状态以及视网膜上的成像,有利于眼科临床方面的应用,需要分别对二者进行光学特性模型的分析与研究。根据角膜和晶状体的光学特性,应用光学设计软件Zemax和有关的数学工具,从光学成像角度,分别对角膜和晶状体模型进行研究:基于结合人眼光学模型的角膜双二次曲面模型的建立方法,统计分析了我国人眼实测角膜参数的数据;通过对晶状体折射率分布特点的分析,分别在轴向和径向上进行了综合分析。最后给出了符合我国人眼特点的角膜面型的统计数值,完善了我国人眼角膜光学模型的建立;获得了形式简单且能够表示晶状体折射率分布一般特征的梯度渐变模型表达式。基于收集的我国人眼实测数据的角膜面型模型和晶状体梯度渐变形式的折射率模型,为解决人眼光学系统研究中的关键问题提供了新的方案和思路。     

Spherical gradient-index polymer lens with low spherical aberration

A sphere lens with a spherical gradient index (GRIN) was prepared by the modified suspension polymerization technique. GRIN spheres with quadratic- and linear-index distributions were obtained by two different methods to confirm the effect of the GRIN profile on the focusing property of the sphere lens. It was confirmed in both theory and experiment that the spherical aberration of such GRIN spheres was remarkably decreased compared with that of a homogeneous sphere.     

轴位移法测定GRIN透镜的折射率分布系数

本文对GRIN透镜的精确光线追迹公式作了推导,介绍了轴位移法对透镜折射率分布的测量方法及相应的数据处理,并对测量误差进行了分析和计算,从而对测定GRIN透镜折射率分布提供了一种精确而且实用的方法。     

Foundations for low-loss fiber gradient-index lens pair coupling with the self-imaging mechanism

A fiber-optic collimator that emits a Gaussian beam with its beam waist at a certain distance after the exit face of the lens is labeled a self-imaging collimator. For such a collimator, the waist of the emitted Gaussian beam and its location are partly dependent on the properties of the gradient-index (GRIN) lens. Parameters for the self-imaging collimator are formulated in terms of the parameters of a GRIN lens (e.g., pitch, core refractive index, gradient index, length) and the optical wavelength. Next, by use of the Gaussian beam approximation, a general expression for the coupling power loss between two self-imaging-type single-mode fiber (SMF) collimators is, for the first time to our knowledge, derived as a function of three types of misalignment, namely, separation, lateral offset, and angular tilt misalignment. A coupling experiment between two self-imaging collimators with changing separation distance is successfully performed and matches the proposed self-imaging mechanism coupling loss theory. In addition, using a prism, lateral offset, as well as angular tilt, misalignments are experimentally simulated for a two self-imaging collimator coupling condition by a single collimator reflective test geometry. Experimental results agree well with the proposed loss formulas for self-imaging GRIN lenses. Hence, for the first time to our knowledge, the mathematical foundations are laid for employing self-imaging-type fiber collimators in SMF-based free-space systems allowing optimal design for ultra-low-loss coupling.     

Gradient-index contact lens

A gradient-index (GRIN) contact lens (CL) is proposed to decrease spherical aberration and to increase the diopter. A plastic radial GRIN rod was successfully obtained by using the vapor-phase diffusion copolymerization technique. The resulting index distribution of the GRIN rod was almost parabolic against the distance from the center axis, and the Δn value was -0.030. The GRIN CL was prepared by grinding and polishing the rod. It is theoretically and experimentally concluded that using the radial GRIN material can significantly improve the optical properties of CL's.     

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.     

玻璃GRIN光学元件光学性质计算模型的适用性

建立在均匀、无应力状态基础上的玻璃性质计算模型,在用于离子交换法制备的GRIN光学元件的折射率变化值(△N)计算时与实验结果有一定的偏差.本文对HSD(Higgins,Sun和Davis)模型和干福熹模型进行了比较,结果表明,干福熹模型用于低应力光学元件的计算时计算结果与实验值偏差较小,但对高应力GRIN光学元件的△N计算偏差很大;HSD模型在计算过程中分别考虑了玻璃组分的极化率因素和体积因素对其折射率的影响,在用于高应力GRIN光学元件△N计算时具有更强的适用性.     

ABCD matrix of the human lens gradient-index profile: applicability of the calculation methods

The applicability of different approximate methods proposed to determine the paraxial properties of the gradient-index (GRIN) distribution resembling that of the human lens, by means of the system ABCD matrix, is tested. Thus, the parabolic-ray-path approximation has been extended to provide the ABCD matrix of a slab lens comprised of a rotationally GRIN medium. The results show that this method has good numerical stability, and it is also the easiest one in determining the Gaussian constants of the human lens GRIN profile.     

Age-dependent variation of the Gradient Index profile in human crystalline lenses

PURPOSE: To reconstruct the gradient index (GRIN) profile of human crystalline lenses ex-vivo using Optical Coherence Tomography (OCT) imaging with an optimization technique and to study the dependence of the GRIN profile with age. METHODS: Cross-sectional images of nine isolated human crystalline lenses with ages ranging from 6 to 72 (post mortem time 1 to 4 days) were obtained using a custom-made OCT system. Lenses were extracted from whole cadaver globes and placed in a measurement chamber filled with preservation medium (DMEM). Lenses were imaged with the anterior surface up and then flipped over and imaged again, to obtain posterior lens surface profiles both undistorted and distorted by the refraction through the anterior crystalline lens and GRIN. The GRIN distribution of the lens was described with three variables by means of power function, with variables being the nucleus and surface index, and a power coefficient that describes the decay of the refractive index from the nucleus to the surface. An optimization method was used to search for the parameters that produced the best match of the distorted posterior surface. RESULTS: The distorted surface was simulated with accuracy around the resolution of the OCT system (under 15 μm). The reconstructed refractive index values ranged from 1.356 to 1.388 for the surface, and from 1.396 to 1.434 for the nucleus. The power coefficient ranged between 3 and 18. The power coefficient increased significantly with age, at a rate of 0.24 per year. CONCLUSION: Optical Coherence Tomography allowed optical, non-invasive measurement of the 2-D gradient index profile of the isolated human crystalline lens ex vivo. The age-dependent variation of the changes is consistent with previous data using magnetic resonance imaging, and the progressive formation of a refractive index plateau.     

Laser differential confocal lens refractive index measurement

A new laser differential confocal lens refractive index measurement is proposed, which uses the absolute zero of the differential confocal axial intensity curve to precisely identify the positions of the objective when the measurement pencil is focused on the vertex of the test lens and the reflector with or without the test lens in the measurement light-path, and then uses aberration compensation and ray tracing facet iterative calculation to obtain the refractive index of the test lens, thereby achieving the high-precision noncontact measurement of lens refractive index. The theoretical analyses and preliminary experiments indicate that the accuracy of the approach can reach about 2.5×10(-4).     

Graded-index fiber lens proposed for ultrasmall probes used in biomedical imaging

The quality and parameters of probing optical beams are extremely important in biomedical imaging systems both for image quality and light coupling efficiency considerations. For example, the shape, size, focal position, and focal range of such beams could have a great impact on the lateral resolution, penetration depth, and signal-to-noise ratio of the image in optical coherence tomography. We present a beam profile characterization of different variations of graded-index (GRIN) fiber lenses, which were recently proposed for biomedical imaging probes. Those GRIN lens modules are made of a single mode fiber and a GRIN fiber lens with or without a fiber spacer between them. We discuss theoretical analysis methods, fabrication techniques, and measured performance compared with theory.     

Gradient-index human lens as a quadratic phase transformer

The correspondence between the linear integral transform and the ray-transfer matrix of a first-order optical system is used to evaluate the transmittance function of gradient-index (GRIN) human lens regarded as a quadratic phase transformer. The size of the GRIN crystalline lens has been considered for redefining the effective transmittance function by the pupil function. The role that the GRIN nature of the human lens plays in the retinal image quality using the point spread function (PSF) is commented on. The simulation results show that the effective radius of the output face of the lens decreases with increasing thickness and that it is higher for flat end surfaces than for curved end faces. On the other hand, the simulation results also show, for small pupil sizes, that the GRIN nature of the human lens is a retinal image degradation source producing the spread of the PSF and that the curved end surfaces of the lens constitute a retinal image quality improve factor contributing to the narrowing of the PSF.     

Retroreflector approximation of a generalized Eaton lens

We extended a previous study of the Eaton lens at specific refraction angles to the Eaton lens at any refraction angle. The refractive index of the Eaton lens is complicated and has no analytical form except at a few specific angles. We derived a more accessible form of the refractive index for any refraction angle with some accuracy using the retroreflector approximation. The finding of this study will be useful for a rapid estimation of the refractive index, and the the design of various Eaton lenses.     

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.