Adaptive model of the gradient index of the human lens. II. Optics of the accommodating aging lens

A simple, parametric model of the gradient refractive index distribution (GRIN) of the human lens with conicoid surfaces able to adapt to individual distributions as well as to the changes of the lens shape and structure with age and accommodation is presented. The first part of this work was published in a companion paper [J. Opt. Soc. Am. A24, 2175 (2007)]. It included the development of the mathematical formulation of the adaptive model; the validation of its customization capability by fitting, sample by sample, a set of in vitro refractive index distributions of lenses of different ages, ranging from 7 to 82 years, from the recent literature; and an average model of the (in vitro) aging crystalline lens. Here we extrapolate that in vitro GRIN model by assuming that the same structural parameters are valid for the living lens. Then, recent data of the changes of the shape of the aging lens with accommodation from the literature are used to build an aging and accommodating lens model. This is straightforward since the GRIN model adapts automatically to the chosen external lens geometry. A strong coupling was found between the adaptive GRIN distributions and the conic constants affecting the refractive power. To account for the lens paradox and the reported changes in lens spherical aberration with age and accommodation, age- and accommodation-dependent functions for the anterior and posterior internal conic constants were derived through optimization.     

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.     

Broadband microwave Luneburg lens made of gradient index metamaterials

Luneburg lenses are able to form perfect focus that is free of aberration. Because of the varying refractive index throughout the lens, incoming electromagnetic waves can travel in a curved path and be guided to focus at the back of the lens. The implementation of Luneburg lenses is often difficult due to the challenges in creating a medium with varying refractive index using normal materials. This problem can be overcome with the use of gradient index metamaterials. We report a two dimensional Luneburg lens made of gradient index metamaterials. It consists of 17 concentric shells with etched patterns on a printed circuit board working in microwave X band frequency. The broad properties of the Luneburg lens are then discussed.     

Determination of refractive indices of biconvex lenses by use of a Michelson interferometer

Measurements of lens parameters such as focal length, radius of curvature, and refractive index are important. We describe a measurement method that utilizes a Michelson interferometer to determine parameters of thin, convex lenses. The real fringe system formed by a Michelson interferometer is used to determine the focal lengths and the radii of curvature of the lenses. The refractive index of the lens material is determined from the thin-lens formula. We were able to determine the refractive indices to an accuracy as great as 99.97%. A detailed theoretical and experimental analysis is given.     

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.     

Influence of spoliation in poly(2-hydroxy ethyl methacrylate) soft contact lens on its free volume and optical transparency

The calcification in poly(2-hydroxy ethyl methacrylate) contact lens was investigated using positron annihilation spectroscopy (PLS). The two poly(2-hydroxy ethyl methacrylate) (PHEMA) lenses of different companies were calcified employing a simple mechanism of calcification in abiotic aqueous solutions. The calcium deposit was analyzed using energy dispersive X-ray spectroscopy (EDS). Calcified lenses showed decrease in ortho-positronium (o-Ps) lifetime and free volume hole size of the lens material suggesting diffusion of Ca²⁺ into these cavities. The change in optical property viz. refractive index of these calcified lenses were measured and correlated with positron results. To find a better correlation, a series of worn spoilt PHEMA lenses of the same power with mainly calcium deposits, were similarly characterized using PLS and refractive index. These results correlate well with the free volume of the material. For hydrophilic lenses this correlation is reported for the first time.     

Analysis of grating lens aberration

A method that approximates a diffraction formula to the Fourier integral formula by the application of a coordinate transformation is proposed, and it is proved to be a superior and more accurate approximation when it is applied to focusing systems under a nonsine condition. Based on this approximation, the condition that gives the diffraction focus position is derived by the application of two methods. The first is applicable to rotationally symmetrical aberrations such as spherical aberration, and the second is a more general method that utilizes the standard deviation of a wave aberration. These methods are actually applied to a grating lens as a way to study both chromatic aberration characteristics and oblique incidence characteristics.     

Effects of the peripheral layers on the optical properties of spherical fish lenses

We created a computational optical model of spherical fish lenses that takes into account the effects of the peripheral layers, which differ in cellular composition from the bulk of the lens. A constant refractive index, except for the lens capsule, in the outer about 6% of lens radius made it possible to uniquely infer the refractive index gradient in more central layers from a known or desired longitudinal spherical aberration curve using the inverse Abel transform. Since the zone of constant refractive index is wider than necessary to make the solution unique and for optimal optical performance of the lens, we propose that its width be set by the metabolic needs of the lens.     

Foamy-dielectric lens transmission lines for millimeter and submillimeter wavelength range

The principles of the construction of transmission lines based on lenses made of a material with a nearly unity refractive index (foamy-dielectric lens transmission line, FDLTL) are formulated. The transmission characteristics of two FDLTL variants for 3-mm wavelength band have been studied. It is shown that FDLTLs possess low damping and feature wave propagation in a single-mode regime.     

Fast response varifocal lenses using KTa(1-x)Nb(x)O3 crystals and a simulation method with electrostrictive calculations

We fabricated cylindrical varifocal lenses with fast responses by using the strong Kerr effect of KTa(1-x)Nb(x)O(3) (KTN) single crystals. We observed focus shifts of up to 87 mm with the assistance of a 250 mm focal length lens, which corresponds to a focus shift from infinity to 720 mm by the KTN lens itself. The response time was as fast as 1 μs. We also present a simulation method for calculating refractive index distributions in KTN single crystals, which is essential when designing the lens. The method is characterized by the strain contribution, which has not conventionally been typical of electro-optic simulations. We used this method to explain the refractive index modulations that are characteristic of the varifocal lenses.     

Measurement of parameters of simple lenses using digital holographic interferometry and a synthetic reference wave

The use of digital holographic intrerferometry in the testing of simple thin lenses is explored. Focal length, radius of curvature, and refractive index are the lens parameters that can be determined using this method. The digital holograms using the lens under test are recorded at various positions of the test lens using off-axis geometry. This is combined with a digitally computed plane wavefront to determine the curvature of the light beam emerging from the test lens. Focal length is the position of the test lens where a single fringe results. The radius of curvature of the test lens is also determined similarly using a long focal length lens to concentrate a collimated beam onto the test lens. The nonuniformities on the lens surface could also be found by using this method. The implementation of the method is shown by using computer simulations in the case of biconvex lenses. The method can be utilized to measure the parameters of plano-convex and concave lenses also.     

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.     

Conditions under which two-element variable power lenses can be created. Part 1. Theoretical analysis

The conditions under which a two-element variable power lens can be created are examined. Such a lens is defined as one in which the functional form of the optical effect created does not change as the elements translate with respect to one another--only the magnitude of the effect changes. It is found that only variable power optical effects that can be described by quadratic functions can be formed by laterally translating two-element variable power lenses. In the case of rotationally translating two-element variable power lenses, possible designs are found by mapping possible laterally translating designs from a Cartesian space to the polar coordinate space of the rotationally translating lens.     

Achromatic fourier processor with holographic optical lenses

An optical Fourier processor that allows the use of broadband light sources and colored inputs is designed, fabricated, and tested. We develop a design technique based on phase manipulation in the Fourier plane to construct an image processor that provides a chromatically corrected image making use of the good aberrations behavior of symmetrical optical systems. Only a small number of diffractive lenses and one achromatic refractive lens are required to obtain a real image. We verify our design experimentally using holographic lenses, which are presented, owing to their versatility, as a good alternative to expensive blazed diffractive elements.     

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.     

Alternative method for measuring effective focal length of lenses using the front and back surface reflections from a reference plate

We present a simple method for measuring the effective focal length without determining the location of principle plane of the lens. The method is based on the measurement of confocal backreflection axial responses from the front and back surfaces of a reference plate with known refractive index and thickness. We proved the concept by measuring the effective focal lengths of thin singlet lenses and complex microscope objectives. The theoretical limit of measurement precision varies depending on the numerical aperture of the lens. This method can provide an alternative focal length measurement method for complex lenses or lenses that are permanently attached to other structures. Measurement errors were analyzed theoretically and improvements in measurement accuracy were discussed.     

Conditions under which two-element variable power lenses can be created. Part 2. Application to specific designs

It was found in Part 1 of this paper [J. Opt. Soc. Am. A 28, 2148 (2011)] that only variable power optical effects that can be described by quadratic functions can be formed by laterally translating two-element variable power lenses. In the case of rotationally translating two-element variable power lenses, possible designs are found by mapping possible laterally translating designs from a Cartesian space to the polar coordinate space of the rotationally translating lens. Several designs that have been manufactured or suggested theoretically are examined in Part 2 to see which ones are true variable power lenses.     

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.     

Synthesis of an inhomogeneous dielectric axisymmetric lens

The problem of determining the index of refraction as a function of the distance from the axis of symmetry of an inhomogeneous dielectric lens is considered for the input and output wave fronts. An analytical method of obtaining a solution for a layered lens is proposed, which is based on the geometric optics approximation and the assumption that the index of refraction within each layer is constant. It is demonstrated that this method is applicable to both layered and gradient lenses (provided that the number of layers is sufficiently large).