Improvements of the ray trace through the generalized Luneburg lens

An analytical expression for the gradient of the generalized Luneburg lens is found and applied to the computation of the profile and the gradient of the refractive index.     

Optical Waveguide Lenses

The development of optical waveguide lenses, i.e. mode-index (including Luneburg), geodesic, Fresnel and Bragg grating lenses is briefly reviewed chronologically. A Bragg lens and a Luneburg lens, fabricated by electron lithography and vacuum deposition respectively, are described. A nearly diffraction-limited lens of the latter type can be obtained by using the shadow mask process.     

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.     

Spherical gradient-index lenses as perfect imaging and maximum power transfer devices

Gradient-index lenses can be viewed from the perspectives of both imaging and nonimaging optics, that is, in terms of both image fidelity and achievable flux concentration. The simple class of gradient-index lenses with spherical symmetry, often referred to as modified Luneburg lenses, is revisited. An alternative derivation for established solutions is offered; the method of Fermat's strings and the principle of skewness conservation are invoked. Then these nominally perfect imaging devices are examined from the additional vantage point of power transfer, and the degree to which they realize the thermodynamic limit to flux concentration is determined. Finally, the spherical gradient-index lens of the fish eye is considered as a modified Luneburg lens optimized subject to material constraints.     

Evolution of graded refractive index in squid lenses.

A lens with a graded refractive index is required for vision in aquatic animals with camera-type eyes. This optical design entails a radial gradient of protein density, with low density in external layers and high density in internal layers. To maintain the optical stability of the eye, different material properties are required for proteins in different regions of the lens. In low-density regions of the lens where slight protein aggregation causes significant light scattering, aggregation must be minimized. Squid lens S-crystallin proteins are evolutionarily derived from the glutathione S-transferase protein family. We used biochemistry, optical modelling and phylogenetics to study the evolution and material properties of S-crystallins. S-crystallins are differentially expressed in a radial gradient, suggesting a role in refractive index. This gradient in S-crystallin expression is correlated with their evolutionary history and biochemistry. S-crystallins have been under positive selection. This selection appears to have resulted in stabilization of derived S-crystallins via mutations in the dimer interface and extended electrostatic fields. These derived S-crystallins probably cause the glassy organization and stability of low refractive index lens layers. Our work elucidates the molecular and evolutionary mechanisms underlying the production and maintenance of camera-like optics in squid lenses.     

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.     

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.     

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.     

Plasmonic Luneburg and Eaton lenses

Plasmonics takes advantage of the properties of surface plasmon polaritons, which are localized or propagating quasiparticles in which photons are coupled to the quasi-free electrons in metals. In particular, plasmonic devices can confine light in regions with dimensions that are smaller than the wavelength of the photons in free space, and this makes it possible to match the different length scales associated with photonics and electronics in a single nanoscale device. Broad applications of plasmonics that have been demonstrated to date include biological sensing, sub-diffraction-limit imaging, focusing and lithography and nano-optical circuitry. Plasmonics-based optical elements such as waveguides, lenses, beamsplitters and reflectors have been implemented by structuring metal surfaces or placing dielectric structures on metals to manipulate the two-dimensional surface plasmon waves. However, the abrupt discontinuities in the material properties or geometries of these elements lead to increased scattering of surface plasmon polaritons, which significantly reduces the efficiency of these components. Transformation optics provides an alternative approach to controlling the propagation of light by spatially varying the optical properties of a material. Here, motivated by this approach, we use grey-scale lithography to adiabatically tailor the topology of a dielectric layer adjacent to a metal surface to demonstrate a plasmonic Luneburg lens that can focus surface plasmon polaritons. We also make a plasmonic Eaton lens that can bend surface plasmon polaritons. Because the optical properties are changed gradually rather than abruptly in these lenses, losses due to scattering can be significantly reduced in comparison with previously reported plasmonic elements.     

Ray-tracing method for isotropic inhomogeneous refractive-index media from arbitrary discrete input

We have developed a ray-tracing simulation procedure for optically isotropic gradient refractive-index media. The procedure can take discrete points of arbitrary distribution for the definition of refractive-index distributions and lens surfaces. It is useful for simulating ray trajectories in real lens systems. The procedure is applied to a ray-tracing simulation of the Luneburg lens and a radial gradient optical fiber. The simulation results are compared with the analytical solutions, and it is shown that they are in precise agreement.     

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.     

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.     

Optical negative refraction by four-wave mixing in thin metallic nanostructures

The law of refraction first derived by Snellius and later introduced as the Huygens-Fermat principle, states that the incidence and refracted angles of a light wave at the interface of two different materials are related to the ratio of the refractive indices in each medium. Whereas all natural materials have a positive refractive index and therefore exhibit refraction in the positive direction, artificially engineered negative index metamaterials have been shown capable of bending light waves negatively. Such a negative refractive index is the key to achieving a perfect lens that is capable of imaging well below the diffraction limit. However, negative index metamaterials are typically lossy, narrow band, and require complicated fabrication processes. Recently, an alternative approach to obtain negative refraction from a very thin nonlinear film has been proposed and experimentally demonstrated in the microwave region. However, such approaches use phase conjugation, which makes optical implementations difficult. Here, we report a simple but different scheme to demonstrate experimentally nonlinear negative refraction at optical frequencies using four-wave mixing in nanostructured metal films. The refractive index can be designed at will by simply tuning the wavelengths of the interacting waves, which could have potential impact on many important applications, such as superlens imaging.     

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.     

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.     

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.     

Micro gradient-index conical lenses: simulation and fabrication methods

In this paper, a micro gradient-index conical lens, which has a larger acceptance angle than a conventional microlens, is presented. Methods on how to simulate these lenses in commercial optical design software CodeV are introduced, and the effects of several index profiles and cone shapes are compared in simulation. Results show that a micro gradient-index conical lens has a four times larger acceptance angle compared with a microlens. Additionally, conical lenses with a Gaussian-index profile show a larger acceptance angle than those with a solid refractive index. Fabricated conical lenses show an acceptance angle of more than 27 degrees for a detection threshold of 50%, which agrees with the simulation result.     

A large numerical aperture structured lens formed by varying high-refractive-index-dielectric square holes

A structured lens with a large numerical aperture for high diffraction efficiency is proposed. This structured lens is formed by varying high-refractive-index-dielectric (HRID) square holes in a metallic film. Compared with previous structured lenses formed by varying air square holes, the modulated wavefront at the exit pupil of the structured lens is smoother and the diffraction efficiency is greatly improved. A dielectric film was introduced as a buffer layer to reduce the energy loss at the interface caused by the great impedance mismatching between air and the high refractive index dielectric. A lens with an aperture of 100 µm and a focal length of 50 µm was designed; its diffraction efficiency has been raised from 22.8% to 47.2% by the finite difference and time domain (FDTD) algorithm. This structured lens has the advantages of not only light weight and high compactness, but also large numerical aperture, high diffraction efficiency, and good imaging quality.     

Diffractive-refractive behavior of kinoform lenses

The connection between the diffractive behavior and the refractive behavior of kinoform lenses is investigated. The image-forming capabilities of the diffractive element are found to be expressed as an interference pattern that is due to a set of associated refractive lenses corresponding to each zone of the kinoform. The coefficient modulating the contribution of each refracting zone is determined. The term linking the point-spread function of a refractive lens and a diffractive lens is also obtained. Spectral and spatial aspects are compared as the diffractive element approaches the refractive limit.