Modeling liquid-crystal gradient-index lenses

The optical properties of one-dimensional gradient-refractive-index lens arrays based on liquid crystals are studied. We find that it is quite possible, using theoretical methods, to predict angular distributions of the light emanating from such arrays when they are illuminated with collimated monochromatic light. We compare four theoretical methods in relation to experiments. The experimental data and the model, based on a combination of eikonal methods and diffraction, are in close correspondence. Features such as maximal beam width and number of extrema in the angular light distribution are discussed and explained theoretically. We also studied dispersion effects, both experimentally and theoretically, with good agreement between the two.     

Spherical aberration of gradient-index rod lenses

Spherical aberration was measured and, by carrying out the computer simulation for ray tracing with the approximate solution of ray equation, the refractive-index distribution parameters of a gradient-index rod lens were determined to most fit the observed data.     

Birefringence measurements in gradient-index rod lenses

Stress birefringence can be found in gradient-index (GRIN) materials because they contain a variation in composition. GRIN glass fabricated by ion exchange may contain stress from two different processes. These include a size difference between the exchange and the diffusing ions and a variation in the thermal-expansion coefficient across the gradient region. The optical properties of the stressed material are polarization dependent, and therefore image quality is directly affected. We examine birefringence in GRIN rod lenses that have lengths more than ten times greater than their diameters. The effects are more easily observed in long rod lenses because of the large optical path lengths.     

Chromatic aberrations of radial gradient-index lenses. II. Selfoc lenses

Simple equations used for analyzing chromatic aberrations of Selfoc lenses were derived in terms of Buchdahl chromatic coordinates and Buchdahl dispersion constants. The equations that employ gradient-index chromatic constants Ψ(1) and Ψ(2) are used for selecting suitable ion-exchange pairs to design an achromatic Selfoc lens.     

Observed and simulated fracture pattern in diametrically loaded discs of rock material

To investigate the influence of a tensile stress gradient on fracture initiation and fracture growth in rock material, a configuration, consisting of a diametrically loaded disc with a hole on the diameter perpendicular to the loaded diameter, is used. The maximum local tensile stresses the material is able to withstand increase as the stress gradient increases. Depending on the diameter and the eccentricity of the hole, the disc splits along the loaded diameter or macro-fracturing starts at the hole. However, the tensile stresses at the top and the bottom of the hole are for nearly all cases considerably higher than the stress along the loaded diameter and than the macroscopic tensile strength of the material, determined by conventional Brazilian tests. To better understand this particular fracturing behaviour, numerical simulations of the experiments are conducted using the boundary element code DIGS, which allows the incorporation of weak elements (flaws), representing defects and weaknesses in the rock material. It is shown that the influence of the stress gradient on the stress concentration at the tip of mobilised defects lies at the origin of the particular fracturing behaviour in the diametrically loaded disc with a hole. The study of the new configuration leads also to a number of conclusions with regard to the failure in diametrically loaded discs in general. Based on the flaw model, fracture initiation in the Brazilian test has to be attributed to fracture growth of a mobilised defect, situated in the area close to one of the platens.     

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.     

Chromatic aberrations of radial gradient-index lenses. I. Theory

Chromatic effects of radial gradient-index materials have been analyzed, and several important conclusions have been derived in terms of material dispersion data. The use of Buchdahl dispersion data, both for base glass materials and ion-exchange pairs, provides some simple relationships for chromatic aberration and helps in selecting suitable materials for producing achromatic radial gradient-index lenses.     

Conoscopic evaluation of the birefringence of gradient-index lenses: infidelity sources

Gradient-index lenses are samples whose special characteristics must be taken into account to design the optical polariscopes that can be applied in the evaluation of their birefringence. We discuss the main infidelity sources that modify the conoscopic patterns when a traditional polariscopic setup is used.     

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.     

Colorless gradient-index cylindrical lenses with high numerical apertures produced by silver-ion exchange

Colorless gradient-index cylindrical lenses with high numerical apertures in slabs 200-1000 μm thick have been fabricated by silver-sodium ion exchange in a specially developed glass. The lenses with numerical apertures of 0.6 are characterized by index profiles and by spherical and chromatic aberration. On-axis focusing properties and the application to the collimation of high-power laser diode bars are discussed.     

Making fast cylindrical gradient-index lenses diffraction limited by using a wave-front-correction element

A wave-front-correction element (WFCE) is produced to make a cylindrical Ag-ion-exchanged gradient-index (GRIN) lens with a high numerical aperture (0.53) diffraction limited (wave-front error, 0.02lambda rms). The wave-front aberrations of the cylindrical GRIN lens are measured by a phase-shifting shearing interferometer, with a conventional microscope objective used as a compensation lens. The continuous surface relief of the WFCE is produced by a lithographic process. The wave-front-corrected GRIN lens is applied to collimate the strongly divergent light (57 degrees full diverging angle measured at 1/e(2) of maximum intensity) emitted by a high-power diode laser. The power irradiated into a full angle of 2 mrad can be enhanced by a factor of 1.8 with the WFCE.     

Birefringence in gradient-index rod lenses: a direct measurement method and interferometric polarization effects

Gradient-index (GRIN) media can contain stress birefringence resulting from the variation in material composition. Anisotropy in a GRIN rod lens affects ray propagation and can degrade image quality. A technique, believed to be new, for measuring birefringence in GRIN rod lenses has been developed. The change in optical path difference (OPD) for orthogonal polarizations is measured directly. With this method, effects on OPD from standard imaging aberrations are excluded. Birefringence measurements for various GRIN rod samples are presented. The data are compared with results obtained previously by use of a Twyman-Green measurement method. Also, the polarization effects on tilt fringes observed with the direct measurement method and the Twyman-Green method are presented and modeled theoretically. Tilt fringes for large birefringence test cases are also modeled.     

General formula for coupling-loss characterization of single-mode fiber collimators by use of gradient-index rod lenses

A general formula for determining the coupling loss between two single-mode fiber collimators with the simultaneous existence of separation, lateral offset and angular tilt misalignments, and spot-size mismatch is theoretically derived by use of the Gaussian field approximation. Based on this general formula, the formulas for coupling losses that are due to the misalignment of insert separation, lateral offset, and angular tilt are given. The formula for the coupling loss that is due to Gaussian spot-size mismatch of two single-mode collimators is also given. Good agreement between these formulas and experimental results is demonstrated with gradient-index rod lens-based fiber collimators operating in the 1300-nm band.