Homogeneous Lens Research Articles

Paraxial reconstruction: conversion of homogeneous lens forms to continuous gradient-index media

We present a technique for the optical design of generalized-distribution GRIN lenses. Multi-element homogeneous lens designs are reconstructed as single GRIN media via smoothing of the homogeneous lens paraxial ray paths. These continuous optical systems successfully replicate the first-order properties of their homogeneous parent lens systems and serve as starting points for further optimization. When the technique is applied at several wavelengths, the chromatic aberration correction of the homogeneous parent lens is also converted. The paraxial reconstruction, finite-ray optimization, and evaluation of several lens designs are demonstrated.

Compact millimeter-wave air-filled substrate-integrated waveguide crossover employing homogeneous cylindrical lens

Compact millimeter-wave air-filled substrate-integrated waveguide crossover employing homogeneous cylindrical lens

Top-shaped asteroids as Lens-shaped bodies

Several asteroids are known to be shaped like toy tops. This paper models Top-Shaped Asteroids (TSAs) as Homogeneous Symmetric Lenses (HSLs), and derives their rotational, self-gravitational, and total energies as functions of their mass, density, and angular momentum. Then we raise, test, and ultimately reject the hypothesis that TSAs take the shape of lowest total energy, subject to the constraint that they keep the same mass, density, and angular momentum, while remaining HSLs. Other processes must control the shapes of TSAs. For completeness, we also describe a Core-Mantle Model for TSAs, as well as an Inverted Core-Mantle Model, and derive their self-gravitational energies, along with their rotational energies. The gravitational potential at the center of an HSL then is derived.

Sequential differential ray tracing formulation for homogeneous and gradient-index lenses using automatic differentiation

A differential ray tracing (DRT) formulation for the analysis and design of arbitrary optical systems is presented. An approach based on sequential rays is proposed to define a functional representation of the ray path, where the gradient information associated with the ray path is obtained using an automatic differentiation technique. The proposed DRT algorithm is neither constrained to a specific optical system geometry nor restricted to a set of surface types or material definition.

Novel Design Methodology for 3D-Printed Lenses for Travelling Wave Antennas

A novel methodology is introduced for designing bespoke homogeneous and graded index lenses for enhancing the gain of travelling wave antenna arrays (TWAs). 3D-printed lenses in the literature are majorly explored with standing wave antennas (SWAs) such as a microstrip or a horn antenna for gain enhancement. As there is progressively less power radiated from each slot in a TWA, as well as the successive phase delay between slots, the existing lens design approaches used for SWAs is not optimal for TWAs. Accordingly, we present a new approach of introducing a curvature to the lens that is derived from studying the power radiation profile of each slot of the TWA. This new methodology is demonstrated on a dielectric filled waveguide (DFW) slot array antenna operating at 26 -30 GHz band. An optimized dielectric graded lens and an optimized homogeneous lens have both been designed, fabricated, and measured with the DFW slot array. The new lens demonstrated a gain enhancement of more than 7 dB compared to less than 4 dB with conventional dielectric lenses. The proposed lens theory has been further verified with a bespoke optimized lens for a periodic stub-loaded microstrip leaky-wave antenna with a beam-scanning of 65∘. Design rules are included that can be applied for any TWA.

A High-Gain Hemispherical Dielectric Lens Antenna Operating Simultaneously in Narrowband or Wideband for X-band Applications

A hemispherical dielectric lens antenna is proposed to provide beam-steering for communication networks operating in X-band. Two different printed antennas are simultaneously used to guarantee high performance and diversity of applications, one a wideband linear polarized antenna, and the other a narrowband circular polarized antenna. To ensure this, the design relies on adding printed angled feeders correctly positioned in relation to the center of a homogeneous dielectric lens. A prototype was simulated, fabricated and tested. The measured results show that the antenna is capable of operating from 8.0 GHz to 12 GHz (gain of 14.2 dBi) or 9.5 GHz to 10 GHz (gain of 17.52 dBi).

Analytical ray transfer matrix for the crystalline lens.

We present the formulation of a paraxial ray transfer or ABCD matrix for onion-type GRIN lenses. In GRIN lenses, each iso-indicial surface (IIS) can be considered a refracting optical surface. If each IIS is a shell or layer, the ABCD matrix of a GRIN lens is computed by multiplying a typically high number of translation and refraction matrices corresponding to the K layers inside the lens. Using a differential approximation for the layer thickness, this matrix product becomes a sum. The elements A, B, C, and D of the approximated GRIN ray transfer matrix can be calculated by integrating the elements of a single-layer matrix. This ABCD matrix differs from a homogeneous lens matrix in only one integration term in element C, corresponding to the GRIN contribution to the lens power. Thus the total GRIN lens power is the sum of the homogeneous lens power and the GRIN contribution, which offers a compact and simple expression for the ABDC matrix. We then apply this formulation to the crystalline lens and implement both numerical and analytical integration procedures to obtain the GRIN lens power. The analytical approximation provides an accurate solution in terms of Gaussian hypergeometric functions. Last, we compare our numerical and analytical procedures with published ABCD matrix methods in the literature, and analyze the effect of the iso-indicial surface's conic constant (Q) and inner curvature gradient (G) on the lens power for different lens models.

Non-Resident, Not Absent: The Caregiving Role Non-Resident Fathers Play in Raising Their Children

ABSTRACT Although there is a strong belief in certain circles that fatherhood is best exercised in the context of a nuclear family, non-residential fatherhood is a growing phenomenon worldwide. In a South African context, the understanding of a family from a homogenous, dual-income, and co-resident lens disregards the emergence and prominence of the so-called non-traditional families. Studies in South Africa have also shown that co-residence is not always plausible or realistic. This article contributes to the growing understanding of the manifold caregiving roles that non-resident biological fathers play in the raising of their children over the life course. In-depth interviews were conducted with 23 black South African non-resident fathers aged 25–64 in Tshwane Municipality. The study found that non-resident fathers play a significant role in raising their children. This study makes an important contribution to dispelling two myths that have dominated fatherhood literature in the last few decades. First, the study dispels the myth that Black South African fathers’ especially non-resident fathers are irresponsible and are deadbeat. Second, the study showed that residence is a poor predictor of paternal involvement. The men in the study make sustained efforts to remain actively involved in the lives of their children. However, the findings also show that non-resident fathers frequently struggle to strike a balance between their many responsibilities and the societal expectations that their role as fathers must meet. Furthermore, non-resident fathers found it difficult to simultaneously adapt to their manifold roles of provider, nurturer, and moral guide.

Paraxial equivalent of the gradient-index lens of the human eye

The lens of the eye has a refractive index gradient that changes as the lens grows throughout life. These changes play a key role in the optics of the eye. Yet, the lens is generally simulated using a homogeneous model with an equivalent index that does not accurately represent the gradient. We present an analytical paraxial model of the gradient lens of the eye that gives the direct relation between refractive index distribution and paraxial characteristics. The model accurately simulates the changes in lens power with age and accommodation. It predicts that a decrease in equivalent index with age is associated with a flattening of the axial refractive index profile and that changes in lens power with accommodation are due primarily to changes in the axial variation of the iso-indicial curvature, consistent with Gullstrand's intracapsular theory of accommodation. The iso-indicial curvature gradient causes a shift of the principal planes compared to the homogeneous equivalent model. This shift introduces a clinically significant error in eye models that implement a homogenous lens. Our gradient lens model can be used in eye models to better predict the optics of the eye and the changes with age and accommodation.

Age-related appearance of lamellar structures in lens capsule of cataractous eyes and its pathological significance.

To determine morphological changes in the lens capsule with aging. Hayashi Eye Hospital, Fukuoka, Japan. Cross-sectional study. 25 eyes from the older patient group (aged 80 years or older) and 25 eyes from the younger patient group (aged 65 years or younger) who were diagnosed with cataract and indicated for surgery were included in the study. After continuous curvilinear capsulorhexis, the anterior lens capsule was collected, immediately fixed, and processed for electron microscopy analysis. Backscattered electron images of the cross-section of the anterior lens capsule were observed under a scanning electron microscope. The ultrastructure of the anterior lens capsule was observed and compared between the groups. Factors associated with the occurrence of the lamellar structure were also identified, with the presence or absence of a lamellar structure as an objective variable and preoperative clinical characteristics as the explanatory variables. 50 eyes of 50 patients were included. In the younger patient group, 20 eyes (80%) had a homogeneous lens capsule, whereas 5 eyes had lamellar structures. By contrast, in the older patient group, 5 eyes had homogeneous structures, while the remaining 20 eyes (80%) had lamellar structures. 1 eye showed capsular delamination. The only significant factor for the occurrence of lamellar structures was age group ( P < .01, nominal logistic regression analysis). Lamellar structures appear in the anterior capsule during aging. The appearance of lamellar structures indicates fragility of the lens capsule, which may, in turn, lead to capsular delamination or lens dislocation in some cases.

Impacts of the gradient-index crystalline lens structure on its peripheral optical power profile

Abstract The crystalline lens makes an important contribution to the peripheral refraction of the human eye, which may affect the development and progression of myopia. However, little has been known about the peripheral optical features of the crystalline lens and its impacts on the peripheral ocular refraction. This study aims to investigate the relationship between the structural parameters of the crystalline lens and its peripheral power profile over a wide visual field. The peripheral power profile is defined with respect to the entrance and exit pupil centers along the chief rays. Analysis is performed by three-dimensional ray tracing through the gradient refractive index (GRIN) lens models built from measurement data. It has been found that the vergence of the wavefronts at the entrance and the exit pupil centers of the lens show an approximate linear correlation to each other for each field angle. The exponent parameters of the axial refractive index profile and the axial curvature profile, and the asphericity of the posterior lens surface are found to be the most influential parameters in the peripheral power profiles. The study also shows that there can be significantly different, sometimes unrealistic, power profiles in the homogeneous lens model compared with its corresponding GRIN model with the same external geometry. The theoretical findings on the peripheral lens properties provide a new perspective for both wide-field eye modelling and the design of intraocular lenses to achieve normal peripheral vision.

Anti-mutation load and error stability of a large aperture lens based on parallel collimation locking.

Based on the lens of spatial filter parallel auxiliary assembly and collimation method, aiming at the unloading and locking problem of the filter lens after collimating, a lens parallel adjustable axial locking structure was proposed. A special elastic chuck structure was designed to realize clamping axial friction locking; each branch realized interference fit, and the adjustable structure of "line-line-line" contact realized equivalent fixation through spring pre-tightening. Compared with radial locking, the abrupt error of the axial locking lens in the unloading process of a collimating auxiliary machine was small, and the posture state maintenance and structural response stability were strong. The established homogeneous transformation matrix (HTM) and lens global error showed that the pose torsion roll error of the radial locking structure was significantly suppressed. The relative stability experiment showed that the angular drift error of the lens axial locking structure was ± 0.2 µrad. The design effectively improved the unloading anti-mutation characteristics and stability of the filter lens, which provided an experimental basis for the application of intelligent collimation assisted assembly and adjustment scheme in the upgrading of a high-power laser device.

Tunable multilayered lens made of PDMS with a biconical surface profile design and manufacture.

A polymer that has been used for the development of optical components and has had a significant impact is polydimethylsiloxane (PDMS) due to its remarkable mechanical and optical properties and easy handling. We present a practical and straightforward technique for designing and manufacturing a tunable graded index, graphical input (GRIN)-type lenses, and tunable lenses with a homogeneous refractive index made of PDMS. Implementing a biconical surface profile in a tunable plane-convex lens is proposed for elaborating both a homogeneous refractive index lens and a multilayered GRIN-type lens with a constant increased variation of 0.014 on its refractive index. Likewise, we introduce a mechanical mounting system that aims to modify their curvatures and therefore their focal lengths through mechanical stimuli applied on the lenses. Simulations of the optomechanical behavior and optical characterization of the lenses are also presented.

Millimeter-Wave Retro-Directive Frequency Coded Lens by Curved One-Dimensional Photonic Crystal Resonator

Innovating passive and chipless coded landmarks have recently emerged for high accuracy self-localization systems. Existing landmarks make use of the combination of retro-directive devices, corner reflectors and lenses, with a coding particle in order to give a high RCS response over a wide angle. In this paper, with a consideration of important practical parameters unappreciated in existing designs, we propose a wide-angle retro-directive frequency-coded lens by a curved one-dimensional Photonic Crystal (PhC) resonator. The proposed frequency-coded lens is made of two parts: a homogenous lens and a curved PhC resonator where the resonator is located along the lens focal line. A frequency coding is used, where the presence or absence of a notch frequency in a specified information channel encodes an information bit. A PhC resonator provides unique advantages over existing coding particles due to its continuity along the lens focal line which creates a stable ID appearance over wide-angle. In addition, the potential of coding in its volume, rather than on the surface, entitles to a high coding capacity. Two frequency-coded lenses with single and dual defect resonators are EM simulated, fabricated, and experimentally validated in the W-band (75 GHz-110 GHz). Simulated results show that a wide detection angle of 170° can be achieved where the tag ID is maintained over all angles. A wide retro-directivity of 80° and 60° is experimentally demonstrated for frequency-coded lenses by a single defect (single notch) and a dual defect (double notch) PhC resonator, respectively.

Ray Transfer Matrix for Onion-Type GRIN Lenses

We present the computation of an ABCD matrix for onion-type GRIN lenses. By applying a differential approximation of the layer thickness, the matrix product of a high number of matrices is synthetized into a single matrix where the elements are integrals. The difference between this ABCD matrix and a homogeneous lens matrix is one integration term in element C, which is the GRIN contribution to the lens power. In the case of the crystalline lens, the analytical approximation to the GRIN lens power provides an accurate and concise solution in terms of Gaussian hypergeometric functions.

Three-Dimensional Broadband and Isotropic Double-Mesh Twin-Wire Media for Meta-Lenses

Lenses are used for multiple applications, including communications, surveillance and security, and medical instruments. In homogeneous lenses, the contour is used to control the electromagnetic propagation. Differently, graded-index lenses make use of inhomogeneous materials, which is an extra degree of freedom. This extra degree of freedom enables the design of devices with a high performance. For instance, rotationally symmetric lenses without spherical aberrations, e.g., the Luneburg lens, can be designed. However, the manufacturing of such lenses is more complex. One possible approach to implement these lenses is using metamaterials, which are able to produce equivalent refractive indices. Here, we propose a new type of three-dimensional metamaterial formed with two independent sets of wires. The double-mesh twin-wire structure permits the propagation of a first mode without cut-off frequency and with low dispersion and high isotropy. These properties are similar to periodic structures with higher symmetries, such as glide symmetry. The variations of the equivalent refractive index are achieved with the dimension of the meandered wires. The potential of this new metamaterial is demonstrated with simulated results of a Luneburg meta-lens.

Modelling the visual world of a velvet worm.

In many animal phyla, eyes are small and provide only low-resolution vision for general orientation in the environment. Because these primitive eyes rarely have a defined image plane, traditional visual-optics principles cannot be applied. To assess the functional capacity of such eyes we have developed modelling principles based on ray tracing in 3D reconstructions of eye morphology, where refraction on the way to the photoreceptors and absorption in the photopigment are calculated incrementally for ray bundles from all angles within the visual field. From the ray tracing, we calculate the complete angular acceptance function of each photoreceptor in the eye, revealing the visual acuity for all parts of the visual field. We then use this information to generate visual filters that can be applied to high resolution images or videos to convert them to accurate representations of the spatial information seen by the animal. The method is here applied to the 0.1 mm eyes of the velvet worm Euperipatoides rowelli (Onychophora). These eyes of these terrestrial invertebrates consist of a curved cornea covering an irregular but optically homogeneous lens directly joining a retina packed with photoreceptive rhabdoms. 3D reconstruction from histological sections revealed an asymmetric eye, where the retina is deeper in the forward-pointing direction. The calculated visual acuity also reveals performance differences across the visual field, with a maximum acuity of about 0.11 cycles/deg in the forward direction despite laterally pointing eyes. The results agree with previous behavioural measurements of visual acuity, and suggest that velvet worm vision is adequate for orientation and positioning within the habitat.

Quantitative analysis and optimization design of the segmented planar integrated optical imaging system based on an inhomogeneous multistage sampling lens array.

The non-uniformity of the lens array in the segmented planar integrated optical imaging system has not been studied. We design an inhomogeneous multistage sampling lens array based on a hierarchical multistage sampling lens array of the segmented planar integrated optical imaging system by changing the radius of the lenslet for frequency-selective sampling to optimize the design of the UV spatial frequency distribution. The signal transfer model of the proposed imaging system was established considering the radius of the lenslet and key waveguide parameters. Simulation results show that the proposed system produces better imaging quality owing to the selective sampling of frequency. Compared with the homogeneous multistage sampling lens array (radius of 1.8 mm), the optimal radius parameters of the inhomogeneous multistage sampling lens array are given for the same longest lens array length. Detailed calculations of the parameters of the waveguide connected to the lens array were performed. Our results provide a theoretical basis for the optimal design and process of a new segmented planar integrated optical imaging system.

Coronavirus (COVID-19) Implications on the Livelihoods of the Farmers

This study aimed at reviewing the literature on Coronavirus (COVID-19) implications on the livelihoods of the farmers. Using a literature review, the paper aimed at understanding how COVID-19 has impacted the livelihoods of the farmers whose survival depends on agriculture. Driven by the following objectives; the paper reviewed firstly, lessons to be learned from the past pandemics, secondly, how COVID-19 has affected farmers’ livelihoods and thirdly, the effects of COVID-19 impacts on communities. The paper argues that, with the enormous public health policies on avoiding the spread of the pandemic, the livelihoods of the farmers have largely been affected due to reduced productivities, challenges in accessing and utilizing the markets for a nutritious and food secure life is leading to increasing poverty levels. Poverty, creates vulnerabilities making farmers susceptible to crime, human trafficking, theft and rape as observed in most African countries in the COVID-19 period. Hence, the need to avoid seeing the world through western homogeneous lenses to context-specific in the fight against the spread of COVID-19.

Distributions of distances and volumes of balls in homogeneous lens spaces

Lens spaces are a family of manifolds that have been a source of many interesting phenomena in topology and differential geometry. Their concrete construction, as quotients of odd-dimensional spheres by a free linear action of a finite cyclic group, allows a deeper analysis of their structure. In this paper, we consider the problem of moments for the distance function between randomly selected pairs of points on homogeneous three-dimensional lens spaces. We give a derivation of a recursion relation for the moments, a formula for the kth moment, and a formula for the moment generating function, as well as an explicit formula for the volume of balls of all radii in these lens spaces.