Axial Gradient-index Research Articles

Symmetric/Asymmetric Vibrational Characteristics of Bi-Directional Functionally Graded Annular Microplates

In this study, a free vibration model of bi-directional functionally graded (BD-FG) annular microplates is established based on the modified couple stress theory (MCST) and the first-order shear deformation theory (FSDT), and the corresponding governing equations are derived from Hamilton’s principle. The annular microplates are assumed to be composed of metal and ceramic materials, and the equivalent material properties are a combination of power-law function and exponential ones. The generalized differential quadrature method (GDQM) is used to obtain the vibration frequencies and mode displacements for the C–C, S–C, C–S, and S–S (S: simply supported; C: clamped) annular microplates. The convergence and validity of the proposed model are verified through selective numerical examples. Further, the effects of gradient index, material length scale parameter, boundary conditions, and geometrical dimensions on the symmetric/asymmetric vibration frequencies of the annular microplate are explored. The modal assurance criterion is applied to estimate the influence of the radial gradient index and material length scale parameter on different-order mode displacements of the annular microplate. The results show that: (1) the vibration frequencies of the annular microplate are sensitive to changes in the geometry and boundary conditions; (2) compared with the axial gradient index, the radial gradient index not only affects the vibration frequencies of the annular microplate, but also impacts the distribution of the highest and lowest points of the vibration mode displacements as well as the vibration mode nodal lines; (3) as the material length scale parameter increases, the vibration frequencies are significantly increased, and the higher-order vibration modes of the annular microplate also change.

Compact Sub‐Semicircular Axial Gradient Index Lens Synthesized by Metasurfaces: A Promising Solution for Miniaturized Antenna and Subwavelength Focusing Systems

AbstractWe present the design, fabrication, and experimental measurements of an Axial Gradient‐index (AGRIN) flat lens synthesized by 2D periodic metallic patches on a grounded dielectric slab, known as metasurfaces. Metasurfaces can be considered as artificial materials that possess unique characteristics, which mainly revolve around the two‐dimensional distribution of the structure. By manipulating the phase and amplitude of the near‐fields over the surface of the metasurface, specific values of the effective permittivity and permeability can be achieved, thereby regulating the overall field behavior by gaining control of the refractive index that is related to the aforementioned two material parameters. By configuring the refractive index, metasurfaces enable the manipulation of the direction in which electromagnetic waves propagate, as dictated by Snell's law. By varying the geometric dimensions of unit cells in each segment that compose the lens, the gradient refractive index profile required for our AGRIN lens can be achieved. A primary feature of this lens lies with its semicircular structure, in contrast to conventional circular ones. Its perimeter comprises a convex boundary and a straight one. In comparison to traditional circular lenses like the Luneburg lens and Maxwell fisheye lens, our proposed AGRIN lens offers the advantage of being more compact. Despite its smaller area, the AGRIN lens maintains its focusing ability. The AGRIN lens is capable of focusing over 71.75% of the energy from the incident wave onto the focal point. This efficient focusing ability, combined with its reduced size, makes the AGRIN lens highly advantageous for various applications.

Investigation of Transverse Vibration Characteristics of Cracked Axially Moving Functionally Graded Beam Under Thermal Load

This article presents a new analysis of the vibration characteristics of open-edge cracks for graded moving beams under thermal load. The material property gradient is based on the distribution of the power law in the direction of the beam thickness. The vibration equation is obtained depending on the precept of Hamilton and resolved by the extension of Galerkin’s approach. A rotational spring is used to represent the cracking in the beam. The effects of the axial velocity, gradient index, thermal load, and cracking parameters on vibration characteristics are observed. Furthermore, the mode shapes of the simple support cracked moving graded beam are determined. The results show that the rise in the axial velocity, the crack depth, and the material property index led to a decrease in the natural frequencies, which indicates the results obtained. HIGHLIGHTS Stability and vibrations analysis of the cracked gradient beams with an axial motion under a uniform temperature rise The effect of axial velocity, gradient index, thermal load, and cracks parameter was studied Galerkin’s technique is developed for vibrational analysis of cracked graded beams with longitudinal motion The Euler-Bernoulli beam model was used taking into account the effect of rotatory inertia GRAPHICAL ABSTRACT

Hygrothermal effects on buckling behaviors of porous bi-directional functionally graded micro-/nanobeams using two-phase local/nonlocal strain gradient theory

Rapid development of technology puts forward higher performance requirements for advanced materials with multi-field coupling stability. The porous functionally graded materials, which is well developed with mature design, tools, can be customized according to specific application through nanostructures. In this work, the influences of hygro-thermo-mechanical coupling loadings on buckling behaviors of porous bi-directional functionally graded (2D FG) Timoshenko nanobeam is investigated through the numerical study. The governing equation and boundary conditions are derived on the basis of two phase local/nonlocal strain gradient theory employing the principle of virtual work, and a processing method for Clamped-Free (C–F) boundary condition is proposed. By employing generalized differential quadrature method (GDQM), the theoretical solution of buckling behavior of the Timoshenko nanobeam is obtained, with the implementation of comparison and convergence studies to demonstrate the accuracy and effectiveness. On the best of the authors’ knowledge, it is novel to investigate the buckling response of 2D FG Timoshenko nanobeam under hygrothermal environment using the two local/nonlocal strain gradient theory. Herein, the influence of nonlocal parameter, strain gradient theory, aspect ratio, axial gradient index, thickness gradient index, hygrothermal loadings and local volume fraction on the critical buckling load under various natural boundary conditions are acquired and discussed. The numerical results indicate that the present model can capture the buckling response of porous 2D FG Timoshenko nanobeams under hygrothermal environment, exhibiting consistent softening/hardening effects under all boundary conditions and modes.

Vibration and Buckling Characteristics of 2D Functionally Graded Microbeams With Variable Cross Sections

Based on the modified couple stress theory and Timoshenko beam theory, the free vibration and buckling mechanics model of a two-dimensional functionally graded microbeam with variable cross-section is established by applying the variational principle. The model contains the intrinsic material length scale parameters of the metal and ceramic components, which can predict the size effect of microbeams. The Ritz method is used to obtain the numerical solution of the vibration frequency and critical buckling load of the microbeams under arbitrary boundary conditions. Numerical examples reveal that when the thickness of the microbeam decreases, the dimensionless first order frequency and the dimensionless critical buckling load increase, and the scale effect increases. The effect of the taper ratio on the dimensionless first order frequency of the microbeam is closely related to the boundary conditions. At the same time, the effect of the taper ratio of the thickness and the width is also significantly different. The dimensionless first order frequency of microbeams increases with the increase of the material length scale parameter ratio of ceramic and metal, and the degree of increase is different under different boundary conditions. The thickness direction and axial material gradient index also have a significant influence on the free vibration and buckling behavior of the microbeam.

Electrically tunable graphene plasmonic quasicrystal metasurfaces for transformation optics.

The past few years have witnessed tremendous achievements of transformation optics applied to metallic plasmonic systems. Due to the poor tunability of metals, however, the ultimate control over surface plasmons remains a challenge. Here we propose a new type of graphene plasmonic (GP) metasurfaces by shaping the dielectrics underneath monolayer graphene into specific photonic crystals. The radial and axial gradient-index (GRIN) lenses are implemented to demonstrate the feasibility and versatility of the proposal. It is found that the designed GP-GRIN lenses work perfectly well for focusing, collimating, and guiding the GP waves. Especially, they exhibit excellent performances in the THz regime as diverse as ultra-small focusing spot (λ0/60) and broadband electrical tunability. The proposed method offers potential opportunities in exploiting active transformational plasmonic elements operating at THz frequencies.

Базовая сферическая линза с исправленной сферической аберрацией путем применения градиентного материала с осевым линейным распределением показателя преломления

Базовая сферическая линза с исправленной сферической аберрацией путем применения градиентного материала с осевым линейным распределением показателя преломления

Design and tolerance analysis of an axial gradient-index singlet broadband laser collimator

A laser collimator is a narrow-field imaging system that is designed for near-diffraction-limited performance when operating at a single wavelength. Although a traditional design might consist of a homogeneous doublet, the design of an axial gradient-index (GRIN) singlet for such an application is described. The analysis begins with third-order aberration thin lens optical design principles in order to examine the effects of spherochromatism from the GRIN. The final design is optimized in CODE V® using GRIN aluminum oxynitride (ALON®), which allows for diffraction-limited operation from the visible through 5 μm. A tolerance analysis is performed on the slope, nonlinearity, and tilt of the GRIN profile in order to assess sensitivity to fabrication errors.

Design and fabrication of a polymer gradient-index optical element for a high-performance eyepiece

A 40-deg full field-of-view high-performance eyepiece design utilizing a polymer spherical gradient-index (GRIN) optical element is presented. In the design process, the GRIN lens material is constrained to current manufacturing capabilities. Several spherical GRIN lens blanks are fabricated from a thermoformable axial GRIN polymethyl methacrylate polystyrene copolymer material. One is diamond turned into a lens for the eyepiece, and the additional blanks are used to characterize the fabrication process. The spherical GRIN profile is evaluated in the original design, and a tolerance analysis is provided.

Quasi-Axial GRIN Lens Implemented in Wedge-Shaped Fiber Coupling With InP-Based PLC

A 3-D equivalent rectangle approximation staircase concatenation method to analyze the optical mode evolution in a wedge-shaped fiber (WSF) is proposed in this paper. A scheme with an axial gradient-index (AGRIN) lens between the WSF and InP-based planar lightwave circuit (PLC) chip, yielding higher transmission efficiency than those with two cascaded lenses and one radial GRIN lenses, by 2.282 and 7.816 dB, respectively, is devised and experimentally demonstrated. The WSF-AGRIN-PLC-single-mode fiber coupling linkage supplies a preferable solution for solving the problem of index mismatching in fiber-to-InP-based PLC assembly to achieve a high-efficiency connection.

Refractive power of a multilayer rotationally symmetric model of the human cornea and tear film

Optical models of the human cornea and tear film typically employ a single homogeneous cornea with an average refractive index. I propose to use a more realistic multilayer model based on morphological data from the literature. The mathematical methodology to derive the refractive power equation of this model is presented. Special attention is given to the axial gradient index of the refraction structure of the stroma layer because of its optical implications. The importance of considering this multilayer model is quantified in a specific example (orthokeratology) with the help of the derived power equation.

Fundamentals of macro axial gradient index optical design and engineering

Homogeneous lens material is characterized by an index of refraction and a point on the glass map nd5f(y d). Gradient refractive index (GRIN) lenses have a spatially varying index and dispersion and are represented by a line on the glass map. GRIN lenses open the door to a wide variety of optical design applications incorporating entire lenses of axial gradient refractive material (macro-AGRIN). Axial gradient mate- rial essentially gives biaspheric behavior to lenses with spherical sur- faces and exhibits a controlled gradient in both index and dispersion. Thus, the applications for this material range from simple singlet lens used for imaging laser light, in which spherical aberration is eliminated, to complex multielement lens systems, where improved overall perfor- mance is desired. The fusion/diffusion process that produces this mate- rial is surprisingly simple, repeatable, and applicable to mass production. The advantages of AGRIN technology coupled with the recent advances in material development and its accessibility in commercially available lens design programs provides optical designers with the opportunity to push the performance of optical systems farther than with conventional optics. © 1997 Society of Photo-Optical Instrumentation Engineers. (S0091-3286(97)00406-6) Subject terms: optical materials; optical systems; gradient index; axial gradient refractive index; gradient refractive index; radial gradient refractive index; macro axial gradient, radial gradient; optical design; glass lines; unique optical systems; high performance; asphere.

The Nature of the Paraxial Approximation II. Systems Containing Media of Radial and of Axial Non-uniform Refractive Index: Invariants

Abstract The methods described in Part I are applied to systems containing media of non-uniform refractive index. For both radial and axial gradients, the parametric equations X = X(Z) and Y = Y(Z), and the values of the direction cosines of the ray-tangent at Z, of any ray are represented by power series in the initial object plane and entrance pupil variables (σ, τ, x y), where the coefficients are functions of the axial coordinate Z. The coefficients of the linear terms in (σ, τ; x y) are denoted by X 1(Z) and Y 1(Z), and paraxial refraction and transfer equations for the rays then appear as algebraic relations between the individual coefficients of the variables (σ, τ; x y) in X 1(Z) and Y 1(Z), and in the linear parts of the direction cosines of the ray-tangent at Z. Explicit forms are given, for both focusing and defocusing, radial and axial gradient-index media.

Third-order aberration theory for weak gradient-index lenses

A set of simple analytic expressions has been developed for calculating the third-order aberration coefficients in weak or nearly thin axial and radial gradient-index lenses.

Gradient-index silver alumina phosphate glasses by exchange of Na^+ for Ag^+

High index of refraction crown glasses containing silver oxide have been cast from a melt and used for ion exchange experiments. High quality axial and radial gradient-index profiles were fabricated and the interdiffusion coefficient calculated. Index changes up to 0.08 were achieved in a base glass with n(d) = 1.61 and V(d) = 57.

Design and tolerancing of aspherical and gradient-index germanium singlets of equal performance

Two different aspherical singlets, four types of axial gradient-index singlet, and one radial gradient-index singlet achieving the same performance are toleranced for thickness, curvatures, base index, and special coefficient (aspheric or gradient) sensitivities. The results show that the singlet with the most relaxed tolerances has a linear axial gradient which extends through the lens, followed in order by a singlet with a parabolic radial gradient, a singlet with an aspheric surface on the front, and a singlet with a parabolic axial gradient crossing the first surface.

Design, fabrication, and testing of a gradient-index binocular objective

A two-element axial gradient-index binocular objective has been designed and fabricated by the ion exchange technique. A prototype system has been assembled and tested.

Gradient-index binocular objective design

A two-element binocular objective with one axial gradient-index element was designed and compared with a three-element homogeneous objective. The index was constrained so that the gradient could be fabricated.

Plastic axial gradient-index lens

An axial gradient-index (GRIN) lens, where the refractive index varies along the optical axis of a convex lens, can reduce the spherical aberration compared with the conventional homogeneous lens. The transparent plastic axial GRIN material was successfully fabricated by chemical copolymerization and diffusion techniques. The depth of the gradient index was ~20 mm with the refractive-index change Δn = 0.07. By carrying out the ray tracing of the axial gradient index, the optimum refractive-index distribution reducing the spherical aberration was determined. If we assume that the axial GRIN lens was constructed from the sample experimentally obtained, it was theoretically confirmed that the spherical aberration of this lens would be remarkably improved.