Gradient-index axicons with spherical symmetry

Abstract

This paper presents the spherically symmetric gradient-index axicon, a new kind of axicon that can be designed, using geometrical optics and conservation of energy in the geometric limit, to produce a variety of different intensity patterns along the axis. The performance of one particular axicon design is analysed using the scalar wave model.     

General method to design spherically symmetric GRIN axicons

Abstract

A general method is presented for designing spherically symmetric gradient-index axicons, which produce a variety of different intensity patterns along the axis, and with a boundary index larger than or equal to the index of the surrounding medium. As the design method is based on geometrical optics, tools to analyse the axicon using wave optics are given.     

Simple lens axicon

We present the design of a cemented doublet-lens axicon made from spherical surfaces only. Compared with diffractive axicons, refractive cone axicons, and earlier lens axicons with aspheric surfaces, this element is inexpensive and easy to manufacture even with large apertures. The lens axicon is based on the deliberate use of the spherical aberration of the surfaces. The design principles of the element and its characterization, numerically and experimentally, are presented in detail. Although performance was traded for simplicity and robustness, the results show that the lens axicon has the main axicon properties: a narrow, extended line focus of relatively constant width.     

Freeform axicon with azimuthal variation

An analytical closed-form equation to design a freeform element that replicates an axicon is presented. In this paper, we get the mentioned analytical closed-form equation step by step. The equation comes from applying both the Fermat principle and the Snell's law at the first refractive surface. This Freeform axicon with azimuthal variation can be applied to produce annular conical beams with the possibility to receive wavefronts not necessarily collimated. In the paper, we test the equation in several scenarios to test the design of the Freeform axicon and the results of the performance of the Freeform axicon were as expected.     

Spatial transformation of Laguerre–Gaussian laser modes

Abstract

A method is described to spatially transform the annular profile of an arbitrary high-order Laguerre-Gaussian (LG) laser mode into an ultranarrow annulus using a combination of an axicon and a lens. The method is shown to conserve the azimuthal phase variation of the illuminating LG mode. The thin annular (hollow) light beam generated possesses orbital angular momentum and is suitable for experimental studies with cold atoms.     

Gaussian beam conversion using an axicon

Abstract

The paper presents a theoretical investigation of laser beam tailoring using a transparent diffracting phase plate. It is shown that a linear axicon is able to transform a Gaussian beam either into a super-Gaussian profile, a ring-shaped profile or a doughnut profile. All these different intensity distributions are easily obtained by only changing the width of the incident Gaussian beam.     

Rotation of microparticles with Bessel beams generated by diffractive elements

Abstract

We show that imaging a non-diverging Bessel beam by a spherical lens leads to the generation of a diverging Bessel beam. Expressions for the projections of the Umov-Poynting vector for a two-dimensional TE-polarized Bessel beam and a three-dimensional paraxial linearly polarized Bessel beam are derived. A fifth-order Bessel beam is produced using a single optical element-a 16-level phase-only diffractive helical axicon fabricated using electron beam lithography. This beam was successfully used to trap and rotate 5-10 μm diameter yeast particles and polystyrene beads of diameter 5 μm.     

Non-diffractive Vector Bessel Beams

Abstract

The non-diffractive vector Bessel beams of an arbitrary order are examined as both the solution to the vector Helmholtz wave equation and the superposition of vector components of the angular spectrum. The transverse and longitudinal intensity components of the vector Bessel beams are analysed for the radial, azimuthal, circular and linear polarizations. The radially and azimuthally polarized beams are assumed to be formed by the axicon polarizers used with the initially unpolarized or linearly polarized light. Conditions in which the linearly polarized Bessel beams can be approximated by the scalar solutions to the wave equation are also discussed.     

Implementation of ordinary and extraordinary beams interference by application of diffractive optical elements

Abstract

We apply diffractive optical elements in problems of transformation of Bessel beams in a birefringent crystal. Using plane waves expansion we show a significant interference between the ordinary and extraordinary beams due to the energy transfer in the orthogonal transverse components in the nonparaxial mode. A comparative analysis of the merits and lack of diffractive and refractive axicons in problems of formation non-paraxial Bessel beams has shown the preferability of diffractive optics application in crystal optics. The transformation of uniformly polarised Bessel beams in the crystal of Iceland spar in the nonparaxial mode by application of a diffractive axicon is investigated numerically and experimentally.     

A modified collocation method and a penalty formulation for enforcing the essential boundary conditions in the element free Galerkin method

The Element free Galerkin method, which is based on the Moving Least Squares approximation, requires only nodal data and no element connectivity, and therefore is more flexible than the conventional finite element method. Direct imposition of essential boundary conditions for the element free Galerkin (EFG) method is always difficult because the shape functions from the Moving Least Squares approximation do not have the delta function property. In the prior literature, a direct collocation of the fictitious nodal values & u circ; used as undetermined coefficients in the MLS approximation, u ^h (x) [u ^h (x)=Φ·& u circ;], was used to enforce the essential boundary conditions. A modified collocation method using the actual nodal values of the trial function u ^h (x) is presented here, to enforce the essential boundary conditions. This modified collocation method is more consistent with the variational basis of the EFG method. Alternatively, a penalty formulation for easily imposing the essential boundary conditions in the EFG method with the MLS approximation is also presented. The present penalty formulation yields a symmetric positive definite system stiffness matrix. Numerical examples show that the present penalty method does not exhibit any volumetric locking and retains high rates of convergence for both displacements and strain energy. The penalty method is easy to implement as compared to the Lagrange multiplier method, which increases the number of degrees of freedom and yields a non-positive definite system matrix.     

Self-referenced interferometry for the characterization of axicon lens quality

A simple interferometer for the characterization of axicon lenses is presented. The phase cone acquired by a wave propagating through an axicon, when interfered with a collinear reference wave, produces a nearly cylindrically symmetric self-referenced interference pattern from which the distortions of the axicon surface may be readily obtained. Comparison with two-dimensional off-axis interferometry is used to validate the self-referenced technique. The measurements are based on retrieval of the accrued spatial phase distribution from interference fringes with on- and off-axis reference beams and are found to be equivalent. We use the ellipticity of the phase maps to qualify axicon lenses, which are expected to exhibit radial symmetry and engage the self-referential capability of the on-axis method to derive deviation maps that characterize the surface quality of the axicons.     

Strain based triangular finite element for plate bending analysis

ABSTRACT

In this work, the formulation of a new triangular finite element is presented for static and free vibration of plate bending. The developed element which contains the three essential external degrees of freedom at each of the three corner nodes is based on the Reissner/Mindlin theory and the strain-based approach. This element is based on the linear variation of the three bending strains and constant transverse shear strains. The present element performances are evaluated through several tests related to moderated thick and thin plates with various shapes where it is found to be numerically more efficient than the bilinear element.     

Uniformization of the axial intensity of diffraction axicons by polychromatic illumination

The axial intensity of axicons illuminated by a coherent wave usually exhibits rapid oscillations from diffraction on the sharp edges of the aperture of the element. These oscillations can be suppressed when the diffractive version of the axicon is illuminated from a polychromatic source. This possibility is examined based on the example of the annular-aperture logarithmic axicon. The estimate for the wavelength interval of the illuminating source required for uniformization is obtained with the help of the stationary-phase method. Furthermore the shape of the radial intensity distribution can be maintained almost unchanged. These findings are confirmed by numerical evaluation of the Fresnel diffraction integral.     

Stresses and phase transformations occurring in quenching of carburized steel gear wheel

Abstract

A method for analysing temperature, metallic structures, stress and strain, and carbon content during the carburized quenching of a steel gear is presented, and the results of calculations using the finite element method are compared with experimentally measured data. Since carburization precedes quenching, a diffusion equation is solved in order to determine the carbon content. Calculated profiles of temperature, stresses, and martensite and pearlite volume fractions during quenching are presented, and the effects of coupling between them are also considered.

MST/19     

Interferometric Measurement of Holographic Lenslets

Abstract

A holographic optical element can either be characterized by the wave aberrations or by the point spread function or alternatively by the modulation transfer function. Here we propose the measurement of the wave aberrations with the help of a Twyman-Green interferometer adapted to the requirements for testing holographic optical elements. The evaluation of the interferograms is done with the phase-shifting technique. The resulting wave aberrations are expressed by Zernike coefficients. In addition to this evaluation, the point spread function and the modulation transfer function are calculated from the wave aberrations. The set-up, the evaluation method and also exemplary results of a tested holographic optical element are presented.     

TORSIONAL STRESS ANALYSIS OF CORD-RUBBER COMPOSITE LAMINATES

SUMMARY

The structural response of two-ply cord-rubber composite laminates subjected to torsional loading is investigated using non-linear three-dimensional finite element analysis. The present model treats a two-ply composite as an equivalent three/six-ply system with rubber and cord-rubber elements. The present finite element model is validated by comparing the results obtained to the existing experimental and analytical solutions. The effects of torque and bias angle on twist angle, axial displacement, coupled strain ratio, and interply and interface shear stresses are presented. Also, the effects of interply thickness and rubber elasticity were studied. The presented results illustrate the torsional behaviour of two-ply cord-rubber composites from three-dimensional analysis.     

Systematic design of an anastigmatic lens axicon

We present an analytical method for systematic optical design of a double-pass axicon that shows almost no astigmatism in oblique illumination compared to a conventional linear axicon. The anastigmatic axicon is a singlet lens with nearly concentric spherical surfaces applied in double pass, making it possible to form a long narrow focal line of uniform width. The front and the back surfaces have reflective coatings in the central and annular zones, respectively, to provide the double pass. Our design method finds the radii of curvatures and axial thickness of the lens for a given angle between the exiting rays and the optical axis. It also finds the optimal position of the reflecting zones for minimal vignetting. This method is based on ray tracing of the real rays at the marginal heights of the aperture and therefore is superior to any paraxial method. We illustrate the efficiency of the method by designing a test axicon with optical parameters used for a prototype axicon, which was manufactured and experimentally tested. We compare the optical characteristics of our test axicon with those of the experimental prototype.     

Spherically symmetric GRIN amplitude formers

Abstract

A new type of spherically symmetric gradient index element is presented. These elements are able to transform an optical field from a point source with a predetermined radiation pattern into an arbitrary amplitude distribution in the target plane. The design method is illustrated with four examples of amplitude formers that produce irradiance distributions that are constant, Gaussian, triangular and ring-shaped.     

Determination of Constitutive Model Parameters and 3-D Finite Element Formulation for Powder Compaction

ABSTRACT

The knowledge of stress-strain distribution of particulate materials during compression is crucial to the powder processing industries. The finite element technology holds the potential to accurately describe the powder's stress-strain (pressure-density) response during compression. At present, most of the FEMs are two-dimensional or axisymmetric. which can not precisely model the compaction process. In this project, a 3-D finite element formulation for powder compression is presented. The material parameters (for an elastoplastic model and an elasto-viscoplastic model) for three selected materials have been measured. The flexible boundary cubical triaxial tester was used to determine the constitutive model parameters. The constitutive models were verified using data from cubical triaxial tests.