Parametric characterization of rotationally symmetric hard-edged diffracted beams

The truncated second-order moments and generalized M2 factor (M(G)2 factor) of two-dimensional beams in the Cartesian coordinate system are extended to the case of three-dimensional rotationally symmetric hard-edged diffracted beams in the cylindrical coordinate system. It is shown that the propagation equations of truncated second-order moments and the M(G)2 factor take forms similar to those for the nontruncated case. The closed-form expression for the M(G)2 factor of rotationally symmetric hard-edged diffracted flattened Gaussian beams is derived that depends on the truncation parameter beta and beam order N. For N --> infinity, the M(G)2 factor equals 4/square root of 3 corresponding to the value of truncated plane waves, which guarantees consistency of the formalism.     

Approximate method for the generalized M2 factor of rotationally symmetric hard-edged diffracted flattened Gaussian beams

On the basis of the truncated second-order moments method in the cylindrical coordinate systems and the expansion of the hard-edged aperture function into a finite sum of complex Gaussian functions, an approximate method used to calculate the generalized beam propagation factor (M2 factor) is proposed. The approximate analytical expressions of the generalized M2 factor for rotationally symmetric hard-edged diffracted flattened Gaussian beams defined by Gori [Opt. Commun. 107, 335 (1994)] and Li [Opt. Lett. 27, 1007 (2002)] are derived, respectively; we show that it depends on the beam order N and the beam truncation parameter delta. Some typical numerical examples are given to illustrate its applications that we compare by using the obtained analytical method and the numerical integration method.     

Laser direct writing of rotationally symmetric high-resolution structures

We present a laser direct writing system for the efficient fabrication of high-resolution axicon structures. The setup makes use of scanning beam interference lithography incorporated with a fringe locking scheme for tight fringe phase control and allows us to fabricate large area structures with a period down to 450 nm.     

Point-spread function for a rotationally symmetric birefringent lens

Imaging properties of a birefringent lens, in which the fast (or the slow) axis is distributed in the radial direction whereas magnitude of birefringence varies as a quadratic function of the pupil radius, are investigated by calculating a point-spread function. It is found that the point image is analytically described by using the Lommel function as well as the zero-order Bessel function, and a localized intensity null surrounded by bright regions in all directions can be realized at a geometrical focus under certain conditions. The magnitude of birefringence that is tolerable in image formations is also discussed, assuming that the lens is applied to microlithography.     

Light scattering by arbitrarily oriented rotationally symmetric particles

We use the T-matrix approach and the analytical orientation-averaging technique to formulate the problem of light scattering by an ensemble of rotationally symmetric particles in arbitrary orientation. The mathematical formulation yields analytical expressions for the elements of the ensemble-averaged scattering matrix that involve no more than four nested summations. An expansion into generalized spherical functions is used in the particular case where the scatterers are partially aligned along the direction of incidence. A computer code that implements the analytical expressions derived is publicly available on the World Wide Web at http://irctr.et.tudelft.nl/ -Skaropoulos/T-matrix.htm.     

Sommerfeld effect in rotationally symmetric planar dynamical systems

Sommerfeld effect concerns the non-linear jump phenomena induced due to the influence of the unbalance response on a non-ideal drive around the critical speed of the excited structure. In this work, we study the influence of external and internal dampings and gyroscopic forces on the Sommerfeld effect in rotationally symmetric planar dynamical systems. The rotational symmetry assumption allows us to obtain neat analytical results for the steady state dynamics. We show that the rotating material or internal damping and the gyroscopic forces influence the spin rate of the non-ideal system and the former changes the system dynamics in an unexpected manner. In particular, we show that the stability threshold may restrict the jump phenomena due to the Sommerfeld effect for larger values of internal damping. Moreover, it is also shown that the Sommerfeld effect would cease to exist under certain conditions. A stability condition for various steady-state equilibriums (branches of steady-state solutions) is derived. A rotor dynamics problem and a structural dynamics problem where the systems interact with a non-ideal source are considered as illustrative examples. A few numerical results are given to validate the analytical solutions.     

Visual instrument image-quality assessment with rotationally symmetric difference of Gaussians

We performed contrast-sensitivity measurements using rotationally symmetric, variable contrast difference-of-Gaussians targets, viewed through specially designed telescopes that presented various amounts of monochromatic aberrations. The contrast-sensitivity ratios between aberrated and unaberrated cases were correlated with instrument-based measures of image quality. The radius that encircles 84% of the energy of the point-spread function was found to give the best correlation with the subjective data.     

Casting methods for the production of rotationally symmetric copper bimetals

ABSTRACT

Multiple-material products are characterised by a complex property profile which is achieved by combining the particular advantages of at least two different materials. Bimetal casting is an energy- and material-efficient technology for the production of multi-metallic objects. This paper describes the development of a semi-continuous casting process for the formation of a rotationally symmetric bimetal with a cohesive bonding character at the interface of a copper–tin alloy (CuSn6) and pure copper (Cu99.5). Initial experiments are conducted by static casting to evaluate the thermal process window. Based on the results of the initial experiments, a vertical semi-continuous compound casting process is developed. A stable cohesive bond between the joining partners is accomplished by forming a solid solution at the interface.

This paper is part of a Thematic Issue on Copper and its Alloys.     

Nonnull testing of rotationally symmetric aspheres: a systematic error assessment

An optical setup for the testing of rotationally symmetric aspheres without a null optic is proposed. The optical setup is able to transfer the strongly curved wave fronts that stem from the reflection of a spherical testing wave front at a rotationally symmetric asphere. By simulation it is proved that the algorithms of the Shack-Hartmann sensor that is used can cope with the steep wave-front slopes (~110lambda/mm) in the detection plane. The systematic errors of the testing configuration are analyzed and separated. For all types of error, functionals are derived whose significance is proved by simulation. The maximum residual errors in the simulations are fewer than lambda/500 (peak to valley).     

Diffraction analysis of rotationally symmetric optical systems using computer-generated aberration polynomials

A ray trace scheme for the automatic generation of optical aberration polynomials to arbitrary orders pioneered by T. B. Andersen is successfully applied to the diffraction analysis of rotationally symmetric optical systems on a desk-top computer. The diffraction-based optical transfer functions at various field positions are computed using the relatively new Winograd Fourier transform algorithm. The coding includes aspherical and reflective surfaces.     

Focusing and the optical transfer function in a rotationally symmetric optical system

Two criteria of focusing in a rotationally symmetric optical system, the focal-amplification ratio and the encircled-energy factor, are expressed by the integrals of the optical transfer function, thus providing a simple relation between these two types of characteristics. The approximation of the optical transfer function of diffractive lenses is made possible on the basis of this relation, as is an estimation of the effect of certain apodizers on the image quality.     

Focusing of axially symmetric flattened Gaussian beams

Abstract

We study the three-dimensional field distribution of a focused axially symmetric flattened Gaussian beam. In particular, exact closed-form expressions for the intensity along the optical axis and at the focal plane are provided, together with a comparison between our results and those pertinent to the case of a converging spherical wave diffracted by a hard-edge circular aperture. Some hints for future investigations are also given.     

Exact calculation of the circle of least confusion of a rotationally symmetric mirror

The position and dimensions of the circle of least confusion (CLC) on axis for a lensless Schmidt camera telescope operating at F(0.82) are calculated. The camera is to be used in the fluorescence detector of the Pierre Auger Observatory. Our analysis was developed for an aspherical mirror for any on-axis position of the point light source. Our technique uses the intersection of the marginal ray from one side of the aperture with the caustic produced by the intermediate rays from the opposite side of the aperture to locate the CLC.     

Wavefront aberrations of rotationally symmetric CGHs fabricated by a polar coordinate laser plotter

Computer-generated holograms (CGHs) were fabricated by a polar coordinate laser plotter. The wavefront aberrations of these CGHs caused by fabrication inaccuracies were measured interferometrically. We tested Fresnel zone plates as typical examples of CGHs with focal ratios of ?/10 and ?/5 with a Fizeau interferometer. The tests comprised both binary phase holograms and binary amplitude holograms. The wavefront aberrations of the CGHs had an rms variation of about λ /50. Two types of aberrations were identified and could be referred to certain writing errors. Their size was deduced from the interferograms.     

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.     

Computer simulation of schlieren images of rotationally symmetric plasma systems: a simple method

Schlieren techniques are commonly used methods for quantitative analysis of cylindrical or spherical index of refraction profiles. Many schlieren objects, however, are characterized by more complex geometries, so we have investigated the more general case of noncylindrical, rotationally symmetric distributions of index of refraction n(r,z). Assuming straight ray paths in the schlieren object we have calculated 2-D beam deviation profiles. It is shown that experimental schlieren images of the noncylindrical plasma generated by a plasma focus device can be simulated with these deviation profiles. The computer simulation allows a quantitative analysis of these schlieren images, which yields, for example, the plasma parameters, electron density, and electron density gradients.     

One-dimensional description of cylindrically symmetric laser beams: application to bessel-type nondiffracting beams

We introduce a new representation of coherent laser beams that are usually described in circular cylindrical coordinates. This representation is based on the decomposition of a laser beam of a given azimuthal order into beams exhibiting Cartesian symmetry. These beams, which we call constituent waves, diffract along only one of their transverse dimensions and propagate noncollinearly with the propagation axis. A cylindrically symmetric laser beam is then considered a coherent superposition of constituent waves and is represented by an integral over an angular variable. Such a representation allows for the introduction of the propagation factor M2, defined in terms of one-dimensional root-mean-square (rms) quantities, in the treatment of two-dimensional beams. The representation naturally leads to the definition of a new rms parameter that we call the quality factor Q. It is shown that the quality factor defines in quantitative terms the nondiffracting character of a laser beam. The representation is first applied to characterize Laguerre-Gauss beams in terms of these one-dimensional rms parameters. This analysis reveals an asymptotic link between Laguerre-Gauss beams and one-dimensional Hermite-Gauss beams in the limit of high azimuthal orders. The representation is also applied to Bessel-Gauss beams and demonstrates the geometrical and one-dimensional characters of the Bessel-Gauss beams that propagate in a nondiffracting regime. By using two separate rms parameters, Q and M2, our approach gives an alternative way to describe laser beam propagation that is especially well suited to characterize Bessel-type nondiffracting beams.