Fraunhofer and Fresnel field distributions from the LP01 mode in a micro-sized hollow optical fibre

Abstract

The electric field distribution of a LP₀₁ mode and properties of the evanescent-wave field in a micro-sized hollow optical fibre under the weakly guiding approximation are analysed. The far- and near-field distributions of the output beam from the LP₀₁ mode and its propagation characteristics in free space are calculated numerically from Fraunhofer and Fresnel diffraction theory. We also derive an analytical expression of the far-field distribution of the LP₀₁ mode and discuss its applicable conditions. Our study shows that the output beam of the LP₀₁ mode in the hollow fibre is a dark hollow divergent beam whose near-field divergent angle is slightly smaller than the far-field divergent angle. The dark spot size (DSS) of the beam in the near field is about equal to the beam radius r ₀, whereas the DSS in the far field is smaller than its beam radius. We analyse the dependences of the far-field divergent angle of the output beam from the LP₀₁ mode on various fibre parameters and briefly discuss potential applications of the dark hollow beam in atomic physics and atomic optics.     

Analytical and Numerical Approximations in Fresnel Diffraction

Abstract

Procedures for the fast and accurate numerical computation of Fresnel diffraction integrals are developed on the basis of geometrical properties of the Cornu spiral. The methods proposed allow the highly oscillatory integrals in Fresnel diffraction to be approximated by means of three simpler integrals and permit the calculation of these final integrals using analytical formulae.     

Generation and selection of laser beams represented by a superposition of two angular harmonics

Abstract

The properties of fields generated by diffractive phase-only optical elements that generate combinations of two angular harmonic fields with different harmonic indices in Fraunhofer and Fresnel regions are investigated theoretically and experimentally. Camomile shaped diffraction patterns are predicted and observed. It is shown that multi-order diffractive phase elements can be used to both generate these beams and to identify the weights of different angular harmonics in a given incident laser beam.     

Reproduction in free space of fields confined by 3-dimensional optical waveguides

Abstract

It is shown by using the Fresnel-Kirchhoff diffraction theory and the method of images that a scalar field confined by a 3-dimensional optical waveguide can be generated in free space by a suitable light source. In the method the boundaries of a waveguide are replaced by virtual sources. This allows one to present the wave guiding as the interference and diffraction of multiple light beams produced in free space by the guide equivalent source (Fresnel waveguide). Thus, the scalar optics of a 3-dimensional waveguide is presented as the free-space beam optics. The method is illustrated by construction of the Fresnel sources for the triangular, rectangular and hexagonal waveguides. The numerical examples demonstrate the diffraction-free and self-imaging propagation in the free-space of the eigenmodes of the Fresnel rectangular-waveguide.     

Fresnel Images of a Binary Diffraction Grating with Open Ratio 0·5

Abstract

Analysis of the Fresnel image of the binary-amplitude diffraction grating with open ratio 0·5 is presented. Fresnel images situated at the distance z = Nd²/8λ (N = 1,2…) from the grating which was illuminated by a plane wave were calculated and investigated by experiment. Forms of intensity distributions in the second grating plane are presented, the second grating cooperating with the self-imaging grating.     

Elliptical inclusion problem in antiplane piezoelectricity: Implications for fracture mechanics

An elliptical piezoelectric inclusion embedded in an infinite piezoelectric matrix is analyzed in the framework of linear piezoelectricity. Using the conformal mapping technique, a closed-form solution is obtained for the case of a far-field antiplane mechanical load and an inplane electrical load. The solution to a permeable elliptical hole problem is obtained as a limiting case of vanishing elastic modulus of the inclusion. This enables the study of the nature of crack tip electric field singularity which is shown to depend on the electrical boundary condition imposed on the crack faces. The energy release rate of a self-similarly expanding slender crack in the presence of electric fields is obtained by using the generalized M-integral. The energy release rate expression indicates that the electric field has a crack-arresting influence. This effect is inferred to have a more fundamental physical origin in the interaction between the applied electric field and the induced surface charges on the crack faces. An experimental result contradicting the theoretical prediction on the crack-arresting effect is also discussed.     

Fractional fourier transform as a tool for analysing the diffraction of misaligned optical systems

Abstract

It is shown that fractional Fourier transform could be viewed as a tool for analysing the diffraction of misaligned optical systems, just as the fractional Fourier transforms are adapted to the mathematical expression of Fresnel diffraction. Some special cases are also discussed.     

Computation considerations and fast algorithms for calculating the diffraction integral

Abstract

One of the most basic optical ‘components’ is free-space propagation. A common approximation used when calculating the resultant field distribution after propagation is the Fresnel integral. This integral can be evaluated in two ways: directly or by using the angular spectrum. In this paper, we estimate the regions in which each mode of evaluation is preferable according to computing efficiency and accuracy considerations. A fast numerical algorithm is introduced for each region. The result is relevant also for the evaluation of the Rayleigh-Sommerfeld diffraction formula.     

Numerical Analysis of Planar Fresnel Lenses

Abstract

The analysis and the comparison between different planar Fresnel lenses has been carried out numerically using the beam-propagation method. This technique allows one to evaluate relevant parameters, such as focal point, spot size, efficiency, paraxial and wide-angle diffraction effects, taking into account the diffraction produced by the refractive-index discontinuities. Comparisons with some of the published experimental results are presented.     

Pupil Effect in a GRIN Lens with Gaussian Illumination

Abstract

Using the propagator method, the quasi-geometric transmittance function and complex amplitude distribution produced by a GRIN lens are evaluated when it is illuminated by a Gaussian beam and the pupil effect is taken into account. The effective pupil is defined and derived by both quasi-geometrical optics and the exact Fresnel diffraction, which it is evaluated in a Rubinowicz representation.     

Fresnel diffraction as an aperture edge integral

Abstract

The standard Fresnel diffraction wave field from an aperture in a plane screen involves an integral over the aperture area. Here it is reduced to an aperture edge integral instead. The reduction really amounts to performing one integral but can be considered as the reduction of the ‘Maggi-Rubinovics’ edge formula from Kirchhoff optics to paraxial (i.e. Fresnel) optics.     

Achromatic White-light Self-imaging Phenomenon: An Approach Using the Wigner Distribution Function

Abstract

The propagation of polychromatic spatially coherent light through an optical system, developed for achromatizing Fresnel diffraction patterns, is discussed using the Wigner distribution function. This approach is applied for obtaining achromatic self-images. The residual chromatic aberration is analysed, and some experimental results are shown.     

Non-paraxial analytical solution for the generation of focal curves

Abstract

A solution for the transmittance of elements forming planar and nonplanar focal curves in the non-paraxial case is proposed in this paper. The treatment is based on the scalar diffraction theory and makes use of the stationary phase method in evaluation of the diffraction integral. The equation of a stationary point enables one to establish mapping relations between the geometrical regions of the element and the corresponding points of the focal curve. These regions turn out to be described by conical curves. The mapping relations can be manipulated by a phase function imposed onto the focal curve. The amplitude factors of the derived transmittances can be omitted and the desired intensity distribution along the focal curve can then be obtained by an appropriate change in the aperture of pure phase elements. The approach is illustrated by examples of elliptical, hyperbolic, conical and circular zone plates. The focus generations for both non-paraxial and paraxial cases of these elements are compared using numerical simulations.     

Focal shifts in focused beams with an elliptical diffracting screen

The approximate analytical formula of the focal shift has been derived for an elliptical diffracting screen that consists of a circular lens and an elliptical aperture. It is found that the focal shift of the beam focused by this elliptical diffraction screen depends not only on the product of the Fresnel number of the focusing geometry and the standard deviation of a mapped function, but also on the ellipticity (the ratio between the minor and the major axes) of the elliptical aperture. The focal shift increases as the decrease in the ellipticity of elliptical aperture. By using an approximate analytical formula and the diffraction integral formula, some numerical simulation comparisons are done. The presented analyses show that the actual analytical expression for the focal shift of a rotationally asymmetric diffraction screen cannot generally be obtained by using the azimuthal average of the screen amplitude transmittance.     

Partial reconstruction of the motional Wigner function of an ensemble of helium atoms

Abstract

We report on double-slit interference of the matter waves of metastable helium atoms in the intermediate regime between Fresnel and Fraunhofer diffraction. From these diffraction data, we partially reconstruct the Wigner function of the atomic motional state immediately behind the double slit by the inverse Radon transformation technique. We were able to observe negative parts of the Wigner function, making evident the quantum nature of the atomic centre-of-mass motion.     

Fresnel Images of a Binary Diffraction Grating with Open Ratio Less than 0·5

Abstract

An analysis of Fresnel images of the amplitude binary diffraction grating with open ratio α/d < 0·5 is presented. Particular attention is paid to images appearing behind the self-imaging grating at distances z = 0 to d ²/λ. An attempt is made at a systematic examination of Fresnel images that have the form of binary gratings with ratio α′/d′ different from that of the original grating.     

Focusing Properties of the Gaussian Beam Generated by Optical Resonators with Gaussian Reflectivity Mirrors

Abstract

This paper presents a parametric study concerning the focusing properties of the light beam generated by optical resonators with Gaussian reflectivity mirrors. The diffraction integral has been evaluated and analytical expressions are derived for intensity patterns near the focus in systems of different Fresnel numbers. Universal curves have been plotted against dimensionless parameters to show the variations in the intensity patterns due to the change in focusing geometry and the incident beam intensity profile. Modification to the well known diffraction formula are considered in order to extend the conventional Gaussian formalism to the case of Gaussian mirror resonators.     

Jones matrix treatment for polarization fourier optics

Abstract

In this work we introduce the use of a Jones matrix method to evaluate the far-field diffraction produced by spatially variant polarization elements. We extend the scalar Fourier optics theory to a vectorial theory by the use of the Jones matrix formalism. With this method it is possible to analyse the diffraction pattern and the local state of polarization in the Fraunhofer approximation by means of the usual Jones matrix calculus.     

Digital Double-pulsed Holographic Interferometry for Vibration Analysis

Abstract

In this paper we describe a digital double-pulsed holographic system. Two separate holograms of an object under test are recorded within a few microseconds using a charge-coupled device camera and stored in a frame grabber. The holograms are digitally reconstructed using a computer, by simulation of the Fresnel diffraction of the hologram illuminated by the reference wave. The phases of the two reconstructed wave fields are calculated from the complex amplitude and the deformation is obtained from the phase difference. Experimental results are presented.     

Real Time Pseudocolour Encoding of Depth Using White Light Processing

Abstract

An interferometer, consisting of a grating and a double exposed speckle pattern, used to perform pseudocolour encoding of depth in real time under white light illumination, is described. A theory of operation based on Fresnel diffraction is also given.