Fraunhofer Diffraction by Koch Fractals: The Dimensionality

Abstract

Some properties concerning the fractal dimension of generalized Koch fractals and their Fraunhofer diffraction patterns are investigated as a continuation of the previous paper by Uozumi et al. The methods are discussed for evaluating the fractal dimension of object fractals from the intensity distribution of their diffraction patterns. Experimental results are shown to demonstrate some properties in this context. The fractal dimension of fractal areas in the Fraunhofer diffraction patterns is also considered.     

Fraunhofer Diffraction from Apertures Bounded by Regular Fractals

Abstract

Some properties of fields diffracted in the Fraunhofer region by apertures bounded by regular fractals are investigated. A recursion relation describing such apertures is introduced and the associated relation in the Fourier transform domain is described. For a triadic Koch aperture whose edge has the fractal dimension of D_s = 1·262, the recursion relation is numerically evaluated. Self-similar structures of intensity distributions in the Fraunhofer region are verified for the present objects. The relationship of the fractal dimension D _s of the fractal edge with the power-law decay of the Fraunhofer diffraction intensities is also verified.     

Fraunhofer Diffraction from Sector Stars

The wavefunction of the Fraunhofer diffraction from sector stars (i.e. radial gratings, spoke targets, Jewell's tests, Siemens' stars) is derived. Particular features of the diffraction patterns are explained by means of both the derived mathematical formulae and the general properties of Fraunhofer diffraction patterns. The graphs and patterns of the sector stars with the symmetry n = 1, 2, 3, 4, 6 and 72 are given.     

Particle Size Distributions from Fraunhofer Diffraction

Generalized concepts of information theory indicate that the numerical solution of the Fredholm equation of the first kind relating a particlesize distribution to its Fraunhofer diffraction pattern can be achieved without ill-conditioning by use of a truncated eigenfunction expansion. We give explicit analytic formulae for the eigenfunctions and eigenvalues of this problem and give numerical values of the latter of use for experimental reductions. The results show how many independent fractions of a particle-size distribution may be determined in the presence of quantified levels of noise.     

Fraunhofer Diffraction Patterns of Titled Planar Objects

Attention is drawn to the changes in the Fraunhofer diffraction patterns encountered when the planar objects become titled with respect to the optical axis. The differences between the far-field images observed in the well-known untilted case and the one investigated are shown and discussed using examples of familiar objects, such as square or rectangular apertures and linear diffraction gratings.     

基于夫琅禾费衍射的刀口直线度检测

现行的刀口直线度测量方法中,由于人眼分辨率和标准光隙带来的不确定度较大,故设计以基于夫琅禾费衍射原理的刀口直线度检测装置以提高检测精度.文章针对目前测量的现状,不仅设计了衍射检测装置并且进行了能力验证分析.     

Fraunhofer Diffraction in Partially Space Coherent Light by Slit Aperture with Amplitude Filters

Fraunhofer diffraction patterns produced by a narrow rectangular aperture having a non-uniform illumination with partially coherent light are presented. Graphs of intensity distributions in the diffraction patterns for three cases of amplitude filters and four forms of the mutual coherence at the slit are given. An increase in the half-width at half-height (H.W.H.H.) and a decrease in the intensity at the centre are observed. Tolerances for the value of the correlation interval in the aperture based on the Strehl intensity criterion have been calculated for various amplitude filters.     

Quasi-geometrical Paraxial Approach to Fraunhofer Diffraction by Thick Planar Objects Using Phase-space Signal Representations

Abstract

Quasi-geometrical techniques for calculating the Fraunhofer diffraction produced by some thick objects are related to phase-space signal representations, such as the Wigner distribution function. In this way, the properties derived from these functions can be applied directly to obtain an adequate display of the corresponding diffraction patterns.     

Fraunhofer Diffraction Effects on Total Power for a Planckian Source

An algorithm for computing diffraction effects on total power in the case of Fraunhofer diffraction by a circular lens or aperture is derived. The result for Fraunhofer diffraction of monochromatic radiation is well known, and this work reports the result for radiation from a Planckian source. The result obtained is valid at all temperatures.     

Fraunhofer Diffraction at Apertures in the Form of Regular Polygons. II

In this second contribution the Fraunhofer diffraction patterns from apertures of the form of an equilateral triangle, a square, a regular pentagon, a hexagon and an octagon are compared with the calculated maps of the intensity distribution. For each shape of the aperture the first few maxima and minima of the intensity are tabulated and the formulae for the wave function are given characterizing the diffraction in the directions of the slowest and the steepest decrease of intensity.     

The Structure of Fraunhofer Diffraction Patterns of Apertures with N-fold Rotational Symmetry

The moment theorem is used to show that the innermost part of the Fraunhofer diffraction pattern of any real aperture with higher than two-fold rotational symmetry is rotationally invariant. Then a formalism is presented in which aperture transmission-functions are represented by series of Zernike circle polynomials and diffracted field-amplitudes by series of Bessel functions, from which it is easily shown that the diffraction patterns of such apertures consist of regions, contained between well-defined values of the radius, whose rotational symmetries are integral multiples of that of the aperture. The central region, extending from = 0 to , N ( measures the diffraction angle, and N is the degree of rotational symmetry of the aperture) is rotationally invariant, and successive circumjacent regions have progressively higher rotational symmetries. The diffraction patterns of sectoral apertures and of rings of pinholes are derived and shown to exemplify these general conclusions. Finally it is shown how the diffraction patterns of some apertures (‘chiral apertures’) with rotational symmetries but no mirror symmetry can be deduced from the diffraction pattern of a related aperture with mirror symmetries, to which a chiral perturbation is applied.     

Fraunhofer Diffraction at Apertures in the Form of Regular Polygons. I.

In this first contribution the symmetry properties of the Fraunhofer diffraction phenomena at the apertures possessing a symmetry are discussed. These properties are used for the derivation of the wave function describing Fraunhofer diffraction at apertures of the form of regular polygons. Besides the general formula expressions are derived giving the wave function in the directions where the diffraction patterns show some particular properties.     

光散射粒度测量中采用Fraunhofer衍射理论或Mie理论的讨论

从理论和实验两方面出发,讨论了颗粒的折射率对光能分布及粒径分布反演结果的影响,同时给出了衍射理论的完整适用条件及基于衍射理论的粒度仪的最佳光靶尺寸。     

Fraunhofer diffraction by a random screen

The stochastic approach is applied to the problem of Fraunhofer diffraction by a random screen. The diffraction pattern is expressed through the random chord distribution. Two cases are considered: the sparse ensemble, where the interference between different obstacles can be neglected, and the densely packed ensemble, where this interference is to be taken into account. The solution is found for the general case and the analytical formulas are obtained for the Switzer model of a random screen, i.e., for the case of Markov statistics.     

Fractals

An elementary introduction to the concept of fractals is given. Some examples of fractals drawn from nature are briefly discussed. It is suggested that fractal geometry may be useful in characterizing the grain size and shape distributions in polycrystalline solids.     

Traitement Optique des Figures de Diffraction de Fraunhofer Pour Une Analyse Avec Une Ligne de Microphotodiodes

On analyse avec une ligne de microphotodiodes la figure de diffraction de Fraunhofer d'un objet opaque pour en déduire le diamètre et la position de celui-ci. On procède à un traitement optique des figures. L'agrandissement de limage, l'éclairage en onde sphérique et le filtrage du fond cohérent sont utilisés dans le même montage. Le contraste et la fréquence spatiale maximale des franges ainsi modifiés deviennent compatibles avec la FTM du détecteur.     

On the Fraunhofer (Far Field) Diffraction Patterns of Opaque and Transparent Objects with Coherent Background

The diffracting objects considered are opaque and semi-transparent objects of known cross section which are situated in an otherwise transparent area and illuminated with a collimated quasi-monochromatic beam of light. The plane in which the resultant diffraction pattern is considered is at a sufficient distance from the object for the approximation of Fraunhofer (far field) diffraction to be made whilst still remaining in the near field of the surrounding transparent area. The resultant intensity distribution is investigated both theoretically and experimentally for objects which present both circular and rectangular cross sections to the beam.     

On the Phase Distributions in Images and Fraunhofer Diffraction Patterns Produced by a Thin,Non-aberrant Lens Illuminated by a Monochromatic Point Source of Light

The paraxial approximation is used to show that the phase distribution in the image plane for an object which itself introduces no phase distortions is that which would be obtained from a monochromatic point source placed in the Fraunhofer diffraction plane and vice versa. The idea of image formation as amplitude modulation of a carrier is suggested.     

Fraunhofer diffraction by cantor fractals with variable lacunarity

Abstract

Optical diffraction by fractal openings is increasingly being studied because it allows the properties and parameters that characterize these objects to be determined. Allain and Cloitre published the first results showing that the resulting analysis of the distribution of intensity obtained from diffraction experiments through fractal deterministic pupils permits the self-similar dimension and other geometrical characteristic of these structures to be obtained directly. In this work the lacunarity effect ?, dimension d and order k of growth of the Cantor fractal about the distribution of intensities of the diffraction pattern are studied, solved analytically and characterized. In particular we note the influence of lacunarity because this is one of the first studies in which this geometric fractal parameter is taken into consideration. The selfsimilarity of the diffraction figure at different orders is also proved. The results of this study allow us to say that an intimate relation exists between the distribution of the diffracted waves and the parameters that describe this kind of fractal geometry.     

Diffraction by dual-period gratings

The dynamical characteristics of dual-period perfectly conducting gratings are explored. Gratings with several grooves (reflection) or slits (transmission) within each period are considered. A scalar approach is proposed to derive the general characteristics of the diffracted response. It was found that compound gratings can be designed to cancel as well as to intensify a given diffraction order. These preliminary estimations for finite gratings are validated by numerical examples for infinitely periodic reflection and transmission gratings with finite thickness, performed using an extension of the rigorous modal method to compound gratings, for both polarization cases.