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.     

Moiré by fractal structures

Abstract

The theoretical and experimental results of the Moiré effect observed by superposing two grids containing fractal Cantor structures are presented in this paper. It is also analysed the equivalence between the information obtained by the Fraunhofer diffraction through those fractal grids and that obtained through Moiré. In a recently published paper [1] it was verified that the diffraction pattern is highly sensitive to variations in dimension, order of growth and lacunarity of the Cantor fractal, becoming a powerful tool to analyse and determine these parameters. So, in this paper it is intended to determine the effect that the order of growth, the dimension, and the lacunarity of one-dimensional fractal Cantor structures have over the Moiré patterns that result from superposing these structures over replicas of themselves which have been rotated through a small angle α. The main goal is to verify if there exists an intimate relation between the resulting Moiré pattern and the parameters that describe this fractal structure, as occurs in the case of the diffraction pattern.     

Two-dimensional penrose-tiled photonic quasicrystals: Diffraction of light and fractal density of modes

Abstract

We report measurements of the diffraction pattern of a two-dimensional photonic quasicrystal structure. Using a set of plane waves defined by the diffraction pattern we introduce a theoretical approach for the calculation of the band structure which captures the rotational and inflational properties of the quasicrystal. Based on this model we find that the density of modes of the quasicrystal displays a fractal character and a depleted region analogous to the band gap in a periodic system.     

Roughness of bainite

Abstract

A bainite sheaf does not have a simple geometry, making it difficult to characterise or calculate its fundamental properties, such as the total amount of interfacial area per unit volume. The sheaf is, in the language of fractals, a rough object in which the area is a function of how the measurements are made. Micrographs taken at a variety of resolutions have been analysed to reveal how the area scales with resolution. It is found that although the interface is rough, it is far less so when compared with what might be expected from a fractal object. In other words, the ideal fractal, where self-similarity propagates over an infinite range of observation does not apply to the bainite sheaf.     

Beam reflection from multilayers with periodic and fractal distributions

Abstract

An impedance model is used to solve the reflection problem from a multilayer with periodic and Cantor bar fractal distributions. It is based on the calculation of the input surface impedance of such a structure. The method consists of replacing the multilayer reflection problem by a simple reflection problem on a single surface. This approach has the additional advantage that only the fields above the surface should be taken into account. Numerical calculation leads to an iterative method which can be implemented for each polarization mode. The results for the input impedance function of the system and for the reflectance are presented for different iteration states. Also to quantify the comparison between both types of structures we introduce a self-similarity function which correlates orders or states of the same type.     

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.     

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.     

Spatial coherence wavelets

Abstract

It is shown that the cross-spectral density at a plane in the Fresnel—Fraunhofer domain can be expressed as a certain diffraction pattern, which is generated by the superposition of second-order spatial coherence wavelets that emerge from the aperture. The amplitude of each coherence wavelet exhibits units of power density (average energy) and the power spectrum at the far zone plane will be the summation of the amplitudes of such wavelets. Thus, the spatial coherence wavelet constitutes a vehicle for both correlation and energy transport in free space. Some simulation results are discussed to illustrate these ideas.     

Finite Aperture Fraunhofer Holograms of Two Co-planar Discs

Abstract

In this paper we present a theoretical model describing the real images replayed from finite aperture Fraunhofer holograms of two identical co-planar objects. We have solved numerically the resulting image equations for the case of two circular disc objects, and compare our predictions with experimental measurements from in-line Fraunhofer holograms recorded on silver-halide emulsions. Three measurement criteria for calculating the disc diameters and separation are described, and their errors discussed. It is found experimentally that a criterion based on average intensity results in the smallest errors due to its insensitivity to the effects of coherent noise.     

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.     

Self-similar and self-affine characteristics of microstructural images of HSLA steel

Abstract

This paper reports a procedure based on discrete Fourier transformation adopted for verifying statistical self-affinity and self-similarity in microstructural images of high strength low alloy steel. The procedure is first applied on synthetic images based on fractional Brownian motion to show the degree of self-similarity in them and finally, applied on microstructural images of three different magnifications. Effects of length scales on power spectral estimates of the microstructural images captured by scanning electron microscopy have been discussed. Importance of appropriate image resolutions and characteristic length for fractal based quantification of microstructural images is highlighted.     

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.     

A STUDY OF THE ACCURACY OF OPTICAL FRAUNHOFER DIFFRACTION SIZE ANALYZER

ABSTRACT

In this paper the discrepance of scattered light between Mie theory and Fraunhofer diffraction is showed at different refractive indexes and sizes of particle. The accuracy of the Malvern particle size analyzer is discussed when it is used to measure small particles and some advice is proposed for its use.     

Parametric homogeneity and non-classical self-similarity. I. Mathematical background

Summary The concept of parametric homogeneity, which includes parametric homogeneous (PH) and parametric-quasi-homogeneous (PQH) functions, PH-sets and corresponding transformations, is a kind of non-classical self-similarity. Introduction and study of the concept is important for understanding scaling properties of natural phenomena. PH-functions and PQH-functions are natural generalisations of concepts of homogeneous and quasi-homogeneous functions when the discrete group of coordinate dilations is considered instead of the continuous group.Some properties of PH- and PQH-functions and some of their applications are studied. Several ways of constructing PH-function of arbitrary degree are described. It is shown that PH-functions often have fractal graphs and can be nowhere differentiable. Such common for fractal geometry objects as the von Koch curve, the Weierstraß-Mandelbrot and Takagi-Hopsen functions, the Cantor staircase and set posses PH-properties. However, the fractal properties of an object and the property of parametric-homogeneity are independent of each other, and the considered scaling law (PH-law) is not a kind of fractal scaling. A nowhere differentiable self-similar function and some associated PH-functions are introduced as examples of non-fractal scaling.     

Generation of self-reproducing fractal patterns using a multiple imaging system with feedback

Abstract

We present a simple optical system for producing self-similar fractal patterns. The key component comprises three adjacent lenses, which form multiple images of a pattern displayed on a monitor. The images are recorded by a camera and displayed as the new pattern on the monitor. Iteration of this process results in an approximation to a self-similar fractal pattern which is independent of the starting image.     

Fraunhofer diffraction of laser beams with amplitude modulations and phase fluctuations by a lens with a central obscure aperture

Abstract

By using the statistical-optics model and generalized Huygens-Fresnel diffraction integral, Fraunhofer diffraction of high-power laser beams with amplitude modulations (AMs) and phase fluctuations (PFs) focused by a lens with a central obscure aperture has been studied. Detailed numerical calculation results have been given, showing the dependence of the intensity distribution at the geometrical focal position and the power (energy) focus-ability of high-power laser beams not only on the obscure ratio, but also on the truncation parameter, AMs and PFs of beams.     

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.     

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.     

Speckle Patterns Generated by Rough Surfaces with a Periodic Component

Abstract

A real diffraction grating is replaced by a coordinate-dependent surface impedance that is constructed as the sum of two contributions: a perfectly periodic contribution and a randomly varying contributon. Numerical results obtained with this model show the presence of diffuse light bands on the speckle pattern that constitutes the diffuse background of the spectrum. We emphasize the role played by the electromagnetic surface waves supported by this structure on the generation of the bands.