Koch fractals in physical optics and their Fraunhofer diffraction patterns

This paper gives an overview of the utilization of fractals in physical optics, especially of Koch fractals and their diffractals. The term fractal itself is defined and some basic characteristics of fractals are mentioned. Constructions of the most typical Koch curves are also depicted. Laser diffraction experiments using regular, random and modified Koch curves are described and the corresponding diffraction patterns (intensity distributions of diffractals) are shown. Some interesting properties of these diffraction patterns are discussed.     

Texture analysis of speckles due to random Koch fractals by lacunarity

We analysed the textures of speckles produced from random Koch fractals to quantitatively confirm the existence of speckle clustering which we investigated in a previous paper. The texture analysis involved using the lacunarity of these speckles, which is defined as fluctuations of the intensity integrated within a box, and comparing this quantity with that for ordinary fully-developed speckles. The lacunarity of clustered speckles was found to be greater than that of ordinary speckles throughout the whole range of the size of the box under examination, indicating the existence of long correlation tails in speckles. We also discuss the physical origin of these long correlation tails.     

Measurement of surface roughness properties using speckle patterns with non-gaussian statistics

The surface roughness properties (i.e. the rms surface roughness and the correlation length) of strong diffuse objects are investigated by using the speckle patterns which obeys the non-gaussian statistics. The intimate linear relation is found to exist between the rms surface roughness of objects and the maximum contrast obtained from the varying curves of the average image speckle contrast as a function of the point spread of an optical imaging system. The correlation length of surface roughness of objects is related to the averaged intensity distribution of speckle patterns produced at the far-field diffraction plane. It now becomes clear that the rms roughness and the correlation length of strong diffuse objects are determined, respectively, from the maximum speckle contrast at the image plane and the averaged speckle intensity distribution at the far-field diffraction plane.     

Diffraction Fields of Fractally Bounded Apertures

Experimental study is performed of optical diffractions by two-dimensional, self-similar mass and surface fractal apertures as well as mass fractal apertures bounded by surface fractals. Self-similar intensity distributions are observed in the Fraunhofer diffraction fields from surface and mass fractal apertures. Power law decays in average intensities of the diffraction patterns are also investigated for the mass, surface and combined fractal apertures in connection with the fractal dimensions of the objects.     

The influence of edge detection algorithms on the estimation of the fractal dimension of binary digital images

The boundary of a fractal object, represented in a two-dimensional space, is theoretically a line with an infinitely small width. In digital images this boundary or contour is limited to the pixel resolution of the image and the width of the line commonly depends on the edge detection algorithm used. The Minkowski dimension was evaluated by using three different edge detection algorithms (Sobel, Roberts, and Laplace operator). These three operators were investigated because they are very widely used and because their edge detection result is very distinct concerning the line width. Very common fractals (Sierpinski carpet and Koch islands) were investigated as well as the binary images from a cancer invasion assay taken with a confocal laser scanning microscope. The fractal dimension is directly proportional to the width of the contour line and the fact, that in practice very often the investigated objects are fractals only within a limited resolution range is considered too.     

Testing for scaling in natural forms and observables

The general procedure of calculating fractal dimensions or other exponents is based on estimating some quantity as a function of scale and on assessing whether or not this function is a power law. This power law manifests itself in a log (quantity) versus log (scale) plot as a linear region (scaling). It has thus become the practice to estimate dimensions by the slope of some linear region in those log-log plots. When we are dealing with exact fractals (the Koch curve, for example) there are no problems. When, however, we are working with natural forms or observables, problems begin to emerge. In such cases the scaling region is subjectively estimated and often is only the result of the generic property of the quantity to increase monotonically or decrease monotonically as the scale goes to zero irrespective of the geometry of the object. Here we discuss these issues and suggest a procedure to deal with them.     

Statistical mechanics of random paths on disordered lattices

The dependence of the universality class on the statistical weight of unrestricted random paths is explicitly shown both for deterministic and statistical fractals such as the incipient infinite percolation cluster. Equally weighted paths (ideal chain) and kinetically generated paths (random walks) belong, in general, to different universality classes. For deterministic fractals exact renormalization group techniques are used. Asymptotic behaviors for the end-to-end distance ranging from power to logarithmic (localization) laws are observed for the ideal chain. In all these cases, random walks in the presence of nonperfect traps are shown to be in the same universality class of the ideal chain. Logarithmic behavior is reflected insingular renormalization group recursions. For the disordered case, numerical transfer matrix techniques are exploited on percolation clusters in two and three dimensions. The two-point correlation function scales with critical exponents not obeying standard scaling relations. The distribution of the number of chains and the number of chains returning to the starting point are found to be well approximated by a log-normal distribution. The logmoment of the number of chains is found to have an essential type of singularity consistent with the log-normal distribution. A non-self-averaging behavior is argued to occur on the basis of the results.     

On the theory of diffraction of light in photonic crystals with allowance for interlayer disordering

Electrodynamic Green’s functions are used to construct an analytical theory of the Bragg diffraction of polarized light in photonic crystals having a close-packed structure. For opal-based photonic crystals, the Bragg diffraction intensity is calculated with allowance for permittivity periodic modulation and for the presence of an optical crystal boundary and interlayer disordering, which usually appears during sample growth. A comprehensive study is made of the effect of the structure disorder caused by the random packing of growth layers on diffraction. For a random constructed twinned fcc structure, the average structure factor and the scattering (diffraction) cross sections (which are dependent on the linear polarization of the incident and scattered waves) are calculated. Numerical examples are used to show that the theory developed can be applied to analyze and process experimental diffraction patterns of real photonic crystals having a close-packed structure disordered in one direction.     

Magnetic correlations on fractals

The critical behavior of magnetic spin models on various fractal structures is reviewed, with emphasis on branching and nonbranching Koch curves and Sierpiriski gaskets and carpets. The spin correlation function is shown to have unusual exponential decays, e.g., of the form exp[-(r/gx) ^x ], and to crossover to other forms at larger distancesr. The various fractals are related to existing models for the backbone of the infinite incipient cluster at the percolation threshold, and conclusions are drawn regarding the behavior of spin correlations on these models.     

Fractal analysis of powder X-ray diffraction patterns

X-ray diffraction (XRD) patterns with broad background are commonly found in the characterization of materials with a certain degree of amorphicity, so the sharp intensity peaks associated with material phases are not well defined. This work used rescaled range (denoted by R/S) analysis, a method intended for fractal analysis of noisy signals, to characterize XRD patterns with broad background. It is found that XRD patterns with broad background are not random at all, but contain information on regularities expressed as autocorrelations of the intensity signal. Sol-gel alumina fired at different temperatures was used as an example to illustrate the applicability of the method. It is shown that fractal R/S analysis is able to locate angular regions that can be associated to ideal International Centre for Diffraction Data Powder Diffraction File (ICDD PDF) lines of diverse alumina phases.     

Phase retrieval using an aperture-array filter under partially coherent illumination

Recently we have proposed a deterministic phase retrieval method using an aperture-array filter to reconstruct a complex-valued object from a single diffraction intensity pattern. We describe here the effect of quasi-monochromatic partially coherent illumination on the object reconstruction by the phase retrieval method, and then present the method of eliminating the effect of the partially coherent illumination from the diffraction intensity pattern via a simple Fourier deconvolution operation provided that the complex degree of spatial coherence of the illuminating beam is known. The usefulness of this method is shown in computer-simulated examples of the object reconstructions under Gaussian Schell-model partially coherent illumination.     

Fractals,coherent states and self-similarity induced noncommutative geometry

The self-similarity properties of fractals are studied in the framework of the theory of entire analytical functions and the q-deformed algebra of coherent states. Self-similar structures are related to dissipation and to noncommutative geometry in the plane. The examples of the Koch curve and logarithmic spiral are considered in detail. It is suggested that the dynamical formation of fractals originates from the coherent boson condensation induced by the generators of the squeezed coherent states, whose (fractal) geometrical properties thus become manifest. The macroscopic nature of fractals appears to emerge from microscopic coherent local deformation processes.     

Atomic structures by direct transform of diffraction patterns

We propose all the diffraction patterns can be directly transformed to provide three-dimensional atomic structures for the system studied. Depending on the scattering process, either the holography or Patterson transform scheme is used. For diffraction patterns which are generated from a localized emitter source or dominated by an inelastic-scattering feature like core-level photoelectron or low-energy Kikuchi electron, holography transform is needed. On the other hand, for diffraction patterns which were dominated by elastic-scattering, like grazing-incidence X-ray diffraction, electron correlated thermal diffuse scattering or low-energy electron diffraction curves, Patterson transform is needed. To prove our point, high-fidelity and artifact-free three-dimensional atomic structures obtained by transform of low-energy Kikuchi electron patterns and low-energy electron diffraction curves are presented. The future of these direct methods by transforming diffraction patterns will be discussed.     

实验研究探测器大小与散焦长度对无透镜鬼衍射的影响

无透镜鬼衍射是一种特殊的非局域关联成像技术,通过对包含物体信息的测试光路和不含物体信息的参考光路的光场强度涨落关联进行测量,在参考光路上可得到物体的衍射图样。利用赝热光源实验研究了探测器尺寸大小和光路散焦长度对无透镜鬼衍射质量的影响。实验以四缝物体为例,定量研究了探测器大小和散焦是如何改变无透镜鬼衍射的成像结果,并和理论模拟进行了比较,获得了比较一致的结果。利用图像相关度定量分析了实验结果与理想无透镜鬼衍射图像的差别,发现这两者都会使无透镜鬼衍射的质量变差。     

Wave scattering on a fractal surface

A generalized Von Koch surface was constructed. On the basis of Freedman' s formulation for wave scattering and by applications of the Lipchitz transform under Holder conditions in fractals, a demonstration was given that the Hausdorff dimension of the solid-angle discontinuity on the scattering surface is the same as the one of the surface itself, and an expression of the scattering strength of the fractal surface has been given. A comparison with the Schulkin-Shaffer empirical formula for the sound scattering from sea surface proposes that, in this situation, the generalized (continuous) Koch surface seems to degenerate into the (discrete) four-two Cantor sets, only the latter make a contribution to the backscattering.     

Singularities in forward scattering through random media

We consider a random medium in which scattering is exclusively in the forward direction. Waves are emitted by an object in the medium and Fourier components of the intensity are shown to propagate independently. At small wavevectors the intensity propagates very simply through increasing thickness, z, of medium, as λ^z, and Fourier components of the object can easily be reconstructed. For wavevectors greater than a critical value, q_c, the intensity changes with z in a more complex fashion making it very difficult to reconstruct the object. They develop a simple model for the singularity and apply it to the reconstruction of an object degraded by passage through a random medium.     

Influence of Blurred Ways on Pattern Recognition of a Scale-Free Hopfield Neural Network

We investigate the influence of blurred ways on pattern recognition of a Barabasi-Albert scale-free Hopfield neural network （SFHN） with a small amount of errors. Pattern recognition is an important function of information processing in brain. Due to heterogeneous degree of scale-free network, different blurred ways have different influences on pattern recognition with same errors. Simulation shows that among partial recognition, the larger loading ratio （the number of patterns to average degree P/ （k） ） is, the smaller the overlap of SFHN is. The influence of directed （large） way is largest and the directed （small） way is smallest while random way is intermediate between them. Under the ratio of the numbers of stored patterns to the size of the network PIN is less than O. 1 conditions, there are three families curves of the overlap corresponding to directed （small）, random and directed （large） blurred ways of patterns and these curves are not associated with the size of network and the number of patterns. This phenomenon only occurs in the SFHN. These conclusions are benefit for understanding the relation between neural network structure and brain function.     

基于CCD尺寸测量中图像边缘退化及恢复技术研究

针对直边衍射效应造成的图像边缘退化问题,建立边缘退化模型。从菲涅尔直边衍射理论出发,基于信号与系统理论的图像边缘退化数学描述方法,分析了造成图像边缘退化的原因。通过分析不同光源波长和成像物距下直边衍射强度分布曲线的特点,构造简单函数实现衍射光强分布的近似,进而得到退化系统传递函数;利用退化系统传递函数构造滤波器对获取图像的边缘进行恢复,从而提高尺寸测量精度。实验结果表明:经边缘恢复方法校正后的测量结果误差约为0.02 mm,相对未校正数据的测量结果误差减小0.04 mm,提高了尺寸测量精度。     

Statistical properties of the dynamic dichromatic laser speckle

The statistical properties of dynamic elongated speckle patterns produced in the far-field diffraction region by a moving diffuse object under illumination of the dichromatic laser beam are experimentally investigated by means of a correlation method. The experimental results show that the correlation length of the normalized autocorrelation function of time-varying dichromatic speckle intensity fluctuations depends strongly on the fibrous radial structure of elongated speckle patterns. The qualitative discussions are given for the experimental results.