Reflections on Refraction (Geometrical Optics)

Opto-Electronics Review, 2009

In this paper we introduce a diffractive structure with a geometry which contains multiple Fresnel zone plates (MFZP) disposed in an arrangement based on a fractal-like rule. The corresponding diffracted intensity in planes perpendicular to the propagation axe presents two or more focal points. Their position and magnitude depend on geometrical parameters of the MFZP and on dimensions in the fractal-like arrangement. In our simulations we also analyze the influence in diffraction pattern of different non-binary phase levels in the MFZP plane. The MFZP structures with different values of geometrical and phase parameters are addressed to an optoelectronic device liquid crystal spatial light modulator (LCSLM), the experimental and simulation results are in a good agreement. The MFZP geometry with better optical parameters in diffraction pattern is then made on glass using electron beam lithography technique.

Applied mathematical sciences, 2014

In this study, we underline the peculiarities of the refraction problem of elastic waves from a layer with a fractal density distribution. The refraction problem is reduced to the system of ordinary differential equations with linear coefficients. Analytic solutions for each of its equations are found. The case for the layer with fractal density distribution is investigated numerically. Characteristic maxima of the reflected wave

Lecture Notes in Mathematics, 2013

Since the refraction angle depends on the frequency of the radiation, we assume radiation is monochromatic.

Journal of The Optical Society of America, 2006

Two compact analytical descriptions of Fresnel diffraction patterns from polygonal apertures under uniform illumination are detailed. In particular, a simple expression for the diffracted field from constituent edges is derived. These results have fundamental importance as well as specific applications, and they promise new physical insights into diffraction-related phenomena. The usefulness of the formulations is illuminated in the context of a virtual source theory that accounts for two transverse dimensions. This application permits calculation of fractal unstable-resonator modes of arbitrary order and unprecedented accuracy.

2009

Many medical and industrial applications require focusing of microwave radiation in the near field of the radiating antenna. Antenna arrays are suitable and flexibel in such applications. Fractal antennas and arrays are finding much interest in recent years for the benefits they offer over conventional ones. The feasibility of deploying the fractal principle to focused arrays is demonstrated. The properties of fractal array whose elements are spatially distributed according to the binary tree fractal are investigated and compared with those of conventional array. Computer simulations results show that focused fractal array can have similar focusing properties while having favorable ones.

International Frontier Science Letters, 2015

A mathematical model of the scattering by a periodically arranged apertures in conducting plates is presented. The boundary value problem of an infinite array of loaded apertures is formulated for an arbitrary incident plane wave. The reflection coefficient for some array geometries is obtained and the calculated values are in good agreement with the measurements in a previously published researches. All the rectangular apertures in the array are assumed to be identical and infinitesimally thin. The mathematical model is based on Floquet's theorem that specifies the requirement of periodicity by the electromagnetic fields.

In this paper, the optical properties of one dimensional fractal structures are investigated. We consider six typical fractal photonic structures: the symmetric dual cantorlike fractal structure, the asymmetric dual cantor-like fractal structure, the single cantorlike fractal structure, the symmetric dual golden-section fractal structure, the asymmetric dual golden-section fractal structure and the single golden-section fractal structure. By using the transfer matrix method the transmission spectra of these structures are simulated. The calculation results shows that the transmission spectrum of the symmetric dual cantor-like fractal structure is self-similar and the peak numbers in the transmission spectra of the SDGSFS also follow the principals of special fractal structures. It is also shown that in the symmetric dual golden-section fractal structure the localization of modes which appears within the stop band increases and getting closer to the middle of the gap by increasing the number of string.

Aeu-international Journal of Electronics and Communications, 2009

The analysis of fractal structures by conventional methods gets harder and harder when the number of length scales rises. In this study, we try to combine the renormalization method and the surface impedance model to help rigorous studying of electromagnetic diffraction of fractal structures at their final (infinitum) iteration. As an application, we applied the method to a couple of fractal structures: Cantor Iris 1D and 2D.

This paper presents an analytical solution to study the reflection and transmission of an electromagnetic wave impinged upon a multilayered structure. The structure is composed of a fractal slab sandwiched by ordinary material on either side. Modified Maxwell equations for fractional dimension space are used to represent the fields in a fractal slab. The electromagnetic characteristics of the structure are studied for different dimensions (D) and numerical results are presented for both the classical (D is integer) and fractal (D is non-integer) slabs. This study provides foundations for investigating the waveguides filled with fractal media and electromagnetic waves propagation in multilayered structures at fractional boundaries.

This paper covered basic investigation of the diffraction and scattered problem of a plane monochromatic wave on pre-fractal grating consist on finite number of the infinite thin perfectly electrically conducting (PEC) strips. Mathematical model of this task is singular integral equation (SIE) with supplementary conditions. Discrete mathematical model based on SIE using specific quadrature formulas of interpolation type with the equidistant grid of the polynomial of order (n-1) for integrals with singular singularities and integrals of smooth functions. Primary strategic course of this work is to use an efficient discrete singularities method (DSM) for the numerical analysis of the diffraction transverse magnetic wave problem.

2001

CHAPTER 1. INTRODUCTION and their various characteristics. The fractals that have been investigated are all deterministic. This thesis contains seven chapters. Here is an overview of the rest of the thesis: chapter 2: This chapter describes the principles of array theory and the basic concepts governing the array beamforming are presented. We begin by representing a space-time signal at the coordinate system. We consider the equation related to wave propagation and establish the wave equation solution, we use these to find a representation for beampatterns. The linear and planar arrays principles are also discussed. Chapter 3: This chapter introduces the basic concepts of fractals and the related geometry concepts. Several examples of fractal geometry are shown. Finally we briefly review some of the applications of fractals, and how the fractal can be applied to antennas and arrays. Chapter 4: In this chapter two types of deterministic fractal linear arrays known as the Cantor linear array and the Koch array are generated and analyzed. Chapter 5: In this chapter special types of fractal planar arrays are designed and their radiation characteristics are investigated. Chapter 6: In this chapter we briefly review an antenna theory related to antenna size, and briefly discuss how the antenna benefit from the reduction of antenna size using fractal design. The applications related to fractal antennas and arrays are proposed, and some practical examples of fractal antennas are shown. Chapter 7: This chapter gives the final conclusion.

Agrociencia, 2009

La variabilidad en el espacio de la salinidad del suelo ha sido documentada en diversos estudios. La reflectancia es una variable fundamental de ellos. En este trabajo se propone un método alternativo para el análisis de la salinidad mediante los datos multiescalares de la reflectancia. El método propuesto extrae la información para describir la estructura del suelo a partir del análisis de las firmas espectrales del suelo utilizando la dimensión fractal (D). La estructura de las costras salinas fue caracterizada a partir del análisis de las firmas fractales (FER) de los espectros de reflectancia, usando las técnicas de rango de reescalado (D R/S) y de ondoletas (D w). Para el análisis estadístico de las FER se aplicó el análisis multivariado y se identificaron las relaciones más significativas entre las FER y las propiedades físicas y químicas del suelo. Se obtuvieron cinco grupos jerárquicos que tuvieron el siguiente orden g1<g2<g3<g4<g5, con base en sus respuestas espectrales evaluadas en la longitud de onda de luz visible (VIS) e infrarrojo (IR) cercano del espectro electromagnético (EM). Se efectuó una prueba comparativa considerando las dimensiones D R/S y D w. Los resultados muestran que la dimensión D R/S para las FER es un mejor indicador de la estructura de las costras salinas en comparación con la dimensión D w. Los valores de la dimensión D R/S para cada grupo presentan un orden 1.73>1.70>1.67>1.66 para los grupos g1, g2, g3 g4; y 1.67 para g5. Se encontró una alta relación significativa entre las dimensiones D R/S al comparar los cinco grupos. La dimensión D w no presentó relación significativa con la morfología de las costras. Por tanto, la D es útil para clasificar la estructura de las costras salinas a partir de las firmas fractales (FER) del suelo extraídas de los espectros de reflectancia.

Journal of The Optical Society of America A-optics Image Science and Vision, 2009

The construction of fractal generalized zone plates (FraGZPs) from a set of periodic diffractive optical elements with circular symmetry is proposed. This allows us to increase the number of foci of a conventional fractal zone plate (FraZP), keeping the self-similarity property within the axial irradiance. The focusing properties of these fractal diffractive optical elements for points not only along but also in the close vicinity of the optical axis are investigated. In both cases analytical expressions for the irradiance are derived. Numerical simulations of the energetic efficiency of FraGZPs under plane wave illumination are carried out. In addition, some effects on the axial irradiance caused by the variation in area of their transparent rings are shown.