Fractal characteristics of far-field diffraction patterns for two-dimensional Thue-Morse quasicrystals

Abstract

Tianjin University of Technology and Springer-Verlag Berlin Heidelberg 2011 C We report a numerical method to analyze the fractal characteristics of far-field diffraction patterns for two-dimensional Thue-Morse (2-D TM) structures. The far-field diffraction patterns of the 2-D TM structures can be obtained by the numerical method, and they have a good agreement with the experimental ones. The analysis shows that the fractal characteristics of far-field diffraction patterns for the 2-D TM structures are determined by the inflation rule, which have potential applications in the design of optical diffraction devices.The fractal phenomenon is easily found in nature, by definition, fractal objects exhibit the self-similarity in their geometric structures which are determined by an iterative rule [1] . The concept of fractals provide a general technique for analyzing the physical phenomena in different fields, for example, the fractal theory used for image inpainting algorithm [2] , the fractal characterization of surface morphology [3] , the fractal structure excitation in soliton-supporting systems [4] and the evolution of self-written waveguides [5] . Although fractals are common in the one-dimensional quasicrystals structures [6] , the fractal characteristics of twodimensional (2-D) quasicrystals structures have not been deeply known. Only a few of significant studies have been carried out, including the transmission resonances in Penrose quasicrystal [7] , the photonic-plasmonic scattering resonances in Fibonacci, Thue-Morse (TM) and Rudin-Shapiro array [8] .In this work, we present a numerical study to demonstrate that the far-field diffraction patterns of the 2-D TM structure, which have the fractal characteristics determined by the inflation rule of TM sequence. In addition, two kinds of 2-D TM chips are fabricated by electron beam lithography (EBL) technique and their far-field diffraction patterns are experimentally recorded by CCD detector. By comparing the numerical far-field diffraction patterns with the experimental photos, it is verified that the numerical method can finely simulate the far-field diffraction results. And, the analysis of fabrication tolerance of the 2-D TM chips shows that the far-field diffraction patterns of 2-D TM structure have high fractal stability.To construct the 2-D TM structure, a one-dimensional binary TM sequence is firstly constructed using the following rule. First of all, let , s give an arbitrary sequence of two symbols, A=0 and B=1, and then a new sequence is formed by replacing each occurrence of A with the pair (A, B) and each occurrence of B with the pair (B, A) [9] . By inflation rule, the one-dimensional TM sequence can be generalized to the two-dimensional TM structure by introducing a substitutional matrix S N+1[9] :