Fractal character istics of far-field diffraction patterns for two-dimensional Thue-Morse quasicrystals ∗ YANG Ming-yang( ) 1 , ZHOU Jun( 骏) 1 ** , L Petti 2 , S De Nicola 3 , and P Mormile 2 1. Institute of Photonics, Faculty of Science, Ningbo University, Ningbo 315211, China 2. Istituto di Cibernetica “E. Caianiello” del Consiglio Nazionale delle Ricerche, Via Campi Flegrei 34, Pozzuoli (Na) 80078, Italy 3. Istituto Nazionale di Ottica del Consiglio Nazionale delle Ricerche, Via Campi Flegrei 34, Pozzuoli (Na) 80078, Italy (Received 13 May 2011) 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 numeri- cal 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 appli- cations in the design of optical diffraction devices. Document code: A Article ID: 1673-1905(2011)05-0346-4 DOI 10.1007/s11801-011-1057-0 OPTOELECTRONICS LETTERS Vol.7 No.5, 1 September 2011 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 algo- rithm [2] , the fractal characterization of surface morphology [3] , the fractal structure excitation in soliton-supporting sys- tems [4] and the evolution of self-written waveguides [5] . Al- though fractals are common in the one-dimensional quasicrystals structures [6] , the fractal characteristics of two- dimensional (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 demon- strate 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 lithogra- phy (EBL) technique and their far-field diffraction patterns are experimentally recorded by CCD detector. By compar- ing the numerical far-field diffraction patterns with the ex- perimental 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 follow- ing 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] : * This work has been supported by the National Natural Science Foundation of China (No.60977048), the International Bilateral Italy-China Joint Projects (CNR/CAS Agreement 2008-2010), the International Collaboration Program of Ningbo (No.2010D10018) and the K. C. Wong Magna Fund in Ningbo University, China. ** E-mail: ejzhou@yahoo.com.cn where N is the order of the 2-D TM structure. According to Eq.(1), the 2-D TM structure will be con- sidered as a binary phase-only diffractive element with the transmittance function: ) 3 , 2 , 1 ( and 1 L = − = ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ = + N S S S S S S N N N N N N N S , (1)