Owing to the particular properties, left-handed materials (LHM) have an extra-important potential of application. The conventional design method which is based on design of the shape and dimension of wires and SRRs have considerable restriction in developing specific LHM. Thus, to research and develop design theory of left-handed materials with high and/or wide frequency band, or with specific band gaps become an important tasks. The main gaol of this project include: (1) Research the discription and evolution method of microstructures based on the topology optimization technique, develop a systemic design theory of microstructure of left-handed materials; (2) Investigate the formulation of topology optimization problem of the left-handed materials with maxmizing frequency or with specific frequency band; (3) Study the multi-scale method for analyzing the propogation of elctro-magnatic waves in heterogeneous materials; (4) Investigate the formulation and the coresponding solving method of materials with specific (negative) permeability, materials with specific (negative) permittivity, as well as materials with negative refraction index.