| Metamaterial is one of the great discoveries on physics and electromagnetics in this century. The permittivity and permeability of metamaterials can be artificially tuned at will. Thus, metamaterials have many extensive applications in optoelectronics and communications. Based on theoretical analysis and numerical simulation methods, we mainly investigate some unusual characteristics of light propagating in metamaterials and design a simple narrowband micro Luneburg lens made by metamaterials. The whole thesis consists of 6 chapters. The main content of this thesis is summarized as follows.(1) Firstly, the background of the thesis is introdued in chapter 1. Next, we present the opical transformation and full-wave numerical simulation methods for designing metamaterials. Lastly, we make a brief comment on the applications of metamaterials.(2) On the basis of paraxial beam propagation theory and angular spectrum representation, the formalisms describing paraxial beams propagating in indefinite media are presented and the effective refraction index for both of the two types of polarized electromagnetic wave, TE and TM waves, are introduced. According to the definition of effective refraction index, we obtain the conditions that paraxial beams pass through the indefinite media. We propose to employ a slab of indefinite media to construct a polarizer or polarizing beam splitter by choosing appropriate anisotropic parameters. Under certain conditions, the indefinite media slab exhibits polarization-selective focusing effect. The expressions for light reflected by anisotropic metamaterial are obtained. Further, the expressions of angular shift for Gaussian beam reflected by metamaterial are derived.(3) Under paraxial approximation, the transfer matrix and imaging formula of left-handed slab focusing system is derived by the law of refraction. Using ABCD law, the propagation expressions for Gaussian beam focused by a left-handed material slab are obtained. On the basis of propagation formalism, the focusing properties of Gaussian beam are analyzed. The beam waist of image Gaussian beam is equal to the beam waist of object Gaussian beam under paraxial approximation, that is to say, the left-handed material slab can not focus Gaussian beam. It's interesting that the focusing for Gaussian beam with a left-handed material coincide with imaging formalism of geometrical optics and do not exist focal shift. (4) We derive the force of the electromagnetic radiation on left-handed materials (LHMs) by a direct application of the Lorentz law of classical electrodynamics. The expressions of radiation force are given for TE-polarized and TM-polarized fields. The numerical results demonstrate that electromagnetic waves exert an inverse lateral radiation force on each edge of the beams, that is, the lateral pressure is expansive for TE-polarized beams and compressive for TM-polarized beams.(5) We present a theoretical study for the simple design of a narrow-band micro Luneburg lens using a gradient refractive index (GRIN) cylinder. The GRIN cylinder is fabricated with metallic rods whose height is tunable. The ray tracing calculations that base on the equation of ray are applied to determine the index of refraction gradient for a given focusing spot. A full-wave finite element method is used to design the required unit cells. The proposed Luneburg lens is constructed by five discrete layers. The numerical results indicate that the metamaterial Luneburg lens can produce a sharp focusing and it is in excellent agreement with the theoretical anticipation.(6) The chapter 6 provides a brief conclusion for the whole thesis and outline some future research topics.
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