We investigate the negative refraction phenomena of electromagnetic waves in photonic crystals. We treat two-dimensional photonic crystals composed of square array of liquid crystal rods in air region. This structure can be prepared by making a periodic array of holes in a silica aerogel plate whose refractive index is about 1.03 and then infiltrating liquid crystal into the holes. Liquid crystals are anisotropic, and thus the refractive index can be varied by rotating their directors. In this thesis, we illustrate some cases of negative refraction for waves of specific frequencies and incident angles, and also for certain director angles of liquid crystals. Those are found by analyzing equi-frequency surface that is obtained from a plane wave expansion method. We design a superlens whose focal length is tunable by controlling the director angle of liquid crystal. However, not all focuses fall at the center axis of superlens due to the anisotropy of the refractive index. Hence we simulate the propagation of waves of the frequency ωa/2 πc = 0.68 at the incident angles $θ_i$=0 ˚, 3 ˚, and 5 ˚ for the director angle π = 90˚ by using finite-difference time-domain(FDTD) method. The width of photonic crystal lens is 50 ㎛ and the length is 20 ㎛. We launch the continuous wave(CW) Gaussian beam of width 20 ㎛. As a result, when the transmitted waves are at the same horizontal position as the source(x=0) for the incident angles of 3 ˚ and 5˚, the z-positions of images are 44.5 ㎛ and 43.8 ㎛, respectively, which can be considered as the same spot within 0.7 ㎛ compared with the beam size of 20 ㎛. Hence we conclude that when the director angle π is 90 ˚, both waves of frequency Φ is 0.68 incident 3 ˚ and 5 ˚ make a focus. Thus the proposed design of our photonic crystal structure can be very useful as a focus-switching lens.
