We show that an electromagnetic (EM) wave undergoes negative refraction at the interface between a positive and negative refractive index material. Finite difference time domain (FDTD) simulations are used to study the time evolution of an EM wave as it hits the interface. The wave is trapped temporarily at the interface and after a long time, the wave front moves eventually in the negative direction. This explains why causality and speed of light are not violated in spite of the negative refraction always present in a negative index material. PACS numbers: 78.20.Ci, 41.20.Jb, Veselago 1 predicted that lossless materials, which possess simultaneously negative permittivity, ǫ, and negative permeability, µ, would exhibit unusual properties such as negative index of refraction, n = − √ ǫµ, antiparallel wave vector, k, and Poynting vector, S, antiparallel phase, v p , and group, v g , velocities, and time-averaged energy flux, S = u v g , opposite to the time-averaged momentum density p = u k/ω, 2 where u is the timeaveraged energy density. Furthermore if these materials are uniform, k, E, H form a left-handed set of vectors. Therefore, these materials are called left-handed materials (LHM) or negative index of refraction materials (NIM). The quantities, S, u, p, refer to the composite system consisting of EM field and material. As a result of k and S being antiparallel, the refraction of an EM wave at the interface between a positive n and a negative n material would be at the "wrong" side relative to the normal (negative refraction). In addition, the optical length, ndl , is negative in a LHM.Following Pendry's suggestions 3, 4 for specific structures which can have both ǫ ef f and µ ef f negative (over a range of frequencies), there have been numerous theoretical and experimental studies.5-8 In particular, Markos and Soukoulis 9 have employed the transfer matrix technique to calculate the transmission and reflection properties of the structure suggested by Pendry 3 and realized experimentally by Smith et al.6, 7 Subsequently, Smith et al.10 proved that the data of Ref.[9] can be fitted by length independent and frequency dependent, ǫ ef f , and µ ef f . They found that in a frequency region both ǫ ef f and µ ef f were negative with negligible imaginary parts. In this negative region, n was found to be unambiguously negative. These unusual results [3][4][5][6][7][8][9][10] have raised objections both to the interpretation of the experimental data and to the realizability of negative refraction.
11, 12In this letter we report numerical simulation results, which clarify some of the controversial issues, especially the negative refraction considered as violating causality and the speed of light.12 Our numerical calculations were performed on a well understood realistic system, which is essentially inherently lossless, namely a 2D photonic crystal (PC). The dielectric constant ǫ is modulated in space and both the permittivity ǫ and permeability µ are locally positive. Specifically, the photonic crystal consists of an ...
