**2. Photonic crystals (PCs) for negative refraction**

Photonic crystals (PCs) are structures with periodic arrangement of dielectrics or metals, which provide the ability to manipulate the propagation of electromagnetic waves. In fact the lattice constant of common materials is 0.2-1 nm, i.e., much shorter than the wavelength of visible light (a few 100 nm). This is the reason why the response of such materials on the electrical and magnetic fields of light wave can be described by macroscopic parameters and µ. In 1987, anomalous refraction properties of PCs were reported based on a numerical analysis with transfer matrix method (Yablonovich, 1987). The light propagation in the PCs can not be considered as an average effect of atoms as in common crystals. In contrary, light propagation in PCs is the result of Bragg diffraction for each atom. Hence the periodic structure of the PCs is very important. The macroscopic constant and µ can not describe the light propagation in PC and the light refraction at the PC boundary. More precisely, light waves in PCs should be considered as the Bloch waves but in the so-called envelope function approximation they may be considered as plane waves.

An effective index of refraction for the crystal is used to describe the overall reflectivity form the photonic crystal:

$$
\eta = c \frac{d\alpha}{dk} \tag{1}
$$

Plasma Photonic Crystal 283

understood from the extensive studies of photonic crystal. Hence in plasma photonic crystal, array of periodic micro plasmas are used at the place of array of dielectrics or metals in the conventional photonic crystals. One or two dimensional layers of array of micro plasmas make forbidden bands for wave propagation are formed beyond the bulk cut of frequency (electron plasma frequency) due to periodicity, where one can refer to such a functional structures as plasma photonic crystal. A photo of plasma photonic crystal is given below in

We know that plane-wave-expansion method has been widely used to analytically derive photonic band diagram of two-and three-dimensional dielectric periodic structures (Ho et al, 1990; Phihal et al, 1991). Dielectric constant of plasma can be obtained considering the field components in electromagnetic waves proportional to exp.[ ( . )] *j t kx* , where *k* and *x* are the complex wave number and spatial position vector, respectively. The dielectric constant as a function of frequency *<sup>p</sup>* inside a cold plasma column with electron plasma

> 2 <sup>1</sup> <sup>1</sup> 1

where *<sup>m</sup>* is the electron elastic collision frequency determined by neutral gas pressure and elastic collision cross section. In metal cases, a similar value ( ) to *m* was used as an inverse of electron relaxation time, and was much smaller than and *pe* (Kuzmiak and 1997) it is also possible that *<sup>m</sup>* is comparable to and *pe* where electron density is around 1013 cm-3 at a gas pressure around atmospheric pressure. Therefore plane-wave expansion method with Drude model in collision plasma has been studied (Sakai et al, 2007) for plasma photonic crystals. Experimental demonstrations have also been performed (Sakai

 In 2 D structures, a mesh type DBD (Dielectric Barrier Discharge) electrode assembly, mounted at 6 mm separation from third electrode (Micro-hollow-cathode-discharge

Array size 33 x 33 lattices, where 33 rows of a micro plasma column with diameter of

*pe <sup>p</sup> <sup>j</sup> <sup>m</sup>* 

et al, 2005 ; Sakaguchi et al, 2007) those typical parameters are summarized below,

One and two dimensional periodic structures have been studied.

Array forms a 4.4 x 4.4 cm2 squire lattice of plasma columns.

(2)

Fig.1.

Fig. 1. Plasma photonic crystal

frequency *pe* is written as

MHCD like configuration).

0.6 mm.

Thus, calculating band structure of a PC numerically leads to calculation of . From the experimental point of view can be calculated by Snell's law. Hence, the negative refraction can be realized also with PCs that is in contrast to the composite metamaterials pave inhomogeneous media with a lattice constant comparable to the wavelength. Although both and µ are positive in dielectric PCs and metallic photonic crystals (MPCs), phenomenon of negative refraction and super resolution can be expected from peculiarities of the dispersion characteristics of certain PCs. The main advantage of PCs over composite metamaterials (CMMs) currently is that they can be more easily scaled to 3D and adapted to visible frequencies (Parimi et al, 2004). Negative refraction at microwave frequencies was observed in both dielectric and metallic PCs, for example, using a square array of alumina rods in air (Cubukcu et al, 2003). 2D and 3D PCs consisting of alumina rods were used for the demonstration of negative refraction in the microwave and millimeter wave range. Two techniques namely, manual assembly of alumina rods and rapid phototyping were used in this study for fabricating low-loss PCs (investigated in the wave range form 26 GHz to 60 GHz). The negative refraction in a metallic PC with hexagonal lattice acting as a flat lens with out optical axis at microwave frequencies was reported at 10.4 GHz for TM mode (Parimi et. al, 2004). Such PC contains cylindrical copper rods, are in triangular lattice, in which negative refraction was found for both TM and TE mode propagation between 8.6 and 11 GHz (TM mode) and between 6.4 and 9.8 GHz (TE mode). Hence, extensive experimental and simulation results were achieved, which pave the way to a variety of well tailored PCs structures. However, the advantages of metallic PC were reported to be highest dielectric constant, low attenuation, and the possibility of focusing. Most of efforts have been dedicated to the engineering and extension of the functionalities of metamaterials or PCs at terahertz (Yen et al, 2004 ; Padilla et al, 2006, Chen et al, 2006) and optical frequencies (Linden et al, 2004; Soukoulis et al, 2007). Negative refraction of surface plasmons was also demonstrated but was confined to a two-dimensional waveguide (Lezec et al, 2007). Three dimensional optical metamaterials have come into focus recently, including the realization of negative refraction in semiconductor metamatetrials and a 3D magnetic metamaterial in the infra red frequencies. However neither of these had a negative index of refraction (Liu et al, 2008 ; Hoffman et al, 2007). Three dimensional optical metamaterial with a negative refractive index has been also demonstrated recently (Valentine et al, 2008). Negative and positive refaction tubability of x-band microwve in MPC have been achieved recently by making defults or holes (Kumar, 2011a).
