## 2 Metamaterials  
It is a widely established fact that when an electromagnetic wave strikes a composite medium, it induces electric and magnetic dipole moments in the inclusions. These dipole moments are closely related to the effective permittivity and permeability of the composite medium. Since the size, density, shape, and orientation of the inclusions can be controlled by the designer, materials with specific electromagnetic response can be synthesized. These artificially designed materials are called metamaterials. When the individual elements are resonant, such materials can possess negative values for both, relative permittivity and permeability. It needs to be emphasized that the negative refractive index is only possible with the resonant individual elements. This is due to the fact that \"resonances have the characteristic that their phase response reverses as frequency changes around the resonance\" [2]. Such materials are referred to in the literature as double negative media (DNG), left-handed media and backward wave media [3; 4; 5]. The electromagnetic properties of the metamaterials can be described by using the [PERSON] classical theory. In this theory, the electron is treated like a damped harmonic oscillator in an electromagnetic field [6; 7]. When the restoring force is negligible, the Lorentz model is reduced to the Drude model. The Drude model allows for the negative values of permittivity and permeability over a wide frequency range. Due to this property, the Drude model is sometimes preferred to than the narrowband Lorentz model for simulations [7]. Metamaterials with resonant individual elements can possess negative relative permittivity and permeability values. These left-handed materials, then according to [PERSON]'s law of refraction, make the refracted angle negative, thus causing the incident and refracted wave to lie on the same side of normal. This phenomenon is called negative refraction. Negative refraction allows the complete control of electromagnetic waves (including light), in all four quadrants of a cartesian plane. Due to this unusual characteristic, metamaterials offer potential applications which would have not been possible by only using the naturally occurring materials [7]. One of the numerous applications of the metamaterials is the phase compensation medium. DNG and positive indexed materials are combined together in such a way that the phase difference across the slab of this medium is zero. By combining double positive and double negative metamaterials, the phase difference can be controlled. It can be shown that it is the ratio of the thicknesses and the refractive index, which cause the phase difference across the medium to be zero and not the total thickness [7]. This shows that the negative index part of the slab compensates for the phase propagation in the positive index part [8]. Another interesting phenomenon which can be observed by combining the positive and negative index materials is the concentrated resonance which occurs at the interface of such two materials [9]. This interface resonance (also known as a surface wave plasmon) can replace the aperture related resonance in a traditional waveguide thus making possible the existence of sub-wavelength thin waveguides. The dispersion relation for such waveguides is also related to the thicknesses ratio and is independent of the total thickness [10; 11]. Metamaterials, with a negative refractive index equal to -1, can also be used to make a superlens [12]. A superlens (also called a perfect lens) breaks the limitations imposed on focussing by wave optics (for a traditional lens, an absolute limit on the area for focussing energy, is a square of wavelength). This is due to the fact that the amplitude of the evanescent waves decays exponentially in a naturally occurring medium, whereas DNG materials enhance their amplitude. The structure still obeys the law of conservation of energy as evanescent waves do not carry any energy.