**1. Introduction**

40 Photonic Crystals – Innovative Systems, Lasers and Waveguides

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A crystal is a periodic arrangement of atoms or molecules. The pattern with which the atoms or molecules are repeated in space is the crystal lattice. The crystal presents a periodic potential to an electron propagating through it, and both the constituents of the crystal and the geometry of the lattice dictate the conduction properties of the crystal.

Importantly, however, the lattice can also prohibit the propagation of certain waves. There may be gaps in the energy band structure of the crystal, meaning that electrons are forbidden to propagate with certain energies in certain directions. If the lattice potential is strong enough, the gap can extend to cover all possible propagation directions, resulting in a complete band gap. For example, a semiconductor has a complete band gap between the valence and conduction energy bands.

The optical analogue is the photonic crystal, in which the atoms or molecules are replaced by macroscopic media with differing dielectric constants, and the periodic potential is replaced by a periodic dielectric function (or, equivalently, a periodic index of refraction). If the dielectric constants of the materials in the crystal are sufficiently different, and the absorption of light by the materials is minimal, then the refractions and reflections of light from all of the various interfaces can produce many of the same phenomena for photons (light modes) that the atomic potential produces for electrons. One solution to the problem of optical control and manipulation is thus a photonic crystal, a low-loss periodic dielectric medium. (Joannnopoulos et al, 2008).

There has been growing interest in the development of easily fabricated photonic band gap materials operating at the optical frequencies. The reason for the interest in photonic band gap materials arises from the possible applications of those materials in several scientific and technical areas such as filters, waveguides, optical switches, cavities, design of more efficient lasers, etc. ( Li et al., 2008; Wang et al., 2008).

The simplest possible photonic crystal consists of alternating layers of material with different dielectric constants: a one-dimensional photonic crystal or a multilayer film. This arrangement is not a new idea. Lord Rayleigh (1887) published one of the first analyses of

<sup>\*</sup> The publishing fee for Andreea Petcu was paid by the National Research and Development Institute for Gas Turbines Bucharest

The Optical Transmission of One-Dimensional

Fig. 1. Schematic representation of a multilayered structure

magnetic fields Em and Hm in the support layer of glass.

*j*

0

0

*j j*

*f* .

*M*

be seen in figure 1.

solved:

where

and <sup>1</sup>

*j j j j d v*

and the angular frequency *ω* is *v v* <sup>2</sup>

Photonic Crystals Containing Double-Negative Materials 43

Let us consider for investigation a stack of m layers perpendicular on the OZ axis as it can

Using the notations given in figure 1, it was considered known the refractive index of the medium from where the beam of light emerges n0=1 (air), the refractive index of the medium in which the beam of light exits ns=1.52 (glass), the intensities of the electric and

To determine the electric and magnetic fields in the air, E0 and H0, the following system was

1 2

cos sin

*M j m*

*E E MM M H H* 

... *<sup>m</sup> m*

<sup>1</sup> , 1.. sin cos

*j j*

the phase variation of the wave passing the layer j. *εj*, *µj* and *dj* are

the electric permittivity, the magnetic permeability, respective the thickness of the layer. *v* is the phase speed and *ω* is the angular frequency. The relationship between the wavelength *λ*

Be multiplying the 1 2 *M* , ,, *M M <sup>m</sup>* matrices we obtain a final matrix of the following shape:

11 12 21 22 *M M*

*M M* 

*j j j j*

*m*

(1)

(2)

(3)

the optical properties of multilayer films. This type of photonic crystal can act as a mirror for light with a frequency within a specified range, and it can localize light modes if there are any defects in its structure. These concepts are commonly used in dielectric mirrors and optical filters. (Joannnopoulos et al., 2008)

Recently, photonic crystals containing metamaterials have received special attention for their peculiar properties (Deng & Liu, 2008). One kind of metamaterials is double-negative materials (DNG) whose electric permittivity ε and magnetic permeability µ are simultaneously negative (Veselago, 1968), which can be used to overcome optical diffraction limit, realize super-prism focusing and make a perfect lens (Pendry, 2000). Another kind of metamaterials is single-negative materials (SNG), which include the mu-negative media (MNG) (the permeability is negative but the permittivity is positive) and the epsilonnegative media (ENG) (the permittivity is negative but the permeability is positive). These metamaterials possess zero-effective-phase gap and can be used to realize easily multiplechanneled optical filters (Zhang et al., 2007).

In this chapter are analzed one-dimensional photonic crystals composed of two layers: A=dielectric material (TiO2) or A=epsilon-negative material (ENG) and B=double negative material (DNG), from the point of view of their optical transmission. In the case in which A is a dielectric material are used the following materials properties: the magnetic permeability μA=1 and the electric permittivity εA=7.0225. To describe the epsilon-negative material (ENG) it is used a transmission-line model (Eleftheriads et al., 2002): the magnetic permeability μA=3 and the electric permittivity εA=1-100/ω2. For the double-negative material (DNG) are used the following material properties: the magnetic permeability μB=1-100/ω2 and the electric permittivity εB=1.21-100/ω2. Ω is the angular frequency measured in GHz.

An algorithm based on the transfer matrix method (TMM) was created in MATLAB and used to determine the optical transmission of the cosidered photonic structures. In the simulations the angular frequency ω takes values from 0 to 9 GHz. In this chapter is analized the influence of various defects upon the optical transmission of the photonic crystal: the type of the material used in the defect layer, thickness and position of the defect layer upon the optical transmission.
