**2. Calculus method**

The transfer matrix method (TMM) is widely used for the description of the properties of stacked layers and is extensively presented in (Born & Wolf, 1999). The transfer matrix method (TMM) provides an analytical means for calculation of wave propagation in multilayer media. This method permits exact and efficient evaluation of electromagnetic fields in layered media through multiplication of 2 × 2 matrices. The TMM is usually used as an efficient tool to analyse uniform and non-uniform gratings, distributed feedback lasers and even onedimensional photonic crystals. The solution of the coupled mode (coupled modes TE and TM) equations is represented by a 2 x 2 transfer matrix which relates the forward and backward propagating field amplitudes. The grating structure is divided into a number of uniform grating sections which each have an analytic transfer matrix. The transfer matrix for the entire structure is obtained by multiplying the individual transfer matrices together.

Such an algorithm was implemeted in MATLAB and used to determine the optical transmission of the cosidered photonic structures. In this approach were considered isotropic layers, nonmagnetic and a normal incidence of the incident light.

Let us consider for investigation a stack of m layers perpendicular on the OZ axis as it can be seen in figure 1.

Fig. 1. Schematic representation of a multilayered structure

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

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

$$
\begin{bmatrix} E\_0 \\ H\_0 \end{bmatrix} = M\_1 M\_2 \dots M\_m \begin{bmatrix} E\_m \\ H\_m \end{bmatrix} \tag{1}
$$

where

42 Photonic Crystals – Innovative Systems, Lasers and Waveguides

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

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 multiple-

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

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

The transfer matrix method (TMM) is widely used for the description of the properties of stacked layers and is extensively presented in (Born & Wolf, 1999). The transfer matrix method (TMM) provides an analytical means for calculation of wave propagation in multilayer media. This method permits exact and efficient evaluation of electromagnetic fields in layered media through multiplication of 2 × 2 matrices. The TMM is usually used as an efficient tool to analyse uniform and non-uniform gratings, distributed feedback lasers and even onedimensional photonic crystals. The solution of the coupled mode (coupled modes TE and TM) equations is represented by a 2 x 2 transfer matrix which relates the forward and backward propagating field amplitudes. The grating structure is divided into a number of uniform grating sections which each have an analytic transfer matrix. The transfer matrix for the entire

Such an algorithm was implemeted in MATLAB and used to determine the optical transmission of the cosidered photonic structures. In this approach were considered

permittivity εB=1.21-100/ω2. Ω is the angular frequency measured in GHz.

structure is obtained by multiplying the individual transfer matrices together.

isotropic layers, nonmagnetic and a normal incidence of the incident light.

optical filters. (Joannnopoulos et al., 2008)

channeled optical filters (Zhang et al., 2007).

layer upon the optical transmission.

**2. Calculus method** 

$$\boldsymbol{M}\_{j} = \begin{bmatrix} \cos \phi\_{j} & \sqrt{\frac{\mu\_{j}}{\varepsilon\_{j}}} \cdot \sin \phi\_{j} \\\\ -\frac{1}{\sqrt{\frac{\mu\_{j}}{\varepsilon\_{j}}}} \cdot \sin \phi\_{j} & \cos \phi\_{j} \\\\ \sqrt{\frac{\mu\_{j}}{\varepsilon\_{j}}} & \end{bmatrix} \prime j = 1 \ldots m \tag{2}$$

and <sup>1</sup> *j j j j d v* 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* and the angular frequency *ω* is *v v* <sup>2</sup> *f* .

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

$$M = \begin{bmatrix} M\_{11} & M\_{12} \\ M\_{21} & M\_{22} \end{bmatrix} \tag{3}$$

The Optical Transmission of One-Dimensional

**4. MATLAB simulation results** 

**4.1.1 Case 1: A is a dielectric material** 

gaps. The values are given in table 1.

(AB)16

(ABBA)8

(ABBABAAB)4

photonic crystal

Crystal's type Band-gaps

Fig. 3. The structure of the photonic crystal with a defect layer

**4.1 Optical transmission of 1D PC without defects** 

in figure 3.

Photonic Crystals Containing Double-Negative Materials 45

the optical transmission of the photonic crystal. The new structure of the crystal is presented

In these experiments the thickness of the layers were kept comstant - *dA* =24mm and *dB* =6mm - and the frequency varied ω∈(0,9)*GHz.* As it is specified earlier there will be two cases: case 1 in which A is a dielectric material and case 2 in which A is an epsilon negative

Figure 4 presents the optical transmission of the three types of the studied photonic crystals versus of the frequency of the light beam so that we can observe easily the band-gaps.

One easily observes that the (AB)16 type photonic crystals have two band gaps, the (ABBA)8 type crystals have three band-gaps and the (ABBABAAB)4 type crystals have four band-

Table 1. The photonic band-gaps for (AB)16, (ABBA)8 and (ABBABAAB)4 one-dimensional

The mid-gap frequency of the gap (GHz)

> 1.8 7

0.8 1.35 7.25

0.58 1.4 3.45 7.25

The width of the band-gaps (GHz)

> 1.2 1

> 0 0.3 0.5

> 0 0.3 3.1 0.5

material. In both cases B is a double negative material (Ramakrishna, 2005).

(GHz)

 (1.2, 2.4) (6.5,7.5)

 0.8 (1.2, 1.5) (7,7.5)

 0.58 (1.25, 1.55) (1.9, 5) (7,7.5)

Using the matrix obtained above we calculate the optical transmission with the following relationship:

$$T = \left(\frac{2}{\left|M\_{11} + M\_{22} + i\left(M\_{12} - M\_{21}\right)\right|}\right)^2\tag{4}$$

The MATLAB code is based on these equations for optical transmission calculus.

#### **3. Structures design**

The studied structures were generated starting from the one-dimensional Thue-Morse sequence (Allouche & Shallit, 2003). The one-dimensional Thue-Morse sequence of N order,*TMN* , is a binary sequence of two symbols 'A' and 'B'. *TMN*1 is generated from *TMN* in which we substitute 'A' with 'AB' and 'B' with 'BA'. Thus *TM*<sup>0</sup> = {A}, *TM*<sup>1</sup> = {AB}, *TM*<sup>2</sup> = {ABBA}, *TM*<sup>3</sup> = {ABBABAAB} etc.

In the simulations were used one-dimensional photonic crystals of the following types: (AB)16, (ABBA)8 and (ABBABAAB)4, where A and B are two isotropic media with the refractive indexes nA and nB.

The following particular cases of one-dimensional photonic crystal composed of two types of layers were considered:


Fig. 2. The one-dimensional analyzed photonic crystal's structure

It was also considered introducing a defect layer in the structure of the analyzed photonic crystals. In the simulations were used both a A type defect layer and a B type defect layer. The position and the thickness of the defect layer will vary so to observe its influence upon

Using the matrix obtained above we calculate the optical transmission with the following

<sup>2</sup> *<sup>T</sup>*

The MATLAB code is based on these equations for optical transmission calculus.

11 22 12 21

*M M iM M* 

The studied structures were generated starting from the one-dimensional Thue-Morse sequence (Allouche & Shallit, 2003). The one-dimensional Thue-Morse sequence of N order,*TMN* , is a binary sequence of two symbols 'A' and 'B'. *TMN*1 is generated from *TMN* in which we substitute 'A' with 'AB' and 'B' with 'BA'. Thus *TM*<sup>0</sup> = {A}, *TM*<sup>1</sup> = {AB},

In the simulations were used one-dimensional photonic crystals of the following types: (AB)16, (ABBA)8 and (ABBABAAB)4, where A and B are two isotropic media with the

The following particular cases of one-dimensional photonic crystal composed of two types

a. The first case: A=dielectric material (TiO2) and B=double negative material with the permeability μA=1, respectively μB=1-100/ω2 and the permittivity εA=7.0225,

b. The second case: A=epsilon negative material and B=double negative material with the permeability μA=3, respectively μB=1-100/ω2 and the permittivity εA=1-100/ω2,

It was also considered introducing a defect layer in the structure of the analyzed photonic crystals. In the simulations were used both a A type defect layer and a B type defect layer. The position and the thickness of the defect layer will vary so to observe its influence upon

2

(4)

relationship:

**3. Structures design** 

refractive indexes nA and nB.

of layers were considered:

respectively εB=1.21-100/ω2.

respectively εB=1.21-100/ω2.

Fig. 2. The one-dimensional analyzed photonic crystal's structure

*TM*<sup>2</sup> = {ABBA}, *TM*<sup>3</sup> = {ABBABAAB} etc.

the optical transmission of the photonic crystal. The new structure of the crystal is presented in figure 3.
