2. Problem formulation

In all this work, the photometric response (transmission and reflection) through the 1D photonic quasicrystal which contains superconductors, are determined by using the Transfer Matrix Method (TMM). We use also the theoretical Gorter-Casimir two-fluid model [13, 14] to describe the properties of the superconductor material (YBa2Cu3O7).

The application of the two-fluid models and Maxwell's equations through, imply that the superconducting materials' electric field equation, obeys to the following equation:

$$\nabla^2 \mathbf{E} + \mathbf{k}\_s \, ^2 \mathbf{E} = \mathbf{0} \tag{1}$$

Where the wave number satisfies the corresponding equality:

$$\mathbf{k}\_s = \sqrt{\frac{\alpha^2}{\mathbf{c}^2} - \frac{\mathbf{1}}{\lambda\_\perp^2}}\tag{2}$$

<sup>p</sup>ffiffiffiffiffiffiffiffiffi with <sup>μ</sup><sup>0</sup> and <sup>c</sup> <sup>¼</sup> <sup>1</sup><sup>=</sup> <sup>μ</sup>0ε<sup>0</sup> denote the permeability and the speed of light in free space, respectively.

As mentioned above, the electromagnetic response of superconducting materials with the absence of an external magnetic field was defined by the Gorter-Casimir two-fluid models (GCTFM) in [13, 14]. According to GCTFM, the complex conductivity of a superconductor satisfies the following expression:

Photonic Quasicrystals for Filtering Application DOI: http://dx.doi.org/10.5772/intechopen.81572

$$\sigma(\boldsymbol{\alpha}) = \frac{-\mathbf{i}\mathbf{e}^2 \mathbf{n}\_s}{\mathbf{m}\boldsymbol{\alpha}} \tag{3}$$

Where ns is the electron density and ω is the frequency of incident electromagnetic wave. Moreover, e and m represent the charge and the mass of electron, respectively. Under the approximation condition indicated in [14], the imaginary part of conductivity is given as follows:

$$\sigma(\mathbf{w}) \approx \frac{-\mathbf{i}}{\imath \mu \mu\_0 \lambda\_\mathbf{L}^2(\mathbf{T})} \tag{4}$$

where λ<sup>L</sup> signifies the term of London penetration depths and satisfies the following equality:

$$
\lambda\_{\rm L}^{2} = \frac{\rm m}{\mu\_{0} \mathbf{n}\_{\rm s} \mathbf{e}^{2}}.\tag{5}
$$

The complex conductivity is given by this formula: σ ¼ σ<sup>1</sup> � jσ2, where σ<sup>1</sup> and σ<sup>2</sup> are the real and imaginary parts of σ. Thus, the complex conductivity satisfies [14]:

$$
\sigma\_2 = \frac{1}{\alpha \mu\_0 \lambda\_L^2(\mathbf{T})},\tag{6}
$$

where ω is the operating frequency. The London temperature-dependent penetration depth is:

$$
\lambda\_{\mathbb{L}}(\mathbf{T}) = \frac{\lambda(\mathbf{0})}{\sqrt{\mathbf{1} - \mathbf{G}(\mathbf{T})}} \tag{7}
$$

Where λð Þ 0 denotesthe London temperaturepenetration depth atT=0K, and G(T) isthe Gorter-Casimirfunction. In this case, G T <sup>ð</sup> <sup>2</sup> ð Þ¼ <sup>T</sup>=Tc<sup>Þ</sup> , where Tc and <sup>T</sup> are the critical and the operating temperatures of the superconductor,respectively.

Based on the Gorter-Casimir theory, we obtain that the relative permittivity of lossless superconductors takes the following equality [14]:

$$
\varepsilon\_s = 1 - \frac{\alpha\_{\rm th}^2}{\alpha^2},
\tag{8}
$$

where ωth is the threshold frequency of the bulk superconductor which satisfies: ω2 th ¼� <sup>c</sup><sup>2</sup>=λ<sup>2</sup> .

Then, the refractive index of the superconductor is written as follows:

$$\mathbf{n}\_{\mathbf{s}} = \sqrt{\mathbf{e}\_{\mathbf{s}}} = \sqrt{\mathbf{1} - \frac{\mathbf{1}}{\alpha^2 \mu\_0 \mathbf{e}\_0 \lambda\_{\mathbf{L}}^2}},\tag{9}$$

In the following, the photometric response through the 1D photonic quasicrystal which contains superconductors, is extracted using the Transfer Matrix Method (TMM). This approach shows that the determination of the reflectance R and the transmittance T depends on refractive indices ns and lower refractive indices nd .

According to TMM, the transfer matrix Cj verifies the following expression [15]:

$$
\begin{bmatrix} \mathbf{E}\_0^+ \\ \mathbf{E}\_0^- \end{bmatrix} = \prod\_{i=1}^{\mathrm{m}} \frac{\mathbf{C}\_j}{t\_j} \begin{bmatrix} \mathbf{E}\_{\mathrm{m}+1}^+ \\ \mathbf{E}\_{\mathrm{m}+1}^- \end{bmatrix}, \tag{10}
$$

For both TM and TE modes, Cj satisfies:

$$\mathbf{C}\_{\mathbf{j}} = \begin{pmatrix} \exp\left(\mathbf{i}\mathfrak{o}\_{\mathbf{j}-1}\right) & \mathbf{r}\_{\mathbf{j}}\exp\left(-\mathbf{i}\mathfrak{o}\_{\mathbf{j}-1}\right) \\\mathbf{r}\_{\mathbf{j}}\exp\left(\mathbf{i}\mathfrak{o}\_{\mathbf{j}-1}\right) & \exp\left(-\mathbf{i}\mathfrak{o}\_{\mathbf{j}-1}\right) \end{pmatrix},\tag{11}$$

Where φ<sup>j</sup> <sup>1</sup> denotes the phase between the two succeed interfaces and it is given by the following formula

$$\mathbf{q}\_{\mathbf{j}-1} = \frac{2\pi}{\lambda} \hat{\mathbf{n}}\_{\mathbf{j}-1} \mathbf{d}\_{\mathbf{j}-1} \cos \theta\_{\mathbf{j}-1} \tag{12}$$

For the two polarizations (p) and (s), the Fresnel coefficients tj and rj take the following equalities [15]:

$$\begin{aligned} \mathbf{r}\_{\text{jp}} &= \frac{\hat{\mathbf{n}}\_{\text{j}-1}\mathbf{cos}\mathbf{\theta}\_{\text{j}} - \hat{\mathbf{n}}\_{\text{j}}\mathbf{cos}\mathbf{\theta}\_{\text{j}-1}}{\hat{\mathbf{n}}\_{\text{j}-1}\mathbf{cos}\mathbf{\theta}\_{\text{j}} + \hat{\mathbf{n}}\_{\text{j}}\mathbf{cos}\mathbf{\theta}\_{\text{j}-1}}; \quad \mathbf{t}\_{\text{jp}} = \frac{2\hat{\mathbf{n}}\_{\text{j}-1}\mathbf{cos}\mathbf{\theta}\_{\text{j}-1}}{\hat{\mathbf{n}}\_{\text{j}-1}\mathbf{cos}\mathbf{\theta}\_{\text{j}} + \hat{\mathbf{n}}\_{\text{j}}\mathbf{cos}\mathbf{\theta}\_{\text{j}-1}} \\ \mathbf{r}\_{\text{js}} &= \frac{\hat{\mathbf{n}}\_{\text{j}-1}\mathbf{cos}\mathbf{\theta}\_{\text{j}-1} - \hat{\mathbf{n}}\_{\text{j}}\mathbf{cos}\mathbf{\theta}\_{\text{j}}}{\hat{\mathbf{n}}\_{\text{j}-1}\mathbf{cos}\mathbf{\theta}\_{\text{j}-1} + \hat{\mathbf{n}}\_{\text{j}}\mathbf{cos}\mathbf{\theta}\_{\text{j}}}; \quad \mathbf{t}\_{\text{js}} = \frac{2\hat{\mathbf{n}}\_{\text{j}-1}\mathbf{cos}\mathbf{\theta}\_{\text{j}-1}}{\hat{\mathbf{n}}\_{\text{j}-1}\mathbf{cos}\mathbf{\theta}\_{\text{j}-1} + \hat{\mathbf{n}}\_{\text{j}}\mathbf{cos}\mathbf{\theta}\_{\text{j}}}, \end{aligned} \tag{13}$$

^ ^ nj and θ<sup>j</sup> are respectively the refractive indices and the angle of incidence nj ^ where in the j th layer which obey to the Snell'<sup>s</sup> law: 1sin ] <sup>j</sup> <sup>1</sup> <sup>¼</sup> njsinj with <sup>j</sup>∈½1; <sup>m</sup> <sup>þ</sup> <sup>1</sup> .

Consequently, the transmittance satisfies [15]:

$$\mathbf{T\_{rs}} = \text{Re}\left(\frac{\hat{\mathbf{n}}\_{\mathrm{m}+1}\cos\Theta\_{\mathrm{m}+1}}{\hat{\mathbf{n}}\_{0}\cos\Theta\_{0}}\right)|\mathbf{t\_{S}}|^{2};\ \mathbf{T\_{rp}} = \text{Re}\left(\frac{\hat{\mathbf{n}}\_{\mathrm{m}+1}\cos\Theta\_{\mathrm{m}+1}}{\hat{\mathbf{n}}\_{0}\cos\Theta\_{0}}\right)|\mathbf{t\_{P}}|^{2},\tag{14}$$
