Repeated {H,L} blocks determined by applying the substitution rule

## Abstract

In this chapter, we study the properties of specific one dimensional photonic quasicrystal (PQCs), in order to design an output multichannel filter. We calculate the transmittance spectrum which exhibits a photonic band gap (PBG), based on the Transfer Matrix Method (TMM) and the two-fluid model. We show that the generalized Thue-Morse (GTM) and generalized Fibonacci GF(m, n) distributions provide a stacking of similar output multichannel with zero transmission when the input was a sharp resonance of peaks at given n = 2 pm where p , is a positive integer. Also, we consider GTM configuration and we apply a deformation y = x h + 1 along the PQC filter, which enhanced the band width of each channel with respect to the number of peaks inside the main transmittance. Here, the coefficient h represents the deformation degree, x and y are thicknesses of the layers before and after the deformation, respectively. This improves the characteristics of PBG.

### Keywords

- hybrid quasiperiodic PC
- superconducting materials
- GTM sequence
- GF sequence
- multichannel optical filters
- deformed 1D photonic quasicrystals

## 1. Introduction

Photonic quasicrystals (PQCs) which are made of alternating dielectric and superconductor layers intervene in numerous researches due to their interesting optical properties [1, 2, 3, 4, 5]. This type of crystal is an artificial super lattice which is built according to quasiperiodic sequences. It is considerably different than photonic crystals (PCs) since it is a non-periodic structure with perfect long-range order and lack translational and it can be considered as an intermediate class between the random and periodic media. Our considered PQC consists of a stack of two different layers A and B which represent building blocks having a self-similarity distribution and long range order with no translational symmetry.

We mention that there are numerous examples of aperiodic chains constructed by a substitution rule. These chains allow forming many deterministic PQCs structures such as: Fibonacci, Thue-Morse, Rudin-Shapiro, Cantor, and Doubly periodic sequences.

Based on PQC heterostructure, many studies have been performed to carry out new optical devices. In this direction, the introduction of superconducting materials into the regular PQC photonic structure has been investigated in [5, 6, 7] in order to improve the characteristics of photonic band gap structures (PBGs) by changing the operating temperature of superconducting layers.

Recently, 1D deterministic multilayered structure including superconducting layers have attracted much attention in developing new kinds of optical filters which make new PQCs devices for optoelectronic system [5, 8, 9, 10, 11]. These quasiperiodic filters have been extended to thermally photonic crystals, including certain cascades superconducting/dielectric layers. It may be used in specific operations as specifying thermal sensors for remote sensing applications. In [12], the authors used superconductors instead of metals within the PC because of the damping of electromagnetic waves in metals. Moreover, the properties of PC including superconductors are mainly depending on the temperature T. In this chapter, based on hybrid dielectric/superconductor photonic quasicrystals, we develop a multi-channel optical filter with tenability around two telecom wavelengths. The main multilayered stacks are organized following quasiperiodic sequences. Hence, a multitude of channel frequencies with zero transmission can be created inside the main photonic band gap (PBG), which offers a resonance state due to the specific defects insert along the structures.

The characteristics of PBGs depend on the parameters of sequences, the thickness of the superconductor and the operating temperature. Furthermore, a multitude of transmission peaks were created within the main PBG which shifted with temperature of superconductors and lattice parameters of the aperiodic sequence.

We also show that, by monitoring the parameters of GTM, the transmission spectrum exhibit at limited gaps a cutoff frequency which is sensitive to the temperature of superconducting layers. The properties of stop channel frequencies can be notably enhanced by applying a whole deformation * x*is the main PQC and

*the structure after deformation. It is found that the gaps broad in with the increase of h. Thus, the main structure can be used to design a useful tunable multichannel filter in the optical information field.*y

## 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 (YBa_{2}Cu_{3}O_{7}).

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:

Where the wave number satisfies the corresponding equality:

with

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:

Where

where

The complex conductivity is given by this formula:

where

Where

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

where

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

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

According to TMM, the transfer matrix C_{j} verifies the following expression [15]:

For both TM and TE modes, C_{j} satisfies:

Where

For the two polarizations (p) and (s), the Fresnel coefficients

where ^{th} layer which obey to the Snell’s law:

Consequently, the transmittance satisfies [15]:

## 3. Generalized quasiperiodic sequences

### 3.1. Generalized Thue-Morse sequence

A one dimensional GTM sequence is called aperiodic because it is more disordered than the quasiperiodic one. In addition, the two different materials included in one dimensional GTM system should be structured by applying the substitution rule: _{k+1} verifying:

Based on GTM sequence

Order of GTM | Organized |
---|---|

1 | HHLL, with S_{0} = H |

2 | HHLLHHLLLLHHLLHH |

3 | HHLLHHLLLLHHLLHHHHLLHHLLLLHHLLHH LLHHLLHHHHLLHHLLLLHHLLHHHHLLHHLL |

The configuration of the proposed 1D photonic dielectric/quasiperiodic superconducting layers which is built according to the GTM sequence is shown in Figure 1.

### 3.2. Generalized Fibonacci sequence

1D Fibonacci quasiperiodic sequences are constructed by applying the inflation rule in [17]: _{k+1} satisfies the recursion relation:

The Fourier transform of Fibonacci class of quasicrystal gives discrete values which represent the significant property of crystals. We note that the eigenvalues of related matrix Fibonacci spectrum are Pisot numbers. For the Fibonacci-type, the material waves interfere constructively in appropriate length. The analysis of Fibonacci quasicrystals submitted to X-ray diffraction shows a multitude of Bragg peaks. Moreover, quasicrystals which are based on the Fibonacci distribution ordered at long distances, show a typical construction without a forbidden symmetry. Hence, the generalized Fibonacci (GF) type gives some basic proprieties which are identical to those given by simple Fibonacci class such as Fourier spectrum with Bragg peaks, inflation symmetry and localized critical modes with zero transmission called pseudo band gaps. In a generic form of the organized multilayers (H, L) through Fibonacci sequence, the four multilayered stacks are grouped in Table 2.

Order of GF | Organized |
---|---|

1 | HHLL, with |

2 | HHLLHHLLHH |

3 | H^{2}L^{2} H^{2}L^{2}HHH^{2}L^{2}H^{2}L^{2}HHH^{2}L^{2}H^{2}L^{2} |

4 | H^{2}L^{2}H^{2}L^{2}HHH^{2}L^{2}H^{2}L^{2}HHH^{2}L^{2}H^{2}L^{2}H^{2}L^{2} H^{2}L^{2}HHH^{2}L^{2}H^{2}L^{2}HHH^{2}L^{2}H^{2}L^{2}H^{2}L^{2}H^{2}L^{2}HH H^{2}L^{2}H^{2}L^{2}HH |

As an example, the third order of GF(m, n) quasiperiodic photonic structure containing alternate dielectric (D) and superconducting layers (S) with m = n = 2 is shown in Figure 2.

## 4. Results and discussion

### 4.1. Multichannel filter narrow bands by using GTM sequence

#### 4.1.1. Effect of GTM(m, n) parameters

In this subsection, we give the transmission properties of GTM and GF quasi-periodic one-dimensional photonic crystals (1DPCs) which contain superconductors. We recall that our one dimensional photonic quasicrystal is made of alternating superconductors and dielectrics (SiO_{2}) with n_{L} = 1.45. In particular, the superconductor is assumed to be YBa_{2}Cu_{3}O_{7} with a critical high-Tc temperature (Tc = 93 K) and a London penetration depth at zero temperature

We adopt TMM approach to exhibit the transmittance, band gaps and characteristics of the hybrid GTM and GF photonic quasicrystals.

Figure 3 shows the transmittance spectrum, at normal incident angle for different n values.

We remark that the spectrum give a stacking of similar channels with zero transmission covering the whole frequency range. We also observe that the number of gaps increases with an increase of the lattice parameter n of GTM.

Also, sharp peaks of transmission appear for specific multiple frequencies. All peaks prohibit the stop band gaps and form a fine zone of propagation wave. This zone constitutes a little region of transmissions with small half bandwidth

#### 4.1.2. Effect of the thickness of superconductor on GTM structure

In this part, the superconductor’s thickness is changed by varying the permittivity of its refractive index. Figure 4 shows that a large PBG augments with an augmentation the thickness. Full gaps were obtained for d_{s} = 80 nm. The amplitude of oscillations around the channels with T = 0 decreases with an increase of d_{s}. Also, a set of peaks is obtained for high values of thickness. Accordingly, the dip of each gap increases when the thickness of YBa_{2}Cu_{3}O_{7} increases, and the pseudo PBG becomes a gap with zero transmission. This improves the characteristics of channel filters.

#### 4.1.3. Quality factor (Q)

In this part, we calculate the quality factor based on the following formula: _{c}is the wavelength of maximum transmission.

Our calculation is summarized in Figure 5 which gives the evolution of quality factor Q versus the frequency center of resonant transmission peak for different superconductor temperatures T. We remark that Q is very sensitive to the position of resonant peaks in 170–171 THz frequency range and it is inversely proportional to superconductor’s temperature T. The FWHM are approximately equal for the lower frequencies and it sharply increase for the higher frequencies range. Then, a high pass filter can be obtained for lower T.

In order to show the consequences of the variation of parameter p of GTM sequence, we determine the transmittance T versus the frequency for p = 7.

As it can be seen from Figure 6, the number of defect modes or channels depends on the superconductor’s thicknesses and the distribution of layers. Moreover, the transmission spectrum exhibit a stacking of narrow gaps without oscillatory behavior. The bandwidth of each gap decreases regularly for an increase of parameter n and it probably forms a great wide PBG covering all telecommunication frequency range. The number of the transmission peaks increases as p increases. The band gaps are symmetrical about the separated transmission due to the symmetry of layers within the GTM structure.

#### 4.1.4. Effect of superconductor temperature on GTM structure

In this subsection, we study the influence of superconductor’s temperature on transmission spectrum of 1D hybrid GTM structure for different incidence levels. Thus, we evaluate the characteristics of multichannel. Indeed, Figure 7 shows that GTM multilayer stack exhibits a specific zone with zero transmission (the yellow area) for different incident angles. In the corresponding band, the propagation wave is prohibited and reached the maximum recovers for θ = 1.5 rad.

Moreover, the spectrum presented a stack of band gaps and separated by sharp transmission peaks (the blue areas) allows the propagation of wave in this specific region of frequencies. The size of propagate zone within all PBG is sensitive to temperature T of YBa_{2}Cu_{3}O_{7.} The width of transmission peak within the channels increases progressively with the increase of T. A large zero of reflection bands is also noticed for T = 80 K, it covers all optical telecommunication frequency range and it constitutes perfect reflectors in these region.

#### 4.1.5. Enhancement of PBGs by applying a particular deformation

In order to improve the characteristics of filtering channels, we apply a particular deformation h satisfying the following low * x*and

*represent the coordinates of the main and the deformed GTM heterostructures, respectively.*y

We recall that in the main structure, two forms of layer, H and L are organized in a GTM sequence, where H and L are the superconductor and dielectric materials, respectively.

Then, the optical phase becomes: * j*and

*and the reference wavelength. According to this notion, Figure 8 illustrates the distributed of H and L with low and high refractive indices of the main and deformed multilayered stack. We take*h

*= 0.1 and m = n = 2.*h

Figure 9 shows the reflectance spectrum for a corresponding deformed GTM heterostructure. For the optimum value of deformation, similar peaks of transmission appear inside all PBGs. This selective channel of transmission is sensitive to parameter n of GTM. The reflection bands form a typical output multichannel. Also, the number of channels and transmission peaks within PBGs increase when n augments. The channel of each PBG becomes narrow as n increases. In this case, m was maintained fix at 2. As a result, the characteristics of PBG are improved by applying the deformation * h*. Consequently, it is possible to improve the filtering properties by varying the suitable configuration of GTM parameters and the deformation

*.*h

In order to improve the characteristics of filtering channels, we apply a deformation to the whole thicknesses of the main GTM structure. Figure 10 shows the distributed of transmission versus frequency for varying deformation * h*. The distribution of electric fields exhibits a stacking of a bending zone with zero transmission (yellow areas) and it is limited by harmonic peaks in blue fine zone. Then, these bending reflection bands are sensitive to superconductor’s temperature. Thus, the PBGs are enhanced for an increase of T. In addition, the zone of transmission increases and the split peaks become narrow when T augments. Thus, the contrast indices of the two materials increases with T. Consequently, the considered factors cause broadening channels. Similarly, the intensity of transmission in all structures is reduced.

### 4.2. Generalized Fibonacci (GF) multichannel filters

#### 4.2.1. The effect of GF(m, n) parameters

In this subsection, we study the properties of filtering through the 1D quasiperiodic GF multilayered stacks which contain superconducting materials. The considered common sequence suggests a typical aperiodic distribution of two alternating layers H and L with high and lower refractive indices, respectively.

The two constituent materials are arranged following the GF(m, n) sequence for m = pn, where p is a positive integer. We found that the transmission spectrum give similar band gaps which depend on the distributed layers initially fixed by the GF parameters (Figure 11). Therefore, the channel with zero transmission becomes narrow when p increases. The hybrid GF heterostructure possess an oscillation transmission around all PBGs. Moreover, the stacking channels are symmetric around the reference frequency.

#### 4.2.2. The effect of contrast indices on hybrid GF(m, n) system

In this subsection, we show the effect of the contrast indices between two alternating materials on the filtering properties. The contrast indices satisfy the following relation: _{s} and n_{d} represent the refractive indices of superconductor and dielectric, respectively. The same conditions are conserved to extract the transmission through the considered GF heterostructure.

Figure 12 gives the transmittance spectrum for different values of contract indices. We mention that the GF form exhibits a large frequency range with zero transmission and shows at limited gap a sharp transition from 0 to 1 at given

The intermediate point between inhibited and propagated waves indicates the cut-off frequency that allows the signal to propagate again, showing itself as a stop band filer. Moreover, we remark that the positions of the two cut-off frequencies f_{cL} and f_{cH} are very sensitive to the contrast indices. As long as

## 5. Conclusion

The filtering properties of the 1D hybrid heterostructure built according to the GTM and GF sequences are investigated in this study. It was observed that the two common quasiperiodic sequences exhibits a multitude of channels with zero transmission for specific values of parameters m and n. In particular, the spectrum of GTM system possesses similar narrow gaps without oscillation beams at a given parameter: m = 2 pm. Indeed, a sharp transmission peak is appreciated in the whole frequency range whose positions are sensitive to superconductor temperature. Therefore, the considered system can be useful as a selective pass band multichannel filter whose narrow bandwidth can be adjusted by temperature. In addition, the main GTM system gives staking gaps which are enhanced by applying a specific deformation. Similarly, the GF heterostructure suggests an identical channel frequencies without transmission as compared to GTM system but their spectrum have particular oscillations around the cut-off frequency. Thus, the properties of filtering change by modifying the type of sequences and the parameters of constituent materials.