# Narrowband Filters Designed from Hybrid One-Dimensional Periodic/Quasiperiodic Photonic Crystals with a Single Defect Layer

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## Abstract

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## 1. Introduction

## 2. Description of the System

#### 2.1. Theoretical Framework

#### 2.2. Simulation Settings

## 3. Results and Discussion

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Schematic diagram of the hybrid periodic/Fibonacci dielectric structures enclosing the defect layer, D, in the filter. The upper plot corresponds to the case of HFB(4,2)-D-HFB(4,2) (see text for notation details), using 4-unit biperiodic Bragg mirrors (BMs) and the second-order Fibonacci sequence. The lower plot shows the case THFB(4,2)-D-THFB(4,2) in which the triperiodic Bragg mirrors (TBMs) with four $ABC$ units are use in the hybrid scheme, with the same Fibonacci FB(2) quasiperiodic part. (

**b**) The variation in the refractive indices for constituent materials with respect to the signal wavelength ($\lambda $) [36,37,38].

**Figure 2.**Distribution of the normalized absolute field amplitude for different biperiodic/Fibonacci heterostructures with a defective layer HFB(4,M)-D-HFB(4,M) [see Figure 1a and the text for a detailed explanation]. Each plot corresponds to a different order of Fibonacci sequence: (

**a**) FB(2)→ HFB(4,3)-D-HFB(4,3); (

**b**) FB(3)→ HFB(4,3)-D-HFB(4,3); (

**c**) FB(4)→ HFB(4,4)-D-HFB(4,4); (

**d**) FB(5)→HFB(4,5)-D-HFB(4,5). The insets show the calculated optical transmission for each structure.

**Figure 3.**The normalized electric field for propagating TM modes located within the photonic gap in the hybrid Bragg/Fibonacci $AB$-based heterostructures with a central defect layer D (see details in the text). (

**a**–

**c**) correspond to Figure 2a, whilst (

**d**–

**f**) correspond to Figure 2b, for the specific values of wavelength shown in the plot labels, together with their respective quality factors Q. In the upper inset of each graphics, the distribution of the normalized field amplitude appears as a density plot.

**Figure 4.**The normalized electric field for propagating TM modes located within the photonic gap in the hybrid Bragg/Fibonacci $AB$-based heterostructures with a central defect layer D (see details in the text). (

**a**–

**c**) correspond to Figure 2c, whilst (

**d**–

**g**) correspond to Figure 2d, for the specific values of wavelength shown in the plot labels, together with their respective quality factors Q. In the upper inset of each graphics, the distribution of the normalized field amplitude appears as a density plot.

**Figure 5.**Distribution of the normalized absolute TM field amplitude for different triperiodic Bragg/Fibonacci hybrid heterostructures with a defective layer (D) THFB(4,M)-D-THFB(4,M) with $M=2,3,4,5$ (see Figure 1). In all cases, four-period Bragg mirrors are used with Fibonacci sequences as follows: (

**a**) FB(2)→THFB(4,2)-D-THFB(4,2); (

**b**) FB(3)→THFB(4,3)-D-THFB(4,3); (

**c**) FB(4)→ THFB(4,4)-D-THFB(4,4); (

**d**) FB(5)→THFB(4,5)-D-THFB(4,5). The insets show the calculated optical transmission for each structure.

**Figure 6.**The normalized electric field for propagating TM modes located within the photonic gap (black in the density plots) in the hybrid Bragg/Fibonacci $ABC$-based heterostructures with a central defect layer D (see details in the text). (

**a**–

**c**) correspond to Figure 5a, for the specific values of wavelength shown in the plot labels, together with their respective quality factors Q. In the upper inset of each graphics, the distribution of normalized field amplitude appears as a density plot.

**Figure 8.**The normalized electric field for propagating TM modes located within the photonic gap in the hybrid Bragg/Fibonacci $ABC$-based heterostructures with a central defect layer D (see details in the text). (

**a**–

**c**) correspond to modes in Figure 5d.

**Figure 9.**(

**a**) The change in the main transmission peak position for the filtering structure formed of hybrid triperiodic/Fibonacci multilayers enclosing a defect layer, as a result of the variation in refractive index of the substance inside it. (

**b**) Linear fitting for the dependence of the peak wavelength with the refractive index.

**Figure 10.**Profiles of the normalized electric field amplitude for the transmission modes corresponding to the different values of refractive index in the data of Figure 9a. (

**a**–

**f**) conrresponding to refractive index $2.565$, $2.589$, $2.610$, $2.639$, $2.655$ and $2.661$, respectively.

n | Creatinine Concentration (μmolL${}^{-1}$) |
---|---|

2.661 | 80.9 |

2.655 | 81.43 |

2.639 | 82.3 |

2.610 | 83.3 |

2.589 | 84.07 |

2.565 | 85.28 |

**Table 2.**Comparison of the values of the resonant wavelength in μm, the quality factor of the proposed filter, and the function of merit (FOM) for the different values of the refractive index of analyte, n, shown in column 1.

n | $\mathit{\lambda}$ (μm) | Q $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{6}$ | FOM × 10${}^{5}$ (RIU${}^{-1}$) |
---|---|---|---|

2.661 | 0.92469 | 2.1 | 2.6 |

2.655 | 0.92283 | 1.9 | 2.4 |

2.639 | 0.92204 | 1.4 | 1.8 |

2.610 | 0.91857 | 1.7 | 2.2 |

2.589 | 0.91607 | 1.8 | 1.4 |

2.565 | 0.91324 | 1.5 | 2.0 |

**Table 3.**Comparison of the numeric values for the sensitivity, FOM, and LOD of the proposed design with previous designs for evaluating the performance of the proposed narrowband filter.

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**MDPI and ACS Style**

Murillo-García, W.; Gómez-Urrea, H.A.; Mora-Ramos, M.E.; Duque, C.A.
Narrowband Filters Designed from Hybrid One-Dimensional Periodic/Quasiperiodic Photonic Crystals with a Single Defect Layer. *Condens. Matter* **2023**, *8*, 50.
https://doi.org/10.3390/condmat8020050

**AMA Style**

Murillo-García W, Gómez-Urrea HA, Mora-Ramos ME, Duque CA.
Narrowband Filters Designed from Hybrid One-Dimensional Periodic/Quasiperiodic Photonic Crystals with a Single Defect Layer. *Condensed Matter*. 2023; 8(2):50.
https://doi.org/10.3390/condmat8020050

**Chicago/Turabian Style**

Murillo-García, Waira, Hernán A. Gómez-Urrea, Miguel E. Mora-Ramos, and Carlos A. Duque.
2023. "Narrowband Filters Designed from Hybrid One-Dimensional Periodic/Quasiperiodic Photonic Crystals with a Single Defect Layer" *Condensed Matter* 8, no. 2: 50.
https://doi.org/10.3390/condmat8020050