# The Determination of the Sensitivity of Refractive Index Sensors

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theory

#### 2.1. Model of the Refractive Index Sensor

#### 2.2. Principal Operating Mechanism of the Sensor

#### 2.3. Absolute and Relative Sensitivities

#### 2.4. Optical Sensor Resolution

## 3. Results and Discussion

#### 3.1. Comparing Absolute to Relative Sensitivity

#### 3.2. Convenient Parameters for Optimizing the Sensitivity of a Sensor Based on 1D PCs with Defects

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Santos, J.L.; Faramarz, F. Handbook of Optical Sensors; CRC Press: Boca Raton, FL, USA, 2014. [Google Scholar]
- Ferreira, M.F.; Castro-Camus, E.; Ottaway, D.J.; López-Higuera, J.M.; Feng, X.; Jin, W.; Jeong, Y.; Picqué, N.; Tong, L.; Reinhard, B.M.; et al. Roadmap on optical sensors. J. Opt.
**2017**, 19, 083001. [Google Scholar] [CrossRef] [PubMed] - Tong, L. Micro/nanofibre optical sensors: Challenges and prospects. Sensors
**2018**, 18, 903. [Google Scholar] [CrossRef] [PubMed] - Kersey, A.D. A review of recent developments in fiber optic sensor technology. Opt. Fiber Technol.
**1996**, 2, 291–317. [Google Scholar] [CrossRef] - Grattan, K.T.; Sun, T. Fiber optic sensor technology: An overview. Sens. Actuators A Phys.
**2000**, 82, 40–61. [Google Scholar] [CrossRef] - Ma, S.; Xu, Y.; Pang, Y.; Zhao, X.; Li, Y.; Qin, Z.; Liu, Z.; Lu, P.; Bao, X. Optical Fiber Sensors for High-Temperature Monitoring: A Review. Sensors
**2022**, 22, 5722. [Google Scholar] [CrossRef] [PubMed] - Wang, Z.; Wang, G.; Kumar, S.; Marques, C.; Min, R.; Li, X. Recent advancements in resonant fiber optic gyro—A review. IEEE Sens. J.
**2022**, 22, 3195502. [Google Scholar] [CrossRef] - Chai, H.; Zheng, Z.; Liu, K.; Xu, J.; Wu, K.; Luo, Y.; Liao, H.; Debliquy, M.; Zhang, C. Stability of metal oxide semiconductor gas sensors: A review. IEEE Sens. J.
**2022**, 22, 5470–5481. [Google Scholar] [CrossRef] - Li, J.; Huang, X.; Tu, L.; Zhang, T.; Wang, L. A review of building detection from very high resolution optical remote sensing images. GISci. Remote Sens.
**2022**, 59, 1199–1225. [Google Scholar] [CrossRef] - Jayawickrema, U.M.N.; Herath, H.M.; Hettiarachchi, N.K.; Sooriyaarachchi, H.P.; Epaarachchi, J.A. Fibre-optic sensor and deep learning-based structural health monitoring systems for civil structures: A review. Measurement
**2022**, 199, 111543. [Google Scholar] [CrossRef] - Efimov, I.M.; Vanyushkin, N.A.; Gevorgyan, A.H.; Golik, S.S. Optical biosensor based on a photonic crystal with a defective layer designed to determine the concentration of SARS-CoV-2 in water. Phys. Scr.
**2022**, 97, 055506. [Google Scholar] [CrossRef] - Luan, E.; Shoman, H.; Ratner, D.M.; Cheung, K.C.; Chrostowski, L. Silicon Photonic Biosensors Using Label-Free Detection. Sensors
**2018**, 18, 3519. [Google Scholar] [CrossRef] [PubMed] - Dutta, H.S.; Goyal, A.K.; Srivastava, V.; Pal, S. Coupling light in photonic crystal waveguides: A review. Photonics Nanostruct. Fundam. Appl.
**2016**, 20, 41–58. [Google Scholar] [CrossRef] - Dutta, H.S.; Suchandan, P. Design of a highly sensitive photonic crystal waveguide platform for refractive index based biosensing. Opt. Quantum Electron.
**2013**, 45, 907–917. [Google Scholar] [CrossRef] - Ahmed, A.M.; Ahmed, M. Ultra-high sensitive 1D porous silicon photonic crystal sensor based on the coupling of Tamm/Fano resonances in the mid-infrared region. Sci. Rep.
**2019**, 9, 6973. [Google Scholar] [CrossRef] [PubMed] - Abdol, S.O.; Babak, A. Novel biosensors based on Weyl semimetals. Phys. Scr.
**2022**, 97, 125502. [Google Scholar] [CrossRef] - El Beheiry, M.; Liu, V.; Fan, S.; Levi, O. Sensitivity enhancement in photonic crystal slab biosensors. Opt. Express
**2010**, 18, 22702–22714. [Google Scholar] [CrossRef] [PubMed] - Wu, F.; Fan, C.; Zhu, K.; Wu, J.; Qi, X.; Sun, Y.; Xiao, S.; Jiang, H.; Chen, H. Tailoring electromagnetic responses in a coupled-grating system with combined modulation of near-field and far-field couplings. Phys. Rev. B
**2022**, 105, 245417. [Google Scholar] [CrossRef] - Bijalwan, A.; Bipin, K.S. Analysis of one-dimensional photonic crystal based sensor for detection of blood plasma and cancer cells. Optik
**2021**, 226, 165994. [Google Scholar] [CrossRef] - Dinodiya, S.; Anami, B. Biosensor Based on One-Dimensional Photonic Crystal for Poliovirus Detection. In Advancement in Materials, Manufacturing and Energy Engineering; Springer Nature: Berlin, Germany, 2021; Volume 1. [Google Scholar]
- Miyan, H.; Agrahari, R.; Gowre, S.K.; Mahto, M.; Jain, P.K. Computational study of a compact and high sensitive photonic crystal for cancer cells detection. IEEE Sens. J.
**2022**, 22, 3298–3305. [Google Scholar] [CrossRef] - Oskoui, A.; Shojaei, S.S.; Abdollahipour, B. Polarization dependent light propagation in WTe 2 multilayer structure. Sci. Rep.
**2023**, 13, 13169. [Google Scholar] [CrossRef] - Suthar, B.; Anami, B. Enhanced optical sensor for waterborne bacteria-based photonic crystal using graded thickness index. Appl. Nanosci.
**2023**, 13, 5399–5406. [Google Scholar] [CrossRef] - Olyaee, S.; Samira, N. A high-quality factor and wide measurement range biosensor based on photonic crystal nanocavity resonator. Sens. Lett.
**2013**, 11, 483–488. [Google Scholar] [CrossRef] - Efimov, I.M.; Vanyushkin, N.A.; Golik, S.S.; Gevorgyan, A.H. Sensor with enhanced performance based on photonic crystal with a defect layer. Comput. Opt.
**2023**, 47, 572–579. [Google Scholar] [CrossRef] - Efimov, I.M.; Vanyushkin, N.A.; Gevorgyan, A.H. Peculiarities of the Electromagnetic Field Distribution Inside a 1D Photonic Crystal with a Defect Layer. Bull. Russ. Acad. Sci. Phys.
**2022**, 86 (Suppl. S1), S60–S65. [Google Scholar] [CrossRef] - Al-Dossari, M.; Awasthi, S.K.; Mohamed, A.M.; Abd El-Gawaad, N.S.; Sabra, W.; Aly, A.H. Bio-alcohol sensor based on one-dimensional photonic crystals for detection of organic materials in wastewater. Materials
**2022**, 15, 4012. [Google Scholar] [CrossRef] [PubMed] - Edappadikkunnummal, S.; Chembra Vasudevan, R.; Dinesh, S.; Thomas, S.; Desai, N.R.; Kaniyarakkal, S. Detection of Hemoglobin Concentration Based on Defective One-Dimensional Photonic Crystals. Photonics
**2022**, 9, 660. [Google Scholar] [CrossRef] - Hu, J.; Daoxin, D. Cascaded-ring optical sensor with enhanced sensitivity by using suspended Si-nanowires. IEEE Photonics Technol. Lett.
**2011**, 23, 842–844. [Google Scholar] - Baraket, Z.; Osswa, S.; Mounir, K. Design of magnetic field direction’s sensor based on a 1D tunable magneto-photonic crystal. Opt. Quantum Electron.
**2022**, 54, 637. [Google Scholar] [CrossRef] - Daher, M.G.; Jaroszewicz, Z.; Zyoud, S.H.; Panda, A.; Hasane Ahammad, S.K.; Abd-Elnaby, M.; Eid, M.M.; Rashed, A.N. Design of a novel detector based on photonic crystal nanostructure for ultra-high performance detection of cells with diabetes. Opt. Quantum Electron.
**2022**, 54, 701. [Google Scholar] [CrossRef] - Almawgani, A.H.; Daher, M.G.; Taya, S.A.; Mashagbeh, M.; Colak, I. Optical detection of fat concentration in milk using MXene-based surface plasmon resonance structure. Biosensors
**2022**, 12, 535. [Google Scholar] [CrossRef] - Yupapin, P.; Trabelsi, Y.; Vigneswaran, D.; Taya, S.A.; Daher, M.G.; Colak, I. Ultra-high-sensitive sensor based on surface plasmon resonance structure having Si and graphene layers for the detection of chikungunya virus. Plasmonics
**2022**, 17, 1315–1321. [Google Scholar] [CrossRef] - Almawgani, A.H.; Daher, M.G.; Taya, S.A.; Colak, I.; Patel, S.K.; Ramahi, O.M. Highly sensitive nano-biosensor based on a binary photonic crystal for cancer cell detection. Opt. Quantum Electron.
**2022**, 54, 554. [Google Scholar] [CrossRef] - Taya, S.A.; Daher, M.G.; Colak, I.; Ramahi, O.M. Highly sensitive nano-sensor based on a binary photonic crystal for the detection of mycobacterium tuberculosis bacteria. J. Mater. Sci. Mater. Electron.
**2021**, 32, 28406–28416. [Google Scholar] [CrossRef] - Panda, A.; Pukhrambam, P.D.; Wu, F.; Belhadj, W. Graphene-based 1D defective photonic crystal biosensor for real-time detection of cancer cells. Eur. Phys. J. Plus
**2021**, 136, 809. [Google Scholar] [CrossRef] - Khani, S.; Hayati, M. Optical biosensors using plasmonic and photonic crystal band-gap structures for the detection of basal cell cancer. Sci. Rep.
**2022**, 12, 5246. [Google Scholar] [CrossRef] [PubMed] - Winn, J.; Fink, Y.; Fan, S.; Joannopoulos, J.D. Omnidirectional reflection from a one-dimensional photonic crystal. Opt. Lett.
**1998**, 23, 1573–1575. [Google Scholar] [CrossRef] [PubMed] - Vinogradov, A.; Dorofeenko, A.V.; Erokhin, S.G.; Inoue, M.; Lisyansky, A.A.; Merzlikin, A.M.; Granovsky, A.B. Surface state peculiarities in one-dimensional photonic crystal interfaces. Phys. Rev. B
**2006**, 74, 045128. [Google Scholar] [CrossRef] - Wu, F.; Lu, G.; Guo, Z.; Jiang, H.; Xue, C.; Zheng, M.; Chen, C.; Du, G.; Chen, H. Redshift gaps in one-dimensional photonic crystals containing hyperbolic metamaterials. Phys. Rev. Appl.
**2018**, 10, 064022. [Google Scholar] [CrossRef] - Wu, F.; Liu, T.; Xiao, S. Polarization-sensitive photonic bandgaps in hybrid one-dimensional photonic crystals composed of all-dielectric elliptical metamaterials and isotropic dielectrics. Appl. Opt.
**2023**, 62, 706–713. [Google Scholar] [CrossRef] - Joannopoulos, J.D. Photonic Crystals: Molding the Flow of Light; Princeton University Press: Princeton, NJ, USA, 1995; 305p. [Google Scholar]
- Yeh, P. Optical Waves in Layered Media; Wiley: New York, NY, USA, 1988. [Google Scholar]
- Yariv, A.; Yeh, P. Optical Waves in Crystals; Wiley: New York, NY, USA, 1984. [Google Scholar]
- Vanyushkin, N.A.; Gevorgyan, A.H.; Golik, S.S. Scattering of a plane wave by an inhomogeneous 1D dielectric layer with gradient refractive index. Opt. Mater.
**2022**, 127, 112306. [Google Scholar] [CrossRef] - Fan, X.; White, I.M.; Shopova, S.I.; Zhu, H.; Suter, J.D.; Sun, Y. Sensitive optical biosensors for unlabeled targets: A review. Anal. Chim. Acta
**2008**, 620, 8–26. [Google Scholar] [CrossRef] - Yao, J. Optoelectronic oscillators for high speed and high-resolution optical sensing. J. Light. Technol.
**2017**, 35, 3489–3497. [Google Scholar] [CrossRef] - Qin, J.; Jiang, S.; Wang, Z.; Cheng, X.; Li, B.; Shi, Y.; Tsai, D.P.; Liu, A.Q.; Huang, W.; Zhu, W. Metasurface micro/nano-optical sensors: Principles and applications. ACS Nano
**2022**, 16, 11598–11618. [Google Scholar] [CrossRef] - Van de Velde, F.; Knutsen, S.H.; Usov, A.I.; Rollema, H.S.; Cerezo, A.S. 1H and 13C high resolution NMR spectroscopy of carrageenans: Application in research and industry. Trends Food Sci. Technol.
**2002**, 13, 73–92. [Google Scholar] [CrossRef] - Wang, K.; Vincent, M. High-resolution photoelectron spectroscopy of molecules. Annu. Rev. Phys. Chem.
**1995**, 46, 275–304. [Google Scholar] [CrossRef] - Vanyushkin, N.A.; Gevorgyan, A.H.; Golik, S.S. Approximation of one-dimensional rugate photonic crystals using symmetric ternary photonic crystals. Optik
**2021**, 242, 167343. [Google Scholar] [CrossRef]

**Figure 1.**Schematic diagram of the structure under investigation. $N$ is the number of unit cells, ${\theta}_{0}$ is the angle of incidence, ${E}_{\mathrm{r}}$ is the reflected wave, ${E}_{0}$ is the incident wave, ${E}_{\mathrm{t}}$ is the transmitted wave, and ${n}_{0}$ and ${n}_{\mathrm{f}}$ are refractive indices of the structure surroundings.

**Figure 2.**Transmission spectra of the defective PC with two different values of ${n}_{d}$. The blue line is the spectrum for a pure defective layer, and the red line is for a defective layer with inclusions.

**Figure 3.**Transmission spectra at ${n}_{\mathrm{d}}=1.75$ and $1.90$ (blue and red lines). The structure parameters are ${n}_{1}=1.5$, ${n}_{2}=2.0$, $N=10$, ${h}_{1}$ = ${h}_{2}=500\text{}k$ nm, and ${h}_{\mathrm{d}}=1050\text{}k$ nm, (

**a**) $k=1,\text{}\left(\mathbf{b}\right)\text{}k=5,\left(\mathbf{c}\right)\text{}k=10$.

**Figure 4.**Dependence of absolute sensitivity on the width of the PBG (

**a**,

**c**,

**e**) and that of the relative sensitivity on the relative width of the PBG (

**b**,

**d**,

**f**). Here, $\mathrm{d}\lambda $ is varied by changing ${h}_{1}$. The green arrows indicate the increase in ${h}_{1}$ over a constant period. The red line marks the fulfilment of the quarter-wave stack. ${\lambda}_{\mathrm{B}}$ is the wavelength of the PBG center. The other parameters are the same as in Figure 3a.

**Figure 5.**Dependence of the relative width of the PBG (

**a**) and relative sensitivity (

**b**) on the refractive index difference. The parameters are as follows: ${n}_{\mathrm{m}}=1.75$ and ${h}_{1}{n}_{1}={h}_{2}{n}_{2}={\lambda}_{\mathrm{B}}/4$. The other parameters are the same as in Figure 3a.

**Figure 6.**Dependence of the difference in the DM position ${\mathsf{\lambda}}_{\mathrm{D}\mathrm{M}}$ and the PBG center ${\mathsf{\lambda}}_{\mathrm{B}}$ on the DL thickness ${h}_{\mathrm{d}}$ and the refractive index difference $\u2206n$. The parameters are the same as in Figure 5b.

${\mathit{S}}_{\mathbf{a}}$$,\text{}\frac{\mathbf{nm}}{\mathbf{RIU}}$ | ${\mathsf{\lambda}}_{0}$$,\text{}\mathbf{nm}$ | ${\mathit{S}}_{\mathbf{r}}$$,\text{}{\mathbf{RIU}}^{-1}$ | Article |
---|---|---|---|

1020 | 5293 | 0.1927 | [11] |

80 | 5295 | 0.0151 | |

347 | 1004 | 0.3456 | |

710 | 2026 | 0.3504 | |

260 | 1560 | 0.1667 | [14] |

197 | 1600 | 0.1231 | |

198 | 1620 | 0.1222 | |

173 | 1640 | 0.1055 | |

80 | 1800 | 0.0444 | |

5018 | 7299 | 0.6875 | [15] |

5092 | 7299 | 0.6977 | |

5031 | 7293 | 0.6899 | |

5013 | 7335 | 0.6834 | |

500 | 523 | 0.9560 | [27] |

496 | 533 | 0.9316 | |

490 | 541 | 0.9058 | |

487 | 547 | 0.8897 | |

475 | 557 | 0.8520 | |

454 | 577 | 0.7875 | |

405 | 611 | 0.6639 | |

145 | 764 | 0.1891 | [28] |

144 | 766 | 0.1881 | |

144 | 769 | 0.1879 | |

144 | 773 | 0.1863 | |

1300 | 1530 | 0.8497 | [29] |

515 | 1550 | 0.3322 |

$\mathit{k}$ | ${\mathit{\lambda}}_{\mathbf{B}}$, nm | ${\mathit{S}}_{\mathbf{a}}$$,\text{}\frac{\mathbf{nm}}{\mathbf{RIU}}$ | ${\mathit{S}}_{\mathbf{r}}$$,\text{}{\mathbf{RIU}}^{-1}$ |
---|---|---|---|

1 | 3500 | 440.33 | 0.1258074 |

2 | 7000 | 880.65 | |

3 | 10,500 | 1320.98 | |

4 | 14,000 | 1761.30 | |

5 | 17,500 | 2201.63 | |

10 | 35,000 | 4403.26 |

$\mathit{k}$ | ${\mathit{S}}_{\mathbf{a}}$$,\text{}\frac{\mathbf{nm}}{\mathbf{RIU}}$ | ${\mathit{S}}_{\mathbf{a}}$$,\text{}\frac{\mathbf{T}\mathbf{H}\mathbf{z}}{\mathbf{RIU}}$ | ${\mathit{S}}_{\mathbf{r}}$$,\text{}{\mathbf{RIU}}^{-1}$ | $\mathbf{Min}\text{}\u2206\mathsf{\lambda},\mathbf{n}\mathbf{m}$ | $\u2206{\mathit{n}}_{\mathbf{d}}$ |
---|---|---|---|---|---|

1 | 440.33 | 680.83 | 0.1258074 | 3.5 | 0.002 |

2 | 880.65 | 340.42 | 7.0 | ||

5 | 2201.63 | 136.17 | 17.5 | ||

10 | 4403.26 | 68.08 | 35.0 |

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**MDPI and ACS Style**

Efimov, I.M.; Vanyushkin, N.A.; Gevorgyan, A.H.
The Determination of the Sensitivity of Refractive Index Sensors. *Photonics* **2024**, *11*, 56.
https://doi.org/10.3390/photonics11010056

**AMA Style**

Efimov IM, Vanyushkin NA, Gevorgyan AH.
The Determination of the Sensitivity of Refractive Index Sensors. *Photonics*. 2024; 11(1):56.
https://doi.org/10.3390/photonics11010056

**Chicago/Turabian Style**

Efimov, Ilya M., Nikolay A. Vanyushkin, and Ashot H. Gevorgyan.
2024. "The Determination of the Sensitivity of Refractive Index Sensors" *Photonics* 11, no. 1: 56.
https://doi.org/10.3390/photonics11010056