# Comparison of the Optical Planar Waveguide Sensors’ Characteristics Based on Guided-Mode Resonance

^{1}

^{2}

^{3}

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## Abstract

**:**

## 1. Introduction

^{−8}is so far the highest value reported for any SPR-based sensor [14]. The variety of fiber sensors is much higher than that based on planar waveguides, but their fabrication is much more complicated, as evidenced by the overwhelming number of theoretical works using numerical methods.

## 2. Spectral and Angular Sensors’ Sensitivities and Their Relationship with Planar Waveguide Parameters

^{−1}for our waveguide parameters, which consist of the grating, the substrate and the tested medium. The difference between the right and left sides of Equations (1) and (3) is less than 0.001 µm

^{−1}. The less the modulation of the grating refractive index, the smaller the difference between the right and left sides of Equation (4) [4]. Discrete propagation constants in the planar dielectric waveguides can be determined by the method described in [34], as well as by the numerical method in the frequency domain, which provides precision calculation accuracy [41,42]. All calculations for three structures shown in Figure 1 were carried out for the resonance wavelength of 1.064 µm.

#### 2.1. Relationship Between the Parameters of the Sensor Based on the Prism Structure and the Properties of the Planar Waveguide Dielectric/Thin Metal Film/Dielectric

#### 2.2. Sensitivity of the Sensor Based on the Metal Grating under Resonant Excitation of the Surface Plasmon–Polariton Wave

#### 2.3. Sensitivity of the Sensor Based on the Dielectric Grating under Resonant Excitation of the Localized Waveguide Mode

#### 2.4. Comparison of the Properties of Sensors based on Planar Waveguides

## 3. Discussion

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Sensitive elements of the sensors, where ${\mathsf{\epsilon}}_{m}$ is the dielectric constant of the metal, ${n}_{a}$ is the test medium refractive index, $d$ is the metal film thickness or the grating thickness, $\mathsf{\Lambda}$ is the grating period, $\mathsf{\beta}$ is the propagation constant of the waveguide modes under resonance. (

**a**) is the prism structure, where $n$ is the prism refractive index, $\mathsf{\theta}$ is the angle of incidence of the light beam. (

**b**) is the metal grating on the metal substrate, where $F$ is the fill factor, which, in our studies, is equal to 0.5. (

**c**) is the dielectric grating-based structure, where ${n}_{g}$ is the average refractive index of the dielectric grating, ${n}_{s}$ is the refractive index of the substrate. $\mathsf{\theta}$ is the angle of incidence of the light beam in the air with the refractive index $n=1$ for Figure 1a,b.

**Figure 2.**Determination of the resonant thickness of the metal film and the angle of incidence on the film with the following parameters = 1.56, n

_{a}= 1.3242, λ = 1.064 μm.

**Figure 3.**Graphical dependences Re$\left(B\left(x,y\right)\right)=0$ (red curve) and Im$\left(B\left(x,y\right)\right)=0$ (green curve) for the following parameters, $\mathsf{\lambda}=1.064\text{}\mu m$, $\mathsf{\theta}=1.040217$ rad, $d=52.697$ nm, $n=1.56,$ ${n}_{a}=1.3242$. The inset shows the progress of the green and red curves on an enlarged scale.

**Table 1.**Analytical expressions connecting the characteristics of the sensitive elements of the three types of sensors with the parameters of the grating and the corresponding waveguide.

Parameter | ${\mathit{S}}_{\mathit{\lambda}}$ | ${\mathit{S}}_{\mathit{\theta}}$ | $\mathit{\delta}{\mathit{\lambda}}_{0.5}$ | $\frac{{\mathit{S}}_{\mathit{\lambda}}}{\mathit{\delta}{\mathit{\lambda}}_{0.5}}=\frac{{\mathit{S}}_{\mathit{\theta}}}{\mathit{\delta}{\mathit{\theta}}_{0.5}}$ |
---|---|---|---|---|

No | 1 | 2 | 3 | 4 |

Figure 1a | $-\frac{\lambda {S}_{n}^{\left(\beta \right)}}{\lambda {S}_{\lambda}^{\left(\beta \right)}+\beta}$ | $\frac{\lambda {S}_{n}^{\left(\beta \right)}}{2\pi n\mathrm{cos}\theta}$ | $\frac{2\pi n\mathrm{cos}\theta}{\lambda {S}_{\lambda}^{\left(\beta \right)}+\beta}\delta {\theta}_{0.5}$ | $\frac{\lambda {S}_{n}^{\left(\beta \right)}}{2\pi n\mathrm{cos}\theta \delta {\theta}_{0.5}}$ |

Figure 1b | $\Lambda $ | $\frac{1}{\mathrm{cos}\theta}$ | $\Lambda \mathrm{cos}\theta \delta {\theta}_{0.5}$ | $\frac{1}{\mathrm{cos}\theta \delta {\theta}_{0.5}}$ |

Figure 1c | $\frac{\lambda \Lambda}{2\pi}{S}_{n}^{\left(\beta \right)}$ | $\frac{\lambda {S}_{n}^{\left(\beta \right)}}{2\pi \mathrm{cos}\theta}$ | $\Lambda \mathrm{cos}\theta \delta {\theta}_{0.5}$ | $\frac{\lambda {S}_{n}^{\left(\beta \right)}}{2\pi \mathrm{cos}\theta \delta {\theta}_{0.5}}$ |

**Table 2.**The parameters of the sensors’ sensitive elements (columns 1–2) and their characteristics calculated numerically (columns 3–10).

Parameter | ${\mathit{n}}_{\mathit{a}}$ | $\mathbf{\Lambda},\text{}\mathbf{nm}$ | $\mathit{d}$ | $\mathsf{\theta},\text{}\mathbf{mrad}$ | $\mathsf{\delta}{\mathsf{\lambda}}_{\mathbf{0.5}},\text{}\mathbf{nm}$ | $\mathsf{\delta}{\mathsf{\theta}}_{\mathbf{0.5}},\text{}\mathbf{mrad}$ | ${\mathit{S}}_{\mathsf{\lambda}},\text{}\mathbf{nm}$ | ${\mathit{S}}_{\mathsf{\theta}},\text{}\mathbf{mrad}$ | $\frac{{\mathit{S}}_{\mathsf{\lambda}}}{\mathsf{\delta}{\mathsf{\lambda}}_{\mathbf{0.5}}}$ | $\frac{{\mathit{S}}_{\mathsf{\theta}}}{\mathsf{\delta}{\mathsf{\theta}}_{\mathbf{0.5}}}$ |
---|---|---|---|---|---|---|---|---|---|---|

No | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |

Figure 1a | 1.3242 | – | 52.7 | 1040.2 | 47.14 | 2.611 | 24,043 | 1327.8 | 510 | 509 |

Figure 1b | 1.3242 | 500 | 10.0 | 896.2 | 0.95 | 3.07 | 500 | 1605 | 526 | 523 |

Figure 1c | 1.5 | 570.35 | 1870 | 349.1 | 0.0044 | 0.008243 | 75.8 | 141.9 | 17,227 | 17,215 |

**Table 3.**The parameters of the sensors’ sensitive elements (columns 1–3) and their characteristics calculated from the obtained equations (columns 6–9).

Parameter | ${\mathit{n}}_{\mathit{a}}$ | $\mathsf{\theta},\text{}\mathbf{mrad}$ | $\mathsf{\delta}{\mathsf{\theta}}_{\mathbf{0.5}},\text{}\mathbf{mrad}$ | ${\mathit{S}}_{\mathsf{\lambda}}^{\left(\mathsf{\beta}\right)},\text{}\mathit{\mu}{\mathbf{m}}^{-\mathbf{2}}$ | ${\mathit{S}}_{\mathit{n}}^{\left(\mathsf{\beta}\right)},\text{}\mathit{\mu}{\mathbf{m}}^{-\mathbf{1}}$ | $\mathsf{\delta}{\mathsf{\lambda}}_{\mathbf{0.5}},\text{}\mathbf{nm}$ | ${\mathit{S}}_{\mathsf{\lambda}},\text{}\mathbf{nm}$ | ${\mathit{S}}_{\mathsf{\theta}},\text{}\mathbf{mrad}$ | $\frac{{\mathit{S}}_{\mathsf{\lambda}}}{\mathsf{\delta}{\mathsf{\lambda}}_{\mathbf{0.5}}}$ | $\frac{{\mathit{S}}_{\mathsf{\theta}}}{\mathsf{\delta}{\mathsf{\theta}}_{\mathbf{0.5}}}$ |
---|---|---|---|---|---|---|---|---|---|---|

No | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 10 | 11 |

Figure 1a | 1.3242 | 1040.2 | 2.611 | – 7.725 | 6.190 | 47.33 | 24,056 | 1326.4 | 508 | 508 |

Figure 1b | 1.3242 | 896.2 | 3.07 | – 7.35 | 5.910 | 0.959 | 500 | 1601 | 521 | 522 |

Figure 1c | 1.5 | 349.1 | 0.008243 | – 8.48 | 0.7859 | 0.00442 | 75.9 | 141.6 | 17,172 | 17,178 |

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**MDPI and ACS Style**

Bellucci, S.; Fitio, V.; Yaremchuk, I.; Vernyhor, O.; Bendziak, A.; Bobitski, Y.
Comparison of the Optical Planar Waveguide Sensors’ Characteristics Based on Guided-Mode Resonance. *Symmetry* **2020**, *12*, 1315.
https://doi.org/10.3390/sym12081315

**AMA Style**

Bellucci S, Fitio V, Yaremchuk I, Vernyhor O, Bendziak A, Bobitski Y.
Comparison of the Optical Planar Waveguide Sensors’ Characteristics Based on Guided-Mode Resonance. *Symmetry*. 2020; 12(8):1315.
https://doi.org/10.3390/sym12081315

**Chicago/Turabian Style**

Bellucci, S., V. Fitio, I. Yaremchuk, O. Vernyhor, A. Bendziak, and Y. Bobitski.
2020. "Comparison of the Optical Planar Waveguide Sensors’ Characteristics Based on Guided-Mode Resonance" *Symmetry* 12, no. 8: 1315.
https://doi.org/10.3390/sym12081315