# Relation between Resonance Parameters of Surface Plasmon-Polariton Waves with Properties of the Dielectric-Metal Film-Dielectric Waveguide

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## Abstract

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## 1. Introduction

## 2. Sensitivity of Prism Structure and its Relation with Properties of the Dielectric–Metal Film–Dielectric Waveguide

## 3. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**The optical scheme of the prism structure, $\mathsf{\beta}$ is propagation the constant of the surface plasmon polariton wave, $\mathsf{\epsilon}$ is the dielectric constant of the prism material, ${\mathsf{\epsilon}}_{m}$ is the dielectric constant of the metal, and ${\mathsf{\epsilon}}_{a}$ is the dielectric constant of the researched medium.

**Figure 2.**Graphical dependencies Re$\left(F\left(x,y\right)\right)=0$ (solid line) and Im$\left(F\left(x,y\right)\right)=0$ (dotted line) at the following parameters of the prism structure: $\lambda =1.064\mathsf{\mu}m$, $\theta =1.040226$ rad, $d=52.55$ nm, ${\mathsf{\epsilon}}_{m}=-57.1915+i1.26097$ [16,20], $n=1.56$ and ${n}_{a}=1.3242$. The reflection coefficient from the metal film is 0.000041 at these parameters. The insertion shows solid and dotted curves on a larger scale.

**Figure 3.**Graphical dependencies Re$\left(F\left(x,y\right)\right)=0$ (solid line) and Im$\left(F\left(x,y\right)\right)=0$ (dotted line) at the prism structure parameters mentioned above.

**Figure 4.**Graphical dependencies Re$\left(r\left(\theta ,\lambda \right)\right)=0$ (solid line) and Im$\left(r\left(\theta ,\lambda \right)\right)=0$ (dotted line) at the following parameters: $d=52.69$ nm, $=1.56,$ ${n}_{a}=1.3242$. The intersection of the solid and dotted curves determines the $\theta $ and $\lambda ,$ at which the reflection coefficient is equal to $R=2.6\times {10}^{-14}.$ The inset shows parts of the intersected curves on a larger scale.

**Figure 5.**Spectral dependence of the reflection coefficient with the following parameters: ${\lambda}_{rez}=1.0645076\mathsf{\mu}\mathrm{m},$ ${\theta}_{rez}=1.04018937,$ $d=52.7$ nm, $n=1.56,$ ${n}_{a}=1.3242$. In this case, $\mathsf{\delta}{\lambda}_{0.5}$ = 0.0471$\mathsf{\mu}\mathrm{m}.$

**Figure 6.**Angular dependence of the reflection coefficient with the following parameters: ${\theta}_{rez}=1.04018937,$ ${\lambda}_{rez}=1.0645076\mathsf{\mu}m,$ $d=52.7$ nm, $n=1.56,$ ${n}_{a}=1.3242$. In this case, $\mathsf{\delta}{\theta}_{0.5}$ = 0.002609 rad.

**Table 1.**Characteristic parameters of the waveguide (${S}_{n}^{\left(\mathsf{\beta}\right)},{S}_{\lambda}^{\left(\mathsf{\beta}\right)}$) and the parameters of the prism structure under the excitation of the surface plasmon–polariton resonance.

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

No. | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |

1 | $6.18685$ (13) * | $-7.7219$ (12) | 0.00261 Figure 6 | 0.0471 Figure 5 | 1.33 (15) | 24.08 (14) | 509 | 511 |

2 | $6.18685$ (13) | $-7.7219$ (12) | 0.00261 Figure 6 | 0.0465 (10) | 1.33 (8) | 23.70 (6) | 510 | 510 |

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**MDPI and ACS Style**

Bellucci, S.; Fitio, V.; Bendziak, A.; Yaremchuk, I.; Bobitski, Y.
Relation between Resonance Parameters of Surface Plasmon-Polariton Waves with Properties of the Dielectric-Metal Film-Dielectric Waveguide. *Materials* **2020**, *13*, 2989.
https://doi.org/10.3390/ma13132989

**AMA Style**

Bellucci S, Fitio V, Bendziak A, Yaremchuk I, Bobitski Y.
Relation between Resonance Parameters of Surface Plasmon-Polariton Waves with Properties of the Dielectric-Metal Film-Dielectric Waveguide. *Materials*. 2020; 13(13):2989.
https://doi.org/10.3390/ma13132989

**Chicago/Turabian Style**

Bellucci, Stefano, Volodymyr Fitio, Andriy Bendziak, Iryna Yaremchuk, and Yaroslav Bobitski.
2020. "Relation between Resonance Parameters of Surface Plasmon-Polariton Waves with Properties of the Dielectric-Metal Film-Dielectric Waveguide" *Materials* 13, no. 13: 2989.
https://doi.org/10.3390/ma13132989