# Nonlocal Soft Plasmonics in Planar Homogeneous Multilayers

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Classical Soft Plasmonics—General Characteristics of Ionic Systems

#### 2.2. Nonlocal Interactions in Dielectric-Electrolyte Multilayered Systems

## 3. Results and Discussion

#### 3.1. Nonlocal Optical Coefficients for Finite Multilayers

#### 3.2. Evaluating Three-Layer Systems

## 4. Conclusions

## Author Contributions

## Funding

^{career}of the Friedrich-Schiller-University Jena (2.11.3-A1/2020-02). This article is the result of P.R.N.’s research project within the framework of the Honours Program for Future Researchers at the Friedrich Schiller University Jena, Germany, funded by the Excellence Strategy of the German State Government and the Länder. The APC was kindly waived for this invited contribution.

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Nonlocal Function F_{l±}

## Appendix B. Nonlocal Transmission and Reflection Coefficients

## References

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**Figure 1.**(

**a**) Surface plasmon polariton (SPP) propagation and its evanescent decay perpendicular to a dielectric-electrolyte interface. (

**b**) In the local response approximation (LRA), a three-layer-system with dielectric constants ${\u03f5}_{0}$, ${\u03f5}_{1}$ and ${\u03f5}_{2}$ shows the common multiple reflections for a single transversal mode. (

**c**) With nonlocal optical response (NOR), multiple reflections of several modes of transversal and longitudinal character occur inside the electrolyte layer. Indices have been simplified.

**Figure 2.**(

**a**) Bulk plasmon frequency ${\omega}_{p}$ of OH${}^{-}$, H${}_{3}$O${}^{+}$, Na${}^{+}$, K${}^{+}$, and Cl${}^{-}$ ions in solution varying with their respective ion mass ratios $m\phantom{\rule{-1.111pt}{0ex}}/\phantom{\rule{-0.55542pt}{0ex}}{m}_{e}$ with ion concentration ${n}_{\pm}=1\mathrm{mol}\phantom{\rule{-1.111pt}{0ex}}/\phantom{\rule{-0.55542pt}{0ex}}{\mathrm{m}}^{3}{N}_{A}$, where ${N}_{A}$ is the Avogadro constant. (

**b**) Dielectric permittivity of Au in a simple Drude model varying with frequency using parameters ${\omega}_{{p}_{Au}}=2.18\times {10}^{15}$ Hz, ${\gamma}_{Au}=1.72\times {10}^{13}$ Hz, ${\u03f5}_{{b}_{Au}}=9$ [50]. (

**c**) Dielectric ionic permittivity of H${}_{2}$O, NaCl, and KCl ions in solution varying with frequency and with parameters as in Table 1 and background permittivity ${\u03f5}_{b}=1.77$.

**Figure 3.**Analytical dispersion ${k}_{\mathrm{SPP}}$ (red curves) and reflected energy flux for (

**a**) a Au-water interface and (

**b**) ionized H${}_{2}$O in a slab of thickness $d=2.5$ mm. The position of ${\omega}_{\mathrm{SPP}}$ and the light line (black curves) are marked for reference. Tabulated data was used for Au [50].

**Figure 4.**(

**a**) Real and imaginary part of the nonlocal wave vector ${q}_{z}$ of ions in solution varying with the parallel wave vector ${k}_{\Vert}$ for frequency $\omega =3\times {10}^{11}$ Hz. (

**b**) Same for Au at frequency $\omega ={10}^{15}$ Hz. (

**c**) Nonlocal skin depth $\delta =\frac{1}{Im\left\{{q}_{z}\right\}}$ of ions in solution.

**Figure 5.**Transmission (upper panel) and reflection spectra (lower panel) of ions in solution in the (

**a**) nonlocal and (

**b**) local case for an electrolyte layer of thickness $d=1\phantom{\rule{4pt}{0ex}}$mm and with parallel momentum ${k}_{\Vert}=2500\phantom{\rule{4pt}{0ex}}$m${}^{-1}$.

**Figure 6.**Transmission (upper panel) and reflection spectra (lower panel) of ionized NaCl in solution in the (

**a**) nonlocal and (

**b**) local case for varying layer thickness d and with parallel momentum ${k}_{\Vert}=2500\phantom{\rule{4pt}{0ex}}$m${}^{-1}$.

**Table 1.**Characteristic material parameters of ionic systems [44].

Ion | Mass ${\mathbf{m}}_{\pm}$ (${\mathbf{m}}_{\mathbf{e}}$) | Velocity ${\mathbf{v}}_{{\mathbf{th}}_{\pm}}$ (m/s) | Frequency ${\mathit{\omega}}_{{\mathit{p}}_{\pm}}$ (${10}^{11}$ Hz) | Damping $\mathit{\gamma}$ (${10}^{11}$ Hz) |
---|---|---|---|---|

OH${}^{-}$ | $31,005$ | $663.63$ | $2.49$ | $2.46$ |

H${}_{3}$O${}^{+}$ | $34,670$ | $627.57$ | $2.35$ | $2.20$ |

Na${}^{+}$ | $41,907$ | $570.82$ | $2.14$ | $3.19$ |

K${}^{+}$ | $64,628$ | $459.65$ | $1.72$ | $2.68$ |

Cl${}^{-}$ | $71,270$ | $437.71$ | $1.64$ | $3.07$ |

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**MDPI and ACS Style**

Ramesh Narayan, P.; David, C.
Nonlocal Soft Plasmonics in Planar Homogeneous Multilayers. *Photonics* **2023**, *10*, 1021.
https://doi.org/10.3390/photonics10091021

**AMA Style**

Ramesh Narayan P, David C.
Nonlocal Soft Plasmonics in Planar Homogeneous Multilayers. *Photonics*. 2023; 10(9):1021.
https://doi.org/10.3390/photonics10091021

**Chicago/Turabian Style**

Ramesh Narayan, Preethi, and Christin David.
2023. "Nonlocal Soft Plasmonics in Planar Homogeneous Multilayers" *Photonics* 10, no. 9: 1021.
https://doi.org/10.3390/photonics10091021