# Design of a Hyperbolic Metamaterial as a Waveguide for Low-Loss Propagation of Plasmonic Wave

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

_{2}O

_{3}films was shown to have a near-zero principal permittivity in the direction parallel to the film surface. Therefore, the third-ordered diffracted wave from a grating with $\mathsf{\lambda}=405\text{}\mathrm{nm}$ and ${\mathrm{k}}_{x}=1.736{\mathrm{k}}_{0}$ was coupled into a horizontally propagating wave in the HMM. The structure is a typical seven-layered metal (M)-dielectric (D) symmetrical film stack MDMDMDM. Figure 1 shows the equivalent model of the film stack. The refractive indices of Al and Al

_{2}O

_{3}are 0.465–4.764i and 1.680, respectively, at a wavelength of 405 nm [23]. Figure 2 shows the IFCs of the proposed HMM, plotted using Equation (1). To obtain the IFCs over a wide range of ${\mathrm{k}}_{x}$, the index of refraction of the cover medium is assumed to have a high value of ten. The refractive index of cover medium is set to be 10 in order to offer enough wave vector component ${\mathrm{k}}_{x}$ then the IFC can be plotted over a wide range of ${\mathrm{k}}_{x}$. On the other hand, the index of cover medium should be large enough to cover several orders of diffracted waves as a grating. The precise IFC reveals the ${\mathrm{k}}_{x}$ that has minimum ${\mathrm{k}}_{z}$ for low-loss horizontally propagation of plasmonic wave. According to the ${\mathrm{k}}_{x}$, the grating can be designed to offer a diffracted wave at a certain order that can propagate horizontally in the waveguide. In order to have the diffracted wave be coupled into the waveguide, the equivalent admittance including the substrate and the multilayered waveguide is considered to have highly efficient light coupling and propagation.

^{m}, where m is the number of symmetrical stacks, or DMDDM…DDMD = (DMD)

^{m}. In both cases, (MDM) or (DMD) is the unit cell in the HMM. Therefore, a basic design for HMM involves the stacking of several cells. The ${\mathrm{E}}_{eq}^{int}$ and the ${\mathrm{k}}_{z}$ versus ${\mathrm{k}}_{x}$ of a multiple cell structure are determined by the unit cell and do not vary with the number of cells. However, in practical applications, whether the system can couple more light into the HMM depends on the layers and medium adjacent to it. Therefore, admittance-matching for the system depends on the number of cells within the HMM. Tuning the cell number is similar to tuning the thickness of an optical coating for antireflection.

## 3. Results

_{2}O

_{3}films was modified to provide an example of a multiple DMD design, such that the thickness of the Al layer was 6 nm. The complex value of ${\mathrm{k}}_{z}$/${\mathrm{k}}_{0}$ was then calculated with the thickness of the Al

_{2}O

_{3}layer from 0 to 90 nm and the incident angle from 0° to 90° using the equivalent refractive index of the correct branch. The black color in Figure 5 indicates the ranges of $\mathsf{\theta}$ and ${\mathrm{d}}_{D}$ that satisfy the conditions of $\left|\mathrm{Re}\left({\mathrm{k}}_{z}/{\mathrm{k}}_{0}\right)\right|\le 0.3$ and $\left|\mathrm{Im}\left({\mathrm{k}}_{z}/{\mathrm{k}}_{0}\right)\right|\le 0.15$. The criteria used for this figure are the upper limits for both the real part and the imaginary part of ${\mathrm{k}}_{z}$. Our purpose is to fine the minimum magnitude of ${\mathrm{k}}_{z}$ to ensure the horizontal propagation and low loss. A wide range of thicknesses of the Al

_{2}O

_{3}layer from 22.9 to 63.4 nm satisfies these conditions at $\mathsf{\theta}=10\xb0$. An incident wave at $\mathsf{\theta}=10\xb0$ in a cover medium with ${\mathrm{N}}_{inc}=10$is considered to offer a wave vector component of ${\mathrm{k}}_{x}=1.736{\mathrm{k}}_{0}$, so the incident light wave is coupled into a horizontally propagating wave. Next, the variations of admittance ${\mathrm{Y}}_{eq}$ with ${\mathrm{N}}_{sub}$ and ${\mathrm{d}}_{D}$ were calculated for different numbers of periods m. ${\mathrm{N}}_{sub}$ was varied in the range of 1.4–1.73 that can be easily achieved by arranging dielectric films on a transparent substrate. Figure 6 shows the absolute difference between the equivalent admittance ${\mathrm{Y}}_{eq}$ and the refractive index of the cover medium ${\mathrm{N}}_{inc}$ for possible ${\mathrm{N}}_{sub}$, ${\mathrm{d}}_{D}$, and numbers of periods, m. The dark blue areas in Figure 6 represent a difference $\left|{\mathrm{Y}}_{eq}-{\mathrm{N}}_{inc}\right|$ of less than 0.5: $\left|{\mathrm{Y}}_{eq}-{\mathrm{N}}_{inc}\right|$ is less than 0.5 only at m = 6. The optimal structure is obtained with ${\mathrm{N}}_{inc}/{\left(\mathrm{DMD}\right)}^{6}/$ ${\mathrm{N}}_{sub}$ with 6 nm thick Al (M), 25.6 nm thick Al

_{2}O

_{3}(D), and ${\mathrm{N}}_{sub}$ of 1.474. Figure 7 plots the IFCs. The ${\mathrm{k}}_{z}/{\mathrm{k}}_{0}$ at ${\mathrm{k}}_{x}=1.736{\mathrm{k}}_{0}$ is 0.140–0.100i, of which the imaginary part is less than that of the previously presented seven-layered structure. The ${\mathrm{Y}}_{eq}$ is 10.077–0.100i, indicating good admittance matching to ${\mathrm{N}}_{inc}=10$. The wave vector component ${\mathrm{k}}_{x}=1.736{\mathrm{k}}_{0}$ in Figure 7 is near the wave vector of surface plasmon wave excited at the interface between Al and Al

_{2}O

_{3}that is presented as ${\mathrm{k}}_{sp}=\sqrt{\left({\mathsf{\epsilon}}_{D}\right)\left({\mathsf{\epsilon}}_{M}\right)/\left({\mathsf{\epsilon}}_{D}+{\mathsf{\epsilon}}_{M}\right)}=1.789+0.024i$, where ${\mathsf{\epsilon}}_{M}$ and ${\mathsf{\epsilon}}_{D}$ are $-22.476+4.429i$ and 2.822 at 405 nm, respectively. Since the metal film is pretty thin, the loss within the metal is small. It is a collective response of all surface plasmon polaritons on each of the interfaces.

## 4. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**(

**a**) Iso-frequency curves and admittance of seven-layered structure composed of 6 nm thick Al and 47 nm thick Al

_{2}O

_{3}films. (

**b**) Iso-frequency curves and admittance around ${\mathrm{k}}_{z}$ = 0.

**Figure 6.**Admittance difference $\left|{\mathrm{Y}}_{eq}-{\mathrm{N}}_{inc}\right|$ for possible ${\mathrm{N}}_{sub}$, ${\mathrm{d}}_{D}$, and numbers of periods m from (

**a**) m = 1 to (

**h**) m = 8.

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**MDPI and ACS Style**

Chang, Y.-C.; Chan, T.-L.; Lee, C.-C.; Jen, Y.-J.; Ma, W.-C.
Design of a Hyperbolic Metamaterial as a Waveguide for Low-Loss Propagation of Plasmonic Wave. *Symmetry* **2021**, *13*, 291.
https://doi.org/10.3390/sym13020291

**AMA Style**

Chang Y-C, Chan T-L, Lee C-C, Jen Y-J, Ma W-C.
Design of a Hyperbolic Metamaterial as a Waveguide for Low-Loss Propagation of Plasmonic Wave. *Symmetry*. 2021; 13(2):291.
https://doi.org/10.3390/sym13020291

**Chicago/Turabian Style**

Chang, Ya-Chen, Teh-Li Chan, Cheng-Chung Lee, Yi-Jun Jen, and Wei-Chieh Ma.
2021. "Design of a Hyperbolic Metamaterial as a Waveguide for Low-Loss Propagation of Plasmonic Wave" *Symmetry* 13, no. 2: 291.
https://doi.org/10.3390/sym13020291