# Arbitrary Super Surface Modes Bounded by Multilayered Metametal

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Analysis and its Comparison with Finite-Difference Time-Domain (FDTD) Modeling

_{2}/d

_{1}, and the axes are setup in Figure 1(a). In the following derivation, we treat ε

_{1}as the insulator and ε

_{2}as the metal. Regarded as a single anisotropic medium, it can be placed next to the semi-space of a dielectric material (ε

_{d}) and form a boundary as host of surface waves. Assuming a fundamental TM-polarized surface wave (super mode) propagating along this boundary and applying proper boundary conditions, the MMI-insulator boundary supports a propagation surface mode with the dispersion relation obtained as

_{x}< 0, or

**Figure 1.**(

**a**) The multilayer metal-insulator (MMI) scheme and definitions of parameters;(

**b**) The q vs. p curve used to analyze different conditions for tuning effective surface plasmon frequency with a semi-space dielectrics ε

_{d}.

_{2}. The solution can be expressed by ε

_{d}, ε

_{1}and the filling ratio η, as

_{x}< 0 applied in the related section of Equation (3). Based on Equation (1), at least one of the two materials in the multilayered medium needs to have negative permittivity to make ε

_{x}< 0. The negative root from Equation 6 gives the largest dielectric constant the metametal could reach at

_{p}is the plasmonic frequency of the filling metal. Here a characteristic frequency for the metametal can be defined as equal to ESPF ( ), which yields . This is analogy to the definition of surface plasmon frequency defined by , in while ε

_{d}describes the dielectric half space. It would be interesting to study the relation between and then, as for the latter, there are many parameters that can be controlled even if the same metal is used in the system. Based on the definition above, the relation between and can be easily appreciated by the ratio of ε

_{2}(–) and ε

_{d}. For simplicity, we introduced a new term p = ε

_{1}/ε

_{d}, and another term q for the ratio of ε

_{2}(–) and –ε

_{d}. From Equation (7), the factor q can be expressed by p and η as

_{d}) will not be able to decrease the free electron oscillation down to , as the “under-relaxed” MMI has somehow averaged or compensated for the relaxation taking place along the substrate boundary. This observation introduces new perspectives into spoof plasmonics, as an essential supplement to the conventional concept such as effective free electron density.

_{d}= 2.5 (which can be regarded as a polymer-based photoresist). We then apply gold-SiO

_{2}(ε = 2.1 for SiO

_{2 }at 633 nm) and gold-Al

_{2}O

_{3}(ε = 3.15 for Al

_{2}O

_{3}at 633 nm) multilayer stacks respectively to this substrate. It is obvious that the filling insulators have been chosen to make sure the ratio p = ε

_{insulator}/ε

_{d}can be less than unity for one case, and larger than unity for the other. Using the Equation (2) and the mode matching condition, we predicted the shifts of ESPF and SPR angle towards different directions (prism n

_{p}= 2.6). In Figure 2(a), the ESPF (upper cut-off for the TM band) is below and above the preset surface plasmon frequency for gold-SiO

_{2}and gold-Al

_{2}O

_{3}case respectively.

**Figure 2.**(

**a**) The analytical dispersion curves calculated by effective medium theory. The upper cut-off frequencies are treated as the effective surface plasmon frequencies (ESPFs) for two MMI cases. Near 633 nm, the shift of wave vectors are shown in the inset for uniform gold (blue), gold-Al

_{2}O

_{3}MMI (green) and gold-SiO

_{2}MMI (red); (

**b**) FDTD simulation for the shift of surface plasmon resonance (SPR) angles based on uniform gold (blue), gold-Al

_{2}O

_{3}MMI (green) and gold-SiO

_{2}MMI (red). All three curves are on top of the same ε = 2.5 substrate. The small arrows mark the calculated angles based on mode matching.

## 3. Numerical and Experimental Demonstration

_{p}= 2.6) to excite the surface waves when the plasmons are neighbored to silicon dioxide (n = 1.45) or silicon nitride (n = 2.01). The dielectric constants of gold (ε = −11.84 + j1.24) and alumina (n = 1.776) are fitted data from [21] and [22] respectively. According to momentum matching condition for SPR, the incident angle can be calculated theoretically as

**Figure 3.**Experimental setup for studying multilayer metal-insulator stacks and the cross-sectional view of the fabricated multilayer sample (SEM). Each individual layer is 10 nm and there are 10 layers (5 pairs) in total.

_{2}and Si

_{3}N

_{4}substrates are deposited via plasma-enhanced chemical vapor deposition (PECVD), while the gold single-layer and gold-alumina multilayer are deposited via e-beam evaporation (Figure 3, right). The thickness of deposition is kept identical to the simulations performed above. The red crosses and the blue triangles in Figure 4 (right axis) denote the results of gold single layer and gold-alumina multilayer respectively. The observed SPR angles shift from 39° to 40° for silicon dioxide substrate, and 68° to 65° for silicon nitride substrate. The shift direction of SPR agrees with the major conclusion regarding the refractive index relation between the substrate and the filling dielectric film. The discrepancy between the exact observed SPR angle and the calculation might be caused by fabrication disorder and the variation of dielectric constants compared to fitted data, but the disagreement of the effective indexes of super modes between the measured and calculated values are all below 2.5% level.

**Figure 4.**Numerical and experimental results of reflection vs. effective index for (

**a**) increased ESPF with SiO

_{2}substrate and (

**b**) decreased ESPF with Si

_{3}N

_{4}substrate. Red and blue lines describe the simulation results for single-layer and multilayer respectively. Red crosses and blue triangles denote the measured results.

_{d}= 2.5, and the MMI system consists of 20 pairs of thin layers (ε

_{1}= 1.25 and ε

_{2}= −1.8) for a filling ratio η = 3. If a Drude metal is used here, the working frequency will locate at approximate 0.6 , larger than the conventional cutoff of . For uniform metal at this frequency, there will be no surface waves supported at the boundary (Figure 5(a)). With the tuning of ESPF from MMI, however, the working frequency can now exceed 0.58 . Here we use COMSOL [22] to simulate the propagation of the super surface mode (Figure 5(b)), showing the subwavelength confinement of the engineered super surface mode bounded and propagated along. Note that in this simulation, the thickness of each repeated unit is 0.04 λ

_{0}(d

_{2}= 0.03 λ

_{0}, η = d

_{2}/d

_{1}= 3), and the EMT theory could well approximate the behavior of the super surface mode. The mesh-size is small enough to resolve the finest layer d

_{1}.

**Figure 5.**H field distribution for: (

**a**) No propagation of bounded surface modes above surface plasmon frequency; (

**b**) Bounded surface wave with subwavelength mode profile beyond the conventional cutoff frequency defined in (a); (

**c**) Bounded surface wave propagated on a single metal-insulator interface; (

**d**) Bounded surface wave on a MMI-insulator boundary with shorter wavelength and manageable mode size compared to (c).

_{d}= 1.0, and the MMI system consists of 35 pairs of thin layers (ε

_{1}= 3.1 and ε

_{2}= −1.6) for a filling ratio η = 3. According to the design rule, a low-index coating will decrease the ESPF as well as the wavelength of the super surface wave. The metal-insulator boundary shown in Figure 5(c) supports the fundamental TM surface wave for a wavelength of 0.61 λ

_{0}, as can also be calculated from Equation 4. When the uniform metal is replaced by “metametal”, the dispersion curve will bend faster away from lightline (Figure 2(b)). Therefore, a larger wave vector plus a decreased wavelength (0.26 λ

_{0}) is expected. According to the dispersion relation, this trend will also shrink the length of the exponential tail in the dielectric side, as can be clearly seen comparing the H field distribution in the ε

_{d}= 1.0 region of Figure 5(c,d). We have also observed the variation of wavelength relative to the thickness of each repeated unit, as have been mentioned in [5,17,18], which indicates the limit of EMT and a general preference of using thin layers to match EMT’s prediction given by Equation (2).

## 4. Summary

## Acknowledgments

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**MDPI and ACS Style**

Yang, R.; Huang, X.; Lu, Z.
Arbitrary Super Surface Modes Bounded by Multilayered Metametal. *Micromachines* **2012**, *3*, 45-54.
https://doi.org/10.3390/mi3010045

**AMA Style**

Yang R, Huang X, Lu Z.
Arbitrary Super Surface Modes Bounded by Multilayered Metametal. *Micromachines*. 2012; 3(1):45-54.
https://doi.org/10.3390/mi3010045

**Chicago/Turabian Style**

Yang, Ruoxi, Xiaoyue Huang, and Zhaolin Lu.
2012. "Arbitrary Super Surface Modes Bounded by Multilayered Metametal" *Micromachines* 3, no. 1: 45-54.
https://doi.org/10.3390/mi3010045