# Design a Stratiform Metamaterial with Precise Optical Property

^{*}

## Abstract

**:**

_{2}five-layered symmetrical film stack shown in previous work is demonstrated to be positive real instead of negative real. The associated type I iso-frequency curve supports negative refraction. In order to extend the operating wavelength of type I metamaterial, the number of the metal-dielectric symmetrical film stack is increased to reduce the thickness of the dielectric film to approach subwavelength requirement.

## 1. Introduction

_{2}and Ag films alternately was claimed to yield an index of negative unity over a wide range of incidence angles at a wavelength of 363.8 nm [7]. Later, whether the negative refraction in the TiO

_{2}-Ag multilayer came from a negative index of refraction was questioned and discussed. The negative refraction also occurs in a hyperbolic metamaterial. Hyperbolic metamaterials have been developed for more than 10 years. In most cases, hyperbolic metamaterials are metal-dielectric multilayers [8,9]. Its property is described by a diagonal permittivity tensor. The principal permittivities are roughly estimated using effective medium approximation [10]. The signs of the two tangential permittivities are the same, but the signs of its tangential and vertical permittivities are opposite. Accordingly, there are two kinds of hyperbolic shaped isofrequency curves (type I and type II) [11]. A previous work on the decomposition of the Bloch wave in the layered structure into a number of harmonics showed that the iso-frequency curve of the main harmonic mode corresponds to the type I hyperbolic metamatrerial [12]. The negative refraction of energy is invoked by the shape of the iso-frequency contour [11]. Therefore, the multilayer is a hyperbolic metamaterial, and light propagation is mainly dominated by right-handed harmonics with a positive phase index.

## 2. Methods

_{eq}were complex. At normal incidence, the equivalent refractive index N

_{eq}is derived from the phase thickness ${\gamma}_{eq}$ through the relationship ${\gamma}_{eq}=2\pi {N}_{eq}\left(2{d}_{A}+{d}_{B}\right)/\lambda $. Both E

_{eq}and N

_{eq}can be tailored separately. Both E

_{eq}and N

_{eq}are represented with the equivalent permittivity ε

_{eq}and the equivalent permeability ${\mu}_{eq}$ as ${E}_{eq}=\sqrt{{\epsilon}_{eq}/{\mu}_{eq}}$ and ${N}_{eq}=\sqrt{{\epsilon}_{eq}{\mu}_{eq}}$, respectively. According to effective medium approximation, the equivalent refractive index and admittance of a metal-dielectric composite are generally complex. In case of oblique incidence and TM polarization state, the equivalent phase thickness should be ${\gamma}_{eq}^{TM}=2\pi {N}_{eq}\left(2{d}_{A}+{d}_{B}\right)\mathrm{cos}{\theta}_{eq}/\lambda $. Figure 1 shows the equivalent scheme of a 3-layered symmetrical film stack ABA. The θ

_{eq}is the equivalent angle of refraction in the film stack. The ${d}_{A}$ and ${d}_{B}$ are thicknesses of film A and film B, respectively. The ${\theta}_{A}$ and ${\theta}_{B}$ are angles of refraction in film A and film B, respectively. The equivalent admittance for TM polarization should be modified as ${E}_{eq}^{TM}={E}_{eq}/\mathrm{cos}{\theta}_{eq}$. The product of the 3 characteristic film matrixes ABA is shown in Equation (2).

_{eq}can be retried from the matrix elements firstly through Equation (3).

_{0}and θ

_{0}are the refractive indexes of the cover medium and angle of incidence, respectively), as shown in Equation (4).

_{eq}shown in Equation (5).

_{eq}and admittance ${E}_{eq}$.

_{11}: ${\gamma}_{eq}={\mathrm{cos}}^{-1}\left({M}_{11}\right)+m2\pi $ (m = 0, $\pm 1,\text{}\pm 2,\dots ).$ The correct solution should satisfy the 3 criteria: (1) The real part of the equivalent admittance should be positive; (2) the imaginary part of the equivalent refractive index should be negative; (3) both $\text{}{N}_{eq}={n}_{eq}-i{k}_{eq}$ and ${E}_{eq}={E}_{r}+i{E}_{i}$ as functions of wavelength or any constitutive parameter should be continuous [19]. It is particular that a specific branch was unable to satisfy the criteria over the whole wavelengths. The continuity of the real part of the equivalent refractive index as a function of wavelength or any constitutive parameter required choosing different branches for regions separated by the discontinuous points. For a metal-dielectric SFS in the form of MDM…DM, a m-th branch was chosen thus that the ${n}_{eq}$ at thickness of dielectric layer $d=0\text{}\mathrm{was}$ equal to the refractive index of metal. The m-th branch was correct for the range form $d=0$ to its first discontinuous point at $d={d}_{c1}$. The ${n}_{eq}$ at thickness d larger than ${d}_{c1}$ relied on a branch next to the m-th branch to connect the m-th branch at $d={d}_{c1}$ to keep the continuous condition till its discontinuous point at $d={d}_{c2}$. The correct branch for the thickness d larger than ${d}_{c2}$ was chosen to keep continuous at the $d={d}_{c2}$.

## 3. Results

_{2}(28 nm)/Ag (30 nm)/TiO

_{2}(28 nm)/Ag (33 nm) in a previous work [7] was examined here. Such SFS was claimed to have a negative index around -1 at a wavelength of 363.8 nm. The refractive indexes of Ag and TiO

_{2}were 0.0785-1.59i and 2.8-0.05i, respectively. The equivalent ${n}_{eq}$ versus thickness of TiO

_{2}from 0 nm to 60 nm at normal incidence with branch (m = 0) is shown in Figure 2a. At the thickness of d = 0 nm, the index of refraction was 0.0785, which was the real part of the refractive index of Ag. It means that the branch m = 0 was correct from d = 0 nm to its first discontinuous point at ${d}_{c1}$ = 22 nm. At the thickness of d = 28 nm in Figure 2a, the index of refraction was −1.058, which was the proposed value of negative index. However, the index of refraction after ${d}_{c1}$= 22 nm needed to be connected with another branch (m = 1), as shown in Figure 2b. For the branch (m = 1), there was a discontinuous point at ${d}_{c2}$ = 37 nm, thus the index after d = 37 nm was connected with branch (m = −1), as shown in Figure 2c. Figure 2d shows the correct real part of ${N}_{eq}$ as a function of d from d = 0 nm to d = 60 nm.

_{2}O

_{5}films was designed at a wavelength of 600 nm. According to our previous work [22], a thin metal film can be inserted between two metal films to perform a huge admittance locus and admittance matching to achieve high transmission property. The method, called modified Fabry-Perot (FP) design, can be extended to any odd-numbered metal–dielectric film stack. At a wavelength of 600 nm, the silver film usually takes 10 nm to 20 nm to achieve the modified FP design. Therefore, each silver film in the film stack was set to have the same thickness of 20 nm for the five-layered symmetrical film stack MDMDM = Ag/Ta

_{2}O

_{5}/Ag/Ta

_{2}O

_{5}/Ag. The optical constants of Ag and Ta

_{2}O

_{5}were adopted from the commercial optical thin-film software (Essential Macleod) [23].

_{2}O

_{5}(100 nm)/Ag (20 nm)/Ta

_{2}O

_{5}(100 nm)/Ag (20 nm), where ${k}_{o}$ was the wave vector in free space. Unlike a smooth and typical hyperbolic curve, the ratio ${k}_{z}/{k}_{o}$ kept around 1.2 within the range of ${k}_{x}/{k}_{o}$ between $+1.56$ and $-1.56$. It means that the reflected or diffracted ray vector was perpendicular to the interface for the parallel-to-interface component of the wave vector during the aforementioned range.

_{2}O

_{5}films. Figure 5a shows ${n}_{eq}$ and ${k}_{eq}$ as functions of ${d}_{D}$. Figure 5b shows ${E}_{r}$ and ${E}_{i}$ as functions of ${d}_{D}$. There are three ranges of ${d}_{D}$ corresponding to low ${k}_{eq}$ (${k}_{eq}$ < 0.1), which were ${d}_{D}:\text{}$(64 nm, 67 nm), (83 nm, 97 nm), and (108 nm, 117 nm). The thickness of the dielectric layer in the seven-layered SFS with a low extinction coefficient was less than that of MDMDM. At ${d}_{D}=\text{}$113 nm, ${E}_{i}$ had a minimum magnitude of 0.1327 and the ${E}_{r}$ was 4.2736. ${E}_{r}$ varied from 1.28 to 4.02 and 2.037 to 9.053 at a low-magnitude ${E}_{i}$ ($\left|{E}_{i}\right|\le 0.3$), which ranges were (${d}_{D}=$84 nm, ${d}_{D}=$92 nm) and (${d}_{D}=$ 110 nm, ${d}_{D}=$ 116 nm), respectively. The ${N}_{eq}$ as a function of ${d}_{D}$ and angle of incidence $\theta $ is shown in Figure 5c. The type I hyperbolic metamaterial occurred at ${d}_{D}$ and ranged from 75 nm to 80 nm. The maximum ${T}_{min}$ of 46.49 % occurred at ${d}_{D}=$ 85 nm, and the corresponding iso-frequency curve is shown in Figure 5d. The optimum thickness for type I iso-frequency curve was shifted from ${d}_{D}=$ 100 nm in MDMDM structure to ${d}_{D}=$ 80 nm here in MDMDMDM structure. Compared with the five-layered case, the ratio ${k}_{z}/{k}_{o}$ keeps around 1.1 within a smaller range of ${k}_{x}/{k}_{o}$ between $+1.7$ and $-1.7$.

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Equivalent ${n}_{eq}$ of branches (

**a**) m = 0, (

**b**) m = 1, and (

**c**) m = −1; (

**d**) The correct ${n}_{eq}$ as a function of d.

**Figure 3.**(

**a**) Correct ${n}_{eq}$ and ${k}_{eq}$ as functions of d, (

**b**) correct ${n}_{eq}$ and ${k}_{eq}$ as functions of angle of incidence at d = 28 nm, (

**c**) the iso-frequency curve, the yellow and purple lines are cover medium and hyperbolic metamaterial, respectively. The dotted arrows and solid arrows represent wave vector and ray vector, respectively.

**Figure 4.**(

**a**) ${n}_{eq}$ and ${k}_{eq}$ as functions of d

_{D}of five-layered MDMDM at a wavelength of 600 nm, (

**b**) ${E}_{r}$ and ${E}_{i}$ as functions of d

_{D}of five-layered MDMDM a wavelength of 600 nm, (

**c**) ${n}_{eq}$ as a function of d

_{D}and angle of incidence ${\theta}_{0}$ at a wavelength of 600 nm, (

**d**) the iso-frequency curve of Ag(20 nm)/Ta

_{2}O

_{5}(100 nm)/Ag (20 nm)/ Ta

_{2}O

_{5}(100 nm)/Ag (20 nm). The dotted arrows and solid arrows represent wave vector and ray vector, respectively.

**Figure 5.**(

**a**)$\text{}{n}_{eq}$ and ${k}_{eq}$ as functions of d

_{D}of seven-layered MDMDM at a wavelength of 600 nm, (

**b**) ${E}_{r}$ and ${E}_{i}$ as functions of d

_{D}of seven-layered MDMDM a wavelength of 600 nm, (

**c**) ${n}_{eq}$ as a function of d

_{D}and angle of incidence ${\theta}_{0}$ at a wavelength of 600 nm, (

**d**) the iso-frequency curve of Ag (20 nm)/Ta

_{2}O

_{5}(85 nm)/Ag (20 nm)/ Ta

_{2}O

_{5}(85 nm)/Ag (20 nm). The dotted arrows and solid arrows represent wave vector and ray vector, respectively.

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Jen, Y.-J.; Liu, W.-C.
Design a Stratiform Metamaterial with Precise Optical Property. *Symmetry* **2019**, *11*, 1464.
https://doi.org/10.3390/sym11121464

**AMA Style**

Jen Y-J, Liu W-C.
Design a Stratiform Metamaterial with Precise Optical Property. *Symmetry*. 2019; 11(12):1464.
https://doi.org/10.3390/sym11121464

**Chicago/Turabian Style**

Jen, Yi-Jun, and Wei-Chin Liu.
2019. "Design a Stratiform Metamaterial with Precise Optical Property" *Symmetry* 11, no. 12: 1464.
https://doi.org/10.3390/sym11121464