# Numerical Study on the Absorption Characteristics of Subwavelength Metallic Gratings Covered with a Lossy Dielectric Layer

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Absorption Characteristics of a Subwavelength Metallic Grating Covered with a Lossy Dielectric Layer

_{g}, respectively, and w is the width of the grating cavity. Aluminum (Al) and indium tin oxide (ITO) are used as the metal and lossy dielectric, respectively. We note that lossy dielectric materials other than ITO are also applicable, which may result in more desirable absorption. The metal substrate is 100 nm thick, thus we can assume there is no transmission of visible light. Transverse magnetic (TM) plane waves are incident with angle θ. Here, we only consider TM polarized incident plane waves since this is necessary to generate GSP modes in the grating cavity at normal incidence (θ = 0).

_{g}= 30 nm, and the corresponding magnitude and phase profiles of the reflection coefficient from ITO to the effective medium (r

_{eff}) are shown in Figure 3c,d. Figure 3e–h show the magnitude and phase profiles of the Fresnel reflection and transmission coefficients at the interface between air and the ITO layer. Optical constants for ITO and Al used to calculate the coefficients are presented in Figure 3b and were determined experimentally from ellipsometry measurements.

_{0}= 2π/λ is the free-space wavenumber, λ is the free-space wavelength, ${\tilde{n}}_{\mathrm{ITO}}$ is the complex refractive index of ITO, and r

_{eff}, r

_{12}, r

_{21}, t

_{12}, and t

_{21}are the coefficients presented in Figure 3c–h, respectively. Figure 4a shows a spectral absorption map calculated using Equation (1) as a function of the upper ITO layer thickness. The red-colored vertical dashed line indicates a wavelength of 590 nm, at which point the grating is highly absorptive. The corresponding spectral absorption map without the grating (w = 0) is presented in Figure 4b for comparison. We find from the two results that the spectral absorbance shown in Figure 4a has both the absorption characteristics of the Al grating as well as that of the ITO resonant cavity on the flat Al substrate. In particular, as can be seen from Figure 4a, strong absorption in the Al grating around 590 nm is preserved, albeit there are slight shifts in the resonance frequency due to the effect of the finitely thick ITO layer. Figure 4c,d show the absorption profiles of the two structures at t

_{d}= 270 nm, as denoted by the dashed lines in Figure 4a,b. There are absorption peaks near 470 nm in both cases, while the absorption peak at 585 nm can only be found when the grating exists. It is clear from these results that the absorption peak at 470 nm primarily stems from the optical losses in the coated ITO layer. On the other hand, most of the light with wavelength near 585 nm is absorbed by the subwavelength grating.

_{g}).

## 3. Conditions for Complete Optical Absorption by Critical Coupling

_{d},λ) as follows:

_{d}and λ that forces the magnitude and phase of D(t

_{d},λ) to be 1 and the integer multiples of 2π, respectively. Figure 7a,b show the magnitude and phase plots of D(t

_{d},λ). In Figure 7a, the white-colored and red-colored dashed lines indicate where the magnitude and phase of D(t

_{d},λ) are 1 and 2πm(m being an integer), respectively. The points where two dashed lines cross correspond to the unity absorption. These points are distributed near 590 nm, at which the magnitude of the reflection coefficient between the ITO and grating (r

_{eff}) is minimized. For comparison, the magnitude and phase of D

_{0}(t

_{d},λ) for a flat Al reflector (w = 0) are presented in Figure 7c,d, respectively. Here, D

_{0}(t

_{d},λ) can be achieved by replacing r

_{eff}as r

_{0}= $\left({\tilde{n}}_{\mathrm{ITO}}-{\tilde{n}}_{\mathrm{Al}}\right)/\left({\tilde{n}}_{\mathrm{ITO}}+{\tilde{n}}_{\mathrm{Al}}\right)$ in Equation (3), where ${\tilde{n}}_{\mathrm{Al}}$ represents the complex refractive index of Al. As can be seen from the plots, perfect absorption is unattainable in our calculation range without the subwavelength grating reflector since the unity magnitude condition on D

_{0}(t

_{d},λ) cannot be fulfilled. By exploiting this tendency, we can determine the structural parameters of visible light absorbers with desired absorption profiles.

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**(

**a**) A subwavelength metallic grating coated with a dielectric layer. The thicknesses of the upper dielectric layer and grating are t

_{d}and t

_{g}, respectively. (

**b**) The optically equivalent model, in which the grating and substrate are replaced by a semi-infinite effective medium.

**Figure 2.**(

**a**) Calculation geometry for determining reflection and phase shift characteristics in the subwavelength Al grating covered with ITO. (

**b**) Spectral reflectance and (

**c**) phase shift of the metallic grating for normally illuminated visible light (θ = 0) with respect to grating thicknesses ranging from 0 to 70 nm.

**Figure 3.**(

**a**) Diagram showing the reflection and transmission coefficients for calculation of the optical response of the metallic grating coated with a lossy dielectric using the numerical effective medium approach. (

**b**) Experimentally measured optical constants of ITO and Al. (

**c**) Magnitude and (

**d**) phase profiles of the reflection coefficient at the interface between ITO and effective medium. (

**e**) Magnitude and (

**f**) phase profiles of the Fresnel reflection coefficients, and (

**g**) magnitude and (

**h**) phase profiles of the Fresnel transmission coefficients at the interface between air and ITO. All coefficients are calculated for normally incident light at visible frequencies.

**Figure 4.**(

**a**) Spectral absorption in the visible region with respect to the thickness of the upper dielectric medium deposited on the Al subwavelength grating (p = 160 nm, w = 40 nm, t

_{g}= 30 nm), and (

**b**) corresponding map for a flat Al reflector (w = 0). Spectral absorption profiles when t

_{d}= 180 nm (

**c**) with and (

**d**) without the grating structure.

**Figure 5.**Distributions of the magnetic field within the structure when (

**a**) 470 and (

**b**) 585 nm transverse magnetic (TM) polarized plane waves illuminate the structure at normal incidence. The structural parameters are p = 160 nm, w = 40 nm, t

_{g}= 30 nm, and t

_{d}= 270 nm. The calculation domain includes two unit cell elements, and the color scale is properly normalized to compare the field magnitudes.

**Figure 6.**(

**a**) The spectral absorption map with respect to the incident angle up to 80°. The structural parameters are the same as those used in Figure 5 (p = 160 nm, w = 40 nm, t

_{g}= 30 nm, and t

_{d}= 270 nm). Absorption modes generated at oblique incidence are marked with the dashed red-colored oval. (

**b**) Absorption profiles for three incident angles of θ = 20, 40, and 60°. Absorption peak shifts of new modes with increasing incidence angle are indicated by green-colored arrows.

**Figure 7.**Plots of (

**a**) magnitude and (

**b**) phase of D(t

_{d},λ) defined in Equation (3) in order to investigate the critical coupling condition for perfect absorption. (

**c**) and (d) show the corresponding results for the case of a flat Al reflector without the grating structure (w = 0). In (

**a**,

**c**), white- and red-colored dashed lines indicate where the magnitude is 1 and the phase is 2πm (m is an integer), respectively.

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**MDPI and ACS Style**

Hwang, C.-Y.; Kim, Y.-H.; Choi, J.H.; Kim, G.H.; Yang, J.-H.; Pi, J.-E.; Kim, H.-O.; Hwang, C.-S.
Numerical Study on the Absorption Characteristics of Subwavelength Metallic Gratings Covered with a Lossy Dielectric Layer. *Appl. Sci.* **2018**, *8*, 1445.
https://doi.org/10.3390/app8091445

**AMA Style**

Hwang C-Y, Kim Y-H, Choi JH, Kim GH, Yang J-H, Pi J-E, Kim H-O, Hwang C-S.
Numerical Study on the Absorption Characteristics of Subwavelength Metallic Gratings Covered with a Lossy Dielectric Layer. *Applied Sciences*. 2018; 8(9):1445.
https://doi.org/10.3390/app8091445

**Chicago/Turabian Style**

Hwang, Chi-Young, Yong-Hae Kim, Ji Hun Choi, Gi Heon Kim, Jong-Heon Yang, Jae-Eun Pi, Hee-Ok Kim, and Chi-Sun Hwang.
2018. "Numerical Study on the Absorption Characteristics of Subwavelength Metallic Gratings Covered with a Lossy Dielectric Layer" *Applied Sciences* 8, no. 9: 1445.
https://doi.org/10.3390/app8091445