# The Design of Optical Circuit-Analog Absorbers through Electrically Small Nanoparticles

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

**k**forms an angle θ with the z-axis. The dielectric spacers have relative permittivity and permeability (ε

_{i}_{i}, μ

_{i}), whereas each reactive layer is characterized by a surface impedance Z

_{s}= R

_{s}+ jX

_{s}, whose real and imaginary parts account for the electromagnetic losses and the reactive behavior of the sheet, respectively.

_{i}and characteristic impedance η

_{i}, whereas each reactive layer is modeled through a shunt lumped impedance Z

_{si}.

_{i}and propagation constant β

_{i}, can be written in the following form:

_{x}is the x-component of the free-space wave vector that is equal to

_{i}is equal to the dielectric wavenumber ${k}_{i}={\omega}^{2}{\mu}_{0}{\epsilon}_{0}{\epsilon}_{i}$.

_{L}at a given z can be calculated as [24]:

_{1}is equal to

_{in}at z = d

_{1}+ d

_{2}+ d

_{3}. Once Z

_{in}is known, the input reflection coefficient can be expressed as follows:

_{0}= 377 Ω is the free-space impedance. The above procedure can be easily extended to a CA absorber made by an arbitrary number of layers N or even to non-homogenous dielectric layers [25,26].

_{x}= r

_{z}and they are made of silver. Their permittivity is expressed through the following size-corrected dielectric function [33]:

_{p}= 9.17 eV, ω

_{L}= 5.27 eV, f = 2.2, Γ

_{L}= 1.14 eV, γ = 0.0023 × ω

_{p}, v

_{F}= 1.39 × 10

^{6}m/s, A = 0.6. It is important to underline that, compared to the conventional Drude model of silver, Equation (7) contains an additional loss contribution that is inversely proportional to the radius of the nanoparticle. This term accounts for the surface dispersion that allows tuning the surface resistance of the nanoparticle array.

_{y}the y-component of the nanoparticle polarizability tensor [22], $k={k}_{0}\sqrt{{\epsilon}_{h}}$ the wavenumber in the hosting material, and, finally, β is the interaction constant among nanoparticles. An approximate expression for β for the case of normal incidence is available, for instance, in Reference [37].

_{s}= 200 Ω) but opposite surface reactance (X

_{s}

_{1}= −1000 Ω/sq and X

_{s}

_{2}= 1000 Ω/sq, respectively) at f = 600 THz. Their dimensions are as follows: (1st) r

_{y}= 16 nm, r

_{x}= 6 nm, a = 55 nm; (2nd) r

_{y}= 22 nm, r

_{x}= 5 nm, a = 55 nm. The frequency behavior of the complex surface impedance for these arrays of nanoparticles is shown in Figure 2b. The ticks represent the numerical values that have been retrieved through full-wave simulations. As can be appreciated, such a configuration allows achieving a non-resonant lossy behavior at a desired frequency of the visible spectrum. Specifically, the surface reactance is mainly related to the nanoparticle eccentricity, whereas the surface resistance can be tuned by acting on the absolute size of the nanoparticles [20]. Thus, by engineering the major and minor axes of the nanoparticles, it is possible to obtain an optical metasurface with a desired amount of losses and either a capacitive or an inductive behavior. It is interesting to observe that, differently from their microwave versions, the surface impedance of an optical metasurface based on nanoparticles is strongly frequency-dispersive, even in its real part. This implies that the maximum achievable bandwidth of a nanoparticles-based optical absorber is unavoidably lower than the one of a corresponding CA absorber working at microwave frequencies.

## 3. Results

_{y}; then, we may want to take into account the limitations of the most common nanofabrication facilities and force a minimum nanoparticle size, i.e., ${r}_{x}^{1},{r}_{x}^{2}>{r}_{\mathrm{min}}$. Moreover, in order to minimize the coupling between metasurfaces that may affect their effective surface impedance, we need to impose that the two nanoparticle arrays have a minimum separation, i.e., d

_{1}, d

_{2}> λ/15, λ being the wavelength inside the dielectric at the central frequency of operation. Finally, we can impose that the overall thickness of the CA absorber is less than or equal to the one of a conventional Salisbury screen, i.e., d

_{1}+ d

_{2}< λ/4.

_{d}= 2.1). It is worth noticing that the choice of the dielectric substrate affects both the thickness of the absorber (equal to λ/4 at the central frequency of operation) and the optimal size of the nanoparticles returned by the optimization process. Assuming a normal incidence, the reflection coefficient (6) reads as

_{d}is the wavenumber in silica, Z

_{s}

^{1}and Z

_{s}

^{2}are the complex surface impedances of the first and the second reactive layer, respectively, and ${Z}_{\times}={Z}_{s}^{1}{Z}_{s}^{2}$, ${Z}_{+}={Z}_{s}^{1}+{Z}_{s}^{2}$. It is important to underline that the first optical reactive layer is completely immersed in silica and, thus, the permittivity of the host medium to be used in (8) is the one of silica. Conversely, as it should be clear from Figure 3a, the second reactive layer is placed between the dielectric and the vacuum. Thus, an average permittivity between these two dielectrics must be used as ε

_{h}.

_{min}= 550 THz, f

_{max}= 650 THz, respectively. This leads to the following optimal parameters: a

_{1}= a

_{2}= 29 nm, r

_{y}

^{1}= r

_{y}

^{2}= 8 nm, r

_{x}

^{1}= r

_{x}

^{1}= 4 nm, d

_{1}= 55 nm, d

_{2}= 30 nm. It is important noticing that, even though the dimensions of the two nanoparticle arrays are the same, their surface reactance (shown in Figure 3b) is different because of the different effective permittivity of their host material. In Figure 3b, we also show the input reactance of the absorber evaluated at $z={d}_{1}^{-}$ and $z={d}_{2}^{-}$, i.e., before the first and the second reactive layer, respectively. From this comparison, it should be clear that the optimization has led to reactive layers, whose surface reactance is engineered to compensate for the intrinsic dispersion of the dielectrics.

_{in}at z = d

_{1}+ d

_{2}. The input impedance is compared to the one of a single-layer optical Salisbury screen, whose resistive sheet is resonant and impedance-matched with the vacuum at 600 THz [20]. As expected, the input reactance of the Salisbury screen is equal to zero only at a single frequency and increases quickly as we move away from the resonance. Conversely, the CA-absorber exhibits an input reactance that is close to zero within a wide range around the design frequency. This is confirmed by the results shown in Figure 4b, where the absorbance of the two absorbers is compared. The CA absorber has a significantly broader absorption bandwidth due to the optimized reactive behavior of the lossy layers. Specifically, assuming an absorbance threshold of 0.9, the fractional bandwidth of the CA absorber is 13.2%, almost 4 times bigger than the one of the optical Salisbury screen. We stress that this comparison has been carried out by considering two absorbers with the same overall thickness.

_{1}= a

_{2}= a

_{3}= 32 nm, r

_{y}

^{1}= 9 nm, r

_{x}

^{1}= 5 nm, r

_{y}

^{2}= 10 nm, r

_{x}

^{2}= 4 nm, r

_{y}

^{3}= 8 nm, r

_{x}

^{3}= 4 nm, d

_{1}= 40 nm, d

_{2}= 25 nm, d

_{3}= 25 nm. The reflectance, absorbance and transmittance of this absorber, obtained through full-wave simulations, are shown in Figure 5a. As can be appreciated, the absorbance stays above 0.9 within the frequency range 540–670 THz, resulting in a fractional bandwidth bigger than 20% with a thickness of only λ/4 at the center frequency of the absorption band. In Figure 5b, we report the value of the absorbance at three different frequencies as the impinging angle θ changes. As can be appreciated, for all the considered frequencies, the absorbance is larger than 0.9 for a wide range of incidence angles, up to 50°.

## 4. Alternative Layout and Sensitivity Analysis

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**(

**a**) A sketch of a three-layered circuit-analog absorber and (

**b**) its equivalent circuit model.

**Figure 2.**(

**a**) The optical metasurface made by an array of ellipsoidal plasmonic nanoparticles arranged in a square lattice; (

**b**) the complex surface impedance of two different optical metasurfaces designed to have the same surface resistance (R

_{s}= 200 Ω) and opposite reactive behaviors (capacitive and inductive). The ticks represent the results of full-wave simulations.

**Figure 3.**(

**a**) The two-layered optical circuit-analogue (CA) absorber. The ground plane is replaced by a thick enough layer of silver; (

**b**) the surface reactance of the two metasurfaces required to maximize the absorption within the frequencies range 550–650 THz. The surface reactances of the metasurfaces are compared with the input reactance of the absorber evaluated before the first and the second layer.

**Figure 4.**(

**a**) The input impedance of the designed CA absorber and of an equivalent Salisbury screen with the same thickness. (

**b**) The analytical and numerical absorbance (absolute value) of the designed CA absorber and of an equivalent Salisbury screen with the same thickness. Ticks represent the results of full-wave simulations.

**Figure 5.**(

**a**) The absorbance, reflectance and transmittance (absolute value) of a three-layered CA optical absorber obtained through full-wave simulations; (

**b**) the value of the absorbance (absolute value) at three different frequencies as the impinging angle changes.

**Figure 6.**(

**a**) The complex surface impedance exhibited by two arrays of nanoellipsoids and nanocylinders with the same size; (

**b**) the absorbance of the three-layered CA absorber made by nanocylinders for different values of the perturbation parameter Δ. In the inset, the equivalent absorber structure for Δ = 6 nm is shown.

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**MDPI and ACS Style**

Monti, A.; Alù, A.; Toscano, A.; Bilotti, F.
The Design of Optical Circuit-Analog Absorbers through Electrically Small Nanoparticles. *Photonics* **2019**, *6*, 26.
https://doi.org/10.3390/photonics6010026

**AMA Style**

Monti A, Alù A, Toscano A, Bilotti F.
The Design of Optical Circuit-Analog Absorbers through Electrically Small Nanoparticles. *Photonics*. 2019; 6(1):26.
https://doi.org/10.3390/photonics6010026

**Chicago/Turabian Style**

Monti, Alessio, Andrea Alù, Alessandro Toscano, and Filiberto Bilotti.
2019. "The Design of Optical Circuit-Analog Absorbers through Electrically Small Nanoparticles" *Photonics* 6, no. 1: 26.
https://doi.org/10.3390/photonics6010026