1. Introduction

Perfect light absorption, as one hot topic in optical physics, is crucial for thermophotovoltaics [1, 2], sensors, cross-talk reduction in optoelectronics [3], photothermal cancer therapy [4] etc. Metamaterial exhibits unusual properties, not found in natural materials, and has grown up into a burgeoning research field in the past decade [5 ,6]. It is well known that metamaterial is comprised of sub-wavelength unit cells, so called “meta-atoms” or “meta-molecules” [7, 8], which can be designed under requirement. N. I. Landy et al proposed a thin layer of metamaterial for perfect absorber at specific frequencies for the first time [9]. In contrast, conventional materials, such as semiconductors, have specific absorption bands with low-energy edge but suffer from the low tunability of the absorption bandwidth. Since then a lot of efforts have been devoted to metamaterial absorber (MA) with working frequency spanning from microwave [10], terahertz (THz) [11–13] to the infrared and visible frequencies [14–23]. Owing to optical resonances [20], absorption band of optical metamaterial can be spectrally selective which is superior to the semiconductor and other conventional materials. However, it is still a great challenge for ultra-wideband optical absorption with PMA. Recently, K. Aydin and his collaborators proposed a kind of nanostructured absorbers for black body by resonant absorption [19, 20]. They show that, a multiple of localized and delocalized surface plasmon polariton (SPP) resonances, generated in the cross-trapezoid shaped metal-insulator-metal (MIM) absorber, could be existed at different frequencies close enough to the neighboring ones, resulting in a broadband (400-700 nm) absorption. A. S. Hall et al demonstrated an approach for broadband absorption in the visible spectrum for both $p$- and $s$-polarized lights based on SPP waves [21]. An efficient solar energy absorber based on layered nanopyramid structure has been numerically studied [24]. The absorber is highly absorptive in the entire solar spectrum with a rather bulky volume. However, for practical applications, a wideband MA covering visible frequencies, which is incident angle independent is still a great challenge.

In this study, we propose fish-scale structure for PMA with wide angle covering all visible frequencies. The fish-scale structure has been investigated extensively as a magnetic mirror [25, 26], a local field concentrator and resonant “amplifier” of losses [25, 27], a polarization transformer [28], even a filter for asymmetric transmission [29]. As demonstrated by N. I. Zheludev and collaborators, the fish-scale structure can also be used for light-harvesting or frequency-selective detection at optical frequencies. We show that the strong dissipative property of our model comes from the hybridization of localized resonances of cascaded fish-scale meta-atoms and delocalized ones arising from their couplings. Within the whole visible spectrum, the proposed PMA has superior performance to the previous studies [19–21].

2. PMA with rectangular fish-scales

Figure 1(a) shows the unit cell and Fig. 1(b) gives a top view of the designed PMA. The PMA is comprised of a fish-scale patterned metal film and a ground metal film which are separated by a dielectric layer. The pattern in each unit cell is comprised of meandering fish-scale metallic strips, which are connected to each other in square loop. As such, the pattern is in four-fold rotational symmetry with respective to the square center, and the absorption is polarization insensitive [30]. Each side of the square loop is comprised of two rectangular fish-scale patterns in different geometric parameters, i.e., two meta-atoms, cascaded by metallic strips as illustrated in Fig. 1(b). Drude model of silver is adopted for the dielectric response of metal in this study. A dielectric constant of 2.1, that of fused Quartz is adopted for the dielectric layer (cyan color part) [14, 15, 31]. Different geometrical parameters are chosen for the two cascaded meta-atoms. Accordingly, the localized resonances and delocalized ones of the meta-atom chains along the square loop are at different frequencies, forming a broad absorption band. To realize this broadband absorption, the period $P$ of the square arrays, the width $l$ of the square loop, the side length $a$ of the inner square loop, the width sizes of silver strips and air-gaps from $g_1$ to $g_5$, and thicknesses of the dielectric layer $t_d$, front silver layer $t_{s1}$ and back silver layer $t_{s2}$ need to be optimized.

 

Fig. 1 (a) Schematic of the unit cell of the PMA and (b) the top view of the unit cell with geometric parameters marked with arrows and letters.

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When the structure of PMA is not in mirror-symmetry, the cross-reflection may occur especially under oblique incidence [32]. For $x$-polarized incidence along the z direction, the absorption of such a PMA shall be expressed as $A_x\left(\omega \right)=1-T_{xx}\left(\omega \right)-T_{yx}\left(\omega \right)-R_{xx}\left(\omega \right)-R_{yx}\left(\omega \right)$, in which $T_{xx}\left(R_{xx}\right)$ and $T_{yx}\left(R_{yx}\right)$ are the co-polarization and cross-polarization transmission (reflection) coefficients [32–37], respectively. These coefficients are defined as $T_{xx}\left(\omega \right)={\left|S_{2x1x}(\omega )\right|}^2$, $T_{yx}\left(\omega \right)={\left|S_{2y1x}(\omega )\right|}^2$, $R_{xx}\left(\omega \right)={\left|S_{1x1x}(\omega )\right|}^2$ and $R_{yx}\left(\omega \right)={\left|S_{1y1x}(\omega )\right|}^2$, in which S_2i1j and S_1i1j are the scattering parameters for transmission and reflection, respectively. Subscripts $i$ and $j$ ($i=x,y$; $j=x,y$) refer to the mode conversion from $j$ to $i$. Since the electromagnetic (EM) waves cannot penetrate through silver substrate of the PMA, $T_{xx}\left(\omega \right)=T_{yx}\left(\omega \right)=0$. We have $A_x\left(\omega \right)=1-R_{xx}\left(\omega \right)-R_{yx}\left(\omega \right)$ for the PMA shown in Fig. 1(a). Similar rules also apply for the absorption $A_y\left(\omega \right)$ provided that the incident wave is $y$-polarized.

We perform finite element method to calculate absorption and reflection of the PMA shown in Fig. 1(a). Periodic boundary conditions are applied along $x$ and $y$ direction with open boundary condition along $z$ direction. For simplicity, normal incidence with polarization along $x$ direction (or $y$ direction) is considered, as shown in Fig. 1(a). The frequency range of our interest is from 300 to 800 THz.

In simulation, dielectric response of silver is described by Drude model [38–40] to fit its realistic property at near-infrared and visible frequencies. The experimental data [31] of the complex dielectric function $\epsilon (\omega )={\epsilon }_1(\omega )+i{\epsilon }_2(\omega )$ and those from Drude model are shown in Fig. 2. We see that the two sets of data are in good agreement from 400 THz to 750 THz.

 

Fig. 2 Complex dielectric function from experiment [31] (hollow squares) and Drude model (solid lines).

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One optimized set of geometrical parameters of the PMA unit cell are shown as following (in nanometers): $P=380$, $l=360$, $g_1=18$, $g_2=22$, $g_3=19$, $g_4=27$, $g_5=20$, $a=270$, $t_d=55$, $t_{s1}=30$ and $t_{s2}=150$. The parameter $t_{s2}$ is chosen to be a fairly large value because the silver back layer should be optically thick enough to block the transmission of the light.

3. Results and discussions

According to the simulation results of absorption shown in Fig. 3, the absorption is higher than 90% for both $x$- and $y$-polarized waves at most frequencies of the visible spectrum (400-700 nm) under normal incidence. The maximum value of absorption is 99.97% at 671.5 THz for $x$-polarized wave and 99.79% at 621.5 THz for $y$-polarized wave, respectively. The absorption, higher than 99%, is nearly perfect in the frequency range of 578.5-691.5 THz ($x$-polarized wave) and 586.5-682.5 THz ($y$-polarized wave). We also note that the absorption is polarization insensitive with $A_x\left(\omega \right)$ and $A_y\left(\omega \right)$ being almost identical within the whole visible spectrum.

 

Fig. 3 The absorption of the PMA at normal incidence for $x$-polarized (A_x) and $y$-polarized (A_y) EM waves.

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In the entire visible spectrum (400-700 nm), there is an absorption dip in the frequency range of 428.5-446.5 THz (672-700 nm) as shown in Fig. 3. We also see that the absorption $A_x\left(\omega \right)$ of $x$-polarized incidence is slightly different from $A_y\left(\omega \right)$ of $y$-polarized incidence in the frequency range of 500-580 THz. More calculations indicate that the minor variation results from the lack of mirror symmetry of the square supercell, different strength of specific SPP modes are excited with respective to the $x$- or $y$-polarized incidence. At short wavelength end, it is diffraction that makes the absorption to have strong vibration, as shown in Fig. 3.

The strong light absorption of the ultrathin PMA relies on the localized and delocalized SPP resonances [19]. These resonances can be traced by the evolution of the absorption bands as a function of the geometric parameters [20]. For example, a delocalized SPP resonance which arises from propagating waveguide modes along the square loop, will have apparent shift with respective to the variation of the period $P$. While a resonance, that is almost independent of the variation of the period $P$, shall be localized surface plasmon resonance (LSPR) or Fabry Pérot like mode which is independent of periodicity. The evolutions of the absorption bands as a function of the period $P$ for the $x$- and $y$-polarized incident waves are shown in Fig. 4(a) and 4(b), respectively. When the period $P$ varies from 180 to 400 nm, there is an optimized set of geometrical parameters of the PMA unit cell which give the best results for every period value. The absorption bands of the PMA exhibit excellent structural tunability as shown in Figs. 4(a) and 4(b).

 

Fig. 4 The absorption of the PMA for the (a) $x$-polarized wave, (b) $y$-polarized wave as a function of period $P$, frequency and wavelength.

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Since the frequencies of resonance modes are very close to the neighboring ones, it is hard to trace them all. Figures 4(a) and 4(b) present four easily observed modes, which are marked as $m_{x1}$ to $m_{x4}$ for the $x$-polarized incident wave ($m_{y1}$ to $m_{y4}$ for the $y$-polarized incident wave). The resonance mode $m_{x1}$ ($m_{y1}$ for the $y$-polarized wave) shows weak dependence on the change of the period $P$ so it is either LSPR mode or Fabry Pérot like mode. As to modes $m_{x2}$-$m_{x4}$ ($m_{y2}$-$m_{y4}$ for the $y$-polarized wave)_, they should be propagating modes as they are sensitive to the variation of the period $P$. Also we can find that the PMA with a period of 380 nm (used in our design) has superior performance for absorption with wide range and strong dissipation.

Figures 5(a)-5(h) present the calculated magnitude distributions of magnetic fields of the propagating mode ($m_{x3}$) at 495.5 THz and localized resonance mode ($m_{x1}$) at 725 THz for the $x$-polarized incident wave. We see that the magnetic fields are highly concentrated within a skin depth between the patterned fish-scale structure and the ground metal plane.

 

Fig. 5 The calculated magnitude distribution of the magnetic field for the $x$-polarized incident wave at 495.5 THz: (a) the top Ag / bottom quartz interface (in the $x$-$y$ plane), (b) the right side (in the $x$-$z$ plane), (c) the top side (in the $z$-$y$ plane), (d) the top quartz / bottom Ag interface (in the $x$-$y$ plane) of the PMA unit cell. 725 THz: (e) the top Ag / bottom quartz interface (in the $x$-$y$ plane), (f) the right side (in the $x$-$z$ plane), (g) the top side (in the $z$-$y$ plane), (h) the top quartz / bottom Ag interface (in the $x$-$y$ plane) of the PMA unit cell.

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The SPP modes could be magnetic, aroused by the interaction of the fish-scale patterns and their images with respective to the ground silver mirror. As shown in Figs. 5 (a)-5(d), the magnetic SPP mode at 495.5 THz, weakly localized to the interface of the metal and the dielectric layer is propagating within the dielectric layer. In contrast, the mode at 725 THz, strongly localized to the interface of the metal and the dielectric layer, is not confined within the dielectric layer (shown in Figs. 5 (e)-5(h)).

From the magnitude distributions of magnetic fields of the propagating modes $m_{x2}$ and $m_{x4}$ (not shown here), we see that the propagating modes with different wavelengths are located in different regions of the structure. These results agree well with the theoretical analysis. It is obvious that a delocalized/localized resonance mode shall lead to the enhancement of the local field within dielectric layer/at the interface of the metal and the dielectric layer, respectively [41]. And the high performance of such a PMA shall be attributed to the excitations of both localized and delocalized SPPs.

As shown in Fig. 6, the absorption of the PMA is polarization insensitive in the whole visible spectrum. The averaged absorption is higher than 90% over the entire visible spectrum when the polarization angle $\theta$ varies from 0° to 90°. When the polarization angle is not 0° ($x$-polarized) or 90° ($y$-polarized), the incident EM wave will also excite the localized and delocalized surface plasmon resonances which are close enough to the neighboring ones. And the broadband absorption characteristics of the PMA can be maintained at different polarization angles which lead to the polarization insensitivity of the PMA.

 

Fig. 6 Absorption of the PMA as a function of polarization angle, frequency and wavelength.

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The PMA also exhibits excellent broadband absorption property for the non-normal incidence (shown in Figs. 7(a) and 7(b)) owing to the multiple localized resonances and propagating modes. For both the transverse electric (TE) and transverse magnetic (TM) incident waves, the absorptions of the PMA are almost independent of the incident angle below 60°, and the averaged value is higher than 70% at 60° incidence in the visible spectrum. When the incident angle increases further, only several absorption peaks still exist. In general, the absorption for the TE incidence at incident angles higher than 60° is lower than the TM incidence due to the weak induction of the magnetic resonances between the top connected rectangular fish-scales and the ground metal film.

 

Fig. 7 Absorption of the PMA for the non-normal incident EM waves: (a) TE mode, (b) TM mode.

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Owing to the rotational symmetry of the PMA, cross-polarization reflection coefficients $R_{yx}$ and $R_{xy}$ are almost zero at normal incidence for the $x$- and $y$-polarized incidence as shown in Figs. 8(a) and 8(b), respectively. The case of oblique incidence is also investigated in detail. With the increase of incident angles, the values of $R_{yx}$ and $R_{xy}$ are not zero at several frequency ranges. The maximum value of $R_{yx}$ is 2.5% at 55° incidence centered at 469 THz. As to the coefficient $R_{xy}$, the maximum value is 3.5% at 25° incidence centered at 575.5 THz.

 

Fig. 8 Cross-polarization reflection coefficients of the PMA for the non-normal incident EM waves: (a) $R_{yx}$, (b) $R_{xy}$.

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4. Conclusions

We propose a fish-scale PMA at visible frequencies, which is polarization insensitive. The fish-scale structure, as thin as 235 nm, is quite suitable for geometrical optimization, which is employed for broadband absorption. Such superior performance of the PMA is attributed to a multiple of localized resonances and propagating ones of the structure. The PMA can operate in a wide range of incident angle. For both TE and TM incidence, the averaged absorption is higher than 70% in the entire visible spectrum at incident angles from 0° to 60°. This work might be helpful for the design of sensors, solar cells and thermophotovoltaic cells.