## Abstract

A dual-band polarization-independent coherent perfect absorber(CPA) based on metal-graphene nanostructure is proposed, which is composed of golden nanorings with different sizes on graphene monolayer. Based on the finite-difference time-domain (FDTD) solutions, coherent perfect absorptions of the metal-graphene CPA are achieved at frequencies of 50.54 THz and 43.60 THz, which are resulted from the excited surface plasmon resonance induced by different size nanorings. Through varying the relative phase of two incident countering-propagating beams, the absorption peaks are all-optically tuned from 98.3 % and 98.4 % to nearly 0, respectively. By changing the gate-controlled Fermi energy of the graphene layer, the resonance frequencies of the CPA are tuned simultaneously without changing the geometrical parameters. And polarization independence of the metal-graphene CPA is revealed due to the center symmetry of nanoring structure. The electrical tunability of resonance frequency and polarization independence enable the proposed CPA to be widely applied in optoelectronic and engineering technology areas for tunable active multiple-band regulation and control.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Perfect absorber, which based on metamaterial system working from single band to multi-band, has attracted much attention due to the excellent prospect for applications in thermal imaging, optoelectronics, photonics, and materials detection [1–3]. Recently, graphene has been introduced to design perfect absorbers because of its excellent properties such as electronical tunability and relatively low plasmonic losses [4–9]. Hybrid graphene-metal perfect absorbers with broadband [10] and dual-band operation [11,12] have been studied, which have outstanding features of widely frequency tunability, relatively high modulation depth and fast modulation speed [10, 13]. Importantly, coherent perfect absorber (CPA) has become a main concern of research due to the characteristic of all-optical modulation [14–16]. The theoretical analysis of the coherent perfect absorption was investigated with the scattering matrix by Stone’s group [17]. Afterwards the first coherent perfect absorber was realized in a silicon slab cavity by Wan’s group [18]. Then many kinds of coherent perfect absorbers with different nanostructures have been investigated [19–24], such as a free-standing corrugated metal film by suppressesing specular order and converting the incident energy into the SPs (surface plasmons) [25], transversely isotropic CPA of circular polarization beams in the chiral metamaterial slab to perform polarization control with constant output intensity [26] and so on. CPA with graphene of electrical tunability has also been explored. By employing a graphene film patterned with a periodical array of holes, tunable coherent perfect absorption in the mid-infrared range has been studied by Zhang’s group [27]. Using a graphene film patterned with periodical split ring, a high-speed gate-tunable CPA is demonstrated in the terahertz regime with the response time of ∼ 36 ps by Hu’s group [28]. By exploiting a suspending monolayer graphene, coherent perfect absorber is realized in a nonresonant manner by Fan’s group [29]. Electronically tunable CPA with graphene brings practical cababilities to optical detection and signal processing.

In this paper, the polarization-independent metal-graphene absorber, which contains golden nanorings of two different sizes on the graphene-covered SiO_{2} substrate, is proposed to realize dual-band coherent perfect absorption in mid-infrared region. The principle of coherent absorption is demonstrated by the scattering matrix and the properties of the metal-graphene CPA are analyzed by the finite-difference time-domain (FDTD) solutions. Due to the interaction of excited surface plasmon induced by golden nanorings with different sizes, the absorption peaks of the dual-band metal-graphene CPA are achieved at frequencies of 43.51 THz and 50.54 THz, which have been both enhanced compared with that of single size golden nanoring only structure. And the center symmetric nanoring structures lead to polarization independence of the metal-graphene CPA. Through changing the relative phase of two countering propagating incident beams, the coherent absorptions are all-optically modulated. By tuning the Fermi energy of graphene from 0.15 eV to 0.75 eV, the coherent absorption peaks of the dual-band metal-graphene CPA induced by big nanoring and small nanorings are modulated sensitively in a relative wide frequency band from 42.64 THz to 45.09 THz and 49.31 THz to 52.01 THz, respectively, which provides a very promising technology with convenience and sensitivity for applications such as tunable multi-band modulators, switches and detectors.

## 2. Structure design and theoretical backgrounds

The schematic of the tunable polarization-independent metal-graphene dual-band coherent perfect absorber is shown in Fig. 1, which contains two different sizes golden nanorings separated with the dielectric substrate SiO_{2} by a monolayer of graphene. The geometric parameters of unit cell of the CPA are shown in Fig. 1(b). The graphene layer is located at z = 0 on the SiO_{2} substrate and then the golden nanorings are put on the graphene layer. The thickness of golden nanoring is 80 nm. The smaller golden nanoring has the inner semi-diameter of *R*_{s–in} = 550 nm and the outer semi-diameter of *R*_{s–out} = 650 nm; the bigger golden nanoring has the inner semi-diameter of *R*_{b–in} = 750 nm and the outer semi-diameter of *R*_{b–out} = 950 nm. The center of small nanorings are put at the focus of the crossed diagonals made by four big nanorings and vice versa. The centers of neighbouring golden nanorings of identical size are all fixed at P = 5000 nm. The refractive index of SiO_{2} is taken to be 1.45. In order to apply voltage on graphene, the top gating method is used [30–33], of which the detailed electrode structure is shown in Fig. 1(a). The double-sided conductive tapes on the graphene layer are used as electrical contacts. Ion gel layer is set on the metal-graphene CPA. The double-sided top Au electrodes with upper SiO_{2} substrate are on the ion gel layer. The top gating scheme with high capacitance ion gel enables high magnitudes of electric fields to be formed near graphene, resulting in a shift in Fermi level from the Dirac point [31,33]. The top Au electrodes can be formed by photolithography [32] and the golden nanorings can be fabricated by electron beam evaporation [34]. Two coherent incident laser beams, *I*_{1} and *I*_{2}, are illuminated on the CPA from opposite sides in the negative and positive z-axis direction, respectively. The relationship between *I*_{1} and *I*_{2} is *I*_{2} = *αI*_{1}*exp* (*iφ* + *ikz*), where *φ* is the phase difference between *I*_{1} and *I*_{2}, *z* indicates the phase reference point of *I*_{1} and *I*_{2}, *α* is the relative amplitude of *I*_{2} compared with *I*_{1}. *O*_{1} and *O*_{2} are the output beams scattering to the lower and upper side of the CPA.

When only one incident beam *I*_{1} is illuminated on the metal-graphene structure in the z-axis direction, the incoherent absorption *A* is derived as [27]:

*η*is the special self-consistent amplitude related to the materials parameters and metal-graphene nanostructure. In the symmetry environment which the refractive indices are the same in upper and lower sides of the CPA, the combined reflection and transmission coeffcients of CPA are derived as

*r*=

*η*and

*t*= 1 +

*η*, respectively. When

*η*= −0.5,

*r*= −0.5,

*t*= 0.5, the incoherent absorption reaches its maximum

*A*= 0.5, which is called incoherent absorption limit.

_{max}When two coherent incident beams *I*_{1} and *I*_{2} impact on the CPA from opposite sides, which is defined as coherent illumination, the relationship between the input beams *I*_{1} and *I*_{2} and output beams *O*_{1} and *O*_{2} is described by scattering matrix:

*O*

_{1}is the interference result between the transmitted light of

*I*

_{1}and the reflected light of

*I*

_{2}, and so does

*O*

_{2}. When the incoherent absorption limit is satisfied,

*r*

_{1,1}=

*r*

_{2,2}= −0.5,

*t*

_{1,2}=

*t*

_{2,1}= 0.5. And utilizing the relationship

*I*

_{2}=

*αI*

_{1}

*exp*(

*iφ*+

*ikz*) with

*z*= 0, the coherent absorption

*A*of the CPA is derived as [27] :

_{co}*A*can be tuned by changing

_{co}*α*and

*φ*. Especially, when

*α*= 1, the coherent absorption

*A*can be varied from the maximum

_{co}*A*

_{co–max}= 1 to the minimum

*A*

_{co−min}= 0 when

*φ*changes from 2

*Nπ*to (2

*N*+ 1)

*π*.

## 3. Simulation results

The FDTD solutions (Lumerical Solutions, Inc.) are employed to numerically study the properties of the tunable polarization-independent metal-graphene dual-band CPA. In the FDTD simulation, an ideal model of the metal-graphene CPA with the semi-infinite SiO_{2} substrate has been utilized in which the ion gel layer has not been included. The simulation unit is a region of x span of 5000 nm, y span of 5000 nm and z span of 40 *μ*m with periodical boundary condition. Two coherent incident beams *I*_{1} and *I*_{2} are illuminated on the ideal CPA model from opposite sides in the negative and positive z-axis direction, and the coherent beam *I*_{2} is set in the SiO_{2} substrate located at z = − 3.5 *μ*m. The mesh accuracy in the simulation is set as 2. Additional precise meshes are set as 50 nm in x and y directions, and 5 nm in the z direction covering the range of the golden nanorings and graphene layer, which include the SiO_{2} substrate with thickness of 20 nm. The complex permittivity of Au is invoked from “Au (Gold) - Palik” material data [35] included in the FDTD database, and the surface conductivity of graphene *σ* is invoked from “C (graphene) - broadband” material model, which is computed from the Kubo formula:

*f*(

_{d}*∊*) = (

*e*

^{(∊–EF)/(kBT)}+ 1)

^{−1},

*E*is the Fermi energy of graphene,

_{F}*ω*is the angular frequency, Γ = 0.00099 eV is the scattering rate. The temperature is

*T*= 300 K.

The spectrum characteristics of big-size golden nanoring only and small-size golden nanoring only structures on graphene layer are simulated by the FDTD solutions with Fermi energy *E _{F}* = 0.35 eV. The unit cells of individual size golden nanoring only structures are shown in Fig. 2(e). The simulated transmission (T), reflection (R) and absorption (A = 1 − R − T) spectra of the big-size golden nanoring only structure are presented as the black, red and blue curves in Figs. 2(a) and 2(b), respectively. The absorption under incoherent illumination climbs to the maximum of

*A*

_{big–max}= 44.9 % at frequency of 43.51 THz shown as the blue curve in Fig. 2(a). The absorption peak is achieved due to the excited surface plasmon resonance induced by big nanorings at frequency of 43.51 THz where the incoherent absorption limit is satisfied with

*η*= −0.5. Under coherent illumination with two countering propagating incident beams

*I*

_{1}and

*I*

_{2}with

*φ*= 0 and

*α*= 1, the absorption climbs to the maximum

*A*

_{co–big–max}= 91.8 % at frequency of 43.51 THz where the frequency is the same as that of incoherent illumination shown as the blue curve. Meanwhile, the transmission and reflection spectra both decrease to near 0 at 43.51 THz shown as the black and red curves in Fig. 2(b). As

*I*

_{1}and

*I*

_{2}illuminate from opposite sides, the surface plasmon resonances are excited by the golden nanorings and interact with each other, leading to the coherent absorption peak. The simulated transmission (T), reflection (R) and absorption (A) spectra of the small-size golden nanoring only structure under incoherent illumination and coherent illumination are shown in Fig. 2(c) and 2(d), respectively. Similarly, coherent absorption climbs to the maximum with

*A*

_{co–small–max}= 94.8 % at frequency of 50.10 THz where incoherent absorption limit is satisfied.

The absorption spectra of metal-graphene dual-band CPA are simulated by the FDTD solutions with Fermi energy *E _{F}* = 0.35 eV. Under coherent illumination of p polarized incident beams with

*φ*= 0 and

*α*= 1, two absorption peaks are achieved at resonance frequencies of 50.54 THz and 43.60 THz in the absorption spectrum, which are shown as the red curve in Fig. 3(a). The resonance frequencies at the double coherent absorption peaks are the same as that of incoherent illumination shown as the black curve in Fig. 3(a), where the incoherent absorption limits are both realized with

*η*= −0.5. The excited surface plasmon resonances induced by small nanorings and big nanorings lead to the double absorption peaks of metal-graphene CPA. Compared with big-size golden nanorings only and small-size golden nanoring only structures, the resonance frequencies both blue shift while the absorption peaks rise up to

*A*

_{co–small–max}= 98.3 % and

*A*

_{co–big–max}= 98.4 %, respectively. As the excited surface plasmon resonance induced by different sizes of nanorings interact with each other, coherent perfect absorption are realized at the resonance frequencies. The simulated absorption spectrum of coherent illumination of s polarized incident beams is also shown by the purple dash curve in Fig. 3(a), which nearly coincide with that of p polarized incident beams. The electrical field distributions of CPA in the xy-plane at resonance frequencies of 50.54 THz and 43.60 THz with p and s polarized incident beams are shown in Figs. 3(b) and 3(c), respectively. The electrical field distributions present as a face-to-face quarter moons shape, of which the direction changes from left-and-right to up-and-down as the polarization of incident beams vary from s to p for both big and small nanorings. Although the electric field distributions vary with different polarization of incident beams, the absorption spectrum remains the same due to the symmetry of the nanoring structure. Therefore the metal-graphene dual-band CPA is a polarization-independent device.

The modulation of coherent absorption of the metal-graphene dual-band CPA with various phase differences *φ* between two countering propagating coherent beams is simulated by the FDTD solutions with Fermi energy *E _{F}* = 0.35eV and the relative amplitude

*α*of the incident beams is set to 1. The absorption spectra of the CPA with the change of phase difference

*φ*from 0 to 2

*π*are shown as the multicolor solid curves in Fig. 4, and the maximum coherent absorption

*A*

_{co–small–max}at resonance frequency of 50.54 THz induced by small nanorings and

*A*

_{co–big–max}induced by big nanorings at resonance frequency of 43.60 THz with various phase difference

*φ*are presented as the black and red dash-dotted curves, respectively. As

*φ*increases from 0 to

*π*, the maximum coherent absorption

*A*

_{co–small–max}and

*A*

_{co–big–max}decline continuously from 98.3 % and 98.4 % to nearly 0, respectively; as

*φ*increases from

*π*to 2

*π*gradually,

*A*

_{co–small–max}and

*A*

_{co–big–max}both go up to nearly 98.0 % again from 0. The maximum coherent absorption at two resonance frequencies give a modulation contrast of 42.94 dB and 43.50 dB with

*φ*, respectively. Especially, when

*φ*= 2

*Nπ*, the coherent perfect absorption are achieved at resonance frequencies of 50.54 THz and 43.60 THz. The maximum coherent absorption at two resonance frequencies both present as a function of cosine with phase difference

*φ*, which is in accord with conclusions derived from Eq. (3). Altering the phase difference

*φ*between two coherent beams leads to all-optical modulation of the dual-band metal-graphene CPA at the doublle resonance frequencies.

The absorption spectra of the metal-graphene dual-band CPA with various inner and outer semi-diameter of the golden nanoring are simulated by the FDTD simulations with Fermi energy *E _{F}* = 0.35 eV. The absorption spectra with the change of semi-diameter of small nanoring are shown in Figs. 5(a) and 5(b), while the dimensions of the big golden nanoring remain unchanged. As the inner semi-diameter of small nanoring

*R*

_{s–in}increases from 510 nm to 570 nm with fixed outer semi-diameter

*R*

_{s–out}= 650 nm, the resonance frequency induced by the small golden nanoring red shifts from 54.04 THz to 49.51 THz and the coherent perfect absorption peak increases from 90.6 % to 99.0 %, which is shown in Fig. 5(a). As the outer semi-diameter of small nanoring

*R*

_{s–out}decreases from 690 nm to 630 nm with fixed inner semi-diameter

*R*

_{s–in}= 550 nm, the resonance frequency induced by the small golden nanoring red shifts from 52.93 THz to 49.94 THz and the coherent perfect absorption peak increases from 86.7 % to 99.2 %, which is shown in Fig. 5(b). When shortening the width of small nanoring

*R*from 140 nm to 80 nm, the coupling of surface plasmon resonance between the inner and outer small nanoring wall enhances [36], the amplitude of the absorption peak increases and the resonance frequency induced by small nanoring red shifts and gets close to that of big nanoring, heightening the interaction of excited surface plasmon resonance between different sizes of nanorings. Therefore, the coherent perfect absorption peak induced by small nanorings increases, and that induced by big nanorings also varies, but not significantly. The absorption spectra with the change of semi-diameter of big nanoring are shown in Figs. 5(c) and 5(d), while the dimensions of the small golden nanoring remain unchanged. Similarly, as the inner semi-diameter of big nanoring

_{s}*R*

_{b–in}increases from 720 nm to 810 nm with fixed

*R*

_{b–out}= 950 nm, the resonance frequency induced by the big golden nanoring red shifts from 44.70 THz to 41.36 THz; as the outer semi-diameter of big nanoring

*R*

_{b–out}decreases from 980 nm to 890 nm with fixed

*R*

_{b–in}= 750 nm, the resonance frequency induced by the big golden nanoring red shifts from 45.03 THz to 42.17 THz. When widening the width of big nanoring

*R*from 140 nm to 230 nm, the coupling of surface plasmon resonance between the inner and outer big nanoring wall weakens [36], the amplitude of the absorption peak decreases and the resonance frequency induced by big nanoring blue shifts and gets close to that of small nanoring, heightening the interaction of excited surface plasmon resonance between different sizes of nanorings. Due to the overall minor size of the small nanoring, the surface plasmon resonance induced by small nanoring is weak and easy to be influenced by the interaction of excited surface plasmon resonance between golden nanorings, resulting that the energy of excited surface plasmon resonance of small nanoring transfer to that of big nanoring. Therefore, the coheret perfect absorption peak induced by big nanorings keeps almost constant and that induced by small nanoring decreases a little, as shown in Figs. 5(c) and 5(d). The change of geometrical parameters of golden nanorings result in the variation of resonance frequency and coherent abosorption, leading to the spectrum positions for

_{b}*η*= −0.5 shifted to maintain the coherent perfect absorption.

The absorption spectra of the metal-graphene dual-band CPA with various Fermi energies of the graphene layer are simulated by the FDTD solutions. As the Fermi energy increases from 0.15 eV to 0.75 eV, the resonance frequencies induced by the small nanorings and big nanorings in the simulated absorption spectra simultaneously blue shift in a relative wide frequency band from 49.31 THz to 52.01 THz and 42.64 THz to 45.09 THz, respectively, as shown in Fig. 6(a). The absorption spectra of the golden nanorings only structure and graphene layer only structure are presented as the red dashed curve and black dashed curve in Fig. 6(b), respectively. The absorption spectrum of the golden nanorings only structure has the similar shape as that in Fig. 6(a) but with different resonance frequencies. The absorption of graphene layer only structure is nearly zero as a function of frequency due to its characteristic of transparency and low absorption of light. In order to further analyze the effect of the introduced graphene layer, the electric field distributions of small and big nanorings along the diameters on the vertical section in the yz-plane which is shown in Fig. 1(a) without graphene layer and with graphene layer of various Fermi energies are simulated at frequencies of 50.00 THz and 43.17 THz, which is shown in Figs. 6(c)–6(h). Without graphene layer, the electric fields of surface plasmon resonance are concentrated around golden nanorings, as shown in Fig. 6(c) and 6(d). As the graphene layer is introduced, the electric fields of surface plasmon resonance enhance both in the graphene layer and around the golden nanorings. When the Fermi energy of graphene increases, the coupling of excited surface plasmon between the graphene layer and golden nanorings are changed, resulting in the further enhancement of electric fields in the graphene layer, as shown in Figs. 6(e)–6(h). Therefore, the resonance frequencies of the dual-band metal-graphene CPA are modulated. By increasing the Fermi energy of graphene layer, the coherent absorptions are tuned effectively in resonance frequency with a wide band without changing in amplitude. The spectrum positions for *η* = −0.5 at double absorption peaks are shifted simutaneously to preserve coherent perfect absorption, which leads to the active electrical tunability of the metal-graphene CPA [7–9].

In our simulations, coherent perfect absorption are achieved at two frequency domains in the mid-infrared region due to the interference of surface plasmon resonance induced by nanorings with two different sizes. By employing golden nanorings with more than two different sizes, coherent perfect absorption at multiple frequency domains will be realized. Meanwhile, by introducing the graphene layer, coherent perfect absorption at double frequency domains is electronically controlled with the Fermi energy, which will be useful and benefical for tunable multi-band optical devices such as sensors, filters and switches.

## 4. Summary

In summary, a dynamically tunable polarization-independent dual-band metal-graphene CPA is proposed in the mid-infrared region by integrating golden nanorings with different sizes on the graphene layer. Based on FDTD solutions, the coherent perfect absorption are achieved at two different frequency domains, which are resulted from the interaction of surface plasmon resonance excited by golden nanorings with different sizses. The coherent absorptions are all-optically modulated through changing the phase difference between coherent beams. By introducing the graphene layer into the nanostructures, electronical tunability of the double resonance frequencies is realized instead of geometrical adjustment, which makes the proposed CPA efficient and convenient in applications for tunable active multiple-band optical modulators, switches and detectors.

## Funding

National Natural Science Foundation of China (NSFC) (61675131, 61378067).

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