Tunable polarization-independent coherent perfect absorber based on a metal-graphene nanostructure

Yaying Ning; Zhewei Dong; Jiangnan Si; Xiaoxu Deng

doi:10.1364/OE.25.032467

1. Introduction

Coherent perfect absorption, which relies on the suppression of the scattering waves, has attracted much attention [1–4] for the all-optical tunability of absorption through the interference of the two counter-propagating coherent incident beams. The coherent perfect absorber (CPA) was first realized in a simple Si slab cavity illuminated in the 500 – 900nm range [5], and has been theoretically analyzed with the scattering matrix by Stone’s group [6]. Then the CPA with different structures have been proposed successively. A free-standing corrugated metal film CPA was proposed by Gupta’s group [7], in which the specular order energy is suppressed and converted into the surface plasmon. The broadband nearly coherent perfect absorber is studied by Luo’s group [8] with the heavily doping ultrathin silicon film. A perfect magnetic conductor (PMC) surface covered by an ultrathin conductive film has been fabricated to accomplish the single-beam illuminated coherent perfect absorption by Wang’s group [9]. With a heterogeneous metal-dielectric structure, the tunability of the absorption resonance wavelength of the CPA is achieved [10], which is further explored in the graphene-based system including the monolayer graphene quasi-CPA and the metal-graphene metamaterial surface quasi-CPA [11–14].

In this paper, a tunable polarization-independent metal-graphene coherent perfect absorber (CPA) with the central-symmetry etched cross array nanostructure is proposed at mid-infrared region. The scattering matrix is used to demonstrate the principle of coherent absorption and the finite-difference time-domain (FDTD) method is employed to analyze the tunability of the CPA. When the incoherent absorption limit is matched, the polarization-independent coherent perfect absorption is achieved at the absorption resonance. The coherent absorption is all-optically modulated from nearly 100% to zero by tuning the phase difference between coherent incident beams. By adding the graphene layer, the absorption resonance of the CPA is electrically tuned with the Fermi energy. And within range of incident angle θ = −12° to θ = 12°, the CPA is angular insensitive. The electric tunability of the absorption resonance and the central-symmetry metal-graphene nanostructure enable the CPA promising in the tunable polarization-independent detections, modulators, transducers and switches at mid-infrared region.

2. Principle

The schematic of the tunable polarization-independent metal-graphene coherent perfect absorber (CPA) is shown in Fig. 1, which is composed of the hollowed-out cross array in the golden film placed on the graphene-covered glass substrate. The geometric parameters of unit cell of the CPA are shown in Fig. 1(b). The length and width of the cross structure is a₁ = 3100nm and a₂ = 360nm, respectively. The thickness of the gold film is 100nm. The thickness of glass substrate is 2mm. The period of CPA unit is P = 8150nm. The refractive index of the environment at the upper and lower side of the CPA is the same, which builds a symmetric environment. The incident laser beams I₁ and I₂ with identical incident polarization and angle impact on the CPA from above and below, respectively. I₁ = αe^iβI₂, where α is the relative amplitude of I₂ compare with I₁, β is the phase difference between I₁ and I₂. O₁ and O₂ are the output beams scattering to the upper and lower side, respectively.

Fig. 1 (a) The schematic of the tunable polarization-independent metal-graphene coherent perfect absorber. (b) The unit cell of the metal-graphene CPA nanostructure

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When only incident beam I₁ impacts on the CPA, which is defined as incoherent illumination, the incoherent absorption of the CPA at the condition that the thickness of CPA is much smaller than the wavelength of incident beam is derived [15]:

A = 1 - (| r |^{2} + | t |^{2}) = 1 - R - T

where r and t are the reflection and transmission coefficients of the CPA, respectively; R and T are the transmittance and reflectivity, respectively. In the symmetric environment, r and t of the CPA are written as [16]:

r = η, t = 1 + η

where η is the self-consistent amplitude depends on the parameter of the geometry and material, incident wavelength and angle. On the basis of Eq. (1) and Eq. (2), the maximum incoherent absorption of the CPA is derived A_max = 0.5 at η = −0.5 and R = T = 0.25, which is called incoherent absorption limit.

When two incident beams I₁ and I₂ oppositely impinge on the CPA, which is defined as coherent illumination, the input beams I₁, I₂ and the output beams O₁, O₂ are related by the scattering matrix S [15]:

[\begin{matrix} O_{1} \\ O_{2} \end{matrix}] = S [\begin{matrix} I_{1} \\ I_{2} \end{matrix}] = [\begin{matrix} r_{1}^{^{'}} & t_{2}^{^{'}} \\ t_{1}^{^{'}} & r_{2}^{^{'}} \end{matrix}] [\begin{matrix} I_{1} \\ I_{2} \end{matrix}]

where

r_{1}^{^{'}}

,

r_{2}^{^{'}}

, and

t_{1}^{^{'}}

,

t_{2}^{^{'}}

are the reflection and transmission coefficients of each side, respectively. In the symmetry environment, the scattering matrix S owns the same transmission and reflection coefficients in both upper and lower direction:

r_{1}^{^{'}} = r_{2}^{^{'}} = r^{'}

,

t_{1}^{^{'}} = t_{2}^{^{'}} = t^{'}

. Utilizing Eq. (3) and I₁ = αe^iβI₂, the coherent absorption of the CPA is derived:

A_{c o} = 1 - \frac{{| O_{1} |}^{2} + {| O_{2} |}^{2}}{{| I_{1} |}^{2} + {| I_{2} |}^{2}} = 1 - \frac{{| r_{1}^{^{'}} I_{1} + t_{2}^{^{'}} I_{2} |}^{2} + {| r_{2}^{^{'}} I_{2} + t_{1}^{^{'}} I_{1} |}^{2}}{{| I_{1} |}^{2} + {| I_{2} |}^{2}} = 1 - \frac{{| r^{'} + t^{'} α e^{i β} |}^{2} + {| t^{'} + r^{'} α e^{i β} |}^{2}}{1 + α^{2}}

When the incoherent absorption limit is satisfied, there is r′ = −0.5 and t′ = 0.5. Hence, Eq. (4) is rewritten as:

A_{c o} = 1 - 1 / 2 \frac{1 + α^{2} - 2 α c o s β}{1 + α^{2}}

From Eq. (5) the maximum coherent absorption A_co₋_max = 1 is obtained with the α = 1 and β = 2Nπ (N is integer). On the basis of Eq. (5), the coherent absorption A_co can be tuned by changing α or β. Especially when α = 1, the coherent absorption of the CPA varies between 0 and 1 with β, therefore the all-optical modulated switch is achieved.

3. Simulation

The characteristics of the tunable polarization-independent metal-graphene CPA are simulated by a 3-dimension finite-difference time-domain(FDTD) solution. A Johnson and Christy model is used to describe the relative permittivity ϵ_g of gold [17]. The conductivity of graphene is computed from Kubo formula:

σ (ω) = \frac{i e^{2} (ω - 2 i Γ)}{π ℏ^{2}} [\frac{1}{{(ω - 2 i Γ)}^{2}} \int_{0}^{\infty} ϵ (\frac{\partial f_{d} (ϵ)}{\partial ϵ} - \frac{\partial f_{d} (- ϵ)}{\partial ϵ}) d ϵ - \int_{0}^{\infty} \frac{f_{d} (- ϵ) - f_{d} (ϵ)}{{(ω - 2 i Γ)}^{2} - 4 {(ϵ / ℏ)}^{2}} d ϵ]

where

f_{d} (ϵ) = {(e^{(ϵ - E_{F}) / (k_{B} T)} + 1)}^{- 1}

is the Fermi-Dirac distribution, ϵ is the permittivity of graphene, k_B is the Boltzman constant, T is the temperature, E_F is the Fermi energy of graphene, ω is the angular frequency and ħΓ = 0.001eV is the scattering rate [18].

The spectrum characteristics of the metal-graphene CPA are simulated by the FDTD solution with Fermi energy E_F = 0.2eV. Under the incoherent illumination with p polarized incident beam, the simulated transmission, reflection and absorption spectra of the CPA are represented as the black, red and blue curve in Fig. 2(a), respectively. The transmission spectrum intersects with the reflection spectrum at λ = 9969nm where the absorption spectrum climbs to the peak. The excited surface plasmon resonance, corresponding to the peak in the absorption spectrum, results in the enhancement of the incoherent absorption, therefore the incoherent absorption limit is satisfied. The simulated transmission, reflection and absorption spectra of the CPA under the coherent illumination with two identical p polarized incident beams are shown by the black, red and blue curve in Fig. 2(b), respectively. The transmission spectrum and the reflection spectrum of the CPA almost overlap totally and both of them decline to near 0 at λ = 9969nm where the coherent perfect absorption is obtained with A_co₋_max = 99.97%. The surface plasmon resonance excited by the incident beams interact with each other which eventually leads to the coherent perfect absorption at λ = 9969nm. The absorption spectrum of coherent illumination with s polarized incident beams shown by the purple dash curve in Fig. 2(b) is also simulated, which is highly fitted with that of with p polarized incident beams. Hence the metal-graphene CPA is a polarization-independent device. The electrical field distribution at the resonance peak with p and s polarized incident beam are shown in Fig. 2(c). The electrical field energy concentrates from the horizontal bar to vertical bar of the hollowed-out cross structure when the polarization of incident beams switches from s to p. Though the electric field distribution with different polarized incident beams are different, the absorption remains the same due to the central symmetry of the hollowed-out cross array structure, resulting the polarization independent of the metal-graphene CPA.

Fig. 2 (a)The simulated transmission, reflection and absorption spectra with the p polarization under the incoherent illumination. (b) The simulated transmission, reflection and absorption spectra with the p polarization under the coherent illumination and the absorption spectrum with the s polarization under the coherent illumination. (c) The simulated electric filed distribution at the absorption resonance of the coherent illumination under the p and s polarization, respectively.

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The modulation of coherent absorption of the metal-graphene CPA with the changed phase difference β between incident beams is simulated with Fermi energy E_F = 0.2eV. The correspond coherent absorption, transmission and reflection of the CPA versus the phase difference between the two incident beams at the absorption resonance λ = 9969nm are shown by the black, red and blue curve in Fig. 3, respectively. The incident beams are p polarized and the relative amplitude of the beams is α = 1. The coherent absorption of the CPA changes as a function of β. At β = 2Nπ, the coherent perfect absorption is achieved with $A_{c o_{m a x}} = 99.97 %$ where the transmission and reflection decline to near 0. At β = (2N + 1)π, the coherent absorption drops to near 0 where the transmission and reflection both rise to around 50%. When the β changes from 2N π to (2N + 1)π, the coherent absorption A_co of the CPA adjusts from the maximum and minimum, and the transmission and reflection move from nearly 0 to 50%, which is fitted with the conclusion derived from the Eq. (5). The interaction of the surface plasmon resonance excited by the incident beams changes with the variation of β, which leads to the modulation of the coherent absorption of the CPA. The modulation depth D versus the wavelength λ is shown in inset. At resonance wavelength λ = 9969nm, the modulation depth is the highest. By controlling the phase difference between the incident beams, the all-optical modulation is achieved, especially, when β = 2Nπ the coherent perfect absorption is accomplished.

Fig. 3 The simulated coherent absorption, transmission and reflection versus the phase difference between two incident beams at the absorption resonance wavelength λ = 9969nm. The inset is the simulated modulation depth D of the metal-graphene CPA.

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The coherent absorption spectra of the metal-graphene CPA with various hollowed-out cross structure size are simulated with the Fermi energy E_F = 0.2eV. When the hollowed-out cross structure length a₁ decrease from 3100nm to 2900nm with fixed a₂ = 360nm, the absorption resonance of the coherent absorption spectrum of the CPA blue shifts from the 9969nm to the 9557nm as shown in Fig. 4(a). When the hollowed-out cross structure width a₂ ranges from 300nm to 420nm with fixed a₁=3100nm, the coherent absorption spectrum has been changed for less than 100nm as shown in the Fig. 4(b). The effective circuits models for CPA has been used [19, 20] to analyse the tunability of the absorption resonance, which the hollowed-out cross in the gold film has been interpreted as the equivalent capacitance C_R and the graphene film has been represented as the equivalent resistance R_G and the equivalent inductance L_G. When the geometric parameter of CPA has been changed, C_R, R_G and L_G have been changed correspondingly, which leads to the variation in the absorption resonance. The coherent absorption is almost unchanged at the absorption resonance with different a₁ and a₂, of which the spectrum position for η = −0.5 is shifted to maintain the coherent perfect absorption.

Fig. 4 The simulated coherent absorption spectra of the metal-graphene CPA with (a) different hollowed-out cross structure length a₁, (b) different hollowed-out cross structure width a₂, (c) different Fermi energies. And (d) is the effective circuit model of the CPA.

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The coherent absorption spectra of the metal-graphene CPA with different Fermi energies E_F are simulated with the fixed hollowed-out cross structure a₁ = 3100nm and a₂ = 360nm. When the Fermi energy of the graphene increases from 0.2eV to 0.8eV, the absorption resonance of the coherent absorption spectrum blue shifts from 9969nm to 9313nm with the maximum coherent absorption of the CPA almost unchanged as shown in Fig. 4(c). By changing the Fermi energy of the graphene through applied voltage, the absorption resonance has been tuned with the varied equivalent inductance L_G and equivalent resistance R_G. The maximum coherent absorption is almost unchanged, where the self-consist amplitude η has been maintained to −0.5 to achieve the coherent perfect absorption [21, 22]. Comparing with the adjustment of the geometric parameter of the CPA, the actively tuning of the absorption resonance wavelength by Fermi energy of graphene layer is much more flexible, high-speed and efficient.

The performance of the metal-graphene CPA dependent on the different incident angles θ are also simulated with E_F = 0.2eV. The simulated coherent absorption spectra with the different incident angles θ are shown in Fig. 5(a). With the incident angle θ moving from −12° to 12°, the simulated absorption spectra are almost unchanged. Meanwhile, the coherent absorption of the CPA versus the phase difference between the incident beams are also simulated with different incident angles θ at λ = 9969, as shown in Fig. 5(b). For different incident angles, the curves all overlap with each other. With fixed material and geometric parameter of the CPA, the η is almost invariable within the incident angle range from −12° to 12° [23, 24].

Fig. 5 (a) The coherent absorption spectra of the metal-graphene CPA with the different incident angles θ. (b) The coherent absorption of the metal-graphene CPA versus the phase difference between the incident beams with different incident angles θ.

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With the central-symmetry metal-graphene nanostructure, the polarization-independent coherent perfect absorption is achieved at the absorption resonance. By changing the phase difference between the incident beams, the coherent absorption is optically modulated. And the absorption resonance wavelength of CPA is electrically tuned with Fermi energy, which is much faster and more convenient than geometric adjustment. Though in our simulation the insensitivity to the incident angle is observed within −12° to 12°, it could be adjusted by the variation of the materials and the geometric parameter of the CPA for different application.

4. Conclusion

In summary, a tunable polarization-independent metal-graphene coherent perfect absorber (CPA) is designed at mid-infrared region. The coherent perfect absorption of the CPA is analyzed with the scattering matrix and simulated with the FDTD solution. The coherent absorption is all-optically tuned by the phase difference between the incident beams, meanwhile the absorption resonance wavelength of the coherent perfect absorption is dynamically modulated with the gate-controlled Fermi energy. Within the range of incident angle −12° to 12°, the metal-graphene CPA is angular insensitive to the incident angle. By introducing the graphene layer, the CPA has the ability to electrically tune the absorption resonance efficiently and conveniently, which is helpful in applications such as the tunable detector, sensor and switches in mid-infrared wavelength.

Funding

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

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Tunable polarization-independent coherent perfect absorber based on a metal-graphene nanostructure

Abstract

1. Introduction

2. Principle

3. Simulation

4. Conclusion

Funding

References and links

Cited By

Figures (5)

Equations (6)

Optics Express