Coherent perfect absorption and transparency in a nanostructured graphene film

Jianfa Zhang; Chucai Guo; Ken Liu; Zhihong Zhu; Weimin Ye; Xiaodong Yuan; Shiqiao Qin

doi:10.1364/OE.22.012524

Optical absorption plays an important role in a variety of applications such as photodetectors and photovoltaics. As a result, there has recently been lots of interest in realizing total light absorption and two different routes have generally been employed. Firstly, one can take advantage of diffusion or multi-scattering of light on very disordered lossy surfaces such as carbon nanotube arrays [1, 2]. The second method exploits the critical coupling effect [3]. When the coupling rate is equal to the dissipative rate of the structure and other energy loss channels such as diffraction and polarization conversion are blocked, critical coupling will lead to perfect absorption [4, 5]. Such kind of perfect absorption has been realized in various micro- and nano-structures such as plasmonic surfaces, metallic particle arrays and metamaterials [6–11].

Most perfect absorbers have only one port and critical coupling happens under the illumination of one beam. A generation to the two port situation leads to the so-called ’Coherent Perfect Absorption (CPA)’ [12]. Similar to the critical coupling effect, CPA relys on the coherent interference of light where two coherent beams are incident on the absorber from opposite sides and the reflected light of one beam interfere with the transmitted light of the other and vice versa [13, 14]. CPA was first realized using a silicon slab cavity [15], and has recently been reported in composite absorptive films, planar metamaterials, waveguides and so on [16–23]. Coherent absorption provides an additional flexibility to tune the absorption by changing the relative phase of two beams and shows promising potentials for applications ranging from nanoscale light manipulations to data processing [19, 21, 24, 25].

Graphene, a single layer of carbon atoms arranged in plane with a honey comb lattice, shows promising potentials in optics and optoelectronics [26] and has recently attracted great attentions in many applications, such as photodetectors, optical modulators, tunable filters and polarizers [27–30]. Even though the optical absorption of monolayer graphene is quite weak (about 2.3% in the visible range), enhanced absorption by unstructured graphene can be realized by placing it in an optical cavity or exploiting the critical coupling effect [31–34]. Moreover, doped and patterned graphene can support localized plasmonic resonances in the infrared and THz ranges which significantly enhance the absorption [35–37]. Specifically, complete light absorption has been reported in an array of doped graphene disks backed by a metallic mirror or a dielectric with high optical permittivity [38]. Graphene plasmonics provide an effective route to enhance light-graphene interactions.

Here we show CPA in a nanostructured graphene film. The graphene film, patterned with periodical arrays of holes, displays a strong plasmonic resonance in the Mid-infrared wavelength range. By controlling the relative phase of the two coherent beams incident on the graphene film, we are able to either enhance or suppress the resonant absorption in the graphene. Besides CPA, its opposite effect, coherent perfect transparency (CPT), can also be realized due to the extremely thin thickness of the graphene sheet which is less than one ten-thousandth of a wavelength thick.

It has been known that there is a universal limit to absorption by a thin layer with its thickness much smaller than the wavelength of light [38, 39]. We consider a thin film at the interface between two media, medium 1 and medium 2, with different refractive indices, n₁ and n₂. If light impinges on the film from medium 1 at normal incidence, the combined reflection and transmission coefficients are

R = r + (1 + r) η, T = t + t η

where η is the self-consistent amplitude by the thin film, while r and t are the Fresnel coefficients of the bare n₁ | n₂ interface

r = \frac{n_{1} - n_{2}}{n_{1} + n_{2}}, t = \frac{2 n_{1}}{n_{1} + n_{2}}

The absorption is

\begin{array}{l} A & = 1 - {| R |}^{2} - \frac{n_{2}}{n_{1}} {| T |}^{2} \\ = 1 - {| r + (1 + r) η |}^{2} - \frac{n_{2}}{n_{1}} {| t + t η |}^{2}] \end{array}

The condition of maximum absorption is ∂A/∂η = 0 (∂²A/∂²η is real and negative) and we get

η = - \frac{1}{2}

which leads to the absorption limit A_max = 1/(1 + χ), where χ = n₂/n₁. The corresponding reflection and transmission coefficients are

R = - \frac{n_{2}}{n_{1} + n_{2}}, T = \frac{n_{1}}{n_{1} + n_{2}}

For any thin layer with symmetric environments, i.e., n₁ = n₂, the limit to maximum absorption is A_max = 0.5. The maximum absorption can be increased in asymmetric environments and most perfect absorbers employ a back mirror to block the transmission channel.

Now let’s consider two counter-propagating coherent beams, beam 1 and beam 2, impinge on the thin film at normal incidence from opposite sides, medium 1 and medium 2, respectively. We assume the electric field amplitude of beam 1 is 1 and that of beam 2 is α. The reflected part of one beam will interfere with the transmitted part of the other one and vice versa. Therefore, the amplitude of scattered light from each side of the thin film will be

s_{1} = R_{1} + T_{2} α e^{i φ}, s_{2} = T_{1} + R_{2} α e^{i φ}

where φ is the phase difference of beam 1 and beam 2. R₁ and T₁, R₂ and T₂ are the reflection and transmission coefficients of beam 1 and beam 2 respectively, which are given by Eq. (5)

R_{1} = - \frac{n_{2}}{n_{1} + n_{2}}, T_{1} = \frac{n_{1}}{n_{1} + n_{2}}

R_{2} = - \frac{n_{1}}{n_{1} + n_{2}}, T_{2} = \frac{n_{2}}{n_{1} + n_{2}}

So the coherent absorption is

A_{coh} = 1 - \frac{{| s_{1} |}^{2} + χ {| s_{2} |}^{2}}{1 + χ α^{2}}

Using Eqs. (6), (7) and (8), we have

A_{coh} = 1 - \frac{χ}{1 + χ} \frac{1 + α^{2} - 2 α cos φ}{1 + χ α^{2}}

Apparently, A_coh gets its maximum of 1 only if α = 1 and φ = 2Nπ (N is an integer). Therefore if a thin film can reach the incoherent absorption limit, coherent perfect absorption can be achieved under the illumination of two counter-propagating coherent beams with equal field amplitudes and a relative phase of zero. Once the amplitude of two coherent beams are fixed to be the same, the coherent absorption depends on their phase difference and can be tuned continuously between $A_{\min}^{coh} = {[(1 - χ) / (1 + χ)]}^{2}$ (at φ = (2N + 1)π) and $A_{coh}^{\max} = 1$ .

In this part, we study numerically the coherent absorption in a nanostructured graphene. The numerical simulations are conducted using a fully three-dimensional finite element technique (in Comsol MultiPhysics). In the simulation, the graphene is modelled as a conductive surface without thickness [38, 40, 41]. The optical conductivity of graphene can be derived within the random-phase approximation (RPA) in the local limit [42, 43]

\begin{array}{l} σ_{ω} = & \frac{2 e^{2} k_{B} T}{π {\bar{h}}^{2}} \frac{i}{ω + i τ^{- 1}} \ln [2 cosh (\frac{E_{F}}{2 k_{B} T})] + \frac{e^{2}}{4 \bar{h}} [\frac{1}{2} + \frac{1}{π} \arctan (\frac{\bar{h} ω - 2 E_{F}}{2 k_{B} T}) - \\ \frac{i}{2 π} ln \frac{{(\bar{h} ω + 2 E_{F})}^{2}}{{(\bar{h} ω - 2 E_{F})}^{2} + 4 {(k_{B} T)}^{2}}] \end{array}

Here k_B is the Boltzmann constant, T is the temperature, ω is the frequency of light, τ is the carrier relaxation lifetime, and E_F is the Fermi level. The first term in Eq. (11) corresponds to intra-band transitions and the second term is related to inter-band transitions. We restrict our calculations to energies below 0.2 eV in order to avoid the contribution of inter-band contribution and to the Fermi level E_F ≫ 2k_BT. Equation (11) reduces to the Drude model if we neglect both inter-band transitions and the effect of temperature (T = 0) [44, 45]

σ_{ω} = \frac{e^{2} E_{F}}{π {\bar{h}}^{2}} \frac{i}{ω + i τ^{- 1}}

where E_F depends on the concentration of charged doping and

τ = μ E_{F} / e v_{F}^{2}

, where v_F ≈ 1 × 10⁶ m/s is the Fermi velocity and μ is the dc mobility. Here we use a moderate measured mobility μ = 10000 cm² ·V⁻¹ · s⁻¹ [46]. The Fermi level of graphene is assumed to be E_F = 0.6 eV which corresponds to an doping density of about 2.6 × 10¹³ cm⁻² and may be realized by electrostatic or chemical doping [46, 47].

Figure 1(a) shows the schematic of coherent absorption in a freestanding graphene film patterned with a periodical array of square holes. Two coherent beams are incident on the graphene film from opposite sides and the coherent absorption is controlled by their relative phase. The geometric parameters of the patterned graphene are shown in Fig. 1(b). The width of hole is L = 220 nm and the period of unit cell is P = 400 nm in both x- and y- directions.

Fig. 1 (a) Schematic of coherent absorption in a nanostructured graphene film. Two coherent optical beams impinge on the graphene film from opposite sides at normal incidence. Part of the energy may be absorbed while others will be scattered from both sides which can be controlled by changing the relative phase of the two beams. (b) A unit cell of the nanostructured graphene film with geometric parameters. The period is P = 400 nm and the size of hole is 220 nm.

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Figure 2 is the simulated spectra under incoherent as well as coherent illuminations. When a single beam of light is illuminated on the patterned graphene film at normal incidence, part of the energy will be absorbed while others will either be reflected or transmitted, as shown in Fig. 2(a). There is a plasmonic resonance centered at λ = 8.476 μm in the studied spectral range with a maximum absorption of A = 49.97%. When two coherent beams with equal intensities are incident on the graphene from opposite sides, they will interfere with each other, leading to coherent absorption. Now the absorption depends on the relative phase of the two counter-propagating beams.

Fig. 2 Incoherent and coherent absorption in the nanostructured graphene film. (a) Simulated reflection, transmission and absorption of the nanostructured graphene film under the illumination of only one beam at normal incidence (or when the two input beams are incoherent). (b) Normalized total scattering output intensities |S|² under the illumination of two counter-propagating coherent beams with the same intensities. The solid blue line and dashed red line show for parity-even mode and parity-odd mode, respectively. The geometric parameters of the patterned graphene is show in Fig. 1(b). Equation (12) is used to describe the conductivity of graphene and the Fermi level of graphene is assumed to beE_F = 0.6 eV.

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Figure 2(b) shows the normalized total output intensities S_tot = 1 − A_tot (total input energy minus absorption) when their relative phase equals 0 and π, respectively. As the two beams have no phase difference, they are symmetric for the graphene film and form an parity-even mode. In this situation, the interference of the two beams suppresses their scattering and the absorption is enhanced (graphene is at the anti-node of the interference pattern formed by the two coherent beams). Particularly, the total output intensity drops more than 30 dB (A = 99.93%) at the plasmonic resonance of the nanostructured graphene film. On the contrary, they form an parity-odd mode when their phase difference is π and show reduced absorption due to constructive interference of scattered light at the two sides of the graphene film (graphene is at the node of the interference pattern formed by the two coherent beams). As a result, the normalized total output intensity is nearly unitary (A < 0.01%) at the whole studied spectral range. Thus, coherent absorption allows control of resonant absorption in the nanostructured graphene film, either an increase to nearly 100% (corresponding to CPA) or a suppression to almost zero (corresponding to CPT).

Figure 3 shows the normalized coherent absorption in the nanostructured graphene as a function of the relative phase. At the resonance wavelength of 8.476 μm, the coherent absorption varies continuously from 99.93% to less than 0.01% as the relative phase changes from 0 to π (see Fig. 3(a)), giving a modulation contrast of 40 dB. At the off-resonance wavelength, coherent perfect absorption cannot be realized and the maximum coherent absorption is about two times of the incoherent absorption (see Fig. 3(b)). Besides the variation of absorption, we also see transfer of energy between scattering at the two sides of the graphene film (S₁ and S₂) as the relative phase changes because of the interference between the reflected part of each beam with the transmitted part of the other.

Fig. 3 Phase modulation of coherent absorption. Two coherent beams with equal intensities impinge on the nanostructured graphene film from opposite sides. (a) At the resonance wavelength of 8.476 μm, the coherent absorption decreases from 99.93% to less than 0.01% as the relative phase changes from 0 to π. (b) At the off-resonance wavelength of 8.4 μm, the coherent absorption (blue solid curve) varies from about 41% to less than 0.01%. Other energy will be scattered from the two sides of graphene (dashed red curve and dashed green curve).

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Coherent absorption in nanostructured graphene is a quite robust phenomenon and can also be implemented for graphene with asymmetric environments, i.e., on a substrate. We consider the same nanostructured graphene on a semi-infinite substrate with a refractive index of n = 1.5. To make our modeling more practical, instead of Eq. (12), here we use Eq. (11) for the conductivity of graphene in which both the intra-band contribution and temperature effect are considered. As in previous sections, we assume E_F = 0.6 eV and μ = 10000 cm² · V⁻¹ · s⁻¹. The temperature is assumed to be 300 K. Because of the substrate, the resonance wavelength redshifts from 8.476 μm to 10.61 μm. As the dielectric environment now becomes asymmetric for graphene, the incoherent absorption of patterned graphene film is 38.16% for light incident on the graphene from the air side and 57.23 % for light incident on the graphene from the substrate side at the resonance, respectively.

Figure 4 shows the interferometric control of coherent absorption in the nanostructured graphene on a substrate at the resonance wavelength of 10.61 μm. Two coherent beams with equal electric field amplitudes (relative intensity I₁/I₂ = 1/1.5) impinge on the graphene film from the air and the substrate side, respectively. The normalized coherent absorption varies from 95.4% to 3.8% as the relative phase changes from 0 to π. Here the minimum coherent absorption is very close to the ideal lower limit of $A_{coh}^{\min} = {[(1 - χ) / (1 + χ)]}^{2} = 4 %$ while maximum absorption is slightly below coherent perfect absorption. This is because the maximum incoherent absorption of our nanostructured graphene at the resonance is a bit lower than the limit, i.e. 40% and 60% for incidence from the air and substrate side, respectively. Coherent perfect absorption may be realized by optimizing the doping level of graphene and the design of the nanostructures. We can also reduce the minimum absorption with a trade off of decreasing the maximum absorption by changing the relative intensities of input beams, which may increase the modulation contrast. For example, the coherent absorption here can be modulated between about 94.4% and 0.96% for two coherent beams with equal intensities (I₁/I₂ = 1/1).

Fig. 4 Phase modulation of coherent absorption in a nanostructured graphene film with asymmetric environments. The graphene is on a semi-infinite substrate with a refractive index of 1.5. Two coherent beams, with equal field amplitudes, are incident on the graphene at normal direction from opposite sides.

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In summary, we have studied coherent absorption in a single layer of nanostructured graphene. The thickness of monolayer graphene is about 3.4 A^o which is less than one ten-thousandth of a wavelength. The coherent absorption can be tuned continuously from nearly 99.93% (corresponding to CPA) to less than 0.01% (corresponding to CPT) depending on the relative phase of the two counter-propagating coherent beams. This phenomenon relies on interplay of plasmonic resonances and the interference of optical beams on the patterned graphene and provides functionality that can be implemented freely across a broad Mid-infrared to THz range by varying the structural design and graphene doping level. Coherent absorption provides a very versatile platform for manipulating the interaction between graphene and light. It may serve applications in optical modulators, transducers, sensors or coherent detectors.

Acknowledgments

This work was supported by National Natural Science Foundation of China [Grant Nos. 11304389, 61177051 and 61205087] and the State Key Program for Basic Research of China [Grant No. 2012CB933501].

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Coherent perfect absorption and transparency in a nanostructured graphene film

Abstract

Acknowledgments

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