GRAPHENE, a honeycomb crystal of carbon atoms, has

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1 320 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 35, NO. 2, JANUARY 15, 2017 Numerical Study on Plasmonic Absorption Enhancement by a Rippled Graphene Sheet He Huang, Shaolin Ke, Bing Wang, Hua Long, Kai Wang, and Peixiang Lu, Fellow, OSA Abstract We investigate the absorption property of a single layer of periodically rippled graphene adhering to dielectric substrate. By varying the geometric parameters of the ripple and chemical potential of graphene, the absorption can remarkably exceed that of planar graphene and reaches over 50%. It is found that the absorption peak undergoes a redshift as the ripple height increases. The variation of the incident angle will basically not influence the absorption wavelength but may yield additional absorption peaks due to higher order oscillation. The study reveals the optical response of graphene in the intrinsically nonplanar profile and will benefit to the design of graphene-based absorbers and photodetectors. Index Terms Absorption, graphene, surface plasmon polaritons. I. INTRODUCTION GRAPHENE, a honeycomb crystal of carbon atoms, has attracted much attention in recent years due to its unique electric and optical properties [1] [3]. The electrons in graphene behave as massless particles due to the linear dispersion relation [4] [6], which leads to ultrahigh charge-carrier mobility and may find applications in ultrafast devices [7] [9]. As a two-dimensional flexible material, graphene is convenient to integrate with other optical nanodevices [10] [12]. Moreover, the surface conductivity of graphene can be tuned via external field or gate voltage [13] [19]. All the fascinating features make graphene a promising material which can be used in many photoelectric devices with great performance. So far large area of stable graphene can be obtained by nanofabrication processes Manuscript received September 23, 2016; revised December 1, 2016; accepted December 5, Date of publication December 6, 2016; date of current version January 9, This work was supported in part by the 973 Program (No. 2014CB921301), in part by the National Natural Science Foundation of China (No ), in part by the Specialized Research Fund for the Doctoral Program of Higher Education of China (No ), and in part by the Natural Science Foundation of Hubei Province (No. 2015CFA040). H. Huang is with the School of Physics and Wuhan National Laboratory for Optoelectronics, Huazhong University of Science and Technology, Wuhan , China, and also with the School of Science and the Laboratory of Optical Information Technology, Wuhan Institute of Technology, Wuhan , China ( heh@hust.edu.cn). S. Ke, B. Wang, H. Long, and K. Wang are with the School of Physics and Wuhan National Laboratory for Optoelectronics, Huazhong University of Science and Technology, Wuhan , China ( keshaolin@hust.edu.cn; wangbing@hust.edu.cn; longhua@hust.edu.cn; kale_wong@hust.edu.cn). P. Lu is with the School of Physics and Wuhan National Laboratory for Optoelectronics, Huazhong University of Science and Technology, Wuhan , China, and also with the Laboratory for Optical Information Technology, Wuhan Institute of Technology, Wuhan , China ( lupeixiang@ hust.edu.cn). Color versions of one or more of the figures in this paper are available online at Digital Object Identifier /JLT Fig. 1. Schematic of periodically rippled graphene adhering to the dielectric substrate. The period and height of the graphene ripple are denoted by W and h, respectively. The incident wave is TM polarized with the magnetic field along the y direction. The incident angle is denoted by θ. The period is fixed at W = 0.1 μm. such as chemical vapor deposition (CVD) method. In practice, an ideal planar graphene is hard to achieve and it tends to be crumpled when adhering to the flat surface of substrate [20] [22]. The rippled graphene used to manifest different optical and electronmagnetic properties [23], [24] [26]. It is well known that a single layer of graphene can absorb 2.3% energy of incident light in visible range. In order to achieve higher absorption in selected frequencies, one can use patterned graphene arrays, such as graphene discs [27], [28], ribbons [29], rings [30], and cross-shaped structures [31]. The absorption of graphene is dramatically enhanced in the infrared and terahertz (THz) region thanks to the resonance of surface plasmon polaritons (SPPs) [32] [35]. In this wok, we shall investigate the absorption properties of rippled graphene adhering to the dielectric substrate. Due to the periodic ripples of graphene, the incident plane wave can be coupled to SPPs in graphene subject to the match of their wave vectors, which can be realized by varying the geometric parameters of the rippled structure, chemical potential of graphene, and incident angles. The resonance of SPPs in the periodical ripple leads to stronger interaction between incident light and graphene and hence the absorption enhancement [31]. The study aims to reveal the optical properties of graphene when it is with the nonplanar profile. II. THE ABSORPTION OF THE RIPPLED GRAPHENE SHEET ARRAY The schematic diagram of the rippled graphene is illustrated in Fig. 1. The structure is composed of a square periodic array on the substrate of SiO 2. The period of the array is denoted as IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See standards/publications/rights/index.html for more information.

2 HUANG et al.: NUMERICAL STUDY ON PLASMONIC ABSORPTION ENHANCEMENT BY A RIPPLED GRAPHENE SHEET 321 Fig. 2. The absorption spectra for different heights of the graphene curved surface at normal incidence, for (a) μ c = 0.5 ev and (b) μ c = 0.8 ev. W and the height of ripple is h. The relative permittivity of SiO 2 is given as ε d =3.9. The ripple follows the curved surface z = h 2 [sin2 (xπ/w )+sin 2 (yπ/w)] (1) The system is illuminated by a TM polarized plane wave with electric field E = {E x, 0, E z } and magnetic field H = {0, H y, 0}. The waves impinge on graphene at an incident angle θ and the incident wavelength is λ in air. In the THz and far-infrared range, the intraband transition dominates and the surface conductivity of graphene follows the Drudelike formula σ g = ie 2 μ c [π 2 (ω + iτ 1 )] 1, where e, μ c, τ, ω, represent the electron charge, chemical potential, relaxation time, and photon frequency, respectively [4]. As the graphene ripple is of large curvature, the geometric potential induced by the bending is not distinctive [36]. The momentum relaxation time is chosen as τ = 0.5 ps, corresponding to a mean free path up to 500 nm at room temperature that coincides with the experiment [37] [39]. All the numerical simulation results are obtained using the finite element solver COMSOL Multiphysics. In the simulation, the graphene is defined as surface currents with the boundary conditions. As the graphene is rippled, the surface current along the three directions should be given by J s =[σ g E x,σ g E y,σ g E z ] [36], [40], with E x, E y, E z the three components of electric field. The absorption is obtained by A(λ) =1 T R with T and R being the transmission and reflection. In order to excite SPPs, the wave vector of incident waves should abide by the matching relation [41], [42] k SPP = k + mg x + ng y, (2) where k SPP is the SPP wave vector and k = k 0 sin(θ), representing the in-plane wave vector component of incident wave. G x and G y are the reciprocal vectors in the x and y directions, respectively. We denote the order of SPP modes by (m, n) with m and n being integers. As the ripple height is small, the dispersion relation of SPPs in the planar graphene is approximately given by [43] k SPP = iω(ε air + ε d )ε 0 /σ g (3) As μ c = 0.5 ev and 0.8 ev, we have k SPP = iμm 1 and μm 1 at λ = 6 μm, which is remarkably larger than the wave vectors of SPPs in air. We firstly study the case of normal incidence as θ =0.Fig.2 plots the absorption spectra as the height of the ripple varies. The absorption peaks in the infrared range correspond to the excitation of the SPP modes (m, n) =(±1, 0). The modes for m = ±1 degenerate is because of the spatial symmetry at normal incidence. As the height h is small, the absorption peaks can be predicted by Eq. (2). For example, the absorption peak locates at λ = 6.9 μm ash = 0.02 μm, as shown in Fig. 2(a). In this case the SPP wave vector is Re(k SPP )=56.8 μm 1 according to Eq. (2), which is close to the reciprocal vectors G x = 2π/W = 62.8 μm 1 as m =1 and n =0. The deviation is due to the difference of SPP dispersion relation between rippled graphene and ideal planar one. As the height of ripple increases, the absorption peak experiences a redshift. The shift can be explained by the resonance condition 2 k SPP L +2δ =2pπ, where k SPP =2πn SPP /λ peak with n SPP being the effective index of SPPs and λ peak the resonant wavelength. The phase change at the ends of the arm is denotes by δ and p is an integer. As the incident wavelength is much larger than the arm length L, we can suppose p =1. Therefore, the resonant wavelength is given by λ peak =2πn SPP L/(π δ). The effective arm length L equals to the propagation length of SPP in a unit cell, that is, L = W for a planer graphene. As the ripple height increases, the SPP experiences a longer propagation length, leading to a larger L. Consequently, the absorption peak shifts to longer wavelength as the ripple height increases. It is also apparent the absorption peak broadens in longer wavelength. The reason lies in the fact that the real part of the surface conductivity of graphene increases at longer wavelengths, arousing larger intrinsic loss of graphene. When the chemical potential increases to μ c = 0.8 ev, the absorption peaks move to shorter wavelengths, as shown in Fig. 2(b). The reason is because of the wavelength of SPPs increases as the chemical potential of graphene increases. The absorption peak reaches maximum value when the impedance of the system matches the impedance of air. In Fig. 2(b), the system impedance first starts to match that of free space and then differs from it as the ripple height increases. Therefore, the peak absorption value at 5.6 μm is higher than that of at 6 μm. Figure 3(a) (c) illustrate the electric and magnetic field distributions at the absorption peak as μ c = 0.5 ev and h = 0.02 μm. The results clearly show the excitation of SPPs since the field is well confined on the graphene surface and the mode experience anti-symmetric field distribution for both E z and H y, which is one of the typical features of graphene SPPs. The wavelength of the mode along the curved surface is about 100 nm, which fairly coincides with the computation result by using Eq. (3). To display the mode profiles more clearly in the rippled graphene, the distributions of electric field magnitude ( E ) at the absorption peaks are shown in Fig. 3(d) (i). One can see clearly that the strongest field is located at the corners, that is, the valleys of graphene ripple. As the height increases, the mode profile will not change and the field maxima remain at the valleys except for the magnitude increases. The magnitude of electric field corresponds to the absorption maximum as shown in Fig. 2. Stronger SPP field excitation tends to a more enhanced absorption. Fig. 4 depicts the absorption spectra as a function of the ripple height h for different chemical potentials μ c.atagiven μ c, two absorption peaks can be observed in the considered region. Different peaks represent different excited SPP modes.

3 322 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 35, NO. 2, JANUARY 15, 2017 Fig. 5. The absorption spectra for different incident angles as h = 0.05 μm. The chemical potentials are (a) μ c = 0.5 ev and (b) μ c = 0.8 ev, respectively. Fig. 3. The field distributions in the unit cell at the absorption peaks. (a)-(c) The electric (E x, E z ) and magnetic field (H y ) components at y = 0.05 μmfor the peak labeled as a in Fig. 2. (d)-(i) The normalized electric field ( E ) for the peaks in Fig. 2. Fig. 6. The normalized electric field distributions (E x ) corresponding to the absorption peaks in Fig. 5. All the fields are plotted 5 nm above the graphene. Fig. 4. The absorption spectra as a function of height h, for(a)μ c = 0.2 ev, (b) μ c = 0.5 ev, and (c) μ c = 0.8 ev. The dashed lines indicate the resonant wavelength of SPPs in the planar graphene for modes (±1, ±1) and (±1, 0), which locate at the wavelengths of (a) 9.14 μm, μm, (b) 5.57 μm, 6.52 μm, and (c) 4.36 μm, 5.16 μm, respectively. The incident angle is θ =0. The peak at longer wavelength implies the generation of the fundamental modes (±1, 0), while the peak at shorter wavelength corresponds to the generation of higher-order modes (±1, ±1). One can see, as the ripple heights approximate to zero, the resonant wavelengths approximate to that in flat graphene predicted by Eqs. (2) and (3). Generally, the fundamental modes exhibit larger absorption than the higher-order ones. As the height of ripple increases, both modes experience a redshift. On the other hand, the absorption peaks undergo a blue shift as the chemical potential μ c increases. The shift due to the change of μ c can be understood as follows. The imaginary part of surface conductivity increases as μ c increases, which will yield a smaller SPP wave vector according to Eq. (3). As the reciprocal vectors remain unchanged, the resonant frequency must increase to generate SPP. Therefore, the absorption peaks shift to smaller wavelength as chemical potential of graphene increases. We then investigate the case of oblique incidence. Fig. 5 shows the absorption spectra for different incident angles. The single peak seen at θ =0 splits into two peaks for nonzero incident angle. The mode splitting could be understood from Eq. (2). At θ = 0, the in-plane wave vector k is zero and the excited (±1, 0) SPP modes are degenerate. As θ increases, k will be added to or subtracted from the reciprocal vectors depending on the SPP propagation direction. As a result, the degeneracy of the modes will be broken. Moreover, as incident angle increases, the splitting gets larger due to the increase of in-plane wave vector. For example, as shown in Fig. 5(a), the splitting wavelengths Δλ of the two peaks are 0.04 μm as incident angle is 40 and 0.1 μmasθ = 80, while the theoretical results are 0.06 and 0.1 μm, respectively. One can see that the center wavelength of absorption peaks undergoes a blue shift as the increase of chemical potential. The center wavelength is λ = 7.17 μmforμ c = 0.5 ev (Fig. 5(a)) and λ = 5.63 μmforμ c = 0.8 ev (Fig. 5(b)). On the other hand, the absorption maximum first increases and then decreases as θ increases. The decrease of absorption is due to the increase of reflection at highly oblique incidence. For the purpose of analyzing the different modes at the peaks in Fig. 5, we plot the corresponding electrical field component E x. For normal incidence, the field distributions (E x ) for degenerate modes (±1, 0) are symmetric along the x direction and the largest fields lie at the center of rippled graphene as shown in Fig. 6(a) and 6(d). For oblique incidence, the degenerate modes turn to be separated. The largest fields lie in different position from the center as shown Fig. 6(b) and 6(c) and displaced in Fig. 6(e) and 6(f) as well. The absorption spectra as a function of incident angle θ for different chemical potentials μ c are analyzed in Fig. 7(a) (c).

4 HUANG et al.: NUMERICAL STUDY ON PLASMONIC ABSORPTION ENHANCEMENT BY A RIPPLED GRAPHENE SHEET 323 Fig. 7. The absorption spectra as a function of the incident angle θ. The chemical potentials are (a) μ c = 0.2 ev, (b) μ c = 0.5 ev, and (c) μ c = 0.8 ev, respectively. In all figures, the height is fixed at h = 0.05 μm. conclude, in order to obtain the largest absorption for a fixed structure, there is an optimum incident angle θ, which relates to the chemical potential μ c. The optimized maximum value of A max can reach as high as 53% when μ c = 1 ev and θ = 45, which can be clearly seen in Fig. 8(b). Figure 8(c) and 8(d) depict the wavelength of absorption peaks λ peak corresponding to Fig. 8(a) and 8(b), respectively. The result shows that the resonant wavelength decreases as chemical potential increases and increases as ripple height increases, while λ peak almost remains unchanged as incident angle changes. Experimentally, the large-area graphene film is first grown using an optimized liquid precursor chemical vapor deposition method and determined to be single-layer through Raman measurements [11]. The ripples can be achieved by using mechanical instabilities to construct periodic wrinkling [21]. The Au electrodes can be used as top gate for electrostatic doping [12]. Fig. 8. (a) The absorption maximum A max varies with chemical potential μ c and height h,whereθ =0. (b) The absorption maximum A max varies with μ c and θ,whereh = 0.05 μm. (c) The wavelength of absorption peak λ peak varies with μ c and h, whereθ =0. (d) The wavelength of absorption peak λ peak as a function of μ c and θ, whereh = 0.05 μm. For a fixed μ c, one can see the center wavelength of two absorption peaks almost remain unchanged as θ increases. At the same time, the absorption peak gradually broadens, which is caused by the splitting of the fundamental mode. On the hand, the absorption peaks experience a blue shift as the chemical potential increases. Fig. 8(a) plot the absorption maximum A max as a function of chemical potential μ c and the ripple height h. The results show that for a fixed height of the ripple, the absorption maximum first increases and then decreases as μ c increases. On the other hand, at a given chemical potential, the variation of absorption maximumissimilartothatforfixedh. That means that a maximum can be obtained by adjusting a proper μ c and h. The optimized absorption maximum A max can reach as high as 34%. The absorption can be further increased by using oblique incidence. Fig. 8(b) illustrates the absorption maximum as a function of chemical potential and incident angle, where the height of ripple is fixed at h = 0.05 μm. One can see the maximum absorption A max firstly increases and then decreases with the increase of μ c at a given θ. When the chemical potential of graphene is fixed, A max increases at first and then decreases as the incident angle increases. Combining the result from Fig. 7, we can III. CONCLUSION In conclusion, we have studied the tunable absorption enhancement in rippled graphene sheet. The rippled graphene act as diffraction grating and are utilized to generate SPPs. Therefore, the absorption in the proposed system can be greatly enhanced with the aid of SPPs. As the height of ripple increases, the absorption peak experiences a red shift. At a given height, the largest absorption can be obtained by choosing optimum chemical potential. The absorption can be further improved by applying proper incident angle and the absorption maximum can reach as high as 53%. At the same time, mode splitting is observed for the oblique incidence. The results are proper with longer wavelength according to the scaling law. 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Lett., vol. 101, 2012, Art. no He Huang is currently working toward the Ph.D. degree in the School of Physics, Huazhong University of Science and Technology, Wuhan, China. His research interests include ultrafast optics, nanophotonics, and fiber laser. ShaoLin Ke is currently working toward the Ph.D. degree in the School of Physics, Huazhong University of Science and Technology, Wuhan, China. His research interests include ultrafast optics, nanophotonics, and fiber laser. Bing Wang received the B.S. and the Ph.D. degrees in physics from Wuhan University, Wuhan, China, in 2002 and 2007, respectively. He is currently a Professor in the School of Physics and Wuhan National Laboratory for Optoelectronics, Huazhong University of Science and Technology, Wuhan, China. His research interests include metal/graphene plasmonics, nanophotonics, nonlinear, and ultrafast optics. Hua Long received the Ph.D. degree in physics from Huazhong University of Science and Technology, Wuhan, China, in She is currently an Associate Professor in the School of Physics, Huazhong University of Science and Technology. Her research interests include theoretical and experimental research for interaction of femtosecond laser and materials. Kai Wang received the Ph.D. degree in physics from Huazhong University of Science and Technology, Wuhan, China, in He is currently an Associate Professor in the School of Physics, Huazhong University of Science and Technology. His research interests include nanophotonics, nonlinear, and ultrafast optics. Peixiang Lu received the B.S. degree in physics from Peking University, Beijing, China, in 1987, and the Ph.D. degree from the Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, Shanghai, China, in He is currently a Professor in the School of Physics and Wuhan National Laboratory for Optoelectronics, Huazhong University of Science and Technology, Wuhan, China. His current research interests include ultrafast optics, laser physics, and nanophotonics. He is a Fellow of the Optical Society of America.