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We numerically demonstrate dynamically tuneable plasmon-induced transparency in a π-shaped metamolecules made of graphene nanostrips by applying external static magnetic field. It is shown that for graphene nanostrips with appropriate Fermi energy, the resonant wavelength, line-shape, and the polarization of transmitted light in the mid-infrared can be effectively controlled by magnetic field. In particular, giant polarization rotation exceeding 20° has been observed in asymmetric graphene metamolecules, which is further enhanced to almost 40° due the Faraday effect in the applied magnetic field, at around 9 μm wavelength, much higher frequency than the Faraday rotation observed in a semi-infinite graphene microribbons. The results offer a flexible approach for the development of compact, tunable graphene-based photonic devices.
© 2015 Optical Society of America
1. Introduction
Electromagnetically induced transparency (EIT) [1] is a quantum interference effect in laser-driven atomic systems, which results in an otherwise opaque medium becoming transparent in a narrow transmission window due to electromagnetic coupling between atoms. Metamaterials can mimic this EIT effect due to coupling between plasmonic resonances of meta-atoms comprising the metamaterial, the so-called plasmon-induced transparency (PIT) [2, 3]. PIT has attracted much attention owing to its potential applications in slow light, all-optical information processing, and enhanced nonlinear effects. PIT effect has been demonstrated with several plasmonic systems, such as asymmetric double bars [4], three-dimensional stacked plasmonic resonators [5], and the planar complimentary metallic aperture structures [6].
Graphene-based nanostrip structures have also been recently used to implement wavelength-tuneable PIT [7, 8], making use of tuneable plasmonic properties of continuous and structured graphene layers [9, 10]. The surface plasmon frequency of graphene depends strongly on the Fermi energy which can be controlled through chemical doping or electrostatic gating [11]. Nanophotonic devices based on continuous graphene sheets or ribbons have been extensively studied [12–14]. However, in many cases, the electrically modulated (via gating voltage) graphene nanophotonic devices with contact electrodes are influenced by the interactions at the interface between electrodes and graphene [15]. Furthermore, with the increase of the Fermi energy, the nonlinearity of the electronic band structure of graphene needs to be also considered [16].
Alternative approach for controlling plasmons in graphene can be based on external static magnetic field. Electrons in graphene are influenced by the external magnetic field due to their small cyclotron mass and relatively low concentrations [17]. The splitting of the plasmon modes of the patterned graphene disks into edge and bulk plasmons has been demonstrated in the applied magnetic field [18]. Large Faraday rotation has also been predicted in arrays of graphene microribbons through the excitation of the magnetoplasmons of individual ribbons [19]. Compared to the electric modulation, tuning the optical response of graphene microstructure by a static magnetic field does not require direct electric contact.
In this work, we propose a mechanism to control the resonant wavelength, line-shape and polarization of the transmitted light in the PIT geometry based on graphene nanostrips by applying external static magnetic field. Our numerical results show that large Faraday rotation can be achieved at much higher resonant frequencies, up to 32.5 THz, than the reported in the graphene ribbon structures [19]. We also show large polarization rotation in transmission through asymmetric graphene metamalolecules even in the absence of magnetic field. These results provide a basis for an efficient and flexible approach to control light via the electromagnetic resonances in plasmonic structures based on graphene nanostrips.
2. Simulation model
In order to realize PIT, metamolecules need to support bright (radiative) and dark (nonradiative) modes. We have studied a metamolecule based on graphene nanostrips similar to one investigated in [7] for electric modulation of PIT [Fig. 1]: a wider and longer single graphene strip plays a role of a radiative element, while the dark element consists of two identical parallel graphene strips. The graphene metamolecules are assumed to be deposited on a glass substrate with refractive index n = 1.45. For simulations, the metamolecules are arranged in a two-dimensional array with the same periodicity along both directions; the periodicity, which is much smaller than the operational wavelength px,y/λ0 ≪ 0.05, may modify the coupling strength between the metamolecules but has a little effect on PIT which is predominantly related to the properties of individual metamolecules.
Fig. 1 Schematic of the metamolecule unit cell composed of graphene nanostrips on a glass substrate: the strip lengths are L1=64 nm and L2=48 nm, the widths are w1=20 nm and w2=12 nm, the separation between the nanostrips in x and y directions are s = 15 nm and d = 8 nm, respectively. The periodicity in x and y directions are px=py=120 nm. The electric field of incident light is along y direction. The static magnetic field B is perpendicular to the graphene plane (xy plane).
If we assume the static magnetic field perpendicular to the graphene plane [Fig. 1], the permittivity tensor of graphene can be written as [20]:
Here, t=0.5 nm denotes the effective thickness of graphene layer in z direction, ε0 is the permittivity of vacuum, and σd does not depend on external magnetic field being σd = σxx(B = 0). The conductivity tensor of graphene can be represented semiclassically as [21]:
where Ef is the Fermi energy, τ is the relaxation time, ω and denote the angular frequency of incident light and the cyclotron frequency of electrons, Vf =106 m/s is the Fermi velocity of the Dirac fermions in graphene, and is the intrinsic relaxation time with μ = 105 cm2/V.s being the carrier mobility. Thus, in the absence of magnetic field, the graphene’s permittivity is isotropic.The numerical simulations were performed using the finite element method (FEM) based on COMSOL Multiphysics software [22]. The three-dimensional simulations were performed for a single unit cell and periodic boundary conditions with y-polarized plane wave normally incident in z direction. The perfectly matched layer (PML) absorbing boundary conditions are applied at either end of the computing space in z direction.
3. Results and discussions
To understand how the static magnetic field influences the electromagnetic interaction in a metamolecule, we first investigate the dependence of the permittivity of graphene on the external magnetic field [Fig. 2]. For Fermi energy Ef = 0.1 eV, the real parts of εxx and εxy are strongly modulated by the external magnetic field. For high applied magnetic field, the real part of εxx has a transition from negative (metallic behaviour) to positive value (dielectric behaviour) at the wavelength around 27 and 22 μm for B= 8 and 10 T, respectively, with the peak values of εxy appearing around these wavelengths [Fig. 2(b)]. With the increase of the Fermi energy, the dependence of permittivity on applied magnetic field becomes weaker [Figs. 2(c) and 2(d)]. For Ef = 0.3 eV, the real part of εxx is always negative in the considered spectral range, whereas its counterparts of Re(εxy) is positive. With further increase of the Fermi energy, the permittivity exhibits similar dependence on the magnetic field. This strong dependence of the graphene’s permittivity on the external static magnetic field is the basis of our proposed magnetic modulation scheme for dynamically tuneable PIT.
Fig. 2 Spectra of the real part of permittivity components ((a),(c)) εxx and ((b),(d)) εxy of graphene in external magnetic field B= 0, 4, 8, and 10 T for ((a),(b)) Ef = 0.1 eV and ((c),(d)) 0.3 eV.
3.1. Dynamically tuneable wavelength and line-shape of PIT
The transmission spectra of the graphene metamolecules shown in Fig. 3 are determined by the Fano interference between bright and dark modes of the metamolecule. They have characteristic Fano-type line-shapes. Strong red-shift of the transmission resonances is observed with the increased magnetic field, accompanied by the redistribution of the amplitudes. The Fano resonance line-shape is known to strongly depend on the asymmetry parameter of the resonant structure [23]. Since graphene in the external magnetic field has intrinsic anisotropy [Eqs. (1) and (2)], the variation of graphene’s permittivity tensor induced by external magnetic field changes the optical anisotropy, analogous to the change of the shape anisotropy in conventional PIT nanostructures. This leads to the redistribution of the electromagnetic fields within the metamolecule and the modifications of the line-shape. Thus, the resonant wavelength shift and the line-shape modification can be controlled via tuning the graphene’s permittivity with the static magnetic field [Figs. 2 and 3]. With the increased magnetic field, the real part of some of the diagonal components changes sign around the resonant wavelength, while off-diagonal components εxy are always positive. These anisotropic properties mean that the plasmonic resonances of a graphene metamolecule are not removed in high magnetic field while being significantly affected.
Fig. 3 Transmission spectra of the array of graphene metamolecules for different external magnetic fields. The Fermi energy is Ef = 0.1 eV. The dips labeled as A0,4,8 and C0,4,8 and the peaks B0,4,8 are the wavelengths related to the Fano line-shape in the external magnetic field B=0, 4, 8 T, respectively. All other parameters are as in Fig. 1.
To further understand the magnetic modulation of the observed PIT, we studied the electric field distributions within the graphene metamaolecules [Fig. 4]. In the absence of magnetic field, the optical near-field distributions clearly demonstrate the coupling between the dipole excited on the left graphene strip and the quadrupole formed on the two right graphene strips [Figs. 4(a)–4(c)]. These near-field features mean that opposite charges oscillate at the edge of graphene strips, indicating it is the second edge mode of the graphene strip [24]. With the increase of static magnetic field, a cyclotron motion of the electrons induces the charge redistribution, and the electric field becomes concentrated at the upper right (left) corner and the lower left (right) corner at the left (right) graphene strips, respectively [Figs. 4(d)–(f) and 4(g)–4(i)]. These field distributions lead to a longer effective length of the strips and a weaker restoring force in each induced dipole in the nanostrip, compared to the metamolecule in the absence of the magnetic field B=0. Therefore, this cyclotron motion and the abrupt variation of graphene permittivity induced by external magnetic field causes a red shift of the resonant wavelengths [Fig. 3].
Fig. 4 Electric field Ez distributions at 4 nm distance from graphene layer corresponding to the transmission spectral features labelled in Fig. 3 for the external magnetic field B =0, 4 and 8 T: (a)–(c) A0, B0, C0; (d)–(f) A4, B4, C4, (g)–(i) A8, B8, C8.
It is interesting to compare the discussed behaviour to the metamolecules of graphene with Ef =0.3 eV for which no metal-insulator transitions were observed in the spectral range of PIT [Fig. 5(a)]. In this case, the transmission window for PIT undergos a small 60 nm red-shift under the static magnetic field of B = 10 T. Although this wavelength modulation is much smaller than that for Ef =0.1 eV, the intensity changes reach ∼60% around the resonant wavelength of ∼13 μm, compared to less than 25% for Ef =0.1 eV in the respective resonance. Our calculations (not shown) confirm that the line-shape modifications also take place for highly-doped graphene strips in sufficiently high external magnetic fields. The dependence of the PIT transmission wavelength on the static magnetic field become weaker with the increase of the Fermi energy [Figs. 5(b)–5(c)]; as expected, for larger concentrations of free carries, the higher magnetic fields are needed to achieve the same modulation.
Fig. 5 (a) Transmission spectra of graphene metamolecules for Ef = 0.3 eV. (b)–(c) The dependence of the PIT wavelength on the applied magnetic field for different Fermi energies. All other parameters are as in Fig. 1.
3.2. Polarization modulation with external magnetic field
The discussed effects provide an efficient way to control light polarization. The polarization of a plane wave is rotated when light passes through a transparent magnetic media in the presence of static magnetic field B, the so-called Faraday rotation. Recently, large Faraday rotation has been explored based on a single graphene sheet or graphene microribbons [19]. The enhancement of the Faraday rotation and possibility to select its resonant wavelength are important issues for applications in optical diodes, sensing and magnetic microscopy [19].
We calculate the Faraday rotation angle in our PIT configuration under the linear y-polarized illumination at normal incidence as [25] θF = 1/2arg[(−ty + itx)/(ty + itx)], where tx and ty are the amplitudes of the transmitted light along x- and y-directions, respectively. The highest Faraday rotation is observed at the resonant PIT frequency [Fig. 6(a)]. Interestingly, the Faraday rotation angle changes from θF ≈ −2.2° to 1.8° around the resonant wavelength of about 16 μm. Similarly, a change of the Faraday rotation sign also occurs in the second resonance at the wavelength of 16.8 μm. Both can be understood considering the modifications of the effective conductivity tensor components [Eq. (4) in [19]].
Fig. 6 Transmission and Faraday rotation spectra for different magnetic fields, Fermi energies, and metamolecule parameters: (a) transmission and Faraday rotation spectra of symmetric metamolecules as in (d) for B=10 T and Ef = 0.2 eV; (b)–(c) Polarization rotation spectra for asymmetric metamolecules as in (e) and (f) with (b) Ef = 0.2 eV and (c) Ef = 0.6 eV, respectively. The separations between the nanostrips in x and y directions are s = 10 nm. (d)–(f) Electric field Ez distributions at the resonant wavelengths λ1, λ2, and λ3, respectively, marked in (a)–(c). The sign “+” and “−” denotes the charge distributions.
To further enhance the Fano resonance and related effects in a graphene metamolecule, an asymmetric metamolecule design can be considered. We achieved this by moving the two right graphene strips up by 16 nm [Fig. 6(e)]. The PIT resonant wavelength is practically not affected by this symmetry breaking. In this case, however, the asymmetry of the metamolecule itself may induce polarization rotation without external magnetic field. The effect of the plasmonic metamolecule asymmetry on the polarization of transmitted and reflected light has been observed with loss-induced symmetry breaking [26]. In Fig. 6(b), the asymmetry-induced polarization rotation at the resonant PIT wavelength 16μm is about 5° in the absence of any magnetic field. With the magnetic field increase, there is an additional polarization rotation associated with a Faraday effect (few degrees in this case, accompanied by the resonance shift to longer wavelength [Fig. 6(b)]. The symmetry breaking results in the increase of the net dipole moment, leading to a larger Faraday rotation. This can be understood by comparing the field and charge distributions in symmetric [Fig. 6(d)] and asymmetric [Fig. 6(e)] metamolecules; the former can be considered as two L-shaped nanoparticles with charges oscillating out-phase, while the later consists of a second order resonance in a similar L-shaped nanoparticle and electric dipole resonance on the lower graphene nanostrips [see the charge signs labeled on the field distributions of Figs. 6(d) and 6(e)]. Thus, with asymmetric metamolecules, there is an opportunity to design large polarization rotation by the symmetry breaking, and use external magnetic field to select a desired wavelength range where it occurs. The angle of the effective Faraday rotation (the polarization rotation at a given wavelength in the presence of magnetic field) is similar to the asymmetry-induced rotation and can be much higher than the intrinsic Faraday effect in symmetric metamolecules. For example, the effective Faraday rotation at around 16.1 μm wavelength is about 6° in the magnetic field of B=5 T [Fig. 6(b)].
With the increase of the Fermi energy, the Faraday rotation is dramatically increased [Fig. 6(c)] at the resonant wavelength (at such high Fermi energies, the shift of the resonance in the magnetic field is negligible [Fig. 5(c)]. The polarization rotation of about 38° is observed around the wavelength of 9.2 μm (corresponding to ∼32.5 THz) in a magnetic field of B=5 T. It has contributions from both the asymmetry of the metamolecule and the Faraday effect. The giant polarization rotation of about 25° takes place at the 9.2 μm resonant wavelength even for B=0 for an asymmetric metamolecule with this Fermi energy. This is further enhanced by more than 10° in the applied magnetic field. This giant Faraday rotation appears in a mid-infrared spectral range at a much shorter resonant wavelength than that in semi-infinite graphene microribbons observed in [19]. The corresponding electric field Ez distribution also confirms the increase of the net dipole moment of the metamolecule [cf. Figs. 6(e) and 6(f)]. The absolute values of both the real and imaginary parts of the effective permittivity of graphene increase with the increase of the Fermi energy: the increase in the real part leads to stronger metallic properties (more electrons) and the imaginary part results in greater absorption within a graphene layer. These changes of the effective permittivity tensor contribute to the enhancement of light-graphene interaction, including the observed Faraday rotation. Thus, using external magnetic field, both the PIT resonant wavelength and polarization of transmitted light can be efficiently controlled with graphene nanostructures.
4. Conclusions
We have demonstrated dynamic modulation of plasmon-induced transparency in graphene-based metamolecules in mid-infrared wavelength range using external static magnetic field. The resonant wavelengths, Fano line-shape and polarization of transmitted light can be controlled by choosing static magnetic field appropriate for a given Fermi energy of graphene. The modification of Fano line-shape and its red-shift are explained by the changes of the components of the effective permittivity tensor of graphene in the external magnetic field, leading to the changes of the charge and filed distributions symmetry of a metamolecule. Further breaking the structural symmetry of graphene metamolecules allows one to additionally enhance polarization rotation in the metamolecules and a possibility to design giant effective Faraday rotation. The described effects pave the way for development of tuneable optical elements for the next generation of plasmonic and nanophotonic on-chip devices such as modulators and active optical filters in the mid-infrared spectral range.
Acknowledgments
This work was supported, in part, by EPSRC (U.K.). J. Q. Liu acknowledge financial supports from the National Natural Science Foundation of China (Grant. 11264021), China Scholarship Council (CSC), and the Key Program for Scientific Research of Jiujiang University ( 2014KJZD005). A. V. Zayats acknowledges support from the Royal Society and the Wolfson Foundation.
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