Polarization dependent plasmonic modes in elliptical graphene disk arrays

Plasmonic modes at mid-infrared wavelengths in elliptical graphene disk arrays were studied. Theoretically, analytical expressions for the modes and their dependence on the size, Fermi energy and the permittivity of substrate materials of the ellipses were derived. Experimentally, the elliptical graphene disks were fabricated and their plasmonic modes were characterized with the polarization-resolved extinction spectra. Both experimental and analytical results show that two electrical dipole modes, whose dipole moments are orthogonal to each other and along the major and minor axis of the ellipse respectively, exist in the elliptical disks. By adjusting the polarization directions of the incident light, the two orthogonal plasmonic modes could be excited either together or separately, showing that the optical properties of elliptical graphene disks are highly polarization dependent. By using ultraviolet illumination to change the Fermi energy of the elliptical graphene disks, the two modes can be tuned dynamically. Moreover, the highly polarization dependent modes are able to couple with the surface phonons of the substrate, leading to polarized plasmon-phonon polaritons. Thus the elliptical graphene disks can provide more degrees of freedom to design the mid-infrared polarization-resolved photonic devices. © 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

ellipse array on a substrate are presented. We found that graphene elliptical nanostructures support two plasmonic electrical dipole modes which are orthogonal to each other with dipole moments along major and minor axis of the ellipse respectively, showing polarization-dependent properties. Furthermore, it was experimentally demonstrated that the optical properties of graphene elliptical plasmonic structures can be controlled through polarization of excitation light, structure size, Fermi energy of graphene and permittivity of the substrate material etc. Compared to graphene ribbons and disks, the results obtained provide more degrees of freedom for manipulation of light-matter interactions in the mid-infrared frequency range.

Theoretical model
The optical conductivity and its equivalent relative dielectric constant ε g of graphene are related with the environmental temperature, Fermi level, relaxation time and thickness of the graphene [39,40]. In this work, temperature and thickness were considered as constant 300 K and 1 nm [41,42] respectively. Compared with 0.33 nm or 0.24 nm thickness of graphene used by other researches [43,44], our results are almost similar with the case of 1 nm. The Fermi level E F , depending on the doping level of graphene, varied from 0.3 to 0.22 eV. The relaxation time τ, depending on the quality of graphene, was taken as 14 fs [6].
To theoretically characterize the polarization dependent plasmonic modes in elliptical graphene disk array, each graphene ellipse was taken as a dipole with polarizability α. The peak frequency of the imaginary part of α corresponds to the resonant frequency of the system. And the value of that represents the extinction, which can be directly measured in experiment. In order to analytically obtain the polarizability of graphene ellipse array, each graphene ellipse is assumed as a flat ellipsoid with the length of semi-major axis a and semi-minor axis b far larger than its thickness. It is assumed the volume of such an ellipsoid V is equal to that of the ellipse, i.e. V = 4/3πabc = πabd g , where c is the length of third semi-axis of the ellipsoid, and d g is the thickness of the graphene ellipse. Therefore, c = 3/4d g . Considering the non-local effect of graphene plasmons, c should be modified with an extra term ∆c, c = 3/4d g + ∆c, (∆c << d g ).
To calculate the polarizability of graphene ellipsoid array, the local field in the center of each ellipsoid should be obtained. Generally, the local field is the sum of the external incident field E inc , the field produced by the surrounding ellipsoids E sur , and the field produced by all the image ellipsoids E img when considering a substrate, E loc = E inc + E sur + E img . In mid-infrared frequency range, the local field on each graphene ellipsoid is assumed as a homogeneous quasi-static field.
For a single graphene ellipsoid, the local field is the external incident field, E loc = E inc . If the homogeneous quasi-static local field is along axis a(x axis) E x loc or axis b(y axis) E y loc , the induced electric field outside the ellipsoid and parallel to the local field, E x ind or E y ind , can be written as [45] Where α 1 and α 2 are the polarizability of the ellipsoid along axis a and axis b, respectively; V is the ellipsoid volume; and are the modified geometrical factors along axis a and b; while ξ is one of the ellipsoidal coordinates which is determined by x 2 (a 2 +ξ) + y 2 (b 2 +ξ) + z 2 (c 2 +ξ) = 1. [45] For simplicity, let For an ellipsoid placed on an isotropic substrate with dielectric constant ε s , as shown in Fig. 1, the influence of the substrate can be replaced by the image ellipsoid. And the local field E loc of the ellipsoid at (0,0,c) is the sum of the incident field and the field produced by its image at (0,0,-c). [46] (0,0,c) where i = 1, 2 is an index representing x and y components of the electric field respectively. According to (1), , where E i loc is the local field at the image ellipsoid (0,0,-c). The continuity boundary conditions of the electric field and electric displacement give E i loc = ε 0 −ε s ε 0 +ε s E i loc , thus [46].
where ε 0 is the vacuum dielectric constant. From (5) and (6), The dipole moment p i of the ellipsoid is where the modified polarizability α i is For an ellipsoid array on a substrate, the local field of each ellipsoid is the sum of the field E inc , E sur and E img . [45] According to (1), E i sur can be written as According to (6), E i img can be written as where j is an index representing different ellipses. From (10), (11) and (12), E i loc can be obtained, therefore the modified polarizability α i can be written as where and are the geometrical factors along axis a and b respectively. and are the correction terms of L 1 and L 2 respectively. Then the extinction cross section C ext can be obtained, C ext = k × Im[ α i ] [46], and the extinction rate E ext is also obtained as E ext = C ext /S period , where k is the wave vector of the incident light, and S period is the area of a unit cell of the ellipse array. Above results show that for each ellipsoid in the array there exist two independent polarizability α 1 and α 2 which are related to the permittivity of graphene ε g and substrate material ε s , therefore related to frequency, Fermi level, relaxation time τ and so on. Mathematically, by ploting Im[ α 1 ] and Im[ α 2 ] versus frequency, it can be shown that both of them exist one resonant frequency. In brief, two electrical dipole modes exist in the ellipsoid, whose dipole moments are orthogonal to each other and along the major and minor axis respectively. Many parameters including the length of major and minor axis, Fermi level of graphene and the permittivity of substrate will affect the modified polarizability, therefore the properties of the two modes.

Experimental results
To show the two orthogonal graphene plasmonic modes experimentally, four graphene ellipse arrays on BaF 2 with different sizes(a = 77.5 nm, b = 30 nm; a = 72.5 nm, b = 25 nm; a = 67.5 nm, b = 20 nm and a = 62.5 nm, b = 15 nm respectively.) were fabricated. For each array the period along major axis and minor axis is 200 nm and 120 nm, respectively. The fabrication process of graphene ellipse arrays is given in the method section. The schematic diagram and the SEM image of these samples are shown in Fig. 2(a). The extinction spectra 1 − T/T 0 of the four structures were measured and shown in Figs. 2(b) and 2(c), where T and T 0 are the transmittance with and without the graphene array respectively, while the polarization  is given in the method section. It can be seen that the theoretical extinction spectra are in good agreement with the experimental spectra both for major axis-polarization mode and minor axis-polarization mode, revealing that the plasmonic modes can be manipulated by adjusting the geometric structures of graphene to meet the specific requirements. As shown in Figs. 2(b) and 2(c), the resonant frequency of minor axis polarization mode is higher than that of major axis polarization mode. For the eigen mode that along the minor axis of the ellipse, the plasmon resonant frequency obviously shows a blueshift, while for the eigen mode that along the major axis, the resonant frequency is almost unchanged. It reveals that the resonant frequency of the mode along minor axis is more sensitive to its length. The polarization-dependent extinction spectra were measured, shown in the upper panels of Figs. 3(b) and 3(c). The 0 degree is defined as polarization direction of the incident light along the minor axis of the ellipses and the 90 degree as the polarization direction along the major axis. It is obvious that by varying the polarization direction of normal incident light, the relative strength between the two orthogonal modes can be tuned. When the polarization direction varies from the minor axis (0 degree) to the major axis (90 degree), the high-frequency mode becomes weaker while the low-frequency mode keeps growing. The intensity of the two orthogonal resonant modes for all 360 degree polarization direction is extracted and plotted in the lower panels of Figs. 3(b) and 3(c), showing significant polarization dependence. The maximum intensity along the major axis is larger than that along the minor axis, showing the stronger light-matter interaction along major axis direction. From the polarization-dependent spectra shown in Fig. 3, it can be seen that the larger the size difference between major and minor axis is, the farther apart the two peak positions are. This approach offers good flexibility for light manipulation.
It is well known that the Fermi energy and then the optical properties of graphene can be tuned by means of applied voltage [7,31,32] and chemical doping via UV light [47]. Here, ultraviolet (UV) light was used. The spectra were measured with unpolarized normal incident light. As shown in Fig. 4(a), illumination of UV (355 nm) on the elliptical graphene disk array for 31.8 seconds, resulted in obvious redshift of both the two resonant peaks. This is because UV irradiation can decrease the adsorption of oxygen molecules on graphene, leading to the reduction of carrier (hole) concentration [47]. Therefore, the irradiation of ultraviolet light can continuously change the Fermi energy of graphene, leading to a redshift of the two plasmonic resonant peaks. The peak position shifts by 173 cm −1 for mode along the minor axis and by 85 cm −1 for mode along major axis. The resonant frequency of the two plasmonic modes versus illumination time of ultraviolet light is plotted in Fig. 4(b) with red dots representing high-frequency modes along minor axis and blue dots representing the low-frequency ones along major axis. By fitting the Exp. Theo. Exp. Theo.
coupled uncoupled coupled uncoupled Fig. 5. Spectra measured on Si 3 N 4 /Si substrate and comparing the theoretical spectral with and without plsmon-phonon coupling. Upper panels: Red and blue lines are experimental extinction spectra with polarization direction of the incident light along minor and major axis respectively at normal incidence. Lower panels: Theoretical comparison of the extinction spectra with and without plsmon-phonon coupling.
two peak positions, the variation of Fermi energy with irradiation time was extracted, and shown in Fig. 4(b). The Fermi level adjusted by illumination of ultraviolet light, from the pristine 0.3 eV to 0.22 eV. With the change of Fermi energy, the resonant frequencies of two plasmonic modes will shift over a wide range. In addition, the same change of Fermi energy results in a wider tuning range for mode along major axis than that along minor axis. Therefore adjustment of E F can dynamically tune the two plasmonic modes. The permittivity of substrate materials will influence the plasmon resonant mode, especially the substrate of polar material with lattice vibration (phonon). Plasmons and phonons represent fundamental collective excitations in solids, and the interaction between them has fundamental implications.
In present experiment, in addition to BaF 2 , polar material silicon nitride deposited on silicon (Si 3 N 4 /Si) was also used as substrate. Upper panels of Figs. 5(a) and 5(b) are spectra measured with polarization of incident light along minor and major axis respectively. Two distinct peaks can be clearly seen for the case along minor axis, while only one peak is observed for the incident lights polarized along the major axis. The increase of extinction with the decrease of wavenumber in low-wavenumber range indicates another peak could be beyond the measurement range. As shown in the lower panel of Figs. 5(a) and 5(b), the theoretical spectra are generally consistent with the experimental results, demonstrating that two modes are generated by the coupling of the plasmonic modes of graphene ellipse array with phonons of the substrate. As a comparison, assuming the imaginary part of the permittivity of the substrate is zero, which means no phonon exists, only one peak at 882 cm −1 and 1163 cm −1 for the two incident polarizations (grey dashed line) exists, obviously different from that with phonon coupling, as shown in the lower panel of Figs. 5(a) and 5(b). The theoretical and experimental results further demonstrate substrate materials will strongly modulate the plasmonic modes of graphene ellipse array.

Methods
Sample fabrications: A large-area CVD grown graphene sheet was first transferred onto a BaF 2 substrate. Graphene structures were fabricated with a 100 × 100 µm 2 area at the center of the graphene layer obtained by standard e-beam lithography followed by oxygen plasma etching. Fabricated samples were then exposed to Br 2 vapor for 30 seconds to increase the doping level.
Spectral measurements: A Bruker Vertex 70 FTIR micro-spectrometer was used for midinfrared spectral measurements. All measurements were performed at room temperature in air. For UV illumination, a 355 nm solid-state laser was used at power density of 200 mW/cm 2 .

Conclusions
In summary, the plasmonic modes of graphene ellipse array have been theoretically and experimentally investigated. Firstly, the feasibility of the theory is verified by comparing the theoretical spectra with the experimental ones. Secondly, by changing the polarized direction of normal incident light, the relative strength between the two orthogonal modes can be tuned. Thirdly, the two plasmonic modes show red shift with the increase of the axis length or decrease of the Fermi level of graphene, respectively. Finally, with a polar substrate, the coupling of graphene plasmons and substrate phonons generate two resonant modes in each axis direction of the ellipse. Our findings propose that the unique properties of anisotropic graphene structures may have many potential applications as polarization-dependent detectors, filters and sensors at infrared wavelengths.