1. Introduction

Surface plasmons are electromagnetic excitations propagating along the interface between dielectrics and conductors [1]. Graphene is a semiconductor with a two-dimensional form of carbon atoms arranged in a honeycomb lattice [2]. Graphene has an extremely high quantum efficiency for light–matter interactions and contains plasmons (GPs) with several superiority properties than the counterparts on noble metal, such as the high field confinement [3] and tunable carrier concentration through electrical grating [4], making graphene a promising material for plasmonic devices.

However, the high field confinement also brings challenges to the excitation procedure of GPs due to the large momentum mismatch between plasmons and incident light [5]. Accordingly, several methods have been proposed to enhance the efficiency of coupling from photons to plasmons, such as through the strongly concentrated fields at apex of metallic near-field probes [6], compressing surface polaritons of tapered bulk materials [7], periodic gratings with sufficient short period [8] or graphene surface elastic vibrations generated by a flexural wave [9]. Recently, a new method of exciting has been proposed [5], in which the incident light is coupled to GPs through the dipole antenna resonances with the plasmons wavefront could be engineered by the extremity shape of antenna, revealing new possibilities of exciting and controlling GPs on graphene through metal antenna arrays.

The interactions between circular polarized light and chiral plasmonics lenses have been extensively studied and applied in the design of circular polarization analyzers [10–14], which could focus circular polarization incidence with one chirality into a solid dot shape, while defocusing the polarization with opposite chirality into a donut shape [14]. Recently, to overcome the shortage of narrow work bandwidth of traditional circular polarization analyzers, we have proposed a tunable circular polarization analyzer based on graphene-coated spiral dielectric gratings architecture [15]. However, in such structure the GPs are excited via guide mode resonances of graphene-coated dielectric gratings and to achieve the functionality of circular polarization analyzer, the occupation ratio and refractive index of grating need to be selectively chosen to satisfy both the demands of plasmons excitation and geometric phase effect, limiting its flexibility in practical applications. An alternative excitation method is in need to reduce the complexity of implementation.

In this paper we propose a graphene circular polarization analyzer based on spiral metal antenna arrays. In Sec. II, the excitation of graphene plasmons via one single metal antenna with triangle shape is investigated under incidence with two orthogonal linear polarized directions. Then we present the excitation of plasmons via linetype antenna arrays under two orthogonal linear polarized incidence through the amplitude and phase field profiles. Afterwards we investigate in Sec. III the tailing of plasmons wavefront through circle and spiral shape antenna arrays via the intensity and phase distributions of field. Then we design a graphene circular polarization analyzer and illustrate the extinction ratio with respect to the incident wavelength, graphene Fermi levels and detect areas. In the end we draw the conclusion.

2. Polarization dependent excitation via single antenna and linetype antenna arrays

We first investigate the excitation of graphene plasmons through single triangle metal antenna on graphene sheet. Sketched in Fig. 1 is the schematic of implementation, in which one triangle metal antenna is deposited on the graphene layer, which is placed on a dielectric layer on metal substrate. The materials of the metal antenna and dielectric are assumed as Au and CaF₂ with the material refractive data derived from the experimental measurements [16, 17]. The graphene layer is characterized by a surface conductivity (σ_g = σ_intra + σ_inter) material model, in which the terms σ_intra and σ_inter are the intraband and interband contributions [18]. The scattering rate of graphene is assumed as Γ = 0.0082 eV with the temperature T = 300K. All the numerical calculations have been performed by the commercial software from Lumerical based on FDTD methods. Since the metal substrate can be regarded as a perfect electric conductor in the mid-infrared region, we set the bottom boundary condition of calculation region as Perfect Electric Conductor (PEC) with other boundaries are assumed as Perfect Matched Layer (PML).

 

Fig. 1 Schematic view of the triangle metal antenna on graphene, the dielectric layer below graphene is CaF₂ and the whole structure is mounted on a metal substrate. The light incidents from z- axis.

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According to the geometric shape of metal antenna, any linear polarization incidence can be decomposed into two basic illumination situations, namely the polarization direction along the base of triangle (i.e. along x axis) and the height (i.e. along y axis). We subsequently show in Figs. 2(a)-2(b) and 2(c)-2(d) the amplitude and phase distributions field component E_z under x and y polarized incidence. The base length and height of antenna are chosen as l_x = l_y = 200 nm. The graphene Fermi level is assumed as E_f = 0.4 eV and incident wavelength is λ = 10 μm.

 

Fig. 2 Distributions of the amplitude (a,b) and phase (c,d) of field component E_z under x-polarized (a,c) and y-polarized (b,d) incidence. The light incident direction is marked in the figures using arrow. The Fermi level is assumed as E_f = 0.4 eV and wavelength is λ = 10 μm.

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As shown in Figs. 2(a) and 2(c), the graphene plasmons could be excited via the dipole resonance of metal antenna, while such excitation only occurs on the metal boundary which is unparallel to the polarized direction (e.g. for x-polarized incidence, it is at both sides rather than the base where plasmons excitation occurs). Meanwhile, due to the intrinsic mechanism of dipole resonance, the plasmons excited at two terminals of triangle antenna have a π-phase difference and therefore in Fig. 2(b) a destructive interference along the symmetry axis of triangle is presented, which has been also observed in the excitation of plasmons on metal through triangle aperture [19]. However, in Fig. 2(d), although a π-phase difference is indeed observed near the excitation point, the phase difference disappears as the plasmons wave propagating. This can be attributed to the different geometric shapes for the two terminals of dipole antenna.

Although previous investigations have proved the possibility of coupling incident light to graphene plasmons by dipole antenna resonances, the energy of excited plasmons is low due to the limited number of antennas and the wavefront is non-planar. The aforementioned disadvantages could be solved by arranging multiple antennas into linetype arrays. Without loss of generality, we focus on two simple combinations of metal triangle antennas and study the excitation of plasmons under linear polarized incidence. The 1st form of combination is the one in which symmetry axes of triangles are collinear [see the inset in Fig. 3(a)] and the 2nd form is the one in which symmetry axes of triangles are parallel to each other [see the inset in Fig. 3(c)]. For each form of combination, two incident linear polarizations with mutually perpendicular polarized directions are taken into account, one is parallel to the base while the other is parallel to the symmetry axis. The base length and height of antenna are chosen as l_x = l_y = 200 nm. The graphene Fermi level is assumed as E_f = 0.4 eV and incident wavelength is λ = 10 μm. Only six periods are illustrated and each period length is 250 nm. The boundaries parallel to y axis are Periodic Boundary Condition while the ones parallel to x axis are Perfect Matched Layer (PML).

 

Fig. 3 Distributions of the normalized amplitude (a-d) and phase (e-f) of field component E_z of linetype antenna arrays under x-polarized and y-polarized incidence. The polarized directions are marked using arrows and the combinations of arrays are shown in the insets of (a-d). The Fermi level is assumed as E_f = 0.4 eV and wavelength is λ = 10 μm.

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As shown in Figs. 3(b) and 3(d), both the two combination form of arrays could excite graphene plasmons under y-polarized incidence. The π-phase difference of plasmons towards + y and –y directions could be clearly observed in Figs. 3(f) and 3(h) with the nearly planar plasmons wavefronts. It is also worth noting that for the 2nd form in Fig. 3(d), the plasmons propagating along + y direction are stronger than the one along -y direction and the ones toward both directions in Fig. 3(b). This could be understood by considering the plasmons along + y direction of 2nd form has longer effective boundary on which plasmons could be excited. For the x-polarized incidence in Fig. 3(a), the 1st form of combination could still excite plasmons without phase difference and the strength of plasmons is much weaker than that under y-polarized incidence. For the 2nd form of combination in Fig. 3(c), no plasmons could be excited due to the destructive interfere of plasmons excited from adjacent triangle sides.

3. Circular polarization analyzer based on spiral antenna arrays

In the previous section, we present that the linetype antenna arrays could couple the energy of incident light into graphene plasmons. Furthermore, if one arranges such arrays into a spiral shape under illumination of circular polarization, then the geometric shape of the arrays would interact with the chirality carried by the circular polarization incidence, producing different field profiles at the center. The mechanism behind this could be explained via the mathematical form of distribution of normal electric field component near the center, which could be expressed by the l_th order Bessel function with the spiral phase profile of lφ [10],

 

Fig. 4 The normalized electric field |E_z|² profiles and phase distributions for the symmetric circular antenna arrays (a,d) and spiral antenna arrays under right-handed (b,e) and left-handed (c,f) circular polarization incidence. The chirality of incidence are marked with white arrows and the combinations of arrays are shown in the insets of (a-b). The Fermi level is assumed as E_f = 0.4 eV and wavelength is λ = 10 μm.

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If the chirality of incidence (e.g. l_incidence = −1) is cancelled by the geometric phase effect of spiral arrays (e.g. l_spiral = 1), the index l would be zero after superposition. Since the zeroth order Bessel function J₀ has its maximum at the center, therefore the field distribution would be a solid dot type. However, if circular polarization incidence and spiral arrays carry the same l (e.g. l_incidence = l_spiral = 1), the index would be l = 2 at center after superposition. In contrast to the zeroth order Bessel function, the high order Bessel function of the first kind J_l (l = 1,2,3,…) has zero intensity at the center, therefore the field distribution would be donut shape near the center. To reveal such mechanism, we present in Fig. 4 the normalized electric field |E_z|² profiles and phase distributions for the circular and spiral antenna arrays under right-handed and left-handed circular polarization incidence. The 2nd form of combination of metal antennas is adopted due to the stronger excited plasmons field (see Fig. 3). The number of antennas is 19 and the radius of circle is r = 600 nm (also is the start radius of spiral). To produce a geometric phase of l_spiral = 1, the end radius is r = 920 nm according to the plasmons wavelength λ_GP = 320 nm near the incidence wavelength λ = 10 μm. The graphene Fermi level is set as E_f = 0.4 eV.

Figures 4(a) and 4(d) show that for the symmetric circular arrays, the field distribution near the center is totally determined by the chirality of circular polarization incidence. It can be clearly observed that the normal field component intensity performs as a donut shape due to the distribution of 1st order of Bessel function of the first kind J₁. Meanwhile, in the Fig. 4(d), the phase varies from –π to π along azimuthal direction near the center. However, if the geometric phase effect is cancelled by the chirality of circular polarization incidence [see Figs. 4(b) and 4(e)], plasmons from all radial directions are in same phase and a solid spot field profile emerges. Conversely, if the geometric phase effect and chirality of circularly polarized light are superposed [see Figs. 4(c) and 4(f)], a phase variation of 4π would occur along azimuthal direction and accordingly, the field profile with a donut shape would emerge due to the distribution of 2nd order Bessel function of the first kind J₂. According to the results presented in Fig. 4, if one near-field probe (e.g. the Si tip in [5]) is placed above graphene at the center of such analyzer, then the measured normal field component intensity could tell the chirality of incident circular polarization, which is the basic principle of circular polarization analyzer.

It is worth noting that the geometric phase l_spiral = 1 is essential to achieve the functionality of circular polarization analyzer. When the metal triangle antennas are placed into symmetric circular arrays [see Figs. 4(a) and 4(d)], there would be no geometric phase effect. For both the right-hand and left-hand circular polarization, a 2π-phase variation exists at the center and accordingly, only donut shape field distributions can be obtained. Under such circumstance, no difference in the field intensity can be detected and the extinction ratio would be close to one. On the other hand, for antenna arrays with geometric phase l_spiral>1 (l_spiral = 2,3,4…), there would be still no solid dot field distribution at center after superposition. For instance, if the distance between start and end radius of spiral antennas arrays is 4π/k_GP, the geometric phase would be l_spiral = 2 and the index after superposition is l = 1 (or l = 3). Both field profiles at center would be donut shape and no difference in the field intensity can be detected.

Since the traditional circular polarization analyzer is designed to work in single wavelength, when the incident wavelength shifts away from the fixed distance between spiral start and end radius, geometric phase effect is not satisfied and the performance of analyzer would be impaired. Such disadvantage can be solved in the propose analyzer by adjusting the graphene Fermi level dynamically to enable the plasmons wavelength to stay unchanged for varying incident wavelengths. To reveal this, we investigate in Fig. 5 the extinction ratio of analyzer under various Fermi levels and incident wavelengths. Such benchmark is denoted by the ratio of integration of normal field component intensity on a square region around the center for opposite circular polarized incidences. The wavelength spans from 8 μm to 12 μm. The Fermi levels are chosen as E_f = 0.3 eV, 0.35 eV and 0.4 eV and the integration region is 50 × 50 nm². The start and end radius of spiral arrays are respectively set as r = 600 nm and r = 850 nm.

 

Fig. 5 The extinction ratios of analyzer with respect to the incident wavelength under several Fermi levels (a) and integrated regions (b). The integration region is 50 × 50 nm² in (a) and the Fermi level is E_f = 0.4 eV in (b).

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As shown in Fig. 5(a), the extinction ratios under various Fermi levels present several distinctive peaks within the whole wavelength regime due to the operation mechanism of analyzer, in which the best performance of analyzer is obtained near the wavelength under which the plasmons wavelength is equal to the difference of start and end radius of spiral arrays. However, by adjusting graphene Fermi level from 0.4 eV to 0.3 eV, the wavelength of best extinction ratio could be tuned from near 9.2 μm to 10.5 μm. Moreover, with the decrease of Fermi level, peak values of extinction ratio also decrease due to the larger plasmons loss under smaller Fermi level [20]. The maximum value of the extinction can be over 300 under E_f = 0.4 eV. On the other hand, the integration area is also critical to the performance of analyzer. To illuminate this, we choose several different areas of integration regions and calculate extinction ratio on each of them. The graphene Fermi level is set as E_f = 0.4 eV with other parameters are the same to Fig. 5(a).

In the figure we could see that, with the enlarging of integration area, the performances of analyzer are impaired due to the highly focused field distribution near center. For instance, extinction ratio over 550 can be achieved for a 25 × 25 nm² integration area, while decreases sharply to below 50 if the integration area is enlarged to 125 × 125 nm². Meanwhile, the peak wavelength shifts with the integration area as well, which could be attributed to the slight shift of focusing center with the incident wavelength. Therefore, in the practical applications, smaller probe tip would be preferable to obtain better performance. Compared to the previously reported circular polarization analyzer [15], the circular polarization analyzer based on metal antennas arrays can also achieve tunable working wavelength while with smaller Fermi levels. Furthermore, only the geometric phase effect of spiral structure needs to be satisfied here while both the occupation ratio and material of dielectric grating should be carefully designed in [15]. In addition, since the excitation of plasmons through single rectangular metal antenna on graphene has already been demonstrated experimentally [5], the proposed analyzer based on metal antenna arrays may possess more advantages in practical applications.

4. Conclusion

In this paper, we propose a graphene circular polarization analyzer based on spiral metal triangle antennas arrays. Through the dipole antenna resonance, the antenna could couple the energy of incident light into graphene plasmons. According to the geometric shape of antenna, the excited plasmons present different amplitude and phase distributions under linear polarized incidence. For the two basic combinations of linetype metal antennas arrays, only the normal linear polarization incidence could efficiently excite plasmons with planar wavefront. For the incident circular polarization with same chirality to the geometric phase effect of spiral metal antennas arrays, the plasmons field is defocused into a donut shape distribution while for the incidence with opposite chirality, the plasmons field is focused into a solid dot distribution, enabling its functionality as a circular polarization analyzer. The extinction ratio of analyzer could be over 550 and the operation wavelength can be tuned by graphene Fermi level. Moreover, we also show that the smaller integration region would be preferable for high extinction ratio. The proposed analyzer may have potential applications in chemistry or biology, such as analyzing the physiological properties of chiral molecules using different circularly polarized waves.