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In this paper we propose a method of unidirectional excitation of graphene plasmons via metal nanoantenna arrays and reveal its application in a circular polarization analyzer. For nanoantenna pairs with orthogonal orientations, the graphene plasmons are excited through antenna resonances with the direction of propagation can be controlled by incident polarization. On the other hand, based on the spiral shape distribution of antenna arrays, a circular polarization analyzer can be obtained via the interaction of geometric phase effect of antenna arrays and the chirality carried by incident polarization. By utilizing the unidirectional excitation of plasmons, the extinction ratio of analyzer can be improved to over 103, which is at least an order of magnitude larger than the result of antenna pairs with same orientations or antenna arrays with closed circular shape formation. The proposed analyzer may find applications in analyzing chiral molecules using different circularly polarized waves.
© 2015 Optical Society of America
1. Introduction
Surface plasmons are electromagnetic excitations propagating along the interface between dielectrics and conductors [1]. Graphene has been known as a semiconductor with a two-dimensional form of carbon atoms arranged in the honeycomb lattice [2]. Graphene possesses an extremely high quantum efficiency for light–matter interactions and supports plasmons (GPs) with superior properties than the traditional surface plasmons on noble metal, such as the high field confinement [3] and the tunable carrier concentration [4], making graphene a promising material for plasmonic devices.
However, the high field confinement also brings challenges to the excitation 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 probes [6], compressed surface polaritons of tapered bulk materials [7], guided mode resonances on dielectric gratings [8] or elastic vibrations on graphene surface 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 resonances of metal antennas on graphene, which provides possibilities of engineering propagation of GPs via the shape of metal antenna arrays.
Due to its advantages in switching applications [10], directional excitations of plasmons have been widely investigated [11–14 ]. For instance, by arranging two columns of aperture arrays with orthogonal orientations on metal surface, the direction of excited plasmons can be controlled by the state of incident circular polarization [13]. Furthermore, such a methodology has been applied in the focusing of linear polarization [15] or circular polarization analyzer [16], which focuses the plasmons into solid dot or donut shape profile at center according to the state of incident circular polarization. Meanwhile, directional excitation of plasmons on graphene has been proposed [10], in which the non-reciprocal dispersions of GPs on magneto-optical material grating are utilized to ensure the excitation condition is only satisfied along one direction. However, the need of external magnetic field and high Fermi level may raise the complexity of implementation.
Inspired by the excitation of graphene plasmons through single metal antenna [5] and unidirectional plasmons on metal surface through aperture arrays with orthogonal orientations [13], we propose the unidirectional excitation of graphene plasmons via metal antenna arrays and then design a graphene circular polarization analyzer with improved performance. In Sec. II, the excitation of graphene plasmons through single column of metal antenna arrays is shown. Then the unidirectional excitation of graphene plasmons are presented through two columns of orthogonal oriented nanoantenna arrays under opposite states of incident circular polarizations. Afterwards we design a circular polarization analyzer in Sec. III utilizing the unidirectional excitation of graphene plasmons and investigate the extinction ratio of analyzer as function of wavelength and integration areas. In the end we draw the conclusion.
2. Circular polarization controlled unidirectional excitation of graphene plasmons
In this section we investigate the unidirectional excitation of graphene plasmons through nanoantenna arrays. Sketched in Fig. 1(a) is the schematic of two columns of nanoantenna arrays with orthogonal orientations. The rectangular nanoantenna arrays are embedded in the dielectric material beneath graphene layer. The incident direction of circular polarized light is along z + axis, making this whole system compatible with near-field scanning optical microscope (NSOM) measurement [13]. All the numerical calculations have been performed by the commercial software Lumerical based on FDTD methods. The graphene layer is characterized by a built-in surface conductivity (σg = σintra + σinter) material model, in which the σintra and σinter are intraband and interband contributions to conductivity [17]. The materials of the metal antenna and dielectric layer below graphene are assumed as Au and CaF2 where the material refractive data is derived from the experimental measurements [18, 19 ]. Within the discussion of Sec. II, the metal antenna arrays are periodically distributed along y axis with the boundaries normal to x axis are set as Perfect Matched Layer. While for the simulations in Sec. III, all the surrounding boundaries are set as Perfect Matched Layer.
Fig. 1 Schematic view (a) of unidirectional excitation of graphene plasmons through two columns of rectangular nanoantenna arrays embedded in the dielectric layer beneath graphene. (b) The geometric parameters of antenna arrays. The light is incident from z+ axis.
The geometric parameters of antennas are shown in Fig. 1(b). The length and width of rectangle is respectively set as W = 30 nm and L = 150 nm. In each column, the metal antennas are spaced with a vertical distance D = 200 nm. The adjacent columns are offset by half a period D/2 along the y axis and separated by a lateral distance S = λGP/4 along the x axis. The λGP is the plasmons wavelength of GPs and can be calculated as [20],
where λ0 is the incident wavelength, e is the unit of electric charge, ℏ is the reduced Planck constant, ω is the angular frequency, τ = 0.5 ps is the carrier relaxation time, Ef = 0.5 eV is the graphene Fermi level, εr1 = 1 is the permittivity of air and εr2 is the permittivity of dielectric substrate.To reveal the mechanism of unidirectional excitation of graphene plasmons, it is essential to first look at the case of excitation of graphene plasmons using single column of metal antenna arrays. For plasmons supported by a metal-dielectric (e.g. graphene-air) interface, the normal component of electric field Ez is dominant in the region above graphene [21], therefore we only focus on this field component in our manuscript. The distribution of the real part of electric field component Ez under left-handed circular polarization is shown in Fig. 2 . The antenna arrays are distributed along y-axis at x = 0. The incident wavelength is chosen as λ = 10 μm. For better presentation, the state of incident polarization (from point of view of the source) is shown by arrows and the format of array distribution is shown in the inset.
Fig. 2 Distributions of the normalized real part of field component Ez under left-handed circular polarization. The state of incident polarization (from point of view of the source) is shown by black arrows and the format of array distribution is shown in the inset. The incident wavelength is λ = 10 μm.
As shown in Fig. 2, the excited graphene plasmons propagate away from both sides of antenna array along x axis, which is perpendicular to the distribution of antenna array. Since the vertical distance D is smaller than the plasmons wavelength λGP, such an array can be regarded as a source of plasmons plane wave. It is worth noting that due to the chirality carried by incident circular polarization, there is a π phase difference between the excited plasmons at both sides of antennas, which can be deduced according to the opposite signs of real parts of field component Ez. Such phenomenon is similar to the unidirectional excitation of plasmons through two columns of apertures with orthogonal orientations on metal surface [13].
Since the resonances in a subwavelength aperture on metal is an approximation of dipole resonances of a metal antenna, if two columns of aperture arrays could be utilized to excite unidirectional plasmons [13] on metal surface, similarly, metal antenna arrays should excite plasmons on graphene directionally. The mechanism of unidirectional excitation relies on the following two aspects. Firstly, the differences of orientations in each column of antennas would lead to a relative phase difference δ = π/2 or δ = -π/2 between the excited plasmons depending on the state of incident circular polarization. Secondly, as the two columns of antennas are placed adjacently with a distance as S = λGP/4, the π/2 phase retardation caused by lateral distance S would be superposed with or cancelled by the relative phase difference δ. In the end, the total phase difference between plasmons towards either side would be π or zero, leading to the destructive or constructive interferences of plasmons and finally the unidirectional excitation.
To reveal the relationship between incident circular polarization and unidirectional excitation of GPs, we place two columns of nanoantenna arrays with orthogonal orientations at x = -S/2 and x = S/2. The lateral distance is chosen as S = 100 nm, which is nearly a quarter of the plasmons wavelength under λ = 10 μm. The distributions of the normalized real part of electric field component Ez under two opposite circular polarizations with different chirality are shown below in Figs. 3(a) and 3(b) . For better comparison, the case of two columns of nanoantenna arrays with same orientations is shown in Fig. 3(c). The states of circular polarizations are shown by arrows and the formats of arrays distribution are shown in the insets.
Fig. 3 Distributions of the normalized real part of field component Ez under left-handed (a) and right-handed (b) circular polarization. For comparison, field profile produced by antenna pairs of same orientation under left-handed circular polarization is also shown (c). The states of incident polarization (from the point of view of the source) are shown by black arrows and the formats of array distribution are shown in the insets. The wavelength is λ = 10 μm.
As shown in Figs. 3(a) and 3(b), the excitation direction of graphene plasmons can be controlled by the state of incident polarization. The plasmons towards right are excited under left-handed circular polarization while the ones towards left are excited under right-handed circular polarization. Meanwhile, compared to the case of same orientation in Fig. 3(c), the field amplitudes in Figs. 3(a) and 3(b) are stronger; this may be attributed to the constructive interference between the plasmons excited on the two columns of nanoantenna arrays.
3. Circular polarization analyzer based on unidirectional excitation of plasmons
In the previous section, we present that the two columns of nanoantenna arrays could enable the unidirectional excitation of graphene plasmons. Moreover, the direction of excited plasmons can be controlled by the state of incident polarization. Previously, the interactions between circular polarization and chiral plasmonics lenses have been extensively studied and applied to design circular polarization analyzers [22–26 ]. By arranging the surface plasmons sources (e.g. aperture arrays on metal surface) into spiral shape, the mathematical form of distribution of electric field component Ez of plasmons near the center can be expressed by the lth order Bessel function with the spiral phase profile of lφ [22],
where kGP = 2π/λGP is the propagation constant of graphene plasmons and (r,φ) is the polar coordinate corresponding to the Cartesian coordinate (x,y). The index l = lincidence + lgeometric denotes the proportional constant of the azimuthal angle φ in the phase around the center, determined by the superposition of the chirality of circular polarization incidence lincidence and the geometric phase effect lgeometric caused by the format of antenna distribution. For the spiral shape of antennas array in which the distance between start and end radius is 2π/kGP, the geometric phase is lgeometric = 1, which means the plasmons excited at the circumference of antenna arrays would arrive at the center carrying a circumferential 2π-phase variation according to the phase retardation caused by the optical path difference. On the other hand, when the antenna arrays are illuminated by right-handed or left-handed circular polarization, the chirality of polarization leads to the |lincidence| = 1. Apparently, for one state of circular polarization, the index l would be l = lincidence + lgeometric = 0, while for the other state the index would be l = 2. The difference of index l leads to the different orders of the first kind of Bessel function and therefore solid dot (for l = 0) or doughnut (for l = 2) profiles at center. The state of incident circular polarization can be detected by measuring the intensity difference of electric field at center, achieving the functionality of a circular polarization analyzer.When the nanoantenna pairs are placed into closed circular arrays, there is no geometric phase effect (i.e. lgeometric = 0). Therefore the right-handed or left-handed circular polarization only leads to l = |lincidence| + lgeometric = 1. According to Eq. (2), such an index l corresponds to the doughnut shape field profile and no difference in the field intensity for two polarization states can be detected at the center. On the other hand, for antenna arrays with geometric phase lgeometric>1 (lgeometric = 2,3,4…), there would be still no solid dot field profile at center after superposition. For instance, if the distance between start and end radius of spiral antenna arrays is 4π/kGP, the geometric phase would be lgeometric = 2 and the index after superposition is l = 1 (or l = 3). Both field profiles at center would still be doughnut shape. Therefore the spiral formations of plasmons sources are essential for the analyzer.
According to the working principle of analyzer, if the constructively interfered plasmons are directionally excited and only propagate towards outside for the donut shape of field distribution, while only propagate towards inside for the solid dot shape of field profile, the extinction ratio of such analyzer should be enhanced. To prove this, we present below the manipulation of graphene plasmons using spiral shape of two columns of nanoantenna arrays with orthogonal orientations. The spiral shape antenna arrays are composed of 23 nanoantenna pairs and each of them possesses the same geometric parameters as those in Figs. 3(a) and 3(b). Specifically, the start radii of the two columns of spiral arrays are 575 nm and 675 nm, respectively. For each column, the distance between start and end radius is chosen as 400 nm [see Figs. 4(a) ], which is nearly the same as the graphene plasmons wavelength under λ = 10 μm. The normalized electric field |Ez| profiles for spiral nanoantenna arrays under right-handed and left-handed circular polarization incidence are shown in Figs. 4(d) and 4(g). For better comparison, we have also shown the normalized electric field |Ez|2 profiles for spiral nanoantenna arrays with swapped orientations. The schematic is shown in Fig. 4(b) with the results are plotted in Figs. 4(e) and 4(h). Meanwhile, as observed in the previous work [13], the magnitudes of plasmons field at the center of circular metal aperture arrays also depend on the chirality of incident circular polarizations due to unidirectional excitation. To reveal the importance of geometric shape of nanoantenna arrays, we present the distribution of field component |Ez| for antenna pairs with closed circular shape. Specially, the number of nanoantenna pairs is same as that of Fig. 4(a) [or Fig. 4(b)] and the radius of circular array is 825 nm. The schematic is shown in Fig. 4(c) and the normalized electric field |Ez| profiles for spiral nanoantenna arrays under right-handed and left-handed circular polarization incidence are shown in Figs. 4(f) and 4(i). All of the field profiles are obtained under wavelength λ = 10 μm.
Fig. 4 The normalized electric field |Ez| distributions for the spiral antenna arrays with different compositions of orientations (a,b) and circular antenna arrays (c) under right-handed (d,e,f) and left-handed (g,h,i) circular polarizations. The chirality of incidence is marked with white arrow (from the point of view of the source). The wavelength is λ = 10 μm.
Although Figs. 4(d) and 4(e) show that for the right-handed circular polarization, the spiral shape of antenna arrays could focus the excited plasmons wave into solid dot profiles at the center of analyzer, the magnitudes of graphene plasmons field inside and outside the spiral nanoantenna arrays show significant difference due to unidirectional excitation of plasmons. Ideally, there should be no plasmons field inside the arrays in the case of Fig. 4(e), however, due to the existence of plasmons attenuation within the lateral distance between two columns [13], the destructive interference is not perfect and small amount of plasmons energy can be observed. As shown in Fig. 4(f), although the plasmons propagating towards center are dominant due to the unidirectional excitation of plasmons, the field of |Ez| component at center clearly presents a doughnut shape due to the lack of geometric phase effect (i.e. lgeometric = 0). Once the illumination is switched from right-handed to left-handed circular polarization, the direction of plasmons excitation is reversed, which can be clearly observed in the Figs. 4(g-i). Meanwhile, since the superposition of geometric phase effect and chirality carried by the incident circular polarization equals to l = 2 rather than l = 0, the plasmons field profile at center becomes doughnut shape.
By comparing the field profiles of Figs. 4(d) and 4(f), obviously the former formation of nanoantenna pairs could achieve higher field amplitude at center. Though the closed circular antenna arrays produce enhanced plasmons field with a doughnut shape [Fig. 4(f)], the magnitude of which is still lower than the one of solid dot field of spiral antenna arrays. Therefore we could conclude both the unidirectional excitation of plasmons and the geometric shape of antenna arrays are essential to enhance the extinction ratio of analyzer. To prove this, we separately illustrate the influence of unidirectional excitation and geometric shape of arrays on analyzer extinction ratio in Figs. 5(a) and 5(b) . The extinction ratio is defined by the ratio of integration of normal field |Ez|2 for opposite states of circular polarizations on a square region around the center. Shown in Fig. 5(a) are the extinction ratios as functions of incident wavelengths for spiral nanoantenna arrays with orthogonal and same orientations. For the same orientations case, graphene plasmons towards inside and outside of arrays are simultaneously excited along radical direction. We compare in Fig. 5(b) the extinction ratios from spiral and closed circular shape of nanoantenna arrays under different integration areas. The sizes of integration regions in Fig. 5(b) are 50 × 50 nm2, 75 × 75 nm2 and 100 × 100 nm2. In contrast, the integration region is fixed at 50 × 50 nm2 in Fig. 5(a). The specific geometric parameters of spiral antenna arrays are the same as those in Fig. 4.
Fig. 5 The extinction ratios of analyzer with respect to the incident wavelength under different orientations (a) and for the two formats of nanoantenna arrays with different integrated regions (b). The integration region is 50 × 50 nm2 in (a).
As shown in Fig. 5(a), the extinction ratio under various incident wavelengths presents a maximum value near 103 at λ = 9.5 μm within the whole wavelength regime. In contrast, for the case with same orientation, the extinction ratio is much smaller and only reaches near 102. This indicates that the unidirectional excitation could indeed improve the performance of analyzer. As for Fig. 5(b), the circular polarization analyzer based on spiral antenna arrays could provide higher extinction ratio than the one based on closed circular antenna arrays. Specially, for small integration region (e.g. 50 × 50 nm2), the spiral format of nanoantenna array has improved the extinction ratio by at least an order of magnitude. Meanwhile, 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 103 can be achieved for a 50 × 50 nm2 integration area, while decreases sharply to below 2 × 102 if the integration area is enlarged to 125 × 125 nm2. Meanwhile, the peak wavelength shifts with the increasing of integration area as well, which could be attributed to the slight shift of focusing center with the incident wavelength.
In the recent experiments of excitation graphene plasmons [5], the authors used a Fermi level of 0.44 eV and a relaxation time of 0.05 ps. Compared with the values in Ref [5], we have used the almost same Fermi energy in our discussion with a higher value relaxation time of 0.5 ps. This value is elevated compared to graphene samples of recent plasmonic studies [6, 27 ], but it allows for a better illustration of plasmons focusing. As for experimental feasibility, the gold nanoantenna arrays can be fabricated with standard electron beam lithography technique [5, 28 ]. The main problem is to improve the electric properties of large-scale graphene grown by CVD method [29]. To improve the qualities of substrate supported graphene layer, new material of substrate can be taken into consideration, such as the crystal hexagonal boron nitride (h-BN) [29].
4. Conclusion
In this paper, we propose a graphene circular polarization analyzer based on unidirectional excitation of plasmons. Through the dipole antenna resonance, the nanoantenna arrays could couple the energy of incident light into graphene plasmons with plane wavefront. When two columns of arrays with orthogonal orientations are placed with a selected lateral distance, the different states of circular polarization would lead to the destructive interference between plasmons towards one side while constructive interference towards the other side, leading to the polarization controlled unidirectional excitation of graphene plasmons. Moreover, when arrays of nanoantenna pairs with orthogonal orientations are placed into a spiral shape format, a circular polarization analyzer can be achieved with the performance can be greatly improved than the antenna pairs with same orientations or the arrays with closed circular shape format, enabling an extinction ratio over 103 under the integration area 50 × 50 nm2, which is at least an order of magnitude larger than the result of counterparts. Moreover, we show that the smaller integration region would be preferable for higher extinction ratio. The analyzer with improved performance may have potential applications in chemistry or biology, such as analyzing the physiological properties of chiral molecules using different circularly polarized waves.
Acknowledgment
This work is supported by the National Natural Science Foundation of China (NSFC) (Grant Nos. 61178008, 61275092) and the Fundamental Research Funds for the Central Universities (Grant No. 2011RC050), China.
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