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We propose a tunable circular polarization analyzer based on a graphene-coated spiral dielectric lens. Spatially separated solid dot shape (or donut shape) field can be achieved if the geometric shape of analyzer and incident circular polarization possess the opposite (or same) chirality. Moreover, distinct from the narrow working bandwidth of a traditional circular polarization analyzer, the focusing and defocusing effects in the analyzer are independent of the chemical potential of graphene, and depend only on the dielectric permittivities and the grating occupation ratio. Combined with the strong tunability of graphene plasmons, the operation wavelength of analyzer can be tuned by adjusting the graphene chemical potential without degrading the performance. The proposed analyzer could be used in applications in chemistry or biology, such as analyzing the physiological properties of chiral molecules based on circular polarization.
© 2015 Optical Society of America
1. Introduction
Infrared surface waves open the way to the development of chemical and biological sensing tools, including lab-on-a-chip integrated biomolecule sensors [1]. However, the diffraction limit challenges the development of highly integrated infrared devices and the study of individual molecules and nanostructures [2]. Surface Plasmons (SPs) are electromagnetic excitations propagating along the interface between a dielectric and a conductor [3], making it possible to achieve highly integrated infrared devices. Graphene is a zero band-gap semiconductor with a two-dimensional form of carbon atoms arranged in the honeycomb lattice [4], leading to unique electronic and optical properties [5, 6]. The carrier density in graphene can be electrically adjusted by over 2 orders of magnitude with a small bias voltage applied on a field-effect transistor [7] and tuning time below a nanosecond [8]. In addition to broad tunability, graphene plasmons (GPs) also exhibit high field confinement [9], making graphene a promising material for active plasmonic devices.
Recently, the interactions between circular polarized light and chiral plasmonics lenses have been extensively studied [10–15]. Considering the extra phase resulting from the geometrical spiral structure, spiral plasmonic lenses focus circular polarization with opposite chirality into a solid spot in the center, while defocusing the polarization with same chirality into a donut shape [11]. Such an effect eliminates the alignment requirement between the singularity centers of illumination and metallic structure [14], demonstrating its capability as circular polarization analyzer [10, 15], as well as parallel imaging with an array format [11]. However, the traditional spiral structure is designed for single wavelength operation [14], in which operation wavelength is not tunable once the geometry is fixed. The analyzer performance degrades as the operation frequency moves away from the designed frequency.
A tunable circular polarization analyzer based on graphene-coated spiral dielectric lens is proposed in this paper. In Section 2, graphene-coated dielectric gratings are presented and shown to be able to excite graphene plasmons which can be tuned using grating period or graphene chemical potential, this is verified by guided mode resonance theory. Then we propose in Section 3 a circular polarization analyzer based on graphene-coated spiral dielectric lens. The focusing and defocusing effects of this device under linear or circular polarization illuminations are shown. Then we theoretically derive that the focusing or defocusing effect only depends on the dielectric permittivities of upper space and substrate and grating occupation ratio, leading to the possibility of tuning operation wavelength by the graphene chemical potential without degrading the performance.
2. Excitation of graphene plasmons by dielectric gratings
We first investigate the excitation of graphene plasmons through graphene-coated dielectric gratings. It has been verified that the incident light can be coupled to graphene plasmons via guided mode resonances of grating-spacer-graphene hybrid systems [16–18]. For simplicity, we first look into a simplified situation of 2D graphene-coated dielectric gratings in which the incident light is assumed to be x polarized. A schematic of the coupling implementation is illustrated in Fig. 1(a).
Fig. 1 Schematic view of the excitation implementation (a) and Left-handed graphene-coated spiral dielectric lens (b) under normal incident, Left-hand circular polarized light.
Figure 1 shows a dielectric grating with period Λ and permittivity εd = 11 on a metal substrate. In this section, the height, H and the occupation ratio, f are respectively fixed at H = 500 nm and f = 0.5. A graphene sheet with uniform chemical potential is placed on top of the dielectric. The graphene is characterized by a surface conductivity (σg = σintra + σinter) [9], in which the σintra and σinter denote the intraband and interband contributions, respectively. The scattering rate of graphene is τ = 6.4 × 10−13 s [17]. Here we assume that the temperature is T = 300 K. The surface-normal permittivity of graphene is assumed as εg,n = 2.5, based on the dielectric constant of graphite [19]. By treating the graphene as an ultrathin layer with thickness d = 1 nm, the tangential component of the effective permittivity of graphene can be expressed as εg,t = 2.5-iσg/ωε0d [20]. The maximum chemical potential of graphene is set as μc = 0.7 eV, which can be implemented through electronic biasing [21] combined with chemical doping [22, 23]. The minimum wavelength is set as λ = 6 μm to avoid the additional loss caused by phonon scattering [24]. All the numerical calculations have been performed by the Finite Difference Time Domain software from Lumerical. Since there is a large dimensional difference between the thickness of the graphene and the grating period, we use non-uniform meshing in the numerical simulations. The mesh size inside the graphene layer is 0.2 nm and gradually increases outside the graphene layer. Since the metal can be regarded as a perfect electric conductor in the mid-infrared region, we analyze one grating period (see Fig. 1(a)) and set the bottom of structure as a PEC boundary with Periodic boundary adopted along x direction. In the mid-infrared region, the dispersion relation of plasmons on graphene can be approximately expressed as [16],
where β is the propagation constant of graphene plasmons, e is the Electric charge, η is the reduced Planck constant and ω is the angular frequency. It should be noticed that there are two propagation constants of GPs, β1 and β2 in our device, which corresponds to the regions of graphene on dielectric gratings or on dielectric substrate, respectively. Since the graphene plasmons are far more highly confined than the plasmons between metal and air interfaces, the effects of substrate permittivities on the propagation constants cannot be neglected. For the former (β1), the εr1 is the effective permittivity of the gratings below graphene, which can be calculated via εr1 = f*εd + (1-f)εr2 [25], while for the latter (β2), the εr1 = 11. In both conditions, the upper space is air with permittivity εr2 = 1. The graphene plasmons can be excited through guided mode resonance (GMR) [26, 27] provided the dielectric grating period Λ satisfies the phase matching equation [16],where β0sinθ is the transverse wave vector of incident light, Λ is the grating period and N∈Z+ is the GMR mode order. Under normal-incidence (θ = 0), each order of resonant mode leads to a dip in the reflection spectrum at the excitation frequency ω, which can be expressed as (e.g. for N = 1) [16],Note that the permittivities of upper space, substrate and grating occupation ratio are fixed here (since the chirality dependence of this analyzer completely depends on these three factors, which we show in Section 3). The reflection spectra with respect to period Λ are presented in Fig. 2(a). While in Fig. 2(b) we compare the resonant wavelength of the 1st GMR mode (N = 1) derived from Eq. (3) (solid line) with those from numerical solutions (red circles). The wavelength spans from 6 μm to 11 μm and the chemical potential is μc = 0.64 eV.
Fig. 2 (a) The dependencies of normalized reflection spectra with different grating periods (Λ = 100 nm, 150 nm, 200 nm and 250 nm). (b) The derived excitation wavelengths of 1st order mode (N = 1) from Eq. (3) and numerical simulations. The wavelength spans from 6 μm to 11 μm. The dielectric permittivity εd = 11, μc = 0.64eV and occupation ratio f = 0.5. Illustrated in the inset of (b) are the Ex profiles of the two orders’ resonant modes in spectrum under Λ = 200 nm (circle marker a for N = 1 and b for N = 2).
Figure 2(a) shows that as the grating period increases, the excitation wavelength of 1st GMR mode shifts from near λ = 7 μm to near λ = 11 μm. For the period Λ = 200 nm and 250 nm, the second reflection dips correspond to the 2nd GMR mode also appear in the figure. For better presentation, we illustrate the modal field Ex profiles of the 1st and the 2nd GMR modes (circle marker a for 1st and b for 2nd) within one grating period Λ in the insets of Fig. 2(b). In such profiles, the electric field phase shifts with 2π or 4π in one grating period for the two orders’ modes. Next we present the tunability of excitation by adjusting graphene chemical potential in Fig. 3. The wavelength spans from 6 μm to 11 μm with gratings period Λ = 150 nm.
Fig. 3 (a) The dependencies of normalized reflection spectra with different chemical potentials are presented. (b) The derived excitation wavelengths of 1st order mode (N = 1) from Eq. (3) and numerical simulations. The wavelength spans from 6 μm to 11 μm with gratings period Λ = 150 nm.
Figure 3 renders that the excitation wavelength of 1st GMR mode can be tuned from wavelength λ = 8 μm to near λ = 10.5 μm. Additionally, the reflection at the dip increases as the chemical potential decreases, indicating the coupling efficiency decreases at lower chemical potential, which is similar to the results in [16]. In Figs. 2(b) and 3(b), the numerical solutions show good agreement with the results from Eq. (3), indicating the excitation wavelength can be well explained using guided mode resonances. It is worth noting that the chemical potential enables the excitation of graphene plasmons to be dynamically tuned over a wavelength range of 2.5 μm (from near 8 μm to 10.5 μm in Fig. 3(a)), which is very promising for actively tunable plasmonics devices.
3. Circular polarization analyzer based on graphene-coated spiral lens
In this section, we investigate the focusing properties of the graphene-coated dielectric spiral lens under the illumination of linear and circular polarized light (Fig. 1(b) shows the latter case). Here spiral dielectric gratings with twofold left-handed Archimedes’ spiral shapes are etched into dielectric layer on a metal substrate to excite graphene plasmons. In cylindrical coordinates, the Left-Handed Archimedes’ spiral can be described as [28],
where r0 is a constant and d = (2n + 1)λspp, where λspp = 2π/β1 is the wavelength of graphene plasmons on the dielectric gratings and n can be zero or any positive integer. The illumination is along the negative z direction. For linear polarized light, GPs are excited from every two opposite points along the diameter at the circumference of gratings, propagating in opposite directions. Therefore, for each pair of two opposite points at the circumference, the difference of the optical path to the center of the spiral lens would be (2n + 1)λspp/2, leading to constructive interference at the center. Based on this point, we calculate the 3D profiles of normalized electric field intensity |E|2 on the surface of the graphene under illumination of linear polarized light along the x or y direction, which are shown in Figs. 4(a) and 4(c). Meanwhile, the distributions along the x axis (for x polarized) or y axis (for y polarized) are plotted in Figs. 4(b) and 4(d). For better presentation, the distributions of normalized |E|2 field profiles on the xy plane are illustrated in the insets. The wavelength is λ = 6.2 μm, the dielectric permittivity is εd = 11, the period is Λ = 150 nm, the chemical potential of graphene is μc = 0.7 eV and occupation ratio f = 0.2.Fig. 4 The electric field intensity |E|2 distributions on the surface of graphene under x (a,b) or y (c,d) linear polarized light. The wavelength is λ = 6.2 μm, the dielectric permittivity is εd = 11, the period is Λ = 150 nm, the chemical potential of graphene is μc = 0.7 eV and occupation ratio f = 0.2.
As illustrated in Fig. 4, strong constructive interferences patterns can be observed along the polarization direction on the surface of graphene. Also in Figs. 4(b) and 4(d) we can see the field intensity is obviously enhanced at the center of the spiral lens. For the y-polarized light the enhancement is one time larger than the one under illumination of x-polarized light, which can be explained by the more efficient excitation of plasmons due to more dielectric grating periods along y-axis.
When a circular polarized beam is focused onto axially symmetric plasmonic structure, unlike linear polarized beams, the entire beam is transverse magnetic (TM) polarized with respect to the interface, enabling plasmons excitation from all directions and homogeneous plasmon focusing through interferences of these plasmon waves. A strongly confined solid spot will be obtained when the geometric phase produced by the spiral dielectric gratings cancels out the vortex wavefront of circular polarized illumination (spiral structure and incident light owe opposite chirality) [13]. Conversely, an electric field with donut shape emerges due to the superposition of geometric phase and the vortex wavefront (spiral structure and incident light have the same chirality). Similarly, the grating-spacer-graphene hybrid system in our design performs a critical role. On one hand, the gratings-spacer-graphene system should satisfy the phase matching condition. The guided mode resonance frequency of such structure should be as close as possible to the working frequency. On the other hand, the geometric structures of dielectric gratings should be Right- or Left-Handed Archimedes’ spiral shape to ensure the focusing or defocusing effect for circular polarized light with opposite chirality.
To simultaneously achieve the phase matching condition in the excitation procedure and the chirality dependent focusing effect of spiral lens, the following relationship need to be satisfied,
where n can be zero or any positive integer. For the metal plasmonic lens [10, 13–15, 28], the equation is satisfied at n = 0 since the comparatively weak confinement of plasmons on the metal and the constants β1 and β2 are nearly the same. While for the graphene plasmons the equation cannot be satisfied for n = 0 since the f is less than one and the divergence between β1 and β2 cannot be neglected [29].We choose n = 1 and the occupation ratio f = 0.2 to ensure Eq. (5) is satisfied. The incident wavelength is λ = 6.2 μm and grating period Λ = 150 nm. Then the structure is illuminated by the Left-handed or Right-handed circular polarized light. The 3D profiles of normalized electric field intensity |E|2 on the graphene surface are shown in Figs. 5(a) and 5(c). Meanwhile, the field profiles along the x axis are plotted in Figs. 5(b) and 5(d). Additionally, the 2D distributions of the normalized |E|2 fields on the xy plane are illustrated in the insets.
Fig. 5 The electric field intensity |E|2 distributions on the surface of graphene under Right-handed (a,b) or Left-handed (c,d) circular polarized light are shown. The wavelength is λ = 6.2 μm, the dielectric permittivity is εd = 11, the period is Λ = 150 nm and f = 0.2, the chemical potential of graphene is μc = 0.7 eV and wavelength λ = 6.2 μm.
As shown in Fig. 5, the Left-handed graphene-coated spiral dielectric lens focuses Right-handed circular (RHC) polarization into a solid spot in the central, while defocusing Left-handed circular (LHC) polarization into a donut shape field. This can be explained by looking at the field distribution function of spiral lens [10]. The LHC spiral lens focuses the RHC polarization into a plasmonic field profile as zeroth-order evanescent Bessel function with a central peak. For a LHC polarization illumination, the field profile presents a distribution as a donut second-order evanescent Bessel function. It is worth noting that the size of the focal spot is around tens of nanometers with field intensity enhancement more than 20 times compared to the field where graphene plasmons is initially excited (inner edge of gratings).
The spatially separated field profiles presented by the plasmonic lens under circular polarized illuminations with opposite chirality leads to its potential application as circular polarization analyzer [13–15]. If a detector with a fixed length a is placed in the vicinity of the focus, the detected signal can be distinct for RHC and LHC polarizations. In Fig. 6(a) we illustrate the circular polarization extinction ratio (defined by the intensity integral on the sensor region of solid dot field divided by the one of donut shape field) of the spiral lens with respect to detector size a under different incident wavelengths. The configurations of spiral gratings are same as those in Fig. 5.
Fig. 6 The circular polarization extinction ratio of the spiral lens with respect to detector size a under various incident wavelengths (a) and with respect to incident wavelength under various chemical potentials (b). The chemical potential is μc = 0.64 eV in (a) and the length is a = 20 nm in (b). All the configurations are same with those in Fig. 5.
As shown in Fig. 6(a), the extinction ratios decrease to near one as the sensor size increases (a >120 nm). The extinction ratios can be nearly 60 when the detector length near a = 20 nm and higher extinction ratio can be obtained with smaller detector size. Additionally, in Eq. (5) we can see the relationship is independent of graphene chemical potential. While on the other hand, in Eq. (3) the excitation wavelength can be tuned by the chemical potential (see Fig. 3). Therefore the chemical potential provides routes to a novel circular polarization analyzer with tunable working frequency and stable performance (e.g. extinction ratio of illumination with chirality). To illuminate this, the extinction ratios as functions of incident wavelengths with different chemical potentials (i.e. μc = 0.5 eV, 0.6 eV and 0.7 eV) are shown in Fig. 6(b). The sensor length is fixed at a = 20 nm.
Firstly, in Fig. 6(b) the curves clearly show that for the plasmonic spiral lens circular polarization analyzer, the performance varies with the incident wavelength due to the different excitation efficiency of graphene plasmons. Although the intrinsic narrow bandwidth restricts the application of circular polarization analyzer, it can be solved by applying various chemical potentials on graphene sheet. For instance, at the chemical potential μc = 0.6 eV, even though the 3dB bandwidth is only 0.7 μm (from 6.9 μm to 7.6 μm), the peak in extinction ratio spectrum can be shifted from 6.7 μm to 8 μm by adjusting chemical potential μc from 0.7 eV to 0.5 eV. Meanwhile, as the spectrum shifts, the maximum extinction ratio stays nearly unchanged, giving rise to the great potential of analyzing chirality illumination with various frequencies.
Although the thick metal back mirror is introduced in the graphene-gratings system of our manuscript. The results from such system show good agreement with the theoretical results in Fig. 2 and Fig. 3, which indicates that the thick metal back mirror does not affect the excitation of graphene plasmons through guided mode resonances. Furthermore, the chirality dependent effect of analyzer relies on the shape of spiral dielectric gratings rather than the thickness of dielectric spacer. Meanwhile, the strain of graphene induced by the dielectric gratings cannot be neglected and therefore should be taken into account in the design [17]. For simplicity, we assume the upper space medium to be air. While in practical applications, other filling media can be chosen (e.g. MgF2) to reduce strain related effects.
4. Conclusion
In this paper, we propose a tunable circular polarization analyzer based on graphene-coated spiral dielectric lens. The incident light is coupled into graphene plasmons through dielectric gratings where the excitation frequency can be adjusted by grating period and graphene chemical potential. The field focusing on graphene surface under linear or circular polarized illumination is presented. For the incident light with same chirality the plasmon field on the graphene is defocused into a donut shape while for light with opposite chirality, the field is focused into a solid dot with nearly 20 times enhancement. Finally we show that the operation frequency of this analyzer can be tuned by chemical potential while maintaining its performance. The proposed analyzer may have potential applications in chemistry or biology, such as analyzing the physiological properties of chiral molecules using circular polarizations.
Acknowledgment
This work is supported in part by the National Natural Science Foundation of China (NSFC) (Grant Nos. 61178008, 61275092).
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