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We propose in this paper a graphene-coated tapered nanowire probe providing strong field enhancement in the infrared regimes. The analytical field distributions and characteristic equation of the supported surface plasmons mode are derived. Based on the adiabatic approximation, analytic methods are adopted in the investigation of field enhancement along the tapered region and show well consistence with the rigorous numerical simulations. Both the numerical and analytical results have shown that the graphene-coated nanowire probe could achieve an order of magnitude larger field enhancement than the metal-coated probes. The proposed probe may have promising applications for single molecule detection, measurement and nano-manipulation techniques.
© 2014 Optical Society of America
1. Introduction
Mid-infrared radiation allows the analysis of a wide range of different material properties, including the chemical composition and structure of matter [1]. Infrared spectroscopy is therefore an essential analytical tool in high-resolution imaging of sample topography properties. However, the diffraction limit challenges the development of highly integrated infrared optical devices [2].
Surface plasmons (SPs) are the electromagnetic excitations propagating along the interface between a dielectric and a conductor, evanescently confined in the perpendicular direction [3], providing solutions for overcoming the diffraction limit. The excitation of enhanced optical near-fields at plasmonic nanostructures [4–7] allows the localization of electromagnetic energy on the nanoscale [8]. At nanotip, this effect has enabled a variety of applications, most prominent amongst them are scanning near-field optical microscopy (SNOM) [9] and tip-enhanced Raman spectroscopy (TERS) [10]. Although various efforts have been made in the visible light band, nanofocusing in the mid-infrared spectral range is a widely unexplored terrain from both fundamental and applied perspectives [9]. Graphene, a two-dimensional form of carbon with the atoms arranged in a honeycomb lattice [11], is a zero band-gap semiconductor, leading to the unique electronic and optical properties [12, 13]. Compared with the SPs on conventional plasmonic metals, graphene surface plasmons (GSPs) have shown better field confinement and could naturally be restricted on curved graphene surfaces [14], making it possible to achieve strong field enhancement on the probe tip at mid-infrared frequencies.
In this paper, we firstly present the confinement of fundamental SPs modes supported on graphene-coated and metal-coated nanowire waveguides. For acquiring deeper physical insight, analytical field distributions of the fundamental GSPs mode are derived and validated by the numerical simulations. Based on adiabatic approximation and numerical simulation, the field enhancement along graphene-coated nanowire waveguide has been discussed and compared with the results of metal-coated waveguides. The graphene-coated nanowire probe is proposed and investigated using the adiabatic theory and numerical simulation. Finally we compare the field localization and enhancement of the probe with circular aperture and the one with graphene-coated spherical tip.
2. Graphene-coated nanowire waveguide
We initially investigate the performances of SPs mode on the graphene-coated nanowire waveguide and compare the results with metal-coated one. The graphene layer has been demonstrated experimentally that can be tightly coated on the nanowire due to van der Waals force [15–17]. The analysis would be conducted for the fundamental TM (transverse magnetic) SPs mode with the magnetic field having only Hϕ component and both electric and magnetic fields are independent of the angle ϕ. It has been already proved that only such mode could be supported on the nanowire with arbitrarily small radius [18, 19].
Our analysis in this paper relies on a classical electromagnetic description of graphene using a surface conductivity σg [20], depending on the frequency f, the temperature T, the chemical potential μc and the external magnetic field B. Here we assume that the environment temperature is T = 300 K with no external magnetic field. The surface-normal and tangential permittivities are εg,n = 2.5 [21] and εg,t = 2.5-iσg/ωε0d, in which the thickness d is 0.34 nm [22]. The optical phonon scattering rate needs to be taken into account for the frequencies ω >ωoph (ћωoph≈0.2 eV, corresponding to the frequency f ≈50 THz) [23], which could increase the real part of the conductivity and decrease the propagation length, thus only the frequencies below ωoph are considered in our paper. A nanowire with permittivity εd is coated by graphene (or 10 nm thickness Au/Ag layer) and surrounded by air. The cross sections are illustrated in insets of Figs. 1(a) and 1(b). The chemical potential of graphene are set from μc = 0.5 eV to μc = 1.0 eV, which can be implemented through ON doping [24] or nitrogen doping [25]. The permittivities of metals are derived from Drude Model [26, 27]. All the numerical calculations have been performed by finite-element software package (COMSOL).
Fig. 1 The normalized mode areas of fundamental SPs mode on the graphene-coated and metal-coated nanowire waveguides as functions of frequency f (a), nanowire radius R (b), nanowire permittivity εd (c) and chemical potential μc (d). Without specified, the f = 48 THz, R = 120 nm, εd = 2.48 and μc = 1 eV. Cross sections of the two waveguides are shown in the insets of (a) and (b). The normalized electric field amplitudes |E| of the two waveguides under R = 100 nm are shown in the inset of (d), the area is 500 × 500 nm2.
For better understanding, the normalized mode area is introduced, which is defined as Aeff = ∬W(x,y)dxdy /(λ2Wmax). Wmax is the maximum of the energy density of the whole waveguide cross section [28]. The performances of GSPs modes depend on the frequency f, radius R, nanowire permittivity εd and chemical potential μc. The normalized mode areas Aeff as functions of the aforementioned parameters are shown in the Fig. 1. For comparison, the results of Au- and Ag-coated nanowire waveguides are shown as well.
Without specified, the parameters are set as f = 48 THz, R = 120 nm, εd = 2.48 and μc = 1 eV. In Fig. 1(a), the normalized mode areas increase as the frequency increases. The differences of mode areas between metal-coated and graphene-coated waveguides are larger at higher frequency. As shown in Fig. 1(b), the normalized mode area increases with nanowire radius R. The normalized electric field |E| (|E|2 = |Ez|2 + |Er|2) on the Ag-coated (left panel) and graphene-coated (right panel) waveguides with radius R = 100 nm are shown in the insets of Fig. 1(d), in which the modal field is tightly confined along graphene layer while much more weakly confined on Ag layer. In Fig. 1(c), the mode areas of metal-coated nanowire waveguides are insensitive to the nanowire permittivity. While for the graphene-coated one, higher confinement is achieved on the nanowire with larger permittivity εd. As illustrated in Fig. 1(d), the mode area of graphene-coated nanowire waveguide decreases with the chemical potential. In most figures, the normalized mode areas of graphene-coated waveguide are much smaller than the metal-coated counterparts, indicating the graphene-coated nanowire waveguide could achieve much better confinement and thus having superior potentials in nanofocusing in the infrared and terahertz frequencies.
3. Graphene-coated tapered nanowire waveguide
We first derive the analytic field distribution and characteristic equation of the fundamental GSPs mode. The analytic field distributions are obtained by solving Maxwell equation in cylinder coordinate. The graphene is regarded as a boundary layer with surface conductivity σg. The Er, Ez and Hϕ field components of the fundamental GSPs mode could be respectively expressed as,
where, , ε1 = εdε0 and ε2 = ε0. The coefficient A is obtained from the integration of time average Poynting vector A2 = P/(2π∫Re((Hϕ)*(Er)/2)rdr) and P is the energy flow through integration area. Due to the existence of graphene surface conductivity, the boundary conditions for the electric and magnetic fields at r = R can be expressed as Ez(r<R) = Ez(r≥R) and Hϕ (r≥R)-Hϕ (r<R) = σgEz(r<R), ensuring the continuity of the tangential components of the electromagnetic field at graphene layer. The propagation constant β could be obtained by solving the characteristic equation,To verify the analytical equations, the electric field |E| (|E|2 = |Ez|2 + |Er|2) on the surface of waveguide from analytical equations and numerical simulations as functions of the frequency f, radius R, nanowire permittivity εd and chemical potential μc are shown in Fig. 2. The energy flow is normalized to P = 1 W. For comparison, the electric fields |E| of metal-coated nanowire waveguides are shown in the Fig. 2. Without specified, the parameters are frequency f = 48 THz, nanowire radius R = 200 nm, permittivity εd = 2.48 and chemical potential μc = 1 eV.
Fig. 2 The electric fields |E| of the graphene-coated and metal-coated nanowire waveguides as functions of frequency f (a), nanowire radius R (b), nanowire permittivity εd (c) and chemical potential μc (d). Without specified, f = 48 ΤΗz, R = 200 nm, εd = 2.48 and μc = 1 eV.
As shown in Fig. 2, the analytical results (red dashed lines) agree well with numerical simulation (blue solid lines). In Fig. 2(a), higher enhancements are achieved at higher frequency. While for the metal-coated waveguides (green solid and black dashed lines), the enhancements are insignificantly affected by frequency and one order of magnitude smaller than graphene-coated waveguide at f = 48 THz. In Fig. 2(b), the electric fields |E| increase as the radius R decreases, indicating that highly enhancement fields could be achieved at end of the tapered nanowire waveguide. Figures 2(c) and 2(d) show that the electric fields |E| of GSPs modes increase with higher nanowire permittivity while lower chemical potential. In Figs. 2(b)-2(d), the field amplitudes of metal-coated counterparts are about an order of magnitude smaller than those of graphene-coated ones.
For the tapered nanowire waveguide, if the variations of TM plasmon wavelength within one plasmon wavelength are negligible, the adiabatic approximation can be used, in which the field distribution is determined at each distance to the tip, assuming uniform nanowire waveguide with local parameters at the considered point [18]. Mathematically, the applicability condition for the adiabatic approximation can be expressed as |d[Re(β)−1]/dz| = 1 [4, 29]. The adiabatic theory assumes there is no reflection from the end of probe, which could not be neglected for numerical simulation [29]. For better comparison, we consider a tapered nanowire waveguide with a uniform nanowire attached to the exit, which is illustrated in inset of Fig. 3(a). The input and output radii are r0 and r1, respectively. The length of the tapered waveguide is L and the taper angle is α. In our discussions, the r0 = 200 nm and L = 800 nm. Figure 3(a) shows the variations of |d[Re(β)−1]/dz| along the tapered waveguide under various taper angles. The corresponding output radii r1 are presented in Fig. 3(a), respectively.
Fig. 3 The left-hand side of inequality (a) and the electric fields |E| (b) on the surface of graphene-coated nanowire waveguide under different taper angles. The electric fields |E| of the Ag-coated waveguide are also presented. The f = 48 ΤΗz, r0 = 200 nm, L = 800 nm, εd = 2.48 and μc = 1 eV.
Next we calculate the electric field distribution of the plasmons at an arbitrary distance z to the end. Assuming the input power is Pz = 1 W at the point z = L (R = r0). As the plasmon propagates from this point z to a point z-dz towards to the end, the overall energy should be reduced by the amount of energy dissipated within the distance dz, which is determined by the Im(β). Then the overall power and corresponding electric field amplitudes at the points z-dz, z-2dz, etc. could be obtained from Eqs. (1)-(4) [19, 30]. Based on this procedure, the electric fields |E| along the surface of the tapered waveguide are determined and compared with numerical simulations in Fig. 3(b). Meanwhile, the electric fields |E| of the Ag-coated waveguide under the same configurations are also presented. The parameters are set as f = 48 ΤΗz, r0 = 200 nm, L = 800 nm and εd = 2.48. Although lower chemical potential μc results in smaller normalized area as well as higher electric field [see Fig. 1(d) and Fig. 2(d)], higher confinement of field on the graphene surface can also introduce larger attenuation. We choose μc = 1eV in the following discussions for the least attenuation.
In Fig. 3(a), we could see that for all the taper angles, adiabatic approximations are satisfied and the propagation constant varies non-monotonically with nanowire radius. As shown in Fig. 3(b), all the analytical results (dashed lines) agree well with the numerical results (solid lines). Although there are still periodical corrugations along the surface of the tapered waveguide (solid lines) due to the reflections of the energy, the corrugations get less significant at smaller taper angle α. As the SPs waves propagate along the tapered waveguide, the electric fields |E| on the waveguide surface get stronger. Besides, the larger taper angle also leads to the smaller output radius r1, and accordingly the stronger electric fields on the surface [see Fig. 2(b)].
Noting that the graphene-coated tapered waveguide could achieve nearly 20 times larger electric field than the metal-coated ones (dot-dashed lines), showing the promising potential of the nanofocusing using graphene-coated nanowire waveguide. The analytical methods are convenient for accurate analysis of plasmon nanofocusing in the tapered graphene-coated nanowire waveguide with long waveguide length and various taper angles, which may be challengeable for the numerical methods due to limited computation resources.
4. Graphene-coated tapered nanowire probe
We subsequently discuss the tapered waveguide with the uniform waveguide replaced by a tip, namely the graphene-coated tapered nanowire probe. The plasmons reflection from tip will enhance the electric field amplitudes on the surface, thus the results from adiabatic theory could be underestimated. For more accurate results, the reflected power from tip needs to be taken into account. Since the size of tip is much smaller than the length of tapered region, the power attenuation on tip is neglected.
On the other hand, as the incident localized plasmon energy can hardly radiate any bulk waves from the rounded tip, actually all the energy of the incident plasmon is reflected back [29]. In this case, the standing wave patterns on the surface of probe formed by counter-propagating plasmons must have an anti-node at the tip. Thus the actual local field amplitude is expected about ~2 times larger than that of the adiabatic theory assuming no reflections from the tip. In Fig. 4(a) we present the distributions of electric field amplitudes |E| under two taper angles along the tapered region from the numerical simulations and adiabatic theories with plasmons reflections taken into account. The numerical results of the Ag-coated probe are shown as well. Since the adiabatic approximation is not satisfied on the tip, we only numerically calculate the electric fields along the surface from point z = -r1 (apex of tip) to z = 0. The schematic of probe and the numerical results are shown in Fig. 4(b).
Fig. 4 The electric fields |E| under two taper angles along the tapered waveguide region (a) and tip region (b). For comparison, the electric fields |E| of Ag-coated probe are shown. Normalized electric fields |E| of the metal-coated (left panel) and graphene-coated (right panel) at the tip are presented in the inset of (a). The schematic of the probe is shown in the inset of (b). The f = 48 ΤΗz, r0 = 200 nm, L = 800 nm, εd = 2.48 and μc = 1 eV.
In Fig. 4(a), obvious standing wave patterns are illustrated (solid lines), indicating the reflections from tip affect overall field distribution on the tapered waveguide. Noting that the results from the adiabatic assumption with reflections taken into account are consistent with those from the numerical results and are an order of magnitude larger [see the inset of (a)] than the ones with Ag-coated (dot-dashed lines). As shown in Fig. 4(b), the maximum fields of graphene-coated probes appear before the apex of tip. On the other hand, the maximum electric field is nearly two times larger than that from the adiabatic theory (see Fig. 3(b)) without considering the plasmons reflections.
In our paper, the spherical tip is completely covered with graphene. While in the fiber-based scanning near-field optical microscopes, the metal-coated tapered optical fiber probes with sub-wavelength sized circular apertures at the tip apex are commonly adopted, which was first proposed in 1991 by Betzic and Trautman [31]. Benefits of aperture probes include the high energy transmission [32] and low level of nonlocal background light, making it easier to achieve a sufficient signal-to-noise ratio [33]. We next compare the field localization and enhancement of the probe with circular aperture and the one with graphene-coated spherical tip. For better comparison, we set the radius of the aperture as r1, which is the same with the radius of spherical tip. The distributions of electric field |E| by numerical simulations for these two kinds of probes are shown in Fig. 5. The insets show the schematic view of the circular aperture probe as well as the normalized field profiles at the ends of two probes.
Fig. 5 The electric field distributions along the tapered region for the two kinds of probes. The schematic view of the circular aperture probe as well as the normalized |E| field profiles at the ends of the probes with spherical tip (left panel) and circular aperture (right panel) are shown in the inset. The f = 48 THz, r1 = 15 nm, L = 800 nm, α = 13°, εd = 2.48 and μc = 1 eV.
As shown in the Fig. 5, the circular aperture causes the strong field concentration (~1010 in amplitude) on the rim while insignificantly influences the field distributions along tapered region, which is similar to the metal-coated aperture probe [33]. In the tapered region, the nearly same standing patterns can be explained by considering surface waves are totally reflected at the rim of aperture (with circular aperture) and the tip apex (with completely coated spherical tip), respectively. However, since the tip dimension is much smaller than the whole probe, the positions where the two surface waves are totally reflected cannot be distinguished spatially, leading to the similar distributions of electric field along the tapered region. As for the localization, the fields are respectively located around the spherical tip apex and the edge of aperture, which can be observed in the insets of Fig. 5.
It is worth noting that in our paper, the loss and dispersion characteristics of the dielectric material in both probes (graphene-coated or metal-coated) are not taken into account. The additional dielectric loss may become possible roadblock in practical applications. Therefore, dielectric materials with low attenuation and dispersion in the infrared region would be preferred, such as MgF2 [34].
5. Conclusion
The graphene-coated tapered nanowire infrared probe with strong field enhancement is proposed. The fundamental GSPs mode supported on graphene-coated nanowire waveguide shows better field confinement than the metal-coated counterparts. Based on the adiabatic approximation, the field enhancements on the surface of graphene-coated tapered probe could be well described and are an order of magnitude stronger than the metal-coated probe. For the graphene-coated nanowire probe with circular aperture, the aperture causes strong field concentration on the rim while insignificantly influences the field distributions along tapered region. The proposed graphene-coated tapered nanowire probe has promising applications in future infrared spectroscopic lab-on-a-chip designs and high-resolution imaging of sample topography and infrared properties.
Acknowledgment
This work is supported in part by the Major State Basic Research Development Program of China (Grant No. 2010CB328206), the National Natural Science Foundation of China (NSFC) (Grant Nos. 61178008, 61275092), and the Fundamental Research Funds for the Central Universities, China.
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