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A planar graphene/dielectric multilayer structure is investigated, where the graphene surface plasmon polariton and the planar waveguide mode are coupled to realize Fano resonances. Few-layer graphene with high doping levels is used to excite surface plasmons at mid-infrared wavelength. Reflectance of the structure is calculated numerically by transfer-matrix method, and tunable Fano resonances with different line shapes are demonstrated by varying doping levels of graphene. Properties of the Fano resonances are discussed qualitatively by calculating electric field distribution in the structure and quantitatively by utilizing an analytical fitting equation. We also calculate Goos-Hänchen shift of the Fano resonances as an example for potential applications, and find that large Goos-Hänchen shift appears for optimized doping levels of graphene.
© 2016 Optical Society of America
1. Introduction
Fano resonances, which were first studied in describing asymmetric autoionization spectra of He atoms [1], are widely studied not only in various quantum systems [2] but also in different optical systems, such as metamaterials [3, 4], photonic crystals [5, 6] and plasmonics [7, 8]. Fano resonances arise when a narrow resonance or discrete state and a broad resonance or continuum state are coupled. Sharp asymmetric line shapes and rapid variation in phase and amplitude are the characteristics of Fano resonances, and can find various applications such as sensing [9], Goos-Hänchen shift(GHS) [10, 11], extraordinary optical transmission (EIT) [12,13] and switching [14].
Fano resonances in nanostructures have been widely studied, where mostly plasmonic resonances of noble metals are utilized. While noble metals can provide well-confined plasmon field at visible frequencies, they suffer from large intrinsic losses and are not appropriate for applications in longer wavelengths, such as infrared and terahertz. Also for plasmonic structures made of noble metals, active tunability is difficult to realize. Recently graphene has emerged as a promising plasmonic material for applications at infrared and terahertz wavelengths [15–20]. Graphene, a single layer of carbon atoms with honeycomb lattice, has been proved to support surface plasmon polariton (SPP) over a wide wavelength range. Comparing to noble metals, graphene SPP has tighter field localization, lower losses, and most appealing property that its conductivity can be tuned dynamically by electrostatic gating. Then it is possible to realize various tunable plasmonic devices [21, 22]. Although tunable Fano resonances using graphene have been reported recently [23,24], graphene SPP was not used as a component of Fano resonances, which is still caused by nanostructures made of noble metals, only tunable conductivity of graphene is utilized. In another work, phonons were used to couple with graphene plasmons and realized Fano type resonances [25].
In this work, we investigate tunable Fano resonances, arising from interference between graphene SPP and a dielectric waveguide mode. The graphene/dielectric interface supports a SPP mode, leading to a broad resonance; while a planar waveguide (PWG) supports a waveguide mode which shows a narrow resonance. If these two resonances are coupled, sharp Fano resonance will appear. Further due to electrically tunable of graphene SPP, active tunable Fano resonance can be realized.
2. Theoretical models and methods
The structure analyzed here is shown in Fig. 1(a) which is a typical Otto geometry. Graphene is marked by thick red line and is separated by a small air gap from coupling prism. Between graphene and substrate, there is a 3-layer PWG structure. In the following discussion, the prism, substrate and PWG core are considered to be Ge with np = ns = n3 = 4, cladding layers of PWG are set to be CaF2 with n2 = n4 = 1.3. Thickness of each layer is set to be d1 = 0.5μm, d2 = d3 = d4 = 2μm, respectively. The incident field is assumed to be transverse magnetic (TM) polarized to excite graphene SPP.
Fig. 1 (a) Schematic of the Otto geometry. (b) Dispersion relations of PWG mode (blue solid) and FLG SPP of 5-layer (red dashed) and 4-layer (black dash-dotted) graphene.
Graphene is modeled as a surface conductive sheet with zero thickness, and its complex surface conductivity σ, which is a function of angular frequency ω, Fermi energy EF, carrier scattering rate Γ, and absolute temperature T of environment, is contributed by intraband and interband σ =σintra+σinter terms, and can be expressed according to the Kubo formula [26,27]:
where e is elementary charge, h is the reduced Plancks constant, kB is the Boltzmann constant. In our work, the graphene carrier scattering rate is assumed to be Γ = 5 THz, the temperature T = 300K, and Fermi energy EF can be tuned dynamically by electrostatic gating. By using top-gate electrical doping in extended graphene, Fermi energys as high as EF = 1eV have been achieved experimentally [28,29]. The Fermi energies can be further increased if patterned graphene is used, such as ribbons or plates. This method would boost the Fermi energy to about 3.2eV [18]. Another method to increase EF is that we could use graphene with a high chemical-doping basepoint, whereby the Fermi energy could be doped above 1eV in the absence of bias [30]. Then extra gating could be used to tune EF around this basepoint [18]. In this work the required EF can be further decreased if longer wavelengthes are considered, especially in THz wavelengths. Few-layer graphene can also be used to decrease the required EF. It is known that graphene can support TM SPP when imaginary part of σ is positive. The excitation of SPP with Otto geometry, requires that refractive index of prism is larger than effective index of SPP. Monolayer graphene (MLG) can only support SPP with very large effective refractive index at infrared wavelength, which is impossible to find a feasible prism to overcome the momentum mismatch. To reduce effective index of graphene SPP, σ should be increased. Either we can excite the SPP in longer wavelengths, such as terahertz, or use few-layer graphene (FLG) instead with higher EF. Increase EF of MLG constantly by higher gating voltage maybe not appropriate as it may be higher than breakdown voltage. Taking individual graphene sheet as non-interacting monolayer, which is reasonable if layer number N<6, the surface conductivity of FLG is Nσ [31]. In the following analysis, FLG with N=5 layers is considered to reduce the momentum mismatch. For the SLG is used, where υF ≈ c/300 and ng is the carrier densities. While for FLG in our work the calculation of EF can be done by assuming the charge is uniformly distributed among N layers giving [32]. The assumption is reasonable if stacked CVD graphene is used, and due to extremely subwavelength length scale of graphene the FLG behaves as an effectively uniform charge sheet.The usual well known dispersion relation describing graphene SPP surrounded by two semi-infinite dielectric layers maybe not appropriate here. The prism, air spacer, graphene and cladding layer 2 compose a 3-layer structure. d2 is chosen to be thick enough to treat this layer as semi-infinite. By matching boundary conditions for the np-n1-FLG-n2 system, the SPP dispersion can be derived as:
where Γ2 = (α1ε2)/(ε1α2)[1+iσα2/(ωε0ε2)], Γp = α1εp / (ε1αp), , j = p,s,1,2,3,4, β is the x component wavenumber, εj is the permittivity.By setting d1=0 and Γp=1, the dispersion relation returns to the wellknown formula [33]: ε1/α1 +ε2/α2 + iσ/(ωε0) = 0. Note that for FLG, σ should be replaced by Nσ.
For the n2-n3-n4 PWG system, d3 is chosen to keep TM0 single mode excitation and d2, d4 are large enough that their further increase will not influence the PWG mode. However d2 should not be too thick that coupling between PWG mode and graphene SPP may be blocked. The dispersion relation of PWG mode can be derived as:
where pj = ε3/εj, . The dispersion relations of Eqs.(3) and (4) can only be solved numerically. And we can calculate their effective refractive index separately by definition of ne ff = β/k0. When effective index of FLG SPP and PWG mode are matched, we can couple two modes and excite Fano resonance. In Fig. 1(b) we show the calculated real part of effective index according to Eqs. (3) and (4), we fix incident light wavelength to be 10.6μm and varying EF utilizing gate-tunability of doped graphene. As EF increases imaginary part of σ increases and results in decreased of FLG SPP as indicated in Fig. 1(b). The blue solid line is of PWG mode which is independent of EF, and can only be varied by varying wavelength or geometric parameters. The red dashed line is of 5-layer graphene SPP, and black dash-dotted line is of 4-layer graphene for comparison. When EF is smaller than 0.6eV, for FLG is even larger than refractive index of prism, FLG SPP excitation is impossible only if prism with higher refractive index and light with large incident angle are used. For EF lager than 0.7eV, for 5-layer graphene is smaller than 3.6, and the required incident angle will be smaller than 64°. As EF increases of 5-layer graphene decreases and approaches that of PWG mode, they finally intersect at point of EF=0.86eV. Around this cross point two modes will be excited simultaneously and mode coupling can occur. As a comparison, we also show of 4-layer graphene in Fig. 1(b), obviously as N decreases the effective index is enhanced. As discussed above, higher makes excitation of FLG SPP more difficult and to couple two modes together higher EF are required. This is also why we choose 5-layer graphene in our following analysis.The multilayer systems, shown in Fig. 1(a), can be solved numerically by standard transfer-matrix method. To excite FLG SPP, TM polarized incident light with incident angle is required, the prism and substrate are treated as semi-infinite layers. The transmission matrix of FLG for TM field can be express as:
which is in fact utilization of boundary conditions for tangential magnetic field and tangential electric field. The validity of this treatment is checked by treating FLG as an ultrathin metallic layer with isotropic permittivity of εg = 1+iσ/(ωε0dg), where dg is the thickness of MLG and is usually set to be 0.34nm. These two kinds of method matches well at infrared and terahertz wavelength, while in shorter wavelength such as communication and visible wavelength, MLG should be treated as an anisotropic material by setting perpendicular permittivity to be 1 and tangential permittivity εg.3. Results and discussions
For a FLG with specific EF, we can get the angular reflection spectra numerically, as shown in Fig. 2, blue solid lines. The red dashed lines are analytical fit of numerical results and will be discussed later. In Fig. 2, angular spectrums with different EF are calculated. In Fig. 2(a), EF=0.7eV, according to the reflection dip both FLG SPP and PWG modes can be excited, and due to large difference in their effective index, two reflection dips separate apart. FLG SPP is excited around 65°, and show a broad reflection dip. PWG mode is excited around 51.5°, and the reflection dip is much narrower. If separation between two modes is reduced, mode coupling and Fano resonance can be expected. In Fig. 2(b), EF is increased to 0.8eV, two modes approach gradually, and sharp asymmetric line shape of reflection appears which is feature of Fano resonance. The resonant angle is around the PWG mode excitation angle, and we can tune the Fano resonance by tuning resonant angle of the broad FLG SPP resonance. Further we increase EF=0.86eV, which is the cross point predicted in Fig. 1(b), nearly symmetric line shape of Fano resonance appears, as is shown in Fig. 2(c). In this case, coupling between two modes is the strongest, a destructive coupling between two modes is expected and the reflection appears as a peak instead. Continue to increase EF=0.9eV, excitation angle of FLG SPP keep moving to smaller angle, the Fano resonance appears at the other side of the reflection dip of FLG SPP and have opposite asymmetry compared to that of Fig. 2(b). Thus by tuning EF of doped FLG, we are able to get Fano resonances with different line shapes and realize active tunability.
Fig. 2 Reflection spectra for different values of EF, in (a) to (d), EF is taken to be 0.7eV, 0.8eV, 0.86eV, 0.9eV. The blue solid lines are numerical results and red dashed lines are fitted curves of Eq. (6).
To further understand the origin of sharp Fano resonance, we calculated electric field distribution inside our structure. The numerical results are shown in Fig. 3, the calculated electric field Ex is normalized by incident electric field E0, and we set the substrate interface to be z=0. We fix EF=0.8eV, the related reflection spectrum is the shown in Fig. 2(b), and vary the incident angle around 51.45° where general asymmetry Fano resonance appears. In Fig. 3(a), the incident angle is 52°(marked by circle in Fig. 2(b)), it shows that the electric field distributs mainly in the FLG layer, due to partly excitation of FLG SPP the reflection is lower than the reflection peak. In Fig. 3(b), the incident angle is 51.45◦(marked by square), it shows that electric field distributes mainly in the PWG, and electric field in FLG layer is depressed, the coupling between two modes is destructive and a reflection peak appears. Further decrease incident angle to be 51.35°(marked by diamond), as is shown in Fig. 3(c), electric field is enhanced both in FLG and PWG layers, coupling between two modes in this region is constructive, and the reflection appears as a dip. Finally we set incident angle to be 51°(marked by triangle), the condition is the same as Fig. 3(a), and we get reflection lower than peak and higher than dip. We notice that above process happened in angular width less than 1 degree, the destructive and constructive interference between two modes in such a narrow angular width is believed to be the origin of sharp Fano resonance.
Fig. 3 Electric field distribution with different incident angles, in (a) to (d), incident angle is taken to be 52°, 51.45°,51.35°, 51°. Different colors of background refer to different layers like in Fig. 1(a).
Above we have described the Fano resonance qualitatively, to understand quantitatively we introduce fitting equation for the reflection expressed as:
where θj = (θ −ϕj)/Δj, ϕ1 and ϕ2 are excitation angle of FLG SPP and Fano resonance, Δj are resonance width, T1 is for nonzero reflection dip of FLG SPP, R2 is the relative strength of Fano resonance and q is the Fano constant. In this equation, we use 1 minus Fano model because the usual Fano model is for transmission coefficient, and for the broad SPP resonance we assume a Lorentzian line shape. The numerical results have been proved to be well fitted by the analytical equation, as is shown in Fig. 2, red dashed lines. We have checked vast numerical results to make sure this equation is generally applicable.In Fig. 4, we show curves of fitting parameters rely on varying value of EF. In Fig. 4(a), we show ϕ1 −ϕ2 as a reference, this value is positive when FLG SPP excitation is on the right side of Fano resonance in angular spectra, and two excitation angles coincide at about EF =0.86eV. Then in Figs. 4(b) and 4(d), we show both strength and width of Fano resonance. As can be seen, two parameters both have maximum at EF =0.86eV. As two resonances separate apart, strength and width of Fano resonance decreases. The width of FLG SPP is much larger and not concerned here. And we also show Fano parameter q in Fig. 4(c), the opposite sign of q indicates that the Fano resonances have opposite asymmetry. When q is 0, a symmetry resonance appears, sometimes called anti-resonance, as shown in Fig. 2(c). As q tends to be very large absolute values, which is not shown in Fig. 4(c) with even larger or smaller values of EF, the Fano resonance gradually evolves to symmetry resonance of PWG mode, as shown in Fig. 2(a). Absolute values of q refer to different types of asymmetry Fano resonance line shapes.
Fig. 4 Fitting parameters with varying EF. (a) Deviation between two resonant angles; (b) Strength of Fano resonance; (c) Fano parameter; (d) Width of Fano resonance.
Sharp Fano resonances may have various potential applications, and in the following we will calculate the GHS as an example which is highly desirable in applications such as sensing and switching. The lateral shift D of the reflected field can be expressed as [34]:
where φ is phase of the reflection coefficient. It is well known that narrower resonance in the angular spectrum will result in large GHS. According to Fig. 4(d), it seems either very large or very small values of EF is required to get the narrowest resonance, i.e. when separation between Fano resonance and FLG SPP is large. However it should be noticed from Fig. 4(b) that if separation between two resonances increases, strength of Fano resonance will also be decreased. So we can imagine that an optimized value of EF will result in the largest GHS. The numerical result of Eq. (7) is shown in Fig. 5, we fix incident angle to be 51.46° and vary EF from 0.86 to 1.1eV. The incident angle is always located between maximum and minimum of Fano resonance to get larger GHS. The numerical results show that the peak GHS, which occurs at EF =0.96eV, is about 500 times the incident wavelength.4. Conclusion
In conclusion, we have demonstrated numerically the excitation of Fano resonances utilizing coupling between a broad FLG SPP and a narrow PWG mode in an Otto geometry. Owing to tunability of FLG SPP, the resulted Fano resonances can be tuned actively by varying EF of doped graphene. We explained the origin of Fano resonances qualitatively by showing the electric field distribution, and also carry out a quantitative analyzation by fitting numerical results with an analytical equation. Finally, for potential applications of the Fano resonances, we calculated the GHS as an example. The result demonstrate a large negative GHS, which is over 500 times of incident wavelength, and an optimized value of EF is required. We believe that our results will find potential applications, such as sensing and switching at infrared wavelength.
Acknowledgments
This work is partially supported by the National Natural Science Foundation of China (NSFC) (Grant Nos. 61505111 and 61490713), and the Guangdong Natural Science Foundation(Grant No. 2015A030313549).
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