Under certain conditions of the incident light polarization direction a Fano resonance arises in small gold nanorods arranged in a H-like configuration. This stems from the coupling between a bright dipole plasmon mode excited in the horizontal rod and dark quadrupole plasmon modes in both vertical rods. We investigate these surface plasmon modes, and analyze the dependence of the Fano resonance on the geometry parameters such as rod size and interparticle separation, and refractive index of embedding medium. To describe the degree of this energy transfer, we introduce a new parameter: the Fano resonance efficiency. We calculate absorption cross-sections for visible and NIR spectrum in each element of the structure, and near-field distributions at different wavelengths. We show that Fano resonance in small H-like structures exhibits high sensitivity with respect to the refractive index of the host medium, outperforming the values for larger plasmonic structures based on nanorods already investigated.
© 2014 Optical Society of America
Fano resonance (FR) in metallic nanoparticles (MNP) has attracted large interest due to its high sensitivity to the optical properties of surrounding media and geometrical configuration of MNPs . It is well known that FR appears namely as a consequence of destructive interference between a bright (superradiant) dipole mode and a dark (subradiant) quadrupole mode. The strong figure of merit (FOM) of the FR favors applications in sensing, optical switching and slow-light systems . Systems composed of small MNPs are of great interest particularly for studying optical properties of large biomolecules in visible range of spectrum. Larger particles (length of the order of hundreds of nm) present resonances in near-infrared (NIR). It was found recently that arrangements of coupled nanoparticles provide more possibilities for the realization and controlling of FR [3, 4]. Some of the first complexes of nanoparticles where FR was experimentally investigated were the dolmen structure and ring-near-disk cavity [5–9]. Later, other configurations of coupled MNPs were proposed, such as two nanorods in linear end-to-end or T-like configurations [10–13]. Whereas in the end-to-end structure the quadrupole mode in the longer rod is excited due to inhomogeneity of the field of the shorter rod, in a T-like configuration it arises in the top rod due to the symmetry of the system when the external field is polarized along the vertical axis. Calculations have shown that, in small nanorods, the near-field coupling between dipolar and quadrupolar modes in T-like configuration is more effective when the length of the longer rod (on top) is approximately twice of the length of the shorter one, for the same diameter . This is in accordance with scaling laws of multipole resonances found in nanorods [14–16]. Numerical calculations accounting retardation effects for larger spheroidal particles and rods using a dielectric function extracted from the experimental data of Johnson and Christy  yield FR for particles with approximately the same aspect ratios [13, 18]. This condition allows the frequency of the bright dipole mode in the vertical rod to approach the frequency of the dark quadrupole mode of the horizontal rod. Therefore, a FR can be observed in this T-like configuration if the incident light is polarized along the vertical rod. We note that longitudinal dipole oscillations in the horizontal rod cannot be excited because of symmetry, and the coupling of transversal and longitudinal modes in both rods is negligible for the particles with large enough aspect ratio.
Other plasmonic structures supporting FR are arrangements of three slabs, or rods in dolmen-like or H-like configuration. In the dolmen-like structure composed of relatively large slab particles, FR appears due to coherent coupling between superradiant and subradiant plasmon modes, studied theoretically in [5, 19, 20], and observed experimentally in [5, 20]. Multiple FR may occur in this case due to the strong transversal-longitudinal coupling as the sizes of the slab components of the structure increase and retardation effects become significant. Moreover, strong spatial inhomogeneity of the electric field causes excitation of higher multipoles in the plasmonic spectrum if the size of a MNP is comparable or larger than the penetration depth of the light. Nanorods in H-like configuration were so far investigated only as a structural elements of metamaterials for plasmonic induced transparency (the analog of electromagnetic induced transparency) in NIR spectral region [20, 21]. A review on FRs excited in plasmonic nanostructures based on nanorods was recently presented .
In this communication we demonstrate the peculiarities of the FR in a H-like configuration of small (compared to the excitation wavelength) Au nanoparticles. The electric field of the incident plane wave polarized along the shorter (middle) rod excites the dipole plasmon mode, and couples to the dark quadrupole modes in both vertical rods. To understand the coupling mechanism leading to FR, we build a simple model of coupled dipoles based on the quasi-static approximation theory. The obtained formulae ease to tailor particle systems with efficient dipole-quadrupole interaction. To describe quantitatively the effectiveness of dipole-quadrupole coupling (DQC) giving rise to FR we introduce a new parameter: FR efficiency. To further analyze the dependence of FR peak position on various parameters of the system such as, relative sizes of component rods, interparticle separation and refractive index of surrounding medium, we perform calculations based on the finite-element method (FEM, COMSOL Multiphysics 4.3b). These calculations reveal that the sensitivity of the H-like configured nanorods is of the order of 500 nm per refractive index unit (RIU) which makes the system suitable for sensing applications. Moreover, our calculations highlight that the absorbed energy in the dark (vertical) rods can be up to 20 times larger than that of the bright (horizontal) rod, due to the DQC at the FR. Other advantage that the H-like structure offers is the strong attenuation of the radiation damping rate due to the excitation of dark modes, as in the case of the T-like structure . These findings are useful towards engineering of efficient particle systems with optimized energy transfer.
2. Analytical modeling of nanorods
In order to clarify the conditions required for DQC in H-like structure we first consider two interacting rods with aspect ratios k1 and k2 and length small enough, where the quasi-static approximation is applicable. Estimation of the optimal ratio k2/k1 for the generation of FR is possible if we replace the rods by nanospheroids, for which analytical expressions of the optical properties are known. This replacement is justified by the well known similarity of surface plasmons (SP) spectra of nanorods and nanospheroids . We show that it is possible to develop an analytical approach allowing to estimate proper sizes of the spheroids providing well-pronounced DQC. This implies to know the relationship between aspect ratios of individual spheroids, which can be determined by analyzing the dependence of eigenfrequencies on aspect ratios for dipole and quadrupole plasmon modes. ωdip and ωquad are the resonance frequencies of dipole and quadrupole SP modes respectively, and are determined by solving the boundary value problem of electrostatics using spheroidal coordinates. If we express the dielectric function of the metal as a function of the angular frequency ε = ε(ω) it is possible to obtain simple expressions for ε(ωdip) and ε(ωquad) by Fig. 1 is plotted the real part of ε(ωdip) (a) and the real part of ε(ωquad) (b) as function of the aspect ratio of an Au spheroid, according to Eq. (1) and experimental data from . The intersection points between any horizontal line and both curves determine the aspect ratios required for the dipole-quadrupole coupling.
For example, for the photon energy E = 1.66 eV, ℜ[ε(ω)] = −20 and εm = 2.25, the corresponding aspect ratios are k1 = 3.9 and k2 = 6.9. Thus, the dipole-quadrupole coupling condition requires k2/k1 = 1.77. This value remains nearly constant in the whole visible range. Beyond the regime of quasi-static approximation, accounting the effect of retardation, as in FEM calculations, the ratio k2/k1 does not change considerably, as it is shown later. We neglect higher order modes since their frequencies are more blue shifted and lie far from the dipole-quadrupole resonance. The dipole-octuple coupling in H-like structure is evident for specific aspect ratios as discussed below.
To understand the induced charge distribution in the H-like structure with two equiradial vertical nanorods about twice longer than the horizontal one, we consider the quasistatic approximation. To be able to neglect the longitudinal-transversal coupling due the large separation between the corresponding frequencies , we further assume that aspect ratios for all three rods exceed 3.0. Ignoring transversal plasmon modes, the charge distribution in the three elements of the configuration is shown schematically in Fig. 2.
Whereas the longitudinal dipole oscillation excited in the horizontal rod is driven by the external field, in the vertical rods the quadrupole mode is the first order plasmon mode allowed by the symmetry (see Fig. 2). In this situation the field acting on the horizontal rod is a superposition of the external field and the near-field of quadrupole oscillations in the vertical rods. The resulting field can thus be strongly suppressed because of the destructive interference on coupled systems presenting FR . On the other hand, this causes a sharp decrease of the dipole moment of the horizontal rod, and gives rise to the Fano dip in the absorption spectrum.
In order to understand the fundamental optical characteristics of a H-like structure we introduce a model of coupled dipoles, ignoring transversal oscillations. In this model the horizontal rod is substituted by a spheroidal particle and each vertical rod by two spheroids. All five particles are identical (see Fig. 3). Effectively, the charge distribution associated to the quadrupole mode in the vertical rods can be substituted by charge distributions of two dipole modes acting in opposite direction. Further simplification can be reached by representing the electric field of each spheroid as the field of a point dipole located at the center of spheroid with the polarizability given by26].
We describe the interaction between point dipoles using the expression of electric field Eij created by i-th dipole interacting with the j-th oneEq. (3) into Eq. (4) leads finally to 27]), we obtain expressions for the absorbed power in the horizontal spheroid N1(ω), for the absorbed power in one of the vertical spheroids N2(ω), and for the total absorbed power in all five spheroids Ntot (ω). They read Eq. (2) and the Drude-Lorentz model for the dielectric function, accounting the contribution of interband transitions , Fig. 4). A Fano resonance arises for λ = 547 nm. The absorbed power at this wavelength for the horizontal spheroid is one order of magnitude smaller than that in vertical rods. This type of plasmonic energy transfer between particles of the coupled system at specific wavelength is the fundamental mechanism of the FR.
3. Absorption spectra in H-like configured rods
To have a complete view of the optical properties of the H-like structures, we have solved Maxwell’s equations numerically by FEM. In all simulations the sizes of the nanorods are modeled according to conditions leading to strong DQC as described above. Figure 5 shows the absorption cross-sections calculated for the elementary parts (horizontal rod, vertical rods, and for the full H-like structure.
Calculations were performed for two different values of separation between vertical and horizontal rods, refractive index of surrounding medium and lengths of rods. The radii of the rods are R = 2.5 nm. Two different lengths for the vertical rods are fixed: h = 40 nm (a) and h = 35 nm (b). In both cases the length of horizontal rod takes the values of l = 13 nm, l = 15 nm, l = 18 nm and the separation between the horizontal and vertical rods is assumed to be s = 2 nm. Optical constants for gold were taken from  and the refractive index of the surrounding media is n = 1.5.
Absorption cross-section spectra are identified by color: horizontal rod (blue curve), two vertical rods (green curve) and full H-like structure (red curve). In contrast to the previous investigation of the FR, the FEM calculations permit not only to obtain absorption spectra in individual parts of the structure separately but also to compare the balance of absorbed energy between composite parts of the structure at many different wavelengths of incident light in far-field with near-field patterns. This analysis permits to reveal clearly the physics underlying the DQC that causes FR in H-like structures. Simulations results nevertheless show that both calculations, i.e. model with five quasistatic spheroids and a more realistic model of the H-like structure achieve similar results (see Figs. 4 and 5). The total scattering cross-sections represent typically only few percent of the total extinction and are therefore not included in the present discussion.
In all cases considered one can see a set of three peaks. The left peak, visible in all absorption spectra of the full H-like structure and for vertical rods of Fig. 5, corresponds to the transversal SP mode in vertical rods and naturally does not depend on l. Note that transversal dipole oscillations in horizontal rod can not be excited because of symmetry of the problem. Calculations for two different radii of rods (R = 2.5 nm and R = 5 nm) show also that the position of this resonance peak remains almost unchanged. This is an expected result for the particles with sufficiently large aspect ratios [24, 26]. At λ = 515 nm, on both (a) and (b) absorption spectra in the horizontal rod is strongly attenuated comparing to absorption in vertical rods, since the excitation wavelength is far from that corresponding to the longitudinal dipole oscillations. However, weak interaction between the horizontal rod and the vertical ones and contributions from higher order multipoles lead to a nonzero absorption at the same wavelength.
The central peaks in the absorption spectra of the horizontal rod (blue curves in Fig. 5) correspond to the longitudinal SP dipolar mode at λdip. It is interesting to note that the absorption in the vertical rods (green curve) has a weak maximum at the this wavelength. The reason for this behavior is the near-field created by the horizontal rod and acting on the vertical ones which undergoes a resonance enhancement. Surprisingly, in both cases (a) and (b), the peak value of the dipolar absorption decreases with increasing rod length l. This effect seems to be anomalous since an increase of l (increase of particle volume) should lead to an increase of dipolar absorption cross-section. However, there are no contradictions in these results, as the absorption peaks of vertical rods at the same wavelength increase (peaks at λdip of the green curves). Moreover, at the FR the total absorption increases with the length of the horizontal rod, mainly in the spectral region between the central and the right spectrum peaks. Thanks to the strong interaction between dipolar and quadrupolar modes in the H-like structure the absorption properties around λdip are fully conditioned by the absorption in the vertical rods.
Peaks at right in the plots (green curves of Fig. 5) correspond to the longitudinal SP quadrupolar plasmon modes, excited in the vertical rods at λquad. It is interesting to verify that there is an absorption maximum for the horizontal rod at the same wavelength as well (blue curves), which again is a consequence of strong DQC. This conclusion is corroborated by results of the absorption cross-section for a H-like structure with double rod separation (s = 4 nm). The weaker coupling leads to a remarkable attenuation of the absorption in the horizontal rod at λquad. The peak values of all maxima at λquad increase with increasing length of the horizontal rod. Moreover, the peak of the resonance of the blue curve at λquad increases faster compared to the peak value of the green curve. In conclusion, an increase of l leads to a pronounced contribution of the quadrupole mode to central peak at λdip and to both dipolar and quadrupolar absorption at λquad. Simulations carried out for longer horizontal rods (up to 24 nm) show that all the features discussed above are retained.
Minima in the absorption cross-sections of the horizontal rod appearing in graphs of Fig. 5 are a manifestation of FR. The pronounced Fano dips at wavelengths around λ = 830 nm (Fig. 5(a)) and λ = 770 nm (Fig. 5(b)), are conditioned by the destructive interference between the dipole mode induced in the horizontal rod and quadrupole modes excited in each vertical rod, due to strong near-field coupling.
For a fixed length h a small increase of l leads to a small red-shit of the FR dip. However, varying h by 5 nm leads to a strong red-shift of the FR wavelength of about 60 nm. This remarkable sensitivity could be exploited in various applications, e.g. chemical and biological sensors. As well, there is a noticeable red-shift of the absorption maxima determined by the longitudinal SP modes in horizontal rods with increasing l.
Figure 6 shows the near-field patterns obtained for the wavelengths corresponding to the most relevant points of the absorption cross-sections. These patterns are associated to the surface charge distribution on the rods. The near-field distribution at the transversal SP wavelength (λ = 515 nm) of the vertical rods is shown in Figure 6(a). The electric field distribution in the vertical rods is approximately homogeneous with small lobes at both sides of the extremities. In the horizontal rod, by contrary, two lobes are located at both ends.
Figure 6(b) shows the electric field distribution at λdip = 690 nm, i.e. the dipole mode of the horizontal rod. The field strength increases strongly comparing to the previous case and the lobes are mainly concentrated at the caps of the horizontal rod. In Fig. 6(c) is shown the near-field at FR (λ = 770 nm), where due to the strong interaction of the DQC the energy is transferred to the vertical rods. Finally, in Fig. 6(d) is shown the near-field at the quadrupole resonance of the vertical rods (λquad = 825 nm), where the near-field reaches its maximum. Note that in all four graphs, the electric near-field distribution around the horizontal rod is the result of the main dipole mode with small contributions from higher order multipoles.
The near-field distribution in the vertical rods at FR wavelength (Fig. 6(c)) also contains an octupolar contribution determined by the attractive forces resulting from charges of the opposite sign accumulated at the caps of vertical rods (see Fig. 2). This interaction leads, namely, to an asymmetry in the middle near-field lobes of the vertical rods. Such behavior is much less pronounced in the near-field distributions at dipolar and quadrupolar wavelengths (Figs. 6(b) and 6(d), respectively).
In Fig. 7 are shown the results of simulations of the absorption cross-sections for the H-like structure embedded in medium of refractive index n = 1.4. All geometrical parameters remain the same of Fig. 5. It is evident a blue-shift of all resonances comparing to those obtained for n = 1.5. It is well known than Fano resonances on plasmonic nanostructures are very sensitive to the refractive index of the medium. Estimations of the sensitivities of FR for the H-like structures considered are of the order of 450 nm (for structures with h = 40 nm) and 500 nm (for h = 35 nm) per RIU, respectively. Simulations for n = 1.3 and n = 1.6 result in resonance shifts of the same type. It is this strong sensitivity to the refractive index of medium which makes the FR on plasmonic nanostructures very attractive for sensing applications.
Simulations for doubled interparticle separation (s = 4 nm) result in similar effects. These are however less pronounced because of the weaker coupling. The calculations were repeated for all the parameters of the H-like structure described in Fig. 5 except the rods radius (R = 5 nm). All features already described remain unchanged.
4. Fano resonance efficiency
The absorption at the FR (plots of Fig. 5 and Fig. 7) in horizontal rod is usually very small comparing to the absorption in the vertical ones. This allows us to introduce a new characteristic of FR - Fano resonance efficiency (FRE, or η) in the following way. The formulation of FRE at the FR wavelength for a H-like plasmonic structure has the form
The value for FRE at Fano dip (λ = 830 nm in Fig. 5(a), λ = 770 nm in Fig. 5(b)) reaches approximately 20. Since the absorption power is proportional to the stored energy, it is clear that FRE represents the ratio of energies stored in dark mode and bright mode. It is noteworthy to mention that in T-like structures, constituted by a short vertical rod and a longer horizontal rod, embedded in medium of refractive index n = 1.5 the FRE ratio reaches only η = 8 (width l = 60 nm, h = 30 nm, R = 10 nm, gap s = 2 nm) . This proves the advantage of using H-like structures to obtain higher DQC efficiency. In simplified models of coupled particles, as in Fig. 3, the value of FRE reaches 12 (Fig. 4).
Interestingly, the introduced definition of FRE can be generalized for other configurations of MNPs. Indeed, in all cases the surface plasmon energy is transferred from the particles supporting radiant modes to those particles that can not be excited by the external field because of the symmetry. In order to have a more detailed description of FR we have also calculated the FOM for a H-like structure using the absorption spectra. For rod lengths l = 18 nm and h = 35 nm the simulations show SP resonances shifts, in particular for the quadrupole mode, dependent on the refractive index of the surrounding medium (see Fig. 8).
As it can be seen in Fig. 8, the SP energy peak is linearly dependent on the refractive index of the medium and presents a red shift with increasing n. The slope of the curve is −0.78 eV/RIU. The full width at half maximum (FWHM) of the quadrupole resonance is ≃ 0.073 eV, which gives a rather large value for the FOM: 11. The large FOM is conditioned by the strong attenuation of radiation damping because of the relatively small size of the horizontal rod (l = 18 nm), where the bright dipole mode is excited. In fact, if we consider larger rods in a T-like structure, like those of Fig. 8 of reference  (lengths of rods L1 = 600 nm and L2 = 300 nm) the FOM reaches only 3. Other plasmonic structures like those of  reach FOM values of the order of 30. However, the fabrication complexity and sizes exceed largely those of the structures investigated here. Surprisingly the FOM for a T-like structure of small rods (l = 18 nm and h = 40 nm) reaches 12. However, its FRE is much smaller than that of a H-like structure as discussed above.
Thus, FRE plays the role of a Q-factor characterizing the quality of FR and, therefore, likely the FOM provides a quantitative description of FR. These two parameters can be used jointly to estimate the strength of FR that appears in wide range of physical phenomena starting from photoionization of atoms and clusters to sensor applications in nanoscale.
Here we present the results of a theoretical investigation of the surface plasmon modes and their optical spectra in H-like configured Au nanorods of relatively small sizes by using the finite-element method (FEM, COMSOL Multiphysics 4.3b). We show that FR arises as the dipole plasmon mode in the horizontal rod couples to the dark quadrupole modes supported in both vertical rods. We calculate absorption cross-sections for the particles supporting bright and dark plasmon modes separately which permits to know the energy distribution in each component of the structure at many wavelengths.
We introduce a new characteristic of FR - Fano resonance efficiency, that along with the FOM provides a quantitative description of the FR. We then show that small H-like nanostructures exhibit superior sensing properties. We remark that near-field coupling in plasmonic nanostructures, mainly fabricated using lithography techniques, always suffer from local defects at the nanoscale which limit, or even prevent their application. Thus, nanostructures based on few plasmonic particles which considerably improve near-field coupling and FR efficiency can be favorably fabricated. Presently, it is challenging to fabricate nanostructures with the dimensions and separation described above in a reproducible way, using lithography techniques. However, recent progresses in the fabrication of plasmonic nanostructures using self-assembly methods of chemically synthesized gold nanoparticles have been promising to obtain very small plasmonic nanostructures. For instance, such a technique based on the self-assembly of DNA strains and gold colloids  presents an alternative route to obtain hybridized structures with various topologies. This requires, however, functionalization of the colloidal particles. Nevertheless, this technique can be very useful in some thermoplasmonics applications such as cancer cell treatment  since such structures can be prepared directly in a desired solution. On the other hand, larger rods can be fabricated easier on solid surfaces by lithography techniques. However, the applicability of these rods is limited to NIR wavelengths because of the shift of their FR resonances.
This work was supported by State Committee of Science of Ministry of Education and Science of the Republic of Armenia in frame of the research project N. SCS - 13-1C353 and by the VolkswagenStiftung Grant # 86933.
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