We report on a theoretical investigation of the coupling between magnetic plasmons (MPs) and Tamm plasmons (TPs) in a metal–dielectric Bragg reflector (DBR) containing a gold nanowire pair array embedded in the low refractive index layer closest to the metal film. Strong coupling between MPs and TPs is observed, manifested by large anticrossings in the dispersion diagram. It creates a narrow-band hybridized MP mode with a Rabi-type splitting as large as 290 meV. Upon the excitation of this hybridized MP mode, a 2.5-fold enhancement of the magnetic field in the center of nanowire pairs is achieved as compared with the pure MP of the nanowire pairs embedded in a bare DBR structure (without the metal film). This result holds a promising potential application in magnetic nonlinearity and sensors.
©2012 Optical Society of America
During the last few years, metallic nanostructures have attracted considerable research efforts in various fields, such as optoelectronics or biophotonics [1, 2]. This relies on the excitation of localized surface plasmons (LSPs), which are spatially confined free electron oscillations in metallic nanostructures . The LSPs are often associated with a large enhancement of the electromagnetic field around nanostructures, and their spectral positions are highly sensitive to the size, shape, and surrounding medium of the nanostructures, which have found interesting applications in surface enhanced Raman spectroscopy , biosensing , solar cells , and so on.
However, due to their efficient coupling to vacuum light modes, the LSPs have a high radiative damping and a large linewidth , which significantly limit the intensity of localized optical fields that can be achieved in the nanostructures. An effective way to reduce the radiative damping lies in coupling the LSPs to a resonance mode with a narrow linewidth. For example, through coupling a broad plasmonic dipole to a narrow plasmonic quadruple resonance, plasmonic analogue of electromagnetically induced transparency can be realized owing to nearly complete suppression of radiative losses . In a similar way, Zentgraf et al. couple the broad-band LSPs to a narrow-band waveguide mode to obtain a quasiparticle mode with dephasing times up to 50 fs . Furthermore, enhanced sensing properties are achieved based on the combination of LSPs in nanostructures and a photonic microcavity .
Up to now, the interactions of electric surface plasmon resonances have been extensively investigated. While in metamaterials comprising subwavelength spaced magnetic resonators, like split ring resonators , stacked nanodisks  and nanowire pairs , electric ring currents can be excited by an external electromagnetic field. These ring currents induce magnetic dipole moments in the perpendicular direction, which produce magnetic resonances at optical frequencies, also termed “magnetic plasmons (MPs)”. Thus, these metallic nanostructures have been widely exploited to mimic magnetic atoms and build negative-permeability metamaterials with peculiar electromagnetic properties. Compared with electric plasmon resonances, the interactions of MPs with other resonances also result in many interesting phenomena, but are still rarely investigated. For example, either the electric (symmetric) or magnetic (antisymmetric) plasmon mode of nanowire pairs can be coupled to the resonator modes of a microcavity depending on the position of the nanowire pairs in the microcavity, and the splitting can be as large as 82 meV for the magnetic and 354 meV for the electric plasmon mode . Furthermore, through coupling to resonance modes with narrow linewidth such as collective surface mode  and Bloch surface wave , enhanced magnetic resonances are achieved at optical frequencies, which have received considerable attention  due to its potential applications in magnetic nonlinearity  and magnetic sensors.
In this paper, we propose a method to enhance the magnetic response of metallic nanowire pairs via coupling with Tamm plasmons (TPs) in a system consisting of a metal-dielectric Bragg reflector (DBR) enclosing a gold nanowire pair array. TPs are electromagnetic surface states formed at the interface between the metal film and the DBR structure [19–21]. Due to the coupling between TP and MP, clear anticrossings of these resonances are observed in the dispersion diagram. The coupling results in a large Rabi-type splitting as large as 290 meV, and the ratio of the splitting to the resonance energy of the MP is two times that achieved in the system with the gold nanowire pair array embedded in a microcavity . The strong coupling also creates a hybridized MP mode with a narrow linewidth, and the maximum magnetic field enhancement induced by this hybridized mode is 2.5 times larger than that induced by the pure MP of the nanowire pairs embedded in a bare DBR structure (without metal film).
2. System description and working principles
The system under consideration is shown in Fig. 1(a) , and assumed to be infinite along the y direction. A gold nanowire pair array (infinite along the y direction and periodic along the x direction) with a period of 200 nm is embedded in a metal-DBR structure. The DBR structure consists of a TiO2 spacer layer with a thickness of ds, then followed by 10-period stack of alternate SiO2 and TiO2 layers with thicknesses and respectively. On top of the DBR structure is a silver film with a thickness of. The gold nanowires have a width of and a thickness of 20 nm. The two nanowires in each nanowire pair are separated by a distance of 20 nm. Both the gold and silver have a wavelength-dependent dielectric index , and the refractive indices of SiO2 and TiO2 are assumed to be constant and taken as 1.5 and 2.2, respectively.
In our analysis, a TE or TM-polarized (see Fig. 1(a) for polarization definitions) incident light illuminates the whole system from the air side at normal incidence. Numerical simulation has been done using rigorous coupled wave analysis (RCWA) [23–25] method. The RCWA was first developed to analyze the diffraction by planar (including binary) gratings , and the detailed formulation for the efficient and stable implementation is well described in . It was then extended to surface-relief and multilevel binary gratings by using the enhanced transmittance matrix approach . The system considered in this paper contains multiple layers, and each layer is uniform in the z direction despite the layer containing the nanowire pairs. In order to apply the RCWA, the layer containing the nanowire pairs is divided into 4 sub-layers separated by the horizontal interfaces of the nanowires indicated by the dashed lines in Fig. 1(a), so that each sub-layer is uniform in the z direction and periodic along the x direction. Thus, our system is also a multilayer grating structure, and it is very feasible and efficient to analyze the system by using the RCWA method. In the RCWA method, the electromagnetic fields and dielectric functions in each layer are expanded as a sum of spatial harmonics, and the electromagnetic fields satisfy Maxwell’s equation within each layer. The overall problem is solved by matching boundary conditions at each of the interfaces.
Before analyzing the plasmon mode interactions in the hybrid structure, we first calculate the reflection spectra of both the bare DBR structure and the Ag-DBR structure without embedding Au nanowires, as shown in Fig. 1(b). Unless otherwise stated, here and in the following, the thickness of spacer layer is taken asand the incident light is normal incidence and TM polarized. It is clearly shown that the bare DBR structure exhibits a bandgap from 880 nm to 1200 nm. When a 20-nm-thick Ag film is deposited onto the DBR structure, one pronounced reflection dip appears in the bandgap, which arises from the excitation of the TP formed at the interface between the Ag film and the DBR structure. Compared to surface plasmons which are TM polarized nonradiative modes, TPs can be formed for both TE and TM polarizations, and are characterized by an in-plane parabolic dispersion inside the light cone. Therefore, the TPs can be produced by direct optical excitation, and their linewidth is also one order of magnitude narrower than that of surface plasmons. Thus, it is very advantageous and straightforward to enhance the magnetic responses of gold nanowire pairs through coupling with TPs. Figure 1(c) shows the magnetic field intensity distribution of the TP mode along the z direction of the Ag-DBR structure (normalized by incident magnetic field intensity). It is observed that the magnetic field is enhanced greatly in the first SiO2 layer from the Ag film [position indicated by the dash line in Fig. 1(c)]. That is the reason why the gold nanowire pair array is embedded in the first SiO2 layer as shown in Fig. 1(a), so that the excitation of the TP mode results in a large increase of the magnetic field in the proximity of the gold nanowire pairs, which extremely benefits the excitation of the MP.
3. The plasmon spectra and mode nature
Figure 2(a) presents the normal-incidence reflection spectrum while a gold nanowire pair array is embedded in the bare DBR structure (without Ag film), and that of the bare DBR structure without embedding Au nanowire pairs is also shown in dashed line as a comparison. A broad reflection dip (labeled as dip 1) centered at is observed in the bandgap while a gold nanowire pair array is embedded in the bare DBR structure (solid line), which is attributed to the excitation of the MP of the gold nanowire pairs. The corresponding magnetic and electric field intensity distributions for the MP (normalized by the incident field intensity) are shown in Fig. 2(b) and 2(c), respectively. It is apparently shown that the magnetic fields are highly confined in the region between the nanowires, and the electric fields are mainly concentrated near the ends of the nanowires, which are exactly the characteristics of the MP of the nanowire pairs . The resonance frequency of the MP is determined by both the geometrical parameters of the nanowires and the distance between the two nanowires of the nanowire pairs. This mode splits into two modes (labeled as dip 2 and dip 3) with narrow linewidths while the gold nanowire pair array is embedded in an Ag-DBR structure, as shown in Fig. 2(d).
To get a deeper insight into the nature of the split modes, the normalized magnetic field distributions for the resonances at dip 2 and dip 3 are shown in Fig. 2(e) and 2(f), respectively. For the resonance at dip2 (centered at), the magnetic field distribution is almost the same as that of the resonance at dip1, but the magnetic fields between the nanowires become weaker due to the presence of the Ag film. On the contrary, the magnetic fields between the nanowires are enhanced greatly when the resonance at dip 3 (centered at) is excited, and a nearly 2.5 times enhancement is achieved compared with those in the nanowire pairs embedded in a bare DBR structure as shown in Fig. 2(b). In addition, the magnetic fields are also enhanced near the Ag/DBR structure interface, which is the signature of TP. Therefore, we deduce that the resonance at dip 3 is a kind of hybridized mode, and the large optical magnetic field enhancement achieved in the nanowire pairs is attributed to the coupling between the MP and the TP. This will be further confirmed in the following.
4. The dependence of reflection on spacer layer thicknesses
As is well known, the resonance wavelengths of the TP mode periodically depend on the spacer layer thickness . Hence, in order to investigate the coupling between the MP mode and the TP mode, we present in Fig. 3 the contour plots of the reflectivity versus wavelength and spacer layer thicknesses ds for the considered structure, illuminated by TE and TM polarized light at normal incidence. For TE polarization [see Fig. 3(a)], the MP of the nanowire pairs cannot be excited, and only multiple TPs are observed. The resonance wavelengths of the unperturbed TPs periodically decrease with increasing spacer layer thickness. While for TM polarization [see Fig. 3(b)], both the MP and the TP are excited, and large anticrossings of the modes are observed each time when the resonance wavelength of the TP approaches that of the MP, which is a signature of the strong coupling between the TP and the MP. In the figure, the resonant positions of the unperturbed TPs [the positions of reflection dips in Fig. 3(a)] and those of the MP are depicted as dashed lines for comparison. Away from the strong coupling regime, the positions of the resonances in the coupled system approximately approach to those of the unperturbed modes. Especially, a Rabi-type splitting as large as 290 meV is achieved for the 1st order TP (indicated by the black arrow). This value is quite large and corresponds to a splitting/resonance energy ratio of 2.2:10, which is two times that achieved in the system with the gold nanowire pairs embedded in a microcavity . We can attribute the larger splitting mainly to the smaller confinement length of the TP with respect to the microcavity mode, because of its extremely sharp decay of magnetic field in the metal. The effective coupling of the TP to the MP is due to the strong magnetic field antinodes of the TP near the nanowire pairs.
In order to better understand the coupling mechanism, a coupled oscillator model  can be used, which describes the coupling behavior of the original “bare” modes in the system. Beside the MP, multi-order TPs exist in the system considered here, and they do not interact directly with each other. From this point, the Hamiltonian of the system at the center of the first Brillouin zone can be written as follows,Eq. (1). In order to get good agreement with the calculated results in the present spacer layer thickness range, seven order TP modes are included in the Eq. (1). The eigenvalues of the Hamiltonian represent the new eigenenergies of the coupled system, and their dependence on the spacer layer thickness is demonstrated by the solid red lines in Fig. 4 . To draw these lines, we have used the energies of the unperturbed TP extracted from Fig. 3(a) and the fitted values: EMP, V1, V2, V3, V4, V5, V6, V7 = 1294, 145, 110, 100, 90, 80, 100, 100, 100 meV. In this case, the open black circles in Fig. 4 summarize the dependence of the positions of the reflection dips in Fig. 3(b) on the ds. The energies of the unperturbed TPs and the MP mode are denoted by the dashed lines. Obviously, the resonance positions predicted from the coupled oscillator model are in good agreement with the calculated reflection spectra dip positions, and reproduce very well the anticrossing behavior in the dispersion diagram. This further confirms that the anticrossing behavior is the result of the TP–MP coupling, and this coupling results in two branches of hybridized MP modes. From the former investigation, we know these hybridized MP modes have the nature of TP and MP simultaneously. Due to the formation of the hybridized MP mode, the light can be more effectively coupled to the MP of the nanowire pairs, leading to a large magnetic field enhancement at optical frequencies.
4. The effect of nanowire width on the coupling
As is well know, the plasmon resonances of the nanowire pairs can be shifted by tuning the nanowire widths. In this section, we examine the effect of nanowire width on the coupling. Figures 5(a) and 5(b) show the contour plots of the reflectivity versus wavelength and nanowire width for the gold nanowire pairs array embedded in a bare DBR structure and in an Ag-DBR structure, respectively. In both cases, the thickness of spacer layer is taken as, and the nanowire width is tuned from to 150 nm. When the gold nanowire pairs array is embedded in a bare DBR structure (without the Ag film), the TPs can’t be formed, and only two branches of plasmon modes of the nanowire pairs are observed in Fig. 5(a). From their field distributions (not shown here), it can be deduced that the branch indicated by the dashed line originates from excitation of the magnetic (antisymmetric) plasmon mode of nanowire pairs, while the branch indicated by the dashed-dot line originates from excitation of electric (symmetric) plasmon mode of nanowire pairs. These two branch of plasmon modes overlap near, and only the magnetic plasmon mode can be excited at this nanowire width. With the increase of the nanowire width, the resonance wavelength of the MP mode can be tuned through the DBR band gap.
On the other hand, when the gold nanowire pairs array is embedded in an Ag-DBR structure, both the plasmon resonances of nanowire pairs and the TP are excited, the contour plot of the reflectivity versus wavelength and nanowire width is shown in Fig. 5(b). The chosen spacer layer thickness allows the excitation of TP at resonance wavelength of , indicated by the dot line in Fig. 5(b). With the increase of the nanowire width, the resonance wavelength of the MP mode increases gradually, and the MP reflection spectrum exhibits an anticrossing behavior with that of the TP mode resonance when the resonance wavelength of the MP approaches that of the TP. This finding further confirms the happening of strong coupling between TP and MP.
In conclusion, we have demonstrated that changing the spacer layer thickness or tuning the nanowire width of the metal–DBR structure enclosing a gold nanowire pair array allow to control the coupling between MPs and TPs. At an appropriate thickness of the spacer layer, strong coupling between MPs and TPs results in large anticrossings in the reflection spectrum, and leads to a narrow-band hybridized MP mode with a large magnetic field enhancement at optical frequencies. The anticrossing behavior of the resonances is also well reproduced by the coupled oscillator model. These findings have potential applications in fields employing enhanced magnetic field, such as magnetic nonlinearity and magnetic sensors.
This work was supported by the National Natural Science Foundation of China (grant 60808027, 61107044, 11176009) and the Ministry of Science and Technology (MOST) of China (973 project 2007CB307001).
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