1. Introduction

Localized surface plasmons (LSP’s) due to spatially confined free electron oscillations dominate the optical response of nanosized noble metal structures and are fundamental ingredients in the emerging field of nano-photonics [1]–[3]. Nanoscale plasmonic systems are also crucial for the development of a new class of engineered optical materials known as metamaterials, of which so-called left-handed media that exhibit negative refraction have emerged as the most prominent possible application [4]. Negative refraction has recently been investigated for frequencies in the optical range [5]–[7] and the phenomenon has been shown to be closely connected to the resonant magnetic response of the plasmonic nanostructures [8]. There are, however, many challenges in designing a metamaterial that exhibits both negative dielectric permittivity and negative magnetic permeability, leading to a negative real part of the complex refractive index, in particular for the visible spectral range. Since negative permittivity is inherent to plasmonic structures at resonance, efforts have been focused on developing structures with a strong magnetic response. This has been achieved in the terahertz frequency regime using split-ring resonators (SRR) [9]–[12], but recently two novel types of lithographically produced nanostructure assemblies - planar pairs of nanopillars and vertical stacks of nanorods - have been shown to exhibit resonant magnetic properties also at visible frequencies. In the former case, pairs of gold nanopillars arranged in a planar array show a strong magnetic field enhancement in the region between the nanopillars as a consequence of locally induced electric currents associated with an antisymmetric plasmon resonance [8]. In the latter case, an array of gold nanowire pairs on a glass substrate support a similar plasmonic excitation, which results in a virtual circular current that induces a magnetic field enhancement effect [7], [13]. However, the structures studied so far have complicated morphologies that require sophisticated nanofabrication tools. In the present communication, we investigate magnetic and electric field- enhancement effects at visible frequencies in a single vertically coupled plasmonic system composed of a pair of gold nanodisks illuminated in the end-fire configuration. This system has recently been realized experimentally through a self-assembly technique suitable for mass production of nanoplasmonic structures [14]. According to the present analysis, this type of sandwich structure could be a suitable basis for future development of left-handed metamaterials for the optical range.

2. Basic concepts and calculation details

In a simple quasistatic picture, a spherical metal nanoparticle illuminated by a plane wave E _inc, can be represented by an induced electrical point dipole. The dipole moment is P =α(ω)E _inc, where α(ω) is the frequency-dependent particle polarizability [15]–[16]. By applying the Clausius-Mossotti relation and a simple Drude model for the dielectric function of the metal particle, one obtains a LSP resonance frequency ${\omega }_0=\frac{{\omega }_p}{\sqrt{3}}$. For a pair of spheres separated by a distance d that is small compared to the resonance wavelength, and oriented along a line parallel to the incident wave vector, i.e. an end-fire configuration, one instead obtains two hybridized LSP modes with resonance frequencies ${\omega }_{\pm }={\omega }_0\sqrt{1\pm {\left(\frac{a}{d}\right)}^3}$, where a is the sphere radius. The pair interaction is here only due to the electrostatic dipolar r ^-3-term; but for a real system, one also need to consider the $\frac{1}{r}$ radiative dipolar term (dominant for $d>~\frac{\lambda }{2\pi }$). For very short distances ($d\ll ~\frac{\lambda }{20}$), all multipolar fields of the system contribute to the interaction [17]. The two resonant modes correspond to in-phase and out-of-phase dipolar coupling, respectively. The high-energy symmetric mode will have an overall electric dipole character, whereas the low-energy asymmetric mode generates a magnetic moment M = I S_A, where I is the virtual current due to the induced dipole oscillations and S_A is the current loop cross-section. The latter results in an enhancement of the magnetic field in the region between the two nanoparticles. Because the magnetic dipole moment is directed perpendicular to the induced electric dipole, it can couple to the magnetic part of the incident electromagnetic field. Thus, a pair of metal spheres illuminated in the end-fire configuration can be expected to have both an electric and a magnetic resonant optical response. Fig. 1 illustrates this concept for a pair of disks in a sandwich configuration. The fundamental LSP mode of each particle is then no longer three-fold degenerate. Instead, we expect two elementary dipole resonances, associated with the long and short axes of the disk. The former will dominate the optical response in the present configuration, and its precise position depends on the absolute size and the aspect ratio of the disk [18]. However, realistic estimates of the electromagnetic response of a sandwich in the size range of relevance to experiments require a treatment beyond the electrostatic point dipole approximation.

 

Fig. 1. Schematic illustration of the studied system. Two closely spaced Au nanodisks (separation d) with degenerate LSP resonance frequencies ω ₀ interact vertically in the “end-fire” illumination configuration. The coupling results in two hybridized LSP modes associated with a net electric dipole moment (ω₊) and a net magnetic dipole moment (ω₊).

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In the present work, we investigate the system illustrated in Fig.1 using the dispersive finite-difference time-domain (D-FDTD) method, which numerically solves the full set of Maxwell’s equations in dispersive media [19]–[20]. We confine ourself to equally sized circularly symmetric gold disks of diameter 88 nm and thickness 24 nm, i.e. dimensions that are accessible through a range of lithography techniques. The disks are situated in vacuum, ε_r = 1.0. The metal is described by a complex dielectric function ε_r(ω), hence no charges or currents appear explicitly in Maxwell’s equations. Several approaches, such as the recursive convolution (RC), auxiliary differential equation (ADE), and Z-transform methods, have been developed to incorporate frequency dispersion into FDTD [20], [21], [22]. In this work, we fitted the experimental dielectric function of Au [23] with a sum of Lorentz-Drude functions [15] that together closely approximates the experimental data in the wavelength range 300-1400 nm. All materials were assumed to be non-magnetic, so that μ = μ₀. We used a uniform cell size with dimensions Δx = Δy = Δz ranging from 2 nm up to 4 nm. The temporal cell size for Δx = 3 nm is Δt = 5 × 10^-3 fs, which satisfies the Courant-Friedrich-Levy (CFL) stability condition[20]. Perfectly matched layers (PML’s), which constitutes an effective absorbing boundary condition (ABC), were used to truncate the computational domain as a non-physical absorbing boundary [19]. The structure is excited by a Gaussian pulse in the form of a modulated plane wave. In the assumed coordinate system, the incident plane wave has E_x and H_y components and propagates in the z-direction, see Fig. 1. After the decay of the transient state, we extract the whole spectral response at a given point in space using a fast-Fourier transform (FFT). We found good agreement between the implemented FDTD code and the results of Mie theory calculations for Au spheres, which verifies the accuracy of the method.

 

Fig. 2. Electric (E_x) and magnetic (H_y) fields around Au nanodisks illuminated in the end-fire configuration (disk diameters - 88 nm, thicknesses - 24 nm). (a) Field enhancement plots for d =300 nm together with intensity enhancement spectra at two different locations (lower graph; blue and red curves for magnetic and electric fields, respectively). (b) Same as in (a), but for an inter-disk separation of d =32 nm.

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3. Results and discussion

For a sandwich structure illuminated in the end-fire configuration, the electric field enhancement is not expected to be localized in between the nanodisks, as is the case for the traditional illumination configuration [17]. Rather, following a conventional dipolar excitation picture, a strong enhancement at the sides of the disks is expected. However, as discussed above, the phase variation of the induced electric fields in the two nanodisks is expected to result in two separate dipole resonances, the in-phase (symmetric) and the anti-phase (asymmetric) modes. Fig. 2 illustrates the emergence of these two modes through electric (E_x) and magnetic (H_y) near-field distributions for non-interacting and strongly interacting nanodisks. For a center-to-center spacing of d =300 nm (Fig. 2a, plotted at 600 nm), the electric and magnetic near-fields of the two nanodisks do not overlap and the field distribution strongly resembles the case of two isolated disks. Intensity enhancement spectra at two different locations in the vicinity of the nanodisks (A (side) and B (top), with lower index indicating either electric field (E) or magnetic field (H)) have a single resonance, as expected for two identical non-interacting particles. The magnetic field intensity-enhancement is small, and only reaches a value of ~8 at the most favorable location at the upper side of the disks (Fig. 2a, lower graph). Decreasing the inter-disk separation in the sandwich down to 32 nm, corresponding to a surface separation of 8 nm, leads to a striking change in the resonant behavior of the structure. The intensity enhancement spectra, plotted for the same points as for the non-interacting disks, now reveal the two aforementioned in-phase and out-of-phase (high- and low-energy, respectively) resonant modes (Fig. 2b, lower graph). The induced magnetic field is dramatically enhanced in the region between the nanodisks (Fig. 2b, left plot), i.e. at the center of the virtual current loop associated with the out-of-phase resonance (see Fig. 1). The magnetic intensity enhancement factor reaches ~ 60 at λ = 700 nm, which is similar to the calculated value for the Au nanopillar pairs recently investigated by Grigorenko et. al. [8]. The electric field also shows a sizable enhancement at λ = 700 nm (Fig. 2b, right plot), but this effect is concentrated to the region around edges of the upper disk. The fact that the E-field distribution is asymmetric in the xz-plane indicates that the out-of-phase mode is activated mainly through finite-size (retardation) effects.

 

Fig. 3. Electric-field phase variation along a line parallel to the z-axis, 4 nm from the side of sandwich in the xy-plane, for the two resonance wavelengths at 575 nm - blue curve, at 700 nm - red curve. The nanodisk position are shown schematically in the background.

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The relative orientation of the induced dipoles in the two disks can be extracted from the phase plot shown in Fig. 3. While the phase of the E-field associated with the high-energy mode (575 nm) stays constant across both nanodisks (blue curve), the corresponding plot for the low-energy mode (700 nm) changes sign from the upper to the lower disk in the sandwich. This behavior thus allows us to conclusively associate the high-energy mode with an in-phase coupled dipolar oscillation and the low-energy mode with an anti-phase oscillation, in agreement with the schematics in Fig. 1. Consequently, the strong magnetic field enhancement in between the disks, characteristic for the low-energy resonant mode (c.f., Fig. 2b) can be rationalized in terms of a virtual current-loop in the sandwich.

The combined resonant response of the electric and magnetic fields thus produces two spectrally resolved hybridized modes in the near-field around the nanosandwich, but the modes can also be detected in far-field scattering simulations. Fig. 4a shows far-field scattering spectra for the same nanodisk configuration (d = 32 nm) as before, along the three coordinate axis of the system. The z-axis coincides with the propagation direction of the incoming (and forward-scattered) radiation, and this spectrum is therefore, according to the optical theorem, equivalent to an extinction spectrum. The x- and y-axes are instead collinear with the electric and magnetic dipoles in the structure, respectively. Hence, we expect that the high-energy electric dipole resonance should contribute strongly to the scattering along y and z, but the intensity should be weak along the x-axis, i.e. along the electric dipole moment. As seen in Fig. 4a, this is indeed the case. By the same argument, the low-energy magnetic resonance contribution should be strong along x and z, but vanish along the y-axis. This is in principle the case, although the scattering intensity along x is obviously much weaker than along the z-axis. This is due to the fact that the virtual current loop associated with the low energy resonance is not circular, as for a classical magnetic dipole, but strongly compressed along the z-axis.

 

Fig. 4. (a) Differential far-field scattering spectra along the three coordinate axis for the same sandwich structure as in Fig. 2b). (b) Spectral peak position and corresponding magnetic intensity enhancement factor at the mid point between the disks vs. inter-disk separation. (c) Magnetic-field intensity-enhancement vs. peak-shift relative to the non-hybridized system.

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It is clear from the preceding results that the strength of the magnetic response of the nanosandwich can be quantified through two important parameters: the magnetic field intensity-enhancement and the spectral position of the out-of-phase mode. Fig. 4b shows the evolution of these parameters with changing inter-disks separation. The data show that the spectral position of the maximum magnetic field intensity enhancement in between the nanodisks red shifts from 575 nm to 830 nm when the spacing decreases from 300 nm down to 28 nm. Similarly, the absolute magnitude of the enhancement dramatically increases. The relation between the peak-shift and the intensity-enhancement is found to be roughly linear at small separations, as can be see in Fig. 4c. The trends seen in Fig. 4b,c strongly resemble the behavior of the hybridized in-phase dipole mode of two nanodisks coupled in a planar pair configuration [17], which suggests that the nanosandwich serves as a magnetic analog of the planar dimer system.

4. Summary and Conclusions

We have investigated the optical properties of simple Au nanosandwiches using FDTD simulations and found a strong magnetic response at visible frequencies due to a hybridized dipolar plasmon mode. The advantages of the proposed structure for metamaterials applications and negative refraction are manifold: The footprint of the structure is small compared to previously investigated structures and it can be manufactured though self-assembly [14]. The structure is circularly symmetric, and thus polarization insensitive at normal incidence. Moreover, although not shown here, the plasmonic response of the structure can be tuned over an extended wavelength range in the visible by varying the aspect ratio of the elementary Au disks [18] or by choosing silver instead of gold [17]. Still, to enable a true metamaterial for the visible, one would need to pack a large number of nanosandwiches within a diffraction limited volume of dimension ~ (λ_res/2)³, where λ_res is the resonance wavelength. This requires further shrinking of the elementary structure, which may be difficult experimentally. A large reduction in size may also compromise the magnetic response through the size dependence of the metal dielectric function [16] and through vanishing retardation effects, which are necessary for the activation of the formally dipole forbidden out-of-phase plasmon. Although preliminary results indicate that the latter problem can be circumvented by letting the sizes of the two disks in the sandwich be different, the actual design of a true metamaterial for the visible obviously requires much further work.