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We present a description of a multiple excitation of localized surface plasmons (LSPs) from an Au nanoparticle (NP) array-based ridge waveguide to create a small optical spot size with an extremely strong intensity. Using a numerical finite-difference time-domain method, we find that the optical intensity of the ridge waveguide with an Au NP array is about 700% higher than that of a simple ridge waveguide. Moreover, the spacing between the NPs plays an important role in the multiple excitation of LSPs. The spot size, calculated at FWHM, is 10 nm × 10 nm at a distance of 5 nm from the exit plane.
©2010 Optical Society of America
1. Introduction
A nanometer-scale optical spot with high intensity beyond the diffraction limit has many applications such as heat-assisted magnetic recording [1–3], optical data storage [4,5], and mask-less nanolithography [6]. Recently, some metallic apertures, such as bowtie antennas [7–10], H- or I- shaped apertures [8,11,12], and C-shaped apertures [13–15] have been studied based on their potential to achieve a high power throughput. (The C-shaped aperture has been described as a “ridge waveguide” [16] in the literature, and we will refer to it by this name.) These next-generation apertures have a high optical transmission and a small spot size, attributes that are usually associated with localized surface plasmon resonance (LSPR) [17–19]. In LSPR, the charge density oscillations are confined to metallic nanostructures [18], and an individual nanoparticle (NP) of a specific metal, such as gold (Au) or silver (Ag), can induce strong field enhancement in the optical wavelength range [20–23]. It is possible to induce the multiple excitation of LSPs using an LSPR-based nanoaperture and a coupled plasmonic mode-based NP array. In this paper, to verify the generation of a very small optical spot with an extremely strong intensity via the multiple excitation of LSPs, we provide the design of a novel structure composed of a ridge waveguide and a dielectric layer with an Au NP array, shown in Fig. 1(a) . Simple ridge waveguides were chosen because they are widely used in microwave and antenna systems due to their low cutoff frequency, wide bandwidth, and low impedance. The performance of nanoscale ridge waveguides has been reported by Shi et al. [13] and Itagi et al. [14,15]. We use the numerical finite-difference time-domain (FDTD) [24,25] method to study the physical mechanisms and resonance conditions that allow for the multiple excitation of LSPs.
Fig. 1 A ridge waveguide with a dielectric layer and an Au NP array-deposited on a dielectric, with (a) a schematic and (b) the overall dimensions illustrated. The dielectric layer that hosts the Au NP array is located under the bottom surface of the ridge waveguide.
2. Design of an Au nanoparticle array-based ridge waveguide
Figure 1(a) shows a schematic of the ridge waveguide with an Au NP array. The metal plate of the ridge waveguide is also assumed to be Au. In general, Au has a large optical conductivity for red and infrared light and can support surface plasmon effects in this wavelength range. It is also chemically inert, and therefore more likely to adequately handle environmental rigors. Although silver (Ag) has a larger optical conductivity than Au throughout most of the visible range, and thus might be expected to provide even better optical results, its chemical reactivity is a problem. Aluminum (Al) might also be considered at the blue end of the spectrum, but its low melting point is a drawback [26]. Therefore, we choose Au to be the material for the ridge waveguide and nanoparticle array. The overall dimensions of this structure are shown in Fig. 1(b). The array of Au NPs is deposited onto the dielectric layer with refractive index = 1.5 and the length (l) of each Au NPs is 10 nm. The 5 nm thick dielectric layer is attached to the bottom surface of the ridge waveguide. The metal surface is normally illuminated by an incident plane wave that is linearly polarized in the x-direction. The wavelength of the incident light is 780 nm, and the incident |E|2 intensity is 1.
To obtain good resolution in numerical simulations, the size of spatial discretization should be < λ/10 (Taflove & Hagness, 2000 [25], ). A spatial discretization scheme chosen according to this criterion not only provides better spatial resolution, but also minimizes the numerical dispersion error. In this simulation work, we use a commercial software package XFDTD 6.5 [27] from Remcom. The dimension of each cell is chosen to be 5 × 5 × 5 nm3, and the discretized Maxwell curl equation is solved in the near-field region below the aperture with good spatial resolution and low numerical dispersion error. The stability condition that relates the spatial and temporal step size s is expressed as
where c represents the velocity of light and Δx, ∆y, and ∆z represent the spatial discretizations in the x, y, and z directions. The electromagnetic fields are calculated in each cell in both space and time for each time step, until the steady state is reached. In the case of a sinusoidal source, such as the one used in this study, the steady state is reached when all scattered fields vary sinusoidally in time. The time step calculated according to the stability criterion of the FDTD algorithm is 9.63 × 10−18 s. The total number of time step is 7000, enough to maintain numerical stability and to sufficiently approach the steady state. Fields are observed at a distance of 5 nm below the exit plane. In addition, a second-order stabilized Liao [28] absorbing boundary condition is used at the six sides of the simulation volume.At optical frequencies, real metals, such as Au and Ag, have complex permittivities that are strongly dependent on the excitation frequency. In order to treat real metals accurately, a modified Debye model [24] is used to describe the frequency dependence of the complex relative permittivity, which is given by
where εs represents the static permittivity, εα is the infinite frequency permittivity, τ is the relaxation time, and σ is the conductivity. Using the experimental refractive index data for Au [29] at a wavelength of 780nm, the parameters of the modified Debye model are determined to be εs = −5349.2, εα = 1, τ = 6.0315e-15 s, and σ = 7854100 s/m.3. Simulation Results
It is well known that Fourier optics is not adequate for analyzing optical properties in real metals, due to finite skin depth, film thickness and surface plasmon effects. Accordingly, this work utilized the numerical FDTD simulation method. Figure 2 shows a comparison of the near-field |E|2 intensity distribution for a simple ridge waveguide and the nanoparticle array-based ridge waveguide.
Fig. 2 The near-field |E|2 intensity distribution of a ridge waveguide (a) in the x-y plane at a distance of 5 nm from the exit plane, (b) in the y-z plane and (c) in the z-x plane; the near-field |E|2 intensity distribution of a ridge waveguide with an Au NP array (d) in the x-y plane at a distance of 5 nm from the exit plane, (e) in the y-z plane and (f) in the z-x plane.
Panels (a), (b), and (c) in Fig. 2 show the |E|2 intensity distribution of the simple ridge waveguide in the x-y plane at a distance of 5 nm from the structure, the y-z plane, and z-x plane, respectively. Panels (d), (e), and (f) in Fig. 2 show the |E|2 intensity distribution of the ridge waveguide with an Au NP array in the x-y plane at a distance of 5 nm from the structure, the y-z plane, and z-x plane, respectively. Even though a nanoscale ridge waveguide was reported to form a well-confined spot with an FWHM diameter of ~100 nm at a distance of 48 nm downstream in the simulation work by Shi et al. [13], two local field enhancements are generated at the corners of the sides of the ridge near the exit, as shown in Fig. 2(a). This phenomenon is a result of LSPR, and the maximum near-field |E|2 intensity of the ridge waveguide is ~340 times higher than the incident light intensity. However, a dielectric layer with an Au NP array can be placed under the bottom surface of only one corner of the ridge waveguide to produce a single optical spot that is much smaller and stronger, as shown in Fig. 1(b).
When the incident light illuminates the top surface of a ridge waveguide with an Au NP array and the electromagnetic (EM) wave propagates along the surface of the ridge, there is strong near-field enhancement at the corners of the ridge, due to LSPR. Then, the strong local fields generated at the corners of the ridge are transmitted to the dielectric layer containing the Au NP array, and the fields are re-amplified between the closely spaced Au NPs such that extremely localized field enhancement occurs. The Au NP array has two possible functions: the re-amplification of surface plasmon resonance at the ridge corner and control over the optical spot diameter using the spacing between Au NPs. The maximum near-field |E|2 intensity of the ridge waveguide with an Au NP array, observed at a distance of 5 nm from the exit plane as shown in Fig. 2(d), is ~2780 times higher than the incident light |E|2 intensity. Figure 2(e) and 2(f) show the near-field |E|2 intensity distribution in the y-z plane and in the z-x plane, respectively. The FWHM of the spot size is 10 nm × 105 nm for the simple ridge waveguide and 10 nm × 10 nm for the ridge waveguide with an Au NP array, as shown in Fig. 3 .
Fig. 3 Maximum |E|2 intensity profile in cross-sections of the two ridge waveguides at a distance of 5 nm from the exit plane; the intensity of the ridge waveguide with the Au NP array is ~700% higher than that of a simple ridge waveguide. The spot size, calculated at FWHM, is 10 nm × 105 nm for the simple ridge waveguide and 10 nm × 10 nm for the ridge waveguide with an Au NP array.
Therefore, through comparison of the simulation results with a ridge waveguide and an Au NP array-based ridge waveguide, we can speculate that the field enhancement based on LSPR at the corners of the ridge is re-excited between closely spaced Au NPs, so that a stronger field is generated at the gap between adjacent Au NPs, a phenomenon we call the multiple excitation of LSPs. We consider this phenomenon to be primarily related with to the coupled plasmonic modes [22,23,30] of the Au NP array, and we discuss the details in section 4.
4. Mechanism of multiple excitation of localized surface plasmon resonance
Metallic particles embedded in a dielectric medium can support SPs [31]. It is well known that very small Au particles exhibit a SP resonance in the visible range. The EM fields are generally most intense at the edges of a metallic structure because the charge density accumulates there. Therefore, in the case of a ridge waveguide with a metallic NP array, the transmitted EM wave inside the ridge waveguide can concentrate the charge density at the ridge corners so that the SPs propagate along the narrow gap region between the Au NPs and excite SPs on the side walls of each pair of adjacent Au NPs. Therefore, we can expect that a high near-field intensity and small spot size will be created near the bottom of Au NPs.
Figure 4 shows the electric field distribution and the power density versus the spacing between the Au NPs. The space, s, between Au NPs is increased from 0 nm to 30 nm in steps of 5 nm. We also define the ratio r = s / l. The spacing between the Au NPs is important for creating a small optical spot with an extremely strong intensity. More specifically, because the near-field interaction between closely spaced metallic NPs induces a coupled plasmon mode [22,23,30,32], the EM field at the corners of the ridge can be re-amplified by the metallic particles. Qualitatively different results were obtained as the spacing between Au NPs was increased, due to changes in resonance. Au NPs in contact with each other (r = 0) do not allow the incident field to transmit in the dielectric because the Au NP array obstructs the propagating field that is enhanced at the corners of the ridge; see Fig. 4(a). At a spacing of r = 0.5, after the enhanced EM field at the corners of the ridge is transmitted to the space between the Au NPs, the near-field interaction between closely spaced Au NPs produces an electric dipole field. Then, the EM energy is transported along adjacent NPs in the longitudinal direction, as shown in Fig. 4(b), such that a coupled electric dipole field is created; this type of NP chain [30] or plasmonic waveguide [23,33,34] has been previously reported. It is difficult to make a single optical spot with a small size and a strong transmission because the electric charge is dispersive. At a spacing of r = 1, maximum power density can be obtained because the enhanced field is not transported along the NP array but confined near the local NPs; see Fig. 4(c). Lastly, as the spacing is increased further (r > 1), the power density decreases because the near-field interaction of coupled NPs is diminished; see Fig. 4(d).
Fig. 4 The near-field |E|2 intensity distribution in the y-z plane versus the space between Au NPs for (a) r = 0, (b) r = 0.5, and (c) r = 1, where r is the ratio between the gap spacing and the size of Au NPs. (d) The power density versus NP spacing at a distance of 5 nm from the exit plane.
Physically, the field enhancement process in a ridge waveguide with an Au NP array can be divided into three SP modes: the dispersive SP mode [see Fig. 5(a) ], overlapping SP mode [see Fig. 5(b)], and localized SP mode [see Fig. 5(c)]. Figure 5 schematically illustrates the surface charge displacements and the electric fields associated with the three SP modes. For the dispersive SP mode (r = 0.5), the EM energy transport along chains of closely spaced metal NPs relies on the near-field electrodynamics interaction between metal particles, which sets up coupled dipole or plasmon modes [32,35], as shown in Fig. 5(a). This field behavior is analogous to the process of resonant energy transfer, which is observed in systems that contain closely spaced, optically excited atoms, molecules, or semiconductor nanocrystals [34,36–38]. Therefore, the dispersive SP mode cannot generate a single strong optical spot but creates many optical spots, due to EM energy transport as shown in Fig. 4(b). For the overlapping SP mode (r = 1), because the EM wave is transmitted along the gap inside the ridge waveguide and the electrical charge is accumulated at the corners of the ridge, strong fields are generated at the local region. The strong field region covers two Au NPs, just at the bottom of the ridge. After travelling through the narrow gap between Au NPs at the bottom of the ridge, the EM energy is transported in the y-direction, and generates coupled plasmonic modes between pairs of adjacent Au NPs, as shown in Fig. 5(b). However, the EM energy cannot be transported along the chain of Au NP because the strong field generated at the ridge corner only exists in the local region. As shown in Fig. 5(b), a coupled plasmonic mode is generated by the overlapping strong field region between two adjacent Au NPs. Our simulation work demonstrates that this coupling effect generates a single strong optical spot when the resonant condition r = 1 is satisfied, and that the enhanced field at the sides of two closely spaced Au NPs supports re-excitation of SPR. (We consider this re-excitation of the SPR to be the multiple excitation of LSP.) Lastly, in the localized SP mode, adjacent metallic particles are decoupled due to the increased gap size between them, and charge density accumulates and remains on each side of each Au NP. No overlapping strong fields exist, and a single strong optical spot cannot be induced. A similar phenomenon was reported in reference [39], in which the amplitude was observed to decrease as the space between coupled metallic particles increased.
Fig. 5 Sketch to illustrate the charge displacement at the metal surface and associated electric fields of SP modes in the y-z plane: (a) Dispersive SP mode, (b) Overlapping SP mode, and (c) Localized SP mode.
Figure 6 shows the near-field |E|2 distribution in the x-y distribution and power density versus the thickness of the dielectric layer. The maximum power density is generated when the bottom surface of the ridge waveguide is just in contact with the Au NPs, i.e., the thickness t of the dielectric layer is 0 nm as shown in Fig. 6(a): (b) t = 5 nm, (c) t = 10 nm, (d) t = 15 nm, (e) t = 20 nm, and (f) t = 35 nm. This configuration optimizes the power density because the local field enhancement at the bottom surface of the ridges’ corners, which is exponentially attenuated with distance, is in the near-field region of the Au NPs. Figure 6(g) demonstrates that this is the configuration for optimal enhancement via multiple excitation of an LSP.
Fig. 6 The near-field |E|2 intensity distribution in the x-y plane at a distance of 5 nm from exit plane versus the thickness of dielectric layer: (a) t = 0 nm, (b) t = 5 nm, (c) t = 10 nm, (d) t = 15 nm, (e) t = 20 nm, and (f) t = 25 nm. (g) The power density at 5 nm away from the exit plane versus thickness of dielectric layer.
In this paragraph, we briefly discuss the effects of different sizes, different numbers of NPs, and the differently shaped nanoapertures in the Au NP array. Since the main features of LSPR-based-resonance are related to the characteristics of the cluster, e.g., size, shape, and spatial distribution [40], changing the size of the NPs changes the resonant condition of r = 1. (i.e., the array will no longer satisfy the resonant condition r = 1). In addition, when the number of NPs increases and many NPs are located on the bottom of the ridge waveguide, optical spots are induced by electric field enhancements at the corners of each NP and their |E|2 intensities are weaker than the |E|2 intensity of a single optical spot. As the number of NPs in our simulation is decreased (excepting the case of a single NP), similar phenomenon like the strong field generation based on multiple excitation of LSP is observed. Furthermore, we simulated the differently shaped nanoapertures with Au NP array, including conventional square-shaped nanoapertures, H- or I-shaped nanoapertures, and Bow-tie shaped nanoapertures, and found that the results (not shown here) regarding the multiple excitation of LSPR were similar to those reported here.
5. Conclusion
In summary, the aim of this work was to verify that the multiple excitation of LSPs could produce a very small optical spot with an extremely strong intensity. It was found that, with the proper Au NP spacing, a ridge waveguide combined with an Au NP array contributes to both the near-field collimation and the field intensity due to the multiple excitation of LSPs. Moreover, we found that the thickness of the dielectric layer plays an important role in the multiple excitation of an LSP. In optimal conditions at resonance, the |E|2 intensity of a ridge waveguide with an Au NP array is ~700% higher than that of a simple ridge waveguide, and the FWHM spot size is 10 nm × 10 nm at a distance of 5 nm from the exit plane.
Acknowledgments
This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean government (MEST) (No. 2009-0086278).
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