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The surface integral equation (SIE) method is utilized to characterize plasmonic waveguide made of two parallel chains of silver nanowires with radius of 25nm fed by a V-shaped funnel at a working wavelength of 600nm. The efficiency of energy transport along the waveguide due to surface plasmonic coupling is investigated for different dimensions and shapes. The opening angle of the V-shaped funnel region for optimum light capturing is included in the investigation as well. A long plasmonic double-chain waveguide of length ∼3.3μm has been analyzed and optimized.
©2007 Optical Society of America
1. Introduction
When a plasmon wave is coupled with a photon at an interface, it is described as surface plasmon polariton (SPP), which offers the possibility of transporting energy with light confined below the diffraction limit [1–4] in nanoscale structures. Therefore, employing SPP excitations in the design of nanophotonic devices, such as plasmonic waveguides that involve optical modes at the interface between metallic and dielectric materials, have attracted much interest in recent years [5–10].
Several 2D studies investigating the propagation of SPP and its confinement within the nanoparticle chain of metallic nanowires (i.e. [11–14]) have been published to date. However, most of them have focused on the propagation characteristics of single-chain metallic nanowires. In this paper we discuss the propagation properties in long plasmonic waveguides consisting of a parallel double-chain of silver nanowire arrays fed by a V-shaped funnel. The main focus here will be on the 2D waveguide characterization excited at a wavelength of 600 nm. The surface integral equation method will be used for this purpose. The main thrust of this investigation is to determine the influence of the waveguide geometry including the funnel-arm, the center-arm and the distance between the two chains on the propagation characteristics of the waveguide. Since the opening angle of the V-shaped funnel is important for optimum light capturing, this parameter is also included in the investigation. Finally, for a given geometry, the structure is illuminate by three different wavelengths to demonstrate that 2D optical waves are well confined and guided along the core of a plasmonic waveguide with length of ∼3.3 μm. As a result of this investigation it is shown, that electric and magnetic fields are separated with the magnetic field strongly enhanced between the two chains and a significant electric field enhancement between the cylinders of each chain.
2. Surface integral equation method
Typically, the dielectric constant of metals at optical wavelengths is a complex number, i.e. εr(ω) = εRe(ω) + jεIm(ω) with a negative real value and strongly dependent on the frequency [15]. Thus, several simulation techniques which are limited to lossless, non-dispersive materials are not applicable to plasmonic devices. In time-domain methods the dispersion properties of metals are approximated by suitable analytical expressions. In most cases the Drude model is invoked to characterize the frequency dependence of the metallic dielectric function [8, 13]. However, the Drude model approximation is only valid over a limited wavelength range. The range of validity of the Drude model can be extended by adding Lorentzian terms to obtain the Lorentz-Drude model. Even though the Lorentz-Drude model extends the range of validity of analytical approximations to metallic dielectric constants it is still only approximately representing the metal dielectric constants obtained from experiment.
In contrast to time domain methods, frequency-domain techniques are single frequency simulations requiring the definition of material dispersion at individual frequencies only. As such these techniques can treat arbitrary material dispersion and also allow direct use of experimental data for the dielectric constants. Therefore, frequency-domain techniques are more suitable candidates for modeling plasmonic devices. The numerical modeling technique used in this paper is based on the surface integral equation (SIE) method. The SIE can be used to accurately study the SPP of arbitrarily shaped homogeneous objects. In contrast to time-domain methods, the SIE method does not require any dispersion model, such as Drude, Lorentz dispersion models or the combination of both to simulate scattering effects in metallic nanostructures. The SIE method can directly use experimental permittivity data for characterizing wave propagation in such structures. The SIE method is formulated by considering the total fields and the boundary conditions at the surface of the object. By expanding the surface currents using pulse basis functions and applying point matching, the SIE method can be converted into a matrix equation which is then solved by a matrix solver [16–18]. The results of the SIE simulations have been validated with high accuracy in comparison with the scattering from a single silver nanowire [19].
3. Model
In order to increase the excitation efficiency of localized surface plasmons in coupled nanowires and efficiently guiding the lightwave in the desired direction, we introduce a funneling array to feed the waveguide. This funneling consists of a double-chain of a silver nanowire array [13]. The exciting light source is included in the simulation to mimic the practical environment as closely as possible [5].
The system under study is shown in Fig. 1. It consists of a V-shaped feed configuration (funnel arm) and the double-chain of silver nanowires. The V-shaped feed opens with an aperture of α (default value set to 90°) and consists of 10 silver nanowires while the two parallel chains consist of 120 silver nanowires. The center-center separations of adjacent cylinders in the funnel-arm and in the center-arm may be different and are represented by dfn, and d, respectively. The two parallel chains are separated by a distance of h (center-center separation of adjacent cylinders). For simplicity, the medium surrounding the waveguide is assumed to be air (εr = 1.0). Throughout the entire paper, the cylindrical silver nanowires have a radius of r = 25 nm. The light source is located between the first cylinders in the funnel region and in the center plane of the double-chain waveguide, as shown in Fig. 1. The structure is illuminated by a TEZ mode, which means that all E-field components are transverse to the z-direction [16], and we assume excitation at 600 nm wavelength. From experimental data [15] the relative permittivity for silver is taken as εr(ω) = -13.98 + j0.95. The computed field intensity is extracted along two planes; one along the horizontal symmetry plane (x-z plane or line AB), that is along the axis of propagation and the other transverse to this plane (z-y plane or line CD) that cuts through the center of the second last cylinder pairs as shown in Fig. 1.
Fig. 1. Plasmonic waveguide using 130 cylindrical silver nanowires. The double-chain waveguide consists of two parts: the funnel-arm and the center-arm. The light source, located at the left, is used to excite the structure with a TEZ-mode.
4. Results and discussion
It is expected that for a given wavelength the confinement of light and its propagation between the parallel double-chain can be significantly enhanced depending on the distance between the parallel chains and the center-center separation of adjacent nanowires. It is also expected that the opening angle of the funnel region will have a direct impact on the amount of light captured. Therefore, we will first investigate the influence on the field distribution as a function of the separation between nanowires in the funnel-arm, dfn, the center-arm, d, and between the double-chain, h. The excitation at different wavelengths will also be investigated as this may reveal some extra information on how the wavelength-dependent permittivity affects the optimum waveguide dimensions. Finally the effect of the opening angle in the funnel region will be determined.
The H-field distribution along the center arm is investigated first for different center-center spacings between the silver nanowires in the funnel region (AgNW), dfn = 2.04r, 2.2r, 2.6r, 3r and 4r, which correspond to actual values of dfn = 51, 55, 65, 75 and 100 nm. Fig. 2 illustrates the relative field intensity along the AB and CD planes (defined in Fig. 1). It is evident that for dfn = 2.2r, 2.6r, 3r and 4r there is little effect on the field distribution along the double-chain region or transverse to the propagation direction at the output of the waveguide [CD, Fig. 2(b)], 3190 nm away from the last pair of cylinders in the double-chain. This changes, however, when two nanowires in the funnel region become too close (or too far), such as for dfn = 2.04r. Figure 2(b) also illustrates that the light is well confined in the center of the double-chain and decays rapidly away from the symmetry plane in the ±y-direction of the waveguide. From both figures a value of dfn = 2.6r can be considered as optimum center-center distance for two adjacent nanowires to obtain maximum energy transport along the double-chain.
Fig. 2. H-field intensity in the double-chain region for different center-center separation of the AgNW in the funnel region with V-shaped aperture α = 90°. (a) Field distribution along the propagation direction in the double-chain region (AB) and (b) field distribution in the cross section of the waveguide just before the last cylinders in the double-chain (CD).
Fig. 3. H-field intensity in the waveguide for different center-center separation of AgNW in the center-arm, with dfn = 2.6r and α = 90°. (a) Field distribution along the propagation direction in the double-chain region (AB) and (b) field distribution in the cross section of the waveguide just before the last cylinders in the double-chain (CD).
Figure 3 illustrates the dependence of the H-field intensity with respect to the center-center separation of AgNW in the center-arm region for a constant dfn = 2.6r. The center-center separation varies over a range of d = 2.04r to 4r. It can be seen from Fig. 3 that, for a spacing of d = 2.04r and d = 4r, the field intensity at the end of the waveguide becomes quite weak with best values obtained for a spacing of d = 2.2r. Judging from the results displayed in Fig. 3 it is evident that the center-center spacing in the double-chain region has a much higher influence on the field distribution than that in the funnel region.
Next we investigate the propagation characteristics under the influence of the distance h separating the parallel chains. The separation between the AgNW in the funnel region and the double-chain region are kept constant at dfn = 2.6r = 65 nm and d = 2.2r = 55 nm, respectively. Figure 4(a) shows that for values of h = 2.04r and 2.2r, the field intensity decays rapidly along the propagation direction and that the intensity at the end of the double-chain drops to zero for h = 2.04r, which corresponds to 5 nm. For gap widths ranging from h = 2.2r to h = 3r the field intensities change only slightly reaching a maximum for h = 3r = 75nm. This value can be considered the optimum distance between the double-chain to obtain the maximum light energy propagating along the structure.
Fig. 4. H-field intensity in the waveguide for different gaps, h, between the two parallel chains in the center-arm, with dfn = 2.6r, d = 2.2r and α = 90°. (a) Field distribution along the waveguide (AB) and (b) field distribution in the cross section of the waveguide just before the last cylinders in the double-chain (CD).
Fig. 5. Normalized H-field intensity distribution in the waveguide for different wavelengths: (a): λ = 480 nm, (b): λ = 600 nm and (c): λ = 830 nm. The field-intensity is strong and well confined within the double-chain waveguide at λ = 600 nm.
We also investigate the field distribution at different excitation wavelengths, chosen as λ = 480, 600 and 830 nm. Qualitative results are shown in Fig. 5 and Fig. 6. Note that the light source is located at the left side of the waveguide. The field intensity distribution along the propagation direction in the center of the waveguide (AB) and that over the cross-section at the output (CD) are plotted in Fig. 6. The dimensions in the funnel region and the double-chain waveguide sections are the same as in Fig. 4. The results in Fig. 5 and Fig. 6 confirm that at λ = 600 nm the wave propagates with the highest H-field intensity in the region between the chains while at λ = 480 nm the intensity rapidly decays the further the wave propagates away from the source. At λ = 830 nm the light intensity at the end of the waveguide is significantly smaller than at λ = 600 nm.
Fig. 6. Field intensity distribution in the center of the waveguide along the propagation direction (a) and transverse to it at the output of the waveguide (b) for different excitation wavelengths: λ = 470, 600 and 830 nm.
It is interesting to note that, for the excitation chosen here single chain propagation would only occur for wavelengths at about 250nm. At 600nm the diameter of the cylinders in the single chain are too small to allow plasmon resonance or field enhancement between neighboring cylinders in the single chain. This is demonstrated in Fig. 7, which shows the double chain waveguide with only 4 cylinders and separated by 100nm. The distance between neighboring cylinders is kept constant at 5nm.
Fig. 7. E-field intensity in the two parallel chains for d = 2.2r (center-center distance between 2 nanowires in the same row is 5 nm) and the gap h = 6r (distance between 2 rows is 100 nm); excitation wavelength λ = 600 nm.
By reducing the distance between both single chains to 25nm (Fig. 8), the cylinders of both chains start to interact and develop a strong E-field between the cylinders in each chain, which is not observed in Fig. 7. Several other effects become visible in the illustration of Fig. 8. Due to the incident field the interaction between the first two cylinders across both chains is strongest which is indicated by a strong Ey-field component [Fig. 8(c)] between the first two cylinders of the double-chain. At the end of the 4-cylinder double-chain the same effect occurs due to the transition from the double-chain to open space. The Ex-field component between the first two cylinders in each chain is weak at first [Fig. 8(b)] and then increases significantly while the Ey-field component decreases. The H-field component is strongly enhanced and occurs at places where the Ey-field component is minimum. The strong H-field enhancement between the double-chain, seen in Fig. 8(d), is due to the fact that the Ex-field component in both chains has probably opposite directions.
Fig. 8. Field intensity in the two parallel chains for d = 2.2r (center-center distance between 2 nanowires in the same row is 5 nm) and the gap h = 3r (distance between 2 rows is 25 nm); excitation wavelength λ = 600 nm.
Another important phenomenon can be noted from Fig. 5. There is obviously a strong standing wave within the funnel region in the vicinity of the source, which also depends on the wavelength. This indicates that the opening angle of the V-shaped funnel region maybe optimized to allow more light to be coupled into the double-chain region.
In Fig. 9 we finally examine the effect of the opening angle of the V-shaped aperture on the H-field intensity along and at the end of the system. It is evident that the opening angle of 90° in our previous analysis is not the optimum one can choose. In fact a smaller opening angle of 30° improves the light coupling into the structure significantly. This confirms our assumption that a smaller opening angle of the funnel region captures more light from the source but also represents a less abrupt transition from the funnel region into the double-chain region and thus reducing back reflection.
Fig. 9. H-field intensity in the waveguide for different V-shaped aperture, α, with dfn = 2.6r, d = 2.2r and h = 3r. (a) Field distribution along the waveguide (AB) and (b) field distribution in the cross section of the waveguide just before the last cylinders in the double-chain (CD).
5. Conclusions
We have presented an optical waveguide structure formed by a parallel double-chain of silver nanowires and a V-shaped funnel feed for transporting energy with light confined below the diffraction limit. The results have demonstrated that guiding of long-range surface plasmon waves and its field confinement depends not only on the waveguide geometry but also on structural dimensions in the funnel region for launching the light into the structure. The SIE approach has been implemented to study the interaction of lightwaves at the nanoscale. Structural parameters like center-center separation of nanowires in the funnel part and in the center-arm, varying gaps between the double-chain and the aperture angle of the feeding section have been included in the investigation. It was found that the double-chain waveguide with separation distance of 25nm can propagate plasmon waves at wavelength at which the single-chain waveguide is unable to do so. It was also found that the opening angle of the funnel region can improve wave propagation significantly.
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