In this paper a new silver (Ag) nanoparticle-based structure is presented which shows potential as a device for front end applications, in nano-interconnects or power dividers. A novel oxide bar ensures waveguiding and control of the signal strength with promising results. The structure is simulated by the two dimensional finite difference time domain (FDTD) method considering TM polarization and the Drude model. The effect of different wavelengths, material loss, gaps and particle sizes on the overall performance is discussed. It is found that the maximum signal strength remains along the Ag metallic nanoparticles and can be guided to targeted end points.
©2009 Optical Society of America
The demand for higher bandwidth, data transfer rate and compactness of devices stimulates the development of nano electronics and photonic miniaturization. In conventional photonics, miniaturization of structures is restricted by the diffraction limit which makes it impossible to exploit the large bandwidth capability of photonic interconnects with nanometer-size electronic devices. Plasmonic-based structures with dimensions significantly below the diffraction limit are seen as an alternative with huge potential in communications but also in biodetection and other sensing applications . The term plasmonics is defined as the collective oscillation of free electrons at the interface of conductors and dielectrics [1–5]. In nanoparticle-based plasmonic structures, each nanoparticle support localized plasmons that interact with the localized plasmons of the neighboring nanoparticles. This interaction enhances the local field in the gap of nanoparticles and leads to resonant modes [6–15]. The signal strength of these resonant modes depends on the direction of signal propagation, gap distance and shape of nanoparticles. This interaction of optical fields between nanaoparticles is the basis for making T junctions or other types of bends, which can be exploited in the design of passive plasmonic devices [8–16]. In addition to resonant modes, recently, plasmonic guided modes in nanoparticle-based structures are also studied . The oscillation of free electrons in nano-metallic structures such as Ag not only causes the enhancement in the local electromagnetic field [5–7], but also the delocalization and transportation of the electromagnetic signals. The signals can be guided along a chain of metallic nanoparticles over a distance of a few hundred nanometers without considerable loss in signal strength when the nanoparticles are closely spaced [18–20]. Besides nanoparticle waveguides, complex nanoparticle structures such as splitter, interferometer, condenser and couplers have been proposed [21–23].
In the following sections of this paper we discuss the analysis and design of a nanoparticle-based power splitter, illustrate the effects of structural measures to improve the transfer characteristics of such devices and study its performance at different wavelengths, with varying nanoparticle diameters and particle distance.
The 2D finite difference time domain (FDTD) method is utilized to analyze the behavior of the Ag-based plasmonic structure of Fig. 1. The FDTD method has been tested in numerous applications as a reliable and general simulation tool [24–26]. To model the dispersive nature of Ag nanoparticles, the frequency dependent permittivity in the visible range of the electromagnetic spectrum is considered through the Drude model
where ωp is the plasma frequency and Γ is the collision frequency. The values of ωp = 9.39×1015 Hz and Γ = 0.3×10 15 Hz. Under consideration of the Drude model Maxwell’s equations are written as:
As an example, the Dx-field in the FDTD formulation can be read as
To convert the frequency dependent permittivity in the above equation into the time domain two approaches are usually used: The z-transform or auxiliary differential equations technique. In this paper the z-transform approach is chosen and Eq. (4) can be written as an example
Where and α = e -ΓΔt
3. Modeling the device under consideration
The Ag nanoparticle-based structure shown in Fig. 1 serves as a power divider. Five different sections guide the wave from the input on the left to the two output ports on the right: These are condenser, collector, waveguide, divider and oxide bar. The structure resembles the front end of a receiver in conventional electronics. The first component from left, the condenser, focuses the field into the collector region or towards center. The shape of the condenser is important to ensure maximum field strength. The field will deviate from the center if nanoparticles of the condenser are not placed in a circular orientation. The second component, the collector, is placed at the center of the condenser and collects the energy focused by the condenser and feeds the waveguide, the third component. At the output of the waveguide a T-section (fourth component) divides the signal into two output ports. In Fig. 1 letters C, U and L indicate the center, upper and lower end of the divider, respectively. To avoid stray coupling of EM waves from any point of the waveguide to any of the two output ports a special oxide bar (fifth component) made of barium titanate nanoparticles is employed. The oxide bar blocks the signal, or, more precisely, if the signal crosses one bar, it will be trapped in the gap between the two rows of nanoparticles and is fed back into the main waveguide. A study of the behavior of the overall structure at different wavelengths, varying diameters of nanoparticles and gaps between them is discussed in the next section.
4. Results and discussion
The nanoparticle size of the first two components in Fig. 1, i.e. the condenser and the collector are 10 nm. Both the condenser and collector dimensions are kept constant through out the analysis because their function is limited to provide input power to the waveguide only. The plane wave is considered as an excitation source. The cell size considered is 1 nm in all directions. Particle and gap sizes of the following two components, the waveguide and the T-junction are changed as they will have the highest influence on the overall device performance. Fig. 2 shows snapshot of the electric field propagating along the nanoparticles with the maximum field strength in close vicinity of the particles. Fig. 2(a) shows the snapshot of the proposed structure without oxide bar, while Fig. 2(b) shows snapshot with oxide bar. The cause of maximum field strength along the nanoparticles is the plasmonic interaction between neighboring nanoparticles. The generation of plasmonics effect is due to the electric dipole nature of nanoparticles at the visible frequency range. At these frequencies, conduction electrons are confined on the metal surface. This produces a restoring force which causes oscillation in the metal dipoles resulting in plasmons excited inside the metal nanoparticles. This interaction between electric dipoles of nanoparticles guides the signal along the nanoparticles to the desired positions.
As expected, the simulation has also shown that by varying the number of nanoparticles in the condenser and the collector region the field intensity can be varied at the input of the waveguide. The snapshot in Fig. 2(b) illustrates this effect with 8 nanoparticles in the condenser and 8 in the collector. In addition, the effect of the oxide bar is also visible: The stray field of Fig. 2(a) (without oxide bar) is now reduced and the field concentration in close vicinity of the waveguide region is significantly enhanced, hence less signal power is lost. Fig.3 shows the ratio between the signal at the output of the collector or the input of the waveguide, Iin, and the signal at center ‘C’ of the divider, IO. This ratio is shown with respect to different nanoparticles size and gaps “d” in between its nanoparticles.
It is clear from Fig. 3 that the electric field intensity decreases either with the increase in the particle size or gap distance, which is a consequence of the decreasing interactions between their electric dipoles. The plot also demonstrates that increasing the size of nanoparticles the electric field intensity is less affected compared to increasing the gap distance.
In the following we present a detailed study of this structure for different wavelengths, size of nanoparticles, and particle gaps to investigate the device performance.
First, we investigate the structure for wavelenghts ranging from 350 to 800 nm. For this case, the center to center distance of waveguide and divider nanoparticles is 15nm, while the particle size is 10 nm. Fig. 4 shows the electric field intensity at the center (C), upper (U) and lower (L) ends of the divider with respect to different wavelengths. It should be noted that the field intensities for both output ports coincide.
The amplitude at each end of the divider is, as expected, approximately half of that at the center for each wavelength of interest. Fig. 4 also shows that with the increase of wavelength, the electric field intensity decreases, both at the T-section (at C, see Fig. 1) and two output ports (curve U and L are identical in Fig. 4). The reason for this decay in amplitude at the T-section and at two ports U and L is the fact that at lower frequencies and given particle size and distance, the field becomes less confined to the waveguide region. The resonance effect between the particles is less pronounced, and, hence, less energy propagates along the longitudinal direction of the waveguide. This reduces the amount of energy available at the T-section and consequently also at the output ports. In the case of Fig. 4, both sides of the divider are of the same length and therefore the same amplitude is observed at both ends of the divider. However, by changing the length of one output arm versus the other, a phase difference between both output ports can be realized.
In the second case, we investigate the structure with respect to different gaps “d” between nanoparticles in the waveguide and divider region. The diameter of the nanoparticles is kept constant at 10 nm throughout this investigation. From the analysis of Fig. 4 at different wavelengths but constant particle gap one might expect similar results for constant wavelength but changing particle gap. This is indeed the case. In addition, a significant time delay can also be observed as illustrated in Fig. 5 for particle separations ranging from 3 nm to 10 nm. The increase in gap width not only delays the signal but also has the same effect as of increasing wavelength in Fig.4, in that the amplitude at the T-junction (point C in Fig.1) decreases. Figure 5(a) shows the normalized electric field with respect to time for different gap size. Figure 5(b) illustrates the fact that the time delay of the electric field vs. gap size at the T-junction (C) behaves almost linear and that the electric field intensity at the output ports shows similar performance as in Fig.4. For the measurement of time delay, the time at d = 3 nm is used as a reference. Careful observation of the latter, however, shows that the field intensity at the T-junction drops faster than that at the output ports. This phenomenon may be attributed to an increasing stray field in the waveguide region with increasing gap size. The field is less confined to the waveguide region and couples directly to the output ports rather than being guided through the T-junction. This stray field can be controlled to some extent by using different dimensions for the oxide bar as illustrated in Fig. 2(b). By doing so the results at output (L and U) of the power divider are roughly 50% of the value at the T-junction (C), as would have been expected in a perfectly guiding structure. Although the structure considered here is symmetrical, other power divider ratios can be realized by varying the structure dimensions in the T-junction region.
In the third case, the structure is analyzed for different diameters of nanoparticles in the waveguide and divider region. The free-space wavelength (λo=785 nm) and gap size (5 nm) is kept constant. The results are shown in Fig. 6, where Fig. 6(a) shows the normalized electric field for particle diameters ranging from 8-15 nm. Also here the signal delay is increasing with increasing particle diameter, which is due to the fact that for the chosen wavelength and particle diameters the resonance effect becomes less pronounced. These results are similar to the case with increasing gap size and are thus not further investigated.
The guided wavelength of the plasmon waveguide changes with particle diameter, just like a dense, homogeneous medium would do to an electromagnetic wave. Fig. 6(b) illustrates the relationship (at 785 nm in this example) with the time delay of the electric field at the T-junction for different nanoparticle diameters by keeping the gap size constant. As evident from the Fig. 6(b) an increase in the particle diameter causes the time delay to increase whilst the width of the propagating wave decreases. In Fig. 6(b), the time delay at 8 nm is taken as a reference, while for the width of the wave (Gaussian shape); variation of the guided wavelength (785 nm) at 15 nm (smallest width from all related simulated results) is taken as a reference. From Fig. 5 and Fig. 6 it is evident that the time delay varies approximately linearly either with an increase in gap size or an increase in particle dimensions (i.e. the increase in particle size or gap size between particles increase the length of structure and as a result time delay also).
Further to observe the influence of the material loss, the relative loss of the proposed structure is measured with respect to different size of nanoparticles and different gaps between nanoparticles.
The relative loss (%) is calculated with the following formula:
where Areference is the value of the field intensity without material loss and Ameasured is the value of the field intensity with material loss of the structure. These measurements are taken at point C (Fig. 1). The relative loss with respect to particle size and with gap size is shown in Fig. 7. It can be observed from these results that the relative loss is highly dependent on size of the nanoparticles and the gap in between them. For example when the gap size in between nanoparticles is 3 nm, the relative loss is around 17%. This loss increases with the increase in gap size and reaches to 50% when the gap size is 10 nm (particle size for different gap size measurements is fixed at 10 nm). Similarly, when the particle size is 8 nm, the relative loss is around 13%. This loss increases with an increase in the size of nanoparticles and it reaches to 58% when the particle size is 15 nm (gap size in these measurements is fixed at 5 nm).
We have analyzed a novel Ag nanoparticle-based power divider structure and will be useful for applications in future nanophotonics/plasmonics such as front-end or short distance interconnects. The device functions significantly below the diffraction limit and can split and redirect light at different power ratios. We have shown that the use of an oxide bar can improve the guiding performance of the device significantly. The relative loss of nanoparticles is also studied to observe the influence of material loss, and loss due to the particle size and gap size variation in between nanoparticles. We observed an almost linear relationship in terms of relative loss with respect to the gap size between nanoparticles and particle size.
We are also thankful for valuable discussion and suggestions of Professor Boris Luk`yanchuk, Data Storage Institute, Agency for Science, Technology and Research, DSI Building 5, Engineering Drive 1, Singapore 117608
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