1. Introduction

Confinement of light in nano-scale region of nanostructures is one of major challenges for nano-scale integrated optical devices, near-field optical research, and nanophotonics. Metallic nanostructures are attractive to scientists and engineers because surface plasmon of metallic nanostructures couples with electromagnetic fields and exhibits interesting optical properties, for examples, large scattering cross section, extraordinarily light transmission[1], nonlinear optical response[2], and highly localized field enhancement in the vicinity of nanostructures [3, 4]. Hence metallic nanostructures (especially for noble metals) have vast potential to become an important element for biosensor [5], nanophotonic devices [6, 7], and plasmonic devices [8].

Interaction of light with single non-regular metallic nanoparticle, like nanoparticle with triangle or pentagon shape, has been investigated by numerical simulations[9] and experiments[10, 11, 12]. Nanoparticle with low symmetric geometric shape exhibits not only complex resonance spectra but also dramatic local-field enhancement in the vicinity of the nanoparticle, and enhanced field amplitude can reach several hundred times of the incident field. A metallic nanowire itself forms a resonant device and Fabry-Perot resonance of surface plasmon in silver nanowires has been observed experimentally by Ditlbacher et al.[13] In addition to investigate the geometry shape dependence of local-field enhancement of individual nanoparticle, over the past few years several studies have been made on enhanced local-fields by near-field coupling between close-spaced metallic nanoparticles with simple geometric shape and found that enhanced electromagnetic fields are confined in the nano-scale gaps between nanoparticles and the surface plasmon resonances exhibit red shift with small particle-particle distance.[14, 15, 16, 17, 18, 19]. The plasmon resonances of two identical silver nanowires have been theoretically investigated by Kottmann et al.[14, 15] and the complex resonances spectra associate with the retardation effect of nanoparticles are observed and the main resonance is red-shifted with decreasing interparticle distance. The enhanced local fields appear in the gap between the nanowires with a few nanometers and can exceed from several hundred to several thousand times the amplitude of incident light. Surface plasmon excitation of two interacting Au nanoparticles has been studied by Rechberger et al. [16] and Su et al. [17] and the red-shift of plasmon resonance is observed with decreasing particle spacing and different polarization direction of incident light.

With increasing number of metal nanoparticles and complex geometry, the surface plasmons of metal nanoparticles become complicated. Local-field enhancement of linear arrays of identical Ag nanoparticles has been studied by Sweatlocak et al. [20] and exhibits 5000-fold local-field intensity enhancement for nanoparticles of diameter of several nanometers. Dramatic surface plasmon field is excited at the extremity of finite chain of gold nanoparticles when illuminated under total internal reflection has been investigated by Ghenuche et al. [21]. These results are closely related to studies of surface enhanced Raman scattering of molecules adsorbed on metallic nanoparticles [22, 23]. Arrays of metallic nanoparticles are also used to transport electromagnetic energy in nano-scale regime [24, 25, 8, 26, 27, 28]. The near-field addressing of a linear chain of metallic nanoparticles with a SNOM tip has been studied numerically by Girard et al. [27, 28]. Man-made nanoparticles could be periodic or orderly arranged, but nanoparticles are usually randomly distributed without special designed processing. Random distributions of silver nanoparticles are possible to support wide-band, multiple-mode surface plasmon resonances. The collective surface plasmon excited around those nanoparticles have been used to explain the superresolution capability of near-field high-density optical disk (AgO_x-type super-resolution near-field structure).[29, 30, 31, 32, 33, 34]

In this paper, we study a nanocylinder structure whose complexity is between a nanoparticle pair and multi-nanoparticle structures. The array of three-pair nanocylinders enables the investigation of particle-particle interaction as well as pair-pair interaction. The enclosure of the pair array forms an open cavity and the electromagnetic field is effectively confined in the gap of each pair to generate high local-field enhancement. Near-field optical responses of three silver nanocylinder pairs interacting with incident plane wave are simulated by finite-difference time-domain (FDTD) method in visible light region. Influences of wavelength of incident light, interpair distance and radius of nanocylinders on local-field enhancement are discussed in our simulations.

2. Simulation model

Two-dimensional FDTD method[35] is used to investigate the near-field optical responses of array of three silver nanocylinder pairs. Figure 1 shows the geometric configuration of three pairs of silver nanocylinders. The interparticle distance d and the interpair distance d _p are set to be 20 nm if not otherwise mentioned. The radius of silver nanocylinders r is varied from 20 nm to 90 nm. The three-pair structures are illuminated with a plane wave of transverse-magnetic (TM) mode, which propagates in the k direction. The direction of the electric field E is perpendicular to k and parallel to the plane of incidence. The dispersion behavior of silver is simulated by the Lorentz dispersion model [36]. The uniaxial perfect matched layer (UPML) is used as radiation boundary condition for simulating the electromagnetic wave propagates in free space [37]. The computational region is 800 by 800 nm and the grid size is 1 nm. The Yee cell scheme in FDTD simulation is not naturally suitable for simulating two very small nanoparticles only several nanometers apart, since the effect of the artificial local-field enhancement on surface of nanoparticles affects the near-field distribution in the gap between two nanoparticles. The influence of grid size on the accuracy of the FDTD results is carefully considered in our simulations. According to our convergence test for two interacting silver nanocylinders with 50 nm radius and 20 nm interparticle distance, the approximation errors of results in FDTD method is nearly proportional to the grid size and the error from FDTD simulation with 1 nm grid size is about 6 percent.

 

Fig. 1. Array of three silver nanocylinder pairs. Two types of gaps are discussed: gap between two nanocylinders in a pair (dot line) and gap between adjacent pairs (dash line). In this work, both d and d_p are set to be 20 nm, except the cases discussing the influence of interpair distance.

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Previous experimental results tell that the imaginary part of the permittivity of metallic nanoparticles is influenced by the size of nanoparticles [38]. However, for nanoparticle whose radius is larger than 20 nm, the experimental result agrees very well with theoretical calculation based on Mie theory using bulk dielectric constant[39]. Since the radii of nanoparticles in this work are larger than 20 nm, the frequency-dependent optical constant of the silver nanoparticles is set using the bulk experimental data from Palik [40]. The wavelength of incident light is varied from 400 nm to 826 nm.

3. Results and discussion

For nanoparticles whose sizes are much smaller than incident wavelength, their optical responses are similar to dipoles and the particle-particle interactions are similar to dipole-dipole interactions.[3] The higher multipole behavior becomes apparent with increasing size of nanoparticles. Therefore we at first study influence of radius of nanocylinders on near-field optical properties of nanocylinder pair arrays. The TM-mode near-field distributions of the three-pair arrays with various radii are shown in Fig. 2(a) for λ=460 nm and in Fig. 2(b) for λ=650 nm. The near-field intensity has maximum on the surface of each cylindered and the near-field intensities inside nanocylinders decay exponentially due to skin effect of metal. The strongest local-field enhancement appears in the gap of the second pair or center pair with r=36 nm at λ=460 nm and r=58 nm at λ=650 nm, respectively, due to localized plasmon resonance. The near-field distributions of Fig. 2(a) and Fig. 2(b) are similar in resonant state, except the field intensity in Fig. 2(b) is stronger than that in Fig. 2(a). Usually the local field vanishes in the gap of single pair of nanocylinders when the direction of the incident electric field is perpendicular to the major axis of the pair [14]. However, in a multi-pair system like this three-pair array, the high local field is not only confined in the gap between two cylinders of a pair but also in the gaps between two closely spaced pairs because of strongly pair-pair interaction (the location of the gap between pairs is denoted by dash line in Fig. 1). It is noted that the E_y component is dominant in the gaps of the nanocylinder pairs, but the E _x component is dominant in the gaps between two closely spaced pairs. For a four-pair array, similar type of enhancement is observed and the strongest local-field enhancement appears in the gap of the third pair. Figure 2 demonstrates that field intensity of local-field in the gaps between nanocylinders is quite sensitive to the radius of nanocylinders.

 

Fig. 2. Movie of TM-mode near-field distributions of three nanocylinder pairs for incident wavelength (a) λ=460 nm, and (b) λ=650 nm, respectively. The radius of nanocylinders is varied from 20 nm to 75 nm. The gap of the second pair exhibites strongest local field enhancement at resonant conditions. [Media 1] [Media 2]

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Fig. 3. Near-field intensities in the gaps of three nanocylinder pairs as a function of nanocylinder radius. The wavelength of incident light is (a) λ=460 nm and (b) λ=650 nm, respectively. The plasmon resonance is excited when the radius of nanocylinder is (a) r=36 nm, and (b) r=58 nm, respectively.

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Figure 3 shows the near-field intensity in the center of the gap of each pair as a function of the radius of nanocylinders. When the illuminating wavelength is 460 nm (shown in Fig. 3(a)), the field intensities in the gap of each pair are almost the same when the radius of nanocylinders is less than 20 nm, since the coupling between nanocylinder pair becomes weaker with decreasing radius of nanocylinders as well as increasing d/r ratio. The field intensity in the gap of the first pair increases when the radius is increased from 20 nm to 36 nm and decreases slightly when radius is increased from 38 nm to 45 nm. The gap intensity keeps almost constant when the radius is larger than 45 nm. The strongest local-field enhancement appears in the gap of the second pair at r=36 nm. It is about 2 times of the intensity between the first pair and 4 times of that between the third pair. The near-field intensities in the gaps of the second pair and third pair decrease rapidly when the radius is increased from 38 nm to 70 nm. The decreasing of intensities in the gaps is associated with shielding effect of the first pair of larger radius. When the illuminating wavelength is changed to 650 nm, Fig. 3(b) demonstrates the same trend as Fig. 3(a), except the resonant peaks shifts to larger radius and the intensities in the gaps become stronger. It means that the resonant wavelength of the three-pair array has red shift for small d/r ratio. This phenomenon has been observed, theoretically and experimentally, in the surface plasmon resonance of nanoparticle pairs in various situations[14, 16, 17] and it generally indicates strong interaction between nanoparticles. These results indicate that confinement of light can be tuned by wavelength of incident light and radius of nanocylinders.

 

Fig. 4. Movie of TM-mode near-field distributions of three pairs cases with different interpair distance d_p. The radius of nanocylinders is 30 nm. The wavelength of incident light is λ=460 nm. The highest local-field appears in the gap of the second pair with d_p=70 nm at resonant conditions. [Media 3]

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The pair-pair interactions in the three-pair array result in several interesting features in Fig. 2 and 3. The pair-pair interactions are closely related with the interpair distance d _p, i.e., the pair-pair interactions become stronger with smaller d _p. The influences of interpair distance d_p on the local-field enhancement in the gap of each pair indicate the impact of the pair-pair interactions. Figure 4 shows the near-field distributions of the three pairs cases with r=30 nm, d=20 nm, and various d _p at λ=460 nm. The field intensity in the gap of each pair is weak with d_p=14 nm although the interpair distance is small and the pair-pair interaction between them is strong. When the interpair distance increased, the local-field in the gap of the second pair becomes stronger and the highest field intensity appears at d_p=70 nm. In this example, the case with larger interpair distance has higher local-field intensity and the high local field in the gap of the second pair is not proportional to the pair-pair interactions. The local field in the gap of the second pair becomes weaker when the interpair distance becomes larger than 70 nm. Apparently a resonant behavior is indicated.

To investigate this resonant property, the near-field intensities in the center of the gap of the second pair as a function of the interpair distance for r 25, 30, and 36 nm are shown in Fig. 5. The main resonant peak shifts to larger interpair distance with decreasing radius of nanocylinders. By adjusting the radius of nanocylinders, highly enhanced local fields appear in the gap of the second pair with larger interpair distance (weaker coupling). The gap intensity of each pair is expected to be the same if the interpair distance approaches to infinity, since the coupling between nanocylinder pair becomes weaker with increasing interpair distance. For off-resonant cases in the weak coupling limitation, the near-field optical responses of nanocylinder pairs become similar and are comparable with the optical responses of single pair. The study of near-field optical responses associated with single pair has been simulated in our previous research work for local-field enhancement of asymmetric silver nanocylinder pair [19]. It is interesting that the resonant interpair distances could be as large as several times of the radius of the nanocylinders. Besides, the broadness of resonance curves in Fig. 5 indicates that the localized plasmon resonance in the gaps of nanocylinder pairs is not sensitive to the interpair distance and this feature is useful for fabrication of nano-scale integrated photonic devices.

 

Fig. 5. Near field intensity in the gap of the second pair as a function of different interpair distance for r=25, 30, and 36 nm. The wavelength of incident light is λ=460 nm. The resonant peak shifts to larger interpair distance with smaller radius of nanocylinders.

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Fig. 6. (a) Radius of nanocylinders and (b) near-field intensity vs resonant wavelength. There is a linear relation between radius and resonant wavelength.

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Unlike the surface plasmon resonance of a plain metal surface or that of a metal sphere, it is difficult to analyze the surface plasmon resonances of the three-pair array of silver nanocylinders. From our numerical simulations, it is possible to illuminate the surface plasmon resonances with chosen variables like the radius of the nanocylinders and to explore how the surface plasmon resonances are modified with different configurations. The plots of nanocylinders radius and the near-field intensity in the gap of each pair vs. resonant wavelength are shown in Fig. 6(a) and Fig. 6(b), respectively. Here the interpair distance is fixed at 20 nm. Figure 6(a) shows that the peak resonance wavelength is nearly proportional to the radius of nanocylinders. Hence the plasmon resonance behavior of this configuration is predictable. On the contrary, the interpair distance is not a sensitive variable of the surface plasmon resonances. The results in Fig. 6(b) shows that the field intensity in the gap of the second pair increased rapidly with longer wavelength and stronger than other two pairs. The linear relation in Fig. 6(b) had the same trend as in Fig. 6(a). The higher local-field in the gap of the second pair can be understood by an open cavity model.

 

Fig. 7. Movie of total near-field distributions (left) and phases of the E_y component (right) for TM illumination at resonant conditions. The radius of nanocylinders and illuminating wavelength are varied from 36 nm to 72 nm and from 460 nm to 775 nm, respectively. [Media 4]

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The predictable and controllable plasmon resonance of the silver nanocylinder pairs in Fig. 6 could be explained by an open cavity model. Three pairs of silver nanocylinders form an open cavity and satisfy resonant condition with appropriate separation distance and radius. The resonant condition of the open cavity can be tuned by illuminating wavelength, interpair distance, and radius of nanocylinders. Phase graphs can help to understand the resonant behavior of the three-pair arrays. Since the dominant component of electromagnetic fields is E _y component for TM illumination, the characteristics of the cavity resonance are demonstrated by the phase behavior of the E_y component. Figure 7 is a movie of the near-field distributions and the phases of E_y component for TM illumination at resonant conditions. The E_y component shows the propagation in the x direction. The phase graph in Fig. 7 tells that the phase of E _y component in the area close to the three-pair array changes more rapidly than the phase in the area away from the array, especially the phase inside the region bounded by the dash line. In all the resonant cases, the phase distributions in the bounded region of the pair arrays are similar. It suggests that they correspond to the same resonant mode.

The virtual boundary of the open cavity for E_y component can be observed from the phase graphs. For the three-pair arrays of increasing nanocylinder radius, the points which maintain their phase at resonant conditions are identified. Two equiphase lines (red and blue line) of the three-pair arrays with various nanocylinder radiuses at resonant conditions are drawn in Fig. 8. In our simulations, the locations of the virtual boundaries (the black lines) are assumed to be the points which keep the same phases while the nanocylinder radius is changed. In the Fig. 8, for the virtual boundary of each side, the physical position and the equiphase line have the same slope with changing radius of nanocylinders. The virtual boundaries (as shown in the inset of Fig. 8) are close to the end of the first pair and the center of the third pair. The asymmetric characteristic may result from the direction of incident light. The equiphase lines correspond to phase values of 0.9 and -2.2 rad, and the phase difference between that two boundaries is close to π. It infers that this resonance is corresponding to the fundamental mode of the open cavity. Hence results in Fig. 8 establish that the fundamental resonance of the three-pair array associates with the open cavity model. The open cavity model is also able to explain the resonant behavior of the interpair distance presented in Fig. 4 and 5, i.e., the nanocylinder array of smaller radius has larger resonant interpair distance. From the inset in Fig. 8, it is clear that this resonant condition means to reach an appropriate size of the open cavity and to excite the fundamental mode.

 

Fig. 8. Equiphase lines (red and blue lines) of the three-pair arrays at the virtual boundaries (black lines) of the open cavity at resonant conditions. The center of the three-pair array is fixed and the positions of the first and the third pairs move outwards as the radius of nanocylinders increases. The inset shows a schematic plot of the three-pair array and the location of the virtual boundary. The phase value are 0.9 and -2.2 rad, respectively, and the phase difference between two boundaries is close to π

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The open cavity model serves as a good first-order approximation to the three-pair array for strong coupling between pairs; however, there are some details features go beyond the first-order approximation. From Fig. 3 and Fig. 6(a), the intensities in the gaps between three pairs reach resonances at close, but not the same, conditions. From Fig 6(a), at the same radius, the intensity in the gap of the first pair reaches the maximum at slightly shorter wavelength than that in the gap of the second pair, and the intensity in the gap of the third pair reaches the maximum at even longer wavelength. The small variations of the resonant wavelengths of those three pairs apparently result from that each pair has individual pair-pair interactions.

4. Conclusion

In this paper, near-field optical properties of arrays of three-pair silver nanocylinders are studied using FDTD simulation at different illumination wavelengths, nanocylinder radius, and nanocylinder spacing. TM-mode near-field distributions show that the strongest local-field enhancement appears in the gap of the second pair due to surface plasmon resonance. The surface plasmon resonance and light confinement of the three-pair arrays can be tuned by radius of nanoparticles, interpair distance, and the wavelength of incident light. The peak resonance wavelength is nearly proportional to the radius of nanocylinders. An open cavity resonance model is proposed to understand the mechanism of these selective plasmon resonant behaviors and the plasmon resonance is controllably excited by changing the cavity resonance condition. This configuration provides a simple and practical way to manipulate local-field enhancement in nano-scale regime.