The study on the optical properties of metallic nano-particles has been an attractive area of research forthe last two decades due to the unique properties ofthese particles, which are different from both indivi-dual atoms and the corresponding bulk structures[1–6]. The effects arising from electrodynamic inter-action of the nanoparticles find important applica-tions in areas like plasmonic waveguides, opticalfilters, surface-enhanced spectroscopies, and biosen-sors [7–11]. Optical properties of the metallic nano-particles depend on the size, shape, structure, andthe medium in which they are embedded [6,12–16].The surface plasmon resonance conditions of nano-particles mainly depend on the effective mass of elec-trons, density of electrons, and the profile of
electronic charge distribution. The embedding med-ium as well as the other nanoparticles in the vicinityalso have a significant influence on the resonancecondition.
Noble metal nanostructures, mainly composed ofsilver, have attained considerable attention due totheir interesting properties and applications [17].Confinement of light in the vicinity of the nanopar-ticle is possible since localized surface plasmons onmetallic nanostructures couple with electromagneticfields [18,19]. It has been shown that clusters ofnanospheres [20–22] can provide large field enhance-ment; similar enhancement was expected and ob-tained in nanocylinder structures [23,24]. Studieswere also performed on nanoshell cylinder structuresmade by forming an air hole or dielectric, which pro-vided higher tunability and greater field enhance-ment [25,26]. The electric potential distributions inthree interacting nanocylinder clusters were also in-vestigated [27]. Trimer systems become interesting
due to the broken linear symmetry in dimers andsince they are the basis of close-packed structures[28,29]. However, to our knowledge a trimer struc-ture with nanoshell cylinders has not yet beenstudied.
In this paper, we investigate the near-field re-sponse of triangular system of shell nanocylinders in-teracting with incident plane wave, using the finitedifference time domain (FDTD) method [30,31]. Theoptical properties can be changed according to theproper choice of particle dimensions and materialparameters. In our simulation, as the inner radius ofthe nanocylinder shell is increased, local fieldenhancements are observed. The study of the depen-dence of surface plasmon modes on interparticledistance revealed that the near-field intensity willbe enhanced as the particles approach one another.The near-field optical properties of the silver shellnanocylinder is controlled by varying the dielectricpermittivity of shell core as well as environment.Further, the effect of illumination direction of the in-cident light on near-field enhancement of the trian-gular system of shell nanocylinders is studied. Thesimulationmethod used to study the optical responseof the system is summarized in Section 2 and thedependence of optical resonances on various param-eters is simulated and discussed in Section 3.
2. Simulation Method
In this study, two-dimensional (2D) FDTD usingopen source software, MEEP [32] was used to inves-tigate the near-field optical properties of triangularsystem of silver shell nanocylinders. Figure 1 showsthe schematic diagram of silver shell nanocylindersarranged in triangular geometry. The infinitely longshell nanocylinders (C1, C2, and C3) are arrangedat the vertices of an equilateral triangle with the
cylinder axis along Z direction. Each shell cylindersconsists of a dielectric core cylinder surrounded by ametallic (silver) shell cylinder. The shell thickness ofthe cylinder shell is the difference between theparticle radius (R1) and the core radius (R2). In thispaper, all the three shell cylinders are considered tobe identical. The shell structure is illuminated by aplane electromagnetic wave propagating in the kdirection with the direction of magnetic field H per-pendicular to k and parallel to the cylinder axis.In our simulation, computation region is 2000nm ×2000nm with a grid size of 1nm. The 2D simulationdomain (XY) containing TM mode with Ex, Ey, andHz as nonzero components is truncated using per-fectly matched layer [33,34]. The silver propertiesof the nanoshell cylinders are modeled using Drudeplus two-pole Lorentzian form:
ϵMðωÞ ¼ ϵ∞−
ω2D
ω2 þ iωγD−
X2
m¼1
gLmω2Lm
Δϵω2
− ω2Lm
þ i2ωγLm
ð1Þ
with ϵ∞¼ 2:3646, ωD ¼ 8:7377 eV, γD ¼ 0:07489 eV,
Δϵ ¼ 1:1831, gL1¼ 0:2663, ωL1
¼ 4:3802 eV, γL1¼
0:28eV, gL2¼0:7337, ωL2
¼5:183eV, γL2¼ 0:5482 eV.
The nanoshell trimer is illuminated by a Gaussianplane wave source [35,36]. The Gaussian source isplaced at an arbitrary location in free space regionof the 2D simulation domain so that excitation ofall resonance modes of the structure within the UV,visible, and IR regions of the spectrum is possible.During the simulation, the field intensity at thereference point is recorded at 10,000 time steps.The field intensity is transformed into the frequencydomain by fast Fourier transform. This field in-tensity is normalized with the Fourier transformedintensity taken at the same point without the struc-ture. The mode distributions are presented in theHDF5 format [37].
3. Results and Discussion
First, the near-field intensity at points A, B, and C asshown in Fig. 1 for a triangular system of shellnanocylinders in vacuum is considered. Points A andC are at the center of the lines joining the shell cy-linders C1C3 and C1C2, respectively. Point B is at the
center of the perpendicular bisector joining the shellcylinder C1 to the base. The center-to-center distanceis fixed at 130nm, the core radius (R2) at 35nm andthe particle radius (R1) at 50nm. Figure 2 shows thevariation of near-field intensity with wavelength atpoints A, B, and C. The strongest field enhancementis observed at point A compared to other points.Thus, for further studies, point A is taken as the re-ference point.
Second, the dependence of near-field intensityon the dielectric constant is considered for three dif-ferent cases. Case 1: the permittivity of the surround-
ing medium is changed with core filled with air,case 2: the permittivity of core medium is variedby keeping the shell cylinder in air, and case 3: thepermittivity of surrounding and core media is variedconcurrently. In Fig. 3, the effect of dielectric con-stant of embedding medium on plasmon resonanceis shown (case 1). Here, the core radius (R2) andsilver cylinder radius (R1) are fixed at 35 and 50nm,respectively. The center-to-center distance is keptconstant at 130nm. The surrounding medium hav-ing permittivity of 1.0, 2.04, and 3.06 is taken intoconsideration. As the permittivity of the embeddingmedium is increased, a distinct redshift is obtainedin the resonance wavelength. This effect can be at-tributed to the effective spatial shortening of the op-tical wavelength with the increase in the dielectricconstant of the environment. As a result, the inducedcharges on the metal–dielectric outer interface de-creases, leading to a reduction of restoring forces act-ing on the polarized electrons [17]. This increases theplasmon coupling strength, resulting in the redshiftof the resonance peak. The resonance peak shows ashift of around 325nm.
Next, the effect of permittivity of the mediumfilling the core region of the nanoshell cylinder onthe plasmon resonance of the shell cylinder trimerplaced in air (case 2) is studied. Figure 4 comparesthe variation of near-field intensity with core permit-tivity as a function of wavelength. Here, a redshift forthe resonance peak wavelength is observed as before.But the shift is much less than that observed pre-viously since the surface area of the boundarybetween the two media is less in comparison to pre-vious case. Further, the dielectric constant of boththe core and the surrounding area is varied togetherin a symmetric manner (case 3). Figure 5 shows themode distributions for wavelengths of 598, 625, and665nm for core permittivities of 1.0, 2.04, and 3.06,respectively. As expected, it is observed that the fieldconcentration in the core increases as the core per-mittivity is increased. The dependence of near-fieldintensity on the permittivity as in cases 1, 2, and 3are summarized in Figs. 6(a) and 6(b). It can be seenthat the plasmon resonance peak shows maximumredshift for case 3, with noticeable broadening.
The dependence of near-field intensities on thedimension of the nanocylinders is shown in Figs. 7and 8. To study the influence of shell thickness on
Fig. 3. (Color online) Near-field intensities at point A as a func-tion of wavelength for different dielectric environments withdielectric constants ϵ ¼ 1, 2.42, and 3.06.
Fig. 4. (Color online) Near-field intensities at point A as afunction of the wavelength for core with dielectric constantsϵ ¼ 1, 2.42, and 3.06.
Fig. 5. (Color online) Normalized field distributions (Hz) of a triangular system with core medium permittivities of 1, 2.04, and 3.06.
the plasmon resonance, the particle radius (R1) isfixed at 50nm and core radius (R2) is varied in stepsof 5nm. The interparticle distance is fixed at130nm. The intensity decreases sharply as the size
of the core is reduced finally settling to the case of asolid nanocylinder trimer. In addition a redshift is ob-tained as the core thickness is increased. Thecoupling between the air–metal and metal–air inter-face increases as the shell thickness is reduced,which results in distinct redshift. The other way tostudy the effect of shell thickness on the plasmon re-sonant condition is to change the particle size (R1)while fixing the core radius (R2). In Fig. 9, R1 is var-ied with R2 fixed at 35nm. Here, the center-to-centerdistance is fixed at 130nm. The large field enhance-ment can be due to two factors, the increase in thesize of the metal part of shell cylinder and the de-crease in the interparticle distance.
Figure 9 shows the simulation results of the near-field intensities as a function of incident wavelengthfor different interparticle distances. The core radius(R2) and particle radius (R1) of the cylinder shellsplaced in air are fixed at 50nm and 35nm, respec-tively. The near-field intensity increases with thedecrease in interparticle distance. In addition to fieldenhancement, a redshift of around 300nm is ob-served due to the larger coupling between the threenanoshell cylinders. Thus, the strength of plasmon
Fig. 6. (Color online) Near-field intensities as a function of wavelength for cases 1, 2, and 3 for dielectric constants (a) 2.04 and (b) 3.06.
Fig. 7. (Color online) Near-field intensities in the gaps of trian-gular system of silver nanocylinders with core radii 5, 10, 20, 30,and 40nm as a function of wavelength.
Fig. 8. (Color online) Near-field intensities in the gaps of trian-gular system of silver nanocylinders with outer radii 50, 55, 60, 61,62, and 62:5nm as a function of wavelength.
Fig. 9. (Color online) Near-field intensities in the gap of silvernanocylinder arranged in triangular geometry as a function ofwavelength for different interparticle distances.
coupling can be effectively tuned by changing the in-terparticle distance of the silver shell nanocylinders.Now, it is clear that for the field enhancement shownin Fig. 9, the significant contribution comes from thevariation in interparticle distance. This type of an op-timization also helps in the design and developmentof plasmonic nanostructures.
The nature of induced polarization charge on theshell cylinders depends on the incident light as wellas mutual interaction between shell cylinders. Bychanging the illumination direction, the polarizationcharge distribution and hence the coupled modeproperties can be altered [18]. The angle is varied insteps of 15° in the anticlockwise direction with re-spect to geometry shown in Fig. 1. Figures 10(a)–10(c) shows the variation of near-field intensity asa function of wavelength at the three reference pointsA, B, and C, respectively, as in Fig. 1. It is observedthat field intensity is maximum at point A for angleθ ¼ 0°, whereas the field intensity is maximum forpoints B and C at angle θ ¼ 30° because point A isshielded as we change the direction of illuminationof incident light. Thus, the dependence of near-fieldintensities on the angle of incidence can be employedin angle selective light extraction [38].
In a typical light scattering experiment, thescattering cross section represents the amount oflight scattered in the far field, which is a measurablequantity. The variation of the scattering cross sec-tion with the wavelength has the same trend asthat of near-field intensity [39]. Thus, the calculatednear-field intensity can be used in the design ofnanoparticle-based antenna or sensor applications.Further, if the trimer cell is being used as a buildingblock of a resonant chain waveguide, the computedfield pattern can give information on the mode cou-pling, dispersion, and propagation loss.
4. Conclusion
The near-field optical properties of silver shellnanocylinders arranged in triangular geometry isstudied using the FDTD method. By varying theinterparticle distance, the dimensions of the nano-shell cylinders and the illumination direction theplasmon resonance is tuned across the UV, visible,and near-IR spectral range. The field intensity can be
enhanced as the interparticle distance is decreased,which facilitates our abilities to design, develop, andoptimize coupled nanoparticle systems. Localizedsurface plasmon resonance and field enhancementcan also be controlled by changing the illuminationdirection of the incident light. The results of ourcalculations showing variation of near-field inten-sity with the dielectric constant of the core, environ-ment, and both core and surrounding area broadenthe range of potential applications in biosensing,chemical sensing, near-field microscopy, and nano-waveguiding.
The authors are grateful for financial supportfrom the Department of Science and Technology,Government of India, through a research grant(SR/S2/CMP-0012/2009).
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