We numerically study the effect of the symmetry breaking on surface plasmon (SP) modes in two-dimensional dense arrays of truncated metal nanoshells (nanocups), by investigating light transmission through the arrays. We show that localized spherelike and voidlike Mie SP modes, and delocalized Bragg-type SP modes in complete nanoshell arrays become progressively weak and finally disappear when the opening angle of nanocups is increased to tens of degrees. Under higher degree of symmetry breaking, however, the coupling between spherelike and voidlike SP modes leads to an enhancement of SP resonances even though these modes are weakly excited, due to the large optical cross section of voidlike modes. Energy variations of the hybridized mode versus the opening angle are well predicted using a plasmon standing wave model. Furthermore, disappeared Bragg-type SP modes could be re as a result of near-field coupling via hot spots around the rims of nanocups.
©2011 Optical Society of America
Optical properties of individual metallic nanoparticles (NPs) and their ensembles have been the subject of intense research because they can support localized surface plasmon (SP) resonances. In general, the optical responses of NPs, such as spectral position, damping, and strength of the dipole and higher-order SP modes, depend sensitively on the particle size, shape and composition, and also on the dielectric constant of the surrounding host . In a striking contrast to solid metallic NPs, metallic nanoshells composed of a spherical dielectric core surrounded by a concentric metal shell, support plasmon resonances whose energies can be engineered by controlling the inner core and outer shell dimensions . When the metallic shell layer is extremely thin, sphere and void plasmons interact with each other, which results in the splitting of the plasmon resonances into interesting hybrid modes . By carefully designing metallic shells , novel optical properties can be observed, such as cloaking .
Recently, remarkable effects caused by the excitation of localized SPs in nanovoids have been studied theoretically and experimentally for metallic nanoporous structures [6–8]. Void plasmons can be excited not only in voids of metallic nanoshells [9,10] but also in spherical voids buried near the surface of a metal substrate [6–8] and furthermore, they show novel optical properties due to the distinctive geometrical confinement of light inside the voids. Making use of the trapped and enhanced electric field, hybrid exciton-plasmonic crystals have been fabricated and coupling between localized Mie plasmons and J-aggregate excitons has been studied and demonstrated , allowing for a potential of plasmonic lasers and organic nanophotonics. Since void Mie plasmons can be efficiently excited over a wide range of incidence angles, the trapped light in metal is eventually dissipated into heat, yielding an omnidirectional and polarization-independent near-total absorption of light in nanoporous metal surfaces .
An interesting and important issue in the field of plasmonics is symmetry-breaking induced effect [13–30]. Many unique optical properties of metallic nanostructures originate from their reduced symmetry [13,14]. For example, when nanoshells are reshaped into nanoeggs or nanocups, an admixture of dipolar components in all plasmon modes of the particles, which is absent in complete nanoshells, becomes observable [15,16]. In recent studies, symmetry-broken semishells [17–29] have aroused increasing interest due to their highly tunable plasmonic properties and interesting plasmon modes, such as the anti-bonding mode which has been experimentally verified . These approaches suggest potential applications, for example, surface-enhanced Raman spectroscopy substrates for molecular detection [22–25,29]. In our previous work, light transmission resonances in monolayer rigid arrays of cavity-controllable metallic mesoparticles were demonstrated experimentally . Also, symmetry-breaking effect in metallic nanovoids (anti-nanoparticles) has been investigated. It was reported that when metallic nanoscale voids are progressively truncated, in contrast to nanoparticles, the additional rim plasmon modes can selectively couple with void plasmons, leading to energy splitting of degenerate radial void modes .
It has been proved that the strong coupling of localized and delocalized SP modes could be exploited to tailor field distributions of the electromagnetic field at a nanoscale. For example, by controlling the interference between delocalized and low Q-factor localized SP modes, light transmission resonances of microstructured metallic films could be manipulated, leading to an enhancement or suppression of the transmission amplitude [31–34]. Moreover, for some metallic nanostructures which support narrow-band localized void SPs, the coupling between void SPs and delocalized SPs exhibits interesting optical phenomena. For example, frequency-selective enhanced absorption of light  and diffraction of light  have been predicted for a periodic lattice of dielectric spheres buried just beneath the surface of a metal substrate.
On the other hand, dense metallic NP arrays also possess rich optical properties, for which multipolar SP modes and their near-field interactions are extremely important [27,37–39]. Our previous work has shown that a monolayer of hexagonal-close-packed (HCP) metallic nanoshells supports a variety of SP excitations, including localized voidlike and spherelike plasmons, and delocalized Bragg-type surface plasmon polaritons (SPPs) . Although the plasmonic properties of single metallic particles with a reduced symmetry have been studied extensively, up to now less is known about the symmetry-breaking effect on both localized and delocalized plasmons in dense NP arrays. In this paper, we numerically study the symmetry-breaking effect in a two-dimensional (2D) HCP metallic nanocup array. We found that for large opening angles of nanocups, strong hybridized resonances due to coupling between localized voidlike and spherelike plasmons can be observed. Caused by the symmetry-breaking of nanoshells, the two localized modes are only weakly excited, if they are not in resonance in spectrum. However, when they are tuned close to each other, an enhanced transmission is mediated as a result of the excitation of a hybridized mode, which can be manipulated by controlling the dielectric constant of the inside and outside media of nanocups. The energy of hybridized modes shows a dependence on the opening angle of nanocups and can be approximately described by an intuitive plasmon standing wave model. Moreover, we have found that although delocalized Bragg-type plasmons vanish when the perturbation to the electromagnetic field distribution becomes large enough due to broken symmetry, they can still be re-established under large opening angles via the excitation of hot spots around the metal rims created by the opening.
2. Numerical simulations
The studied structure is shown schematically in Fig. 1 . A 2D HCP array of symmetry-broken silver nanoshells with an opening is modeled via truncating each nanoshell with an inverted cone whose vertex is located at the center of the nanoshell. The size of the opening of nanocups can be described using the cone angle θ. Three-dimensional numerical simulations are carried out with a commercial finite-element method based package (COMSOL Multiphysics). In the simulations, the subdomains of metallic shells are restricted to a maximum mesh size of 47 nm with the element growth rate of 1.45. Fine meshes (maximum mesh size: 105 nm and element growth rate: 1.45) are applied to the remaining subdomains. The number of degrees of freedom is estimated to be ~3.9×105. A Drude mode is adopted for the relative permittivity of silver shells: ε = 1-ωp2/[ω(ω + iωc)], where bulk plasma frequency ħωp = 9.2 eV. For understanding the essence of the underlying physics, we consider the cases in which metal is lossless by choosing ħωc = 0 eV. The transmission and reflection spectra can be retrieved from the S parameter analysis. The transmission peaks of the arrays exactly correspond to the reflection dips, which means that same conclusions can be drawn either using transmittance or reflectance. For simplicity, in the following we only investigate the transmission spectral features of the nanostructures. Here without loss of generality, the truncated nanoshells are assumed to have an inner and outer radius r1 = 200 nm and r2 = 250 nm, and the lattice period (a) of the nanoshell array being a = 520 nm.
3. Results and discussion
3.1 Localized SPs
3.1.1Spherelike and voidlike SPs
First, a typical transmission spectrum of a complete silver nanoshell array embedded in air under normal incidence is shown in Fig. 2(a) in which a series of transmission peaks with 100% transmittance are observed due to the excitation of different types of SP modes . Basically, the resonance at 479 nm is caused by a delocalized Bragg-type SP mode [41–43] that relies on a strong near-field interaction on adjacent nanoshells. It is well known that SPPs waves propagating along a planar surface of metal cannot be directly coupled to incoming light. But with periodic arranged coupling elements, for example, holes [41,42], voids [7,43], the incident optical field is Bragg scattered and couple to the SPPs, leading to Bragg-type plasmons. The sharp resonance at 532 nm is due to the excitation of the dipolar void mode of individual metallic nanoshells. The resonances at 614 nm and 810 nm can be assigned as the quadrupolar and hexapolar spherelike SP resonances respectively, as they are related to the quadrupolar and hexapolar SP resonances of single metallic nanoshells in a good approximation . In contrast to the voidlike mode, the field of spherelike modes is distributed on the outer surfaces of the nanoshells and thus they suffer more radiation losses and appear so broad .
Figure 2(b) shows how these SP resonances evolve when circular openings on the metallic nanoshells are created and further enlarged. It is seen that as the particle symmetry is reduced, both localized and delocalized SPs can sustain up to large openings. For example, the energy and intensity of the two wide-band spherelike SP resonances keeps almost unchanged when the opening is smaller than 45°. Then they become gradually weak in intensity and finally disappear when the opening is sufficient large [see Fig. 2(b)]. For the dipolar voidlike mode, the perturbation is quite small for θ < 25°. For θ > 30°, this voidlike mode is gradually weakened with a clear red-shift in its location. When the truncation angle is larger than 90°, the voidlike mode becomes coupled to the hexapolar spherelike mode to form a new wide band hybrid mode which displays a blue-shift as the opening angle is increased, as will be explained below. The Bragg-type mode is also observed to keep its narrow-band characteristics over a large range of opening angles. It is interesting to note that although this Bragg-type mode could be suppressed at certain values of opening angles, it could be re-established by further truncating the nanoshells, see Fig. 2 (b) when opening angle is in the region 30° < θ < 100°.
Shown in Fig. 3 are representative electric field distributions of the spherelike SP modes [Figs. 3(a)–(f)] and the voidlike modes [Figs. 3(g)–(i)], for a complete and truncated nanoshell arrays with different opening angles. Note that for a complete nanoshell array, both the quadrupolar [Fig. 3(a)] and hexapolar [Fig. 3(d)] spherelike SP resonances (the order of these spherelike modes can be identified by field vector arrows in figures) have a highly symmetrical electric field distribution, both with a maximum field enhancement between the gaps of adjacent nanoshells. The quadrupolar mode has four hot spots on the surface of nanoshells [Fig. 3(a)], while the hexapolar mode has six ones [Fig. 3(d)] . Since there is very little field enhancement around the top of nanoshells for the spherelike modes, it is thus understandable that these modes can sustain in truncated nanoshells with opening angles of tens of degrees, although small perturbations are indeed observed from the field distribution of the nanocup arrays [Figs. 3(b) and 3(e)]. The field calculations shown in Figs. 3(a) and (d) also help to explain why the quadrupolar spherelike mode can survive in truncated nanoshells with wider range of opening angles than the hexapolar spherelike mode [see Fig. 2(b)], because the latter mode has more nodes on the shells which make them more sensitive to truncations.
Quite different from the spherelike modes, the voidlike mode in 2D arrays of complete nanoshells is highly concentrated within the metallic voids . When the opening windows are relatively small, the characteristic of a typical dipolar void mode is kept well within the symmetry-reduced nanoshell arrays, because leaky energies from the opening to the outside of the nanoshells are negligible [Fig. 3(h)].A clear effect of band broadening of the voidlike dipolar mode is observed when the opening angle is increased beyond 30°, accompanied with decreasing transmission intensity [Fig. 2(b)], which is the result of increasing radiation losses caused by the openings . This is confirmed by the electric field distribution for a truncated nanoshell array with θ = 50° shown in Fig. 3(i) in which the field within the void is nearly uniform.
To more clearly demonstrate the nature and evolution of these plasmon modes, similar to the method used in Ref , we have studied their dispersion properties by calculating the transmittance spectra of the nanocup array under off-normal incidence of light, from which its dispersion properties can be identified. Figure 4(a) plots the transmittance spectra of the nanocup array with an opening angle of θ = 10° as a function of the incident angle under p-polarization at the azimuth angle of array φ = 0° (the angle between the x axis and xy-plane component of incident wave vector). As is clearly seen, for the broad spherelike modes (centered at 810 nm and 614 nm, respectively), they still sustain a wide-band peak with small dispersion, which confirms their localized nature. Also, for the dipolar voidlike mode (at 532 nm), it is found to be independent of the incident direction of light due to its highly localized nature. However, for the Bragg-type mode, we can see that its transmission resonance at λ = 498 nm under normal incidence splits into three branches at oblique incidence whose positions are all angle-dependent. The black dashed lines in Fig. 4(a) represent the dispersion relations of the Bragg-scattered SP modes on a smooth metal-air interface calculated by matching the momentum of a SP with the reciprocal vector Gmn of a 2D hexagonal lattice [40–43]. The split transmission peaks at oblique incidence follow the dispersion lines to a good approximation (especially for the line labeled as G-10G-11) indicating the delocalized nature of this mode. Furthermore, to verify that near-field coupling is essential for this delocalized mode, we have enlarged the array period shown in Fig. 4(b). It is seen that with the increasing nanocup separations, the delocalized mode (at 479 nm) finally disappears when the separation is too large. Under this circumstance, the near-field coupling between adjacent nanocups is suppressed which demonstrates the crucial role of inter-cup coupling in forming this Bragg-type mode.
3.1.2. Hybridization between spherelike and voidlike SPs
In Fig. 2(b), a significantly broad transmission band is observed when the opening angle of nanocups is quite large. In this section, we will discuss the formation of this mode which is in fact the result of a coupling effect of localized voidlike mode with the quadrupolar spherelike mode. To demonstrate this, the normalized electric field distributions that correspond to three marked points 10-12 in Fig. 2(b) are plotted in Figs. 5(a)–(c) . In Fig. 5(a), although an enhancement of field within the cavity near the opening can still be identified, the field distribution shows a mixed pattern of the dipolar voidlike and hexapolar spherelike modes, suggesting a strong hybridization of the two modes. Therefore, a further broadening effect of voidlike mode is understandable, considering its hybridization with originally broad spherelike modes . Note that such a coupling sustains even when the nanoshells are reduced to nearly half-shells for which the field becomes localized between the gaps as well as on the outer surface of nanocups [see Fig. 5(c)].
Here, the strong coupling between the dipolar voidlike and hexapolar spherelike modes can be manipulated by controlling the relative dielectric permittivity within [Fig. 6(a) ] and outside the nanocups [Fig. 6(b)]. For example, as the dielectric constant of the medium (ε2) outside truncated nanocups (θ = 130°) is increased from 1.0 to 2.5, the hexapolar spherelike mode moves towards a longer wavelength; however, the voidlike mode is expected to remain unchanged in its position, although it becomes difficult to identify the position of the voidlike mode when it has coupled to the spherelike mode. In fact, the broad transmission peak centered at about 540 nm for ε2 = 1.0 becomes weaker with ε2 being increased to 2.5, as is shown by red arrows in Fig. 6(a), due to a reduced coupling strength between the voidlike and spherelike modes. In addition, more peaks appear in the visible range with increasing ε2. Among them, the sharp peaks arise from lattice resonance  signaled by Wood anomaly , while the others can be attributed to the higher-order spherelike modes. With the increasing dielectric constant of outer environment, spherelike modes will redshift, making these modes densely distributed in the low energy regime. Note that a tunable coupling effect between localized voidlike and delocalized Bragg-type plasmon modes has been discussed in nanostructured metal surfaces comprised of periodically arranged truncated spherical voids [35,43]. The above strong hybridized resonance could be attributed to the large optical cross section of the dipolar voidhike Mie mode . As the opening angle increases, the dipolar voidlike SP mode is continuously redshifted. Although the hexapolar spherelike mode is weakly excited for θ > 56°, it becomes strongly excited at wavelengths for which the two modes are in resonance.
Figure 6(b) shows the transmission spectra of the nanocup arrays with the same structural parameters as in Fig. 6(a) but with the core having different dielectric constant ε1 = 1.0, 1.5 and 2.0 with ε2 = 1.0 being fixed. For ε1 = 1.0, the main transmission peak labeled as λ1 is attributed to a hybridization of the dipolar voidlike and hexapolar spherelike modes, as illustrated from the corresponding electric field distributions in the inset in Fig. 6(b). When ε1 = 1.5, this intensity of the resonance is decreased due to a weakened coupling as a result of the redshift of the dipolar voidlike mode; simultaneously, a broad transmission peak centered at ca. 740 nm in the long-wavelength region arises. When ε1 is further increased to 2.0, a broad transmission peak at 790 nm labeled as λ2 is observed, which is apparently caused by the strong coupling between the redshifted dipolar voidlike mode and the quadrupolar spherelike mode, as is confirmed from the electric field distributions mapping shown as inset in Fig. 6(b). Meanwhile, with increasing ε1 the higher-order voidlike modes [9,30,40] also occur a redshift, and may couple with the spherelike SP modes. For example, when ε1 is increased to 2.0, a hybridized resonance at λ3 is produced as a result of the coupling between the quadrupolar voidlike and hexapolar spherelike mode [Fig. 6(b)]. The corresponding field distributions show that this resonance has a mixed character of the quadrupolar voidlike and hexapolar spherelike mode.
The hybridized modes can be described as a linear combination of voidlike modes (ψl v) and spherelike modes (ψl′ s): ψ = c1ψl v (θ) + c2ψl′ s(θ, a). For large opening angles, |c1|2 can be negligible as compared to |c2|2 due to a poor capability of energy confinement in such a notched nanoshell structure. Consequently, we just need to know the eigenvalues of the states ψl′ s. Following the method in Ref , a plasmon standing wave model can be adopted to qualitatively predict the resonant wavelength of the hybridized modes as a function of the opening angle of nanocups. In this model, it is assumed that the strong coupling between the two types of localized SP modes leads to the formation of SP standing waves on the outer surface of nanocups. The standing waves have a certain number of nodes and are pinned at the nanocup rims, whose wavelengths satisfy the condition: L = n·λs/2 (n = 1, 2, 3…) . Here, n is the number of half wavelength that fits the curved rim-to-rim distance L, as shown in the inset of Fig. 7 .
Using ks = 2π/λs, the wave vector ks of the standing wave is obtained: ks = n/[2·(1-θ/360)·r2]. The energy of the hybridized modes E(ω) = ħcks[1+ε−1(ω)]1/2 can be obtained, where ε is the relative permittivity of the shell metal, and c is light speed in air. Solutions for n = 4 to the above plasmon standing wave formed as a result of the hybridization of dipolar voidlike and hexapolar spherelike modes are plotted in Fig. 7, where numerical data extracted from Fig. 2(b) are also displayed. Strictly speaking, the effect of the lattice period on the energy of the hybridized modes should be taken into account because the hybridized modes are partially dependent on the near-field interactions between NPs [46,47]. Nevertheless, in our case the plasmon standing wave model gives a good first approximation of the resonant wavelength of the hybridized mode, and is found to be a good agreement with the numerical data, in which the small difference is caused by the neglect of voidlike modes (ψl v) in our model.
3.2 Delocalized SPs
In section 3.1, the localized voidlike and spherelike SP modes and their couplings are investigated. In this section, we will study the influences of the opening size on delocalized Bragg-type SP modes. In Fig. 8 we plot normalized electric field distributions on the xoz plane at transmission resonances with the corresponding wavelengths and opening angles marked as points 13-16 in Fig. 2(b). For the Bragg-type mode at λ = 479 nm of a complete nanoshell array (θ = 0°), the field extends into the air with a longer decay length and has more hot spots than spherelike modes [see Fig. 8(a)]. Among these hot spots, two of which are located close to the top and bottom of nanoshells, suggesting that this mode is more sensitive than the spherelike mode due to the openings on the top of nanoshells. From Fig. 2(b), it is seen that this delocalized mode vanishes for 24° < θ < 30°. This is consistent with the electric field distribution in Fig. 8(b) for a nanocup array with an opening angle θ = 28° (point 14 marked in Fig. 2(b)), from which one can see that the feature of a Bragg-type mode is totally lost under this degree of truncation.
Interestingly, the Bragg-type mode could be re-excited when the opening angle is further increased, although it could vanish for the truncated nanoshell array [see Fig. 2(b)]. This can be confirmed from the field distribution calculation of two nanocup arrays with opening angles θ = 50° and 90°, which are presented in Figs. 8(c) and 8(d). It is seen that the field patterns are very similar to that of Fig. 8(a) for the complete nanoshell array. We believe that the large field enhancement around the opening rims plays the same role as the top hot spots on the complete nanoshell array and thus mediates the formation of the field pattern of Bragg-type modes.
To confirm the narrow band discussed above is the delocalized Bragg-type SP mode, following the method in Fig. 4(a) we have further calculated the transmission spectra of the nanocup array under off-normal incidence of light. Figure 9(a) plots the transmittance spectra of the nanocup array with a typical opening angle of θ = 50° as a function of the incident angle. We can clearly see that the transmission resonance at λ = 498 nm under normal incidence splits into three branches at oblique incidence whose positions are also all angle-dependent. The split transmission peaks at oblique incidence follow the dispersion lines to a good approximation (especially for the line labeled as G-10G-11), further supporting that the narrow transmission band appeared for θ > 30° in Fig. 2(b) are due to the excitation of the delocalized Bragg-type modes. To demonstrate that re-excited mode also relies on the inter-cup near-field coupling, Fig. 9(b) is plotted in which the array period is gradually enlarged from 520 nm to 560 nm. When the period is increased to 560 nm, it is seen that this mode (at 498 nm) disappears, suggesting the essence of near-field coupling of this mode. Thus, we can conclude that via the hot spots around the metal rims created by the opening, the nanocup arrays with certain opening angles can support the Bragg-type mode as the complete nanoshell arrays do.
In this paper, we have investigated variations of optical transmission properties as a 2D HCP metallic nanoshell array is progressively evolves into a semi-shell array. Details on the opening angle dependent transmission spectra are displayed. We have shown that due to the symmetry breaking, both voidlike and spherelike SP resonances in complete nanoshell arrays show a gradual weakening effect as the opening angle is increased. Meanwhile, the voidlike plasmon resonance can strongly couple to the spherelike modes at large opening angles and as a result, a hybridized SP resonance is formed. By varying the opening angle, the relative permittivity of the media within or outside the nanocups, the coupling strength between voidlike and spherelike plasmons can be controlled. Although the delocalized Bragg-type plasmons become weakly excited due to the symmetry breaking as compared to the complete metal nanoshell array, it can be excited again under large opening angles via the near field coupling mechanism. This approach enables us to better understand the symmetry-breaking effect in 2D dense NP arrays. Understanding these plasmonic structures allows for the engineering of devices tailored for a wide range of surface enhanced Raman spectroscopy and sensing application.
We greatly acknowledge financial support from the State Key Program for Basic Research of China under grant Nos. 2007CB613204 and 2012CB921500 and NSFC under grant Nos. 10734010, 11021403. P. Z. and C. J. T. acknowledge partial support from NSFC under Grant Nos. 0804044 and 11104136.
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