1. Introduction

Surface plasmon polaritons (SPPs) have been studied intensively in this decade [1, 2, 3] due to their potential applications in nanophotonics [4], thermal-photovoltaic devices [5, 6, 7], biophotonics and bio-sensing [8], as well as data storage and imaging [9]. For device realization, in particular, how to excite and control SPPs in a desirable way has always been the main concern. Since SPP is highly geometry-dependent, in analogy to the field of photonic crystals, focus has been shifted towards the use periodic plasmonic arrays for guiding and manipulating the excitable resonance modes. To date, several examples including plasmonic Bragg reflectors, lenses and waveguides have been demonstrated by using one-dimensional (1D) and two-dimensional (2D) metallic arrays [10, 11]. For 2D arrays, since the discovery of extraordinary transmission [12], majority interest has been centered on studying cylindrical hole arrays in which subwavelength hollow cylinders are formed periodically onto a flat metal film using lithographic methods [13, 14]. However, Kelf et al. [15] have revealed that the shape of the individual hole also plays a dominant role in controlling the excitation of SPPs. Different shapes could lead to different resonance because the holes can act like plasmonic cavities for confining the electric field and thereby give rise to strong localized resonance [2, 16, 17]. To date, not toomany studies are on this topic. Acquiring the know-how on designing the hole shape to tailor the electric field enhancement could lead to applications in surface enhanced Raman scattering, thermal radiation, etc.

In a recent work, we have reported the fabrication of 2D arrays on gold surface nanobottle cavity interference lithography (IL) [18]. In this article, we study the plasmonic properties of the gold nanobottle arrays by using finite-difference time-domain (FDTD) simulation [19] methodology. In particular, we emphasize on the dependence of resonance coupling, field localization, as well as polarization dependence of SPPs on the profile of bottle neck. It is found that the shape of bottle neck strongly affect not only the field enhancement but also localization for different polarizations.

2. Finite-difference time-domain simulations and fabrication of the Au nanobottle

The FDTD simulations are performed by using the MIT MEEP package [20, 21]. A multi-lorentzian function is used to model the dielectric function of Au [22]. Figure 1 shows the simulation cell used in the FDTD calculations and Figs. 2(a)–(b) shows the SEM images of a sample nanobottle array. A model Au metallic slab of thickness t=1 µm is used in our simulations. The bottle-shape cavities are embedded in the Au slab, with a period a=575 nm. Periodic boundary condition is imposed in x- and y- directions, and perfectly matched layers (PML) are set at the top and bottom of the simulation cell. The bottles are of height h=325 nm, with bottom radius (w) and rim radius (r) equal to 150 nm and 75 nm respectively, and the bottleneck (l) varies from l=0, 150, 200, 250 to 325 nm. Here l=0 nm and l=325 nm correspond to large and small cylindrical holes while the other l’s represent bottle-like cavity with different bottlenecks.

 

Fig. 1. The cross-section of the unit cell defined for FDTD simulations. The cell has thickness t=1 µm and other parameters are defined in the text.

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Fig. 2. (a) Cross-section of the nanobottle array with an aperture of 160 nm, the red line outlines the bottle shape. (b) Plane view SEM image of the nanobottle array.

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3. Plasmonic band structures of different polarizations and bottle geometries

The dispersion relations, or the plasmonic band structures, in Γ-X direction of the first Brillouin zone, for p-polarized (H_y) and s-polarized (E_y) polarized lights are shown in Figs. 3(a)–(f). We first look at the calculated band structures for the p-polarized incident light [Figs. 3(a)–(c)]. Apart from negligible numerical errors, the band structures for p-polarization are essentially identical for different values of bottleneck (l=0, 150, 250 nm). As the model systems are considered to be optically thick, bulk photonic effects such as Fabry-Pérot resonance and guiding mode resonance [23] are negligible. For the p-polarized dispersion relations, the origins of resonant modes should be attributed to Bragg scattering of the SPPs at Au/air interface waves and Wood’s anomalies [13]. To confirm our proposition, we consider Wood’s anomalies estimated in the first Brillouin zone

 

Fig. 3. (a)–(c) Band structures for p-excited SP modes of l=0, 150 and 250 nm, open circles are the excited resonances. Frequency and in-plane wavevector are given in normalized units, where a ₀=1µm. Lines are for visualization purpose, dotted lines are Wood’s anomalies given by Eq. (1), solid lines are given by surface plasmon dispersion relation [Eq. (2)], plasmon excitations are joined by red lines. Both (±1, 0) and (0,±1) SP modes are excited. (d)–(f) Band structures for s-excited SP modes, only (0,±1) SP mode is found.

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where n_x and m_x are integers and ω and k_x denote respectively the frequency and the wavevector parallel to the interface. The dotted lines shown in Figs. 3(a)–(f) derived from Eq. (1) match very well with some of the calculated eigenmodes, indicating they in fact arise from Wood’s anomalies. Moreover, the Bragg scattering of SPPs, which depends solely on the periodicity of the lattice structure, is evident from the solid lines given as:

where ε _Au and ε _air are the dielectric function of Au and air. Equation (2) (solid lines) display similar asymptotic behavior as some of the resonant modes (open circles fitted with red lines). Therefore, the various branches of resonances should be related to excitation of propagating SP resonances for (n_x,n_y)=(±1, 0) and (0,±1). The existence of bottleneck modifies the plasmonic band gap of ω~1.72 at the Γ-point, but the growth of bottleneck from l=150 nm to l=250 nm does not introduce noticeable changes in the band structures, and the band structure for l=325 nm (not shown here) is almost identical to that for l=250 nm. In view of that, we conclude the p-polarized band structures mainly depend on the periodicity of lattice but not the individual shape of cavity.

We next consider the band structures of s-polarized (E_y) incident light [Figs. 3(d)–(f)]. It is well known that s-polarized light do not excite surface plasmon resonance. In fact, most of the resonant modes found can be fitted with Wood’s anomaly equation given by Eq. (1) for all bottle neck lengths. In particular, only Wood’s anomalies are found in the case of l=0 nm. There are, however, some eigenmodes found at ω≈1.70-1.76 when the bottleneck is above 150 nm. These should correspond to the (0,±1) plasmonic mode as reported by Barnes et al. [13], and it is evident from the band diagrams that excitation of such (0,±1) modes depends critically on the presence of bottleneck. On the contrary, further increase of bottleneck does not have a pronounced effect on the band structure. As a result, we conclude that the bottleneck should play an important role in determining the s-excited band structures, while for the p-polarization case the bottleneck only shows insignificant effect on the band structures.

A direct comparison between p-excited and s-excited band structures may provide further insights into the nanobottle array. As (±1, 0) excitations are found only in p-band structures, (0,±1) modes exist in both p- and s- band structures at similar frequencies. The p-excited (0,±1) mode is less reported, and it is generally believed that p-plasmonic excitations should be propagating in nature while s-plasmonic modes are related to localization of energy. However, band structures do not give information on the physical origins of the p- and s- (0,±1) modes and further examination on the p- and s- excited (0,±1) modes is needed.

4. Geometric dependence and near electric field enhancement

As the band structures of (0,±1) mode for p-polarization and s-polarization display similar behaviors, in order to investigate the physical origins of the observed s- and p- plasmonic resonances, and understand how these resonances depend on the geometry of the Au nanobottle cavity, we perform FDTD simulations at resonant conditions to see how the incident light interacts with the nanobottle arrays. For each calculation, a narrow Gaussian source of central frequency ω is operated for a duration t ₀ > 1000 simulation time steps ≈25 µm/c (in seconds). After the source is turned off, the electric field E(r,t) at each position was recorded for a duration of 2T, where T=2π/ω; the resultant spectral component E _ω(r) is the Fourier transform of the electric field. We first consider p-polarized light, shown in Figs. 4(a)–(h) are the spectral field density of electric field |E _ω(r)|₂ on the x-y and y-z planes in the vicinity of the cavity at surface plasmon resonance ω~1.75 and k_x=0.4 for different bottlenecks. As the nanobottle arrays process D ₄ symmetry, the electromagnetic (EM) field of the excited eigenmodes should either be symmetric or anti-symmetric about y=0 and y=±a/2, which correspond to p-polarization and s-polarization respectively. This results in the y-symmetric pattern of |E _ω(r)|². The asymmetry in x-direction is a general property of Bloch modes with a non-zero k_x, which is an indication of propagating waves. The field strength can be adjusted by tuning the bottleneck. For example, by increasing the bottleneck to 150 nm, the maximum field density decreases substantially from ~80 (l=0 nm) to ~20 (l=150 nm) and remains weak at l=325 nm. The decrease of field strength arises from the reduced aperture of the cavity, which makes the scattering of surface waves become inefficient. This results in weak coupling of SP modes to the incident light. Furthermore, our FDTD calculations show that the field is not localized within the bottle. In fact, the incident light cannot enter the bottleneck except for the l=0 case, and the electric field will radiate once the incident source is turned off. As a result, the resonances are due to excitation of propagating SPs. Our calculations thus confirm unambiguously that the (0,±1) resonances excited by p-polarized light should be propagating SP modes rather than localized SP modes. This is, as we shall see, in contrast to the case of s-polarization.

 

Fig. 4. Spectral field density of the p-excited (0,±1) resonances on z=0 and x=0 planes calculated by FDTD for different bottlenecks: l=0 [(a), (b)], 150 [(c), (d)], 250 [(e), (f)], and 325 nm [(g),(h)]. The pink solid lines outline the cross-section and aperture of the nanobottle.

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Fig. 5. Spectral field density of the s-excited (0,±1) resonances for the same cavities shown in Figs. 4(a)–(h).

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To further explore the relationship between the geometry of nanobottle cavity and s-excited (0,±1) modes, we calculate the spectral field density at k_x=0.4 and ω~1.73 for l=0,150,250 and 325 nm [Figs. 5(a)–(h)]. At first glance, |E _ω(r)|² is symmetric in y-direction and have nodal planes located at y=0 and y=±a/2, this is consistent with s-excited EM field, which should be anti-symmetric about the mirror planes. Moreover, |E _ω(r)|² is also symmetric about x=0, as the incoming signal is asymmetric about x=0 (k_x≠0), this symmetry implies that the excited resonance does not have a strong coupling with the in-plane wavevector of the incident light. In addition, the anti-symmetry of E at y=0 implies induction of a strong effective dipole moment p=p_yŷ at the opening of cavity. In the quasistatic limit and dilute limit (a≫r), the effective dipole moment is proportional to

whereα is given by cosh α=a/2r. The summation is due to the multiple images dipole induced along the y-direction [24]. In addition, as (sinhnα)^-2 decays rapidly as n increases, p_y depends strongly on the factor (ε _air-ε _Au)/(ε _air+ε _Au), where its magnitude should be much greater than 1 for metals. The resultant strong effective dipole moment implies there should be an accumulation of induced surface charges near the opening of cavity, which is responsible for the strong local field.

We now consider the dependency of field strength on the bottleneck, in the case for l=0, which is in fact a cylindrical cavity of radius 150 nm. As seen, the electric field is weak with a maximum value of |E _ω(r)|²~2.5, and the field density is zero within the cavity. This is consistent with the calculated band structure [Fig. 3(a)] as the (0,±1) mode is not excited in such condition. However, as the bottleneck is elongated to about half of the bottle height (l=150 nm), the field pattern tends to concentrate near the rim of aperture and the strongest field density is enhanced almost by a factor of 6 (|E _ω(r)|²~16) when compared to that of l=0 nm. Further increase of bottleneck provide additional enhancement of field until a maximum value of |E _ω(r)|²~40 is reached for l=325 nm. More importantly, Our FDTD calculations reveal that the electric field remains strong even after the source is turned off for 1000 time steps, where the incident signal has already been reflected. Therefore, energy is localized within the nanobottle and the (0,±1) resonant modes found in s-band diagram should be considered as localized SP modes. The strong field located at the rim is due to accumulation of free charges induced by the incident wave. Moreover, the position of local field and the field strength can be manipulated by controlling the length of bottleneck. Figure 5(c) shows the lateral field pattern for a bottle with a short bottleneck (l=150 nm). It appears that small amount of electric field energy has “leaked” into the nanobottle and is mainly concentrated at the top and bottom part of bottleneck. In addition, energy is not stored in the body of the bottle. This can be explained by the shape resonance in analogy to the excitation of optical cavity. In other words, the bottleneck can be treated as a resonant wave guide, which allow coupling of plasmon modes corresponding to the upper part (opening) and lower part (void) of the bottle. Similar observation can be found for the case of l=250 nm [Fig. 5(e)]. With the length of bottleneck increases gradually from 150 to 250 nm, energy stored inside the neck rises accordingly. On the other hand, the coupling between the upper and lower openings becomes weak and the field is concentrated at the rim of the bottle. Further increase of bottleneck does not enhance the localized field. When the bottleneck continues to increase, and becomes equal to the height of the cavity, both the field pattern and field density remain almost the same as in the case of l=250 nm. In view of such observation, the bottleneck should play an important role in the field localization of s-polarized light. Conclusively, the excitation of localized SPs is mainly governed by the geometry of cavity. By changing the length of bottleneck, the field strength and field pattern can be adjusted accordingly.

5. Discussion and conclusion

Here, a few comments are in order. According to the band diagrams, the eigenmodes of the p-polarized and s-polarized lights in this study have similar (0,±1) dispersion relations. However, their physical origins are very different from each other. For the case of p-polarization, the resonances are due to excitation of propagating SPs, whereas for the s-polarization case, the effects are due merely to cavity resonance, or in other words, excitation of localized SPs. The effects of bottleneck on p- and s- polarized lights appear to be complementary to each other. While in the case of p-polarization, the field strength decreases gradually with a longer bottleneck, the field for s-polarization will substantially be enhanced. On the other hand, a longer bottleneck causes the s-polarized light to move inside the cavity, which is opposite to the case of p-polarized light, where field is non-vanishing in the nanobottle only when l=0. Consequently, by adjusting the bottleneck, the proposed nanobottle cavity permits a flexible control of field strength and field localization. Last but not the least, it should be noted that the nanobottle array has similar dispersion relations with other 2D metallic surfaces. For instance, our nanobottle cavity is similar to the “nanohole” as reported by Barnes et al. [13]. Yet, the emphasis on this work is on the field manipulation by the bottleneck, which provides a new tunable parameter for SP control. It would be interesting to investigate how to utilize the nanobottle and bottleneck to control other well-known SP phenomena such as extraordinary transmission.

In summary, we have studied the optical characteristics of the nanobottle cavity for both s-and p- polarizations and its bottleneck plays an important role in the manipulation of field strength and energy localization. The nanobottle may be useful in applications of surface-enhanced Raman scattering, second harmonic generation, as well as thermal emission devices. What’s more, our study may provide new insights into the fabrications of novel plasmonic devices.