1. Introduction

Resonances of individual metal nanowires (nanorods) have recently attracted a great deal of attention due to the possibility of designing resonator and field enhancement effects by properly choosing the nanowire length (Refs. [1–4] and references therein). However, a simpler (two-dimensional) configuration of finite-width metal strips has not yet been analyzed for realizing resonators and achieving strong local fields. It is well known that metal-dielectric interfaces support surface plasmon polaritons (SPPs) which are surface electromagnetic waves being bound to and propagating along the interfaces [5]. A nm-thin metal film embedded in a dielectric supports a short-range SPP (SR-SPP) exhibiting a very strong field confinement to the metal, though at the expense of a large propagation loss [5 6]. For thinner films, the confinement is stronger and the SR-SPP mode index is larger, deviating more from the refractive index of the dielectric. The latter has also the effect of increasing the SR-SPP reflection by strip terminations, a feature that plays a key role in the resonant behaviour of metal nano-strips analyzed in this paper.

A possible application of the metal nano-strip resonators is to utilize the field enhancement that can be achieved near the strip, e.g. for Raman spectroscopy of molecules placed near the surface. Consider a laser illumination of metal nano-strips immersed in an aqueous solution of molecules. Such a scattering configuration is in fact equivalent to a more practical one, in which the nano-strips are placed on a Cytop (a fluorinated polyether) substrate and covered with the solution, because the Cytop refractive index is very close to that of water [7]. Another natural choice of the surrounding dielectric is glass, since a glass substrate can be covered with index-matching liquid containing molecules to be analyzed. These are two configurations considered in the reported work. Throughout the paper we use the dielectric index of either 1.5 (glass) or 1.33 (Cytop/water) and the experimental data [8] for silver and gold indexes. The presented theoretical results were obtained by making use of a Green’s function surface-integral-equation method for the magnetic field, which is quite similar to that found in the literature (see e.g. Ref. [9]). In the calculations all strip corners were rounded by 2nm for numerical reasons.

2. Nano-strip resonators

Let us first consider the scattering of a p-polarized plane wave (propagation in the xy-plane, electric field in the xy-plane, amplitude E ₀) being incident at an angle θ=45° with respect to the x-axis on 10-nm-thick silver strips of different widths w [Fig. 1(a)].

 

Fig. 1. (a). Normalized scattering cross section vs wavelength for 10-nm-thin silver strips of widths w embedded in glass or water. The angle of incidence is θ=45° with respect to the x-axis. (b). Electric field magnitude along the centre line of the silver strip in water at resonance for the widths w=96nm (λ=672nm, θ=90°), w=244nm (λ=680nm, θ=41°) and w=392nm (λ=684nm, θ=90°).

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Note that the considered silver strips of different widths exhibit scattering peaks (or shoulders) at the same wavelength of ~680nm [Fig. 1(a)]. Our approach to choose these specific widths, e.g. for water, was to take as the starting point the scattering-peak free-space wavelength of 672nm for the strip width of 96nm and increase it in steps by λ _SR/2=148nm, λ _SR being the SR-SPP wavelength found with the appropriate dispersion relation [6] for a 10-nm-thin silver film immersed in water. The matching of scattering peaks in wavelength demonstrates that the peaks are related to (standing-wave) resonances in the strips formed by forward and backward travelling SR-SPPs. This conclusion is further strengthened by calculations of the electric-field magnitude through the silver strip at resonance for the three strip widths 96nm, 244nm and 392nm [Fig. 1(b)], where the standing-wave patterns formed by the forward and backward propagating SR-SPPs are clearly seen. The peaks and shoulders in the scattering spectra reveal the wavelengths of resonance excitation of standing-SR-SPP waves in the strip resonators. Note that, although the scattering cross section is relatively small at λ=684nm for w=392nm (water), the field strength is comparable to that in the resonators with smaller w (96nm and 244nm) having much larger (normalized) scattering cross sections. We believe that smaller values of the normalized scattering cross sections for higher-order resonances are due to more complex radiation patterns (producing less radiated powers) compared to the dipole-like radiation of the 1^st-order resonance. Similar features were also observed for the silver strips embedded in glass. In the case of glass, the resonance peaks are found at the wavelength of ~680nm but now for the strip widths of 77, 198 and 318nm, again with the strip width difference being equal to half of the corresponding SR-SPP wavelength. Note that, in the case of glass, the normalized scattering cross section decreases faster with the strip width than in the case of water, which may be a consequence of the larger SR-SPP propagation loss for the former than for the latter dielectric.

 

Fig. 2. Normalized scattering cross section vs wavelength for 10-nm-thin gold strips of widths w embedded in glass or water. The angle of incidence is θ=45° with respect to the x-axis. (b) Electric field magnitude along the centre line of the gold strip in water at resonance for the widths w=75nm (λ=684nm, θ=90°), w=194nm (λ=688nm, θ=41°) and w=313nm (λ=688nm, (θ=90°).

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The scattering spectra and field distributions in the case of gold strips (Fig. 2) exhibit the same features as those discussed above for the case of silver strips (Fig. 1). The differences in the resonant strip width are simply related to the difference in the SR-SPP wavelengths (due to the difference in the metal refractive indexes). In general, all resonances are weaker due to stronger damping and shorter propagation length of SR-SPP supported by gold strips, again due to the difference in the dielectric constants of gold and silver.

In the scattering spectra considered above [Figs. 1(a) and 2(a)], we chose the angle of incidence θ=45° because this incidence allows excitation of all resonances albeit with different efficiencies. It is clear that the excitation efficiency is determined by the symmetry of the resonant SR-SPP field distribution. For example, the SR-SPPs cannot be excited at all with the plane-wave incidence if θ =0°. Note that the resonant fields shown in Figs. 1(b) and 2(b) have different symmetry across the strip resonator (which cannot be directly seen since only the field magnitude is displayed). The effect is that the 1^st and 3^rd -order resonances can be efficiently excited when using the angle of incidence θ=90°. The 2^nd -order resonance cannot be excited at this angle because of symmetry mismatch with the incident field. In order to investigate in more detail the importance of the angle of incidence we have calculated the angular dependencies of the scattering cross section at resonance for 1^st and 2^nd order resonances for both silver and gold strips in water (Fig. 3). In this case, the normalization factor w sin θ was chosen for the scattering cross section, since this factor approximates the width of the strip projection on the wave front of the incident plane wave (we considered θ in the range from 1° to 90° to avoid dealing with a singularity in the limit of θ=0°). It is seen that the 1^st-order resonance for silver and gold have the most efficient scattering peak (for this normalization) when θ=90°, and that the scattering is very weak when θ≈1° in agreement with our expectations based on the symmetry arguments. Similarly, for the 2^nd -order resonances, one notices that the scattering is rather weak for θ=90° and that, in this case, the angle θ=41° is optimum. The presence or not of extinction peaks depending on angle of incidence was observed experimentally for gold and silver nanowires of finite length in Ref. [10].

 

Fig. 3. Normalized scattering cross section vs angle of incidence θ of the plane wave with respect to the x-axis for 10nm-high gold or silver strips. The strip widths correspond to the 1^st or 2^nd-order resonances for standing SR-SPP waves.

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The SR-SPP has the property that it becomes more tightly bound to a metal film as the film thickness h decreases, and equivalently the SR-SPP mode-index deviates more from the refractive index of the surrounding medium. This means that the backward and forward propagating SR-SPPs will be reflected more efficiently from the ends of a thin metal strip with decreasing h. On the other hand, the propagation loss of SR-SPPs increases when decreasing the film thickness h. In order to achieve a high quality factor of the strip resonator, it is desirable to have high reflection and low propagation loss for SR-SPPs, and thus one might expect to find an optimum film thickness h. Our simulations concerning the 1^st-order resonance showed however that this issue is somewhat more complicated (Fig. 4).

 

Fig. 4. Normalized scattering cross section vs wavelength for metal strips made of (a) silver and (b) gold of different thickness h and widths w.

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It is seen that, for silver strips in the investigated range of thicknesses, smaller thickness results in a stronger and narrower peak in the scattering spectrum [Fig. 4(a)]. The reason is that smaller thicknesses result also in shorter strip lengths (corresponding to the 1^st-order resonance) attenuating thereby the effect of the increase in the propagation loss. We did not consider strips thinner than 10nm, because thinner strips are progressively more difficult to fabricate. Ultimately, for very small h, the usage of classical electrodynamics (which we rely on here) becomes problematic, e.g. because of quantum-size effects. For gold strips, it appears that the peak cross section decreases when using h=10nm instead of 15nm while the quality factor slightly increases (due to the decrease in the resonance width). This indicates that the aforementioned compensation of propagation loss by the decrease in the resonance strip length depends on the material dispersion and might have different effects for the same metal in different wavelength ranges. It should be also borne in mind that the scattering cross section normalized in this way can increase a little with h without a resonance being more efficiently excited simply because a thicker strip is a stronger scatterer, which is generally seen for wavelengths away from the resonance. A more detailed study of the near-field distributions (not shown) revealed that in all the considered cases the field magnitude through the centre of the strips [cf. Fig. 1(b)] decreased when increasing the strip thickness.

4. Nano-strip resonators with gaps

Let us consider again the field magnitude distributions along the strip centre line for the 1^st-order resonance [Figs. 1(b) and 2(b)]. It is seen that the field magnitude peaks in the centre (x=0) of the metal strip. Furthermore, because of the SR-SPP nature, this electric field (being inside the metal) is primarily longitudinal, i.e., oriented mainly along the propagation direction (x-axis). Consequently, if one splits the metal strip in the middle by a small gap, say δ=5nm [see the inset of Fig. 5(a)], one may expect to find an increase (in the gap) of the electric field magnitude proportional to the ratio ε_m/ε_d between metal and dielectric permittivities (due to the electromagnetic boundary conditions at the metal-dielectric interface), provided that the fields will not change much inside the metal strips compared to the case of no gap (δ=0). For example, this ratio, for silver and water, amounts to |ε _m/ε _d |≈12 at λ=680nm and |ε _m/ε _d |≈8 at λ=580nm.

 

Fig. 5. (a). Normalized scattering cross section vs wavelength for a structure consisting of two 10-nm-thick silver (Ag) or gold (Au) strips of widths w separated by the distance δ. (b). The corresponding field magnitude distributions (marked with the same colours) along the strip centre line at resonance.

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We conducted the appropriate simulations of this field enhancement effect and found that the 5-nm-wide gap results in a significant blue-shift of the resonance wavelength, which in the case of silver is ~92nm, and in the case of gold ~60nm. Such a shift is not surprising since splitting a strip in the middle modifies the scattering structure precisely at the position of maximum field strength of the resonator mode. It is interesting though that the shift appears to be independent of whether the background medium is water or glass. The field magnitude distributions along the strip centre line at the resonance wavelengths [Fig. 5(b)] show that the field magnitude is indeed significantly enhanced in the gap region (e.g., by a factor of ~12 for the 96-nm-wide wide silver strip in water). The enhancement is larger when using water as compared to glass because the electric field jump at the strip-gap interface, which is proportional to |ε_m/ε_d|, is larger for smaller ε_d. Our analysis of similar structures in air [11] showed that the field enhancement in that case can even reach the factor of ~30. Note that the field magnitude that can be obtained in the gap centre increases with the decrease in the strip thickness h, since the overall field along the strip centre line increases with the decrease of h. In addition, for the SR-SPPs, the magnitude of the longitudinal electric field (x-) component increases relative to the field y-component.

 

Fig. 6. Field magnitude distributions for 10-nm-thin silver strips embedded in water for the angle of incidence θ=90°. (a) Single strip with w=96nm, λ=672nm and (b) two strips with w=48nm separated by a 5-nm-wide gap, λ=580nm.

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In order to give a more complete picture of the field distributions in the strip resonator with and without the gap we have made plots of the field magnitude in the xy-plane for the case of a 96-nm-wide silver strip in water (Fig. 6). It is seen that, for the strip without a gap, the largest fields are found at the corners of the strip with the field maximum being limited by the 2-nm-radius of corner rounding. Finally, we believe that our analysis of field enhancement in a gap provides a qualitative understanding for the mechanisms responsible for experimentally observed large fields in “resonant optical antennas” [12].

5. Conclusion

In conclusion, we have considered optical resonators formed by rectangular gold or silver nano-strips embedded in glass or water. Increasing the strip width by half of the corresponding SR-SPP wavelength resulted in a structure exhibiting the same resonance wavelength in scattering spectra. This observation along with the field distributions at the resonance wavelengths allowed us to conclude that the peaks in scattering spectra are in many cases related to the excitation of standing SR-SPP waves. We found that the angle of incidence of the plane wave on the strip is important for the efficiency of excitation of the resonator modes. Thus, the angle of 90° is optimum for exciting the 1^st-order resonances but not suitable for exciting the 2^nd-order resonances. It was shown the 1^st-order resonance peak broadens as the strip thickness increases due to less efficient reflection of SR-SPPs at the ends of the strip resonator. We have also shown that splitting a metal strip into two strips of equal width separated by a small gap results in a ~10% blue shift of the resonance wavelength and, more importantly, in a significant field enhancement in the gap, e.g. ~12 for a silver strip in water. Finally, it should be stressed that the considered field enhancement in the gap between two strips is solely due to the boundary conditions and not due to gap modes or electrostatic resonances, a circumstance that promises this effect to be well defined and robust.