1. Introduction

The properties of surface plasmons (SPs) in metals and semiconductors [1] have opened new exciting avenues for nanoscale manipulation of light, which represents the new frontier for optics. Since the beginning of research into plasmonics, localized SPs have primarily attracted the interest addressing sensing applications (e.g. by exploiting surface-enhanced Raman scattering spectroscopy), whereas propagating SPs, i.e. SP polaritons (SPPs), have been mostly exploited for the manipulation of optical signals, driving thereby the plasmonic research along two almost independent paths, viz. sensing and waveguiding [2]. With respect to advanced sensing capabilities like enhanced sensing, chemical identification spectroscopy, or bio-molecule detection, probably the most intensive investigations were concerned with nanometer-sized particles exhibiting resonant scattering and field enhancement due to localized SPs (LSPs). With the LSP field confined far below the optical wavelength, a quasi-static approach can be applied to derive general properties of the LSP resonances [3], usually referred to as electrostatic resonances. Different particle shapes, such as spheres, triangles, cubes, and nano-shells have been thoroughly investigated (see Ref. [2] and references therein). However the requirements to geometrical parameters (radius of curvature, shell thickness, etc.) are rather stringent and result in large variations of the observed resonant behaviour for nominally identical structures. Furthermore, the challenging task of assembling these particles poses some limitations to design and development of nanoparticle-based devices.

Recently, another type of plasmonic resonant structures, viz. thin metal wires of finite length (nanorods), has attracted considerable attention [4, 5, 6, 7, 8]. Finite-length nanorods, taking advantage of both the SP field confinement and guiding effects (bringing in a sense together the two paradigms of plasmonics), exhibit retardation-based resonances that occur due to constructive interference of SPP waves propagating back and forth and being reflected by structure terminations. The corresponding SPP mode persists existing in the limit of infinitely small rod radii, in which the mode wavelength tends to zero as well [9]. These (retardation-based) resonances can therefore be realized, similarly to electrostatic resonances, with extremely small structures achieving thereby nanoscale field confinement (along with its enhancement). More recently, thin metal strips (nano-strips) have been also proposed to exploit retardation-based resonance effects [10]. Numerical simulations [11, 12] and optical experiments [13] have demonstrated that the nano-strip structures (which can be viewed as two-dimensional analogues of nanorods) exhibit pronounced resonant scattering and field enhancement effects as well as size-dependent tunability of resonant wavelength. Similarly to the case of nanorods, these features are related to the resonant excitation of tightly bounded SPP modes propagating along nanostrips. In addition, the geometrical simplicity of the strip configuration results in reliability of the fabrication process, with the potential of assembling several nano-strips of different size (and resonant properties) onto the same substrate according to a desired pattern (e.g. for sensing applications). However, in order to address such a design capability a complete investigation of the size-dependent tunability, resonance bandwidth and maximum achievable field enhancement should be carried out.

In this paper we consider the spectral properties of silver and gold nano-strips using numerical simulations with a surface-integral equation method for the magnetic field [14] (see also [15, 16]). Both first-order and second-order resonances are investigated and characterized in terms of the peak wavelength, resonance quality Q-factor, and field enhancement achievable in a very wide range of wavelengths, from the visible to the near-infrared (400–1700 nm). Furthermore, a detailed comparison of a Fabry-Perot type resonance condition with numerically computed data was also carried out, allowing for an estimation of the phase change due to internal reflection of SPPs at strip terminations.

2. Spectral properties of short-range SPPs in thin metal films

Surface plasmon polaritons (SPPs) are electromagnetic waves bound to and propagating along a metal-dielectric interface [1]. When two or more metal-dielectric interfaces are placed close to each other the coupling between SPPs sustained by individual interfaces give rise to SPP super-modes exhibiting very interesting properties [17]. In particular, for the simple case of a thin metal film of thickness t (i.e. two metal-dielectric interfaces displaced by a distance t) two SPP super-modes exist: a short-range SPP (SR-SPP) and a long-range SPP (LR-SPP) mode. This nomenclature accounts for the dramatic difference in propagation length (namely the propagation distance after which the plasmonic field is reduced by a 1/e factor), which results to be in the range of tens of µm and few mm for SR-SPP and LR-SPP respectively. Another nomenclature is also adopted in the literature, namely slow-SPP and fast-SPP modes respectively, accounting for another remarkable difference: phase velocity. In fact, it is well known that SR-SPPs are more tightly confined to the metal-dielectric interface, with a significant field component inside the metal (in a sense, SR-SPPs are more electronic in nature), with a stronger coupling of the field to the oscillatory motion of conductive electrons. This accounts not only for large propagation losses but also causes a significant slowing in the propagation speed, thus an increase in the effective mode index.

 

Fig. 1. (a) Effective index and propagation length of SR-SPP modes sustained by silver films of different thickness t. Lines: exact numerical computation. Points: values from approximate analytical formula given by Eq. (1) [12]. (b) Same as (a) for gold films. Peaks in the propagation length derive from discontinuities in the slope of the metal refractive index curve taken from experimental data [18].

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These general features of SR-SPP modes stem at the base of the interesting properties exhibited by metal nano-strips. Therefore, to address a spectral investigation of these structures, we calculated the effective index and the propagation length of SR-SPPs in a wide range of wavelengths for silver and gold films of different thickness. We assumed metal refractive index data reported by Johnson and Christy [18]. Results are reported in Fig. 1.

Note that the effective index is strongly wavelength-dependent and decreases with increasing wavelength. As an example, for the 10 nm thick silver film n_eff decreases from 3.46 at 390 nm to 1.07 at 1550 nm. On the contrary, the propagation length scales more than linearly with wavelength, and results to be about four times longer in silver with respect to gold, as expected in view of lower silver conduction losses. Note also that thinner films results in slower (higher effective index) and more lossy (shorter propagation length) SR-SPP modes.

In Fig.1 (marker points) we also show effective index and propagation length calculated as n_eff=Re{k_srsp/k ₀} and L=[ 2Im{k_srsp}]^-1 respectively, where k ₀=2π/λ and k_srsp is the SR-SPP propagation constant given by the analytical formula [12]:

It is worth pointing out that these values are in very good agreement with numerically computed data. This demonstrates that the analytical approach to the general properties of SR-SPP modes given in [12] is valid in a huge range of wavelengths. Note also that, in the limit of very thin films, eq. (1) gives the following expression for the effective index: n_eff≃-2/(Re{ε_m}k ₀ t), where ε_m is the metal dielectric constant. Considering that, for noble metals, the real part of the dielectric constant increases in magnitude almost quadratically with increasing wavelength, the previous formula well synthesizes the dispersion properties of SR-SPP noticed above.

3. Nano-strip resonators based on SR-SPP modes

A schematic of the metal nanostrip investigated is presented in Fig. 2(a). We considered both silver and gold strips of different thickness t and width w surrounded by air (ideally vacuum). The strip length along the z-axis was assumed to be much longer than w (ideally infinite), thus allowing a rigorous 2D-modelling in the xy plane. The structure was excited by an incident p-polarized plane wave propagating at an angle of 45° with respect to the x axis, and the strip scattering cross section as a function of wavelength was computed. Fig. 2(b) shows samples of these scattering spectra for three silver strips of 10 nm thickness. Note several peaks emerging from the background, clearly revealing the intrinsic resonant behaviour of these structures.

As recently suggested [10, 11, 12], these resonances can be ascribed to constructive interference between forward and backward travelling SR-SPP modes experiencing high reflection at strip terminations. According to this picture, the metal nanostrip acts approximately as a two-mirror-like Fabry-Perot resonator for SR-SPP modes, with a resonance condition given by the following equation:

n_eff is the effective index of a SR-SPP mode bound to and propagating along a metal film with the same thickness as the strip, m=1,2,3,… accounts for the order of the resonance, and ϕ is a phase change (modulus π) due to reflection at the edges. Typically, in very short strips only the first order resonance is sustained and a single peak appears in the scattering cross-section, as for the 100 nm wide silver strip of Fig. 2(b) exhibiting a quite sharp resonance at 535 nm. On the contrary, longer structures exhibit resonances of different order. As an example, the 300-nm wide strip of Fig. 2(b) shows two resonant peaks in the scattering spectrum, one at 968 nm and another at 589 nm. Note that, in any case, the first order resonant wavelength is much longer than twice the strip width. This particular behaviour is precisely due to the high effective index exhibited by SR-SPP modes, which causes Eq. (2) to be fulfilled for wavelengths much longer than the geometrical size.

 

Fig. 2. (a) Schematic of the metal (silver or gold) nanostrip. The strip is illuminated with a p-polarized plane wave propagating at an angle of 45° with respect to the x axis. (b) Scattering cross section (normalized to the strip width) as a function of wavelength for 10 nm thick silver strips of three different widths w. Black and white arrows indicate first and second order (harmonic) resonances, respectively.

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4. Broad-band wavelength tunability of nano-strip resonators

The resonance condition given by eq. (2), though providing an elegant explanation of the resonant behaviour and much physical insight, doesn’t allow to predict the position of resonant scattering peaks unless the phase parameter ϕ, which is also expected to be wavelength-dependent, is identified. Unfortunately, an evaluation of this parameter starting from first principles is quite complicated and still lacking. Furthermore, the accuracy of Eq. (2) for very short (w<100 nm) and very long (w~1 µm) nano-strips has been never checked.

 

Fig. 3. Strip width w as a function of the desired (first or second order) resonance wavelength for silver strips of different thickness t. Lines represent linear fitting of the data.

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For these reasons, a complete set of simulations was carried out by changing the strip width w in a huge range of values, comprised between 40 nm to 1200 nm, for silver and gold nanostrips of thickness t=5, 10, 15 nm. For each width w the scattering cross-section was calculated similar to Fig. 2 and first and second order resonance wavelengths were identified with 0.5 nm accuracy. Results are collected in Fig. 3 (for silver) and Fig. 4 (for gold) and presented according to a design point of view by showing the strip width as a function of the desired resonant wavelength.

 

Fig. 4. Strip width w as a function of the desired (first or second order) resonance wavelength for gold strips of different thickness t. Lines represent linear fitting of the data.

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Note that both first and second harmonic resonances can be tuned in a huge range of wavelengths, starting from 400 nm for silver and 530 nm for gold structures, by simply changing the strip width according to a fairly linear relation (see linear fitting curves). A detailed analysis of this linear behaviour and comparison to resonance condition given by Eq. (2) are reported in section 6. It is worth pointing out here that thinner strips exhibit longer resonance wavelengths for a given strip width and resonance order. Comparing this behaviour with that of nanorod resonators/ antennas [8], we would like to note that the effective index of SPP mode supported by a nanorod increases with the decrease in the nanorod diameter [9], quite similar to the SR-SPP effective index increase with the decrease in the film thickness [12]. For this reason, thinner nanorods are also expected to exhibit longer resonance wavelengths for a given nanorod length [8]. Note also that even 5 nm thin strips can’t be precisely described according to the electrostatic approximation, namely by using the approximate formula n_eff≃-2/(Re{ε_m}k ₀ t) instead of Eq. (1). In fact, according to this approximation, a factor of 2 decrease in the slope of the tuning curve is expected by moving from 10 nm to 5 nm thick structures. On the contrary, Fig. 3 and Fig. 4 clearly show that the slope decrease is actually quite small.

5. Q-factor and field enhancement

5.1. Q-factor

Peak resonances in the scattering spectra of Fig. 2(b) exhibit different sharpness. An interesting point to be addressed is to investigate how this sharpness changes as a function of the resonant wavelength. We computed the resonance Q-factor of the first and second order resonances according to the formula Q=λ_P/FWHM, where λ_P is the peak wavelength and FWHM is the Full Width at Half Maximum of the resonance in the scattering spectrum [see Fig. 2(b)].

Results are presented in Fig. 5 for the case of 10-nm thick silver and gold nano-strips. Note that for silver structures both first and second order resonance Q-factor monotonically decreases with increasing resonant wavelength. This is in agreement with the dispersion curves of Fig. 2. In fact at longer wavelengths the effective index is lower, and this results in lower reflection at strip terminations, thus in a lower Q-factor. Note also that since the propagation length is typically much longer than the strip width, the reduction in propagation loss per unit length attained for longer wavelengths is negligible. Furthermore, even when the strip width is of the same order as the propagation length, i.e. in the short wavelength range, an increase in resonance wavelength implies an increase in the strip width also (to fulfill the resonant condition), thus even if propagation loss per unit length is reduced, the overall contribution to propagation losses may increases because of the longer structure, thus contributing anyway to a decrease in the Q-factor. Figure 5 also shows that second order resonance Q-factors are almost twice as high as first order resonance Q-factors at a given wavelength. Actually, second order resonances are attained in much wider strips and this results in much higher energies stored in the resonator thus contributing to an increase in the Q-factor. Indeed, the Q-factor is proportional to the resonator length and inversely proportional to the round-trip loss [19]. In the considered situation with the reflection loss being dominant (over the propagation loss), one should expect the Q-factor to be simply proportional to the strip width for a given thickness and wavelength, which is quite close to our observations (cf. Fig. 3 and Fig. 5).

 

Fig. 5. Q-factor of the first and second order resonances for silver (black) and gold (red) strips of 10 nm thickness.

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If we look at gold structures, the result is similar, with an exception occurring for the first order resonance, showing a non-monotonic trend, with a maximum approximately located at 620 nm. Note that this peak wavelength is very close to the minimum of the imaginary part of the gold dielectric constant (~660 nm, see [18]). In view of the fact that the propagation length is significantly lower for gold compared to silver, especially below 600 nm (see Fig. 1), propagation losses dominate the Q-factor in the short wavelength range, thus giving rise to an optimum value.

5.2. Field Enhancement under resonant excitation

As already demonstrated, metal nano-strips are capable of a significant enhancement of the electric field [10, 11, 12, 13]. We report here a spectral investigation of the field enhancement under resonant excitation (i.e. the incident wave was precisely tuned to resonance scattering peaks).

We computed the field enhancement (namely the ratio |E/E ₀| with E ₀ being the electric field of the incident plane wave) at the maximum of the standing wave pattern of the electric field inside the resonator (point M in the inset of Fig. 6). Though this point is actually not experimentally accessible, nevertheless it is interesting to inspect the point by numerical simulations, because it reveals how the field enhancement is connected to the resonant behaviour. As can be seen from the results of Fig. 6, both silver (black solid squares) and gold (red solid triangles) field enhancement curves precisely resemble the ones of the resonant Q-factor.

We also computed the field enhancement at the strip edge (point D in the inset of Fig. 6), where the field is expected to be significantly higher in view of the discontinuity of the dielectric constant across the metal-dielectric interface. The field value was taken 0.5 nm from the strip edge for numerical reasons. This position anyway also results in a more physical estimation of the maximum sensing field providable to nm-sized particles placed in contact with the strip, in view of possible nano-sensing applications. A field enhancement as high as 10 and 13 was obtained for first and second order resonances respectively. Note that enhancement values for silver and gold are comparable and almost wavelength independent in the long wavelength range. On the contrary, they result to be significantly different in the short wavelength range, where again SR-SPP propagation losses in gold become dominating, and reduce the field enhancement at the edges. Note that for first order resonances the field enhancement can be significantly higher if the incident field propagates along the y-axis instead of 45° relative to the x-axis, even if anti-symmetric field configurations (i.e. odd order resonances) can’t be excited in that case (see [11]).

 

Fig. 6. (a) Field enhancement as a function of the first order resonance wavelength for silver (black) and gold (red) strips of 10 nm thickness, computed at the maximum of the field inside the metal (point M in the inset) and in the dielectric just outside the strip edge (point D in the inset). Inset reproduces a typical cross section of the electric field magnitude along the x-axis through the center of the strip. (b) Same as (a) for second order resonance. Inset shows field enhancement under detuning condition with respect to the resonant wavelength for 10-nm thick, 300 nm wide silver (black squares) and gold (red triangles) nanostrips.

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We also checked the sensitivity of the field enhancement to the resonance condition by inspecting its variation as the excitation wavelength is tuned out of resonance. Typically we observed a significant reduction in the field within a detuning range as large as the FWHM of the corresponding resonance peak, even if the maximum seems to be slightly red-shifted by about 5 nm [see inset of Fig. 6(b), showing detuning effect on second order resonance for 300 nm wide silver and gold strips]. The 5nm red-shift of the maximum means that maximum field enhancement and maximum scattering are not found at the exact same wavelength. Anyway field enhancement at the edges results to be mostly of a resonant nature.

6. Estimation of the phase change due to reflection

Previous simulations (see Fig. 3 and Fig. 4) attested a linear tunability of the resonance with the strip width, despite of the strong nonlinear dispersion exhibited by SR-SPPs. This interesting feature is here investigated in detail, by using numerically computed resonance wavelengths to predict the phase using Eq. (2). We considered 10-nm thick silver strips, and ultra-short structures (w=20 nmand w=30 nm) were also comprised in the analysis. Figure 7(a) shows the strip width as a function of the resonant wavelength for first, second and third order resonance (points). Numerically computed data have been fitted (according to the least-squares method) by linear equations: w_m(λ)=a_mλ+b_m; m being the resonance order [see dashed lines in Fig. 7(a)]. Fitting parameters resulted to be: a ₁=0.4786, a ₂=1.0231, a ₃=1.5446 and b ₁=-155.57 nm, b ₂=-305.45 nm, b ₃=-436.88 nm. The squared correlation coefficient (R²) was greater than 0.999, thus ascertaining a fairly linear behaviour, which is certainly beneficial from a design point of view.

 

Fig. 7. (a) Strip width w as a function of the desired (first, second and third order) resonance wavelength for 10 nm thick silver strips. Points: exact numerically computed data. Dashed lines: linear fitting of exact data. Solid curves: prediction according to Eq.(2) and assuming a wavelength independent phase. (b) Estimated phase change due to reflection at strip terminations. Points: phase estimation from exact data. Curves: phase estimation from linear fitting data.

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According to Eq. (2), for a given wavelength λ, the strip width fulfilling the m-th order resonant condition is expected to be given by w(λ)=(mπ-ϕ)λ/[2πn_eff(λ)]. As previously stated, the phase term ϕ due to reflection at the edges is unknown, but its value can be estimated by fitting numerical data. In Fig. 7(a) we report such a prediction where we assume (wrongly) that the phase is independent of the wavelength (solid curves). Note significant deviations from exact data (points) and linear-fitting lines (dashed lines) too, even if the general trend is well reproduced. Furthermore, the estimated values for the phase resulted to be different for different orders, namely, ϕ ₁=0.72 rad, ϕ ₂=0.9 rad and ϕ ₃=1.2 rad for first, second and third order resonance respectively. Though surprising, this behaviour is in agreement with what was reported in recent theory and experiment [13] for gold nano-strips.

Actually, in light of the strongly nonlinear dispersion exhibited by SR-SPP modes (see. Fig. 1), ϕ is expected to be strongly wavelength-dependent too. An estimation of this wavelength-dependent phase can be directly obtained from Eq. (2) according to the following equation:

Results are presented in Fig. 7(b) assuming for w_m(λ) the exact data (points) or the linear fitting curves (i.e. assuming for w_m(λ) the linear expressions given by numerical fitting of the exact data). Note that the phase (especially for higher order resonances) exhibits a strong dispersion in the short wavelength range, where also the effective index of SR-SPP is mostly non linear. Generally, the phase difference between the first and second order resonance results to be slightly larger than between second and third order resonances. Note also that higher order resonances exhibit a better agreement between phase estimation from exact data and phase estimation from linear fitting data.

7. Conclusion

We investigated the resonant properties of plasmon-polariton metal nano-strips in a huge range of wavelengths, from the visible to the near infrared, by using a numerical tool based on the surface integral equation method. Scattering spectra have been calculated for silver and gold strips of different thickness, and revealed a fairly linear tunability of the resonance by acting on the strip width. First and second order resonances have been also characterized both in terms of Q-factor and field-enhancement. Most of the nano-strip features reported in this numerical analysis can be attributed to general properties of SR-SPPs. It is also found that, while the Q-factor decreases for longer wavelengths due to the SR-SPP becoming closer to the light line, the field enhancement depending also on the metal susceptibility magnitude remains largely unaffected. Finally, a detailed comparison of the numerically computed data with the resonant condition given in [13] was carried out for 10-nm thick silver strips, and allowed for a spectral estimation of the phase change due to SR-SPP reflection at strip terminations.