1. Introduction

Recent advances in nanofabrication techniques have enabled the realization of a wide variety of metallic nanostructure shapes and sizes [1]. Such structures provide a unique and effective means to concentrate and manipulate light at the nanoscale through the excitation of collective electron oscillations known as surface plasmons (SPs). In order to take full advantage of this class of objects, it would be desirable to formulate simple design rules that can capture the essence of their operation, while hiding much of the internal complexity [2, 3]. Deep subwave-length metallic nanoparticles exhibit localized SP excitations which are electrostatic in nature and fairly well understood [4, 5]. Such particles are already extensively used to concentrate light and have enabled a wide variety of optical sensing and spectroscopy techniques, most notably Surface Enhanced Raman Spectroscopy (SERS) [6, 7, 8].

More recently, metallic waveguide structures, such as wires and strips, have also gained significant interest for their ability to support propagating SPs known as surface plasmonpolaritons (SPPs). These SPPs are electron density waves propagating at a metal-dielectric interface that exhibit a strong coupling to electromagnetic fields. The metallic nanostructures that support SPPs thus can serve as miniature optical waveguides and their use for chipscale optical information transport has been suggested [9, 10, 11, 12, 13]. In addition, it has recently been shown that retardation-based optical resonators can be constructed by truncating such waveguides to wavelength-scale dimensions [14, 15, 16]. Local field-enhancements can be achieved in these truncated structures through the excitation and constructive interference of SPP waves that propagate back and forth between the terminations [16]. Because of their high radiation efficiency, these resonators can also be thought of as optical analogs to traditional microwave antennas [17]. As these structures behave as both resonators and antennas, they can logically be termed plasmonic resonator antennas (PRAs) in analogy to dielectric resonator antennas [18]. This type of resonator antenna has effectively been used for Raman spectroscopy and non-linear optics applications [19, 20]. The resonant lengths of these PRAs are strongly dependent on the SPP wavelength and reflection phase [21]. While the calculation of the SPP wavelength can be accomplished using well-established techniques [22, 23], the direct calculation of reflection phase has only been performed more recently for the case of SPPs on thick metal films and for gap SPPs [24, 25]. In addition, recent work has provided indirect estimates of reflection phase for SPPs on strips from their resonant response to optical excitation [21, 26].

In this paper, we directly calculate both the reflection amplitude and phase for thin metal film terminations using an intuitive Fresnel reflection model and full-field electromagnetic simulations. To our knowledge this is the first direct calculation of these reflection parameters for an SR-SPP reflecting off a terminated thin metal film. We then develop a Fabry-Pérot model that uses these parameters to predict the peak position and spectral shape of the field-intensity resonances for metallic strips (truncated films) of different width, thickness, and optical material properties.

2. Optical properties of metallic films and strips

A thin metallic film supports two distinct types of SPP modes: a long range SPP (LR-SPP) and a short range SPP (SR-SPP) [22]. These two modes result from the coupling of the SPPs supported by the two individual surfaces. In this study our focus is on the SR-SPPs, which exhibit substantially increased mode indices and increased field-confinement for films that are much thinner than the wavelength of light. These characteristics naturally give rise to strong reflections off metal film terminations and can result in large local field enhancements in wavelength-scale structures due to constructive interference effects. To illustrate these useful properties of SR-SPPs we start by investigating their behavior on extended films of a thickness, t, with a metal dielectric constant, ε_m, which is embedded in a dielectric with ε_d. Fig. 1(a) shows the significant increase in the real part of the SR-SPP effective index, n′_spp, with decreasing film thickness for a silver film in air. This increase in n′_spp for thinner films is related to a reduced mode size and increased overlap of the SR-SPP mode with the metal. The plot shows similar trends in t/λ_o for three different free-space wavelengths (λ_o=500 nm, 600 nm, and 700 nm). With decreasing excitation wavelength n′_spp increases; the SR-SPP is slowed down due to a decrease in the magnitude of the the metal dielectric constant and an increase in the mode overlap with the metal. Fig. 1(b) shows the rapid decrease in the SR-SPP mode size with decreasing film thickness. Here, mode size is defined as the distance between the points in the two dielectric cladding regions where the electric field decays to 1/e of its peak value. The calculations are based on the analytic solutions to the SR-SPP mode [22] and the optical properties of silver are taken from experimental data [27].

 

Fig. 1. Trends in the guiding and reflection properties of short range surface plasmonpolaritions (SR-SPPs) supported by truncated silver films with normalized metal thickness, t/λ_o. The blue dotted, green dashed, and red solid curves show the trends for ε_m values that correspond to silver at 500, 600, and 700 nm free-space wavelengths respectively. (a) Real part of the effective index, n′_spp, of SR-SPP mode. (b) Mode size of the SR-SPP. (c) Reflection amplitude for an SR-SPP reflecting off of a silver film truncation. (d) Phase pickup for an SR-SPP reflecting off of a silver film truncation. (c) & (d): The thin lines were calculated with the Fresnel reflection model and the thick lines were obtained from full-field simulations.

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In contrast to SR-SPPs, the LR-SPP modes are poorly confined, exhibit low mode indices, and provide little field enhancement [16]. LR-SPPs are ideally suited for the realization of low-loss plasmonic components [28], but not for strip PRAs and thus these modes are not further considered in this paper.

3. Fabry-Pérot resonator model

A metallic strip, such as shown in Fig. 3(a), can be generated by truncating a silver film of thickness t to a width w. When top-illuminated with the electric field polarized along the x-direction, SR-SPPs will be excited at the truncations and start propagating back and forth between the antenna end-faces. When the SR-SPP encounters an end-face it will partially reflect and partially scatter into free-space modes. For a properly chosen strip width, constructive interference of multiply-reflected SR-SPPs will occur and generate resonantly enhanced fields in the vicinity of the strip.

It was recently suggested that the metal strip can be treated as a Fabry-Pérot resonator for SR-SPPs where the reflection phase can dramatically affect the resonance condition [16]. In this study we will provide further evidence that wavelength-scale metallic structures behave as SR-SPP resonators. This is a valuable conclusion as it allows for a more intuitive way of thinking about this class of structures. Since the properties of SR-SPPs can be captured in just a few normalized geometric and materials parameters (w/λ_o, t/λ_o, and ε_m/ε_d), it also enables a complete description of the resonant optical properties of metal strips in terms of these normalized quantities. For example, the resonant width of a metal strip can be written in terms of the SR-SPP wavelength (λ _spp=λ_o/n′_spp) and reflection phase which only depend on t/λ_o and ε_m/ε_d. On resonance, the round trip phase must be equal to an integer multiple of 2π. For the case of the metal strip this implies that (2π/λ _spp)2^w _res,m+2ϕ=m2π where m is the order of the resonance. This then leads to

From the equation, it is clear that a larger n′_spp or a larger ϕ will result in a shorter w_res,m. It is worth noting that for structures exhibiting large reflection phases, our microwave intuition that suggests w _res,1=λ_o/2 is insufficient to predict their resonant widths. Below we will show that reflection phases exceeding π/2 can be expected in this system which can cause shifts in the resonance width, w _res,1, from λ _spp/2 to less than λ _spp/4.

One of the most exciting characteristics of nano-strip resonators is their ability to generate local field enhancements for the resonant widths, w _res,m. With the Fabry-Pérot model, the dependence of the field intensity at the end-faces, |E _end|², can be found by summing contributions from the multiply-reflected SR-SPPs that were launched onto the strip. A simple addition of these fields gives

In this equation we express |E _end|² as a proportionality rather than an equality since the (possibly frequency-dependent) coupling efficiency of the incident wave into SR-SPPs is unknown. Note that for symmetry reasons only odd modes (m=1,3,5…) can be excited in the considered top-illumination geometry. Through a careful comparison to full-field simulations, we will show that the frequency dependence of the coupling efficiency is weak and the Fabry-Pérot resonances occur where exp[i2k _spp ^w+i2ϕ] is close to unity, i.e. where Eq. (1) holds, and the quantity in Eq. (2) is maximized. Note that the numerator in Eq. (2) gives rise to an asymmetric line-shape as a function of w and slightly shifts (by less than 5%) the resonance maximum from the w _res,m predicted by Eq. (1). Interestingly, the weak frequency dependence of the coupling also enables an accurate prediction of the resonant line-shape and thus the quality factor, Q.

4. Determination of reflection amplitude and phase

In order to test the validity of the Fabry-Pérot model, we first estimate the amplitude and phase of SR-SPP reflection off of a film truncation using a Fresnel reflectivity model. In this simple estimate, we treat the metal film as a uniform medium with an effective complex refractive index n _spp=k _spp/k_o, where k _spp is the in-plane wave vector of the SR-SPP supported by the film. Reflections from a termination are then obtained by considering a plane wave propagating in a uniform medium of n _spp and reflecting off a dielectric with an index equal to that of the embedding medium (n=1 in our case). The reflection amplitude, |r|, and phase pickup, ϕ, for the SR-SPP wave can now be calculated based on this effective index contrast using the well-known Fresnel equations [29]:

The thin lines in Fig. 1(c) show the dependence of the reflection amplitude on thickness for the same ε_m values as in Fig. 1(a) and (b). For sufficiently thick films, the reflection parameters asymptote as the SR-SPP becomes more like a SPP on a semi-infinite metal film. However, as the film thickness is decreased, the increase in the SR-SPP mode index leads to a larger reflection amplitude that tend towards unity for films that are thinner than just a few percent of the free-space wavelength. Within the confines of the Fresnel reflection model, the reflection phase would be equal to zero for the case of a lossless metal. However, real metals exhibit loss and give rise to a non-negligible phase pickup. For low-loss noble metals this Fresnel reflection model predicts a small phase pickup as seen in Fig. 1(d) for silver.

The Fresnel reflection model (Eq. (3)) is expected to underestimate the reflection amplitude and phase because it does not take into account the significant mode mismatch between the SR-SPP mode and the free-space modes it couples to at the end-faces. Fig. 1(b) shows that the mode size can drop well below the free-space wavelength and modifications of the reflection properties should be expected. To analyze these effects and more accurately predict the end-face reflection properties, we performed full-field finite-difference frequency-domain (FDFD) simulations [30] to calculate the reflection amplitude and phase. FDFD allows for the use of frequency-dependent optical constants determined from experiments [27]. Fig. 2(a) shows the simulated geometry consisting of a semi-infinite metal film with an abrupt truncation at which the SR-SPP will partially reflect and partially scatter into free-space modes. To calculate |r| and ϕ at a frequency and film thickness of interest, we launch an analytically-derived SR-SPP mode [22] from x_o towards the end-face of the slab at x_e and monitor its reflection at x_m.

 

Fig. 2. Full-field simulation of an SR-SPP reflecting off of a metal film truncation. (a) Schematic of the simulation geometry showing the truncated metal film with the SR-SPP launch point (x_o), end-face (x_e), and measurement point (x_m). (b) Incident, (c) total, and (d) reflected (total minus incident) tangential electric field, E_x, for a 30 nm thick silver film excited with a SR-SPP mode with a wavelength of λ _spp=465 nm (free-space λ_o=550nm).

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To illustrate the procedure, Fig. 2(b) shows the tangential electric field distribution of the incident, forward-propagatingwave, E_x,i(x,y) for a 30 nmthick silver film at an excitation (free-space) wavelength of 550 nm. Upon reaching the end-face, the SR-SPP scatters and produces a total field E _x,tot(x,y) as shown in Fig. 2(c). Fig. 2(d) shows that near the strip the difference field, E _x,tot(x,y)-E_x,i(x,y), is dominated by contributions from a backward-propagating, reflected SR-SPP, as is expected for such a strongly bound mode. For this reason we call this difference field the reflected field E_x,r(x,y). Sufficiently far from the end-face and within the metal film E_x,r only has contributions from the reflected SR-SPP wave. In those locations, (x_m,y_m), the phase and amplitude of the SR-SPP reflections can be directly computed from our simulated field patterns. To determine the reflection amplitude, |r|, and phase, ϕ, we first write out the incident, E_x,i, and reflected, E_x,r, SR-SPP fields:

It is then straightforward to show that the complex reflection coefficient, r, for the SR-SPP is given by the ratio of the incident and reflected fields as

The thick lines in Fig. 1(c) and (d) show the dependence of the simulated reflection amplitude and phase on film thickness. These curves exhibit similar qualitative behavior as those obtained from the Fresnel reflection model (thin lines). However, the full-field simulations properly take into account the significant decrease in mode size with decreasing film thickness (Fig. 1(b)) and thus predict stronger reflections and substantially larger reflection phases. The rapid decrease in |r| with increasing film thickness can be explained by both the decrease in the SPP effective index and the concurrent spreading of the SR-SPP mode into the surrounding dielectric. The increased mode overlap of the now larger SR-SPP mode gives rise to better coupling to freespace modes and thus a reduced reflection.

Unlike the Fresnel reflection model, these full-field simulations show a very significant ϕ, in some cases exceeding π/2. Similar to the Fresnel reflection model, the reflection phase pickup again tends towards zero as the metal thickness is decreased. This limit can be explained by the fact that the reflection becomes almost perfect (r → 1). For large film thicknesses (t/λ _o) the reflection phase increases and approaches the phase pickup of a single-interface (semi-infinite) SPP reflection as the surface modes decouple.

5. Fabry-Pérot resonances in local field enhancement

Using the methods to determine in |r| and ϕ in the previous section we can now assess the usefulness of the Fabry-Pérot model for predicting the spectral response of metallic strips. To this end we will directly compare full-field simulations of the field enhancement near strips with the Fabry-Pérot model. For our simulations we consider a generic silver strip, shown schematically in Fig. 3(a), with a given thickness (t=30nm) in y, a width (w) in x, and is infinite in z. The strip is top-illuminated under normal incidence by a plane wave with an E-field parallel to the x-axis. Fig. 3(f) shows the simulated local field intensity enhancement, |E _end/E_o|², 4 nm outside of the end of the strip as a function of the strip width and incident wavelength. As predicted by the Fabry-Pérot model, this near-field intensity map indeed shows first-, third- and fifth-order resonances. Representative horizontal and vertical cuts of this resonance map (dashed green lines in Fig. 3(c) and (e) respectively) were made for a detailed comparison to the Fabry-Pérot model. The field intensity distributions corresponding to the resonant peaks in Fig. 3(c) show the odd order resonant modes and the high field enhancement at the PRA end-faces (Fig. 4).

 

Fig. 3. Resonance behavior of 30 nm thick silver strips and comparison with Fabry-Pérot models. Dashed blue lines correspond to the Fresnel reflection Fabry-Pérot model and solid red lines correspond to Fabry-Pérot model with phase and reflection coefficient from fullfield simulations. The dashed green indicates cuts from the full-field strip resonance map. (a) Schematic of the simulated strip geometry. (b) Slice of resonance map and (c) Fabry-Pérot model vs. strip width, w, at λ_o=550 nm. (d) Slice of resonance map and (e) Fabry-Pérot model vs. excitation wavelength, λ_o, at w=0.8µm. (f) Resonance map of 30 nm thick silver strips from full-field simulations. |E _end|² is shown as a density plot, overlayed with predicted resonance peaks from Fabry-Pérot models.

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Fig. 4. Field intensity distributions (|E|²/|E_o|²) for the lowest order resonances (m=1,3, 5) of 30 nm thick silver strips at an illumination wavelength of λ_o=550 nm. For this excitation wavelength w _res,1=130 nm is shown in (a), w _res,3=575 nm is shown in (b), and w _res,5=1040 nm is shown in (c).

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Figure 3(c) shows three resonances in the local field enhancement. From Eq. (1) the spacing between these odd order resonances in w _res,m is expected to match the SR-SPP wavelength of λ _spp=465 nm for the considered excitation wavelength of λ_o=550 nm. The differences between w _res,1 and w _res,3 is 445 nm, just 4.3% smaller than the SR-SPP wavelength. This similarity to the surface plasmon wavelength was used previously to argue that metal strips behave as SR-SPP resonators [31]. It is worth noting that the difference between the m=3 and m=5 resonance widths is less than 1 nm from the SR-SPP wavelength. This indicates that for wide strips the local field intensity enhancement can fully be attributed to resonating SR-SPPs and that narrow strips are dominantly, but not purely SR-SPP resonators.

In Fig. 3(b) and (d) we have also calculated the local intensity enhancement obtained the Fabry-Pérot model (Eq. (2)). The dashed blue lines were calculated using |r| and ϕ from the Fresnel reflection model. For this simple model the Fabry-Pérot resonance peaks occur at too large a width or too short a wavelength when compared to the full-field simulations because this model significantly underestimates the phase pickup upon reflection. The solid red curves in these Figs. were obtained by taking |r| and φ from full-field reflection simulations as shown in Fig. 2. With these more accurate reflection parameters the Fabry-Pérot model predicts peak positions and spectral shapes that are in agreement with full-field simulations of strips. The small observed deviations in the line-shape and peak position may be attributed to minor contributions on the local intensity from other modes as well as to the frequency-dependence of the coupling efficiency for free-space waves into SR-SPPs. The close agreement between the full-field simulations and the Fabry-Pérot model further confirm that wavelength-scale strips behave as resonators for SR-SPPs. It also shows that intuitive Fabry-Pérot models can effectively be used to predict not only the resonance positions, but also their line-shapes.

6. Conclusions

This study on the optical properties of metallic strips has provided further evidence that wavelength-scale metallic structures behave as SR-SPP resonators. This is a valuable conclusion as the properties of SR-SPPs can be captured in terms of just a few normalized geometric and materials parameters (w/λ_o, t/λ_o, and ε_m/ε_d). Moreover, it allows for the SR-SPP reflection amplitudes and phases to be described in terms of these parameters. This in-turn enables the construction of an intuitive Fabry-Pérot model capable of predicting the position and line-shapes of resonant metallic structures for a wide variety of choices for the metal, surrounding dielectric, and structure geometries. Although we have only verified applicability for strips, it is expected that these models can effectively be extended to other wavelengths-scale structures of different cross-sectional shape. We anticipate that the presented concepts will provides optical engineers with a powerful framework for designing the properties of this exciting new class of resonators.