1. Introduction

Over the last few decades, surface plasmons, i. e. collective oscillations of free electrons at metal-dielectric interfaces, have attracted much scientific attention and been studied for various potential applications, such as light manipulation [1], biomedical sensing [2], energy conversion [3], emission enhancement [4], nonlinear optics [5], etc. Depending on parameters and geometry of the system, different forms of surface plasmons can be excited, ranging from localized surface plasmons (LSP) observed in individual metal nanoparticles to propagating surface plasmon polaritons (SPPs) [6,7] at continuous metal dielectric interfaces. For the same frequency of the electromagnetic wave, the k-vector of SPP, k_spp, is higher than k vector of the photon, k₀, and can be found as [7]

Metal surfaces with periodically modulated profiles, such as a continuous sine-wave grating, can provide matching conditions for SPP excitation [6] at

2. Experiment

In our experiments we use an array of gold nanostrips with periodicity d_s = 358 ± 1 nm, height h_s = 100 ± 10 nm and width w_s = 179 ± 1 nm. For comparison purposes, we use continuous gold gratings with sine-wave profile modulation with various periodicity d_g, depth of modulation h_g = 40 - 60 nm, and thickness of gold film l_s = 60 ± 3 nm. The schematic of the strips and gratings is shown in Figs. 1(a) and 1(b) correspondingly.

 

Fig. 1 Schematics of (a) strips array, (b) sine-wave grating, (c) SEM image of the strips array.

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The sinusoidal gratings were fabricated using the laser interference lithography technique [18,19]. The technique relies on recording the interference of two coherent beams with the photosensitive compound (photoresist). First, a glass substrate was spin coated with S1805 photoresist at a thickness of ~100 nm followed by 1 minute soft bake at 98$\text{°}$ C. The sample was exposed to an interference pattern using He-Cd laser (λ = 332 nm) at an exposure dose of ~1.2 J/cm². Then the sample was developed by MF-26 for 30 second to create the sinusoidal polymer grating. Finally, a thin film of gold (60 nm) was deposited with the thermal evaporation method. Arrays of gold strips on glass substrates were fabricated using a similar method. First with the interferometric technique a photoresist pattern in SU-8 was formed on top of a metal film and then the pattern was transferred into the metal with a reactive ion etching (dry etch).

The reflectivity of the samples, R, is measured using He-Ne laser at the wavelength of 632.8 nm and p polarization. The samples are oriented with the direction of strips/grooves vertical (perpendicular to the incidence plane, see the setup in Fig. 2(a)). The reflected beam intensity is recorded as a function of the incidence angle θ. In the continuous gratings, a dip is observed, Fig. 2(b). This dip is a typical signature of the SPP excitation [7]. Note that in our case d_g < λ and the SPP is excited “backward”, in the direction opposite to the projection of the optical k-vector on the sample plane. The angular position of the dip is well described with the Eq. (2) with n = - 1 as

 

Fig. 2 (a) Setup for angular reflectivity measurement; Reflectivity vs angle of incidence in (b) gratings with d_g = 463 nm (red) and 350 nm (blue), and (c) strips; (d) Narrow feature in the reflectivity in more detail: experiment (points), fitting with Lorentzian line with 2 $\delta =$0.03° (solid trace).

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In the strips, the angular dependence of the reflectivity demonstrates quite a different behavior. No pronounced minimum is observed: Instead, the reflectivity demonstrates a maximum which can be approximated with a sum of a broad peak at ~44-46$\text{°}$ and an additional extremely narrow peak at 49.77$\text{°}$ (see Figs. 2(c) and 2(d)). The angular width of the narrow feature obtained from the fitting with the Lorentzian line with the angular halfwidth of 0.03$\text{°}$, which is on the order of the divergence angle (~1 mrad) for a standard He-Ne laser [21]. Note that a very narrow feature of the opposite character (a sharp dip in reflection) is predicted in the continuous square-wave gratings [10] resulting from strong enhancement of plasmonic fields at certain conditions of excitation. However, we believe that in our case, the origin of the sharp peak is associated with the bright Wood-Rayleigh anomaly expected at the condition [15].

A peak of this origin can be seen in the gratings as well, see the reflectivity of the gratings with d_g = 463 nm, Fig. 2(b), however with a much smaller magnitude and a broader width.

In nanohole arrays in gold and palladium films [14,15], intense peaks in optical transmission spectra with a relatively narrow spectral width of 45 nm were observed and ascribed to the combined effect of the Rayleigh anomaly and SPP resonance on the opposite sides of the metal film. In order to better understand the origin of the peaks in our strips, we measure the spectral dependence of the absorption in the strip array and continuous grating with similar periodicity, d_g = 350 nm, collecting all reflected, scattered and transmitted light. The reflection spectra at different angles are obtained with the sample placed in the center of the integrating sphere (Fig. 3(a)) at the horizontal light polarization and vertical orientation of the strips/grooves. Typical results are presented in Figs. 3(b) and 3(c). No narrow features are observed. Spectral dependences of the total collected intensity show relatively broad dips in the both grating and strips, associated with an increased absorption at resonance conditions. From the width of the dips, Q-factors of the resonance are estimated and presented in Fig. 3(d). For the strips they are approximately half that of the grating for the most of optical range. From the dependence of the dip position on the angle of incidence, one can find the dispersion relationship $\omega$ (k) for the plasmonic waves in the strips and gratings, estimating its k-vector as k = G - k₀ sin θ.

 

Fig. 3 (a) Schematic of the setup in spectral measurements; total reflection spectra for (b) gratings and (c) strips; (d) Q-factor and (e) dispersion relationship for the grating and strips as indicated. Solid curve is the theoretical prediction for a flat gold-air interface.

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In Fig. 3(e), the dispersion curve in the continuous grating is very similar to what is theoretically calculated for a flat gold-air interface using standard parameters for gold [20]. However for the strips, the dependence is noticeably different, featuring a lower slope (slower propagation of the plasmonic mode).

3. Theory

In our numerical simulations we consider the continuous sine-wave structure and the strips array having the same period of 358 nm.

We use the coordinate transformation method [22] to calculate electromagnetic fields in continuous sine-wave gratings, presenting the fields E and H as functions of coordinates u = x and v = z - a(x), where a(x) = h_g sin(Gx) /2. The fields are calculated for different modulation depths, h_g, as the functions of angle of incidence, θ, and photon frequency, $\omega$. Figure 4(a) shows the reflectivity coded as color in the dependence of θ and $\omega$ obtained in the grating with 40 nm modulation which corresponds to our experimental sample. The SPP curve (seen as a relatively narrow feature in yellow, spreading from (1.95 eV, ~50$°$) to (2.3 eV to ~25$°$)) corresponds to the decreased reflectivity and can be well described with Eq. (2). The reflectivity profiles at 42$°$ normalized to the reflectivity of flat gold, R_flat, are shown in Fig. 4(b).

 

Fig. 4 (a) and (b) Reflectivity in gratings: (a) R as the function of incidence angle and photon frequency at 40 nm depth modulation; (b) spectral profiles normalized to that in gold at various modulation depths (curves are shifted vertically). (c) and (d) Reflectivity in strips: (c) R as the function of $\theta$ and $\omega$ at h_s = 100 nm; (d) spectral profiles of R at different strip heights, (e), and (f) snap shot of magnetic fields at the resonance conditions indicated with arrows.

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The combined coupled-wave/transfer matrix approach [23] is employed to numerically simulate fields in the array of strips. The calculated reflectivity is shown in Fig. 4(c) for our experimental strip height h_s = 100 nm. In comparison with continuous gratings (Fig. 4(a)), it shows quite an opposite character. Instead of the SPP-related decrease in reflectivity clearly observed in Fig. 4(a), a curve with almost the same shape and position indicates an increase in R, Fig. 4(b). In similarity with the grating, it corresponds to the propagation of a plasmonic mode along the metal-air interface. The cross-section of the graph at $\theta$ = 42$\text{°}$ for different h_s is shown in Fig. 4(d). As one can see, starting from moderate heights, h_s = 30 nm and higher, the reflectivity maximum is predicted. A “snap-shot” of the magnetic fields at the resonance conditions in the strips with h_s = 100 nm is shown in Fig. 4(e). Calculations predict another branch of plasmons in the strips; propagating along gold-glass interface, see Fig. 4(f), which corresponds to an increased reflectivity as well (Fig. 4(c), low angles and frequencies).

Thus, the results of numerical simulations correspond to experimental observations and predict an enhanced reflection in the strips at the resonance conditions. The physical origin of this effect can be viewed in terms of the resonance-related enhancement of the extinction cross-section [24]. This effect can also be discussed in terms of an equivalent circuit model, which considers the energy exchange between free space and a plasmonic mode in terms of the energy exchange between a transmission line and a resonance cavity. As shown in [25], this model provides a reasonably good description of losses associated with plasmonic modes in sine-wave gratings, assuming the coupling constant proportional to the modulation amplitude squared. Here we consider a plasmonic mode as a single resonance cavity and make an attempt to account for additional channels of energy leakage related to the transmission, scattering and diffraction from the sample, including an additional active impedance, R₂ as shown in Fig. 5. The reflectivity can be found as [26]

 

Fig. 5 (a) Equivalent circuit; (b) spectral profiles calculated at Q = 90, R₂ = 0, and various M²/L₁R₀ = 0.0015 n², where n = 1, 2, 3, 4 as indicated; (c) and (d) calculations at Q = 40, M²/L₁R₀ = 0.0375 and R₂/R₀ = 0, 0.05, 0.3, 0.5 and 1.

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In Fig. 5(b) the reflectivity is calculated at a relatively high Q-factor = 90, an absence of additional losses, R₂ = 0, and a different coupling ratio M²/L₁R₀ varied in the proportion 1:4:9:16. The shape of the resonance dip is similar to what is observed and calculated for sine-wave gratings at the different surface profile modulation depths (compare with the curves presented in Fig. 4(b), where the modulation depth varied as 1:2:3:4). The major difference between curves in Figs. 4(b) and 5(b) is seen in the spectral range higher than the frequency of decoupling of the −1st diffraction order, which was not taken into account in the equivalent circuit calculations.

In order to model effects in the strips, let us assume a Q-factor = 40 and a high mutual inductance, M²/L₁R₀ = 0.0375. An additional energy leakage (related to transmission, scattering and diffraction), is taken into account assuming various R₂ >0. In Fig. 5(c) adding of a small active resistance R₂ = 0.05 R₀ (low energy leakage) leads to an overall decrease in the reflectivity demonstrating a common resonance-related dip in the reflection. However, at high R₂/R₀, instead of a dip, a peak in reflection back to the transmission line is expected at the resonance conditions (Fig. 5(d)). Thus, this simple model predicts the existence of the resonance peak from the resonance conditions themselves without an account for the bright Rayleigh anomaly. In order to explain the origin of the sharp feature in the frames of the equivalent circuit model, one can speculate that in the strips, the major energy leakage mechanism is related to diffraction. (In fact, in our experiment, up to 55% of incoming power is diffracted to the −1th order of diffraction.) If this leakage channel does not exist (at the conditions defined by Eq. (4)), in the equivalent circuit model this situation corresponds to R₂ = 0 and a high reflectivity only at this particular condition (as schematically shown with the dotted line in Fig. 5(d)).

In conclusion, we present detailed experimental and theoretical studies of optical effects in a discontinuous system of gold nanostrips in close comparison with a continuous sine-wave grating of the same periodicity. In spite of a great similarity between collective plasmonic modes in gold continuous gratings and those in arrays of strips, their optical signatures can be significantly different. In the strips, SPPs can be excited at both front and back surfaces, and manifest themselves as a maximum in reflectivity as the opposite to the common SPP-related dip in reflectivity in continuous films. The Q-factor of these modes varies from 10 to 66 in the visible range. The presence of an additional strong and extremely narrow (with the angular width of 0.03°) peak in the strips reflectivity is a very interesting feature and can present new opportunities for applications. The experiments are supported with numerical simulations, and a simple equivalent circuit model is considered which provides a qualitative description of the observed effects.