1. Introduction

Integrated plasmonics operates with surface plasmon polaritons (SPPs) and is hoped to substitute integrated optics, which suffers from its components being bulkier than electronic ones, but in turn much faster than those [1,2]. The success of development of this relatively young branch of optics depends on the variety of active and passive components available for the integration. Plasmonics, having features of both photonics and electronics, in terms of operational elements, bears more resemblance to optics since Bragg mirrors, waveguides, beam splitters, and interferometers are used [3–6]. So, the vast majority of elements used in plasmonics have their ancestors in integrated optics.

One of the most efficient and widely used means to control the propagation of SPPs is the exploiting of a band-gap, or Bragg-grating effect [4,5,7]. Although flexible and suitable for many applications, those structures suffer from considerable out-of-plane scattering [8], which gets significantly smaller for the wavelength longer than the band gap [9]. Another elegant method for controlling the SPP propagation is borrowed from conventional optics. It introduces elements of higher refractive index to define a spatial change in the optical path length [10]. Hohenau et al. use a silica layer (below the cutoff thickness of the guided modes in silica) on top of a metal surface to shift the SPP dispersion relation. This method allows simple structures like lenses and prisms to be made. The possibility of effective and subwavelength SPP waveguiding in such structures has recently been demonstrated at both near-infrared [11] and telecom [12] wavelengths. Finally, the realization of fiber-accessible plasmon waveguides composed of a two-dimensional array of metal nanoparticles on a silicon membrane exploits the periodicity of the structure to tailor the dispersion relation of the SPP modes to match the refractive index of the waveguide with that of the silica fiber [13].

In this work we utilize a similar principle: a 100-nm-period square lattice of gold nanoparticles on top of a gold film is used to form variously shaped structures. The SPP waves propagating along the surface inside the periodic arrays experience an increase in the effective refractive index (ERI) with respect to that of a flat surface SPP. In this case, since the period is considerably smaller than the wavelength, out-of-plane scattering is relatively weak [9]. We show that those structures possess the ERI of about 1.08 and, if shaped appropriately, feature properties of prisms, lenses, and optical-fiber waveguides.

2. Structures and experimental technique

The parameters of all studied periodic arrays are identical, only the shape is different. Arrays are based on 50-nm-high gold bumps (rounded squares with the width of about 50–60 nm) arranged in a square lattice with a period of 100 nm. The structures were fabricated by electron-beam lithography on a resist layer spun on top of a 50-nm-thick gold film on a quartz substrate, evaporation of a second gold layer, and subsequent lift-off. A typical fabricated periodic pattern is shown in Fig. 1(a).

 

Fig. 1. (a) Scanning electron microscope image of the periodic (triangular shaped) structure along with the excitation ridge. (b) LRM image of the free SPP mode excited on a 180-nm-wide straight (gold) ridge. SPP propagation direction is shown by a solid line. (c) LRM image of the SPP beam propagating through a triangular-shaped periodic structure (same as in Fig. 1(a)). SPP propagation direction is shown by a solid line. Dashed line shows the propagation direction of the SPP beam in Fig. 1(b). Note that all the experimental LRM images of the SPP intensity distributions are presented so that the left image border coincides with the excitation ridge position.

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To launch SPPs, we directly illuminated a 180-nm-wide straight gold ridge by a focused laser beam. The ridge was placed 5 µm away from each structure (Fig. 1(a)). We used tuneable (wavelength range of 700–860 nm) Ti:Sapphire laser to make a 5-µm-diameter spot polarized perpendicular to the ridge, thereby defining the direction of the SPP propagation. Note that the divergence of the excited SPP beam is determined mainly by the diffraction and can roughly be estimated as λ/d, where λ is the SPP wavelength and d is the waist diameter of the SPP Gaussian beam. With the 5-µm-diameter waist of the beam, the divergence is appreciable, whereas it is more convenient to have a narrower, self-focusing beam when determining the direction of SPP propagation. We used a defocused excitation in this case. Normally, one would focus the laser beam by means of an objective onto the sample surface, so that the waist position of an emerging SPP Gaussian beam coincides with its excitation point. However, if one moves the focal plane of the objective behind the sample’s surface, the SPP beam is excited with the defocused (converging) Gaussian laser beam and continues to converge upon excitation. This method was used to narrow down the SPP beam.

Mapping of the SPP fields was accomplished via leakage radiation microscopy (LRM) [14,15], a well established technique described in details elsewhere [16].

3. Experimental results

Prism effect. First, we show the refractive property of a nanoparticle array by demonstrating a prism effect. For that purpose, a triangular array of bumps was investigated (Fig. 1(a)). It is an isosceles right-angled triangle placed with one of its catheti parallel to the excitation ridge. Figure 1(b) shows the propagation of a reference beam on a smooth gold film without any structure. Note that the sample was initially rotated several degrees counter clockwise, so that the excitation ridge was not exactly vertical, therefore the reference SPP beam looks as propagating to the right and a bit upwards. Note also that the defocused laser beam was used here to excite SPPs as described above. That narrows the SPP beam allowing determining its propagation direction more precisely. Figure 1(c) shows the SPP beam passing through the structure, which is denoted by the white triangle. The image is obtained at the free-space wavelength λ₀ = 800 nm. Inside the structure the beam is poorly visible. This is due to the array of bumps, which makes the average gold layer in that area thicker and thus the leakage of SPP power into the substrate (where it is collected by an objective) lower. Passing through the triangular array of bumps, the SPP beam is declined towards the base of that structure as it happens with a light beam passing through a glass prism. The propagation direction of the reference SPP beam is shown by the dashed line in Fig. 1(c), and the magnitude of deflection is easily seen. We estimated the deflection angle to be approximately 4.7 degrees, which gave the value of ERI of the structure to be equal to n _eff≈1.08.

 

Fig. 2. LRM image of a SPP beam scattered by (a–c) a 7.5-µm-diameter circular-shaped periodic structure and (d–f) a 15-µm-diameter circular-shaped periodic structure. The images are recorded at free-space wavelengths of (a,d) 730 nm, (b,e) 800 nm, and (c,f) 860 nm. Images (b) and (e) are linked with the movies (629KB and 574KB) showing images of scattering of a SPP beam on the corresponding structures passing across. [Media 1][Media 2]

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Lensing effect. The observed deflection seems sufficient for focusing of SPP beams. Indeed, circular-shaped periodic arrays can be seen as a set of prisms with their angles, particularly near the edges, being considerably larger than 45 degrees. We fabricated two such arrays: one of 7.5 µm and the other of 15 µm diameter. Figures 2(a–c) show scattering of a SPP beam with the first of those two structures recorded at different free-space wavelengths. The images clearly demonstrate the focusing property of the structure (cf. with Fig. 1 in [17]), however neither of the figures shows a SPP focusing to a spot. This has two reasons: (i) the incident beam is not a plane wave, but rather a divergent Gaussian beam, and (ii) there is a sufficient spherical aberration due to the nonparaxial rays participating in the image formation. This set of images demonstrates also a wavelength dispersion of the periodic structure with the highest ERI corresponding to the lowest wavelength. Yet, it is difficult to give a quantitative estimate to the dispersion or to the ERI of the SPP inside this array. This was investigated using 15-µm-diameter circular-shaped array of bumps. With this structure the focusing effect was not clearly observed because of the lower optical strength of the lens due to the decreased curvature. Instead, a well pronounced deflection of a part of SPP beam impinging the edge of the array (Figs. 2(d–f)) was observed. Beside the increased propagation length of SPPs, one can notice a slight decrease of the scattering angle with the increasing wavelength, which indicates also the decrease of the ERI. We estimated those angles to be 24°, 22° and 19° for the wavelengths 730 nm, 800 nm and 860 nm, respectively.

From the images obtained, we evaluated that the border area responsible for the SPP deflection is approximately 2-µm-wide. Thus, the geometrical shape of the array involved in the deflection process is a 2-µm-high circular segment, which has an angle adjacent to the chord of ~30°. For the simplicity of our estimation, we substitute that circular segment with an isosceles 120° prism (180°-30°-30°) and assume a symmetric passage of the beam when the refractive index n of the prism is given by a well-known formula: n = sin(α/2+θ/2)/sin(α/2), where α is the prism angle, and θ is the deflection angle. This gives the ERI values of n = 1.10 at 730 nm, n = 1.09 at 800 nm, and n = 1.08 at 860 nm. Even though it is a rough estimate, it gives an order of magnitude of the ERI dispersion. It also demonstrates the consistency of the ERI value obtained with the triangular structure.

Sets of scattering images when the structures were moved across the stationary SPP beam were also recorded (movies linked to Figs. 2(b) and 2(e)). The sets of images are obtained at the free-space wavelength of 800 nm.

Waveguiding. A simple estimation for the possibility of waveguiding inside rectangular arrays of nanoparticles can be given by determining the V-parameter (normalized frequency) of the planar waveguide and by comparing this parameter with the dispersion curves [18]. In our case of propagating SPPs, the V-parameter is defined as follows: $V=\mathrm{kd}{\left(n_1^2-n_2^2\right)}^{\frac{1}{2}}\approx \mathrm{kd}\sqrt{2n_2\Delta n}$ , where k is the propagation constant of SPP at a flat surface, d is the waveguide width, n ₁ and n ₂ are the effective refractive indices of the waveguide and the outer region, respectively. With n ₂ being 1 and Δn being 0.08, one gets V = 2.98 for a 1-µm-wide planar waveguide at the free-space wavelength λ₀ = 860 nm, which is enough to have one guided mode [18] with about 83% of power concentrated inside the waveguide [19]. A 2-µm-wide planar waveguide is already double-mode (V = 5.95) with about 96% of power of the first and 80% of power of the second mode concentrated inside the waveguide. We note, though, that with the symmetrical illumination of the waveguide, only the first mode is excited since the overlap integral with the second mode is zero.

 

Fig. 3. LRM image of a SPP beam propagating along (a) a smooth gold film and (b–f) a gold film covered with funnel waveguides of the width (b) 1 µm, (c) 2 µm, (d) 3 µm, (e) 4 µm, and (f) 5 µm shown by a white contour line. The funnel region is an equilateral triangle with the side length of 10 µm. The total length of each waveguide (including the funnel region) is 25 µm.

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To prove the possibility of SPP waveguiding by periodic patterns of nanoparticles, we fabricated five arrays having a shape of waveguides of width 1, 2, 3, 4 and 5 µm and of length 25 µm. To facilitate the coupling of a smooth gold film SPP to the guided modes, the input of each waveguide was made in a form of a funnel which is an equilateral triangle with the side length of 10 µm. Figure 3(a) shows the reference SPP beam used to test the funnel waveguides, and Figs. 3(b–f) illustrate the propagation of this beam through the structures under consideration (wavelength λ₀ = 860 nm). As it is expected from the estimation, the narrowest structure supports a mode, which is evidenced by the SPP beam coming out of the waveguide (Fig. 3(b)). In fact, this beam is strongly divergent because the waveguide width is about the wavelength. This divergent SPP beam interferes with the plasmons leaking out of the funnel and then propagating outside, along the waveguide. Three beams discernible at the output of the waveguide are the result of this interference. For wider waveguides the output SPP beam diverges less, and less power leaks out of the funnel. As a result, with the increasing width of the waveguide, the interference pattern converges to a single SPP beam (Figs. 3(c–f)). Figure 3(f) demonstrates almost perfect (without leakage from the funnel region) coupling of the incident SPP to guided modes and a well confined plasmon beam coming out of the waveguide.

4. Numerical simulations

The SPP beam scattering by periodic arrays of particles was numerically simulated using total Green’s tensor formalism and the dipole approximation for multiple SPP scattering by nanoparticles [20–22]. The bumps forming the arrays were approximated by spherical particles, considered as point-dipole scatterers suspended 40 nm above the surface, and characterized with isotropic free-space polarizability obtained in the long-wavelength approximation. First, the dipole moments of particles illuminated with a Gaussian SPP beam were calculated self-consistently, and then the total electric field distribution was determined 130 nm above the sample surface (using in both cases the total Green’s dyadic) [21]. The radius of the particles was a fitting parameter chosen to match the experimental images.

To optimize the calculation resources, we reduced the total amount of scatterers in the modeled array by increasing their periodicity up to 150 nm. As long as the array period is sufficiently smaller than the incident SPP wavelength, we consider this method as a one giving the correct result, provided that the size of the particles is adjusted appropriately. We found that experimental images are best fitted in simulations if the diameter of the bumps is set to 60 nm. Note that this is the value we had in the experiment, even though the array period was smaller. This is because the size of the particles used in simulations is only relevant within the model and influences particles’ polarizability, but does not bear a direct relation to the size of the bumps in the experiment.

Figure 4(a) shows the zoom into the dashed bar in Fig. 2(b). This is the experimental LRM image that we tried to fit in calculations shown in Fig. 4(b). One can see a very good agreement with the main features perfectly reproduced, which confirms the validity of the developed numerical approach.

With the diameter of bumps having been fitted, we were able to model a SPP beam deflection by a periodic triangular structure, similar to that shown in Fig. 1(a), to determine the ERI of the structure. Figure 4(c) shows the result of the modeling. As an incidentwave, we used a Gaussian SPP beam having the waist situated 5 µm apart (before) the array. The divergent SPP beam is partially reflected (inside the structure) by the horizontal cathetus of the triangle. This creates an interference pattern in form of several dark lines along the passed through the structure SPP beam (the lowermost one is clearly pronounced). To estimate the magnitude of the beam deflection, we used its upper part: the dashed line in Fig. 4(c) shows the assumed direction of SPP beam propagation after the periodic structure. The deflection angle was estimated to be approximately 3 degrees, which gives the ERI of the structure n _eff≈1.05. This value is very close to the one obtained experimentally and is somewhat smaller, which can be partially explained by our underestimation of the SPP beam deflection magnitude due to its divergent behaviour. Thus, we showed that the developed numerical approach gives consistent results and can be used to model SPP scattering on various structures prior to their fabrication. Moreover, we showed that the value of the ERI estimated in the experiment is fairly reliable.

 

Fig. 4. (a) The zoom into the dashed rectangle shown in Fig. 2(b) and (b) the electric field intensity distribution calculated at the height of 130 nm above the surface plane in the same region. (c) The electric field intensity distribution calculated (130 nm above the surface) for the SPP beam incident onto a periodic array of bumps having the shape of a right-angle triangle. The dashed line shows the propagation direction of the SPP beam after its scattering by the array and indicates the deflection of the SPP beam by such a triangular prism.

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5. Conclusion

Summarizing, we showed that a periodic pattern of gold nanoparticles with a period considerably smaller than the wavelength of the propagating SPP possesses an effective refractive index of about 1.08. We demonstrated the possibility of creating variously shaped arrays that feature useful properties such as prism effect, lensing, and waveguiding using the total internal reflection. Even though the similar properties had already been observed on dielectric-coated gold surfaces [10], we believe that the periodic patterns are more flexible, since the ERI can be controlled by the filling factor of the structure. In addition, the refractive index can be sufficiently higher at the wavelength close to the resonance of the individual nanoparticle, thus giving rise to non-trivial dispersion properties that can lead to new interesting phenomena.