A fractal is a geometrical shape whose parts resemble, at least statistically, the whole [1]. This often makes fractals self-similar, i.e. parts look like each other and like the whole object. As an example, Fig. 1 shows the construction of the Sierpinski carpet, one of the most studied fractal curves, named after Waclaw Sierpinski who first described it in 1916 [2]. The construction starts with a square of side length l (Fig. 1(a)). This square is then divided into a grid of 3 × 3l/3-sided squares and the central sub-square is removed (Fig. 1(b)). The same procedure is then recursively applied to the remaining 8 squares, in principle ad infinitum. The result of the first four iteration is shown in Fig. 1(c). Importantly, each iteration in the process adds new length-scales to the resulting figure: the length scale of the original square is only l, while at any of the n-th iteration the length scale l/3n² is also added.

 

Fig. 1 Construction of the Sierpinski carpet. (a) First, (b) second, and (c) fourth step of the construction process of the fractal.

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Multiple-scale properties of fractal geometries have proven useful in studying phenomena with recurring features at progressively smaller scales, from eroded coastlines to snowflakes and from Internet networks to galaxy formation [3]. Moreover, the design of fractal geometries has been employed in the development of multiband compact fractal antennas for mobile telecommunication devices and radio-frequency (RF) infrastructures [4]. In the microwave, mid- and far-infrared regime, metallic materials, obtained by cutting fractal slits into metal films, have also shown to resonate at multiple self-similar wavelengths [5–7] or to be able of confining waves below the limit of diffraction [8, 9]. In the visible, nonlinear light propagation has been shown in fractal waveguide arrays [10]. However, the potential of engineered plasmonic nanostructures featuring a fractal geometry has just started being explored.

The interaction between light and matter leads to a wide range of physical phenomena. Surface plasmons (SPs) are light waves confined at an interface between a dielectric and a metal; they arise as a result of the interaction of an incident electromagnetic field with the induced oscillation of the electron plasma of the metal [11]. In recent years, SPs have received increasing attention because of their ability of concentrating light in truly subwavelength volumes [12], thus opening new opportunities to enhance the interaction of photons with small amounts of matter down to the molecular level. In particular, thanks to recent progress in numerical modeling and the spreading of nanofabrication techniques, it is now possible to accurately engineer the optical properties of surface plasmons for specific applications. For instance, tailored plasmonic nanostructures have been proposed for data storage [13], microscopy [14], solar cells [15], optical trapping [16–18], and molecular sensing [19]. Using the frequency invariance of Maxwell’s equations, it has been possible to rescale several concepts of RF antennas down to the optical regime [20], e.g. gap antennas [21] and Yagi-Uda antennas [22], even though this down-scaling raises some technological challenges mainly due to nanofabrication issues and to the high absorption losses of metals at optical frequencies.

On the one hand, the optical properties of fractal self-organized metal clusters, in terms of spectral broadening, light confinement, and enhancement at the nanometer scale, have been studied in detail both theoretically [23, 24] and experimentally [25, 26]. Recently, for example, it has been shown how the statistical distribution of the local density of optical states on disordered semi-continuous metal films exhibits a maximum in a regime where fractal clusters dominate the film surface [27]. On the other hand, because of the same random nature of the cluster formation process, these previous approaches suffer from poor control over the accurate positioning of the fractal nanostructures as well as over their fractal degree of complexity.

In this article, we show that the combination between plasmonics and engineered fractal geometries has far-reaching properties that go beyond rescaling the concepts behind RF fractal antennas down to shorter wavelengths, e.g. the possibility of supporting multiple resonances [7]. We study a specific fractal geometry known as the Sierpinski carpet. We show, in particular, some of its interesting emerging features at visible wavelengths due to an accurate control of the degree of its fractal complexity, i.e. broad spectral responses and sub-diffraction focusing. The application of the latter to optical manipulation is also briefly discussed.

The simulations are performed using the Green dyadic method [28]. In its discretized form, this formalism allows us to calculate the electric field scattered by a nanoparticle upon a given illumination at any position in space, both as a near-field intensity distribution and as a far-field scattering spectrum. The metal volume was discretized into 5nm-cubic meshes and singularity issues were solved using the approach described in [29]. All the calculated spectra are normalized to the volume of the gold of the structure under study, so that they are on a comparable scale.

Ideally the self-similarity of the plasmonic structure should be iterated ad infinitum; practically, however, we limit it to the minimum feature size feasible with state-of-the-art fabrication techniques, i.e. about 20 nm for the minimum structure size and few nanometers for the gaps between adjacent structures. For example, electron beam lithography on positive or negative electron sensitive resists combined with thermal evaporation of gold layers is a process that has been extensively used to fabricate arrays of plasmonic nanostructures, with remarkable results in terms of minimum gaps (a few nanometers) between adjacent structures [19, 30]. This method suffers from the multi-crystalline character of the gold film produced by vapour deposition, where the precision of the fabricated structure is ultimately limited by the size of the vapour grains, typically from about 20 nm [31]. Colloidal metal particles usually offer better crystallinity, but the accurate positioning of colloids onto a substrate still remains challenging, even though, recently, chemically grown single-crystalline gold flakes, after immobilization on a substrate, have been proposed as a platform for focused-ion beam (FIB) milling and other top-down nanofabrication techniques for the fabrication of high-definition nanostructures on a substrate [31].

As is shown in Fig. 2, the Sierpinski carpet-like geometry features a broadband far-field spectrum, which may prove useful to enhance the coupling to a broadband light source, e.g., for solar cell applications [32]. In order to construct our Sierpinski nanocarpet, we follow a bottom-up approach, starting with a 50×50×25nm³ gold pad as basic element (pink structure in Fig. 2(a)) lying on a glass substrate (ɛ_gold is taken from [33], ɛ_glass = 2.3 and ɛ_air = 1). Its resonance upon plane-wave illumination with linear polarization along the x-axis is at 532 nm with 59 nm full-width-half-maximum (FWHM) (pink curve in Fig. 2(b)). The first fractal level can be obtained arranging 8 of these nanopads on a square frame grid leaving a separation of 5nm between them (red structure in Fig. 2(a)). It is resonant at 680 nm with 56 nm FWHM (red curve in Fig. 2(b)). Even though the FWHM slightly decreases, the red-shift of the resonance is indicative of a strong near-field coupling of the nanopads forming the first fractal level. The second fractal level is composed by a frame of the first-order fractals separated by 15nm gap (blue structure in Fig. 2(a)). It resonates at 760 nm with 140 nm FWHM (blue curve in Fig. 2(b)). Finally, the third fractal level is built applying the same rule to the second-order fractals separated by 45nm gap (black structure in Fig. 2(a)). It is resonant at 720 nm with 295 nm FWHM (black curve in Fig. 2(b)), thus overlapping the solar radiation reaching the surface of the Earth [32]. Interestingly, the wavelength of the resonance red-shifts while increasing the complexity of the fractal stucture because of the strong near-field coupling between the elements forming the different fractal levels.

 

Fig. 2 Broadeining of the resonance in a plasmonic Sierpinsky nanocarpet. (a) First three fractal orders of the nanostructure, respectively in red, blue and black, and (b) relative far-field scattering spectra upon plane-wave excitation polarized along the x-axis. In pink, far-field spectrum of the basic building block of the fractal, a 50 × 50 × 35nm³ gold pad. All the spectra are in a logarithmic scale and normalized to the volume of the gold of the respective substructure.

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In Fig. 3 we show that the Sierpinski nanocarpet is capable of focusing light beyond the diffraction limit and might, therefore, prove beneficial for applications that require a tight light spot, such as sub-diffraction imaging. Alternatively, sub-diffraction focusing can be achieved by employing left-handed metamaterials [34]; however, such technologies are still challenging at optical frequencies because of the dimished magnetic permeability of metals in this regime of frequencies [35]. Fig. 3(a) shows the intensity of the electric near-field associated to the second fractal level of the structure of Fig. 2(a) (in blue) illuminated at its resonance, namely λ = 760nm. The evanescent waves induced by the gold nanopads interfere constructively at the center of the fractal and induce a sub-diffraction focus, whose size is about λ/8; for comparison, the focus of a Gaussian beam (NA = 1.25, λ = 760nm) is plotted in Fig. 3(b). Various plasmonic nanostructures have been employed to concentrate the light into a subdiffraction spot, e.g. gap antennas [36] or bow-tie [37]; however, such spot systematically lies in very close proximity to the metal. Interestingly, in the case of the Sierpinsky nanocarpet the center of the focus is ≈ 80nm away from the metal surface and, due to the symmetry of the Sierpinski nanacarpet, this result is nearly independent from the polarization of the incident plane wave. This provides an open region accessible to objects for enhanced light-matter interaction, which is not available in existing standard geometries. For example, gap antennas can focus light in the gap area between their two arms below the limit of diffraction [21,36], but, if the two arms are separated by the same distance of the fractal case (160nm), there is no significant interaction between the two arms and consequently no hot spot (Fig. 3(c)). Moreover, the formation of a hot spot in the gap is limited only to an incident illumination polarized along the longitudinal axis of the gap antenna.

 

Fig. 3 Sub-diffraction focusing by a plasmonic Sierpinsky nanocarpet. (a) Normalized intensity of the electric near-field generated by the second fractal level of the Sierpinsky nanocarpet (blue structure in Fig. 2(a)) upon plane-wave illumination at 760 nm. (b) Intensity (arbitrary units) at the focus of a Gaussian beam (NA = 1.25, λ = 760nm. (c) Normalized intensity in a gap antenna geometry with the same separation as the open region in the fractal geometry. (d–e) Normalized intensities of the electric near-field of two symmetric non-fractal nanstructures obtained by removing elements from the structure in (a). The intensities presented in (a), (c), (d) and (e) are normalized to the intensity of the incident field (enhancement factor). The white scale bar represents 100nm and is in the direction of the polarization of the incident fields.

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The superfocusing properties of the plasmonic Sierpinski nanocarpet can be attributed to the fractal nature of the interference more than to the symmetry of the system. In Fig. 3(d) and 3(e) two different cases of symmetric non-fractal structures, obtained by removing elements from the Sierpiski carpet are presented. In both these cases no high-intensity spot is formed at the center of the structure, far from the metal.

One possible use of the superfocusing properties of the Sierpinski nanocarpet is in optical trapping near surfaces. Even though optical forces arising from surface plasmons and plasmonic optical manipulation have already been demonstrated [16–18], some limitations arise because the trapping spots, which are near the maxima of the electromagnetic field intensity, are typically at the surface and in the proximity of the metallic nanostructures. However, it would be beneficial for many applications, e.g., single emitter manipulation, to have a subwavelength focus removed from the surface and relatively far from the metallic nanostructures. The focus produced by the plasmonic Sierpinski carpet (Fig. 3(a)) has these desirable features. The resulting optical forces [38] are shown in Fig. 4(a) for an incident power on the structure of 10¹² W/m² and a particle of 40nm (ɛ_particle = 2.5), which is notoriously difficult to trap with such low power. The equilibrium position of the force field is ≈ 15nm above the dielectric surface of the substrate. The resulting optical potential well (black line in Fig. 4(b)) is several times deeper than the thermal energy k_BT, allowing for a stable trapping. Also the potentials corresponding to a 30nm (red) and a 25nm (blue) are deep enough to allow at least a metastable trapping.

 

Fig. 4 Trapping forces in the subwavelength focus at the center of the Sierpinski nanocarpet (cf. Fig. 3(a)). (a) 3D distribution of the force field and (b) trapping potential for particles (ɛ_particle = 2.5) of different radii R.

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In conclusion, the design of engineered plasmonic fractal geometries is still a largely unexplored territory that has just started to show its potential. In this article, in particular, we have studied numerically some properties of a plasmonic fractal nanostructure, namely the Sierpinsky nanocarpet. In particular, we have shown that it features a controlled broadband response and superfocusing. The latter, in particular, can be achieved far from the metallic surfaces and the dielectric surface opening new possibilities, e.g., for plasmonic optical manipulation.