Fractal plasmonic metamaterials for subwavelength imaging

We show that a metallic plate with fractal-shaped slits can be homogenitized as a plasmonic metamaterial with plasmon frequency dictated by the fractal geometry. Owing to the all-dimensional subwavelength nature of the fractal pattern, our system supports both transverse-electric and transverse-magnetic surface plasmons. As a result, this structure can be employed to focus light sources with all-dimensional subwavelength resolutions and enhanced field strengths. Microwave experiments reveal that the best achievable resolution is only, and simulations demonstrate that similar effects can be realized at infrared frequencies with appropriate designs.

Surface plasmon polaritons (SPPs) are elementary electromagnetic (EM) excitations bounded at metal/dielectric interfaces, and attracted considerable attention recently [1]. For a natural material, its plasmon frequency ( p ω ) is fixed by the electron density, so that many SPP-based applications only work at a single frequency.
Recently, people showed that Bragg scatterings can modulate the SPPs significantly, and found high optical transmissions in a silver film drilled with periodic holes [2][3].
However, the Bragg mechanism can only fold the SPP bands into the first Brillouin zone, but can not change the p ω of a material. In 2004, Pendry et al. demonstrated that a metallic plate with periodic square holes can mimic a plasmonic material in terms of SPP properties, with effective p ω being the waveguide cut-off frequency of the hole [4][5]. This opens up a way to design artificial plasmonic metamaterials at any desired frequencies. However, to make the idea work, one has to fill the holes with high-index materials [4][5] which is not easy to realize in practice, particularly at higher frequencies. Very recently, Shin et al. [6] showed that high-index insertion is not necessary if the square holes are replaced by closely packed narrow rectangle holes . Indeed, such a hole has a cut-off wavelength 2 y b λ = much longer than the periodicity along x direction (~x b since the holes are closely packed) [6]. However, since such holes are subwavelength only along one direction, we will show that the generated SPPs on such structures only have transverse-magnetic (TM) polarization travelling along one (x) direction. This limitation restricts the applications of such structures in many cases.
Here, we demonstrate that a metallic plate drilled with fractal-shaped slits exhibits SPPs with p ω dictated by the fractal geometry. Without using high-index insertions [4][5] and distinct from the narrow rectangle hole case [6], here the fractal pattern is subwavelength along all dimensions at resonance. Therefore, we found such a system can be homogenitized as a plasmonic metamaterial to support TM and transverse-electric (TE) polarized SPPs simultaneously. We further show by both experiments and finite-difference-time-domain (FDTD) simulations that our structure works as a super lens to focus light sources with all dimensional subwavelength resolutions (best achievable resolution~/15 λ ), with physical mechanism different from other lenses discussed previously [7][8]. We performed FDTD simulations [9] with dispersive Ag ε given by [10] to calculate the SPP band structure of the designed system. Since this system shows no x-y symmetry, we depicted in Fig. 2 This is intriguing at first sight, since a complex structure at resonance usually exhibits complicated local field distribution, making it difficult to identify the eigenmode polarizations. We can understand this point in the spirit of "metamaterial". As the probing wavelength is much longer than the unit-cell size, one can perform field average to homogenitize the complex structure as an effective medium. Thanks to the high symmetry of the fractal geometry, the averaged field exhibits well-defined polarization characteristic so that identifying the SPP mode polarization is possible.
The crucial advantage of our structure is clear. Whereas a flat Ag film supports only TM polarized SPP [11], our system supports simultaneously TM and TE polarized SPPs related to each resonance. We also studied a metal plate with narrow rectangular holes [6], and depicted its SPP dispersions in the insets to Fig. 2 Indeed, this structure exhibits a SPP band even without high-index insertions.
However, in sharp contrast to our case, such a system supports only a single TM-like SPP band traveling along x direction [ Fig. 2(a)]. This is because the rectangle shape is subwavelength along only one (x) direction, and therefore, the SPP band along y direction can not be formed since the subwavelength condition is not satisfied [4][5].
In contrast, our fractal pattern is subwavelength along all directions and possesses multiple resonances, so that for each resonance, SPP bands along both x and y 3 directions can be formed (see Fig. 2). FDTD simulations show that the plasmon frequencies can be changed via adjusting the fractal geometry and scaling the unit cell size [12]. Therefore, we can in principle design a plasmoinc metamaterial at any desired frequency.
One can employ our structures to realize many SPP-based applications. It was shown both theoretically [13] and experimentally [14] that a silver film works as a lens to focus near field light sources with subwavelength resolutions. However, such a super lens does not function at a frequency other than silver's natural SPP frequency, and since silver only supports TM-polarized SPPs, the source has to be carefully designed to emit p waves only [14]. In what follows, we show that our structures can collect both s and p waves emitted from a source to form an all-dimensional We first performed microwave experiments to demonstrate this idea. We designed a plasmonic metamaterial [picture given in Fig. 3(a)] with unit-cell shown schematically in the right panel of Fig. 3(a) λ ∼ in the present case [12].
Similar effects can be realized at infrared frequencies using the fractal structure designed for Fig. 2. As a comparison, we also adopted the rectangle-hole structure (same as that for Fig. 2) as a lens to focus light sources. We considered two types of source, i.e., two x-polarized dipoles working at 41 THz separated by1 m μ either in x direction (case 1) or in y direction (case 2). For these two cases, we show the FDTD calculated images formed without any lenses in Fig. 4(a) and 4(d), those with a 0.5 m μ -thick rectangle-hole structure lens in Fig. 4(b) and 4(e), and those with our fractal structure lens in Fig. 4(c) and 4(f), correspondingly. Here, the source (image) plane is 0.1 m μ above (below) the lens [see the right panel in Fig. 1]. Since the sources are located within a subwavelength region, two sources cannot be clearly These are all caused by the fact that this structure does not support TE-polarized SPPs (see Fig. 2). With our lens, however, two sources are clearly distinguishable in both cases, and the formed images are subwavelength along all directions, with much enhanced field strength (see the E-field scales in Fig. 4).
To explore the underlying physics, we assume the source taking a simple form 0 ( ) i t J r P r ê ( , ) t x ω δ − = , and calculate the EM fields by a Green's function method [18]. We find the fields on the image plane as || ( cos sin ) 2 plasmonic lens is a, which is much less than the working wavelength.
Our mechanism is different from many others [7][8]. In the mechanisms described in Refs. [7] and [8], the operation frequencies depend the lens thickness. In contrast, our working frequency is independent of the lens thickness, demonstrated both experimentally and theoretically in Fig. 3. Such a unique property makes our structure a good candidate for far field imaging.
In short, we showed that a metal plate with fractal-shaped holes can be homogenitized as a plamonic metamaterial to support SPPs in both TE and TM  the rectangle-hole structure and fractal structure are the same as those in Fig. 2