Hyperbolic Polaritonic Crystals Based on Nanostructured Nanorod Metamaterials

and enhanced non-linearities. [ 22 ] Self-assembled arrays of gold nanorods have recently attracted signiﬁ cant interest as modular optical metamaterials, mainly as a result of their ease of fabrication and inherent optical anisotropy, presenting the opportunity to tailor the frequency onset of hyperbolic dispersion throughout the visible and infrared spectral range. [ 16,23 ] In this paper


DOI: 10.1002/adma.201501325
and enhanced non-linearities. [ 22 ] Self-assembled arrays of gold nanorods have recently attracted signifi cant interest as modular optical metamaterials, mainly as a result of their ease of fabrication and inherent optical anisotropy, presenting the opportunity to tailor the frequency onset of hyperbolic dispersion throughout the visible and infrared spectral range. [ 16,23 ] In this paper, we combine the functionality provided by metamaterials and photonic-crystal-type structures to demonstrate metamaterial-based photonic crystals, termed hyperbolic polaritonic crystals (HPCs). By employing a fully fl exible plasmonic nanorod metamaterial platform exhibiting hyperbolic dispersion and tunable epsilon-near-zero (ENZ) frequency [ 15,24 ] determining the plasmonic behavior of the metamaterial similar to the plasma frequency for bulk metals, [ 25 ] we show how both the resonant response of HPCs and their mode confi nement capability can be controlled. The unique approach described here provides the opportunity to tailor the HPC's optical response both extrinsically, through nanostructuring, and intrinsically through the metamaterial's effective permittivity design, including its hyperbolic dispersion, by dimensional nanorod array parameterization. As a result, HPCs comprised mainly of dielectric patches separated by metamaterial areas only a few nanorods across, provide a fl exible alternative to conventional plasmonic metals in applications requiring on-demand engineering of plasmonic behavior. They may also be used for light coupling to, and extraction from, waveguided modes of hyperbolic metamaterials, inaccessible via conventional or total-internal refl ection illumination due to their very high modal effective index, but which play a signifi cant role in conditioning both the non-linear response of hyperbolic metamaterials and their spontaneous emission properties.
We consider metamaterials composed of plasmonic nanorod arrays ( Figure 1 a) fabricated in self-assembled anodic aluminum oxide (AAO) templates (see Methods in the Supporting Information). The typical extinction spectrum of the metamaterial under plane wave illumination shows two resonances linked to the plasmonic response of the metamaterial for the electromagnetic fi eld polarized along either short or long nanorod axis (Figure 1 a). [ 24 ] These optical properties result from the anisotropy of the metamaterial, which can be described using effective medium theory (EMT) by the effective permittivity tensor xx yy zz ( ) ( ) ( ) ε ω ε ω ε ω = ≠ (see the Supporting Information for details). The spectral dispersion of the permittivity tensor results from the coupling between the plasmonic resonances of the individual nanorods in the array. Hyperbolic dispersion, where ε xx , ε yy > 0 and ε zz < 0, is observed for frequencies lower than the effective plasma frequency (where Re( ε zz ) = 0) of the metamaterial (Figure 1 b). In this regime, the metamaterial has The optical properties of metallic nanoparticles, fi lms, and surfaces are determined by surface plasmon excitations [ 1 ] and provide the framework for the manipulation of light on the nanoscale, [ 2,3 ] including generation, absorption, confi nement, and guiding. [ 4,5 ] Physical limitations imposed by the permittivity of the metal restrict the dispersive properties of surface plasmon polaritons (SPPs) and thus the design of SPP-based components, mostly compounded by the narrow choice of metals with useful plasmonic properties throughout the visible and near infrared spectral range. While signifi cant effort is currently devoted to the development of novel plasmonic materials, [ 6,7 ] the structuring of conventional plasmonic metals also provides a route to control plasmonic waves. Periodically nanostructured metallic fi lms, termed plasmonic crystals, facilitate coupling between photons and SPPs and demonstrate unique and controllable optical properties determined by a combination of design geometry (period and elementary cell), the surrounding dielectric permittivity, and the choice of metal. [ 8,9 ] As a result, the optical resonances of plasmonic crystals, exhibiting strong electromagnetic fi eld enhancement, are possible at frequencies ranging from the ultraviolet to the near infrared, [ 7,10 ] an approach which may be extended to even lower frequencies by utilizing so-called spoof plasmons. [ 11 ] In the same context, the development of anisotropic optical metamaterials, [12][13][14][15][16] exhibiting hyperbolic dispersion, provides the ability to achieve signifi cant control over the effective permittivity and has already led to the design of many new functionalities including subwavelength imaging, [ 14,15,17 ] increased spontaneous emission, [ 18,19 ] high-performance sensing, [ 20,21 ] plasmonic properties for extraordinary waves, i.e., waves with an electric fi eld polarized along the nanorod axes. It should be noted that nonlocal effects, which can be observed in nanorod based metamaterials, [ 26,27 ] were not observed experimentally or numerically in the metamaterial considered, where the bulk-plasma frequency is close to the Au interband absorption and local EMT was therefore applied throughout.
The attenuated total refl ection (ATR) spectra of the metamaterial slab exhibit features clearly illustrating the existence of guided modes, manifested as minima in the refl ection spectrum for both transverse electric (TE) and transverse magnetic (TM) polarizations (Figure 1 c,e). In the asymmetric surroundings of the metamaterial slab, these waveguided modes become leaky between light lines in the superstrate (air) and the substrate (glass), making them accessible to free-space radiation from the substrate, and thus to ATR measurements.
These modes extend throughout the studied spectral range with a high-frequency cut-off determined by the position of the effective plasma frequency p eff ω ≈ 2.17 eV of the metamaterial [ 25 ] (Figure 1 b). For frequencies below p eff ω , TM modes are hyperbolic waveguided modes (HWMs), resulting from the 1D confi nement of hyperbolic plasmon-polaritons supported by the unbounded metamaterial. Both phase and group velocities, as well as group velocity dispersion (GVD) of HWMs are determined by the geometry of the metamaterial, such as nanorod diameter, length, separation, as well as the permittivities of both the metal and embedding medium. TE modes have a greater group velocity than TM modes due to the different effective permittivity of the metamaterial seen by these modes (Figure 1  metamaterial geometry and boundary conditions determine the structure of the supported modes, which have been identifi ed from both the numerically calculated electromagnetic fi eld distributions ( Figure S3, Supporting Information) and analytical theory (see Methods in the Supporting Information).
In order to form the HPC, the metamaterial slab is periodically structured in a square lattice with periodicities ranging from 600 to 800 nm ( Figure 2 a,b) using a procedure described in the Methods in the Supporting Information. This results in the metamaterial being periodically punctured with dielectric patches ranging from 150 nm × 150 nm to 300 nm × 300 nm in size (Figure 2 b). The dielectric patches are separated by metamaterial areas comprised of only 3-4 nanorods for a period of 600 nm (300 nm square dielectric patch) and 8-9 nanorods for a 800 nm period structure (150 nm square dielectric patch). The periodic structuring of the metamaterial generates a family of Bragg-scattered waveguided modes leading to the formation of hyperbolic waveguided Bloch modes, in analogy to the modes of 2D dielectric photonic crystals or plasmonic crystals, where the periodic structuring of a continuous metallic fi lm with a period comparable to the SPP wavelength leads to the formation of SPP Bloch modes on the nanostructured interface. [ 8 ] The periodic structuring ensures momentum matching to couple free-space radiation to the metamaterial Bloch modes as well as coupling to HWMs of the surrounding unstructured metamaterial, in analogy to grating couplers for plasmonic or dielectric waveguides.
In comparison to the unstructured metamaterial, the extinction spectra of HPCs reveal additional resonances in the region of hyperbolic dispersion, having spectral locations dependent on the period of the structure (Figure 2 c). This is consistent with the folding of the modes in the Brillouin zones of the fabricated 2D crystal lattice (Figure 1 c-e). One can see the striking difference between TE and TM Bloch modes related to the difference in the dispersion of the TE and TM modes: TE modes, before folding, have positive dispersion while TM modes show negative dispersion at high wavevectors ( Figure 1 and Figure S4 and S5, Supporting Information). The mode folding due to the periodic structure brings the region of negative dispersion within the light cone, allowing access to these unusual modes under conventional illumination of the HPC, modes which would not be accessible in the unstructured metamaterial. Despite the small number of constituent nanorods between adjacent dielectric patches comprising the metamaterial, hyperbolic waveguided modes are still observed. For small dielectric patches (150 nm × 150 nm, corresponding to the removal of approximately four nanorods), the low scattering cross-section of the patches reduces the Bragg effi ciency of the HPC (Figure 2 d). As the size of the patch, determining the lattice basis, is increased, a small but distinct blue-shift of the resonance wavelengths is observed (Figure 2 d), similar to the blue-shift the SPP band gap edge of the plasmonic crystal with the increasing hole size. [ 28 ] Due to the reduced number of nanorods (3-4 nanorods) forming the metamaterial area between dielectric patches, the Adv. Mater. 2015, 27, 5974-5980 www.advmat.de www.MaterialsViews.com effective permittivity of the metamaterial in the structured area no longer precisely corresponds to the effective permittivity of the metamaterial before structuring. This is due to a modifi cation in the inter-rod electromagnetic coupling depending on local environment, which is different in areas with fewer interacting nanorods. As a result, it is apparent that as dielectric patch size increases, so does the coupling effi ciency of the incident free-space photons to the hyperbolic waveguided Bloch modes supported by the HPC, as evidenced by the increased extinction at the resonant frequencies, and is here related to an increase in the scattering cross-section of the dielectric patches compared to the surrounding media. The ability to couple to these modes via conventional illumination, and in particular at normal incidence, opens up new opportunities to couple free-space radiation to high index, highly confi ned modes, inaccessible through conventional prism coupling and critical for slow light applications and the control of spontaneous emission. Additionally, the introduction of periodic nanostructuring affects the modes both by introducing spectral broadening, resulting from Bragg-induced radiation losses and spectral mode dispersion, due to modifi ed inter-rod interactions ( Figure 3 ). As the in-plane wavevector of the modes increases, their folding at the Brillouin zone edges leads to an observable splitting for TE polarized modes only. For TM modes, the counter-propagating diffracted orders (±1,0) cannot be resolved due to a much lower group velocity of the modes in the range of mode indices probed here. It was theoretically shown that in 1D hyper-crystals formed by natural materials with hyperbolic dispersion Dirac-type dispersion can also be observed. [ 29 ] The unstructured nanorod metamaterial layer supports a variety of TM modes (Figure 1 ). When compared to the dispersion of SPPs at a smooth metal-air interface, it is clear that the metamaterial provides extensive fl exibility in the design of modal effective index, as illustrated in Figure S5 (Supporting Information), where extending the effective medium approach to examine the modal behavior beyond the experimentally measurable range of wavevectors recovers the experimental observations made in Figure 1 c,e for wavevectors corresponding to leaky modes, present between the light lines in the substrate and superstrate. However, the modes of the metamaterial slab exhibit negative group velocity when the modal index is either greater than that of the substrate (fully guided regime) or smaller than that of the superstrate (unbounded modes), in which case, the metamaterial slab essentially acts as a low Q-factor Fabry-Perot resonator. The characteristic electromagnetic fi eld distributions for both TE and TM modes are shown in Figure 4 c-f and Figure S3c-f (Supporting Information) and agree with conventional waveguiding theory. [ 30 ] To illustrate the fl exibility of the system for light guiding at a particular wavelength, the full complex modal index has been calculated using eigenmode analysis for weakly ( n eff < 1.5) and strongly ( n eff > 1.5) confi ned modes, highlighting the ability to choose the desired modal index within a narrow spectral bandwidth. The unstructured nanorod metamaterial layer supports a variety of TM modes with in-plane momentum which can be well-approximated within the effective medium theory as: [ 25 ] c q L zz zz xx where permittivities are assumed frequency-dependent, q is a nonzero positive integer referring to the mode order, and L is the thickness of the metamaterial layer, leading to an effective index for each mode q expressed as: where n xx zz xx zz , eff , eff ε = . When compared to the dispersion of SPPs at a smooth metal-air interface, it is clear that the metamaterial provides extensive fl exibility in the design of the modal effective index with the ability to achieve similar effective index Adv. Mater. 2015, 27, 5974-5980 www.advmat.de www.MaterialsViews.com simultaneously at multiple frequencies as well as narrowband multimodal behavior. For the geometry considered here, effective indices exceeding n q Re 5 eff _ TM ( )= can be achieved simultaneously at multiple frequencies, dictated by the mode order q, throughout both visible and infrared spectral ranges. HWMs show strong fi eld confi nement within the metamaterial layer, a property that scales with n q eff _ TM and that is largely preserved after Bloch mode formation as attested in both the far-fi eld measurements, where HWMs correspond to minima/maxima in refl ectance/extinction, and more directly in the calculated near-fi eld maps (Figure 4 and Figure S3, S5, and S6, Supporting Information). The penetration depth of TM electromagnetic modes on a fl at interface between the nanorod metamaterial and a medium of refractive index n can be expressed as: Similarly, for SPP modes on a fl at Au/dielectric interface, this corresponds to: For a free space wavelength of 1.5 µm, these values are ≈1.25 µm for an SPP at an Au/air interface and ≈50 nm for the HPC TM fundamental mode q = 1 with n 2.9 1 eff _ TM ≈ for a metamaterial thickness of 400 nm. Thus, HPC modes provide more than 20 times stronger confi nement. For the q = 1 TE mode in the metamaterial, the effective refractive index is much lower n 1.6 1 eff _ TE ≈ with a penetration depth of about 100 nm. Despite the small quantity of metal present in the HPC (dielectric patches are separated by the metamaterial areas of only a few plasmonic nanorods across), the waveguided modes of these structures at telecom wavelengths have confi nement ten times superior to conventional surface plasmon polaritons on a smooth Au/air interface. In the case of SPPs on a metallic fi lm, such high mode indices are diffi cult to achieve due to loss-induced back bending of the SPP dispersion, [ 10 ] notwithstanding the fl exibility afforded by the metamaterial system here, in choosing the wavelength of operation and mode order with unprecedented fl exibility.
Numerical simulations confi rm the HPC Bloch mode formation and the observed optical properties of structured metamaterials ( Figure 4 and Figure S6, Supporting Information). Despite the inference that periodic structuring of the nanorod Adv. Mater. 2015, 27, 5974-5980 www.advmat.de www.MaterialsViews.com www.advmat.de www.MaterialsViews.com array perturbs the effective index of the metamaterial, this effect only results in a small additional frequency shift of the features in the extinction spectra, and both the EMT approach and the full 3D simulations demonstrate excellent qualitative and quantitative agreement with each other, and with the resonances associated with the Bloch modes measured experimentally. The simulated electric fi eld distributions near the extinction peak in the elliptic regime (position A) and HWMs (positions B-D) are shown in Figure 4 a-d and Figure S6 (Supporting Information). The electromagnetic fi eld distributions of the HWMs exhibit the expected Bloch mode structure and confi rm the waveguided origin of the modes within the structured layer, encompassing modes TM3-TM5 in direct analogy with the fi eld distributions for the unstructured area ( Figure S3, Supporting Information). These modes exhibit a typical Fabry-Perot like structure verifying that Bragg coupling to waveguided modes has indeed been achieved by periodic nanostructuring. The electromagnetic fi eld distributions, calculated for the periodically nanostructured area using both the full 3D (Figure 4 c) and effective medium approach ( Figure S6c, Supporting Information), are in close agreement with differences arising from fi eld confi nement effects due to either the presence of nanorods or empty pores in the AAO matrix. [ 31 ] As can be observed on examination of Figure 4 c-f, the full 3D simulation reveals regions of high fi eld confi nement with a length scale on the order of the rod diameter (≈ 40 nm), in both the metamaterial and the dielectric patches.
In conclusion, we have fabricated hyperbolic polaritonic crystals via wavelength-scale structuring of plasmonic nanorodbased hyperbolic metamaterials. We have experimentally demonstrated the formation of a designed Bloch-mode band-structure of hyperbolic guided waves and the inherent coupling of free-space radiation to high-index modes of the metamaterial. Strikingly, despite being comprised of areas of only a few (as few as 3) nanorods across, the metamaterial behavior of the hyperbolic polaritonic crystals is preserved. These hyperbolic Bloch modes have several signifi cant advantages over either modes of dielectric waveguides or surface plasmon polaritons modes; they provide strong confi nement in deep-subwavelength waveguides with the modal indices reaching 5, not achievable in dielectrics, and provide fl exible design of the dispersion via metamaterial properties, which would be otherwise fi xed by the plasmonic metal properties. Several distinct modes exist throughout the visible and infrared region of the electromagnetic spectrum providing multiplexing capabilities within nanoscale optical components. In particular, this approach benefi ts from the combined tunability afforded by the dual design freedom of the metamaterial's optical dispersion from both the bottom-up (nanorod parameters, thickness) and by top-down nanostructuring. Moreover, the high refractive index sensitivity exhibited by the waveguided modes in the nanorod metamaterial [ 20 ] may be advantageously exploited in designer nanostructures. Nanoscale non-linear optical devices may be created either with bare nanorods [ 22 ] or when hybridized with active media, [ 24 ] and these materials also present an ideal opportunity to develop highly integrated, multiwavelength lab-on-a-chip devices with enhanced sensing capabilities. Both of these applications can be further enhanced by the ability to employ sub-attolitre controlled core-shell geometry. [ 32 ] Such highly tunable metamaterials represent a new class of plasmonic platform with enormous potential in future active and passive nanophotonic applications.

Supporting Information
Supporting Information is available from the Wiley Online Library or from the author.