Designer Bloch Plasmon Polariton Dispersion in Hyperbolic Meta-Gratings

Hyperbolic metamaterials (HMMs) represent a novel class of fascinating anisotropic plasmonic materials, supporting highly confined propagating plasmon polaritons in addition to surface plasmon polaritons. However, it is very challenging to tailor and excite these modes at optical frequencies by prism coupling because of the intrinsic difficulties in engineering non-traditional optical properties with artificial nanostructures and the unavailability of high refractive index prisms for matching the momentum between the incident light and the guided modes. Here, we report the mechanism of excitation of high-k Bloch-like Plasmon Polariton (BPPs) modes with ultrasmall modal volume using a meta-grating, which is a combined structure of a metallic diffraction grating and a type II HMM. We show how a 1D plasmonic grating without any mode in the infrared spectral range, if coupled to a HMM supporting high-k modes, can efficiently enable the excitation of these modes via coupling to far-field radiation. Our theoretical predictions are confirmed by reflection measurements as a function of angle of incidence and excitation wavelength. We introduce design principles to achieve a full control of high-k modes in meta-gratings, thus enabling a better understanding of light-matter interaction in this type of hybrid meta-structures. The proposed spectral response engineering is expected to find potential applications in bio-chemical sensors, integrated optics and optical sub-wavelength imaging.


Introduction
Manipulation of photons at the nanoscale, well beyond the diffraction limit of light [1][2][3][4], has become a topic of great interest for the prospect of real-life applications [5], such as efficient energy harvesting and photosensitive chemical reactions [6][7][8], subwavelength waveguides [9], nanocavity lasers [10], opto-electronics [11], biochemistry [12] and nanomedicine [13]. Since the last decade conventional metallic materials have been molded with the most advanced nanofabrication techniques in order to create electromagnetically coupled nanostructured systems, dubbed metamaterials, with novel optical properties emerging from the subwavelength confinement of light [14]. Structured meta-surfaces enable an unprecedented control of the propagation direction of optical excitations on their surface. In this framework, hyperbolic metamaterials (HMMs) have received great attention from the scientific community due to their unusual properties at optical frequencies that are rarely or never observed in nature [15][16][17]. Furthermore, HMMs have been shown to enable negative refraction [18][19][20][21], resonant gain singularities [22], nanoscale light confinement [23], optical cloaking [24], as well as extreme biosensing [25,26], nonlinear optical phenomena [27], super resolution imaging and superlensing effects [28,29], plasmonic-based lasing [30], artificial optical magnetism [31], full control of decay channels on the nanoscale [32], etc. They display a hyperbolic iso-frequency surface [33][34][35], which originates from one of the principal components of their electric or magnetic effective tensor, having the opposite sign to the other two principal components. When considering the dielectric tensor, HMMs can be divided into two types: type I has one negative component in its permittivity tensor and two positive ones. In contrast, a type II HMM has two negative components and one positive. In practical terms type II appears as a metal in one plane and as a dielectric in the perpendicular axis, while type I is the opposite. Such anisotropic materials can sustain propagating modes with very large wave vectors and longer lifetime and propagation length in comparison to classic plasmonic materials [36] and exhibit diverging density of states [37], leading to a strong Purcell enhancement of spontaneous radiation [38,39]. Beyond the so-called natural hyperbolic materials, it is possible to mimic hyperbolic properties, for instance of type II, using a periodic stack of metallic and dielectric layers [40] that can support surface plasmons with large wave vectors [41] and whose effective permittivities for different polarizations have different signs [35,42].

Results and discussion
In this work we focus on a specific type of artificial hyperbolic material, namely a n HMM of type II made of alternating layers of Au and Al 2 O 3 , which has been coupled to a one-dimensional (1D) plasmonic diffraction grating, also known as plasmonic crystal (PC). A sketch of the metagrating concept is presented in Figure 1. In the top panel, we have made a simple sketch of a 1D-PC made of gold, which is illuminated with a transverse magnetic (TM, p-polarized) light wave at an angle  ≠ 0°. The experimental reflectance as a function of the wavelength of the incident light impinging at  = 60° on a 1D-PC with grating period 450 nm and PMMA stripe dimensions of 250 nm (width) and 60 nm (thickness) with a 20 nm gold layer is plotted on the right. As can be seen by looking at the reflectance spectrum, no special features appear in the wavelength range 1000-1750 nm. Similarly, if we consider the experimental reflectance of an HMM crystal (middle-left panel) made of 8 alternating layers of Au (15 nm) and Al 2 O 3 (30 nm) with on top an additional dielectric spacer (Al2O3, thickness 10 nm), we can see that no features are present in the same spectral range (middle-right panel). Furthermore, the sample is highly reflective in the NIR region, as already demonstrated in previous works on similar systems [43]. If we add the 1D-PC on top of our HMM crystal, we construct a meta-grating, and sharp and intense modes appear in the spectral region of interest. Moreover, the mode associated with the grating at 850 nm remains in the same position for the meta-grating. It is worth mentioning here that the 1D-PC does not support any diffraction-coupled plasmonic mode in the 1000-1750 nm range, as demonstrated by the experimental measurement reported in Figure 1. In straight contrast, the two systems, if coupled through a dielectric spacer, display an interesting behavior under the same excitation mechanism, namely a TM polarized light wave. It is important to note that with transverse electric (TE, s-polarized) incident light, no modes are observed in the meta-grating (see Supporting Figure S1). Although the 1D-PC itself does not support any propagating plasmon in this range of wavelengths, HMM can indeed support confined modes, which are not excitable by directly coupling the HMM with external radiation. To demonstrate this effect we have calculated, using a transfer matrix method (see Methods for additional details), the dispersion of the modes that can be excited in the HMM multilayer in the absence of a diffraction grating, Figure 2a. Under certain experimental conditions (such as the prism-coupling technique or high-energy electron beams) we can excite two Surface Plasmon Polariton (SPP) modes, indicated with S1 and S2, one at the interface with air and one at the interface with the glass substrate, respectively. Nevertheless, these modes fall outside the spectral region we are interested in. We observe other extremely confined and high-index propagating modes, which are within the multilayered structure known also as Bloch Plasmon Polaritons (BPPs), labelled BX, where X = 1, 2, 3, …, etc. These modes are the reported eigenmodes of an HMM multilayer [41]. We now consider the well-known dispersion relation of SPP in a grating, that is k SPP = sin + mG, where  = 2/ ( is the wavelength of light), G = 2/a (a is the grating period),  the angle of incidence and m is an integer number (positive or negative). If we calculate the geometrical dispersion of a grating with period 250 nm, and an angle of incidenc e of 60° for different values of m, for instance -1, 1 and 2, we end up with the red lines in Figure 2a. As it can be noticed, these curves intersect the BPP mode dispersion curves at precise energy values in the NIR spectral range, which we highlight with colored dots in Figure 2a. If we focus our attention on the grating mode (-1,0) we observe that it crosses the curves of the first four BPPs at 1370 nm (red dot -B1), 1520 nm (green dot -B2), 1680 nm (orange dot -B3) and 1840 nm (purple dot -B4). These results have also been confirmed through a scattering matrix method [44]. By considering also the grating structure on top of the HMM multilayer, it results in the blue curve in In this geometry and spectral range, B3 was the clearest example of the same BPP mode that can be excited via two different diffraction orders at two different wavelengths (see Figures 2a, 2e-h).
Finally, it is worth noticing that there is also an additionally (-1,0) contribution from the grating mode close to the multilayer/air interface, which might come from the SPP mode excited by the grating and which is clearly present in the experimental curve shown in Figure 1a. This is the mode at around 900 nm close to the B3 mode at 870 nm.
A more detailed mode analysis unveils that actually the geometrical diffraction mode (-1,0) supported by the 1D-PC, upon excitation with the far-field TM radiation, couples with the BPP modes supported by the HMM at certain wavelengths. In Figure 3a we show a representative SEM image of a FIB-milled cross section on the experimentally realized 1D-PC introduced on top of a HMM.  This proves that, although no SPP modes are supported by the 1D-PC in this wavelength range, the fact that a diffraction order is present allows the excitation of BPP modes which are actua lly impossible to excite with direct coupling to far-field infrared radiation. It might be that localized plasmons are excited within the 1D-PC through a diffraction mechanism. If we look at Figure 3c, we can also observe that the field distribution at the grating level is almost the same for all four the cases, indicating that i) it is only one grating mode which is actually contributing to the excitation of the BPP modes and ii) the mode is almost localized.
To prove the latter hypothesis, we have also calculated the reflectance of systems where we considered different type of gratings on top of the HMM. In Figure 4a we plot the four geometries considered, along with the related calculated reflectance spectra at  = 60° and the near-field distributions for the B2 mode.   Figure 4b, where we plot the related calculated absorption efficiency spectra in the wavelength range where the B2 mode is excited using these four different diffraction gratings. By playing with the grating composition, one can pass from a rather negligible absorption (pure gold grating case) to a huge absorption (> 9 9%) in the case of the Au film deposited on top of a PMMA stripe, which is the same of our experimental case, although there we reach an absorption efficiency of ≈ 90% due to structural defects. Similar effects were also recently studied in the visible spectral region by Azzam et al., who observed the formation of very sharp and high-quality factor hybrid photonic-plasmonic modes when coupling a 1D-PC with a dielectric optical waveguide [47]. In that study the plasmonic propagating modes can be excited directly by coupling with far-field radiation and hybridize with the cavity modes in the waveguide, while in our case it is crucial to couple the two systems, which are not displaying any bright mode if taken separately.
We have also studied how the thickness of different building blocks (either the HMM multilayer or the PMMA/Au grating on top), such as the PMMA bar, the Al 2 O 3 spacer or the metal/dielectric layers, can affect the dispersion of the modes. We then calculated the dependence of the BPP modes dispersion on the thickness of the PMMA bar. We left the Au film thickness constant and equal to 20 nm in all the cases. In Figure 6 we report the reflectance of the meta-grating for  = 50° and for different values of the PMMA bar thickness, as a function of the wavelength of the incident light and of the Al 2 O 3 spacer thickness.
As it can be inferred, to achieve a proper coupling between the metallic grating and the HMM multilayer, the PMMA bar thickness has to be larger than the Au film thickness, that is, we need to separate well the top part of the grating from the bottom part. Indeed, only for thicknesses above 40 nm we start to see the appearance of the four BPP modes reported in Figure 1. It is also interesting to note that for PMMA bar thicknesses above 60 nm (as in the experimental case), the dispersion of the four BPP modes is not strongly modified. Moreover, it is also important to note that the Al 2 O 3 spacer thickness should not be larger than 20-30 nm, because for thicknesses beyond this threshold the reflectance dips become less intense, which is a signature that the coupling between the grating and the HMM multilayer is not optimal.
It is also important to note that the BPP modes supported by the HMM display different refractive indices depending on the order of the mode [41]. These indices are known as modal indices, and affect both the propagation and penetration distance of a surface wave, in our case the BPP modes.
In Figure 7a and 7b are reported the real and imaginary part, respectively, of the modal index of the first seven modes supported by the HMM. For clarity we plot also the same quantities for the two SPP modes at the air and glass interfaces (blue and yellow lines, respectively). tunable life-times (thus losses) can be exploited in the system, and our architecture might represent an interesting candidate to achieve this functionality. Finally, it is worth mentioning here that highindex and slow photonics modes like these can be used for enhancing nonlinear interactions as well as in optical circuitry, especially for buffering, switching and time -domain processing of optical signals [48].

Conclusion
We have demonstrated the excitation mechanism of Bloch Plasmon Polariton modes in the radiation continuum using a meta-grating approach. Due to the strong coupling of the hyperbolic and the plasmonic modes, the nature of the modes is exchanged at given spectral points, giving rise to sharp modes with geometry-tailored quality factor and tunable spectral position. These modes are characterized by high absorption efficiency (> 99%) in the near-infrared spectral region.
The system studied exhibits an exquisite set of phenomena including the formation of far-field radiation coupled high-index propagating modes with tunable life-times and group velocities, photonic bandgap engineering and slow light with broad spectral robustness. We believe that these hybrid structures are perfect candidates for applications in tunable-threshold lasers, sharp spectral filters, perfect absorbers, enhancement of nonlinear phenomena as well as biochemical sensors.

Sample fabrication:
The hyperbolic metamaterial samples were produced on a 1 mm thick glass slide (float glass, BK7 equivalent) cleaned with acetone, isopropanol and oxygen plasma. A 30 nm thick layer of aluminum oxide, confirmed with ellipsometry, was produced with an atomic layer deposition (ALD) process using trimethylaluminum and oxygen plasma as precursors at 80 °C in an Oxford Instruments FlexAL reactor. The sample was transferred with a vacuum shuttle to a electron beam evaporation chamber (Kenosistec KE300ET), and a gold layer with a thickness of 15 nm as measured by ellipsometry and atomic force microscopy was deposited at room temperature. These two deposition processes were repeated eight times to produce an alternating multilayer on the glass, with an additional 10 nm ALD-produced Al 2 O 3 as the final layer for the HMM. By using the shuttle, the sample was kept continuously in vacuum between the deposition processes to prevent the introduction of impurities between the layers. The grating on the multilayer was produced by electron beam lithography (EBL), starting with a spin coated 60 nm polymethylmetacrylate (PMMA 950) resist layer, which was exposed using a Raith 150 Two EBL system with a 20 kV beam to expose grating lines with a 450 nm period. The resist was developed using a mix of 1 part methylisobutylketone and 3 parts isopropanol for 40 s. Subsequently 20 nm of gold was evaporated (KE300ET) on the sample with neodymium permanent magnets near the sample to prevent charged particles from reaching the surface. No lift-off was done, leaving the PMMA layer as part of the grating. The corresponding control sample was made on a similar glass slide with a 20 nm indium tin oxide (ITO) layer produced with a Kenosistec KS500C sputter coater at room temperature using an ITO target at 250 °C with 1 sccm O 2 flow in the chamber.
Subsequently PMMA coating, EBL and 20 nm Au coating were performed identically as with the hypergrating sample. The samples were also characterized by scanning electron microscopy after FIB milling with a 30 kV gallium ion beam using a FEI Helios NanoLab 650 dual beam system.
Optical measurements: Spectroscopic ellipsometry with a J.A. Woollam V-VASE system was done to acquire measurements for fitting refractive index functions and thicknesses of single layers to be used for simulations. Reflection spectra of the meta-gratings and control samples were taken with the same system. Specular reflectance spectra had typically a wavelength range of 270 nm to 1800 nm with a resolution of 2 nm, and were measured sequentially with a xenon lamp and a monochromator. The spectra were acquired with a focused probe accessory with lenses for both incident and reflected beams from areas less than 0.5 mm in diameter.
Simulation details: A homemade transfer matrix method code has been used to calculate the mode dispersion in Figure 2a, while the full optical response has been calculated by using the scattering matrix method [44] by considering the full structure and also by using the finite element method  Figure S1. Specular reflection spectra at TE polarization measured from a meta-grating with 250 nm wide PMMA stripes. Incident angles from 20° to 80° in relation to sample normal. Wavelength / nm 20deg 30deg 40deg 50deg 60deg 70deg 80deg