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Van der Waals (vdW) materials supporting phonon polaritons (PhPs) - light coupled to lattice vibrations - have gathered significant interest because of their intrinsic anisotropy and low losses. In particular, $\alpha$-MoO$_3$ supports PhPs with in-plane anisotropic propagation, which has been exploited to tune the optical response of twisted bilayers and trilayers. Additionally, various studies have explored the realization of polaritonic crystals (PCs) - lattices with periods comparable to the polariton wavelength -. PCs consisting of hole arrays etched in $\alpha$-MoO$_3$ slabs exhibit Bragg resonances dependent on the angle between the crystallographic axes and the lattice vectors. However, such PC concept, with a fixed orientation and size of its geometrical parameters, constrains practical applications and introduces additional scattering losses due to invasive fabrication processes. Here we demonstrate a novel PC concept that overcomes these limitations, enabling low-loss optical tuning. It comprises a rotatable pristine $\alpha$-MoO$_3$ layer located on a periodic hole array fabricated in a metallic layer. Our design prevents degradation of the $\alpha$-MoO$_3$ optical properties caused by fabrication, preserving its intrinsic low-loss and in-plane anisotropic propagation of PhPs. The resulting PC exhibits rotation of the Bloch modes, which is experimentally visualized by scanning near-field microscopy. In addition, we experimentally determine the polaritons momentum and reconstruct their band structure. These results pave the way for mechanically tunable nanooptical components based on polaritons for potential lasing, sensing, or energy harvesting applications.

Phonon polaritons (PhPs) (light coupled to lattice vibrations) in the highly anisotropic polar layered material molybdenum trioxide (\alpha-MoO_3) are currently the focus of intense research efforts due to their extreme subwavelength field confinement, directional propagation and unprecedented low losses. Nevertheless, prior research has primarily concentrated on exploiting the squeezing and steering capabilities of \alpha-MoO_3 PhPs, without inquiring much into the dominant microscopic mechanism that determines their long lifetimes, key for their implementation in nanophotonic applications. This study delves into the fundamental processes that govern PhP damping in \alpha-MoO_3 by combining ab initio calculations with scattering-type scanning near-field optical microscopy (s-SNOM) and Fourier-transform infrared (FTIR) spectroscopy measurements across a broad temperature range (from 8 to 300 K). The remarkable agreement between our theoretical predictions and experimental observations allows us to identify third-order anharmonic phonon-phonon scattering as the main damping mechanism of \alpha-MoO_3 PhPs. These findings shed light on the fundamental limits of low-loss PhPs, a crucial factor for assessing their implementation into nanophotonic devices., Comment: 24 pages, 4 figures

Polaritons are a hybrid class of quasiparticles originating from the strong and resonant coupling between light and matter excitations. Recent years have witnessed a surge of interest in novel polariton types, arising from directional, long-lived material resonances, and leading to extreme optical anisotropy that enables novel regimes of nanoscale, highly confined light propagation. While such exotic propagation features may also be in principle achieved using carefully designed metamaterials, it has been recently realized that they can naturally emerge when coupling infrared light to directional lattice vibrations, i.e., phonons, in polar crystals. Interestingly, a reduction in crystal symmetry increases the directionality of optical phonons and the resulting anisotropy of the response, which in turn enables new polaritonic phenomena, such as hyperbolic polaritons with highly directional propagation, ghost polaritons with complex-valued wave vectors, and shear polaritons with strongly asymmetric propagation features. In this Review, we develop a critical overview of recent advances in the discovery of phonon polaritons in low-symmetry crystals, highlighting the role of broken symmetries in dictating the polariton response and associated nanoscale-light propagation features. We also discuss emerging opportunities for polaritons in lower-symmetry materials and metamaterials, with connections to topological physics and the possibility of leveraging anisotropic nonlinearities and optical pumping to further control their nanoscale response.

Polaritons in strongly anisotropic thin layers have recently captured the attention in nanophotonics because of their directional propagation at the nanoscale, which offers unique possibilities for nanooptical applications. However, exploiting the full potential of anisotropic polaritons requires a thorough understanding of their properties, including field confinement, energy and phase propagation direction and losses. Here we fill this critical gap by providing fundamental insights into the propagation of anisotropic polaritons in thin biaxial layers. In particular, we introduce a novel methodology that allows us to represent isofrequency curves of polaritons in strongly anisotropic materials considering that the real and imaginary parts of the wavevector are not parallel. In fact, we analytically show that the direction of the imaginary part of the wavevector is parallel to the group velocity, which can have different, even perpendicular or opposite, directions with respect to the phase velocity. This finding is crucial for understanding polaritonic phenomena in anisotropic media, yet it has so far been widely overlooked in the literature. Additionally, we introduce a criterion for classifying the polaritonic modes in biaxial layers into volume and surface categories, and analyze their dispersion, field structure, and losses. Finally, we discover the existence of anisotropic transverse electric modes, which can exhibit natural canalization. Taken together, our results shed light on hitherto unexplored areas of the theory of electromagnetic modes in thin biaxial layers. Although exemplified for van der Waals MoO3 layers, our findings are general for polaritons in other strongly anisotropic biaxial hyperbolic crystals., Comment: 25 pages, 5 figures, including supplementary information

The terahertz (THz) frequency range is key to study collective excitations in many crystals and organic molecules. However, due to the large wavelength of THz radiation, the local probing of these excitations in smaller crystalline structures or few-molecular arrangements, requires sophisticated methods to confine THz light down to the nanometer length scale, as well as to manipulate such a confined radiation. For this purpose, in recent years, taking advantage of hyperbolic phonon polaritons (HPhP) in highly anisotropic van der Waals (vdW) materials has emerged as a promising approach, offering a multitude of manipulation options such as control over the wavefront shape and propagation direction. Here, we demonstrate the first THz application of twist-angle-induced HPhP manipulation, designing the propagation of confined THz radiation between 8.39 and 8.98 THz in the vdW material alpha-molybdenum trioxide ($\alpha-MoO_{3}$), hence extending twistoptics to this intriguing frequency range. Our images, recorded by near-field optical microscopy, show the frequency- and twist-angle-dependent change between hyperbolic and elliptic polariton propagation, revealing a polaritonic transition at THz frequencies. As a result, we are able to allocate canalization (highly collimated propagation) of confined THz radiation by carefully adjusting these two parameters, i.e. frequency and twist angle. Specifically, we report polariton canalization in $\alpha-MoO_{3}$ at 8.67 THz for a twist angle of 50{\deg}. Our results demonstrate an unprecedented control and the manipulation of highly-confined collective excitations at THz frequencies, offering novel possibilities for nanophotonic applications., Comment: 25 pages, 4 figures, article, submitted

The discovery of quantum emitters in two-dimensional materials has triggered a surge of research to assess their suitability for quantum photonics. While their microscopic origin is still the subject of intense studies, ordered arrays of quantum emitters are routinely fabricated using static strain-gradients, which are used to drive excitons toward localized regions of the 2D crystals where quantum-light-emission takes place. However, the possibility of using strain in a dynamic fashion to control the appearance of individual quantum emitters has never been explored so far. In this work, we tackle this challenge by introducing a novel hybrid semiconductor-piezoelectric device in which WSe2 monolayers are integrated onto piezoelectric pillars delivering both static and dynamic strains. Static strains are first used to induce the formation of quantum emitters, whose emission shows photon anti-bunching. Their excitonic population and emission energy are then reversibly controlled via the application of a voltage to the piezoelectric pillar. Numerical simulations combined with drift-diffusion equations show that these effects are due to a strain-induced modification of the confining-potential landscape, which in turn leads to a net redistribution of excitons among the different quantum emitters. Our work provides relevant insights into the role of strain in the formation of quantum emitters in 2D materials and suggests a method to switch them on and off on demand., Comment: 13 pages, 4 figures

Polaritons - hybrid light-mater excitations - are very appealing for the confinement of light at the nanoscale. Recently, different kinds of polaritons have been observed in thin slabs of van der Waals (vdW) materials, with particular interest focused on phonon polaritons (PhPs) - lattice vibrations coupled to electromagnetic fields in the mid-infrared spectral range with - in biaxial crystals, such as e.g. MoO3. In particular, hyperbolic PhPs - having hyperbola-like shape of their isofrequency curves - in MoO3 can exhibit ultra-high momenta and strongly directional in-plane propagation, promising novel applications in imaging, sensing or thermal management at the nanoscale and in a planar geometry. However, the excitation and manipulation of in-plane hyperbolic PhPs have not yet been well studied and understood. Here we propose a technological platform for the effective excitation and control of in-plane hyperbolic PhPs based on polaritonic crystals (PCs) - lattices formed by elements separated by distances comparable to the PhPs wavelength -, twisted with respect to the natural vdW crystal axes. In particular, we develop a general analytical theory valid for an arbitrary PC made in a thin biaxial slab. As a practical example, we consider a twisted PC formed by rectangular hole arrays made in MoO3 slab and demonstrate the excitation of Bragg resonances tunable by the twisting angle. Our findings open novel avenues for both fundamental studies of PCs in vdW crystals and the development of mid-infrared sensing and photodetection applications

In general insurance companies, a correct estimation of liabilities plays a key role due to its impact on management and investing decisions. Since the Financial Crisis of 2007-2008 and the strengthening of regulation, the focus is not only on the total reserve but also on its variability, which is an indicator of the risk assumed by the company. Thus, measures that relate profitability with risk are crucial in order to understand the financial position of insurance firms. Taking advantage of the increasing computational power, this paper introduces a stochastic reserving model whose aim is to improve the performance of the traditional Mack's reserving model by applying an ensemble of Recurrent Neural Networks. The results demonstrate that blending traditional reserving models with deep and machine learning techniques leads to a more accurate assessment of general insurance liabilities.

Negative reflection occurs when light is reflected towards the same side of the normal to the boundary from which it is incident. This exotic optical phenomenon, which provides a new avenue towards light manipulation, is not only yet to be visualized in real space but remains largely unexplored both at the nanoscale and in natural media. Here, we directly visualize nanoscale-confined polaritons negatively reflecting on subwavelength mirrors fabricated in a low-loss van der Waals crystal. Our near-field nanoimaging results unveil an unconventional and broad tunability of both the polaritonic wavelength and direction of propagation upon negative reflection. Based on these findings, we introduce a novel device in nano-optics: a hyperbolic nanoresonator, in which hyperbolic polaritons with different momenta reflect back to a common point source, enhancing its intensity. These results pave the way to realize nanophotonics in low-loss natural media, providing a novel and efficient route to confine and control the flow of light at the nanoscale, key for future optical on-chip nanotechnologies.

Terahertz (THz) electromagnetic radiation is key to optically access collective excitations such as magnons (spins), plasmons (electrons), or phonons (atomic vibrations), thus bridging between optics and solid-state physics. Confinement of THz light to the nanometer length scale is desirable for local probing of such excitations in low dimensional systems, thereby inherently circumventing the large footprint and low spectral density of far-field THz optics. For that purpose, phonon polaritons (PhPs, i.e., light coupled to lattice vibrations in polar crystals) in anisotropic van der Waals (vdW) materials have recently emerged as a promising platform for THz nanooptics; yet the amount of explored, viable materials is still exiguous. Hence, there is a demand for the exploration of novel materials that feature not only THz PhPs at different spectral regimes, but also exhibit unique anisotropic (directional) electrical, thermoelectric, and vibronic properties. To that end, we introduce here the semiconducting alpha-germanium(II) sulfide (GeS) as an intriguing candidate. By employing THz nano-spectroscopy supported by theoretical analysis, we provide a thorough characterization of the different in-plane hyperbolic and elliptical PhP modes in GeS. We find not only PhPs with long life times ($\tau$ > 2 ps) and excellent THz light confinement ($\lambda_0/\lambda$ > 45), but also an intrinsic, phonon-induced anomalous dispersion as well as signatures of naturally occurring PhP canalization within one single GeS slab.

Phonon polaritons (PhPs) -- lattice vibrations coupled to electromagnetic fields -- in highly anisotropic media display a plethora of intriguing optical phenomena (including ray-like propagation, anomalous refraction, and topological transitions, among others), which have potential for unprecedented manipulation of the flow of light at the nanoscale. However, the propagation properties of these PhPs are intrinsically linked to the anisotropic crystal structure of the host material. Although in-plane anisotropic PhPs can be steered (and even canalized) by twisting individual crystal slabs in a van der Waals (vdW) stack, active control of their propagation via external stimuli presents a significant challenge. Here, we report on a technology in which anisotropic PhPs supported by biaxial vdW slabs are actively tunable by simply gating an integrated graphene layer. Excitingly, we predict active tuning of optical topological transitions, which enable controlling the canalization of PhPs along different in-plane directions in twisted heterostructures. Apart from their fundamental interest, our findings hold promises for the development of optoelectronic devices (sensors, photodetectors, etc.) based on PhPs with dynamically controllable properties.

Events such as the Financial Crisis of 2007-2008 or the COVID-19 pandemic caused significant losses to banks and insurance entities. They also demonstrated the importance of using accurate equity risk models and having a risk management function able to implement effective hedging strategies. Stock volatility forecasts play a key role in the estimation of equity risk and, thus, in the management actions carried out by financial institutions. Therefore, this paper has the aim of proposing more accurate stock volatility models based on novel machine and deep learning techniques. This paper introduces a neural network-based architecture, called Multi-Transformer. Multi-Transformer is a variant of Transformer models, which have already been successfully applied in the field of natural language processing. Indeed, this paper also adapts traditional Transformer layers in order to be used in volatility forecasting models. The empirical results obtained in this paper suggest that the hybrid models based on Multi-Transformer and Transformer layers are more accurate and, hence, they lead to more appropriate risk measures than other autoregressive algorithms or hybrid models based on feed forward layers or long short term memory cells.

Phonon polaritons (PhPs),light coupled to lattice vibrations,with in-plane hyperbolic dispersion exhibit ray-like propagation with large wavevectors and enhanced density of optical states along certain directions on a surface. As such, they have raised a surge of interest as they promise unprecedented possibilities for the manipulation of infrared light with planar circuitry and at the nanoscale. Here, we demonstrate, for the first time, the focusing of in-plane hyperbolic PhPs propagating along thin slabs of MoO3. To that end, we developed metallic nanoantennas of convex geometries for both the efficient launching and focusing of the polaritons. Remarkably, the foci obtained exhibit enhanced near-field confinement and absorption compared to foci produced by in-plane isotropic PhPs. More intriguingly, foci sizes as small as lamdap/5 =lamda0/50 were achieved (lamdap is the polariton wavelength and lamda0 the photon wavelength). Focusing of in-plane hyperbolic polaritons introduces a first and most basic building block developing planar polariton optics utilizing in-plane anisotropic van der Waals materials and metasurfaces.

Recent discoveries of polaritons in van der Waals (vdW) crystals with directional in-plane propagation, ultra-low losses, and broad spectral tunability have opened the door for unprecedented manipulation of the flow of light at the nanoscale. However, despite their extraordinary potential for nano-optics, these unique polaritons also present an important limitation: their directional propagation is intrinsically determined by the crystal structure of the host material, which imposes forbidden directions of propagation and hinders its control. Here, we theoretically predict and experimentally demonstrate that directional polaritons (in-plane hyperbolic phonon polaritons) in a vdW biaxial slab (alpha-phase molybdenum trioxide) can be steered along previously forbidden directions by inducing an optical topological transition, which naturally emerges when placing the slab on a substrate with a given negative permittivity (4H-SiC). Importantly, due to the low-loss nature of this topological transition, we are able to visualize in real space exotic intermediate polaritonic states between mutually orthogonal hyperbolic regimes, which permit to unveil the unique topological origin of the transition. This work provides new insights into the emergence of low-loss optical topological transitions in vdW crystals, offering a novel route to efficiently steer the flow of energy at the nanoscale.

Phonon polaritons (PPs) in van der Waals (vdW) materials can strongly enhance light-matter interactions at mid-infrared frequencies, owing to their extreme infrared field confinement and long lifetimes. PPs thus bear potential for achieving vibrational strong coupling (VSC) with molecules. Although the onset of VSC has recently been observed spectroscopically with PP nanoresonators, no experiments so far have resolved VSC in real space and with propagating modes in unstructured layers. Here, we demonstrate by real-space nanoimaging that VSC can be achieved between propagating PPs in thin vdW crystals (specifically h-BN) and molecular vibrations in adjacent thin molecular layers. To that end, we performed near-field polariton interferometry, showing that VSC leads to the formation of a propagating hybrid mode with a pronounced anti-crossing region in its dispersion, in which propagation with negative group velocity is found. Numerical calculations predict VSC for nanometer-thin molecular layers and PPs in few-layer vdW materials, which could make propagating PPs a promising platform for ultra-sensitive on-chip spectroscopy and strong coupling experiments.

Currently, legal requirements demand that insurance companies increase their emphasis on monitoring the risks linked to the underwriting and asset management activities. Regarding underwriting risks, the main uncertainties that insurers must manage are related to the premium sufficiency to cover future claims and the adequacy of the current reserves to pay outstanding claims. Both risks are calibrated using stochastic models due to their nature. This paper introduces a reserving model based on a set of machine learning techniques such as Gradient Boosting, Random Forest and Artificial Neural Networks. These algorithms and other widely used reserving models are stacked to predict the shape of the runoff. To compute the deviation around a former prediction, a log-normal approach is combined with the suggested model. The empirical results demonstrate that the proposed methodology can be used to improve the performance of the traditional reserving techniques based on Bayesian statistics and a Chain Ladder, leading to a more accurate assessment of the reserving risk.

Electromagnetic field confinement is crucial for nanophotonic technologies, since it allows for enhancing light-matter interactions, thus enabling light manipulation in deep sub-wavelength scales. In the terahertz (THz) spectral range, radiation confinement is conventionally achieved with specially designed metallic structures - such as antennas or nanoslits - with large footprints due to the rather long wavelengths of THz radiation. In this context, phonon polaritons - light coupled to lattice vibrations - in van der Waals (vdW) crystals have emerged as a promising solution for controlling light beyond the diffraction limit, as they feature extreme field confinements and low optical losses. However, experimental demonstration of nanoscale-confined phonon polaritons at THz frequencies has so far remained elusive. Here, we provide it by employing scattering-type scanning near-field optical microscopy (s-SNOM) combined with a free-electron laser (FEL) to reveal a range of low-loss polaritonic excitations at frequencies from 8 to 12 THz in the vdW semiconductor ${\alpha}-MoO_3$. We visualize THz polaritons with i) in-plane hyperbolic dispersion, ii) extreme nanoscale field confinement (below ${\lambda}_o/75$) and iii) long polariton lifetimes, with a lower limit of > 2 ps.

Recent discoveries have shown that when two layers of van der Waals (vdW) materials are superimposed with a relative twist angle between their respective in-plane principal axes, the electronic properties of the coupled system can be dramatically altered. Here, we demonstrate that a similar concept can be extended to the optics realm, particularly to propagating polaritons, hybrid light-matter interactions. To do this, we fabricate stacks composed of two twisted slabs of a polar vdW crystal (MoO3) supporting low-loss anisotropic phonon polaritons (PhPs), and image the propagation of the latter when launched by localized sources (metal antennas). Our images reveal that under a critical angle the PhPs isofrequency curve (determining the PhPs momentum at a fixed frequency) undergoes a topological transition. Remarkably, at this angle, the propagation of PhPs is strongly guided along predetermined directions (canalization regime) with no geometrical spreading (diffraction-less). These results demonstrate a new degree of freedom (twist angle) for controlling the propagation of polaritons at the nanoscale with potential for nano-imaging, (bio)-sensing, quantum applications and heat management.

The biaxial van der Waals semiconductor $\alpha$-phase molybdenum trioxide ($\alpha$-MoO$_3$) has recently received significant attention due to its ability to support highly anisotropic phonon polaritons (PhPs) -infrared (IR) light coupled to lattice vibrations in polar materials-, offering an unprecedented platform for controlling the flow of energy at the nanoscale. However, to fully exploit the extraordinary IR response of this material, an accurate dielectric function is required. Here, we report the accurate IR dielectric function of $\alpha$-MoO$_3$ by modelling far-field, polarized IR reflectance spectra acquired on a single thick flake of this material. Unique to our work, the far-field model is refined by contrasting the experimental dispersion and damping of PhPs, revealed by polariton interferometry using scattering-type scanning near-field optical microscopy (s-SNOM) on thin flakes of $\alpha$-MoO$_3$, with analytical and transfer-matrix calculations, as well as full-wave simulations. Through these correlative efforts, exceptional quantitative agreement is attained to both far- and near-field properties for multiple flakes, thus providing strong verification of the accuracy of our model, while offering a novel approach to extracting dielectric functions of nanomaterials, usually too small or inhomogeneous for establishing accurate models only from standard far-field methods. In addition, by employing density functional theory (DFT), we provide insights into the various vibrational states dictating our dielectric function model and the intriguing optical properties of $\alpha$-MoO$_3$.

Anisotropic crystals have recently attracted considerable attention because of their ability to support polaritons with a variety of unique properties, such as hyperbolic dispersion, negative phase velocity, or extreme confinement. Particularly, the biaxial crystal $\alpha$-MoO$_3$ has been demonstrated to support phonon polaritons, light coupled to lattice vibrations, with in-plane anisotropic propagation and unusually long lifetime. However, the lack of theoretical studies on electromagnetic modes in biaxial crystal slabs impedes a complete interpretation of the experimental data, as well as an efficient design of nanostructures supporting such highly anisotropic polaritons. Here, we derive the dispersion relation of electromagnetic modes in biaxial slabs surrounded by semi-infinite isotropic dielectric half-spaces with arbitrary dielectric permittivities. Apart from a general dispersion relation, we provide very simple analytical expressions in typical experiments in nano-optics: the limits of short polaritonic wavelength and/or very thin slabs. The results of our study will allow for an in-depth analysis of anisotropic polaritons in novel biaxial van der Waals materials., Comment: 16 pages, 4 figures

The appearance of single photon sources in atomically thin semiconductors holds great promises for the development of a flexible and ultra-compact quantum technology, in which elastic strain engineering can be used to tailor their emission properties. Here, we show a compact and hybrid 2D-semiconductor-piezoelectric device that allows for controlling the energy of single photons emitted by quantum emitters localized in wrinkled WSe2 monolayers. We demonstrate that strain fields exerted by the piezoelectric device can be used to tune the energy of localized excitons in WSe2 up to 18 meV in a reversible manner, while leaving the single photon purity unaffected over a wide range. Interestingly, we find that the magnitude and in particular the sign of the energy shift as a function of stress is emitter dependent. With the help of finite element simulations we suggest a simple model that explains our experimental observations and, furthermore, discloses that the type of strain (tensile or compressive) experienced by the quantum emitters strongly depends on their localization across the wrinkles. Our findings are of strong relevance for the practical implementation of single photon devices based on two-dimensional materials as well as for understanding the effects of strain on their emission properties.

Enhanced light-matter interactions are the basis of surface enhanced infrared absorption (SEIRA) spectroscopy, and conventionally rely on plasmonic materials and their capability to focus light to nanoscale spot sizes. Phonon polariton nanoresonators made of polar crystals could represent an interesting alternative, since they exhibit large quality factors, which go far beyond those of their plasmonic counterparts. The recent emergence of van der Waals crystals enables the fabrication of high-quality nanophotonic resonators based on phonon polaritons, as reported for the prototypical infrared-phononic material hexagonal boron nitride (h-BN). In this work we use, for the first time, phonon-polariton-resonant h-BN ribbons for SEIRA spectroscopy of small amounts of organic molecules in Fourier transform infrared spectroscopy. Strikingly, the interaction between phonon polaritons and molecular vibrations reaches experimentally the onset of the strong coupling regime, while numerical simulations predict that vibrational strong coupling can be fully achieved. Phonon polariton nanoresonators thus could become a viable platform for sensing, local control of chemical reactivity and infrared quantum cavity optics experiments.

Light properties in the mid-infrared can be controlled at a deep subwavelength scale using hyperbolic phonons-polaritons (HPPs) of hexagonal boron nitride (h-BN). While propagating as waveguided modes HPPs can concentrate the electric field in a chosen nano-volume. Such a behavior is at the heart of many applications including subdiffraction imaging and sensing. Here, we employ HPPs in heterostructures of h-BN and graphene as new nano-optoelectronic platform by uniting the benefits of efficient hot-carrier photoconversion in graphene and the hyperbolic nature of h-BN. We demonstrate electrical detection of HPPs by guiding them towards a graphene pn-junction. We shine a laser beam onto a gap in metal gates underneath the heterostructure, where the light is converted into HPPs. The HPPs then propagate as confined rays heating up the graphene leading to a strong photocurrent. This concept is exploited to boost the external responsivity of mid-infrared photodetectors, overcoming the limitation of graphene pn-junction detectors due to their small active area and weak absorption. Moreover this type of detector exhibits tunable frequency selectivity due to the HPPs, which combined with its high responsivity paves the way for efficient high-resolution mid-infrared imaging.

The response of an electron system to electromagnetic fields with sharp spatial variations is strongly dependent on quantum electronic properties, even in ambient conditions, but difficult to access experimentally. We use propagating graphene plasmons, together with an engineered dielectric-metallic environment, to probe the graphene electron liquid and unveil its detailed electronic response at short wavelengths.The near-field imaging experiments reveal a parameter-free match with the full theoretical quantum description of the massless Dirac electron gas, in which we identify three types of quantum effects as keys to understanding the experimental response of graphene to short-ranged terahertz electric fields. The first type is of single-particle nature and is related to shape deformations of the Fermi surface during a plasmon oscillations. The second and third types are a many-body effect controlled by the inertia and compressibility of the interacting electron liquid in graphene. We demonstrate how, in principle, our experimental approach can determine the full spatiotemporal response of an electron system., Comment: 8 pages, 4 figures

Terahertz (THz) fields are widely applied for sensing, communication and quality control. In future applications, they could be efficiently confined, enhanced and manipulated - well below the classical diffraction limit - through the excitation of graphene plasmons (GPs). These possibilities emerge from the strongly reduced GP wavelength, lp, compared to the photon wavelength, l0, which can be controlled by modulating the carrier density of graphene via electrical gating. Recently, GPs in a graphene-insulator-metal configuration have been predicted to exhibit a linear dispersion (thus called acoustic plasmons) and a further reduced wavelength, implying an improved field confinement, analogous to plasmons in two-dimensional electron gases (2DEGs) near conductive substrates. While infrared GPs have been visualised by scattering-type scanning near-field optical microscopy (s-SNOM), the real-space imaging of strongly confined THz plasmons in graphene and 2DEGs has been elusive so far - only GPs with nearly free-space wavelength have been observed. Here we demonstrate real-space imaging of acoustic THz plasmons in a graphene photodetector with split-gate architecture. To that end, we introduce nanoscale-resolved THz photocurrent near-field microscopy, where near-field excited GPs are detected thermoelectrically rather than optically. The on-chip GP detection simplifies GP imaging, as sophisticated s-SNOM detection schemes can be avoided. The photocurrent images reveal strongly reduced GP wavelengths (lp = l0/66), a linear dispersion resulting from the coupling of GPs with the metal gate below the graphene, and that plasmon damping at positive carrier densities is dominated by Coulomb impurity scattering. Acoustic GPs could thus strongly benefit the development of deep subwavelength-scale THz devices.

Controlling, detecting and generating propagating plasmons by all-electrical means is at the heart of on-chip nano-optical processing. Graphene carries long-lived plasmons that are extremely confined and controllable by electrostatic fields, however electrical detection of propagating plasmons in graphene has not yet been realized. Here, we present an all-graphene mid-infrared plasmon detector, where a single graphene sheet serves simultaneously as the plasmonic medium and detector. Rather than achieving detection via added optoelectronic materials, as is typically done in other plasmonic systems, our device converts the natural decay product of the plasmon---electronic heat---directly into a voltage through the thermoelectric effect. We employ two local gates to fully tune the thermoelectric and plasmonic behaviour of the graphene. High-resolution real-space photocurrent maps are used to investigate the plasmon propagation and interference, decay, thermal diffusion, and thermoelectric generation.