Search

Cathodoluminescence (CL) microscopy has emerged as a powerful tool for investigating the optical properties of materials at the nanoscale, offering unique insights into the behavior of photonic and plasmonic materials under electron excitation. We introduce an atlas of bulk CL spectra for a range of materials widely used in photonics and plasmonics. Through a combination of experimental CL spectroscopy and Monte Carlo simulations, we characterize electron penetration depth and energy deposition, offering a foundational reference for interpreting CL spectra and understanding material behavior under electron excitation. By capturing CL signal from a diverse range of materials, this atlas provides insights into the intrinsic emission properties essential for material selection and design in photonic and plasmonic device engineering.

Common techniques for measuring refractive indices, such as ellipsometry and goniometry, are ineffective for van der Waals crystal flakes because of their high anisotropy and small, micron-scale, lateral size. To address this, we employ near-field optical microscopy to analyze the guided optical modes within these crystals. By probing these modes in MoS$_2$ flakes with subwavelength spatial resolution at a wavelength of $1570\,\mathrm{nm}$, we determine both the in-plane and out-of-plane permittivity components of MoS$_2$ as $16.11$ and $6.25$, respectively, with a relative uncertainty below $1\%$, while overcoming the limitations of traditional methods.

We investigate how the near-field enhancement of self-similar nanolenses, made of three plasmonic nanospheres with decreasing sizes and separations, is affected by quantum corrections in the optical response of the metal, and by fabrication imperfections related to the positioning of the spheres in the nanolens. In particular, we integrate mesoscopic phenomena, such as electron spill-in and -out and surface-enabled Landau damping, via the surface response formalism, focusing particularly on the role of spill-out in alkali metals. In addition, we take realistic imperfections in the nanofabrication process into account, through numerical averaging of both far- and near-field spectra for large collections of nanolenses. Statistical analysis of hundreds of trimers shows that inevitable deviations from the perfectly aligned chain only slightly, if at all, impair the field enhancement, as long as the average misplacement is kept within 1 nm from the ideal situation. Wishing to explore whether such imperfections can be harvested for practical applications, we probe the potential for triggering chiral response. Our results verify that imperfect nanolenses can display considerable circular dichroism, while providing a means to manipulate the optical chirality density. Nonetheless, we emphasize that quantification of chiral light-matter interactions can be largely affected by mesoscopic phenomena, which cannot be ignored when near-field quantities like optical chirality density are investigated.

In this paper we study the topology of the bands of a plasmonic crystal composed of graphene and of a metallic grating. Firstly, we derive a Kronig-Penney type of equation for the plasmonic bands as function of the Bloch wavevector and discuss the propagation of the surface plasmon polaritons on the polaritonic crystal using a transfer-matrix approach considering a finite relaxation time. Second, we reformulate the problem as a tight-binding model that resembles the Su-Schrieffer-Heeger (SSH) Hamiltonian, one difference being that the hopping amplitudes are, in this case, energy dependent. In possession of the tight-binding equations it is a simple task to determine the topology (value of the winding number) of the bands. This allows to determine the existense or absence of topological end modes in the system. Similarly to the SSH model, we show that there is a tunable parameter that induces topological phase transitions from trivial to non-trivial. In our case, it is the distance d between the graphene sheet and the metallic grating. We note that d is a parameter that can be easily tuned experimentally simply by controlling the thickness of the spacer between the grating and the graphene sheet. It is then experimentally feasible to engineer devices with the required topological properties. Finally, we suggest a scattering experiment allowing the observation of the topological states., Comment: 17 pages; significant changes in the presentation of the results (unchanged)

Scattering-type scanning near-field optical microscope (s-SNOM) has become an essential tool to study polaritons - quasiparticles of light coupled to collective charge oscillations - via direct probing of their near field with a spatial resolution far beyond the diffraction limit. However, extraction of the polariton complex propagation constant from the near-field images requires subtle considerations that have not received necessary attention so far. In this study, we discuss important yet overlooked aspects of the near-field analysis. First, we experimentally demonstrate that the sample orientation inside the s-SNOM may significantly affect the near-field interference pattern of mid-infrared polaritons, leading to an error in momentum measurement up to 7.7% even for the modes with effective index of 12.5. Second, we establish a methodology to correctly extract the polariton damping rate from the interference fringes depending on their origin - the s-SNOM nano-tip or the material edge. Overall, our work provides a unified framework for the accurate extraction of the polariton momentum and damping from the near-field interference fringes., Comment: 15 pages, 4 figures

Surface plasmon polaritons (SPPs) carry transverse optical spin within the evanescent field, which has enabled the demonstration of various chiral light-matter interactions in classical and quantum systems. To achieve high spin selectivity in the interactions, the elliptical polarization of the evanescent field should be made circular, but the engineering principle of the degree of circular polarization (DOCP) of SPPs has been lacking. In this study, we theoretically and numerically investigate the scaling behavior of the DOCP of the SPP field with respect to the modal effective refractive index (neff). The DOCP of the SPP field exhibits power-law scalability to the effective refractive index in the 1D layered system, regardless of the material, structural geometry, and excitation wavelength. The power-law scalability is also confirmed in 2D waveguide structures for in-plane and out-of-plane SPP fields, but the scaling exponents vary depending on the distance from the waveguide boundaries by the reduced symmetry of the given system. Due to Lorentz reciprocity, the power-law scalability can be extended to the coupling directionality of chiral emitters towards the plasmonic waveguide. To this end, we propose a chiral photonic platform for enhanced light-valley interaction, which utilizes simultaneous enhancement of the DOCP and coupling directionality. An incident SPP can excite a chiral emitter with high spin selectivity that unidirectionally couples the emitted light into the plasmonic waveguide depending on the valley polarization of excitons in 2D material. Our work provides a ground rule for designing chiral nanophotonic systems and paves the way for the exploration of scale-free phenomena of electromagnetic waves.

We demonstrate the existence of previously unreported magnetic modes with record-high magnetic Purcell factors in topological-insulator nanospheres. Focusing on bismuth selenide (Bi$_{2}$Se$_{3}$), and based on full electromagnetic Mie theory, we find magnetic modes arising from both displacement current loops in the bulk, and surface currents due to delocalized surface states, induced by electronic transitions between topologically protected states within the Dirac cone and discretized due to the sphere finite size. Furthermore, we discuss how Dirac plasmon polaritons, resulting from the interaction between THz photons and Dirac electrons, dramatically influence both the magnetic and the electric transitions of quantum emitters placed near Bi$_2$Se$_3$ nanospheres, significantly enhancing the corresponding Purcell factors. These findings position Bi$_{2}$Se$_{3}$ nanospheres, whose optical response is related to a richness of physical mechanisms, among the most promising candidates for enhancing light--matter interactions in nanophotonics and THz technologies.

We show that hexagonal boron nitride (hBN), a two-dimensional insulator, when subjected to an external superlattice potential forms a new paradigm for electrostatically tunable excitons in the near- and mid-ultraviolet (UV). The imposed potential has three consequences: (i) it renormalizes the effective mass tensor, leading to anisotropic effective masses; (ii) it renormalizes the band gap, eventually reducing it; (iii) it reduces the exciton binding energies. All these consequences depend on a single dimensionless parameter, which includes the product of strength of the external potential with its period. In addition to the excitonic energy levels, we compute the optical conductivity along two orthogonal directions, and from it the absorption spectrum. The results for the latter show that our system is able to mimic a grid polarizer. These characteristics make one-dimensional hBN superlattices a viable and unexplored platform for fine-tuned polaritonics in the UV to visible spectral range.

We discuss the possibility of self-hybridisation in high-index dielectric nanoparticles, where Mie modes of electric or magnetic type can couple to the interband transitions of the material, leading to spectral anticrossings. Starting with an idealised system described by moderately high constant permittivity with a narrow Lorentzian, in which self-hybridisation is visible for both plane-wave and electron-beam excitation, we embark on a quest for realistic systems where this effect should be visible. We explore a variety of spherical particles made of traditional semiconductors such as Si, GaAs, and GaP. With the effect hardly discernible, we identify two major causes hindering observation of self-hybridisation: the very broad spectral fingerprints of interband transitions in most candidate materials, and the significant overlap between electric and magnetic Mie modes in nanospheres. We thus depart from the spherical shape, and show that interband--Mie hybridisation is indeed feasible in the example of GaAs cylinders, even with a simple plane-wave source. This so-far unreported kind of polariton has to be considered when interpreting experimental spectra of Mie-resonant nanoparticles and assigning modal characters to specific features. On the other hand, it has the potential to be useful for the characterisation of the optical properties of dielectric materials, through control of the hybridisation strength via nanoparticle size and shape.

Topological insulators generally have dielectric bulk and conductive surface states. Consequently, some of these materials have been shown to support polaritonic modes at visible and THz frequencies. At the same time, the optical properties of topological insulators in the mid-infrared (IR) remain poorly investigated. We employ near-field imaging to probe the mid-IR response from the exfoliated flakes of bismuth (Bi) / selenide (Se) / telluride (Te) / antimony (Sb) crystals with varying stoichiometry - Bi2Se3, Bi2Te2Se, and Bi1.5Sb0.5Te1.7Se1.3 - in pristine form as well as covered by thin flakes of hexagonal boron nitride (hBN). Contrary to theoretical expectations, all three materials exhibit a dielectric response with a high refractive index and with a loss below the experimental detection limit. Particularly, the near-field mapping of propagating phonon-polaritons in hBN demonstrates that these van der Waals crystals act as a practically lossless dielectric substrate with an ultra-high refractive index of up to 7.5 in Bi2Te2Se. Such a unique dielectric crystal would be of great advantage for numerous nanophotonic applications in the mid-IR.

Within the framework of the T-matrix method, we present a modeling tool that predicts the optical response from the Nanosphere-on-Mirror (NSoM) construct. The nonclassical effects in metals are accounted for by the nonlocal hydrodynamic Drude model (NLHDM) or the surface response model (SRM). Two essential elements in the T-matrix method, i.e., the T-matrix of the sphere and the R matrix accounting for the effects of the mirror, have been fully upgraded to include longitudinal waves for the NLHDM and the augmented interface conditions for the SRM. The proposed tool is quantitatively validated both in the near and the far field by an in-house developed BEM solver for the NLHDM where the gap between the sphere and the mirror is as small as 1 nm. Two physical checks are performed, where the results from the classical local response model are compared with the ones from the NLHDM and the SRM. The observed shifts in resonances and reduced field enhancements in the gap region agree well with previous physical findings. The proposed tool may not only serve as a reference tool for other numerical methods, but also provides an ideal platform for investigating nonclassical optical processes in the NSoM, hence paving a semi-analytical way to understand the extreme optics at very small scales.

We present an analytic, Mie theory-based solution for the energy-loss and the photon-emission probabilities in the interaction of spherical nanoparticles with electrons passing nearby and through them, in both cathodoluminescence and electron energy-loss spectroscopies. In particular, we focus on the case of penetrating electron trajectories, for which the complete fully electrodynamic and relativistic formalism has not been reported as yet. We exhibit the efficiency of this method in describing collective excitations in matter through calculations for a dispersive and lossy system, namely a sphere described by a Drude permittivity. Subsequently, we use the analytic solution to corroborate the implementation of electron-beam sources in a state-of-the-art numerical method for problems in electrodynamics, the discontinuous Galerkin time-domain (DGTD) method. We show that the two approaches produce spectra in good mutual agreement, and demonstrate the versatility of DGTD via simulations of spherical nanoparticles characterized by surface roughness. The possibility of simultaneously employing both kinds of calculations (analytic and numerical) facilitates a better understanding of the rich optical response of nanophotonic architectures excited by fast electron beams., Comment: 16 pages, 8 figures

The lowest energy states in transition metal dichalcogenide (TMD) monolayers follow valley selection rules, which have attracted vast interest due to the possibility of encoding and processing of quantum information. However, these quantum states are strongly affected by the temperature-dependent intervalley scattering causing complete valley depolarization, which is hampering any practical applications of TMD monolayers at room temperature. Therefore, for achieving clear and robust valley polarization in TMD monolayers one needs to suppress parasitic depolarization processes, which is the central challenge in the growing field of valleytronics. Here, in electron-doping experiments on TMD monolayers, we demonstrate that strong doping levels beyond $10^{13}$~cm$^{-2}$ can induce 61\% and 37\% valley contrast at room temperature in tungsten diselenide and molybdenum diselenide monolayers, respectively. Our results indicate that charged excitons in TMD monolayers can be utilized as quantum units in designing of practical valleytronic devices operating at 300 K.

Nonlocal and quantum mechanical phenomena in noble metal nanostructures become increasingly crucial when the relevant length scales in hybrid nanostructures reach the few-nanometer regime. In practice, such mesoscopic effects at metal-dielectric interfaces can be described using exemplary surface-response functions (SRFs) embodied by the Feibelman $d$-parameters. Here we show that SRFs dramatically influence quantum electrodynamic phenomena -- such as the Purcell enhancement and Lamb shift -- for quantum emitters close to a diverse range of noble metal nanostructures interfacing different homogeneous media. Dielectric environments with higher permittivities are shown to increase the magnitude of SRFs calculated within the specular-reflection model. In parallel, the role of SRFs is enhanced in nanostructures characterized by large surface-to-volume ratios, such as thin planar metallic films or shells of core-shell nanoparticles. By investigating emitter quantum dynamics close to such plasmonic architectures, we show that decreasing the width of the metal region, or increasing the permittivity of the interfacing dielectric, leads to a significant change in the Purcell enhancement, Lamb shift, and visible far-field spontaneous emission spectrum, as an immediate consequence of SRFs. We anticipate that fitting the theoretically modelled spectra to experiments could allow for experimental determination of the $d$-parameters., Comment: 9 pages, 5 figures

Classical electrodynamics describes the optical response of systems using bulk electronic properties and infinitesimally thin boundaries. However, due to the quantum nature of electrons, interfaces have a finite thickness. Non-classical surface effects become increasingly important as ever smaller nanoscale systems are realized and eventually dominate over volume-related phenomena. Investigating the response of surface electrons in such systems, therefore, becomes imperative. One way to gain control over non-classical interface effects and study them is through electrical gating, as the static screening charges reside exclusively at the surface. Here, we investigate the modulation of the surface response upon direct electric charging of a single plasmonic nanoresonator by measuring the resulting changes in resonance. We analyze the observed effects within the general framework of surface-response functions and provide a basic model derived from electron spill-out within the local-response approximation (LRA). Our observed change in resonance frequency is well accounted for by assuming a modulation of the in-plane surface current. Surprisingly, we also measure a change in the resonance width, where adding electrons to the surface leads to a narrowing of the plasmonic resonance, i.e., reduced losses. The description of such effects requires considering nonlocal effects and the inclusion of a possible anisotropy of the perturbed surface permittivity. Our experiment, therefore, opens a vast field of investigations on how to gain control over the surface response in plasmonic resonators and to develop ultrafast and extremely small electrically driven plasmonic modulators and metasurfaces by leveraging electrical control over non-classical surface effects., Comment: 10 pages, 4 figures, 15 pages Supplementary

Impurity-vacancy centers in diamond offer a new class of robust photon sources with versatile quantum properties. While individual color centers commonly act as single-photon sources, their ensembles have been theoretically predicted to have tunable photon-emission statistics. Importantly, the particular type of excitation affects the emission properties of a color center ensemble within a diamond crystal. While optical excitation favors non-synchronized excitation of color centers within an ensemble, electron-beam excitation can synchronize the emitters and thereby provides a control of the second-order correlation function $g_2(0)$. In this letter, we demonstrate experimentally that the photon stream from an ensemble of color centers can exhibit $g_2(0)$ both above and below unity. Such a photon source based on an ensemble of few color centers in a diamond crystal provides a highly tunable platform for informational technologies operating at room temperature.

Promising applications in photonics are driven by the ability to fabricate crystal-quality metal thin films of controlled thickness down to a few nanometers. In particular, these materials exhibit a highly nonlinear response to optical fields owing to the induced ultrafast electron dynamics, which is however poorly understood on such mesoscopic length scales. Here, we reveal a new mechanism that controls the nonlinear optical response of thin metallic films, dominated by ultrafast electronic heat transport when the thickness is sufficiently small. By experimentally and theoretically studying electronic transport in such materials, we explain the observed temporal evolution of photoluminescence in pump-probe measurements that we report for crystalline gold flakes. Incorporating a first-principles description of the electronic band structures, we model electronic transport and find that ultrafast thermal dynamics plays a pivotal role in determining the strength and time-dependent characteristics of the nonlinear photoluminescence signal, which is largely influenced by the distribution of hot electrons and holes, subject to diffusion across the film as well as relaxation to lattice modes. Our findings introduce conceptually novel elements triggering the nonlinear optical response of nanoscale materials while suggesting additional ways to control and leverage hot carrier distributions in metallic films., Comment: 20 pages, 6 figures, 64 references

We introduce and theoretically analyze the concept of manipulating optical chirality via strong coupling of the optical modes of chiral nanostructures with excitonic transitions in molecular layers or semiconductors. With chirality being omnipresent in chemistry and biomedicine, and highly desirable for technological applications related to efficient light manipulation, the design of nanophotonic architectures that sense the handedness of molecules or generate the desired light polarization in an externally controllable manner is of major interdisciplinary importance. Here we propose that such capabilities can be provided by the mode splitting resulting from polaritonic hybridization. Starting with an object with well-known chiroptical response -- here, for a proof of concept, a chiral sphere -- we show that strong coupling with a nearby excitonic material generates two distinct frequency regions that retain the object's chirality density and handedness, which manifest most clearly through anticrossings in circular-dichroism or differential-scattering dispersion diagrams. These windows can be controlled by the intrinsic properties of the excitonic layer and the strength of the interaction, enabling thus the post-fabrication manipulation of optical chirality. Our findings are further verified via simulations of the circular dichroism of a realistic chiral architecture, namely a helical assembly of plasmonic nanospheres embedded in a resonant matrix.

We propose an efficient approach for actively controlling the Rabi oscillations in emitter-cavity hybrids based on the presence of an element with optical gain. Inspired by recent developments in parity-time ($\mathcal{PT}$)-symmetry photonics, we show that nano- or micro-cavities where intrinsic losses are partially or fully compensated by an externally controllable amount of gain offer unique capabilities for manipulating the dynamics of emitters. In particular, one can drastically modify the dynamics of the system, increase the overall occupation numbers, enhance the longevity of the Rabi oscillations, and even decelerate them to the point where their experimental observation becomes less challenging. Furthermore, we show that there is a specific gain value that leads to an exceptional point, where both emitter and cavity occupation oscillate practically in phase, with occupation numbers that can significantly exceed unity. By revisiting a recently-introduced Rabi-visibility measure, we provide robust guidelines for quantifying the coupling strength and achieving strong-coupling with adaptable Rabi frequency via loss compensation.

The nonlocal response of plasmonic materials and nanostructures is usually described within a hydrodynamic approach which is based on the Euler-Drude equation. In this work, we reconsider this approach within linear response theory and employ Halevi's extension to this standard hydrodynamic model. After discussing the impact of this improved model, which we term the Halevi model, on the propagation of longitudinal volume modes, we accordingly extend the Mie-Ruppin theory. Specifically, we derive the dispersion relation of cylindrical surface plasmons. This reveals a nonlocal, collisional damping term which is related to earlier phenomenological considerations of limited-mean-free-path effects and influences both, peak width and amplitude of corresponding resonances in the extinction spectrum. In addition, we transfer the Halevi model into the time-domain thereby revealing a novel, diffusive contribution to the current which shares certain similarities with Cattaneo-type currents and analyze the resulting hybrid, diffusive-wave-like motion. Further, we discuss the relation of the Halevi model to other approaches commonly used in the literature. Finally, we demonstrate how to implement the Halevi model into the Discontinuous-Galerkin Time-Domain finite-element Maxwell solver and are able to identify an oscillatory contribution to the diffusive current. The Halevi model thus captures a number of relevant features beyond the standard hydrodynamic model. Contrary to other extensions of the standard hydrodynamic model, its use in time-domain Maxwell solvers is straightforward -- especially due its affinity to a class of descriptions that allow for a clear distinction between bulk and surface response. This is of particular importance for applications in nano-plasmonics where nano-gap structures and other nano-scale features have to be modeled efficiently and accurately., Comment: 20 pages, 7 figures, 1 tables

We analyse the emergence of unphysical superluminal group velocities in Su--Schrieffer--Heeger (SSH) parity-time ($\mathcal{PT}$) symmetric chains, and explore the origins of such a behaviour. By comparing the band structure of an infinite loss-gain SSH chain with that of a one-dimensional Bragg stack, we first exclude insufficient coupling consideration in the tight-binding description as the cause of group-velocity divergence. We then focus on material dispersion, and show that indeed, restoring causality in the description of both the lossy and the gain components resolves the problem and recovers finite group velocities, whose real part can only exceed the speed of light in vacuum when accompanied by a significant imaginary part. Our analysis introduces thus the required practical limits in the performance of common $\mathcal{PT}$-symmetric systems., Comment: 9 pages, 3 figures

Cathodoluminescence spectroscopy performed in an electron microscope has proven a versatile tool for analysing the near- and far-field optical response of plasmonic and dielectric nanostructures. Nevertheless, the transition radiation produced by electron impact is often disregarded in the interpretation of the spectra recorded from resonant nanoparticles. Here we show, experimentally and theoretically, that transition radiation can by itself generate distinct resonances which, depending on the time of flight of the electron beam inside the particle, can result from constructive or destructive interference in time. Superimposed on the eigenmodes of the investigated structures, these resonances can distort the recorded spectrum and lead to potentially erroneous assignment of modal characters to the spectral features. We develop an intuitive analogy that helps distinguish between the two contributions. As an example, we focus on the case of silicon nanospheres, and show that our analysis facilitates the unambiguous interpretation of experimental measurements on Mie-resonant nanoparticles.

Giant shell CdSe/CdS quantum dots are bright and flexible emitters, with near-unity quantum yield and suppressed blinking, but their single photon purity is reduced by efficient multiexcitonic emission. We report the observation, at the single dot level, of a large blueshift of the photoluminescence biexciton spectrum ($24\pm5$ nm over a sample of 32 dots) for pure-phase wurtzite quantum dots. By spectral filtering, we demonstrate a 2.3 times reduction of the biexciton quantum yield relative to the exciton emission, while preserving as much as 60% of the exciton single photon emission, thus improving the purity from $g_2(0)=0.07\pm0.01$ to $g_2(0)=0.03\pm0.01$. At larger pump fluency the spectral purification is even more effective with up to a 6.6 times reduction in $g_2(0)$, which is due to the suppression of higher order excitons and shell states experiencing even larger blueshift. Our results indicate the potential for synthesis engineered giant shell quantum dots, with further increased biexciton blueshift, for quantum optical applications requiring both high purity and brightness., Comment: Supporting information is available

Cathodoluminescence spectroscopy in conjunction with second-order auto-correlation measurements of $g_2(\tau)$ allows to extensively study the synchronization of quantum light sources in low-dimensional structures. Co-existing excitons in two-dimensional transition metal dichalcogenide monolayers provide a great source of identical quantum emitters which can be simultaneously excited by an electron. In this article, we demonstrate large photon bunching with $g_2(0)$ up to $156\pm16$ of a tungsten disulfide monolayer, exhibiting a strong dependence on the electron-beam current density. To further improve the excitation synchronization and the electron-emitter interaction, we show exemplary that the careful selection of a simple and compact geometry -- a thin, monocrystalline gold nanodisk -- can be used to realize a record-high bunching $g_2(0)$ of up to $2152\pm236$. This approach to control the electron excitation of excitons in a \ce{WS2} monolayer allows for the synchronization of quantum emitters in an ensemble, which is important to further advance quantum information processing and computing technologies., Comment: Supporting information is available upon request to authors

Historically, the field of plasmonics has been relying on the framework of classical electrodynamics, with the local-response approximation of material response being applied even when dealing with nanoscale metallic structures. However, when approaching the atomic-scale confinement of the electromagnetic radiation, mesoscopic effects are anticipated to become observable, e.g., those associated with the nonlocal electrodynamic surface response of the electron gas. We investigate nonlocal effects in propagating gap surface plasmon modes in ultrathin metal--dielectric--metal planar waveguides, exploiting monocrystalline gold flakes separated by atomic-layer-deposited aluminum oxide. We use scanning near-field optical microscopy to directly access the near-field of such confined gap plasmon modes and measure their dispersion relation (via their complex-valued propagation constants). We compare our experimental findings with the predictions of the generalized nonlocal optical response theory to unveil signatures of nonlocal damping, which becomes appreciable for smaller dielectric gaps., Comment: Supporting information is available upon request to authors

Polaritonic modes in low-dimensional materials enable strong light-matter interactions and provide a platform for light manipulation at nanoscale. Very recently, a new class of polaritons has attracted considerable interest in nanophotonics: image polaritons in van der Waals crystals, manifesting when a polaritonic material is in close proximity to a highly conductive metal, so that the polaritonic mode couples with its mirror image. Image modes constitute an appealing nanophotonic platform, providing an unparalleled degree of optical field compression into nanometric volumes while exhibiting lower normalized propagation loss compared to conventional polariton modes in van der Waals crystals on non-metallic substrates. Moreover, the ultra-compressed image modes provide access to the nonlocal regime of light-matter interaction. In this Review, we systematically overview the young yet rapidly growing field of image polaritons. We discuss their dispersion properties, showcase the diversity of image modes in various van der Waals materials, and highlight the experimental breakthroughs owing to the unique properties of image polaritons.

Near-field mapping has been widely used to study hyperbolic phonon-polaritons in van der Waals crystals. However, an accurate measurement of the polaritonic loss remains challenging because of the inherent complexity of the near-field signal and the substrate-mediated loss. Here we demonstrate that large-area monocrystalline gold flakes, an atomically-flat low-loss substrate for image polaritons, provide a platform for precise near-field measurement of the complex propagation constant of polaritons in van der Waals crystals. As a topical example, we measure propagation loss of the image phonon-polaritons in hexagonal boron nitride, revealing that their normalized propagation length exhibits a parabolic spectral dependency. Further, we show that image phonon-polaritons exhibit up to a twice lower normalized propagation loss, while being 2.4 times more compressed compared to the case of dielectric substrate. We conclude that the monocrystalline gold flakes provide a unique nanophotonic platform for probing and exploitation of the image modes in low-dimensional materials.

The suite of highly confined polaritons supported by two-dimensional (2D) materials constitutes a versatile platform for nano-optics, offering the means to channel light on deep-subwavelength scales. Graphene, in particular, has attracted considerable interest due to its ability to support long-lived plasmons that can be actively tuned via electrical gating. While the excellent optoelectronic properties of graphene are widely exploited in plasmonics, its mechanical flexibility remains relatively underexplored in the same context. Here, we present a semi-analytical formalism to describe plasmons and other polaritons supported in waveguides formed by bending a 2D material into a parabolic shape. Specifically, for graphene parabolas, our theory reveals that the already large field confinement associated with graphene plasmons can be substantially increased by bending an otherwise flat graphene sheet into a parabola shape, thereby forming a plasmonic waveguide without introducing potentially lossy edge terminations via patterning. Further, we show that the high field confinement associated with such channel polaritons in 2D parabolic waveguides can enhance the spontaneous emission rate of a quantum emitter near the parabola vertex. Our findings apply generally to 2D polaritons in atomically thin materials deposited onto grooves or wedges prepared on a substrate or freely suspended in a quasi-parabolic (catenary) shape. We envision that both the optoelectronic and mechanical flexibility of 2D materials can be harnessed in tandem to produce 2D channel polaritons with versatile properties that can be applied to a wide range of nano-optics functionalities, including subwavelength polaritonic circuitry and bright single-photon sources., Comment: 12 pages, 4 figures

The field of 2D materials-based nanophotonics has been growing at a rapid pace, triggered by the ability to design nanophotonic systems with in situ control, unprecedented degrees of freedom, and to build material heterostructures from bottom up with atomic precision. A wide palette of polaritonic classes have been identified, comprising ultra confined optical fields, even approaching characteristic length scales of a single atom. These advances have been a real boost for the emerging field of quantum nanophotonics, where the quantum mechanical nature of the electrons and-or polaritons and their interactions become relevant. Examples include, quantum nonlocal effects, ultrastrong light matter interactions, Cherenkov radiation, access to forbidden transitions, hydrodynamic effects, single plasmon nonlinearities, polaritonic quantization, topological effects etc. In addition to these intrinsic quantum nanophotonic phenomena, the 2D material system can also be used as a sensitive probe for the quantum properties of the material that carries the nanophotonics modes, or quantum materials in its vicinity. Here, polaritons act as a probe for otherwise invisible excitations, e.g. in superconductors, or as a new tool to monitor the existence of Berry curvature in topological materials and superlattice effects in twisted 2D materials., Comment: 27 pages, 7 figures

We obtain an analytical expression for the linewidth of the 1s-exciton as a function of temperature in transition metal dichalcogenides. The total linewidth, as function of temperature, is dominated by three contributions: (i) the radiative decay (essentially temperature independent); (ii) the phonon-induced intravalley scattering; (iii) the phonon-induced intervalley scattering. Our approach uses a variational \emph{Ansatz} to solve the Wannier equation allowing for an analytical treatment of the excitonic problem, including rates of the decay dynamics. Our results are in good agreement with experimental data already present in the literature and can be used to readily predict the value of the total linewidth at any temperature in the broad class of excitonic two-dimensional materials., Comment: 6 pages; suplemental material is available from the authors upon reasonable request

Surface-response functions are one of the most promising routes for bridging the gap between fully quantum-mechanical calculations and phenomenological models in quantum nanoplasmonics. Within all the currently available recipes for obtaining such response functions, \emph{ab initio} calculations remain one of the most predominant, wherein the surface-response function are retrieved via the metal's non-equilibrium response to an external perturbation. Here, we present a complementary approach where one of the most appealing surface-response functions, namely the Feibelman $d$-parameters, yield a finite contribution even in the case where they are calculated directly from the equilibrium properties described under the local-response approximation (LRA), but with a spatially varying equilibrium electron density. Using model calculations that mimic both spill-in and spill-out of the equilibrium electron density, we show that the obtained $d$-parameters are in qualitative agreement with more elaborate, but also more computationally demanding, \emph{ab initio} methods. The analytical work presented here illustrates how microscopic surface-response functions can emerge out of entirely local electrodynamic considerations., Comment: Supplementary material is available upon request to authors

Plasmons in nanostructured metals are widely utilized to trigger strong light--matter interactions with quantum light sources. While the nonclassical behavior of such quantum emitters (QEs) is well-understood in this context, the role of quantum and surface effects in the plasmonic resonator is usually neglected. Here, we combine the Green's tensor approach with the Feibelman $d$-parameter formalism to theoretically explore the influence of quantum surface effects in metal-dielectric layered nanostructures on the relaxation dynamics of a proximal two-level QE. Having identified electron spill-out as the dominant source of quantum effects in jellium-like metals, we focus our study on sodium. Our results reveal a clear splitting in the emission spectrum, indicative of having reached the strong-coupling regime, and, more importantly, non-Markovian relaxation dynamics of the emitter. Our findings establish that strong light--matter coupling is not suppressed by the emergence of nonclassical surface effects in the optical response of the metal., Comment: 6 pages, 5 figures

Noble metals with well-defined crystallographic orientation constitute an appealing class of materials for controlling light-matter interactions on the nanoscale. Nonlinear optical processes, being particularly sensitive to anisotropy, are a natural and versatile probe of crystallinity in nano-optical devices. Here we study the nonlinear optical response of monocrystalline gold flakes, revealing a polarization dependence in second-harmonic generation from the {111} surface that is markedly absent in polycrystalline films. Apart from suggesting an approach for directional enhancement of nonlinear response in plasmonic systems, we anticipate that our findings can be used as a rapid and non-destructive method for characterization of crystal quality and orientation that may be of significant importance in future applications., Comment: 5 pages, 4 figures, color

Noble metal nanostructures are ubiquitous elements in nano-optics, supporting plasmon modes that can focus light down to length scales commensurate with nonlocal effects associated with quantum confinement and spatial dispersion in the underlying electron gas. Nonlocal effects are naturally more prominent for crystalline noble metals, which potentially offer lower intrinsic loss than their amorphous counterparts, and with particular crystal facets giving rise to distinct electronic surface states. Here, we employ a quantum-mechanical model to describe nonclassical effects impacting the optical response of crystalline noble-metal films and demonstrate that these can be well-captured using a set of surface-response functions known as Feibelman $d$-parameters. In particular, we characterize the $d$-parameters associated with the (111) and (100) crystal facets of gold, silver, and copper, emphasizing the importance of surface effects arising due to electron wave function spill-out and the surface-projected band gap emerging from atomic-layer corrugation. We then show that the extracted $d$-parameters can be straightforwardly applied to describe the optical response of various nanoscale metal morphologies of interest, including metallic ultra-thin films, graphene-metal heterostructures hosting extremely confined acoustic graphene plasmons, and crystallographic faceted metallic nanoparticles supporting localized surface plasmons. The tabulated $d$-parameters reported here can circumvent computationally expensive first-principles atomistic simulations to describe microscopic nonlocal effects in the optical response of mesoscopic crystalline metal surfaces, which are becoming widely available with increasing control over morphology down to atomic length scales for state-of-the-art experiments in nano-optics., Comment: 20 pages, 7 figures

A quantitative understanding of the electromagnetic response of materials is essential for the precise engineering of maximal, versatile, and controllable light--matter interactions. Material surfaces, in particular, are prominent platforms for enhancing electromagnetic interactions and for tailoring chemical processes. However, at the deep nanoscale, the electromagnetic response of electron systems is significantly impacted by quantum surface-response at material interfaces, which is challenging to probe using standard optical techniques. Here, we show how ultra-confined acoustic graphene plasmons (AGPs) in graphene--dielectric--metal structures can be used to probe the quantum surface-response functions of nearby metals, here encoded through the so-called Feibelman $d$-parameters. Based on our theoretical formalism, we introduce a concrete proposal for experimentally inferring the low-frequency quantum response of metals from quantum shifts of the AGPs' dispersion, and demonstrate that the high field confinement of AGPs can resolve intrinsically quantum mechanical electronic length-scales with subnanometer resolution. Our findings reveal a promising scheme to probe the quantum response of metals, and further suggest the utilization of AGPs as plasmon rulers with \r{a}ngstr\"{o}m-scale accuracy., Comment: 7 pages, 4 figures

We examine the far-field optical response, under-plane wave excitation in the presence of a static magnetic field, of core-shell nanoparticles involving a gyroelectric component, either as the inner or the outer layer, through analytic calculations based on appropriately extended Mie theory. We focus on absorption and scattering of light by bismuth-substituted yttrium iron garnet (Bi:YIG) nanospheres and nanoshells, combined with excitonic materials such as organic-molecule aggregates or two-dimensional transition-metal dichalcogenides, and discuss the hybrid character of the modes emerging from the coupling of the two constituents. We observe the excitation of strong magneto-optic phenomena and explore, in particular, the response and tunability of a magneto-transverse light current, indicative of the photonic Hall effect. We show how interaction between the Bi:YIG and excitonic layers leads to a pair of narrow bands of highly directional scattering, emerging from the aforementioned hybridization, which can be tuned at will by adjusting the geometrical or optical parameters of the system. Our theoretical study introduces optically anisotropic media as promising templates for strong coupling in nanophotonics, offering a means to combine tunable magnetic and optical properties, with potential implications both in the design of all-dielectric photonic devices but also in novel clinical applications., Comment: 10 pages, 8 figures

We show that the Higgs mode of a superconductor, which is usually challenging to observe by far-field optics, can be made clearly visible using near-field optics by harnessing ultraconfined graphene plasmons. As near-field sources we investigate two examples: graphene plasmons and quantum emitters. In both cases the coupling to the Higgs mode is clearly visible. In the case of the graphene plasmons, the coupling is signaled by a clear anticrossing stemming from the interaction of graphene plasmons with the Higgs mode of the superconductor. In the case of the quantum emitters, the Higgs mode is observable through the Purcell effect. When combining the superconductor, graphene, and the quantum emitters, a number of experimental knobs become available for unveiling and studying the electrodynamics of superconductors., Comment: 8 pages, 5 figures

The recent generalised nonlocal optical response (GNOR) theory for plasmonics is analysed, and its main input parameter, namely the complex hydrodynamic convection-diffusion constant, is quantified in terms of enhanced Landau damping due to diffusive surface scattering of electrons at the surface of the metal. GNOR has been successful in describing plasmon damping effects, in addition to the frequency shifts originating from induced-charge screening, through a phenomenological electron diffusion term implemented into the traditional hydrodynamic Drude model of nonlocal plasmonics. Nevertheless, its microscopic derivation and justification is still missing. Here we discuss how the inclusion of a diffusion-like term in standard hydrodynamics can serve as an efficient vehicle to describe Landau damping without resorting to computationally demanding quantum-mechanical calculations, and establish a direct link between this term and the Feibelman $d$ parameter for the centroid of charge. Our approach provides a recipe to connect the phenomenological fundamental GNOR parameter to a frequency-dependent microscopic surface-response function. We therefore tackle one of the principal limitations of the model, and further elucidate its range of validity and limitations, thus facilitating its proper application in the framework of nonclassical plasmonics.

Dark plasmonic modes have interesting properties, such as a longer lifetime and a narrower linewidth than their radiative counterpart, as well as little to no radiative losses. However, they have not been extensively studied yet due to their optical inaccessibility. Using electron-energy loss (EEL) and cathodoluminescence (CL) spectroscopy, the dark radial breathing modes (RBMs) in thin, monochrystalline gold nanodisks are systematically investigated in this work. It is found that the RBMs can be detected in a CL set-up despite only collecting the far-field. Their visibility in CL is attributed to the breaking of the mirror symmetry by the high-index substrate, creating an effective dipole moment. The outcoupling into the far-field is demonstrated to be enhanced by a factor of 4 by increasing the thickness of the supporting SiN membrane from 5 to 50 nm due to the increased net electric dipole moment in the substrate. Furthermore, it is shown that the resonance energy of RBMs can be easily tuned by varying the diameter of the nanodisk, making them promising candidates for nanophotonic applications.

Plasmon-exciton polaritons provide exciting possibilities to control light-matter interactions at the nanoscale by enabling closer investigation of quantum optical effects and facilitating novel technologies based, for instance, on Bose-Einstein condensation and polaritonic lasing. Nevertheless, observing and visualising polaritons is challenging, and traditional optical microscopy techniques often lead to ambiguities regarding the emergence and strength of the plasmon-exciton coupling. Electron microscopy offers a more robust means to study and verify the nature of plexcitons, but is still hindered by instrument limitations and resolution. A simple theoretical description of electron beam-excited plexcitons is therefore vital to complement ongoing experimental efforts. Here we apply analytic solutions for the electron-loss and photon-emission probabilities to evaluate plasmon-exciton coupling studied either with the recently adopted technique of electron energy-loss spectroscopy, or with the so-far unexplored in this context cathodoluminescence spectroscopy. Foreseeing the necessity to account for quantum corrections in the plasmonic response, we extend these solutions within the framework of general nonlocal hydrodynamic descriptions. As a specific example we study core-shell spherical emitter-molecule hybrids, going beyond the standard local-response approximation through the hydrodynamic Drude model for screening and the generalised nonlocal optical response theory for nonlocal damping. We show that electron microscopies are extremely powerful in describing the interaction of emitters with the otherwise weakly excited by optical means higher-order plasmonic multipoles, a response that survives when quantum-informed models are considered.

The ability to control scattering directionality of nanoparticles is in high demand for many nanophotonic applications. One of the challenges is to design nanoparticles producing pure high-order multipole scattering (e.g., octopole, hexadecapole), whose contribution is usually negligible compared to strong low-order multipole scattering (i.e., dipole or quadrupole). Here we present an intuitive way to design such nanoparticles by introducing a void inside them. We show that both shell and ring nanostructures allow regimes with nearly pure high-order multipole scattering. Experimentally measured scattering diagrams from properly designed silicon rings at near-infrared wavelengths (~ 800 nm) reproduce well scattering patterns of an electric octopole and magnetic hexadecapole. Our findings advance significantly inverse engineering of nanoparticles from given complex scattering characteristics, with possible applications in biosensing, optical metasurfaces, and quantum communications., Comment: Submitted to ACS Photonics

We use mono-crystalline gold platelets with ultra-smooth surfaces and superior plasmonic properties to investigate the formation of interference patterns caused by surface plasmon polaritons (SPPs) with scattering-type scanning near-field microscopy (s-SNOM) at 521~nm and 633~nm. By applying a Fourier analysis approach, we can identify and separate several signal channels related to SPPs launched and scattered by the AFM tip and the edges of the platelet. Especially at the excitation wavelength of 633~nm, we can isolate a region in the center of the platelets where we find only contributions of SPPs which are launched by the tip and reflected at the edges. These signatures are used to determine the SPP wavelength of $\lambda_{SPP}=606$ nm in good agreement with theoretical predictions. Furthermore, we were still able to measure SPP signals after 20~$\upmu$m propagation, which demonstrates impressively the superior plasmonic quality of these mono-crystalline gold platelets., Comment: Main 20 pages, 8 figures, Supplemental Information 5 pages, 3 figures

Singular graphene metasurfaces, conductivity gratings realized by periodically suppressing the local doping level of a graphene sheet, have recently been proposed to efficiently harvest THz light and couple it to surface plasmons over broad absorption bands, achieving remarkably high field enhancement. However, the large momentum wavevectors thus attained are sensitive to the nonlocal behaviour of the underlying electron liquid. Here, we extend the theory of singular graphene metasurfaces to account for the full nonlocal optical response of graphene and discuss the resulting impact on the plasmon resonance spectrum. Finally, we propose a simple local analogue model that is able to reproduce the effect of nonlocality in local-response calculations by introducing a constant conductivity offset, which could prove a valuable tool in the modelling of more complex experimental graphene-based platforms., Comment: 8 pages, 5 figures

Rooted in quantum optics and benefiting from its well-established foundations, strong coupling in nanophotonics has experienced increasing popularity in recent years. With nanophotonics being an experiment-driven field, the absence of appropriate theoretical methods to describe ground-breaking advances has often emerged as an important issue. To address this problem, the temptation to directly transfer and extend concepts already available from quantum optics is strong, even if a rigorous justification is not always available. In this Review we discuss situations where, in our view, this strategy has indeed overstepped its bounds. We focus on exciton--plasmon interactions, and particularly on the idea of calculating the number of excitons involved in the coupling. We analyse how, starting from an unfounded interpretation of the term N/V that appears in theoretical descriptions at different levels of complexity, one might be tempted to make independent assumptions for what the number N and the volume V are, and attempt to calculate them separately. Such an approach can lead to different, often contradictory results, depending on the initial assumptions (e.g. through different treatments of $V$ as the -- ambiguous in plasmonics -- mode volume). We argue that the source of such contradictions is the question itself -- How many excitons are coupled?, which disregards the true nature of the coupled components of the system, has no meaning and often not even any practical importance. If one is interested in validating the quantum nature of the system -- which appears to be the motivation driving the pursuit of strong coupling with small N -- one could instead focus on quantities such as the photon emission rate or the second-order correlation function.

Light-matter interactions at the nanoscale constitute a fundamental ingredient for engineering applications in nanophotonics and quantum optics. To this regard electromagnetic Mie resonances excited in high-refractive index dielectric nanoparticles have recently attracted interest because of their lower losses and better control over the scattering patterns compared to their plasmonic metallic counterparts. The emergence of several resonances in those systems results in an overall high complexity, where the electric and magnetic dipoles have significant overlap in the case of spherical symmetry, thus concealing the contributions of each resonance separately. Here we show, experimentally and theoretically, the emergence of strong light-matter coupling between the magnetic and electric-dipole resonances of individual silicon nanodisks coupled to a J-aggregated organic semiconductor resonating at optical frequencies, evidencing how the different properties of the two resonances results in two different coupling strengths. The energy splittings observed are of the same order of magnitude as in similar plasmonic systems, thus confirming dielectric nanoparticles as promising alternatives for localized strong coupling studies. The coupling of both the electric and magnetic dipole resonances can offer interesting possibilities for the control of directional light scattering in the strong-coupling regime and the dynamic tuning of nanoscale light-matter coupled states by external fields.