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Nonlinear optical waveguides, particularly those harnessing the optical Kerr effect, are promising for advancing next-generation photonic technologies. Despite the Kerr effect`s ultrafast response, its inherently weak nonlinearity has hindered practical applications. Here, we explore free-electron-induced Kerr nonlinearities in all-semiconductor waveguides, revealing that longitudinal bulk plasmons (inherently nonlocal excitations) can generate exceptionally strong Kerr nonlinearities. We specifically develop a nonlinear eigenmode analysis integrated with semiclassical hydrodynamic theory to compute the linear and nonlinear optical responses originating from the quantum behavior of free electrons in heavily doped semiconductors. These waveguides achieve ultrahigh nonlinear coefficients exceeding 10$^7$ W$^{-1}$km$^{-1}$ and support long-propagating modes with propagation distances over 100 $\mu$m. Additionally, we confirm the robustness of the nonlinear response under realistic conditions by considering viscoelastic and nonlinear damping mechanisms. Finally, we implement our all-semiconductor waveguides in a Mach-Zehnder interferometer, demonstrating efficient nonlinear modulation of the transmittance spectrum via the free-electron Kerr effect. This work evidences the transformative potential of free-electron nonlinearities in heavily doped semiconductors for photonic integrated circuits, paving the way for scalable on-chip nonlinear nanophotonic systems.

The confinement of electromagnetic radiation to subwavelength scales relies on strong light-matter interactions. In the infrared (IR) and terahertz (THz) spectral ranges, phonon polaritons are commonly employed to achieve extremely subdiffractional light confinement, with much lower losses as compared to plasmon polaritons. Among these, hyperbolic phonon polaritons in anisotropic materials offer a highly promising platform for light confinement, which, however, typically plateaus at values of {\lambda}0/100, with {\lambda}0 being the free-space incident wavelength. In this study, we report on ultraconfined phonon polaritons in hafnium-based dichalcogenides with confinement factors exceeding {\lambda}0/250 in the terahertz spectral range. This extreme light compression within deeply sub-wavelength thin films is enabled by the unprecedented magnitude of the light-matter coupling strength in these compounds, and the natural hyperbolicity of HfSe2 in particular. Our findings emphasize the critical role of light-matter coupling for polariton confinement, which for phonon polaritons in polar dielectrics is dictated by the transverse-longitudinal optic phonon energy splitting. Our results demonstrate transition metal dichalcogenides as an enabling platform for THz nanophotonic applications that push the limits of light control.

Phonon polaritons (PhPs) -- light coupled to lattice vibrations -- in polar van der Waals (vdW) crystals are promising candidates for controlling the flow of energy at the nanoscale due to their strong field confinement, anisotropic propagation, and ultra-long lifetime in the picosecond range \cite{ref1,ref2,ref3,ref4,ref5}. However, the lack of tunability in their narrow and material-specific spectral range -- the Reststrahlen Band (RB) -- severely limits their technological implementation. Here, we demonstrate that the intercalation of Na atoms in the vdW semiconductor $\alpha$-V$_2$O$_5$ enables a broad spectral shift of RBs, and that the PhPs excited exhibit ultra-low losses (lifetime of $4 \pm 1$~ps), similar to PhPs in the non-intercalated crystal (lifetime of $6 \pm 1$ ps). We expect our intercalation method to be applicable to other vdW crystals, opening the door for the use of PhPs in broad spectral bands in the mid-infrared domain., Comment: 15 pages, 5 figures

We propose a microscopic mechanism to electrically reconfigure the Kerr nonlinearity by modulating the concentration of free electrons in heavily doped semiconductors under a static bias. Our theory incorporates electrostatic and hydrodynamic frameworks to describe the electronic dynamics, demonstrating electrically tunable linear and nonlinear modulations. The power threshold of achieving optical bistability shows unprecedented tunability over two orders of magnitude, reaching values as low as 10 $\mu$W through surface charge control. These findings offer new insights into understanding and actively controlling Kerr nonlinearities, paving the way for efficient refractive index engineering as well as the development of advanced linear and nonlinear electro-optical modulators., Comment: 6 pages, 4 figures

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

We demonstrate nanoscale optical bistability at an exceptionally low power threshold of 1 mW by leveraging Kerr-type hydrodynamic nonlinearities due to the heavily doped semiconductor's free carriers. This high nonlinearity is enabled by a strong coupling between metallic nanopatch surface plasmons and longitudinal bulk plasmons (LBP) that arise in heavily doped semiconductors due to the nonlocality. Through the coupling, an efficient, near-unity conversion of far-field energy into LBP states could be achieved. These findings offer a viable approach to experimentally probe LBPs and lay the groundwork for developing efficient and ultrafast all-optical nonlinear devices., Comment: 6 pages, 3 figures

Canalization is an optical phenomenon that enables unidirectional propagation of light in a natural way, i.e., without the need for predefined waveguiding designs. Predicted years ago, it was recently demonstrated using highly confined phonon polaritons (PhPs) in twisted layers of the van der Waals (vdW) crystal alpha-MoO3, offering unprecedented possibilities for controlling light-matter interactions at the nanoscale. However, despite this finding, applications based on polariton canalization have remained elusive so far, which can be explained by the complex sample fabrication of twisted stacks. In this work, we introduce a novel canalization phenomenon, arising in a single vdW thin layer (alpha-MoO3) when it is interfaced with a substrate exhibiting a given negative permittivity, that allows us to demonstrate a proof-of-concept application based on polariton canalization: super-resolution (up to ~{\lambda}0/220) nanoimaging. Importantly, we find that canalization-based imaging transcends conventional projection constraints, allowing the super-resolution images to be obtained at any desired location in the image plane. This versatility stems from the synergetic manipulation of three distinct parameters: incident frequency, rotation angle of the thin vdW layer, and thickness. These results provide valuable insights into the fundamental properties of canalization and constitute a seminal step towards multifaceted photonic applications, encompassing imaging, data transmission, and ultra-compact photonic integration., Comment: 11 pages, 3 figures

The vast repository of van der Waals (vdW) materials supporting polaritons offers numerous possibilities to tailor electromagnetic waves at the nanoscale. The development of twistoptics - the modulation of the optical properties by twisting stacks of vdW materials - enables directional propagation of phonon polaritons (PhPs) along a single spatial direction, known as canalization. Here we demonstrate a complementary type of directional propagation of polaritons by reporting the visualization of unidirectional ray polaritons (URPs). They arise naturally in twisted hyperbolic stacks with very different thicknesses of their constituents, demonstrated for homostructures of $\alpha$-MoO$_3$ and heterostructures of $\alpha$-MoO$_3$ and $\beta$-Ga$_2$O$_3$. Importantly, their ray-like propagation, characterized by large momenta and constant phase, is tunable by both the twist angle and the illumination frequency. Apart from their fundamental importance, our findings introduce twisted asymmetric stacks as efficient platforms for nanoscale directional polariton propagation, opening the door for applications in nanoimaging, (bio)-sensing or polaritonic thermal management., Comment: 15 pages, 5 figures

The assembling of twisted stacks of van der Waals (vdW) materials had led to the discovery of a profusion of remarkable physical phenomena in recent years, as it provides a means to accurately control and harness electronic band structures. This has given birth to the so-called field of twistronics. An analogous concept has been developed for highly confined polaritons, or nanolight, in twisted bilayers of strongly anisotropic vdW materials, extending the field to the twistoptics realm. In this case, the emergence of a topological transition of the polaritonic dispersion at a given twist angle (photonic magic angle) results in the propagation of nanolight along one specific direction (canalization regime), holding promises for unprecedented control of the flow of energy at the nanoscale. However, there is a fundamental limitation in twistoptics that critically impedes such control: there is only one photonic magic angle (and thus canalization direction) in a twisted bilayer and it is fixed for each incident frequency. Here, we overcome this limitation by demonstrating the existence of multiple spectrally robust photonic magic angles in reconfigurable twisted vdW trilayers. As a result, we show that canalization of nanolight can be programmed at will along any desired in-plane direction in a single device, and, importantly, within broad spectral ranges of up to 70 cm-1. Our findings lay the foundation for robust and widely tunable twistoptics, opening the door for applications in nanophotonics where on-demand control of energy at the nanoscale is crucial, such as thermal management, nanoimaging or entanglement of quantum emitters., Comment: 16 pages, 3 figures

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

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.

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.

As one of the most fundamental optical phenomena, refraction between isotropic media is characterized by light bending towards the normal to the boundary when passing from a low- to a high-refractive-index medium. However, in anisotropic media, refraction is a much more exotic phenomenon. The most general case of refraction between two anisotropic media remains unexplored, particularly in natural media and at the nanoscale. Here, we visualize and comprehensively study refraction of electromagnetic waves between two strongly anisotropic (hyperbolic) media, and, importantly, we do it with the use of polaritons confined to the nanoscale in a low-loss natural medium, alpha-MoO3. Our images show refraction of polaritons under the general case in which both the direction of propagation and the wavevector are not collinear. As they traverse planar nanoprisms tailored in alpha-MoO3, refracted polaritons exhibit non-intuitive directions of propagation, enabling us to unveil an exotic optical effect: bending-free refraction. Furthermore, we succeed in developing the first in-plane refractive hyperlens, which yields foci as small as lamdap/6, being lamdap the polariton wavelength (lamda0/50 with respect to the wavelength of light in free space). Our results set the grounds for planar nano-optics in strongly anisotropic media, with potential for unprecedented control of the flow of energy at the nanoscale.

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.

Functional linear models are one of the most fundamental tools to assess the relation between two random variables of a functional or scalar nature. This contribution proposes a goodness-of-fit test for the functional linear model with functional response that neatly adapts to functional/scalar responses/predictors. In particular, the new goodness-of-fit test extends a previous proposal for scalar response. The test statistic is based on a convenient regularized estimator, is easy to compute, and is calibrated through an efficient bootstrap resampling. A graphical diagnostic tool, useful to visualize the deviations from the model, is introduced and illustrated with a novel data application. The R package goffda implements the proposed methods and allows for the reproducibility of the data application., Comment: 7 pages, 2 figures

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$_3$.

A new series expansion for the the Airy function is presented here that stems from the method of steepest descents and can be related to the Hadamard expansions as presented in prevous works cited in the manuscript, and which is convergent for all values of the complex variable. Hadamard expansions were introduced as an extension of the method of steepest descents and are defined in terms of a large number of non-systematic integration path subdivisions. Unlike them, the expansions in the present work originate in the splitting of the steepest descent in a number of segments that is not only finite but very small, and which are defined on the basis of the location of the branch points. One of the segments reaches to infinity and this gives rise to the presence of upper incomplete Gamma functions. This is one of the most important differences with the Hadamard series as defined in the aforementioned references, where all the incomplete Gamma functions are of the lower type. The theoretical interest of the new series expansion is twofold. First of all, it shows how to convert an asymptotic series into a convergent one with a finite splitting of the steepest descent path. Secondly, the inverse of the phase function that is part of the Laplace-type equation is Taylor-expanded around branch points to produce Puiseux series when necessary. In addition to this, the proposed analysis shows again how the Stokes phenomenon for the Airy function is related to the transition of the steepest descent paths at $\arg z = \pm 2 \pi/3$ from one to two. In regard to its computational application, these series expansions require a relatively small number of terms for each of them to reach a very high precision., Comment: 27 pages, 8 figures. Some typos have been corrected

The Functional Linear Model with Functional Response (FLMFR) is one of the most fundamental models to assess the relation between two functional random variables. In this paper, we propose a novel goodness-of-fit test for the FLMFR against a general, unspecified, alternative. The test statistic is formulated in terms of a Cram\'er-von Mises norm over a doubly-projected empirical process which, using geometrical arguments, yields an easy-to-compute weighted quadratic norm. A resampling procedure calibrates the test through a wild bootstrap on the residuals and the use of convenient computational procedures. As a sideways contribution, and since the statistic requires a reliable estimator of the FLMFR, we discuss and compare several regularized estimators, providing a new one specifically convenient for our test. The finite sample behavior of the test is illustrated via a simulation study. Also, the new proposal is compared with previous significance tests. Two novel real datasets illustrate the application of the new test., Comment: 24 pages, 2 figures, 10 tables. Suplementary material: 2 pages, 1 figure

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