In-Plane Hyperbolic Dispersion

• In-plane hyperbolic dispersion is a phenomenon in anisotropic media where the iso-frequency contour becomes an open hyperbola due to sign inversion of in-plane permittivities.
• This dispersion enables large-momentum states and highly directional wave propagation, significantly enhancing photonic density of states for advanced light-matter interactions.
• Material realizations include vdW crystals, electrides, and metamaterial structures that offer tunable subwavelength focusing, negative diffraction, and tailored optical functionalities.

In-plane hyperbolic dispersion denotes the regime in anisotropic media where the iso-frequency contour (IFC) for propagating waves in a principal 2D plane becomes an open hyperbola, rather than a closed ellipse as found in isotropic or conventionally birefringent materials. This phenomenon occurs when the permittivity (or, in general, the combined permittivity–permeability tensor product) exhibits sign-inversion between in-plane components: specifically, for propagation in the plane of the material (e.g., the $xy$ plane), the condition $$\mathrm{Re}[\epsilon_x]\cdot\mathrm{Re}[\epsilon_y]<0$$ must hold. In-plane hyperbolic IFCs support large-momentum states, enable highly directional propagation, and greatly enhance the photonic density of states, giving rise to a wide range of nano-photonic, imaging, and light–matter interaction effects.

1. Fundamental Theory: Permittivity Tensors and Hyperbolic Isofrequency Contours

For an anisotropic medium described by a diagonal permittivity tensor $$\hat{\epsilon}=\mathrm{diag}(\epsilon_x,\epsilon_y,[\epsilon_z])$$, the general plane-wave dispersion for TM polarization is

$$\frac{k_x^2}{\epsilon_y} + \frac{k_y^2}{\epsilon_x} = \left(\frac{\omega}{c}\right)^2\,.$$

When $\epsilon_x\,\epsilon_y<0$, the IFC becomes a two-sheet hyperbola. This contrasts with conventional elliptic dispersion ($\epsilon_x,\epsilon_y>0$ or both $<0$) resulting in closed contours. In-plane hyperbolic dispersion is distinct because arbitrarily large $|k|$ are permitted along one axis, which fundamentally alters confinement, propagation, and available electromagnetic states (Guo et al., 2019, Martín-Sánchez et al., 2021).

Hyperbolicity can occur in various material platforms:

• Biaxial van der Waals crystals (e.g., $\alpha$-MoO$_3$) with $\mathrm{Re}[\epsilon_x]<0$, $\mathrm{Re}[\epsilon_y]>0$, and $\mathrm{Re}[\epsilon_z]>0$ over certain infrared bands (Zheng et al., 2021).
• Electrides (non-cubic, charge-localized materials) where the highly anisotropic electronic structure and interband transitions yield sign-inversion between $\epsilon_{xx}$ and $\epsilon_{yy}$, producing broad in-plane hyperbolic windows (Hao et al., 22 Nov 2025, Guan et al., 2017).
• Metamaterials: multilayer or nanowire architectures with effective medium engineering, for which sign control over each tensor component allows tailored regions of in-plane hyperbolicity (Guo et al., 2019, Hong et al., 2020, Yan et al., 2012).

2. Guided-Wave and Slab Waveguide Dispersion

In the canonical symmetric slab geometry, an isotropic dielectric core is sandwiched between claddings with indefinite permittivity tensor (e.g., $\epsilon_o<0$ normal to slab, $\epsilon_e>0$ in-plane). Solving Maxwell's equations for TE and TM modes with continuity boundary conditions yields

$$\frac{k_z^2}{\epsilon_e} + \frac{k_x^2}{\epsilon_o} = k_0^2$$

for the propagation constant $k_z$. For TM modes, the guided dispersion exhibits two cutoffs (one at $k_z=0$ and another at $k_z=k_0\sqrt{\epsilon_e}$), with no modes above $k_z^2>\epsilon_e$. The total number of TM branches is finite, unlike the infinite modal spectrum for elliptic waveguides. For both polarizations, the longitudinal Poynting vector ($S_z$) approaches zero as $k_z\rightarrow0$, enabling slow-light or frozen-mode regimes (Lyashko et al., 2015).

3. Microscopic Origin and Material Realization

Natural and Artificial Materials

• Natural non-cubic electrides: Interstitial quasi-atom (ISQ) charge localization yields extremely flat bands along specific axes, dramatically increasing effective mass $m^*$ and reducing plasma frequencies $\omega_{p,x}\neq\omega_{p,y}$. This produces sign-opposite $\epsilon_{xx}$ and $\epsilon_{yy}$, and hence hyperbolic IFCs over extensive spectral windows (e.g., $\Delta\omega\sim0.1$–$0.9$ eV depending on material) (Hao et al., 22 Nov 2025).
• Layered vdW materials: $\alpha$-MoO$_3$, $\alpha$-V$_2$O$_5$, and related compounds exhibit two Reststrahlen bands in the infrared. Overlap regions (e.g., $545$–$960$ cm$^{-1}$) realize $\epsilon_x<0$, $\epsilon_y>0$, so that polariton rays propagate hyperbolically (Zheng et al., 2021).
• Hyperbolic metamaterials (HMMs): Metal-dielectric multilayers (effective medium theory) or nanowire arrays permit independent engineering of $\epsilon_x$ and $\epsilon_y$. Biaxial HMMs fabricated via oblique-angle deposition (OAD) realize a controlled $\sim$7 nm visible in-plane hyperbolic band due to dual ENZ crossings in $\epsilon_x$ and $\epsilon_y$ (Hong et al., 2020).

Conductivity and Shear Metasurfaces

Ultrathin metasurfaces with tailored dipolar resonator arrays—detuned in orientation and resonance—produce effective surface conductivity tensors with real, opposite-sign eigenvalues. A shear twist angle modulates the principal axes of the hyperbola in $k$-space, yielding axially programmable anisotropic hyperbolic dispersion and loss asymmetry (Renzi et al., 24 May 2024).

4. Ray-Like Propagation, Focusing, and Beam Dynamics

Group velocity and energy transport in in-plane hyperbolic media are strictly normal to the IFC rather than parallel to $\mathbf{k}$. Launching a dipole (or nanoantenna) on a hyperbolic slab (e.g., MoO$_3$) results in multi-modal excitation collimated along the hyperbola’s asymptotes, forming sharply focused, ray-like spots. Analytical ray tracing yields focal lengths $f = R\sqrt{1-\epsilon_x/\epsilon_y}$, and optimized trapezoidal antenna geometries achieve spot sizes down to $\lambda_0/50$ (Martín-Sánchez et al., 2021, Zheng et al., 2021). Compared to isotropic systems, hyperbolic polariton focusing yields tighter confinement, larger field enhancement, and greater absorption (by factors $2$–$4$), due to diverging photonic density of states near the asymptotes.

Negative diffraction emerges as a direct consequence of the negative effective mass $m^*$ in hyperbolic media; beams self-focus rather than broaden, and spatial diffraction compensation can be achieved in layered stacks (“time reversal” of beam spreading) (Alberucci et al., 2016).

5. Surface Waves, Hybrid Plasmons, and Boundary Phenomena

At interfaces between isotropic and uniaxial hyperbolic media, guided surface waves (Dyakonov plasmons) propagate with hyperbolic IFCs: $\frac{k_z^2}{a^2} - \frac{k_y^2}{b^2} = 1$ where $a^2$ and $b^2$ depend on the in-plane permittivities, and the dispersion locus is sharply directional. Hybrid-polarized hyperbolic surface plasmon modes exist even when only one principal permittivity is negative, and propagation directions can be electrically tuned by adjusting composition or field (Takayama et al., 2015).

At dielectric–metal interfaces beneath hyperbolic slabs, multimodal ray-like excitations exhibit asymmetric reflection: narrow rays striking the metal edge produce destructive interference across all transmitted channels, leading the reflected power $R\to1$ as the ray width $\Gamma\to0$ (Sheinfux et al., 2021).

6. Topological Transitions, Anti-Crossing, and High-$k$ Physics

In magnetically tunable biaxial media, external field modulation shifts principal permittivities through zero, toggling the in-plane IFC between elliptic and hyperbolic. Realistic losses round off the otherwise open hyperbola into finite contours; increasing magnetic damping or carrier collision frequency contracts and eventually collapses the hyperbolic region (Tuz et al., 2020).

Beyond bi-hyperbolic phases, tri- and tetra-hyperbolic optical regimes may arise from Bloch-mode hybridization, anti-crossing splitting, and stratified unit-cell designs. These introduce multiple intersecting hyperbolic branches and additional high-density-of-states directions for advanced wavevector filtering, imaging, and directional emission control (Durach, 2020).

Hydrodynamic nonlocal response further regularizes the infinite high-$k$ singularity (and LDOS enhancement) typical of local hyperbolic media, imposing a fundamental cutoff $k_c\propto\omega/v_F$ determined by the Fermi velocity of the metal constituent, and endowing even subwavelength unit cells with memory of microscopic scale (Yan et al., 2012).

7. Representative Applications and Dispersion Engineering

• Nanofocusing and planar optics: Nanoantenna launchers (disks, rods) in vdW hyperbolic slabs realize tunable, sub-diffractional focusing for mid-IR optical circuitry (Martín-Sánchez et al., 2021, Zheng et al., 2021).
• Hyperlensing and subwavelength imaging: Planar or rolled multilayer HMMs convert near-field evanescent waves into freely propagating, high-$k$ channelized rays, breaking classical diffraction limits (Guo et al., 2019, Hong et al., 2020).
• Spontaneous emission engineering: The diverging photonic density of states near the hyperbola asymptotes results in orders-of-magnitude Purcell-factor enhancement and sharp directional emission profiles (Renzi et al., 24 May 2024).
• Negative refraction and beam steering: Lossless electrides and engineered HMMs enable all-angle negative refraction, with refraction angle determined analytically in terms of tensor eigenvalues (Guan et al., 2017, Hao et al., 22 Nov 2025).
• Dispersion compensation and soliton formation: Spatial stacking of hyperbolic and elliptic slabs enables complete reversal of diffraction and unique nonlinear phenomena (e.g., reversed sign for Kerr-soliton formation) (Alberucci et al., 2016).

Tuning in-plane dispersion through strain, composition, magnetic field (in gyrotropic superlattices), or metasurface engineering provides a versatile toolbox for on-demand control over bandwidth, resonance, confinement, and optical pathway (Hao et al., 22 Nov 2025, Tuz et al., 2020, Renzi et al., 24 May 2024).

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In-plane hyperbolic dispersion is thus a unifying paradigm in advanced photonic materials—governing ray-like transport, confinement, negative diffraction, and mode filtering—and is increasingly available both through artificial metamaterial architectures and newly discovered naturally occurring platforms. The central feature remains the sign inversion of in-plane permittivity tensor components, which underpins unique propagation physics and enables unprecedented control over light at the nanoscale.