Hyperbolic Shear Polaritons Overview

• Hyperbolic shear polaritons are highly confined polaritonic modes arising from low-symmetry crystals with off-diagonal dielectric responses, leading to frequency-dependent optical axis rotation.
• They exhibit open, skewed isofrequency contours and asymmetric dissipation that enable spectral steering and enhanced near-field phenomena such as tunable heat transfer.
• Experimental platforms including β-Ga₂O₃, gypsum thin films, and engineered metasurfaces demonstrate controllable HShP behavior via isotopic tuning, twist effects, and source-induced asymmetry.

Hyperbolic shear polaritons (HShPs) are a class of highly confined polaritonic modes that arise when hyperbolic dispersion is combined with low-symmetry, shear-like constitutive response. In the canonical case of monoclinic polar crystals, the in-plane dielectric tensor carries substantial off-diagonal components, the principal optical axes rotate with frequency, and the real and imaginary parts of the tensor are not simultaneously diagonalizable. The result is a distinctive polaritonic phenomenology: open hyperbolic isofrequency contours, frequency-dispersive optical axes, asymmetric propagation, and asymmetric dissipation. Subsequent work has extended the concept beyond bulk natural crystals to exfoliable thin films, engineered metasurfaces, twist-controlled elastic analogs, and source-structured excitations that reproduce key HShP signatures (Jia et al., 2022, Renzi et al., 2024).

1. Tensorial origin and defining criteria

The defining electromagnetic structure of intrinsic HShPs is the in-plane response of a low-symmetry crystal. In monoclinic systems such as $\beta$-Ga$_2$O$_3$, the relevant tensor is

$$\boldsymbol{\varepsilon}_{xy}(\omega)= \begin{pmatrix} \varepsilon_{xx}(\omega) & \varepsilon_{xy}(\omega)\ \varepsilon_{xy}(\omega) & \varepsilon_{yy}(\omega) \end{pmatrix},$$

with $\varepsilon_{xy}\neq 0$. In this setting, hyperbolicity is not referenced to a fixed orthogonal crystal frame. Instead, one diagonalizes the real part of the tensor in a rotated basis $(m,n)$, and the hyperbolic regime is identified by

$$\mathcal{R}(\varepsilon_{mm})\,\mathcal{R}(\varepsilon_{nn})<0.$$

The optical-axis angle is frequency dependent; a conventional expression is

$$\gamma(\omega)=\frac{1}{2}\,\mathrm{atan}\!\left( 2\,\frac{\mathcal{R}(\varepsilon_{xy})}{\mathcal{R}(\varepsilon_{xx})-\mathcal{R}(\varepsilon_{yy})} \right),$$

while an eigenvector-tracking construction is preferred in practice because it avoids artificial discontinuities and the $90^\circ$ ambiguity of the arctan form (Carini et al., 28 Jul 2025).

In the rotated frame, the real part is diagonal by construction, but the imaginary part generally is not. This is the tensorial core of the shear effect. For gypsum thin films, the rotated frame satisfies $\Re\{\varepsilon_{mn}\}=0$ while $_2$0; equivalently, the propagation axes and the loss axes do not coincide. HShPs are therefore not merely hyperbolic modes in an anisotropic medium. They are hyperbolic modes in which non-orthogonal resonances and off-diagonal response enforce a frequency-dependent rotation of the optical axes and a nontrivial redistribution of dissipation (Díaz-Núñez et al., 31 Jan 2025).

A useful complementary quantity is the hyperbola opening angle,

$_2$1

which directly encodes the angular spread of the long and short polaritonic rays in real space. Together, $_2$2 and $_2$3 determine the frequency-dependent orientation and aperture of HShP propagation cones in low-symmetry crystals (Carini et al., 28 Jul 2025).

2. Dispersion topology, axial dispersion, and asymmetric damping

The immediate spectral signature of HShPs is an open, skewed isofrequency contour. Unlike the symmetric hyperbolae of orthorhombic hyperbolic media, the HShP contour is rotated relative to the crystallographic axes and changes its orientation with frequency. In $_2$4-Ga$_2$5O$_2$6, lossless isofrequency surfaces at $_2$7 and $_2$8 are open hyperboloids whose $_2$9-$_3$0 projections have no mirror symmetry about the crystal axes; when losses are included, the corresponding real-space fields exhibit tilted wavefronts and asymmetric Fourier-space lobes (Jia et al., 2022).

The term axial dispersion refers to this frequency-dependent rotation of the effective optical axis. In intrinsic HShPs it is produced by the changing relative weights of non-orthogonal phonon resonances. In practice, axial dispersion means that the preferred propagation direction of the polariton beam can be steered spectrally without changing the sample geometry. The same tensor structure also produces asymmetric dissipation: one branch of the hyperbolic contour may be long lived while its mirror counterpart is strongly damped, because the eigenvectors of the Hermitian and anti-Hermitian tensor parts are misaligned (Jia et al., 2022, Díaz-Núñez et al., 31 Jan 2025).

Gypsum thin films make the topology especially explicit. In the first Reststrahlen band, around $_3$1–$_3$2, the in-plane isofrequency contours are hyperbolic and support hyperbolic shear phonon polaritons. Near $_3$3, where $_3$4, the contour becomes nearly flat and the system enters a canalization regime. At $_3$5, the contour is closed and elliptical, yet the propagation remains shear-like because the loss asymmetry persists. This establishes a topological transition from hyperbolic shear to elliptical shear polaritons through a canalized intermediate state (Díaz-Núñez et al., 31 Jan 2025).

The same experiments quantify how strongly slowed these modes can become. For propagation perpendicular to the $_3$6 axis, the extracted group velocity spans approximately $_3$7 to $_3$8 in the $_3$9–$$\boldsymbol{\varepsilon}_{xy}(\omega)= \begin{pmatrix} \varepsilon_{xx}(\omega) & \varepsilon_{xy}(\omega)\ \varepsilon_{xy}(\omega) & \varepsilon_{yy}(\omega) \end{pmatrix},$$0 range, with lifetimes between $$\boldsymbol{\varepsilon}_{xy}(\omega)= \begin{pmatrix} \varepsilon_{xx}(\omega) & \varepsilon_{xy}(\omega)\ \varepsilon_{xy}(\omega) & \varepsilon_{yy}(\omega) \end{pmatrix},$$1 and $$\boldsymbol{\varepsilon}_{xy}(\omega)= \begin{pmatrix} \varepsilon_{xx}(\omega) & \varepsilon_{xy}(\omega)\ \varepsilon_{xy}(\omega) & \varepsilon_{yy}(\omega) \end{pmatrix},$$2. For propagation perpendicular to the $$\boldsymbol{\varepsilon}_{xy}(\omega)= \begin{pmatrix} \varepsilon_{xx}(\omega) & \varepsilon_{xy}(\omega)\ \varepsilon_{xy}(\omega) & \varepsilon_{yy}(\omega) \end{pmatrix},$$3 axis, $$\boldsymbol{\varepsilon}_{xy}(\omega)= \begin{pmatrix} \varepsilon_{xx}(\omega) & \varepsilon_{xy}(\omega)\ \varepsilon_{xy}(\omega) & \varepsilon_{yy}(\omega) \end{pmatrix},$$4 lies between $$\boldsymbol{\varepsilon}_{xy}(\omega)= \begin{pmatrix} \varepsilon_{xx}(\omega) & \varepsilon_{xy}(\omega)\ \varepsilon_{xy}(\omega) & \varepsilon_{yy}(\omega) \end{pmatrix},$$5 and $$\boldsymbol{\varepsilon}_{xy}(\omega)= \begin{pmatrix} \varepsilon_{xx}(\omega) & \varepsilon_{xy}(\omega)\ \varepsilon_{xy}(\omega) & \varepsilon_{yy}(\omega) \end{pmatrix},$$6 in $$\boldsymbol{\varepsilon}_{xy}(\omega)= \begin{pmatrix} \varepsilon_{xx}(\omega) & \varepsilon_{xy}(\omega)\ \varepsilon_{xy}(\omega) & \varepsilon_{yy}(\omega) \end{pmatrix},$$7–$$\boldsymbol{\varepsilon}_{xy}(\omega)= \begin{pmatrix} \varepsilon_{xx}(\omega) & \varepsilon_{xy}(\omega)\ \varepsilon_{xy}(\omega) & \varepsilon_{yy}(\omega) \end{pmatrix},$$8, with a lifetime of about $$\boldsymbol{\varepsilon}_{xy}(\omega)= \begin{pmatrix} \varepsilon_{xx}(\omega) & \varepsilon_{xy}(\omega)\ \varepsilon_{xy}(\omega) & \varepsilon_{yy}(\omega) \end{pmatrix},$$9. The confinement factors reach about $\varepsilon_{xy}\neq 0$0 and $\varepsilon_{xy}\neq 0$1 for the two directions, respectively (Díaz-Núñez et al., 31 Jan 2025).

3. Material platforms and experimental observation

Several distinct platforms now anchor the experimental study of HShPs and closely related shear-polaritonic regimes.

Platform	Key manifestation	Citation
$\varepsilon_{xy}\neq 0$2-Ga$\varepsilon_{xy}\neq 0$3O$\varepsilon_{xy}\neq 0$4	Monoclinic HShPs with frequency-dispersive optical axes and asymmetric azimuthal reflectance	(Jia et al., 2022)
$\varepsilon_{xy}\neq 0$5O/$\varepsilon_{xy}\neq 0$6O $\varepsilon_{xy}\neq 0$7-Ga$\varepsilon_{xy}\neq 0$8O$\varepsilon_{xy}\neq 0$9	Near-field imaging of HShPs and isotopic redshift of $(m,n)$0	(Carini et al., 28 Jul 2025)
Gypsum thin films	Hyperbolic, canalized, and elliptical shear phonon polaritons in an exfoliable monoclinic mineral	(Díaz-Núñez et al., 31 Jan 2025)

The first detailed HShP platform discussed in the literature surveyed here is monoclinic $(m,n)$1-Ga$(m,n)$2O$(m,n)$3. In Otto-prism coupling, the reflectance as a function of azimuth angle and frequency is asymmetric under $(m,n)$4, directly revealing the absence of mirror symmetry in the in-plane dispersion. The same system exhibits tilted wavefronts in real space and rotated lobes in Fourier space, all rooted in the full monoclinic tensor with $(m,n)$5 (Jia et al., 2022).

A later advance was isotopically engineered $(m,n)$6-Ga$(m,n)$7O$(m,n)$8. Near-field optical microscopy on $(m,n)$9O bGO films homo-epitaxially grown on a $$\mathcal{R}(\varepsilon_{mm})\,\mathcal{R}(\varepsilon_{nn})<0.$$0O bGO substrate demonstrated a spectral redshift of $$\mathcal{R}(\varepsilon_{mm})\,\mathcal{R}(\varepsilon_{nn})<0.$$1 in the HShP response. The experiment used Au discs as launchers; $$\mathcal{R}(\varepsilon_{mm})\,\mathcal{R}(\varepsilon_{nn})<0.$$2 discs preferentially excited ray-like high-$$\mathcal{R}(\varepsilon_{mm})\,\mathcal{R}(\varepsilon_{nn})<0.$$3 HShPs, while $$\mathcal{R}(\varepsilon_{mm})\,\mathcal{R}(\varepsilon_{nn})<0.$$4 discs additionally revealed wavefront fringes. A central point of that work is that the HShP spectral shift can be estimated directly from near-field images, without requiring prior knowledge of the dielectric tensor, and then cross-validated by far-field measurements and ab initio calculations (Carini et al., 28 Jul 2025).

Gypsum extends the field beyond the previously demonstrated bulk non-vdW crystals. It is an exfoliable monoclinic sulphate mineral, and thin films of $$\mathcal{R}(\varepsilon_{mm})\,\mathcal{R}(\varepsilon_{nn})<0.$$5 and $$\mathcal{R}(\varepsilon_{mm})\,\mathcal{R}(\varepsilon_{nn})<0.$$6 thickness support shear phonon polaritons that can be imaged by s-SNOM. Edge-launched measurements reveal guided polaritons along directions normal to the $$\mathcal{R}(\varepsilon_{mm})\,\mathcal{R}(\varepsilon_{nn})<0.$$7 and $$\mathcal{R}(\varepsilon_{mm})\,\mathcal{R}(\varepsilon_{nn})<0.$$8 axes, while disk launchers directly visualize the transition from hyperbolic to canalized to elliptical shear topologies. The significance is not only materials diversification; it is the demonstration that shear polariton physics survives in an ultrathin, mechanically integrable, low-symmetry platform (Díaz-Núñez et al., 31 Jan 2025).

4. Spectral tuning and topological transitions

The narrow spectral windows that support intrinsic HShPs are set by phonon resonances, so spectral tuning is central. In $$\mathcal{R}(\varepsilon_{mm})\,\mathcal{R}(\varepsilon_{nn})<0.$$9-Ga$$\gamma(\omega)=\frac{1}{2}\,\mathrm{atan}\!\left( 2\,\frac{\mathcal{R}(\varepsilon_{xy})}{\mathcal{R}(\varepsilon_{xx})-\mathcal{R}(\varepsilon_{yy})} \right),$$0O$$\gamma(\omega)=\frac{1}{2}\,\mathrm{atan}\!\left( 2\,\frac{\mathcal{R}(\varepsilon_{xy})}{\mathcal{R}(\varepsilon_{xx})-\mathcal{R}(\varepsilon_{yy})} \right),$$1, oxygen isotopic substitution from $$\gamma(\omega)=\frac{1}{2}\,\mathrm{atan}\!\left( 2\,\frac{\mathcal{R}(\varepsilon_{xy})}{\mathcal{R}(\varepsilon_{xx})-\mathcal{R}(\varepsilon_{yy})} \right),$$2O to $$\gamma(\omega)=\frac{1}{2}\,\mathrm{atan}\!\left( 2\,\frac{\mathcal{R}(\varepsilon_{xy})}{\mathcal{R}(\varepsilon_{xx})-\mathcal{R}(\varepsilon_{yy})} \right),$$3O redshifts the relevant $$\gamma(\omega)=\frac{1}{2}\,\mathrm{atan}\!\left( 2\,\frac{\mathcal{R}(\varepsilon_{xy})}{\mathcal{R}(\varepsilon_{xx})-\mathcal{R}(\varepsilon_{yy})} \right),$$4 transverse-optical phonons by about $$\gamma(\omega)=\frac{1}{2}\,\mathrm{atan}\!\left( 2\,\frac{\mathcal{R}(\varepsilon_{xy})}{\mathcal{R}(\varepsilon_{xx})-\mathcal{R}(\varepsilon_{yy})} \right),$$5–$$\gamma(\omega)=\frac{1}{2}\,\mathrm{atan}\!\left( 2\,\frac{\mathcal{R}(\varepsilon_{xy})}{\mathcal{R}(\varepsilon_{xx})-\mathcal{R}(\varepsilon_{yy})} \right),$$6 for the higher-frequency modes, consistent with the approximate $$\gamma(\omega)=\frac{1}{2}\,\mathrm{atan}\!\left( 2\,\frac{\mathcal{R}(\varepsilon_{xy})}{\mathcal{R}(\varepsilon_{xx})-\mathcal{R}(\varepsilon_{yy})} \right),$$7 scaling. For mode $$\gamma(\omega)=\frac{1}{2}\,\mathrm{atan}\!\left( 2\,\frac{\mathcal{R}(\varepsilon_{xy})}{\mathcal{R}(\varepsilon_{xx})-\mathcal{R}(\varepsilon_{yy})} \right),$$8, $$\gamma(\omega)=\frac{1}{2}\,\mathrm{atan}\!\left( 2\,\frac{\mathcal{R}(\varepsilon_{xy})}{\mathcal{R}(\varepsilon_{xx})-\mathcal{R}(\varepsilon_{yy})} \right),$$9 and $90^\circ$0, yielding the $90^\circ$1 shift that propagates directly into the HShP bands (Carini et al., 28 Jul 2025).

This spectral tuning alters both the hyperbolic window and the angular response. In the rotated $90^\circ$2 basis, the $90^\circ$3O and $90^\circ$4O samples display corresponding red-shifted curves for $90^\circ$5 and $90^\circ$6: the overall functional form is preserved, but the onset, center, and cutoff of the HShP band move to lower frequency in the heavier isotope. Because the ray geometry can be read directly from near-field images, isotopic substitution provides a chemically clean tuning knob for directional polariton steering in a material system that remains structurally monoclinic (Carini et al., 28 Jul 2025).

Gypsum reveals a distinct, topology-centered tuning mechanism. In its first and second Reststrahlen bands, the sign structure of $90^\circ$7, $90^\circ$8, and $90^\circ$9 changes such that the in-plane response traverses hyperbolic type I, hyperbolic type II, elliptical, and then returns to hyperbolic type II and type I. The experimentally emphasized segment is the transition from hyperbolic shear to elliptical shear via canalization. Here the tuning parameter is simply frequency, yet the response changes qualitatively: open IFCs in the $\Re\{\varepsilon_{mn}\}=0$0–$\Re\{\varepsilon_{mn}\}=0$1 range, nearly flat IFCs around $\Re\{\varepsilon_{mn}\}=0$2, and closed IFCs near $\Re\{\varepsilon_{mn}\}=0$3 (Díaz-Núñez et al., 31 Jan 2025).

A useful caution follows from the broader shear-polariton literature: shear signatures are not synonymous with hyperbolicity. In engineered analogs, asymmetric loss can survive even when the contour is no longer hyperbolic. This suggests that the broader category of shear polaritons includes hyperbolic, canalized, and elliptical regimes, with HShPs constituting the hyperbolic subset (Yves et al., 2023).

5. Engineered, twisted, and source-induced realizations

Engineered metasurfaces have generalized HShP physics from fixed crystal lattices to symmetry-designed surfaces. In electromagnetic hyperbolic shear metasurfaces, two detuned in-plane resonances are arranged at a relative angle $\Re\{\varepsilon_{mn}\}=0$4, producing an effective conductivity tensor

$\Re\{\varepsilon_{mn}\}=0$5

The off-diagonal terms vanish at $\Re\{\varepsilon_{mn}\}=0$6 and grow as the resonators become non-orthogonal. The corresponding rotation of the hyperbolic optical axes is

$\Re\{\varepsilon_{mn}\}=0$7

with extreme axial dispersion near the critical frequency defined by $\Re\{\varepsilon_{mn}\}=0$8. In that regime, the metasurface exhibits geometry-controlled, ultra-confined, low-loss hyperbolic surface waves and broadband Purcell enhancement (Renzi et al., 2024).

A mechanically realized analog appears in twist-induced hyperbolic shear metasurfaces. There, a twisted bilayer elastic metasurface formed by two anisotropic pillar-loaded plates reproduces the full HShP template with an elastodynamic response tensor $\Re\{\varepsilon_{mn}\}=0$9 in place of $_2$00. The twist angle tunes hyperbolic dispersion, axial dispersion, and asymmetric loss at will. Experiments reported a hyperbolic band around $_2$01–$_2$02, axial steering of about $_2$03 over a frequency span of approximately $_2$04 around $_2$05, and a shear factor approaching $_2$06 for small twist angles near $_2$07. The same platform demonstrated reflection-free negative refraction and diffraction-free non-destructive testing based on frequency-direction encoding (Yves et al., 2023).

A separate line of work shows that HShP-like asymmetry can be induced by the excitation source even when the material tensor is diagonal. In a diagonal hyperbolic medium modeled on $_2$08-Ga$_2$09O$_2$10, a vortex source with topological charge $_2$11 generates left-skewed or right-skewed hyperbolic patterns because the complex azimuthal angle $_2$12 makes the vortex factor $_2$13 act as a direction-dependent gain or attenuation term. When a purely imaginary off-diagonal element $_2$14 is then introduced, the system undergoes controlled transitions between left-skewed, symmetric, and right-skewed HShPs. The critical symmetry-restoring values scale with charge: for $_2$15, symmetry occurs around $_2$16; for $_2$17, around $_2$18; and the required sign reverses for negative $_2$19 (Xue et al., 2022).

Orbital angular momentum can also emulate HShP phenomenology in a high-symmetry orthorhombic crystal. In $_2$20-MoO$_2$21, vortex-beam excitation of a gold microdisk launcher produces Airy-like hyperbolic phonon polaritons whose Fourier-space intensity is concentrated on one side of an otherwise symmetric hyperbola. The real-space main lobe deviation $_2$22 increases monotonically with $_2$23: $_2$24 for $_2$25, $_2$26 for $_2$27, $_2$28 for $_2$29, and $_2$30 for $_2$31. These modes are not intrinsic tensorial HShPs in the strict monoclinic sense, but they reproduce the asymmetric branch population and skewed energy flow that define HShP phenomenology in practice (Bai et al., 2023).

6. Functional implications and broader outlook

HShPs matter because they combine deep subwavelength confinement with spectrally steerable directionality and nontrivial dissipation control. In engineered metasurfaces, this yields broadband Purcell enhancements for nearby emitters, with enhancement strongly amplified as the resonator angle $_2$32 is reduced from the orthogonal configuration. In the elastic analog, the same tensorial logic enables reflection-free negative refraction and frequency-multiplexed directional inspection, indicating that HShP physics is not restricted to optics or to phonon polaritons in the narrow sense (Renzi et al., 2024, Yves et al., 2023).

Near-field heat transfer is one of the clearest application domains in natural low-symmetry materials. For $_2$33-Ga$_2$34O$_2$35-based radiative heat transfer, the heat transfer coefficient reaches $_2$36 at a gap of $_2$37 and zero twist. Introducing a scaling factor $_2$38 for the off-diagonal permittivity and defining the twist-induced modulation coefficient as

$_2$39

the maximum $_2$40 reaches $_2$41 at $_2$42 and $_2$43, which is $_2$44 higher than in the shear-free case. This directly links the off-diagonal tensor component to enhanced and twist-tunable near-field thermal transport (Jia et al., 2022).

The concept is also expanding beyond classical phonon-polaritonic media. Charge-neutral graphene nanoribbon metasurfaces under a perpendicular magnetic field support quantum hyperbolic magnetoexciton polaritons with field-tunable elliptic-to-hyperbolic IFC transitions and a canalization regime in which the IFC becomes nearly flat. For armchair ribbons of width $_2$45, periods $_2$46–$_2$47, and magnetic fields $_2$48–$_2$49, the isolated-ribbon mode reaches an effective index $_2$50. This system has been proposed as a quantum, magnetically tunable blueprint for HShP-like canalization (Domina et al., 30 Jun 2025).

A persistent misconception is that HShPs are synonymous with any asymmetric hyperbolic pattern. The literature distinguishes three levels more carefully. Intrinsic HShPs are tied to low-symmetry constitutive tensors with off-diagonal response and frequency-dispersive optical axes. Engineered HShPs reproduce the same tensor physics in metasurfaces or twisted bilayers. Source-induced realizations reproduce the same asymmetric branch selection and skewed propagation through structured excitation, even when the background tensor is diagonal. Taken together, these strands define HShPs less as a single material system than as a robust wave phenomenon at the intersection of hyperbolicity, non-orthogonal resonances, and non-coincident propagation and loss axes (Xue et al., 2022, Renzi et al., 2024).

The field has therefore moved from a material-specific discovery in low-symmetry polar crystals to a broader platform concept. Bulk monoclinic oxides established the phenomenon, isotopic substitution demonstrated materials-level spectral control, gypsum showed that exfoliable thin films can host hyperbolic, canalized, and elliptical shear regimes, and metasurfaces demonstrated that shear can be designed rather than merely inherited from the lattice. The unifying theme is the same throughout: frequency-dispersive optical axes and asymmetric dissipation convert hyperbolic polaritons into shear polaritons, and that conversion produces unusually directional, tunable, and often slow surface-wave transport (Carini et al., 28 Jul 2025, Díaz-Núñez et al., 31 Jan 2025).