Progress In this paper, firstly, the theory for hyperbolic PhPs is described. In the equation of dispersion and dielectric constant, it can be concluded that the hyperbolic propagation originates from the different dielectric constant. In addition, the angle between the edge orientation and the crystallographic direction significantly affects the optical response, and serves as a key tuning parameter in tailoring the polaritonic patterns. Secondly, the hyperbolic PhPs propagation characteristics including the focusing, tunability, and optical topological transformation, are discussed. These exotic properties pave the way for the nanophotonic device. Thirdly, we conclude and outlook the development of in-plane hyperbolic PhPs. With the ability of locally controlling light at the nanoscale, it can be expected that the in-plane hyperbolic PhPs have a broad prospect in nanophotonics.

Conclusions and Prospects In this work, we summarize the propagation characteristics of in-plane hyperbolic PhPs based on natural metamaterials. We demonstrate that the relationship between the dielectric constant and dispersion of in-plane hyperbolic PhPs and the angle between the edge orientation and the crystallographic direction are key parameters for achieving in-plane hyperbolic PhPs. The methods to realize reflection, diffraction, focusing, optical field enhancement, and optical topological transformation of in-plane hyperbolic PhPs are summarized. This work can promote the theory and technology development of PhPs in nanophotonics.

The research of in-plane hyperbolic PhPs based on natural metamaterials is still in its infancy. At present, the tunable propagation of in-plane hyperbolic PhPs has made some progress. Some optical devices such as focusing lenses, prisms, and gratings have been realized. However, it is still at the beginning stage of a single optical device function. More complex and multi-functional optical systems are required. In addition, the currently reported natural in-plane hyperbolic metamaterials are mainly α-V₂O₅ and α-MoO₃, in which the PhPs spectra are in the infrared band. The in-plane hyperbolic PhPs in other bands are needed. Therefore, more natural metamaterials need to be discovered to realize the application of multi-band in-plane hyperbolic PhPs and nanophotonics.

Phonon polaritons (PhPs) can strongly confine the free-space light field deeply below the incidence wavelength. In comparison with the well-documented metallic plasmon polaritons, PhPs can achieve improved light confinements, significantly reduced optical losses, and much higher quality factors. These characteristics enable a wealth of potential applications, such as sensing, thermal management, and ultramicroscopy. Recently, hyperbolic PhPs in thin slabs of the van der Waals materials have been intensively investigated as they exhibit extreme field confinement and exotic ray-like propagation, with potential for hyperlensing and focusing of mid-infrared light at the nanoscale. However, hyperbolic PhPs exhibiting out-of-plane hyperbolic propagation limits the development of hyperbolic nano-optics compatible with on-chip optical devices. In this regard, hyperbolic PhPs with in-plane propagation have recently been demonstrated in artificial metasurfaces and in the natural van der Waals crystals α-V₂O₅ and α-MoO₃. Compared with the complex photolithography technology from artificial metasurfaces, the natural metamaterials have great prospects in planar configuration nanophotonic devices. Hence, this work reviewed the research progress of in-plane hyperbolic PhPs based on natural layered metamaterials in recent years.