A perspective of twisted photonic structures

Moire superlattices-twisted van der Waals (vdW) structures with small angles-are attracting increasing attention in condensed matter physics, due to important phenomena revealed therein, including unconventional superconductivity, correlated insulating states, and ferromagnetism. Moire superlattices are typically comprised of atomic layers of vdW materials where the exotic physics arises from the quantum electronic coupling between adjacent atomic layers. Recently, moire electronics has motivated their photonic counterparts. In addition to vdW materials, twisted photonic systems can also be comprised of metamaterials, metasurfaces, and photonic crystals, mediated by interlayer electromagnetic coupling instead. The interplay between short-ranged interlayer quantum and long-ranged electromagnetic coupling in twisted structures are expected to yield rich phenomena in nano-optics. This perspective reviews recent progress in twisted structures for nanophotonics and outlooks emerging topics, opportunities, fundamental challenges, and potential applications.


Introduction
When a graphene layer is laid upon another with a small twist angle, the composite bilayer generally reveals a moiré pattern with a period much larger than the lattice constants. The twist angle controls the interlayer quantum coupling that strongly influences electronic and optical properties of the composite bilayer. Particularly, when bilayers are twisted at the so-called magic angles, moiré flat bands would emerge and lead to unconventional superconductivity and strongly correlated electronic states in twisted bilayer graphene [1][2][3][4].

Twisted monolayers
We begin with an introduction of plasmonics in twisted bilayer graphene [ Fig. 2]. Using scattering-type scanning near-field optical microscopy (s-SNOM) and dark-field transmission electron microscopy (TEM) [ Fig. 2(a)], one can visualize the periodic variation of local surface conductivity across the moiré superlattice over the domain wall (soliton) network in bilayer graphene with small twist angles [24]. These s-SNOM results originate from the modified electronic structure by interlayer coupling and atomic reconstruction that affected plasmonic responses in bilayer graphene with small twist angles. Hence, the twisted bilayer graphene acts as a lithographyfree photonic crystal for propagating surface plasmon polaritons. Recently, Ref. [24] visualized the interference pattern of propagating surface plasmon polaritons over these soliton networks [ Fig. 2(b)]. Ref. [24] also predicted the formation of a plasmonic bandgap in twisted bilayer graphene that can eliminate the propagation of surface plasmon polaritons in a similar rule of the bandgap in photonic crystals [42]. However, the formation of plasmonic bandgaps in twisted bilayer graphene requires strong scattering of surface plasmon polaritons from solitons that is not yet achieved in experiments. In addition, the soliton network in twisted bilayer graphene separates AB and BA stacking domains and therefore leads to rich nonlocal responses over the moiré superlattices [39][40][41]. For example, by incorporating nanoelectronics measurements with the s-SNOM, Ref. [39] showed that local photocurrent varies over the moiré superlattices [ Fig. 2(c)] and can be altered by electrostatic gating [39]. Chiral materials are ubiquitous in nature with right-and left-handed counterparts under the mirror symmetry.
However, natural chiral materials generally have weak chiral light-matter interactions, e.g., circular dichroism.
Surface plasmon polaritons can become nonreciprocal (without external magnetic field) if an in-plane driving electric current is applied. This leads to the so-called plasmonic Doppler effect [71][72][73][74][75] that has been observed in recent experiments [76,77]. The conventional plasmonic Doppler effect originates from the relative motion between the surface plasmon polaritons and the electric current, but the induced nonreciprocity is generally weak [71]. Ref. [71] predicted that the plasmonic nonreciprocity can be greatly enhanced in graphene moiré superlattices via the quantum plasmonic Doppler effect, which is governed by the strong electron-electron interactions in moiré superlattices where electron bands are flat with extremely low Fermi velocity [71]. This quantum plasmonic Doppler effect in graphene moiré superlattices is therefore important and is promising for nonreciprocal photonic applications.
In addition to stacked bilayers, twisted monolayers can be separated where the interlayer electromagnetic coupling surpasses the quantum electronic coupling and dominates. Under this scenario, these twisted monolayers can support surface waves with exotic isofrequency contours. To facilitate the discussion, we briefly introduce the calculation of surface wave dispersion in twisted monolayers. Without loss of generality, each monolayer is considered to be uniaxial with a conductivity tensor ̿ s . That is, we have ̿ s,1 = [ . By enforcing the electromagnetic boundary conditions, the dispersion of surface waves can be rigorously calculated [46]. The surface waves on twisted monolayers are hybrid TM-TE modes. However, for highly squeezed surface waves, the TM wave components dominate [46]. This way, the dispersion for highly-squeezed surface waves in twisted monolayers can be written as , and rj is the relative permittivity of region ( = 1, 2 or 3).
Based on equation (1), Ref. [43] showed that when two separated black phosphorus monolayers are  [44]. The specific twist angle at topological transition is referred to as the photonic magic angle. Both Refs. [43,44] showed the rich dispersion engineering in twisted monolayers, which can be exploited to induce the broadband field canalization [44,78]. The tunable topological transition via the control of chemical potential by electrostatic gating is expected to offer opportunities for active on-chip nanophotonic devices.

Twisted slabs
Topological transitions revealed in separated monolayers [43,44] can also be observed in twisted slabs of

Twisted photonic crystals
Photonic crystals are periodic structures hosting weakly confined guided waves and can be used for electromagnetic hybrids [42]. Specifically, twisted photonic crystal [ Fig. 1(d)] is a promising platform to explore the photonic analogue of moiré electronics in twisted bilayer graphene. Since twisted photonic crystals are generally quasiperiodic with a very large supercell, the computation cost is very high especially at small twist angles. This calculation difficulty impedes the systematic analysis of both photonic and electronic moiré structures. To mitigate this issue, Ref. [49] developed a high-dimensional plane wave expansion method for analyzing scattering properties of stacked photonic crystals with arbitrary twist angles [ Fig. 4(a)]. This method [49] surpasses the limitations of supercell approximation and reveals strongly tunable resonant chiral behaviors in twisted photonic crystals. In particular, the concept of moiré light line has been proposed [49] and is waiting for experimental verification. In essence, the moiré light line represents the phase boundary between regions with strong and weak circular dichroism in the parameter space of twist angle and frequency [ Fig. 4(b)].
The method from Ref. [49] can be used to investigate other phenomena of moiré photonics, such as photonic moiré flat bands from twisted 2D or 1D photonic crystals. The calculation of flat bands in twisted photonic structures is complex [50,51]. Refs. [50,51] showed the photonic band structures of twisted photonic crystals can be engineered in a similar fashion to the electronic counterpart in twisted bilayer graphene. They further discovered the photonic flat bands as a result of the interaction between in-plane and out-of-plane electromagnetic coupling [ Fig. 4(c)]. In this way, the separation between photonic crystals offers a degree of freedom to tune photonic moiré bands without high pressure [50,51]. "Magic distances" corresponding to the emergence of photonic flat bands over the whole Brillouin zone [ Fig. 4(d)] has also been predicted [51].
Since twisted photonic crystals are associated with aperiodic structures and natural crystals, they can offer a feasible route to explore commensurate to incommensurate transitions [ Fig. 4(e)]. Recently, reconfigurable twisted photonic crystals with controllable parameters and symmetry were created in a photorefractive crystal (e.g., strontium barium niobate (SBN): 61 crystal) by super-positioning two periodic patterns of light [52,53].
Using these commensurable and incommensurable twisted photonic crystals, Ref. Ref. [53] reported the formation of solitons in these moiré photonic crystal structures that smoothly evolve from fully periodic geometries to aperiodic ones, where the soliton formation is attributed to photonic flat-band physics.

Innovative constituent components
One promising direction for twisted photonics is to explore additional constituent components.  [107,108], superlensing effect [109], ghost polaritons [86], and diverging photonic density-of-states [110]. Particularly, hyperbolic metamaterials can enhance the recombination rate of a nearby emitter over a broad spectral range due to the high photonic density-of-states. However, this does not directly lead to a high quantum efficiency since the excited high-momentum eigenmodes are generally confined in the metamaterial. Ref. [111] overcomes this issue by building photonic hypercrystals which possess combined virtues of large broadband photonic density-of-states in hyperbolic metamaterials and strong light outcoupling in photonic crystals [ Fig.   5(a)-(b)]. Correspondingly, the large enhancement of light emission was observed [111]. Moreover, Ref. [112] theoretically reported that the broadband enhancement of on-chip photon extraction can also be achieved by This facilitates the smooth conversion between the eigenmodes in hyperbolic metamaterials and the guided waves in nanofibers. Due to these advances, photonic hypercrystals [111,113,114] and tilted hyperbolic metamaterials [112] can be an attractive component for further research in twisted photonic structures.
Particularly, these innovative twisted hyperbolic metamaterials are expected to host unconventional far-field phenomena related to chiral optics, different from twisted bilayer α-MoO3 slabs where the near-field phenomena dominate. In addition, topological transition for surface plasmon polaritons or phonon polaritons may be explored in free-space photon or other wave systems (e.g., acoustic waves [115][116][117][118][119]) in twisted structures comprised of corresponding components.

Twisted multilayers or multi-slabs
Another prospective direction of twisted photonic structures is to investigate twisted multilayers or multislabs [ Fig. 1(f)]. For twisted multilayers, each twist angle and distance between adjacent layers govern the interlayer interactions and therefore offer more degrees of freedom to tailor light-matter interactions in twisted photonic structures. These twisted multilayer structures can be utilized to engineer different photonic properties that are otherwise not possible in twisted bilayers, in a similar fashion as their electronic counterparts (e.g., twisted trilayer graphene, twisted double bilayer graphene) [92][93][94][95][96][97][98][99].
In this perspective, we have reviewed the recent progress in twisted photonic structures. One enticing short- Another enticing long-term goal is to reshape the radiation feature of conventional light sources (such as moving charged particles, composite magnetic and electric dipoles, and quantum emitters) by exploiting the exotic properties of twisted photonic structures. The unique interactions between light sources and twisted photonic structures may induce a plethora of attractive yet unexplored phenomena of electromagnetic radiation [ Fig. 6], such as Cherenkov radiation [137][138][139][140][141], transition radiation [142,143], Smith-Purcell radiation [144][145][146][147], spin-related near-field directionality [59,67,69,70], and anomalous Doppler effect [60,61], which may enable novel on-chip applications for conventional light sources.

Acknowledgement
The work at Zhejiang University was sponsored by the National Natural

Data availability
Data sharing is not applicable to this article as no new data were created or analyzed in this study.   that with hole arrays [111]. One typical photonic hypercrystal is shown in (a), with the corresponding SEM image of the fabricated sample with different pitches in (b) [111]. (c)-(d) Tilted hyperbolic metamaterial integrated with a nanofiber [112]. The optical axis of hyperbolic metamaterial forms a tilted angle with respect to the norm vector of interface, as shown in (c). The normalized on-chip photon extraction rate fiber / 0 of a quantum emitter, which is positioned close to the hyperbolic metamaterial, is plotted as a function of the tilted angle in (d) [112]. Both photonic hypercrystals and tilted hyperbolic metamaterials can be used to enhance  dipoles (e.g., circularly polarized dipole, Huygens dipole and Janus dipole [59]), and quantum emitters. For conceptual illustration, this figure shows the free-electron radiation when a charged particle perpendicularly penetrates through a twisted multilayer.