PERSPECTIVES
Moiré photonic crystals
How does light respond to these multiple periodicities? It depends on the distance separating the two layers. Qualitatively, if the interlayer spacing is larger than the operating wavelength, light will propagate through the two slab photonic crystals successively without noticing the moiré pattern. The observed effects are simply the combined influences of both layers individually, just as two successive gratings on an optical bench diffract light in two directions to form a 2D lattice of points in the far field. However, when the two layers are placed in near-field proximity, such that the interlayer distance is smaller than the wavelength, new behaviours emerge. In this configuration, light within the bilayer system not only perceives the periodicities of individual monolayers, but also experiences the moiré potential landscape, resulting in new photonic modes. In other words, the modes localized in each layer will interact with the( tilted) ones from the opposing layer. If the interlayer coupling between modes in different layers is of the same order of magnitude as the intralayer coupling within a single layer, the modes from each layer will strongly hybridize to form new bilayer modes( see Figure 3 for intra and interlayer couplings). These“ moiré modes” inherit the characteristics of the moiré pattern, displaying a spatial profile that reflects the moiré( pseudo-) periodicity and are highly sensitive to the twist angle.
What has attracted attention on moiré photonic crystals is that, for a few very specific twist angles, some
Figure 2. Graphene-like moiré photonic crystal.( a) Bilayer geometry.( b) Moiré lattice and reciprocal lattice depending on the twist angle.
moiré modes see their field being strongly localized to small spots in the bilayer superlattice. This behaviour is unusual because the geometry does not contain any heterostructure that could confine the modes so tightly. Instead, this effect is related to the moiré modulated interaction between the two layers. More specifically, it results from the interplay between the intra and interlayer coupling strengths, which determine the spatial profile of the bilayer moiré modes. The sharply localized modes are easy to identify in the band diagram of the bilayer photonic crystal: they are characterized by the emergence of a flat band, ∀k →, ω( k →) = ω 0, that signifies zero group velocity across all wavevectors. This is why people refer to“ slow light” in the context of moiré photonic systems. The twist angle plays a crucial role in the emergence of these slow modes. Changing the twist
Figure 3. Intra and interlayer couplings in a bilayer photonic crystal. The modes can be guided modes living in each layer for example. angle effectively shifts both lattices relative to each other, hence modifying the way two modes located in different layers interact together. Consequently, a slight change in the twist angle has a pronounced impact on the interlayer coupling, giving rise to“ magic angles” that correspond to a particular balance between intra and interlayer coupling strengths. While looking at the formation process of the flat band as a function of the twist angle, it can be seen that it originates from the interaction between the slightly shifted dispersions of the two layers in k-space. Analogous to electrons in graphene, photonic dispersion in a honeycomb photonic crystal exhibits Dirac cones, a characteristic feature marked by a linear dispersion close to the K points, where the conduction and valence bands touch. At“ magic angles”, the Dirac cones from the two layers strongly hybridize and merge into a quasi-flat band dispersion( Figure 4).
PART 2: NEW MOIRÉ GEOMETRIES The first studies on moiré photonic crystals with honeycomb lattices were very similar to those on 2D materials. While retaining the concept of the moiré patterns, photonic crystals physics allows much more freedom in the geometry and shape of the moiré patterns than condensed matter physics, which is constrained by the availability and stability of existing materials. Consequently, the bilayer photonic crystal platform opens the door to a wide variety of moiré patterns, as there is no reason to restrict our imagination to bilayer graphene-like geometries at small twist angles [ 2 ].
The first deviation from bilayer graphene geometry lies in the order of magnitude of the twist angle. In photonics, magic angles are not always small, but can be as large as 22 ° for example [ 3 ]. This difference might be related to the interlayer coupling being stronger between slab
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