Moiré photonic crystals PERSPECTIVES visual rendering of colour printing if the frames from the different colours have a particular alignment, or the display of pictures on LED screens due to the mismatch between the screen and the picture resolutions. On the other hand, the hypnotic aspect of their shapes has attracted artists’ attention, and took part in new artistic movements in the 1960s such as the so-called“ optical art” or phase music. Moiré patterns also found applications in precise measurement: just as beating allows to tune two music instruments, moiré patterns can be used to align precisely two objects, or to measure their misalignment. Following this idea, they were applied to deformation and stress measurement, and to topography and 3D imaging with the technique called“ projection fringes”. The level of precision that can be achieved is so high that similar methods are even used for precise alignment in nanofabrication processes such as some lithography techniques. Finally, moiré patterns were found to be useful for measuring thread density in fabrics using a type of striped ruler called lunometer, an invention that closed the loop with the textile industry.
All the moiré patterns that have been mentioned so far are macroscopic. The underlying geometry is much larger than the wavelength of light in the visible range, and the ray optics framework applies. However, when the size is reduced down to the subwavelength scale, completely different and fascinating phenomena occur. This scale reduction was first introduced in a completely different field of physics. In 2018, it was discovered that, when two graphene sheets are stacked with a slight angular misalignment between their crystal lattices, bilayer graphene electronic properties could be tuned by varying the twist angle. More importantly, for some small and very precise angles( ≲1 °) known as“ magic angles”, twisted bilayer graphene becomes superconductive [ 1 ]. This discovery gave rise to twistronics, a new research field dedicated to twisted 2D materials. Since electrons in condensed matter crystals exhibit a wave-like character and behave similarly to photons in photonic crystals, parallels with photonics were quickly drawn, and research on moiré photonic crystals was launched.
PART 1: MAGIC MOIRÉ PHOTONIC CRYSTALS The exploration of moiré patterns in photonics started by emulating twisted bilayer graphene. Following this idea, each graphene sheet was modelled by a slab photonic crystal with a honeycomb lattice, consisting of a thin dielectric membrane that provides in-plane light confinement, and contains a subwavelength periodic modulation of the refractive index within the plane. When the two layers are stacked with a small angular offset, a new large-scale hexagonal pattern appears: this corresponds to the moiré pattern( Figure 2. a).
What happens geometrically can be viewed as a kind of spatial beating phenomenon. Beating in acoustics occurs when two sine waves of slightly different frequencies ω 0 ± δω are superposed. It results in an average sine wave at frequency ω 0 slowly modulated at frequency δω. Similarly, in moiré photonic crystals, the temporal frequency offset is replaced by a spatial offset arising from the twist angle between layers. This offset generates a long-range modulation that forms the moiré superlattice, whose period increases as the twist angle decreases. Just as with acoustic beats, the smaller the offset, the slower the modulation. This new periodicity is reflected in reciprocal space where a new Brillouin zone corresponding to the superlattice can be defined. It is smaller than the monolayers’ Brillouin zones, and shrinks as the twist angle is reduced( Figure 2. b). However, it is important to notice that it is not always correct to talk about a“ perfect” periodicity for the moiré pattern. In the general case, a moiré pattern is only quasi-periodic. This can be illustrated by the example of two square lattices twisted by 45 °, for which no translational invariance exists due to √— 2 being irrational. Moiré patterns are strictly periodic only at specific discrete twist angles known as commensurate angles. At these angles, the perfect superposition of two lattice nodes at one point implies the existence of other perfectly aligned nodes elsewhere in the lattice. In such cases, the smallest distance between two points of perfect superposition defines the true lattice parameter of the superlattice. Nevertheless, it is always possible to refer to the moiré lattice corresponding to the pseudo-periodicity of the beating pattern.
Figure 1. Macroscopic moiré patterns.( a) Moiré fabrics.( b) Moiré patterns in curtains.( c) Resolution mismatch moiré pattern.( d) Stripped moiré pattern resulting from the superposition of the bridge barriers.( e) Moiré pattern on the back of a chair.
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