Moir´e Fringe Induced Gauge Field in Photonics

We realize moir´e fringe induced gauge field in a double-layer photonic honeycomb metacrystal with mismatched lattice constants. Benefitting from the generated strong effective gauge field, we report direct measurement of the band diagrams of both Landau level flat bands and intermagnetic-domain edge states. Importantly, we observe the correlation between the momentum and orbital position of the Landau modes, serving as an evidence of the noncommuteness between orthogonal components of the momentum. Without complicated time driving mechanics and careful site-by-site engineering, moir´e superlattices could emerge as a powerful means to generate effective gauge fields for photonics benefiting from its simplicity and reconfigurability, which can be applied to nonlinearity enhancement and lasing applications at optical frequencies.

When two periodic patterns with mismatched lattice constants are combined, they form a beating pattern with longer range quasiperiodicity, which are called moiré superlattices (MS). The atomic scale MS have been utilized to modify the electronic properties and have been exploited for generating strong electron correlations, Hofstadter spectrum, fractional quantum Hall effect and van Hove singularities [1][2][3][4][5]. Very recently, it was discovered that magic-angle twisted bilayer graphene possess flat band and function as correlated Mott insulators, providing a gateway towards high temperature superconductors [6,7]. Theoretical studies have attributed the flat band of magic-angle bilayer graphene to the effective gauge field EGF [8][9][10]. In optoelectronics, the versatility of the MS in tuning the electronics band structures has facilitated the interactions of electrons with photons resulting in formation of nanoscale waveguiding and IR plasmons [11]. Despite the normally small relative angle between the bilayer van der Waals heterostructures, the electronic band structure is radically altered, resulting in various novel physical properties such as topological helical states [12,13] and topological crystalline insulators [14].
MS have been also explored in classical systems. In photonics, MS have been used to create effective gratings to control the diffractions of light [15]. Plasmonic double layer MS are found to induce a strong circular dichroism for light [16]. Very recently, flat bands in double layer photonic crystals are explored for their capabilities in localizing and delocalizing light [17]. Meanwhile, effective gauge field (EGF) has been a long-sought effect in both quantum and classical systems such as photonics and acoustics. Using methods including straining and deforming the lattice, giant EGF has been created in graphene [18][19][20][21]. In photonics and optomechanical systems with EGF, nonreciprocal phase shift, Lorentz force, Landau energy levels have been revealed [22][23][24][25][26][27][28][29][30]. Lasing utilizing the Landau level modes has also been proposed in photonic graphene [31]. In acoustic systems with EGF, Landau levels were found in phononic crystals with gradual deformations of the crystal inclusions [32], and chiral zero mode was observed in acoustic Weyl systems [33].
Here we report the creation of EGF, or more specifically, effective magnetic field (EMF), in a moiré bilayer metacrystal (MBM) [ Fig. 1(a)], which leads to experimental observation of the band diagrams of the photonic Landau level modes and intermagnetic-domain edge states in the microwave regime. Our microwave near field measurements provide a direct and definite evidence of the gauge field induced flat band and the surface states. In addition, we directly observe the correlation between the momentum and the orbital location of the Landau modes, which indicates noncommuteness between the orthogonal components of momentum introduced by the EMF. Our findings provide a new platform for studying moiré pattern related effects in classical wave systems and may find applications in flat band lasing, nonlinear optics.
The bilayer metacrystal consists of two parallel twodimensional hexagonal-lattice metallic ring metacrystals separated by a thin dielectric layer (ε r ≈ 10). A relative lattice constant mismatch in the y direction is introduced to the bilayer system to form MBM, with a quasiperiodicity Λ given by l 2 y =δl in which δl is the difference in their lattice constants along the y direction. An illustration of the MBM whose metallic rings represented by colored dots are given in Fig. 1(b). As will be shown later, due to the spatially varying local misalignment between the two layers in the MBM, an inhomogeneous gauge field is introduced, leading to generation of effective magnetic field for each Dirac cone [ Fig. 1(c)].
To understand the origin of the artificial magnetic field, we first study two identical hexagonal metacrystals aligned to each other. The primitive vector of each layer has a length of 5 mm and the inner and outer radius of the rings are 2.1 and 2.4 mm, respectively. Each metacrystal layer alone supports spoof surface plasmons [34][35][36][37][38][39], which exhibit Dirac cones around 12.5 GHz at K and K 0 points of the Brillouin zone due to the C 3v symmetry [40][41][42] (Supplemental Material [43], Fig. S1). By positioning the two metacrystals close to each other separated by a thin dielectric layer, they strongly interact with each other and form hybridized modes. The Dirac cones experience a strong energy splitting due to the near-field interaction. Single Dirac cone dispersion similar to that of a single layer is found at 12.65 GHz (Supplemental Material [43],  Fig. 1(a)]. The lower excitation dipole is fixed at the center of the sample, while the upper receiving dipole performs a 2D grid raster scanning. After a 2D Fourier transformation of the measured field, we obtain the equifrequency contours (EFC) at different frequencies. The Dirac cone dispersion of the EFC are clearly visualized (Supplemental Material [43], Fig. S1) through the shrinking and expanding of the EFC when the frequency sweeps through the Dirac cone.
When a relative shift Δ is introduced in y between the two layers [ Fig. 2(b)], the MBM experiences a symmetry reduction from C 3v to C 2h that shifts the Dirac cones along the KðK 0 Þ − M high symmetry line [ Fig. 2(c)], rather than opening topological valley gaps [44][45][46][47][48][49]. The Dirac cone dispersion can be fully described by a low energy linear Hamiltonian that can be derived from the C 2 rotational symmetry and the parity-time-reversal symmetry (Supplemental Material [43], Note 1): where ω 0 is the frequency of the Dirac points, v x;y the fermi velocities in the x (y) directions, T the tilting parameter The red layer has a normal hexagonal lattice whose lattice constant is a ¼ 5 mm. The lower (blue) layer is stretched in the y direction relative to the upper layer, and a moiré superlattice is formed with the periodicity indicated by the black dashed lines. κ is the deviation of the excitation dipole from the moiré pattern center. The gray and the green wave forms represent the Landau level flat band modes (0th and AE1th order) and the orange one in the middle represents the interdomain surface states. These modes are localized in the y direction and propagating in the x direction. (b) Illustration of the 1D moiré pattern formed between two lattice-constant mismatched metacrystals. The red and blue dots represent the metallic rings in the upper and lower layers, respectively. Two effective magnetic field domains D and D 0 are identified. The moiré quasiperiodicity is Λ. (c) Top: Gauge field represented by the location of the Dirac cones measured locally within one moiré pattern. Bottom: the distribution of the effective magnetic field given by the gradient of the gauge field. The sign switching leads to formation of two magnetic domains in each moiré periodicity. [40], δk the location of the Dirac cone that plays the role of the gauge field. All the above parameters can be retrieved from the band structure obtained by full wave simulation. The numerically extracted location and ω 0 of the Dirac points are given in Fig. 2(d). Note that beyond Δ ¼ 0.2l y , the two Dirac points annihilate each other, and a band gap is formed.
The moiré fringe is equivalent to a periodic nonuniform shift Δ between the two layers as a function of y. The system has a mirror symmetry M y at locations where the two layers are perfectly aligned, the EGF ⃗ A x is, hence, symmetric about this mirror plane. Therefore, for each Dirac point the moiré periodicity can be separated into two magnetic domains [D (y > 0) and D 0 (y < 0), respectively, in Fig. 1(b)] with positive and negative EMF [ Fig. 1(c)]. The EMF satisfies To observe the Landau levels experimentally, a MBM sample is prepared with a mismatch δl=l y ¼ 2.5% and a quasiperiodicity 346.4 mm. Snapshots of the top and bottom layers of the MBM are given side by side in Fig. 2(e). The field distribution is measured with the excitation dipole located at several locations with different displacements (κ) from the magnetic domain wall [ Fig. 1(a)]. Because of the lack of translational symmetry in the y direction, k x remains the only good quantum number. By combining all the measurement results at different excitation positions, the full dispersions of the photonic landau levels are obtained [ Fig. 3(a)], which are in excellent agreement with that obtained by full wave simulation [ Fig. 3(b)]. The band structure shows relatively flat bands of double degeneracy around the middle of the Brillouin  zone. At large k x , the two modes split and acquire a large positive and negative slope, respectively. We show the experimental results of δl=l y ¼ 8% in the Supplemental Material [43], Note 3. Note that the Landau level modes remain even under much smaller moiré periodicity (Ref. [43], Note 5).
Next, we show that the observed Landau levels can be explained by the gauge field present in Eq. (1). Because of merging and annihilation of the two Dirac points and formation of a band gap at locations far from the moiré center, a linear Dirac Hamiltonian that only accounts for a single Dirac cone is insufficient to describe the whole system. Instead, a quadratic Hamiltonian is utilized to take into account the shift and annihilation of both Dirac cones: where α; β; E; Ω 0 and v y as functions of Δ, hence y. Derivation of the quadratic Hamiltonian from the linear Hamiltonian and the retrieval of the parameters are detailed in Supplemental Material [43], Note 2. Specifically, the gauge fields for the two Dirac points are given as AE ffiffiffiffiffiffiffiffi ffi E=α p , respectively. The numerically calculated spectrum based on the Hamiltonian Eq. (2) is presented in Fig. 3(c), which nicely captures the main feature of the Landau levels including the doubly degenerate flat bands, confirming the moiré-fringe-induced EMF as the origin of the flat bands. The double degeneracy originates from the two symmetrical Landau level orbitals residing in the two magnetic domains D and D 0 within each moiré period Λ. When k x approaches the Brillouin zone boundary, the Landau level orbitals in each magnetic domain gradually move towards the magnetic wall, resulting in coupling between them and, consequently, an energy splitting and a large dispersion in the Landau levels. The correlation between the orbital location in the y direction and k x can be revealed by the microwave near-field scanning experiment.
As shown in Figs. 3(d)-3(f), for the excitation dipole with κ equals 0, 39, and 65 mm, different parts of the band structure are excited preferentially. More specifically, with a particular location of the excitation dipole, only the modes with orbitals strongly overlapping with the excitation position can be efficiently excited. For each Landau level, these modes are centered at a particular k x [marked by the white arrows in Figs. 3(d)-3(f)], indicating the presence of correlation between k x and the orbital center in the y direction. This can be understood by a simple harmonic oscillator model, wherein the location of the potential minimum is determined by k x [50]. As shown in Figs. 4(a) and 4(b), for deviations of 65 and 39 mm, we observe localized states centered around the excitation dipole due to the diminishing group velocity, representing the almost flat band states around k x ¼ 0 and k x ¼ 0.4π=a.
On the other hand, when the dipole is placed right at the MBM's domain wall (κ ¼ 0) [Fig. 4(c)], the real space field distribution reveals an interdomain guided mode propagating in x, but confined in y.
This correlation is further illustrated by the experimental data of eigenfield distributions extracted for various k x [ Fig. 4(d)] (Supplemental Material [43], Note 4). For small k x , the orbitals are located far from the domain wall, where the local relative shift between the two metacrystal layers is around 0.2l y and the Dirac points are expected to meet at the M point. Consequently, the orbitals have negligible interactions and are doubly degenerate in frequency. As k x increases, the orbitals in each magnetic domain move towards the domain wall, dictated by the k x dependent potential minima in the Landau gauge. When k x approaches the location of the original Dirac point (2π=3a) the orbitals strongly interact with each other, forming odd and even modes (defined by E x ) at different frequencies. In experiments, the even mode extracted from the measurement data is illustrated in Fig. 4(d) using the lower frequency branch after the two Landau levels split in frequency, while the odd mode is described in the Supplemental Material [43], Fig. S7. The measured y-k x correlation fits well with the theoretical result obtained using Eq. (2) [Fig. 4(e)] and that obtained from the full wave simulation (Supplemental Material [43], Fig. S8). Finally, for the sake of completeness we also show the theoretically calculated eigenfield distribution for the odd mode [ Fig. 4(f)], which only shows noticeable difference from that of the even mode at large k x when the field distribution is close to the domain wall.
In conclusion, we have realized effective magnetic fields in moiré bilayer metacrystals, observed the corresponding photonic Landau levels, and demonstrated the correlation between momentum and orbital location. The spoof plasma modes employed in the construction of the Dirac points allow strong interaction between the layers, hence creating a strong effective gauge field. Our experiments provide a direct and definite evidence of the Landau level zero bands and the interdomain surface states in the moiré pattern system and thus provide a platform for studying moiré pattern effects in classical wave systems. Our finding may find various applications in photonics, such as flat band lasing and enhancement of nonlinear optical effects.