Fourier modal method for moiré lattices

In recent years twisted bilayers of two-dimensional (2D) materials have become very popular in the field due to the possibility to totally change their electronic properties by simple rotation. At the same time, in the wide field of photonic crystals, this idea still remains almost untouched, and only some particular problems have been considered. One of the reasons is the computational difficulty of accurate consideration of moiré superlattices that appear due to the superimposition of misaligned lattices. Indeed, the unit cell of the complex lattice is typically much larger than the original crystals and requires many more computational resources for the computations. Here, we propose a moiré-adapted Fourier modal method (MA-FMM) in the form of scattering matrices for the description of twisted one-dimensional (1D) gratings' stacks. We demonstrate that MA-FMM allows us to consider sublattices in close vicinity to each other and account for their interaction via the near field. In the developed numerical scheme, we utilize the fact that each sublattice is only 1D periodic and therefore simpler than the resulting 2D superlattice, as well as the fact that even a small gap between the lattices filters out high Fourier harmonics due to their evanescent origin. Such approach accelerates the computations from 1 up to 3 or more orders of magnitude for typical structures depending on the number of harmonics. In turn, the high computational speed paves the way for rigorous study of almost any photonic crystals of the proposed geometry and demonstration of specific moiré-associated effects.

Figure 1

Schematic of the twisted stack consisting of two arbitrary 1D-periodic lattices twisted around a common axis.

Figure 2

Sketches of the (a) upper and (b) lower sublattices and (c) their stack in real and reciprocal spaces. The left column in (a) and (b) schematically shows the structure of the rotated 1D-periodic sublattices and their reciprocal lattices with respect to the coordinate axes and wave vector of the incident light ${\mathbf{k}}_{\parallel }$. The right column in (a) and (b) formally considers the same sublattices as structures with mutual 2D periodicity. Each sublattice binds harmonics of either a common row or column. (c) shows the superimposing of 1D sublattices resulting in the formation of the true 2D-periodic structure, which substantially connects all the harmonics. The dashed circle in the bottom plot contains the harmonics that fulfill condition (26) $|{\mathbf{k}}_{\parallel }+n{\mathbf{G}}^1+m{\mathbf{G}}^2|<\sqrt{{\varepsilon }_{\mathrm{gl}}{\omega }^2/c^2+{(-ln\mathrm{\Theta }/H)}^2}$ for real ${\varepsilon }_{\mathrm{gl}}$ and therefore are used for the calculation of twisted lattices coupling.

Figure 3

Scheme of the 1D lattice sandwiched between homogeneous adjacent layers. Long-dashed lines separate the periodic part of the structure from the outer dielectric environment and lie in homogeneous layers; short-dashed lines denote the boundaries of the unit cells. Arrows illustrate the incoming and outgoing harmonics that are connected via the scattering matrix.

Figure 4

Study of the computation convergence with the number of harmonics used to calculate the absorption of normally incident $x$-polarized light in the plasmonic stack depicted in (a). (b) shows the dependence of the absorption on the total number of harmonics taken into account in a logarithmic scale. The inset employs the linear scale for the square root of the harmonics to demonstrate the details of the oscillations near the expected limit. (c) shows the calculation time scaling with the number of harmonics for the different approaches. The green line and circles correspond to the reference FMM, and the yellow line and circles correspond to our approach without filtration, whereas red and blue ones correspond to different filtration cutoff levels.

Figure 5

Absorption maps of $p$-polarized 1200-nm-wavelength light in four sketched ${\mathrm{SiO}}_2$-embedded plasmonic lattices illuminated through the prism ($k_x/k_0$ and $k_y/k_0$ are in-plain wave vector components normalized by the wave vector in vacuum $k_0=\omega /c$; $x-y$ projections show the bottom-up view, which is in accordance with other figures). (a) and (b) demonstrate the hyperboliclike behavior of plasmons in solitary lattices placed at different distances from the prism. (c) and (d) show the mode hybridization in stacks of (c) perpendicularly and (d) ${60}^{\circ }$-angle-crossed identical plasmonic lattices. Green dashed lines indicate the boundaries of the 1D lattices Brillouin zones, whereas the cyan lines are moiré-pattern-associated boundaries.

Figure 6

(a) Absorption and (b) emission of 10-$\mu \mathrm{m}$ light from the twisted diamond photonic crystal slab with graphite inclusions (insets). (a) shows the absorption of the right- and left-handed circularly polarized light as well as the degree of circular polarization in absorption as a function of the rotation angle. (b) demonstrates the angle dependence of the upwards and downwards emission in the normal directions for the circularly polarized ${\sigma }^{\pm }$ dipoles. The bottom panel shows the routing efficiencies ${\eta }^{-}=-{\eta }^{+}$, which almost reach the ultimate unitary value.