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Abstract
Twistronics has been studied for manipulating electronic properties through a twist angle in the formed moiré superlattices of two dimensional layer materials. In this paper, we study twistoptics for manipulating optical properties in twisted moiré photonic patterns without physical rotations. We describe a theoretic approach for the formation of single-layer twisted photonic pattern in square and triangular lattices through an interference of two sets of laser beams arranged in two cone geometries. The moiré period and the size of unit super-cell of moiré patterns are related to the twist angle that is calculated from the wavevector ratio of laser beams. The bright and dark regions in moiré photonic pattern in triangular lattices are reversible. We simulate E-field intensities and their cavity quality factors for resonance modes in moiré photonic pattern in square lattices. Due to the bandgap dislocation between the bright and dark regions, the resonance modes with very high quality-factors appears near bandgap edges for the moiré photonic pattern with a twist angle of 9.5 degrees. At the low frequency range, the resonance modes can be explained as Mie resonances. The cavity quality factor decreases for resonance modes when the twist angle is increased to 22.6 degrees.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The moiré phenomenon occurs when repetitive structures are superposed against each other with a relative rotation angle [1]. Recently, twisted bilayers of two-dimensional (2D) materials have ignited significant interest due to the ability in experiments to control the relative twist angle between successive layers [2–59]. This approach was referred as “twistronics” [2,32,57] for manipulating their electronic properties through a twist angle in the formed moiré superlattices. Manipulating the twist angle between the two layers of 2D materials has resulted in a fine control of the electronic band structure [2], magic-angle flat-band superconductivity [3–7], the formation of moiré excitons [8–12], interlayer magnetism [13,39,58], topological edge states [8,14–18], and correlated insulator phases [20–22]. Research has progressed to moiré superlattices between two layers of van der Waals materials [10,13,18] including twisted bilayer transition metal dichalcogenides [48–50].
Very recently, several groups have studied twistoptics: studies of optical properties in twisted moiré lattice [60–62]. A moiré photonic lattice has been generated in a photorefractive crystal by a shallow modulation of the refractive index induced by two mutually twisted, periodic square lattices generated by laser interference (both sublattices interfere in one plane) [61,62]. The study of twistoptics is still in the very early stage [60–62].
In this paper, we describe an approach for the formation of single-layer moiré photon pattern through an interference of two sets of interfering beams. The twist angle of the moiré interference pattern is determined by the wavevector ratio of these laser beams. We simulate the cavity quality factors for resonance modes in a wide frequency range for the moiré photonic pattern in square lattices with a twist angle of 9.5 degrees and in a certain frequency range for the one with a twist angle of 22.6 degrees. The high cavity quality factors are observed for resonance modes near photonic bandgap edges.
2. Theoretical approach for single-layer moiré photonic pattern for twistoptics
We introduce the moiré concept into the laser interference pattern in a square lattice. When laser beams (with wave-vectors of km and kn) interfere, the intensity of the interference pattern is modulated by a cosine function with a period and orientation that can be characterized by a wave-vector difference of (km-kn). Figure 1(a) shows an arrangement of wave-vectors (k1, k2, …, k8) for eight laser beams. (k1,k2,k3,k4) have the same magnitude and (k5,k6,k7,k8) also have the same magnitude that is much smaller than that for (k1,k2,k3,k4). The wave-vector difference (kn+1-kn) among (k1,k2,k3,k4) determines square photonic lattices with a small periodicity. As shown in Fig. 1(a), the direction of wavevector difference (k1-k8) is twisted from (k3-k8) by a rotation angle α. Thus the formed square lattices by (k1-k6), (k2-k7), (k3-k8), and (k4-k5) is twisted by a rotation angle α from those by (k1-k8), (k2-k5), (k3-k6), and (k4-k7).
Fig. 1. (a) Schematic of arrangement of 8 wave-vectors of interfering beams for the generation of a twist angle α; (b) Schematic of optical setup for 8 interfering laser beams; (c-d) moiré interference patterns with a twist angle of 7.2 degrees at different z-location=0 and 0.27Pz, respectively. The unit super-cell, moiré period Pmoiré, lattice period Psmall, bright and dark regions are indicated in figures. (e) x-z cross section of moiré interference pattern.
In experimental setup, these wave-vectors (k1, k2, …, k8) can be achieved through the configuration of interfering angles [63,64]. As shown in Fig. 1(b), one set of interfering beams are arranged in a cone with a cone angle of 2β1 with their wave vectors of: {k1…4}={k(sin(β1)cos(45+n×90)°, sin(β1)sin(45+n×90)°, cos(β1)), n=0,1,2,3. The other set of interfering beams are arranged in a cone with a much smaller cone angle of 2β2 than 2β1 with wave vectors of {k5…8}={k(sin(β2)cos(45+n×90)°, sin(β2)sin(45+n×90)°, cos(β2)), n=0,1,2,3. In this case, the twist angle α in Fig. 1(a) is determined by Eq. (1):
The interference intensity of these 8 beams, I(r), is calculated by Eq. (2):
For bilayer graphene with a twist angle of θ, the moiré period equals a/[2sin(θ/2)], where a is the lattice constant of graphene [20]. From both Eq. (3) and the equation in above line for the moiré period for the twisted bilayer graphene, the moiré period and the size of unit super-cell increase with decreasing twist angle (rotational angle).
Fig. 2. (a) Schematic of the arrangement of 12 wave-vectors of interfering beams for the generation of a twist angle α indicted by dashed blue line and dotted purple line; (b-c) moiré interference patterns with a twist angle of 9.9 degrees at z-location=0 and 0.55Pz, respectively. The moiré period Pmoiré, and bright regions are indicated. Dashed red circles indicate the interface between two distinct pattern regions.
The moiré period Pmoiré, small period Psmall (lattice period) and unit super-cell are indicated in Fig. 1(c) for a moiré pattern with a twist angle of 7.2 degrees. In Eq. (2), there is an interference intensity modulation in z-direction by cos(k(cos(β1)-cos(β2))z). The formed moiré photonic lattice has a feature distance between bright zones of Pz=λ/(cos(β1)-cos(β2)) in z-direction as indicated in Fig. 1(e). At location z=0 as indicated by the solid white line in Fig. 1(e), the pattern in Fig. 1(c) has a bright region at the center of the solid red square and dark region at the corners of the square, corresponding to AA and AB stacking region in twisted bilayer 2D layered materials [26], respectively. Figure 1(d) shows a moiré interference pattern at a location z=0.27Pz as indicated by the dashed white line in Fig. 1(e). Dual lattices appear at the central region inside the unit super-cell in the figure. The bright and dark regions have been described as almost-uniform and graded regions and the photonic patterns have been assigned as graded photonic super-crystals [63,64]. A pattern at a different z-location can be selected for the study of twistoptics. As shown in Fig. 1(e), we can get the clearest moiré pattern at z=0.
We can also generate moiré photonic patterns in triangular lattice. As shown in Fig. 2(a), one set of interfering beams have six beams and are arranged in a cone with a cone angle of 2β1 with their wave vectors of: {k1…6}={k(sin(β1)cos(n×60)°, sin(β1)sin(n×60)°, cos(β1)), n=0,1,..,5. The other six interfering beams are arranged in a cone with a much smaller cone angle of 2β2 than 2β1 with wave vectors of {k7…12}={k(sin(β2)cos(n×60)°, sin(β2)sin(n×60)°, cos(β2)), n=0,1,…5. The wave-vector difference (kn+1-kn) among (k1,…,k6) determines triangular photonic lattices with a small periodicity Psmall ${\cong} $ 2π/(2ksin(β1)sin60). As indicated by dashed blue lines in Fig. 2(a), the direction of vector difference (k1-k9) is twisted from (k4-k9) by a rotation angle α. The direction of vector difference (k1-k8) is also twisted from (k4-k8) by the same rotation angle α as indicated by dashed purple lines. Thus the formed triangular lattices by (k1-k9), (k2-k10), (k3-k11), (k4-k12), (k5-k7) and (k6-k8) is twisted by a rotation angle α from those by (k4-k9), (k5-k10), (k6-k11), (k1-k12) and (k2-k7). Similarly, the formed triangular lattices by (k1-k8), (k2-k9), (k3-k10), (k4-k11), (k5-k12) and (k6-k7) is twisted by the same rotation angle α from those by (k4-k8), (k5-k9), (k6-k10), (k1-k11) and (k2-k12). α=a1+a2 as shown in Fig. 2(a). These angles can be determined by following equations:
Figure 2(b) and 2(c) show moiré interference patterns with a twist angle of 9.9 degrees at z-location=0 and 0.55Pz, respectively. These two patterns are different due to the modulation from k(cos(β1)-cos(β2)z in the cosine function in Eq. (2). The pattern inside the dashed red circle in Fig. 2(b) has motif of cylinders in a triangular lattice orientated in directions of 0, 60 and 120 degrees. Outside the dashed red circle, the motifs are mostly in triangular shape in a triangular lattice orientated in 30, 90, and 150 degrees. In Fig. 2(c), the triangular lattices inside and outside the dashed red circle have a same orientation. There are air ring structures surrounding the cylinders inside the dashed red circle while there are only cylinder patterns outside. Thus there are two distinct photonic materials inside and outside the dashed red circles in Fig. 2(b) and 2(c). After half Pz, the bright and dark regions are switched. The bright regions are inside the dashed red circles in Fig. 2(b) while the bright regions are outside in Fig. 2(c).
3. Cavity quality factors and resonance modes in moiré photonic lattices with different twisted angles
In this section, we simulate cavity quality factors and electric field (E-field) intensity distributions for resonance modes in silicon moiré photonic pattern using the harmonic inversion function included in the MIT MEEP software [65,66]. The interference intensity function, I(r), in Eq. (2), is replaced by a binary Si/air structures by comparing I(r) with a threshold intensity, Ith, that is set to be 28% of maximum of I(r). A step function is used: ɛ(r) = 1 (for air) when I < Ith, and ɛ(r) = 12 (for Si) when I > Ith. Simulated E-field intensity distributions as shown in Fig. 3(a-l), is performed for the moiré photonic pattern with a twist angle of 9.5 degrees, a thickness of 2Psmall, and a unit cell size of 12Psmall×12Psmall, the same size as the one used for photonic band gap calculation [63,67]. The dashed red squares indicate the unit super-cell. The bright (graded) and dark (almost-uniform) regions are indicated by the dashed blue and green squares, respectively. The frequency Psmall/λ and Q-factor are labelled for each figure. At frequency Psmall/λ= (0.576, 0.531, 0.503, 0.491, 0.478, 0.432, 0.380, 0.343, 0.333, 0.267, 0.255, and 0.238), we obtain Q-factor=(291256, 402479, 32690, 863601, 521476, 254402, 417278, 13306, 1491234, 69204, 26000, and 167471), respectively.
Fig. 3. Simulated electric field intensity for resonance modes in moiré photonic lattice with a twist angle of 9.5°. Frequency Psmall/λ and Q-factor are labelled in each figure (a-l).
Figure 4 plots Q-factors at different frequency for resonance modes in moiré Si photonic pattern in Fig. 3 with a twist angle of 9.5 degrees. The red rectangle in Fig. 4 indicates the location of photonic band gap for the bright (graded) region in Fig. 3 while the purple rectangle indicates the common photonic band gap in both bright (graded) and dark (almost-uniform) regions [63,67]. Within the frequency range indicated by these rectangles, there is no strong resonance mode in Fig. 4. At the top edge of band gap at Psmall/λ = 0.478, and bottom edge at 0.432, the resonance modes show up just outside the dashed blue squares in Fig. 3(e) and 3(f). A resonance mode in Fig. 3(g) appears just inside the blue square at Psmall/λ = 0.478, near the edge of photonic band gap in dark region. In Fig. 3(h), the beam propagates in the dark region and has weak resonance. The strong modes in Fig. 3(i) can be considered as a resonance at a point-cavity. At higher frequencies, resonance modes cover both bright (graded) and dark (almost-uniform) regions in Fig. 3(a-d). The resonance modes look like square shape orientated 45° and 0° relative to the horizontal direction. In Fig. 3(c), the resonance modes separate and appear in both bright and dark regions, however, the oscillation direction is rotated by 45 degrees in Fig. 3(d).
Fig. 4. Calculated Q-factors at different frequencies for resonance modes in Si moiré photonic pattern in Fig. 3 with a twist angle of 9.5 degrees. The red and purple rectangles indicate the locations of simulated photonic band gaps.
Resonances in Fig. 3(j-l) occur at Psmall/λ between 0.267 and 0.238. Assuming the cylinder diameter D in the moiré Si photonic pattern is half of Psmall, we can calculate:
From Eq. (8) and (9), the ratio of the size of Si cylinder over the wavelength is near 1, thus these resonances in Fig. 3(j-l) can be assigned to Mie resonances [68].
Figure 5 shows simulated E-field intensity distributions for the Si moiré photonic pattern with a twist angle of 22.6 degrees and a thickness of 2Psmall. The dashed blue squares indicate the unit super-cell. The frequency Psmall/λ and Q-factor are labelled in each figure. At frequency Psmall/λ= (0.333, 0.343, and 0.355), we obtain Q-factor=(1273, 8131 and 37877), respectively. These Q-factors in Fig. 5 is much smaller than these in Fig. 3(h-i). It is due to the coupling of resonance modes in the photonic pattern with a small unit super-cell, i.e. 5Psmall×5Psmall in Fig. 5. We have not performed the simulation of Q-factors in other frequency ranges in Fig. 3. Further simulations are needed in order to obtain full knowledge of twistoptics. From Fig. 5(a), the resonance modes at the corners of the blue square (dark regions) oscillate in a direction 45 degrees relative to the horizontal, couples to weaker modes in the top and bottom dark regions with a vertical oscillations, and to weaker modes in the left and right dark regions with a horizontal oscillations. The resonance modes in Fig. 5(b) have already appeared in the bright regions outside the blue square during the simulation decay times. At a frequency of Psmall/λ=0.355 in Fig. 5(c), the mode is localized inside the blue square with a weaker coupling to bright regions outside and has a higher Q-factor compared with these modes in Fig. 5(b).
Fig. 5. Simulated electric field intensity for resonance modes at the dark regions (a), bright regions (b) and highly localized mode at the bright region (c) in Si moiré photonic pattern with a twist angle of 22.6 degrees. The unit super-cell is indicated by dashed blue squares.
4. Discussion
The moiré photonic structure can be fabricated by either E-beam lithography or holographic lithography on a substrate of silicon-on-insulator. When E-beam lithography is used, the laser interference pattern from Eq. (2) is used as an input for the E-beam pattern writing program.
So far we picked up symmetric patterns with a desired moiré period Pmoiré. For Pmoiré=16, 12 and 5Psmall, the corresponding pattern has a twist angle of 7.2, 9.5 and 22.6 degrees, respectively. Further studies can include these asymmetric patterns generated through any twist angles. There are many moiré photonic patterns after combining different twist angles and z-locations. To address this challenge of huge work load, the machine learning method [69] can be applied to perform the research work on twistoptics.
5. Conclusion
We have presented an approach for the generation of moiré photonic patterns with different twist angles through interference of two sets of interfering laser beams, without physical rotations. The moiré photonic patterns have been demonstrated in square and triangular lattices. Resonance modes and cavity Q-factors have been simulated in these Si moiré photonic patterns with twist angles of 9.5 and 22.6 degrees. High Q-factors have appeared for these resonance modes near the edge of the photonic band gaps. The presented approach can lead toward a study of twistoptics for the manipulating of optical properties in moiré patterns with different twist angles.
Funding
National Science Foundation (1661842).
Acknowledgments
This work is supported by research grants from the U.S. National Science Foundation under Grant No. CMMI-1661842.
Disclosures
The authors declare no conflict of interest related to this paper.
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