Experimental probe of twist angle–dependent band structure of on-chip optical bilayer photonic crystal

Abstract

Recently, twisted bilayer photonic materials have been extensively used for creating and studying photonic tunability through interlayer couplings. While twisted bilayer photonic materials have been experimentally demonstrated in microwave regimes, a robust platform for experimentally measuring optical frequencies has been elusive. Here, we demonstrate the first on-chip optical twisted bilayer photonic crystal with twist angle–tunable dispersion and great simulation-experiment agreement. Our results reveal a highly tunable band structure of twisted bilayer photonic crystals due to moiré scattering. This work opens the door to realizing unconventional twisted bilayer properties and novel applications in optical frequency regimes.

Optical bilayer photonic crystals are tuned via twist angle.

INTRODUCTION

There are emerging interests in using moiré physics to engineer optical dispersion. For example, moiré-patterned single-layer (1–15) and twisted-bilayer (16–35) photonic structures exhibit ultra-flat bands with no dispersion. The moiré pattern created by twisting two photonic structures relative to each other gives rise to distinctive optical properties, including nonlinear enhancement (36) and anisotropic dispersion (37). Using a pair of photonic crystal slabs (38) that are twisted relative to each other provides a large number of degrees of freedom—choice of material, lattice symmetry, feature size, twist angle, and interlayer gap—and permits tailoring the optical properties of the material. In particular, recent theoretical work shows that twisted bilayer photonic crystal (TBPhC) structures exhibit slow light (39), facilitating the study of strong light-matter interactions and Purcell enhancement (40) and frequency filtering (41). To date, however, there has not been any demonstration of TBPhC devices in the optical frequency range.

Here, we report on the fabrication of dielectric TBPhC structures that work in the optical frequency range and on the measurement of their momentum space–resolved optical response and optical band structure. We also compare the band-folding and band-hybridization phenomena observed in the measurement to numerical and analytical results. The results presented in this paper open the door to experimentally exploring theoretically predicted unconventional physics, such as nontrivial topological phenomena (18) and bound states in the continuum (42). They also allow potential applications such as adaptive (43, 44) and compressive (43, 44) sensing and on-chip, real-time configurable optical filters, polarimeters, switches, and lasers.

RESULTS

We start by analyzing the optical scattering properties of a single-layer photonic crystal slab and of the TBPhC slabs. A photonic crystal slab is a dielectric structure that is finite in the z direction and periodic in xy plane. The periodicity can be represented by a lattice, either in real space or in reciprocal space. For example, a square lattice in the reciprocal space is a set of points ${\mathcal{G}}_1\equiv \{(i\hat{\mathbf{x}}+j\hat{\mathbf{y}})2\mathrm{\pi }/a\mid i,j\in \mathbb{Z}\}$, where $\hat{\mathbf{x}}$, $\hat{\mathbf{y}}$ are the unit vectors in the xy plane, and a is its period in real space. When two identical square lattices are twisted against each other (Fig. 1A), they create moiré lattices in real space (Fig. 1B). The reciprocal lattice of TBPhC involves a large number of wave vectors due to the scattering by the periodic structure in both layers (17, 21) (see section S2). The lattice of the twisted layer with a twist angle θ in the reciprocal space would be ${\mathcal{G}}_2\equiv \{(i{\hat{\mathbf{x}}}^{\mathrm{\prime }}+j{\hat{\mathbf{y}}}^{\mathrm{\prime }})2\mathrm{\pi }/\mathrm{\theta }\mid i,j\in \mathbb{Z}\}$, where ${\hat{\mathbf{x}}}^{\mathrm{\prime }}$, ${\hat{\mathbf{y}}}^{\mathrm{\prime }}$ are the unit vectors in the twisted xy plane, namely, ${\hat{\mathbf{x}}}^{\mathrm{\prime }}=\cos (\mathrm{\theta })\hat{\mathbf{x}}+\sin (\mathrm{\theta })\hat{\mathbf{y}}$, ${\hat{\mathbf{y}}}^{\mathrm{\prime }}=\cos (\mathrm{\theta })\hat{\mathbf{y}}-\sin (\mathrm{\theta })\hat{\mathbf{x}}$. The reciprocal lattice for TBPhC is the sum of the reciprocal lattices for each layer, i.e.,

$$\begin{array}{rl}{\mathcal{G}}_{\mathrm{TB}}& \equiv \{(i\hat{\mathbf{x}}+j\hat{\mathbf{y}}+k{\hat{\mathbf{x}}}^{\mathrm{\prime }}+l{\hat{\mathbf{y}}}^{\mathrm{\prime }})2\mathrm{\pi }/a\mid i,j,k,l\in \mathbb{Z}\}\\ & \equiv \{{\mathbf{G}}_{(i,j,k,l)}\mid i,j,k,l\in \mathbb{Z}\}\end{array}$$ (1)

Fig. 1. Schematic of the experiment and the process of moiré scattering.

(A) TBPhC slabs with circular holes in a square lattice illuminated by focusing Gaussian incident light and mapping the transmitted light to its momentum space. (B) Moiré lattice from top-down view. (C) The twisted bilayer reciprocal lattice and basis are involved in the scattering process between the two layers. The reciprocal spaces of the first and second layer are marked by the center of the blue and orange spots, respectively. Larger and brighter spots indicate stronger scattering. The square mesh and shaded area indicate the first-order moiré Brillouin zone. A theoretical analysis of the moiré scattering strength is in section S2.

However, only a small subset of wave vectors needs to be considered for light scattering. For TBPhC, the subset of wave vectors of particular interest, which is closely related to the moiré pattern, is

$$\begin{array}{rl}{\mathcal{G}}_{\mathrm{m}}& \equiv \{(i\hat{\mathbf{x}}+j\hat{\mathbf{y}}-i{\hat{\mathbf{x}}}^{\mathrm{\prime }}-j{\hat{\mathbf{y}}}^{\mathrm{\prime }})2\mathrm{\pi }/a\mid i,j\in \mathbb{Z}\}\\ & =\{{\mathbf{G}}_{(i,j,-i,-j)}\mid i,j\in \mathbb{Z}\}\\ & \equiv \{{\mathbf{G}}_{\mathrm{m},(i,j)}\mid i,j\in \mathbb{Z}\}\end{array}$$ (2)

where G_m,(i,j) are moiré wave vectors. In particular, the properties of TBPhC that are strongly dependent on the twist angle arise from a scattering process in which incident light with in-plane wave vector k_inc is scattered by a moiré wave vector G_m,(i,j). Such scattering process gives rise to resonances when k = k_inc + G_m,(i,j) matches with the wave vector of the resonant modes of the system (see section S1). The first-order moiré wave vectors G_m,(±1,0) or G_m,(0,±1) are determined by the shortest single-layer wave vectors G_(±1,0,0,0) and G_(0,0,±1,0) or G_(0,±1,0,0) and G_(0,0,0,±1) (see Fig. 1C). The next higher-order moiré wave vectors within imaging range G_m,(±1,±1) and G_m,(±1,∓1) are determined by G_{(±1,±1,0,0)} and G_{(0,0,±1,±1)} (see section S2). In our experiment, only the scatterings by a collection of moiré wave vectors ${\mathcal{G}}_{\mathrm{m}}^{\mathrm{\prime }}=\{{\mathbf{G}}_{\mathrm{m},(\pm 1,0)},{\mathbf{G}}_{\mathrm{m},(0,\pm 1)},{\mathbf{G}}_{\mathrm{m},(\pm 1,\pm 1)},{\mathbf{G}}_{\mathrm{m},(\pm 1,\mp 1)}\}$ are observable. Other higher-order moiré wave vector scatterings are negligible when the low-frequency range and small twist angles are considered (21, 41). The first moiré Brillouin zone of the TBPhC slabs is then defined by first-order moiré wave vectors G_m,(±1,0) and G_m,(0,±1) as shown in Fig. 1C.

We use two fully suspended nanostructured silicon nitride membranes to fabricate the TBPhC slabs (Fig. 2A). A 150-nm silicon oxide (SiO₂) layer is first deposited and patterned on the silicon wafer, after which a 439-nm low-stress silicon nitride layer (Si₃N₄) is deposited on top of the patterned SiO₂. A 400 × 400 large photonic crystal is patterned into the silicon nitride layer through e-beam lithography: The square lattice photonic crystal has a lattice constant of a = 1220 nm and circular air holes with a radius of r = 502 nm. The photonic crystal is shallowly etched for an etch depth of t_PhC = 320 nm. To suspend the photonic crystal slab, the silicon wafer layer underneath is fully etched away from the back side until the etching reaches the SiO₂ layer. The SiO₂ layer is then removed through wet etching. A spacing and adhesion layer (SU8 2002) is patterned on top of the Si₃N₄ layer in one wafer. Finally, two wafers are bonded together through flip chip bonding techniques that control the relative angle within a precision of 0.1°. There is a 550 ± 50 nm air gap between the two photonic crystal membranes. A moiré pattern appears in the microscope image after the flip chip bonding (see sections S3 and S4).

Fig. 2. Device fabrication process and experimental setup.

(A) Fabrication process of the TBPhC slabs. The zoomed-in pictures are a scanning electron microscope (SEM) image of fabricated single-layer PhC slabs and a 50× microscope image of TBPhC slabs with a visible moiré pattern. (B) Schematic of the measurement setup. The red light line represents the incident light and its direct transmission and radiation-induced scattering from the bilayer lattice. P, polarizer; O, objective lens; CCD, charge-coupled device.

We measure the momentum space–resolved optical response of the sample through a free-space band structure measurement setup (see Fig. 2B) (45, 46). Two lasers are used for the frequency domain and momentum space measurements: the SuperK continuum laser with SuperK SELECT filter ranging from 1100 to 1700 nm with a frequency resolution of 6.4 to 19.8 nm, and the TSL-550 Santec tunable laser ranging from 1500 to 1630 nm with a frequency resolution of 0.003 nm. The laser is sent through the first polarizer (P1) before the light enters the back focal plane of an infinity-corrected objective lens (O1). The incident light focuses on the sample plane and excites the eigenmodes of TBPhC slabs at the same wavelength. These eigenmodes are scattered by the reciprocal lattices, which create resonances with finite linewidths, as well as by fabrication disorder (38), which further broadens the linewidth of the resonances. A confocal objective lens (O2) collects transmitted light and projects it into momentum space onto its back focal plane. After passing through a 4-f system with second polarizer (P2) that is cross-polarized with P1, the momentum space image is magnified 1.33 times and imaged on a monochromatic charge-coupled device (CCD) camera. P2 is used to block directly transmitted light from P1 and pass TBPhC’s resonances with different polarization states (see section S5). This imaging system provides a k-space resolution of 1.48 × 10⁴ m⁻¹/pixel. The k-space imaging range and resolution, in units of the Brillouin zone, depend on the numerical aperture of the objective lens, the incident wavelength, and the lattice constant. For example, when the incident wavelength is 1550 nm and the single-layer PhC lattice constant is a = 1220 nm, the k-space imaging range in length covers 0.53b, and the resolution is 0.003b/pixel, where b = 2π/a is the single-layer reciprocal lattice constant. As for the 10.0° twist bilayer structure, the k-space imaging range in length covers 5.6b_m, and the resolution is 0.02b_m/pixel, where b_m = 2π/a_m = 4πsin(θ/2)/a is the moiré reciprocal lattice constant (see section S6).

Analytically, the resonant frequencies of TBPhC can be derived as the eigenfrequencies of a Hamiltonian (38, 47) that describes various slab modes and their interactions. We set the unperturbed uniform slab modes (i.e., waveguide modes) as bases and treat the lattice-facilitated scattering as coupling terms in the Hamiltonian matrix. For an incident beam of light with in-plane wave vector k_inc, the relevant slab modes that need to be considered as the bases for the Hamiltonian can be denoted as ∣G_(i,j,k,l)〉_h, or ∣i, j, k, l〉_h for short, with the corresponding field distribution:

$$\mathbf{E}(\mathbf{r})\equiv {\mathrm{\varphi }}_h(z)e^{-i({\mathbf{G}}_{(i,j,k,l)}+{\mathbf{k}}_{\mathrm{inc}})\cdot {\mathbf{r}}_{\parallel }}$$ (3)

where h = 1,2 denotes which layer the mode is localized in and ϕ_h(z) is the mode profile in the z direction for the frequency range of interest. Many of these bases have a strongly angle-dependent resonant frequency, which follows from the angle dependence of their in-plane wave vectors. As a result, the eigenfrequencies of the Hamiltonian matrix also have a strong twist angle dependence. Examples of the theoretical band structures are shown in Fig. 3A. Because the first-order moiré scattering is dominant, to construct the Hamiltonian, we choose a set of bases that includes those with wave vectors from individual layer’s reciprocal lattice, such as ∣G_(±1,0,0,0)〉_h and ∣G_(0,±1,0,0)〉_h, as well as first-order moiré bases, such as ∣G_m,(±1,0)〉_h and ∣G_m,(0,±1)〉_h, where h ∈ {1,2} labels which layer the mode is localized in (see section S7). However, the measurement, which is the frequency domain optical response, does not reflect all the eigenfrequencies in the Hamiltonian calculation. The eigenmodes with more ∣G_{(i, j,0,0)}〉₁ components in the first layer have stronger coupling strength to the input light. Similarly, eigenmodes with more ∣G_(0,0,k,l)〉₂ components in the second layer have stronger coupling strengths to the output light. Eigenmodes with fewer components of ∣G_(i,j,0,0)〉₁ and ∣G_(0,0,k,l)〉₂ are expected to manifest less in the transmission spectrum (see section S7). In Fig. 3A, the color bars of the transverse magnetic (TM)- and transverse electric (TE)-like modes represent the coupling strength. Eigenmodes in semitransparent color are invisible to the measurement. In all following TBPhC band structures, the x axis is scaled by the single-layer photonic crystal reciprocal lattice constant, while the directions of Γ → M and Γ → X refer to the high symmetry axis of the moiré lattice. The experimental TBPhC isofrequency contours show good quantitative agreement with the Hamiltonian results (Fig. 3B). Their frequencies (f₁ = 189.7 THz, f₂ = 192.7 THz, f₃ = 194.7 THz, f₄ = 201.2 THz) are shown as dashed lines in the analytical band structure in Fig. 3A. The dotted lines in the first experiment isofrequency contour in Fig. 3B indicate moiré lattice. The isofrequency contour in the first moiré Brillouin zone repeats itself by a translation through the first-order moiré vectors G_m,(±1,0) and G_{m, (0,±1)}. Similar but much weaker repetitions of isofrequency contours also appear through the translation of G_m,(±1,±1) and G_m,(±1,∓1), indicating a reduced scattering intensity of this order (see section S2). The repeated isofrequency contours are not perfectly identical, which coincides with the quasicrystalline feature of the moiré lattice (see section S1).

Fig. 3. Hamiltonian band structure compared with the isofrequency contours in measurement.

(A) Hamiltonian band structure of the TBPhC slabs with a twist angle of θ = 10.0°. TM- and TE-like modes are shown in red and blue, with each color bar indicating the coupling strength. The dashed lines indicate the frequencies of isofrequency contours. (B) Comparisons of Hamiltonian (left) and measured (right) isofrequency contours. In the analytical isofrequency contours, TM- and TE-like bands are again shown in red and blue, respectively, and they use the same color bars as in (A). In the measured isofrequency contours, both TM- and TE-like resonances are imaged together, with the color bar indicating the total transmitted intensity received by the CCD. The dotted line in the first measurement of the isofrequency contour indicates the first moiré Brillouin zone. Experimental images are processed using distortion correction and high dynamic range exposure (see section S6).

The observed band structures reflect band folding and band hybridization of single-layer photonic crystal bands. To model the observed band structure of the TBPhC slabs, we use a rigorous coupled-wave analysis (RCWA) (48). In contrast to the Hamiltonian approach, RCWA seeks to describe the system from first principles using Maxwell’s equations, and the result of RCWA contains more information, including resonance line width and intensity. We compare Hamiltonian calculations (third column), RCWA calculations (fourth column), and experimental measurements (last column) of the single-layer photonic crystal slab (first row), bilayer photonic crystal slabs (second row), and TBPhC slabs at a twist angle of 10.0° (last row) in Fig. 4. In the second column of Fig. 4, we illustrate the behavior of the TM-like parabolic dispersive bands, which is a prominent feature in the band structure of this system. In the bilayer photonic crystal slabs where the twist angle θ = 0°, the parabolic bands are doubled into two sets vertically (solid→dashed) as a result of interlayer coupling and band hybridization. In the TBPhC slabs, bands are folded back to the first moiré Brillouin zone (17), and the second moiré Brillouin zone can be visualized in the field of view of our setup. The frequency of bands also changes due to the band hybridization (39, 49) caused by the interlayer coupling. More features can be visualized in the following three columns. In the Hamiltonian, both the TM-like (red) and TE-like (blue) modes are plotted, and the color bars indicating the coupling strength are the same as Fig. 3. In the RCWA calculations, the cross-polarization filter was applied to remove the transmission background. In the measurement, region 1 is measured by the SuperK select laser, and the resolution is lower; region 2 is measured by the Santec TSL tunable laser, and the resolution is higher. The following results were observed from Hamiltonian, RCWA, and measurement results: For the single-layer PhC slab, there are four TM-like bands (red lines). For the bilayer photonic crystal slab, four TM-like bands split into two sets. The gap between each set is Δ_b = 2.2 THz for the upper parabolic bands and ${\text{Δ}}_{\mathrm{b}}^{\mathrm{\prime }}=2.8$ THz for the bottom parabolic bands, which corresponds to the coupling strength in band hybridizations (39, 49). Notice that the gap of the perfectly aligned bilayer photonic slab and the misaligned bilayer photonic slab are the same (see section S8). For the TBPhC slab, eight more TM-like bands emerge at G_m(0,−1), which originated from two sets of TM-like bands in the bilayer PhC slab. The gap is Δ_t = 7.0 THz for the upper parabolic bands and ${\text{Δ}}_{\mathrm{t}}^{\mathrm{\prime }}=7.3$ THz for the bottom parabolic bands. The gap is changed in the TBPhC slab because of the change of the interlayer coupling strength, which is caused by the lattice mismatch. The error between RCWA and measurement result is ±1 THz at Γ-point (see section S9). Because of the fabrication disorder, the measurement band structures have wider line widths compared to RCWA. From this comparison between Hamiltonian, RCWA, and experiment, we see a high degree of consistency between the analytical, numerical, and experimental results, which verifies our understanding of the origin of the guided resonances in this system.

Fig. 4. Schematic, analytical, first-principle, and measured band structures of single-layer, bilayer, and TBPhC slabs with a twist angle of θ = 10.0°.

(First column) Photonic crystal slab configurations. (Second column) Schematic band structures. (Third column) TM-like (red) and TE-like (blue) Hamiltonian band structures. The gray lines are eigenmodes that are invisible in the measurement. (Fourth column) Rigorous coupled-wave analysis (RCWA) calculation. (Fifth column) Experimental measurements. Band structures in region 1 are measured by the SuperK laser, and band structures in region 2 are measured by the Santec TSL tunable laser.

One major attraction of TBPhC lies in its twist angle–tunable guided resonances. Resonances in TBPhC are typically a mixture of angle-independent and angle-dependent resonances. The angle-dependent resonances are strongly associated with the moiré wave vector G_m and therefore the twist angle (16, 41). In Fig. 5, we compared the measured and calculated band structures of TBPhCs with different twist angles 8.0°, 10.0°, 12.7°, and 14.0° (Fig. 5A). The measured isofrequency contours for these four angles are in section S10. As illustrated in Fig. 5, the twisted bilayer structure has a set of parabolic bands centering the moiré wave vector G_m,(1,0). As the angle gets larger, the magnitude of the moiré wave vector G_m becomes larger, and a shift of the whole band towards larger k is observed. The band edge is marked by red crosses and stars in Fig. 5A for clarity. The photodetector measured band edge resonance modes are in section S11. To better see how the band edge moves as the twist angle is varied, we plot the transmission as a function of wave vector and twist angle for a fixed frequency of 190.5 THz in Fig. 5B, where the points marked by red stars correspond to the same parameters as those marked in Fig. 5A. In this parameter range, the tunable resonance can be well explained by the band-folding picture, where the parabolic bands are moved according to the corresponding moiré wave vector. We also provide the angle dependence of the resonance frequency at k = 0.189b (Fig. 5C), where the transmission dips around frequency of 190.3 THz also follow a parabolic shape. For higher frequencies, more complicated twist angle dependence can be observed. The angle dependency of guided resonances helps identify the tunability of scattering properties in the TBPhC slabs and foresee the potential of next-generation bilayer reconfigurable devices.

Fig. 5. Angle dependence of optical resonances.

(A) Experimental (left column) and RCWA (right column) band structures of TBPhC slabs with different twist angles. The band edge is traced by red crosses in the experiment and red stars in RCWA. (B) Wave vectors of TBPhC as a function of twist angle in RCWA, where f = 190.5 THz. (C) Resonant frequencies of TBPhC as a function of twist angle in RCWA, where k = 0.189b. The color bar represents the intensity.

DISCUSSION

The observed strong twist angle dependency of guided resonances underlies the tunability of the scattering properties of the TBPhC slabs and forms the foundation for next-generation bilayer reconfigurable devices. Applications include tunable filtering, tunable thermal emission, and beam steering. For tunable filtering and thermal emission, the twist angle dependence of the resonance frequency is exploited to realize different filter line shapes by varying the twist angle. In beam steering, the twist angle dependence of both resonance frequency and resonance wave vector can be exploited, leading to different frequency responses and diffraction angles when the twist angle is varied. In these applications, the structure needs to be further optimized according to the desired operational regime. The frequencies of the resonances are primarily determined by both the periodicity in each layer and the twist angle. A larger periodicity and a smaller twist angle lead to a larger length scale of the moiré pattern and, therefore, a larger resonant wavelength. The linewidth of the tunable resonance is determined by both the interlayer coupling and the scattering strength of the individual layers. A smaller slab thickness, as well as a larger gap distance between the two slabs, leads to smaller resonance linewidth. In all the applications mentioned above, a clean spectrum with a small number of resonances in the operating frequency range is typically desired, which can be fulfilled by operating in a small twist angle regime where the moiré length scale differs drastically from each layer’s periodicity. In the tunable filter application, the desired filter lineshape could require tuning the background transmission in the spectrum, in addition to resonance frequency and lineshape. This can be achieved by adding uniform slab layers sandwiching the patterned layer, which allows changing the lineshape without introducing extra angle-dependent resonances.

In conclusion, we show here how the twist angle between two photonic crystal slabs can be used to tune the optical band structures of the assembly. We built TBPhC structures through nanofabrication and developed a fundamental understanding of their complex, unconventional optical properties. Specifically, we theoretically demonstrated and experimentally measured the band structure of TBPhC slabs in the optical frequency range. In particular, we especially analyzed the first-order moiré scattering behavior that is observable from isofrequency contours along with the band-hybridization and band-folding behavior from the band structure. Our work establishes the basis for engineering electromagnetic wave propagation in twisted bilayer photonic structures, generating a new suite of optical properties through the creation of synthetic moiré systems controllable by twist angle and interlayer coupling. The band structure tunability provides a starting point for the understanding of other optical properties in the dielectric twisted bilayer systems, including bound state in continuum, quasicrystalline optics, chirality, polarimetry, nontrivial-topological modes, and superscattering. The fabrication, measurement, and analysis approaches presented here will also be a platform for building complex bilayer nanomaterials and controlling electromagnetic waves with mechanical reconfigurability, such as interlayer gap, twist angle, and sliding distance. Our experiment also serves as a foundation for developing bilayer flat-optical devices such as tunable filters, thermal radiation, beam steering, tunable lasers, adaptive sensors, and LiDAR.

MATERIALS AND METHODS

The sample was prepared to start with two bare silicon wafers, first coated with silicon oxide. Using low-pressure chemical vapor deposition, the wafers are coated with a low-stress silicon nitride film. The photonic crystal slab patterns are written using e-beam lithography, and then the silicon nitride is etched with reactive ion etching. One of the two wafers is patterned with SU8-2002 posts through photolithography, with height controlled by the spin-coating speed. The back sides of both wafers are patterned into disks and etched to the front side, leaving the silicon nitride photonic crystal fully suspended. Finally, the two wafers are bonded using a wafer bonding stage with rotary, vertical, and translational axes. One wafer is picked up by hexamethyldisilane (HMDS), where the photonic crystal can be seen through the deep etching hole. For detailed information, please see section S3.

Supplementary Materials

This PDF file includes:

Sections S1 to S11

Figs. S1 to S11