Optical moiré bound states in the continuum

ABSTRACT Trapping electromagnetic waves within the radiation continuum holds significant implications in the field of optical science and technology. Photonic bound states in the continuum

(BICs) present a distinctive approach for achieving this functionality, offering potential applications in laser systems, sensing technologies, and other domains. However, the simultaneous

achievement of high Q-factors, flat-band dispersions, and wide-angle responses in photonic BICs has not yet been reported, thereby impeding their practical performance due to laser direction

deviation or sample disorder. Here, we theoretically demonstrate the construction of moiré BICs in one-dimensional photonic crystal (PhC) slabs, where high-Q resonances in the entire moiré

flat band are achieved. Specifically, we numerically validate that the radiation loss of moiré BICs can be eliminated by aligning multiple topological polarization charges with all

diffraction channels, enabling the strong suppression of far-field radiation from the entire moiré band. This leads to a slow decay of Q-factors away from moiré BICs in the momentum space.

Moreover, it is found that Q-factors of the moiré flat band can still maintain at a high level with structural disorder. In experiments, we fabricate the designed 1D moiré PhC slab and

observe both high-Q resonances and a slow decrease of Q-factors for moiré flat-band Bloch modes. Our findings hold promising implications for designing highly efficient optical devices with

wide-angle responses and introduce a novel avenue for exploring BICs in moiré superlattices. SIMILAR CONTENT BEING VIEWED BY OTHERS APPLICATIONS OF BOUND STATES IN THE CONTINUUM IN PHOTONICS

Article 06 October 2023 BRILLOUIN ZONE FOLDING DRIVEN BOUND STATES IN THE CONTINUUM Article Open access 17 May 2023 ANGULAR DISPERSION SUPPRESSION IN DEEPLY SUBWAVELENGTH PHONON POLARITON

BOUND STATES IN THE CONTINUUM METASURFACES Article Open access 16 May 2025 INTRODUCTION Bound states in the continuum (BICs), which are localized states with energies lying in the continuum

of radiating modes, were first proposed by Neumann and Wigner in electronic systems1. Later, it was discovered that BICs are a general phenomenon resulting from wave interference. In this

case, through well-designed artificial structures, BICs have been extensively observed in various classical wave systems, such as

photonic2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22, acoustics23, and others24,25,26,27,28. Photonic BICs enable the achievement of optical resonance modes with ultra-high

Q-factors, making them excellent candidates for creating exotic light-matter interactions5. For instance, BICs have facilitated advancements like low-threshold lasers6,7,29,30,

ultra-sensitive nano-sensors8,31,32,33, and surface-enhanced nonlinear frequency conversion34,35. Moreover, BICs in PhC slabs can carry topological charges corresponding to far-field

polarization singularities36, enabling the efficient generation of vortex beams and chiral light37,38,39. Despite the significant advance achieved thus far in photonic BICs, certain

challenges remain unresolved in practical applications, including the large dispersive effect of Bloch bands with BICs and the rapid attenuation of Q-factors for quasi-BICs in _k_-space.

These limitations impose constraints on the performance of BICs when subjected to a wide-angle illumination at a fixed frequency since most incident fields with mismatched wavevectors or

frequencies only result in weak interactions with designed nano/microstructures. In addition, practical Q-factors of photonic BICs are often limited by sample disorder, rendering the pursuit

of high Q-factors unattainable in practical applications. Recently, the high Q-factors in a wider range of _k_-space have been realized by merging multiple BICs in PhC slabs10,11,12.

However, the availability of Bloch modes with Q-factors exceeding 5000 is still limited to a finite range of k-space (about ±25° for the incident angle), accompanied by the significant band

dispersion effect. Consequently, the construction of BICs capable of facilitating high-Q resonances, flat-band dispersions, and wide-angle responses (±90° for the incident angle) remains an

outstanding issue. Inspired by the intriguing properties of moiré superlattices in twisted bilayer van der Waals structures40,41, there has been a significant interest in investigating moiré

physics in photonic systems42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57. For instance, the experimental realization of localization-delocalization transition has been achieved through

the utilization of reconfigurable photonic moiré lattices43,44,45. Twist-enabled topological transitions of iso-frequency curves have been observed in bilayer α-MoO3 flakes46,47,48.

Moreover, the twisted bilayer PhC slabs supporting moiré flat bands are also widely reported49,50,51,52,53,54,55,56. It is worth noting that a recent theoretical investigation has

illustrated the design of quasi-BICs in moiré flat bands using twisted bilayer PhC slabs57. However, due to the breaking of inversion and mirror symmetries, such twist-enabled moiré

quasi-BICs at the center of the Brillouin zone always exhibit non-zero leaky losses towards higher-order diffraction channels. Moreover, the Q-factors of moiré flat-band eigenmodes

significantly decay with their _k_-vectors moving away from the quasi-BIC at Γ point. Therefore, it is interesting to ask whether moiré BICs can be constructed and what novel properties,

unreported in any previous works, can be unveiled. Here, we theoretically demonstrate the construction of moiré flat-band BICs in one-dimensional (1D) PhC slabs by tuning multiple

topological charges to align with all radiation channels. In comparison to other Bloch modes surrounding conventional BICs, the decay rate of Q-factors for those around moiré BICs is

significantly decreased due to the combined effect of the band folding17 and the presence of multiple far-field topological charges. In addition, we find that Q-factors of the moiré flat

band can still maintain at a high level even in the presence of structural disorder. Furthermore, we experimentally fabricate the designed 1D moiré PhC slab and observe high-Q resonances and

slow decreasing of Q-factors for Bloch modes away from moiré BICs. Our findings have significant implications for designing highly efficient optical devices and offer a novel platform for

exploring BICs in moiré superlattices. RESULTS THE THEORETICAL DESIGN OF MOIRÉ BICS IN 1D PHC SLABS We start to design a 1D moiré PhC slab, which supports a BIC at Γ point of a moiré flat

band. As shown in Fig. 1a, the 1D moiré PhC slab is constructed by merging two silicon gratings with identical thickness _d_ but different periods and filling fractions (_a_1 and _F_1 for

the top layer, _a_2 and _F_2 for the bottom layer). To achieve the periodic moiré superlattice, these two periods must satisfy the commensurate condition expressed as

\({a}_{1}/{a}_{2}=\left(N+1\right)/N\) with _N_ being an integer. The period of the moiré unit is given by \(A={a}_{1}N={a}_{2}\left(N+1\right)\), comprising _N_ periods of the top layer and

_N_ + 1 periods of the bottom layer. In addition, each moiré unit possesses mirror symmetry with respect to the central perpendicular along the _y_-axis. Numerical results of TE-like moiré

bands are presented in Fig. 1b, with geometric parameters being _d_ = 116 nm, _A_ = 2965 nm, _F_1 = 0.351, _F_2 = 0.651, and _N_ = 8. The refractive indexes of the background medium and

silicon gratings are set as _n__b_ = 1.45 and _n__Si_ = 3.438. We find that several moiré bands with nearly flat dispersions appear, as highlighted by red, blue, green, and orange lines. In

Supplementary Note 1, we propose an effective Hamiltonian to elucidate the formation of moiré flat bands, which arise from the band-folding effect and the coupling between folded bands

induced by the moiré potential. Figure 1c illustrates the spatial profile of an eigenstate at Γ point in the moiré flat band marked by the red line in Fig. 1b. It is shown that the

eigen-field is concentrated at the center of the moiré unit, resulting in a narrower bandwidth compared to moiré bands with extended eigen-fields. In Supplementary Note 2, we further

demonstrate that the flatness of moiré Bloch bands can be enhanced by appropriately engineering the period number and filling fraction of the 1D moiré PhC slab. To characterize moiré BICs in

our designed 1D PhC slab, we investigate the far-field radiation of moiré flat-band eigenmodes. It is noted that BICs in conventional PhC slabs are typically found at frequencies below the

diffraction limit, where a single topological charge can lead to the non-radiative behavior of BICs by blocking the zero-order diffraction channel36. However, it is worth noting that the

eigenfrequencies of moiré Bloch bands are always above the diffraction limit, resulting in multiple independent diffraction channels of moiré eigenstates. Therefore, to realize BICs in a

moiré flat band, multiple topological charges should exist in the _k_-space to match with wavevectors associated with all diffraction channels. To further clarify the radiational property of

moiré flat bands, we provide a detailed description of diffraction channels for moiré eigenstates. The eigenstate \({{{\boldsymbol{E}}}}_{\widetilde{{k}_{x}},{k}_{z}}({{\boldsymbol{r}}})\)

of a 1D moiré PhC slab satisfies the Bloch theorem given by:

$${{{\boldsymbol{E}}}}_{\widetilde{{k}_{x}},{k}_{z}}\left({{\boldsymbol{r}}}\right)={e}^{-i\widetilde{{k}_{x}}x}{e}^{-i{k}_{z}z}{{{\boldsymbol{u}}}}_{\widetilde{{k}_{x}},{k}_{z}}\left(x,y\right),$$

(1) where \(\widetilde{{k}_{x}}\) and \({k}_{z}\) represent the Bloch wavevector and out-of-plane wavevector, respectively. Here, \({{{\boldsymbol{u}}}}_{\widetilde{{k}_{x}},{k}_{z}}(x,y)\)

is a periodic function along _x-_axis. By performing the Fourier expansion on:

$${{{\boldsymbol{u}}}}_{\widetilde{{k}_{x}},{k}_{z}}\left(x,y\right)={\sum}_{n}{{{\boldsymbol{C}}}}_{n,\widetilde{{k}_{x}},{k}_{z}}\left(y\right){e}^{-i{{\rm{n}}}{Gx}}$$ (2) with \(G=2\pi

/A\) being the reciprocal vector of the moiré unit, the moiré eigenstate \({{{\boldsymbol{E}}}}_{\widetilde{{k}_{x}},{k}_{z}}({{\boldsymbol{r}}})\) can be expressed as a superposition of

plane waves as:

$${{{\boldsymbol{E}}}}_{\widetilde{{k}_{x}},{k}_{z}}\left({{\boldsymbol{r}}}\right)={\sum}_{n}{{{\boldsymbol{C}}}}_{n,\widetilde{{k}_{x}},{k}_{z}}\left(y\right){e}^{-i{k}_{z}z}{e}^{-i{k}_{x}^{n}x}\,\left({k}_{x}^{n}=\widetilde{{k}_{x}}+{nG},\,n=0,\,\pm

1,\,\pm 2,\ldots \right),$$ (3) where the expansion coefficient is written as: $${{{\boldsymbol{C}}}}_{n,\widetilde{{k}_{x}},{k}_{z}}\left(y\right)=\frac{1}{A}{\int

}_{A}{{{\boldsymbol{u}}}}_{\widetilde{{k}_{x}},{k}_{z}}\left(x,y\right){e}^{{inGx}}{dx}$$ (4) From this perspective, the _n_th-order diffraction channel of

\({{{\boldsymbol{E}}}}_{\widetilde{{k}_{x}},{k}_{z}}({{\boldsymbol{r}}})\) corresponds to a plane wave with the free-space wavevector \({k}_{x}^{n}\) and electric-field strength

\({{{\boldsymbol{C}}}}_{n,\widetilde{{k}_{x}},{k}_{z}}{(}y{)}\). In the following, we use: $${{{\boldsymbol{C}}}}_{{k}_{x},{k}_{z}}\left(y\right){{=}}\frac{1}{A}{\int

}_{A}{{{\boldsymbol{u}}}}_{{k}_{x},{k}_{z}}\left(x,y\right){e}^{{inGx}}{dx}$$ (5) with \({k}_{x} =\widetilde{{k}_{x}}+{nG}\), which unifies

\({{{\boldsymbol{C}}}}_{{{\boldsymbol{n}}}{{,}}\widetilde{{k}_{x}},{k}_{z}}{{(}}y{{)}}\) from different-order Brillouin zones in the form of a continuous wavevector \({k}_{x}\), to

characterize the far-field radiation. Thus, the projection of \({{{\boldsymbol{C}}}}_{{k}_{x},{k}_{z}}\) into the _x-z_ plane is defined as the far-field polarization vector36. In this case,

the intensity of radiation can be defined as \({IOR}\left({k}_{x},{k}_{z}\right)=|{{{\boldsymbol{C}}}}_{{k}_{x},{k}_{z}}|.\) It is noted that only when the wavevector

\({k}_{y}^{n}=\sqrt{{k}^{2}-({{k}_{x}^{n}})^{2}-{k}_{z}^{2}}\) is real, the moiré Bloch mode with eigenfrequency being \(\omega={kc}/{n}_{b}\) can leak into free space from the _n_th-order

diffraction channel. Based on the reciprocal vector of a moiré unit (G) and the eigenfrequency (_f_) of the considered moiré flat-band Bloch mode at Г point, we identify that there are five

independent diffraction channels satisfying the relationship of \(\sqrt{{(2\pi f{n}_{b}/c)}^{2}-({{nG}})^{2}} \, > \, 0\) with _n_ = 0, ±1 and ±2. Therefore, at least five topological

charges are required to suppress the far-field leaking of the moiré BIC from all diffraction channels. To illustrate the distribution of topological charges in k-space, we calculate the IOR

of the considered moiré flat band (the red line in Fig. 1b), as shown in Fig. 1d. Black arrows in three enlarged views display the distributions of far-field polarization vectors

\({{{\boldsymbol{C}}}}_{{k}_{x},{k}_{z}}\) around −2nd-order, −1st-order, and 0th-order diffraction channels in the momentum space. Due to the existence of mirror symmetry, spatial profiles

of far-field polarization vectors around +2nd-order and +1st-order diffraction channels are consistent with those around −2nd-order and −1st-order diffraction channels. Numerical results

clearly show that there are seven topological charges with zero IORs in the moiré flat band. Specifically, the polarization vortex located at the center of the Brillouin zone carries a

single topological charge, which is protected by the mirror symmetry of the 1D moiré PhC slab. Additionally, there are other six topological charges situated away from the Γ point, which

arise from the destructive interference between two (or more) sets of waves expanded from moiré eigenmodes, resembling the previously revealed off-Γ-point BICs2. We note that the positions

for these off-Γ topological charges can be adjusted by varying system parameters (see Supplementary Note 3 for details on the creation, annihilation, separation, and shift for topological

charges with different geometric parameters). By tuning the geometric parameters of the 1D moiré PhC slab, we can precisely position four off-Γ topological charges at ±2nd-order and

±1st-order diffraction channels with \(\left({k}_{x},{k}_{z}\right)=\left(\pm 2G,0\right)\) and \(\left(\pm G,0\right)\), respectively. In combination with the symmetry-protected 0th-order

topological charge, a BIC can emerge at the center of the moiré flat band. To further confirm the existence of moiré BIC, Fig. 1e illustrates the numerical results on the variation of

Q-factors for all moiré bands as a function of the Bloch vector. The red line highlights the result of the moiré flat band with a moiré BIC. It is clearly shown that the calculated Q-factor

at Γ point of the moiré flat band reaches to infinity, showing the radiation of moiré BIC can be eliminated in theory. In Supplementary Note 4, we further calculate the mode profile excited

by a point source, which numerically verifies the complete suppression of the radiation. It is worth noting that there is a trade-off between the realization of moiré BICs and the desired

flatness of the moiré band. To improve the flatness, the moiré period needs to be enlarged, thereby resulting in an increase of diffraction channels. Thus, the k-vectors of additional

topological charges should be independently tuned to match to those of all diffraction channels. In this case, the moiré BIC is not achievable when the number of tunable parameters of moiré

grating is fewer than that of independent diffraction channels. MOIRÉ BICS-ENABLED HIGH-Q RESONANCES IN THE ENTIRE MOMENTUM SPACE In this part, we show that the moiré BIC can give rise to

the high-Q resonances in the entire momentum space. To elucidate this characteristic, we calculate the IORs of two representative Bloch bands: one corresponding to the moiré flat band with a

BIC and the other possessing a Γ-point BIC belonging to a conventional 1D PhC slab (the bottom layer in Fig. 1a), as shown in Fig. 2a, b, respectively. Black arrows highlight positions of

topological charges with zero-valued IORs. It can be seen that the IOR of the Bloch band in the conventional 1D PhC slab rapidly increases away from the topological charge. While, as for the

moiré flat band with a BIC, there is a drastic decrease in the maximum IOR compared to that associated with a single topological charge, indicating the enhanced Q-factors of moiré flat-band

eigenmodes in the entire momentum space. In addition, the _k_-vectors with a sudden decrease of IORs for both moiré and conventional 1D PhC slabs correspond to the positions of light-cone

lines (black dash lines). Furthermore, it should be noted that tuning geometric parameters of the 1D moiré PhC slab can further minimize the IOR within moiré flat bands (see Supplementary

Note 5 for details). We note that the high Q-factors of the entire moiré flat band are resulting from the combined effect of the moiré potential-induced band folding and multiple topological

charges (See Supplementary Note 6 for details). In addition, it is known that the Q-factors of conventional 1D PhC slabs always decrease monotonically as the level of structural disorder

increases. In the following, we show that our moiré gratings can exhibit an oscillating behavior in terms of Q-factors at large k-vectors when subjected to varying degrees of disorder. To

demonstrate this effect, we introduce the structural disorder into both the position (dx) and width (dw) of each silicon grating, where dx and dw are limited within the range of [-_δ_, _δ_],

which characterizes the disorder strength. Red, orange, yellow, and green lines in Fig. 2c display numerical results of moiré flat-band Q-factors with _δ_ = 0 nm, 1 nm, 3 nm, and 5 nm,

respectively. Here, one hundred times of configuration averaging is performed to eliminate accidental results. It is shown that the Q-factors around the center of the Brillouin zone are

significantly decreased with increasing the disorder strength. This arises from the fact that the structural perturbation can break the C2 symmetry and destroy the integer topological

charges, leading to a decrease in Q-factors around the center of the Brillouin zone. However, due to the high Q-factors in the entire moiré band, the scattering loss induced by structural

disorder can be effectively suppressed at the boundary of the Brillouin zone, making the corresponding Q-factors still maintain at the same level compared to those of moiré gratings without

any disorder, as presented in the left inset of Fig. 2c. In addition, moiré flat bands and their associated localized eigenstates persist despite structural disorder (see Supplementary Note 

7). From the above results, we can see that the Q-factors in the entire moiré flat band can still maintain a high level under disorder. For comparisons, we also calculate the Q-factors in a

conventional 1D grating (used in Fig. 2b) with structural disorder, as shown in Fig. 2c. Gray solid and dashed lines correspond to results with _δ_ = 0 nm and 5 nm, respectively. The right

inset shows the enlarged view around the band edge. It is evident that the Q-factors in the entire Bloch band are significantly decreased under the influence of structural disorder. In

particular, the maximum Q-factor of the conventional grating with _δ_ = 5 nm is even smaller than the minimum Q-factor of moiré flat-band eigenmodes with the same strength of disorder. In

Supplementary Note 8, we further increase the value of _δ_ to 9 nm to probe the evolution of Q-factors with higher disorder strengths. It is shown that Q-factors of conventional 1D PhC slabs

consistently decrease monotonically as the strength of disorder increases, but our moiré gratings can exhibit an oscillating behavior in terms of Q-factors at large _k_-vectors when

subjected to varying degrees of disorder. EXPERIMENTAL RESULTS OF ONE-DIMENSIONAL MOIRÉ PHOTONIC CRYSTAL SLABS In this section, we experimentally observe the moiré flat band and verify the

slow decrease of corresponding Q-factors far away from the moiré BIC. The designed 1D moiré PhC slab is fabricated in a silicon layer with 116 nm, which is deposited on the silica substrate,

using a combination of _e_-beam lithography and reactive ion etching techniques. The fabricated 1D moiré PhC slab is approximately 500 × 500 μm2, encompassing 170 moiré units along the

_x_-axis. Figure 3a presents a scanning electron microscope image of the sample, with an enlarged view provided in the right chart. Each silicon waveguide and gap-slot within a moiré unit

are labeled by black and white numbers from one to eight, and their widths are also provided. It is noted that structural parameters of the fabricated sample are consistent with those

employed in simulations. To establish an optically symmetric environment, we immerse the sample into an optical liquid that matches the refractive index of the silica substrate. To detect

the moiré flat band, we measure the transmission spectrum of the sample from 1420 nm to 1530 nm, as depicted in the intensity-plot of Fig. 3b (the experimental details are provided in

“Methods” section). The accessible range of _k_-vectors in our applied polarization-resolved momentum-space imaging spectroscopy is confined within the first Brillouin zone. The gray

colormap quantifies the measured transmissivity, and moiré bands are indicated by dark regions. For comparisons, we also calculate the moiré band structure within enlarged _k_-vector and

eigenfrequency ranges, as plotted in Fig. 3b by discrete points. It is shown that a good consistence between simulated and measured band dispersions is obtained. Specifically, a moiré flat

band (the simulation counterpart is marked by red dots) with bandwidth being 41.2 nm is observed, where the ratio between bandwidth and averaged eigen-frequency equals 0.0283. Such a small

ratio manifests the flat-band characteristic. It is worth noting that the remarkably low transmissivities of the moiré flat band around the Γ-point suggest the presence of a topological

charge localized at the zero-order diffraction channel. Such a topological charge can be directly observed by measuring the far-field polarization of transmission waves (see Supplementary

Note 9 for details). Then, we measure Q-factors of moiré flat-band eigenmodes located at different _k_ vectors. The scattering wave method under a laser excitation has been adopted to

measure the Q-factors of moiré eigenmodes (see “Methods” section for details). Figure 3c presents the experimental iso-frequency contour of the 1D moiré PhC slab at 1466 nm. Red, pink and

orange dashed circles display three groups of in-plane _k_-vectors, where the angle (\({{\rm{\theta }}}\)) between wavevector

\({{{\boldsymbol{k}}}{{=}}k}_{x}{{{\boldsymbol{e}}}}_{x}+{k}_{y}{{{\boldsymbol{e}}}}_{y}+{k}_{z}{{{\boldsymbol{e}}}}_{z}\) and _y_-axis are 3.6°, 6.4° and 8.3°, respectively. It is noted

that due to the limitation of our experimental system (the iso-frequency contour around Γ point cannot be resolved in k-space), we cannot directly measure the Q-factor of moiré BIC. Three

subplots in Fig. 3d show experimental scattering spectra (orange dots) with in-plane wavevectors being located at three discrete points in _k_-space (the red square, pink star and orange

circle, belonging to red, pink and orange circles in Fig. 3c), respectively. The Q-factors of these resonance peaks can be obtained by fitting scattering spectra using Lorentzian functions,

as shown by green lines. In this case, the fitted Q-factor at _θ_ = 3.6° is equal to 9042, being about ten times lower than that from numerical simulation (about 1e5). The observed tenfold

reduction in the measured Q-factor compared to the theoretical result is reasonable and should be mostly attributed to the fabrication imperfection and surface roughness58,59. In

Supplementary Note 10, we develop a multi-channel temporal couple mode theory (TCMT) to quantitatively analyze the reduction of Q-factors. Furthermore, the experimental Q-factors of 6051 and

1094 are achieved at _θ_ = 6.4° and 8.3°, respectively, which are about 2 to 10 times lower than theoretical predictions. We note that our experimental Q-factors at large _k_-vectors are

much larger than that of conventional gratings sustaining BICs22. Consequently, our experimental results can indeed illustrate the key properties of moiré BICs, including the flat-band

dispersion and relatively large Q-factors across a wide range of k space. However, even with these perturbations, the measured Q-factors at _θ_ = 3.6°, 6.4°, and 8.3° are still much larger

than the theoretical Q-factors of Bloch eigenmodes within a Bloch band sustaining conventional BICs at the same value of \({{\rm{\theta }}}\). It is important to note that the _k_-vector at

the boundary of the first Brillouin zone corresponds to \({{\rm{\theta }}}=18.5\)°. In this case, because of the periodicity of Brillouin zone, large-valued Q-factors of moiré flat-band

resonances can repeatedly appear with the increase of \({{\rm{\theta }}}\). Thus, the above experimental results clearly demonstrate the enhanced wide-angle resonance facilitated by moiré

BICs. To further illustrate the influence of structural disorder on Q-factors of moiré flat band at large k-vectors, we measure Q-factors on other two samples with designed fabrication

errors (see Supplementary Note 11 for details). It is shown that, despite the existence of designed fabrication errors, the measured Q-factors can still maintain at the same level with that

of moiré grating without disorder. DISCUSSION In this paper, we have presented, to the best of our knowledge, the first theoretical design of a moiré flat-band BIC using a 1D PhC slab. By

precisely tuning the positions of multiple far-field topological charges in the momentum space, we eliminate radiation losses of the moiré flat band eigenmode at the center of Brillouin

zone, resulting in the disappearance of electromagnetic wave leakage from all possible diffraction channels. The existence of this moiré BIC is further confirmed by its infinite Q-factor.

Moreover, due to the combined effect of the band folding and the multiple far-field topological charges within the moiré flat band, our proposed moiré structure exhibits a slow decay rate

for Q-factors compared to conventional Bloch bands with BICs. Additionally, the Q-factors at large k-vectors of moiré flat bands can exhibit an oscillating behavior in terms of Q-factors at

large k-vectors when subjected to varying degrees of disorder, distinguishing them from conventional 1D PhC slabs where Q-factors consistently decrease monotonically as the strength of

disorder increases. In Supplementary Note 12, we have compared the performance of our moiré BIC with other types of BICs discussed in recent works. Although some recent works demonstrate

promising performance in some aspects, none of previous studies can simultaneously achieve the high Q-factor, flat-band dispersion and wide-angle response exhibited by our proposed moiré

BIC. Furthermore, we experimentally fabricated designed 1D moiré PhC slabs and directly observed the moiré flat band. The slow decreasing of Q-factors of the moiré flat band with BIC has

also been verified. Our proposed moiré BICs enable omnidirectional and high-Q optical resonance at a single frequency and hold significant potential for designing strong light-matter

interactions with wide-angle responses. METHODS SAMPLE FABRICATION A layer of Si film with thickness of 116 nm was grown with the plasma enhanced chemical vapor deposition (PECVD) method on

a 500 um-thick silica substrate. The intrinsic loss of PECVD Si is negligible in the near-infrared range. Moreover, when compared to low-pressure chemical vapor deposition (LPCVD) Si (about

3.76), the real part of refractive index for PECVD Si (measuring at 3.31) exhibits good consistency with our theoretical design. A layer of 350 nm ZEP520A was spun-coated on the Si film as

e-beam photoresist. After exposure and development, resist patterns were formed. Then these patterns were transferred to Si with an anisotropic plasma of HBr gas, using ZEP520A as an etching

mask. MEASUREMENT SYSTEM The transmission spectra were measured using a home-made polarization-resolved momentum-space imaging spectroscopy built based on a Nikon micro-scope. The

corresponding schematic diagram of the optical setup is provided in Supplementary Note 13. Specifically, the source is a tungsten lamp and the incident light is focused on to sample by an

objective (20_magnitude, NA 0.4). The transmission light from the sample was imaged onto the entrance slit of imaging spectrometer (Princeton Instruments IsoPlane-320) through a series of

convex lens. Iso-frequency contours of 1D PhC samples can be visualized directly by leveraging resonance-enhanced photon scattering. Matching the frequency and incident angle of incoming

light to a resonance mode of the 1D PhC slab leads to the scattering of photons by fabrication defects within the sample, diverting them toward other resonance modes. These resonance modes

share identical frequencies with the incident light, enabling the replication of iso-frequency contour patterns in the far-field region. The schematic of the optical setups is provided in

Supplementary Note 13. A near-infrared laser with continuous tunability serves as the incident light source in this setup. The laser beam undergoes spatial filtering and is adjusted to match

the sample size. Employing a polarizer, the laser polarization is tuned to align with the excitation k-point of the iso-frequency contours in the sample (Pol y). Following this, lens L1

focuses the light onto the back focal plane of an infinitely corrected objective. By displacing L1 in the x-y plane, the incident angle of the laser is controllable, allowing for oblique

incidence excitation at any angle within the numerical aperture range. After traversing the sample, the scattered light is gathered by another objective and ultimately directed onto a CCD

through successive lenses L2, L3, and a 4f system. The CCD corresponds to the Fourier plane of the sample. In the receiving optical path, an additional polarizer, oriented orthogonally to

the incident polarization direction (Pol x), is utilized to suppress direct transmission through the sample, thereby enhancing the signal-to-noise ratio of the scattered light. When the

wavelength and incident angle of the incident light match the resonant modes supported by the sample, iso-frequency contours can be observed on the CCD. At this point, with the position of

lens L1 fixed (i.e., without changing the incident angle), adjusting the laser can obtain the scattering signals corresponding to different wavelengths. When the wavelength matches the

resonant mode, the iso-frequency contour is illuminated. As the wavelength gradually deviates from the resonance wavelength, the iso-frequency contour gradually dims. Thus, we can obtain the

scattering spectrum curve of the intensity at any point in momentum space as a function of wavelength, and the Q-factor of which can be obtained by numerically fitting the scattering

spectrum with a Lorentzian function. NUMERICAL SIMULATIONS All numerical simulations performed in this paper are based on the finite element methods. Due to the existence of continuously

translational symmetry along _z_-axis, the 2D model in the _xy_-plane can be used for simulations. The refractive index of the structure and background are 3.43 and 1.45, respectively. The

Floquet periodic boundary conditions are applied to two boundaries along _y_-axis, and the perfectly matched layers are added to top and bottom sides of the simulation domain in the

eigen-frequency calculation. In addition, the IOR is proportional to the amplitude of Bloch modes in 1D PhC slabs. Thus, for the IOR calculation of a band, we need to normalize all

eigen-modes within the band with different Bloch wavevectors. In addition, as for the comparison between IORs from different structures, such as IORs in Fig. 2a, b in the main text, all of

the corresponding Bloch modes are needed to be normalized. For this purpose, we normalize the eigenmode of the electric field \({{{\boldsymbol{E}}}}_{{k}_{x},{k}_{z}}\left(x,y\right)\) by

dividing it with the norm of the corresponding eigen-mode, defined as \({({\int }_{S}{|{{{\boldsymbol{E}}}}_{{k}_{x},{k}_{z}}(x,y)|}^{2}{dxdy})}^{\frac{1}{2}}\), where the integration area S

represents the total simulation domain. DATA AVAILABILITY All data are displayed in the main text and Supplementary Information. REFERENCES * Von Neumann, J. & Wigner, E. Uber

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CAS  PubMed  Google Scholar  Download references ACKNOWLEDGEMENTS This work was supported by National Natural Science Foundation of China (Nos. 12234004, 12422411, 12074420, 12204527) and

National Key Basic Research Program (No. 2022YFA1404800). The authors also acknowledge support from the Synergic Extreme Condition User Facility (SECUF), China. AUTHOR INFORMATION Author

notes * These authors contributed equally: Haoyu Qin, Shaohu Chen, Weixuan Zhang, Huizhen Zhang. AUTHORS AND AFFILIATIONS * Key Laboratory of Advanced Optoelectronic Quantum Architecture and

Measurements of Ministry of Education, School of Physics, Beijing Institute of Technology, 100081, Beijing, China Haoyu Qin, Weixuan Zhang, Huizhen Zhang & Xiangdong Zhang * Beijing Key

Laboratory of Nanophotonics and Ultrafine Optoelectronic Systems, School of Physics, Beijing Institute of Technology, 100081, Beijing, China Haoyu Qin, Weixuan Zhang, Huizhen Zhang & 

Xiangdong Zhang * Key Laboratory of Micro- and Nano-Photonic Structures (Ministry of Education), Department of Physics, Fudan University, Shanghai, 200433, China Shaohu Chen, Lei Shi & 

Jian Zi * Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, 100190, Beijing, China Ruhao Pan & Junjie Li Authors * Haoyu Qin

View author publications You can also search for this author inPubMed Google Scholar * Shaohu Chen View author publications You can also search for this author inPubMed Google Scholar *

Weixuan Zhang View author publications You can also search for this author inPubMed Google Scholar * Huizhen Zhang View author publications You can also search for this author inPubMed 

Google Scholar * Ruhao Pan View author publications You can also search for this author inPubMed Google Scholar * Junjie Li View author publications You can also search for this author

inPubMed Google Scholar * Lei Shi View author publications You can also search for this author inPubMed Google Scholar * Jian Zi View author publications You can also search for this author

inPubMed Google Scholar * Xiangdong Zhang View author publications You can also search for this author inPubMed Google Scholar CONTRIBUTIONS H. Qin and W. Zhang finished the theoretical

scheme and designed the moiré BIC. S. Chen performed the experiment with the help of H. Zhang, J. Li, R. Pan, L. Shi, and J. Zi. W. Zhang, H. Qin, and X. Zhang wrote the manuscript. X. Zhang

initiated and designed this research project. CORRESPONDING AUTHORS Correspondence to Weixuan Zhang, Lei Shi or Xiangdong Zhang. ETHICS DECLARATIONS COMPETING INTERESTS The authors declare

no competing interests. PEER REVIEW PEER REVIEW INFORMATION _Nature Communications_ thanks Meng Xiao, and the other, anonymous, reviewers for their contribution to the peer review of this

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continuum. _Nat Commun_ 15, 9080 (2024). https://doi.org/10.1038/s41467-024-53433-9 Download citation * Received: 04 January 2024 * Accepted: 11 October 2024 * Published: 21 October 2024 *

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