Quasi-bound flat bands in the continuum

Domain: physics
Published: 2025-12-02
Editorial outcome: major_revision

Abstract
Bound states in the continuum (BICs) are widely known spatially localized states experimentally implemented as quasi-BICs. Although they emerged as a promising solution for achieving high-quality resonances in photonic structures, quasi-BICs are confined to a very narrow range in k-space and are highly sensitive to disorder. Here, we introduce quasi-bound flat bands in the continuum (quasi-BFICs)—a class of optical states where Bloch modes are found within a photonic flat band, leading to a quasi-BIC behaviour at every k-point above the light line. We analytically and numerically demonstrate the origin of quasi-BFICs from the disorder-induced band folding, mode localization and multiple topological charges in k-space, and identify the optimal strength of structural disorder to maximise their generation probability. Angle-resolved transmission and Q-factor measurements confirm the existence of quasi-BFICs, opening new avenues for designing devices with high quality factor and wide-angle response, presenting a counterintuitive strategy that leverages disorder to enhance optical performance.

Introduction
Trapping electromagnetic waves in micro- and nanostructures with high quality factors (Q-factors) is critical for enhancing light-matter interactions and advancing novel photonic devices. Over the past decade, bound states in the continuum (BICs)1–8—a unique class of localized states whose energies reside within the continuum of radiating modes—have garnered significant attention as a promising approach to the design of photonic structures with exceptionally high Q-factors2–52, enabling various innovations such as BIC lasers9–13, ultra-sensitive sensors14–17, and more18. In 2013, Hsu et al. made a ground-breaking experimental realization of photonic BICs in two-dimensional (2D) photonic crystal (PhC) slabs2, demonstrating both symmetry-protected and accidental BICs at distinct k points with theoretically infinite Q-factors. However, despite these progress, several issues remain in translating BICs to many practical applications. First, quasi-BICs with high theoretical Q-factors are typically confined to a narrow range in momentum space. This means that efficiently exciting quasi-BICs requires a precise excitation angle, as even slight deviations can lead to a sharp drop in the Q-factor. Second, quasi-BICs are highly dispersive in PhCs, limiting their performance under the wide-angle illumination with a fixed frequency. Lastly, the Q-factors of experimentally realized BICs are constrained by fabrication imperfections, such as surface roughness and structural disorder. These imperfections introduce additional radiation losses, reducing the Q-factors of quasi-BICs in practical applications.

To address these long-standing issues, several novel types of optical BICs have been proposed and realized. For example, a merged BIC20–22, in which multiple topological charges19 are tuned to lie very close to each other, can broaden the momentum range of quasi-BICs with high Q-factors and suppress out-of-plane-scattering losses caused by structural disorder. In addition, by merging a symmetry-protected BIC with two Friedrich-Wintgen quasi-BICs, researchers have developed a super BIC, which features an enhanced Q-factor and near-zero group velocity across an extended region in the Brillouin zone23,24. Although merging multiple BICs or quasi-BICs can locally broaden the k-space range of high-Q resonances and reduce band dispersion, their performances remain limited in the entire k-space. More recently, a moiré BIC, protected by multiple topological charges locating at different orders of diffraction channels, has been theoretically proposed and experimentally demonstrated25. This approach further extends the momentum range of quasi-BICs while improving band flatness. However, the Bloch modes that are distant from moiré BICs still exhibit low Q-factors. Building on these advances and addressing the remaining limitations, a promising direction is the construction of a photonic flat band, where the Bloch mode at each k-point above the light line behaves as a quasi-BIC. We refer to this unique optical state as a quasi-bound flat band in the continuum (quasi-BFIC), characterized by its capacity to sustain high-Q resonances across the full range of excitation angles (Fig. 1a). In contrast, current quasi-BICs are confined to a very narrow angular range, where the radiation loss of nearby guided resonances increases rapidly as they move away from the BIC (Fig. 1b). This distinctive feature of quasi-BFICs is expected to open new opportunities for enhancing light-matter interactions under wide-angle illumination, thereby broadening the scope of BIC-based applications. A crucial question remains: how can such an innovative optical state be effectively realized?

On the other hand, the field of photonics systems has witnessed a remarkable surge of interest in exploring disorder effects, with numerous studies demonstrating that disorder can both induce localization phenomena (such as random laser emission53 and Anderson localization of light54) and enhance wave transport55,56. While, concerning BICs in PhC slabs, all existing experimental observations have unveiled substantial scattering of Bloch waves due to disorder2,20, which compromises the performance of BICs. In this work, we challenge the detrimental view of disorder on BICs and reveal that quasi-BFICs can be randomly generated in PhC slabs through mirror-symmetric disorder within each supercell. Using both analytical and numerical methods, we demonstrate that random quasi-BFICs arise from the combined effects of band folding, mode localization and multiple topological charges induced by disorder. Additionally, we identify an optimal strength of structural disorder that maximizes the probability of generating random quasi-BFICs, which results from the balance between disorder-induced mode localization, band flatness, and the radiative Fourier coefficients of the localized eigenmode. In experiments, we fabricate a randomly created PhC slabs and observe quasi-BFICs through both angle-resolved transmission spectra and iso-frequency contour image of the scattering light. Our findings open new avenues for the design of highly efficient optical devices with wide-angle responses, while also suggesting a counterintuitive strategy that leverages disorder to improve the optical functionality.

Results
The theoretical framework for the quasi-bound flat bands in the continuum
We start with a one-dimensional (1D) silicon PhC slab, as shown in Fig. 2a, with the system parameters detailed in the figure caption. Figure 2b presents the calculated band structure of TE-like modes, with the colormap representing the corresponding Q-factors. We find that an isolated BIC with infinite Q-factor is identified at the center of momentum space, while other Bloch modes located within the light cone (the white region), experience a rapid decrease in Q-factor as their k-vectors deviate from the BIC. In addition, Bloch modes below the light line are non-radiative, exhibiting infinite Q-factors. Now, we construct a supercell with thirteen fundamental units (A=13a0) by adding mirror-symmetric disorder. Then, the supercell is repeated in periodic direction to construct an infinite photonic crystal slab. Specifically, as depicted in Fig. 2c, the widths of silicon pillars within each supercell are randomly assigned within the range of [W−δ, W+δ], where W is the original width and δ/W characterizes disorder strength. We emphasize that that the mirror symmetry of each supercell is essential for constructing the quasi-BFIC, which is fully discussed later. To maintain the mirror-symmetry of each supercell, pairs of silicon pillars positioned symmetrically about the mirror plane share the same width. We mention that the proposed structure is different from the chirped system with gradually varying structural modulations and exhibit local rather than global periodicity57–59. The disorder adopted in the random design breaks the local periodicity but preserves the global periodicity, as the supercell is periodically repeated to form the final structure.

To assess the impact of mirror-symmetric disorder with different strengths, we randomly generate 800 different structures at each value of δ/W, ranging from 0.05 to 0.4. We then calculate the bandwidth and averaged Q-factor of the first TE-like band above the low-frequency bandgap (marked by the dash lines in Fig. 2b) of these structures. Notably, our considered band is isolated from other TE-like bands due to band folding and coupling induced by the structural disorder within each supercell (Supplementary Note 1). The bandwidth is defined as the difference between the maximum and minimum eigen-frequencies of the targeted band (Supplementary Note 2). And, the averaged Q-factor is calculated as the mean of the Q-factors for all Bloch modes within the band. We present three typical statistical distributions with disorder strengths being δ/W=0.05, 0.15 and 0.4 (Fig. 2d). When δ/W=0.05, most randomly created structures exhibit high averaged Q-factors of the targeted band, but also relatively large band dispersions. As δ/W increases to 0.15, contrary to conventional expectations that disorder would reduce the Q-factors, we observe that some randomly generated structures can exhibit both high averaged Q-factors and narrow bandwidths, as indicated by discrete points within the blue region. However, when the disorder strength is further increased to δ/W=0.4, the averaged Q-factors of all generated structures drops significantly. The probability of generating PhC slabs with high Q-factor flat bands (dots in the blue region) reaches maximum at δ/W=0.15 (Fig. 2e).

Because when δ/W is relatively small, the eigenmodes of these randomly generated structures always exhibit weak localization within each supercell, accompanied by significant band dispersion. Conversely, if the disorder strength is too large, the strongly localized eigenmodes within each supercell can induce the substantial far-field radiation. Therefore, the optimal strength of structural disorder is determined by balancing the disorder-induced mode localization and band flatness, as well as the amplitude of Fourier coefficients above the light line (the far-field leakage) of the localized eigenmodes (see details below).

To further illustrate the disorder-induced high-Q flat bands, we calculate the band structure of a randomly generated PhC slab with δ/W=0.15 (marked by the green star in Fig. 2d) sustaining the high-Q flat band (Fig. 2f). Notably, two flatbands are highlighted in the eigen-spectrum, with the colormap representing their Q-factors. Other dispersive Bloch modes are plotted by gray lines. Remarkably, flat band at 1520 nm exhibit exceptionally high Q-factors, reaching up to 200000 throughout the entire k-space, respectively. We emphasize that such angle-insensitive Q-factors can substantially enhance robustness against scattering losses from fabrication defects (Supplementary Note 3). Another flat band at 1935 nm has the relatively low Q-factor (about 8000). Both flat-band eigenmodes exhibit strong localization within the supercell (Fig. 2g). These spatially localized eigenstates induce the formation of the flat-band dispersions.

To further elucidate the above phenomena, we calculate the distribution of Fourier coefficients for flat-band eigen-fields in momentum space19. It is defined as
u_n(k_x) = |\int A^{-1} A E(k_x, x, y) e^{i(k_x + n 2\pi/A)x} dx| (1)
with the Bloch modes E(k_x, x, y) = E(k_x, x+A, y). For n=0, ±1, ..., we consider the E(k_x, x, y) in the near-field region, where the radiative and non-radiative components of flat-band eigenfields in k-space can be obtained. If the integral plane is set in the far-field region, Eq. (1) only quantifies the far-field radiation amplitude19 (Supplementary Note 4). Details on the calculation of u(k_x) using the finite element method is presented in the Methods section. It is shown that, for the high-Q flat band at 1520 nm the dominate components of u(k_x) are concentrated in the non-radiative region, with extremely low values of u(k_x) appearing in the radiative domain (Fig. 2h). This suggests that only a small portion of the Fourier coefficients for the flat-band eigenmode is able to radiate away from the PhC slab.

Additionally, the suppression of radiative components above the light cone is further enhanced by the presence of multiple zero points of u(k_x). When we magnify the view of u(k_x) in the radiative area (Fig. 2i), multiple zero-amplitude points (labeled as α, β, γ, and δ) are observed at discrete k points. These zero points are characterized by ill-defined far-field polarization, corresponding to the polarization singularities with topological charges equaling to ±1 (Fig. 2j). The robustness of the topological charges is proved by their evolution under continuous parameter variation (Supplementary Note 5). We note that, beyond preventing leakage at specific wavevectors, topological charges also suppress the radiation of Bloch modes at nearby wavevectors, which explains the low values of u(k_x) in the radiative region. In this case, protected by the existence of multiple zero points of u(k_x), each Bloch mode within the high-Q flat band can be regarded as a quasi-BIC. Consequently, this high-Q flat band behaves as a quasi-BFIC. For comparison, we further calculate the distribution of u(k_x) for the low-Q flat band at 1935 nm, as shown in Fig. 2k with enlarged plot within the radiative region. Unlike the high-Q flat band, there is only one topological charge within the radiative region, resulting in the significantly increased values of u(k_x) in this area. Consequently, this flat band possesses a large radiation loss and exhibits a much low averaged Q-factor.

To further elucidate the above phenomena, we truncate the near-field distribution at y=100 nm for the eigenmode presented in the upper plot of Fig. 2g, as shown by black dots in Fig. 3a. This flat-band eigen-field exhibits an exponentially localized spatial profile combined with an oscillating phase within a supercell, taking the form of E(x, y=0) = E_0 e^{-\alpha x^2} sin(k_s x). The mirror-symmetric disorder ensures that the eigen-field is an even or odd function with respective to the mirror plane of the supercell. Here, the fitting parameter α determines the strength of the spatial localization and k_s describes the wavevector of the oscillating phase. The red line in Fig. 3a shows the fitted profile of this flat-band eigen-field, with fitting parameters being α=4.40×10^{11} [1/m^2] and k_s=8.44×10^6 [1/m]. It is shown that the fitted eigenmode reveals an excellent agreement with the numerical counterpart.

Next, we demonstrate how these two fitting parameters play key roles in the generation of quasi-BFICs. It is important to note that the spatial profile of flat-band eigenfields are nearly independent of the Bloch vector due to the strong localization within each supercell. In this context, the amplitude distribution of Fourier coefficients for flat-band eigen-fields in momentum space can be re-expressed as
u(k_x) = |\int A^{-1} A E(x, y=0) e^{i(k_x + n 2\pi/A)x} dx| (2)
with the out-of-slab eigen-field E(x, y=0) being independent of the Bloch vector. Figure 3b presents the analytical results of u(k_x), which closely align with the numerical findings in Fig. 2h, showing two permissible values of n as u(k_x) = u(k_x + n 2\pi/A). Additionally, the widths of these two peaks are determined by the localization strength α, with larger values of α leading to broader peaks. For the flat band to exist, a sufficiently large α is required, ensuring that the eigenfields are strongly localized. However, if α is too large, two peaks in the non-radiative region broaden significantly, causing a substantial portion of u(k_x) to spill into the radiative domain and thereby reducing the Q-factors (Supplementary Note 6). Thus, there exists an optimal range of the localization strength α, which is determined by the disorder strength (as the disorder strength increases, the value of α is also increased), allowing for the coexistence of the flat-band dispersion with localized eigen-modes and narrow peak-widths of u(k_x) in the non-radiative domain. To further illustrate the radiative properties, we present an enlarged view of u(k_x) around the center of k-space (Fig. 3c). There are multiple equally spaced topological charges within the light cone and zero points of u(k_x) in the non-radiative region near the light line, ensuring that the flat band possesses the low-valued radiation loss in the entire k-space. Interestingly, we find that the number of topological charges depends not only on the fitting parameters α, but also on the period A of the supercell. To verify this phenomenon, we calculate the variation in the number of topological charges (within the light cone) and the averaged values of u(k_x) above the light line <u(k_x)> |k_x| < k_b (k_b is the background wavevector) for the flat band as a function of the supercell period A (Fig. 3d), with fitting parameters α and k_s identical to those in Fig. 3a. The results indicate that the number of topological charges increases with the supercell period, accompanied by a significant reduction in radiation loss. This suggests that expanding the supercell size could be a viable way to approaching ideal BFICs.

Additionally, we note that besides the spatial profile described by E_0 e^{-\alpha x^2} sin(k_s x), other types of flat band eigenfields with varied distributions can also be randomly generated with mirror-symmetric disorder within each supercell. It is shown that these different spatial profiles of localized flat-band eigenmodes can also give rise to quasi-BFICs under suitable fitting parameters (Supplementary Note 7). Moreover, the importance of the mirror-symmetry on the emergence of quasi-BFICs is also theoretically verified by considering the flat-band eigen-field without any even or odd symmetry, where no topological charge is found above the light line (Supplementary Note 8).

The experimental evidence for the quasi-bound flat bands in the continuum
In this section, we provide the experimental results, including the angle-resolved transmission spectra, iso-frequency contour, and the direct Q-factor measurement of the quasi-BFIC resonances. To verify our findings, the photonic crystal samples are fabricated according to the theoretical parameters. Figure 4a, b shows a scanning electron microscope image of the fabricated sample, with an enlarged side view in Fig. 4b. A 241-nm-thick silicon layer is firstly deposited on a silica substrate. Then, the 1D PhC slab with mirror-symmetric disorder in each supercell is fabricated in the silicon layer using a combination of electron-beam lithography and reactive ion etching techniques (see Methods section for details). The fabricated PhC slab measures approximately 500×500 μm² and contains ~100 supercells with each supercell possessing a length of 4940 nm. Such sample size is sufficient to exhibit the high-Q properties associated with quasi-BFICs (Supplementary Note 9). To create an optically symmetric environment, the fabricated sample is covered with PMMA, which matches the refractive index of the silica substrate.

At first, we measure the angle-resolved transmission spectra using the optical setups shown in Fig. 4c. Details are provided in Methods section. The result is shown in Fig. 4d where the colormap qualifies the strength of transmissivity. In Supplementary Note 10, the line spectra with specific incident angles are presented. The accessible ranges of the incident angle and wavelength in our applied polarization-resolved momentum-space imaging spectroscopy are confined within -40 to 40 degree and 1410 to 1580 nm, respectively. For comparison, we also numerically calculate the transmission spectra of this 1D PhC slab (Fig. 4e). The scanning steps for the incident wavelength and wavevector are matched to their experimental resolution limits. We find that the measured dispersions of Bloch bands agree well with the simulation results, confirming the successful fabrication of the sample with mirror-symmetric disorder within each supercell. Notably, we find that for both the experimental and numerical transmissivities, the high-Q flat band vanish entirely across the full range of incident angles. As a matter of fact, this observation serves as indirect validation for the presence of quasi-BFICs. The disappearance of the entire band shares the same origin as the disappearance of small segment of the band observed in previous BIC studies2,24,25,29,45.

To directly measure the Q-factors of quasi-BFICs, we employ an additional optical setup (Fig. 5a), which enables band structure visualization through iso-frequency contour observation in momentum space. Details are provided in the Methods section. The experimental results of iso-frequency contours at 1520 nm, 1515 nm, and 1510 nm can be obtained through precise control of the incident point (the large bright spot in the upper right corner), where two distinct sets of iso-frequency contours are observed (Fig. 5b–d). The flat-band iso-frequency contours appear as parallel lines that gradually separate with decreasing wavelength. The second set of contours corresponds to the dispersive TM-like mode. For reference, numerical simulations of the iso-frequency contours are displayed in Fig. 5e–g, where gray and red lines represent the eigenmodes of the quasi-BFICs and dispersive bands, respectively. Excellent agreement between experimental results and theoretical predictions confirms the observation of flat bands.

The observation of flat bands enables direct extraction of the resonance Q-factor through scattering spectrum linewidth measurements. We select four resonances centered at 1515 nm (marked by colored symbols in Fig. 5c), corresponding to incident angles ranging from 18.3 deg to 47.4 deg. By scanning the incident wavelength from 1513 nm to 1517 nm in 0.05 nm steps, narrow intensity peaks are observed with symmetric Lorentzian line shapes (Fig. 5h–k), where Lorentzian fitting of these scattering spectra yields the Q-factor of the resonances. Remarkably, the Q-factor remains stable at approximately 8000 across the entire angular range. This contrasts sharply with traditional BICs, where the Q-factor drops rapidly to ~1×10³ with just 2 deg deviation from normal incidence2. Our proposed quasi-BFIC structure thus demonstrates exceptional angular insensitivity of the high-Q property.

Methods
Sample fabrication
A 241-nm-thick silicon film was grown on a 500-µm-thick silica substrate using plasma-enhanced chemical vapor deposition (PECVD). This silicon layer exhibits the negligible intrinsic loss within the wavelength range from 1300 nm to 2000 nm, making it highly suitable for optical measurements. A 350 nm layer of ZEP520A electron-beam resist was then spin-coated on the silicon film to serve as a photoresist. Following exposure and development, the resist patterns were defined and subsequently transferred to the silicon film using an isotropic etching with HBr plasma. The ZEP520A layer acted as an etching mask during this process, ensuring precision patterning on the silicon film.

Angle-resolved transmittance spectrum measurement system
Transmittance spectra were obtained using a home-built, polarization-resolved momentum-space imaging spectroscopy system integrated with a Nikon microscope. Specifically, we employ a tungsten lamp as a broadband light source, whose emitted light is focused onto the sample through a series of lenses. The focused light interacts with and transmits through the sample, producing mixed transmission signals containing different wavelengths and wavevectors. It is needed to point out that, as we are dealing with a linear system, light components with different wavelengths and wavevectors couple with the sample independently, making post-coupling resolution feasible. The transmitted light was then directed through a series of convex lenses which resolves different wavevector components by imaging the back focal plane - this plane contains the momentum-space (Fourier-space) information of the sample’s radiation field. The wavevector-resolved transmission light was directed to the entrance slit of an imaging spectrometer (Princeton Instruments IsoPlane-320) for wavelength resolution. To isolate the resonant response from system artifacts, we normalize all measurements by dividing them by a background transmission spectrum obtained through an unetched, uniform Si layer. The transmission spectrum presented in the main text represent these normalized results, enabling clearer resolution of the band structure features.

Iso-frequency contour and scattering spectrum measurement system
An 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 were conducted using the finite element method with commercial software COMSOL Multiphysics. To ensure the accuracy of the numerically calculated results, the mesh size used in the FEM simulation is small enough to guarantee the convergence of the results, which means a higher mesh density leads to nearly the same results. Due to the continuous translational symmetry along the z-axis, our simulations can be reduced to a 2D model within the xy-plane. Floquet periodic boundary condition was applied along the x-axis, while perfectly matched layers were implemented on the top and bottom boundaries of the simulation domain. To calculate the amplitude distribution of Fourier coefficients for flat-band eigenfields in momentum space, u(k_x), we first perform an eigenfrequency calculation of the PhC slab. This allows us to obtain the Bloch modes E(k_x, x, y) in the near-field region. It is noted that u(k_x) is proportional to the amplitude of the Bloch modes within the 1D PhC slabs. Thus, each eigenmode is normalized across the target band. Finally, the normalized Bloch modes are substituted into Eq. (1) to calculate u(k_x) in Fig. 2.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "I have carefully reviewed the manuscript “Quasi-bound flat bands in the continuum”submitted by the authors from Beijing\nInstitute of Technology. This paper builds upon their previous work, “Optical moiré bound states in the continuum,” and\nfollows a similar author list and structure. The core idea of this new manuscript is the introduction of symmetric random\nperturbations to achieve high-Q values across the entire band, which is an interesting and potentially innovative concept.\nIf one focuses only on the first half of the manuscript, it appears to be a well-motivated and creative study, as the authors\npresent an attractive schematic of their expected results. However, upon reading the second half of the manuscript, it\nbecomes apparent that the authors fail to convincingly demonstrate their claims. Both the numerical simulations and\nexperimental validations are not rigorous enough to support their conclusions. The paper sets up ambitious expectations but\nultimately fails to deliver compelling evidence. Below are my detailed concerns for your consideration:\n---\n### **Major Concerns**\n1. **Definition and Reproducibility of the Symmetric Random Structure**\n- The authors claim to introduce a “random structure” that is also “symmetric” to achieve\nbroadband BIC (Bound States in the Continuum). However, how to publish a paper with a random generated structure and require this randomy has even\nsymmetry? the concept of a symmetric random structure is contradictory and difficult to reproduce in subsequent research.It\nis highly recommend that the authors change their description.\n- From this reviewer's understanding, the authors essentially introduce small width variations into an existing BIC grating,\nand the collective effect of these width modulations leads to broadband BIC formation. Instead of describing it as a “random\nerror,” the authors should consider referring to existing works on chirped photonic crystals, such as the research by **T.\nBaba on chirped photonic crystal waveguides**. This terminology is more precise, better aligns with established literature,\nand makes it easier for future researchers to understand and replicate the work.\n- Also, does the symmetry is really neccessary in the structure?\n2. **Unreliability of the Simulation Results (Figure 2d)**\n- The simulation results presented for the randomly perturbed widths (Figure 2d) are questionable. No matter what\nelectromagnetic simulation methods the authors use (FDTD, FDE, FEM, coupled-mode theory), each of these methods\ninherently involves numerical errors.\n- In particular, mesh discretization errors could significantly affect the simulated Q-factor, especially when modeling\nsmall width variations. How do the authors ensure that the observed changes in Q-factor are due to the structural perturbations\nrather than numerical artifacts from the simulation? This is a critical issue that needs to be addressed.\n3. **Insufficient Details in Experimental Methods**\n- The experimental section lacks crucial details regarding how light is coupled into different wavevectors:\n- The authors claim to use a **tungsten lamp** as a broadband light source. However, how do they control the coupling of\ndifferent wavelengths and wavevectors into the structure?\n- How is the transmitted light collected?\n- How do the authors ensure that only the transmitted light from the grating is measured and that background light\n(unperturbed by the structure) is not interfering with the results?\n- More information on the experimental setup is necessary to evaluate the validity of their claims.\n4. **Mismatch Between the Title and the Final Experimental Results**\n- The title of the paper emphasizes *quasi-BIC*, and the theoretical analysis in the first half of the manuscript aims to\ndemonstrate high-Q behavior across the entire band.\n- However, the final experimental results (even if we assume they are correct) only show **low-Q** flat bands. the authors\neven not disclose how low the Q value is in their experiment. This should be disclosed to properly assess their contribution.\n- Since the defining characteristic of BIC states is their high Q-factor, how can the authors justify calling their results “BIC” if\nthey only obtain low-Q flat bands? This is a fundamental inconsistency that must be addressed.\n5. **Difficulty in Achieving High-Q Flat Bands vs. Low-Q Flat Bands**\n- The challenge of realizing **high-Q** flat bands is significantly greater than that of achieving **low-Q** flat bands. The\nauthors should clarify why their experimental results fall short of their theoretical predictions and provide a detailed\ndiscussion on the limitations of their fabrication and measurement techniques.\n---\n### **Recommendation**\nWhile the idea of using random perturbations to generate broadband BIC is interesting, the paper in its current form lacks the\nnecessary rigor in simulation and experimental validation. The authors need to:\n1. Clearly define and justify their use of “random” vs. “chirped” structural modifications.\n2. Address concerns regarding numerical accuracy in their simulations.\n3. Provide detailed explanations of their experimental setup and methodology.\n4. Reconcile the discrepancy between their theoretical predictions and experimental results.\n5. Justify the use of “BIC” in their title and discussion, given the observed low-Q results.\n6. Try to get a high Q over the flat bandwidth before they resubmitted.\nUnless these major concerns are addressed, I do not believe this manuscript is ready for publication in *Nature\nCommunications*. I would recommend either **major revisions** or **rejection**, depending on the authors' ability to provide\nconvincing solutions to these issues.", "reviewer_2": "In this work, the authors present an interesting approach for realizing high-quality factor optical resonances by introducing\ncontrolled disorder into photonic crystal structures. The core element of the work is the formation of quasi-bound flat bands,\ni.e., modes that maintain their quality even at large incidence angles. While the initial premise is intriguing and the\nmanuscript presents both numerical and experimental results, several aspects require significant clarification and further\njustification before the work can be considered suitable for publication in Nature Communications.\n(a) The manuscript claims a \"novel approach\" and \"counterintuitive strategy.\" However, the extent to which this work truly\nrepresents a radical departure from existing approaches in disordered photonics and BIC engineering is not convincingly\ndemonstrated. The introduction lacks a detailed discussion of existing literature in disordered photonic structures and their\npotential for tailoring band structures and localization. The claimed \"counterintuitive\" nature of using disorder likewise needs\nstronger justification. The influence of disorder on resonance properties and localization is a well-established concept in\ncondensed matter physics and photonics, and the specific \"counterintuitive\" aspect here is not clearly articulated. The\nmanuscript needs to more rigorously position its novelty against established literature on disorder engineering in photonic\nsystems.\n(b) The manuscript attributes flat band formation to disorder-induced band folding, but the precise mechanism by which\nmirror-symmetric disorder leads to flat bands of quasi-BICs is not rigorously explained. Is it truly just band folding? Are there\nspecific symmetry considerations or interference effects at play that are essential for this phenomenon? The manuscript\nneeds a more detailed theoretical framework to establish the underlying physics.\n(c) The discussion of topological charges appears somewhat disjointed from the main message of the manuscript. The\nevidence for topological charges relies on limited simulations of Fourier coefficient distributions. The manuscript needs to\ndemonstrate more convincingly that these topological charges are indeed topological in nature (i.e., robust to perturbations)\nand that they play a causal role in the suppression of radiation and the quasi-BIC behavior. The presented analysis seems\ndescriptive rather than providing deep mechanistic insight.\n(d) In the introduction, the authors argue that BICs are constrained by “fabrication imperfections, such as surface roughness\nand structural disorder.” They later argue that their approach (which deliberately introduces disorder) overcomes these\nchallenges. This is problematic. Surface scattering from roughness induced by nanofabrication can’t be avoided and will\ndampen resonances. Also, even when the authors introduce a specific pattern of disorder, this will be overlaid with an\nadditional disorder pattern introduced by the fabrication inaccuracies. Therefore, it is not accurate to say that the authors’\nconcept overcomes these imperfections. These aspects should be clarified and discussed in detail in the manuscript.\n(e) In Figure 2d, the bandwidth is plotted. How exactly is this bandwidth defined? It could be useful to include a simple\nsketch of the resonance and the important resonance parameters in the figure.\n(f) As a general point, the manuscript heavily relies on plotting band diagrams and color coded dispersion diagrams (such as\nFigure 4c,d). What is really missing are more plots of some examples of the actual transmittance spectra. It is very difficult to\nevaluate parameters like line shape, etc. from the color plots. I suggest to add transmittance spectra for different incidence\nangles for both simulation and experiment to the manuscript, or at least to the supporting information.\n(g) Can the authors justify the choice of the blue area in Figure 2? The exact choice of the “cut-off values” for bandwidth and\nQ factor will heavily influence the statistical evaluation in Figure 2f.\n(h) How many supercells are needed in the experimental realization? The size of the photonic crystal seems extremely large\nat 500 µm x 670 µm. What is the lower limit of supercells required to form the flat band mode? How do Q factor and\nbandwidth scale as a function of supercell number? The authors should analyze this point, ideally with additional\nexperiments or numerical simulations.\n(i) The authors mention that “the fabricated sample is covered with PMMA, which matches the refractive index of the silica\nsubstrate”. Is an optically symmetric environment always needed for the proposed flat band design? Encapsulation with\nPMMA prevents many practical applications of the structure, including the sensing experiments that are proposed later.\n(j) The high-Q flat bands are not observed in experiment and only the low-Q flat band appears. Because the high-Q bands\nare designed to have Q > 100.000, any slight surface roughness will dampen them. It is very problematic to try to explain this\nbehavior with the spectral resolution of the experimental setup as the authors state: “Specifically, the linewidths of the\ncorresponding transmission spectrum (~0.005 nm) fall well below the wavelength resolution of the CCD (0.35 nm) used in\nthe experiments”. Because of the experimental losses, these high-Q modes were never present in the system, no matter the\nspectral resolution. Therefore, the following statement is unfortunately also completely wrong: “This complete disappearance of the transmission signal provides direct evidence of the existence of quasi-BFICs with remarkably high Q factors”. The\nabsence of the resonances in the spectrum DOES NOT prove their existence. All of these arguments have to be clarified\nand revised in the manuscript.\n(k) The sensing comparison part at the beginning of the discussion section should be removed in its entirety. Claiming better\nsensing performance based only on simulations is highly problematic, since the final sensing performance will depend on a\nmultitude of different factors. Also, looking at Figure S6, it appears to me that the authors have not chosen a fair “traditional”\nBIC geometry (blue line) for comparison. The traditional design shows worse sensing performance even at normal incidence\n(theta = 0), which is very unlikely. The traditional BIC geometry can be engineered to have extremely high Q factors as well,\nso sensing performance should at least be equal. Finally, extreme angle independence of the sensor signal is not generally a\nlarge issue in optical sensors. Many existing BIC designs are angle-stable up to incidence angles of around 10 degrees, which\ncoincides with the NA of low magnification objectives commonly used to read out metasurface-based sensors in many\npractical optical sensing devices."}, "author_response": "Response Letter to Reviewers\nWe are grateful for the constructive comments on this manuscript (NCOMMS-24-84707-T) from\ntwo reviewers. In the text below, reviewer comments are quoted in blue and followed by our detailed\nresponse. We have also revised the manuscript and the Supplemental Note based on the reviewer’s\ncomments, and these updates are highlighted in red in those files. In the text below, these updates\nare also highlighted in Italics.\nResponse to comments of the reviewer #1\nI have carefully reviewed the manuscript“Quasi-bound flat bands in the continuum”submitted by\nthe authors from Beijing Institute of Technology. This paper builds upon their previous work,\n“Optical moiré bound states in the continuum,”and follows a similar author list and structure. The\ncore idea of this new manuscript is the introduction of symmetric random perturbations to achieve\nhigh-Q values across the entire band, which is an interesting and potentially innovative concept. If\none focuses only on the first half of the manuscript, it appears to be a well-motivated and creative\nstudy, as the authors present an attractive schematic of their expected results. However, upon reading\nthe second half of the manuscript, it becomes apparent that the authors fail to convincingly\ndemonstrate their claims. Both the numerical simulations and experimental validations are not\nrigorous enough to support their conclusions. The paper sets up ambitious expectations but\nultimately fails to deliver compelling evidence. Below are my detailed concerns for your\nconsideration:\nReply: We thank the reviewer for the thorough evaluation of our manuscript and for recognizing\nthe potential innovation in our proposed concept. However, we acknowledge the reviewer's concern\nregarding the need for more rigorous numerical simulations and experimental validations to support\nour conclusions. In response to these concerns, we have undertaken comprehensive revisions to\nstrengthen our manuscript, as outlined below:\n Enhanced explanation of structural design: We have expanded the discussion of our\nstructural design, with a particular emphasis on clarifying the role of mirror-symmetric random\nperturbations within each supercell. Furthermore, we now explicitly distinguish our approach\nfrom conventional chirped structures, ensuring a more accurate understanding of the\nunderlying design principle.\n Examination of numerical accuracy: To verify the reliability of our numerical results, we\nconducted a systematic mesh-convergence analysis, particularly for structures exhibiting high\nQ-factors. This analysis confirms that the observed Q-factor variations stem from structural\nmodifications rather than numerical artifacts.\n New experimental validation of high-Q flat bands: To provide experimental support for the\nexistence of high-Q flat bands, we have designed and fabricated new samples and performed\nadditional measurements using the momentum-space imaging spectroscopy with a tunable\nlaser around 1550 nm. These measurements reveal iso-frequency contours that directly confirm\nthe presence of high-Q flat bands, with Q-factors reaching approximately 8000 across a wide\nangular range—significantly exceeding those of frequency bands embedded with BICs. These\nempirical results further strengthen our conclusions.\nWe believe these revisions have significantly improved the manuscript, addressed the\nreviewer's concerns, and provided stronger support for our conclusions.\n**Definition and Reproducibility of the Symmetric Random Structure**\n- The authors claim to introduce a“random structure”that is also “symmetric” to achieve\nbroadband BIC (Bound States in the Continuum). However, how to publish a paper with a random\ngenerated structure and require this random has even symmetry? the concept of a symmetric random\nstructure is contradictory and difficult to reproduce in subsequent research. It is highly recommend\nthat the authors change their description. From this reviewer's understanding, the authors essentially\nintroduce small width variations into an existing BIC grating, and the collective effect of these width\nmodulations leads to broadband BIC formation. Instead of describing it as a“random error,”the\nauthors should consider referring to existing works on chirped photonic crystals, such as the research\nby **T. Baba on chirped photonic crystal waveguides**. This terminology is more precise, better\naligns with established literature, and makes it easier for future researchers to understand and\nreplicate the work.\nReply: We appreciate the reviewer's comment regarding the definition and reproducibility of our\nstructure. We acknowledge that the initial description in the manuscript may have led to some\nconfusion. In our study, the random generation process is employed as a design strategy for\nconstructing the supercell configuration. Thus, it is reasonable to apply a symmetric constraint when\nintroducing the disorder. Below, we provide a detailed explanation of our structural design and\naddress the reviewer's concerns from two aspects.\n Clarification of structural design: Our structure is not purely random in the conventional\nsense but is systematically designed based on a 1D photonic crystal with a supercell approach. The\ndesign process involves the following steps:\n1. Supercell Definition: We begin by defining a supercell composed of 13 fundamental unit\ncells of a bound state in the continuum (BIC) grating (see Fig. R1a).\n2. Introduction of mirror-symmetric random perturbations: Within the supercell, random\nwidth perturbations are introduced to individual grating elements. Crucially, these\nperturbations are applied in a manner that preserves mirror symmetry about the center of\nthe supercell (see Fig. R1b). This constraint ensures that the global symmetry of the\nsupercell is maintained, which is essential for the emergence of flat bands and polarization\nvortices in our structure.\n3. Periodic arrangement of the supercell: Once a symmetric random supercell is generated,\nit is then repeated periodically to form the final structure (see Fig. R1c). Thus, the entire\nsystem can be viewed as a periodic repetition of identical supercells.\nFigure R1. Schematic illustration of the structural design process with A=13𝒂 .\n𝟎\n(a) Definition of a\nsupercell composed of 13 fundamental unit cells of a conventional grating. (b) Introduction of width\nperturbations exhibiting mirror symmetry within the supercell. Pillars with the same width are assigned\nthe same color. The disorder is applied to the left half of the supercell and then symmetrically mirrored\nonto the right half, thereby preserving mirror-plane symmetry. (c) Periodic arrangement of the perturbed\nsupercell to form the final structure, consisting of a periodic repetition of identical supercells.\n Distinction from chirped photonic crystals: We appreciate the reviewer’s suggestion to\ncompare our work with chirped photonic crystals, such as those studied by T. Baba. We note that\nour design fundamentally differs from chirped structures, and therefore a new terminology is needed\nto clearly distinguish between these concepts. Unlike chirped systems, which feature gradually\nvarying structural modulations and exhibit local rather than global periodicity, our design\nincorporates mirror-symmetric disorder within each supercell. This disorder breaks the local\nperiodicity but preserves global periodicity, as the supercell is periodically repeated to form the final\nstructure. This periodic arrangement with an enlarged supercell supports flat bands, whose\neigenstates exhibit strong localization within each supercell (as shown in Fig. R2), leading to the\nformation of a full-band quasi-BIC. This distinction is essential, and we have cited three references\non chirped photonic crystals in the revised manuscript to provide further context and clarify this\npoint.\nFigure R2. Eigenfield distribution in the grating with periodic supercells (A=13𝒂 ). The eigenfield\n𝟎\npattern is repeated within each supercell. This periodic structure, composed of enlarged supercells,\nsupports flat bands whose eigenstates exhibit strong localization within each supercell.\nAction taken:\n In page 4 of the revised manuscript, we have modified the description of the structure\ngeneration to avoid the misunderstanding: “Now, we construct a supercell with thirteen\nfundamental units (A=13𝑎 ) by adding mirror-symmetric disorder. Then, the supercell is\n0\nrepeated in periodic direction to construct an infinite photonic crystal slab. Specifically, as\ndepicted in Figure 2c, the widths of silicon pillars within each supercell are randomly assigned\nwithin the range of [W-δ, W+δ], where W is the original width and δ/W characterizes disorder\nstrength. We emphasize that that the mirror symmetry of each supercell is essential for\nconstructing the quasi-BFIC, which is fully discussed later. To maintain the mirror-symmetry\nof each supercell, pairs of silicon pillars positioned symmetrically about the mirror plane share\nthe same width. We mention that the proposed structure is different from the chirped system\nwith gradually varying structural modulations and exhibit local rather than global periodicity.\nThe disorder adopted in the random design breaks the local periodicity but preserves the\nglobal periodicity, as the supercell is periodically repeated to form the final structure.”\n Following papers are cited in the revised manuscript:\n[57]. Baba, T. Slow light in photonic crystals. Nat. Photon. 2, 465–473 (2008).\n[58]. Mori, D., Kubo, S., Sasaki, H. & Baba, T. Experimental demonstration of wideband dispersion\ncompensated slow light by a chirped photonic crystal directional coupler. Opt. Express 15, 5264–5270\n(2007).\n[59]. Mori, D. & Baba, T. Dispersion-controlled optical group delay device by chirped photonic crystal\nwaveguides. Appl. Phys. Lett. 85, 1101–1103 (2004).\n- Also, does the symmetry is really necessary in the structure?\nReply: We thank the reviewer for the comment regarding the necessity of symmetry for high-Q flat\nbands. The mirror symmetry of the supercell plays a critical role in sustaining high-Q flat bands, as\nit is a necessary condition for the formation of topological charges.\nThe requirement of mirror symmetry for high-Q flat bands can be understood through\nanalytical theory describing the far-field radiation of supercell photonic crystal slabs. Specifically,\nwe consider the case in which the flat-band eigenfield lacks either even or odd symmetry with\nrespect to the mirror plane of the supercell. In such cases, no topological charge appears above the\nlight line, and strong far-field radiation is observed (see revised Supplementary Note 8 for details).\nThis highlights the essential role of mirror symmetry in each supercell for the emergence of quasi-\nbound flat bands in the continuum.\nFurthermore, to support this conclusion, we provide additional numerical results on the\nstatistical distribution of Q-factors for randomly generated supercell gratings lacking mirror\nsymmetry. Fig. R3 shows the variation of the average Q-factor with disorder strength for structures\nwith and without mirror symmetry (A = 13𝑎 ). The results demonstrate that the Q-factors are\n0\nsignificantly lower than those of mirror-symmetric disordered supercells. Together, these numerical\nand theoretical findings highlight the crucial role of mirror symmetry in achieving high-Q flat bands\nin the supercell design.\nFigure R3. Numerical results of the Variation of the average Q-factor with disorder strength for\nstructures with and without mirror symmetry. These numerical results, combined with analytical far-\nfield radiation theory (revised Supplementary Note 8), demonstrate the critical role of mirror symmetry\nin individual supercells for realizing high-Q flat bands in photonic crystal slabs.\nAction taken:\n In page 11 of the revised manuscript, we have added “Furthermore, in Supplementary Note 8,\nwe also consider the flat-band eigen-field without any even or odd symmetry relative to the mirror\nplane of the supercell, where no topological charge is found above the light line, showing the\nimportance of the mirror-symmetry on the emergence of quasi-BFICs.”.\n In the revised supplementary Note, we have added the section 8 to demonstrate the crucial role\nof the mirror symmetry in the formation of the quasi-BFIC.\n2. **Unreliability of the Simulation Results (Figure 2d)**\n- The simulation results presented for the randomly perturbed widths (Figure 2d) are questionable.\nNo matter what electromagnetic simulation methods the authors use (FDTD, FDE, FEM, coupled-\nmode theory), each of these methods inherently involves numerical errors. In particular, mesh\ndiscretization errors could significantly affect the simulated Q-factor, especially when modeling\nsmall width variations. How do the authors ensure that the observed changes in Q-factor are due to\nthe structural perturbations rather than numerical artifacts from the simulation? This is a critical\nissue that needs to be addressed.\nReply: We thank the reviewer for their insightful comment. We fully agree that numerical accuracy\nis crucial, especially for simulating high-Q photonic structures where minor perturbations can lead\nto significant variations.\nThe numerical results in our manuscript were obtained using COMSOL Multiphysics 6.0,\nwhich employs the finite element method (FEM). In our structure, the super-narrow slit induces a\nmuch more drastic field variation than in free space, potentially affecting simulation accuracy. To\nmitigate this issue, we applied a higher mesh density to properly resolve the smooth variations of\nthe continuous field, ensuring reliable results.\nTo ensure the accuracy of our results, we performed a systematic mesh convergence analysis\nfor high-Q structures, confirming that the observed Q-factor variations stem from structural\nmodifications rather than numerical artifacts. Specifically, we conducted finite element method\n(FEM) simulations with progressively refined meshes and monitored Q-factor convergence. For\ninstance, in a representative random configuration (same structure used in the Fig. 2f in the revised\nmanuscript), we incrementally increased the mesh density (see insets) until the calculated Q-factor\nstabilized (Fig. R4). The mesh size employed in our simulations is indicated by the orange star.\nFigure R4. Mesh convergence analysis for high-Q structures (A=13𝒂 ). Left: Convergence of the Q-\n𝟎\nfactor with progressive mesh refinement in a representative random configuration. The orange star\nindicates the mesh size adopted in our simulations. Right: Mesh distribution profiles for different mesh\nsize.\nAction taken:\n In the method section of the revised manuscript, we have added the numerical details regarding\nthe mesh size of the simulation: “All numerical simulations were conducted using the finite\nelement method with commercial software COMSOL Multiphysics. To ensure the accuracy of the\nnumerically calculated results, the mesh size used in the FEM simulation is small enough to\nguarantee the convergence of the results, which means a higher mesh density leads to nearly the\nsame results.”\n3. **Insufficient Details in Experimental Methods**\n- The experimental section lacks crucial details regarding how light is coupled into different\nwavevectors: The authors claim to use a **tungsten lamp** as a broadband light source. However,\nhow do they control the coupling of different wavelengths and wavevectors into the structure?\nHow is the transmitted light collected? How do the authors ensure that only the transmitted light from the\ngrating is measured and that background light (unperturbed by the structure) is not interfering with\nthe results? More information on the experimental setup is necessary to evaluate the validity of their\nclaims.\nReply: We thank the reviewer for their valuable comment. In response, we provide a comprehensive\nexplanation of the experimental methodology for angle-resolved transmission spectroscopy,\naccompanied by a schematic diagram of the optical measurement setup (Fig. R5).\nWe employ a tungsten lamp as a broadband light source, whose emitted light is focused onto\nthe sample through a series of lenses. The focused light interacts with and transmits through the\nsample, producing mixed transmission signals containing different wavelengths and wavevectors.\nIt is needed to point out that, as we are dealing with a linear system, light components with different\nwavelengths and wavevectors couple with the sample independently, making post-coupling\nresolution feasible. The transmitted light is collected by lens L4, which resolves different\nwavevector components by imaging the back focal plane - this plane contains the momentum-space\n(Fourier-space) information of the sample's radiation field. The wavevector-resolved transmission\nlight was directed to the entrance slit of an imaging spectrometer (Princeton Instruments IsoPlane-\n320) for wavelength resolution. To isolate the resonant response from system artifacts, we normalize\nall measurements by dividing them by a background transmission spectrum obtained through an\nunetched, uniform Si layer. The transmission spectrum presented in the main text represent these\nnormalized results, enabling clearer resolution of the band structure features.\nFigure R5. Schematic diagram of the optical measurement setup for the measurement of the angle-\nresolved momentum-space imaging spectroscopy.\nAction taken:\n In the method section of the revised manuscript, we have added the experimental details\nregarding the measurement of the transmittance spectra.\n The optical measurement setup for the angle-resolved transmittance spectra measurement is\nincluded in the Figure 4 of the revised manuscript.\n**Mismatch Between the Title and the Final Experimental Results**\n- The title of the paper emphasizes *quasi-BIC*, and the theoretical analysis in the first half of the\nmanuscript aims to demonstrate high-Q behavior across the entire band.\n- However, the final experimental results (even if we assume they are correct) only show **low-\nQ** flat bands. the authors even not disclose how low the Q value is in their experiment. This should\nbe disclosed to properly assess their contribution. - Since the defining characteristic of BIC states is\ntheir high Q-factor, how can the authors justify calling their results“BIC”if they only obtain low-\nQ flat bands? This is a fundamental inconsistency that must be addressed.\nReply: We sincerely appreciate the reviewer's insightful comments regarding the experimental\nvalidation of our results. We fully agree that direct measurement of the high-Q flat bands would\nsignificantly strengthen our conclusions. To address the reviewer's concern, we designed and\nfabricated new samples and performed additional measurements using momentum-space imaging\nspectroscopy with a tunable laser (centered at 1550 nm). This system provides significantly higher\nwavelength resolution, enabling direct observation of the high-Q flat bands. [Nature 574, 501-505\n(2019)]. These new experiments directly measure the Q-factor of the high-Q flat band, yielding\nvalues around 8000 across a wide angular range (towards to 47.4°). This empirical evidence further\nsupports our conclusions. Below, we provide a detailed description of the new experimental setup\nand results.\nTo directly measure the Q-factor of the flat bands, we employ an additional optical setup (Fig.\nR6). This configuration enables band structure visualization through iso-frequency contour\nobservation in momentum space. Specifically, the iso-frequency contours of PhC samples are\ndirectly imaged using resonance-enhanced photon scattering. When the frequency and incident\nangle of incoming light match a resonance mode of the PhC slab, fabrication-induced defects scatter\nphotons into other resonance modes. Since these modes share the same frequency as the incident\nlight, they reproduce the iso-frequency contour patterns in the far field.\nFigure R6. The isofrequency contour imaging system.\nA near-infrared laser with continuous tunability serves as the incident light source in this setup.\nThe laser beam undergoes spatial filtering and is adjusted to match the sample size. Employing a\npolarizer, the laser polarization is tuned to align with the excitation k-point of the iso-frequency\ncontours in the sample (Pol y). Following this, lens L1 focuses the light onto the back focal plane\nof an infinitely corrected objective. By displacing L1 in the x-y plane, the incident angle of the laser\nis controllable, allowing for oblique incidence excitation at any angle within the numerical aperture\nrange. After traversing the sample, the scattered light is gathered by another objective and ultimately\ndirected onto a CCD through successive lenses L2, L3, and a 4f system. The CCD corresponds to\nthe Fourier plane of the sample. In the receiving optical path, an additional polarizer, oriented\northogonally to the incident polarization direction (Pol x), is utilized to suppress direct transmission\nthrough the sample, thereby enhancing the signal-to-noise ratio of the scattered light.\nWhen the wavelength and incident angle of the incident light match the resonant modes\nsupported by the sample, iso-frequency contours can be observed on the CCD. At this point, with\nthe position of lens L1 fixed (i.e., without changing the incident angle), adjusting the laser can obtain\nthe scattering signals corresponding to different wavelengths. When the wavelength matches the\nresonant mode, the iso-frequency contour is illuminated. As the wavelength gradually deviates from\nthe resonance wavelength, the iso-frequency contour gradually dims. Thus, we can obtain the\nscattering spectrum curve of the intensity at any point in momentum space as a function of\nwavelength, and the Q-factor of which can be obtained by numerically fitting the scattering\nspectrum with a Lorentzian function.\nThe experimental results are presented in Fig. R7. Through precise control of the incident point\n(marked by the large bright spot in the upper right corner), two distinct sets of iso-frequency\ncontours are observed. As shown in Fig. R7a-c, these correspond to incident wavelengths of 1520\nnm, 1515 nm, and 1510 nm, respectively. The flat-band iso-frequency contours appear as parallel\nlines (due to x-direction flatness) that gradually separate with decreasing wavelength. The second\nset of contours corresponds to the dispersive TM-like mode. For reference, numerical simulations\nof the iso-frequency contours are displayed in Fig. R7d-f, where gray and red lines represent the\neigenmodes of the high-Q flat bands and dispersive bands, respectively. Excellent agreement\nbetween experimental results and theoretical predictions confirms the observation of flat bands.\nThe observation of flat bands enables direct extraction of the resonance Q-factor through\nscattering spectrum linewidth measurements. We selected four resonances centered at 1515 nm\n(marked by colored symbols in Fig. R7b), corresponding to incident angles ranging from 18.3° to\n47.4°. As demonstrated in Fig. R7g-j, scanning the incident wavelength from 1513 nm to 1517 nm\nin 0.05 nm steps reveals narrow intensity peaks exhibiting symmetric Lorentzian line shapes.\nLorentzian fitting of these scattering spectra yields the Q-factor of the resonances. Remarkably, the\nQ-factor remains stable at approximately 8000 across the entire angular range. This contrasts sharply\nwith traditional BIC, where the Q-factor drops rapidly to ~1×10³ with just 2° deviation from normal\nincidence [Nature 499, 188-191 (2013)]. Our proposed quasi-BIC structure thus demonstrates\nexceptional angular insensitivity.\nFigure R7. Direct observation of quasi-BIC flat bands. (a)-(c) Experimental iso-frequency contours\nat 1520 nm, 1515 nm, and 1510 nm, respectively. The high-Q flat bands appear as diverging parallel lines\nwith decreasing wavelength. (d)-(f) Corresponding numerical simulations at 1485 nm, 1480 nm, and\n1475 nm, respectively, with gray and red lines representing the eigenmodes of high-Q flat bands\nand dispersive bands. (g)-(j) Experimental intensity spectra (dots) with Lorentzian fits (lines) for four\nresonances at the marked k-points in (b).\nAction taken:\n In the revised manuscript, we have added the latest experimental results, including the iso-\nfrequency contour and directly measured Q-factors of the quasi-BFICs, which is primarily\npresented in Figure 5 of the revised manuscript.\n To enable direct comparison between theory and experiment, all theoretical results in the\nrevised manuscript were updated according to the parameters of the latest sample design.\n In the method section of the revised manuscript, we have added the experimental details\nregarding the measurement of the iso-frequency contour and the scattering spectra.\n The optical measurement setup for the measurement of the iso-frequency contour and the\nscattering spectra is included in Figure 5 of the revised manuscript.\n5. **Difficulty in Achieving High-Q Flat Bands vs. Low-Q Flat Bands**\n- The challenge of realizing **high-Q** flat bands is significantly greater than that of achieving\n**low-Q** flat bands. The authors should clarify why their experimental results fall short of their\ntheoretical predictions and provide a detailed discussion on the limitations of their fabrication and\nmeasurement techniques.\nReply: We thank the reviewer for their insightful comment. We fully acknowledge that realizing\nhigh-Q flat bands presents significantly greater challenges compared to low-Q implementations. In\ndirect response to this observation, we have performed additional experiments using an enhanced\nmomentum-space imaging spectroscopy system featuring a tunable laser (centered at 1550 nm).\nThis advanced setup, adapted from established high-resolution techniques [Nature 574, 501-505\n(2019)], enables direct observation and characterization of the high-Q flat bands.\nOur measurements demonstrate Q-factors of approximately 8000 maintained across an\nexceptionally wide angular range (see Fig. R7). While these experimental values remain below\ntheoretical predictions, we identify three primary contributing factors: (1) scattering losses from\nstructural disorder, (2) intrinsic material absorption, and (3) surface roughness effects. Additional\nmeasurement constraints include the fundamental signal-to-noise limitations (addressed by\nmaximizing source intensity), finite resolution constraints of the spectroscopic system and practical\ntrade-offs between measurement duration and precision.\nThe obtained Q-factors nevertheless represent significant progress, exceeding conventional\nangular-sensitive designs by nearly an order of magnitude across comparable angular ranges.\n**Recommendation**\nWhile the idea of using random perturbations to generate broadband BIC is interesting, the paper in\nits current form lacks the necessary rigor in simulation and experimental validation. The authors\nneed to:\n1. Clearly define and justify their use of “random” vs. “chirped” structural modifications.\n2. Address concerns regarding numerical accuracy in their simulations.\n3. Provide detailed explanations of their experimental setup and methodology.\n4. Reconcile the discrepancy between their theoretical predictions and experimental results.\n5. Justify the use of “BIC” in their title and discussion, given the observed low-Q results.\n6. Try to get a high Q over the flat bandwidth before they resubmitted.\nUnless these major concerns are addressed, I do not believe this manuscript is ready for publication\nin *Nature Communications*. I would recommend either **major revisions** or **rejection**,\ndepending on the authors' ability to provide convincing solutions to these issues.\nReply: We sincerely appreciate the reviewer's valuable recommendations, which have significantly\nimproved our manuscript. In response to the reviewer's comments, we have implemented the\nfollowing revisions:\n1. We have provided a more detailed explanation of our structural design, with particular\nemphasis on the role of random perturbations within each supercell and a clear distinction\nfrom chirped structures.\n2. We have demonstrated the numerical accuracy of our simulations to further substantiate\nour findings.\n3. We have included a more detailed description of the experimental setup and methodology\nto enhance the rigor of our validation.\n4. We have fabricated new samples and conducted additional measurements using\nmomentum-space imaging spectroscopy with a tunable laser (around 1550 nm wavelength\nrange) to directly measure the Q-factor of quasi-BFIC, showing a good agreement with\ntheoretical predictions.\n5. Based on the direct observation of high Q-factors in a wide angle range for our sample, it\nis reasonable to use “quasi-BIC” in the title and discussion.\n6. We have obtained the high-Q flat band in our new sample.\nWe believe these revisions have addressed all the reviewers' concerns and have significantly\nstrengthened our manuscript.\nResponse to comments of the reviewer #2\nIn this work, the authors present an interesting approach for realizing high-quality factor optical\nresonances by introducing controlled disorder into photonic crystal structures. The core element of\nthe work is the formation of quasi-bound flat bands, i.e., modes that maintain their quality even at\nlarge incidence angles. While the initial premise is intriguing and the manuscript presents both\nnumerical and experimental results, several aspects require significant clarification and further\njustification before the work can be considered suitable for publication in Nature Communications.\nReply: We thank the reviewer for their thorough review and constructive feedback, as well as their\npositive assessment of the quasi-bound flat bands in the continuum. We fully acknowledge the need\nfor further clarification and justification regarding several aspects, as highlighted by the reviewer.\nIn response to these comments, we have implemented the following revisions to strengthen the\nmanuscript:\n1. Novelty clarification: We have more rigorously positioned our work's novelty within the\ncontext of existing literature on disorder engineering in photonic systems, emphasizing its\nunique contributions and distinguishing features from prior approaches.\n2. Theoretical framework: We have expanded the theoretical framework to provide a clearer\nand more comprehensive explanation of the underlying physics governing quasi-bound flat\nbands in the continuum formation.\n3. Topological charges: We have explicitly demonstrated both the robustness of topological\ncharges against perturbations and their causal role in radiation suppression, thereby\nreinforcing the quasi-BIC behavior in our system.\n4. New experimental data on high-Q flat bands: To experimentally observe the high-Q flat\nbands, we designed and fabricated new samples and performed additional measurements\nusing the momentum-space imaging spectroscopy with a tunable laser around 1550 nm.\nThese new experiments directly observe the high-Q flat band through iso-frequency contour,\nwith measured Q-factors of ~8000 across a wide angular range.\n5. Removal of sensing comparison: As suggested by the reviewer, we removed the sensing\ncomparison section to maintain focus on the core findings of the work.\nWe believe these revisions have significantly enhanced our manuscript. We hope the revised\nversion meets the standards of Nature Communications and look forward to the reviewer’s\nrecommendation for publication.\n(a) The manuscript claims a \"novel approach\" and \"counterintuitive strategy.\" However, the extent\nto which this work truly represents a radical departure from existing approaches in disordered\nphotonics and BIC engineering is not convincingly demonstrated. The introduction lacks a detailed\ndiscussion of existing literature in disordered photonic structures and their potential for tailoring\nband structures and localization. The claimed \"counterintuitive\" nature of using disorder likewise\nneeds stronger justification. The influence of disorder on resonance properties and localization is a\nwell-established concept in condensed matter physics and photonics, and the specific\n\"counterintuitive\" aspect here is not clearly articulated. The manuscript needs to more rigorously\nposition its novelty against established literature on disorder engineering in photonic systems.\nReply: We thank the reviewer for the comment and acknowledge the need to more rigorously\nposition our work within the context of existing literature on disordered photonic systems. In\nresponse, we have expanded the introduction to provide a more comprehensive discussion of prior\nstudies in this field. In the revised manuscript, we highlight the remarkable surge of interest in\nexploring disorder-induced localizations in photonic systems, encompassing phenomena such as\nrandom laser emission [Nature Phys. 4, 359-367 (2008)] and Anderson localization of light [Nature\n446, 52-55 (2007)]. Furthermore, numerous studies have demonstrated that disorder can\nalso enhance wave transport, as evidenced by disorder-induced Anderson topological photonic\ninsulators [Nature 560, 461–465 (2018)] and disorder-enhanced light transport [Nat. Rev. Phys. 2,\n141-158 (2020)].\nWe then specifically review the effects of disorder on BICs in photonic crystal slabs, noting\nthat prior studies consistently demonstrate that structural disorder leads to the degradation of high-\nQ BICs [Nature 499, 188–191 (2013), Nature 574, 501–504 (2019)]. The \"counterintuitive\" aspect\nof our work lies in the fact that, contrary to the conventional understanding that disorder disrupts\nhigh-Q BICs, we show that introducing controlled disorder of appropriate strength within each\nsupercell can instead induce the formation of quasi-bound flat bands in the continuum. Specifically,\nour approach introduces disorder within each supercell while maintaining an overall periodic\nsupercell arrangement. This unique strategy enables the periodic localization of Bloch waves within\neach supercell, thereby generating flat bands, topological charges and quasi-bound flat bands in the\ncontinuum.\nWe believe these clarifications and additions more clearly articulate both the novelty and\ncounterintuitive nature of our approach, and its distinction from established literature on disorder\nengineering in photonic systems.\nAction taken:\n In page 3-4 of the revised manuscript, we have added a detailed discussion of existing literature\nin disordered photonic structures and their potential for tailoring band structures and\nlocalization: “On the other hand, the field of photonic systems has witnessed a remarkable surge\nof interest in exploring disorder effects, with numerous studies demonstrating that disorder can both\ninduce localization phenomena (such as random laser emission [53] and Anderson localization of\nlight [54]) and enhance wave transport [55, 56]. While, concerning BICs in PhC slabs, all existing\nexperimental observations have unveiled substantial scattering of Bloch waves due to disorder [2,\n20], which significantly compromises the performance of BICs. In this work, we challenge the\ndetrimental view of disorder on BICs and reveal for the first time that quasi-BFICs can be randomly\ngenerated in PhC slabs through mirror-symmetric disorder within each supercell.”.\n(b) The manuscript attributes flat band formation to disorder-induced band folding, but the precise\nmechanism by which mirror-symmetric disorder leads to flat bands of quasi-BICs is not rigorously\nexplained. Is it truly just band folding? Are there specific symmetry considerations or interference\neffects at play that are essential for this phenomenon? The manuscript needs a more detailed\ntheoretical framework to establish the underlying physics.\nReply: We would like to thank the reviewer for the comment. It is true that the formation of the flat\nband arises from the disorder-induced band folding and eigenmode localization, while high Q-\nfactors of the flat band require the mirror symmetry. Below, we provide a detailed theoretical\nframework by separately illustrating the formation of the flat bands and full-band quasi-BIC, which\njointly lead to the quasi-bound flat bands in the continuum.\n Flat bands: We mention that the band folding theory provides a general understanding of the\nband structure of all randomly generated structures, regardless of the flatness and Q-factor of the\nband. Among all randomly generated structures, the existence of the flat band further requires the\nlocalization of the mode within the supercell. The localization of the mode will extremely weaken\nthe interaction between adjacent supercells, resulting in the alignment of the flat band\neigenfrequency with that of an isolated unit cell, i.e., flat band. In this work, the localization of the\nmode induced by disorder can be understood from Anderson localization, which ensures that a\ncertain proportion of the generated structures have strong localization. Besides, we mention that the\nintroduction of mirror symmetry is to increase the Q-factor of the flat band by constructing\ntopological charges in momentum space, which has nothing to do with the flat band. As proof, in\nFig. R8a, we provide the average bandwidth of the randomly generated structure (same structure\nused in the Fig. 2f in the revised manuscript) without mirror symmetry. It is shown that the\nbandwidth of the bands also considerably decreases with the increase of disorder strength.\n Full-band quasi-BICs: In the main text, we have discussed the conditions required for\nachieving a high-Q band: a suitable field distribution. The introduction of mirror symmetry ensures\nthat the eigen-field follows even or odd symmetry, which is critical for the existence of topological\ncharges, as analytically demonstrated in revised Section 8 of the Supplementary Note. In Fig. R8b,\nwe further present the evolution of the average Q-factor with respect to the disorder strength with\nred and blue lines corresponding to cases with and without mirror-symmetry in each supercell,\nillustrating the role of mirror symmetry in the formation of quasi-BFICs.\nFigure R8. Numerical illustration of the role of the mirror symmetry in the formation of the quasi-\nBFICs. (a) The variation of the bandwidth with respect to the disorder strength for structure without\nmirror symmetry (A=13𝑎 ). The bandwidth of the band considerable decreases with the increase of the\n0\ndisorder strength. (b) The variation of the average Q-factor with respect to the disorder strength for both\nstructures with and without the mirror symmetry.\nAction taken:\n In page 11 of the revised manuscript, we have added “Furthermore, in Supplementary Note 8,\nwe also consider the flat-band eigen-field without any even or odd symmetry relative to the\nmirror plane of the supercell, where no topological charge is found above the light line,\nshowing the importance of the mirror-symmetry on the emergence of quasi-BFICs.”.\n In the revised supplementary Note, we have added the section 8 to demonstrate the crucial role\nof the mirror symmetry in the formation of the quasi-BFIC.\n(c) The discussion of topological charges appears somewhat disjointed from the main message of\nthe manuscript. The evidence for topological charges relies on limited simulations of Fourier\ncoefficient distributions. The manuscript needs to demonstrate more convincingly that these\ntopological charges are indeed topological in nature (i.e., robust to perturbations) and that they play\na causal role in the suppression of radiation and the quasi-BIC behavior. The presented analysis\nseems descriptive rather than providing deep mechanistic insight.\nReply: We appreciate the reviewer’s comment and acknowledge the importance of convincingly\ndemonstrating the topological nature of the charges and their role in radiation suppression and quasi-\nBIC behavior. To address this concern, we have performed additional simulations and analyses to\nfurther validate our findings and provide enhanced mechanistic insights.\nFirst, we performed systematic simulations to analyze the evolution of topological charges,\ndemonstrating their robustness to perturbations. Specifically, starting from the structure used in Fig.\n2f of the revised manuscript, we added an additional width Δ to each pillar. Seven topological\ncharges are found in the radiative 𝑘 -space. Due to the mirror symmetry in the x-direction, only\n𝑥\nfour of them evolve independently, as displayed in Fig. R9. By continuously varying Δ, we can track\nthe evolution of the topological charges. With the increase in the additional width Δ, the symmetry-\nprotected charge α remains at the center of k-space, while the charges β and γ continuously move\naway from the center, and charge δ moves toward the center. These results confirm that the\ntopological charges persist under moderate variations in system parameters, thereby reinforcing\ntheir intrinsic topological nature. This robustness is a key hallmark of topological phenomena and\nunderscores the validity of our claims.\nFigure R9. Numerical results of the evolution of topological charges in k-space. As the additional\nwidth Δ increases, the symmetry-protected charge α remains at the center of the 𝑘-space, while charges\nβ and γ continuously move away from the center, and charge δ moves toward it.\nTo demonstrate the radiation suppression effect of the topological charges, we examine the\nfollowing two structures. These structures are nearly identical, differing only by an additional width\nΔ of 2 nm. As shown in Fig. R10a, the additional 2 nm width in Structure B causes charges β and γ\nin Structure A (which carry opposite topological charges) to meet and annihilate. Consequently,\nStructure B exhibits 4 topological charges in the radiative kₓ-space, compared to 8 in Structure A.\nThis observation helps demonstrate the causal role of topological charges in radiation suppression.\nIn Fig. R10b, we compare the Q-factors between these two structures. As shown, the structure with\nmore topological charges exhibits a significantly higher Q-factor. Furthermore, as explained\npreviously regarding symmetry requirements (Fig. R8b), the key difference between symmetric and\nasymmetric structures lies in the existence of topological charges, which results in a lower Q-factor\nfor the asymmetric case.\nFigure R10. Demonstration of the causal role of topological charges in radiation suppression. (a)\nComparison of k-space polarization fields between Structure A and B. The addition of a 2 nm width\ndifference causes charges β and γ in Structure A to converge and ultimately annihilate. (b) Numerical\ncomparison of Q-factors between the structures. The structure possessing more topological charges\ndemonstrates a significantly higher Q-factor.\nAction taken:\n In page 8 of the revised manuscript, we have added “In Supplementary Note 5, we present the\nevolution of topological charges under continuous parameter variation, confirming their\nrobustness”.\n In the revised supplementary Note, we have added the section 5 to demonstrate the robustness\nof topological charges.\n(d) In the introduction, the authors argue that BICs are constrained by“fabrication imperfections,\nsuch as surface roughness and structural disorder.”They later argue that their approach (which\ndeliberately introduces disorder) overcomes these challenges. This is problematic. Surface\nscattering from roughness induced by nanofabrication can’t be avoided and will dampen resonances.\nAlso, even when the authors introduce a specific pattern of disorder, this will be overlaid with an\nadditional disorder pattern introduced by the fabrication inaccuracies. Therefore, it is not accurate\nto say that the authors’concept overcomes these imperfections. These aspects should be clarified\nand discussed in detail in the manuscript.\nReply: We thank the reviewer for their insightful comment. We agree that our original statement\nregarding robustness against fabrication imperfections requires more precise formulation and\nadditional demonstration. Fabrication imperfections can indeed be categorized into two distinct\ntypes: (1) intrinsic limitations such as surface roughness, finite sample size, and material absorption,\nwhich are fundamentally unavoidable as the reviewer correctly notes; and (2) extrinsic scattering\nlosses induced by additional fabrication disorder. We would like to emphasize that our designed\nquasi-BFIC demonstrates superior robustness against the second type of disorder compared to\nconventional quasi-BICs. This advantage originates from the broadened Q-factor peak (manifested\nas uniformly high Q-factor in our system), a phenomenon that has been previously demonstrated in\nmerging BIC configurations [Nature 574, 501–504 (2019)].\nAs numerical verification, Fig. R11 compares the Q-factor attenuation under introduced\ndisorder for two structures: (1) a regular grating supporting a BIC (upper plot in Fig. 2a of the\nrevised manuscript) and (2) our random structure supporting a quasi-BIC (structure used in Fig. 2f\nof the revised manuscript). For both structures, each pillar was assigned an additional width\nmodulation 𝛥∈[−5 𝑛𝑚,5 𝑛𝑚] . We generated and numerically analyzed one hundred\nrandomized configurations for each case, calculating the average Q-factor for relevant Bloch modes.\nThe results demonstrate that under comparable disorder levels, the quasi-BIC maintains its high Q-\nfactor with minimal degradation, while the conventional BIC exhibits significant Q-factor reduction.\nFigure R11. Numerical illustration of the robustness for quasi-BFIC against the structural disorder.\nEach pillar was assigned an additional width modulation 𝛥∈[−5 𝑛𝑚,5 𝑛𝑚]. One hundred structures\nare generated and numerically calculated and the average Q-factor of the Bloch modes with different\nBloch modes are considered.\nAction taken:\n In page 6 of the revised manuscript, we have added “We emphasize that such angle-insensitive\nQ-factors can substantially enhance robustness against scattering losses from fabrication\ndefects, as demonstrated in Supplementary Note 3.”.\n In the revised supplementary Note, we have added the section 3 to demonstrate the robustness\nof Q-factors against the structural disorder.\n(e) In Figure 2d, the bandwidth is plotted. How exactly is this bandwidth defined? It could be useful\nto include a simple sketch of the resonance and the important resonance parameters in the figure.\nReply: We sincerely appreciate the reviewer's insightful comment. To clarify, we define the\nbandwidth as the spectral difference between the maximum and minimum eigenfrequencies of the\nband. For the flat band analysis, we specifically examine two crucial resonance characteristics: (1)\nthe frequency shift (bandwidth) as a function of excitation angle (which maps to wavevector space\nin the Brillouin zone), and (2) the resonance linewidth governed by the Q-factor. To enhance\nconceptual clarity, we have added a schematic diagram (Fig. R12) that visually summarizes these\nresonance parameters and their interrelationships. This illustration will help readers better\ncomprehend the connection between bandwidth, angular-dependent frequency shifts, and Q-factor\nin our system's response.\nFigure R12. Sketch of the resonance profile. (a) The definition of the bandwidth. Line shapes under\ndifferent excitation angles (𝜃 to 𝜃 ) are arranged by the center frequency from minimum (𝑓) to the\n1 𝑛 1\nmaximum (𝑓 ). The bandwidth is defined as the difference between 𝑓 and 𝑓. (b) The definition of the\n𝑛 𝑛 1\nQ-factor of the single resonance.\nAction taken:\n In page 5 of the revised manuscript, we have modified the definition of the bandwidth: “The\nbandwidth is defined as the difference between the maximum and minimum eigen-frequencies\nof the targeted band (Details are shown in Supplementary Note 2).”\n In the revised supplementary material, we have added section 2 to provide the sketch of the\nresonance and the important resonance parameters.\n(f) As a general point, the manuscript heavily relies on plotting band diagrams and color coded\ndispersion diagrams (such as Figure 4c,d). What is really missing are more plots of some examples\nof the actual transmittance spectra. It is very difficult to evaluate parameters like line shape, etc.\nfrom the color plots. I suggest to add transmittance spectra for different incidence angles for both\nsimulation and experiment to the manuscript, or at least to the supporting information.\nReply: We sincerely appreciate the reviewer's valuable suggestion regarding the need for more\ndetailed spectral data to complement our band structure and dispersion analysis. In direct response\nto this insightful feedback, we have performed additional experimental measurements and\nnumerical simulations to provide a comprehensive set of angle-resolved transmittance spectra. The\ncorresponding experimental results are presented in Fig. R13a, while the numerical simulation\nresults are shown in Fig. R13b for comparison. We note that the scanning steps for the incident\nwavelength and wavevector are matched to their experimental resolution limits. To isolate the resonant\nresponse from system artifacts, we normalize both experimental and numerical results by dividing\nthem by a background transmission spectrum obtained through an unetched, uniform silicon layer.\nWe find that the measured dispersions of Bloch bands agree well with the simulation results, confirming\nthe successful fabrication of the sample with mirror-symmetric disorder within each supercell. The\nresonance peak positions are indicated by yellow stars in the figures. The widened resonance peaks\nindicate additional losses originating from fabrication imperfections. Importantly, the proposed quasi-\nBFIC are not found in the transmittance spectra because of the low resolution of the CCD.\nFigure R13. The line transmittance spectra with fixed angles. (a) Experimentally measured\ntransmittance spectra. The smoothing results are presented with shaded regions indicating the fluctuation\nrange. (b) Numerically simulated transmittance spectra. Both panels show angle-dependent responses for\nincident angles of 5°, 10°, 15°, 20°, 25°, and 30°, presented as color maps to enable direct comparison.\nResonance peak positions are marked by yellow stars.\nAction taken:\n In page 11 of the revised manuscript, we have added: “In Supplementary Note 10, the line\nspectra with specific incident angles are presented.”\n In the revised supplementary material, we have added section 10 to provide the spectra with\nspecific incident angles.\n(g) Can the authors justify the choice of the blue area in Figure 2? The exact choice of the“cut-off\nvalues”for bandwidth and Q factor will heavily influence the statistical evaluation in Figure 2f.\nReply: We sincerely appreciate the reviewer's insightful comment concerning the selection criteria\nfor the blue region in Figure 2 and its implications for our statistical analysis. We acknowledge that\nthis region's delineation is indeed non-unique, being inherently dependent on the target bandwidth\nand Q-factor requirements for our designed structures. To comprehensively address this point, we\nhave conducted additional analyses evaluating different cutoff thresholds for both bandwidth and\nQ-factor, as presented in Fig. R14. These extended results confirm that while the precise optimal\ndisorder strength shows some variation depending on the specific selection criteria, the core physical\ntrend remains robust: the system consistently exhibits an optimal disorder strength that maximizes\nthe probability of obtaining the desired photonic properties.\nFigure R14. The probability distribution varies with disorder strength for different cutoff criteria.\n(a) Q-factor>5E4 and bandwidth<0.5 THz. (b) Q-factor>8E4 THz and bandwidth<0.4. (c) Q-factor>1E5\nand bandwidth<0.3 THz. While the precise optimal disorder strength depends on the specific selection\ncriteria, our results consistently reveal an optimal disorder strength that maximizes the probability of\nobtaining structures with the desired photonic properties.\n(h) How many supercells are needed in the experimental realization? The size of the photonic crystal\nseems extremely large at 500 µm x 670 µm. What is the lower limit of supercells required to form\nthe flat band mode? How do Q factor and bandwidth scale as a function of supercell number? The\nauthors should analyze this point, ideally with additional experiments or numerical simulations.\nReply: We sincerely appreciate the reviewer's insightful comment. We provide numerical results on\nthe evolution of Q-factor and bandwidth as a function of the supercell number, as shown in Fig.\nR15a. It is shown that the Q-factor converge to ~180,000 when the supercell number reaches N=50.\nThe corresponding bandwidth shows a weak dependence on the supercell number. This comes from\nthe strong localization in each supercell, which results in weak coupling between adjacent supercells.\nThis picture suggests that the flat-band mode can be effectively understood as a superposition of\nmodes from individual supercells. Consequently, the eigen-field distribution of the periodic\nstructure is nearly identical to that of a single supercell structure (Fig. R15b). Due to the nearly\nidentical field patterns, the two resonances in Fig. R15b exhibit closely matched eigenfrequencies\nof 199.96 THz and 199.94 THz, respectively. Thus, the flat band mode in finite structure will always\nalign with the eigenfrequency of the single supercell structure, leading to the stable bandwidth.\nFigure R15. (a) The variation of the Q-factor and bandwidth with increasing unit supercell number. (b)\nField profile comparison between the ideal infinite periodic structure and an isolated single supercell.\nAction taken:\n In page 11 of the revised manuscript, we have added: “As investigated in Supplementary Note\n9 regarding the influence of sample size on quasi-BFIC characteristics, our sample size is\nsufficient to exhibit the high-Q properties associated with quasi-BFICs.”.\n In the revised supplementary material, we have added section 9 to provide the spectra with\nspecific incident angles.\n(i) The authors mention that “the fabricated sample is covered with PMMA, which matches the\nrefractive index of the silica substrate”. Is an optically symmetric environment always needed for\nthe proposed flat band design? Encapsulation with PMMA prevents many practical applications of\nthe structure, including the sensing experiments that are proposed later.\nReply: We sincerely appreciate the reviewer's insightful comment concerning the requirement for\noptical symmetry in our quasi-BIC design. The up-down mirror symmetry indeed plays a crucial\nrole in aligning the upward and downward radiation patterns, thereby significantly simplifying the\nradiation engineering process. In direct response to the reviewer's suggestion, we have performed\nadditional calculations for an air-encapsulated structure (n = 1), with the corresponding band\nstructure presented in Fig. R16a. In Fig. R16b, we compare the Q-factor between both structures,\nwhich reveals that replacing the PMMA encapsulation with air reduces the average flat-band Q-\nfactor from 2×10⁵ to 6×10⁴, while the bandwidth remains consistently narrow.\nTo investigate the decrease of the Q-factor, the k-space radiation features were also calculated,\nas shown in Figs. R16c and 16d, respectively showing the radiation distribution of the PMMA-\nencapsulated structure and air-encapsulated structure. It is shown that, compared to the PMMA-\nencapsulated structure, the decrease in effective refractive index results in increased eigenfrequency\nand widened radiative 𝑘 -space, finally leading to decreased Q-factor. Moreover, breaking the up-\n𝑥\ndown mirror symmetry eliminates the topological charges, thereby releasing the radiation from their\nconstraint. To sum up, the structure without symmetry matching will result in the decrease of the Q-\nfactor, although the random generation structure can still suppress the radiation to a certain extent,\ndepending on the specific requirements.\nFigure R16. Numerical comparison of the radiation feature between the PMMA-encapsulated\nstructure and air-encapsulated structure. (a) The numerically calculated band structure of the air-\nencapsulated structure. (b) Numerical comparison of the Q-factor between the PMMA-encapsulated and\nair-encapsulated structures. (c) Radiation distribution of the PMMA-encapsulated structure in k-space\nwith enlarged plot in the bottom. (d) Radiation distribution of the air-encapsulated structure in k-space\nwith enlarged plot in the bottom. Radiation to the air and substrate sides is shown in blue and black,\nrespectively.\n(j) The high-Q flat bands are not observed in experiment and only the low-Q flat band appears.\nBecause the high-Q bands are designed to have Q > 100.000, any slight surface roughness will\ndampen them. It is very problematic to try to explain this behavior with the spectral resolution of\nthe experimental setup as the authors state: “Specifically, the linewidths of the corresponding\ntransmission spectrum (~0.005 nm) fall well below the wavelength resolution of the CCD (0.35 nm)\nused in the experiments”. Because of the experimental losses, these high-Q modes were never\npresent in the system, no matter the spectral resolution. Therefore, the following statement is\nunfortunately also completely wrong: “This complete disappearance of the transmission signal\nprovides direct evidence of the existence of quasi-BFICs with remarkably high Q factors”. The\nabsence of the resonances in the spectrum DOES NOT prove their existence. All of these arguments\nhave to be clarified and revised in the manuscript.\nReply: We acknowledge the reviewer’s concern and agree that the original explanation regarding\nspectral resolution was not sufficient to establish the existence of high-Q quasi-BFICs. To address\nthe reviewer's concern, we have designed and fabricated new samples and performed additional\nmeasurements using the momentum-space imaging spectroscopy with a tunable laser around 1550\nnm, which provides much higher wavelength resolution, allows us to directly observe the high-Q\nflat bands [Nature 574, 501-505 (2019)]. These new experiments directly measure the Q-factor of\nthe high-Q flat band, yielding Q-factors around 8000 over a wide-angle range (towards to 47.4°).\nThis empirical evidence further substantiates our conclusion.\nThe experimental results are presented in Fig. R17. Through precise control of the incident\npoint (marked by the large bright spot in the upper right corner), two distinct sets of iso-frequency\ncontours are observed. As shown in Fig. R17a-c, these correspond to incident wavelengths of 1520\nnm, 1515 nm, and 1510 nm, respectively. The flat-band iso-frequency contours appear as parallel\nlines (due to x-direction flatness) that gradually separate with decreasing wavelength. The second\nset of contours corresponds to the dispersive TM-like mode. For reference, numerical simulations\nof the iso-frequency contours are displayed in Fig. R17d-f, where gray and red lines represent the\neigenmodes of the high-Q flat bands and dispersive bands, respectively. Excellent agreement\nbetween experimental results and theoretical predictions confirms the observation of flat bands.\nThe observation of flat bands enables direct extraction of the resonance Q-factor through\nscattering spectrum linewidth measurements. We selected four resonances centered at 1515 nm\n(marked by colored symbols in Fig. R17b), corresponding to incident angles ranging from 18.3° to\n47.4°. As demonstrated in Fig. R17g-j, scanning the incident wavelength from 1513 nm to 1517 nm\nin 0.05 nm steps reveals narrow intensity peaks exhibiting symmetric Lorentzian line shapes.\nLorentzian fitting of these scattering spectra yields the Q-factor of the resonances. Remarkably, the\nQ-factor remains stable at approximately 8000 across the entire angular range. This contrasts sharply\nwith traditional BIC, where the Q-factor drops rapidly to ~1×10³ with just 2° deviation from normal\nincidence [Nature 499, 188-191 (2013)]. Our proposed quasi-BIC structure thus demonstrates\nexceptional angular insensitivity.\nFigure R17. Direct observation of quasi-BIC flat bands. (a)-(c) Experimental iso-frequency contours\nat 1520 nm, 1515 nm, and 1510 nm, respectively. The high-Q flat bands appear as diverging parallel lines\nwith decreasing wavelength. (d)-(f) Corresponding numerical simulations at 1485 nm, 1480 nm, and\n1475 nm, respectively, with gray and red lines representing the eigenmodes of high-Q flat bands\nand dispersive bands. (g)-(j) Experimental intensity spectra (dots) with Lorentzian fits (lines) for four\nresonances at the marked k-points in (b).\nAction taken:\n In the revised manuscript, we have added the latest experimental results, including the iso-\nfrequency contour and directly measured Q-factors of the quasi-BFICs, which is primarily\npresented in Figure 5 of the revised manuscript.\n To enable direct comparison between theory and experiment, all theoretical results in the\nrevised manuscript were updated according to the parameters of the latest sample design.\n In the method section of the revised manuscript, we have added the experimental details\nregarding the measurement of the iso-frequency contour and the scattering spectra.\n The optical measurement setup for the measurement of the iso-frequency contour and the\nscattering spectra is included in Figure 5 of the revised manuscript.\n(k) The sensing comparison part at the beginning of the discussion section should be removed in its\nentirety. Claiming better sensing performance based only on simulations is highly problematic,\nsince the final sensing performance will depend on a multitude of different factors. Also, looking at Figure\nS6, it appears to me that the authors have not chosen a fair “traditional” BIC geometry (blue line)\nfor comparison. The traditional design shows worse sensing performance even at normal incidence\n(theta = 0), which is very unlikely. The traditional BIC geometry can be engineered to have\nextremely high Q factors as well, so sensing performance should at least be equal. Finally, extreme\nangle independence of the sensor signal is not generally a large issue in optical sensors. Many\nexisting BIC designs are angle-stable up to incidence angles of around 10 degrees, which coincides\nwith the NA of low magnification objectives commonly used to read out metasurface-based sensors\nin many practical optical sensing devices.\nReply: We appreciate the reviewer’s insightful comments and acknowledge their concerns\nregarding the comparison of sensing performance. The superiority of the Γ point in random\nstructures stems from its higher sensitivity, which reflects a more pronounced resonance peak shift\nper unit refractive index change. We fully agree that asserting superior sensing performance based\nsolely on simulations is problematic, as the actual sensing capability depends on numerous\nadditional factors. In response, we have removed the sensing comparison discussion from the\nmanuscript.\nReference.\n1. Wiersma, D. S. The physics and applications of random lasers. Nat. Phys. 4, 359–367 (2008).\n2. Segev, M., Silberberg, Y. & Christodoulides, D.N. Anderson localization of light. Nat. Photon. 446,\n52-55 (2007).\n3. Stützer, S., Plotnik, Y., Lumer, Y. et al. Photonic topological Anderson insulators. Nature 560, 461–\n465 (2018)\n4. Yoon, S., Kim, M., Jang, M. et al. Deep optical imaging within complex scattering media. Nat. Rev.\nPhys. 2, 141-158 (2020)\n5. Jin, J. et al. Topologically enabled ultrahigh-Q guided resonances robust to out-of-plane scattering.\nNature 574, 501–504 (2019).\n6. Hsu, C. W. et al. Observation of trapped light within the radiation continuum. Nature 499, 188–191\n(2013)."}, {"round": 2, "reviewer_comments": {"reviewer_1": "The authors have failed in providing their replies to address reviewers' comments. This reviewer is not convinced by the\nnovelty and the correctness of this work, eg. the misleading define of mirror-symmetric disorder, the Unreliability of the\nSimulation results, the orders of magnitude difference between their simulation and experimental results. They announced\ntheir simulation is accurate enough when mesh size is 50nm which is only 1/8 of effective wavelength thus may not sufficient\nfor high Q cavity simulation and problomatic, not to mention the FEM method has its own simulation\nlimitation/error/deviations and better to be benchmarked by a different EM simulation method. As a result, this reviwer can\nnot suggest the acceptance of this work.", "reviewer_2": "The authors have undertaken a tremendous amount of additional numerical and experimental work to address the\ncomments of the reviewers. Taken together with the extensive additional explanations and justifications, I believe the\nmanuscript has been greatly improved and I now support its publication without further changes."}, "author_response": "Response Letter to Reviewers\nWe are grateful for the constructive comments on this manuscript (NCOMMS-24-84707A-Z) from\ntwo reviewers. In the text below, reviewer comments are quoted in blue and followed by our detailed\nresponse. We have also revised the manuscript and the Supplemental Note based on the reviewer’s\ncomments, and these updates are highlighted in red in those files. In the text below, these updates\nare also highlighted in Italics.\nResponse to comments of the reviewer #1\nThe authors have failed in providing their replies to address reviewers' comments. This reviewer is\nnot convinced by the novelty and the correctness of this work, eg. the misleading define of mirror-\nsymmetric disorder, the Unreliability of the Simulation results, the orders of magnitude difference\nbetween their simulation and experimental results. They announced their simulation is accurate\nenough when mesh size is 50nm which is only 1/8 of effective wavelength thus may not sufficient\nfor high Q cavity simulation and problomatic, not to mention the FEM method has its own\nsimulation limitation/error/deviations and better to be benchmarked by a different EM simulation\nmethod. As a result, this reviwer can not suggest the acceptance of this work.\nReply: We would like to thank the reviewer for the comments. We demonstrate that introducing\nmirror-symmetric disorder into the supercell enables the photonic crystal slab to support quasi-BFIC\nmodes—a finding thoroughly verified by theoretical, numerical, and experimental results. To ensure\nthe reliability of our numerical simulations, a convergence analysis confirmed that a 50-nm mesh\nsize is sufficiently accurate, as further refinement yields no change in the results.\nResponse to comments of the reviewer #2\nThe authors have undertaken a tremendous amount of additional numerical and experimental work\nto address the comments of the reviewers. Taken together with the extensive additional explanations\nand justifications, I believe the manuscript has been greatly improved and I now support its\npublication without further changes.\nReply: We would like to thank the reviewer for the recommendation of publication, as well as the\nconstructive feedback throughout the review process."}]

Decision letter