A Magnon-photon interface based on Van der Waals Magnetic semiconductor

Domain: physics
Published: 2026-01-22
Editorial outcome: major_revision

Abstract
Coherent control of light with electron spins represents a fundamental goal across multiple research fields, such as opto-spintronics, chiroptics, quantum information. In a magnetic-ordered material, coherent magnon gives rise to temporally and spatially modulated spin structure that may stimulate intriguing spin-photon interactions as well as opto-spintronic applications. However, the development of such magnon-photon interface is hampered due to the lack of magnetic photonic structure that embraces strong light-matter interaction. Here, we present such magnon-photon interface in a CrSBr metasurface by showing the formation of a magnetic exciton polariton bound states in the continuum, whose energy and the radiative properties can be tuned with an external magnetic field. We further show that the exciton polaritons are modulated by the coherent magnon excitation, exhibiting a nontrivial magnon mode- and k-dependent behavior which suggests the formation of hybrid magnon-exciton polariton states. Our results shed light on the interplay of spin, magnon and photon in the magnetic metasurface and provide a new platform for exploring spin functional photonic and quantum devices.

Introduction
The electron spin carries non-volatile quantum information, which has been widely explored for information storage and processing1–5. Interfacing the spin states with photons allows communication and detection of spin quantum information over long distance6–9. Such spin-photon interface is therefore critical to connecting spatially separated quantum nodes and has been recognized as a vital part of the vision for scalable spin quantum information.

Significant progress has been made for non-magnetic spin-photon interface. This includes development of high-fidelity spin-photon entangled states, photons switches, routers and efficient chiral emitters controlled by electron spin in III-V semiconductor quantum dots7,9,10, transition metal dichalcogenide11–13 as well as scalable on-chip integration of optically active defect spins in diamond and SiC8,14,15. The non-magnetic semiconductor nanostructures and color centers are benefited from the optically active materials and long spin coherence time16,17. Whereas, magnetic materials, although being the backbone of contemporary spintronic research and memory industry, are less explored due to optically inactive band-edge transitions18,19, degraded optical properties20,21, as well as limited spin relaxation and coherence time22,23.

Recently, the discovery of magnetic semiconductors, e.g., CrSBr, with strong light-matter interaction and intriguing magneto-electronic coupling effect has revived the hope for developing a spin-photon interface that allows harvesting the versatility of spin manipulation and magnetic phase engineering in a magnetic material24–30. Additionally, magnon arises as a unique wave-like spin excitation in magnetically ordered materials31. In CrSBr, magnon is found strongly coupled to the excitonic states, reshaping its optical properties in the ultrafast timescale32,33. Despite the development, two major obstacles persist. First of all, studies of magnon-exciton interactions in 2D magnetic semiconductors have only been conducted in structureless samples which mostly involving interaction with free-space photons. Carefully designed photonic structures, e.g., photonic cavity or metasurfaces with high quality factor or strong magneto-optical response, will likely not only enhance the spin-photon interaction but also provide extra tunability34–36. In particular, photonic structures hosting bound states in the continuum (BICs) have attracted great attention due to their potential to achieve infinite-high cavity Q factors and sensitive to external perturbations, giving rise to intriguing light-matter interaction that facilitate the low threshold lasing, ultrafast optical switches, etc37–39. Coupling of the non-local BICs mode with the local excitation in the magnetic materials, e.g., excitons, may also give access to significant tunability through interplay of the local and non-local transition of the system40,41. Secondly, the fundamental physics underlying the coupling between collective spin waves and photons is not well understood. Although the propagation of magnons in CrSBr has been confirmed42,43, its interaction with the excitonic states is probed in a momentum-integrated manner. A momentum-resolved perspectives are therefore highly appreciated to shed light on the exotic nature of the magnon-photon coupling.

In this work, we demonstrate a magnon-photon interface based on a 1D metasurface of the magnetic semiconductor CrSBr. The CrSBr metasurface hosts BICs that are strongly coupled with excitons forming exciton polaritons (EPs). Owing to the magneto-electronic coupling effect of CrSBr and photonic singularity of the BICs mode, both the energy and the reflectivity of the half-light, half-matter state can be efficiently tuned with an external magnetic field. Furthermore, by performing the transient angle-resolved reflectance spectroscopy on the CrSBr metasurface, we find that the coherent magnon introduces a non-trivial k-dependent oscillation of EP energies that behaves differently for acoustic and optical magnons. These results highlight the wave nature of the magnon-photon coupling in the CrSBr metasurface, which establishes an innovative approach for controlling the photonic or polaritonic devices with electron spins.

Results
Magnon-photon interface combining magneto-optical effect and photonic BICs

Figure 1a illustrates the proposed magnon-photon interface made from a CrSBr grating, in which the reflected light of frequency ωph and momentum k is coupled to the transverse electric (TE) mode of the grating and is modulated by a launched coherent magnon. The bulk CrSBr is an A-type antiferromagnetic semiconductor with Néel temperature TN = 132 K. The orientation of the antiferromagnetically coupled macrospins S1 and S2 in the adjacent layers has strong impact on the excitonic structure as well as optical response of the material. By applying an external magnetic field of B = 3 T along the c-axis, S1 and S2 switch from anti-parallel (AP) to parallel (P) spin configuration as shown in Fig. 1b. As a result, the b-axis dielectric function εb = ε1 + iε2 of CrSBr is modified drastically as shown in Fig. 1c, d. Here, εb is extracted from the reflectance spectra with details given in Supplementary Note 1 and Fig. S1. The multi-resonance reflectance features in Fig. S1 is due to the multi-order Fabry-Perot resonances arise from the planar cavity formed between the top and bottom surface of the CrSBr flake and is therefore not reflected by the fitted εb in Fig. 1c, d. The giant magneto-optical response is caused by spin mediated interlayer coupling, leading to a prominent redshift of the 1s exciton state from X(0) = 1.3643 eV to X(B) = 1.3485 eV as marked in Fig. 1d. Such magneto-electronic effect stems from the exchange interaction and is expected to have a stronger impact than typical magneto-optical effects, e.g., Faraday, Kerr rotation, etc., which are received from the spin orbital interaction44. In Fig. 1e, we show the calculated TE mode dispersion of the proposed CrSBr grating. Owing to the strong light-matter interaction, the shown eigenmodes are essentially EPs hybridized between the 1s exciton and guided resonances (GRs). The Γ point of the TE1 mode as marked in Fig. 1e is a BICs with topological charge C = +1, evidenced by divergent Q factor and the far-field polarization map as shown in Fig. 1f. The corresponding mode distribution can be found in Supplementary Fig. S2. The BICs mode is often considered as photonic singularity which is inherited with strong tunability. However, to our knowledge, its role in a magnetic metasurface as the one shown here has not yet been explored.

Strong coupling of exciton and BICs mode in a CrSBr 1D grating

The optical and atomic force microscopy (AFM) image of the exfoliated CrSBr flake is shown in Fig. 2a, b. The polarization-resolved photoluminescence (PL) spectra in Fig. 2c reveal emissions from a series of self-hybridized EPs that are linearly polarized along the b-axis. The coupling strength is estimated to be ~ 167 meV by fitting the emission peak position with a coupled oscillator model (COM) as shown in Fig. 2d (see Supplementary Note 2 and Fig. S3).

Next, the CrSBr metasurface is fabricated by patterning the CrSBr flakes with e-beam lithography followed by dry etching. The grating grooves are aligned along the b-axis of the crystal to maximize the coupling strength. In Fig. 2e, the scanning electron microscope (SEM) image shows that the bottom part of the CrSBr flake is still connected, allowing the continuous propagation of spin wave throughout the whole structure. Figure 2f shows the angle-resolved reflectance (ARR) spectrum taken from the CrSBr metasurface at 10 K, which is nicely matched to the simulated ARR spectrum in Fig. 2g. In Supplementary Fig. S4, we show the ARR spectrum from a CrSBr metasurface with the same structure parameters while the grating grooves are along a-axis. In this case, the photonic modes are decoupled from the exciton, giving rise to markedly different dispersion as in Fig. S4. Furthermore, we have carried out analysis based on COM and the simulation results are shown in Fig. S5. The COM suggests that a coupling strength of ~ 186 meV is needed to reproduce the observed polariton dispersion and the strong coupling is also reached in the CrSBr grating sample. We note that the slightly increase of the fitting values for the coupling strength as well as exciton oscillator strength (see table S1) for the CrSBr grating as compared to the bulk sample is likely artificially caused by the Lorentzian dielectric model25,26 which neglects the energy dependence of the background dielectric constant εb0. A more accurate determination of the coupling strength requires a more realistic modeling of the dielectric function over large spectral range. Nevertheless, the obtained comparable exciton oscillator strength for the etched and unetched sample suggest that the optical quality of the CrSBr grating is not compromised by etching. The EPs at around 1.27 eV in Fig. 2f features a bright and dark state (BS and DS) at Γ point (θ = 0) separated by a gap of ~ 8 meV. The same spectral feature is observed in the angle-resolved PL (ARPL) spectrum in Fig. 3a. The EPs emission intensity diminishes at the vicinity of the DS, accompanied by the increase of the Q factor as shown in Fig. 3b. The DS and BS are identified as the TE1 and TE2 EP. The two modes are slightly redshifted compared from the eigenmode calculation in Fig. 1e due to the presence of the SiO/Si substrate. The absence of the PL and reflectance contrast of the DS is due to the symmetry-forbidden emission of the BICs at θ = 0°. In the following content, we will refer to the exciton polariton formed by the strong coupling between the BICs mode and CrSBr excitons as EP BICs for short.

Tuning the exciton polariton BICs with magnetic field

The formation of EP BICs facilitates the control of its optical response by manipulating the spin configuration. Applying an out-of-plane magnetic field tilts the macrospins S1 and S2 as illustrated in Fig. 3c. The change of spin configuration modulates the energy of the EPs. As shown in Fig. 3d, the BS redshifts with magnetic field and reaches saturation at B = 2.1 T. The total energy change of the BS ΔEBS = 13.1 meV is huge considering the field induced total energy shift of 1s exciton is about 4Eex = 15.8 meV. The large energy shift is consistent with the substantial exciton fraction of the EP which is estimated |X|2 = (|X|2 + |C|2) = 0.39 using the coupled oscillator model. Here, X and C are the exciton and photon part of the Hopfield coefficients for the lower polariton branch45.

Interestingly, The EP BICs which is completely dark in the ARR measurement in the absence of magnetic field, start to red shifts and develops a reflectance feature with applying magnetic field. The brightening of EP BICs is further illustrated in Fig. 3e, where two resonant features are identified at B = 3 T and fitted with Fano lineshape. The dashed and solid arrows mark the fitted resonant energies of the bright and dark state. Table S2 in the supplementary information summarizes the fitting parameters for the two resonances. As guided by the fitting curves, the dark state locates at about 8 meV above the bright state and exhibits a highly asymmetric lineshape. As a comparison, only the resonant feature of the BS is observed and fitted for spectrum measured at B = 0 T in Fig. 3e.

An out-of-plane magnetic field contributes to a sizable off-diagonal dielectric tensor element εab = ε0ab + iε0ab (see Supplementary Note 3 and Fig. S6)46. εab introduces coupling between GRs and broad FP resonances, that can potentially affects the reflectance spectrum in field. Ideally the Γ-point BICs is strictly radiatively forbidden. The robustness of the BICs is protected by the C2 symmetry which holds even in the presence of magnetic field. This is shown with simulated reflectance spectrum in Fig. 3f in which the reflectance features of BICs is absent at different ε0ab for a perfect grating with infinite periods. However, the translation symmetry is broken due to limited sample size or white light spot size. As a result, the reflectance at θ = 0° mixes in the contributions from near-Γ quasi-BICs at finite k. Considering system dimension of ~ 20 μm limited by beam size, it leads to a k-space blurriness of Δθ ~ 1.4° as estimated with the uncertainty relation Δk · Δx ≥ 1/2. Such near-Γ contribution is simulated in Fig. 3g, h for θ = 0.3° and 0.6°. It is evident that the reflectance features of the quasi-BICs state (DS) gradually build up with the increase of ε0ab within the uncertainty limit of Δθ, which is consistent with the experimental observation of the brightened BICs in Fig. 3d and e. In Fig. S7, we further show from simulation that the field-induced reflectance features from the quasi-BICs depends on both real and imaginary part of εab and generally leads to asymmetric Fano lineshape as observed in the experiment. The observed intriguing magneto-optical effect emerges exclusively at the magnetic EP-BICs, which is due to the combined effect of the magnetic field induced εab and the fact that the quasi-BICs is sensitive to such external perturbation. In Supplementary Fig. S8, as a comparison, we show that the reflectance amplitude is barely changed for other EPs that are far apart from the BICs point.

k-dependent magnon-exciton polariton interaction

Next, we turn to the EPs that are closer to the exciton resonance which maximize the magnon-photon interaction. A transient ARR (tr-ARR) spectroscopy is employed to study the interaction between the EPs and coherent magnons in the CrSBr metasurface. The tr-ARR monitors the time evolution of the pump induced ARR change ΔR/R(θ, Eprobe) which are function of both the probe light energy Eprobe and θ. Our measurement configuration is shown in Fig. 4a. A constant magnetic field of 0.48 T is applied to cant the interlayer spins which facilitate exciton-magnon coupling25. Figure 4b shows a typical tr-ARR spectrum measured at 14 K and t = 47 ps. Here, the ΔR/R is mainly caused by pump induced redshift of the EPs, which is opposite to the typical blueshift observed for EPs nonlinearity in non-magnetic materials47,48. The effect is pronounced for EPs derived from the FP mode at around 1.35 eV and multiple GRs in between 1.3-1.34 eV. We attribute the ΔR/R signal to the excitation of incoherent magnon, which contributes to the decreases of EP energies and decays slowly with time as shown in Supplementary Fig. S9. Apart from the incoherent component, the coherent magnon excitation is also observed as the temporal oscillation of the ΔR/R. In the upper panel of Fig. 4c, the coherent oscillation of the tr-ARR (CoTR) is made pronounced by taking the time derivative of ΔR/R and is plotted as function of Eprobe and t for θ = 0°. The oscillations take opposite phase for Eprobe above and below the EP resonances. It suggests that the EP energies are periodically modulated through interaction with coherent magnon, which is consistent with previous report in pristine CrSBr flakes32. In the lower panel of Fig. 4c, CoTR at probe energy (Eprobe = 1.3536 eV) slightly above the 1.35-eV EP is plotted against time, exhibiting clear oscillations with a beating amplitude.

To gain in-depth understanding of the coherent magnon-EP dynamics, we performed fast Fourier transformation (FFT) of the CoTR signal for each Eprobe and the results for different θ are shown in Fig. 4e. In the FFT spectra, two dominant oscillation frequencies at 30 GHz and 22 GHz are identified and attributed to the acoustic and optical magnon modes (marked as Ac. and Op.). The corresponding mode character featured by the in-phase and out-of-phase precession of the macrospin S1 and S2 of the neighboring antiferromagnetically coupled layers is illustrated in Fig. 4d. In Fig. 4f, the FFT magnitude at the two magnon mode frequencies is plotted against θ. The FFT magnitude at the frequency of optical magnon decreases at higher |θ|, whereas the acoustic magnon contribution remains at a comparable level for all θ in our measurements. The k-dependent coupling between EP and optical magnon modes cannot be otherwise explained by any k-space dependence of the EP, which is shown to be similar for both magnon modes. It has been suggested that the transient strain field generated by the pump light plays important roles for launching magnon in CrSBr and the k-space distribution of the coherent magnon should be limited by the Fourier transformation of the real-space transient strain field42. In our measurement, the pump laser was defocused to a large laser spot size of ~ 10 μm. It therefore generates a broad real-space heat/strain profile. The dashed line in Fig. 4f estimate the contribution of heat/strain by showing the Fourier transformation of the real-space laser spot profile, which is too narrow to account for the observation of the broad k-space profile for the optical magnon. It suggests that the EP oscillations ΔE(t) ∝ Δm(t) ∝ Δm(t) where ΔE(t) is the relative angle between S1(t) and S2(t) for the exciton energy difference between ferromagnetic and antiferromagnetic phase. The magnon-photon interactions observed here are beyond the conventional heat/strain-mediated mechanism which is often responsible for magnon-photon interaction in magnetic materials or devices. Furthermore, the optical magnon has a similar profile as the angular distribution of the pump density (Fig. 4f). It is worth to know that such angular distribution of the pump density deviates from the Fourier transformed real-space profile of the pump beam due to the defocused pump laser beam setting. This is a strong indication for a k-conserved coupling between EP and the optical magnons such that the momentum is transferred to the magnon mode upon optical excitation. However, such momentum conservation law does not hold for acoustic magnons. We note that the similar k-conserved behavior of the optical magnon is also observed for measurement in pristine CrSBr as shown in Fig. 4g, h. However, a noteworthy difference in the unpatterned CrSBr is that the acoustic mode is absent in the FFT spectra at all angles.

To shed light on the different k-space behavior of the optical and acoustic magnons, we solved the Landau-Lifshitz equation describing the time evolution of sublattice magnetization mi=1,2(t) of bulk CrSBr as illustrated in Fig. 5a,

dmi(t)/dt = -γμ0 mi(t) × Hi,eff (1)

Here, γ and μ0 are gyromagnetic ratio and vacuum permeability. Hi,eff is the effective magnetic field of mi=1,2(t) and includes contributions from Zeeman interaction, interlayer exchange coupling and the magnetic anisotropy (detailed derivation is given in Supplementary Note 4). Assuming the reported interaction parameters, Fig. 5b shows the calculated mode frequencies for acoustic and optical magnons as function of the c-axis magnetic field32. It is established that the modulation of the CrSBr exciton and exciton polariton energies ΔE is proportional to the time-varying angle between S1(t) and S2(t). Figure 5c, d calculate the temporal modulation of ΔE at zero and finite magnetic field assuming either acoustic or optical magnon is excited. At Hext = 0, neither acoustic nor optical magnon is coupled to the exciton or EPs, as evidenced by the absence of ΔE at the magnon frequency ω in Fig. 5c. However, weak higher order frequency component at 2ω is identified for both magnons. A finite magnetic field, e.g., μ0Hext = 0.48 T, initiates the coupling between the exciton and the optical mode as shown by the emerging oscillation of ΔE at ω in Fig. 5d. Whereas, the acoustic mode remains decoupled at frequency ω. Such distinct selection rules for acoustic and optical magnons in bulk CrSBr nicely explain the absence of the acoustic magnon in the FFT spectra for the unetched CrSBr sample (Fig. 4g, h). The observation of disallowed acoustic magnon in the FFT spectra for CrSBr grating (Fig. 4e, f) then must arise from the etched edges and sidewalls which have low symmetry. At such localized site, the acoustic magnon-exciton polariton interaction therefore does not conserve k and naturally leads to an insensitive dependence on angle. On the other hand, the selection-rules-allowed optical mode requires match of momentum owing to the preserved translation symmetry and therefore leads to an angle-sensitive behavior for both the pristine CrSBr flake and the CrSBr grating. The mode- and k-dependent magnon-EP interaction may indicate the formation of strongly coupled EP-magnon hybrid such that a k-selected excitation of the EPs directly leads to the k-dependent pumping of the coherent magnons which in turn modulates the EPs energies. Such mechanism is therefore sensitive to the character of the magnons in the way that favors the direct optical magnon-EP coupling through the induced angle change between adjacent interlayer macrospins.

We note that controlling light with electron spins has strong implications in spin quantum information and spintronics, which is generally difficult for conventional meta-surfaces or polaritonic devices based on non-magnetic materials. Towards such goal, we have demonstrated both static and ultrafast control of exciton polariton with electron spin or collective modes of spins in a magnetic photonic structure of CrSBr, by harnessing the synergistic interplay between strong light-matter interactions and magneto-electronic effects. Furthermore, the observation of 2D magnon mode- and k-dependent EP interaction provides a compelling evidence of direct momentum transfer from pump photons to the optical magnons, suggesting the formation of strongly coupled magnon-EP hybrid state. These findings provide valuable information for the spin- and magnon-photon interactions in the 2D limit and open a new gateway for exploring spin functional photonic and quantum devices with 2D magnetic semiconductors.

Methods
Fabrication of CrSBr metasurface
The commercially available CrSBr crystal (from Nanjing MKNANO) was mechanically exfoliated onto a silicon substrate with a 300-nm oxide layer in the glove box, and the thickness was determined using AFM. Electron beam lithography and ICP-RIE were used to partially etch CrSBr, forming a one-dimensional grating. The etched area of the CrSBr flake is 30 μm × 30 μm. The grating structural parameters and roughness of the measured grating sample is characterized by SEM and AFM in Fig. S10.

Angle-resolved reflectance and PL spectroscopy
The optical measurements are conducted with samples loaded into a low-vibration closed-cycle cryostat. For the PL measurement, a 532-nm continue-wave laser was used for excitation. A halogen lamp was used for the reflectance measurement. The PL emission and the reflected light were collected with a long-working-distance objective lens with a numerical aperture of 0.42. The angle-resolved reflectance/PL spectroscopy consists of reflectance/PL spectra for different incidence angle θ, which are performed by projecting the back focal plane of the objective lens to a spectrometer (FHR640) equipped with a Si charge coupled device (CCD) camera.

Transient angle-resolved reflectance spectroscopy
The tr-ARR measurements are performed with a home-built set-up, which is schematically shown in supplementary Fig. S11. A 1-kHz 800-nm pulsed laser is used as pump light. The pulse width is about 150 fs. The probe light is the white light generated by pumping a sapphire plate with the fs laser. The delay time t between the pump light and the white probe light is precisely controlled to yield the time evolution of ΔR/R. The pump energy density is kept low at 5.1 pJ/μm2 to avoid compromising the coherence of the exciton-magnon coupling due to high carrier/exciton density. The pump light intensity is modulated at the frequency of 20 Hz and the k-space image of the pump induced reflectance change is dispersed and recorded with an EMCCD camera (from Raptor Photonics) that is synchronized to the modulation speed. The pump and probe light spot size is about 10 μm.

Numerical simulations
The finite element method (FEM) is employed to simulate the eigenvalues and reflectance spectra of CrSBr photonic structures using a commercially available software (COMSOL Multiphysics). Periodic boundary conditions are applied in the x- and y-directions, while perfectly matched layers (PMLs) are implemented in the z-direction to minimize reflections. The optical anisotropy of the material is modeled using a dielectric tensor,

ε = [εa 0 0; 0 εb 0; 0 0 εc]

where εa = 12, εc = 3.7. To simulate the eigen-energy of polariton bands, the εb is approximated with the Lorentzian mode:

εb(ω) = εb0 + f/(ω2 - iωΓ - E2X/ℏ2) + f*/(ω2 - iωΓ* - E2X*/ℏ2)

with εb0 = 11.1. The contribution of exciton resonance is taken into account with following parameters, EX = 1.364 eV, EX* = 1.379 eV, ℏΓ = 0.7 meV, ℏΓ* = 7.5 meV, ℏf = 1.304 eV, ℏf* = 0.707 eV. The wave equation,

∇ × (∇ × E(ω)) = k20ε(ω)E(ω)

is then solved numerically by matching the frequency of the eigen mode E(ω) with the dielectric function dispersion ε(ω).

To simulate the magneto-optical effect, an off-diagonal gyromagnetic term is introduced into the dielectric tensor as46,

ε = [εa εab 0; -εab εb 0; 0 0 εc]

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "The manuscript presents an intriguing and mostly experimental study on magnetic exciton-polaritons in a CrSBr\nmetasurface. Unlike prior works where CrSBr serves solely as a source of magnetic excitons coupled to external photonic\nstructures or Fabry-Pérot modes in bulk, this work employs CrSBr itself as both the excitonic medium and the photonic\nlattice. This dual role enables in-situ engineering of the photonic modes and offers a compelling route for tunable exciton-\npolariton properties via external magnetic fields and photonic lattice design. The authors demonstrate tunable reflectance\naround the BIC and observe exciton-polariton modulation by magnons using transient angle-resolved reflectance\nspectroscopy. These results are likely to attract broad interest and make meaningful contributions to the study of hybrid\nmagnon-polariton systems in magnetic semiconductors.\nHowever, several points require clarification or further support before the manuscript can be considered for publication:\n1. Figure 3e shows the experimental results of reflectance at θ=0 to illustrate the brightening of dark BIC mode with applied\nmagnetic field. This observation appears inconsistent with the statement that \"the robustness of the BIC is protected by the\nC₂ symmetry, which holds even in the presence of magnetic field.\" This inconsistency should be addressed more clearly.\nMoreover, although the role of the off-diagonal dielectric tensor element εab is invoked to explain the effect, simulations are\nonly shown at a finite angle (θ=2.3). What is the predicted effect of εab at Γ (θ = 0°)? Does it agree with the experimental data\nshown in Fig. 3e?\n2. In Fig. 4f, the authors present two very different k-space distributions for the pump: one as the \"Fourier transform of the\nreal-space pump pulse\" and another as the \"measured normalized k-space distribution of the pump light.\" The notable\ndiscrepancy between these two curves is confusing and should be clearly explained. Additionally, the authors should clarify\nhow this relates to their argument on the impact of heat/strain and direct momentum transfer mechanisms.\n3. The FFT amplitude of the optical and acoustic magnon modes shows different dependence on momentum. Could the\nauthors provide a more detailed physical explanation of why the optical magnon exhibits stronger k-dependence and why\nacoustic magnon does not?\nMinor Comments and Suggestions\n1. The text description of Fig. 3e seems to refer to Fig. 3f. Please check and revise accordingly.\n2. The color bar scale in Fig. 3f uses only \"max\" and \"min\" labels. Providing numerical values would better illustrate the\nabsolute reflectance change attributed to εab.\n3. Consider plotting optical and acoustic magnon magnitudes separately in Supplementary Material to more clearly reveal\ntheir different k-dependences.", "reviewer_2": "In this work, the authors report strong exciton-photon coupling modulated by the magnon-exciton coupling in a CrSBr\ngrating. Formation of polaritons with the grating bright and dark (BIC) modes have been experimentally extracted with large\ncoupling strength of 167 meV, consistent with previous reports in this material. The authors then demonstrated tuning of\nthese polariton modes with an applied external magnetic field in both static and pump-probe measurements. While the\nattempt of fabricating nanophotonic devices that engineers the photonic band structures is applaudable, there are some\nexperimental observations that are potentially intriguing and worth further investigation, the data quality and analysis of the\ncurrent work need to be significantly improved. In particular, I think that the figures of the manuscripts are weakly connected\nand there lacks a clear conclusion of how such devices can bring novel properties to the field and what are the implications.\nFirst of all, in fig. 2 f&g, the experimental and simulation data are quite different in terms of the mode energy, linewidth, and\nbrightness, what is the reason of that? The fabrication process involved etching of the material, what is the quality of the\nstructure, such as roughness, homogeneity, sidewall angle, etc.? How are they deviates from the design and simulation\nparameters? How was the exciton and other material properties affected by the etching? Are these the causes of the\ndeviation between the experiment and simulation?\nSecondly, the phenomenon demonstrated in figure 3 is quite confusing. First, it is very hard to see the “Fano-like” feature the\nauthors mentioned that indicates the lightening up of the dark state, in either Fig. 3 d and e. It is quite confusing what are the\narrows pointing in the figure, as they are not aligned to any peaks or dips. Also, the experimental data is not in accordance with\nthe theory. As the authors mentioned, the dark mode at the Gamma point remained dark upon applying magnetic field,\nas it is protected by the C2 symmetry, and thus the simulation in Fig. 3f corresponds to the reflectance data at 2.3 degree. In\nthis case, why the Gamma point data is shown in Fig. 3 d&e? Why is brightening of the dark state expected in these data?\nFinally, what is the implication of this experiment?\nFigure 4 demonstrates different angle-dependence of the acoustic and optical magnons, which is quite interesting. However,\nthe angle-resolved pump-probe reflectance data is noisy and needs improvements. The explanation about the difference\nbetween the acoustic and optical magnons is quite preliminary and oversimplified. Is there any further theory or calculation\nthat can support the authors’ claim about the difference in momentum conservation in the optical and acoustic magnons?\nAlso, the authors mentioned that the optical magnons exhibit stronger coupling to excitons while the acoustic magnons has\nmuch weaker coupling. However, in Fig. 4d, it seems that at near zero angles the signals originating from two types of\nmagnons have similar intensity, while at large angles the acoustic phonon dominates. Therefore the authors may want to\nrevise to a more precise description.", "reviewer_3": "Overall, this manuscript presents a timely and original contribution to the emerging field of spin–photon interfaces based on\nvan der Waals magnetic semiconductors. By employing a CrSBr metasurface, the authors demonstrate that high-Q\nresonances and exciton polariton bound states in the continuum (BICs) can be utilized to substantially enhance magnon–\nphoton interactions. The combination of steady-state optical spectroscopy and transient angle-resolved reflectance\nmeasurements provides convincing evidence for magnon–exciton–photon hybridization and reveals mode- and momentum-\ndependent dynamics that highlight direct momentum transfer from pump photons to magnons. These findings advance the\nfundamental understanding of the interplay between spins, magnons, and photons in two-dimensional magnetic systems\nand establish a promising platform for tunable, spin-dependent photonic and quantum devices.\nThat said, the current version requires further refinement before it can meet the publication standards of Nature\nCommunications. In particular, aspects of the experimental methodology, the theoretical interpretation, and the\ncontextualization within the broader literature should be clarified and strengthened.\n1. Quoting “The CrSBr metasurface hosts an exciton polariton (EP) bound state in the continuum (BIC). Owing to the\nmagneto-electronic coupling effect of CrSBr and photonic singularity of the BIC mode, both the energy and the reflectivity of\nthe half-light, half-matter state can be efficiently tuned with an external magnetic field.” This statement in the Introduction\nappears imprecise and somewhat misleading. Specifically, based on the one-dimensional CrSBr grating, the authors\ninvestigate polaritons arising from the strong coupling between high-Q quasi-BICs and magnon–excitons in momentum\nspace, rather than a generic BIC-hosted polariton. To improve clarity, it would be important to provide a concise but accurate\nintroduction to the nonlocal BICs physics, with emphasis on how quasi-BIC resonances enhance light–matter interactions. In\nparticular, since exciton polaritons represent a local response while grating resonances are inherently nonlocal, the authors\nshould expand the Introduction to highlight the content of from local to nonlocal high-Q metasurfaces, supported by relevant\nliterature, so as to place this work more convincingly in the broader context of recent advances in photonic and polaritonic\nmetasurfaces.\n2. Continuning my last Question. Also, NOT “bound state in the continuum (BIC)”, but “bound states in the continuum (BICs)”.\n3. In line 86, the authors mention that the relative permittivity along the b-axis is derived from the reflectance spectrum shown\nin Fig. S1. Near the exciton state X, three distinct resonant valleys are visible in the reflectance spectrum; however, the real\n(ε₁) and imaginary (ε₂) parts of the relative permittivity in Fig. 1 do not exhibit multi-order oscillations near the exciton state X.\nPlease explain this inconsistency.\n4. I am wondering the size of the fabricated sample. No information about the grating size.\n5. The authors use a coupled oscillator model to explain the strong coupling between the FP mode and the exciton in a\nuniform 150 nm-thick CrSBr flake (Figs. 2 and S3). However, does the quasi-BIC mode in the one-dimensional CrSBr\nmetasurface also form strong coupling with the exciton? Although the energy of the quasi-BIC mode is far from that of the\nexciton mode—making its coupling features susceptible to interference from other modes closer to the exciton and difficult to\ndirectly observe experimentally—the authors should provide supporting theoretical analysis to substantiate the claim of\nstrong coupling between the BIC and the magnon–exciton in the following contents.\n6. In lines 146–154, the authors discuss changes in the reflectance spectra of the quasi-BIC mode and the guided\nresonances (GRs) under an external magnetic field. They claim that the pronounced amplitude response unique to the\nquasi-BIC reflectance spectrum indicates a stronger magneto-optical effect compared to the GR mode that is far away from\nthe BIC position. However, this comparison is not convincing, as amplitude modulation primarily arises from the high\nsensitivity of radiative losses near the BIC to external perturbations. In contrast, the larger energy shift of the GR mode under\na magnetic field modulation (Fig. S6b) better reflects the magneto-optical interaction. The authors are advised to provide a\nmore convincing explanation.\n7. In line 139, the authors claim that Fig. 3f shows that an external magnetic field can brighten the dark mode. However, the\ncorresponding experimental data are presented in Fig. 3e, not Fig. 3f. Moreover, a reflection peak at the DS position under B\n= 3 T is not clearly identifiable from the fitting curves in Fig. 3e. The authors should provide a more detailed explanation.\n8. In line 179–180, the authors state: “In the FFT spectra, two dominant oscillation frequencies at 22 GHz and 30 GHz are\nidentified and attributed to the acoustic and optical magnon mode (marked as Ac. and Op.).” However, in subsequent\ndescriptions, the acoustic and optical magnons are assigned to 30 GHz and 22 GHz, respectively. Please ensure\nconsistency in the description to avoid ambiguity.\n9. Generally, in magnetic semiconductors, the optical magnon mode has a higher frequency than the acoustic magnon\nmode. However, the assignment in Fig. 4d appears to be reversed. Please explain this discrepancy.\n10. The authors study k-dependent magnon–exciton polariton coherent interactions based on the FP mode in the CrSBr\nmetasurface. However, such coherent interactions also exist in uniform CrSBr thin flakes. It is therefore necessary to discuss\nthe differences between the two platforms—for instance, whether the metasurface enhances the strength of the magnon–\nexciton polariton coherent interaction.\n11. A femtosecond pulsed laser centered at 800 nm is used as the pump to generate a transient strain field. How would\nchanging the central wavelength of the pump laser affect the experimental results?"}, "author_response": "We appreciate the reviewer for his/her critical review of our manuscript and for recognizing\nour effort to contribute to the field of hybrid magnon-polariton with magnetic semiconductors.\nThe reviewer has made several suggestions and comments regarding the presentation of the\nexperimental and theoretical results, which we found excellent. Below, please find our detailed\nresponse to the reviewer’s comments.\n1. Figure 3e shows the experimental results of reflectance at θ=0 to illustrate the\nbrightening of dark BIC mode with applied magnetic field. This observation appears\ninconsistent with the statement that \"the robustness of the BIC is protected by the C₂\nsymmetry, which holds even in the presence of magnetic field.\" This inconsistency should be addressed more clearly. Moreover, although the role of the off-diagonal dielectric tensor element εab is invoked to explain the effect, simulations are only shown at a finite angle (θ=2.3). What is the predicted effect of εab at Γ (θ = 0°)? Does it agree with the experimental data shown in Fig. 3e?\nResponse:\nWe apology for the unclear description. Theoretically, considering perfect grating structure\nwith infinite periods, the BIC at Γ (θ = 0°) is strictly protected by the C symmetry, which is\n2\nalways dark even in the presence of magnetic field. This effect is shown in Fig. R1b\nconsidering a Bloch boundary condition for CrSBr grating structure shown in Fig. R1a.\nHowever, the translation symmetry is weakly broken by e.g. limited white light spot size or\nsample size, leading to a mixture of at-Γ BIC with near-Γ quasi-BIC states at finite k. In our\nmeasurements, the white light spot size is ~20 μm, which leads to a k-space blurriness of\nΔθ~1.4° as estimated with the uncertainty relation Δ𝑘⋅Δ𝑥 ≥ 0.5. Therefore, any quasi-BIC\nstate within |θ|<1.4° will contribute to the reflectance at Γ (θ = 0°). However, this effect is not\nwell captured in simulations considering the Bloch boundary condition in which the k-point\nis ideally defined. To illustrate such contribution, one therefore needs to simulate the effect at\nsmall but finite θ within the k uncertainty limit (e.g. θ = 0.3° and 0.6° as shown in Fig. R1c\nand d). It is evident that the brightening of the quasi-BIC state (DS) immediately build up\nwithin the angular uncertainty Δθ~1.4°. The above analysis and simulations show that the\nbrightening of the dark state can be well captured by the theory once more realistic\nexperimental condition is considered.\nRevision:\nFollowing the above argument, we have now added following sentences on Page 8 of the\nmain text.\n“Ideally the Γ-point BICs is strictly radiatively forbidden. The robustness of the BICs is\nprotected by the 𝐶 symmetry which holds even in the presence of magnetic field. This is\n2\nshown with simulated reflectance spectra in Fig. 3f in which the reflectance features of BICs\nis absent at different 𝜀′ for a perfect grating with infinite periods. However, the translation\n𝑎𝑏\nsymmetry is broken due to limited sample size or white light spot size. As a result, the\nreflectance at 𝜃 = 0° mixes in the contributions from near-Γ quasi-BICs at finite k.\nConsidering system dimension of ~20 μm limited by beam size, it leads to a k-space\nblurriness of Δθ~1.4° as estimated with the uncertainty relation Δ𝑘⋅Δ𝑥 ≥ 1/2. Such near-Γ\ncontribution is simulated in Fig. 3g and h for θ = 0.3° and 0.6°. It is evident that the\nreflectance features of the quasi-BICs state (DS) gradually build up with the increase of 𝜀′\n𝑎𝑏\nwithin the uncertainty limit of Δθ, which is consistent with the experimental observation of\nthe brightened BICs in Fig. 3d and e.”\nWe have also replaced Fig. 3f with the simulation results of Fig. R1b-d in the manuscript to\nprovide a comprehensive illustration of the brightening of BIC within angle uncertainty Δθ.\nFigure R1 a, The simulated CrSBr grating structure. b, Simulated reflection spectra for 𝜃 =\n0° as the function of field-induced off-diagonal dielectric tensor element 𝜀′ . c,d, Simulated\n𝑎𝑏\nreflection spectra for 𝜃 = 0.3° and 0.6°. The selected angle is within the angle uncertainty\nΔ𝜃~1.4° defined by the white light spot size of 20 μm.The bright state (BS) and dark state\n(DS, quasi-BIC) are marked.\n2. In Fig. 4f, the authors present two very different k-space distributions for the\npump: one as the \"Fourier transform of the real-space pump pulse\" and another as the\n\"measured normalized k-space distribution of the pump light.\" The notable\ndiscrepancy between these two curves is confusing and should be clearly explained. Additionally, the authors should clarify\nhow this relates to their argument on the impact of heat/strain and direct momentum transfer mechanisms.\nResponse:\nWe apology for the unclear description of our measurement scheme. The “Fourier transform\nof the real-space pump pulse” refers to 2D Fourier transform of the real-space Gaussian beam\nlateral profile 𝐼(𝑧;𝑥,𝑦) at position 𝑧, denoted as ℱ[𝐼(𝑧;𝑥,𝑦)]. Whereas, \"measured\nnormalized k-space distribution of the pump light\" refers to the angular distribution of the\npump density, denoted as 𝐼(𝑘 ,𝑘 ). For a focused Gaussian beam, ℱ[𝐼(𝑧;𝑥,𝑦)] at the focus\n𝑥 𝑦\nplane coincides with 𝐼(𝑘 ,𝑘 ) as shown in Fig. R2 at 𝑧 = 𝑧 . However, this is not the case at\n𝑥 𝑦 0\n𝑧 = 𝑧 as evident in Fig. R2c and d. At positon away from the focus plane, 𝐼(𝑘 ,𝑘 )\n1 𝑥 𝑦\ncorresponding to the angular distribution of the light beam. Whereas, ℱ[𝐼(𝑧;𝑥,𝑦)] inversely\nreflects the real-space beam profile. In our measurements, the pump laser beam is arranged\nsuch that the laser focal plane does not coincide with the sample surface. This is a necessary\nsetting to create large enough pump spot size of ~10 μm for the collective pumping of the\nphotonic modes. It also dramatically reduces the pump fluent density, which is also a critical\ncondition to pump coherent magnon as well as to avoid sample damage. As a result,\nℱ[𝐼(𝑧 ;𝑥,𝑦)] at the sample position is therefore different from that of 𝐼(𝑘 ,𝑘 ).\n1 𝑥 𝑦\nOur experiments thus distinguished a direct momentum transfer effect from a heat/strain\neffect. Since the heat/strain mechanism is expected to follow the real-space absorption profile\nas suggested in [Phys. Rev. B 109, 104401], its effect should have a k-space profile similar to\nℱ[𝐼(𝑧 ;𝑥,𝑦)]. Whereas, a direct momentum transfer scheme tend to give rise to a k-space\n1\ndistribution proportional to 𝐼(𝑘 ,𝑘 ). The observation of the k-space optical magnon profile\n𝑥 𝑦\ncoincides with 𝐼(𝑘 ,𝑘 ) therefore support our claim of a direct momentum transfer\n𝑥 𝑦\nmechanism.\nRevision:\nWe have now added following sentences on Page 10 of the main text to avoid\nmisunderstanding of the measurement schema.\n“In our measurement, the pump laser was defocused to a large laser spot size of ~10 μm. It\ntherefore generates a broad real-space heat/strain profile. The dashed line in Fig. 4f estimate\nthe contribution of heat/strain by showing the Fourier transformation of the real-space laser\nspot profile, which is too narrow to account for the observation of the broad k-space profile\nfor the optical magnon.” “Furthermore, the optical magnon has a similar profile as the\nangular distribution of the pump density (Fig. 4f). It is worth to know that such angular\ndistribution of the pump density deviates from the Fourier transformed real-space profile of\nthe pump beam due to the defocused pump laser beam setting.”\nFigure R2 a, Illustration of the beam profile of a focused Gaussian beam, 𝑧 and 𝑧\n1 0\ncorrespond to the sample plane and focus plane for the Gaussian beam. b, The field profile of\nthe Gaussian beam evaluated at 𝑧 and 𝑧 . c, Illustration of the Fourier transform of I(𝑧 ;𝑥,𝑦)\n1 0 1\nand I(𝑧 ;𝑥,𝑦). d, Illustration of the angle distribution of the pump density.\n0\n3. The FFT amplitude of the optical and acoustic magnon modes shows different\ndependence on momentum. Could the authors provide a more detailed physical explanation of why the optical magnon exhibits stronger k-dependence and why\nacoustic magnon does not?\nResponse:\nThe different k-dependence of the optical and acoustic magnon modes is rooted in the mode\ncharacter. Fig. R3c and d show the precession of the sublattice spins for the acoustic and\noptical magnon modes for bulk CrSBr in the absence of magnetic field (𝐻 = 0), which is\n𝑒𝑥𝑡\nobtained by solving the Landau-Lifshitz equation. Here, 𝛿𝑚⃗⃗ (𝑡) and 𝛿𝑚⃗⃗ (𝑡) are time-\n1 2\nvarying part of the sublattice spins 𝑚⃗⃗ (t) = 𝑚⃗⃗ +𝛿𝑚⃗⃗ (𝑡) and 𝑚⃗⃗ (t) = 𝑚⃗⃗ +𝛿𝑚⃗⃗ (𝑡). The\n1 1 1 2 2 2\nchange of the angle between 𝑚⃗⃗ (t) and 𝑚⃗⃗ (t) leads to the temporal modulation of exciton\n1 2\nenergy in bulk CrSBr, which is given by Δ𝐸 ∝ 𝑚⃗⃗ (t)⋅𝑚⃗⃗ (t)/|𝑚⃗⃗ (t)||𝑚⃗⃗ (t)|. At 𝐻 = 0,\n1 2 1 2 𝑒𝑥𝑡\none can tell from Fig. R3c and d that neither acoustic or optical magnon can modulate the\nexciton energy at the corresponding magnon frequency 𝜔. This is simulated in Fig. R3e\nwhere the time evolution of Δ𝐸 does not have frequency component at 𝜔 for both acoustic\nand optical magnon excitation. Although weak higher order frequency component at 2𝜔 can\nbe identified. When an out-of-plane external magnetic field is applied, 𝑚⃗⃗ and 𝑚⃗⃗ is tilted\n1 2\naway from 𝑎 −𝑜−𝑏 plane, which breaks the symmetry and initiates the coupling between\nthe exciton and the optical magnon. This is simulated in Fig. R3f, where a strong oscillation\nof Δ𝐸 at the corresponding magnon frequency 𝜔 is observed only for the optical magnon.\nWhereas, the acoustic mode remains decoupled at frequency 𝜔. Such distinct “selection rule”\nfor acoustic and optical magnon is observed in our measurement on unetched CrSBr flakes as\nshown in Fig. R4 c and d, where the acoustic mode is absent in the FFT spectra at all angles.\nThe observation of disallowed acoustic magnon in the FFT spectra from the CrSBr grating\n(Fig. R4 a and b) then must arise from the etched edges and sidewalls which have low\nsymmetry. At such localized site, the acoustic magnon-exciton polariton interaction therefore\ndoes not conserve k and naturally leads to a insensitive dependence on angle. On the other\nhand, the selection-rules-allowed optical mode requires match of momentum owing to the\npreserved translation symmetry and therefore leads to an angle-sensitive behavior for both the\npristine CrSBr flake and the CrSBr grating.\nRevision:\nTo provide a more comprehensive physical picture of the different k-dependence of the\noptical and acoustic magnon modes, we have now included our new measurement results on\nthe unetched CrSBr flake (Fig. R4c and d) in Fig.4. We have also included simulation results\nfor exciton-magnon coupling (Fig. R3a, b, e and f) as the new Fig. 5 in the manuscript. We\nhave also added the above discussion on Page 11-12 of the main text and a detailed derivation\nof the magnon-exciton coupling as supplementary note 4.\nFigure R3 a, Illustration of the sublattice spin orientation 𝑚⃗⃗ and 𝑚⃗⃗ of CrSBr in the\n1 2\nprecense of 𝐻⃗⃗ . b, Simulation of the magnon frequency as function of 𝐻 /𝐻 . 𝐻 is\n𝑒𝑥𝑡 𝑒𝑥𝑡 𝑠𝑎𝑡 𝑠𝑎𝑡\nthe saturation magnetic field. c, d, Simulated acoustic and optical magnon modes for 𝐻 =\n𝑒𝑥𝑡\n0. e, f, Simulated temporally modulated exciton energy due to the coherent magnon\nexcitation for acoustic and optical magnon modes for zero and finite magnetic field. The\nmagnon amplitude is set to |𝛿𝑚⃗⃗ | = |𝛿𝑚⃗⃗ | = 0.1.\n1 2\nFigure R4 a, FFT spectra of CoTR measured from the CrSBr grating as function of 𝐸\n𝑝𝑟𝑜𝑏𝑒\nfor different 𝜃. The frequency of acoustic (Ac.) and optical (Op.) magnon is marked. b, The\nFFT magnitude at optical and acoustic magnon frequency is plotted as function of 𝜃 for\nCrSBr grating. The measured k-space pump density profile and the estimation of angular\nprofile for heat/strain effect are also shown. c, FFT spectra of CoTR measured from the\npristine CrSBr flake as function of 𝐸 for different 𝜃. d, The FFT magnitude at optical\n𝑝𝑟𝑜𝑏𝑒\nand acoustic magnon frequency is plotted as function of 𝜃 for CrSBr flake.\nMinor Comments and Suggestions\n1. The text description of Fig. 3e seems to refer to Fig. 3f. Please check and revise\naccordingly.\nResponse and Revision:\nWe have now corrected the typos and made a thorough check throughout the manuscript to\navoid similar mistakes.\n2. The color bar scale in Fig. 3f uses only \"max\" and \"min\" labels. Providing\nnumerical values would better illustrate the absolute reflectance change attributed\nto εab.\nResponse and Revision:\nWe have now replaced all the “max” and “min” labels in the reflectance spectra in both Fig. 2\nand Fig. 3 with numerical values to provide a better presentation of the experimental and\nsimulation results.\n3. Consider plotting optical and acoustic magnon magnitudes separately in\nSupplementary Material to more clearly reveal their different k-dependences.\nResponse and Revision:\nThe suggestion is excellent. We have now plotted angle-dependent FFT magnitudes\nseparately for optical and acoustic magnon in Fig. R4b and d here as well as Fig. 4f and h in\nthe main text for better presentation of the k-dependent behavior of the two magnon modes.\nReviewer #2 (Remarks to the Author):\nIn this work, the authors report strong exciton-photon coupling modulated by the\nmagnon-exciton coupling in a CrSBr grating. Formation of polaritons with the grating\nbright and dark (BIC) modes have been experimentally extracted with large\ncoupling strength of 167 meV, consistent with previous reports in this material. The authors\nthen demonstrated tuning of these polariton modes with an applied external magnetic\nfield in both static and pump-probe measurements. While the attempt of fabricating\nnanophotonic devices that engineers the photonic band structures is applaudable,\nthere are some experimental observations that are potentially intriguing and worth\nfurther investigation, the data quality and analysis of the current work need to be\nsignificantly improved. In particular, I think that the figures of the manuscripts\nare weakly connected and there lacks a clear conclusion of how such devices can bring\nnovel properties to the field and what are the implications.\nResponse 1:\nWe appreciate the reviewer’s careful reads of our work. The reviewer has a general concern\nregarding the implication of our magnetic photonic device based on CrSBr. We note that\ncontrolling light with electron spins has strong implications in spin quantum information and\nspintronics. Such goal is difficult for conventional meta-surfaces or polaritonic devices based\non non-magnetic materials. Our work filled the gap by demonstration of both static and\nultrafast control of exciton polariton with electron spin or collective modes of spin in a\nmagnetic photonic structure. We believe such device has advanced the fundamental\nunderstanding of the interplay between spins, magnons, and photons in 2D magnetic systems\nand establish a promising platform for tuneable, spin-dependent photonic and quantum\ndevices.\nRevision:\nWe have now included following sentences on Page 12 of the manuscript to provide a clearer\nimplication of our device.\n“We note that controlling light with electron spins has strong implications in spin quantum\ninformation and spintronics, which is generally difficult for conventional meta-surfaces or\npolaritonic devices based on non-magnetic materials. Towards such goal, we have\ndemonstration of both static and ultrafast control of exciton polariton with electron spin or\ncollective modes of spins in a magnetic photonic structure of CrSBr, by harnessing the\nsynergistic interplay between strong light-matter interactions and magneto-electronic effect in\nthe 2D magnetic semiconductor.”\nResponse 2:\nThe reviewer also shared his/her comments on the analysis in current work, which we found\nmostly stems from our inadequate description of our theory and insufficient characterization.\nFollowing the reviewer’s comments and suggestions, we have made a thorough revision by\nincluding new experimental data and in-depth theoretical analysis. Please find our one-to-one\nresponse below.\nFirst of all, in fig. 2 f&g, the experimental and simulation data are quite different\nin terms of the mode energy, linewidth, and brightness, what is the reason of that?\nThe fabrication process involved etching of the material, what is the quality of the\nstructure, such as roughness, homogeneity, sidewall angle, etc.? How are they\ndeviates from the design and simulation parameters? How was the exciton and other\nmaterial properties affected by the etching? Are these the causes of the\ndeviation between the experiment and simulation?\nResponse:\nFollowing the reviewer’s suggests, we have now made careful structural characterization of\nthe fabricated CrSBr grating to resolve the discrepancy between simulation and experiments.\nWe found out the problem is mainly caused by the our AFM measurement, which\noverestimate the on/off ratio and thickness of the CrSBr grating. Fig. R1a and b show the\nrecent SEM measurement, from which we extract the correct structural parameters, e.g.\ngrating period, on/off ratio, sidewall angle, etc. for simulation. In Fig. R1c, we have also\nmade careful characterization of the surface roughness at the etched surface of the sample.\nAlthough the residual PMMA is rough, the etched surface of CrSBr is quite flat with RMS\nroughness of 0.57 nm. Fig. R1d and e compare the measured angle-resolved reflectance\nspectra and our updated simulation after correction of the structural parameters. The result is\nsubstantially improved with nicely matched experimental and simulation data in term mode\nenergy, linewidth and brightness. The experimental linewidth is only marginally broadened\ncompared with simulation, which is likely contributed by the surface roughness.\nTo get a hint on whether the material properties is affected by the etching, we compared the\nparameters of the dielectric function which are used for simulation of the reflectance spectra\nof the bulk and grating sample in the following table.\nSample 𝜀 𝑏0 𝐸 𝑋 𝐸 𝑋∗ ℏΓ ℏΓ∗ ℏ𝑓 ℏ𝑓∗\n(eV) (eV) (meV) (meV) (eV) (eV)\nPristine\n15.7 1.3643 1.3787 0.50 6.1 1.1233 0.4278\nCrSBr\nCrSBr\n10 1.3640 1.3787 0.50 6.1 1.28 0.55\ngrating\nWe notice that the simulation parameters for the unetched and etched CrSBr samples are\ncomparable with only minor deviation (e.g. smaller 𝜀 and larger oscillator strength ℏ𝑓 for\n𝑏0\nCrSBr grating). This result suggests that the dielectric and optical properties of CrSBr is\nnot noticeably affected by the etching.\nRevision:\nWe have now updated all the numerical simulation (including Fig 1e and f, Fig. 2g and Fig.\nS2) in the manuscript with the correct structural parameters. We also make sure that the\nconclusion and discussion in the manuscript is not affected but strengthen by the revision. We\nfurther included the accurate characterization of the fabricated structures (Fig. R1a-c) as Fig.\nS10 and the above table in the table S1 of the supplementary information.\nFigure R1 a, SEM image of the measured CrSBr grating, exhibiting uniformity over large\narea. b, Representative cross-section SEM image of a period of the measured CrSBr grating.\nThe CrSBr, residual PMMA and SiO substrate are marked. The two sidewall angles are\n2\nmarked. c, AFM profile measured on the residual PMMA (upper panel) and etched surface of\nCrSBr (lower panel). The RMS roughness of the etched surface of CrSBr is estimated to be\n0.57 nm. d, The measured ARR spectra of the CrSBr grating. e, The simulated ARR spectra\nimplementing the structure parameters in b. Both the experimental and simulated spectra are\nobtained by dividing the reflectance spectra of CrSBr grating (R ) with that from substrate\nCrSBr\n(R ).\nsub\nSecondly, the phenomenon demonstrated in figure 3 is quite confusing. First, it is\nvery hard to see the “Fano-like” feature the authors mentioned that indicates the\nlightening up of the dark state, in either Fig. 3 d and e. It is quite confusing what\nare the arrows pointing in the figure, as they are not aligned to any peaks or dips.\nAlso, the experimental data is not in accordance with the theory. As the authors\nmentioned, the dark mode at the Gamma point remained dark upon applying magnetic\nfield, as it is protected by the C2 symmetry, and thus the simulation in Fig. 3f\ncorresponds to the reflectance data at 2.3 degree. In this case, why the Gamma point\ndata is shown in Fig. 3 d&e? Why is brightening of the dark state expected in these\ndata? Finally, what is the implication of this experiment?\nResponse 1:\nWe understand the confusion of the reviewer, which is the result of our insufficient and\nuncompleted presentation of our theory.\nThe reviewer find the reflectance feature of the dark state in Fig. 3e hard to recognize, which\nmay be due to the strong asymmetric Fano lineshape. To show the reflectance feature for both\ndark state and bright state, the reflectance spectra in Fig. 3e are fitted with Fano lineshape of\n𝐼 ∝ (𝑞+𝜖)2/ (1+𝜖2), where 𝜖 = (𝐸 −𝐸 )/𝑤 is the unitless energy with respect to the\n0\nresonant energy 𝐸 . 𝐸 and 𝑤 are the photon energy and resonant linewidth. 𝑞 is the Fano\n0\nparameter. The dashed and solid arrows in Fig. 3e mark the fitted 𝐸 of the bright and dark\n0\nstate, which may seem not aligned exactly to the peak or dip position. We have listed all the\nother fitting parameters in the table below.\nB Bright state Dark state\n(T) 𝐸 𝑤 𝑞 𝐸 𝑤 𝑞\n0 0\n(eV) (meV) (eV) (meV)\n0 1.267 2.754 2.644 N.A. N.A. N.A.\n3 1.254 5.820 0.9869 1.263 5.272 -3.536\nWe note the strongly asymmetric line shape is also seen in the simulation when both the\nimaginary and real part of the off-diagonal dielectric tensor element 𝜀 = 𝜀′ +𝑖𝜀′′ is\n𝑎𝑏 𝑎𝑏 𝑎𝑏\nconsidered. For instance, Fig. R2b-d shows the simulated lineshape of the brightened quasi-\nBIC state and Fano parameters change as the function of Arg[𝜀 ].\n𝑎𝑏\nRevision 1:\nWe have now included following discussion of the fitting and the lineshape of the reflectance\nfeature of the dark state in Fig. 3e on Page 7 of the main text.\n“The brightening of EP BICs is further illustrated in Fig. 3e, where two resonant features are\nidentified at 𝑩 = 3 𝑇 and fitted with Fano lineshape. The dashed and solid arrows mark the\nfitted resonant energies of the bright and dark state. Table S2 in the supplementary\ninformation summarizes the fitting parameters for the two resonances. As guided by the\nfitting curves, the dark state locates at about 8 meV above the bright state and exhibits a\nhighly asymmetric lineshape. As a comparison, only the resonant feature of the BS is\nobserved and fitted for spectrum measured at 𝑩 = 0 𝑇 in Fig. 3e.”\nWe have also included Fig. R2 as Fig. S7 in the supplementary information to further\nillustrate the Fano reflectance lineshape obtained in the simulation.\nResponse 2:\nNext, the reviewer mentioned that the experimental observed brighten of dark state occurs at\nΓ (θ = 0°), which is may not be reflected by our simulation at small but finite k. We note that\nin perfect grating structure with infinite periods, the BIC at Γ (θ = 0°) is strictly protected by\nthe C symmetry, which is always dark even in the presence of magnetic field. This effect is\n2\nshown in Fig. R3b for CrSBr grating in Fig. R3a. However, the translation symmetry is\nweakly broken by e.g. limited white light spot size or limited sample size, leading to a\nmixture of at-Γ BIC with quasi-BIC states at finite k. In our measurements, the white light\nspot size is ~20 μm, which leads to a k-space blurriness of Δθ~1.4° as estimated with the\nuncertainty relation Δ𝑘 ⋅Δ𝑥 ≥ 0.5. Therefore, any quasi-BIC state within |θ|<1.4° will\ncontribute to the reflectance at Γ (θ = 0°). However, this effect is not well captured in\nsimulations considering the Bloch boundary condition in which the k-point is ideally defined.\nTo illustrate such contribution, one therefore needs to simulate the effect at small but finite θ\nwithin the k uncertainty limit (e.g. θ = 0.3° and 0.6° as shown in Fig. R3c and d). It is evident\nthat the brightening of the quasi-BIC state (DS) immediately build up within the angle\nuncertainty Δθ~1.4°.\nRevision 2:\nWe have now added following sentences to strengthen the discussion of the field-induced\nbrightening of the quasi-BIC on Page 8 of the main text. We have also included Fig. R3b-d in\nFig. 3 of the manuscript.\n“Ideally the Γ-point BICs is strictly radiatively forbidden. The robustness of the BICs is\nprotected by the 𝐶 symmetry which holds even in the presence of magnetic field. This is\n2\nshown with simulated reflectance spectra in Fig. 3f in which the reflectance features of BICs\nis absent at different 𝜀′ for a perfect grating with infinite periods. However, the translation\n𝑎𝑏\nsymmetry is broken due to limited sample size or white light spot size. As a result, the\nreflectance at 𝜃 = 0° mixes in the contributions from near-Γ quasi-BICs at finite k.\nConsidering system dimension of ~20 μm limited by beam size, it leads to a k-space\nblurriness of Δθ~1.4° as estimated with the uncertainty relation Δ𝑘⋅Δ𝑥 ≥ 1/2. Such near-Γ\ncontribution is simulated in Fig. 3g and h for θ = 0.3° and 0.6°. It is evident that the\nreflectance features of the quasi-BICs state (DS) gradually build up with the increase of 𝜀′\n𝑎𝑏\nwithin the uncertainty limit of Δθ, which is consistent with the experimental observation of\nthe brightened BICs in Fig. 3d and e.”\nFigure R2 a, The simulated CrSBr grating reflection spectra at (θ) = 0.6° for different off-\ndiagonal dielectric matrix element. Arg[𝜀 ] = 0 is assumed. The bright state (BS) and dark\n𝑎𝑏\nstate (DS, quasi-BIC) are marked. b, The simulated CrSBr grating reflection spectra at (θ) =\n0.6° for different Arg[𝜀 ] with |𝜀 | fixed to 1.5. c, Representative simulated reflectance\n𝑎𝑏 𝑎𝑏\nspectra of the dark state in b, showing the lineshape change. d, Fitting values of the Fano\nparameter for different curves in b, showing that the Fano parameter varies and changes signs\nwith different 𝜀 .\n𝑎𝑏\nFigure R3 a, The simulated CrSBr grating structure with C symmetry. b, c, d, Simulated\n2\nreflection spectra for (θ) = 0, 0.3° and 0.6° as function of field-induced off-diagonal dielectric\ntensor element. The bright state (BS) and dark state (DS, quasi-BIC) are marked.\nFigure 4 demonstrates different angle-dependence of the acoustic and optical magnons,\nwhich is quite interesting. However, the angle-resolved pump-probe reflectance data\nis noisy and needs improvements. The explanation about the difference between the\nacoustic and optical magnons is quite preliminary and oversimplified. Is there any\nfurther theory or calculation that can support the authors’ claim about the\ndifference in momentum conservation in the optical and acoustic magnons? Also, the\nauthors mentioned that the optical magnons exhibit stronger coupling to excitons\nwhile the acoustic magnons has much weaker coupling. However, in Fig. 4d, it seems\nthat at near zero angles the signals originating from two types of magnons have\nsimilar intensity, while at large angles the acoustic phonon dominates. Therefore the\nauthors may want to revise to a more precise description.\nResponse:\nIn the angle-resolved pump-probe reflectance measurement, the temporal oscillation of the\nreflectance spectra has been recorded simultaneously for different angles and wavelengths.\nSuch measurement schema is critical for us to yield reliable comparison between different\nangles. It results in large data set and is difficult to further improve the data quality. However,\nit does not compromise the observation and analysis for the general trend of the different k-\nspace behavior of the two magnon modes.\nThe different k-dependence of the optical and acoustic magnon modes is rooted in the mode\ncharacter. We have now included more experimental and theoretical results in the following\nto provide comprehensive understanding. Fig. R4c and d shows the precession of the\nsublattice spin for the acoustic and optical magnon modes for bulk CrSBr in the absence of\nmagnetic field (𝐻 = 0), which is obtained by solving the Landau-Lifshitz equation. Here,\n𝑒𝑥𝑡\n𝛿𝑚⃗⃗ (𝑡) and 𝛿𝑚⃗⃗ (𝑡) are time-varying part of the sublattice spins 𝑚⃗⃗ (t) = 𝑚⃗⃗ +𝛿𝑚⃗⃗ (𝑡) and\n1 2 1 1 1\n𝑚⃗⃗ (t) = 𝑚⃗⃗ +𝛿𝑚⃗⃗ (𝑡). The change of the angle between 𝑚⃗⃗ (t) and 𝑚⃗⃗ (t) leads to the\n2 2 2 1 2\ntemporal modulation of exciton energy in bulk CrSBr, which is given by Δ𝐸 ∝ 𝑚⃗⃗ (t)⋅\n1\n𝑚⃗⃗ (t)/|𝑚⃗⃗ (t)||𝑚⃗⃗ (t)|. At 𝐻 = 0, one can tell from Fig. R4c and d that neither acoustic or\n2 1 2 𝑒𝑥𝑡\noptical magnon can modulate the exciton energy at the corresponding magnon frequency 𝜔.\nThis is simulated in Fig. R4e where the time evolution of Δ𝐸 does not have frequency\ncomponent at 𝜔 for both acoustic and optical magnon excitation. Although weak higher order\nfrequency component at 2𝜔 can be identified. When an out-of-plane external magnetic field\nis applied, 𝑚⃗⃗ and 𝑚⃗⃗ is tilted away from 𝑎−𝑜−𝑏 plane, which breaks the symmetry and\n1 2\ninitiates the coupling between the exciton and the optical magnon. This is simulated in Fig.\nR4f, where a strong oscillation of Δ𝐸 at the corresponding magnon frequency 𝜔 is observed\nfor only the optical magnon. Whereas, the acoustic mode remains decoupled at frequency 𝜔.\nSuch distinct “selection rule” for acoustic and optical magnon is observed in our\nmeasurement on unetched CrSBr flakes as shown in Fig. R5 c and d, where the acoustic\nmode is absent in the FFT spectra at all angles. The above theoretical analysis and\nmeasurement on unpatterned CrSBr strongly support our claim of stronger coupling of the\noptical magnon, which is also questioned by the reviewer.\nThe observation of disallowed acoustic magnon in the FFT spectra from the CrSBr grating\n(Fig. R5 a and b) then must arise from the etched edges and sidewalls which have low\nsymmetry. At such localized site, the acoustic magnon-exciton polariton interaction therefore\ndoes not conserve k and naturally leads to an insensitive dependence on angle. On the other\nhand, the selection-rules-allowed optical modes requires match of momentum owing to the\npreserved translation symmetry and therefore leads to an angle-sensitive behavior for both the\npristine CrSBr flake (Fig. R5c and d) and the CrSBr grating (Fig. R5 a and b).\nRevision:\nTo provide a more comprehensive physical picture of the different k-dependence of the\noptical and acoustic magnon modes, we have now included our new measurement results on\nthe unetched CrSBr flake (Fig. R5c and d) in Fig.4. We have also included simulation results\nfor exciton-magnon coupling (Fig. R4a, b, e and f) as the new Fig. 5 in the manuscript. We\nhave also added the above discussion on Page 11-12 of the main text and a detailed derivation\nof the magnon-exciton coupling as supplementary note 4.\nFigure R4. a, Illustration of the sublattice spin 𝑚⃗⃗ and 𝑚⃗⃗ of CrSBr in the precense of 𝐻⃗⃗ .\n1 2 𝑒𝑥𝑡\nb, Simulation of the magnon frequency as function of 𝐻 /𝐻 . 𝐻 is the saturation\n𝑒𝑥𝑡 𝑠𝑎𝑡 𝑠𝑎𝑡\nmagnetic field. c, d, Simulated acoustic and optical magnon modes for 𝐻 = 0. e, f,\n𝑒𝑥𝑡\nSimulated temporally modulated exciton energy due to the coherent magnon excitation for\nacoustic and optical magnon modes for zero and finite magnetic field. The magnon amplitude\nis set to |𝛿𝑚⃗⃗ | = |𝛿𝑚⃗⃗ | = 0.1.\n1 2\nFigure R5 a, FFT spectra of CoTR measured from the CrSBr grating as function of 𝐸\n𝑝𝑟𝑜𝑏𝑒\nfor different 𝜃. The frequency of acoustic (Ac.) and optical (Op.) magnon is marked. b, The\nFFT magnitude at optical and acoustic magnon frequency is plotted as function of 𝜃 for\nCrSBr grating. The measured k-space pump density profile and the estimation of angular\nprofile for heat/strain effect are also shown. c, FFT spectra of CoTR measured from the\npristine CrSBr flake as function of 𝐸 for different 𝜃.d, The FFT magnitude at optical and\n𝑝𝑟𝑜𝑏𝑒\nacoustic magnon frequency is plotted as function of 𝜃 for CrSBr flake.\nReviewer #3 (Remarks to the Author):\nOverall, this manuscript presents a timely and original contribution to the emerging\nfield of spin–photon interfaces based on van der Waals magnetic semiconductors. By\nemploying a CrSBr metasurface, the authors demonstrate that high-Q\nresonances and exciton polariton bound states in the continuum (BICs) can be utilized to\nsubstantially enhance magnon–photon interactions. The combination of steady-state\noptical spectroscopy and transient angle-resolved reflectance measurements provides\nconvincing evidence for magnon–exciton–photon hybridization and reveals mode- and\nmomentum-dependent dynamics that highlight direct momentum transfer from pump photons\nto magnons. These findings advance the fundamental understanding of the interplay\nbetween spins, magnons, and photons in two-dimensional magnetic systems and establish\na promising platform for tunable, spin-dependent photonic and quantum devices.\nThat said, the current version requires further refinement before it can meet the\npublication standards of Nature Communications. In particular, aspects of the\nexperimental methodology, the theoretical interpretation, and the\ncontextualization within the broader literature should be clarified and strengthened.\nResponse:\nWe appreciate for the careful reads of our manuscript and excellent constructive comments\nand suggestions. We have included new experimental results and theoretical analysis to\naddress the reviewer concerns and to place this work in the broader literature. Below, please\nfind the one-to-one response to the reviewer’s comments and revision of the manuscript.\n1. Quoting “The CrSBr metasurface hosts an exciton polariton (EP) bound state in the\ncontinuum (BIC). Owing to the magneto-electronic coupling effect of CrSBr and\nphotonic singularity of the BIC mode, both the energy and the reflectivity of the\nhalf-light, half-matter state can be efficiently tuned with an external magnetic\nfield.” This statement in the Introduction appears imprecise and somewhat\nmisleading. Specifically, based on the one-dimensional CrSBr grating, the authors\ninvestigate polaritons arising from the strong coupling between high-Q quasi-BICs and\nmagnon–excitons in momentum space, rather than a generic BIC-hosted polariton. To\nimprove clarity, it would be important to provide a concise but accurate introduction\nto the nonlocal BICs physics, with emphasis on how quasi-BIC resonances enhance\nlight–matter interactions. In particular, since exciton polaritons represent a local\nresponse while grating resonances are inherently nonlocal, the authors should expand\nthe Introduction to highlight the content of from local to nonlocal high-Q\nmetasurfaces, supported by relevant literature, so as to place this work more\nconvincingly in the broader context of recent advances in photonic and polaritonic\nmetasurfaces.\nResponse and Revision:\nFollowing the reviewer’s suggestion, we changed the sentences mentioned by the reviewer on\nPage 4 to the following to avoid misunderstanding.\n“The CrSBr metasurface hosts bound states in the continuum (BICs) that are strongly coupled\nwith excitons forming exciton polaritons (EPs).”\nWe have also added sentences “In the following content, we will refer to the exciton polariton\nformed by the strong coupling between the BICs mode and CrSBr excitons as EP BICs for\nshort.” on Page 7 and changed the subsection of “Formation of the exciton polariton BICs in\nCrSBr 1D grating” to “Strong coupling of exciton and BICs mode in a CrSBr 1D grating” on\nPage 6 to avoid similar confusion.\nWe further included following sentences on the introduction section on Page 4 and the listed\nreferences to give a broader context for the relevant development.\n“In particular, photonic structures hosting bound states in the continuum (BICs) have\nattracted great attention due to their potential to achieve infinite-high cavity Q factors and\nsensitive to external perturbations, giving rise to intriguing light-matter interaction that\nfacilitate the low threshold lasing, ultrafast optical switches, etc.37-39. Coupling of the non-\nlocal BICs mode with the local excitation in the magnetic materials, e.g. excitons, may also\ngive access to significant tunability through interplay of the local and non-local transition of\nthe system40,41.”\n37.Huang, C. et al. Ultrafast control of vortex microlasers. Science 367, 1018–1021 (2020).\n38.Wu, X. et al. Exciton polariton condensation from bound states in the continuum at room\ntemperature. Nat Commun 15, 3345 (2024).\n39.Koshelev, K., Lepeshov, S., Liu, M., Bogdanov, A. & Kivshar, Y., Asymmetric\nmetasurfaces with high-Q resonances governed by bound states in the continuum. Phys. Rev.\nLett. 121, 193903 (2018).\n40.Liang, Y., Tsai, D. P. & Kivshar, Y. From Local to Nonlocal High-Q Plasmonic\nMetasurfaces. Phys. Rev. Lett. 133, 053801 (2024).\n41.Yuan, Y. et al. Robust Purcell Effect of CsPbI Quantum Dots Using Nonlocal Plasmonic\n2. Continuning my last Question. Also, NOT “bound state in the continuum (BIC)”,\nbut “bound states in the continuum (BICs)”.\nResponse and Revision:\nWe have now corrected the terminology and check thoroughly to avoid misunderstanding.\n3. In line 86, the authors mention that the relative permittivity along the b-axis is\nderived from the reflectance spectrum shown in Fig. S1. Near the exciton state X,\nthree distinct resonant valleys are visible in the reflectance spectrum; however,\nthe real (ε₁) and imaginary (ε₂) parts of the relative permittivity in Fig. 1 do not\nexhibit multi-order oscillations near the exciton state X. Please explain this\ninconsistency.\nResponse:\nWe do not expect multi-order oscillations in the relative permittivity as shown in Fig.1. The\nmulti-order oscillations in the reflectance spectra in Fig. S1 originates from the Fabry-Perot\nresonances due to the vertical dielectric confinement. We note that similar reflectance spectra\nand fitting of the relative permittivity is also demonstrated in CrSBr flakes in earlier report\n[Nat Commun 14, 5966 (2023)].\nRevision:\nWe have now included following sentences on page 5 of the manuscript.\n“The multi-resonance reflectance features in Fig. S1 is due to the multi-order Fabry-Perot\nresonances arise from the planar cavity formed between the top and bottom surface of the\nCrSBr flake and is therefore not reflected by the fitted 𝜀 in Fig. 1c and d.”\n𝑏\n4. I am wondering the size of the fabricated sample. No information about the grating\nsize.\nResponse and Revision:\nThe size of the grating sample is 30 μm × 30 μm. We have now added the information in the\nmethod section.\n5. The authors use a coupled oscillator model to explain the strong coupling between\nthe FP mode and the exciton in a uniform 150 nm-thick CrSBr flake (Figs. 2 and S3).\nHowever, does the quasi-BIC mode in the one-dimensional CrSBr metasurface also form\nstrong coupling with the exciton? Although the energy of the quasi-BIC mode is far\nfrom that of the exciton mode—making its coupling features susceptible to\ninterference from other modes closer to the exciton and difficult to directly observe\nexperimentally—the authors should provide supporting theoretical analysis to\nsubstantiate the claim of strong coupling between the BIC and the magnon–exciton in\nthe following contents.\nResponse:\nAs point out by the reviewer, strong coupling features of quasi-BICs mode is difficult to\ndirectly observe experimentally. To support the strong coupling between BICs and the exciton\nin CrSBr grating, we have now included a coupled oscillator model (COM) for the CrSBr\ngrating in the following. The dielectric function of CrSBr ε is decomposed a background\nb\ncontribution of ε and excitonic contribution. The photonic mode is then simulated for the\nb0\nCrSBr grating structure in which the dielectric function of CrSBr is replaced by ε and the\nb0\nguided resonance energies 𝐸𝑝ℎ(𝑘) for TE and TE are shown in Fig. R1d and e. The exciton\n1 2\npolaritons are formed from coupling of the two excitonic states (X and X*) and the photonic\nmodes. Due to the much larger oscillator strength of X, we consider a simplified COM with\nonly the contribution of X. The upper and lower polariton (UP and LP) dispersion for TE\n1\nand TE are then simulated assuming the coupling strength 𝑔 = 186 𝑚𝑒𝑉 and 194 𝑚𝑒𝑉\n2\nrespectively. The results are shown in Fig. R1d and e. The dispersion of the LP from TE and\n2\nspectra in Fig. R1a as well as the FEM simulation result considering a full dielectric function\nin Fig. R1 b and c. We then verified that the strong coupling condition 2𝑔 > [ℏΓ+𝜖]/2 =\n2.9 𝑚𝑒𝑉 is fulfilled at the BICs.\nRevision:\nWe have now included following sentences covering the theoretical analysis of the coupling\nstrength for quasi-BICs mode on Page 6 of the main text. We have also included the\nsimulation results of the COM (Fig. R1) as Fig. S5 in the supplementary information.\n“Furthermore, we have carried out analysis based on COM and the simulation results are\nshown in Fig. S5. The COM suggests that a coupling strength of ~186 𝑚𝑒𝑉 is needed to\nreproduce the observed polariton dispersion and the strong coupling is also reached in the\nCrSBr grating sample. ”\nFigure R1 a and b are the measured and simulated Angle-resolved reflectance spectra from\nthe CrSBr grating. c, The FEM Simulation of the EP mode dispersion in the CrSBr 1D\ngrating on the SiO /Si substrate considering the full dielectric function of CrSBr. d, e\n1 2\ncoupled-oscillator model. The coupling strength 𝑔 is assumed to be 186 meV and 194 meV\nrespectively for TE and TE to match the LP dispersion to the FEM simulation results in c. f,\n1 2\nclose-up of the LP dispersion calculated with the coupled-oscillator model.\n6. In lines 146–154, the authors discuss changes in the reflectance spectra of the\nquasi-BIC mode and the guided resonances (GRs) under an external magnetic field. They\nclaim that the pronounced amplitude response unique to the quasi-BIC reflectance\nspectrum indicates a stronger magneto-optical effect compared to the GR mode that is\nfar away from the BIC position. However, this comparison is not convincing, as\namplitude modulation primarily arises from the high sensitivity of radiative losses\nnear the BIC to external perturbations. In contrast, the larger energy shift of the\nGR mode under a magnetic field modulation (Fig. S6b) better reflects the magneto-\noptical interaction. The authors are advised to provide a more convincing\nexplanation.\nResponse:\nThe reviewer maybe misunderstood our discussion regarding the pronounced magnetic\nresponse of the quasi-BICs reflectance. The magneto-optical effects in CrSBr including 1) the\nmagneto-electronic effect, which is essentially a change of the electronic structure driven by\nthe magnetic phase transition, 2) as well as the conventional magneto-optical effect (e.g.\nmagnetic Kerr/Faraday effect) which is the result of spin-orbital coupling. The former is\nresponsible for the energy shift of the exciton or exciton polariton and is contributed by\nmainly the magnetic field response of the diagonal dielectric tensor element 𝜀 . Whereas the\nb\nlatter is reflected by the magnetic-field-induced off-diagonal element 𝜀 and is responsible\n𝑎𝑏\nfor the change of the reflectance contrast of the quasi-BICs. Therefore, one may not use the\nredshift of the quasi-BICs to estimate the magneto optical interaction that is responsible for\nthe brightening of the quasi-BICs. In fact, our argument for a more pronounced modulation of\nthe BICs reflectance contrast is in line with the reviewer, which is due to the combined effect\nof the magnetic field induced 𝜀 and the fact that the quasi-BICs is sensitive to external\n𝑎𝑏\nperturbation.\nRevision:\nWe have now revised the manuscript with following sentences on Page 8 to give a more\ndirect description of the magneto-optical effect.\n“The observed intriguing magneto-optical effect emerges exclusively at the magnetic EP-\nBICs, which is due to the combined effect of the magnetic field induced 𝜀 and the fact that\n𝑎𝑏\nthe quasi-BICs is sensitive to such external perturbation.”\n7. In line 139, the authors claim that Fig. 3f shows that an external magnetic field\ncan brighten the dark mode. However, the corresponding experimental data are\npresented in Fig. 3e, not Fig. 3f. Moreover, a reflection peak at the DS position\nunder B = 3 T is not clearly identifiable from the fitting curves in Fig. 3e. The\nauthors should provide a more detailed explanation.\nResponse:\nWe apology for the typos. We have now checked throughout the manuscript to avoid such\nmistake.\nThe reviewer find the reflectance feature of the dark state in Fig. 3e hard to recognize, which\nmay be due to the strong asymmetric Fano lineshape. To show the resonance feature more\nclear, the reflectance spectra in Fig. 3e are fitted with Fano lineshape of 𝐼 ∝ (𝑞 +𝜖)2/ (1+\n𝜖2), where 𝜖 = (𝐸 −𝐸 )/𝑤 is the unitless energy with respect to the resonant energy 𝐸 . 𝐸\n0 0\nand 𝑤 are the photon energy and resonant linewidth. 𝑞 is the Fano parameter. The dashed and\nsolid arrows in Fig. 3e mark the fitted 𝐸 of the bright and dark state, which may seem not\n0\naligned exactly to the peak or dip position due to the asymmetric lineshape. We have listed all\nthe other fitting parameters in the table below.\nB Bright state Dark state\n(T) 𝐸 𝑤 𝑞 𝐸 𝑤 𝑞\n0 0\n(eV) (meV) (eV) (meV)\n0 1.267 2.754 2.644 N.A. N.A. N.A.\n3 1.254 5.820 0.9869 1.263 5.272 -3.536\nWe also note that the strongly asymmetric line shape is also seen in the simulation when both\nthe imaginary and real part of the off-diagonal dielectric tensor element 𝜀 = 𝜀′ +𝑖𝜀′′ is\n𝑎𝑏 𝑎𝑏 𝑎𝑏\nconsidered. Fig. R2a shows the reflectance of the quasi-BICs state at θ = 0.6° is enhanced at\nfinite |𝜀 |. Fig. R2b-d shows the lineshape of the brightened quasi-BICs state and Fano\n𝑎𝑏\nparameters changes as function of the Arg[𝜀 ]. The fitting of experimental data and the\n𝑎𝑏\nsimulation therefore justify our assignment of the peaks in Fig. 3d and e.\nRevision:\nTo present our assignment of the reflectance feature in a better manner, we have now\nincluded the full fitting parameters for the resonance peaks as table S2 in the supplementary\ninformation. Fig. R2 is now included as Fig. S7 in the supplementary information to confirm\nthe asymmetric Fano lineshape from the simulation perspective. We have also added\nfollowing sentences on Page 7 of the main text.\n“The brightening of EP BICs is further illustrated in Fig. 3e, where two resonant features are\nidentified at 𝑩 = 3 𝑇 and fitted with Fano lineshape. The dashed and solid arrows mark the\nfitted resonant energies of the bright and dark state. Table S2 in the supplementary\ninformation summarizes the fitting parameters for the two resonances. As guided by the\nfitting curves, the dark state locates at about 8 meV above the bright state and exhibits a\nhighly asymmetric lineshape. As a comparison, only the resonant feature of the BS is\nobserved and fitted for spectrum measured at 𝑩 = 0 𝑇 in Fig. 3e. ”\nFigure R2 a, The simulated CrSBr grating reflection spectra at (θ) = 0.6° for different off-\ndiagonal dielectric matrix element. Arg[𝜀 ] = 0 is assumed. The bright state (BS) and dark\n𝑎𝑏\nstate (DS, quasi-BICs) are marked. b, The simulated CrSBr grating reflection spectra at (θ) =\n0.6° for different Arg[𝜀 ] with |𝜀 | fixed to 1.5. c, Representative simulated reflectance\n𝑎𝑏 𝑎𝑏\nspectra of the dark state in b,showing the lineshape change. d, Fitting values of the Fano\nparameter for different curves in b.\n8. In lines 179–180, the authors state: “In the FFT spectra, two dominant\noscillation frequencies at 22 GHz and 30 GHz are identified and attributed to the\nacoustic and optical magnon mode (marked as Ac. and Op.).” However, in subsequent\ndescriptions, the acoustic and optical magnons are assigned to 30 GHz and 22 GHz,\nrespectively. Please ensure consistency in the description to avoid ambiguity.\nResponse and Revision:\nWe apology for the ambiguity due to typos. We attribute the higher energy mode (30 GHz) to\nacoustic magnon while the lower energy one (22 GHz) to the optical mode. We now correct\nthe typos and make sure that the description is consistence through the manuscript.\n9. Generally, in magnetic semiconductors, the optical magnon mode has a higher\nfrequency than the acoustic magnon mode. However, the assignment in Fig. 4d appears\nto be reversed. Please explain this discrepancy.\nResponse and Revision:\nWe note that the energy ordering of the optical and acoustic magnon mode depends on the\nexchange and anisotropic parameters. In CrSBr , it has been reported that the acoustic\nmagnon has a higher frequency ([Phys. Rev. B 109, 104401]).\nTo avoid ambiguity in the assignment of the magnon modes, we have now provided a\ndetailed derivation of the bulk magnon modes in supplementary note 4 by solving the\nLandau-Lifshitz equation. By adopting the interaction parameters from [Nature 609, 282–286\n(2022)], we have simulated the energy ordering for the two magnon modes in Fig. R3b,\nwhich is included as Fig. 5b in the manuscript.\nFigure R3 a, Illustration of the sublattice spin orientation 𝑚⃗⃗ and 𝑚⃗⃗ of CrSBr in the\n1 2\nprecense of 𝐻⃗⃗ . b, Simulation of the magnon frequency as function of cosθ which is\n𝑒𝑥𝑡\nproportional to 𝐻 . c, d, Simulated acoustic and optical magnon modes for 𝐻 = 0. e,\n𝑒𝑥𝑡 𝑒𝑥𝑡\nSimulated temporally modulated exciton energy due to the coherent magnon excitation for\nacoustic and optical magnon modes. The magnon amplitude is set to |𝛿𝑚⃗⃗ | = |𝛿𝑚⃗⃗ | = 0.1\n1 2\nand a finite magnetic field of 𝐻 /𝐻 = 0.48/2.1 is applied.\n𝑒𝑥𝑡 𝑠𝑎𝑡\n10. The authors study k-dependent magnon–exciton polariton coherent interactions\nbased on the FP mode in the CrSBr metasurface. However, such coherent interactions\nalso exist in uniform CrSBr thin flakes. It is therefore necessary to discuss the\ndifferences between the two platforms—for instance, whether the metasurface enhances\nthe strength of the magnon–exciton polariton coherent interaction.\nResponse:\nWe thank the reviewer’s excellent suggestion. We have performed new experiments on\nuniform CrSBr thin flakes. The angle-dependent FFT spectra and the FFT amplitude for\nacoustic/optical magnon modes for the unpatterned CrSBr are shown in Fig R4c and d, which\nare compared with the results the CrSBr grating (Fig R4a and b). We note that the similar k-\nconserved behavior of the optical magnon is also observed for measurement in the\nunpatterned CrSBr, evidenced by the observation of similar k-space distribution of FFT\namplitude at the optical magnon frequency. However, a noteworthy difference in the\nunpatterned CrSBr is that the acoustic mode is absent in the FFT spectra at all angles.\nThe comparison between CrSBr grating and unpattern sample strengthens our discussion for\nthe mode- and k-dependent magnon-EP interactions. We argued in the previous version of the\nmanuscript that the optical magnon couples stronger with the exciton polariton comparing\nwith the acoustic mode. This is now supported by our theoretical analysis shown in Fig R3e\nshowing that the optical magnon can induce strong temporal modulation of the exciton\nenergy in a bulk CrSBr. Furthermore, such distinct selection rules for acoustic and optical\nmagnon in bulk CrSBr is nicely reflected by our experimental result showing the absence of\nthe acoustic magnon in the FFT spectra for the unetched CrSBr sample (Fig R4c and d).\nHence, observation of the disallowed acoustic magnon in the FFT spectra for CrSBr grating\n(Fig R4a and b) then must arise from the etched edges and sidewalls which have low\nsymmetry. At such localized site, the acoustic magnon-exciton polariton interaction therefore\ndoes not conserve k and naturally leads to an insensitive dependence on angle. On the other\nhand, the selection-rules-allowed optical modes requires match of momentum owing to the\npreserved translation symmetry and therefore leads to an angle-sensitive behavior for both the\npristine CrSBr flake and the CrSBr grating.\nRevision:\nWe have now included the experimental results from the unpattern CrSBr sample (Fig R4c\nand d) in Fig. 4 of the manuscript and the above discussion on Page 11-12 of the manuscript.\nFigure R4 a, FFT spectra of CoTR measured from the CrSBr grating as function of 𝐸\n𝑝𝑟𝑜𝑏𝑒\nfor different 𝜃. The frequency of acoustic (Ac.) and optical (Op.) magnon is marked. b, The\nFFT magnitude at optical and acoustic magnon frequency is plotted as function of 𝜃 for\nCrSBr grating. The measured k-space pump density profile and the estimation of angular\nprofile for heat/strain effect are also shown. c, FFT spectra of CoTR measured from the\npristine CrSBr flake as function of 𝐸 for different 𝜃.d, The FFT magnitude at optical and\n𝑝𝑟𝑜𝑏𝑒\nacoustic magnon frequency is plotted as function of 𝜃 for CrSBr flake.\n11. A femtosecond pulsed laser centered at 800 nm is used as the pump to generate a\ntransient strain field. How would changing the central wavelength of the pump laser\naffect the experimental results?\nResponse:\nWe thank the reviewer for the nice suggestion, which may help us gain deeper understanding\nof the energy relaxation and dephasing dynamics that are currently beyond the scope of this\nwork. But unfortunately our pump laser has a fixed wavelength at 800nm, which cannot be\nchanged."}, {"round": 2, "reviewer_comments": {"reviewer_1": "The authors have responded to my previous review and made revisions to the paper clarifying the questions that I had\nraised. The manuscript now provides sufficient evidence and explanation to support the key claims. I support its publication\nin the journal.", "reviewer_2": "In this version of revision, the authors have extensively revised the manuscript and addressed all my questions. I believe that\nthe manuscript has been greatly improved.\nI have only one minor comment. The fitting of the polariton spectrum now matches well with the experiment. However, it is\nstrange that the oscillator strength of the exciton increases after the etching. Can the authors check the fitting parameters or\ngive a physical interpretation on theis point?"}, "author_response": "We appreciate the review’s excellent comments and suggestions which has led to the\nsubstantial improvement of our work. Following the reviewer’s new comment, we also\nnoticed that the oscillator strength in the fitting values in table S1 (listed below) for the CrSBr\ngrating is slightly enhanced comparing to the bulk sample.\nSample B\n∗ ∗ ∗\n(T) 𝜀𝑏0 (e𝐸V 𝑋 ) (𝐸e 𝑋 V) (mℏeΓV) (mℏeΓV) (ℏeV𝑓) (ℏe𝑓V)\nPristine 0 15.7 1.3643 1.3787 0.50 6.1 1.1233 0.4278\nCrSBr 3 15.3 1.3485 1.3666 0.85 8.1 1.1719 0.3092\nCrSBr\n0 10 1.3640 1.3787 0.5 6.1 1.28 0.55\nGating\nWe note that parameters for the bulk CrSBr are obtained by fitting the reflectance spectra\nwith the spectral range from 1.3eV to 1.38eV. Whereas, the results for CrSBr grating are\nobtained by fitting of the spectra with a larger spectral range from 1.216eV to 1.361eV. Both\nfittings assume the Lorentzian dielectric model which has been used extensively for exciton\npolariton in CrSBr [Nature 620, 533–537 (2023), Nat Commun 14, 5966 (2023), Adv. Funct.\nMater. 34, 2411589 (2024)]. It consisting of two exciton resonances and an energy-\nindependent background of . However, we note that the realistic dielectric function might\nslightly deviate from the Lorentzian model especially for energies far away from the exciton\n𝜀𝑏0\nresonance. We see such indication in the above table that the fitting values of is smaller\nfor CrSBr grating as compared to bulk CrSBr to accommodate the polariton modes with large\n𝜀𝑏0\ndetuning energies. A more accurate model would therefore need to consider further the\nenergy dependence of . Such effect may lead to the slightly overestimation of the exciton\noscillator strength in CrSBr grating. Nevertheless, the obtained comparable excitonic\n𝜀𝑏0\noscillator strength for etched and unetched sample suggests that the optical quality of the\nCrSBr grating is not compromised by etching.\nRevision:\nWe have now added following sentences on Page 6 of the manuscript to discuss the\nenhancement of the fitting values of the exciton oscillator strength.\n“We note that the slightly increase of the fitting values for the coupling strength as well as\nexciton oscillator strength (see table S1) for the CrSBr grating as compared to the bulk\nsample is likely artificially caused by the Lorentzian dielectric model25,26 which neglects the\nenergy dependence of the background dielectric constant . A more accurate determination\nof the coupling strength requires a more realistic modelling of the dielectric function over\n𝜀𝑏0\nlarge spectral range. Nevertheless, the obtained comparable exciton oscillator strength for the\netched and unetched sample suggests that the optical quality of the CrSBr grating is not\ncompromised by etching.”"}, {"round": 3, "reviewer_comments": {"reviewer_2": "The authors have answered my question and made the corresponding revision. I therefore recommend its publication."}, "author_response": ""}]

Decision letter