Mie-enhanced microfocused Brillouin light scattering for full wave vector resolution of nanoscale spin waves

Krčma et al., Sci. Adv. 11, eady8833 (2025) 31 October 2025 Science AdvAnceS | ReSeARch ARticle 1 of 9 CONDENSED MATTER PHYSICS Mieenhanced microfocused Brillouin light scattering for full wave vector resolution of nanoscale spin waves Jakub Krčma1, Ondřej Wojewoda2*, Martin Hrtoň1,2, Jakub Holobrádek2, Jon Ander Arregi2, Jaganandha Panda2, Ekaterina Pribytova1, Michal Urbánek1,2* Magnons, the quanta of spin waves, are magnetic excitations of matter spanning through the entire crystal’s Brillouin zone and covering a wide range of frequencies ranging from subgigahertz to terahertz. Magnons play a crucial role in many phenomena, such as the reduction of saturation magnetization with increasing temperature or the BoseEinstein condensation. However, established experimental techniques cannot resolve magnons with wave vectors between 30 and 300 rad μm−1. We address this gap by tailoring the Brillouin light scattering process with dielectric periodic nanostripes hosting Mie resonances. This approach enables access to the previously unmeasurable wave vector range while providing at the same time full wave vector resolution, all within a tabletop setup. Filling this gap can stimulate further experimental investigations of the fundamental phenomena associated with magnons as well as applications in computational and microwave devices. In addition, the same methodology can be applied to other excitations of matter, such as phonons, opening up possibilities in, e.g., mechanobiological studies. INTRODUCTION The field of magnonics has emerged as a promising area of research, focusing on the collective magnetic excitations known as spin waves and their associated quasiparticles, magnons. Magnonics can be categorized into two areas of interest: mesoscopic magnonics and research of fundamental properties of matter at the atomic scale. The field of mesoscopic magnonics searches for energy efficient computation paradigms, devices, or studies interesting behavior of matter at microand nanoscale (1–7). These studies are usually performed on spin waves with their wave vectors spanning only from 0 to approximately 100 rad μm−1, and use variety of magnetooptical techniques such as kresolved Brillouin light scattering (kresolved BLS), microfocused BLS (μBLS), timeresolved magnetooptical Kerr effect (TRMOKE), timeresolved scanning transmission xray microscopy (TRSTXM), nitrogenvacancy microscopy (limited only to the very low frequencies and thus not relevant for the following discussion), or electrical techniques such as propagating spinwave spectroscopy (PSWS) (8–18). In contrast, to investigate fundamental properties of matter, such as chiral splitting in altermagnets (19,20) or spinorbital separation (21), one usually needs to investigate magnons across the whole Brillouin zone and at hundreds of gigahertz to terahertz frequencies. These characterization needs can fulfilled by inelastic neutron scattering, resonant inelastic xray scattering (RIXS), or spinpolarized highresolution electron energyloss spectroscopy (SPHREELS) (22–25). However, there is no single experimental technique that covers the whole range of interest, with each technique being confined to a particular wave vector and frequency range resulting from various technical and fundamental constraints (seeFig.1A). Concerning wave number, imaging techniques are limited by the probing spot size to approximately 10 rad μm−1 for μBLS and TRMOKE and 100 rad μm−1 for TRSTXM (although the theoretical limit of the STXM is approximately 250 rad μm−1, it has never been experimentally demonstrated, most likely due to the problems with the low amplitude of the excited shortwavelength spin waves and insufficient signal sensitivity of the detection) (10, 26). Kresolved BLS is limited by the conservation of momentum to approximately 30 rad μm−1 (8), whereas PSWS is restricted by the excitation efficiency to approximately 100 rad μm−1 (27–29). Regarding frequency, μBLS has no limitation in the range of interest as various types of spectrometers can be used, such as (tandem) FabryPerot interferometers (30), virtual phase arrays (31), or grating monochromators (32). Contrary to that, techniques that rely on coherent excitation (pumpprobe, electrical detection) are limited by the used electronics, typically to tens of gigahertz. Techniques used to measure spin waves across the whole Brillouin zone (inelastic neutron scattering, SPHREELS, and RIXS) are limited in resolving lower wave numbers and frequencies by their resolution (typically over 300 rad μm−1 and 100 GHz). In between these two areas, a relevant experimental blind spot becomes apparent, which spans approximately from 100 to 300 rad μm−1 for low frequencies, and from 30 to 300 rad μm−1 for high frequencies (seeFig.1A). Various attempts to fill this gap were made by using nearfield enhancement of the μBLS signal, which can lift the restriction posed by the conservation of momentum law by introducing an imaginary part to the momentum of the incident electromagnetic wave. This principle was used to enhance the accessible wave number range by using plasmon resonances; however, in practical applications, the signal yield is very low (33–35). Recently, we have reported that dielectric nanoresonators hosting Mie resonances enhance both the wave vector sensitivity and the signal yield even for phaseresolved BLS measurements (36,37). However, without coherent excitation, individual nanoresonators lack wave vector resolution and thus can only provide limited information about the studied excitation. Note, that recently, a new technique using extreme ultraviolet transient magnetization gratings (EUVTMG) has been introduced that allows the study of shortwavelength spin waves (38). However, this technique is limited only to materials with 1institute of Physical engineering, Brno University of technology, technická 2, Brno 616 69, czech Republic. 2ceitec BUt, Brno University of technology, Purkyňova 123, Brno 612 00, czech Republic. *corresponding author. email: ondrej. wojewoda@ vutbr. cz (O.W.); michal. urbanek@ ceitec. vutbr. cz (M.U.) copyright © 2025 the Authors, some rights reserved; exclusive licensee American Association for the Advancement of Science. no claim to original U.S. Government Works. distributed under a creative commons Attribution license 4.0 (cc BY). Downloaded from https://www.science.org at Brno University of Technology on December 03, 2025 Krčma et al., Sci. Adv. 11, eady8833 (2025) 31 October 2025 Science AdvAnceS | ReSeARch ARticle 2 of 9 accessible absorption edges in the used energy range and can only study spin waves excited by the femtosecond EUV pulses. Moreover, the spatial resolution is limited to hundreds of micrometers. Here, we address these limitations by introducing a periodic array of dielectric nanostripes on top of the magnetic system of interest (a NiFe film) and measuring it in a standard μBLS setup. In this configuration, the light’s wave vector is superimposed with the lattice vector of the nanostripe array. The concept draws inspiration from wave propagation in periodic potentials, such as spin waves in artificial magnonic crystals (39,40). However, since the nanostripes are nonmagnetic, they do not modify the dispersion relation, allowing us to probe spin waves in an otherwise unaltered material system (41). As a result, we achieve full wave vector resolution, both magnitude and direction, across a previously inaccessible kvector range, a capability not attainable with individual dielectric nanoresonators (36,37). This approach goes far beyond the experiments from the previous studies, as we now show that it is even possible to resolve wave vectors of incoherent magnons (e.g., thermally excited or parametrically pumped). RESULTS AND DISCUSSION To measure inelastically scattered spectra, we use a standard BLS setup (seeFig.1B). The laser light is focused by a high–numerical aperture (NA) objective lens ( NA =0.75 ) onto a 440nm spot on the sample (36). The backscattered light is then collected by the objective lens and analyzed by the multipass tandem FabryPerot interferometer, which measures the spin wave frequency spectrum (30). The sample consists of a (27.6±0.1) nmthick permalloy film deposited on a Si substrate (for thickness measurement details see note S1). Periodic Si nanostripes, acting as Mie resonators, were fabricated on top of the magnetic film by electron beam lithography, ion beam sputtering, and a liftoff process. The stripes form gratings with varying periodicities, each grating covering 6.5×6.5 μm2 of the sample, seeFig.1C. To describe the Mieenhanced BLS process quantitatively, it is essential to understand how the incident light interacts with the dielectric nanoresonators and the magnetic material. When the driving incident electric field ( Edr ) interacts with the periodic dielectric nanostripes array with the lattice constant A, it generates a strongly localized electric field (hot spots) at the edges of the individual nanostripes within the illumination Gaussian spot (see Fig. 2A). k k - k k r r e e s s o o l l v v e e d d B L S PSWS µ µ µ µ µ µ µ µ µ µ µ µ µ µ µ µ µ B B B B B B B B B B B B B B B B B L L L L L L L L L L L L L L L L L S S S S S S S S S S S S S S S S S S 1 0 0 × / 0 . 7 5 L A S E R D E T Frequency (GHz) BLS signal (cts) NiFe Si substrate E M x y z S i B AC C Frequency (GHz) 10 1 10 2 10 3 10 4 10 5 Wave n u m b e r ( r a d / µ m ) π / a 10 1 10 2 10 3 1 1 T F P i Neutron scattering SPHREELS RIXS TR-MOKE Mie-enhanced µBLS (this work) EUVTGM TR-STXM A Fig. 1. Comparison between experimental techniques, and schematics of Mieenhanced BLS measurement. (A) Schematics of a spin wave dispersion relation in a permalloy thin film spanning over the whole Brillouin zone in an external magnetic field of 50 mt. various experimental techniques are highlighted with their corresponding accessible frequency and wave number ranges. the range achieved using Mieenhanced μBlS in this work is emphasized. (B) experimental setup for μBlS measurements. Green laser light (532 nm) is guided and focused onto the sample through an objective lens ( NA =0.75 ). the backscattered light from the sample is collected by the same lens and analyzed using a tandem FabryPerot interferometer to measure the spin wave frequency spectrum. (C) Sample and measurement geometry. the sample consists of Si nanostripes arranged with a periodicity A on top of a permalloy 27.6nmthick film deposited onto a Si substrate. 0400 400 0 00.06 E2(V2/m 2) x (nm) y(nm) A BC E2(V2/m 2) 0200 400 90 45 0 z(nm) x(nm) 10 10 101 E 2(V2/m 2) NiFe Si substrate Si 0 100 E2(V2/m 2) k x ) ky) 10 10 0200 10 10 Fig. 2. FDTD simulations of the electric field distribution of the incident laser light on the magnetic layer. (A) logarithmicscale electric field distribution in the x,zplane, showing a crosssectional view of the dielectric grating. Strong electric field localization is observed at the edges of the stripes (hot spots). White dotted lines indicate the interfaces. (B) electric field distribution in the x,yplane at the middle of the permalloy layer. Periodic field modulation is visible within the Gaussian beam spot. White dotted lines indicate the stripe positions. (C) Reciprocal electric field distribution in the x,yplane, obtained via Fourier transformation of the data from (B). the 1d plot (lower panel) shows the logarithmic intensity profile at ky = 0 , highlighting the field’s periodic distribution in the reciprocal space. Downloaded from https://www.science.org at Brno University of Technology on December 03, 2025 Krčma et al., Sci. Adv. 11, eady8833 (2025) 31 October 2025 Science AdvAnceS | ReSeARch ARticle 3 of 9 These hot spots are the result of excitation of Mie resonances within the nanoresonators, which modulate the field distribution inside the magnetic layer (Fig.2B). This modulation leads to a periodic electric field distribution, enabling the light to couple only to states with wave numbers corresponding to multiples of 2π∕A , even beyond the free space light accessible boundaries given by momentum conservation (Fig.2C) (34,36). At the same time, the high symmetry of periodic dielectric nanostripes in the y direction (seeFig.2C) leads to a strong localization of the electric field along the ky direction in the reciprocal space. The widths of the k=0 peak and all other higherorder peaks are determined by the width of the illumination laser spot. This modulated electric field interacts inelastically with the dynamic magnetization (spin waves) via magnetooptical coupling, inducing polarization where 𝛘m( r,ω m) is the dynamic magnetic susceptibility tensor (42), Edr(r,ω) the field generated in the layer by the incident laser beam (driving field), and ωm the frequency of spin waves. Since the frequency of the incoming radiation is much larger than that of the spin waves, we omit its dependency on the induced polarization. The polarization acts as a local source of radiation and its outcoupling from the magnetic layer and subsequent propagation toward the detector can be described using the dyadic Green’s function formalism (10). However, because of the broken spatial symmetry caused by the presence of the dielectric periodic nanostripes, the dyadic Green’s function is not known analytically and it needs to be derived from numerical simulations. Recalling that it represents the impulse response of the system to a point source, the task of finding the dyadic Green’s function amounts to running a large set of simulations, where the position of the point source is varied within a sufficient range and with an adequate resolution. Alternatively, one can switch to reciprocal space and attempt to reconstruct the angular spectrum representation of the dyadic Green’s function (36), i.e., an impulse response to a plane wave, but since the dielectric nanostripes do not have radial symmetry, the number of simulations to obtain the dyadic Green’s function with a sufficient precision becomes prohibitively high. To overcome these limitations, we exploited the reciprocity theorem of electromagnetism, drawing an inspiration from a similar approach for the scattering process in scanning nearfield optical microscopy (43). The reciprocity theorem simplifies the scattering problem by introducing the concept of a virtual source located at the detector and providing a clear mathematical link between the electric field generated by this virtual source and the radiation emitted by an actual source, later collected by the detector. In accordance with our BLS detection scheme, the virtual source is represented by an electric dipole pv oriented perpendicularly to the incident laser light. Denoting Ev(r) the electric field generated by the virtual source within the magnetic layer and Em( r det) the electric field produced by the polarization source Pm(r) at the position of the detector rdet , the reciprocity theorem enforces the following relation where the integration spans, in principle, the entire magnetic layer, but in practice is limited to the illumination spot. Assuming that the induced polarization has the form given byEq.1 and that only a single magnon with a lateral wave vector km⊥z is being excited (for details, see Materials and Methods), the detected BLS signal can be expressed as where the transfer function Tij( k m ,z ) defined as effectively determines the range of spin wave wave vectors that the system is able to couple with, regardless of whether these spin wave states are occupied or not. Note that the vector nature of the electric fields makes the transfer function a rank two tensor. Furthermore, the result of the integration over the vertical coordinate z will generally depend on the type of magnon that is being excited, but given the relatively large thickness of our metallic layer with respect to the penetration depth of the electric field, so that it effectively probes only the top portion of the magnetic layer, we simplified our analysis by recording the field and evaluating the transfer function only within a single plane. AsEq.3 suggests, the problem is reduced to finding the driving field ( Edr ), the virtual field ( Ev ), and the dynamic magnetic susceptibility ( χm ). The electric fields are obtained by performing two finitedifference timedomain (FDTD) simulations with polarizations perpendicular to each other, and the dynamic susceptibility is obtained by a semianalytical approach assuming only the firstorder magnetooptical coupling (10). We apply this model to obtain the transfer function of Si nanostripes with the periodicity A=150 nm, with materials and geometry matching with the fabricated sample (seeFig.3A). As expected, the transfer function shows maxima at the multiples of the periodicity, i.e., 2π∕A=42 rad μm−1, 4π∕A=84 rad μm−1. Also, these maxima show strong directionality, as they occur only for the close to zero y–wave vector component (along the long axis of the stripes), which is in the agreement with reciprocal distribution of the electric field, seeFig.2C. This behavior can be exploited to gain directional and magnitude sensitivity in the BLS process. The transfer matrix components Txy and Tzy , which couple to the mz and mx magnetization components, respectively, are approximately two orders of magnitude larger than the remaining transfer matrix components. However, depending on the exact geometry of the incident electric field, the direction of the static magnetization, the coupling mechanism and the ellipticity of the spinwave mode, the absolute contribution to the resulting signal can vary. For example, if the static magnetization is aligned along the x direction and only linear magnetooptical coupling is assumed, the contribution of the Tzy component of the transfer matrix function to the resulting signal would be zero. The width of the illumination spot determines the wave vector resolution (the width of the periodic peaks in the transfer matrix components). The larger the illuminated area, the better wave number resolution can be achieved. In our experiment, we have used a 440nmwide Gaussian spot, resulting in a wave vector resolution of approximately 10 rad μm−1. The resolution is independent of the order of the peak of transfer function and of the periodicity of the Pm( r,ω+ω m) =𝛘 m( r,ω m) E dr (r,ω ) (1) pv(rdet)⋅Em(rdet)= ∫ dr3Pm(r)⋅Ev(r ) (2) σ( km,ωm ) ∼| | |pv ( rdet ) ⋅Em ( rdet ) | | | 2 = =| | | | | | ∑ i,j∫dzχij m(z,ωm)Tij(km,z)| | | | | | 2(3) T ij(km,z)= ∫ dr2 ∥Ei v(r)Ej dr(r)eikm⋅r ∥ (4) Downloaded from https://www.science.org at Brno University of Technology on December 03, 2025 Krčma et al., Sci. Adv. 11, eady8833 (2025) 31 October 2025 Science AdvAnceS | ReSeARch ARticle 4 of 9 nanostripe array. The increase in wave vector resolution can only be achieved at the expense of spatial resolution and vice versa. InFig.3B, this transfer function has been used to model the BLS signal and compare it directly with the experiment. We can observe the formation of peaks at the frequencies given by the dispersion relation (seeFig.1A) matching with kvector sensitivity imposed by the periodicity of the dielectric nanostripes ( A=150 nm) (see Figs.2C and3A). The intensity of the μBLS accessible mode on a bare film ( k≲10 rad μm−1, n=0 ) is strongly suppressed compared to the peak with matching periodicity to the periodic dielectric nanostripes ( k=2π∕A and n=0 ). The intensity of the matching higherorder peaks ( k=4π∕A , k=6π∕A , etc.) decays exponentially, so only the first two orders are visible in this representation. However, thanks to the high contrast of the tandem FabryPerot interferometer (150 dB), even very small contributions can be resolved experimentally. Also note that the process is robust and fabrication imperfections do not noticeably affect the resulting spectra; see micrographs of nanostripe arrays in the sample preparation section in Materials and Methods. At higher frequencies, we can also observe peaks caused by the firstorder perpendicular standing spin waves (PSSWs; n=1 ). There, we can see a small frequency offset between the model and the experimental data, which will be discussed later in the text. To verify the origin of the signal, we measured the BLS spectra on the same dielectric stripes ( A=150 nm) in different external magnetic fields and compared them with a standard μBLS measurement on the bare film (seeFig.4A). On the bare film (Fig.4A, left), only spin waves around the center of the Brillouin zone ( k=0 ) are visible for both PSSW modes. However, as mentioned before, with the periodic dielectric nanostripes, we resolve spin waves with wave numbers corresponding to the first two multiples of 2π∕A . For both experiments, we can clearly observe the expected shift of the spin wave frequencies when increasing the magnetic field, which agrees well with the analytical formula for spin wave dispersion and does not change by the introduction of the nanostripe array (44,45). Also note, that the intensity of the modes up to fourth order is higher or comparable to the signal strength measured on the bare film. To extract the dispersion relation of the NiFe film, we performed a similar experiment as inFig.3B. We used multiple nanostripe arrays with periodicities ranging from 150 to 300 nm and measured BLS signal in low (50 mT) and high (550 mT) magnetic fields. The arrays with different periodicities have a robust optical response across different wavelengths and their dimensions are limited only by nanofabrication capabilities (see note S2). Also, note that the measured wave vector of the spin waves is determined by the periodicity of the nanostripe array and is not influenced by the illumination light wavelength or NA of the used objective lens. By analyzing the measured peak positions, we were able to reconstruct the dispersion relation (Fig.4B). Please note that if more peaks coincide in the frequency, we were not able to resolve them and we only associated its frequency position with the wave number with the highest expected signal; for details, see Materials and Methods. With just eight separate measurements, we were able to obtain a wide wave number range spanning from dipolar to exchange dominated spin waves. The experimentally obtained results were fitted with theoretical dispersion relations (44,45). The wide range of measured wave numbers allowed us to extract from a single fit the saturation magnetization Ms=(700 ±20) kA/m, the gyromagnetic ratio γ∕2π=(30.4 ±0.4) GHz/T, and also usually strongly correlated material parameters, the exchange constant Aex =(8.1 ±0.3) pJ/m and the film thickness t=(26.0 ±0.3) nm. However, the model has a frequency offset (approximately 1.5 GHz) from the experimental data for modes n=1 and k=0 . This may be due to more complicated boundary conditions than totally unpinned spins used in the modeled dispersion relation. To demonstrate the directional sensitivity of the BLS process on periodic dielectric nanostripes, we rotated the sample along the z axis while keeping the direction of the external magnetic field and the incident electric field constant, and acquired the azimuthal dependence of BLS spectra. Because of the induced periodicity of the electric field in the direction of the periodic nanostripes, the direction of the detected wave vector followed the angle of the rotation of the sample. However, it should be noted that minor changes in intensity and resolution might occur (see note S3). For each angle, the frequency positions of magnon modes with specific wave numbers were extracted (Fig.4C). Regarding the fundamental spin wave mode ( n=0 , solid line) we can observe a strong anisotropy in the dispersion relation caused by magnetic dipolar interaction. For the first PSSW ( n=1 , dashed line), where the dipolar interaction is much weaker compared to the exchange interaction, the angular dependence of spin wave frequencies is almost isotropic. The obtained data agree well with the analytical models (44,45). In conclusion, we have demonstrated a method for measuring magnons in a previously inaccessible wave vector range using 01.210 kx −1)kx −1)kx −1) ky −1)ky −1)ky −1) mz 02.910 my 200 0 01.610 mz A 01.210 mx 0200 200 0 07.510 my 0200 01.510 mx 200 0 0 0 0200 B 0 10 20 30 40 50 20 40 60 80 100 120 103 BLSsignal(cts) Frequency(GHz) Exp. data Sim. data non-magnetic k=0 n=0 (FMR) kA n=0 k A n=0 k=0 n=1 k A n=1 k A n=1 Fig. 3. FDTD simulation data and results for a dielectric array with periodicity A=150 nm. (A) transfer function tensor components, with each panel labeled by the corresponding dynamic magnetization component that it interacts with (assuming only linear magnetooptical coupling). the diagonal components are not computed, as they do not contribute to the final signal. (B) Simulated BlS spectra compared with experimental results. Peaks are annotated with their corresponding wave numbers and perpendicular standing spin wave (PSSW) modes. Downloaded from https://www.science.org at Brno University of Technology on December 03, 2025 Krčma et al., Sci. Adv. 11, eady8833 (2025) 31 October 2025 Science AdvAnceS | ReSeARch ARticle 5 of 9 Mieenhanced microfocused Brillouin light scattering. By integrating periodic dielectric nanostripes into a standard μBLS experiment, we achieved full wave vector resolution, enabling the detection of magnons with wavelengths down to 30 nm (wave vectors up to 200 rad μm−1). This technique bridges a longstanding experimental gap, providing opportunities for investigating fundamental spinwave physics across the entire Brillouin zone or for imaging spinwave propagation in nanoscale spinwave devices. Our results establish a robust method for characterizing nanoscale magnons in space, wave number, and frequency with unprecedented resolution, validated through both experiments and theoretical modeling. Beyond magnonics, the methodology developed here is also applicable to other collective excitations, such as phonons, potentially opening avenues in fields ranging from ultrafast magnetism to mechanobiology (46,47). These can include, for instance, wave vector resolved measurement of spin currents or of the mechanical properties of tissues and cells in the range of considerably larger wave vector values, which is crucial for modeling across all material parameters. Moreover, this technique could enable the investigation of action potential propagation in a membrane, as it allows measuring acoustic excitations with shorter wavelengths than with standard μBLS. The ability to resolve excitations with nanoscale precision and directional sensitivity could drive further advancements in the scope of wave vector–resolved spectroscopy in condensed matter physics. MATERIALS AND METHODS Microfocused Brillouin light scattering Brillouin light scattering is a process in which incident photons are inelastically scattered by quasiparticles such as magnons or phonons, resulting in the creation of photons with different energy and momentum. The process obeys conservation of energy and inplane momentum, which can be expressed as where ωi and ki are the angular frequency and inplane wave vector of the incident photon, ωf and kf correspond to the scattered photon, and ωm and km to the magnon. In a typical μBLS experiment, the accessible magnon wave vector range in thinfilm measurements is determined by the wavelength λ ωf=ω i±ω m (5) kf=ki±km (6) 04080120 160200 0 10 20 30 40 50 0 10 20 30 40 50 60 70 80 90 0 5 10 15 20 25 30 25 30 35 40 45 50 Frequency(GHz) Wave number n=0 n=1 k=0 k A k A k A k A k A k A TR-STXM k-resolvedBLS 100200 300400 500 Magnetic field(mT) 0 311 BLSsignal (cts) A=150 nm kA,n= 0 kA,n=0 kA,n= 1 kA,n= 1 100 200300 400500 5 10 15 20 25 30 35 40 45 Magneticfield (mT) Frequency(GHz) k= 0, n= 1 k= 0, n= 0 Bare film Frequency(GHz) AB deg B ext || k Bext k Bextk k C Frequency(GHz) Bext=50mT Bext=550 mT Bext 1000 100504030 Wavelength (nm) Fig. 4. Experimental results measured on arrays with varying periodicity A. (A) comparison of 2d BlS spectra (frequency versus magnetic field) for a bare magnetic film and a dielectric array. the presence of the array leads to additional accessible magnon wave numbers and their higherorder thickness modes. the curves represent theoretical calculations using the material constants obtained from (B). (B) Magnon dispersion relation at 550 mt (top panel) and 50 mt (bottom panel) obtained from arrays with periodicities ranging from 150 to 300 nm, showing discrete modes (data points) fitted with theoretical dispersion curves using the KalinikosSlavin model (44,45). the color code of the experimental points represents order of multiples of the periodicity of the analyzed peak. the periodicity of the nanostripe array can be deduced from its order and from the position of the point. (C) Angular dependence of the frequency of magnons with specific wave number on the angle between the external magnetic field and the magnon wave vector. the data were acquired on the grating with A=150 nm. the polarization of the incident laser light was perpendicular to the magnetic field. the curves represent theoretical predictions calculated using the fitted material parameters from (B). Downloaded from https://www.science.org at Brno University of Technology on December 03, 2025 Krčma et al., Sci. Adv. 11, eady8833 (2025) 31 October 2025 Science AdvAnceS | ReSeARch ARticle 6 of 9 of the probing light and the NA of the focusing optics. Together they define the range of incident angles. However, the wave vector range predicted by this geometric consideration does not fully match our experimentally observed detection sensitivity. Instead, the detection is better described by an analytical expression based on the Fourier transform of the beam spot, as discussed in (10), and approximated by where HWTM is the halfwidth at 10th maximum of the incident laser intensity in reciprocal space, roughly corresponding to the detection range. In this paper, we used a customdeveloped optical setup to measure Brillouin light scattering spectra (10,36). A singlemode laser (COBOLT Samba) with a wavelength of 532 nm was used as the light source. The spectral purity of the laser light was improved by a FabryPerot filter (TCF2, Table Stable). The incident power on the sample was 3 mW, which did not introduce any visible nonlinear phenomena or heating of the sample. An optical microscope with active stabilization was used to compensate the mechanical drifts of the sample (THATec Innovation). The light was focused and collected through the same objective (Zeiss LD EC EpiplanNeofluar 100 ×/0.75 BD). The inelastic frequency shift was measured with a tandem FabryPerot interferometer (TFP2HC interferometer, Table Stable) (30). To generate the magnetic field, we used a watercooled GMW 5403 electromagnet powered by two KEPCO BOP2020DL power supplies and a predefined current field calibration at the sample position. FDTD simulations Simulations were conducted using the Ansys Lumerical FDTD Solutions software. The simulation region was set to 8000×8000×1150 nm3, with the shortest dimension aligned along the optical axis. The model consisted of a semiinfinite Si substrate coated with a 27.6-nmthick permalloy film. On top, amorphous Si nanostripes with a periodicity of 150 nm were arranged, covering the entire simulation area. Amorphous Si was selected because of its closer optical resemblance to sputtered Si. The global mesh was set to conformal variant 0 (mesh order 4), with a refined 2nm mesh in the central simulation region (1024× 1024×96 nm3) in the vicinity of the nanostripes, approximately corresponding to the laser spot size. Perfectly matched layers were used as boundary conditions, and appropriate symmetry conditions were applied. A Gaussian source, modeled using the thinlens approximation with a NA of 0.75, was focused on the top of the center stripe. The source polarization was set either perpendicular or parallel to the nanostripes. The dielectric functions for the Si substrate, amorphous Si, and permalloy were obtained from (48–50). Electric field data were collected using a field monitor positioned in the middle of the permalloy film and analyzed with MATLAB 2024a. Sample preparation The sample was fabricated by electron beam lithography and a liftoff process. We started with a Si(100) substrate with a native SiO2 layer. A Ni80Fe20 film (approximately 30 nm thick) was then deposited on the sample by ebeam evaporation at room temperature. For the lithography process, two layers of polymethylmethacrylate resist were spincoated: a 200nm layer of Allresist ARP 649.04 followed by a 60 nm layer of Allresist ARP 679.02. Pattern exposure was performed using a RAITH 150two electron beam writer, followed by development of the exposed resist. Oxygen reactive ion etching was performed to improve adhesion for the subsequent Si deposition. Si was deposited by ionbeam sputtering at room temperature using a crystalline Si target. The final liftoff process involved immersing the sample in an acetone bath for approximately 24 hours, followed by rinsing with acetone and isopropanol to complete the fabrication. The scanning electron microscope images of nanostripe arrays with periodicities of 150 and 200 nm are shownFig.5 (A and B). Spectra analysis The BLS spectra taken on the periodic dielectric nanostripes yields multiple peaks at various frequency positions, which correspond to the different orders of detection or spin wave modes, seeFig.3B. Therefore, the spectrum needs to be carefully analyzed. First, the frequency positions of all peaks are found and labeled with the corresponding order of detection by periodic array and spin wave mode. For wellseparated, high signaltonoise ratio peaks (Fig.6A), we fit individual Gaussian curve to each peak without any further processing. We used a Gaussian curve instead of a Lorentzian one to model the spectra, because its shape is mainly determined by the Gaussian profile of the probing beam spot. However, for overlapping low signaltonoise ratio peaks (above 35 GHz;Fig.6B), we fit the sum of multiple Gaussian curves and an exponential background to account for quasielastic scattering and localized acoustic modes. It is important to note that the spectra inFig.6 had a 24hour acquisition time. The highest measured wave number (approximately 200 rad μm−1) has still a signaltonoise ratio bigger than 3 and thus can be distinguished. We estimate that with substantially shorter acquisition times (tens of minutes), it would not be clearly distinguishable. After determining peak positions, we proceeded to match them to their corresponding spin wave mode and wave number. A critical consideration in this process is that signals corresponding to lower wave numbers (lower order of detection) typically appear stronger in BLS spectra (this does not apply to ferromagnetic resonance and PSSW at k=0 ). Using this knowledge, we compared the observed peak positions with the predicted dispersion relation of the material to establish frequency–wave number pairs. HWTM ≈6.72 NA λ (7) AB 3 µm Fig. 5. Scanning electron microscope images of fabricated structures. the periodicity is 150 (A) and 200 nm (B). Downloaded from https://www.science.org at Brno University of Technology on December 03, 2025 Krčma et al., Sci. Adv. 11, eady8833 (2025) 31 October 2025 Science AdvAnceS | ReSeARch ARticle 7 of 9 This matching process becomes considerably more complex in thicker films due to the emergence of higherorder PSSW modes. This complexity accounts for the gaps present in our final dispersion relation (Fig.4B). When two different wave numbers, one corresponding to n=0 and another to n=1 , produced overlapping peaks at similar frequencies that could not be clearly distinguished, we attributed the peak to the lower inplane wave number magnon ( n=1 ). BLS spectra modeling using the reciprocity theorem of electromagnetism The task of calculating the intensity of the BLS signal collected by a detector can be elegantly completed using the reciprocity theorem (43), where we exploit a close link between the fields generated by an electric dipole ( p ) (emulating the polarization current induced within the magnetic layer) and a virtual source located at the position of the detector, see supplementalFig.7. Mathematically, this connection can be written down as where the subscripts differentiate whether the particular field or dipole moment is associated with the magnetic source (m) or the virtual source (v). Note that the two dot products are evaluated at different positions in space corresponding to the respective locations of the sources (detector and magnetic sample). The above relation can be translated into a simple statement: If a photon emitted from the detector is able to reach a particular spot on the sample, then it should work also the other way around. Although this is not true in general [e.g., in gyrotropic materials (51)], the materials in our sample are linear and isotropic (at least with respect to the outcoupling process that follows the inelastic scattering event) and the reciprocity theorem is valid. While the amplitude of the virtual dipole source can be set arbitrarily, its orientation should match the experimental conditions under which the BLS signal is collected. In presented experiments and calculations, we use a crosspolarized scheme, which is sensitive to the spin wave BLS signal in linear case (10). The virtual electric field should capture all the effects arising from the propagation between the detector and the sample, including any interaction with the Si stripes or surrounding media. We assume that the collection and illumination spots are identical with a width of 440 nm and a Gaussian shape (36). This is justified, because both branches share the same objective lens and are perfectly aligned. Taking all this information as new settings for our FDTD model, we calculated the distribution of the virtual electric field within the magnetic layer. To account for the continuous nature of the induced polarization Pm (r)=𝛘 ( r,ω m) E dr (r ) (implicitly assuming linear magnetooptical coupling), the righthand side ofEq.8 should be replaced by a volume integral over the magnetic layer Restricting our analysis to laterally propagating magnons, the magnetic susceptibility tensor can be cast as where km denotes the magnon wave vector and the symbol ∥ indicates the lateral (inplane of the magnetic sample) spatial coordinates. pv( r det) ⋅E m( r det) =p m (r)⋅E v (r ) (8) pv(rdet)⋅Em(rdet)= ∫ dr3Pm(r)⋅Ev(r) = =∑ i , j ∫dr3χij m(r)Ei v(r)Ej dr(r) (9) 𝛘m( r,ω m) =𝛘 m( z,ω m) e ir m⋅ r∥ (10) 35.0 37.5 40.0 42.5 45.0 47. 55 0.0 0 500 1000 1500 2000 2500 3000 Frequency(GHz) Nonmagnetic Background Summation /A,n=1 /A,n=1 /A,n=0 BLSsignal(cts) 10 20 30 40 50 500 140,000 BLSsignal(cts) FMR /A,n=0 /A,n=0 ,n=0 Frequency(GHz) A B B /A,n=1 /A,n=1 Fig. 6. Measured and analyzed BLS spectrum on a grating with A=150 nm. Peaks are fitted with Gaussian curves and matched with corresponding magnon wave numbers. (A) shows high signaltonoise ratio peaks, while (B) shows low signaltonoise ratio peaks. SAMPLE A DETECT OR pv(rdet) LASER Edr(r ) Ev(r) Em(rdet) Pm(r)=χ(r)Edr(r) VIRTUAL SOURCE INDUCED POLARIZATION Fig. 7. Illustration of the reciprocity theorem concept. it highlights the key physical quantities and their polarization states, parallel, perpendicular, and arbitrary (elliptic). the letter A denotes an analyzer. Downloaded from https://www.science.org at Brno University of Technology on December 03, 2025 Krčma et al., Sci. Adv. 11, eady8833 (2025) 31 October 2025 Science AdvAnceS | ReSeARch ARticle 8 of 9 The variation of the susceptibility tensor with respect to the vertical coordinate z will generally depend on the thickness of the magnetic layer and the PSSW order. Since we are studying metallic layers (NiFe), where the penetration depth of light ( ≈20 nm) is comparable or shorter than the thickness of the sample, we neglect this dependence in our analysis. InsertingEq.10 intoEq.9 and recalling that the detected BLS signal is proportional to the intensity of the collected light, we find the contribution from a single magnon with a wave vector km to be The above equation indicates that the strength of the BLS signal is largely determined by the overlap integral between the driving and virtual electric field distributions modulated at the frequency of the magnon. It can be perceived as a transfer function of the system that determines the range of detectable magnons and its inspection can be of great value for both the design process and the subsequent analysis of experimental measurements. Supplementary Materials This PDF file includes: note S1 to S3 Figs.S1 to S3 REFERENCES AND NOTES 1. Q. Wang, G. csaba, R. verba, A. v. chumak, P. 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Funding: this research was supported by the project no. cZ.02.01.010022_0080004594 (teRAFit) and by the Grant Agency of the czech Republic, project no. 2304120l. czechnanolab project lM2023051 is acknowledged for the financial support of the measurements and sample fabrication at ceitec nano Research infrastructure. Author contributions: conceptualization: O.W., M.U., and J.K. data curation: J.K. Formal analysis: J.K. and O.W. Funding acquisition: M.U. investigation: J.K., O.W., and J.A.A. Methodology: O.W., M.h., and J.K. Project administration: M.U. Resources: J.K., J.h., O.W., J.P., and e.P. Software: J.K., M.h., and O.W. Supervision: M.U. validation: J.K. and O.W. visualization: J.K. Writing–original draft: O.W., J.K., M.h., and J.A.A. Writing–review and editing: M.U., O.W., J.K., M.h., J.A.A., and J.h. Competing interests: the authors declare that they have no competing interests. Data and materials availability: All data and code used to generate the presented figures are available in the Zenodo repository (dOi: 10.5281/zenodo.14801809). All other data needed to evaluate the conclusions in this paper are present in the paper and/or the Supplementary Materials. Submitted 9 May 2025 Accepted 26 September 2025 Published 31 October 2025 10.1126/sciadv.ady8833 Downloaded from https://www.science.org at Brno University of Technology on December 03, 2025