Abstract
Local probing of dynamic excitations such as magnons and phonons in materials and nanostructures can bring new insights into their properties and functionalities. For example, in magnonics, many concepts and devices recently demonstrated at the macro and microscale now need to be realized at the nanoscale. Brillouin light scattering (BLS) spectroscopy and microscopy has become a standard technique for spin wave characterization, and enabled many pioneering magnonic experiments. However, the conventional BLS cannot detect nanoscale waves due to its fundamental limit in maximum detectable quasiparticle momentum. Here we show that optically induced Mie resonances in nanoparticles can be used to extend the range of accessible quasiparticle’s wavevectors beyond the BLS fundamental limit. These experiments involve the measurement of thermally excited as well as coherently excited high momentum magnons. Our findings demonstrate the capability of Mieenhanced BLS and significantly extend the usability of BLS microscopy for magnonic and phononic research.
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Introduction
Advances in Brillouin light scattering microscopy (BLS)^{1,2,3}, and spectroscopy^{4} significantly helped magnonics to become one of the most promising alternatives to complementary metaloxidesemiconductor (CMOS) technologies^{5,6}. For device miniaturization, a shift towards shortwavelength, highmomentum magnons is necessary. So far, the BLS techniques could not satisfy this requirement due to their fundamental limit in maximum detectable magnon momentum^{6,7,8,9,10}. This limit is given by the law of conservation of momentum in the Stokes process: k_{i} = k_{s} + k_{mag}, where k_{i} and k_{s} are kvectors of the incident and scattered light and k_{mag} is the kvector of the magnon on which the light is being scattered. It means that in a typical BLS experiment in backscattering geometry, the maximal detectable kvector of spin waves equals twice the kvector of the incident light. For the laser wavelength λ_{i} = 532 nm, for example, the maximum kvector that can be theoretically detected is \({k}_{{{{{{{{\rm{mag}}}}}}}}}^{\max }=23.6\,{{{{{{{\rm{rad}}}}}}}}\) μm^{−1}, which corresponds to a minimum spinwave wavelength \({\lambda }_{{{{{{{{\rm{mag}}}}}}}}}^{\min }={\lambda }_{{{{{{{{\rm{i}}}}}}}}}/2=266\,{{{{{{{\rm{nm}}}}}}}}\)^{1,2}. Taking an inspiration from tip and surfaceenhanced Raman scattering spectroscopy^{11,12,13}, nanosized apertures or other plasmonic structures made of metals have been used to locally enhance the electromagnetic field and increase the range of the accessible kvectors^{14,15,16}. Unfortunately, the efficiency of the plasmonic approach is severely limited by high optical losses in metallic structures which makes it unsuitable for convenient magnon measurements (see^{16} and Supplementary Fig. 1). However, recent advances in nanophotonics suggest that plasmonic structures made of metals can be substituted by structures made of dielectric materials. Such dielectric nanoresonators have an advantage in reduced dissipative losses and associated heating, while their high refractive index still enables comparatively strong light confinement^{17,18,19,20,21,22,23}.
Here we show that optically induced Mie resonances in dielectric nanoparticles can be used in magnonic BLS experiments to increase the magnon signal and to enhance the range of accessible kvectors beyond the fundamental limit of conventional BLS microscopy setup (μBLS)^{1}. We use simple silicon disks, which support Mie resonances^{24,25,26,27} with strong and localized electric fields (hot spots). When the incident light with momentum k_{i} is restricted to the subdiffraction hot spots, its momentum becomes complex \({k}_{{{{{{{{\rm{nf}}}}}}}}}={k}_{{{{{{{{\rm{nf}}}}}}}}}^{{\prime} }+{{{{{{{\rm{i}}}}}}}}{k}_{{{{{{{{\rm{nf}}}}}}}}}^{{\prime\prime} }\) and thus its real part \({k}_{{{{{{{{\rm{nf}}}}}}}}}^{{\prime} }\) can be larger than the momentum of the freespace light^{28} (see Fig. 1a). This way, the fundamental limit of BLS in maximum detectable magnon momentum can be overcome.
In our experiments, we have investigated the fundamental mode of spin waves in a 30 nm thick permalloy film, on top of which we fabricated arrays of 60 nm thick silicon disks and the diameters ranging from 100 nm to 1.5 μm. The sample was measured on a standard μBLS, using a microscope objective lens with NA = 0.75 to illuminate it by λ_{i} = 532 nm coherent laser light (Fig. 1b). The scattered photons were collected by the same objective lens and guided to a tandem FabryPerot interferometer (TFPi). Prior to the measurement on the silicon disk, we verified the kvector detection limit of our setup on a bare permalloy film. In contrast to the theoretically predicted value of \({k}^{\max }={{{{{{{\rm{NA}}}}}}}}\cdot 23.6\,{{{{{{{\rm{rad}}}}}}}}\) μm^{−1} = 17.7 rad μm^{−1}, the estimated value for our setup turned out to be \({k}^{\max }=10\,{{{{{{{\rm{rad}}}}}}}}\) μm^{−1}, in agreement with the experimentally measured laser spot size (440 nm, see Supplementary Fig. 2). A more detailed description of the sample fabrication and the μBLS setup is given in the Methods section.
Results
Improvement of highk magnon detection sensitivity
The dramatic improvement of highk magnon detection sensitivity in the presence of a 175 nm wide silicon disk is visible in Fig. 2. We can see that at low magnetic field of 50 mT (Fig. 2a) the BLS signal increases, and the spin wave band broadens towards higher frequencies. At high magnetic field of 550 mT (Fig. 2b), we can again see the increase of the BLS signal, and the spin wave band now broadens to both sides.
Different broadening of the spin wave band at high and low magnetic fields can be explained by different shapes of the spin wave dispersion relations of the permalloy thin film for different directions of kvector k with respect to the direction of magnetization vector M^{29,30}. The upper part of the spin wave band is limited by the DamonEshbach mode (k⊥M, DE, Fig. 2a, b light blue solid line), which rises at both values of the external field, and for exchange dominated (highk) spin waves converges towards quadratic dependence of frequency f ∝ k^{2}. Hence, the shift of the right edge of the detected spinwave band towards higher frequencies always means an enhanced sensitivity to spin waves with higher kvectors. The left edge of the spin wave band is limited by the backward volume mode (k∥M, BV, Fig. 2a, b, dark blue dashed line), which first decreases in frequency for dipolar (lowk) spin waves and then increases for exchange dominated (highk) spin waves (and again converges towards quadratic dependence f ∝ k^{2}). In the low magnetic field, the exchange interaction prevails already for the spin waves with k ≈ 10 rad μm^{−1} and the drop in the frequency for the BV mode is only 0.2 GHz. This results in a sharp increase of the BLS signal at the left edge of the spin wave band. The sharp increase is the same for both measured spectra (with the silicon disk and on the bare film). The μBLS even without the presence of the dielectric nanoresonator can still detect spin waves with kvectors around the mode minimum at k ≈ 10 rad μm^{−1} and thus the complete lower part of the spin wave band is captured in both cases. In the case of high magnetic field, the onset of the exchange dominated spin waves occurs at much higher values of k, at approx. 30 rad μm^{−1}. Here, the BV mode is very pronounced, and the mode frequency decreases approx. 1.5 GHz down from the ferromagnetic resonance frequency (FMR, k = 0 rad μm^{−1}) before it starts rising again (Fig. 2b, dark blue dashed line). In this case, the μBLS on the bare film cannot detect spin waves above k = 10 rad μm^{−1} and capture the whole lower part of the spin wave band.
To quantify the enhancement of the BLS signal, we use a simple phenomenological model which assumes a Gaussian detection function Γ(k) in the kspace (see “Methods”)
The detection function has two parameters which we use for the quantification of the enhancement. Amplitude A, which represents the strength of the BLS signal, and halfwidthattenthmaximum HWTM, which represents the maximum detectable kvector. The third parameter bg representing the background signal is not used in further analysis. By multiplying this detection function by the spinwave density of states in the magnetic thin film \({{{{{{{\mathcal{D}}}}}}}}(f,{k}_{x},{k}_{y})\), obtained from a micromagnetic simulation, we can model the acquired BLS signal σ_{BLS}(f)
By fitting the parameters A and HWTM, we can obtain very good agreement between the model and the experimentally measured spectra for both the bare film and the silicondiskenhanced measurement (see black and red solid lines in Fig. 2a, b). In Fig. 2c, the detection function Γ(k_{x}) resulting from the fit to the experimental data measured at 50 mT is plotted for the bare film (black line) and for the measurement on the silicon disk (red line). This figure gives us a direct visualization of the enhancement of the detection sensitivity caused by the presence of the dielectric nanoresonator. The HWTM (i.e. the maximum detectable k) increased from 9.5 ± 1.0 rad μm^{−1} for bare film to 47 ± 3 rad μm^{−1} for silicon disk, whereas the A increased from 245 ± 6 cts to 259 ± 19 cts. Note that the increase in the amplitude of the Gaussian function does not represent the total increase of the integrated signal. If we integrate the detection function, we obtain the value of 2720 ± 170 cts rad μm^{−1} for the case of the bare film and the value of 14300 ± 1600 cts rad μm^{−1} for the silicondiskenhanced signal. This gives us an enhancement factor of 5.3.
Mie resonances are strongly dependent on the geometry of nanoresonators. To investigate this dependency, we have measured a series of silicon disks with diameters ranging from 100 to 300 nm. Figure 3a shows relative scattering intensities acquired by darkfield optical spectroscopy. We can see a characteristic red shift of the Mie resonances with increasing disk diameter. For 175 nm disk the peak resonance wavelength perfectly matches with the laser in our μBLS setup (532 nm). To quantify the dependence of the enhancement of the measured BLS spectra on the disk diameter, we fitted the HWTM (Fig. 3b), amplitude of the detection function A, and calculated integrated detection function (Fig. 3c) parameters for each disk diameter. From these data, we can see that the enhancement of both parameters starts appearing for disk diameters beyond approx. 125 nm and reaches its maximum for the diameters between 170 and 200 nm. When the disk diameter exceeds 200 nm, we observe a sharp decrease of both parameters.
Theoretical description of Mieresonance enhanced BLS
To further understand the mechanism of the enhancement, we implemented the socalled continuum model of inelastic scattering^{31}. In this model, the driving electric field E_{d} probes the dynamic modulation of the susceptibility via magnetooptical coupling, which gives the polarization inside the magnetic material as^{32,33,34}
where \(\hat{{{{{{{{\boldsymbol{\chi }}}}}}}}}={\hat{{{{{{{{\boldsymbol{\chi }}}}}}}}}}_{{{{{{{{\rm{mat}}}}}}}}}+{\hat{{{{{{{{\boldsymbol{\chi }}}}}}}}}}_{{{{{{{{\rm{SW}}}}}}}}}(t,{{{{{{{\bf{r}}}}}}}})\) is a sum of the static material susceptibility \({\hat{{{{{{{{\boldsymbol{\chi }}}}}}}}}}_{{{{{{{{\rm{mat}}}}}}}}}\) and of the additional dynamic contribution caused by spin waves \({\hat{{{{{{{{\boldsymbol{\chi }}}}}}}}}}_{{{{{{{{\rm{SW}}}}}}}}}(t,{{{{{{{\bf{r}}}}}}}})\). It should be stressed that the modulations in the susceptibility caused by magnons are on a vastly different time scale than the optical cycle of the probing photons. As a consequence, there is mixing of frequencies both in temporal and spatial domains, namely
where ω denotes the frequency of the induced polarization, k_{p} stands for its inplane wavevector (parallel to the permalloy layer), while ω_{m} and k_{m} represent their magnon counterparts.
The polarization vector in (4) acts as a local source of radiation that eventually forms the detected BLS signal. The contribution of a particular spatial frequency to this signal is determined by its ability to efficiently couple to the free space continuum and pass through the optical setup towards the detector. In the case of a bare permalloy layer without any particles, the range of spatial frequencies that can reach the detector is mostly limited by the numerical aperture of the used objective. Presence of a silicon disk (or any other perturbation) broadens this frequency space by providing an extra momentum that is required for highk_{p} fields to be scattered into the free space continuum. This scattering process thus represents an additional channel by which the information from local sources can reach the detector.
The transition from nearfield to farfield can be mathematically expressed using Green’s function formalism
The dyadic Green’s function \(\hat{{{{{{{{\bf{G}}}}}}}}}(\omega ,{{{{{{{{\bf{k}}}}}}}}}_{{{{{{{{\rm{p}}}}}}}}},{{{{{{{{\bf{k}}}}}}}}}_{{{{{{{{\rm{p}}}}}}}}}^{{\prime} })\) embodies the response of a system to a local source and it accounts fully for the presence of any scattering object or substrate effects. By inserting (4) into (5) and integrating over all spatial frequencies supplied by the driving electric field, the farfield angular spectrum becomes
Another important aspect of the BLS detection process is the limited area from which the signal is collected Assuming that the collection spot has a Gaussian spatial profile \(h(x,y)={{{{{{{{\rm{e}}}}}}}}}^{({x}^{2}+{y}^{2})/{w}_{{{{{{{{\rm{c}}}}}}}}}^{2}}\), the detectable portion of the farfield radiation amounts to
where the integration limits reflect the restrictions placed on the spatial frequencies by the numerical aperture of the objective lens.
Finally, to estimate the strength of the BLS signal at a particular frequency ω_{m}, one has to sum up contributions from all magnons (i.e. integrate over k_{m}). Hence, the modeled BLS signal reads
Inspecting the above expression, there are apparently many factors that can affect the resulting shape of the measured BLS spectrum, but the ability to access highk magnons has only two possible origins: scattering of highk field components into the free space continuum represented by \(\hat{{{{{{{{\bf{G}}}}}}}}}({{{{{{{{\bf{k}}}}}}}}}_{{{{{{{{\rm{p}}}}}}}}},{{{{{{{{\bf{k}}}}}}}}}_{{{{{{{{\rm{p}}}}}}}}}^{{\prime} })\) or the fact that silicon disks broaden the range of spatial frequencies making up the driving electric field E_{d}(k_{p}). While one of these processes occurs before and the other one after the magnon scattering event, both of them are facilitated by the silicon disks and their nearfields.
To obtain the distribution of the driving electric field inside the sample, we employed finitedifferencetimedomain (FDTD) simulations. We used Gaussian illumination with experimentally obtained beam parameters (see Supplementary Fig. 2). In the bare film, the electric field has a 2D Gaussian distribution (Fig. 4a), and its Fourier transform is shown in Fig. 4c. In the case of silicon disk, we observe the field localized in subdiffraction regions due to Mie resonance (see Fig. 4b). Consequently, the electric field spans over a larger area of the kspace exceeding values of 120 rad μm^{−1} (see Fig. 4d). We can observe higher localization (and thus higher kvalues) along the light’s polarization axis.
This model allows us to directly calculate the theoretical BLS signal (see Fig. 4e–f). In the case of bare film, the model perfectly matches to the experimental spectra. Figure 4b shows that silicon disk supports dipole Mie resonance (see also Supplementary Fig. 4 for disk cross sections). In reciprocal space (Fig. 4d) we can see how the electric field localization provides an extra momentum that is required for highk magnon detection. The modeled BLS spectra (4f) is in a good agreement with the experimental data until the frequency reaches approx. 30 GHz (which corresponds to k=30 rad μm^{−1}), above this value the model overestimates the BLS signal strength. The lower sensitivity to higher kvectors in real experiments can be caused e.g. by unstable laser spot position and imperfect polarization of the incoming laser beam.
In resonant photonic structures, the areas with locally enhanced electromagnetic field are often called hot spots as the temperature in their vicinity can rise by hundreds of Kelvins upon illumination. Such large increase in the temperature could change the magnetic properties of the studied material. This behavior is typical for plasmonic structures made of metals. To investigate how much can the silicon disk heat up the neighboring magnetic layer, we iteratively solved heat transfer and Maxwell equations. This calculation reveals, that for the used incident laser power of 3 mW, the temperature increase in the hot spots is max. 30 K (see Supplementary Fig. 6). This reduces the saturation magnetization of the permalloy by 0.6 percent, which in turn shifts down the spin wave band by approx. 20 MHz, which is far below the resolution of our TFPi with the mirror spacing set to 3 mm^{35}.
Directional sensitivity
To estimate how the laser spot positioning influences the measurement, we performed 2D mapping of (1 × 1) μm^{2} area around the nanoresonator, see Supplementary Fig. 5. To achieve the maximum enhancement, the laser spot has to be aligned with sub100 nm precision on the center of the nanoresonator.
To explore the role of the nanoresonators’ edges, we have measured BLS spectra with the laser spot focused on the right edge and then on the top edge of a large, 1500 nm wide, disk (see Fig. 5a). We can see that compared to the measurement on a bare film, the overall BLS signal is lower at both laser spot positions. Nevertheless, the enhancement of maximum detectable kvector is still present. Interestingly, when the laser beam is positioned on the right edge of the disk we can see broadening of the spin wave band towards higher frequencies (Fig. 5a, red line). When the laser beam is positioned on the top edge of the cylinder we can see broadening of the spin wave band towards lower frequencies (Fig. 5a, green line). This experiment suggests that it is possible to change the sensitivity to different spin wave kvector directions. On the right edge, we were more sensitive to DE spin waves (k⊥M), whereas on the top edge, we were more sensitive to BV spin waves (k∥M).
To quantify this directional sensitivity, we performed a BLS line scan from the disk center across the right edge of the disk (Fig. 5b). Then we fitted the measured data with Eq. (2), considering different HWTM parameters for k_{x} and k_{y}. The fitted detection function in Fig. 5e indeed confirms that the HWTM parameter and integrated detection function is enhanced only in the vicinity of the disk edge and only in the xdirection (Fig. 5c, d).
Detection of coherently excited spin waves
So far, we have investigated only thermal spin waves, but for many experiments in magnonics one needs to probe coherently excited spin waves with a defined kvector. To prove that our approach works also for coherent spin waves, we fabricated a sample with 180 nm wide microwave (RF) antenna in the vicinity of 200 nm wide silicon disks on the top of the permalloy thin film (see Supplementary Fig. 7). We connected the antenna to a RF generator with the excitation power set to 10 dBm, and swept the excitation frequency from 5 GHz to 17 GHz. The external magnetic field for this experiment was set to 50 mT and we probed the spin waves at the distance 1 μm from the antenna. In the case of the bare film (Fig. 6a), the excitation efficiency of the antenna is broader than the detection sensitivity, which is confirmed by the fact that the signal from coherent spin waves ends at the same frequency as the thermal background. Contrary, in the case of the silicon disk (Fig. 6b), the thermal background exceeds the coherent spin waves, which means that the coherent spin waves were limited by the excitation efficiency of the antenna. For the antenna with the width w = 180 nm the frequency cutoff is approximately w/2π which corresponds to the kvector of 29 rad μm^{−1}^{36,37}. This is consistent with the maximum detected frequency of 16 GHz, corresponding to k = 30 rad μm^{−1} (see Fig. 6c).
Conclusion
Our results show that with the use of dielectric nanoresonators it is possible to use the μBLS technique to detect thermal and coherently excited magnons with kvectors exceeding 50 rad μm^{−1} (which corresponds to spin wave wavelength λ = 125 nm). This is enhancement by a factor of 5 from the standard μBLS technqiue (k ≈ 10 rad μm^{−1}). We assert that achieving detection of spin waves with even higher kvectors is just a signaltonoise issue, as the simulated Fourier images of nanoresonatorenhanced electric field distributions show nonzero intensity even for k > 120 rad μm^{−1} (see Fig. 4). Spin waves with kvectors higher than 50 rad μm^{−1} could be measured using a source of coherent spin waves with better excitation efficiency at high kvectors^{7,38,39,40,41} as this should substantially improve the signaltonoise ratio. Also, the geometry and the material of the dielectric nanoresonator can be further optimized in order to shrink the hot spot size, or to excite higher order Mie or anapole resonances. This should lead to further increase of the enhancement factor^{42,43,44}. These findings elevate μBLS to the forefront of the nanoscale magnonics research and the possibility to probe materials with high momentum photons is relevant also for other applications, e.g. for phononic studies^{45} or even mechanobiology experiments^{46}.
Methods
Sample fabrication
Samples with silicon disks were fabricated by electron beam lithography and a liftoff process. We started with room temperature deposition of a 30 nm thick permalloy film onto a Si(100) substrate using ebeam evaporation from Ni_{80}Fe_{20} (at. %) pellets. Then we spin coated a doublelayer polymethyl methacrylate resist (200 nm thick Allresist ARP 649.04 200K and 60 nm thick ARP 679.02 950K, this resist combination provides sufficient undercut for liftoff). The patterns were written with RAITH 150two and Tescan Mira ebeam writers. Silicon film with the thickness of 60 nm was consequently deposited onto the patterned sample by RF magnetron sputtering from a crystalline silicon target at room temperature. The liftoff procedure consisted of immersing the sample in acetone for 8.5 h, followed by 30 s isopropanol rinse and blowdrying by nitrogen gas. The antennas for coherent excitation were prepared in a second lithographic step, again using the doublelayer resist and the liftoff process. The antennas consisted of a (10 nm SiO_{2}/85 nm Cu/10 nm Au) multilayer stack deposited all at once in the ebeam evaporator.
The sample with silver nanospheres was prepared by dropcasting from a commercial solution (nanoComposix) to a 30 nm thick permalloy layer covered by 2 nm of an insulating Al_{2}O_{3} spacer prepared by atomic layer deposition. We deposited 30 μl droplet of the solution containing silver nanospheres with the diameter of 200 nm to the sample surface and let the sample dry for 120 min. After that, the solution was rinsed in deionized water and blowdried by clean dry air.
After the fabrication, all samples were checked for their exact shape, size and uniformity by scanning electron microscopy (Tescan Lyra) and atomic force microscopy (Bruker Dimension Icon). The scanning electron microscope image of the sample with 175 nm wide silicon disks and 200 nm wide RF excitation antenna is shown in Supplementary Fig. 7.
BLS experiments
We used Cobolt Samba solid state laser with the wavelength of 532 nm and maximal optical power of 300 mW. The laser output pinhole was protected against the backreflections with Faraday optical isolator. Spectral purity of the laser light was improved by a FabryPerot filter (TCF2, Table Stable). After the filter, a small portion of the light was guided towards the TFPi to ensure its stabilization during the measurement. The rest of the light passed through a λ/2 waveplate and a polarizer. The λ/2 waveplate was mounted on a motorized stage and served for setting the incident laser beam power. Unless stated otherwise, we always set the optical power to the value of 3 mW. We used an optical microscope with active stabilization for compensation of the mechanical drifts of the sample (THATec Innovation). The incident laser light was guided through two 50:50 beam splitters towards Zeiss LD EC EpiplanNeofluar 100 × /0.75 BD objective lens. The scattered light (including the portion which underwent the BLS process) was collected by the same objective lens and guided through the same 50:50 beamsplitters back towards other λ/2 waveplate and then into the TFPi. We used TFP2HC interferometer (Table Stable)^{35} with the resolution of approx. 250 MHz with mirror spacing set to 3 mm. We used 450 μm input and 700 μm output pinholes. For the thermal spectra we acquired 2000 scans, while in the case of coherent excitation we measured 2928 scans. In both cases, the acquisition time was 1 ms for a single frequency bin. The acquisition of one thermal spectrum took approx. 10 minutes. To generate the magnetic field, we used a watercooled electromagnet GMW 5403 driven by two current sources KEPCO BOP2020DL. The magnetic field in the sample position was measured by a LakeShore 450 Hall probe.
Quantification of the BLS signal enhancement
To quantify the enhancement of the BLS signal, we developed a simple phenomenological model based on the following equation
where \({{{{{{{\mathcal{D}}}}}}}}(f,{k}_{x},{k}_{y})\) is the density of states of spin waves, Γ(k_{x}, k_{y}) is an instrumental detection function and σ_{BLS}(f) is the measured signal. We assume that the detection function Γ has a Gaussian form
where A is the strength of the measured signal, k_{x}(k_{y}) is the spinwave kvector in x(y) direction, and HWTM_{x}(HWTM_{y}) is half width at tenth of maximum of the detection sensitivity for spinwave kvectors in x(y) direction. The bg stands for the background signal, which could be caused by a dark current in the detector, inelastic scattering on the phonon modes or by stray light. To quantify the enhancement of the BLS signal in our data, we only fit A and HWTM.
The density of states \({{{{{{{\mathcal{D}}}}}}}}(f,{k}_{x},{k}_{y})\) was obtained from a micromagnetic simulation, using MuMax3 micromagnetic solver^{47}. The simulation size was set to 7680 × 7680 × 34.8 nm^{3} with cell size of 3.75 nm in inplane and 4.35 nm in outofplane direction. Periodic boundary conditions with 32 repetitions in both inplane directions were used. Saturation magnetization (M_{s} = 741 kA m^{−1}), gyromagnetic ratio (γ = 29.5 GHz T^{−1}), and exchange constant (A_{ex} = 16 pJ m^{−1}) used in the simulation were extracted from a ferromagnetic resonance measurements (see Supplementary Fig. 3). The initial magnetization m and the external field pointed in the x direction. The dynamics of the magnetization was excited by a 3D sinc pulse B_{sinc}(x, y, t), which is depicted in Supplementary Fig. 8a, b. The cutoff frequency was set to 60 GHz and the cutoff kvector to 150 rad μm^{−1}. The amplitude of the sinc function was 1 mT, and the pulse was delayed by 100 ps.
We let the magnetization evolve for 5 ns with the sampling interval of 8 ps. The obtained m_{z} is shown in Supplementary Fig. 8c, d. All three magnetization vector components were taken from the topmost layer, so we acquired a 3D array m(x, y, t). The used script code is available at^{48}. The m_{z} component was then transformed to the reciprocal space using a builtin FFT function in Matlab2021a. The obtained dispersion relation was compared with analytical calculation using KalinikosSlavin formula^{29,49}, as it is shown in Supplementary Fig. 8e, f. No windowing nor detrending was used. The resulting density of states was then obtained as
where n(f) is BoseEinstein distribution, which can be calculated as
This distribution at room temperature is shown in Supplementary Fig. 8g. Following Eq. 2 the detection function (Γ) with free parameters A, HWTM_{x}, and HWTM_{y} was multiplied by the density of states (\({{{{{{{\mathcal{D}}}}}}}}\)), integrated over kspace and fitted to the experimental data using Nonlinear Fitting Tool in Matlab2021a. The used function is available at^{48}. For all experiments, where the beam spot was positioned in the center of the silicon disk, we set k_{x} = k_{y} and fitted only one HWTM parameter.
Analytical modeling of the BLS spectra
To analytically model the BLS signal generated by a system containing a scatterer (e.g. a silicon disk), one requires several key ingredients that need to be combined and evaluated using Eq. (8): the distribution of the driving electric field E_{d}(r), the susceptibility tensor \(\hat{{{{{{{{\boldsymbol{\chi }}}}}}}}}({\omega }_{{{{{{{{\rm{m}}}}}}}}},{{{{{{{{\bf{k}}}}}}}}}_{{{{{{{{\rm{m}}}}}}}}})\), and the dyadic Green’s function \(\hat{{{{{{{{\bf{G}}}}}}}}}({{{{{{{{\bf{k}}}}}}}}}_{{{{{{{{\rm{p}}}}}}}}},{{{{{{{{\bf{k}}}}}}}}}_{{{{{{{{\rm{p}}}}}}}}}^{{\prime} })\). The complexity of the evaluation itself depends on a number of parameters, namely the numerical aperture of the objective lens (NA = 0.75), the size of the collection spot (w_{c} = 440 nm), and the size of the reciprocal space involved in the integrations over \({{{{{{{{\bf{k}}}}}}}}}_{{{{{{{{\rm{p}}}}}}}}}^{{\prime} }\) and k_{m}. Note that special care must be taken when choosing integration steps and limits, otherwise the demands on computational power can easily skyrocket.
The computational complexity can be partially mitigated by adopting certain simplifications. The vertical profile of the dynamic magnetization (along z), for example, largely depends on the exact geometry of the system and should be taken into account for precise calculations. For the purpose of our qualitative estimate of the BLS signal, it can be, however, disregarded. Similarly, the full distribution of the driving field E_{d}(r_{∥}, z) can be, for simplicity, replaced by its mean value over z (possibly augmented by a phase factor due to retardation). Since ω ≫ ω_{m}, one can also drop the explicit dependence of the driving field on ω_{m}. These approximations are reflected in the transition between Eqs. (4) and (5).

1.
The first key ingredient of our analytical model, the spatial map of the driving electric field E_{d}(r), is extracted from finitedifference timedomain (FDTD) simulations. Its further processing—such as the calculation of its Fourier transform E_{d}(k_{p}, z)—is then carried out in Matlab2021a.

2.
The dynamic susceptibility tensor \(\hat{{{{{{{{\boldsymbol{\chi }}}}}}}}}({\omega }_{{{{{{{{\rm{m}}}}}}}}},{{{{{{{{\bf{k}}}}}}}}}_{{{{{{{{\rm{m}}}}}}}}})\) acts as the source of inelastic scattering and captures the interaction between spin waves and photons. If we neglect higher order terms like the CottonMouton effect (which are usually small in magnetic metals^{50}), we obtain the following tensor
$$\hat{{{{{{{{\boldsymbol{\chi }}}}}}}}}({\omega }_{{{{{{{{\rm{m}}}}}}}}},{{{{{{{{\bf{k}}}}}}}}}_{{{{{{{{\rm{m}}}}}}}}})={Q}_{0}\left(\begin{array}{ccc}0&i{m}_{z}&i{m}_{y}\\ i{m}_{z}&0&i{m}_{x}\\ i{m}_{y}&i{m}_{x}&0\end{array}\right),$$where Q_{0} is the Voigt constant, and (m_{x}, m_{y}, m_{z}) are components of a normalized magnetization vector. Since we are not interested in the absolute values but rather in the shape of the signal, we can consider Q_{0} = 1. The magnetization vector is obtained from micromagnetic simulations in the same manner as for the phenomenological model (see Supplementary Fig. 8).

3.
The dyadic Green’s function \(\hat{{{{{{{{\bf{G}}}}}}}}}(\omega ,{{{{{{{{\bf{k}}}}}}}}}_{{{{{{{{\rm{p}}}}}}}}},{{{{{{{{\bf{k}}}}}}}}}_{{{{{{{{\rm{p}}}}}}}}}^{{\prime} })\) is unique for each geometry and it possesses a closed form analytical representation only in a handful of cases. One can, however, reconstruct it approximately from numerical simulations, as the dyadic Green’s function can be also viewed as a response of the system to a single plane wave excitation with a propagation vector \({{{{{{{{\bf{k}}}}}}}}}_{{{{{{{{\rm{p}}}}}}}}}^{{\prime} }\). The accuracy of this approximation then depends on the number of plane waves used for its reconstruction (in other words the size of the space spanned by \({{{{{{{{\bf{k}}}}}}}}}_{{{{{{{{\rm{p}}}}}}}}}^{{\prime} }\)). The clear drawback is that each value of \({{{{{{{{\bf{k}}}}}}}}}_{{{{{{{{\rm{p}}}}}}}}}^{{\prime} }\) corresponds to one simulation. Since our silicon disks possess a rotational symmetry, the total number of simulations is, fortunately, substantially reduced. Practical implementation of this approach in Lumerical FDTD—although one can employ, in principle, any simulation software—requires the use of a custom source (at least for \({k}_{{{{{{{{\rm{p}}}}}}}}}^{{\prime} } \, > \,{k}_{0}\)). The source is inserted into the silicon substrate and its amplitude and spatial profile are set in such a way that the injected wave has the desired \({{{{{{{{\bf{E}}}}}}}}}_{{{{{{{{{\bf{k}}}}}}}}}_{{{{{{{{\rm{p}}}}}}}}}^{{\prime} }}{{{{{{{{\rm{e}}}}}}}}}^{{{{{{{{\rm{i}}}}}}}}{{{{{{{{\bf{k}}}}}}}}}_{{{{{{{{\rm{p}}}}}}}}}^{{\prime} }\cdot {{{{{{{\bf{r}}}}}}}}}\) dependence after it enters the space above the permalloy film. The dyadic Green’s function \(\hat{{{{{{{{\bf{G}}}}}}}}}({{{{{{{{\bf{k}}}}}}}}}_{{{{{{{{\rm{p}}}}}}}}},{{{{{{{{\bf{k}}}}}}}}}_{{{{{{{{\rm{p}}}}}}}}}^{{\prime} })\) can then be identified as the Fourier component E(k_{p}) of the electric field distribution detected above the silicon disk.
Besides the parameters and input quantities listed above, there is another important aspect of the analytical model that should be addressed: the order of integration over k_{m} with respect to the square modulus operation occurring in Eq. (8). It is determined by the coherence properties of the magnon population and in the case of thermal magnons—which are inherently incoherent—the proper procedure is to add intensities of fields derived from magnons with different k_{m}.
Finitedifferencetimedomain electromagnetic simulation
Finitedifference timedomain (FDTD) calculations were performed using Lumerical’s FDTD Solutions software. The 3D simulation region spanned 5.3 × 5.3 × 1.36 μ m^{3}, with the shorter side oriented along the optical axis. Each model included a semiinfinite silicon substrate covered by a 30 nm thick permalloy thin film on top of which was a 60 nm thick silicon disk of varying diameter located at the simulation center. Staircase meshing (mesh order 4) was adopted everywhere except in the vicinity of the silicon disk, where it was fixed to 3 nm cells in all directions. Boundary conditions in the form of perfectly matched layers were used at all simulation boundaries, while the computation was accelerated by application of appropriate symmetry conditions. Gaussian source, implemented using the thin lens approximation, was focused onto the diskpermalloy interface from the air side with the waist diameter set to roughly 600 nm. The dielectric function of permalloy was taken from^{51} and the dielectric function of silicon was taken from^{52}. For the substrate, we used values corresponding to crystalline silicon, while amorphous silicon was assigned to disks to account for the different optical properties of sputtered silicon employed in lithography. The resulting electric field vector components were recorded by field monitors and further processed in Matlab2021a^{48}.
COMSOL multiphysics simulation of heat transfer and electromagnetic wave
The Gaussian beam scattering calculation on the substrate consists of 4 segregated interconnected parts:

1.
Calculation of the Gaussian beam in the air. In the background, a Gaussian beam with E_{x} polarization is excited. The solution is transferred to the next step using General Extrusion, where it generates the Gaussian beam at the in interface.

2.
Calculation of the Gaussian beam passing through the permalloy layer and the substrate. This wave serves as the background wave in the calculation of Gaussian beam scattering. The electric field strength of the Gaussian beam determined in Part 1 is prescribed at the in interface. The Scattering Boundary Condition is prescribed at the out interface, ensuring that the wave is transmitted out of the domain without reflection. The boundary condition Perfect Electric Conductor is prescribed on the vertical boundaries (they are far enough away from the spatially limited Gaussian beam). The solution, extended to the perfectly matched layer (PML) domain (where zero electric intensity is artificially prescribed), is used as the Background Field for the next calculation step.

3.
Calculation of the Gaussian beam scattering on the silicon disk on the permalloy layer and the substrate. PML is used to attenuate the scattered wave. Total Power Dissipation Density is used as the heat source in the next calculation step.

4.
Calculation of the temperature field in the silicon substrate, permalloy layer and silicon disk. Temperature of 20 ^{∘}C is prescribed at the out interface, the other interfaces are isolated (zero heat flux), and radiation to the surroundings is not considered either. The temperature field is again artificially extended to air and PML, where a temperature of 20 ^{∘}C is prescribed. The temperature field affects the value of the dielectric function (real and imaginary part) of the silicon.
These 4 steps are solved using the FrequencyStationary Solver. Quadratic elements are used in all calculations. The tetrahedral mesh in each domain has a step of λ/5, where λ is the wavelength of the wave in that domain.
Ferromagnetic resonance measurement
The fundamental mode of the ferromagnetic resonance (FMR) was measured by a broadband ferromagnetic resonance technique using a vectornetwork analyzer (Rohde&Schwarz ZVA50). We used socalled flipchip geometry, where the sample is placed on the excitation antenna with the magnetic layer facing down^{53}. We measured the scattering parameter S21 while sweeping the external magnetic field from 0.4 to 0 T. Then we subtracted the median value of the S21 parameter measured at all fields from the spectra measured at each single field^{36}. In these processed spectra, we were able to find the position of the ferromagnetic resonance by fitting the Lorentzian function, and to obtain the material parameters by fitting the HerringKittel formula^{30}.
Scattering measurement
Scattering intensity was measured by a singleparticle darkfield confocal spectroscopy. A hollow cone of white light from a halogen lamp was focused onto the sample by a darkfield objective lens (Olympus LMPLANFLN 100 × NA = 0.8). The backscattered light was collected by the same objective lens and then spatially filtered by a 200 μ m pinhole of a multimode optical fiber which subsequently guided the scattering signal to an entrance slit of a spectrometer (Andor Shamrock 303i equipped with an iDus DU420ABU camera). The relative scattering intensity was calculated as I_{sca} = (I_{nd} − I_{bg})/(I_{ref} − I_{bg}), where I_{nd} is the darkfield signal collected from a silicon disk, I_{bg} is the background, and I_{ref} is the signal from a spectrally uniform diffuse reflectance standard (Labsphere Inc.).
Data availability
The datasets generated during and/or analyzed during this study are available from the corresponding authors upon reasonable request.
Code availability
The Matlab code used to evaluate the BLS enhancement and to model the BLS spectra using the input from FDTD simulations is available on Github together with COMSOL mph file^{48}. The code used for the dispersion relation calculation is available on Github in a separate repository^{49}.
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Acknowledgements
CzechNanoLab project LM2018110 is gratefully acknowledged for the financial support of the measurements and sample fabrication at CEITEC Nano Research Infrastructure. O.W. was supported by Brno PhD talent scholarship. O.W. and J.H. acknowledge support from the project Quality Internal Grants of BUT (KInG BUT), Reg. No. CZ.02.2.69 / 0.0 / 0.0 / 19_073 / 0016948, which is financed from the OP RDE. F.L. acknowledges the support by the Grant Agency of the Czech Republic (2129468S). We acknowledge R. Dao and Nenovison company for help with AFM measurements.
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O.W. and M.U. designed the experiments, performed and evaluated the BLS measurements. M.H., O.W., J.Kr. and T.S. developed the theoretical scattering model. J.Kl., M.D., K.D., and M.S. prepared the samples and characterized them by SEM and AFM. F.L. and T.S. performed scattering measurements. J.H. and O.W. performed VSM and FMR measurements. J.Z. performed COMSOL multiphysics simulations. All authors contributed in writing and reviewing the paper.
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Wojewoda, O., Ligmajer, F., Hrtoň, M. et al. Observing highk magnons with Mieresonanceenhanced Brillouin light scattering. Commun Phys 6, 94 (2023). https://doi.org/10.1038/s4200502301214z
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DOI: https://doi.org/10.1038/s4200502301214z
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