Fast acquisition of spin-wave dispersion by compressed sensing
aa r X i v : . [ phy s i c s . op ti c s ] F e b Fast acquisition of spin-wave dispersion by compressed sensing
Ryo Kainuma , Keita Matsumoto , , and Takuya Satoh Department of Physics, Tokyo Institute of Technology, Tokyo 152-8551, Japan Department of Physics, Kyushu University, Fukuoka 819-0385, Japan (Dated: February 11, 2021)For the realization of magnonic devices, spin-wave dispersions need to be identified. Recently, the time-resolved pump-probe imaging method combined with the Fourier transform was demonstrated for obtaining thedispersions in the lower-wavenumber regime. However, the measurement takes a long time when the samplingrate is sufficiently high. Here, we demonstrated the fast acquisition of spin-wave dispersions by using thecompressed sensing technique. Further, we quantitatively evaluated the consistency of the results. Our resultscan be applied to other various pump-probe measurements, such as observations based on the electro-opticaleffects.
Spin waves are collective modes of spin precession in mag-netically ordered materials. They are considered promisinginformation carriers in the field of magnonics, because theycan propagate over a long distance without Joule heating [1–4]. Various devices such as spin wave switches [5], magnonic-logic circuits [6, 7], spin wave-assisted recorders, [5, 8], andlow-magnetic-field sensors [9] require the spatial control ofspin waves.The propagation characteristics of spin waves are manifestedin their dispersion relation. The higher-wavenumber regimeis governed by exchange interactions. In contrast, lower-wavenumber spin waves are governed by magnetic dipole inter-actions and are called magnetostatic waves [10–13]. Since themagnetostatic waves are suitable for long-distance propaga-tion, further investigation of dispersion relations in the lower-wavenumber regime is indispensable [2].Experimental techniques for acquiring dispersion rela-tions of spin waves are being actively studied. For exam-ple, inelastic neutron scattering [14, 15] and spin-polarizedelectron energy loss spectroscopy [16] have been demon-strated. However, these methods are suitable for observing thehigher-wavenumber region, rather than observing the lower-wavenumber region of the dispersion.Recently, a method called spin-wave tomography (SWaT),which used time-resolved pump-probe measurements and theFourier transform to visualize the dispersion relations of spinwaves in the lower-wavenumber region was demonstrated[17, 18]. Further, similar measurements in metals were per-formed using the magneto-optical Kerr effect [19]. In theSWaT method, a pump pulse is used to impulsively excite thespin wave, and a probe pulse with a time delay is used todetect the change in magnetization. By using an ultrashortpulsed laser as a pump pulse, spin waves in a wide frequencyrange can be excited simultaneously. Moreover, by focusingthe pulses, spin waves in a wide wavenumber range can beexcited. Wavenumber-resolved measurements can be madeby spatially scanning a sample with the focused probe pulses[19] or imaging a large area without focusing the probe pulses[17, 18]. Therefore, this method is useful for observing the dis-persions over a wide region in the wavenumber ( 𝑘 )–frequency( 𝑓 ) space.For observing the dispersion, a sufficiently high samplingrate must be maintained for the time-resolved measurement.This is because of the risk of folding noise due to the nature of the discrete Fourier transform (DFT). As a result, the measure-ment time can range from 10 hours to several days. To searchfor novel photo-induced dynamics in innumerable materials,the measurement time must be reduced.In recent years, in experiments such as the terahertz imag-ing [20], the NMR spectroscopy [21], and the scanning tun-neling microscopy and spectroscopy [22], it has been shownthat a method called compressed sensing can reduce the mea-surement time. Compressed sensing is a signal processingtechnique that allows the estimation of a signal from a smallamount of data [23]. The signal estimation can be achieved bythe least absolute shrinkage and selection operator (LASSO)method, a commonly used form of sparse regression [24].In this letter, we demonstrate the fast acquisition of the spin-wave dispersions by time-resolved pump-probe measurementsusing compressed sensing. Moreover, we quantitatively eval-uated the effect of reducing the number of measurements incompressed sensing on the results of observations.Our experimental setup is shown in Fig. 1. Our samplewas a single crystal of (111)-oriented 150 𝜇 m thick bismuth-doped rare earth iron garnet (Gd / Yb / BiFe O ) grown ona gadolinium gallium garnet substrate by the liquid-phase epi-taxy method. This magnetic material has been widely used forinvestigating laser-induced spin dynamics due to its strongmagneto-optical coupling [25–31]. The magnetic field of 𝐻 ext = 𝑥 -direction.The light pulse for this pump-probe measurement was gen-erated by a Ti:sapphire regenerative amplifier with a pulseduration of 70 fs and a repetition rate of 1 kHz. A circularlypolarized pump pulse with a central wavelength of 1300 nmwas focused along a line parallel to the 𝑦 -axis with a cylin-drical lens with a fluence of 80 mJ cm − . This pump pulseproduced an effective magnetic field in the 𝑧 -direction via theinverse Faraday effect [32], and the magnetization saturatedin the 𝑥 -direction tilted in the 𝑦 direction. Since the effectivemagnetic field was instantaneous, the magnetization then be-gan to precess in the 𝑦 - 𝑧 plane. The spin precession excitedalong the line-shaped pumping spots propagated perpendic-ular to the line via magnetic dipole interactions. A linearlypolarized pulse with a central wavelength of 800 nm was usedto probe the 𝑧 -component of magnetization 𝑚 𝑧 ( r , 𝑡 ) via theFaraday effect. The Faraday rotation angle was determinedfrom the angle of the analyzer, which minimized the intensityof the transmitted probe pulse detected by a complementary (cid:43)(cid:58)(cid:51) (cid:43)(cid:58)(cid:51)(cid:42)(cid:55)(cid:51) (cid:42)(cid:55)(cid:51)(cid:52)(cid:58)(cid:51)(cid:38)(cid:92)(cid:79)(cid:76)(cid:81)(cid:71)(cid:85)(cid:76)(cid:70)(cid:68)(cid:79)(cid:3)(cid:79)(cid:72)(cid:81)(cid:86) (cid:40)(cid:79)(cid:72)(cid:70)(cid:87)(cid:85)(cid:82)(cid:80)(cid:68)(cid:74)(cid:81)(cid:72)(cid:87)(cid:54)(cid:68)(cid:80)(cid:83)(cid:79)(cid:72) (cid:36)(cid:81)(cid:68)(cid:79)(cid:92)(cid:93)(cid:72)(cid:85) (cid:55)(cid:72)(cid:79)(cid:72)(cid:86)(cid:70)(cid:82)(cid:83)(cid:72)(cid:3)(cid:9)(cid:3)(cid:38)(cid:48)(cid:50)(cid:54) y xz FIG. 1. Setup of our pump-probe experiment. The circularly polar-ized pump pulse was focused along a line by a cylindrical lens. Thelinearly polarized probe pulse was irradiated on the entire samplewithout focusing. HWP: half-wave plate, GTP: Gran Taylor prism,and QWP: quarter-wave plate. metal–oxide semiconductor (CMOS) camera. The time delaybetween the pump and probe pulses was achieved by a variableoptical path difference using a delay stage. This was changedin increments of Δ 𝑡 = .
01 ns. The maximum value of the delaywas set to 𝑇 = .
60 ns. Then, we obtained the spatiotempo-ral waveform of the spin wave. The waveform was integratedalong the 𝑦 -direction to create a one-dimensional waveform 𝑚 𝑧 ( 𝑥, 𝑡 ) . The power spectrum that depicts the dispersion curvein the 𝑘 – 𝑓 space was obtained by the two-dimensional DFTof the spatiotemporal waveform. Further, micromagnetic sim-ulations were performed to confirm our experimental results(See supplementary data).Let 𝑁 = 𝑇 / Δ 𝑡 be the number of samples in the time domainof the time-resolved pump-probe measurement. Owing to thenature of the DFT, the frequency resolution of the spectrumis Δ 𝑓 = / 𝑇 . In addition, according to Nyquist’s theorem,the frequency components above 1 /( Δ 𝑡 ) cannot be observed,and they appear as folding noise. Therefore, with regard toreducing the measurement time, decreasing 𝑇 leads to a poorresolution, and increasing Δ 𝑡 increases the risk of the appear-ance of folding noise.Compressed sensing is a method of reducing 𝑁 by taking Δ 𝑡 randomly, without changing 𝑇 , and estimating the spectrumfrom such data. Since Δ 𝑡 is not a constant, the DFT does notwork well, resulting in spectral leakage. Instead, the spectrumcan be estimated by the LASSO method.In the LASSO method, the spectrum estimation is treatedas an inverse problem. Let y = ( 𝑦 ( 𝑡 ) , 𝑦 ( 𝑡 ) , · · · , 𝑦 ( 𝑡 𝑁 )) T be awaveform sampled with discrete time 𝑡 𝑖 ( 𝑖 = , , · · · , 𝑁 ) and x = ( 𝑎 ( 𝜔 ) , 𝑎 ( 𝜔 ) , · · · , 𝑎 ( 𝜔 𝑀 ) , 𝑏 ( 𝜔 ) , 𝑏 ( 𝜔 ) , · · · , 𝑏 ( 𝜔 𝑀 )) T are the cosine and sine components of the spectrum fordiscrete frequencies 𝜔 𝑗 ( 𝑗 = , , · · · , 𝑀 ) . The spectrum tobe estimated is the solution ˆ x of the following minimization problem ˆ x = arg min x (cid:26) 𝑁 || y − 𝐴 x || + 𝜆 || x || (cid:27) . (1)Arg min {·} denote the argument of the minimum, an elementthat minimizes the value in the brackets. || · || 𝑝 is a term calledthe 𝑙 𝑝 norm of the vector and is defined as || x || 𝑝 = (cid:0)Í 𝑖 𝑥 𝑝𝑖 (cid:1) / 𝑝 .The first term on the right-hand side corresponds to the methodof least squares. 𝐴 is a 𝑁 × 𝑀 matrix, which corresponds tothe inverse Fourier transform: 𝐴 𝑖 𝑗 = ( cos ( 𝜔 𝑗 𝑡 𝑖 ) ( 𝑗 = , , · · · , 𝑀 ) sin ( 𝜔 𝑗 − 𝑀 𝑡 𝑖 ) ( 𝑗 = 𝑀 + , 𝑀 + , · · · , 𝑀 ) . (2)The second term on the right-hand side in Eq. (1) imposesa sparsity constraint on the solution ˆ x , and 𝜆 is a parame-ter that adjusts the sparsity. We determined 𝜆 via five-foldcross-validation [33]. In this method, the elements of y wererandomly divided into five data sets. Four of these sets wereused to obtain ˆ x , and the other set was used to evaluate thewaveform reproduced from ˆ x .Generally, the output of the LASSO is not strictly unique,and it depends on the random sampling [34]. Cosine similari-ties (CS) were used to evaluate the consistency of results fromdifferent dataset. The CS of two dispersions is given byCS = f · g | f || g | (3)where f and g are vectorized data of the dispersion relations.Here, f is the dispersion calculated by LASSO from the 𝑁 =
361 data as the most ideal dispersion available from the presentdata and g is the dispersion with reduced 𝑁 to be compared.For the DFT method, 𝑁 was reduced by taking Δ 𝑡 as 0 .
01 nsmultiplied by the divisors of 360 without changing 𝑇 . For theLASSO method, random sampling with 𝑁 = , , · · · , 𝑡 = , . , .
60 ns were always sampled to maintain thefrequency resolution and the Nyquist frequency.Figure 2(a) shows the entire spatiotemporal waveform of thespin wave that we observed in our experiments with 𝑁 = Δ 𝑡 = .
01 ns. Figure 2(b) shows a waveformdataset with 𝑁 =
46 by taking Δ 𝑡 = .
08 ns, and Fig. 2(c) rep-resents a dataset with 𝑁 =
46 by taking Δ 𝑡 at random. Further,Figs. 2(d)–(f) shows the simulated waveforms correspond toFigs. 2(a)–(c).Figure 3 shows the dispersion relations corresponding tothe dataset shown in Fig 2. Figure 3(a) was obtained fromthe data shown in Fig. 2(a) by DFT in the time and spacedomain. Moreover, Fig. 3(b) was obtained by DFT from thedata shown in Fig. 2(b). The data in Fig. 3(c) were estimatedvia the LASSO method instead of DFT in the time domain.Figure 3(b) shows that the information was only available up to6 GHz due to the insufficient sampling rate of the data in Fig.2(b). Therefore, signals appearing to exhibit the dispersionrelation are folding noises bounded by the Nyquist frequency.In Fig. 3(c), the same curve can be observed as in Fig. 3(a), FIG. 2. Spatiotemporal waveforms of spin wave observed by time-resolved pump-probe magneto-optical imaging method. (a) the whole datawith 𝑁 = 𝑁 =
46 by taking Δ 𝑡 = .
08 ns. (c) waveform sampled from (a) at random in time domainwith 𝑁 =
46. (d)–(f) waveforms simulated via MuMax3 corresponding to (a)–(c), respectively. which implies that sufficient information could be extractedfrom random sampling as shown in Fig. 2(c). Figures 3(d)–(f)show the dispersion relations calculated from the data shownin Figs. 2(d)–(f), the waveforms calculated by MuMax3. Theresults of the simulation and the experiment were found to bein good agreement.The excited modes were the backward volume magneto-static waves, which are mainly dominated by magnetic dipoleinteractions, and they have a negative gradient of dispersion[10, 12, 13]. The observed dispersion was in good agreementwith the lowest order mode shown in Fig. 3 with red lines,and there were no peaks corresponding to the higher ordermodes. This is because the higher order modes have nodesin the thickness direction, and the Faraday rotation caused bythe higher modes was mostly cancelled out through the lighttransmittance. The multiple branches seen in Figs. 2(a) and(d) are due to the spin-wave echoes [35].Figure 4 shows the 𝑁 -dependence of the CSs between thedispersion obtained from the data with 𝑁 =
361 and the dis-persions obtained from the data with reduced 𝑁 . For the DFTmethod, the points at 𝑁 ≤
46 correspond to the conditionswith Δ 𝑡 ≥ .
08 ns, and their CSs were almost zero because theNyquist condition was not satisfied. In contrast, the CSs of theresults of the LASSO method were above 0 .
90, even when 𝑁 was reduced to 50. Note that in the LASSO method, 𝑁 was reduced without decreasing the Nyquist frequency.It is necessary to discuss in what systems LASSO can beapplied. Empirically, samplings 2–5 times the number ofsparse coefficients is sufficient to reconstruct the spectrumusing the 𝑙 norm [36]. Based on this, even for a systemwith multiple modes, the number of required samples canbe roughly estimated from the number of predicted peaks.Furthermore, for a system with strong damping, the linewidthof the spectrum may be underestimated. This can be improvedby setting the sampling time range to 𝑇 ≈ /( 𝛼 𝑓 ) where 𝑓 isthe center frequency.In conclusion, we demonstrated the fast acquisition of spin-wave dispersion by using compressed sensing. Further, wequantitatively evaluated the effects of random sampling on theresults of the LASSO method. This technique significantlyreduced the measurement time for acquiring the dispersion re-lations. Moreover, this method of applying compressed sens-ing to time-resolved pump-probe measurements is not limitedto the magneto-optical imaging of spin waves, to various ex-periments based on pump-probe measurements, such as obser-vations via the electro-optical effects and the refractive indexmodulations. FIG. 3. (a)–(c) Dispersion relations obtained from spatiotemporal waveforms shown in Figs. 2(a)–(c). (d)–(f) dispersion relations obtained byanalyzing the simulated waveforms corresponding to Figs. 2(d)–(f). The red lines are the theoretical curves of the dispersion relation of thebackward volume magnetostatic wave. The horizontal axis is the wavenumber multiplied by the thickness of the sample.
ACKNOWLEDGMENTS
We would like to thank K. Hukushima and T. Ishikawafor valuable suggestions. This study was supported by the Japan Society for the Promotion of Science (JSPS) KAK-ENHI (Grants No. JP19H01828, No. JP19H05618, No.JP19J21797, No. JP19K21854, and No. JP26103004). K.M. would like to thank the Research Fellowship for YoungScientists by the JSPS. [1] A. G. Gurevich and G. A. Melkov,
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