Fast acquisition of spin-wave dispersion by compressed sensing
Ryo Kainuma1 , Keita Matsumoto1,2 , and Takuya Satoh1
1 Department of Physics, Tokyo Institute of Technology, Tokyo 152-8551, Japan
2 Department of Physics, Kyushu University, Fukuoka 819-0385, Japan(Dated: February 14, 2021)
For the realization of magnonic devices, spin-wave dispersions need to be identiﬁed. Recently, the timeresolved 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 sampling
rate 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 results
can 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 magnetically ordered materials. They are considered promisinginformation carriers in the ﬁeld of magnonics, because they
can propagate over a long distance without Joule heating [1–4]. Various devices such as spin wave switches [5], magnoniclogic 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 manifested
in their dispersion relation. The higher-wavenumber regime
is governed by exchange interactions. In contrast, lowerwavenumber spin waves are governed by magnetic dipole interactions and are called magnetostatic waves [10–13]. Since themagnetostatic waves are suitable for long-distance propagation, further investigation of dispersion relations in the lowerwavenumber regime is indispensable [2].
Experimental techniques for acquiring dispersion relations of spin waves are being actively studied. For example, inelastic neutron scattering [14, 15] and spin-polarized
electron energy loss spectroscopy [16] have been demonstrated. However, these methods are suitable for observing thehigher-wavenumber region, rather than observing the lowerwavenumber 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 performed using the magneto-optical Kerr eﬀect [19]. In the
SWaT 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 frequency
range can be excited simultaneously. Moreover, by focusing
the pulses, spin waves in a wide wavenumber range can be
excited. 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 dispersions over a wide region in the wavenumber (𝑘)–frequency( 𝑓 ) space.
For observing the dispersion, a suﬃciently high sampling
rate 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 measurement time can range from 10 hours to several days. To search
for novel photo-induced dynamics in innumerable materials,
the measurement time must be reduced.
In recent years, in experiments such as the terahertz imaging [20], the NMR spectroscopy [21], and the scanning tunneling microscopy and spectroscopy [22], it has been shownthat a method called compressed sensing can reduce the measurement time. Compressed sensing is a signal processing
technique that allows the estimation of a signal from a small
amount of data [23]. The signal estimation can be achieved by
the 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 spinwave dispersions by time-resolved pump-probe measurements
using compressed sensing. Moreover, we quantitatively evaluated the eﬀect of reducing the number of measurements incompressed sensing on the results of observations.
Our experimental setup is shown in Fig. 1. Our sample
was a single crystal of (111)-oriented 150 𝜇m thick bismuthdoped rare earth iron garnet (Gd3/2 Yb1/2 BiFe5 O12 ) grown on
a gadolinium gallium garnet substrate by the liquid-phase epitaxy method. This magnetic material has been widely used forinvestigating laser-induced spin dynamics due to its strong
magneto-optical coupling [25–31]. The magnetic ﬁeld of
𝐻ext = 2440 Oe was applied in the 𝑥-direction.
The light pulse for this pump-probe measurement was generated by a Ti:sapphire regenerative ampliﬁer with a pulse
duration of 70 fs and a repetition rate of 1 kHz. A circularly
polarized pump pulse with a central wavelength of 1300 nm
was focused along a line parallel to the 𝑦-axis with a cylindrical lens with a ﬂuence of 80 mJ cm−2 . This pump pulse
produced an eﬀective magnetic ﬁeld in the 𝑧-direction via theinverse Faraday eﬀect [32], and the magnetization saturated
in the 𝑥-direction tilted in the 𝑦 direction. Since the eﬀective
magnetic ﬁeld was instantaneous, the magnetization then began to precess in the 𝑦-𝑧 plane. The spin precession excitedalong the line-shaped pumping spots propagated perpendicular to the line via magnetic dipole interactions. A linearlypolarized pulse with a central wavelength of 800 nm was used
to probe the 𝑧-component of magnetization 𝑚 𝑧 (r, 𝑡) via the
Faraday eﬀect. The Faraday rotation angle was determinedfrom the angle of the analyzer, which minimized the intensityof the transmitted probe pulse detected by a complementary

2z

7HOHVFRSH &026$QDO\]HU
12x̂ = arg min||y − 𝐴x|| 2 + 𝜆||x|| 1 .
2𝑁xx(OHFWURPDJQHW

y
*73
+:3

problem

6DPSOH&\OLQGULFDOOHQV

4:3
*73
+:3

FIG. 1. Setup of our pump-probe experiment. The circularly polarized pump pulse was focused along a line by a cylindrical lens. The
linearly polarized probe pulse was irradiated on the entire sample
without focusing. HWP: half-wave plate, GTP: Gran Taylor prism,
and QWP: quarter-wave plate.

metal–oxide semiconductor (CMOS) camera. The time delay
between the pump and probe pulses was achieved by a variable
optical path diﬀerence using a delay stage. This was changed
in increments of Δ𝑡 = 0.01 ns. The maximum value of the delay
was set to 𝑇 = 3.60 ns. Then, we obtained the spatiotemporal waveform of the spin wave. The waveform was integrated
along the 𝑦-direction to create a one-dimensional waveform𝑚 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 simulations were performed to conﬁrm our experimental results(See supplementary data).
Let 𝑁 = 𝑇/Δ𝑡 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 = 1/𝑇. In addition, according to Nyquist’s theorem,
the frequency components above 1/(2Δ𝑡) cannot be observed,
and they appear as folding noise. Therefore, with regard to
reducing the measurement time, decreasing 𝑇 leads to a poor
resolution, and increasing Δ𝑡 increases the risk of the appearance of folding noise.
Compressed sensing is a method of reducing 𝑁 by taking
Δ𝑡 randomly, without changing 𝑇, and estimating the spectrum
from such data. Since Δ𝑡 is not a constant, the DFT does notwork well, resulting in spectral leakage. Instead, the spectrum
can be estimated by the LASSO method.
In the LASSO method, the spectrum estimation is treatedas an inverse problem. Let y = (𝑦(𝑡 1 ), 𝑦(𝑡 2 ), · · · , 𝑦(𝑡 𝑁 )) T be a
waveform sampled with discrete time 𝑡𝑖 (𝑖 = 1, 2, · · · , 𝑁) andx = (𝑎(𝜔1 ), 𝑎(𝜔2 ), · · · , 𝑎(𝜔 𝑀 ), 𝑏(𝜔1 ), 𝑏(𝜔2 ), · · · , 𝑏(𝜔 𝑀 )) T
are the cosine and sine components of the spectrum fordiscrete frequencies 𝜔 𝑗 ( 𝑗 = 1, 2, · · · , 𝑀). The spectrum tobe estimated is the solution x̂ of the following minimization(1)

Arg min{·} denote the argument of the minimum, an elementthat minimizes the value in the brackets. || · || 𝑝 is a term called 1/ 𝑝Í
the 𝑙 𝑝 norm of the vector and is deﬁned as ||x|| 𝑝 = 𝑖 𝑥𝑖𝑝.
The ﬁrst term on the right-hand side corresponds to the methodof least squares. 𝐴 is a 𝑁 × 2𝑀 matrix, which corresponds tothe inverse Fourier transform:(cos(𝜔 𝑗 𝑡 𝑖 )( 𝑗 = 1, 2, · · · , 𝑀)
𝐴𝑖 𝑗 =(2)sin(𝜔 𝑗−𝑀 𝑡 𝑖 ) ( 𝑗 = 𝑀 + 1, 𝑀 + 2, · · · , 2𝑀).
The second term on the right-hand side in Eq. (1) imposesa sparsity constraint on the solution x̂, and 𝜆 is a parameter 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 x̂, and the other set was used to evaluate thewaveform reproduced from x̂.
Generally, the output of the LASSO is not strictly unique,and it depends on the random sampling [34]. Cosine similarities (CS) were used to evaluate the consistency of results from
diﬀerent dataset. The CS of two dispersions is given by
CS =

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 present
data and g is the dispersion with reduced 𝑁 to be compared.
For the DFT method, 𝑁 was reduced by taking Δ𝑡 as 0.01 ns
multiplied by the divisors of 360 without changing 𝑇. For the
LASSO method, random sampling with 𝑁 = 10, 20, · · · , 360
was performed in ten ways each, and the mean and standard
deviation of the CSs were calculated. Moreover, the points
𝑡 = 0, 0.01, 3.60 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 = 361
at increments of Δ𝑡 = 0.01 ns. Figure 2(b) shows a waveform
dataset with 𝑁 = 46 by taking Δ𝑡 = 0.08 ns, and Fig. 2(c) represents a dataset with 𝑁 = 46 by taking Δ𝑡 at random. Further,
Figs. 2(d)–(f) shows the simulated waveforms correspond to
Figs. 2(a)–(c).
Figure 3 shows the dispersion relations corresponding to
the 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 estimated
via the LASSO method instead of DFT in the time domain.
Figure 3(b) shows that the information was only available up to
6 GHz due to the insuﬃcient sampling rate of the data in Fig.
2(b). Therefore, signals appearing to exhibit the dispersion
relation are folding noises bounded by the Nyquist frequency.
In Fig. 3(c), the same curve can be observed as in Fig. 3(a),3FIG. 2. Spatiotemporal waveforms of spin wave observed by time-resolved pump-probe magneto-optical imaging method. (a) the whole datawith 𝑁 = 361. (b) waveform sampled from (a) with 𝑁 = 46 by taking Δ𝑡 = 0.08 ns. (c) waveform sampled from (a) at random in time domain
with 𝑁 = 46. (d)–(f) waveforms simulated via MuMax3 corresponding to (a)–(c), respectively.

which implies that suﬃcient information could be extracted
from 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. The
results of the simulation and the experiment were found to be
in good agreement.
The excited modes were the backward volume magnetostatic waves, which are mainly dominated by magnetic dipole
interactions, 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 nodes
in the thickness direction, and the Faraday rotation caused by
the 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 dispersions obtained from the data with reduced 𝑁. For the DFT
method, the points at 𝑁 ≤ 46 correspond to the conditions
with Δ𝑡 ≥ 0.08 ns, and their CSs were almost zero because the
Nyquist 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 spectrum
using the 𝑙 1 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 ≈ 1/(𝛼 𝑓0 ) where 𝑓0 is
the center frequency.
In conclusion, we demonstrated the fast acquisition of spinwave dispersion by using compressed sensing. Further, wequantitatively evaluated the eﬀects of random sampling on the
results of the LASSO method. This technique signiﬁcantly
reduced the measurement time for acquiring the dispersion relations. Moreover, this method of applying compressed sensing to time-resolved pump-probe measurements is not limited
to the magneto-optical imaging of spin waves, to various experiments based on pump-probe measurements, such as observations via the electro-optical eﬀects and the refractive index
modulations.

4

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 the
backward volume magnetostatic wave. The horizontal axis is the wavenumber multiplied by the thickness of the sample.

We would like to thank K. Hukushima and T. Ishikawa
for valuable suggestions. This study was supported by theJapan Society for the Promotion of Science (JSPS) KAKENHI (Grants No. JP19H01828, No. JP19H05618, No.
JP19J21797, No. JP19K21854, and No. JP26103004). K.
M. would like to thank the Research Fellowship for Young
Scientists by the JSPS.

[1] A. G. Gurevich and G. A. Melkov, Magnetization Oscillationsand Waves (CRC press, Boca Raton, 1996).
[2] D. D. Stancil and A. Prabhakar, Spin Waves: Theory and Applications (Springer, New York, 2009).
[3] V. V. Kruglyak, S. O. Demokritov, and D. Grundler, J. Phys. D:
Appl. Phys. 43, 264001 (2010).
[4] A. V. Chumak, V. I. Vasyuchka, A. A. Serga, and B. Hillebrands,Nat. Phys. 11, 453 (2015).
[5] B. Lenk, H. Ulrichs, F. Garbs, and M. Münzenberg, Phys. Rep.
507, 107 (2011).
[6] T. Schneider, A. A. Serga, B. Leven, B. Hillebrands, R. L.
Stamps, and M. P. Kostylev, Appl. Phys. Lett. 92, 022505 (2008).
[7] N. Kanazawa, T. Goto, K. Sekiguchi, A. B. Granovsky, C. A.
Ross, H. Takagi, Y. Nakamura, H. Uchida, and M. Inoue, Sci.
Rep. 7, 7898 (2017).
[8] T. Seki, K. Utsumiya, Y. Nozaki, H. Imamura, and K. Takanashi,Nat. Commun. 4, 1726 (2013).

[9] I. Lee, Y. Obukhov, G. Xiang, A. Hauser, F. Yang, P. Banerjee,
D. V. Pelekhov, and P. C. Hammel, Nature 466, 845 (2010).
[10] R. W. Damon and J. R. Eshbach, J. Phys. Chem. Solids 19, 308(1961).
[11] R. W. Damon and H. Van De Vaart, J. Appl. Phys. 36, 3453
(1965).
[12] M. J. Hurben and C. E. Patton, J. Magn. Magn. Mater. 139, 263(1995).
[13] M. J. Hurben and C. E. Patton, J. Magn. Magn. Mater. 163, 39(1996).
[14] H. A. Mook and D. M. Paul, Phys. Rev. Lett. 54, 227 (1985).
[15] X. Liu, M. M. Steiner, R. Sooryakumar, G. A. Prinz, R. F. C.
Farrow, and G. Harp, Phys. Rev. B 53, 12166 (1996).
[16] R. Vollmer, M. Etzkorn, P. S. A. Kumar, H. Ibach, andJ. Kirschner, Phys. Rev. Lett. 91, 147201 (2003).
[17] Y. Hashimoto, S. Daimon, R. Iguchi, Y. Oikawa, K. Shen,
K. Sato, D. Bossini, Y. Tabuchi, T. Satoh, B. Hillebrands,

ACKNOWLEDGMENTS

5

1.0
0.8

CS

0.6
0.4
0.2
LASSO
DFT
0.0
0

50

100

150

200

250

300

350

N
FIG. 4. Cosine similarities of the dispersions at reduced 𝑁 to thedispersion at 𝑁 = 361. Blue and orange dots represent the results of
LASSO method and DFT, respectively. The error bars of blue dots
represent the standard deviations of the CSs of 10 diﬀerent sets ofrandom sampling.

G. E. W. Bauer, T. H. Johansen, A. Kirilyuk, Th. Rasing, and
E. Saitoh, Nat. Commun. 8, 15859 (2017).
[18] Y. Hashimoto, T. H. Johansen, and E. Saitoh, Appl. Phys. Lett.
112, 072410 (2018).
[19] A. Kamimaki, S. Iihama, Y. Sasaki, Y. Ando, and S. Mizukami,
Phys. Rev. B 96, 014438 (2017).
[20] W. L. Chan, K. Charan, D. Takhar, K. F. Kelly, R. G. Baraniuk,and D. M. Mittleman, Appl. Phys. Lett. 93, 121105 (2008).
[21] K. Kazimierczuk and V. Y. Orekhov, Angew. Chem. Int. Ed. 50,
5556 (2011).

[22] Y. Nakanishi-Ohno, M. Haze, Y. Yoshida, K. Hukushima,
Y. Hasegawa, and M. Okada, J. Phys. Soc. Jpn. 85, 093702(2016).
[23] E. J. Candès, J. Romberg, and T. Tao, IEEE Trans. Inf. Theory
52, 489 (2006).
[24] R. Tibshirani, J. R. Stat. Soc. Ser. B Methodol. 58, 267 (1996).
[25] T. Satoh, Y. Terui, R. Moriya, B. A. Ivanov, K. Ando, E. Saitoh,
T. Shimura, and K. Kuroda, Nat. Photonics 6, 662 (2012).
[26] S. Parchenko, A. Stupakiewicz, I. Yoshimine, T. Satoh, and
A. Maziewski, Appl. Phys. Lett. 103, 172402 (2013).
[27] I. Yoshimine, T. Satoh, R. Iida, A. Stupakiewicz, A. Maziewski,
and T. Shimura, J. Appl. Phys. 116, 043907 (2014).
[28] I. Yoshimine, Y. Y. Tanaka, T. Shimura, and T. Satoh, Europhys.
Lett. 117, 67001 (2017).
[29] A. L. Chekhov, A. I. Stognĳ, T. Satoh, T. V. Murzina, I. Razdolski, and A. Stupakiewicz, Nano Lett. 18, 2970 (2018).
[30] K. Matsumoto, T. Brächer, P. Pirro, T. Fischer, D. Bozhko,
M. Geilen, F. Heussner, T. Meyer, B. Hillebrands, and T. Satoh,Jpn. J. Appl. Phys. 57, 070308 (2018).
[31] K. Matsumoto, I. Yoshimine, K. Himeno, T. Shimura, and
T. Satoh, Phys. Rev. B 101, 184407 (2020).
[32] A. V. Kimel, A. Kirilyuk, P. A. Usachev, R. V. Pisarev, A. M.
Balbashov, and Th. Rasing, Nature 435, 655 (2005).
[33] G. James, D. Witten, T. Hastie, and R. Tibshirani, An Introduction to Statistical Learning, Vol. 112 (Springer, New York,
2013).
[34] A.-C. Haury, P. Gestraud, and J.-P. Vert, PLOS ONE 6, e28210(2011).
[35] A. A. Serga, A. V. Chumak, and B. Hillebrands, J. Phys. D:
Appl. Phys. 43, 264002 (2010).
[36] M. Lustig, D. Donoho, and J. M. Pauly, Magn. Reson. Med. 58,
1182 (2007).