PHYSICAL REVIEW B 101, 184407 (2020)

Observation of evanescent spin waves in the magnetic dipole regime
Keita Matsumoto ,1,2 Isao Yoshimine ,3,4 Kosei Himeno,2 Tsutomu Shimura,4 and Takuya Satoh

1,2,4,*

1

Department of Physics, Tokyo Institute of Technology, Tokyo 152-8551, Japan2
Department of Physics, Kyushu University, Fukuoka 819-0385, Japan
3
RIKEN Center for Advanced Photonics, RIKEN, Sendai 980-0845, Japan
4
Institute of Industrial Science, The University of Tokyo, Tokyo 153-8505, Japan(Received 12 August 2019; revised manuscript received 15 February 2020; accepted 31 March 2020;
published 5 May 2020)
We observed spin-wave transmission through an air gap that works as a prohibited region. The spin waves
were excited by circularly polarized pump pulses via the inverse Faraday effect, and their spatial propagation was
detected through the Faraday effect of probe pulses using a pump-probe imaging technique. The experimentallyobserved spin-wave transmission was reproduced using numerical calculations with a Green’s function method
and micromagnetic simulation. We found that the amplitude of the spin waves decays exponentially in the airgap, which indicates the existence of evanescent spin waves in the magnetic dipole regime. This finding will
pave the way for controllable amplitudes and phases of spin waves propagating through an artificial magnonic
crystal.
DOI: 10.1103/PhysRevB.101.184407
I. INTRODUCTION

When an electromagnetic plane wave propagates from amore refractive medium towards a less refractive medium
at an incidence angle exceeding the critical angle, the electromagnetic wave cannot propagate in the second medium
and is totally internally reflected. Nevertheless, waves exist
within the second medium, which are referred to as evanescent waves [1]. The amplitude of an electromagnetic field
decreases exponentially along an interface normal and can be
expressed as ∼ exp(−Akw). Here, A is of the order of unity
when the incidence angle far exceeds the critical angle, k is
the wave number, and w is the distance from the interface.
This exp(−Akw) characteristic of evanescent waves should
exist in every type of wave and not just electromagnetic
waves.
Spin waves are propagating waves of precessing magnetization in a magnetically ordered material. They have beenstudied extensively because a spin wave can propagate ininsulators with a long propagation length [2] with dispersion that is controllable using external conditions [3–5]. Theinterference of multiple spin waves can be applied for thecreation of novel logic gates [5]. In recent years, it has been reported that spin-wave transmission through artificial periodic
magnetic inhomogeneities known as magnonic crystals has aspin-wave forbidden band gap in wave-number space [6,7].
Analogous to the electromagnetic case, evanescent spin waves
are expected where the spin-wave propagation is prohibited inreal space. Previous research [8,9] reported the existence of aspin-wave tunneling effect through an air gap. However, to ourknowledge, evanescent spin waves with the exp(−Akw) characteristic have not been verified, because of the requirementof high temporal and spatial resolution.

*

satoh@phys.titech.ac.jp2469-9950/2020/101(18)/184407(6)An all-optical pump-probe technique with femtosecond
laser pulses provides sufficient temporal resolution and yields
the phase information of coherent oscillation in particular.
This technique has been widely used for spin-wave experiments [10,11] in a noncontact manner. Spin precession canbe excited nonthermally by a pump pulse via the inverse
Faraday effect [12], where a pump pulse with circular polarization induces an effective magnetic field within a material. This effect causes no heating because the excitation
process is based on impulsive stimulated Raman scatteringin a nonresonant condition [13]. The spatial distribution ofa generated magnetic field is proportional to the intensity
profile of the pump spot on the sample surface [14]. This
impulsive field exerts a torque on magnetization, leading tospin precession that propagates as a spin wave out of the pumpspot [14–17]. The time-resolved imaging of spin-wave propagation has been performed using magneto-optical effects[14,16,18–25].
In this paper, we show the dynamics of spin-wave transmission through an air gap using a time-resolved pump-probemagneto-optical imaging technique. The results are comparedwith numerical calculations using a Green’s function method[8,9,26] and micromagnetic simulation [27] describing longrange magnetic dipole interaction. We interpreted the transmission effect by drawing an analogy with evanescent phenomena.
II. METHOD

Our sample was an epitaxially grown single crystalof a (111)-oriented 110-μm-thick ferrimagnetic insulator,
Gd3/2 Yb1/2 BiFe5 O12 . Its Curie temperature was 573 K. This
sample is a suitable magnetic material for the optical observation of spin-wave transmission through an air gap because
it shows large magneto-optical interaction and long spin-wave
propagation [14–16,20,28,29].

184407-1©2020 American Physical Society

KEITA MATSUMOTO et al.
(a)

z
x

y

PHYSICAL REVIEW B 101, 184407 (2020)
(b)

Probe

analyzer

electromagnet

Hu is the uniaxial anisotropic field, Ms is the saturation
magnetization, Hext is the externally applied magnetic field,
ω(k) is the angular frequency with dispersion of the backward
volume magnetostatic wave (BVMSW) in the lowest order[3,4,30], and γ is the gyromagnetic ratio. All the parameters
for the Green’s function method are described in Appendix A.
The first term on the right-hand side of Eq. (1) expresses
the dipole interaction by the precession of magnetization,
whereas the second term represents the incident spin wave.
It is assumed that the spin wave is uniformly excited along thesample thickness (z direction), with its validity discussed inAppendix B. The Green’s function G(x, k) is described as ∞1exp(−ik  x)
G(x, k) =dk  ,(3)
2π −∞ W (k  ) − W (k) − iv

Pump

CCD camera

telescopeyz

Probe 800 nm

x
Hext

samplecylindrical lens

(c)

Pump
Probe

λ/4 plateGTP
λ/2 plateGTP
λ/2 plate

Pump 1300 nm

yz

x
Hext

FIG. 1. The experimental setup and sample geometries. (a) The
optical system used in our experiments. λ/2 plate: Half-wave plate.
GTP: Glan-Taylor prism. λ/4 plate: Quarter-wave plate. The electromagnet applies an in-plane magnetic field, and a CCD camera detects
the transmitted probe light that is guided by a telescope. Sample andpump-probe geometry: (b) Gap-free and (c) finite-gap cases.

where W (k) = [exp(−|k|L) − 1]/(|k|L), L is the sample
thickness, and v is a phenomenological parameter.
Because the first term of Eq. (1) diverges in the air gap(χ = 0), we transformed Eq. (1) using the dipole field hz =mz /χ as ∞χ0 − χ
χ hz (x, k) =hz (x  , k)dx G(x − x  , k)
χ0−∞+G(x − x0 , k).

The experimental setup is shown in Fig. 1(a). The lightpulse for the pump-probe measurement was generated by a
Ti:sapphire regenerative amplifier with a pulse duration of
120 fs. A circularly polarized pump light pulse with a central
wavelength of 1300 nm was focused on the sample in a
linear shape with a width of 20 μm using a cylindrical lens.
This leads to the excitation of a spin wave via the inverse
Faraday effect and one-dimensional propagation of the spinwave along an external magnetic field of 1000 Oe. We utilized
a linearly polarized probe pulse with a central wavelength of
800 nm. The polarization rotation of the transmitted probe
light enables us to obtain the phase and amplitude informationof the spin wave via the Faraday effect, which is sensitive
to the out-of-plane component of magnetization mz . The
spatially resolved profile of the propagating spin wave wasobtained by a CCD camera [20]. All the measurements were
performed at room temperature.
III. NUMERICAL CALCULATIONIn the Green’s function method, the out-of-plane component of the magnetization profile mz (x, k) at position x forwave number k in the presence of an air gap is calculated as[8,9,26]


 ∞
1
1
mz (x  , k)dx mz (x, k) =G(x − x  , k)
−
χ
χ0
−∞
+ G(x − x0 , k).

(1)Here, G(x, k) is the magnetostatic Green’s function of the spinwave, x0 is the excitation position in the left sample (x0 = 0
for our calculation), and the magnetic susceptibility χ is zeroin the air gap and equal to χ0 otherwise, whereχ0 =4π Ms Hext.
Hext (Hext − Hu ) − ω(k)2 /γ 2

(2)

(4)

This self-consistent equation takes into account multiple reflections of the dipole field in the gap.
We applied the Green’s function method to the dynamics
of spin-wave transmission through an air gap. The timedependent magnetization mz (x, t ) can then be represented by
the superposition of mz (x, k) in the k space [14], 2 2k r
mz (x, t ) = exp − 0 exp[iω(k)t]2
× exp[−αω(k)t]mz (x, k)dk.

(5)Here, r0 is the spatial width of the pump light and α is theGilbert damping constant.
IV. RESULTS

We investigated two different geometries, i.e., the gapfree case [Fig. 1(b)] and the finite-gap case [Fig. 1(c)]. The
obtained spatiotemporal maps of spin waves represented as
an out-of-plane component of magnetization mz with thesegeometries are shown in Figs. 2(a) and 2(b), respectively.
First, for the gap-free case, Fig. 2(a) indicates the excitation and propagation of spin waves with magnetic dipole
characteristics—BVMSW. The direction of phase velocity
is opposite to that of group velocity because the dispersion
curve has a negative slope [14]. Figure 2(c) demonstrates
the results obtained by the Green’s function method. The
agreement between Figs. 2(a) and 2(c) confirms that theobserved spin waves in experiments were magnetic dipoledominated BVMSW with the lowest order. In addition, theGreen’s function G(x, k) as expressed in Eq. (3) was proved
to be valid up to k ≈ 0.1 rad/μm, which was the upper limit
for the excited spin wave in the experiment.
Second, the results for the finite gap with width w =40 μm, obtained by experiment and the numerical calculationby the Green’s function method, are shown in Figs. 2(b)184407-2

OBSERVATION OF EVANESCENT SPIN WAVES IN THE …

(b)
1.0

3

Time [ns]Faraday rotation [arb. units](a)

PHYSICAL REVIEW B 101, 184407 (2020)

2

0.0
1

0

-400

0

400

0

400

800

-1.0

Distance from the pump spot [ m]

(d)

(c)

1.0

mz [arb. units]Time [ns]

3

2

0.0
1

0

-1.0
-400

0

400

0

400

800

wave number reaches the edge first, following which the wave
with a higher wave number reaches the edge.
It is worth noting that the transmitted wave was discernible
in the right sample. In addition, a higher transmission wasobserved for a lower wave number. These experimental characteristics [Fig. 2(b)] were reproduced well by the numericalcalculation [Fig. 2(d)] qualitatively and quantitatively.
To confirm the validity of the calculation by the Green’s
function method, we also performed a micromagnetic simulation using MUMAX3 [27] (for parameters, see Appendix A)
as shown in Figs. 2(e) and 2(f) for the gap-free and finite-gap
cases, respectively. The results reproduced the calculation by
the Green’s function method. Here, only the dipole interaction
was taken into account and the exchange stiffness was set tozero in the micromagnetic simulation. The waveform exhibited a negligible change even if we set the actual exchange
stiffness. The excellent agreement among the results of theexperiment, the numerical calculations of the Green’s function
method, and the micromagnetic simulation indicates that thetransmission of the spin wave through the air gap originatesfrom long-range magnetic dipole interaction between the left
and right samples.

Distance from the pump spot [ m](e)V. DISCUSSION

(f)
1.0

2
0.0
1

0

mz [arb. units]Time [ns]

3

-1.0
-400

0

400

0

400

800

Distance from the pump spot [ m]FIG. 2. The experimental and calculated results of spin-wave
dynamics as a function of distance from the pump spot along the xaxis and time. Spatiotemporal maps for the gap-free system [(a), (c),(e)] and the finite-gap system [(b), (d), (f)]; these are Faraday rotation
maps obtained in the experiment [(a), (b)], the perpendicular magnetization component calculated using the Green’s function method[(c), (d)], and that calculated by micromagnetic simulation [(e), (f)].
In (b), (d), and (f), the gap ranges from x = 200 to 240 μm inthe x axis.

and 2(d), respectively (also see the movie in the Supplemental
Material [31]). The air gap is located at x = 200−240 μm
as displayed in Fig. 1(c). The noisy signal near the air gap
in Fig. 2(b) is due to the random reflection of probe light at
the sample edge. On the left side of the pump spot (x < 0)in Figs. 2(b) and 2(d), the propagation characteristic wasalmost identical to that of the gap-free case in Figs. 2(a) and2(c). On the right side of the pump spot in the left sample
in Figs. 2(b) and 2(d), the spin wave exhibits a standing
wave (0 < x < 200 μm), and its node point of interference
approaches the sample edge over time. This is explained asfollows. The spatially focused pump pulse excites a spin-wave
packet with a broadband wave number. Due to the dispersionof the BVMSW with negative slope, the wave with a lower

The numerical calculation with the Green’s function
method enables us to evaluate the spin-wave transmission
phenomena quantitatively. We calculated the spin-wave transmittance defined as T = |mz,t (x = xfar , k)/mz,i (x = xfar , k)|,where mz,i (x, k) and mz,t (x, k) are the waveforms of the incident and the transmitted spin waves, respectively. In thecalculation of the transmittance, the excitation source (x0 = 0)
was sufficiently far to the left of the air gap, and xfar was
sufficiently far to the right of the air gap, i.e., xfar is of theorder of centimeters, where T converges to a constant value.
The phase shift is defined as arg[mz,t (xfar , k)/mz,i (xfar , k)] andis the phase difference between the incident and transmitted
spin waves.
Figures 3(a) and 3(b) show the transmittance and phaseshift of the spin wave as a function of gap width w for various
wave numbers k. Figure 3(a) indicates that a spin wave witha smaller wave number exhibits a higher transmission, which
reproduced the experimental finding shown in Fig. 2(d). We
replot in the insets of Figs. 3(a) and 3(b) the transmittanceand the phase shift as a function of kw. Surprisingly, allcurves almost coincide, forming one universal curve. For
kw ∼ π , transmittance is approximated as ∼ exp(−Akw),
where A is of the order of unity. These results are reminiscentof the evanescent effect because the transmittance of electromagnetic waves can be written as exp(−Akw), where A is of
the order of unity.
The propagation of electromagnetic waves is generally
described with the Huygens-Fresnel principle, which statesthat each point on an electromagnetic wave front is a sourceof spherical waves. The superposition of these waves forms
the forward wave front. When a plane wave is incident on
an interface between two media with a finite incidence angle,
these spherical waves have a phase shift along the intersectionof the incident plane and the interface. The superposition ofthese phase-shifted waves then yields an evanescent wave if

184407-3

KEITA MATSUMOTO et al.

(a)

PHYSICAL REVIEW B 101, 184407 (2020)
TABLE I. Parameters for numerical calculations.

1

Parameters
Uniaxial anisotropy Hu
Saturation magnetization 4π Ms
Externally applied field Hext
Gyromagnetic ratio γ /(2π )Gilbert damping α
Spatial width of pump light r0

Transmittance

1
Transmittance

0.1

0.1

0.01

0

0.001

π

20
40
gap width [μm]60gap width and spin-wave wave number. This behavior is analogous to the evanescent phenomena of the electromagnetic
wave. Our findings can be useful for the future development of
magnonic crystals with multiple gaps (magnonic crystal). The
observation of an evanescent spin wave will pave the way toapplications in surface-sensitive devices as used in near-fieldspin-wave spectroscopy.

0

ACKNOWLEDGMENTS-π/2

The authors thank H. Shimizu, S. Tamaru, K. Sawada, Y.
Ozeki, and B. Hillebrands for valuable discussions. We arealso grateful to M. P. Kostylev for answering our inquiry. This
study was supported by Japan Society for the Promotion of
Science (JSPS) (Grants No. JP15H05454, No. JP19H01828,
No. JP19H05618, No. JP19J21797, No. JP19K21854, and
No. JP26103004), JST-PRESTO, and JSPS Core-to-Core
Program (A. Advanced Research Networks). K.M. thanks
Kyushu University QR Program and Research Fellowship for
Young Scientists by JSPS. K.M. and I.Y. contributed equallyto this work.

-π
-3π/2
-2π
-5π/2
-3π
-7π/2
0

0
Phase shift [rad]Phase shift [rad]

(b)

0

π/2
kw [rad]k [rad/μm]
0.003
0.01
0.02
0.04
0.06
0.08
0.1

600 Oe1158 G
1000 Oe
2.8 MHz/Oe
0.02
20 μm

-π/2

π

-3π/2
0

π/2
kw [rad]π2040gap width [μm]k [rad/μm]
0.003
0.01
0.02
0.04
0.06
0.08
0.1

60

FIG. 3. Calculated transmittance and phase shift of the spinwave. (a) Transmittance as a function of the gap widths w for various
k and as a function of kw in the inset. The straight line in the inset
indicates 0.4 exp(−0.64kw) as a guide to the eye. (b) Phase shift asa function of the gap widths w for various k and as a function of kw
in the inset.

the incidence angle is larger than the critical angle, above
which total internal reflection occurs [32]. In the present caseof spin waves, however, the incidence angle is zero. Therefore,
an evanescent spin wave resulting from a phase shift along
the intersection is unlikely. Instead, we argue that the phaseshift along the propagating direction is responsible for theevanescent wave because of the long-range dipole nature of
BVMSW.
VI. CONCLUSION

In conclusion, we measured spatiotemporally resolved
spin-wave transmission through an air gap using an all-opticalpump-probe technique and found excellent agreement with
the Green’s function method and micromagnetic simulation.
Furthermore, we found that the transmittance calculated by
the Green’s function method decayed exponentially with the

APPENDIX A: SIMULATION CONDITIONS

We performed numerical calculations with the Green’s
function method and the micromagnetic simulation (MUMAX3 )by using the parameters listed in Table I.
For the Green’s function method, the phenomenologicalparameter v = 2×10−5 was used, which is sufficiently small
but still finite to ensure the convergence of the numericalcalculation.
For the micromagnetic simulation, a periodic boundary
condition was applied along the y direction [along the sample
width (see Fig. 1)] because we expect the one-dimensional
propagation of the spin wave along the x axis (the propagation
direction). To simulate a spin-wave signal via the inverse
Faraday effect, we consider the perturbative deviation ofmagnetization from the equilibrium position as mz (x, y) ∝exp [−x 2 /(2r02 )] [33]. Then, the magnetization at each cell
begins precessing in accordance with the Landau-LifshitzGilbert equation. The obtained magnetization profile wasaveraged along the z axis for the purpose of reproducing
the experimental data in the transmission geometry. Thus,
we obtained the space- (along the x axis) and time-resolved
magnetization.
APPENDIX B: SPATIAL PROFILE OF THE SPIN WAVE
IN THE THICKNESS DIRECTION

In the calculation using the Green’s function method, we
assumed that the spin wave is excited uniformly along the184407-4

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PHYSICAL REVIEW B 101, 184407 (2020)

FIG. 4. The spin-wave profiles for (a) the gap-free case and(b) the 40-μm-gap case at 3 ns after pump excitation along the z
direction, which are calculated by a micromagnetic simulation.

thickness direction (z). It is interesting to calculate the spatial
profile of the spin wave in the thickness direction by amicromagnetic simulation, which is shown in Figs. 4(a) and4(b) at 3 ns after the pump excitation for the gap-free and40-μm-gap cases, respectively. In Fig. 4(a), the dominant spinwave corresponds to the lowest order with an even function,
and a nearly uniform profile was confirmed. In Fig. 4(b), in
addition to the lowest order, the thickness profile exhibits an
odd-function behavior, corresponding to the next order [3].
The odd behavior partially results from the demagnetizationnear the sample edge, and it vanishes if we average along
the thickness. Therefore, we can safely assume that the spinwave is excited uniformly along the thickness direction in theGreen’s function method.

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(b)

Gap freeGap 40 m

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10.1103/PhysRevB.101.184407 for the experimental result of

the spin-wave transmission through the air gap, where theshaded region represents the air gap.
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