# Functionalized high-speed magnon-polaritons resulting from the magnetic
antenna effect

Kenta Kato  Department of Material Engineering Science, Graduate School of
Engineering Science, Osaka University, 1-3 Machikaneyama, Toyonaka, Osaka
560-8531, Japan  Tomohiro Yokoyama<EMAIL_ADDRESS>Department of Material Engineering Science, Graduate School of Engineering
Science, Osaka University, 1-3 Machikaneyama, Toyonaka, Osaka 560-8531, Japan
Hajime Ishihara<EMAIL_ADDRESS>Department of Material Engineering
Science, Graduate School of Engineering Science, Osaka University, 1-3
Machikaneyama, Toyonaka, Osaka 560-8531, Japan  Department of Physics and
Electronics, Graduate School of Engineering, Osaka Prefecture University, 1-1
Gakuen-cho, Naka-ku, Sakai, Osaka 599-8531, Japan  Center for Quantum
Information and Quantum Biology, Osaka University, 1-3 Machikaneyama,
Toyonaka, Osaka 560-8531, Japan

###### Abstract

Magnon-polaritons (MPs) refer to a light–magnon coupled state and can
potentially act as information carriers, possibly enabling charge-free
computation. However, the light–magnon coupling is inherently weak. To achieve
sufficiently strong coupling, a large ferromagnet or coupling with a microwave
cavity is necessary. Herein, we theoretically propose a fundamental platform
for magnonic and magnon–optical information storage devices and discuss the
transport properties of MP’s. The proposed multi-layered structure overcomes
the aforementioned issues. Owing to the waveguide modes, magnons placed in a
nanometer-thin layer are strongly coupled with light, exhibiting rich
functionalities of thick-layer MPs via the ‘magnetic antenna effect’. Thus,
the thin-layer MPs are faster, and the direction is switchable. The results of
this study will enable the integration of ferromagnetic micro and
nanostructures for MP-based information devices without any restrictions due
to cavities.

In magnetic materials, elementary excitation arises from collective motion of
spin precession; this phenomenon, which is equivalent to a quasiparticle, is
called a spin wave or magnon. The magnon is an potentially viable candidate
for low-energy-consumption devices because it can carry Kajiwara10 ; Chumak14
; Chumak15 ; Oyanagi19 , process Chumak14 ; Tabuchi15 ; Lachance-Quirion19 ,
and store XZhang15  the information of spin without charge carrier transport.
The spin current from magnons is proportional to the density of excited
magnons $\rho_{\rm m}$ and the group velocity of the magnons $v_{\rm
g}=(1/\hbar)(\partial\varepsilon/\partial k)$. Thus, there exist two
significant issues affecting the efficiency of magnonic devices: (i) the
increase in magnon current with increasing density and velocity, and (ii)
switchable control over the current direction. However, typical magnon
densities are significantly lower than the typical electron density in metals.
The typical velocity of a magnon is also considerably lower than the Fermi
velocity of electrons. To overcome this issue, recent studies have focused on
the engineering of magnon velocity CLiu18 ; CLiu19  and transport Onose10 ;
Matsumoto14 ; Murakami17 . One method to increase the velocity of magnons is
to form magnon-polaritons (MPs) via strong coupling between magnons and
microwaves YCao15 ; Tabuchi14 ; XZhang14 . The spin-light coupling is
significantly weaker than the coupling between an electric dipole and light.
Recent studies have achieved ultra-strong coupling for MPs using optical
cavities, and significant splitting of the dispersion relation of magnetic
microspheres has been reported Tabuchi14 ; XZhang14 . Such cavity-based MPs in
magnetic microspheres exhibit rich properties, such as bistability YPWsng18 ;
Hyde18 , coherent perfect absorption DZhang17 ; XZhang19 , level attraction
Harder18 ; PCXu19 ; WYu19 ; Rao19NJP , the microwave Hall effect Rao19NC , and
information communication between magnetic segments in cavities Lambert16 ;
Bai17 ; Rameshti18 ; ZZhang19 ; JLi19 . However, to ensure information
transport via magnons, it is necessary to achieve strongly coupled MPs, which
we refer to as ‘strong MPs’, in one- or two-dimensional integrable magnetic
materials.

Several systems have been studied with the aim of discovering new applications
for strong MPs using cavities. To the best of our knowledge, however,
contemporary studies have not exploited the potential of the interplay between
the spatial degrees of freedom of the magnons and the radiation field. To
incorporate this ingredient, we propose a multi-layered two-dimensional
magnetic film, where the spatial profiles of the magnon wave and radiation
field and their interplay are considered key components. Subsequently, a
nonlocal treatment for the microwave response from magnons is formulated. This
approach, which considers the nonlocal response of magnons to the radiation
field, provides a new application prospect for strong MPs and is expected to
offer several degrees of freedom in the spatial design of sample structures.
Our formulation of a nonlocal theory is based on the quantum microscopic
perspective of magnons. (This scheme can be extended to nonlinear magnon
response in a straightforward manner.) The spatial interplay between magnons
and the field was designed under both dielectric and magnetic environments.
Thus, a multi-layered structure of magnetic and dielectric materials is
proposed, as shown in Fig. 1. The structure comprises thin ($\sim$ 100 nm) and
thick ($\sim 1\mathrm{mm}$) magnetic layers. A dielectric layer with an
appropriate thickness separates the magnon wavefunctions in the two magnetic
layers. Figure 1(a) depicts the model considered in the present calculations,
while (b) shows an imaginary processed structure. This structure is based on
three important concepts. First, the thin magnetic layer can be processed to
fabricate integrable magnonic circuits. Second, the ‘waveguide modes’ formed
by the interfaces significantly enhance the strength of the MPs in the
magnetic layers. Third, an interlayer magnon–magnon interaction results in the
‘antenna effect’, which transfers the rich functionalities of the thick-layer
MPs (such as the switching function of the transport direction) to the thin-
layer MPs. Importantly, the separation of magnons by the dielectric layer is
essential for the antenna effect and magnon circuit fabrication, as discussed
in subsequent sections. Moreover, the cavity-less structure offers significant
advantages for further development, such as MP network fabrication on a
substrate and MP control using a small magnet tip. The formulated nonlocal
theory (refer to the Methods section) for magnons indicates that a spatial
correlation between the magnons and waveguide modes results in strong MPs (as
discussed in the following sections). A combination of these aspects enables
the realization of highly efficient and switchable solid-state devices,
without the requirement for charge carrier transport.

Figure 1:  Schematic of the proposed multi-layer structure for magnonic
devices. (a) Proposed structure comprising thin yttrium iron garnet (YIG),
thin gadolinium gallium garnet (GGG), and thick YIG layers. The thick YIG
layer acts as an antenna to capture photons via the formed waveguide mode. The
photons are coupled with the magnons in the thin YIG layer via the
magnon–magnon interaction. Thus, the thin-layer magnons can form polaritons.
(b) Imaginary processed structure. It is expected that even when using a thin
YIG layer to fabricate circuits, the MPs in the thin YIG layer will
demonstrate high-speed and controllable magnon transport in circuits, without
any leakage of the magnon current. This is due to the functionality of MPs.
(Refer to the text immediately above the Summary section.) To form polaritons
using the waveguide mode, a microwave with a prism or fiber coupling is
employed. As an example of circuit fabrication, an X-shaped junction is
presented.

##  I Results and Discussion

For the proposed multi-layered structure shown in Fig. 1(a), we assume
$100\,\mathrm{nm}$ and $1\,\mathrm{mm}$ yttrium iron garnet (YIG) layers.
These two layers are separated by a $1\,\mathrm{\mu m}$ gadolinium gallium
garnet (GGG) layer, which is the paramagnetic insulator and standard material
for heterostructures with YIG. A schematic of this structure is presented in
Fig. 2(a). Thick YIG films are favorable for stronger MP coupling, which
results in larger magnon currents. However, to ensure suitable fabrication
quality, it is important to verify that even a considerably thinner
(nanometer-order) YIG film can function suitably under the antenna effect of
the thick layer. Thus, we select a thickness of 100 nm, which can be easily
achieved and processed using standard fabrication techniques.

The saturation magnetization is $\mu_{0}M_{\rm s}\simeq 175\,\mathrm{mT}$ for
YIG Daniel , where the frequency corresponding to magnetization is
$\omega_{\rm M}=g\mu_{\rm B}\mu_{0}M_{\rm s}/\hbar\simeq 4.89\times
2\pi\,\mathrm{GHz}$. $\mu_{\rm B}$ is the Bohr magneton, the g-factor is
$g=2$, and the coherence length of the exchange interaction is $\lambda_{\rm
ex}\simeq 1.76\times 10^{-8}\mathrm{m}$. The dielectric constant is
$\epsilon_{\rm M}=15$ for YIG Sadhana  and $\epsilon_{\rm b}=12$ for GGG Lal77
. The magnon is excited in each YIG layer, and the interlayer magnon–magnon
interaction is attributed to the dipole–dipole interaction. An incident
microwave $h_{\rm inc}$ is introduced onto the surface of the thick layer. The
microwave is applied to the multi-layered structure even for an in-plane
wavenumber $k_{\parallel}$ larger than the wavenumber $q_{0}=\omega/c$ in
vacuum. This can be realized by utilizing an evanescent prism mode on the
structure, or a fiber coupling at the edge surface. For the calculation of the
fields inside and outside the sample, we place a field inside the thick YIG
layer and consider the boundary conditions of the microwave through Green’s
function to determine the self-consistent fields in other layers (refer to the
Methods section). We apply an in-plane gradient magnetic field with
$H_{0}^{(1)}=H_{\rm M}$ and $H_{0}^{(2)}=1.5H_{\rm M}$ to the thin and thick
layers, respectively, to obtain the level (anti-)crossing of thin-layer
magnons and thick-layer MPs. Here, $H_{\rm M}\equiv\hbar\omega_{\rm
M}/(g\mu_{\rm B}\mu_{0})$. Such a spatial gradient of the magnetic field can
be realized using micromagnets or current-induced fields. (The use of
different magnetic materials for the two layers also affords similar level
crossings.)

Figure 2:  (a) Schematic of the cross-section of the proposed multi-layer
structure with $100\,\mathrm{nm}$ YIG/$1\,\mathrm{\mu m}$ GGG/$1\,\mathrm{mm}$
YIG layered structure. (b,c) Spatial profiles of the transverse (blue) and
longitudinal magnetic fields (red lines) originating from (b) the thick and
(c) thin-layer magnons. The field amplitude is normalized by the incident
field $h_{\rm inc}$. Other parameters are as follows: $\mu_{0}M_{\rm
s}=175\,\mathrm{mT}$, $\lambda_{\rm ex}=1.76\times 10^{-8}\,\mathrm{m}$,
$\epsilon_{\rm M}=15$, $\epsilon_{\rm b}=12$. $H_{0}^{(1)}=H_{\rm M}$, and
$H_{0}^{(2)}=1.5H_{\rm M}$. The unit of the magnetic field is $H_{\rm
M}\equiv\hbar\omega_{\rm M}/(g\mu_{\rm B}\mu_{0})=M_{\rm s}$.

Here, we emphasize the role of the GGG layer. The GGG spacer layer enables
engineering dispersion. Even for a system without a GGG layer, namely a single
$100\,\mathrm{nm}+1\,\mathrm{mm}$ YIG film, circuit fabrication is possible.
However, the magnon wavefunctions spread along the longitudinal direction.
Hence, the spatial gradient of the applied magnetic field, which is
indispensable for manipulating dispersion crossing, becomes ineffective. In
addition, the magnon becomes insensitive to the fabricated circuit, because
the hundred-nanometer-scale modulation of thickness is not significant for the
magnons in the millimeter-scale YIG film. Therefore, the GGG layer plays a
critical role in the functioning of the proposed system. It disconnects the
magnon wavefunction in the thin layer from that in the thick layer, and a
thickness of 1 $\mu$m is sufficient for this purpose. The decay length of the
spin dipole–dipole interaction for the magnon wavenumber under consideration
is significantly greater than the aforementioned thickness. Hence, a slight,
micrometer-order modulation of the thickness for the GGG layer does not affect
the antenna effect significantly.

In the proposed structure, the ‘waveguide modes’ and the ‘antenna effect’
impart the thin-layer MP transport with functionalities that are essential for
practical applications. We consider the waveguide mode in a simple YIG layer
with thickness $d$. The two waveguide modes, i.e., the transverse electric
(TE) and magnetic (TM) modes, are formed within $q<k_{\parallel}<q_{0}$ in the
YIG layer, which decays exponentially in vacuum, where
$q=q_{0}/\sqrt{\epsilon_{\rm M}}$. The waveguide modes satisfy the condition
$k_{z}d+{\rm Arg}(r_{\xi})=n\pi$ for the $\xi=\text{TE, TM}$ modes. Here,
$k_{z}=\sqrt{q^{2}-k_{\parallel}}$. The reflection coefficient $r_{\xi}$ at
the interfaces depends on the wavenumber $(k_{\parallel},k_{z})$. Here, $n$
denotes the index of the waveguide mode, and the modes require millimeter-
scale thickness, $d\simeq\pi/k_{z}\sim\mathrm{mm}$. Although this condition is
slightly different for multi-layer structures, the behavior of the mode
remains qualitatively unaffected.

In the proposed structure, the magnons in the thick and thin YIG layers induce
both a longitudinal magnetic field and a transverse electromagnetic field
(Figs. 2(b) and (c)). The longitudinal field results in an interlayer
magnon–magnon interaction. The fields from the thick-layer magnons are almost
symmetric in the $z$-direction, whereas the transverse field from the thin-
layer magnons exhibits non-symmetric behavior owing to the difference of
dielectric constants. Hence, considering the nonlocal response of magnons is
essential. We formulate a general nonlocal response theory for magnons
kinoshita19 ; book:Ishihara , which considers a self-consistent relation
between Maxwell’s equations and the constitutive equation with $\bm{E}$,
$\bm{H}$, $\bm{M}$, $\bm{P}$, and nonlocal susceptibility (see Methods
section).

Engineering the dispersion relation enables control over magnon propagation.
To demonstrate this, we consider three types of layered systems with YIG:
Structure I: $1\mathrm{mm}$ YIG film, Structure II: $100\mathrm{nm}$
YIG/$1\mathrm{mm}$ GGG, and Structure III: $100\mathrm{nm}$ YIG/$1\mathrm{\mu
m}$ GGG/$1\mathrm{mm}$ YIG (the proposed structure). The YIG layers exhibit
in-plane magnetization. The antenna effect originates from the dispersion of
the thick YIG layer, and the waveguide modes require the system to possess a
certain thickness. Figures 3(a) and (b) present the dispersion relations of
the MPs in Structures I and II, respectively. In Structure I, the bare magnon
exhibits a negative slope Kalinikos . The thick film spontaneously forms the
TE and TM waveguide modes, which results in strong MPs and three dispersions.
We refer to these as the thick-layer upper, middle, and lower MPs (denoted as
thick-UMP, thick-MMP, and thick-LMP, respectively). At
$0.1<k_{\parallel}d<0.5$, the group velocity of the three MPs is significantly
higher than that of the bare magnon. The in-plane wavenumber $k_{\parallel}$
can be tuned by adjusting the incident angle. With an increase in the incident
angle, the propagation direction of the thick-LMP changes around
$k_{\parallel}d=0.6$. In the proposed structure (Structure III), the
functionalities of the thick-layer MPs are transferred to the thin-layer
magnons via the antenna effect. By contrast, in Structure II, the dispersion
exhibits direct coupling between the thin-layer magnons and the TE/TM modes.
Their strengths are $\Delta_{\rm TE}\simeq 3.3\times 10^{-3}\omega_{\rm M}$
and $\Delta_{\rm TM}\simeq 1.7\times 10^{-3}\omega_{\rm M}$, respectively. If
the $100\,\mathrm{nm}$ YIG layer is isolated from the GGG layer, where the
waveguide mode is not formed, the coupling strength between the magnon and
photon is negligible. In this case, the dispersion of the bare magnon should
be almost flat.

The properties of the thin-layer MPs in the proposed structure are provided by
the waveguide modes (transverse field) and the antenna effect, based on the
interlayer magnon–magnon interaction (longitudinal field). Figure 3(c) shows
the dispersions of the MPs in Structure III. The thick YIG layer captures the
microwaves and provides functionality to the thin-layer magnons via the
antenna effect. The functionalities imparted to the thin-layer magnons can be
tuned using the external magnetic fields. As shown in Fig. 3(c), the thin-
layer magnons couple with the thick-MMP at $k_{\parallel}d\approx 0.2$ and
with the thick-LMP at $k_{\parallel}d\approx 0.4$ and $1.2$. Interestingly, at
the latter two anti-crossings, the MPs exhibit forward and backward
propagation, respectively.

We evaluate the magnon current in the proposed structure. The group velocity
is evaluated based on the dispersion, whereas the magnon density is evaluated
based on the magnetization amplitude of the deviation from the saturation
magnetization, $\bm{{\rm m}}(\bm{{\rm r}})=\bm{{\rm M}}(\bm{{\rm r}})-\bm{{\rm
M}}_{\rm s}$. Figures 3(d) and (e) present the spatial integral of the induced
magnetization in the thin YIG layer,
$\bar{m}=\sqrt{\frac{1}{S_{0}d_{0}}\int_{d_{0}}dz\int_{S_{0}}dxdy|\bm{{\rm
m}}(\bm{{\rm r}})|^{2}}$, for Structures II and III as functions of $\omega$
and $k_{\parallel}$, respectively. The enhancement in $\bar{m}$ implies the
generation of a large magnon density. Here, $S_{0}$ and $d_{0}$ denote the
area of the YIG unit cell and the thickness of the thin layer, respectively.
In Structure II, although the induced magnetization
$\bar{m}(k_{\parallel},\omega)$ follows the dispersion of the bare magnon, the
amplitude is not large. However, in Structure III, $\bar{m}$ is considerably
increased owing to the interlayer magnon–magnon interaction appearing on the
dispersions of the thin layer of the bare magnon and thick-LMP. This large
$\bar{m}$ results in a large magnon current in the thin YIG. Along the
dispersion $\omega^{(\alpha)}(k_{\parallel})$ in Structure III, we plot the
wavenumber dependence of the induced magnetization $\bar{m}^{(\alpha)}$ and
the group velocity $v_{\rm g}^{(\alpha)}$, as shown in Figs. 4(a) and (b),
respectively. $\alpha$ is an index of the MP branches
$\omega^{(\alpha)}(k_{\parallel})$. The magnon density in the thin YIG layer
is enhanced owing to the strong coupling with the thick-LMP. The MP velocity
is increased up to $0.01$ times the light speed in YIG. This velocity is in
the order of $10^{6}\mathrm{m/s}$, which is $10^{3}$ times the typical
(exchange) magnon velocity HXia98 ; HYu12 ; Yamanoi13 ; CLiu18 .

Figure 3:  (a,b,c) Dispersion relations $\omega(k_{\parallel})$ of MPs in the
single $1\mathrm{mm}$ YIG film (a, Structure I); $100\,\mathrm{nm}$
YIG/$1\,\mathrm{mm}$ GGG structure (b, Structure II); and proposed structure,
$100\,\mathrm{nm}$ YIG/$1\,\mu\mathrm{m}$ GGG/$1\,\mathrm{mm}$ YIG (c,
Structure III). The corresponding structures are shown above the panels. The
wavenumber is normalized by the thickness of the system, $d=1\,\mathrm{mm}$.
The YIG layers exhibit in-plane saturation magnetization. The other parameters
are the same as those in Fig. 2. The external static magnetic fields are
applied along the in-plane magnetization. The solid lines indicate the MPs in
panels (a), (b), and (c). In (a), the black dotted line indicates the bare
magnon, and the blue and green dotted lines represent the TE and TM modes,
respectively. In (b) and (c), the colors of the solid lines are clearly
distinguishable. The insets of (b) and (c) present zoomed views. (d,e)
Magnetization amplitude $\bar{m}$ deviated from the saturation magnetization
of the thin YIG in Structure II (d) and III (e), respectively. The amplitude
is normalized by the incident magnetic field $h_{\rm inc}$. The magnetization
$\bar{m}$ is enhanced on the dispersion of MPs.

Figure 4(c) shows the magnon currents in the MPs, which are expressed as

$I_{\rm m}^{(\alpha)}(k_{\parallel})=v_{\rm
g}^{(\alpha)}(k_{\parallel})\times\bar{m}^{(\alpha)}(k_{\parallel},\omega^{(\alpha)}(k_{\parallel})).$
(1)

Following the peaks of $\bar{m}^{(\alpha)}$ and the sign of $v_{\rm
g}^{(\alpha)}$, the magnon current in the thin YIG layer exhibits peak and dip
behaviors at $k_{\parallel}d\approx 0.4$ and $1.2$, respectively. For the
applied static magnetic fields, $H_{0}^{(1)}=H_{\rm M}$ and
$H_{0}^{(2)}=1.5H_{\rm M}$, it is necessary to tune the incident angle of the
microwave from $k_{\parallel}d\approx 0.4$ to $1.2$ to switch the magnon
current direction. The distance between the peak and dip is rather large in
this case. However, this distance can be reduced by controlling the shifts in
the thin-layer magnon energy. To this end, two anti-crossing points can be
modulated by tuning the applied field $H_{0}^{(1)}$ in the thin layer. We
examine a sweep of the thin-layer magnon energy by $H_{0}^{(1)}$ up to the
dispersion edge of the thick-LMP. Figs. 4(d) and (e) demonstrate the magnon
current $I_{\rm m}^{(\alpha)}$ from the two MP branches, respectively. With a
slight increase in $H_{0}^{(1)}$, the distance between the peaks and dips of
$I_{\rm m}^{(\alpha)}$ can be reduced. At $H_{0}^{(1)}>1.02H_{\rm M}$, the
peaks and dips are unified, and the magnon current is reduced significantly.
The thin lines and the inset of Fig. 4(c) indicate the magnon current and
dispersion, respectively, when $H_{0}^{(1)}\approx 1.02H_{\rm M}$. Here, the
peak and dip are sufficiently detectable, and the sign can be altered by a
slight shift in $k_{\parallel}$. This condition can be deemed appropriate for
magnon current switching. When $H_{0}^{(1)}$ decreases, the distance between
the peaks and dips increases. For a large $k_{\parallel}d$, the magnon–magnon
interaction is suppressed owing to the exponential decay of the longitudinal
field for short-wavelength microwaves. Then, the dip disappears when
$H_{0}^{(1)}$ is reduced.

Figure 4:  Magnon current in thin YIG layer of Structure III. (a) $\bar{m}$ in
Fig. 3(e) along dispersion of two MPs in Fig. 3(c). (b) Group velocity $v_{\rm
g}$ of thin-layer MP dispersion, as shown in Fig. 3(c). The velocity is
normalized by the velocity of light in vacuum, $c$. (c) (Normalized) magnon
current $I_{\rm m}^{(\alpha)}$ from thin-layer MPs when external fields are
$H_{0}^{(1)}=H_{\rm M}$ (thick lines) and $\approx 1.02H_{\rm M}$ (thin
lines). The red and blue lines in (a), (b), and (c) denote the lowest and
second branches in Fig. 3(c), respectively. The green and magenta lines and
inset show $I_{\rm m}^{(\alpha)}$ and the dispersion relation when
$H_{0}^{(1)}\approx 1.02H_{\rm M}$, respectively. (d,e) The magnon currents,
$I_{\rm m}$, from the second (d) and lowest (e) branches of thin-layer MPs in
a plane of the external magnetic field are represented as functions of
$H_{0}^{(1)}$ and the in-plane wavenumber, $k_{\parallel}d$.

The dispersion exhibits anti-crossing with the thick-MMP at
$k_{\parallel}d\approx 0.2$ (see Fig. 3(c)). The peak and dip of $I_{\rm
m}^{(\alpha)}$ for the thick-LMP are broad, whereas the peak of the thick-MMP
is extremely sharp and isolated (outside the window in Fig. 4(c)). However, we
do not consider this peak, because the characteristics of such a sharp peak
are not useful for practical applications. If the magnon circuit is fabricated
in the thin layer, as shown in Fig. 1(b), the internal magnetic field and the
magnetization are slightly varied. The sharp peak is highly sensitive to such
changes and is difficult to control.

For the processed structures illustrated in Fig. 1(b), some parts of YIG are
removed. In such areas, the dispersion of MPs is almost equivalent to that
shown in Fig. 3(a). The thick-LMP may induce a large magnon current. However,
at the dispersion edge ($k_{\parallel}d\approx 0.65,\omega\approx
1.76\omega_{\rm M}$), the magnon current is suppressed due to the low
velocity. Thus, the leakage magnon current is expected to be reduced. This is
also an important advantage of the proposed structure. Details regarding the
behaviors of magnons in the processed thin layer cannot be obtained only from
the present study; these behaviors should be elucidated based on corresponding
models. We are currently investigating this issue using numerical techniques
based on nonlocal response theory.

##  II Conclusions

The magnon is a promising candidate for low-energy-consumption devices. To
ensure effective device operation, the magnon current should be increased by
increasing the density and velocity and acquiring switchable control over the
direction of the current. One method to increase the velocity of the magnon is
the formation of MPs. However, the light–magnon coupling is inherently weak.
To address this bottleneck, we proposed a multi-layered magnetic structure to
demonstrate a large and switchable magnon current in the integrable thin
magnetic layer. We considered two crucial mechanisms: the ‘waveguide mode’ and
‘antenna effect’. The large magnon current and its switchable functionality
are supported by these two phenomena. The formation of the waveguide mode is
attributed to the spatial structure of the dielectric constant, which
considerably enhances the light–magnon coupling caused by the spatial
interplay between the magnons and microwaves. The antenna effect, originating
from the magnon–magnon interaction (longitudinal field), maps the
functionality of the thick-layer MPs to the nanoscale thin-layer magnons. By
analyzing the dispersion relations of the MPs in this system and numerically
calculating the magnon currents in the nanometer-scale thin film based on
nonlocal response theory, we demonstrated that the proposed multi-layered
structure can successfully exhibit the desired features mentioned above. The
antenna effect arising from the thick-LMP produces a broad peak and dip in the
magnon current as a function of the incident angle. This is desirable for
circuit fabrication in the thin layer. Our proposed approach thus presents the
intriguing possibility of developing advanced magnonic technologies.

This work is supported in part by JSPS KAKENHI 16H06504 in Scientic Research
on Innovative Areas Nano-Material Optical-Manipulation”, JSPS KAKENHI 18K13484
and 18H01151. T.Y. is supported by the Toyota Riken Scholar from the Toyota
Physical and Chemical Research Institute.

The authors declare no conflicts of interest.

##  III Methods

###  III.1 Magnon

The spin wave in magnetic materials is described by the exchange interaction
$\hat{\mathcal{H}}_{\rm ex}=-\sum_{\langle i,j\rangle}J_{ij}\hat{\bm{{\rm
S}}}_{i}\cdot\hat{\bm{{\rm S}}}_{j}$. Here, we consider a uniform
ferromagnetic film, $J_{ij}=J_{0}>0$, and apply the Holstein–Primakoff
transformation HP40 , $\hat{S}^{z}_{i}=S-\hat{b}_{i}^{\dagger}\hat{b}_{i}$,
$\hat{S}^{-}_{i}=\sqrt{2S}\hat{b}_{i}^{\dagger}(1-\frac{\hat{b}_{i}^{\dagger}\hat{b}_{i}}{2S})$,
$\hat{S}^{+}_{i}=\sqrt{2S}(1-\frac{\hat{b}_{i}^{\dagger}\hat{b}_{i}}{2S})\hat{b}_{i}$.
On removing the ground state energy of the ferromagnet, we obtain the magnon
Hamiltonian $\hat{\mathcal{H}}_{\rm{sw}}$, which can be rewritten in terms of
the boson operators $\hat{b}_{\bm{{\rm k}}}$ and $\hat{b}_{\bm{{\rm
k}}}^{\dagger}$ as $\hat{\mathcal{H}}_{\rm ex}\simeq\hat{\mathcal{H}}_{\rm
sw}-\delta NJS^{2}$ with

$\hat{\mathcal{H}}_{\rm sw}=2S\sum_{\bm{{\rm
k}}}\left[\tilde{J}(0)-\tilde{J}({\bm{{\rm k}}})\right]\hat{b}_{\bm{{\rm
k}}}^{\dagger}\hat{b}_{\bm{{\rm k}}}.$ (2)

Here, $\hat{b}_{\bm{{\rm
k}}}=\frac{1}{\sqrt{N}}\sum_{i}\hat{b}_{i}e^{-i\bm{{\rm k}}\cdot\bm{{\rm
R}}_{i}}$. $\tilde{J}({\bm{{\rm k}}})=\sum_{i^{\prime}}J_{0}e^{-i\bm{{\rm
k}}\cdot\bm{{\rm R}}_{i^{\prime}}}$ is the summation of all neighboring sites
$\bm{{\rm R}}_{i^{\prime}}$ of the localized electrons at $\bm{{\rm R}}_{i}$;
$\delta$ and $N$ represent the number of neighboring sites and the total
number of sites, respectively. $S$ is the size of the localized spin at each
site. This boson describes the quantum of the spin wave, i.e., the so-called
magnon. A constant term can be reduced without a loss of generality. The
exchange interaction originates from the interaction between the localized
electrons, which is a part of the longitudinal electric fields. However, the
dipole–dipole interaction is considered in Maxwell’s equations.

The magnetic fields also contribute to the magnon under the Zeeman effect:
$\hat{\mathcal{H}}_{\rm Z}=-g\mu_{\rm B}\mu_{0}\sum_{i}\hat{\bm{{\rm
S}}}_{i}\cdot\bm{{\rm H}}(\bm{{\rm r}}_{i};t)$, where $\mu_{\rm B}$ is the
Bohr magneton and $g$ is the $g$-factor. The magnetic field $\bm{{\rm
H}}(\bm{{\rm r}};t)$ comprises an external static field, $\bm{{\rm H}}_{0}$,
an incident microwave, $\bm{{\rm h}}_{\rm inc}(\bm{{\rm r}};t)$, and the
response field, $\bm{{\rm h}}_{\rm res}(\bm{{\rm r}};t)$. The response field
is generated by the induced magnons. We consider $\bm{{\rm h}}(\bm{{\rm
r}};t)=\bm{{\rm h}}_{\rm inc}(\bm{{\rm r}};t)+\bm{{\rm h}}_{\rm res}(\bm{{\rm
r}};t)$ and define the non-perturbative and perturbative Hamiltonians as
$\hat{\mathcal{H}}_{0}=\hat{\mathcal{H}}_{\rm sw}+g\mu_{\rm B}\mu_{0}|\bm{{\rm
H_{0}}}|\sum_{\bm{{\rm k}}}\hat{b}_{\bm{{\rm k}}}^{\dagger}\hat{b}_{\bm{{\rm
k}}}$ and $\hat{\mathcal{H}}^{\prime}=-g\mu_{\rm
B}\mu_{0}\sum_{i}\hat{\bm{{\rm S}}}_{i}\cdot\bm{{\rm h}}(\bm{{\rm r}}_{i};t)$,
respectively. For $\hat{\mathcal{H}}_{0}$, the energy dispersion of the
magnons is

$E_{\bm{{\rm k}}}=2S[\tilde{J}(0)-\tilde{J}({\bm{{\rm k}}})]+g\mu_{\rm
B}\mu_{0}|\bm{{\rm H}}_{0}|\simeq\hbar\omega_{\rm M}\lambda_{\rm
ex}^{2}k^{2}+\hbar\omega_{\rm H},$ (3)

which is shifted by the external magnetic field $|\bm{{\rm H}}_{0}|$. Here,
$\lambda_{\rm ex}=\sqrt{2S|\nabla_{k}^{2}J(k)|/(\hbar\omega_{M})}$ is the
exchange length of the spin wave. $\hbar\omega_{\rm
M}=\hbar\gamma\mu_{0}|\bm{{\rm M}}_{\rm s}|$ is the magnetization energy,
where $\gamma=g\mu_{\rm B}/\hbar$ is the gyromagnetic ratio and the saturation
magnetization is $\bm{{\rm M}}_{\rm s}$.

###  III.2 Nonlocal response of magnon

Based on the quantum description of magnons, we formulate the microscopic
theory of magnon response. The constitutive equation for magnetic response is
$\bm{{\rm m}}(\bm{{\rm r}}_{i};\omega)=\sum_{j}\bar{\chi}(\bm{{\rm
r}}_{i},\bm{{\rm r}}_{j};\omega)\bm{{\rm h}}(\bm{{\rm r}}_{j};\omega)$ with
the nonlocal susceptibility book:Cho . Here, $\bm{{\rm m}}$ is the deviation
of the magnetization from the saturation magnetization, and $\bm{{\rm
m}}(\bm{{\rm r}};t)=\bm{{\rm M}}(\bm{{\rm r}};t)-\bm{{\rm M}}_{\rm s}$. The
susceptibility is obtained from the Hamiltonian
$\hat{\mathcal{H}}_{0}+\hat{\mathcal{H}}^{\prime}$, based on the linear
response theory Kubo57 :

$\bar{\chi}(\bm{{\rm r}}_{i},\bm{{\rm
r}}_{j};\omega)=-\frac{\mu_{0}}{v}\sum_{n}\left[\frac{\bm{{\rm\mu}}_{0n}(\bm{{\rm
r}}_{i})\\{\bm{{\rm\mu}}_{n0}(\bm{{\rm r}}_{j})\\}^{\rm
t}}{E_{n}-\hbar\omega-i\eta}+\frac{\bm{{\rm\mu}}_{n0}(\bm{{\rm
r}}_{i})\\{\bm{{\rm\mu}}_{0n}(\bm{{\rm r}}_{j})\\}^{\rm
t}}{E_{n}+\hbar\omega+i\eta}\right].$ (4)

Here, $\bm{{\rm\mu}}_{nm}(\bm{{\rm r}}_{i})=-g\mu_{\rm B}\langle\bm{{\rm
k}}_{n}|\hat{\bm{{\rm S}}}_{i}|\bm{{\rm k}}_{m}\rangle$. The index $n$
represents the eigenstate $|\bm{{\rm k}}_{n}\rangle$ of
$\hat{\mathcal{H}}_{0}$, $\eta$ is an infinitesimal positive value due to
causality, and $v$ is the volume of the unit cell for each spin site.

We consider the magnetic nonlocal response of magnons in a thin ferromagnetic
film. At the surfaces, the spin waves are confined owing to the pinning
effect. For the magnon wavefunction, we obtain $\sqrt{\frac{1}{v}}\langle
0|\hat{\bm{{\rm S}}}_{i}|{n,\bm{{\rm
k}}_{\parallel}}\rangle=\sqrt{\frac{S}{2}}\frac{1}{L}\exp(i\bm{{\rm
k}}_{\parallel}\cdot\bm{{\rm\rho}}_{i,{\parallel}})\phi_{n}(z_{i})$, where
$\phi_{n}(z_{i})=\sqrt{\frac{2}{d}}\sin(K_{n}z_{i})$ and $K_{n}=n\pi/d$. Here,
$d$ and $L$ are the thickness of the film in the $z$-direction and the size of
the system in the $xy$-plane, respectively. $\bm{{\rm k}}_{\parallel}$ and
$\bm{{\rm\rho}}_{i,{\parallel}}$ are the in-plane wavenumber and position
vectors, respectively, and $S$ is the value of the spin. We apply the
continuous approximation for the magnon wavefunctions. For the translational
symmetry in the $xy$-plane, the field and magnetization are partially Fourier-
transformed as follows: $\bm{{\rm
h}}(\bm{{\rm\rho}}_{\parallel},z;\omega)=(1/2\pi)\int{\rm d}^{2}\bm{{\rm
k}}_{\parallel}\ \tilde{\bm{{\rm h}}}(\bm{{\rm
k}}_{\parallel},z;\omega)e^{i\bm{{\rm
k}}_{\parallel}\cdot\bm{{\rm\rho}}_{\parallel}}$. The constitutive equation is
also rewritten as $\tilde{\bm{{\rm m}}}(\bm{{\rm
k}}_{\parallel},z;\omega)=\int{\rm d}z^{\prime}\bar{\chi}(\bm{{\rm
k}}_{\parallel},z,z^{\prime},\omega)\tilde{\bm{{\rm h}}}(\bm{{\rm
k}}_{\parallel},z^{\prime};\omega)$, where the susceptibility is represented
by

$\bar{\chi}(\bm{{\rm
k}}_{\parallel},z,z^{\prime};\omega)=-\frac{\mu_{0}}{v}\sum_{n}\left[\frac{\tilde{\bm{{\rm\mu}}}_{n}(z)\tilde{\bm{{\rm\mu}}}_{n}^{\dagger}(z^{\prime})}{E_{n,\bm{{\rm
k}}_{\parallel}}-\hbar\omega-i\eta}+\frac{\tilde{\bm{{\rm\mu}}}_{n}^{*}(z)\tilde{\bm{{\rm\mu}}}_{n}^{\rm
t}(z^{\prime})}{E_{n,\bm{{\rm k}}_{\parallel}}+\hbar\omega+i\eta}\right].$ (5)

where $\tilde{\bm{{\rm\mu}}}_{n}(z)=g\mu_{\rm
B}\sqrt{\frac{S}{2}}\phi_{n}(z)(1,-i,0)^{\rm t}$. For multi-layered systems,
the susceptibility is expressed as a combination of Eq. (5) for the respective
layers.

###  III.3 Maxwell’s equations

The coupling between the magnon and the electromagnetic field (photon) to form
the MP is described by a self-consistent relation with the nonlocal
constitutive equation and the microscopic Maxwell’s equations, where the
spatial correlation between the magnon wavefunction and the electromagnetic
field are treated appropriately. The microscopic Maxwell’s equations are
$\bm{{\rm\nabla}}\cdot\bm{{\rm
E}}=-(1/\varepsilon_{0})\bm{{\rm\nabla}}\cdot\bm{{\rm P}}$,
$\bm{{\rm\nabla}}\times\bm{{\rm E}}=-\partial\bm{{\rm B}}/\partial t$,
$\bm{{\rm\nabla}}\cdot\bm{{\rm B}}=0$, and $\bm{{\rm\nabla}}\times\bm{{\rm
B}}=\varepsilon_{0}\mu_{0}\partial\bm{{\rm E}}/\partial
t+\mu_{0}(\bm{{\rm\nabla}}\times\bm{{\rm M}}+\partial\bm{{\rm P}}/\partial
t)$. Here, $\bm{{\rm E}}$, $\bm{{\rm B}}$, $\bm{{\rm M}}$, $\bm{{\rm P}}$,
$\varepsilon_{0}$, and $\mu_{0}$ denote the total electric field, total
magnetic flux density, magnetization, polarization, dielectric constant in
vacuum, and permeability in vacuum, respectively. The magnons are described in
terms of $\bm{{\rm M}}(\bm{{\rm r}})$ with nonlocal susceptibility in Eq. (5),
while the polarization is represented by the dielectric constant of each
layer, $\bm{{\rm P}}(\bm{{\rm r}})=\varepsilon_{0}\chi_{\rm E}(z)\bm{{\rm
E}}(\bm{{\rm r}})$. The dielectric constant $\varepsilon_{0}\chi_{\rm E}(z)$
forms the TE and TM waveguide modes. We set the field propagation direction
along the $xz$-plane, $\bm{{\rm k}}=(k_{\parallel},0,k_{z})^{\rm t}$.
Subsequently, the $y$-component of the magnetic (electric) field determines
the reflection and transmission of the TM (TE) mode.

For the magnetic field $\bm{{\rm H}}=\bm{{\rm B}}/\mu_{0}-\bm{{\rm M}}$, the
Maxwell’s equations are $\bm{{\rm\nabla}}\times\bm{{\rm\nabla}}\times\bm{{\rm
H}}+\mu_{0}\varepsilon_{0}(1+\chi_{\rm E})\partial^{2}\bm{{\rm H}}/\partial
t^{2}=-\mu_{0}\varepsilon_{0}(1+\chi_{\rm E})\partial^{2}\bm{{\rm M}}/\partial
t^{2}+\varepsilon_{0}(\bm{{\rm\nabla}}\chi_{\rm E})\times\partial\bm{{\rm
E}}/\partial t$. The systems considered in this study exhibits transverse
symmetry in the $xy$-direction. We consider partial Fourier transformations
with $\bm{{\rm k}}_{\parallel}$ and $\omega$ for the magnetic fields.
$\tilde{\bm{{\rm H}}}(\bm{{\rm
k}}_{\parallel},z;\omega)=\frac{1}{(2\pi)^{3/2}}\int{\rm d}x{\rm d}y\int{\rm
d}t,\bm{{\rm H}}(\bm{{\rm r}};t)e^{i(k_{x}x+k_{y}y)}e^{-i\omega t}$. The
electric field is also Fourier-transformed in the same manner. For the TM
mode, $\tilde{\bm{{\rm H}}}^{\rm(TM)}=\left(0,\tilde{H}_{y},0\right)^{\rm t}$,
the solution is given as

$\tilde{\bm{{\rm H}}}^{\rm(TM)}(\bm{{\rm
k}}_{\parallel},z;\omega)=\tilde{\bm{{\rm H}}}^{\rm(TM)}_{\rm inc}(\bm{{\rm
k}}_{\parallel},z;\omega)+\int{\rm d}z^{\prime}\bar{G}^{\rm(TM)}(\bm{{\rm
k}}_{\parallel},z,z^{\prime};\omega)\tilde{\bm{{\rm M}}}(\bm{{\rm
k}}_{\parallel},z^{\prime};\omega)$ (6)

with a dyadic Green’s function

$\bar{G}^{\rm(TM)}(\bm{{\rm
k}}_{\parallel},z,z^{\prime};\omega)=-\left(\begin{matrix}0&&\\\
&q(z^{\prime})^{2}&\\\ &&0\end{matrix}\right)\frac{ig(\bm{{\rm
k}}_{\parallel},z,z^{\prime};\omega)}{2\sqrt{q(z^{\prime})^{2}-k_{\parallel}^{2}}}.$
(7)

Here, $g(\bm{{\rm k}}_{\parallel},z,z^{\prime};\omega)$ is a scalar Green’s
function describing a scalar wave propagation from position $z^{\prime}$ in a
layer to $z$ in another layer. The $(\bm{{\rm\nabla}}\chi_{\rm
E})\times\tilde{\bm{{\rm E}}}$ term is considered by the boundary conditions
for $g(\bm{{\rm k}}_{\parallel},z,z^{\prime};\omega)$. In the $n$-th layer,
$z_{n-1}<z<z_{n}$, the wavenumber is $q(z)=q_{n}=\chi_{{\rm E},n}\omega/c$.
When $z_{n-1}<z<z_{n}$ and $z_{m-1}<z^{\prime}<z_{m}$, the scalar Green’s
function is given as

$g(\bm{{\rm
k}}_{\parallel},z,z^{\prime};\omega)=\left\\{\begin{matrix}\left[e^{ik_{n}(z-z_{n-1})}+e^{-ik_{n}(z-z_{n})}\tilde{r}_{n+1,n}e^{ik_{n}d_{n}}\right]\tilde{t}_{n,m}A_{m}(z^{\prime})&(n>m)\\\
e^{ik_{m}|z-z^{\prime}|}+e^{ik_{m}(z-z_{m-1})}B_{m}(z^{\prime})+e^{-ik_{m}(z-z_{m})}C_{m}(z^{\prime})&(n=m)\\\
\left[e^{-ik_{n}(z-z_{n})}+e^{ik_{n}(z-z_{n-1})}\tilde{r}_{n-1,n}e^{ik_{n}d_{n}}\right]\tilde{t}_{n,m}D_{m}(z^{\prime})&(n<m)\end{matrix}\right.$
(8)

where $\tilde{r}_{n\pm 1,n}$ and $\tilde{t}_{n,m}$ are the generalized
reflection and transmission coefficients, respectively book:Chew . Here,
$k_{n}=\sqrt{q_{n}^{2}-k_{\parallel}^{2}}$ is the wavenumber in the
perpendicular direction of layer $n$. The factors are

$\displaystyle A_{m}(z^{\prime})$ $\displaystyle=$ $\displaystyle
M_{m}\left[e^{ik_{m}(z_{m}-z^{\prime})}+e^{ik_{m}d_{m}}\tilde{r}_{m-1,m}e^{-ik_{m}(z_{m-1}-z^{\prime})}\right],$
(9) $\displaystyle B_{m}(z^{\prime})$ $\displaystyle=$ $\displaystyle
M_{m}\tilde{r}_{m-1,m}\left[e^{-ik_{m}(z_{m-1}-z^{\prime})}+e^{ik_{m}d_{m}}\tilde{r}_{m+1,m}e^{ik_{m}(z_{m}-z^{\prime})}\right],$
(10) $\displaystyle C_{m}(z^{\prime})$ $\displaystyle=$ $\displaystyle
M_{m}\tilde{r}_{m+1,m}\left[e^{ik_{m}(z_{m}-z^{\prime})}+e^{ik_{m}d_{m}}\tilde{r}_{m-1,m}e^{-ik_{m}(z_{m-1}-z^{\prime})}\right],$
(11) $\displaystyle D_{m}(z^{\prime})$ $\displaystyle=$ $\displaystyle
M_{m}\left[e^{-ik_{m}(z_{m-1}-z^{\prime})}+e^{ik_{m}d_{m}}\tilde{r}_{m+1,m}e^{ik_{m}(z_{m-1}-z^{\prime})}\right],$
(12)

and
$M_{m}=\left[1-\tilde{r}_{m+1,m}\tilde{r}_{m-1,m}e^{i2k_{m}d_{m}}\right]^{-1}$.

For the electric field, we obtain
$\bm{{\rm\nabla}}\times\bm{{\rm\nabla}}\times\bm{{\rm
E}}+\mu_{0}\varepsilon_{0}(1+\chi_{\rm E})\partial^{2}\bm{{\rm E}}/\partial
t^{2}=-\mu_{0}\bm{{\rm\nabla}}\times\partial\bm{{\rm M}}/\partial t$. For the
TE mode, $\tilde{\bm{{\rm E}}}^{\rm(TE)}=\left(0,\tilde{E}_{y},0\right)^{\rm
t}$, is related with $\tilde{\bm{{\rm H}}}^{\rm(TE)}$ through Faraday’s law.
The $\bm{{\rm\nabla}}\bm{{\rm\nabla}}\cdot\bm{{\rm E}}$ term from the left-
hand side is incorporated with the boundary conditions. Thus, the solution of
$\tilde{\bm{{\rm
H}}}^{\rm(TE)}=\left(\tilde{H}_{x},0,\tilde{H}_{z}\right)^{\rm t}$ is

$\tilde{\bm{{\rm H}}}^{\rm(TE)}(\bm{{\rm
k}}_{\parallel},z;\omega)=\tilde{\bm{{\rm H}}}^{\rm(TE)}_{\rm inc}(\bm{{\rm
k}}_{\parallel},z;\omega)+\int{\rm d}z^{\prime}\bar{G}^{\rm(TE)}(\bm{{\rm
k}}_{\parallel},z,z^{\prime};\omega)\tilde{\bm{{\rm M}}}(\bm{{\rm
k}}_{\parallel},z^{\prime};\omega)$ (13)

with

$\bar{G}^{\rm(TE)}(\bm{{\rm
k}}_{\parallel},z,z^{\prime};\omega)=-\left(\begin{matrix}\partial_{z}\partial_{z^{\prime}}&&ik_{\parallel}\partial_{z}\\\
&0&\\\
-ik_{\parallel}\partial_{z^{\prime}}&&k_{\parallel}^{2}\end{matrix}\right)\frac{ig(\bm{{\rm
k}}_{\parallel},z,z^{\prime};\omega)}{2\sqrt{q(z^{\prime})^{2}-k_{\parallel}^{2}}}-\left(\begin{matrix}0&&\\\
&0&\\\ &&1\end{matrix}\right)\delta(z-z^{\prime}).$ (14)

The scalar Green’s function $g(\bm{{\rm k}}_{\parallel},z,z^{\prime};\omega)$
is the same as that for the TM mode. The $xx$-component of the second term is
cancelled by the elimination of singularity owing to the
$\partial_{z}\partial_{z^{\prime}}$ term.

Finally, we obtain the Green’s function as the sum of Eqs. (7) and (14),
$\bar{G}(\bm{{\rm
k}}_{\parallel},z,z^{\prime};\omega)=\bar{G}^{\rm(TM)}(\bm{{\rm
k}}_{\parallel},z,z^{\prime};\omega)+\bar{G}^{\rm(TE)}(\bm{{\rm
k}}_{\parallel},z,z^{\prime};\omega)$.

###  III.4 Self-consistent treatment

The electromagnetic fields $\tilde{\bm{{\rm H}}}$ and $\tilde{\bm{{\rm E}}}$,
the magnetization $\tilde{\bm{{\rm M}}}$, and the polarization
$\tilde{\bm{{\rm P}}}$ are determined self-consistently based on the
constitutive and Maxwell’s equations. The self-consistent relation is
rewritten in the matrix form as follows: For the total and incident fields,
the coefficients are introduced as

$\displaystyle F_{n}$ $\displaystyle=$ $\displaystyle\int{\rm
d}z\frac{\tilde{\bm{{\rm\mu}}}_{n}^{\dagger}(z)\tilde{\bm{{\rm
h}}}(z)}{E_{n,\bm{{\rm k}}_{\parallel}}-\hbar\omega-i\eta},$ (15)
$\displaystyle f_{n}$ $\displaystyle=$ $\displaystyle\int{\rm
d}z\frac{\tilde{\bm{{\rm\mu}}}_{n}^{\rm t}(z)\tilde{\bm{{\rm
h}}}(z)}{E_{n,\bm{{\rm k}}_{\parallel}}+\hbar\omega+i\eta},$ (16)
$\displaystyle X_{n}$ $\displaystyle=$ $\displaystyle\int{\rm
d}z{\tilde{\bm{{\rm\mu}}}_{n}^{\dagger}(z)\tilde{\bm{{\rm h}}}_{\rm inc}(z)},$
(17) $\displaystyle x_{n}$ $\displaystyle=$ $\displaystyle\int{\rm
d}z{\tilde{\bm{{\rm\mu}}}_{n}^{\rm t}(z)\tilde{\bm{{\rm h}}}_{\rm inc}(z)}.$
(18)

The associated vectors are $\bm{{\rm F}}=(\\{F_{n}\\},\\{f_{n}\\})^{\rm t}$
and $\bm{{\rm X}}=(\\{X_{n}\\},\\{x_{n}\\})^{\rm t}$. The dyadic Green’s
function yields the factors of the matrices:

$\displaystyle A_{n,n^{\prime}}$ $\displaystyle=$
$\displaystyle-\frac{\mu_{0}}{v}\int{\rm d}z\int{\rm
d}z^{\prime}\tilde{\bm{{\rm\mu}}}_{n}^{\dagger}(z)\bar{G}(\bm{{\rm
k}}_{\parallel},z,z^{\prime};\omega)\tilde{\bm{{\rm\mu}}}_{n^{\prime}}(z^{\prime}),$
(19) $\displaystyle a_{n,n^{\prime}}$ $\displaystyle=$
$\displaystyle-\frac{\mu_{0}}{v}\int{\rm d}z\int{\rm
d}z^{\prime}\tilde{\bm{{\rm\mu}}}_{n}^{\rm t}(z)\bar{G}(\bm{{\rm
k}}_{\parallel},z,z^{\prime};\omega)\tilde{\bm{{\rm\mu}}}_{n^{\prime}}^{*}(z^{\prime}),$
(20) $\displaystyle W_{n,n^{\prime}}$ $\displaystyle=$
$\displaystyle-\frac{\mu_{0}}{v}\int{\rm d}z\int{\rm
d}z^{\prime}\tilde{\bm{{\rm\mu}}}_{n}^{\dagger}(z)\bar{G}(\bm{{\rm
k}}_{\parallel},z,z^{\prime};\omega)\tilde{\bm{{\rm\mu}}}_{n^{\prime}}^{*}(z^{\prime}),$
(21) $\displaystyle w_{n,n^{\prime}}$ $\displaystyle=$
$\displaystyle-\frac{\mu_{0}}{v}\int{\rm d}z\int{\rm
d}z^{\prime}\tilde{\bm{{\rm\mu}}}_{n}^{\rm t}(z)\bar{G}(\bm{{\rm
k}}_{\parallel},z,z^{\prime};\omega)\tilde{\bm{{\rm\mu}}}_{n^{\prime}}(z^{\prime}).$
(22)

We find

$\bar{S}\bm{{\rm F}}=\bm{{\rm X}}$ (23)

with

$\bar{S}\equiv\left(\begin{matrix}(\bar{E}-\hbar\omega-i\eta)\bar{1}+\bar{A}&\bar{W},\\\
\bar{w}&(\bar{E}+\hbar\omega+i\eta)\bar{1}+\bar{a}\end{matrix}\right).$ (24)

$\bar{E}$ is a diagonal matrix based on eigenenergies. Notably, the
interaction between magnons is described in the matrix through the Green’s
function. The determinant of matrix $\bar{S}$ yields the dispersion relation
of the MPs.

Even for magnetic materials with magnon excitation, the dielectric constant
contributes to the microwave reflection and refraction. The nonlocal self-
consistent theory proposed herein for the magnons and the radiative field
plays a significant role, although we focus on the linear response of magnons.

## References

  * (1) Y. Kajiwara, K. Harii, S. Takahashi, J. Ohe, K. Uchida, M. Mizuguchi, H. Umezawa, H. Kawai, K. Ando, K. Takanashi, S. Maekawa, and E. Saitoh, Nature 464, 262 (2010).
  * (2) A. V. Chumak, A. A. Serga, and B. Hillebrands, Nature Commun. 5, 4700 (2014).
  * (3) A. V. Chumak, V. I. Vasyuchka, A. A. Serga, and B. Hillebrands, Nature Phys. 11, 453 (2015).
  * (4) K. Oyanagi, S. Takahashi, L. J. Cornelissen, J. Shan, S. Daimon, T. Kikkawa, G. E. W. Bauer, B. J. van Wees, and E. Saitoh, Nature Commun. 10, 4740 (2019).
  * (5) Y. Tabuchi, S. Ishino, A. Noguchi, T. Ishikawa, R. Yamazaki, K. Usami, Y. Nakamura, Science 349, 405 (2015).
  * (6) D. Lachance-Quirion, S. P. Wolski, Y. Tabuchi, S. Kono, K. Usami, Y. Nakamura Science 367, 425 (2019).
  * (7) X. Zhang, C.-L. Zou, N. Zhu, F. Marquardt, L. Jiang, and H. X. Tang, Nature Commun. 6, 8914 (2015).
  * (8) C. Liu, J. Chen, T. Liu, F. Heimbach, H. Yu, Y. Xiao, J. Hu, M. Liu, H. Chang, T. Stueckler, S. Tu, Y. Zhang, Y. Zhang, P. Gao, Z. Liao, D. Yu, K. Xia, N. Lei, W. Zhao, and M. Wu, Nature Commun. 9, 738 (2018).
  * (9) C. Liu, S. Wu, J. Zhang, J. Chen, J. Ding, J. Ma, Y. Zhang, Y. Sun, S. Tu, H. Wang, P. Liu, Ch. Li, Y. Jiang, P. Gao, D. Yu, J. Xiao, R. Duine, M. Wu, C.-W. Nan, J. Zhang, and H. Yu, Nature Nanotech. 14, 691 (2019).
  * (10) Y. Onose, T. Ideue, H. Katsura, Y. Shiomi, N. Nagaosa, and Y. Tokura, Science 329, 297 (2010).
  * (11) R. Matsumoto, R. Shindou, and S. Murakami, Phys. Rev. B 89, 054420 (2014).
  * (12) S. Murakami and A. Okamoto, J. Phys. Soc. Jpn. 86, 011010 (2017).
  * (13) Y. Cao, P. Yan, H. Huebl, S. T. B. Goennenwein, and G. E. W. Bauer, Phys. Rev. B 91, 094423 (2015).
  * (14) Y. Tabuchi, S. Ishino, T. Ishikawa, R. Yamazaki, K. Usami, and Y. Nakamura, Phys. Rev. Lett. 113, 083603 (2014).
  * (15) X. Zhang, C.-L. Zou, L. Jiang, and H. X. Tang, Phys. Rev. Lett. 113, 156401 (2014).
  * (16) Y.-P. Wang, G.-Q. Zhang, D. Zhang, T.-F. Li, C.-M. Hu, and J. Q. You, Phys. Rev. Lett. 120, 057202 (2018).
  * (17) P. Hyde, B. M. Yao, Y. S. Gui, G.-Q. Zhang, J. Q. You, and C.-M. Hu Phys. Rev. B 98, 174423 (2018).
  * (18) D. Zhang, X.-Q. Luo, Y.-P. Wang, T.-F. Li, and J. Q. You, Nature Commun. 8, 1368 (2017).
  * (19) X. Zhang, K. Ding, X. Zhou, J. Xu, and D. Jin, Phys. Rev. Lett. 123, 237202 (2019).
  * (20) M. Harder, Y. Yang, B. M. Yao, C. H. Yu, J. W. Rao, Y. S. Gui, R. L. Stamps, and C.-M. Hu, Phys. Rev. Lett. 121, 137203 (2018).
  * (21) P.-C. Xu, J. W. Rao, Y. S. Gui, X. Jin, and C.-M. Hu, Phys. Rev. B 100, 094415 (2019).
  * (22) W. Yu, J. Wang, H. Y. Yuan, and J. Xiao, Phys. Rev. Lett. 123, 227201 (2019).
  * (23) J. W. Rao, C. H. Yu, Y. T. Zhao, Y. S. Gui, X. L. Fan, D. S. Xue, and C.-M. Hu, New J. Phys. 21, 065001 (2019).
  * (24) J. W. Rao, S. Kaur, B. M. Yao, E. R. J. Edwards, Y. T. Zhao, X. Fan, D. Xue, T. J. Silva, Y. S. Gui, and C.-M. Hu, Nature Commun. 10, 2934 (2019).
  * (25) N. J. Lambert, J. A. Haigh, S. Langenfeld, A. C. Doherty, and A. J. Ferguson, Phys. Rev. A 93, 021803(R) (2016).
  * (26) L. Bai, M. Harder, P. Hyde, Z. Zhang, C.-M. Hu, Y. P. Chen, and J. Q. Xiao, Phys. Rev. Lett. 118, 217201 (2017).
  * (27) B. Z. Rameshti and G. E. W. Bauer, Phys. Rev. B 97, 014419 (2018).
  * (28) Z. Zhang,M. O. Scully, and G. S. Agarwal, Phys. Rev. Research 1, 023021 (2019).
  * (29) J. Li and S.-Y. Zhu, New J. Phys. 21, 085001 (2019).
  * (30) D. D. Daniel and A. Prabhalar, Spin Waves: Theory and Application, Springer (2009).
  * (31) K. Sadhana, R. S. Shinde, and S. R. Murthy, Int. J. Mod. Phys. B 23, 3637 (2009).
  * (32) K. Lal and H. K. Jhans, J. Phys. C: Solid State Phys. 10 1315 (1977).
  * (33) T. Kinoshita, T. Matsuda, T. Takahashi, M. Ichimiya, M. Ashida, Y. Furukawa, M. Nakayama, and H. Ishihara, Phys. Rev. Lett. 122, 157401 (2019).
  * (34) H. Ishihara, Comprehensive Nanoscience and Nanotechnology 2nd Edition Vol. 4, p323-346 “Nanoscale Optical Response” (Academic Press) 2019.
  * (35) B. A. Kalinikos and A. N. Slavin, J. Phys. C: Solid State Physics, 19, 7013 (1986).
  * (36) H. Xia, P. Kabos, R. A. Staudinger, and C. E. Patton, Phys. Rev. B 58, 2708 (1998).
  * (37) H. Yu, R. Huber, T. Schwarze, F. Brandl, T. Rapp, P. Berberich, G. Duerr, and D. Grundler, Appl. Phys. Lett. 100, 262412 (2012).
  * (38) K. Yamanoi, S. Yakata,T. Kimura, and T. Manago, Jpn. J. Appl. Phys. 52, 083001 (2013).
  * (39) T. Holstein and H. Primakoff, Phys. Rev. 58, 1098 (1940).
  * (40) K. Cho, Optical Response of Nanostructures: Microscopic Nonlocal Theory Springer Series in Solid-State Sciences (Springer-Verlag, Tokyo) 2003.
  * (41) R. Kubo, J. Phys. Soc. Jpn., 12, 570 (1957).
  * (42) W. C. Chew, Waves and Fields in Inhomogeneous Media IEEE PRESS Series on Electromagnetic Waves (IEEE Press, New York) 1995.