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1
Losses of Interface Waves in Plasmonic and Gyrotropic Structures
A. Schuchinsky
University of Liverpool, L3 5TQ, Liverpool, UK, a.schuchinsky@liverpool.ac.uk
Abstract – The loss mechanisms of slow interface waves in the layered resonant media are
examined and illustrated by the examples of (i) surface plasmon polaritons in an isotropic plasma
layer, (ii) magnetoplasmons in magnetised plasma and (iii) spin waves in ferrimagnetic layers. It
is shown that losses of all these interface waves grow at the same rate of Im
~ Re
3, where
is the
wavenumber. These abnormal losses are caused by vortices of the power flow of the interface
waves near their resonance cut-off. The basic properties of the slow interface waves discussed in
the paper are inherent to the waves of hyperbolic type in the layered resonant media.
I. INTRODUCTION
Slow electromagnetic waves guided by the layers and interfaces of the resonant plasmonic and ferrimagnetic
structures represent a distinct class of the surface waves. These waves are of hyperbolic type and exist only in the
finite frequency bands, being resonantly absorbed at their upper frequency cut-offs. Their main properties are
discussed in this paper by the examples of interface waves (IWs) such as surface plasmon polaritons [1]-[7],
magnetoplasmons [8]- [13] and spin waves [14]-[17]. The slow bulk waves (BWs) like magnetoplasmons and spin
waves exist when the magnetic bias has components directed along the wave propagation or normal to the guiding
interface. The mechanisms of the BW propagation, dissipation and power flow are somewhat similar to the IWs
and they are not discussed in detail here.
The properties, functionality and applications of the slow IWs and BWs in the resonance media have been
extensively studied in the literature, see e.g., [1]-[17] and references therein. In contrast to the conventional surface
waves guided by dielectric layers, IWs and BWs are slower than the plane waves in the constituent media of the
guiding structure. Therefore, these waves of the hyperbolic type cannot be described by a basic superposition of
plane waves. The waves in the hyperbolic metamaterials have recently attracted increased attention [18]-[26].
Their various applications have been proposed and explored in the literature [19], [20], [23]-[26], including the
promise of enhancing the sub-wavelength resolution [19], [20] and realising “slow light” [27], [28]. However, the
published practical demonstrators exhibited high losses, which were notably higher than in the conventional
devices based on dielectric waveguides and resonators, and optical fibres. Therefore, the detailed analysis of the
loss mechanisms of the slow IWs and BWs in imperfect hyperbolic media is essential for their practical use.
The effects of the medium losses on the properties of IWs and their Poynting vector are examined and quantified
in this work. The properties of the IW propagation and dissipation are elucidated with the examples of the waves
guided by the interfaces of dielectric layers with isotropic and magnetised plasma and ferrimagnetic layers. The
paper scope includes the analysis of
- the basic modes of the slow IWs,
- the dispersion and attenuation characteristics of the IWs, and
- the effect of the power flow vorticity on the IW propagation and resonance losses.
The main properties of the IWs are discussed by the three examples:
(i) Surface plasmon polaritons (SPPs) in isotropic plasma layer,
(ii) Magnetoplasmons (MPs) in tangentially magnetised plasma layer and
(iii) Spin waves (SWs) in tangentially magnetised ferrimagnetic layer.
The results of this work demonstrate that the anomalous losses are inherent to the IWs. In contrast to the
conventional surface waves, the IWs are of the hyperbolic type, and their attenuation constants are proportional to
acube of their propagation constants. This is why the IWs exhibit strong attenuation in the proximities of their
high frequency resonance cut-offs. It is shown that both the slow propagation and the high losses of the IWs are
intrinsically linked to vorticity of the power flow of the IWs in the hyperbolic medium, and the examples of the
three types of IWs illustrate this effect.
The paper is organised as follows. The canonical 3-layer structure, used for the analysis of the SPPs, MPs and
SWs, is described in Section II. SPPs in thin metallic layers are discussed in Section III. MPs in the tangentially
magnetised plasma layer are considered in Section IV, and spin waves in Section V. The main properties of the
IWs, their dissipation and power flow are summarised in Conclusion.
2
II. CANONICAL STRUCTURE
The basic 3-layer planar structure shown in Fig. 1 is used for the
study of IWs bound to the central layer of thickness a0= 2ac. Two
isotropic dielectric layers have relative permittivities
1,2 and
thicknesses a1,2. The whole structure is bounded by the perfect electric
conductor1(PEC) walls located at y=(a1+ac), -(a2+ac). The central
layers are of the following types
(i) an isotropic plasmonic layer with Drude scalar permittivity
p(
),
(ii) a gyrotropic plasma slab, magnetised along the x-axis and
described by Voigt permittivity tensor
m(
),
(iii) a ferrimagnetic layer, magnetised along the x-axis and described
by a scalar permittivity
gand Polder permeability tensor g(
).
These types of the central layers have negative effective permittivity or permeability in the finite frequency
bands limited by the intrinsic resonances of the medium. They support propagation of the IWs of hyperbolic type,
which have the high frequency resonance cut-offs. In the proximities of the cut-off frequencies, losses of the IWs
rapidly grow in the resonant manner, and the attenuation constants of the IWs are proportional to a cube of the
propagation constants. It is shown below that the high losses of the IWs is their inherent property related to vorticity
of their Poynting vector. The main mechanisms of losses and power flow of the IWs are elucidated by the examples
of SPPs, MPs and SWs in isotropic and gyrotropic (magnetised plasma and ferrimagnetic) layers.
III. SURFACE PLASMON POLARITONS IN PLASMONIC LAYER
Let us consider the 3-layer structure shown in Fig. 1 where the central layer is the isotropic plasma with Drude
permittivity
p(
) defined as
2
1p
p L
(1)
where =
-j
,
is angular frequency,
pand are the plasma and collision frequencies, respectively, and
L
is the background permittivity.
Eigenwaves in a planar structure shown in Fig. 1 include TE and TM waves [29]. TE waves with the field
components Ex,Hyand Hzare the ordinary surface waves. They are the guided modes of the plasmonic layer only
at Re
p(
) > max(
1,
2) and layer thickness a0about a half wavelength. For thin layers, these conditions are
fulfilled only at very high frequencies
>>
p. Therefore, TE waves are not considered here.
TM waves with the field components Hx,Eyand Ezare the extraordinary modes of the plasmonic layer. They
have been extensively studied in the literature, see, e.g., [2]-[7], [10] and references therein. The dispersion
equation (DE) of TM waves with the wave propagator exp{j(
t-
z} is readily obtained by enforcing the boundary
conditions of the tangential field continuity at the layer interfaces and at PEC enclosure. The DE can be presented
as follows
2
1 2
0
0
sinh
p
p
K K a
(2)
where
2 2 2 2
0 0 0
coth , tanh , , 1, 2; ;
m
m p m p p m m m m m p p
m
K V a V a k m k
and k0are longitudinal and free space wavenumbers, respectively. The main features of the TM wave dispersion
and attenuation are illustrated in Fig. 2, obtained by numerical solution of (2).
Spectrum of the fundamental TM modes in the plasmonic layer includes the conventional surface waves and
SPPs. The surface waves are the bound modes only at
>
p, see Fig. 2, when the plasmonic layer acts as a
dispersive dielectric waveguide with Re
p(
) > max(
1,
2). The propagation constants of these surface waves vary
in the range
1 2 0
max , Re Re p L
k
. At 0 < Re
p(
) < max(
1,
2), the eigenwaves are not
bound to the plasmonic layer and leak into a dielectric layer with a higher permittivity.
At frequencies
<
p, Re
p(
) < 0 and only SPPs with wavenumbers 0,Re
m m
k
m= 1, 2 are guided by
the plasmonic layer. SPPs are the IWs and their fields decay exponentially from the layer surfaces. In the case of
1The PEC enclosure is used here for a sole purpose of making the eigenwave spectrum discreet. This facilitates a rigorous
analysis of the complete spectrum including complex and leaky waves unbounded to the centre layer.
a0
a1
a
2
p(
)/m(
)/{
g,g(
)}
1
2
z
y
x
Fig. 1. Canonical 3-layer structure in PEC
enclosure. Magnetic bias H0is applied only to
plasma and ferrimagnetic centre layers.
H0
3
dielectric layers with unequal permittivities,
1
2, the dispersion characteristics of SPPs and cut-off frequencies
r1and
r2differ. From the definition of
pand
min (2), it is evident that magnitude of the SPP fields decay much
faster in the plasmonic layer than in the surrounding it dielectric layers. Therefore, the effect of the non-guiding
interface is exponentially small and the last term in (2) can be neglected. Then the DEs of the two SPPs at the
opposite interfaces are separated
0
tanh coth 0, 1, 2.
p
mm
m m p
p m p
Ka a m
(3)
where
mand
pare defined in (2). The dispersion characteristics of SPPs in Fig. 2 show that away from the SPP
resonances at
r1and
r2, the dispersion curves of SPP1and SPP2are fairly well correlated with those of the SPP'1
and SPP'2in the lossless plasmonic layer. However, this is not the case at
r2<
<
p, as the backward SPP'3of
the lossless plasmonic layer, becomes a strongly attenuated complex wave SPP3in the lossy plasmonic layer. Thus,
losses qualitatively alter the properties of SPPs in the proximities of the plasmonic resonances
r1,2.
To quantify the effect of losses, let us examine the asymptotic solutions of (3) at 0
Re , 1, 2
m m
k m
. The
closed form approximations of the SPP propagation, Re
m, and attenuation, Im
m, constants at each interface are
2
0
02
2
30
2 2
2
0
1
Re 2
2 Im
Im Re
1
pm
m m
p m m
p
m m
m
p m
k
k O
k
O
k
m= 1,2 (4)
where
2Re , 1, 2
p m p
m
p p m
m
. In the proximity of plasmonic resonances,
2
2
1 1
L
m
p m p
O
,
i.e.,
mis small at
(1+
L/
m) <
p. Equations (4) reveal the fundamental property of SPPs that their attenuation
has the anomalous growth rate of Im
m~ (Re
m)3near the resonance cut-off, and this clearly seen in Fig. 2. This
is why the SPPs decay much faster than the conventional surface waves with the attenuation constants ~Re
m.
Fig. 2. Dispersion (solid lines) and attenuation (dashed lines) of SPPs guided by isotropic metallic film in a planar
waveguide of Fig. 1. The structure parameters: a1=a2= 40 m, a0= 20 m,
L= 13.1,
1= 18.0,
2= 7,
p= 135 GHz
,
= 0.05
p. Dotted and dash-dotted lines show the dispersion characteristics of SPP'1and SPP'3in the lossless case
(
= 0). Side panel shows the magnified characteristics of the SPPs near the cut-off frequency
r2. The dispersion
curves in the grey-shaded area at 0 2
Re k
are of the leaky waves, which are not bound to the plasmonic layer.
/
p
SPP2
(SPP'2)
Re
/k0, -Im
/k0
Re(
/k0)
Im(
/k0)
0
0.5
1.0
1.5
10 20 30 40 50 60
r1/
p
r2/
p
SPP'1
SPP
1
SPP
3
SPP
'
3
Leaky
wave
SPP2
SPP'3
p
0.8
0.7
0.9
1.0
0510
SPP3
Re
/k0, -Im
/k0
4
The abnormal rate of the SPP dissipation is directly related to vorticity of the power flow. Fig. 3 shows cross-
sectional distributions of the fields and Poynting vector in the asymmetric structure containing a plasmonic layer
(gold film with Re
p(
) < 0) between lossless dielectric layers with permittivities
1=1.5 and
2=1. The normalised
longitudinal component of Poynting vector, Pz, in dielectric and plasmonic layers has the following form
2
1
2
0
2
0
cosh 1
1 1 , 0 , 1 , 1,2
cosh
Re
cosh sinh
,0 ,
cosh sinh
m
m m c
m m
e p c
m m m
z
p p
e p c
p p c p c
a a y
W y a m
a
P y Q
ky y
W y a
a a
(5)
where
0and k0are the free space impedance and wavenumber, respectively, Qis the normalised magnitude
of the magnetic field, and
2 1
2 1
2
,2 coth
e
p p c
V V
WV V a
(6)
The distinctive feature of the Poynting vector distribution is that at Re
p(
) <0, its longitudinal component Pz(y)
inside and outside the plasmonic layer has opposite signs, see (5). This is particularly evident in the case of V1(
) =
V2(
) when W(
p, 0) = 0. When permittivities of the dielectric layers are unequal,
1
2, the field and power flow
distributions of the SPPs guided by the opposite interfaces of the plasmonic layer are asymmetric because
V1(
)V2(
) in (6). Plots of RePzand RePyin Fig. 3 show that the counter-directed Poynting vectors in the guiding
plasmonic layer and surrounding it dielectric layers form vortices of power flow of SPPs. Vorticity of the SPP
power flow at an air interface of lossless plasma half-space had been mentioned first in [30] but its effect on the
abnormally high losses of SPPs was not recognised then.
The properties of SPPs and their Poynting vector distributions are strongly influenced by thicknesses of dielectric
layers, a1and a2. At k0a1,2 >> 1, SPPs are forward waves, and their power flows in the dielectric layers is greater
than in the plasmonic layer, as evident in Fig. 3. When the dielectric layer at the guiding interface becomes thinner,
SPP in the lossless plasmonic layer turns into a backward wave SPP'3shown in Fig. 2. However, losses
qualitatively alter the eigenwave properties in the proximity of SPP resonance. At the result, the backward wave
SPP'3turns into a complex wave SPP3, which is very strongly attenuated at
>
r2due to its high losses in the
plasmonic layer. This effect is illustrated by Fig. 2, where the attenuation constant of backward SPP3exceeds the
propagation constant at
>
r2.
As frequency
approaches the SPP resonances at
r1and
r2, the SPPs slow down. Their Re
and Im
grow
and vorticity of the power flow increases. At the cut-off frequencies, the oppositely directed power flows become
equal in the plasmonic and dielectric layers, and vortices of Poynting vectors are trapped. Thus, vorticity the
Poynting vector at the layer interfaces causes the anomalous losses of SPPs at their resonance cut-off.
Fig. 3. Normalized cross-sectional distributions of the SPP fields Hx,Ey,Ez, and Poynting vector Pz,Py
components in gold film located at |y|<a0/2 (shaded area) at frequency f= 591 THz;
=k0(1.87-j0.38).
Layer thicknesses: a0=a1= a2= 50 nm and permittivities:
1= 1.5 and
L=
2= 1.
y/a0
|Hx| |Ey| |Ez| RePzRePy
1.5 0.5 0.5 1.5
0
0.5
1
1.5 0. 5 0.5 1.5
0
0.5
1
1.5 0.5 0.5 1. 5
0
0.5
1
1.5 0. 5 0.5 1.5
1
0
1
1.5 0.5 0.5 1.5
1
0
1
a1
1
a2
2
a0
p
5
IV. MAGNETOPLASMONS IN MAGNETISED PLASMA LAYER
When a biasing dc magnetic field H0is applied along the x-axis in the planar structure shown in Fig. 1, relative
permittivity of a plasma layer is described by the tensor
g p t a
j
ε x x I x x x I (7)
where
2 2 2
2 2 2 2
1 , 1 , ,
p p p c
p L t L a L
c c
(8)
Lis the background permittivity
is angular frequency, =
-j
,
and
pare the collision and plasma
frequencies,
c=
H0is cyclotron frequency and
≈ 17.588 MHz/Oe.
The lowest eigen modes in a magnetised plasma layer are TE and TM waves. TE waves are the ordinary waves,
which are not affected by gyrotropy of the plasma layer. Therefore, only TM waves, which exhibit the resonance
and nonreciprocal behaviour, are considered here. The DE of the TM waves with the propagator exp{j(
t-
z)} can
be represented in the form similar to that for SPPs
2
1 2
0
0
sinh
e
e
L L a
(9)
where
0
coth 1 m
a
m e m e e
t
L V a
,m=1,2; Vmare defined in (2);
2 2
0e e
k
,
2 2
2
2 2
1p p
a
e t L
tp c
is the effective permittivity, k0and
are free space and
longitudinal wavenumbers. It is noteworthy that in the absence of magnetic bias
c= 0 and
a= 0. This results in
Lm(
) being reduced to
m(
) and DE (9) becoming (2) for SPPs in isotropic plasmonic layer.
Spectrum of TM waves in a magnetised plasma layer includes the dynamic waves and MPs [9]-[12]. The
dynamic waves are the conventional surface waves guided by the central layer at Re
e(
) > max(
1,
2) only. This
condition is satisfied at frequencies
1u<
<
qu and
>
2u>
qu, where
2
2
12 2
u p c c p
,
2 2
qu p c
and
2
2
2
2 2
u p c c
. At
<
1uand
qu <
<
2u, Re
e(
) is negative, and only
MPs are guided by the magnetised plasma layer.
Nonreciprocity is the distinctive feature of MPs. In DE (9) it is described by the last term of Lm(
) dependent on
a. In the case of identical dielectric layers, V1=V2, wavenumbers of the oppositely directed waves are the same.
But the field distributions differ due to the nonreciprocal field displacement dependent on sign
. MPs exist in the
two frequency bands and have the resonance cut-offs at the upper bound of each band. In these two bands, MPs
travelling in the same direction are guided by the opposite interfaces of the magnetised plasma layer due to the
opposite signs of Re{
a
t
[12]. For example, if a MP is attached to the upper interface of the plasma layer
shown in Fig. 1 at
<
1u, MP of the same direction is displaced to the lower interface in the frequency band
qu <
<
2u.
As frequency
approaches the MP resonances,
1uor
2u, the MP wavenumbers grow similar to those of SPPs.
At
1,20
| | ma
x ,
L
k
, the asymptotic solution of DE (9) has the same form as (4), where
pmust be replaced
by
1m
c
m t a
. Then the propagation and attenuation constants of MPs are approximated as follows
2
0
02
2
30
2 2
0
Re 1
2
2 Im
Im Re
1
c
m m
m m
c
m m m
c
m
m m cm
m m
k
k O
k
O
k
(10)
6
where c
m
depends on frequency
and
2Re , 1, 2
c c
m m m
mc c
m m m
m
. It is important to note that in the proximity
of the MP resonances
2
2
1 1
1
L
mm
m
c
O
and
mare small at
<<
±
c, similar to SPPs.
The dispersion characteristics of MPs in the proximities of their resonances are shown in Fig. 4. They illustrate
two cases permittivities of dielectric layers are (a) the same,
1=
2= 4.7, and (b) different,
1= 4.7,
2= 2.4. The
propagation and attenuation of MPs in Fig. 4 demonstrate that in the proximities of the resonance cut-offs,
attenuation constants Im
mgrow much faster than the propagation constants Re
mas predicted by (10). It is
noteworthy that permittivity of a dielectric layer at the guiding interface influences the cut-off frequencies of MPs,
and they slightly increase at lower
1,2. At frequencies above the MP resonances, the dispersion curves of TM
waves are in the shaded areas at the left from the dotted lines where only the fast waveguide modes of the dielectric
layers exist. These waves are guided by the dielectric layers and are not the bound to the plasma layer.
The fact that MP losses in (10) grow at the same rate as SPPs in (4), i.e., Im
m~ (Re
m)3, suggests that dissipation
of MPs is also the result of their power flow vorticity. The Poynting vector distribution of MPs in (11) shows that
it differs from that for SPPs by the dependence on
a/
t
Fig. 4. Asymptotic dispersion characteristics of magnetoplasmons in the three-layer structure
of Fig. 1. Propagation constants (solid lines) and attenuation constants (dashed lines) in the
cases of dielectric layers with the same permittivities,
1=
2= 4.7 (red and blue lines) and
different permittivities with
1= 4.7,
2= 2.4 (red and green lines). In the shaded areas to the
left from dotted lines, where no interface waves are guided by the
plasmonic layer. Parameters of the plasmonic layer:
L= 13.1,
p= 2.17
c,
= 0.02
c.
Re
/k0, Im
/k0
/
c
0 2 4 6
3
2
1
1u/
c
2u/
c
qu/
c
1= 4.7
2= 4.7
*2= 2.4
7
2
1
2
0
2
0
cosh 1
1 1 , , 1 , 1,2
cosh
Re
cosh sinh
1 , , , ,
cosh sinh
m
m m c
m m
a
e e c
m t m m
z
p p
a a a
e e e e c c
e t t p c t p c
a a y
W y a m
a
P y Q
ky y
S y W W y a
a a
(11)
where
0,k0,Q,
e(
) are defined in (5), (9),
tanh tanh
,tanh tanh
p c p
p
p c p
W a y
S y W
a W y
, and nonreciprocity of
the MP power flow is described by the terms dependent on We(
e,
a/
t) defined in (6).
Poynting vector distribution Pz(y) in (11) shows that at Re
e(
) < 0, power flows of MPs inside and outside
plasma layer are counter-directional, similar to SPPs in the isotropic plasma layer. Therefore, vorticity of the power
flow is the main propagation mechanisms of the MPs responsible for their anomalous losses. It is necessary to note
that despite the similarities in the power flows of SPPs and MPs, Poynting vector distribution is more intricate due
to gyrotropy of the magnetised plasma layer [31], [32].
Nonreciprocity is the distinctive property of MPs. It manifests itself not only in asymmetry of the field and
power flow distributions but also in the nonreciprocity of the cut-off frequencies. Therefore, the propagation and
attenuation constants of MPs, displaced to the opposite interfaces of the magnetised plasma layer, differ when the
dielectric layers are not the same. It is necessary to note that vorticity of the power flow is stronger in MPs than in
SPPs due to the effect of the nonreciprocal field displacement. As the result, losses of MPs are higher than SPPs,
despite the cubic relation between their attenuation and propagation constants remains the same.
V. SPIN WAVES IN FERRIMAGNETIC LAYERS
Let us consider a ferrimagnetic layer located in the middle of a planar structure shown in Fig. 1. It is magnetised
to saturation along the x-axis (Voigt configuration) and characterised by a scalar relative permittivity
gand Polder
tensor of relative permeability g[33]
g t a
j
μ x x I x x x I
(12)
where
2 2
2 2 2 2
,M
t a
H H
,
0, 4 , ,
H M s B H M H B
H j H M
,H0
is internal DC magnetic bias, His the ferrimagnetic resonance linewidth, 4
Msis the saturation magnetization,
and the gyromagnetic ratio
= 2.8 MHz/Oe.
Spectrum of eigenwaves in the ferrimagnetic layer with the tensor permeability gincludes TE and TM waves.
TM waves with the field components Hx,Ey,Ezare the ordinary waves, which are not affected by the layer
gyrotropy. Therefore, they are not considered here. TE waves with the field components Ex,Hy,Hzare the
extraordinary waves, which strongly interact with the ferrimagnetic medium described by the tensor permeability
g. The spectrum of TE waves includes the dynamic waves and surface spin waves (SSWs). Properties of the TE
waves have been extensively explored in the literature and are used in the nonreciprocal devices [14]-[17], [34],
[35]. This is particularly concerned of the dynamic waves, which are the workhorse of the passive ferrite devices
such as circulators, isolators and phase shifters, etc., see e.g., [36]-[40]. Applications of spin waves has been limited
by their high losses. Despite significant efforts in mitigating SW losses, they remain high. It is shown below that
the attenuation constants of SSWs are proportional to a cube of the propagation constant due to vorticity of the
power flow, similar to SPPs and MPs.
Let us consider the TE waves with the propagator exp{j(
t-
z)}. Their DE can be presented in the form similar
to the DE for SPPs and MPs
2
1 2
0
0
sinh
g
g
M M a
(13)
where
2 2
0 0
coth 1 , coth , , 1,2;
ma
m e m g g m m m m m m
t
M U a U a k m
2 2 2
0
, ;
g g e e t a
t
k
k0and
are the free space and the longitudinal wavenumbers.
The solutions of DE (13) include the dynamic waves and SSWs. The dynamic waves are the conventional
surface waves, which exist only at
t·
e> 0, i.e., at frequencies
<
Hand
>
B, where
H= ReHand
8
B= ReBare the ferrimagnetic resonance and plasma frequencies [33]. In contrast to the dynamic waves, SSWs
are the IWs. They exist in the finite frequency band
H<
<
Bonly, and at a1,2 > 0 experience the high frequency
cut-off at the SSW resonance
sw = (
H+
B)/2. Similar to MPs, SSWs are the nonreciprocal waves. Their
nonreciprocity is described by the last term in Mm(
) proportional to
a/
t. In the case of identical dielectric
layers, U1=U2and wavenumbers of the oppositely directed SSWs are the same. But their field distributions differ
due to nonreciprocity of their field displacement, which depends on the propagation direction, i.e., the sign
.
Examples of the dispersion characteristics and field patterns of SSWs in the planar structure of Fig. 1 are shown
in Fig. 5 at several thicknesses of the dielectric layers.
The properties of SSWs in the presence of magnetic losses were studied in detail in [15]-[17], and it was found
that at frequencies
close to
sw, the wavenumbers of the SSWs grow similar to those of SPPs and MPs. The
asymptotic solutions of DE (13) at 0
g
k
can be also represented in the form of (10) where c
m
and
mare
replaced by cr t a
and
2Re
1
cr cr
cr cr
, respectively. Then the asymptotic expansions of the SSW
propagation and attenuation constants are approximated as follows
2
0
02
2
30
2 2
0
Re 1
1
Im
Im Re
2 1
g
cr
cr
g
cr
cr
k
k O
k
O
k
m= 1, 2 (14)
where
cr and
are frequency dependent. Then at the SSW resonance frequency,
2
2
41
H H
MM
O
and
<< 1 at
H<<
M. The asymptotic expansions of
in (14) shows that Re
and Im
do not depend on sign(Re
)
in the proximity of the SSW resonance. This means that SSW nonreciprocity is small at 0
Re g
k
because
the frequencies of SW resonances are weakly affected the permittivities of dielectric layers,
1,2. This is illustrated
by the dispersion curves of SSWs in Fig. 5 at a1,2 2a0. Therefore, despite the nonreciprocity, the propagation and
attenuation constants of SSWs remain practically the same for both propagation directions at 0
Re g
k
.
50-25-0 25 50
Re
/k
0
a2= 0
a2= 0.1a0
a2= 2a0
a2= 0.5a0
a2=a0
a1= 10a0
a1= 2a0
a1=a0
a1= 0.1a0
/
B
Re
/
B
1
a2= 10a0
1
-2.5 0 2.5
-
1
0
1
1
2
g
g
2
y/a0
2
y/a0
-
2.5
0
0.5
0
1
-1
1
1
2
g
g
Fig. 5
. Dispersion characteristics of spin waves (SWs) at different thicknesses of dielectric layers with the
permittivities
1=
2= 1. Cross-
sectional distributions of Poynting vector of SWs are shown in the side panels
at frequency
2for forward waves (a1=a2= 10a0) and at frequency
1for
a combination of forward and
backward waves (a1= 10a0,a2= 0.5a0). Red dots on the dispersion diagram and red lines in the field patterns
are for the waves travelling in the positive direction of the z-axis, and the
blue dots and lines are for the waves
travelling in the negative direction of the z-axis. Arrows show the directions of the phase velocities.
9
An important feature of the asymptotic dispersion relations in (14) is that the SSW attenuation constants grow
at the same rate, Im
~ (Re
)3, as that for SPPs and MPs, see (4) and (10). This means that the mechanisms of the
SSW propagation and losses should be similar to the other types of the IWs and be related to the vortices of power
flow at the layer interfaces. Indeed, distribution of the longitudinal component of the normalised Poynting vector
in dielectric and ferrimagnetic layers shows that it has the same form as that for the magnetised plasma layer
considered earlier, viz.
2
1
20
0
sinh 1
Re 1 1 , , 1 , 1, 2
sinh
cosh sinh
Re 1 , , ,
cosh sinh
m
m m c
m m
a
m e c
t m m
z
p p
a a a
m e m e
e t t p c t p c
a a y
W y a m
a
k
P y Q
y y
S y W W
a a
2
,c
y a
(15)
where
0,k0,
e,
t,
aand S(y, W) are defined in (5), (12) and (11), and nonreciprocity of the SW power flow is
determined by Wm(
e,
a/
t) which depends on
2 1
2 1
2
,2 coth
m
p p c
U U
W
U U a
(16)
Comparison of (15) and (16) with (11) shows that similar to MPs, power flows of SSWs in the ferrimagnetic
layer and surrounding it dielectric layers are directed oppositely. This is illustrated by power flow distributions in
the side panels in Fig. 5. Indeed, they show that vorticity of the power flow at the guiding interfaces is responsible
for the high losses of SSWs [17]. Similar to MPs, SSW power flows are asymmetrically shifted to the guiding
interfaces due to the nonreciprocal field displacement in the ferrimagnetic layer even when the surrounding
dielectric layers are identical. Right panel in Fig. 5 illustrates the power flow distribution at frequency
2in the
symmetric structure with a1=a2= 10a0. In this case, SSWs are attached to the guiding interfaces, and power flows
in the dielectric layers are marginally larger than in the ferrimagnetic layer. This results in the normal dispersion
of SSWs. When thicknesses of dielectric are layers reduced, e.g., a1= 10a0and a2= 0.5a0in the left panel of Fig.
5, power flows of the waves travelling in opposite directions of the z-axis are still displaced to the opposite
interfaces at frequency
1. However, at the right interface, power flow of the SSW inside the ferrimagnetic layer
becomes larger than in the adjacent dielectric layer. This results in the anomalous dispersion (blue dot on the
dispersion diagram) and the higher losses of the backward SW.
At other directions of the magnetic bias H0, the field distributions of the SWs change. Namely, the SW travelling
along the direction of the in-plane magnetisation (H0||0z) are volume backward waves, and at the magnetic bias
normal to the layer surface (H0||0y), the SWs are forward volume waves. But their main features, cubic relation
between their attenuation and propagation constants and vortices of power flow remain unchanged. These common
properties of the SWs are the result of their resonance cut-off at the intrinsic ferrimagnetic resonances of the central
layer. Despite the differences in the types of SW dispersion and field distributions (volume or interface) inside the
ferrimagnetic layer, the mechanisms of their cut-off remain similar for both SSWs and volume SWs as the SWs
of any type experience the resonance absorption when vortices of their power flow are trapped.
VI. CONCLUSION
The dispersion and attenuation properties of IWs and their relations to the power flow distributions have been
discussed with the examples of SPPs, MPs and SSWs. It is shown that these IWs are the waves of hyperbolic type,
and the fundamental mechanisms of their propagation and dissipation are similar. This is evidenced by that
similarity of the dispersion equations (2), (9), (13) for all three types of the IWs. The main results of this work are
- the attenuation of the IWs grows at the rate of [Re
(
)]3in the proximity of the resonance cut-offs, and
- the high losses of the IWs are caused by the vorticity of Poynting vector inherent to hyperbolic media.
The mechanisms of losses, discussed in the paper by the examples of SPPs, MPs and SSWs, are common for any
slow waves in the hyperbolic media with intrinsic resonances at finite frequencies. It is necessary to stress that the
nonreciprocity of MPs and SWs is asymptotically small in the proximity of their resonance cut-offs and does not
affect the cubic relation between the propagation and attenuation constants in eqns. (10) and (14).
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