Hidden Symmetry Protection and Topology in Surface Maxwell Waves Yosuke Nakata Graduate School of Engineering Science Osaka University Osaka 560-8531 Japan and

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Hidden Symmetry Protection and Topology in Surface Maxwell Waves

Yosuke Nakata∗

Graduate School of Engineering Science, Osaka University, Osaka 560-8531, Japan and

Center for Quantum Information and Quantum Biology, Osaka University, Osaka, 560-8531, Japan

Toshihiro Nakanishi

Department of Electronic Science and Engineering, Kyoto University, Kyoto 615-8510, Japan

Ryo Takahashi

Advanced Institute for Materials Research (AIMR), Tohoku University, Miyagi 980-8577, Japan

Fumiaki Miyamaru

Department of Physics, Faculty of Science, Shinshu University, Nagano 390-8621, Japan

Shuichi Murakami

Department of Physics, Tokyo Institute of Technology, Tokyo 152-8551, Japan

(Dated: December 12, 2022)

Since the latter half of the 20th century, the use of metal in optics has become a promising

plasmonics ﬁeld for controlling light at a deep subwavelength scale. Surface plasmon polaritons

localized on metal surfaces are crucial in plasmonics. However, despite the long history of plasmonics,

the underlying mechanism producing the surface waves is not fully understood. This study unveils

the hidden symmetry protection that ensures the existence of degenerated electric zero modes.

These zero modes are identiﬁed as physical origins of surface plasmon polaritons, and similar zero

modes can be directly excited at a temporal boundary. In real space, the zero modes possess

vector-ﬁeld rotation related to surface impedance. Focusing on the surface impedance, we prove the

bulk–edge correspondence, which guarantees the existence of surface plasmon polaritons even with

nonuniformity. Lastly, we extract the underlying physics in the topological transition between metal

and dielectric material using a minimal circuit model with duality. The transition is considered the

crossover between electric and magnetic zero modes.

I. INTRODUCTION

Metal has been one of the fundamental materials for

producing optical elements, such as mirrors, for over 5000

years [1]. However, despite its extensive history, wave

propagation inside metals has received little attention,

because electromagnetic waves are attenuated in metals

owing to their negative response. Because free electrons

in a metal are sensitive to oscillating electric ﬁelds, the

electric ﬁeld and induced current have opposite phases.

Since the 20th century, researchers have investigated ex-

traordinary light propagation enabled by a negative re-

sponse. One prominent example is the discovery of a

negative refractive index, which can be realized in a

medium with simultaneous negative responses for electric

and magnetic ﬁelds [2]. Remarkably, a negative refractive

index can be applied to realize a ﬂat lens, which can help

overcome the diﬀraction limit [3]. Because there is no

natural material with a negative refractive index, the dis-

covery stimulated the development of artiﬁcial materials

called metamaterials [4, 5], and negative refraction was

eventually demonstrated in a metamaterial [6]. These

ﬁndings demonstrate the potential capability of the neg-

ative response in optics.

∗y.nakata.es@osaka-u.ac.jp

The negative response impacts not only spatial wave

propagation but also surface-wave formation. In fact, a

metallic surface supports surface plasmon polaritons, i.e.,

hybridized waves comprising a plasmonic electron oscil-

lation and an electromagnetic wave [7, 8]. As surface

plasmon polaritons can be squeezed into a deep subwave-

length volume, they play essential roles in nanophotonics

toward the miniaturization of optics, and the research

ﬁeld involving surface plasmon polaritons is called plas-

monics [9]. Although surface plasmon polaritons have

been studied for over half a century, the investigation

of their origin only began recently. In the paradigm of

topological physics, integers are used to characterize bulk

materials, where the bulk–edge correspondence predicts

the existence of a boundary mode between two materials

with diﬀerent topological numbers [10, 11]. The bulk–

edge correspondence provides a powerful guiding princi-

ple; however, it is often empirical and requires exact proof

for each case. For plasmonic systems, diﬀerent integers

are used to distinguish between a metal and dielectric

material, such as the Z4number describing the winding

of the complex helicity spectrum [12] and the Zak phase

[13]. The existence of surface plasmon polaritons is indi-

cated by the bulk–edge correspondence. However, these

seminal works are limited to simply observing surface-

wave formations and lack rigorous proof of the bulk–edge

correspondence.

arXiv:2210.13637v3 [physics.optics] 9 Dec 2022

In this study, special zero modes were identiﬁed as the

origin of surface Maxwell waves, and a general bulk–edge

correspondence is rigorously established that explained

the existence of surface plasmon polaritons on metals.

The ﬁrst approach is based on symmetry protection. It

guarantees the existence of localized states under certain

symmetry. In Sec. II, we formulate symmetry-protected

electric zero modes in electrostatics, which are the origin

of surface plasmon polaritons. The robustness of symme-

try protection in some nonuniform systems is conﬁrmed.

Additionally, we demonstrate that analogous zero modes

can be experimentally excited at a temporal boundary.

In Sec. III, we investigate real-space topological polariza-

tion rotation in the electric zero modes. Keller–Dykhne

self-duality is identiﬁed as the physical origin of polar-

ization rotation. Furthermore, we reveal the relationship

between the polarization rotation and surface impedance.

In Sec. IV, we deﬁne topological integers based on the

surface impedance and establish the bulk–edge corre-

spondence, which provides another way to understand

surface plasmon polaritons even with nonuniformity. In

Sec. V, we propose a minimal circuit model to explain

the underlying physics of the continuous transition be-

tween metal and dielectric. From a physical standpoint,

the transition is due to the crossover between electric and

magnetic zero modes.

II. SYMMETRY PROTECTION

Certain symmetry often guarantees the existence of

topological end or boundary states. For instance, in the

Su–Schrieﬀer–Heeger (SSH) model [14, 15], the sublat-

tice symmetry works as a chiral symmetry and protects

end states at zero energy [10]. Nevertheless, the general-

ization of the SSH model in a continuous system simply

leads to the Dirac electron [16]. The topological bound-

ary state of the Dirac electron appears at the boundary

between the positive and negative mass regions in the

Jackiw–Rebbi model, and its energy is maintained to be

zero [17].

Herein, we construct symmetry-protected electric zero

modes in electrostatics and identify them as the robust

origin of surface plasmon polaritons. Additionally, we

show that the analogous zero modes can be excited at a

temporal boundary.

A. Mechanism

We construct symmetry-protected electric zero modes

in electrostatics and identify their boundary degree of

freedom. Finally, we discuss the origin of the singular

charge response near a surface with symmetry.

a. Setup. Consider an electrostatic problem de-

scribed by a scalar permittivity ε(x, y, z). Without free

charge, the fundamental equations are given as follows:

∇ · D= 0,∇ × E= 0,(1)

where Dand Erepresent the electric displacement and

electric ﬁeld, respectively. The constitutive relation is

expressed as follows:

D(x, y, z) = ε(x, y, z)E(x, y, z).(2)

We mainly focus on a particular distribution of ε(x, y, z)

that satisﬁes the following equation:

ε(−x, y, z) = −ε(x, y, z).(3)

For simplicity, we assume that ε(x, y, z)>0 in x≥

0+and that there is no free charge unless noted oth-

erwise. To handle discontinuous functions, such as

ε, we sometimes distinguish the positive side of zero

(0+= limx→0, x>0x) from the negative side (0−=

limx→0, x<0x).

b. Symmetry Operations. We introduce two symme-

try operations to characterize Eq. (3). First, we consider

the mirror reﬂection Mxwith respect to the plane x= 0.

Under the Mxoperation, a polar-vector ﬁeld F(x, y, z) =

[Fx(x, y, z)Fy(x, y, z)Fz(x, y, z)]Ttransforms into

F0(x, y, z) = [−Fx(−x, y, z)Fy(−x, y, z)Fz(−x, y, z)]T.

This transformation can also be expressed as F0=MxF.

To preserve Eqs. (1) and (2) under Mx, the permittivity

should be expressed as follows when considering trans-

formed ﬁelds E0=MxEand D0=MxD:

ε0(x, y, z) = (Mxε)(x, y, z) = ε(−x, y, z) (4)

The second operation is on the internal degree of free-

dom between Eand D. Consider the following conjuga-

tion operation Cfor (E,D):

C(E,D) = (E,−D).(5)

The transformed (E0,D0) = C(E,D) satisﬁes Eq. (1). To

preserve Eq. (2) under the Coperation, the permittivity

changes as follows:

ε0(x, y, z) = (Cε)(x, y, z) = −ε(x, y, z).(6)

The combined operation CMxinduces permittivity

transformation (CMx)ε(x, y, z) = −ε(−x, y, z). Thus,

Eq. (3) represents the CMxsymmetry. Evidently,

(CMx)2is identical to the operation Id. Therefore, the

solutions of a system with CMxsymmetry are classiﬁed

as follows: symmetric (S) and antisymmetric (A) ﬁelds:

CMx(ES,DS) = (ES,DS),(7)

CMx(EA,DA) = −(EA,DA).(8)

Here, we regard −(E,D) := (−E,−D).

c. Antisymmetric Solution. Furthermore, we show

that there is no CMx-antisymmetric ﬁeld. Owing to

the antisymmetry and tangential continuity condition,

Ey=Ez= 0 on x= 0 must follow. By con-

trast, antisymmetry yields Ex(0−, y, z) = Ex(0+, y, z)

and Dx(0−, y, z) = −Dx(0+, y, z). Owing to the assump-

tion of no free charge, Dx(0−, y, z) = Dx(0+, y, z) = 0

holds true. Therefore, Eand Don x= 0 must be zero.

Additionally, all ﬁelds must vanish. We can physically

justify this statement as follows: The solution in x≥0+

can be safely connected to a vacuum in x≤0−, which

has E= 0 and D= 0. Because we have assumed that

there is no source in x≥0+with ε > 0, we can conclude

that all ﬁelds in the entire space vanish.

d. Symmetric Solution. ACMx-symmetric ﬁeld

(E,D) has a unique feature that always satisﬁes the

boundary condition on x= 0. This is the most funda-

mental characteristic of a CMx-symmetric system. Note

that (E,D) can be any continuous ﬁeld and does not need

to satisfy Maxwell’s equations. Let us check this special

property. Because Eis symmetric under Mx,Eyand

Ezare continuous on x= 0. Conversely, the electric dis-

placement is antisymmetric under Mx. Therefore, Dx

is continuous on x= 0. Thus, both of the boundary

conditions are automatically satisﬁed.

e. CMxSymmetrization. The above remarkable

continuity of a CMx-symmetric ﬁeld can be used to ob-

tain a whole-space solution from a half-space solution.

If we have a solution (E,D) satisfying Eqs. (1) and (2)

of an electrostatic problem only in x≥0+, the ﬁeld in

x≤0−is constructed via CMxsymmetrization:

(E,D)(x, y, z) = CMx(E,D)(−x, y, z) (x≤0−).

(9)

Here, we abbreviate (E,D)(x, y, z) =

(E(x, y, z),D(x, y, z)). From the above ﬁeld conti-

nuity, the boundary condition is automatically satisﬁed.

f. CMxPoint and Dipole Fields. We introduce the

fundamental ﬁelds with CMxsymmetry. Consider a half-

space ε(x, y, z)>0 in x≥0+. We begin with the Mx-

symmetrized permittivity, which is deﬁned as follows:

εM(x, y, z) = (ε(x, y, z) (x≥0)

ε(−x, y, z) (x≤0) (10)

First, we place a point charge qat (x, y, z) = (0−, Y, Z)

in εM. The CMxsymmetrization is applied to the ﬁeld

in x≥0+to eliminate the point source. Under sym-

metrization, the permittivity becomes CMx-symmetric.

The obtained ﬁeld is called a CMxpoint ﬁeld and is de-

noted as (E(pt)

R,D(pt)

R) with R= [0 Y Z]T. The most

straightforward situation with a uniform ε(x, y, z) = ε0

is shown in Fig. 1. The second example is the dipole

ﬁeld. Consider a dipole with the dipole moment (p, 0,0)

at (x, y, z) = (0−, Y, Z) in εM. The CMxsymmetriza-

tion for the ﬁeld in x≥0+removes the dipole source.

The obtained ﬁeld is called a CMxdipole ﬁeld and is

denoted as (E(dp)

R,D(dp)

R) with R= [0 Y Z]T. The CMx

dipole ﬁelds for a uniform ε(x, y, z) = ε0are shown in

Fig. 2.

g. Boundary Degree of Freedom. We show that the

degree of freedom of CMx-symmetric ﬁelds is represented

by either a CMxpoint ﬁeld or a CMxdipole ﬁeld. Con-

sider a half-space electrostatic potential ϕin x≥0+with

ε(x, y, z)>0. Assuming that there is no free charge in

ε0

−ε0

(a) ε0

−ε0

(b)

FIG. 1. CMxpoint ﬁelds: (a) Electric ﬁeld. (b) Electric

displacement.

ε0

−ε0

(a) ε0

−ε0

(b)

FIG. 2. CMxdipole ﬁelds: (a) Electric ﬁeld. (b) Electric

displacement.

x > 0, the boundary charge or dipole may appear at

x= 0. Let ϕSand ϕAbe symmetric and antisymmet-

ric extensions of ϕin the whole space, respectively. We

deﬁne ϕSand ϕAas follows:

ϕS(x, y, z) = (ϕ(x, y, z) (x≥0+)

ϕ(−x, y, z) (x≤0−)(11)

ϕA(x, y, z) = (ϕ(x, y, z) (x≥0+)

−ϕ(−x, y, z) (x≤0−)(12)

These potentials satisfy ∇·εM∇ϕ= 0 in x6= 0 with the

Mx-symmetrized permittivity deﬁned in Eq. (10). To

ensure the boundary condition on x= 0 for ϕS, there

should be a boundary charge σon x= 0 satisfying the

following equation:

σ(y, z) = −2ε(0, y, z)∂ϕ

∂x (0+, y, z).(13)

On x= 0, the tangential component Etis continuous,

whereas the normal component Dxis discontinuous indi-

cated by σ. In fact, Dx(0+, y, z) = σ(y, z)/2 holds true.

A nonuniform σmay also contribute to the tangential

component. Conversely, ϕAhas discontinuity on x= 0,

which indicates the existence of a double layer. This is

described as follows:

τx(y, z) = 2ε(0, y, z)ϕ(0+, y, z).(14)

For the double layer on x= 0, the normal compo-

nent Dxis continuous, whereas the tangential com-

ponent Etexhibits discontinuity by Et(0+, y, z)−

ε1

ε2

D(1)

(a) ε1

ε2

D(2)

(b)

ϵ1q

ϵ2q

−q

FIG. 3. Electric-displacement ﬁelds with symmetries: (a)

D(1) and (b) D(2). Here, it is assumed that ε1>0 and

ε2>0.

Et(0−, y, z) = −∇[τx(y, z)/ε(0, y, z)] [18]. Thus, we ob-

tain Et(0+, y, z) = −(1/2)∇[τx(y, z)/ε(0, y, z)], which

agrees with Eq. (14). Additionally, a nonuniform τx

may contribute to the normal component via electric-

ﬁeld leakage to outside the double layer. From the obser-

vation of characteristics of the single and double layers

thus far, we can conclude that either σor τxcan be used

to construct the ﬁeld in x≥0+. This statement is con-

sistent with the treatment of a conventional boundary-

value problem: the solution to an electrostatic problem

is uniquely determined by applying either the Dirichlet

or Neumann boundary conditions for each boundary [19].

Now, consider Φ = (ϕS+ϕA)/2. This ﬁeld is expressed

as follows: Φ = ϕin x≥0+and Φ = 0 in x≤0−; i.e.,

the ﬁeld in x≥0+is generated from σ/2 and τx/2 on

x= 0 in a Mx-symmetrized system. By contrast, Φ van-

ishes in x≤0−. To obtain a CMx-symmetric solution,

we apply the CMxsymmetrization for Φ, which makes

both Dxand Etcontinuous on x= 0. All sources then

vanish, whereas the ﬁeld remains.

h. Singular Response. We characterize the singular

response of a CMxsystem. Consider a system with

permittivities of ε1(x, y, z) in x≥0+and ε2(x, y, z) in

x≤0−. These permittivities do not need to have CMx

symmetry. To construct two modes similar to symmet-

ric and antisymmetric modes, we assume the following

condition:

ε2(−x, y, z)

ε1(x, y, z)= Const.(x≥0+),(15)

which is always satisfactory for a uniform ε1and ε2. Now,

we place charges q1and q2at (x, y, z)=(a, 0,0) and

(x, y, z)=(−a, 0,0), respectively, with a > 0.

First, consider weighted charges q1=1(r0)qand

q2=2(−r0)qwith r0= [a0 0]T, as shown in Fig. 3(a),

where the relative permittivity is given as i=εi/ε0.

The electric-displacement ﬁeld at r= [x y z]Tis given as

follows:

D(1)(r) = 





1(r0)q

4πr−r0

|r−r0|3+r+r0

|r+r0|3(x≥0+)

2(−r0)q

4πr−r0

|r−r0|3+r+r0

|r+r0|3(x≤0−)

(16)

D(1) satisﬁes Maxwell’s equations with charge in x > 0

and x < 0. Because Dx= 0 holds on x= 0, the normal

component Dxis continuous on x= 0. The electric ﬁeld

fulﬁlls the tangential continuity condition on x= 0, ow-

ing to the appropriate choice of the charge weights with

Eq. (15). Second, we consider q1=qand q2=−q, as

shown in Fig. 3(b). The electric displacement is given as

follows:

D(2)(r) = q

4πr−r0

|r−r0|3−r+r0

|r+r0|3.(17)

Equation (17) satisﬁes Maxwell’s equations in x > 0 and

x < 0 and is antisymmetric with respect to Mx; there-

fore, the normal continuity condition of Dxon x= 0 is

satisﬁed. The corresponding electric ﬁeld does not have

a tangential component on x= 0; hence, the tangential

continuity condition on x= 0 is satisﬁed.

D(1) and D(2) represent two solutions for 2(−r0)6=

−1(r0). By combining Eqs. (16) and (17), we can calcu-

late the ﬁeld for a single charge (e.g., q1=qand q2= 0).

However, 2(−r0) = −1(r0) results in CMxsymmetry,

and the weighted charge distribution for D(1) becomes

exactly the same as that for D(2); i.e., the charge dis-

tribution does not uniquely determine the CMxﬁeld.

Therefore, 2(−r0) = −1(r0) leads to the singular re-

sponse for free charge. When the limit of a→0 is taken,

D(1) and D(2) provide the CMxpoint and dipole ﬁelds,

respectively. Here, we can exclude the source by remov-

ing the slab region {(x, y, z)|x∈[−a, a], y, z ∈R}and

joining x=−a−0+and x=a+ 0+under CMxsym-

metry.

B. Surface Plasmon Polaritons Originating from

Symmetry Protection

In this section, we show that surface plasmon polari-

tons originate from the symmetry-protected electric zero

modes. Consider a boundary between the uniform per-

mittivities of ε1>0 in x≥0+and ε2<0 in x≤0−. The

whole space is assumed to have vacuum permeability µ0.

We focus on the transverse-magnetic (TM) surface mode

with angular frequency ωand wavenumber kyalong the

ydirection. The surface impedance on x= 0 for x≥0+

and x≤0−is denoted as Z1and Z2, respectively, which

are expressed as follows:

Zi=−jqky2−εiµ0ω2

ωεi

.(18)

The derivation is summarized in Appendices A–C. The

resonance condition Z1+Z2= 0, which is equivalent to

the continuity conditions of the electric ﬁeld, provides

the well-known dispersion relation as follows:

k0=ω

=kyr1+2

12

,(19)

where c0= 1/√ε0µ0,k0=ω/c0, and irepresent the

speed of light in vacuum, vacuum wavenumber, and rel-

ative permittivity i=εi/ε0, respectively.

Equation (19) gives the ﬂat zero band for 2=−1.

The zero modes accompany Z1=−j∞and Z2= +j∞

at ω→0+, indicating that only the electric ﬁeld appears

owing to the electromagnetic decoupling at direct current

(DC) limit. The origin of the zero modes is the CMx-

symmetrized modes shown in Figs. 1 or 2. There are de-

generated modes located at diﬀerent positions on x= 0.

Because these zero modes do not couple with each other,

they form the ﬂat zero band. The eigenfunction with a

wavenumber kyalong yis obtained by summing CMx

point or dipole ﬁelds with the weight of the exp(−jkyy)

factor like Rx=0 dSE(pt)

Rexp(−jkyY), where we use R=

[0 Y Z]Tand the point charge is replaced with the charge

density. For ky6= 0, Rx=0 dSE(pt)

Rexp(−jkyY) and

Rx=0 dSE(dp)

Rexp(−jkyY) yield the same eigenmode, be-

cause it has both Dx6= 0 and Ey6= 0 components. From

these observations, we can conclude that the CMxpoint

and dipole ﬁelds coalesce. The square-root function in

Eq. (19) is multi-valued in the complex plane. This multi-

value characteristic indicates that 2=−1is the excep-

tional point where the two modes typically coalesce [20].

For ky= 0, point and dipole ﬁelds are decoupled, yield-

ing two modes that produce two waves localized at x= 0

and ∞.

Now, we can state that CMx-protected zero modes are

the origin of surface plasmon polaritons. Starting from

2=−1, we decrease 2to 2<−1. The ﬂat zero band

then becomes a ﬁnite frequency band. As the deforma-

tion induces unbalanced Poynting vectors in x≥0+and

x≤0−, the energy can propagate with a nonzero group

velocity. This CMx-broken mode is usually observed as

surface plasmon polaritons in experiments.

C. Robustness of Symmetry Protection

The CMx-symmetry protection works for both con-

stant and nonuniform permittivities. Here, we provide

two examples to support the robustness of the CMxpro-

tection for a nonuniform permittivity conﬁguration.

a. Layered media. Consider a layered system with

metallic and dielectric materials, as shown in Fig. 4(a).

The binary dielectric layers are periodically aligned in

x≥0+, whereas the binary metals are periodically ar-

ranged with the period ain x≤0−. The thicknesses dA

and dBof the layers are a/2. In x≤0−,A2=−A

and B2=−Bwith A= 4 and B= 2 are assumed.

Conversely, we set the relative permittivities in x≥0+

as A1= (1 −ξ)A+ξand B1= (1 −ξ)B+ξ, using

the parameter ξ. At ξ= 0, the CMxsymmetry holds,

whereas ξ6= 0 breaks the symmetry. We focus on nonra-

diative localized TM surface waves with wavenumber ky

along y. Let Z1and Z2be surface impedances at x= 0

for x≥0+and x≤0−, respectively. These impedances

can be calculated as the Bloch impedances, as described

in Appendix D. The surface-wave resonant condition is

represented by Z1+Z2= 0. For a given discretized ky,

we numerically evaluate the resonant angular frequency

ωfor purely imaginary Z1and Z2.

Figure 4(b) shows real bands of TM surface waves for

ξ= 0.01, 0.4, and 0.8. The ﬁrst band (represented by or-

ange circles) distinctly originates from the ﬂat zero modes

that are CMx-protected at ξ= 0. As ξincreases from 0,

the CMxsymmetry is broken, and the ﬁrst band is raised

from zero. The higher bands are sometimes broken be-

cause the wave becomes leaky and propagates into inﬁn-

ity. Conversely, the plasmonic ﬁrst band is nonradiative,

and it remains continuous under the perturbation by ξ.

This plasmonic dispersion agrees well with the solid red

line calculated via the eﬀective-medium approximation,

as described in Appendices E and F.

b. Corrugated system. Next, we consider nonunifor-

mity in the ydirection. Figure 5(a) shows the unit cell of

a corrugated plasmonic/photonic system with a parame-

ter ξ. The dielectric–metal boundary is located on x= 0.

The corrugation is periodic in the ydirection and is given

by the cos function. The geometric parameters are set

as b/a = 6, w/a = 0.25, hPML/a = 3, and d/a = 0.1.

The permittivity in III is set as ε1/ε0= 5. At ξ= 1, the

system is CMx-symmetric. We calculated the complex

Bloch wavenumber k(Bloch)

y(ω) of TM surface waves for a

given angular frequency ωusing the conventional ﬁnite-

element solver COMSOL Multiphysics [21]. To simplify

the plot, we restrict kysatisfying Re[k(Bloch)

y]≥0. As

the ﬁnite-element eigenmode analysis suﬀers from un-

physical modes near the perfectly matched layers (PMLs)

[22], we ﬁlter the physical modes localized near the sur-

face, using RI|˜

Hz|2dS/ RIII∪IV |˜

Hz|2dS < 0.12 with the

complex amplitude of the zcomponent of the magnetic

ﬁeld ˜

Hz. Here, the complex amplitude ˜

His deﬁned as

H=˜

Hexp(jωt) + c.c.for the real magnetic ﬁeld H,

where trepresents time.

Figure 5(b) presents the calculated complex disper-

sion relations for ξ= 5, 1.5, and 1.05. The wavenum-

ber becomes complex above the light line because the

diﬀraction by corrugation leads to energy leakage to in-

ﬁnity. This observation validates the results. At ξ= 1,

there is a CMx-protected zero mode at each point on

the dielectric–metal boundary x= 0. These degenerated

modes produce inﬁnite bands in ξ > 1. Therefore, the

CMx-protected zero modes are regarded as the sources

of inﬁnite bands. When ξapproaches 1, the frequencies

of all plasmonic bands decrease to zero. Note that lower

eigenfrequencies are missing due to simulation limitation

i.e., these modes are weakly conﬁned and aﬀected by the

ﬁnite simulation domain.

To illustrate the CMxprotection more explicitly, we

present the electric-ﬁeld amplitude |˜

E|of the ﬁrst band

at approximately k(Bloch)

ya/π ≈0.5 in Fig. 6. When ξ

approaches 1, the ﬁeld becomes symmetric with respect

to the dielectric–metal boundary x= 0. This result sup-

ports the crucial role of CMxsymmetry in the forma-

tion of surface plasmon polaritons even if the system has

nonuniformity.

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HiddenSymmetryProtectionandTopologyinSurfaceMaxwellWavesYosukeNakataGraduateSchoolofEngineeringScience,OsakaUniversity,Osaka560-8531,JapanandCenterforQuantumInformationandQuantumBiology,OsakaUniversity,Osaka,560-8531,JapanToshihiroNakanishiDepartmentofElectronicScienceandEngineering,KyotoUniversity...

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Hidden Symmetry Protection and Topology in Surface Maxwell Waves Yosuke Nakata Graduate School of Engineering Science Osaka University Osaka 560-8531 Japan and

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