ОПТИКА И СПЕКТРОСКОПИЯ, 2015, том 118, № 2, с. 322-326
^ ФИЗИЧЕСКАЯ ^^^^^^^^^^^^^^
ОПТИКА
УДК 535.36
SCATTERING FROM A TOPOLOGICAL INSULATOR ELLIPTIC CYLINDER
© 2015 г. Yanyan Zhao, Chen Guiyun, and Zneg Lunwu
College of Engineering, Nanjing Agricultural University, Nanjing 210031, China E-mail: zhaoyanyan@njau.edu.cn Received June 5, 2014
The electromagnetic scattering properties of topological insulator elliptic cylinder is studied. Utilizing Maxwell equations, electromagnetic field components in elliptical coordinate frame are derived. According to the boundary conditions of topological insulator, scattering electric fields and magnetic flux densities are obtained. Numerical results show when time reversal symmetry is broken, the scattering widths are influenced by the topological magneto-electric polarizability.
DOI: 10.7868/S003040341502021X
INTRODUCTION
Topological insulator is a new state of quantum matter, which has been investigated extensively [1— 17]. The concept of the topological insulator can be defined using both the topological field theory [4—8] and the topological band theory [9—13]. Time-reversal invariant topological insulators have a bulk energy gap, but have gapless excitations with an odd number of Dirac cones on the surface. When a topological insulator is coated with a very thin magnetic layer or perturbed by a weak magnetic field [17], time reversal symmetry is broken, and an energy gap also opens up at the surface. In this case, the spectrum of the topo-logically protected surface states acquire gap. As a result, the system becomes an insulator, both at the bulk and on the surface. However, the surface is not an ordinary insulator; it is rather a quantum Hall insulator [6]. This quantized Hall Effect on the surface is the physical origin of the topological magneto-electric effect. Under an applied electric field, a quantized Hall current is induced on the surface, which in turn generates a magnetic polarization and vice versa, namely, co-polarization and cross-polarization. Cross-polarization has been widely investigated in topological insulator [18] and perfect electromagnetic conductor [19—24]. In this paper, we investigated the properties of electromagnetic wave scattering by a time reversal perturbation topological insulator elliptic cylinder and obtain the relation between scattering widths and the topological insulator parameter. Numerical results show when time reversal symmetry is broken, the scattering cross sections are influenced by the topological insulator parameter.
FORMULATION
According to topological field theory, the effective Lagrangian of a topological insulator is [7],
L = L0 + L0 = 11 eE2 -1B2 I + -0-aEB, 0 0 8 ^ | ) 2n 2n
where E and B are the electric and magnetic induction, e and | are permittivity and permeability, respectively, a = e2/hc is the fine structure constant, in which h is Planck constant h divided by 2n, c is the speed of light in vacuum, e is the electric charge, and 0 is a parameter describing the insulator or topological magneto-electric polarizability (TMEP) [14—16]. Under the time reversal transformation, E ^ E, B ^ -B. So for a periodic system, there are only two values of 0 (modulo 2n), namely 0 = 0 (e.g. vacuum) and 0 = n (e.g. topological insulator), which in accordance with a time reversal symmetry theory [6, 7]. When time reversal symmetry is broken, the TMEP is quantized in odd integer values of n such that 0 = (2n + 1)n (n e Z) [14-16]. The value of n is determined by the nature of the time reversal symmetry breaking perturbation, which could be controlled experimentally by covering the topolog-ical insulator with a thin magnetic layer or magnetic perturbed [17]. In this paper, we investigate the electromagnetic scattering of a time reversal symmetry perturbation topological insulator elliptic cylinder. The geometry of the problem used for the analysis is shown in Fig. 1. It contains an infinite topological insulator elliptic cylinder, which is perturbed by a weak magnetic field or covered by a very thin layer (assume that its scattering property is not influenced by the perturbation). For simplicity, we assume that the axis of the elliptic cylinder is coincident with z-axis of the coordinate system. The region outside the elliptic cyl-
inder is termed region 0, and the region inside the elliptic cylinder is termed region 1. The constitutive relations of the topological insulator are
D = e E + a-B, n
H =1B
I
a-E n
[5, 14—16], where D and H are the electric flux density and the magnetic field intensity, respectively. Maxwell's equations can be written as:
VxE = -iroB, VxB = /roe|iE.
(1) (2)
In this paper, we have used e'ra time dependence and it is suppressed throughout the analysis. When the polarization of the incident electric field is parallel to the elliptic cylinder axis, the incident electric field can be written as [24]
Region 0
%
Einc
Fig. 1. Electromagnetic wave scattered by a topological insulator elliptic cylinder.
liptical cylinder can be expressed in terms of a sum series of Mathieu function as [24]
pine _ T-ï ik0(x cos ^ +y sin
Ez = E 0e ,
(3)
where
E?C = X AmÄco, K)Sem(c0, n) +
m=0
+ X AomRoln(c0, K)Som(c0, n)
Aem = Eo^ Sem(Co,eos ^ ),
om N em(C 0) om
(4)
om 2n
Nem(Co) = j
om 0
Sem(c0, n)
dv,
= coshu, n = cosv, c0 = k0F, F is the semi focal length of the elliptical cross section, Sem and Som are the even and odd angular Mathieu functions of order m,
respectively, R.fm and r02 are the even and odd radial Mathieu functions of the first kind and order m, and Nem and Nom are the even and odd normalized functions of order m. The electric field inside and outside the topological insulator elliptic cylinder can be obtained by solving the Helmholtz equation in the elliptical coordinate system using separation of variables technique. The scattered electric field outside the el-
Ez = X BemR-em (c0, ^)Sem(c0, n)
+
where E0 is the amplitude of the incident electric field, and k0 = is the wave number of region 0. In an
elliptical coordinate system (u,v, z) having the same origin as that of the Cartesian coordinate system, the incident electric field can be expanded in terms of Mathieu functions as [24]
m=0
(5)
+ X Bom-om'(C0, Ç)Som(C0, n),
77=1
where R^m and rO2 are the even and odd Mathieu functions of the fourth kind and order m. Similarly, electric field inside the elliptic cylinder can be written as
Ez = X CemRem(cb ^)Sem(c1, n)
+
m=0
(6)
+
X ComRom (c1, K)Som (c1, n),
m=1
where cx = kxF, and kx = ^/e^, £j and ^ are the permittivity and permeability of topological insulator, respectively. The magnetic flux density component can be obtained by using Maxwell's equations,
Bv =
i dEz
roh du
(7)
where h = Wcosh2 u - cos2 v. Substituting Eqs. (4)— (6) into Eq. (7), we obtain
n' = ik0 Bv =--7
roh
X AemR<ein (c0, ^)Sem(c0, n)
+
+ X AomRom'(C0, ^)Som(C0, n)
m=1
m
0
m
OnTHKA H CnEKTPOCKOnHa TOM 118 № 2 2015
ns _ ikQ Bv _---
roh
£ BemRi42'(c0, fySem(c0, n)
m_Q
+ £ BomROm'CCQ, ^)S0m(CQ, n)
+
According to the constitutive relation of topological insulator, the boundary conditions can be written as
m_1
(9) -^Bl
- Q
+ - BS 6 c t , (17)
-1 n
B
_ _ iki roh
£ CemRe^ (c1, %)Sem(c1, n)
+
—Bzs 6 !7 t _ a-Ez
- Q n
(18)
_Q
+ £ C^C, ^)S0m(c1, n)
m_1
(1Q)
In the topological insulator, there are cross polarized field components in addition to the co-polarized field components. The cross polarized components of the magnetic flux density of the scattered electromagnetic fields can be written as [24]
Bs _ £ BemRem(cQ, %)Sem(cQ, n)
m_0
+ £ B'0mR{ol(cQ, K)Som(cQ, n),
+
(11)
W_1
B[ _ £ CeRta, ^)Sem(c1, n) +
m_Q
+ £ ComR0m(cb %)Som(cb n).
m_1
Substituting Eqs.(11)—(12) into Eq.(7), we obtain
£ BemRem (cQ, %)Sem(cQ, n) +
(12)
EV _ _-
ikQ
m_Q
rohe Q-Q
+ £ BomRom (cQ, %)Som(cQ, n)
(13)
Et _ __ ik
£ CemRem (cb ^)Sem(c1, n)
m_Q
ro he1-1
+ £ ComRom (cu ^)Som(c1, n)
+
_1
(14)
The tangential components of E shall be continuous on the surface, so
l^Q EV\
(Ei + Esz )|
_ EZ
(15)
(16)
Substituting Eqs. (4)-(6), Eqs. (8)-(14) into Eqs. (15)-(18), we obtain
£ AemRim(cQ, %Q )Sem(cQ, n) +
m_Q
+ £ BemRem)(cQ, %Q)Sem(cQ, n) _ (19)
m=Q
_ £ CemRem(c1, % Q)Sem(c1, n),
m=Q
£ AomR0i))(cQ, % Q)Som(cQ,n) +
m=1
+ £ BomRom (cQ, % Q)Som(cQ, n) _ m=1
_ £ ComRo)))(c1, % Q )Som (c1, n)
m=1
Vë1-1£ BemRem)'(CQ, %Q^em^ ^ _
m=Q
_ VeQ—Q £ CemRe) (c1 %Q^em^ n), m=Q
Vë1-1£ B0mR0m)'(CQ, %Q^om^ n)] _
m=1
o mRom (C1, % Q)Som(c1, n)],
m=1
nQ£ AemRim (cQ, %Q)Sem(cQ, n) +
m=Q
+ nQ£ BemRe) (cQ, %Q)Sem(cQ, n) _
m=Q
_ n1£ CemRe())'(C1, %Q^mfe, n) +
m=Q
+ a £ CemR<ern (c1, %Q)Sem(cb n)
m=Q
(2Q)
(21)
(22)
(23)
OnTHKA H CnEKTPOCKOnHtf tom 118 № 2 2Q15
m
9/Rad
Fig. 2. Co-polarized components of the far-zone normalized bistatic width versus scattering angle 6 = 0 (1), 21n (2), 41n (3).
Fig. 3. Cross-polarized components of the far-zone normalized bistatic width versus scattering angle 9: 6 = 0 (1), 21n (2), 41n (3).
Ло X AomRom (c0, S 0)Som(c0, Л) + m=1
+ ПоX BomR0m'(Co, S0)Som(C0, П) =
m=1
= niX Corned, S0)Som(Cl,П) +
m=l
nei^i m=i
+ a X ComRoin (c1> S 0 )Som (c1, n)
X BemRem(c0, S 0)Sem(c0, n) =
' m=0
X ^m^em^b S 0)Sem(cb n) +
^ m=0
+ a^ X CemR^(C1, S0)Sem(C1, П),
m=0
X BomRom(c0, S0)Som(c0, n) =
^0 m=1
= X C omRo)^n(c1, S0)Som(cb n) +
(24)
(25)
(26)
and C'om are cross-polarized expansion coefficients. Solving Eqs.(19)—(26), we obtain eight expansion coefficients and the scattering electromagnetic fields and
magnetic flux density components ES, BS, Esv, Bsv.
NUMERICAL RESULTS AND DISCUSSION
The numerical results are based on the above analytical formulations for a topological insulator elliptic cylinder. According to Ref [24], we can write a far zone scattering expression for the normalized bistatic scattering width as
c
co
X
+
c c
X
X imBemSem(c0, n) +X imB om^ om (c0, n)
m=0 m=1
X i CemS e
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