Scattered field from a dielectric-topological insulator rough interface using perturbation theory

Journal Pre-proof Scattered field from a dielectric-topological insulator rough interface using perturbation theory Nelum Andlib, Muhammad Sajid Hanif, Muhammad Arshad Fiaz

PII: DOI: Reference:

S0030-4018(19)31040-5 https://doi.org/10.1016/j.optcom.2019.124958 OPTICS 124958

To appear in:

Optics Communications

Received date : 31 August 2019 Revised date : 12 November 2019 Accepted date : 13 November 2019 Please cite this article as: N. Andlib, M.S. Hanif and M.A. Fiaz, Scattered field from a dielectric-topological insulator rough interface using perturbation theory, Optics Communications (2019), doi: https://doi.org/10.1016/j.optcom.2019.124958. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier B.V.

Journal Pre-proof

Scattered field from a dielectric-topological insulator rough interface using perturbation theory

of

Nelum Andlib, Muhammad Sajid Hanif and Muhammad Arshad Fiaz∗

pro

Department of Electronics, Quaid-i-Azam University, Islamabad, Pakistan. [email protected] , [email protected], [email protected] *Corresponding author: [email protected] Abstract

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Scattering from a dielectric-topological insulator (TI) rough interface is studied using perturbation theory (PT). Two models characterized by magneto-electric pseudo scalar ψ and surface admittance γ for TI material are incorporated in the analysis. The zeroth order solution describes the reflection and transmission properties of the flat TI interface. Results are reported for various values of ψ and γ. The Brewster angle is shifted while it occurs at 58o for the dielectric case. The cross polarized reflection coefficients become small as the value of γ is increased. The first order scattering coefficients represent the contribution by superimposed roughness on the flat interface. Numerical results show that for γ-model, the first order co-polarized scattered field increases as γ is increased while the cross polarized field decreases. It is noted that the model defined by γ can better describe the scattering properties of the TI rough interface both theoretically and practically.

Keywords: Scattering, topological insulator material, rough surface, perturbation theory.

1

Introduction

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Topological insulators (TIs) [1] being a new quantum states of matter have been predicted theoretically and the experimental realization is found in Sb2 T e3 , Bi2 T e3 and Bi2 Se3 . The characteristic features which enable them to be different from the conventional insulator are the presence of a full insulating gap in the bulk and gapless conducting surface states protected by 1

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time-reversal symmetry [2]. To break the time-reversal symmetry, a thin magnetic coating is used on the surface of TI [3]. It can also be broken by creating perturbation using magnetic field. Both field theory and band theory are used to define the topological insulators [4]-[9]. Liu et al. discussed the electromagnetic properties of a 3D topological insulator(TI) [10]. The polarization conversion from incident s-polarization into reflected p-polarization is reported. Based on the analytical formulation and numerical simulations, the change of polarization is related to the strong Kerr effects in TI. The Kerr effect [11] is associated with the rotation of polarization of reflected field. It is also observed that elliptical polarization can be modulated to linear or circular polarization. These behaviors are originated from the magneto-electric coupling in TI. The possible applications of TIs are polarized devices and polarization splitters. Recently, Sobia et al. [12] discussed the TI-chiral flat interface. The expressions of reflection and transmission coefficients are derived for parallel and perpendicular polarizations. Lakhtakia and Mackay [13] studied the reflection and transmission properties of topological flat interface. Two models described by magneto-electric pseudoscalar ψ and admittance γ are incorporated for the analysis. It is concluded that ψ and γ appear identically in the final expression of the reflected and transmitted fields. Moreover, since ψ is not present in the final simplified form of the Maxwell’s equations in the TI medium, the model described by γ is suggested to be used for the analysis. We name them ψ-model and γ-model. The ψ-model defines the TI using magneto-electric pseudoscalar ψ being its surface is charge free and current free. The constitutive relations for a isotropic TI media can be defined in terms of permittivity , permeability µ and magnetoelectric pseudoscalar ψ as [14] D = E + ψB H = µ−1 B − ψE

(1) (2)

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and α is the fine structure constant. The value of θ = 0 where ψ = αθ π and θ = π describe conventional dielectric material and time symmetric TI material respectively. The γ-model is defined by γ = (2n + 1)α/η0 ; n ∈ Z by introducing the surface charge and current density. In this paper, the work presented in [14] is extended to study the reflection and transmission properties of a TI rough interface. 2

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A rough surface can be characterized as slightly rough or rough surface. The amount of roughness can be described using Rayleigh criteria given by h > (λ/8 cos θi ). Slightly rough surfaces can be examined using PT [15, 16]. In PT, the Rayleigh hypothesis is used to express the scattered and transmitted fields. It is applicable for small heights compared to wavelength of the incident field. Johnson[17] reported the third order PT to calculate the scattered field from a dielectric rough surface. The brightness temperature is calculated by using scattering coefficients. The extended boundary condition method (EBCM) [18] is also used to study the rough surface scattering. The method is applicable to such surfaces whose height is of the order of the wavelength of the incident field. The periodic surfaces had been studied by P.C Waterman [19] using the EBCM. A waveguide with rough interface is discussed by Chuang and Koing in [20]. The diffraction efficiency is reported and compared with the experimental data for a metallic grating. The Kirchhoff approximation (KA) is also used to investigate the scattering phenomenon [21]-[22] for those surfaces whose radius of curvature is large compared to the wavelength of the incident field. The region of validity of KA and PT was examined in [23]-[24]. In this paper, scattering from a dielectric-TI rough interface is discussed. Section 2 contains the theoretical formulation of the problem. Results are reported in Section 3.

Theoretical formulation

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Consider the geometry shown in figure 1. The surface perturbations are defined by z = f (x). The mediums above and below the interface are termed dielectric and TI respectively. The permittivity of the TI material is 1 and is characterized by either γ or ψ having the units of admittance. The incident fields are given by   kix kiz ˆz ) ei(kix x−kiz z) ˆy + ap (−ˆ −e (3) E i = as e ex ko ko   kiz kix −1 ˆy + as (ˆ ˆz ) ei(kix x−kiz z) Hi = ηo ap e ex +e (4) ko ko The field scattered due to rough surface can be written as  Z ∞ kz kx ˆz ) ei(kx x+kz z) dkx Es = rs (kx )ˆ ey + rp (kx )(−ˆ ex + e k ko o −∞ 3

(5)

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z

Es

e o , mo

pro

TI-Dielectric rough interface

of

E inc

e1 , mo

x

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Figure 1: A dielectric-TI rough interface The corresponding magnetic field is Hs =

Z∞ 

ηo−1

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−∞

 kz kx ˆz ) ei(kx x+kz z) dkx (6) −rp (kx )ˆ ey + rs (kx )(−ˆ ex + e ko ko

The expressions for the transmitted E and H fields are: Et =

Z∞ 

−∞

Ht =

η1−1

 kx k1z ˆz ) ei(kx x−k1z z) dkx −e ts (kx )ˆ ey + tp (kx )(−ˆ ex k1 k1

Z∞ 

−∞ Z∞

−ψ

−∞



(7)

 k1z kx ˆz ) ei(kx x−k1z z) dkx +e tp (kx )ˆ ey + ts (kx )(ˆ ex k1 k1

 kx k1z ˆz ) ei(kx x−k1z z) dkx ts (kx )ˆ ey + tp (kx )(−ˆ ex −e k1 k1

(8)

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The unknown scattering amplitudes can be determined by using the following boundary conditions. n ˆ × [Ei + Es − Et ] = 0 n ˆ × [Hi + Hs − Ht ] = Js

(9)

where Js = −γ n ˆ × Et and n ˆ = zˆ − xˆ ∂f is the normal to the surface. To apply ∂x the PT, the following steps are performed: 4

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• All the unknown amplitudes are represented as an infinite series:

ts (kx ) =

rsn (kx )

n=0 ∞ X

and

tns (kx )

rp (kx ) =

∞ X

rpn (kx )

of

rs (kx ) =

∞ X

and

tp (kx ) =

n=0 ∞ X

tnp (kx )

n=0

pro

n=0

• The exponentials having z=f(x) are expanded as a Taylor series by assuming that the height and slope are small: ∂f (x) | << 1 ∂x

re-

|kzi f (x)| << 1 , |

Following the above mentioned procedure and applying the boundary conditions at z = f (x), we obtain

q=0

Z∞ X ∞ X ∞

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as

∞ X (−ikiz f (x))q

q!

eikix x +

rsn (kx )

−∞ q=0 n=0 Z∞ ∞ ∞

=

XX

−∞ q=0 n=0

(ikz f (x))q ikx x e dkx q!

tns (kx )

(−ik1z f (x))q ikx x e dkx q!

∞ ∞ kiz X (−ikiz f (x))q ikx x kix ∂f X (−ikiz f (x))q ikx x e − ap e ko q=0 q! ko ∂x q=0 q! Z∞ X ∞ X ∞ kx (ikz f (x))q ikx x ∂f + rpn (kx ) e dkx ∂x ko q! q=0 n=0

−ap

−∞

Jo

−

Z∞ X ∞ X ∞

rpn (kx )

−∞ q=0 n=0 Z∞ X ∞ X ∞

=−

∂f ∂x

−∞ q=0 n=0

kz (ikz f (x))q ikx x e dkx ko q!

tnp (kx )

kx (−ik1z f (x))q ikx x e dkx k1 q! 5

(10)

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−∞ q=0 n=0

ηo−1 ap −η1−1

∞ X (−ikiz f (x))q

q!

q=0 Z∞ X ∞ X ∞

tnp (kx )

k1z (−ik1z f (x))q ikx x e dkx k1 q!

ikx x

−

e

ηo−1

Z∞ X ∞ X ∞

rsn (kx )

(12)

re-

q!

=0

eikx x + ηo−1 as

∞ ∂f kix X (−ikiz f (x))q ikx x e ∂x ko q=0 q!

kz (ikz f (x))q ikx x e dkx ko q!

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−ηo−1

(ikz f (x))q ikx x e dkx q!

(−ik1z f (x))q ikx x e dkx q!

∞ X kiz (−ikiz f (x))q

ko q=0 Z∞ X ∞ X ∞

rpn (kx )

−∞ q=0 n=0

−∞ q=0 n=0 Z∞ X ∞ X ∞ (−ik1z f (x))q ikx x +G tns (kx ) e dkx q! q=0 n=0 −∞

ηo−1 as

(11)

of

tnp (kx )

pro

−

Z∞ X ∞ X ∞

−∞ q=0 n=0 Z∞ X ∞ X ∞ k1z −1 tns (kx ) −η1 k1 −∞ q=0 n=0 Z∞ ∞ ∞

−G

XX

−∞ q=0 n=0 Z∞ ∞

+ηo−1

−η1−1

∂f ∂x

∂f ∂x

tnp (kx )

∞ XX

−∞ q=0 n=0 Z∞ X ∞ X ∞

(−ik1z f (x))q ikx x e dkx q!

k1z (−ik1z f (x))q ikx x e dkx k1 q!

rsn (kx )

kx (ikz f (x))q ikx x e dkx ko q!

tns (kx )

kx (−ik1z f (x))q ikx x e dkx k1 q!

Jo

−∞ q=0 n=0 Z∞ X ∞ X ∞ ∂f kx −G tnp (kx ) ∂x k1 −∞ q=0 n=0

(−ik1z f (x))q ikx x e dkx = 0 q!

The zeroth-order terms of the above equations are: 6

(13)

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as eikix x +

rs0 (kx )eikx x dkx =

Z∞

ηo−1 ap eikix x − ηo−1

kiz as eikix x − ηo−1 ko ηo

Z∞

+G

Z∞

(15)

t0p (kx )eikx x dkx

t0s (kx eikx x dkx

kz rs0 (kx ) eikx x dkx = η1−1 ko Z∞

k1z ikx x e dkx k1

(14)

−∞

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−∞

−∞

t0p (kx )

−∞

rp0 (kx )eikx x dkx = η1−1

−∞ Z∞

−G

=

Z∞

pro

−∞

kz rp0 (kx ) eikx x dkx ko

re-

Z∞

t0s (kx )eikx x dkx

−∞

−∞

kiz ap eikix x + ko

Z∞

of

Z∞

Z∞

−∞

t0s (kx )

(16)

k1z ikx x e dkx k1

t0p (kx )eikx x dkx

(17)

−∞

where

G=ψ+γ

Taking the Fourier transform of the above equations and after some algebra, the following expressions are obtained: −rs0 (kx ) + t0s (kx ) = as δ(kx − kix )

kz 0 r (kx ) + σr t0p (kx ) = ap δ(kx − kix ) kiz p 1 rp0 (kx ) − Gηo t0s (kx ) + t0p (kx ) = ap δ(kx − kix ) ηr kz 0 σr rs (kx ) + t0s (kx ) + Gσr ηo t0p (kx ) = as δ(kx − kix ) kiz ηr

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−

7

(18) (19) (20) (21)

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

 as     = δ(kx − kix )W0−1  ap   ap   as

of

where

written as 

(22)

pro

The above equations can be  0 rs (kx )  rp0 (kx )  0  ts (kx ) t0p (kx ) 

 −1 0 1 0  0 − kkz 0 σr  iz  W0 =  1  0  1 −Gηo ηr kz σr 0 Gηo σr kiz ηr

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and

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ko k1z σr = k1 kiz r o µ η1 ηr = = µo η0

After solving the above matrix equations, the zeroth-order coefficients obtained for s and p-polarizations are given below 1 [(ηr − σr )(1 − ηr σr ) − (Gηo )2 ηr2 σr ]as δ(kx − kix ) D0 −1 [2Gηo ηr2 σr ]ap δ(kx − kix ) rps = D0 1 tss = [2ηr (1 + ηr σr )as ]δ(kx − kix ) D0 −1 [2Gηo ηr2 σr ap ]δ(kx − kix ) tps = D0 −1 rpp = [(ηr + σr )(1 − ηr σr ) + (Gηo )2 ηr2 σr ]ap δ(kx − kix ) D0 1 rsp = [2Gηo ηr2 σr ]as δ(kx − kix ) D0 1 tpp = [2ηr (ηr + σr )ap ]δ(kx − kix ) D0 1 tsp = [2Gηo ηr2 as ]δ(kx − kix ) D0

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rss =

8

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The denominator is defined as:

of

D0 = (ηr + σr )(1 + ηr σr ) + (Gηo )2 ηr2 σr

= −i

Z∞

+i

Z∞

f (x)rs0 (kx )kz eikx x dkx

−∞

Z∞

f (x)t0s (kx )k1z eikx x dkx +

−∞

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−∞

∂f =− ∂x −

Z∞

−∞

kz2 ko

f (x)eikix x dkx −

Jo i η1

−∞

−∞

(23)

Z∞

−∞

rp1 (kx )

−∞

Z∞

−∞

rp0 (kx )

kx ikx x e dkx ko

kz ikx x e dkx ko

t0p (kx )

2 k1z f (x)eikx x dkx k1

k1z ikx x e dkx k1

ikiz −ap f (x)eikix x − i ηo +

Z∞

kix t0p (kx ) eikx x dkx + i k1

t1p (kx )

−∞

Z∞

rs1 (kx )eikx x dkx

−∞

Z∞

Z∞

Z∞

t1s (kx )eikx x dkx

∂f kix ikix x ∂f ik 2 e + ap iz f (x)eikix x − ap ko ∂x ko ∂x

−i

+

re-

−as ikiz f (x)e

ikix x

pro

Putting the reflection and transmission coefficients in equation (5) and (7), the zeroth-order field is obtained.It is in accord with the expressions reported in [15, 16]. Balancing the first order terms of the equations (10−13), we get

Z∞

−∞

(24)

kz 1 rp0 (kx ) f (x)eikx x dkx − ηo ηo

t0p (kx )k1z f (x)eikx x dkx − iG

Z∞

−∞

9

Z∞

rp1 (kx )eikx x dkx

−∞

t0s (kx )k1z f (x)eikx x dkx

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Z∞

t1p (kx )eikx x dkx − G

Z∞

t1s (kx )eikx x dkx

−∞

−∞

1 ∂f − η1 ∂x −G

k1z k1 η1

−∞ Z∞

−∞ Z∞

kx ikx x i e dkx + ko η1

Z∞

kx t0s (kx ) f (x)eikx x dkx + iG k1

t0p (kx )

1 kx ikx x e dkx = k1 ηo

t1s (kx )ekx x dkx + G

−∞

−∞

t0s (kx )

−∞

Z∞

Z∞

Z∞

kz2 f (x)eikx x dkx ko

2 k1z f (x)eikx x dkx k1

t0p (kx )

−∞

rs1 (kx )

−∞

t1p (kx )

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+

∂f ∂x

−∞ Z∞

rs0 (kx )

rs0 (kx )

2 k1z f (x)ekx x dkx k1

re-

1 ∂f ηo ∂x

Z∞

Z∞

pro

ik 2 kix ∂f ikix x i −as iz f (x)eikix x + as e − ko ηo ko ηo ∂x ηo +

(25)

of

1 = η1

−∞

kiz ikx x e dkx ko

k1z f (x)ekx x dkx k1

(26)

Putting the zeroth-order reflection and transmission coefficients and taking the Fourier transform. The above equations can be written as:

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−rs1 (kx ) + t1s (kx ) = iL1 F (kx − kix ) k1z kz rp1 (kx ) − t1p (kx ) = iL2 F (kx − kix ) ko k1 1 1 rp1 (kx ) − Gt1s (kx ) + t1p (kx ) = iL3 F (kx − kix ) ηo η1 kz 1 k1z 1 k1z 1 r (kx ) + t (kx ) + G tp (kx ) = iL4 F (kx − kix ) ko ηo s k1 η s k1

10

(27) (28) (29) (30)

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where

re-

pro

of

t L1 = −as kiz + rsf (kx )kiz + tfs (kx )k1z 1 1 L2 = (ko2 − kx kix )ap − (ko2 − kx kix )rpf (kx ) ko ko 1 + (kx kix − k12 )tfp (kx ) k1 t kiz kiz f k1z t f L 3 = − ap − rp + tf − Gk1z ts (kx ) ηo ηo η1 p 1 1 (kx kix − ko2 )as + (kx kix − ko2 )rsf (kx ) L4 = ko ηo ko ηo 1 G + (k12 − kx kix )tfs (kx ) + (kx kix − k12 )tfp (kx ) k1 η1 k1

(31)

(32) (33)

(34)

In the above equations, the known zeroth order coefficients are sampled at kx = kix and they are given by 1 [(ηr − σri )(1 − ηr σri ) − (Gηo )2 ηr2 σri ]as D0i −1 [2Gηo ηr2 σri ap ] i D0 1 [2ηr (1 + ηr σri )as ] D0i −1 [2Gηo ηr2 σri ap ] i D0 −1 [(ηr + σri )(1 − ηr σri ) + (Gηo )2 ηr2 σri ]ap D0i 1 [2Gηo ηr2 σri as ] i D0 1 [2ηr (ηr + σri )ap ] D0i 1 [2Gηo ηr2 as ] D0i

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f rss = f rps =

tfss =

tfps =

f rpp =

f rsp =

tfpp =

Jo

tfsp =

where D0i is defined by D0i = (ηr + σri )(1 + ηr σri ) + (Gηo )2 ηr2 σri 11

Journal Pre-proof

and t ko k1z k1 kiz r o µ ηr = µo

of

σri =

where



re-

pro

The first order system of equations can be written in a matrix form    1  rs (kx ) L1   rp1 (kx )   −1  L2    1  ts (kx )  = iF (kx − kix )W1  L3  t1p (kx ) L4 −1  0 W1 =   0

kz ko 1 ηo

0

1 0 −G

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kz ηo ko

0

k1z η1 k1

0

−k1z k1 1 η1 G kk1z1

   

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By solving the above system of equations, the unknowns can be obtained as n 1 rs1 (kx ) = iF (kx − kix ) (1 + σr ηr )(L4 ηr − L1 σr ) D1 o 2 2 2 + ηr (L2 − L3 − L1 G η0 σr ) (35) n   1 rp1 (kx ) = iF (kx − kix ) L2 ηr 1 + G2 η0 ηr σr + L2 σr D1 o + ηr σr [(L1 + L4 )Gη0 ηr + L3 (ηr + σr )] (36) n o 1 1 iF (kx − kix ) ηr [(L1 + L4 )(1 + ηr σr ) + (L2 − L3 )Gσr ηr η0 ] ts (kx ) = D1 (37) n o 1 t1p (kx ) = iF (kx − kix ) ηr [(L1 + L4 )Gη0 ηr + (L3 − L2 )(ηr + σr )] (38) D1 where

  D1 = ηr σr ηr (1 + Q2 η02 ) + σr2 + (σr + ηr ) 12

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3

Results

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In this section, numerical results are presented for a flat as well as rough interface. Two models: ψ-model and γ-model defined by magneto-electric pseudo scalar ψ and surface admittance γ respectively are used for analysis. The zeroth-order solution is considered first which is the reflection and transmission due to flat TI interface shown in figure (2) for s-polarization. For comparison purpose, result for dielectric interface (θ = 0) is reported while θ = π represent the time reversal symmetry TI material. Since we use ψ-model for this simulation, it is considered that γ = 0. As incident angle increases, the co-polarized reflection coefficient approaches to 1 while the co-polarized transmission coefficient becomes zero. Figure (3) shows the results for p-polarization. The Brewster angle can be observed at φi = 58o for the dielectric case while it occurs at φi = 820 for TI (θ = π). Figures (4) and (5) show the results utilizing γ-model with ψ = 0. For s-polarization, both models give almost similar behavior while in case of ppolarization, the Brewster angle moves forward as γ is increased. This fact is not highlighted in the ψ-model. It is worth mentioning that only discrete values of θ are possible while γ can assume continuous values. Although, the two models are different practically yet γ-model can be considered a complete alternative to ψ model to study the scattering properties. Now, we present the numerical results for the dielectric-TI rough interface by selecting a Gaussian surface characterized by power spectral density W (kx ) [25], 2l kx c h2 lc W (kx ) = √ e−( 4 ) 2 π

(39)

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where lc and h are the correlation length and the root mean square (rms) height of the rough surface respectively. The bi-static scattering coefficients

13

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σ(kx ) for s-polarization are defined as 2 kz2 W (kx − kix ) rs1 (kx ) kiz 2 k2 σps (kx ) = z W (kx − kix ) rp1 (kx ) kiz 2 kz2 1 σT ss (kx ) = W (kx − kix ) ts (kx ) kiz 2 k2 σT ps (kx ) = z W (kx − kix ) t1p (kx ) kiz

pro

of

σss (kx ) =

(40) (41) (42) (43)

Jo

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re-

By using the ψ-model, bi-static scattering coefficients are evaluated and the result for s-polarization is shown in figure (6) while figure (7) shows the result for p-polarization. The co and cross-polarized incoherent bi-static scattering coefficients are plotted versus different values of φs . The scattered field from a TI rough interface is different from that of the dielectric rough interface. Now, we consider γ-model and the co and cross-polarized incoherent bistatic scattering coefficients are plotted in figure (8) for s-polarization. As γ is increased, the co-polarized scattering increases and cross-polarized scattering tends to decrease. In figure (9), p-polarization is reported and a similar trend is noted. To analyze the fact whether a further increase in γ gives the increasing scattered field or not, backscattering coefficients are plotted versus γ for s and p-polarization in figure (10). It is noted that as γ increases, the co-polarized scattering does not increase further.

14

0.4

pro

1

0.3

R0ps

0.8

R0ss

of

Journal Pre-proof

0.6

0.4

=0 =

0.2

0.1

0.2

re-

=0 =

0

0

20

40 i

60

(deg)

0.8

0

urn al P

(a)

80

20

40 i

60

(b)

0.4

=0 =

0.4

0.2

0 0

20

40 i

60

80

(deg)

(c)

=0 =

0.3

T0ps

T0ss

0.6

80

(deg)

0.2

0.1

0 0

20

40 i

60

80

(deg)

(d)

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Figure 2: Zeroth-order reflected and transmitted fields for different values of θ. (s-polarization)

15

0.4

=0 =

0.8

0.3

R0sp

0.6 0.4

=0 =

0.2

0.1

0

re-

0.2

0

0

20

40

60

(deg)

i

0.25 0.2

T0pp

0.15 0.1 0.05

0

urn al P

(a)

80

20

40 i

60

80

(deg)

(b)

0.4

0.3

T0sp

R0pp

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=0 =

0.2

0.1

=0 =

0 0

20

40 i

60

80

(deg)

(c)

0 0

20

40 i

60

(deg)

(d)

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Figure 3: Same as figure 2 except that p-polarization is considered

16

80

0.4

pro

1

R0ps

0.3

0.6 =0.001 =0.005 =0.01 =0.05

0.2 0

20

40 i

60

(deg)

0.8

0

80

0

0.4

0.2

0 0

20

40 i

60

80

(deg)

(c)

20

40 i

60

80

(deg)

(b)

0.4

=0.001 =0.005 =0.01 =0.05

0.6

0.2

0.1

urn al P

(a)

=0.001 =0.005 =0.01 =0.05

re-

0.4

=0.001 =0.005 =0.01 =0.05

0.3

T0ps

R0ss

0.8

T0ss

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0.1

0 0

20

40 i

60

80

(deg)

(d)

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Figure 4: Zeroth-order reflected and transmitted fields for different values of γ. (s-polarization)

17

0.4

=0.001 =0.005 =0.01 =0.05

0.8

R0sp

0.6

0.3

0.4

=0.001 =0.005 =0.01 =0.05

0.2

0.1

0

re-

0.2

0

0

20

40

60

(deg)

i

0.25 0.2 0.15

0

urn al P

(a)

80

T0pp

=0.001 =0.005 =0.01 =0.05

0.1 0.05 0 0

20

40 i

60

80

(deg)

(c)

20

40 i

60

80

(deg)

(b)

0.4 =0.001 =0.005 =0.01 =0.05

0.3

T0sp

R0pp

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0.2

0.1

0 0

20

40 i

60

(deg)

(d)

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Figure 5: Same as figure 4 except that p-polarization is considered

18

80

-15

pro

-10

[dB]

-20

1 ps

-30 -40 -50

20

40

60

(deg)

[dB] 1 Tss

-40 -50 -60 -70

80

-80 -60 -40 -20

urn al P

(a)

-30

re-

0 s

-20

-30

-40

-80 -60 -40 -20

=0 =

-80 -60 -40 -20

0

s

20

40

60

80

(deg)

(c)

0 s

20

40

60

80

40

60

80

(deg)

(b)

-12 -14

[dB]

-60

=0 =

-25

-35

=0 =

1 Tps

1 ss

[dB]

-20

-10

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=0 =

-16 -18 -20 -22 -24 -80 -60 -40 -20

0 s

20

(deg)

(d)

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Figure 6: Incoherent bi-static scattering coefficients for different values of θ, where h = 0.05, lc = 0.35 (s-polarization)

19

-15

pro

-10

-20

[dB]

1 sp

-30 =0 =

-40

40

60

(deg)

[dB] 1 Tpp

-60 -70 -80

80

-80 -60 -40 -20

urn al P

(a)

-50

re-

20

=0 =

-80 -60 -40 -20

0

s

20

40

60

80

(deg)

(c)

0 s

20

40

60

(b)

-20 -30 -40 -50 -60 =0 =

-70 -80 -80 -60 -40 -20

0 s

20

40

60

(deg)

(d)

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Figure 7: Same as figure 6 except that p-polarization is considered

20

80

(deg)

-10

[dB]

0 s

-40

=0 =

-40

-80 -60 -40 -20

-30

-30 -35

-50

-20

-25

1 Tsp

1 pp

[dB]

-20

-10

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-10

pro

0

-20

[dB]

-20

1 ps

-30 =0.001 =0.005 =0.01 =0.05

-50

-40

=0.001 =0.005 =0.01 =0.05

-50 -60

-80 -60 -40 -20

0 s

20

40

60

(deg)

0 -20 -40 -60 -80 -100

-80 -60 -40 -20

urn al P

(a)

80

=0.001 =0.005 =0.01 =0.05

-80 -60 -40 -20

0

s

20

40

60

80

(deg)

(c)

0 s

20

40

60

80

40

60

80

(deg)

(b)

-10 -15

[dB]

-60

[dB]

-30

re-

-40

1 Tps

1 ss

[dB]

-10

1 Tss

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=0.001 =0.005 =0.01 =0.05

-25 -30 -35 -80 -60 -40 -20

0 s

20

(deg)

(d)

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Figure 8: Incoherent bi-static scattering coefficients for different values of γ, where h = 0.05, lc = 0.35 (s-polarization)

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-10

pro

0

[dB]

-20

1 sp

-20 =0.001 =0.005 =0.01 =0.05

-40

-40

=0.001 =0.005 =0.01 =0.05

-50 -60

-80 -60 -40 -20

0 s

20

40

60

(deg)

0 -20 -40 -60 -80 -100 -120

-80 -60 -40 -20

urn al P

(a)

80

=0.001 =0.005 =0.01 =0.05

-80 -60 -40 -20

0

s

20

40

60

80

(deg)

(c)

0 s

20

40

60

80

40

60

80

(deg)

(b)

0

-20

[dB]

-50

[dB]

-30

re-

-30

1 Tsp

1 pp

[dB]

-10

1 Tpp

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-60

-80 -80 -60 -40 -20

0 s

20

(deg)

(d)

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Figure 9: Same as figure 8 except that p-polarization is considered

22

-15

pro

-5

-20

[dB]

-15

-25

s cross

[dB]

sp

-20

-10

s co

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-30

ps

-35

-25

-40

re-

ss

pp

-30

-45

0

0.02

0.04

0.06

0.08

-10

-50 -60 -70 0

0.02

0.04

0.06

0.08

0.1

(c)

0.04

0.06

0.08

0.1

(b)

T sp T ps

-25

[dB]

-40

0.02

-20

-30

T cross

-30

[dB]

0

T ss T pp

-20

T co

urn al P

(a)

0.1

-35 -40 -45 0

0.02

0.04

0.06

(d)

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Figure 10: First order backscattering coefficients for different values of γ

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Conclusions

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A TI rough interface characterized by magneto-electric pseudoscalar ψ and admittance γ is considered to observe the scattering pattern. The reflection and transmission properties of the flat TI interface are studied using zeroth order coefficients obtained by the perturbation theory. It is noted that the location of the Brewster angle depends upon the value of ψ/γ. The cross polarized reflection becomes small as the value of γ is increased. In γ-model, first order co-polarized scattered field increases while the cross-polarized scattered field decreases for large value of γ. This is also confirmed by observing the backscattered field as a function of γ. Since the scattering properties are completely specified by using the γ-model, it is an alternative to the ψ model and should be used to study the scattering problems involving TI.

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References

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[1] M. Z. Hasan and C. L. Kane, Topological insulators, Rev. Mod. Phys. 82(4), 3045-3067, 2010.

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[2] M. C. Chang and M. F. Yang, Optical signature of topological insulators, Phys. Rev. B 80(11), 113304, 2009. [3] L. Fu and C. L. Kane, Topological insulators with inversion symmetry, Phys. Rev. B 76(4), 045302, 2007.

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[4] B. A. Bernevig, T. L. Hughes and S. C. Zhang, Quantum spin Hall effect and topological phase transition in HgTe quantum wells, Science 314, 1757-1761, 2006. [5] M. K¨onig, S. Wiedmann, C. Br¨ une, A. Roth, H. Buhmann, L. W. Molenkamp, X. L. Qi and S. C. Zhang, Quantum spin Hall insulator state in HgTe quantum wells, Science 318, 766-770, 2007.

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[6] R. Roy, Topological phases and the quantum spin Hall effect in three dimensions, Phy. Rev. B 79(19), 195322, 2009. [7] C. X. Liu, X. L. Qi, X. Dai, Z. Fang and S. C. Zhang, Quantum Anomalous Hall Effect in Hg1−y M ny T e Quantum Wells, Phys. Rev. Lett. 101, 146802, 2008. [8] X. L. Qi, T. L. Hughes and S. C. Zhang, Topological field theory of time-reversal invariant insulators, Phys. Rev. B 78, 195424, 2008. [9] X. L. Qi and S. C. Zhang, The quantum spin Hall effect and topological insulators, Physics Today 63, 33-38, 2010. [10] F. Liu, J. Xu and Y. Yang, Polarization conversion of reflected electromagnetic wave from topological insulator, J. Opt. Soc. Am. B 31(4), 735741, 2014.

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[11] W. K. Tse and A. H. MacDonald, Magneto-optical Faraday and Kerr effects in topological insulator films and in other layered quantized Hall systems, Phys. Rev. B 84(20), 205327, 2011.

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[12] S. Shoukat and Q. A. Naqvi, Scattering from electromagnetic plane wave from a perfect electric conducting strip placed at interface of topological insulator, Optics Communications 381, 77-84, 2016.

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[13] A. Lakhtakia and T. G. Mackay, Classical electromagnetic model of surface states in topological insulators, Journal of Nanophotonics 10, 033004, 2016. [14] L. Zeng, R. Song and X. Jian, Scattering of electromagnetic radiation by a time reversal perturbation topological insulator circular cylinder, Mod. Phys. Lett. B 27, 1350098, 2013.

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[15] S. O. Rice, Reflection of electromagnetic waves from slightly rough surfaces, Comm. Pure Appl. Math. 4, 351-378, 1951.

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[16] M. Sanamzadeh, L. Tsang, J. T. Johnson, R. J. Burkholder and S. Tan, Scattering of electromagnetic waves from 3D multilayer random rough surfaces based on the second-order small perturbation method: energy conservation, reflectivity, and emissivity, J. Opt. Soc. Am. A 34(3), 395409, 2017. [17] J. T. Johnson, Third-order small-perturbation method for scattering from dielectric rough surfaces, J. Opt. Soc. Am. A 16(11), 2720-2736, 1999. [18] L. Guo and Z. Wu, Application of the extended boundary condition method to electromagnetic scattering from rough dielectric fractal sea surface, J. Electromagn. Waves and Appl. 18(9), 1219-1234, 2004. [19] P. C. Waterman, Scattering by periodic surfaces, J. Aco. Soc. Am. 57(4), 791-802, 1975. [20] S. L. Chuang and J. A. Kong, Wave scattering and guidance by dielectric waveguides with periodic surfaces, J. Opt. Soc. Am. 73(5), 669-679, 1983.

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[21] Z. S. Wu, J. J. Zhang, and L. Zhao, Composite electromagnetic scattering from the plate target above a one-dimensional sea surface: Taking the diffraction into account, Progress In Electromagnetics Research 92, 317-331, 2009. 26

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[23] E. Thorsos, The validity of the Kirchhoff approximation for rough surface scattering using a Gaussian roughness spectrum, J. Aco. Soc. Am. 83(1), 78-92, 1988. [24] M. F. Chen and A. K. Fung, A numerical study of the regions of validity of the Kirchhoff and small-perturbation rough surface scattering models, Radio Science. 23(2), 163-170, 1988.

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[25] L. Tsang, J. A. Kong and K. H. Ding, Scattering of Electromagnetic Waves: Theories and Applications, John Wiley & Sons Inc., 2000.

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