Modeling Magnetized Graphene in the Finite-Difference Time-Domain ...

This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TAP.2017.2768081, IEEE Transactions on Antennas and Propagation IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. XX, NO. XX, XX 2017

1

Modeling Magnetized Graphene in the Finite-Difference Time-Domain Method using an Anisotropic Surface Boundary Condition Mina Feizi, Student Member, IEEE, Vahid Nayyeri, Senior Member, IEEE, and Omar M. Ramahi, Fellow, IEEE

Abstract—A cost effective approach to the finite-difference time-domain (FDTD) modeling of magnetized graphene sheets as a dispersive anisotropic conductive surface is proposed. We first introduce a novel method for implementation of anisotropic conductive surface boundary condition in the FDTD method. Then, by applying the surface conductivity matrix of magnetized graphene, we present modeling magnetized graphene as an infinitesimally thin conductive sheet in the FDTD method. The applicability, accuracy, and stability of the method are demonstrated through numerical examples. The proposed approach is validated by comparing the results with existing analytic solution. Index Terms—anisotropic conductive sheet, finite-difference time-domain (FDTD) method, magnetized graphene, surface boundary condition.

I. I NTRODUCTION

R

ECENTLY, there has been great interests for electromagnetic (EM) simulation of graphene due to its various applications from microwaves to the terahertz regime [1]– [3]. Basically, in EM theory, graphene, which is a planar monoatomic layer of carbon, is modeled as an infinitesimallythin, two-sided sheet characterized by a surface conductivity [4]. Accordingly, two approaches have been applied for EM simulation of graphene: 1) boundary condition approach and 2) volumetric approach [5]. In the first, the infinitesimallythin conductive sheet is modeled as a boundary condition at two sides of the sheet using the conductive surface boundary condition (CSBC) [6]–[9]. Applying CSBC in the boundary element method (BEM) which constructs a 2-D mesh over surfaces is straightforward [6]–[8]. However, considering that the applicability of the BEM is restricted to the problems for which Green’s functions can be formulated, for many problems, BEM is not applicable or it is significantly less efficient than numerical methods based on volumetric discretization such as finite-element methods, finite-difference methods, and finite-volume methods. The implementation of the CSBC in the volumediscretization methods requires special modification to the algorithm. In the volumetric modeling approach proposed by Vakil and Engheta [10], the graphene sheet is considered as a very thin layer (with thickness around 1 nm) and the M. Feizi and V. Nayyeri are with the Antenna and Microwave Research Laboratory, Iran University of Science and Technology, Tehran, Iran (e-mail: [email protected], [email protected] O. M. Ramahi is with the Department of Electrical and Computer Engineering, University of Waterloo, ON, Canada (e-mail: [email protected])

surface conductivity of graphene is converted to volumetric conductivity. Due to the simplicity of the implementation and its applicability to most numerical methods and also to commercial EM solvers, this technique has been commonly used in most recent works. In this approach the space inside the graphene layer is finely meshed requiring significant memory resources and computation time. This is most pronounced in the finite-difference time-domain (FDTD) method [11], [12] where fine spatial mesh calls for extremely fine time discretization to guarantee stability. Some techniques have been proposed to improve the efficiency of volumetric modeling of graphene in the FDTD method such as applying non-uniform mesh [13], thin layer formulation [14], [15], and implicit unconditionally stable algorithms [16]–[18]. Several types of impedance boundary conditions have been implemented in the FDTD method [19]–[26]. In 2013, Nayyeri et al., proposed modeling of conductive infinitesimally thin sheets in the FDTD method by splitting the magnetic fields tangential to the sheet and applying the CSBC [9]. Subsequently, this technique has been applied for modeling of graphene in the FDTD method [9], [27]. The main advantage of modeling graphene as a infinitesimally thin surface in the FDTD method (i.e., the CSBC approach) is to avoid meshing the graphene layer. Therefore, this approach eliminates the use of very fine FDTD cell size and consequently the need to have extremely fine time steps. In an alternative approach, the surface conductivity of graphene is represented by a Dirac delta function [28]. Then, by applying an integral form of Ampere law along a path enclosing the graphene sheet, the updating equation for the tangential electric fields were obtained. However, this approach leads to transforming the surface conductivity of the graphene into an equivalent volumetric conductivity (or permittivity) in a way similar to the volumetric modeling approach. Magnetized graphene, which is basically graphene sheets biased by a magneto-static field, behaves as anisotropic conducting sheet characterized by a surface conductivity matrix [29]. The magnetically biased graphene shows gyrotropy and nonreciprocity, which make it applicable for design and realization of components such as circulators and isolators [30], [31]. FDTD modeling of magnetized graphene has been addressed in earlier works [32], [33]. In [32], a FDTD approach was developed by transforming the surface conductivity of graphene into an equivalent volumetric tensor and implementing it by using the auxiliary differential equation

0018-926X (c) 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TAP.2017.2768081, IEEE Transactions on Antennas and Propagation IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. XX, NO. XX, XX 2017

anisotropic conductive sheet

x

Ex Ey

z

K

H x

H x

H y

H y

K 1 2

Ex Ey K 1

Fig. 1. 1-D FDTD cell including a anisotropic conducting sheet at grid K + 12

(ADE) and the matrix exponential method. The volumetric discretization of the graphene sheet requires fine mesh and increases the computation time [34], [35]. Equivalent lumped elements circuits were used to model graphene sheet in the FDTD [33]; however, the stability of this method required much smaller time steps than those in the classical FDTD method. In this work we first present a general approach for implementation of an anisotropic CSBC in the FDTD method. By applying the surface conductivity of graphene, we model magnetized graphene in the FDTD method as an infinitesimally thin sheet. The proposed method has the key advantage of providing direct implementation of the CSBC in the FDTD method, thereby avoiding the need for fine spatial discretization of the graphene sheet. As a consequence, the method achieves long-term numerical stability without any particular restriction on the time step. This unique advantage makes the proposed method highly suitable for modeling highly resonant structures involving graphene sheets where long-term simulation is needed for accurate resonance capture. Validation of the method is presented using numerical examples and comparison to analytic solution where available. II. I MPLEMENTATION OF A NISOTROPIC C ONDUCTING S URFACE B OUNDARY C ONDITION IN T HE FDTD M ETHOD A. Implementation in 1-D FDTD Algorithm Let us assume an anisotropic conducting sheet located parallel to the xy plane at the center of a 1-D FDTD cell, at grid K + 21 as shown in Fig. 1. The surface conductivity of the sheet is given by   σd −σo ¯ σs = , (1) σo σd where σd and σo are constants. As shown in Fig. 1, similar to the Yee’s FDTD algorithm, the electric and magnetic fields are located at the grids with integer (i.e., K and K+1) and integerand-a-half (i.e., K + 21 ) indexes, respectively. However, due to the possibility of a surface current on the conductive sheet, unlike the conventional 1-D FDTD method, two magnetic fields with superscripts − and + immediately to the left and to the right of the sheet are considered to account for different surface currents on both sides of the sheet. Furthermore, unlike a conventional 1-D FDTD cell that includes only a pair of (Ex , Hy ) or (Ey , Hx ) (because Maxwell’s equation for a 1D problem can be decomposed into two separate cases, one

2

includes only Ex , and Hy and the other includes Ey and Hx ), the cell shown in Fig. 1 includes both pairs because an anisotropic surface with non-diagonal elements provides a full coupling between the fields components. To derive updating equations for the magnetic field components at the sheet boundaries, Maxwell equation representing Faraday law, namely ∂B/∂t = −∇ × E, is discretized at the boundary grid K + 12 in a way that the central difference ∂ scheme is used for the time derivatives ( ∂t ) and the backward and forward difference schemes are used for the spatial deriva∂ ) on the left and right sides of tives along the z direction ( ∂z the interface, respectively:   o n (n) (n) = δzb Ey K+ 1 , Hx− ( 2)   n o (n) (n) c + f µ2 δt Hx = δz Ey K+ 1 , ( 2)   n o (n) c − (n) b µ1 δt Hy = −δz Ex K+ 1 , ( 2)   n o (n) (n) µ2 δtc Hy+ = −δzf Ex K+ 1 . ( 2) µ1 δtc

(2)

In (2), µ1 and µ2 are the permeability of the media to the left and right sides of the sheet, respectively, and δ b , δ c , and δ f are the backward, central, and forward difference operators, respectively, defined as F (x) − F (x − ∆x/2) , ∆x/2 F (x + ∆x/2) − F (x − ∆x/2) δxc {F } = , ∆x F (x + ∆x/2) − F (x) δxf {F } = . ∆x/2 δxb {F } =

The electric field components at the sheet boundary (grid K + 21 ) are required in (2); however, according to Fig. 1, they are not located at this grid. To obtain these components, the anisotropic conducting sheet boundary condition,   − ¯ ˆ z × H+ t − Ht = Js = σs . Et

(3)

is applied where ˆ z is the unit vector normal to the sheet, − H+ and H are the tangential magnetic fields immediately t t to the right and left sides of the conductive sheet, Js is the surface current density on the conductive sheet and Et is the tangential electric field at the interface. Substituting (1) in (3) and decomposing the equation, we have, 

Hy− − Hy+ Hx+ − Hx−



 =

σd σo

−σo σd

"

Ex (K+ 1 ) 2 Ey (K+ 1 )

# .

(4)

2

Rearranging (4) and enforcing the equation at the nth time

0018-926X (c) 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TAP.2017.2768081, IEEE Transactions on Antennas and Propagation IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. XX, NO. XX, XX 2017

3

step, the electric field components are obtained as

anisotropic conductive sheet

i, j, K 1

Ex

(n) (K+ 12 )

E field

= n+ 1 Hx+ ( 2 )

σo d

+ 2

Hx− (

n+ 12

− 

σd  Hy+ − d

x

n− 12

)

!

(

Fig. 2. A 3-D FDTD cell including an anisotropic conducting sheet at grid K + 12

) + H + (n− 12 ) y

2 Hy− (

+ Hy− ( 2

n− 12 )

and

 ,

n+ Hx+ ( 2 ) 1

(n− 12 ) (n− 12 ) + c2 Hx+ − c1 Hy− (n− 12 ) (n) + c1 Hy+ − 2 Ey (K) ,

= c5 Hx−

(n)

= c2 Hx−

+ 2

(n)

= c1 Hx−

n+ 12 )

−

(n)

= − c1 Hx−

F2

n− 1 Hx+ ( 2 )

Hx− (

+ Hx− ( 2

n− 12 )

!

F3

 (n+ 12 ) + H + (n− 21 ) σo  Hy+ y + d 2 Hy− (

n+ 12 )

−

F4

+ Hy− ( 2

n− 12 )

(n− 12 )

(n)

F1 (5)

(n− 21 )

(n− 12 ) (n− 12 ) + c6 Hx+ + c1 Hy− (n− 12 ) (n) − c1 Hy+ + 2 Ey (K+1) , (n− 12 ) (n− 12 ) − c1 Hx+ + c5 Hy− (n− 12 ) (n) + c2 Hy+ + 2 Ex (K) ,



(n− 12 )

(n− 12 ) (n− 12 ) + c1 Hx+ + c2 Hy− (n− 12 ) (n) + c6 Hy+ − 2 Ey (K+1) ,

and

,

where d = σd 2 + σo 2 is the determinant of σ¯s . Notice that in (5), the magnetic field components at the time step n are interpolated by their values at n + 21 and n − 12 . Finally, substituting (5) in (2), the following system of equations is obtained,

σo , d µ1 ∆z + c2 ∆t c3 = , ∆t µ1 ∆z − c2 ∆t c5 = , ∆t

σd , d µ2 ∆z + c2 ∆t c4 = , ∆t µ2 ∆z − c2 ∆t c6 = . ∆t

(n+ 12 )

Hx+

(n+ 21 ) −1

¯ C 1

1

(n)

[F1

1

(n)

F2

(n)

F3

(n)

F4 ]T

(6)

where

c2 =

(9)

Solving (6) for Hx− , Hx+ , Hy− , and Hy+ at time step n + 12 , the updating equations for the magnetic fields at grid K + 21 are obtained as [Hx−

¯ . [H − (n+ 2 ) H + (n+ 2 ) H − (n+ 2 ) H + (n+ 2 ) ]T = C x x y y

(8)

(n− 21 )

c1 =

1

y x

i 1, j, K 

 i, j , K 

= σd d

z

2

n+ 21

− (n) (K+ 12 )

i, j 1, K 

) + H −(

n+ 12 )

Ey

H field

n− 1 Hx+ ( 2 )

Hy−

(n+ 12 )

(n)

. [F1

Hy+ (n)

F2

(n+ 12 ) (n)

]T =

(n)

(10)

F3 F4 ]T

By having the magnetic field components at the boundary grid, the electric field components at grids K and K + 1 are updated by applying the classic Yee’s algorithm and using the magnetic field components at the left and right sides of the sheet, respectively. B. Implementation in 3-D FDTD Algorithm



c3  −c2 ¯  C= −c1 c1

−c2 c4 c1 −c1

c1 −c1 c3 −c2

 −c1 c1  , −c2  c4

(7)

Fig. 2 shows a 3-D FDTD cell including an anisotropic conducting sheet positioned parallel to the xy-plane at the z grid K + 12 . As shown in the figure, Hx , Hy , and Ez are defined immediately to the bottom and top sides of the sheet. Moreover, in addition to the filed components defined in the classical 3-D FDTD cell, this cell includes some auxiliary field

0018-926X (c) 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TAP.2017.2768081, IEEE Transactions on Antennas and Propagation IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. XX, NO. XX, XX 2017

components at the interface which are Hx at i + 12 , j, K +
1 1 and Hy at i, j + 2 , K + 2 .

1 2




4

is obtained where F1 , F2 , F3 , and F4 are defined as (n)

F1 To derive the updating equations for the magnetic field components at the top and bottom of the sheet, in a similar way to the 1-D case, we discretize ∂B/∂t = −∇ × E in a way that the central difference scheme is used for the spatial derivatives along x- and y-directions and for the time derivative. On the other hand, the backward and forward difference schemes are used for the spatial derivatives along the z direction on the bottom and top sides of the
interface, respectively. For instance, at grid i + 21 , j, K + 12 we have

(n− 12 ) (n− 1 ) + c2 Hx+ i+ 12,j,K+ 1 (i+ 12 ,j,K+ 12 ) ( 2 2) 1 1 (n− ) (n− ) − c1 Hy− i+ 12,j,K+ 1 + c1 Hy+ i+ 12,j,K+ 1 ( 2 ( 2 2) 2) ∆z (n) − 2 Ey i+ 1 ,j,K + × ( 2 ) ∆y i h (n) (n) Ez− (i+ 1 ,j+ 1 ,K+ 1 ) − Ez− (i+ 1 ,j− 1 ,K+ 1 ) , = c5 Hx−

2

(n) F2

=

2

2

2

(n− 1 ) c2 Hx− i+ 12,j,K+ 1 ( ) 2

−

(n− ) c1 Hy+ i+ 12,j,K+ 1 ( )

2

1

+

2

2

1

2

(n) 2Ey i+ 1 ,j,K+1 ( 2 )

2

+

2

(n− ) c1 Hy− i+ 12,j,K+ 1 ( )

2

(n− 1 ) c6 Hx+ i+ 12,j,K+ 1 ( ) 2

2

∆z + × ∆y

+ h i (n) (n) Ez+ (i+ 1 ,j+ 1 ,K+ 1 ) − Ez+ (i+ 1 ,j− 1 ,K+ 1 ) , 2

(n) F3

2

2

2

=

(n− 1 ) c1 Hx− i+ 12,j,K+ 1 ) (

+

(n− ) c5 Hy− i+ 12,j,K+ 1 ) (

2

+

(n− ) c2 Hy+ i+ 12,j,K+ 1 ) (

2

2

2

(12)

2

1

2

(n) Ex i+ 1 ,j,K ) ( 2

2

−

1

o n (n) µ1 δtc Hx− (i+ 1 ,j,K+ 1 ) = 2 2   o n (n) (n) b δz Ey i+ 1 ,j,K+ 1 − δyc Ez− (i+ 1 ,j,K+ 1 ) , ( 2 2 2 2) n o (n) c + µ2 δt Hx (i+ 1 ,j,K+ 1 ) = 2 2   n o (n) (n) f − δyc Ez+ (i+ 1 ,j,K+ 1 ) , δz Ey i+ 1 ,j,K+ 1 ( 2 2 2 2)   (11) (n) µ1 δtc Hy− i+ 1 ,j,K+ 1 = ( 2 2)   n o (n) (n) b − δz Ex i+ 1 ,j,K+ 1 + δxc Ez− (i+ 1 ,j,K+ 1 ) , ( 2 2 2 2)   (n) = µ2 δtc Hy+ i+ 1 ,j,K+ 1 ( 2 2)   o n (n) (n) − δzf Ex i+ 1 ,j,K+ 1 + δxc Ez+ (i+ 1 ,j,K+ 1 ) , ( 2 2 2 2)

2

(n− 1 ) c1 Hx+ i+ 12,j,K+ 1 ) ( 2

2

∆z + × ∆x

+2 h i (n) (n) Ez− (i+1,j,K+ 1 ) − Ez− (i,j,K+ 1 ) , 2

(n) F4

2

(n− ) −c1 Hx− i+ 12,j,K+ 1 ( )

(n− 12 ) i+ 12 ,j,K+ 12 ) ( 2 2 1 n− n− ( 1) ( ) + c2 Hy− i+ 12,j,K+ 1 + c6 Hy+ i+ 12,j,K+ 1 ) ( 2 ( 2 2 2) ∆z (n) × − 2 Ex i+ 1 ,j,K+1 + ( 2 ) ∆x h i 1

=

(n)

+ c1 Hx+

(n)

Ez+ (i+1,j,K+ 1 ) − Ez+ (i,j,K+ 1 ) . 2

2

Notice that, in (12), Ez+ and Ez− at the
grids
i + 12 , j ± 21 , K
+ 12 , and Ey at the grids i + 12 , j, K and i + 12 , j, K + 1 are observed but they are not defined in the cell shown in Fig. 2. Therefore, a second-order spatial interpolation is used to approximate these components [36],
[37]; for instance, Ez+ at i + 12 , j + 12 , K + 12 is substituted by (n)

where µ1 and µ2 are the permeability of the media above and below the sheet, respectively. The field components to the bottom and top sides of the sheet are identified by superscripts − and + , respectively. In (11), Ex and Ey at the grid
i + 12 , j, K + 21 are required; however these components are not located at that grid in the cell shown in Fig. 2. Therefore, by applying the boundary condition (3), they are expressed in
terms of Hx− , Hx+ , Hy− , and Hy+ , all at grid i + 12 , j, K + 12 using updating equations similar to (5) (not provided here for brevity). Then, by following a procedure similar to that used in the 1-D formulation, a system of equations, identical to (6),

Ez+ (i+ 1 ,j+ 1 ,K+ 1 ) = 2 2 2 h 1 (n) (n) (13) × Ez+ (i,j,K+ 1 ) + Ez+ (i+1,j,K+ 1 ) 2 2 4 i (n) (n) +Ez+ (i,j+1,K+ 1 ) + Ez+ (i+1,j+1,K+ 1 ) 2

2

Finally the magnetic field components at the grid
i + 21 , j, K + 21 are updated using an equation identical to (10). In a similar way, a same system of equations can be obtained for updating Hx− , Hx+ , Hy− , and Hy+ at the
1 1 grid i, j + 2 , K + 2 . Once the magnetic field components at both sides of the sheet are obtained, they can be used for updating Ez− and Ez− by applying the classic leapfrog

0018-926X (c) 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TAP.2017.2768081, IEEE Transactions on Antennas and Propagation IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. XX, NO. XX, XX 2017

algorithm, ∆t (n+1) (n) Ez− (i,j,K+ 1 ) = Ez− (i,j,K+ 1 ) + × 2 2 1      1 1 c − (n+ 2 ) c − (n+ 2 ) − δy Hx i,j,K+ 1 , δx Hy i,j,K+ 1 ( ( 2) 2) (14) ∆t (n+1) (n) Ez+ (i,j,K+ 1 ) = Ez+ (i,j,K+ 1 ) + × 2 2  1     1 (n+ 12 ) c + (n+ 2 ) − δyc Hx+ i,j,K+ , δx Hy i,j,K+ 1 1 ( ( 2) 2) where 1 and 2 are the permittivity of the media above and below the sheet, respectively. It is worth mentioning that the proposed method only changes field updating equations at the boundary grids. Yet, a classical FDTD algorithm is applied on the fields components at other grids. III. A PPLICATION TO M ODELING M AGNETIZED G RAPHENE S HEETS An effective technique to incorporate the physics of magnetically biased graphene in electromagnetic wave scattering problem at the macroscopic scale is to replace the graphene with an anisotropic conducting sheet with a frequency dispersive surface electric conductivity matrix given by [4]   σd (ω) −σo (ω) ¯ σs (ω) = (15) σo (ω) σd (ω) where σd (ω) and σo (ω) are derived from the Drude model [38] σd (ω) = σ0 σo (ω) = σ0

1 + jωτ 2

2

(ωc τ ) + (1 + jωτ ) ωc τ

(16)

2 2. (ωc τ ) + (1 + jωτ )

where   2e2 τ kB T µc σ0 = ln 2 cosh , 2kB T π¯h2

(17)

τ is the phenomenological scattering time, ωc = eB0 vF2 /µc is the cyclotron frequency, B0 is the amplitude of the static magnetic field bias, e is the electron charge, kB is the Boltzmann constant, T is the room temperature, vF is the Fermi velocity, µc is the chemical potential, and ¯h is the reduced Plancks constant. Using the surface conductivity matrix, σ¯s , the electromagnetic boundary condition at the magnetized graphene sheet can be written in the frequency domain as   − ¯ n ˆ × H+ (18) t (ω) − Ht (ω) = Js (ω) = σs (ω) . Et (ω) where n ˆ is the unit vector normal to the graphene sheet, H+ t − and Ht are the tangential magnetic fields at two sides of the sheet, Js is the surface current density on the graphene, and Et is the tangential electric field at the sheet. To incorporate the graphene sheet into the FDTD method, we consider a graphene sheet positioned parallel to the xyplane and biased by a magneto-static field, B0 , directed in

5

the z-direction. Substituting (15) in (18) and decomposing the equation, we have   1  ωc τ Hx+ (ω) − Hx− (ω) Ex (ω) = σ0   − (1 + jωτ ) Hy+ (ω) − Hy− (ω) , (19)   1  Ey (ω) = (1 + jωτ ) Hx+ (ω) − Hx− (ω) σ0   + ωc τ Hy+ (ω) − Hy− (ω) ,

where (Ex , Ey ), Hx− , Hy− and Hx+ , Hy+ are the com+ ponents of Et , H− t and Ht , respectively. Converting (19) ∂ A(t), then into time-domain equations using jωA(ω) → ∂t discretizing the time-domain equations, and subsequently enforcing the discrete time-domain equations at the discrete time step n, we have ( " n+ 1 n− 1 Hx+ ( 2 ) + Hx+ ( 2 ) 1 (n) ωc τ Ex = σ0 2 # n+ 1 n− 1 Hx− ( 2 ) + Hx− ( 2 ) − 2  n+ 1 n− 1 Hy+ ( 2 ) + Hy+ ( 2 )  − 2  n− 1 n+ 1 Hy− ( 2 ) + Hy− ( 2 )  − 2  n− 1 n+ 1 Hy+ ( 2 ) − Hy+ ( 2 ) −τ ∆t  n− 1 n+ 1 Hy− ( 2 ) − Hy− ( 2 )   , −  ∆t (20) (" 1 1 + (n− 2 ) + (n+ 2 ) + H 1 H x x Ey(n) = σ0 2 # n+ 1 n− 1 Hx− ( 2 ) + Hx− ( 2 ) − 2 " n+ 1 n− 1 Hx+ ( 2 ) − Hx+ ( 2 ) +τ ∆t # n+ 1 n− 1 Hx− ( 2 ) − Hx− ( 2 ) − ∆t  1 n+ n− 1 Hy+ ( 2 ) + Hy+ ( 2 )  + ωc τ 2  n+ 1 n− 1 Hy− ( 2 ) + Hy− ( 2 )   , −  2 where the magnetic fields at time step n are approximated by their values at n + 12 and n − 21 . The updating equations for the fields components at two sides of the graphene sheet can be straightforwardly derived by applying the boundary condition (20) (instead of (5)) in the

0018-926X (c) 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TAP.2017.2768081, IEEE Transactions on Antennas and Propagation IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. XX, NO. XX, XX 2017

FDTD Analytic



−c1 c4 + c1 c5 −c5

c3 + c1  ¯ =  −c1 C  −c5 c5

c5 −c5 c3 + c1 −c1

 −c5  c5 , −c1  c4 + c1

(21)

tra [V/m] Etra x [V/m] E x

0.3 0

0

-0.3 -0.6 0

-0.1 1

2

3

4

Time [psec]

and (n)

F1

1 (n− 12 ) + (n− 2 ) + c H 2 1 1 1 x (i+ 2 ,j,K+ 2 ) (i+ 2 ,j,K+ 12 ) 1 1 (n− ) (n− ) − c5 Hy− i+ 12,j,K+ 1 + c5 Hy+ i+ 12,j,K+ 1 ( 2 ( 2 2) 2) ∆z (n) − 2c6 Ey i+ 1 ,j,K − c6 × ) ( 2 ∆y i h (n) (n) Ez− (i+ 1 ,j+ 1 ,K+ 1 ) − Ez− (i+ 1 ,j− 1 ,K+ 1 ) ,

= (c3 − c2 ) Hx−

2

(n)

F2

(n)

2

2

2

2

( ) ( ) + (c4 − c2 ) Hx+ ( ) ( (n− 1 ) (n− 1 ) + c5 Hy− i+ 12,j,K+ 1 − c5 Hy+ i+ 12,j,K+ 1 ( 2 ( 2 2) 2) ∆z (n) + 2c6 Ey i+ 1 ,j,K+1 − c6 × ( 2 ) ∆y i h (n) (n) Ez+ (i+ 1 ,j+ 1 ,K+ 1 ) − Ez+ (i+ 1 ,j− 1 ,K+ 1 ) , = c2 Hx−

2

F3

2

n− 12 i+ 12 ,j,K+ 21

2

n− 12 i+ 12 ,j,K+ 12

2

2

2

)

(n)

Ez− (i+1,j,K+ 1 ) − Ez− (i,j,K+ 1 ) ,

(n) F4

2

(n− ) −c5 Hx− i+ 12,j,K+ 1 ( )

(n− 21 ) (i+ 21 ,j,K+ 12 ) 2 2 1 (n− ) (n− 1 ) + c2 Hy− i+ 12,j,K+ 1 (c4 − c2 ) Hy+ i+ 12,j,K+ 1 ( 2 ( 2 2) 2) ∆z (n) − 2c6 Ex i+ 1 ,j,K+1 + c6 × ( 2 ) ∆x i h 1

=

(n)

+ c5 Hx+

(n)

Ez+ (i+1,j,K+ 1 ) − Ez+ (i,j,K+ 1 ) , 2

2

(22) and c1 = ∆t + 2τ c3 = µ1 σ0 ∆z c5 = ωc τ ∆t

Fig. 3. The x- and y- components of the transmitted electric field through a magnetically biased infinite graphene sheet illuminated by an x-polarized electric field having a Gaussian waveform.

IV. M ODEL VALIDATION AND S IMULATION R ESULTS As a first example, we simulate plane-wave transmission through a magnetically biased infinite graphene sheet with µc = 0.5 eV and τ = 0.5 ps at T = 300◦ K. An xpolarized plan wave normally incident on a graphene sheet placed parallel to the xy-plane and biased by a static magnetic field, B0 . The temporal form of the incident field is a Gaussian waveform described by 2

/β 2

(24)

2

1 (n− 12 ) + (n− 2 ) − c H 5 1 1 1 x (i+ 2 ,j,K+ 2 ) (i+ 2 ,j,K+ 12 ) 1 (n− ) (n− 1 ) + (c3 − c2 ) Hy− i+ 12,j,K+ 1 + c2 Hy+ i+ 12,j,K+ 1 ( 2 ( 2 2) 2) ∆z (n) + 2c6 Ex i+ 1 ,j,K + c6 × ) ( 2 ∆x i h 2

5 -12 x 10

(a)

Exinc (t) = e−π(t−t0 )

= c5 Hx−

(n)

0.1

tra

0.6

E [V/m] Eyytra [V/m]

approach outlined in section II and results in update equations identical to (10) and (14) aside for the following substitutions:

6

, c2 = ∆t − 2τ , c3 = µ2 σ0 ∆z , c6 = σ0 ∆t.

(23)

We emphasize that the stability of the proposed method was tested for different values of σ0 , B0 , τ and mesh sizes. It was demonstrated that stability of the method was guaranteed by
√ satisfying the CFL stability criteria (i.e., ∆t = ∆/ Ndim c0 where Ndim is the dimension of the problem, c0 is wave velocity and ∆ = min {∆x, ∆y, ∆z}). Finally we mention that, the proposed method is capable of simulating structures with not only one layer of graphene but also several graphene layers separated by a distance.

where t0 = 1 ps and β = 0.1 ps. The problem domain consists of 10 × 10 × 20 uniform cells corresponding to 3 µm × 3 µm × 3 µm (notice that the shortest wave length of interest is set to λmin = 30 µm ), was truncated with the periodic boundary condition (PBC) in the x and y directions and an 8-cell PML in the z direction. In order to satisfy the √ CFL-stability condition,
the time step was set to ∆t = ∆z/ 3c0 . The simulations were run for 100,000 time steps for different values of the magneto-static field bias, B0 = 0.5, 2, and 10 T. Fig. 3 shows the computed values of the co- and cross-polarized components of the transmitted electric field for B0 = 2 T. The co- and cross-polarized transmission coefficients were obtained using discrete Fourier transforms, as presented in Fig. 4. Figure 4(c) shows the computed Faraday rotation angle defined as  tra  Ey −1 θf = tan . (25) Extra Clearly in Fig 4 (c), the rotation angle for all values of B0 converges to zero at very high frequencies as expected. (At very high frequency, both σd and σo approach zero, therefore the graphene sheet totally transmits the incident wave without rotating its polarization.) To validate the proposed method, the analytic results (see the Appendix) are compared with the FDTD results in Fig. 3 and Fig. 4. Excellent agreements is observed between the two results. A second example is considered to demonstrate the applicability of the proposed method for finite graphene sheets and non-conical geometries. We consider the problem of a plane wave transmission through a periodic graphene micropatch array analysed recently [39]. As shown in Fig. 5(a), a periodic array of graphene square patches supported on a

0018-926X (c) 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TAP.2017.2768081, IEEE Transactions on Antennas and Propagation IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. XX, NO. XX, XX 2017

7

graphene patch

z

incident plane wave

y B0

x SIO2 substrate

t

(a) (a)

Exinc PML

k

graphene patch 400 cells

z (b)

20 cells

y

x

PML

40 cells

40 cells (b) Fig. 5. (a) Plane wave incident on a periodic graphene micro-patch array supported by a SiO2 substrate. (b) Simulation computational domain. The PML layer is 8 cells deep. Periodic boundary conditions were placed on the yz and xz planes of the computational space. The graphene patch is centered in the computational space. The schematic is not drawn to scale.

(c) Fig. 4. The magnitude of the co- (a) and cross- (b) components of the transmission coefficient for normally incident plane wave on a magnetically biased graphene infinite sheet. (c) Faraday rotation angle.

SiO2 substrate with dielectric constant of 3.9 and thickness of t = 1 µm is normally illuminated by an x-polarized plane wave. The patches are located in the xy-plane and biased by a static magnetic field, B0 along the z-direction. The width of the patches and the distance between two adjacent patches were assumed to be w = 1 µm and d = 1 µm, respectively. The chemical potential, phenomenological scattering time and room temperature were considered identical to the previous example. Fig. 5(b) shows the computational space consisting of 40 × 40 × 400 uniform cells with a size of 20 nm × 20 nm × 20 nm. The computational space is terminated by a periodic boundary condition in the x and y directions and 8 cells of PML in the z direction. The structure was illuminated by a plan wave with a Gaussian waveform given in (24). The FDTD simulations were run for B0 = 0.5, 2, and 10 T. Applying the DFT, the magnitudes of the co- and crosspolarized transmission coefficients were obtained as shown in Fig. 6(a) and Fig. 6(b) respectively. Resonances with different intensities are observed in Fig.

6. These are attributed to the excitation of different modes of the structure [39]. It can be seen that increasing the magnetic field bias leads to an increase in the number of resonances, while decreasing the intensity of the main resonance of the structure. We also observe from Fig. 6(b) that the maximum value of the magnitude of the cross- transmission coefficient increases with increasing B0 . V. C ONCLUSION We introduced a FDTD algorithm for modeling magnetized graphene sheet by using an anisotropic surface boundary condition. First, a new technique for implementation of anisotropic conductive surface boundary condition in the FDTD method was presented in which the updating equations for magnetic field components at two sides of the sheet were coupled. Next, the surface conductivity matrix of magnetized graphene sheet was imposed. The main advantage over the earlier approaches which consider graphene sheet as a thin volumetric layer is that in the proposed approach, graphene is modeled as a boundary surface, avoiding the need for very fine mesh. A numerical examples was provided to validate the proposed method where comparison was made with fields calculated from analytical formulations. A second example was provided to show the strength of the method in addressing practical and emerging applications where biased graphene can be used to affect transmission through thin screens. The long-term stability feature

0018-926X (c) 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TAP.2017.2768081, IEEE Transactions on Antennas and Propagation IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. XX, NO. XX, XX 2017

where

8



α1  −α2 ¯ A=  α3 −α2

α2 α1 α2 α3

α3 −α2 α1 −α2

 α2 α3   α2  α1

(29)

where α1 = η0 + σd /d, α2 = σd /d and α3 = σ0 /d, and ¯ = [−η + σ /d − σ /d σ /d − σ /d]T , B (30) 0 d 0 d 0 (a)

and d = σd 2 + σo 2 is the determinant of σ¯s . Finally by solving the matrix equation (28), the co- and cross-polarized transmission and reflection coefficients are obtained. R EFERENCES

(b) Fig. 6. Magnitude of the co- (a) and cross- (b) components of the transmission coefficient for normally incident plane wave on a periodic array of magnetically biased graphene patches supported by a thin SiO2 substrate.

of the proposed method that is achieved without incurring additional computational burden is highly suited for modeling highly resonant structures. A PPENDIX This appendix provides the analytic solution of plane-wave transmission through magnetically biased graphene infinite sheets. Assuming an infinite graphene sheet in the xy plane and illuminated by an x-polarized incident plane-wave propagating in +z direction. The electric and magnetic fields of ˆ E0 e−jβz the incident wave can be expressed as Einc = x inc −jβz ˆ (1/η0 )E0 e and H = y . The transmitted and reflected electric and magnetic fields can be written as ˆ T co E0 e−jβz + y ˆ T cr E0 e−jβz Etra = x 1 co 1 cr ˆ ˆ Htra = y T E0 e−jβz − x T E0 e−jβz η0 η0 (26) ˆ Γ co E0 ejβz + y ˆ Γ cr E0 ejβz Eref = x 1 co 1 cr ˆ Href = −ˆ y Γ E0 ejβz + x Γ E0 ejβz η0 η0 where T co , T cr , Γ co , and Γ cr are co-polarized transmission, cross-polarized transmission, co-polarized reflection, and cross-polarized reflection coefficients. Enforcing the anisotropic conducting sheet boundary condition ((3)), we have
  Hyinc + Hyref − Hytra = Hxtra − Hxref z=0    tra  (27) Ex σd −σo . Eytra z=0 σo σd Finally, enforcing the continuity of the tangential electric field at the graphene sheet boundary, we have the following system of equations T ¯ ¯ A.[Γ (28) x Γy Tx Ty ] = B ,

[1] A. K. Geim, “Graphene: status and prospects”, Science, vol. 324, no. 5934, pp. 1530-1534, Jun. 2009. [2] K. S. Novoselov, V. I. Falko, L. Colombo, P. R. Gellert, M. G. Schwab, and K. Kim, “A roadmap for graphene”, Nature, vol. 490, no. 7419, pp. 192-200, Oct. 2012. [3] P. Avouris and F. Xia, “Graphene applications in electronics and photonics”, Mrs Bulletin, vol. 37, no. 12, pp. 1225-1234, Dec. 2012. [4] G. W. Hanson, “Dyadic Greens functions and guided surface waves for a surface conductivity model of graphene”, J. Appl. Phys., vol. 103, no. 6, p. 064302, Mar. 2008. [5] V. Nayyeri and O. M. Ramahi, “Electromagnetics Simulation of Graphene”, Mediterranean Microwave Symposium, Marrakesh, Morocco, 2014. [Online]. Available: https://www.youtube.com/watch?v=RFW1WCo7jTc. [6] M. Tamagnone, J. S. Gomez-Diaz, J. R. Mosig, and J. PerruisseauCarrier, “Analysis and design of terahertz antennas based on plasmonic resonant graphene sheets”, J. Appl. Phys., vol. 112, no. 11, p. 114915, Dec. 2012. [7] I. Llatser, C. Kremers, A. Cabellos-Aparicio, J. M. Jornet, E. Alarcn, and D. N. Chigrin, “Graphene-based nano-patch antenna for terahertz radiation”, Photon. Nanost.Fundament. Applic., vol. 10, no. 4, pp. 353358, Oct. 2012. [8] A. Fallahi and J. Perruisseau-Carrier, “Design of tunable biperiodic graphene metasurfaces”, Physical Review B, vol. 86, no. 19, p. 195408, Nov. 2012. [9] V. Nayyeri, M. Soleimani, and O. Ramahi, “Modeling graphene in the finite-difference time-domain method using a surface boundary condition”, IEEE Trans. Antennas Propag., vol. 61, no. 8, pp. 41764182, Aug. 2013. [10] A. Vakili and N. Engheta, “Transformation optics using graphene”, Science, vol. 332, no. 6035, pp. 1291-1294, Jun. 2011. [11] K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag., vol. 14, no. 3, pp. 302 -307, May 1966. [12] A. Taflove and S. C. Hagness, “Computational electrodynamics: the finite-difference time-domain method”, Norwood, MA: Artech, 2000. [13] W. Gao, J. Shu, C. Qiu, and Q. Xu, “Excitation of plasmonic waves in graphene by guided-mode resonances”, ACS Nano, vol. 6, no. 9, pp. 7806-7813, Aug. 2012. [14] G. D. Bouzianas, N. V. Kantartzis, C. S. Antonopoulos, and T. D. Tsiboukis, “Optimal modeling of infinite graphene sheets via a class of generalized FDTD schemes”, IEEE Trans. Magn., vol. 48, no. 2, pp. 379-382, Feb. 2012. [15] R. M. S. de Oliveira, N. R. N. M. Rodrigues, and V. Dmitriev, “FDTD formulation for graphene modeling based on piecewise linear recursive convolution and thin material sheets techniques”, IEEE Antennas Wireless Propag. Lett., vol. 14, pp. 767-770, Dec. 2014. [16] I. Ahmed, E. H. Khoo, and E. Li, “Efficient modeling and simulation of graphene devices with the LOD-FDTD method”, IEEE Microw. Compon. Lett., vol. 23, no. 6, pp. 306-308, Jun. 2013. [17] X. H. Wang, W. Y. Yin, and Z. Z. Chen, “One-step leapfrog ADI-FDTD method for simulating electromagnetic wave propagation in general dispersive media”, Optics Express, vol. 21, no. 18, p. 20565, Sep. 2013. [18] J. Chen and J. Wang, “Three-dimensional dispersive hybrid implicitexplicit finite-difference time-domain method for simulations of graphene”, Computer Physics Communications, vol. 207, pp. 211-216, Oct. 2016. [19] J. G. Maloney and G. S. Smith, “The use of surface impedance concepts in the finite-difference time-domain method”, IEEE Trans. Antennas Propag., vol. 40, no. 1, pp. 3848, Jan. 1992.

0018-926X (c) 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TAP.2017.2768081, IEEE Transactions on Antennas and Propagation IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. XX, NO. XX, XX 2017

[20] J. H. Beggs, R. J. Luebbers, K. S. Yee, and K. S. Kunz, “Finitedifference time-domain implementation of surface impedance boundary conditions”, IEEE Trans. Antennas Propag., vol. 40, no. 1, pp. 4956, Jan. 1992. [21] S. Van den Berghe, F. Olyslager, and D. De Zutter, “Accurate modeling of thin conducting layers in FDTD”, IEEE Microw. Wireless Comp. Lett, vol. 8, no. 2, pp. 75-77, Feb. 1998. [22] M. Sabrina Sarto, “A new model for the FDTD analysis of the shielding performances of thin composite structures”, IEEE Trans. Electromagn. Compat., vol. 41, no. 4, pp. 298-306, Nov. 1999. [23] M. Feliziani, F. Maradei, and G. Tribellini, “Field analysis of penetrable conductive shields by the finite-difference time-domain method with impedance network boundary conditions (INBCs)”, IEEE Trans. Electromagn. Compat., vol. 41, no. 4, pp. 307-319, Nov. 1999. [24] V. Nayyeri, M. Soleimani and O. M. Ramahi, “A method to model thin conductive layers in the finite-difference time-domain method”, IEEE Transactions on Electromagn. Compat., vol. 56, no. 2, pp. 385-392, Apr. 2014. [25] V. Nayyeri, M. Soleimani, and M. Dehmollaian, “Modeling of the perfect electromagnetic conducting boundary in the finite difference time domain method”, Radio Science, vol. 48, no. 4, pp. 453-462, Jul. 2013. [26] M. Feizi, V. Nayyeri, and A. Keshtkar, “Finite-difference time-domain implementation of tensor impedance boundary conditions”, IET Microwaves, Antennas & Propagation, vol. 11, no. 7, pp. 1064-1070, Jun. 2017. [27] V. Nayyeri, M. Soleimani, and O. Ramahi, “Wideband modeling of graphene using the finite-difference time-domain method”, IEEE Trans. Antennas Propag., vol. 61, no. 12, pp. 6107-6114, Dec. 2013. [28] G. D. Bouzianas, N. V. Kantartzis, T. Yioultsis, and T. D. Tsiboukis, “Consistent study of graphene structures through the direct incorporation of surface conductivity”, IEEE Trans. Magn., vol. 50, no. 2, pp. 161-164, Feb. 2014. [29] G. W. Hanson, “Dyadic Green’s functions for an anisotropic, non-local model of biased graphene”, IEEE Trans Antennas Propag., Vol. 56, no. 3, pp. 747757, Mar. 2008. [30] D. L. Sounas and C. Caloz, “Gyrotropy and nonreciprocity of graphene for microwave applications”, IEEE Trans. Microw. Theory Techn., vol. 60, no. 4, pp. 901-914, Apr. 2012. [31] B. Zhu, G. Ren, Y. Gao, B. Wu, Q. Wang, C. Wan, and S. Jian, “Graphene plasmons isolator based on non-reciprocal coupling”, Optic Express, vol. 23, no. 12, pp. 1607116083, Jun. 2015. [32] X. H. Wang, W. Y. Yin, and Z. Chen, “Matrix exponential FDTD modeling of magnetized graphene sheet”, IEEE Antennas Wireless Propag. Lett., vol. 12, pp. 1129-1132, Sep. 2013. [33] B. Salski, “An FDTD model of graphene intraband conductivity”, IEEE Trans. Microw. Theory Techn., vol. 62, no. 8, pp. 1570-1578, Aug.2014. [34] P. Li and L. J. Jiang, “Modeling of magnetized graphene from microwave to THz range by DGTD with a scalar RBC and an ADE”, IEEE Trans. Antennas Propag., vol. 63, no. 10, pp. 44584467, Oct. 2015. [35] Y. S. Cao, L. Jiang and A. E. Ruehli, “The derived equivalent circuit model for magnetized anisotropic graphene”, IEEE Trans. Antennas Propag., vol. 65, no. 2, pp. 948-953, Feb. 2017. [36] J. Schneider and S. Hudson, “The finite-difference time-domain method applied to anisotropic material”, IEEE Trans. Antennas Propag., vol. 41, no. 7, pp. 994999, Jul. 1993. [37] V. Nayyeri, M. Soleimani, J. Rashed Mohassel, and M. Dehmollaian, “FDTD modeling of dispersive bianisotropic media using Z-transform method”, IEEE Trans. Antennas Propag., vol. 59, No. 6, pp. 2268-2279, Jun. 2011. [38] V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte, “On the universal ac optical background in graphene”, New J. Phys., vol. 11, p. 095013, Sep. 2009. [39] P. K. Khoozani, M. Maddahali, M. Shahabadi, and A. Bakhtforouz, “Analysis of magnetically biased graphene-based periodic structures using a transmission-line formulation”, JOSA B, vol. 33, no. 12, pp. 2566-2576, Nov. 2016.

9

Mina Feizi (S’17) was born in Tehran, Iran, in 1990. She received the B.Sc. degree in electrical engineering and M.Sc. degree in electrical engineeringfields, waves and electromagnetics from the Imam Khomeini International University, Qazvin, Iran, in 2013 and 2016, respectively. Her current research interests include theoretical and computational electromagnetics.

Vahid Nayyeri (S’08-M’14-SM’16) was born in Tehran, Iran. He received the B.Sc. degree from the Iran University of Science and Technology (IUST), Tehran, Iran, in 2006, the M.Sc. degree from the University of Teheran, Tehran, Iran, in 2008, and the Ph.D. degree from the IUST in 2013, all in electrical engineering. From 2007 to 2010, he worked as an RF-circuit designer at the IUST satellite research center. He then was the technical manager of three research and industrial projects at the Antenna and Microwave Research Laboratory at IUST. In Jun 2012 he joined the University of Waterloo, Waterloo, ON, Canada, as a visiting scholar. Presently, he is an assistant professor of the department of satellite engineering, IUST, and also serves as vice-director of Antenna and Microwave Research Laboratory. He has authored and coauthored one book (in Persian) and over 30 journal and conference technical papers. He has been collaborating with the Advanced Electromagnetics Research Laboratory, University of Waterloo, since 2012. His research interests include applied and computational electromagnetics and microwave circuit design. In 2014, Dr. Nayyeri received the ”Best Ph.D. Thesis Award” from the IEEE Iran Section for his research on the modeling of complex media and boundaries in the finite-difference time-domain method. He is a senior member of IEEE and has served as reviewer to several journals and conferences.

Omar M. Ramahi (Fellow, IEEE, 09) was born in Jerusalem, Palestine. He received the BS degrees in Mathematics and Electrical and Computer Engineering (Highest Honors) from Oregon State University, Corvallis, OR. In 1990, he was awarded the Ph.D. degree in Electrical and Computer Engineering from the University of Illinois at UrbanaChampaign (UIUC). He then worked at Digital Equipment Corporation (presently, HP), where he was a member of the Alpha Server Product Development Group. In 2000, he joined the faculty of the James Clark School of Engineering at the University of Maryland at College Park as an Assistant Professor and later as a tenured Associate Professor. At Maryland he was also a faculty member of the CALCE Electronic Products and Systems Center. Presently, he is a Professor in the Electrical and Computer Engineering Department, University of Waterloo, Canada. He has authored and co-authored over 390 journal and conference technical papers on topics related to the electromagnetic phenomena and computational techniques to understand the same. He is a co-author of the book EMI/EMC Computational Modeling Handbook, (first edition: Kluwer, 1998, Second Ed: Springer-Verlag, 2001. Japanese edition published in 2005). Professor Ramahi is the winner of the 2004 University of Maryland Pi Tau Sigma Purple Cam Shaft Award. He won the Excellent Paper Award in the 2004 International Symposium on Electromagnetic Compatibility, Sendai, Japan, and the 2010 University of Waterloo Award for Excellence in Graduate Supervision. From 2010 to 2012, he served as the IEEE EMC Society Distinguished Lecturer and presently is a member of the IEEE EMC Society Respected Speakers Bureau. In 2012, Professor Ramahi was awarded the IEEE Electromagnetic Compatibility Society Technical Achievement Award.

0018-926X (c) 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.