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Full-wave finite-difference time-domain simulation of electromagnetic cloaking structures

Yan Zhao, Christos Argyropoulos, and Yang Hao

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Abstract

This paper proposes a radial dependent dispersive finite-difference time-domain method for the modeling of electromagnetic cloaking structures. The permittivity and permeability of the cloak are mapped to the Drude dispersion model and taken into account in dispersive FDTD simulations. Numerical simulations demonstrate that under ideal conditions, objects placed inside the cloak are ‘invisible’ to external electromagnetic fields. However for the simplified cloak based on linear transformations, the back scattering has a similar level to the case of a PEC cylinder without any cloak, rendering the object still being ‘visible’. It is also demonstrated numerically that the simplified cloak based on high-order transformations can indeed improve the cloaking performance.

©2008 Optical Society of America

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Supplementary Material (3)

Media 1: MOV (2756 KB)     
Media 2: MOV (2840 KB)     
Media 3: MOV (2763 KB)     

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Figures (6)
Fig. 1.
Fig. 1. The comparison between the analytical (2) and numerical material parameters (23) for different FDTD spatial resolutions for the case of εr =0.1 (lossless).

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Fig. 2.
Fig. 2. A two-dimensional (2-D) FDTD simulation domain for the case of plane-wave incidence on the cloak.

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Fig. 3.
Fig. 3. (a) Material parameters for an infinite ideal cylindrical cloak [3] where all εr , εϕ and µz are radial dependent. (b), (c) and (d) Field distributions from dispersive FDTD simulations of the cloak: (b) x-component of the electric field, (c) y-component of the electric field and (d) the magnetic field. [Media 1]

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Fig. 4.
Fig. 4. (a) Material parameters for an infinite simplified cylindrical cloak using a linear transformation [4] where only εr is radial dependent. (b), (c) and (d) Field distributions from dispersive FDTD simulations of the cloak: (b) x-component of the electric field, (c) y-component of the electric field and (d) the magnetic field. [Media 2]

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Fig. 5.
Fig. 5. (a) Material parameters for an infinite simplified cylindrical cloak using a high-order transformation [5] where only εr and εϕ are radial dependent. (b), (c) and (d) Field distributions from dispersive FDTD simulations of the cloak: (b) x-component of the electric field, (c) y-component of the electric field and (d) the magnetic field. [Media 3]

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Fig. 6.
Fig. 6. Power flow diagrams for (a) the ideal cloak [3], (b) the simplified cloak based on the linear transformation [4] and (c) the simplified cloak based on the high-order transformation [5]. (d) Comparison of the scattering patterns for different cloaks and for the case of the PEC cylinder without cloak: black solid line - PEC cylinder; yellow dot-dashed line - the simplified cloak based on the linear transformation; blue dashed line - the simplified cloak based on the high-order transformation; red solid line - the ideal cloak.

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Equations (52)

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ε r = μ r = r − R 1 r , ε ϕ = μ ϕ = r r − R 1 , ε z = μ z = ( R 2 R 2 − R 1 ) 2 r − R 1 r ,
ε r ( ω ) = 1 − ω p 2 ω 2 − j ω γ ,
∇ × E = − ∂ B ∂ t ,
∇ × H = ∂ D ∂ t ,
B n + 1 = B n − Δ t · ∇ ˜ × E n + 1 2 ,
D n + 1 = D n + Δ t · ∇ ˜ × H n + 1 2 ,
[ ε xx ε xy ε yx ε yy ] = [ ε r cos 2 ϕ + ε ϕ sin 2 ϕ ( ε r − ε ϕ ) sin ϕ cos ϕ ( ε r − ε ϕ ) sin ϕ cos ϕ ε r sin 2 ϕ + ε ϕ cos 2 ϕ ]
ε 0 [ ε xx ε xy ε yx ε yy ] [ E x E y ] = [ D x D y ] ⇔ ε 0 [ E x E y ] = [ ε xx ε xy ε yx ε yy ] − 1 [ D x D y ] ,
[ ε xx ε xy ε yx ε yy ] − 1 = 1 ε r ε ϕ [ ε r sin 2 ϕ + ε ϕ cos 2 ϕ ( ε ϕ − ε r ) sin ϕ cos ϕ ( ε ϕ − ε r ) sin ϕ cos ϕ ε r cos 2 ϕ + ε ϕ sin 2 ϕ ] .
{ ε r ε ϕ ε 0 E x = ( ε r sin ⁡ 2 ϕ + ε ϕ cos ⁡ 2 ϕ ) D x + ( ε ϕ − ε r ) sin ⁡ ϕ cos ⁡ ϕ D y ε r ε ϕ ε 0 E y = ( ε r cos ⁡ 2 ϕ + ε ϕ sin ⁡ 2 ϕ ) D y + ( ε ϕ − ε r ) sin ⁡ ϕ cos ⁡ ϕ D x ,
{ ε 0 ε ϕ ( ω 2 − j ω γ − ω p 2 ) E x = [ ( ω 2 − j ω γ − ω p 2 ) sin ⁡ 2 ϕ + ε ϕ ( ω 2 − j ω γ ) cos ⁡ 2 ϕ ] D x + [ ε ϕ ( ω 2 − j ω γ ) − ( ω 2 − j ω γ − ω p 2 ) ] sin ⁡ ϕ cos ⁡ ϕ D y , ε 0 ε ϕ ( ω 2 − j ω γ − ω p 2 ) E y = [ ( ω 2 − j ω γ − ω p 2 ) cos ⁡ 2 ϕ + ε ϕ ( ω 2 − j ω γ ) sin ⁡ 2 ϕ ] D y + [ ε ϕ ( ω 2 − j ω γ ) − ( ω 2 − j ω γ − ω p 2 ) ] sin ⁡ ϕ cos ⁡ ϕ D x .
j ω → ∂ ∂ t , ω 2 → − ∂ 2 ∂ t 2 ,
ε 0 ε ϕ ( ∂ 2 ∂ t 2 + γ ∂ ∂ t + ω p 2 ) E x = [ ( ∂ 2 ∂ t 2 + γ ∂ ∂ t + ω p 2 ) sin ⁡ 2 ϕ + ε ϕ ( ∂ 2 ∂ t 2 + γ ∂ ∂ t ) cos ⁡ 2 ϕ ] D x
+ [ ε ϕ ( ∂ 2 ∂ t 2 + γ ∂ ∂ t ) − ( ∂ 2 ∂ t 2 + γ ∂ ∂ t + ω p 2 ) ] sin ⁡ ϕ cos ⁡ ϕ D y .
∂ 2 ∂ t 2 → δ t 2 ( Δ t ) 2 , ∂ ∂ t → δ t Δ t μ t , ω p 2 → ω p 2 μ t 2 ,
δ t F ∣ m x , m y , m z n ≡ F ∣ m x , m y , m z n + 1 2 − F ∣ m x , m y , m z n − 1 2 ,
δ t 2 F ∣ m x , m y , m z n ≡ F ∣ m x , m y , m z n + 1 − 2 F ∣ m x , m y , m z n + F ∣ m x , m y , m z n − 1 ,
μ t F ∣ m x , m y , m z n ≡ F ∣ m x , m y , m z n + 1 2 + F ∣ m x , m y , m z n − 1 2 2 ,
μ t 2 F ∣ m x , m y , m z n ≡ F ∣ m x , m y , m z n + 1 + 2 F ∣ m x , m y , m z n + F ∣ m x , m y , m z n − 1 4 .
ε 0 ε ϕ [ δ t 2 ( Δ t ) 2 + γ δ t Δ t μ t + ω p 2 μ t 2 ] E x = { [ δ t 2 ( Δ t ) 2 + γ δ t Δ t μ t + ω p 2 μ t 2 ] sin ⁡ 2 ϕ
+ ε ϕ [ δ t 2 ( Δ t ) 2 + γ δ t Δ t μ t ] cos ⁡ 2 ϕ } D x + { ε ϕ [ δ t 2 ( Δ t ) 2 + γ δ t Δ t μ t ]
− [ δ t 2 ( Δ t ) 2 + γ δ t Δ t μ t + ω p 2 μ t 2 ] } sin ⁡ ϕ cos ⁡ ϕ D y .
ε 0 ε ϕ [ E x n + 1 − 2 E x n + E x n − 1 ( Δ t ) 2 + γ E x n + 1 − E x n − 1 2 Δ t + ω p 2 E x n + 1 + 2 E x n + E x n − 1 4 ]
= sin ⁡ 2 ϕ [ D x n + 1 − 2 D x n + D x n − 1 ( Δ t ) 2 + γ D x n + 1 − D x n − 1 2 Δ t + ω p 2 D x n + 1 + 2 D x n + D x n − 1 4 ]
+ ε ϕ cos ⁡ 2 ϕ [ D x n + 1 − 2 D x n + D x n − 1 ( Δ t ) 2 + γ D x n + 1 − D x n − 1 2 Δ t ]
+ sin ⁡ ϕ cos ⁡ ϕ { ε ϕ [ D y n + 1 − 2 D y n + D y n − 1 ( Δ t ) 2 + γ D y n + 1 − D y n − 1 2 Δ t ]
− [ D y n + 1 − 2 D y n + D y n − 1 ( Δ t ) 2 + γ D y n + 1 − D y n − 1 2 Δ t + ω p 2 D y n + 1 + 2 D y n + D y n − 1 4 ] } .
E x n + 1 = [ a x D x n + 1 + b x D x n + c x D x n − 1 + d x D y ¯ n + 1 + e x D y ¯ n + f x D y ¯ n − 1 − ( g x E x n + h x E x n − 1 ) ] l x .
a x = sin 2 ϕ [ 1 ( Δ t ) 2 + γ 2 Δ t + ω p 2 4 ] + ε ϕ cos 2 ϕ [ 1 ( Δ t ) 2 + γ 2 Δ t ] ,
b x = sin 2 ϕ [ − 2 ( Δ t ) 2 + ω p 2 2 ] − ε ϕ cos 2 ϕ 2 ( Δ t ) 2 ,
c x = sin 2 ϕ [ 1 ( Δ t ) 2 − γ 2 Δ t + ω p 2 4 ] + ε ϕ cos 2 ϕ [ 1 ( Δ t ) 2 − γ 2 Δ t ] ,
d x = { ε ϕ [ 1 ( Δ t ) 2 + γ 2 Δ t ] − [ 1 ( Δ t ) 2 + γ 2 Δ t + ω p 2 4 ] } sin ϕ cos ϕ ,
e x = { ε ϕ [ − 2 ( Δ t ) 2 ] − [ − 2 ( Δ t ) 2 + ω p 2 2 ] } sin ϕ cos ϕ ,
f x = { ε ϕ [ 1 ( Δ t ) 2 − γ 2 Δ t ] − [ 1 ( Δ t ) 2 − γ 2 Δ t + ω p 2 4 ] } sin ϕ cos ϕ ,
g x = ε 0 ε ϕ [ − 2 ( Δ t ) 2 + ω p 2 2 ] ,       h x = ε 0 ε ϕ [ 1 ( Δ t ) 2 − γ 2 Δ t + ω p 2 4 ] ,       l x = ε 0 ε ϕ [ 1 ( Δ t ) 2 + γ 2 Δ t + ω p 2 4 ] .
E y n + 1 = [ a y D y n + 1 + b y D y n + c y D y n − 1 + d y D x ¯ n + 1 + e y D x ¯ n + f y D x ¯ n − 1 − ( g y E y n + h y E y n − 1 ) ] l y .
a y = cos ⁡ 2 ϕ [ 1 ( Δ t ) 2 + γ 2 Δ t + ω p 2 4 ] + ε ϕ sin ⁡ 2 ϕ [ 1 ( Δ t ) 2 + γ 2 Δ t ] ,
b y = cos ⁡ 2 ϕ [ − 2 ( Δ t ) 2 + ω p 2 2 ] − ε ϕ sin ⁡ 2 ϕ 2 ( Δ t ) 2 ,
c y = cos ⁡ 2 ϕ [ 1 ( Δ t ) 2 − γ 2 Δ t + ω p 2 4 ] + ε ϕ sin ⁡ 2 ϕ [ 1 ( Δ t ) 2 − γ 2 Δ t ] ,
d y = { ε ϕ [ 1 ( Δ t ) 2 + γ 2 Δ t ] − [ 1 ( Δ t ) 2 + γ 2 Δ t + ω p 2 4 ] } sin ⁡ ϕ cos ⁡ ϕ ,
e y = { ε ϕ [ − 2 ( Δ t ) 2 ] − [ − 2 ( Δ t ) 2 + ω p 2 2 ] } sin ⁡ ϕ cos ⁡ ϕ ,
f y = { ε ϕ [ 1 ( Δ t ) 2 − γ 2 Δ t ] − [ 1 ( Δ t ) 2 − γ 2 Δ t + ω p 2 4 ] } sin ⁡ ϕ cos ⁡ ϕ ,
g y = ε 0 ε ϕ [ − 2 ( Δ t ) 2 + ω p 2 2 ] , h y = ε 0 ε ϕ [ 1 ( Δ t ) 2 − γ 2 Δ t + ω p 2 4 ] , l y = ε 0 ε ϕ [ 1 ( Δ t ) 2 + γ 2 Δ t + ω p 2 4 ] .
D x ¯ ( i , j ) = D x ( i , j ) + D x ( i , j + 1 ) + D x ( i − 1 , j ) + D x ( i − 1 , j + 1 ) 4 ,
D y ¯ ( i , j ) = D y ( i , j ) + D y ( i + 1 , j ) + D y ( i , j − 1 ) + D y ( i + 1 , j − 1 ) 4 ,
μ z ( ω ) = A ( 1 − ω pm 2 ω 2 − j ω γ m ) ,
H z n + 1 = 1 A { [ 1 μ 0 ( Δ t ) 2 + γ m 2 μ 0 Δ t ] B z n + 1 − 2 μ 0 ( Δ t ) 2 B z n + [ 1 μ 0 ( Δ t ) 2 − γ m 2 μ 0 Δ t ] B z n − 1 + A [ 2 ( Δ t ) 2 − ω pm 2 2 ] H 2 n − A [ 1 ( Δ t ) 2 − γ m 2 Δ t + ω pm 2 4 ] H z n − 1 } [ 1 ( Δ t ) 2 + γ m 2 Δ t + ω pm 2 4 ] . .
E n = E e j n ω Δ t , D n = D e j n ω Δ t ,
ε r ~ = ε 0 [ 1 − ω p 2 ( Δ t ) 2 cos ⁡ 2 ω Δ t 2 2 sin ⁡ ω Δ t 2 ( 2 sin ⁡ ω Δ t 2 − j γ Δ t cos ⁡ ω Δ t 2 ) ] .
ω p 2 ~ = 2 sin ⁡ ω Δ t 2 [ − 2 ( ε r ′ − 1 ) sin ⁡ ω Δ t 2 − ε r ″ γ Δ t cos ⁡ ω Δ t 2 ] ( Δ t ) 2 cos ⁡ 2 ω Δ t 2 , γ ~ = 2 ε r ″ sin ⁡ ω Δ t 2 ( ε r ′ − 1 ) Δ t cos ⁡ ω Δ t 2 .
ε r = ( R 2 R 2 − R 1 ) 2 ( r − R 1 r ) 2 , ε ϕ = ( R 2 R 2 − R 1 ) 2 , μ z = 1 .
ε r = ( r ′ r ) 2 , ε ϕ = [ ∂ g ( r ′ ) ∂ r ′ ] − 2 , μ z = 1 .
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