Chapter 2 Fast Algorithms for the Electromagnetic Scattering from ...

h
s s
(E ,H )
PEC
Γ
i i
(E ,H )
θ
y
c c
Ω
ε(x,y)
Γ Γ
µ
0
r r
(E ,H )
ε µ
ka
k a
ka
k 2 h
h 1m × 1m
h ≈ 10 −4 m 10 8
0
x
0

i.e.
O(M 2 ) M × M

∇ × E = iωµH,
∇ × H = −iωɛE,
ω ɛ µ
z
µ = µ0
ɛ Im(ɛ) ≥ 0
z E
∆u + k 2 u = f(x, y), (x, y) ∈ Ω ∪ R + 2 ,
u = 0, (∂Ω\Γ) ∪ Γ c ,
√ ∂us s
lim r − ik0u = 0,
r→∞ ∂r
u s R + 2 Γ = (0, a)
r = x 2 + y 2 k 2 = ω 2 ɛµ = k 2 0 ɛrµr k0 = ω √ ɛ0µ0
ɛr = ɛ/ɛ0 µr = µ/µ0
f = 0
f = 0

1
∇ ·
∇u
ɛr
+ k 2 0µru = f(x, y), (x, y) ∈ Ω ∪ R + 2
,
∂u
= 0, (∂Ω\Γ) ∪ Γc
∂n
z H
n
i.e.
u s
Gd(x, x ′ ) = i
4
H (1)
0
(kr) − H(1)
0 (k¯r)
,
∆Gd + k 2 0Gd = −δ(x, x ′ ), x, x ′ ∈ R + 2 ,
Gd = 0, y = 0,
√
lim r
r→∞
∂Gd
∂r
− ik0Gd = 0,
x = (x, y) x ′ = (x ′ , y ′ ) r = |x − x ′ | ¯r = |x − ¯x¯x¯x ′ | ¯x¯x¯x ′ = (x ′ , −y ′ ) x ′
x
u s
(x) =
Γ
∂Gd(x ′ , x)
∂y ′
u s (x ′ )
y ′ =0 +
∂Gd(x ′ , x)
∂y ′
y ′ =0 +
= ik0y
2r H(1)
1 (k0r),
r = (x − x ′ ) 2 + y 2 y y → 0 +
∂u s (x)
∂y
y=0 +
= ik0
2
Γ
dx ′ ,
1
|x − x ′ | H(1)
1 (k0|x − x ′ |)u s (x ′ , 0)
∂us ∂y = IM(u s ) := ik0
2
=
1
ds(x ′ ).
Γ |x − x ′ | H(1) 1 (k0|x − x ′ |)u s (x ′ , 0)dx ′ , x ∈ (0, a),
= Γ

IM(u s ) = ik0
2
R
k 2 0 − ξ2 F[ũ s ]e ξx dξ,
ũs us H1/2 (R) F
⎧⎨
∆u + k
⎩
2u = 0, (x, y) ∈ Ω,
u = 0, ∂Ω\Γ,
∂us ∂n = IM(us ), Γ.
uI = e i(αx−βy) α = k0 sin θ β =
k0 cos θ −π/2 < θ < π/2 y
u s
u s = u − uI + e i(αx+βy) ,
u u s
f = 0 u
⎧
⎨
⎩
∆u + k 2 u = 0, (x, y) ∈ Ω,
u = 0, ∂Ω\Γ,
∂u
∂n = IM(u) + gM(x), Γ,
gM(x) = −2βe iαx
u s
Gn(x, x ′ ) = i
H
4
(1)
0 (k0r) + H (1)
0 (k0¯r)
∆Gn + k 2 0Gn = −δ(x, x ′ ), x, x ′ ∈ R + 2 ,
∂Gn
∂n
= 0, y = 0,
√
lim r
r→∞
∂Gn
∂r
− ik0Gn = 0.
u s
(x) = −
Γ
Gn(x ′ , x) ∂us (x ′ )
∂y ′
y ′ =0 +
dx ′ .

u s (x, 0) = IE( ∂us
i
) := − H
∂y 2 Γ
(1)
0 (k0|x ′ − x|) 1
ɛr
∂u s (x ′ )
∂y ′
y ′ =0 −
IE( ∂us
i 1 ∂us ) = − F (y, 0) e
∂n 2π R k2 0 − ξ2 ∂y iξx dξ.
dx ′ .
uI = ei(αx−βy)
u(x, y)| y=0 + = u(x, y)| y=0 −
∂u
∂n
y=0 +
= 1
ɛr
∂u
∂n
y=0 −
⎧ ⎪⎨
⎪⎩
1 ∇ · ɛr ∇u
+ k2 0 µru = 0, (x, y) ∈ Ω,
∂u
∂n
u = IE( ∂u
∂n ) + gE, Γ,
= 0, ∂Ω\Γ,
,
gE(x) = 2e iαx
Im(k 2 ) ≥ 0
{xi, yj} M+1,N+1
i,j=0 Ω = [0, a]×[−b, 0] xi+1 −xi =
hx yj+1 − yj = hy uij (xi, yj)

ui−1,j − 2uij + ui+1,j
h 2 x
+ ui,j−1 − 2uij + ui,j+1
+ k 2 (xi, yj)uij = 0,
u0j = uM+1,j = ui0 = 0, i = 1, 2, . . . , M, j = 1, 2, . . . , N,
h 2 y
(A D x ⊗ IN + IM ⊗ A D y + D)U1 + (IM ⊗ aN+1)u:,N+1 = 0,
⊗ IM M ×M
A D x = 1
h 2 x
⎡
⎢
⎣
−2 1
1 −2 1
1 −2
⎤
⎥
⎦ , ADy = 1
h2 y
⎡
⎢
⎣
−2 1
1 −2 1
1 −2
D = (D1, D2, . . . , DM) , Di = (k 2 (xi, y1), k 2 (xi, y2), . . . , k 2 (xi, yN)),
aN+1 = 1
h2 eN, eN = (0, . . . , 0, 1)
y
T ,
U1 = (u11, u12, . . . , u1N, u21, u22, . . . , u2N, . . . , uM1, . . . , uMN) T ,
u:,j = (u1j, u2j, . . . , uMj) T .
∂u
∂y
i.e.
ui,N+1 − uiN
hy
=
M
l=1
1
u:,N + ( 1
h 2 y
hy
g M il ul,N+1 + gM(xi), i = 1, 2, . . . , M,
⎤
⎥
⎦ ,
GM − 1
h2 IM)u:,N+1 = −
y
1
gM,
hy
GM = (g M ij )M i,j=1
(GM)
k (ɛr(x, y)) ≥ 0
(GM)

A M 11
A M 21
AM 12
1
hy GM − 1
h2 IM
y
U1
u:,N+1
=
0
− 1
hy gM
A M 11 = A D x ⊗ IN + IM ⊗ A D y + D,
A M 12 = IM ⊗ aN+1, A21 = (A M 12) T .
,
i.e. k 2 = k 2 (y)
D = IM ⊗ Dk
Dk = (k 2 (y1), k 2 (y2), . . . , k 2 (yN)).
A D x
SMA D x SM = Λ =
SM
S 2 M
SM =
2
M + 1
sin lmπ
M + 1
M
⎡
⎢
⎣
λ1
λ2
, λl = −
l,m=1
λM
⎤
⎥
⎦ ,
4(M + 1)2
a 2 sin 2 lπ
2(M + 1) ,
= I
Λ ⊗ IN + IM ⊗ (A D y + Dk) Ū1 + (IM ⊗ aN+1)ū:,N+1 = 0,
Ū1 = (SM ⊗ IN) U1 = (ū11, . . . , ū1N, ū21, . . . , ū2N, . . . , ūM1, . . . , ūMN) T ,
ū:,j = SMu:,j = (ū1j, ū2j, . . . , ūMj) T .
1
ū:,N + SM( 1
h 2 y
hy
GM − 1
h2 IM)SMū:,N+1 = ¯gM,
y
¯gM = − 1
hy SMgM
(AD y + λiIN + Dk)ūi,: + aN+1ūi,N+1 = 0, i = 1, 2, . . . , M,
1
h2 ū:,N + SM(
y
1
hy GM − 1
h2 IM)SMū:,N+1 = ¯gM,
y

ūi,: = (ūi1, ūi2, . . . , ūiN) T , i = 1, 2, . . . , M.
A D y + λiIN + Dk = LiRi, i = 1, 2, . . . , M,
A D y + λiIN + Dk Li
Riūi,: + L −1
i aN+1ūi,N+1 = 0, i = 1, 2, . . . , M,
Ri = (ri pq)
ri NN ūiN + ri N,N+1ūi,N+1 = 0, i = 1, 2, . . . , M,
1
h2 ū:,N + SM(
y
1
hy GM − 1
h2 IM)SMū:,N+1 = ¯gM,
y
r i N,N+1 L −1
i aN+1 ū:,N ū:,N+1
k 2
h
r i NN = 0, i = 1, 2, . . . , M.
ū:,N
( 1
SMGMSM −
hy
1
h2 IM −
y
1
h2 E)ū:,N+1 = ¯gM,
y
( 1
GM −
hy
1
h2 IM −
y
1
h2 SMESM)u:,N+1 = −
y
1
gM,
hy
E =
⎡
⎢
⎣
r 1 N,N+1 /r1 NN
r M N,N+1 /rM NN
ū:,N+1 Γ
(A D y + λiIN + Dk)ūi,: = −aN+1ūi,N+1, i = 1, 2, . . . , M,
1
ū:,N + SM( 1
h 2 y
hy
⎤
⎥
⎦ .
GM − 1
h2 IM)SMū:,N+1 = ¯gM
y

GM
Ri E
¯gM
ū:,N+1 u:,N+1
A = 1
hy
SMGMSM − 1
h2 IM −
y
1
h2 E.
y
P = − 1
h2 (IM + E)
y
AP −1 v = ¯gM P ū:,N+1 = v.
¯gM − AP −1 v p 2
¯gM − AP −1 v 0 2
v p p v 0
GM
IM(u) GM
12M log M A P
12M log M
u:,N+1
p
p
≤ δ,

u f = 0
O(M)
3MN
2M log M
O(12pM log M)
5MN
O(MN) + 12pM log M
1
1
hx ɛr(xi+1/2, yj)
+ 1
1
hy ɛr(xi, yj+1/2) ui+1,j − uij
hx
ui,j+1 − uij
hy
−
1
ɛr(x i−1/2, yj)
−
1
ɛr(xi, y j−1/2)
uij − ui−1,j
hx
uij − ui,j−1
+k 2 0(xi, yj)µruij = 0, i = 1, 2, . . . , M, j = 1, 2, . . . , N,
u1j − u0j
hx
uMj − uM+1,j
hx
ui1 − ui0
hy
= 0, j = 1, 2, . . . , N,
= 0, j = 1, 2, . . . , N,
= 0, i = 1, 2, . . . , M.
i.e.
ui,N+1 = 1
hy
M
l=1
1
ɛr(xl, y 1
N+ )
2
gE il (ul,N+1 − ulN) + gE(xi), i = 1, 2, . . . , M,
1
GEDɛru:,N + (IM − 1
GEDɛr)u:,N+1 = gE,
hy
Dɛr = (1/ɛr(x1, y N+ 1
2
hy
), 1/ɛr(x2, y N+ 1
2
hy
), . . . , 1/ɛr(xM, y 1
N+ )),
2
GE = [g E ij ]M i,j=1
GE
ɛr = ɛr(y)
A E 11
A E 21
1
hy G−1
E
A E 12
1 − h2 Dɛr
y
U1
u:,N+1
=
0
1
hy G−1
E gE
,

A ND
y
= 1
h 2 y
A N x = 1
h 2 x
⎡
⎢
⎣
⎡
⎢
⎣
− 1
ɛr(y
1+ 1 )
2
,
1
ɛr(y 2− 1 2
A E 11 = A N x ⊗ ˜ Dɛr + IM ⊗ A ND
y + k 2 0µrI,
A E 1
12 =
h2 yɛr(yN+1/2) IM ⊗ eN, A E 21 = (A E 12) T ,
) , −
−1 1
1 −2 1
1 −1
1
ɛr(y
1+ 1 )
2
1
ɛr(y
2− 1 )
2
+ 1
ɛr(y
2+ 1 )
2
⎤
,
1
ɛr(y
2+ 1 )
2
1
ɛr(y N− 1 2
) , −( 1
ɛr(y
N− 1 )
2
+
1
ɛr(y
N+ 1 )
2
)
⎥
⎦ Dɛr
˜ = (1/ɛr(y1), 1/ɛr(y2), . . . , 1/ɛr(yN)).
k Im(ɛr(x, y)) ≥ 0
Im(GE)
u ∈ H 1 M(Ω) := {u ∈ H 1 (Ω) ∩ H 1/2
00 (Γ), u = 0 ∂Ω\Γ}
a(u, v) = −(gM, v)Γ,
a(u, v) = − ∇u · ∇vdxdy +
Ω
(gM, v)Γ =
Γ
Ω
k 2
u¯vdxdy + IM(u)¯v(x, 0)dx,
Γ
gM(x)¯v(x, 0)dx,
H 1/2
00 (Γ) = {u ∈ H1/2 (Γ) : ũ ∈ H 1/2 (R) ũ = 0 R\Γ u = ũ}.
{xi, yj}
Th Ω
⎤
⎥ ,
⎥
⎦

K ⊂ Ω Ks Γ V M h ∈ H1 M (Ω)
vh ∈ V M h
V M h = {vh ∈ H 1 M(Ω) : vh|K ∈ (P1 × P1)(K) K ∈ Th}.
vh(x, y) =
ξi(x)
AM BM
B T M
TM
M
i=1
N+1
j=1
u
ub
vijξi(x)ξj(y),
0
=
a M
′
mn = − ξ i1
Ω
(x)ξj1 (y)ξ′ i2 (x)ξj2 (y) + ξi1 (x)ξ′ j1 (y)ξi2 (x)ξ′ j2 (y) dxdy
+ k
Ω
2 ξi1 (x)ξj1 (y)ξi2 (x)ξj2 (y)dxdy, i1, i2 = 1, 2, . . . , M; j1, j2 = 1, 2, . . . , N,
b M
′
m,i2 = − ξ i1
Ω
(x)ξj1 (y)ξ′ i2 (x)ξN+1(y) + ξi1 (x)ξ′ j1 (y)ξi2 (x)ξ′ N+1(y) dxdy
+ k
Ω
2 ξi1 (x)ξj1 (y)ξi2 (x)ξN+1(y)dxdy, i1, i2 = 1, 2, . . . , M; j1 = 1, 2, . . . , N,
t M
′
i1,i2 = − ξ i1
Ω
(x)ξN+1(y)ξ ′ i2 (x)ξN+1(y) + ξi1 (x)ξ′ N+1(y)ξi2 (x)ξ′ N+1(y) dxdy
+ k
Ω
2 ξi1 (x)ξN+1(y)ξi2 (x)ξN+1(y)dxdy
+ IM(ξi2
Γ
)ξi1 (x)dx, i1, i2 = 1, 2, . . . , M,
f M
i1 = − gMξi1
Γ
(x)dx, i1 = 1, 2, . . . , M,
m = (i1 − 1)N + j1, n = (i2 − 1)N + j2.
A1 = hxA D x A2 = hyA D y
C1 = hx
⎡
⎢
⎣
fM
,
AM = A1 ⊗ C2 + C1 ⊗ A2 + C1 ⊗ C2M,
BM = hy
6 A1 ⊗ eN + 1
C1 ⊗ eN + cN+1C1 ⊗ eN,
2/3 1/6
1/6 2/3 1/6
1/6 2/3
⎤
hy
⎡
⎥
⎦ , C2
⎢
= hy ⎢
⎣
2/3 1/6
1/6 2/3 1/6
1/6 2/3
⎤
⎥
⎦ ,

C2M
0
(C2M)ij =
−b
k 2 (y)ξi(y)ξj(y)dy, cN+1 =
0
−b
k 2 (y)ξN(y)ξN+1(y)dy.
ψ = ∂u
∂y
ψ = ɛ −1
r
H 1 e (Ω) = {u ∈ H 1 (Ω), ∂u
∂n
y=0 +
∂u
∂y
.
y=0 −
∈ H−1/2
00
.
(Γ), u = IE( ∂u
) Γ},
∂n
Heb(Γ) = {φ ∈ H −1/2 (Γ) : v ∈ H 1 e (Ω) ∂v
∂y
= φ Γ}.
u ∈ H 1 e (Ω) ψ ∈ Heb(Γ)
−(ɛ −1
r ∇u, ∇v) + (k 2 0µru, v) + (ψ, v)Γ = 0, v ∈ H 1 e (Ω),
(u, φ)Γ − (IE(ψ), φ)Γ = (gE, φ)Γ, φ ∈ Heb(Γ).
u ψ
V E h ∈ H1 e (Ω) W E h ∈ Heb(Γ)
vh ∈ V E h φh ∈ W E h
V E h = {vh ∈ H 1 e (Ω) : vh|K ∈ (P1 × P1)(K) K ∈ Th},
W E h = {φh ∈ Heb(Γ) : φh|Ks ∈ P1(Ks) Ks ∈ Γ}.
vh(x, y) =
φh(x) =
M+1
i=0
M+1
i=0
N+1
j=0
vijξi(x)ξj(y), (x, y) ∈ Ω,
φiξi(x), x ∈ Γ,
ξi(x) Γ
vh(x, 0) =
M+1
vi,N+1ξi(x).
i=0

a E
mn = −
+
Ω
b E m,i2 =
Ω
AE BE
B T E TE
u
ψ
0
=
ɛ −1 ′
r ξ i1 (x)ξj1 (y)ξ′ i2 (x)ξj2 (y) + ξi1 (x)ξ′ j1 (y)ξi2 (x)ξ′ j2 (y) dxdy
fE
,
k 2 0µrξi1 (x)ξj1 (y)ξi2 (x)ξj2 (y)dxdy, i1, i2 = 0, 1, . . . , M + 1; j1, j2 = 0, 1, . . . , N + 1,
ξi1
Γ
(x)ξi2 (x)dx, i1, i2 = 0, 1, . . . , M + 1,
t E i1,i2 = IE(ξi2
Γ
(x′ ))ξi1 (x)dx, i1, i2 = 0, 1, . . . , M + 1,
f E i1 =
gE(x)ξi1
Γ
(x)dx, i1 = 0, 1, . . . , M + 1,
m = i1(N + 2) + j1, n = i2(N + 2) + j2.
ɛr = ɛr(y)
A N 1 = hxA N x
C N 1 = hx
⎡
⎢
⎣
AE = A N 1 ⊗ ˜ C2E + C N 1 ⊗ A ND
2E + k 2 0µrC N 1 ⊗ C N 2 ,
BE = C1 ⊗ eN+2,
1/3 1/6
1/6 2/3 1/6
1/6 2/3 1/6
1/6 1/3
A ND
2E ˜ C2E
(A ND
0
2E )ij = −
−b
⎤
⎥
⎦
, C N 2 = hy
⎡
⎢
⎣
ɛ −1
r (y)ξ ′ i(y)ξ ′ j(y)dy, ( ˜ C2E)ij =
1/3 1/6
1/6 2/3 1/6
1/6 2/3 1/6
1/6 1/3
0
−b
ɛ −1
r (y)ξi(y)ξj(y)dy.
GM
GE TM TE
Jν(z) Yν(z) H (1)
1
H (1)
1 (z) = J1(z) + iY1(z).
(z)
⎤
⎥ ,
⎥
⎦

J1(z) ∼ O(z), Y1(z) ∼ O( 1
),
z
z → 0,
IM(u)
J1(z) = J1(z)/z
(IM(u)(xi, 0)) = k0
2
a
0
1
|xi − x ′ | J1(k0|xi − x ′ |)u(x ′ , 0)dx ′ ≈ (g M ij )u(xj, 0),
(g M ij ) = k2 0 hx
2 J1(k0|xi − xj|),
e.g.
a
(IM(u)) = −1
=
2 0
Y1(k0|x − x ′ |)u(x ′ , 0)
(x − x ′ ) 2 dx ′ ≈ (g M ij )u(xj, 0),
Y1(z) = zY1(z) (gM ij ) = −αij Y1(k0|xi − xj|)/2
αij =
⎧
⎪⎨
⎪⎩
1 (1 − ln 2), |i − j| = 1,
hx
− 2 , i = j,
hx
1 (j−i)2
h ln x (j−i) 2 , .
−1
O(hx)
IE(v)(x) = I (1)
E (v)(x) + I(2)
E (v)(x),
I (1)
1
E (v)(x) =
2 Γ
I (2)
i
E (v)(x) = −
2 Γ
J0(z) I (1)
E
I (1)
E
I (1)
E (v)(xi) = 1
2
(v)(x)
Γ
1
ɛr(x ′ ) Y0(k0|x − x ′ |)v(x ′ )dx ′ ,
1
ɛr(x ′ ) J0(k0|x − x ′ |)v(x ′ )dx ′ .
Y0(z) ∼ ( 2 z
ln
π 2 )J0(z), z → 0,
1
ɛr(x ′ ) Y0(k0|xi − x ′ |)v(x ′ )dx ′ + 1
2
(v)(x) I(2)
E (v)(x)
Γ
1
ɛr(x ′ )
2
π ln k0|xi − x ′ |
2
J0(0)v(x ′ )dx ′ ,

Y0(z) = Y0(z) − ( 2
π
1
2
Γ
ln z
2 )J0(0)
1
ɛr(x ′ ) Y0(k0|xi − x ′ |)v(x ′ )dx ′ ≈
i.e.
1
2 Γ
1
ɛr(x ′ )
2
π ln k0|xi − x ′ |
2
J0(0)v(x ′ )dx ′ ≈ J0(0)
π
j
I (1)
E (v)(xi) ≈
j
j
hx
2 Y0(k0|xi − xj|) v(xj)
ɛr(xj) .
v(xj)
ɛr(xj) Γ
(g E ij) v(xj)
ɛr(xj) ,
(g E ij) = hx
2 Y0(k0|xi − xj|) + J0(0)
xj+1
π xj−1
ln k0|(xi − x ′ )|
ξj(x
2
′ )dx ′
,
ln k0|xi − x ′ |
ξj(x
2
′ )dx ′ .
I (2)
E (v)(x)
I (2)
E (v)(xi) = − 1
2 Γ
1
ɛr(x ′ ) J0(k0|xi − x ′ |)v(x ′ )dx ′ ≈
j
(g E ij) v(xj)
ɛr(xj)
(g E ij) = − hx
2 J0(k0|xi − xj|).
(g M ij )
(g E ij )
(gM ij )
GM = Gre M + iGim M GE = Gre E + iGim E Gre M = Gim M =
(gM ij )
Gre E =
(gE ij )
Gim E =
(gE ij )
k Im(ɛr(x, y)) ≥ 0
GE Gim M Gim E
G im M
G im M G im E
TM TE
H (1)
1 (z)
z
= H (1) d
0 (z) −
dz H(1) 1
(z) = H(1)
0
d2
(z) + H(1)
dz2 0 (z),

t M ij = ik0
2
= ik2 0
2
− i
2
H (1)
1 (k0|x − x ′ |)
|x − x ′ |
Γ Γ
xi+1 xj+1
xi−1 xj−1
xi+1 xj+1
xi−1
xj−1
ξi(x)ξj(x ′ )dxdx ′
H (1)
0 (k0(x − x ′ ))ξi(x)ξj(x ′ )dxdx ′
H (1)
0 (k0(x − x ′ ))ξ ′ i(x)ξ ′ j(x ′ )dxdx ′ .
H (1)
0 (z)
t E ij = − i
xi+1 xj+1
H
2 xi−1 xj−1
(1)
0 (k0|x − x ′ |)ξi(x)ξj(x ′ )dxdx ′ ,
δ = hx/5
1
0.25 a = 1.0 b = 0.25
ɛr = 1.0 ɛr = 4 + i
⎧
⎨ ɛr0 , −b/3 ≤ y < 0,
ɛr(x, y) = 1.5ɛr0 , −2b/3 ≤ y < −b/3,
⎩
2ɛr0 , −b ≤ y < −2b/3.
k0 = 2π
σ = 4
|P (φ)| 2 ,
φ P
P (φ) = k0
2
k0
sin φ
Γ
u(x, 0)e ik0x cos φ dx.
k0 = 2π
θ = 0

PHASE
RCS σ/λ (dB)
MAGNITUDE
3
2
1
−100
(a)
0
0 0.2 0.4 0.6 0.8 1
x
(b)
50
0
−50
0 0.2 0.4 0.6 0.8 1
x
(c)
20
0
−20
−40
−60
−80
100 120 140
φ (degree)
160 180
PHASE
RCS σ/λ (dB)
MAGNITUDE
3
2
1
−100
(d)
0
0 0.2 0.4 0.6 0.8 1
x
(e)
50
0
−50
0 0.2 0.4 0.6 0.8 1
x
(f)
20
0
−20
−40
−60
−80
100 120 140
φ (degree)
160 180
k0 = 2π ɛr = 1.0
ɛr = 4 + i
k0 = 16π ɛr = 4+i
Γ M = N = 1024
θ = 0
k0
N = M = 2048 M
x MN
u:,N+1 O(M 2 ) M × M

MAGNITUDE
2
1.5
1
0.5
(a): k 0 =16π,ε r =4+i,θ=0
0
0 0.2 0.4 0.6 0.8 1
x
2
1.5
1
0.5
(b): k 0 =16π,inhomogeneous medium,θ=0
0
0 0.2 0.4 0.6 0.8 1
k0 = 16π
ɛr ɛr = 4 + i
MAGNITUDE
x

δ = hx/5
k0
M × N
k0 = 4π 256 2
512 2
1024 2
2048 2
k0 = 8π 256 2
512 2
1024 2
2048 2
k0 = 16π 256 2
512 2
1024 2
2048 2
k0 = 32π 256 2
512 2
1024 2
2048 2
k0 = 4π 256 2
ɛr = 4 + 512 2
1024 2
2048 2
k0 = 8π 256 2
512 2
1024 2
2048 2

δ = hx/5
k0
M × N
k0 = 4π 256 2
512 2
1024 2
2048 2
k0 = 8π 256 2
512 2
1024 2
2048 2
k0 = 16π 256 2
512 2
1024 2
2048 2
k0 = 32π 256 2
512 2
1024 2
2048 2
k0 = 4π 256 2
ɛr = 4 + 512 2
1024 2
2048 2
k0 = 8π 256 2
512 2
1024 2
2048 2

δ = hx/5
k0
M × N
k0 = 4π 256 2
512 2
1024 2
2048 2
k0 = 8π 256 2
512 2
1024 2
2048 2
k0 = 16π 256 2
512 2
1024 2
2048 2
k0 = 32π 256 2
512 2
1024 2
2048 2
k0 = 4π 256 2
ɛr = 4 + 512 2
1024 2
2048 2
k0 = 8π 256 2
512 2
1024 2
2048 2