H
|ρ| =
n1 > n 2
1
0 < ρ < 1
||
n1 < n 2
|ρ| = 0
2 E = 0
(
2
+ κ2 )E = 0
κ
∂ 2 ∂ 2 ∂ 2 + 2 + 2 + κ2 E x = 0 2 ∂x ∂y ∂z ∂ 2 ∂ 2 ∂ 2 + 2 + 2 + κ2 E y = 0 2 ∂x ∂y ∂z ∂ 2 ∂ 2 ∂ 2 + + + κ2 E z = 0 2 2 2 ∂x ∂y ∂z
E
x
∂ 2 ∂ 2 ∂ 2 + 2 + 2 + κ2 E x = 0 2 ∂x ∂y ∂z E
∂ 2 ∂ 2 + 2 + κ2 E x = 0 2 ∂y ∂z z
E =
E 0 (z )ax
d2 E x + κ2 E x = 0 2 dz
E x (z ) = C 1 e− jκz + C 2 e jκz
∂ E ∂x x
=0
2 E = 0
(
2
+ κ2 )E = 0
κ
∂ 2 ∂ 2 ∂ 2 + 2 + 2 + κ2 E x = 0 2 ∂x ∂y ∂z ∂ 2 ∂ 2 ∂ 2 + 2 + 2 + κ2 E y = 0 2 ∂x ∂y ∂z ∂ 2 ∂ 2 ∂ 2 + + + κ2 E z = 0 2 2 2 ∂x ∂y ∂z
E
x
∂ 2 ∂ 2 ∂ 2 + 2 + 2 + κ2 E x = 0 2 ∂x ∂y ∂z E
∂ 2 ∂ 2 + 2 + κ2 E x = 0 2 ∂y ∂z z
E =
E 0 (z )ax
d2 E x + κ2 E x = 0 2 dz
E x (z ) = C 1 e− jκz + C 2 e jκz
∂ E ∂x x
=0
C 1
C 1,2 = C 1,2 e jϕ 1,2
| |
C 2 E (t, z )
= [ C 1 cos(ωt cos(ωt
− κz + ϕ ) + |C | cos(ωt cos(ωt + κz + ϕ )]a |C | cos(ωt cos(ωt − κz + ϕ ) |C | cos(ωt cos(ωt + κz + ϕ )
| |
1
2
1
z
1
x
1
1
1
z ∆z/∆ z/ ∆t
∆t ) ω(t + ∆t
− κ(z + ∆z ∆z ) ν p =
∆ ω ω = = = ∆t κ ω µε
√
ν p = √ µ10 ε0 = c = 299792458 ∆z
κ=
ν p ωt
− κz =
√1µε ν p 2π λ κ∆ = κλ = 2π
2π λ
z = ctte.
H
×E − ωµ
H = H
= =
−
−
−
H =
×
1 ∂ ∂ ∂ ax + ay + az jωµ jω µ ∂x ∂y ∂z 1 ( jκ) jκ ) C 1 e− jκz C 2 e jκz jωµ jω µ
(az
−
E )
× µ ε
= az
E
ay
× E η
η = µ/ε 120π 120π = 377 Ω
η = η0 =
C 1 e− jκ ax C 1 − jκ H = e ay η E =
z
µ0 /ε0 =
0
ε = ε β π/2
≤ ≤
− jε = |ε|−
µ = µ
− jµ = |µ|e− κ = ±ω |ε|− |µ|e− − j sin κ = ±ω |ε||µ| cos κ = κ − jκ jα
jβ
α+β
α+β
2
2
jα
jβ
0
=
α π/2 α β − j + ε µe 2
≤ ≤ ± | || | ω
κ , κ > 0
E x (z) = C 1 e−κ z e− jκ z + C 2 eκ z e jκ z
E (t, z)
= [ C 1 e−κ z cos(ωt
| |
κ z
− κz + ϕ ) + |C |e 1
2
cos(ωt + κ z + ϕ1 )]ax κ
κ L
E (z) = κ ∆z L = ln E (z + ∆z)
E (z) = 20 log eκ L = 20 log E (z + ∆z)
∆z
= κ ∆z20loge δ
δ = (κ )−1
C 1 e−κ z e− jκ z ax C 1 −κ z − jκ z H = e e ay η E =
η=
φ = (β
− α)/2
| | | | µ e− jβ = ε e− jα
|| −45◦ ≤ φ ≤ 45◦
µ e ε
||
= C 1 e−κ z cos(ωt C 1 −κ z cos(ωt e H(t, z) = η E (t, z)
| | | | ||
−j
(β −α) 2
= η e− jφη
||
− κz + ϕ )a − κz + ϕ + φ )a 1
1
x
η
y
µ
× H = J + ωεE × H j
ε∼ = ε
tg∆ =
0
= [ jω(ε
jε ) + σ]E σ j ε + ω
−−
= jω ε
= jωεE ε = |ε|e−
ε = 0
µ
σ ε + ω ε
σ =0
E
=
σ
1, 1,
σ ωε
; . σ =0
=0
− − − ··· ≈ − ≈
κ=ω =ω
µ (
)
µ 1
1 = ω µ 1 + 2 8 ω µ 1 2 µ
η
2
1 + 16
3
1+ 2
= 0
− − √ − ≈ − ≈ −
κ=ω = =
σ ω σ ω 2 µ ω ωµ (ω σ) ωµσ µ
ωµσ (1 2
(1 + x)n = 1 + nx +
n(n−1)
2!
x2 +
n(n−1)(n−2)
3!
x3 + . . .
+
)
n
σ
ω
η
κ=κ
≈
ωµ (1 + ) 2σ
− κ
η=
κ ω µ ω µ0 ω µ ω µ
κ ω µ ω µ0 0
ϕη
R + X = |η|e
√√ √√ R X √ √ ω
2
ωµσ
2
ω ω
µ µ0
µ µ ωµ 2σ
µ ωµσ
2
ω ω 0
2
µ µ0
µ ωµ 2σ
x z
Ae− jκz ax A − jκz H = ay e η E =
αi βi
αi βi
γ i
i
γ i
i
∈ {1, 2, 3}
∈ {1, 2, 3}
Ae− jκ(x cos γ 1 +y cos γ 2+z cos γ 3 ) (cos α1 ax + cos α2 ay cos α3 az ) A − jκ (x cos γ 1+y cos γ 2 +z cos γ 3) H = (cos β1 ax + cos β2 ay + cos β3 az ) e η E =
x cos γ 1 + y cos γ 2 + z cos γ 3 = ctte. κ κ
= κ(cos γ 1 ax + cos γ 2 ay + cos γ 3 az )
r
r
·r
= xax + y ay + z az z
E (t)
= F −1
{E (ω)}
E (ω) ∞
1 jωt E (ω)e dω E (t) = 2π −∞
ˆ
E (t)
∆ω ω0
E (t) κ = κ(ω)
E (t, z)
E (t, z)
= Re
ˆ
ω0 + ∆2ω
1 π
ω0
− ∆2ω
z
E (ω
κ
0
−
dω
κ(ω) κ0 = κ(ω0 ) dκ κ(ω) = κ0 + dω
E (t, z)
j [ωt κ(ω )z ]
− ω )e
= Re
ˆ
ω0 + ∆2ω
1 π
ω0
− ∆2ω
E (ω
(ω
−ω )+ 0
ω0
j ωt
− ω )e 0
κ0 +
−
|
dκ (ω dω ω 0
−ω0 )+
z
dω
∆ω
E (t, z)
= Re = Re
ˆ ˆ 1 π 1 π
ω0 + ∆2ω
ω0
−
∆ω 2
ω0 + ∆2ω
ω0
− ∆2ω
E (ω
E (ω
F −1 X (ω)e− ωt0
{
ν gr E (t, z)
j ωt
− ω )e 0
jω (t
0
=
E
t
|
dκ (ω dω ω 0
|
dκ z) dω ω 0
1
dωe
−ω0 )
− j
κ0
z
−
dω
|
dκ ω0 dω ω 0
z
} = x(t − t ) F − {X (ω − ω )} = x(t)e dκ/dω | ν = dω/dκ 0
− − E
κ0 +
−
−
− ω )e ω0
= Re
t
z ν gr
z ν gr
ω0 t
0
gr
jω 0 (t
e
cos(ω0 t
− νgrz )e− j (κ0− νωgr0 )z
− κ z) 0
z ν gr
ν gr ν gr κ(ω)
(ε1 , µ1 )
(ε2 , µ2 )
1 E 1 H 1
2 E 2 H 2
= C 3 e− κ2 z ax C 3 = e− κ2 z ay η2 1
◦
E
◦
H
= C 1 e− κ1z ax C 1 − κ1 z = e ay η1
1 −
E
−
H
= C 2 e κ1 z ax C 2 κ1 z = e ay η1
−
= C 1 e− κ1z + C 2 e κ1 ax 1 = C 1 e− κ1 z C 2 e κ1 ay η1
−
2 = C 3 e− κ2z ax C 3 − κ2 z ay = e η2
+
E
+
H
E 1 (0) = E 2 (0) H 1 (0) = H 2 (0)
C 1 + C 2 = C 3 C 1 C 2 C 3 = η1 η1 η2
−
ρ
E − (0) C 2 = ρ= ◦ E (0) C 1 E + (0) C 3 = τ = ◦ E (0) C 1
1 + ρ = τ 1 ρ τ = η1 η1 η2
−
−
η2 η1 η1 + η2 2η2 τ = η1 + η2 ρ=
1 E 1 H 1
= C 1 e− κ1 z + ρe κ1z C 1 − κ1 z = e ρe κ1z η1
2 E 2 H 2
−
= C 1 τ e− κ2 z ax C 1 τ − κ2z ay = e η2
ax ay
τ
C 1
2 2
1
1.5 1 0.5 ρ
0
1
τ
−0.5
η / η 2
−1 0
5
10
15
ρ
1
η2 /η1
20
ρ
τ
−1
η2 /η1
1
−1 ≤ ρ ≤
|ρ| = 1
0
τ
1
1
|ρ| = 0
0< ρ <1
||
|ρ| = 0 η2 = η1
1 C 2 = 0
|ρ| = 1
η2 = 0
|ρ| = 1
2
η2 /η1
→∞
2 0
≤ τ ≤ 2
1 1
ρ= E 1
= C 1 [e− κ1z =
E 1 (t, z)
= =
−1 κ1 z
−e
] ax
H 1
−2 C sin(κ z)a 1
1
x
ωt
{E e } 2 |C | sin(κ z)sin(ωt + φ )a 1
1
1
1
=
C 1 η1
=
2C 1
H1 (t, z)
x
η1
[e− κ1 z + e κ1 z ] ay cos(κ1 z)ay
ωt
=
{H e }
=
2 C 1
1
| | cos(κ z)cos(ωt + φ )a 1 1 y η1
C 1 = C 1 e φ1
| |
1 E H 2 2 C 1 2 = sin(κ1 z)cos(κ1 z)az η1
S =
× ∗ − | |
0< ρ <1
||
E 1
= C 1 [e− κ1 z + ρe κ1z ] ax = C 1 [(τ
|ρ| = 0, 1
|ρ|
− ρ)e−
κ1 z
+ ρe κ1 z ] ax
= C 1 τ e− κ1 z ax + 2 C 1 ρ sin(κ1 z)ax
H 1
=
C 1 η1
[e− κ1 z
=
C 1 η1
[(τ
=
C 1 − κ1z τ e ay η1
κ1 z
− ρe
] ay
− ρ)e− − ρe κ1 z
κ1 z
] ay
− 2C ρ cos(κ z)a η 1
1
y
1
Z (z)
z E x ( ) H y ( ) e κ1 + ρe− κ1 = η1 κ1 (e ρe− κ1 ) η2 cos(κ1 ) + η1 sin(κ1 ) = η1 η1 cos(κ1 ) + η2 sin(κ1 )
Z (z) z=− =
|
− −
−
Z ()
λ1 /2 η2 cos[(2π/λ1 )(λ1 /2)] + η1 sin[(2π/λ1 )(λ1 /2)] η1 cos[(2π/λ1 )(λ1 /2)] + η2 sin[(2π/λ1 )(λ1 /2)] η2 ( 1) + η1 (0) = η1 η1 ( 1) + η2 (0) = η2
Z (λ1 /2) = η1
− −
1 Z (λ1 /2)
ε µ
σ
∂ E 1 = ∂t ε 1 ∂ H = ∂t µ
× H − σε E − × E
1 ∂E x = ∂t ε 1 ∂E y = ∂t ε 1 ∂E z = ∂t ε
∂H z ∂y ∂H x ∂z ∂H y ∂x
− − −
∂H y ∂z ∂H z ∂x ∂H x ∂y
− − −
E =
∂E x = ∂t ∂H y = ∂t
1 ∂H x = ∂t µ 1 ∂H y = ∂t µ 1 ∂H z = ∂t µ
σ E x ε σ E y ε σ E z ε
E x(t, z)ax
∂E y ∂z ∂E z ∂x ∂E x ∂y
− − −
∂E z ∂y ∂E x ∂z ∂E y ∂x
σ − 1ε ∂H − E ∂z ε − µ1 ∂E ∂z y
x
x
ax
ay
az
∆z ∆t K
T K
× T
E xn (κ) = E x (n∆t, κ∆z) E x
...,κ
−
−
...,n H y 1 1 + 2, . . . 2, κ E x
1 2, n
n κ
−
1 2,
{. . . , κ , κ + 1, . . .}
n + 12 , . . .
{. . . , n , n + 1, . . .}
E x H y
H y H y
E x
n+ 12
E x
− E − n x
(κ)
1 2
(κ)
=
∆t
H yn+1 (κ + 1/2) H yn (κ + 1/2) = ∆t
−
n+ 12
E x n+ 12 E x (κ)
−
1 = 1+
σ ∆t 2ε σ ∆t 2ε
−
1 H yn (κ + 1/2) H yn (κ ∆z ε
−
1 E x µ
n+ 12
− 12
n E x
(κ)
n+ 12
∆t − µ1 ∆z
− 12
n
(κ) = E x
ε0
µ0
−
ε
− 12
n
+ E x 2
(κ)
−
(κ)
η0 =
1 ∆t H yn (κ + 1/2) σ∆t 1 + 2ε ∆z
−ε
n+ 12
E x
ε µ
E x
n+ 12
(κ + 1) E x ∆z
n+ 12 (κ) x
− 1/2) − σ E
H yn+1 (κ + 1/2)
(κ)
H yn+1 (κ + 1/2) = H yn (κ + 1/2)
−
n+ 12
(κ + 1)
E x
n y
− H (κ − 1/2)
(κ)
σ
∆t − 1ε ∆z
H yn (κ + 1/2)
n y
− H (κ − 1/2)
E x
µ0 /ε0 ˜= E
ε0 E µ0
µ = µ0 n+ 12
˜x E
−
1 (κ) = 1+
σ ∆t 2ε σ∆t 2ε
H yn+1 (κ + 1/2) = H yn (κ + 1/2)
√
c0 = 1/ µ0 ε0
1
˜xn− 2 (κ) E
−
−εc
r 0
∆t − c1 ∆z 0
1 1+ 1
σ ∆t 2ε
˜xn+ 2 (κ + 1) E
∆t H yn (κ + 1/2) ∆z 1
˜xn+ 2 (κ) E
n y
− H (κ − 1/2)
εr
(κ)
∆t
∆t
∆t
≤ √∆z nc
0
n=1 ∆t
n
∆t =
1 ˜xn+ 2 (κ) E
∆z 2c0
− − − −
1 = 1+
σ ∆t 2ε σ ∆t 2ε
1
˜xn− 2 (κ) E
H yn+1 (κ + 1/2) = H yn (κ + 1/2)
1/2 εr 1 + σ2∆ε t
1 ˜ n+ 12 E x (κ + 1) 2
∆z 10
z =0 E xn (0) = E xn−2 (1)
H yn (κ + 1/2) 1
˜xn+ 2 (κ) E
n y
− H (κ − 1/2)
E
x
2 ε
ε
0
2
,σ
2
1
0
−1
−2
20
40
60
80
100 z
120
140
160
180
200
120
140
160
180
200
H
y
2 ε
ε
0
2
,σ
2
1
0
−1
−2
20
40
60
80
100 z
E x
H y
E
x
2 ε
ε
0
2
,σ
2
1
0
−1
−2
20
40
60
80
100 z
120
140
160
180
200
120
140
160
180
200
H
y
2 ε
ε
0
2
,σ
2
1
0
−1
−2
20
40
60
80
100 z
E x
H y
E
x
2 ε
ε
0
2
,σ
2
1
0
−1
−2
20
40
60
80
100 z
120
140
160
180
200
120
140
160
180
200
E x
H y
H
y
2 ε
ε
0
2
,σ
2
1
0
−1
−2
20
40
60
80
100 z
E
x
2 ε
ε
0
2
,σ
2
1
0
−1
−2
20
40
60
80
100 z
120
140
160
180
200
120
140
160
180
200
E x
H y
H
y
2 ε
ε
0
2
,σ
2
1
0
−1
−2
20
40
60
80
100 z
z◦ z− z+
z
z◦ z− z+ ϕ
ψ
z ϕ
θ
θ
z z◦ z−
z+
γ 1◦ = z ◦ x = 90◦ γ 2◦ = z ◦ y = 90 ◦ γ 3◦ = z ◦ z = ϕ
−
γ 1− = z − x = 90◦ ϕ γ 2− = z − y = ψ 90◦ γ 3− = z − z = ψ
−
γ 1+ = z + x = 90 ◦ γ 2+ = z + y = 90◦ γ 3+ = z + z = θ
f ◦ (y, z) = e− jκ 1(y sin ϕ+z cos ϕ)
−θ
ψ
f − (y, z) = e− jκ1 (y sin ψ+z cos ψ) f + (y, z) = e− jκ2 (y sin θ+z cos θ) f ◦ f − f +
y f ◦ (y, 0) = f − (y, 0) = f + (y, 0) f ◦ (y, 0) = f − (y, 0) f ◦ (y, 0) = f + (y, 0)
0 180◦
−ψ =ϕ
≤ ϕ ≤ 90◦
90◦
⇒ sin ϕ = sin ψ ⇒ κ sin ϕ = κ sin θ 1
2
≤ ψ ≤ 180◦
sin θ κ1 = sin ϕ κ2 n=
√ε µ r
r
sin θ n1 = = n12 sin ϕ n2 n12 = n1 /n2
1 n1 > n 2
2
n1 < n 2 n1 > n2 ϕ n12
ϕ
↑⇒ θ ↑
ϕc ϕc = sin−1
cos θ =
± − 1
ϕ > ϕc θ sin2 θ
±
cos θ = j
n12
n1 n2
↑
sin θ > sin ϕ θ = 90◦
⇒θ↑
n2 n1
sin θ = (n1 /n2 )sin ϕ > 1
2
sin2 ϕ
− 1 = ± jα(ϕ)
ϕc
xy
xy
ϕ > ϕc
ϕ = ϕc
xy ϕ > ϕc
n1 > n 2
β(ϕ) = sin θ =
n1 n2
sin ϕ
± jα(ϕ) f + (y, z) = e−κ2 zα(ϕ) e− jκ2 yβ (ϕ)
f + n1 < n2 n1 < n2
n1 sin θ n1 = sin ϕ n2
n
1⇒
2
n1 n2
→0⇒θ→0
o
E
o
H
−
E
−
H
+
E
+
H
A B
= Be − jκ 1(y sin ψ+z cos ψ) ae B = e− jκ1 (y sin ψ+z cos ψ) ah η1
−
−
= Ce− jκ2 (y sin θ+z cos θ) ae+ C = e− jκ 2 (y sin θ+z cos θ) ah+ η2
C aeo,−,+
aho,−,+
aen ahn
n
= Ae− jκ 1(y sin ϕ+z cos ϕ) ae ◦ A = e− jκ1 (y sin ϕ+z cos ϕ) ah ◦ η1
= cos α1n = cos β1n
+ cos α2n n ax + cos β2 ax
+ cos αn3 n ay + cos β3 ay
az az
∈ {◦, −, +} o
E
o
H
= Ae− jκ 1 (y sin ϕ+z cos ϕ) ax A = e− jκ1 (y sin ϕ+z cos ϕ) (cos ϕay η1
− sin ϕa ) z
x
, ,+
y
◦ −
α1◦ α2◦ α3◦ β1◦ β2◦ β3◦
−
α− 1 α− 2 α− 3 − β1 β2− β3−
0◦ 90◦ 90◦ 90◦ ϕ ◦ 90 + ϕ
,
,+
◦ −
0◦ α1+ 90◦ α2+ 90◦ α3+ 90◦ β1+ 180◦ ϕ β2+ 270◦ ϕ β3+
0◦ 90◦ 90◦ 90◦ θ ◦ 90 + θ
− −
= Be − jκ1 (y sin ψ+z cos ψ) ax B − jκ 1 (y sin ψ+z cos ψ) = (cos ϕay + sin ϕaz ) e η1
E
−
−
H
+
E
+
H
= Ce− jκ 2(y sin θ+z cos θ) ax C = e− jκ 2 (y sin θ+z cos θ) (cos θay η2
− sin θa )
E ◦ (y, 0) + E − (y, 0) = E + (y, 0) H y◦ (y, 0) + H y− (y, 0) = H y+ (y, 0)
1 (A η1
−
ρ⊥
A + B = C 1 B)cos ϕ = C cos θ η2 τ ⊥
E − (y, 0) B = ρ⊥ = ◦ E (y, 0) A E + (y, 0) C = τ ⊥ = ◦ E (y, 0) A κ1 sin ϕ = κ1 sin ψ = κ2 sin θ
z
1 (1 η1
−
1 + ρ⊥ = τ ⊥ 1 ρ⊥ )cos ϕ = τ ⊥ cos θ η2
η2 cos ϕ η1 cos θ η2 cos ϕ + η1 cos θ 2η2 cos ϕ τ ⊥ = η2 cos ϕ + η1 cos θ
−
ρ⊥ =
sin ψ = sin ϕ cos ψ = cos ϕ
−
z<0 = E o + E − = Ae− jκ 1y sin ϕ (e− jκ 1z cos ϕ + ρ⊥ e jκ 1z cos ϕ )ax A = H o + H − = e− jκ 1 y sin ϕ e− jκ1 z cos ϕ ρ⊥ e jκ 1 z cos ϕ cos ϕay η1 e− jκ1 z cos ϕ + ρ⊥ e jκ1 z cos ϕ sin ϕaz
E 1 H 1
−
z>0 E 2 H 2
= E + = Aτ ⊥ e− jκ2 (y sin θ+z cos θ) ax A = H + = τ ⊥ e− jκ 2(y sin θ+z cos θ) (cos θay η2
E
− sin θa ) z
H ⊥
⊥
a
◦,−,+
e ◦ ◦ ,−,+ ,−,+ α1,2,3 β1,2,3
o
E
o
H
H
ah◦,−,+
= Ae− jκ1 (y sin ϕ+z cos ϕ) (cos ϕay A − jκ1 (y sin ϕ+z cos ϕ) ax = e η1
−
−
x y
z
− sin ϕa ) z
−E
⊥
x
, ,+
y
◦ −
α1◦ α2◦ α3◦ β1◦ β2◦ β3◦
−
90◦ ϕ ◦ 90 + ϕ 180◦ 90◦ 90◦
α− 1 α− 2 α− 3 − β1 β2− β3−
−
,+
90◦ α1+ 180◦ ϕ α2+ 270◦ ϕ α3+ 180◦ β1+ 90◦ β2+ 90◦ β3+
− −
jκ 1 (y sin ψ +z cos ψ )
−Be− B = − e− η =
E
,
◦ −
(cos ϕay + sin ϕaz )
jκ 1 (y sin ψ +z cos ψ )
H
90◦ θ ◦ 90 + θ 180◦ 90◦ 90◦
ax
1
+
E
+
H
= Ce− jκ2 (y sin θ+z cos θ) (cos θay C − jκ 2 (y sin θ+z cos θ) ax = e η2
− sin θa )
−
E y◦ (y, 0) + E y− (y, 0) = E y+ (y, 0) H ◦ (y, 0) + H − (y, 0) = H + (y, 0)
(A
− B)cos ϕ = C cos θ
1 C (A + B) = η1 η2 ρ
τ
E − (y, 0) B = ρ = ◦ E (y, 0) A + E (y, 0) C = τ = ◦ E (y, 0) A
−
κ1 sin ϕ = κ1 sin ψ = κ2 sin θ
z
η2 cos ϕ η1 cos θ η2 cos ϕ + η1 cos θ η2 cos θ η1 cos ϕ ρ = η2 cos θ + η1 cos ϕ
2η2 cos ϕ η2 cos ϕ + η1 cos θ 2η2 cos ϕ τ = η2 cos θ + η1 cos ϕ
−
ρ⊥ =
τ ⊥ =
(1 + ρ )cos ϕ = τ cos θ 1 1 (1 ρ ) = τ η1 η2
−
η2 cos θ η1 cos ϕ η2 cos θ + η1 cos ϕ 2η2 cos ϕ τ = η2 cos θ + η1 cos ϕ
−
ρ =
sin ψ = sin ϕ cos ψ = cos ϕ
−
z<0 E 1
= E o + E − = Ae− jκ1 y sin ϕ
= H o + H −
e− jκ1 z cos ϕ + ρ e jκ1 z cos ϕ cos ϕay
e− jκ 1z cos ϕ ρ e jκ1 z cos ϕ sin ϕaz A − jκ1 y sin ϕ − jκ 1 z cos ϕ = (e e ρ e jκ1 z cos ϕ )ax η1
− H 1
−
−
−
z>0 E 2 H 2
= E + = Aτ e− jκ 2 (y sin θ+z cos θ) (cos θay A − jκ 2 (y sin θ+z cos θ) = H + = τ e ax η2
− sin θa ) z
−
ϕB η2 cos θ = η1 cos ϕB
ρ = 0 n1 cos θ = n2 cos ϕB
n1
n2
sin θ = sin ϕB θ = π2 ϕB
n1 n2
sin θ cos θ = sin ϕB cos ϕB
−
n1 cos θ = n2 sin ϕB ρ = 0 n1 sin ϕB = n2 sin θ
sin2 θ + cos2 θ = 1 = sin 2 ϕB + cos2 ϕB sin2 θ
cos2 θ n21 sin θ = 2 sin ϕB n2 n22 2 cos θ = 2 cos ϕB n1 2
n21 n22 sin ϕB + 2 cos ϕB = sin2 ϕB + cos2 ϕB 2 n2 n1
n2 ϕB = tan−1 n1
1 η2
2
η1
σ2
→0 ρ⊥ = ρ =
−1 −1
2 E = H =
κT = κ1 cos ϕ
τ ⊥ = 0 τ = 0
1
− j2A sin(κ
− jκ y ax
T z)e
2A cos(κT z)e− jκ y cos ϕay + j sin(κT z)e− jκ y sin ϕaz η1
κ = κ1 sin ϕ
→∞⇒
3.5
3
2.5
2
1.5
1
0.5
0 0
2
4
6
8
10
12
14
16
18
20
zy
E (y , z , t)
= 2A sin(κT z) sin(ωt κ y)ax 2A [cos(κT z)cos(ωt κ y)cos ϕay H(y , z , t) = η1 sin(κT z)sin(ωt κ y)sin ϕaz ]
− −
−
κT
−
ΛT =
κ
2π κT
− − −
2π z sin ωt ΛT 2π H(y , z , t) = H y cos z cos ωt ΛT 2π H z sin z sin ωt ΛT E (y , z , t)
= E x sin
−
Λ = 2π y Λ 2π y Λ 2π y Λ
2π κ
ax ay
az
S 1,2 sin Λ2πT z
z
H
y
ϕ
x
κT,
ΛT,
cos Λ2πT z
E H
y
κT
↑
ΛT
↑
↓
ϕ 0 < ϕ <
π
2
y y
λT /2 y
ΛT D =
n Λ2T
D n = 1, 2, 3 . . .
E
H
z
2λ
3.5
3
2.5
2
1.5
1
0.5
0 0
2
4
6
8
10
12
14
16
18
20
zy
κT D = nπ nπ κ1 cos ϕ = D nc cos ϕ = 2Df n = 1, 2, . . . , N N
D = 2
◦
◦
◦
◦
f = 40
◦
n = 1, 2, 3, 4, 5
N = 5
2
1
−2A j sin(κ z)e− 2 jA cos(κ z)e− H = − η E =
1
T
T
jκ y
jκ y
cos ϕay + cos(κT z)e− jκ y sin ϕaz
ax
E (y , z , t)
= 2A [sin(κT z)sin(ωt κ y)cos ϕay cos(κT z)cos(ωt κ y)sin ϕaz ] 2A cos(κT z)sin(ωt κ y)ax H(y , z , t) = η1
−
−
−
−
λT /2 y
f + (y, z) = e− jκ (y sin θ+z cos θ) κ=
ωσµ
2
(1
− )
z
2λ
×10 ×10 ×10 ×10
κ0 sin ϕ = κ sin θ sin θ
7 7
κ0 sin ϕ
7 7
κ0 sin ϕ =
κ
sin θ
f + (y, z) =e− jκ (y sin θ+z cos θ)
− j =e
yκ0 sin ϕ+z
√
κ2 κ20 sin2 ϕ
−
=e−Bz z e− j(Ay y+Az z) Ay = κ0 sin ϕ Az = γ
− κ2
κ20 sin2
ϕ
Bz =
− κ2
2 0
κ
− − κ2
κ2
tan γ =
Ay Az
≈0
2
sin2 ϕ Ay
κ20 sin2 ϕ
A
z
sin ϕ
y γ = tan−1 A Az
e−Bz z κ2
κ20
ϕ
≈ ±κ
κ20 sin2 ϕ
≈κ
γ
ϕ
γ ≈ 0◦ 1 E =
E + (0)e− κC z at
H = az
× ηE
C
κC ηC at
E + (0) = E t◦ (0)+E t− (0)
ε = ε0 µ = µ0
σ = 6,1 20◦ 80◦
τ
τ ⊥
ρ
ρ⊥ 0,5 J
−
θ = arcsin sin ϕ
√
κ1 = ω µ0 ε0
κ2 =
ϕ 20◦ 80◦
ϕ 20◦ 80◦
ϕ 20◦ 80◦
ωµ0 σ2
2
(1
j)
θ 3,926 10−4 (1 + j)◦ 1,13 10−3 (1 + j)◦
× ×
τ 4,006 10−5 (1 + j) (4,006 + j4,005) 10−5
×
− −
5
× 10−− − × 10 − 4
× ×
τ ⊥ 10−5 (1 + j) 10−6 (1 + j)
ρ⊥ 1 + j3,765 1 + j6,957
κ1 sin ϕ
sin θ κ2 cos θ = κ2
E +
E
× 10−− × 10
5 6
= E 0 e− jκ 2 (y sin θ+z cos θ) a+ e
+
κ2 sin θ
3,765 6,957
×
ρ 1 + j4,263 1 + j2,307
E
κ2
κ1 κ2
= E 0 e−Bz z e− j(Ay y+Az z) a+ e
− 1
sin2 θ
× 10
7
f = 0,88
Ay = κ2 sin θ Az = Real(κ2 cos θ)
Bz = Im(κ2 cos θ) θ∗
Ay ay + Az az
az
θ∗ = arctan ϕ 20◦ 80◦
≈ ≈
Ay Az
θ∗ 0[◦ ] 0[◦ ]
ϕ +
E
= 0,5e−Bz z e− j (Ay y+Az z) a+ e +
+
H
= az
× E η
2
J
ϕ = 20◦ +
E
= σE +
= 0,5e−460300z e− j (6,308y+460300z) a+ e
+
= 33,127(1 + j)e−460300z e− j(6,308y+460300z) a+ h
J
= 3,05
H
ϕ = 80◦ +
E
7
460300z
× 10 e−
e− j (6,308y+460300z) a+ e
= 0,5e−460300z e− j(18,163y+460300z) a+ e
+
= 33,127(1 + j)e−460300z e− j (18,163y+460300z) a+ h
J
= 3,05
H
2
7
× 10 e−
460300z
e− j(18,163y+460300z) a+ e
2
κ = κ (f )
κ = κ (f ) κ = κ (f )
κ = κ (f )
κ = κ
ν p = ν p (f ) ∆ = ∆(f )
≈
κ
ωµ 0 σ
≈ ω√µ ε 0
κ
≈ ω√µ ε
2
ν gr = ν gr (f ) tan ∆ = tan ∆(f )
δ = δ(f )
H (ω) = e− κ(ω)∆z ∆z
κ(ω) E i (ω) E i (ω) = F i (t)
E (t) ∆z E (t)
{E }
i
i
o
1
E (t) = F − {E (ω)H (ω)} } = F − {E (ω)e− F { } o
1
[]
FFT
IFFT
i i
κ(ω )∆z
[]
ε (ω)
E (t)
N
i
FFT
F −1
{}
ε (ω)
0
ε
2ε
N
H (ω) N
IFFT
E (t) o
ε = ε (ω)
δ
ε = ε (ω)
ω1 E i (t)
H (ω1 ) = e − κ(ω1)δ
Ei (t) e
|E (t)| = | | E (t) o
i
E (t) o
∆z
ω0
E (t) i
