WAVES IN MEDIA
Radio Communications
Satellite
Ionosphere
(plasma)
Cloud, Rain
Refraction, moist or dense air
Troposcatter
Blue sky, red sunset
Reflection
Optical Fibers
Optoelectronics on chips
θ
Devices
Circular
polarization
Linear
L8-1
WAVES IN MEDIA – Constitutive Relations Vacuum:
D = εoE
∇i D = ρ
E
f
ρf = free charge density
+
+
+
+
+
Dielectric D = εE = ε E + P
o
Materials:
∇ i ε o E = ρ f + ρp
∇iP = −ρp
P
+
+
+
+
+
-
polarization charge density ρp
P = “Polarization Vector”
Magnetic
Materials:
-
∇iB = 0
B = μo H in vacuum
B = μH = μo (H + M )
M = “Magnetization Vector”
e
+
+
e
e
B
e B
+
L8-2
TYPES OF MEDIA
Properties are function of:
Designation:
Field direction
Anisotropic D = εE, B = μH
Position
Inhomogeneous
Time: ≠ f(t)
Stationary
≠ f(history)
Amnesic
Frequency
Dispersive
E or H
Non-linear
Temperature
Temperature dependent
Pressure
Compressive
L8-3
ANISOTROPIC DIELECTRICS
D =
Dx =
Dy =
Dz =
εE
ε xx E x + ε xy E y + ε xz Ez
ε yx E x + ε yy E y + ε yz Ez
ε zx E x + ε zy E y + ε zz Ez
⎡ε x
⎢
Let ε = ⎢ 0
⎢0
⎣
0
εy
0
0⎤
⎥
0⎥
ε z ⎦⎥
y
Dy = εyEy
EY
x,y,z are “principal axes”
0
D
E
EX
Dx = εxEx
x
Note: When ε x ≠ ε y ≠ ε z , D // E iff E // x,y,
ˆ ˆ or zˆ
Real ε, μ ⇒ Lossless medium
L8-4
MAKING ANISOTROPIC MATERIALS
V +A [m2]
+Q
ε
ε >> εo
d
-Q
<
εe
εo
ε
εo
ε A
C≅ o
( d 2)
ε ( A 2)
C≅
d
εeff ≅ ε 2
Atom or molecule
εo
A
-
εeff A
C=
d
εeff = ε
C = Q/V
d/2
A
εeff ≅ 2εo
“Uniaxial Medium”
εx = εy = εo “ordinary”
“extraordinary”
εz = εe
z
εo
>
εe
y
x
L8-5
WAVES IN UNIAXIAL MEDIA
Derive wave equation:
∇ × E = − jωB
∇iD = ρf = 0
∇ × H = jωD
∇iB = 0
(
)
(
)
2
Assume:
⎡εe 0 0⎤
⎥
D = =ε E
= ⎢
ε = ⎢ 0 ε 0⎥
==μ
σ = 0, μ
o
⎢ 0 0 ε⎥
⎣
⎦
2
∇ × ∇ × E = ∇ ∇ iE − ∇ E = − jωμ∇ × H = ω μεE
Can show ∇iE = 0 (can also test final solution)
Therefore:
⎡ ∂2 ∂2 ∂2 ⎤
2
+
+
xE
+
yE
+
zE
+
ω
μεE = 0
ˆ
ˆ
ˆ
[
]
⎢ 2
x
y
z
2
2⎥
⎣ ∂x ∂y ∂z ⎦
Assume = 0 (assume UPW in z direction)
Yields 3 decoupled equations (x,y,z components)
L8-6
BIREFRINGENT MEDIA
Decoupled wave equations:
⎡ 2
⎤
∂
2
e
⎢
+ ω με ⎥ Ex = 0 , k e = ω μεe
⎥
⎢ ∂z2
e 2⎦
⎣
(k )
(x-polarization equation)
⎡ 2
⎤
∂
2
⎢
+ ω με ⎥ Ey = 0 ,
2
⎢ ∂z
o 2⎦⎥
⎣
(k )
(y-polarization equation)
k o = ω με
Solutions:
⎧ e
e
“extraordinary” velocity
v
=
1
με
⎪
E x ∝ e− jk z = e− j( ω / v )z where ⎨
o
⎩⎪v = 1 με “ordinary” velocity
Thus x- and y-polarized waves propagate
independently at different velocities
e
e
If ve < vo then ve → “slow-axis velocity”
L8-7
BIREFRINGENT MEDIA
Example:
Input:
E 1 = E o ( xˆ + yˆ ) ⇒ 45o linear polarization
− jk e d
− jk o d
e
o
−
j
φ
−
j
φ
Output: E2 = Eo (xe
ˆ
ˆ
+ ye
)
What pol.?
X
d
Δφ φe − φo = (k e − k o )d
π /2 “Quarter-wave plate”
= π “Half-wave plate”
=
LHC π/2
Δφ
Output:
π
Say, “slow axis”
Z
45°
Y
E2
E1
0
Linear pol.
RHC 3π/2
L8-8
MIT OpenCourseWare
http://ocw.mit.edu
6.013 Electromagnetics and Applications
Spring 2009
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