Monochromatic Radiation is Always 100% Polarized
Polarization Ellipse
y
y
θ
“Right-Hand”
Polarization
x
z
b
–vx
a
ϕ
x
z
E(t)
Propagation
–vy
3 Parameters Specify Ellipse
e.g. a, b, ϕ Also, (need “+” or “–”
a, ϕ, θ to ⇒ right or left elliptical)
vx, vy, θ
Lec13a.3-1
1/12/01
V1
Polarization of Narrowband Radiation
Let E( t ) = xˆ v x ( t ) cos[ωt + φ( t )] + yˆ v y ( t ) cos[ωt + φ( t ) + δ( t )]
vx(t) and vy(t) are slowly varying and random;
⟨vx⟩, ⟨vy⟩, and ⟨δ⟩ may be non-zero
“Stokes Parameters”
I ≡ So
[
v (t) +
≡
Q ≡ So
2
x
]
2 ηo
[
v (t) − v (t) ]
≡
U ≡ S2 ≡
V ≡ S3 ≡
Lec13a.3-2
1/12/01
2
v y (t)
2
x
2
y
2 ηo
2 v x ( t ) • v y ( t ) cos δ( t )
2 ηo
2 v x ( t ) • v y ( t ) sin δ( t )
2 ηo
-2
[W m ] total power
“x-ness”
“45°-ness”
“circularity”
V2
100% Polarized Narrowband Waves
Let E( t ) = xˆ v x ( t ) cos[ωt + φ( t )] + yˆ v y ( t ) cos[ωt + φ( t ) + δ( t )]
vx(t) and vy(t) are slowly varying and random;
⟨vx⟩, ⟨vy⟩, and ⟨δ⟩ may be non-zero
δ( t ) = δo and v x v y ( t ) = constant ⇒ fixed ellipse, variable size
y
E(t)
x
Also : So2 = S12 + S22 + S32
Therefore, any 3 Stokes parameters specify polarization
Lec13a.3-3
1/12/01
V3
Partially Polarized Narrowband Radiation
“Stokes Parameters”
2
2
[⟨v x (t)⟩ + ⟨v y (t)⟩]
2ηo
I ≡ So ≡
2
[⟨v x (t)⟩
Q ≡ S1 ≡
[W m-2] total power
2
⟨v y (t)⟩]
–
2ηo
U ≡ S2 ≡ 2⟨vx (t) • vy(t) cos δ(t)⟩
2ηo
“x-ness”
V ≡ S3 ≡ 2⟨vx (t) • vy(t) sin δ(t)⟩
2ηo
“circularity”
“45°-ness”
Note: For 2 uncorrelated waves superimposed (A+B), we have
Si
= Si + Si where i = 0, 1, 2, 3
A+B
A
B
For 0% polarization, Stokes: So; S1 = S2 = S3 = 0
Therefore, for partially
polarized wave:
Lec13a.3-4
1/12/01
[So, S1, S2, S3] = [Su, 0, 0, 0] + [So – Su, S1, S2, S3]
2
2
2
where (So – Su)2 = S 1 + S2 + S 3
So – Su
Define percentage polarization =
• 100%
So
∆
= m, 0 ≤ m ≤ 1
V4
Coherency Matrix J
⎡ E x E∗x
∆ 1 ⎢
J=
ηo ⎢ E∗x E y
⎣
e.g.
Lec13a.3-5
1/12/01
{
}
{
E x E∗y ⎤ where E(t) = xR E ( t )e jωt + yR E ( t )e jωt
x
y
e
e
⎥
E y E∗y ⎥ where E x ( t ), E y ( t ) vary slowly
⎦
X-polarization
⎡ 1 0⎤
Jx = 2So ⎢
⎥
⎣0 0⎦
RCP (right-circular)
⎡1 − j⎤
JRC = So ⎢
⎥
j
1
⎦
⎣
Unpolarized
⎡ 1 0⎤
Ju = So ⎢
⎥
⎣0 0⎦
}
V5
Finding Orthogonal Polarization JRC
e.g.
X-polarization
RCP (right-circular)
Unpolarized
Note : Jx + Jy = 2Ju
⎡ 1 0⎤
Jx = 2So ⎢
⎥
0
0
⎣
⎦
⎡1 − j⎤
JRC = So ⎢
⎥
j
1
⎦
⎣
⎡ 1 0⎤
Ju = So ⎢
⎥
⎣0 0⎦
JRC + JLC = 2Ju
JA ⊥ JB , and⎫
If
⎬ then JA + JB = 2Ju
Tr JA = Tr JB ⎭
Therefore, we can find orthogonal polarization JB = 2Jn − JA
Lec13a.3-6
1/12/01
V6
Polarized Antennas
Far Fields
Define
Gij (θ, φ)
G(θ, φ)
=
A ij (θ, φ)
A(θ, φ)
=
Ei E j
E x E∗x + E y E∗y
e.g. {i, j} = {x, y}, {r, }, {a, b} (b ⊥ a )
⎡ A xx
A=⎢
⎣ A yx
Lec13a.3-7
1/12/01
t ⎤
1 ⎡
A xy ⎤
Prec = Tr A • Jinc [W ] for incident
;
claim
⎥⎦
2 ⎣⎢
A yy ⎥⎦
plane wave
[m2] [Wm2]
W1
Polarized Antennas
⎡ A xx
A=⎢
⎣ A yx
t ⎤
1 ⎡
A xy ⎤
Prec = Tr A • Jinc [W ] for incident
;
claim
⎥⎦
2 ⎣⎢
A yy ⎥⎦
plane wave
[m2] [Wm2]
1
So Prec = [A11J11 + A12J12 + A 21J21 + A 22J22 ]
2
for incident uniform plane wave J on antenna A
t
1
⎡
For Ω s ≠ 0 : Prec = ∫ Tr A (θ, φ)J (θ, φ)⎤ dΩ
⎥⎦
2 4 π ⎢⎣
Lec13a.3-8
1/12/01
W2
To Measure Polarization
Measure 4 powers; use 4 antennas
e.g.
⎡ A11a
⎡Ma ⎤
⎢A
⎢M ⎥
1
⎢ b ⎥ = ⎢ 11b
⎢Mc ⎥ 2 ⎢ A11c
⎢A
⎢M ⎥
⎣ d⎦
⎣ 11d
A12a
A12b
•
•
A 21a
•
•
•
(
A 22a ⎤ ⎡ J11 ⎤
• ⎥ ⎢ J12 ⎥
⎥⎢
⎥
• ⎥ ⎢ J21 ⎥
A11d ⎥⎦ ⎢⎣J22 ⎥⎦
−1
1
M = A J , so Ĵ = 2A M Jˆ is estimate
2
)
Is A singular?
Lec13a.3-9
1/12/01
W3
To Measure Polarization
(
−1
1
M = A J , so Ĵ = 2A M Jˆ is estimate
2
)
Is A singular?
For x, y, RC, LC POL:
⎡1 0 0
⎢0 0 0
A=⎢
⎢1 − j j
⎢1 j − j
⎣
For x, 45°, RC, LC:
⎡1 0 0
⎢1 1 1
A=⎢
⎢1 − j j
⎢1 j − j
⎣
0⎤
1⎥
⎥
1⎥
1⎥⎦
0⎤
1⎥
⎥
1⎥
1⎥⎦
Can not distinguish
det A = 0
Lec13a.3-10
1/12/01
vs
det A ≠ 0 " ok"
W4
Example of a Polarimeter
Right Circular
Diplexer
201.5 MHz
231.5 MHz
Local
Oscillator
30 MHz
[Cohen, Proc. IRE, 1, 1958]
Left Circular
201.5 MHz
30 MHz
25.0 MHz
Local
Oscillator
5 MHz
( )2
∫dt
Lec13a.3-11
1/12/01
25.001 MHz
Local
Oscillator
4.999 MHz
1 KHz
⟨ρ2⟩
⟨ρrρ ⟩
⟨ρ 2r ⟩
4 measurements ↔ 4 Stokes parameters
Phase
Comparator
⟨γ ⟩
W5
Antenna Phase Errors
Systematic antenna phase errors:
phase front
1)
2)
3)
poor design and fabrication
gravity, wind, thermal (gravity and
thermal limits near 1 arc minute)
feed offset
Random antenna phase errors:
1)
2)
3)
Lec13a.3-12
1/12/01
machine tolerances, surface roughness
adjustment errors
feed offset
X1
Examples of Antenna Phase Errors
Random antenna phase errors:
1)
2)
3)
matching tolerances, surface roughness
adjustment errors
feed offset
300-ft parabolic reflector antenna at NRAO, Greenbank, West Virginia
1)
systematic sag ⇒ fix backup; footprints on mesh
2)
steamrolled mesh ⇒ long waves
~
new panels: θB > 0.5 − 1 arc minute
3)
Lec13a.3-13
1/12/01
X2
Types of optical and radio propagation phase errors
Systematic:
h
velocity of light
T(h)
ρ(h)
c
c´ < c
Earth
c´
c
Earth
Random phase:
+ amplitude?
~
RMS < λ 2π , weak fluctuations
RMS >> λ 2π , strong fluctuations
∆ pathlength = ∆L ≥ nλ
⇒ interference and nulls
vs
Lec13a.3-14
1/12/01
ionosphere
Thin screen
(constant amplitude)
Thick screen
X3
Effect of Phase Variation on Directivity
x
For x-polarization:
aperture
~ E x (ϕ , ϕ )
E x ( x, y ) ↔
x y
ϕx
ϕy
↓
z
RE x (τ )
↓
~
↔
E x (ϕ)
2
∝ D(ϕ), G(ϕ)
y
[
−j
D( f , θ, φ) = π(1 + cos θ)
Lec13a.3-15
1/12/01
2
2
λ
2π
(ϕ• τ )
λ
dτ
RE x (τ )e
x dτ y
∫
A
]•
2
E
(
x
,
y
)
dxdy
∫A x
X4
Effect of Phase Variation on Directivity
−j
[
D( f , θ, φ) = π(1 + cos θ)
E{D( f , θ, φ)} =
2
2
λ
RE x (τ )e
dτ y
x
∫
]• A E (x, y) 2 dxdy
∫A x
2
π(1 + cos θ)
2
λ
∫A E x ( x, y )
2
2π
(ϕ• τ )
λ
dτ
dxdy
{
}
• ∫ E RE x (τ ) e
A
−j
2π
(ϕ• τ )
λ
dτ
() ∗
A
jγ (r )
Eo (r ) e
∫ E x r E x (r − τ )dr
{
Therefore E RE x (τ ) = REo (τ ) E e jγ (r )− jγ (r − τ )
{
}
{
} {
}
Spatial stationarity : E e jγ (r )− jγ (r − τ ) = E e jγ(o )− jγ (r )
Lec13a.3-16
1/12/01
x dτ y
}
X5
Definition of “Characteristic Function”
It is the Fourier transform of probability distribution p(x)
(also called the moment-generating function)
[ ]
E e j ωx ≡
∞
j ωx
p
(
x
)
e
dx = F.T.[p( x )]
∫
−∞
∆
= Γ(ω ; x) = “characteristic function of p(x)”
One use of the Fourier transform of p(x) is when
we seek p(x1 + x 2 + ... + xn ) =
⎧n
⎫
p(x1) ∗ p(x 2 ) ∗ ... ∗ p(xn ) = F.T.⎨ π F.T.[p(xi )]⎬
⎩i=1
⎭
Lec13a.3-17
1/12/01
X6
Computation of E{ RE x (τ) }
{
Thus Γ(ω1, ω2 ; γ(0), γ (τ )) = E e jω1γ(0 )+ jω2 γ (τ )
}
Recall : If γ1, γ 2 are JGRV , then
Γ(ω 1, ω 2 , γ1, γ 2 )
1
− [ω1ω2 ]
=e 2
⎡ γ γ∗
⎢ 1 1∗
⎢⎣ γ 2 γ1
γ1γ ∗2 ⎤ ⎡ ω 1 ⎤
⎥
∗ ⎢ω ⎥
γ 2 γ 2 ⎥⎦ ⎣ 2 ⎦
Here, γ1 = γ(0) , γ 2 = γ (τ )
∆
{
∆
}
Therefore : E e jγ(0)- jγ (τ ) = Γ(ω1 = 1, ω2 = −1; γ(0), γ (τ ))
Lec13a.3-18
1/12/01
X7
Computation of E{ RE x (τ) }
Γ(ω1, ω2 , γ1, γ 2 )
1
− [ω1ω2 ]
=e 2
Here, γ1 = γ(0) , γ 2 = γ (τ )
∆
∆
{
⎡ γ γ∗
⎢ 1 1∗
⎢⎣ γ 2 γ1
γ1γ ∗2 ⎤ ⎡ ω1 ⎤
⎥
∗ ⎢ω ⎥
γ 2 γ 2 ⎥⎦ ⎣ 2 ⎦
}
Therefore : E e jγ(0)- jγ (τ ) = Γ(ω1 = 1, ω2 = −1; γ(0), γ (τ ))
∗
∗
(τ) = φ(τ)
γ
(
0
)
γ
(0)
=
φ
(0)
γ
(
0
)
γ
Since:
γ (τ )γ ∗ (τ ) = φ(0) ← by stationarity
∆
(ω1 = 1, ω2 = −1; γ(0), γ(τ))
∆
1
− [1−1]
=e 2
⎡φ(0) φ( τ) ⎤ ⎡ 1 ⎤
⎢φ∗ ( τ) φ(0)⎥ ⎢− 1⎥
⎣
⎦⎣ ⎦
= eφ(τ )− φ(0 )
Lec13a.3-19
1/12/01
Therefore E RE x (τ ) = REo (τ ) • eφ(τ )− φ(0 )
{
}
X8
Computation of Expected Directivity
⎡
⎤
2π
-j ϕi τ
⎢ π(1 + cos θ)2 ⎥
φ( τ ) − φ(0)
⎡
E {D( f, θ, φ)} = ⎢
•
e
REo ( τ ) ⎤ e λ
d τ x dτ y
⎥
∫
2
2
⎦
⎢ λ ∫ E ( r ) da ⎥ A ⎣
⎢⎣ A
⎥⎦
()
()
correlation length L of
phase irregularities
()
φ τ = γ(0 )γ ∗ τ
φ τ − φ(0 ) 0
τ
2
σ ≡ φ(0 )
τ
0
e
L
1
φ( τ ) − φ( 0 )
∆
e
0
Lec13a.3-20
1/12/01
1− e
− σ2
τ
L
− σ2
= B(τ )
=
e
+
0
τ
L
0
− σ2
τ
X9
Solution to Expected Directivity
1− e
∆
− σ2
= B( τ )
eφ( τ)− φ(0 ) =
e
+
0
τ
L
− σ2
τ
0
⎡
⎤
2π
−
ϕ• τ
j
⎢ π(1 + cos θ)2 ⎥ ⎡ − σ 2
⎤
+ B(τ )⎥ • REo(τ ) • e λ
dτ x dτ y
⎥ • ∫ ⎢e
E{D( f , θ, φ)} = ⎢
2
⎦
⎢ λ2 ∫ E(r ) da ⎥ A ⎣
⎢⎣ A
⎥⎦
E{D( f , θ, φ)} = e
−σ2
Do ( f , θ, φ) + B(ϕ) ∗ Do ( f , θ, φ)
gain
degradation
Lec13a.3-21
1/12/01
sidelobe
increase
B(ϕ)
0 λ/L
ϕ
X10
Examples of Random Antenna Surface
Let b = RMS surface tolerance of reflector antenna
On-axis gain of random antenna
G′o = Goe
−σ2
= Goe
− (2b•2π λ )2
= Goe
− (b 4 π λ )2
If b = λ 4π ⇒ Go • e −1
b = λ 16 ⇒ Go • 0.54
b = λ 32 ⇒ Go • 0.9
new
log G
(power shifts to sidelobes)
Any aperture
antenna, fixed
illumination
log G = −2 log λ + log 4 πA e
-2
Lec13a.3-22
1/12/01
~minimum
useful wavelength
log λ
X11