electromagnetic wave polarization
TRANSCRIPT
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Electromagnetics:
Electromagnetic Field Theory
Electromagnetic Wave Polarization
Lecture Outline
•Definition of Electromagnetic Wave Polarization
•Orthogonality & Handedness•Polarization Classification: Linear, Circular & Elliptical•Poincaré Sphere•Polarization Explicit Form•Dissection of a Plane Wave
Slide 2
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Definition of Electromagnetic Wave
Polarization
Slide 3
What is Polarization?
Slide 4
Polarization is that property of a electromagnetic wave which describes the time‐varying direction and relative magnitude of the electric field vector.
jk rE r Pe
Polarization Vector
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Why is Polarization Important?
• Different polarizations can behave differently in a device•Orthogonal polarizations will not interfere with each other• Polarization becomes critical when analyzing devices on the scale of a wavelength
• Focusing properties of lenses can depend on polarization• Reflection/transmission can depend on polarization
• Frequency of resonators can depend on polarization• Cutoff conditions for filters, waveguides, etc., can depend on polarization
Slide 5
Orthogonality & Handedness
Slide 6
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Plane Waves in LHI Media are TEM
Slide 7
In linear, homogeneous and isotropic (LHI) media, the electric field 𝐸, magnetic field 𝐻, and direction of the wave 𝑘 are all perpendicular to each other. These are called transverse electromagnetic (TEM) waves.
E k H
Further 𝐸, 𝐻, and 𝑘 follow a right‐hand rule.
Proof That 𝐸 ⊥ 𝑘
Slide 8
Start with Gauss’ law.
0D
Put this in terms of the electric field intensity .E
0E 0E
0E
Substitute in expression for a plane wave.
0jk rPe
0jk rjk Pe
0k P
According to the dot‐product test, if then .0k P
k P
E k
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Proof That 𝐻 ⊥ 𝑘
Slide 9
Start with Gauss’ law for magnetic fields.
0B
Put this in terms of the magnetic field intensity .H
0H 0H
0H
Substitute in expression for a plane wave.
0 0jk rH e
0 0jk rjk H e
0 0k H
According to the dot‐product test, if then .0 0k H
0k H
H k
Proof That 𝐸 ⊥ 𝐻
Slide 10
Start with Faraday’s law.
E j H
Substitute in expression for a plane wave.
0 0jk r jk rE e j H e
00
k EH
0 0jk r jk rjk E e j H e
0 0k E H
According to the properties of cross products, 𝐻will be perpendicular to both 𝑘 and 𝐸.
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Polarization Classification:
Linear, Circular & Elliptical
Slide 11
Linear Polarization (LP)
Slide 12
An electromagnetic wave has linear polarization if the electric field oscillation is confined to a single plane.
0 ˆ jkzxE E a e
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Linear Polarization (LP)
Slide 13
0 ˆ jkzyE E a e
An electromagnetic wave has linear polarization if the electric field oscillation is confined to a single plane.
Linear Polarization (LP)
Slide 14
0 ˆ ˆcos sin jkzx yE E a a e
An electromagnetic wave has linear polarization if the electric field oscillation is confined to a single plane.
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Circular Polarization (CP)
Slide 15
Suppose two LP waves are brought together…
1 0 ˆ jkzxE E a e
2 0 ˆ jkzyE E a e
Circular Polarization (CP)
Slide 16
…and then one LP wave is phase shifted by 90.
0 0
1
0
2
0
Before phase shift (LP)
After phase shift (Rˆ C
ˆ
ˆ
ˆ P)
jkzx
z
z
jky
jkzjkx y
EEE a e
E a e
a e
jE a eE
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Circular Polarization (CP)
Slide 17
The overall electric field will appear to rotate with respect to both time and position.
1 2 0 ˆ ˆ jkzx yE E E a ja e
Circular Polarization (CP)
Slide 18
An electromagnetic wave has circular polarization if the direction of the electric field rotates with time and has a constant magnitude.
0 ˆ ˆ jkzx yE E a ja e
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Right Circular Polarization (RCP)
Slide 19
An electromagnetic wave has right circular polarization if the electric field rotates clockwise when viewed from behind.
0 ˆ ˆ jkzx yE E a ja e
Left Circular Polarization (LCP)
Slide 20
0 ˆ ˆ jkzx yE E a ja e
An electromagnetic wave has left circular polarization if the electric field rotates counterclockwise when viewed from behind.
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Elliptical Polarization (EP)
Slide 21
An electromagnetic wave has elliptical polarization if the electric field rotates with time to form an ellipse.
Continuum of LP+LP Polarizations
Slide 22
As the phase difference between the two LP waves changes, the resulting polarization…
00 ˆ ˆLPLP j jkzjkzx xy yE a e E e a e
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Continuum of LP+LP Polarizations
Slide 23
As the phase difference between the two LP waves changes, the resulting polarization oscillates between RCP, LCP, LP45, LP‐45, and EP.
0
0
0
ˆ
ˆL L
ˆ1
ˆPP
j jkz
j jkzy
kyx
x y
j zx E e a
E a e a e
eE a e
Continuum of CP+CP Polarizations
Slide 24
0 0ˆ ˆR P ˆ ˆLC 1 1C P j jkzjkzx y x yE a ja E a ja e ee
As the phase difference between the two CP waves changes, the resulting polarization is...
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Continuum of CP+CP Polarizations
Slide 25
As the phase difference between the two CP waves changes, the resulting polarization is LP where the phase translates into a tilt angle.
0 0
0 ˆ ˆcos si
ˆ ˆRCP 1 1
n
ˆ ˆLCP
jkz
yjkz
x yj jkz
x
x y
E a j E a ja e
E a
a e e
a e
Poincaré Sphere
Slide 26
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Poincaré Sphere
Slide 27
The polarization of a wave can be mapped to a unique point on the Poincaré sphere.
Points on opposite sides of the sphere are orthogonal.
See Balanis, Chap. 4.
RCP
LCP
+45° LP0° LP
90° LP‐45° LP
Continuum of All Polarizations
Slide 28
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Polarization Explicit Form
Slide 29
k
a
bˆˆa bp a bP p
Possibilities for Wave Polarization
Slide 30
Recall that 𝐸 ⊥ 𝑘 so the polarization vector 𝑃 must fall within the plane perpendicular to 𝑘.
The polarization can be decomposed into two orthogonal directions, 𝑎 and 𝑏.
Rules for Choosing and a b
1. Orthogonality
2. Handedness
ˆa k b
ˆ ˆa b k
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Explicit Form to Convey Polarization
Slide 31
The expression for the electromagnetic wave can now be written as
ˆˆjk r jk ra bE r Pe p a p b e
pa and pb are in general complex numbers in order to convey the relative phase of each of these components.
a bj ja a b bp E e p E e
ˆˆ b a aj j jk ra bE a E e b e e
Substituting pa and pb into the wave expression gives
Interpret a as the phase common to both pa and pb.
b a a
Interpret b – a as the phase difference between pa and pb.
ˆˆ j j jk ra bE r E a E e b e e
The final expression is:
ˆˆa bj j jk ra bE r E e a E e b e
Determining Polarization of a Wave
Slide 32
To determine polarization, it is most convenient to write the expression for the wave that makes polarization explicit.
ˆˆ j j jk ra bE r E a E e b e e
ˆamplite along
ˆamplite along
phase difference
common phase
a
b
E a
E b
Polarization Designation Mathematical Definition
Linear Polarization (LP) = 0°
Circular Polarization (CP) = ± 90° and Ea = EbRight‐Hand CP (RCP) = + 90° and Ea = EbLeft‐Hand CP (LCP) = - 90° and Ea = Eb
Elliptical Polarization Everything else
Given 𝐸 , 𝐸 , and 𝛿 the following table can be used to identify the polarization of the wave…
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Dissection of aPlane Wave
Slide 33
Dissection of a Plane Wave
Slide 34
0
0
, cos
, cos
H r t H t k r
E r t E t k r
0
0
E
H
Impedance
Wavelength in Medium
2k
Free Space Wavelength
00c f
0
0
0
0ˆ ˆ
ˆ
H k P
E P
H
E
Complex Amplitudes
Ordinary Frequency
2 f
Direction
kk
k
Refractive Index
0 0
vn
c
Speed
v f
Constitutive Parameters
0
rr r
r
rr
0
0
0
nn
n
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Reading Off the Parameters
Slide 35
9ˆ ˆ, 3 7 cos 2.4 10 5.1 3.5 V mx yE r t a a t x z
Vˆ ˆ3 7
mx yP a a
Polarization Vector Angular Frequency9 rad
2.4 10 sec
Wave Vector
Step 1 – Factor out negative sign.
, cos V mE r t t rP k
Standard Form
9 9cos 2.4 10 5.1 3.5 cos 2.4 10 5.1 3.5t x z t x z Step 2 – Set expression in parentheses equal to x y zk r k x k y k z
5.1 3.5x y zk x k y k z x z
Step 3 – Read off components of k
1ˆ ˆ5.1, 0, 3.5 5.1 3.5 mx y z x zk k k k a a