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8/12/2019 L04 Polarization
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Lecture 4
Polarization of EM Waves
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Field Polarization
definition: the locus traced by the extremity of a harmonic time-varying field vector at a given observation point
plane of polarization is orthogonal to direction of propagation
there are three types of polarization: linear, circular, and elliptic
E E Ex x x
y y y
z z z
linear circular elliptic
Nikolova 2012 2LECTURE 04: POLARIZATION
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Field Polarization: Time Domain Analysis
field of any polarization can be represented in terms of 2 orthogonallinearlypolarized components:
( ) cos( ) and ( ) cos( )x x y yE t m t E t m t
0( )He t
0( )He t
linearly (horizontally) polarized field V
( ) cos( ) cos( ) x yt m t m t
E x y
Nikolova 2012 3
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Field Polarization: Time Domain Analysis (2)
linearly (vertically) polarized field
0( )Ve t
0( )Ve t
V
Nikolova 2012 4LECTURE 04: POLARIZATION
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5
Field Polarization: Time Domain Analysis (3)
case 1: linear polarization of arbitrary directionn
example: direction at 45 deg. ( = 0, mH= mV)
0t
0t
no constraints on magnitudes
V
( ) cos( ) cos( ), 1,2, x y
t m t m t n n E x y
arctan V
H
m
m
Nikolova 2012 LECTURE 04: POLARIZATION
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Field Polarization: Time Domain Analysis (4)
/ 2L
case 2: circular polarization
example: clockwise circularly polarized vector, L= /2
magnitudes must fulfill mH= mV= m
1t
2 12
t t
(0) 1
(0) 0
H
V
E
E
(90 ) 0
(90 ) 1
H
V
E
E
V
( ) cos( ) cos( / 2), 1,2, t m t m t n E x y
Nikolova 2012 LECTURE 04: POLARIZATION
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Field Polarization: Time Domain Analysis (5)
case 3: elliptic polarizationelliptic polarization is any polarization different from linear
or circular
(0) 1(0) 0
H
V
EE
2 , / 2H V Lm m example: counter-clockwise elliptically polarized vector
(90 ) 0
(90 ) 0.5H
V
E
E
V
Nikolova 2012 7LECTURE 04: POLARIZATION
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Field Polarization and Wave Propagation
linear horizontal linear vertical
linear 45 deg. circular
LH or RH?
Nikolova 2012 8LECTURE 04: POLARIZATION
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Field Polarization in Terms of CW-CP and CCW-CP Terms
any field can be represented by 2 circularlypolarized components
one clockwise and one counter-clockwise
example: linearly polarized wave decomposed into two CP components
Nikolova 2012 9LECTURE 04: POLARIZATION
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Field Polarization Phasor Analysis
work with decomposition into 2 orthogonal linear components
( ) ( ) ( ) x y x yt E t E t E E E x y E x y
( ) cos
( ) cos
x xx x jx yj
y y y y
E mE t m t kzm m e
E t m t kz E m e
E x y
drop the propagation factor, exp(jkz), because it is common to
both components and we can always choosez = 0
Nikolova 2012 10LECTURE 04: POLARIZATION
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Field Polarization Phasor Analysis Linear Polarization
, 0,1,2,n n
( ) cos( ) cos( )
x y
x y
t m t m t n
m m
E x y
E x y
arctan yx
mm
2
0
n
x
y
E
xE
yE
(2 1)
0
n
x
y
xE
yEE
Nikolova 2012 11LECTURE 04: POLARIZATION
sign sign
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Field Polarization Phasor Analysis Circular Polarization
and / 2 , 0,1,2,x ym m m n n
( ) cos( ) cos[ ( / 2 )] ( ) t m t m t n m j E x y E x y
RH (CW) for LH (CCW) for
u zu z
LH (CCW) for RH (CW) for
u z
u z
x
y
0t
4
t
2t
3
4t
t
5
4t
3
2t
7
4t
( )
/ 2
m j
E x y
zx
y
0t
4t
2t
3
4t
t
5
4t
3
2t
7
4
t
( )
/ 2L
m j
E x y
z
Nikolova 2012 12LECTURE 04: POLARIZATION
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Field Polarization Phasor Analysis Circular Polarization (2)
sense of rotation is determined when looking along the direction ofpropagation (caution: in optics, the agreement is opposite)
(CCW) LH-CP wave is
( ( ) ) jkzz em j x yE
(CW) RH-CP wave is
( )( ) jkzz em j E x y
if wave propagates along +z
if wave propagates along z
Nikolova 2012 13LECTURE 04: POLARIZATION
x
y
0t
4t
2t
3
4t
t
5
4t
3
2t
7
4t
2
( )
/
m j
E x y
z
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Field Polarization Phasor Analysis Elliptic Polarization
phase difference can be any
magnitude ratio mx/my can be any
the term elliptic polarization is used to indicate polarizations other
than linear or circular
( ) cos( ) cos( )
x y
jx y
t m t m t
m m e
E x y
E x y
elliptic polarization has sense of rotation the sign of determinesthe sense of rotation
Nikolova 2012 14LECTURE 04: POLARIZATION
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Field Polarization in Terms of Circularly Polarized Components
Any field can be written in terms of two CP terms one RHCP, theother LHCP (wave propagating along +z)
( ) ( ) R LE j E j E x y x y
Bearing in mind that this field can also be represented in terms of twoLP terms
x yE E E x y
find the equations allowing for the calculation ofER andEL ifEx and
Ey are known.
Nikolova 2012 15LECTURE 04: POLARIZATION
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Field Polarization
What is the angular speed of rotation of the field vectors in the caseof a circularly polarized wave of radian frequency ?
( ) cos( ) cos( / 2), 1,2, m mt e t e t n E x y
Nikolova 2012 16LECTURE 04: POLARIZATION
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Summary: Field Polarization
Nikolova 2012 17LECTURE 04: POLARIZATION
harmonic plane waves are characterized by their polarizationpolarization is defined by the locus traced by the tip of the E-field
vector as time flows
polarization can be linear, circular, or elliptic (do not identify withthe other two!)
circular and elliptic polarizations can be right-hand (clockwise) or
left-hand (counter-clockwise)
in microwave and antenna engng: the sense of rotation is
determined by looking alongthe direction of propagation (thumb
points along direction of propagation)
every plane-wave field can be decomposed into: (1) two mutually
orthogonal linearly polarized terms, or (2) two circularly polarized
terms of opposite sense of rotation