( Chapter 7.3- Wave Polarization)
April 9, 2013
What is polarization?
The polarization of a uniform plane wave describes the shape and
locus of the tip of the E vector (in the plane orthogonal to the direction
of propagation) at a given point in space as a function of time.
Right screw
sense in space
Transmitting
antenna
Left sense of
rotation in plane
The polarization state of a wave traveling in the z-direction is determined
by tracing the E(z,t) as a function of time in a plane orthogonal to the
direction of wave travel.
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Linear
Circular
Elliptical


~
E ( z, t )  e E ( z )e jt  xˆax cos(t  kz)  yˆa y cos(t  kz   )
Intensity:



1/ 2
2
2
E ( z, t )  Ex ( z, t )  E y ( z, t )

 a x2 cos 2 (t  kz)  a y2 cos 2 (t  kz   )

1/ 2
E(z1,t1)
Inclination angle:
 E y ( z, t ) 

 ( z, t )  tan 
 Ex ( z, t ) 
1
Page 270


~
E ( z, t )  e E ( z )e jt  xˆax cos(t  kz)  yˆa y cos(t  kz   )
A wave is said to be linearly polarized when Ex(z,t) and Ey(z,t) are
in phase   0
or
out of phase   

E ( z, t )  ( xˆax  yˆa y ) cos(t  kz)

E ( z, t )  ( xˆax  yˆa y ) cos(t  kz)
Intensity



2
2 1/ 2
E ( z, t )  ax  a y cos(t  kz)
Inclination Angle

 E y ( z, t ) 
a
1 
1  y 

  tan 
 tan  
 E ( z, t ) 
 ax 
 x


 E y ( z, t ) 
 ay 
1 
1 


  tan 
 tan 
 E ( z, t ) 
 ax 
 x

ψ, is independent of z and t  E moving along a line
Page 270


~
E ( z, t )  e E ( z )e jt  xˆax cos(t  kz)  yˆa y cos(t  kz   )
ax = ay =a
Left-hand Circular (LHC)
δ=π/2
E ( z , t )  a  xˆ cos(  t  k z )  yˆ sin(  t  k z ) 
Inclination
Angle
Intensity
e.g. z=0
 E y ( z, t ) 
  t  kz  t    
 ( z, t )  tan 
 Ex ( z, t ) 

2
2
E z, t   a cos t  kz   a sin t  kz  a
1
Right-hand Circular (RHC)
δ=-π/2
E ( z , t )  a  xˆ cos(  t  k z )  yˆ sin(  t  k z ) 
Inclination
Angle
Intensity
e.g. z=0
 E y ( z, t ) 
  t  kz 
 ( z, t )  tan 
 Ex ( z, t ) 

E z, t   a
1
t   
Page 271
Right-hand or Left-hand circularly polarized wave?
Right screw
sense in space
Left sense of
rotation in plane
Left-hand Circularly Polarized Wave
Page 272
(Tip of E traces an ellipse in the x-y plane)
y
η
Auxiliary angle
ay
aη
ξ
ψ0
z
γ Rotation angle
aξ
ax
Minor
Axis
x
 /4    /4
auxiliary angle
Find:
Ψ0
γ
Χ
R
Auxiliary angle
Rotation angle
Ellipticity angle
Axial Ratio



 
2
2
Axial ratio
Circular
polarization
1  R  a / a  1 / tan   
Linear
polarization
Page 273
0
tan(2 )  tan 2 0 cos 
tan 0 
tan(2 )  sin 2 0 sin 
ay
 /4    /4
ax
0  0 

2





 
2
2
R  1 / tan 
Find:
R
a  a x sin   a y cos 
a  ax cos   a y sin 
Ψ0
γ
Χ
R
Auxiliary angle
Rotation angle
Ellipticity angle
Axial Ratio
 0
 0
 0
Left circular polarization
Linear polarization
Right circular polarization
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