Download POLARIZATION OF THE UNIFORM PLANE WAVE We define the

POLARIZATION OF THE UNIFORM PLANE WAVE
We define the polarization of a uniform plane wave as the locus of the
tip of the electric field vector in time at a given point in space. If this
locus is a straight line the wave is said to be linearly polarized. If this
locus is a circle we have circular polarization and if it is an ellipse we
have elliptical polarization.
Actually, linear and circular polarizations are special cases of elliptical
polarization.[1]
Consider the electric field vector of a uniform plane wave propagating
in the  z direction:
E  z   E0e jk0 z
Hence,
E0 is
a vector lying in the x-y plane can be written as:
E0  E0x aˆ x  E0 y aˆ y
Where E0 and
x
E0 y are constants. Then the real physical field:
E  z, t   Re  E0e jk0z e jt 
E  z, t   E0x cos t  k0 z   E0 y cos t  k0 z 
Ex  z, t   E0x cos t  k0 z 
Ey  z, t   E0 y cos t  k0 z 
E y  z, t 
Ex  z , t 

E0 y
E0x
 cons.
The electric field vector lies along the straight line of slope tan  , where
 E0 y
 E0
 x
  tan 1 

 . This is true for all z .

Next consider the superposition of two linearly polarized uniform plane
waves; One polarized in the x  direction and the other in the y 
direction lagging 900 in time.
The phasor for this field is
E  z   aˆ x E0x e jk0 z  jaˆ y E0 y e jk0 z
The instantaneous expression of E is:
E  z, t   aˆ x E0x cos t  k0 z   aˆ y E0 y sin t  k0 z 
To examine the direction change of E  z, t  w.r.t. t set k0 z  0 ;
E  0, t   aˆ x E0x cos t  aˆ y E0 y sin t

As, t  0, ,  ,
2
3
, 2 , the tip of the vector E  0, t  will traverse an
2
elliptical locus in the counter-clockwise direction. Analytically, we have:
Ex  0, t   Ex0 cos t
Ey  0, t   Ey0 sin t
cos t 
sin t 
Ex  0, t 
Ex0
E y  0, t 
E y0
In order to eliminate t:
sin 2  t  cos2  t  1
 E y  0, t    Ex  0, t  

  
  1
E
E
y0
x0

 

2
2
This is the equation of an ellipse.
If E0  E0 , E is polarized elliptically.
x
y
If E0  E0 , E is polarized circularly. E (0, t ) , rotates at a uniform rate
x
y
with an angular velocity  in a counter-clockwise direction.
IEEE Convention: If the right hand fingers follow the direction of E and if
thumb points to the direction of propagation then we have right-hand
polarization (R.H.P.). If on the other hand the thumb points to the
negative direction of the propagation then we have left-hand
polarization (L.H.P.).
Consider,
E  z, t   aˆ x E0x cos t  k0 z   aˆ y E0 y sin t  k0 z 
For
k0 z  0
E  0, t   aˆ x E0x cos t  aˆ y E0 y sin t
Type of polarization is CIRCULAR. Let:
E1  E  0,0   E0aˆ x
  
E2  E  0,
  E0aˆ y
2



E1 E2
If, E  E  aˆn , we have R.H.C.P.
1
2
E1 E2
If, E  E  aˆn , we have L.H.C.P.
1
2
The convention for the elliptical polarization is the same.
Similarly the following formulas can be derived to find the sense of the
polarization of a u.p.w.:
E1 E2
If, E  E . aˆn  1 , we have R.H.P.
1
2
E1 E2
If, E  E . aˆn  0 , we have L.H.P.
1
2