Tuesday, April 4, 2017

Stokes Parameters and Their Use In Radio Astronomy

No automatic alt text available.

In the ideal world of radio astronomy observations, one's choice source would likely be one for which the radiation is monochromatic or single frequency. In this case one deals with completely polarized waves and the associated constants (E1, E2,  δ ) are easily computed, with E1, E2 the E-field amplitudes. 

In the real world of radio astronomy, the objects detected display partial polarizations which must be further analyzed to reveal reliable information. Electromagnetic waves are  polarized when their E- field  components are preferentially oriented in a particular direction.

Other forms of polarization include:

Linearly or horizontally polarized: I.e. the E- vector is confined to one  (horizontal) plane


---------à E

Vertically polarized: I.e. the E- vector is confined to one  (vertical) plane

^ E

!
!
!

Circular: The E-vector rotates through 360 degrees

Elliptic: any polarization not circular or plane.

In addition to source polarization, the orientations of the sources in space relative to the observer must be taken into account using something called "the Poincare sphere".  This is where the Stokes parameters enter. Thus, emission from celestial radio sources within any finite bandwidth D u  consists of the superposition of a large number of statistically independent waves for a variety of polarizations. The resulting wave is then "randomly polarized" and the electric field components may be written:

E x  = E1 (t) sin  (wt)

E y  = E2 (t)  sin (wt  +   δ (t))

where all the time functions are taken as independent. Also, we have the time variations of E1(t), E2(t) and δ (t)  as slow compared to that of the mean frequency, i.e.  2 p   , being of the order of the bandwidth  D u  .

The most general state is one for which the wave is partially polarized. Waves emitted from celestial sources are generally of the partially polarized form. In traditional radio astronomy, one deals with partial polarization by making use of the Stokes parameters, first introduced by Sir George Stokes in 1852. Reference may be made here to the attached diagram in order to fix ideas on how the parameters are developed and used.   We can write in the first instance:

E x  = E1  sin (wt -    δ1)

y  = E1  sin (wt -    δ2)

where ( δ1 -   δ2 )  = the phase difference of  E x   and  E y  .

Expanding the two preceding equations and equating the  sin wt   and  cos wt   terms with results from two other equations, one obtains:
Image may contain: text
With each of the Stokes parameters, I, Q, U and V now quantitatively defined. Again, Z is the intrinsic impedance of the medium, while S x  denotes the Poynting vector for the wave polarized in the x direction, and  S y  denotes the Poynting vector for the wave polarized in the y- direction. It follows from this development that: 

I2  =    Q2  +  U 2   +   V2

And:

U/ Q  =    tan 2 t

And:

V/ S  =   sin  2 e   =    V/   Ö (Q2  +  U 2 +  V2)

Note the above relations apply to a completely polarized wave.   Three specific cases are worth considering to fix ideas (AR is aspect ratio):

Case (1) Left Circularly Polarized Wave:

We have:  S x   = S y  ,   AR = 1, I = S, Q = U = 0,   V = S,  e   = 45 deg   


Case (2):  Right Circularly Polarized Wave:



I = S, Q = U = 0,   V = -S





Case (3)  Linearly Polarized Wave:



We have:   t    =   0,  S x   = S,  S y  =  0 ,  AR = oo,    e   = 0

I = S,  Q = S,  U = V = 0

For a completely unpolarized wave we have:  S x   = S y    and   E1   and  E2   are uncorrelated. The condition: Q = U = V = 0 is also a requirement.  Hence, a nonzero value for any of these quantities indicates a polarized component in the wave.  Also important is the degree of polarization, defined as the ratio of the completely polarized power to the total power or:

d  =  Ö (Q2  +  U 2 +  V2) /  I

(For 0 < d < 1.)


It can be seen from the above, that d = 1 for a completely polarized waves and d = 0 for a completely unpolarized wave. (If two waves have identical Stokes parameters the waves are identical). It is also sometimes relevant to express a presence of an unpolarized component, i.e. if the Stokes parameters are expressed in both, say with appropriate subscripts u, p..  Thus one may find:

I =  S u  +    S p      =  S u  +  S xp   +   S yp

Q  =       S xp   -    S yp

U   =     ( S xp   +   S yp )  tan 2 e  


V  =    ( S xp   +   S yp ) sec  2 t   tane  

  Normalizng the Stokes parameters entails dividing the time-averaged ([ ] ) equation:

I   =   [E12] / -   [E22 ] / Z



By: V =   2/ z {  E1 E2 sin δ ) = S sine 


Then we have for the normalized Stokes parameters:

0  =   I / S   =   1  

1  =   Q / S   =  (S x   -  S y) /    S  =  [ cos 2t cose ]


2  =   U / S   =  2/ Z  (E1 E2 cos δ  ) / S =  [ sin2t cose ]


3  =   V  / S   =  2/ Z  (E1 E2 sin δ  ) / S =   [  sin2e  ]


Where 2 e  denotes latitude  and  2t  longitude on the Poincare sphere, while the new symbols ( s 0  .....s 3 ) are now introduced to represent the normalized Stokes parameters. The Stokes parameters can then be expressed as matrices of the column type, e.g.

S[  s i ]  =

S [s 0]
   [s 1]
   [s 2]
   [ s 3]     i=0, 1, 2, 3

Where S is the total Poynting vector in W/ m2

And:   s 0  =      1

1  =   [E12]  -   [E22  ] /  ZS

2  =   2 / ZS  [E1 E2 cos δ ]

3  =   2 / ZS  [E1 E2 sin δ ]

If a partially polarized wave can be regarded as the sum of a completely polarized wave  and a completely unpolarized wave  we can write:  S[  s i ]  =

S [s 0]
   [s 1]
   [s 2]
   [ s 3]

= S [1  -  d]    +
       [ 0]
       [0 ]
       [0 ]

S  [  d ]
    [ d  cos 2t cos]
    [d  sin2t cose ]    
    [d  sin2e  ]

where d is now the degree of polarization given in terms of the normalized Stokes parameters:

So, for a partially polarized wave:

  =   1

1    =  d  cos 2t cose 

2    =  d  sin2t cose

3    =  d  sin2e

Note that circularly polarized wave components may also be used which will then rely on the quantity AR or the ratio of the major axis to the minor axis for the polarization ellipse (top diagram).. Then, e.g. for a completely polarized wave:

AR  =   (E L  +  E R ) /  (E L  -   E R )       =  cot e

Where the numerator and denominator shows left and right circularly polarized E-field components. If then, E L  >  E R     the AR is positive (left-handed polarization) while if  E L  <  E the AR is negative, or right handed polarization.  Then from the first AR equation we can see:

cose  =      2 E L  E R  / (E L 2 +  E R 2    =   (AR2    -  1) / (AR2   +  1)

Also:

sine  =  (E L 2 -   E R 2 ) /  (E L 2 +  E R 2 ) =    2AR/  (AR2   +  1)


There are many occasions for which the "coherency matrix" C will also be used. Here we are concerned with the response of the detection system, antenna  -receiver etc. and treat the detector as one for radio photons. (In regular optics the applicable matrices are called coherency matrices.

The coherency matrices and Stokes parameters for different types of radio wave are indicated below (the Stokes parameter will be given as a row matrix to save space but it is understood it must be rotated to a column form to be proper):

1) Unpolarized wave:

Stokes parameters:   [1..0..0..0]

Coherency matrix:  C =


½ [1…..0]

     [0…..1]


2)  Right circularly polarized wave:

Stokes parameters:  [1..0..0..-1]

Coherency matrix:  C =

½ [1…..-j]

    [j…..1]

3) Left circularly polarized wave:

Stokes parameters:  [1..0..0..1]

Coherency matrix: C =

½ [1…..j]

    [-j…..1]


4) Partially polarized wave (d = 1/3, or completely polarized wave plus part linearly polarized with   t =  45 deg)

Stokes parameters:  [1..0..1/3..0]

Coherency matrix:  C =

½ [1…..1/3]

     [1/3…..1]


Example Problem:

If a radio wave has characteristics d = 1 and  AR = 1, find the normalized Stokes parameters and the coherency matrix.

We have for the normalized Stokes components:

  =   1

1    =  d  cos 2t cose 

2    =  d  sin2t cose

3    =  d  sin2e

And:   cose  =   (AR2    -  1) / (AR2   +  1)

=  (1  - 1)/ (1  + 1) =  0/ 1 = 0

Also:   sine  =  2AR/  (AR2   +  1)   =   2(1)/  (1 + 1) = 1

Then:

  =   1

1    =  d  cos 2t cose  =   0

2    =  d  sin2t cose  =   0

3    =  d  sin2e    =  (1) (1) =  1

So;  Stokes parameters are:  [1..0..0..1]

For which coherency matrix: C =

½ [1…..j]

    [-j…..1]


Problems:

1) Write an expression  for the degree of polarization given in terms of the normalized Stokes parameters.

2)  Write a matrix equation for a completely unpolarized radio wave using a left circularly polarized wave and a right circularly polarized wave.

3) The coherency matrix of some individual radio wave is given by:

½ [1…..0]

     [0…..0]

Show how the  resultant unpolarized wave's coherency matrix may be obtained by showing the C-matrix for the other wave needed to combine with the individual wave above.

4) Four radio waves are detected and analyzed and found to have the characteristics shown below:

a) d = 0

b) d =  ½      AR = 4     and   t=  135 deg

c)  d =  ½      AR = 4     and   t=  - 135 deg

d)  d =  ½      AR = 4     and   t=  45 deg

Find the normalized Stokes parameters and the coherency matrices for these waves.

No comments: