On Classical Electromagnetic Fields (cont.)

VI. Descriptions of the Polarization of Light (pdf copy) [1]

Consider a totally coherent wave propagating in the positive direction
      [ VI-1a ]
      [ VI-1b ]
For later reference, we note that the full(non-normalized) Jones vector representation[2] of this field is given by
      [ VI-2 ]
We can be easily shown that
      [ VI-3a ]
      [ VI-3b ]
Thus the locus of time sequence of fields in a plane perpendicular to the direction of propagtion follows an ellipse -- viz
      [ VI-4 ]

 

For an electric field vector "seen" to be rotating in a clockwise direction by an observer receiving the radiation (i.e., ), the polarization is said to be right-handed. For rotation in the anticlockwise sense (i.e., ), the polarization is said to be right-handed. Although this is a complete decription of the coherent field, it is not a convenient one. It is useful to transform this equation to its principal axes form by the following transformation
       [ VI-5a ]
       [ VI-5b ]
where we choose f so that
       [ VI-6a ]
       [ VI-6b ]
       [ VI-6c ]
Where the upper and lower signs are for, respectively, left-handed and right-handed polarizations. After quite a bit of algebraic manipulation, we find a more elegant and convenient description of polarization in terms of the following set of relationships
       [ VI-7a ]
       [ VI-7b ]
       [ VI-7c ]
       [ VI-7d ]
These relations are greatly simplified if we introduce the two auxiliary angles
       [ VI-8a ]
       [ VI-8b ]
(Note:  specifies the, so called, ellipticityof the vibrational ellipse.). In terms of these auxiliary angles, we may then write
       [ VI-9a ]
       [ VI-9b ]
       [ VI-9c ]
Probably, the most powerful representation of polarization is found in the famous Stokes vector or parameters -- viz.[3]
       [ VI-10a ]
       [ VI-10b ]
       [ VI-10c ]
       [ VI-7d ]
so that a polarized field may be represented by the vector
      [ VI-11a ]
or its transpose
      [ VI-11b ]
These vector components give a complete geometric description of the vibrational ellipse  -- viz.
 
 
Size
I
Azimuth 
Shape
Handedness 
Sign of S

As can be seen from Equations [ VI-10 ], for a completely polarized field only three of these parameters or vector components are independent, since

      [ VI-12 ]
and {M, C, S} can be interpreted as the Cartesian coordinates of a sphere of radius I -- the Poincaré sphere. [4]  Thus, from Equations [ VI-8 ] and [ VI-10 ] we obtain the coordinates of the Poincaré sphere or representation as
       [ VI-13a ]
       [ VI-13b ]
       [ VI-13c ]
       [ VI-13d ]

 
 
The Poincaré Sphere

 


Footnotes

[1] The best references on this subject are the following: 1.) William A. Shurcliff, Polarized Light: Production and Use, Harvard University Press (1962); 2.) D. Clarke and J.F. Grainger, Polarized Light and Optical Measurement, Pergamon Press (1971); 3.) Max Born and Emil Wolf, Principles of Optics, Pergamon Press (Particularly Section 1.4).

[2]R. Clarke Jones, "New calculus for the treatment of optical systems. I. Description and discussion of calculus," J. Opt. Soc. Amer. 31, 488 (1941).

[3]G. G. Stokes, "On the composition and resolution of streams of polarized light from different sources," Trans. Cambridge Phil. Soc. 9, 399 (1852).

[4] H. Poincaré, Théorie Mathématique de la Lumière, Vol. 2, (1892) Chap. 12.
 

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Last updated February 17, 2000