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

[ VI1a ] 

[ VI1b ] 
For later reference, we note that the full(nonnormalized)
Jones
vector representation[2]
of this field is given by

[ VI2 ] 
We can be easily shown that

[ VI3a ] 

[ VI3b ] 
Thus the locus of time sequence
of fields in a plane perpendicular to the direction of propagtion follows
an ellipse  viz

[ VI4 ] 
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 righthanded. For rotation in the
anticlockwise sense (i.e.,
), the polarization is said to be righthanded. 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

[ VI5a ] 

[ VI5b ] 
where we choose f so that

[ VI6a ] 

[ VI6b ] 

[ VI6c ] 
Where the upper and lower signs are for, respectively, lefthanded
and
righthanded 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

[ VI7a ] 

[ VI7b ] 

[ VI7c ] 

[ VI7d ] 
These relations are greatly simplified if we introduce the
two
auxiliary angles

[ VI8a ] 

[ VI8b ] 
(Note:
specifies the, so called, ellipticityof
the vibrational ellipse.). In terms of these auxiliary angles, we may then
write

[ VI9a ] 

[ VI9b ] 

[ VI9c ] 
Probably, the most powerful representation of polarization
is found in the famous Stokes vector or parameters  viz.[3]

[ VI10a ] 

[ VI10b ] 

[ VI10c ] 

[ VI7d ] 
so that a polarized field may be represented by the vector

[ VI11a ] 
or its transpose

[ VI11b ] 
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 [ VI10 ], for a completely polarized
field only three of these parameters or vector components are independent,
since

[ VI12 ] 
and {M, C, S} can be interpreted as the Cartesian coordinates
of a sphere of radius I  the Poincaré
sphere.^{ }[4] Thus, from
Equations [ VI8 ] and [ VI10 ] we obtain the coordinates of the Poincaré
sphere or representation as

[ VI13a ] 

[ VI13b ] 

[ VI13c ] 

[ VI13d ] 
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.
,Back to top
This page was prepared and is maintained
by R. Victor Jones, jones@deas.harvard.edu
Last updated February 17, 2000