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

From: http://www.emeraldgraphics.com/Poincare.html

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^Footnotes

[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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