Description of the radiation field, the coherence matrix

A coherent fully polarised radiation field is represented by a 2 dimensional complex vector field:
 
which depends on 3 physical parameters, plus one irrelevant phase. In general a radiation field is an incoherent superposition of waves with different polarisations, it is then more convenient to describe the radiation field statistically through its hermitian ``coherence'' matrix :
 
Because of the mean value that enter its definition, the coherence matrix depends on 4 independent physical parameters, which reduce to 3 for a fully polarised radiation. These four physical parameters can be chosen to be the Stokes parameters:
 
where matrices are the usual Pauli matrices :

The reason why we choose to index them in this way should be obvious from relations (3) and (4). The Stokes parameters:

can be deduced from the coherence matrix as:
 
and satisfy the inequality:

which expresses the fact that the polarised energy cannot exceed the total energy. It becomes an equality for a fully polarised radiation, thus reducing the number of physical parameters to 3, as expected.