Stokes parameters and partial polarization

Use a common transverse basis and the same negative-sign time factor as in Jones calculus. Define $J_{ab}=\langle E_aE_b^*\rangle$, including the common impedance conversion so the entries have irradiance units. Diagonal entries are nonnegative component intensities; off-diagonal entries retain their mutual coherence. Hermitian symmetry follows directly from complex conjugation. Positivity follows because every projected intensity is the average squared magnitude of a projected field.

For one fixed coherent polarization, $J$ is an outer product and has rank one. For an incoherent mixture, add the coherency matrices weighted by their intensities or probabilities; do not add Jones amplitudes as though relative phase were stable. An equal incoherent mixture of orthogonal linear polarizations has zero off-diagonal entries. A coherent equal-amplitude sum of those same directions is instead linearly polarized at forty-five degrees and has a nonzero off-diagonal entry.

The total irradiance is $S_0$, while the length of the remaining Stokes vector measures its polarized part. Positivity bounds that length by $S_0$. A nonzero beam on the boundary is fully polarized, and a beam at the centre has no preferred polarization at this order of description. Interior points represent partial polarization. Normalizing the three polarization coordinates by their own length would incorrectly move every partially polarized state onto the fully polarized boundary.

The two eigenvalues of the coherency matrix are half the total intensity plus or minus half the polarization-vector length. They describe an incoherent decomposition into two orthogonal polarization states. An unpolarized contribution can also be separated from a fully polarized contribution, but neither decomposition means that individual photons carry a classical hidden polarization label. These are equivalent descriptions of measured second-order statistics, not unique microscopic histories.

An ideal linear analyzer at angle $\alpha$ transmits half the sum of total intensity and the appropriate double-angle projection of the linear Stokes components. Measuring orthogonal pairs at zero and ninety degrees, then forty-five and one hundred thirty-five degrees, determines $S_0,S_1,S_2$. Circular analyzers determine $S_3$ after their transmitted circular basis states have been defined. Circular-sign conventions vary, so a table without a basis definition is incomplete.

These Stokes parameters describe polarization at one location and within a chosen temporal or spectral averaging window. They do not determine mutual coherence between separated points or separate beams. A fully polarized source can still have limited temporal coherence; a partially polarized field can contain a coherent polarized component. The four-parameter transverse description also presumes one propagation direction or a suitable paraxial collection mode. Strongly nonparaxial, angularly unresolved fields may require a larger correlation description.