Degree of polarization for optical near fields

Abstract

We investigate an extension to the concept of degree of polarization that applies to arbitrary electromagnetic fields, i.e., fields whose wave fronts are not necessarily planar. The approach makes use of generalized spectral Stokes parameters that appear as coefficients, when the full 3 x 3 spectral coherence matrix is expanded in terms of the Gell-Mann matrices. By defining the degree of polarization in terms of these parameters in a manner analogous to the conventional planar-field case, we are led to a formula that consists of scalar invariants of the spectral coherence matrix only. We show that attractive physical insight is gained by expressing the three-dimensional degree of polarization explicitly with the help of the correlations between the three orthogonal spectral components of the electric field. Furthermore, we discuss the fundamental differences in characterizing the polarization state of a field by employing either the two- or the three-dimensional coherence-matrix formalism. The extension of the concept of the degree of polarization to include electromagnetic fields having structures of arbitrary form is expected to be particularly useful, for example, in near-field optics.

Classical distinguishability as an operational measure of polarization

“…Although this question has been considered for many years, no satisfactory solution has thus far been found. Indeed, there are several contradictory claims made in the literature on this subject [6][7][8][9][10][11][12][13][14][15]. The divergences occur because notions that are equivalent for the 2D case lead to different definitions when extrapolated to the 3D limit.…”

Dimensionality of random light fields

“…Since for polarized light all field components are completely correlated [20], i.e., |μ αβ (r, ω)| = 1, we can express the polarization matrix in Eq. (1) …”

“…We remark that an expression formally similar to Eq. (8), with (r, ω) replacing (r, ω), has been employed to characterize the degree of polarization of random 3D light fields [20,21]. The polarimetric dimension is thus a real number that obeys 1 ≤ D(r, ω) ≤ 3.…”

Measures of quantum state purity and classical degree of polarization

“…We have modified the coefficients from those in ref. [11] to ease calculation. The eight coefficients r i together form a (generalized) Bloch vector r. One can use eq.…”

“…The measure of purity due to Setälä, Shevchenko, Kaivola, and Friberg [11] starts by writing the 3 × 3 density matrix ρ as a linear combination of some basis matrices in an expression similar to eq.(7). We write ρ as…”