For random electromagnetic beams, Hassinen, Tervo and Friberg [Opt. Lett. 38, 1221 (2013)] introduced 'pure partially polarized fields' — fields whose polarization state remains invariant in linear diffractive systems, in particular Young's two-pinhole interferometer — and derived necessary and sufficient purity conditions in both the space–frequency and space–time domains, showing that the frequency-domain condition is the more general one and connecting polarization purity to nonquantum entanglement between polarization and spatial or temporal correlations. The derivation is restricted to paraxial two-component fields and assumes quasi-monochromatic, stationary light in the time-domain treatment. The open problem is to complete this purity theory: establish necessary and sufficient conditions under which the polarization state is invariant in interference and diffraction for strongly nonparaxial (three-component) fields and for broadband or nonstationary fields in the time domain. Later literature (2019–2026) has substantially developed the adjacent notion of cross-spectral purity for nonstationary vector fields and a modal 'optical cross-purity', but the nonparaxial purity conditions and the broadband time-domain polarization-state purity conditions remain unestablished.

Background

Random light is described statistically, through correlation functions of the complex analytic field. At a single point and a single frequency, the polarization of a paraxial beam is encoded in the 2×22\times2 coherency matrix JJ (also called the polarization matrix), introduced for partially polarized radiation by Wolf in 1959; its degree of polarization (DoP) — a number between 0 (unpolarized) and 1 (fully polarized) — together with the principal polarization state constitutes the 'polarization state' of the beam. When light from two points of a random beam is superposed in Young's double-pinhole interferometer, the polarization of the superposed field generally varies across the observation screen (polarization modulation), as described by the electromagnetic theory of partial coherence in the space–frequency domain (Tervo, Setälä and Friberg, 2004). For scalar fields, Mandel (1961) introduced cross-spectral purity: the normalized spectrum of the two-point superposition equals the component spectra precisely when the normalized cross-spectral density factorizes into a purely spatial and a purely spectral factor (a reduction formula). The vector analogues were developed by Hassinen, Tervo and Friberg in a trilogy: cross-spectral purity of electromagnetic fields (Opt. Lett. 2009), cross-spectral purity of the Stokes parameters (Appl. Phys. B 2011), and purity of partial polarization (Opt. Lett. 2013) — the source of the present question, with the time- and frequency-domain degrees of polarization of random beams reviewed by Setälä, Nunziata and Friberg (2010). A field is a pure partially polarized field if its polarization state remains invariant when it passes through linear diffractive devices, in particular Young's interferometer; the 2013 paper derived necessary and sufficient purity conditions in the space–frequency domain and in the space–time domain — where the two-time correlation matrix is linked to the spectral description by a Wiener–Khintchine-type transform — showed that the frequency-domain condition is strictly more general, and related polarization purity to nonquantum entanglement, i.e. the inseparability of the polarization degree of freedom from the spatial or temporal degrees of freedom in the field's correlation structure, a classical-optics analogue of entanglement. The 2013 treatment rests on restrictions inherent to its formalism: (i) paraxial geometry — only the two transverse electric-field components are retained and diffraction enters through slowly varying scalar geometrical factors; and (ii) quasi-monochromatic, stationary fields in the time-domain analysis, so the generalized Wiener–Khintchine relation applies. Outside these restrictions the concept is undefined. For strongly nonparaxial fields — near fields of small apertures, strongly focused beams, large-angle propagation — all three electric-field components are significant, the coherency matrix becomes 3×33\times3, and the polarization state itself has non-unique generalizations: several inequivalent three-dimensional degrees of polarization exist (Alonso, 2023; Sheppard et al., 2021; Gil et al., 2018, where 'purity' means the polarimetric purity of a mixed polarization state, a different notion from interference-invariance purity). For broadband and nonstationary (e.g. pulsed) light, the correlation structure is two-time and the quasi-monochromatic link between domains breaks down (Koivurova, Laatikainen and Friberg, 2024). Completing the purity theory of partial polarization in these regimes is the unresolved target.

Problem Statement

Complete the purity theory of partial polarization by removing the restrictions under which the 2013 conditions were derived. The single objective is a full set of necessary and sufficient conditions for the polarization state of a random electromagnetic field to be invariant under superposition and linear diffractive propagation, in particular in Young-type interferometry, valid without the paraxial two-component and quasi-monochromatic assumptions. Two jointly exhaustive components express this objective, each relaxing one of the two assumptions: (1) Nonparaxial purity conditions: for fields in which all three electric-field components are significant (near fields of small apertures, strongly focused beams, large-angle propagation), formulate necessary and sufficient conditions on the two-point, two-frequency coherency (polarization) matrices under which the polarization state is invariant in Young-type superposition and in general linear diffractive systems; because the three-dimensional polarization state is described by non-equivalent generalizations of the degree of polarization, the answer must specify which polarization descriptor(s) the invariance is defined against, and the conditions must reduce to the known paraxial purity conditions in the small-angle limit. (2) Broadband time-domain purity: establish necessary and sufficient time-domain purity conditions for fields of arbitrary bandwidth and nonstationary (e.g. pulsed) statistics, beyond the quasi-monochromatic link between the space–time and space–frequency descriptions used in the 2013 treatment; the relation between these conditions and the reduction formulas established for cross-spectral purity of the Stokes parameters of nonstationary vector fields may be clarified along the way, but determining that relation is not itself a completion requirement.

References

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