The so-called coherence Poincaré sphere was recently introduced for geometrical visualization of the state of two-point spatial coherence of a random electromagnetic beam. The formalism and its interpretation strongly utilized a specific decomposition of the Gram matrix of the cross-spectral density (CSD) matrix. In this work, we show that the interpretation of the coherence Poincaré sphere is obtained exclusively and straightforwardly via the singular value decomposition of the CSD matrix.The so-called coherence Poincaré sphere was recently introduced for geometrical visualization of the state of two-point spatial coherence of a random electromagnetic beam. The formalism and its interpretation strongly utilized a specific decomposition of the Gram matrix of the cross-spectral density (CSD) matrix. In this work, we show that the interpretation of the coherence Poincaré sphere is obtained exclusively and straightforwardly via the singular value decomposition of the CSD matrix.

The degree of coherence (DOC) function that characterizes the second-order correlations at any two points in a light field is shown to provide a new degree of freedom for carrying information. As a rule, the DOC varies along the beam propagation path, preventing from the efficient information recovery. In this paper, we report that when a partially coherent beam carrying a cross phase propagates in free space, in a paraxial optical system or in a turbulent medium, the modulus of the far-field (focal plane) DOC acquires the same value as it has in the source plane. This unique propagation feature is employed in a novel protocol for far-field imaging via the DOC, applicable to transmission in both free-space and turbulence. The advantages of the proposed approach are the confidentiality and resistance to turbulence, as well as the weaker requirement for the beam alignment accuracy. We demonstrate the feasibility and the robustness of the far-field imaging via the DOC in the turbulent media through both the experiment and the numerical simulations. Our findings have potential applications in optical imaging and remote sensing in natural environments, in the presence of optical turbulence.

We suggest tailoring of the illumination’s complex degree of coherence for imaging specific two- and three-point objects with resolution far exceeding the Rayleigh limit. We first derive a formula for the image intensity via the pseudo-mode decomposition and the fast Fourier transform valid for any partially coherent illumination (Schell-like, non-uniformly correlated, twisted) and then show how it can be used for numerical image manipulations. Further, for Schell-model sources, we show the improvement of the two- and three-point resolution to 20% and 40% of the classic Rayleigh distance, respectively.