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Both the intensity distribution and the degree of coherence between pairs of points along the propagation axis (z-coherence) are derived in closed form for a phenomenon of self-focusing produced by circularly coherent light. The first confirms results previously obtained numerically, while the second exhibits new complex features. The physical interpretation is obtained by a suitable pseudo-modal expansion that suggests an analogy with a simple two-mode structure.

Diffractive optical elements that divide an input beam into a set of replicas are used in many optical applications ranging from image processing to communications. Their design requires time-consuming optimization processes, which, for a given number of generated beams, are to be separately treated for one-dimensional and two-dimensional cases because the corresponding optimal efficiencies may be different. After generalizing their Fourier treatment, we prove that, once a particular divider has been designed, its transmission function can be used to generate numberless other dividers through affine transforms that preserve the efficiency of the original element without requiring any further optimization.

The degree of coherence and the intensity distribution on the axis of the beam radiated by a planar partially coherent source of the Schell-model type are investigated. We present an expression for the on-axis cross-spectral density which is valid for a very general Schell-model source, with the only constraint that the intensity distribution across the source is Gaussian. Furthermore, we show that such an expression takes very simple analytical forms for several commonly used degrees of coherence of the source.

Partially coherent electromagnetic sources with cylindrical symmetry and infinite extent radiating outwards are introduced. Their 3x3 cross-spectral density matrix is given through expansions of the field components in terms of basis functions related to the Hankel functions. The spectral density and the three-dimensional degree of polarization of such sources and the fields they radiate are examined. Several examples are presented and discussed. Among them, a class of cylindrical sources whose coherent vector modes coincide with the above basis functions is defined and studied.

The on-axis cross-spectral density (CSD) of a beam radiated by a stationary source with a circular coherence state and a Gaussian spectral density is obtained in the closed form. It is revealed that the on-axis CSD is expressed via the Laplace transform of the source's degree of coherence or the Hilbert transform of the corresponding pseudo-mode weighting function. Such relations enable efficient tailoring of the on-axis spectral density, as we show with a slew of numerical examples.

Partially coherent electromagnetic sources with cylindrical symmetry and infinite extent radiating outwards are introduced. Their $3\times3$ cross-spectral density matrix is given through expansions of the field components in terms of basis functions related to the Hankel functions. The spectral density and the three-dimensional degree of polarization of such sources and the fields they radiate are examined. Several examples are presented and discussed. Among them, a class of cylindrical sources whose coherent vector modes coincide with the above basis functions is defined and studied.