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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.

Surface plasmon polaritons have received much attention over the last decades in fields such as photonics or nanotechnology due to their inherent high sensitivity to metal surface variations (e.g., presence of adsorbates or changes in the roughness). Because of this important property, it is expected that they will find promising major applications in widely cross-disciplinary areas ranging from material science to medicine. Motivated by this ever-growing interest, here we introduce a novel theoretical framework suitable to design new types of structured paraxial surface plasmon beams and to control their propagation along the interface between a dielectric and a lossy metal. This is possible by virtue of a convenient Hermite-Gaussian mode expansion, which constitutes a complete basis set upon which new types of structured paraxial plasmon beams can be generated. The beams obtained from this method present a rather peculiar feature, as they exhibit local intensity maxima at different propagation distances. This way of creating tightly confined structured light allows us control where to place the beam energy. This, thus, opens up worth exploring pathways to manipulate light propagation along metal surfaces at the nanoscale. As a proof-of-concept, we provide numerical evidence of the feasibility of the method by considering the analysis of the propagation of Airy-based surface plasmon polaritons along an air-silver interface., Comment: 20 pages, 3 figures (plus 6 pages with supplemental information)

The propagation of Airy beams in free space is characterized by being non dispersive, which warrants the shape invariance of their intensity distribution, and self-accelerating along the transverse direction. These distinctive traits are still present in partially coherent Airy beams as long as the reach of their back tail (and hence their energy content) is not importantly reduced. To investigate the effects associated with the decrease of the beam coherence and its power content (by smoothly reducing the reach of their back tails), here we introduce a novel and insightful methodology based on a generalization of the concept of flux trajectory for paraxial partially coherent beams. This methodologies emphasizes the role of phase relations, thus helping to clarify why and how the beam smears out spatially along its propagation. This formalism, though, is general enough to tackle other types of structured light beams with whatever degree of partial coherence, from full coherence to total incoherence., Comment: 16 pages, 7 figures

In practice, Airy beams can only be reproduced in an approximate manner, with a limited spatial extension and hence a finite energy content. To this end, different procedures have been reported in the literature, based on a convenient tuning of the transmission properties of aperture functions. In order to investigate the effects generated by the truncation and hence the propagation properties displayed by the designed beams, here we resort to a new perspective based on a trajectory methodology, complementary to the density plots more commonly used to study the intensity distribution propagation. We consider three different aperture functions, which are convoluted with an ideal Airy beam. As it is shown, the corresponding trajectories reveals a deeper physical insight about the propagation dynamics exhibited by the beams analyzed due to their direct connection with the local phase variations undergone by the beams, which is in contrast with the global information provided by the usual standard tools. Furthermore, we introduce a new parameter, namely, the escape rate, which allow us to perform piecewise analyses of the intensity distribution without producing any change on it, e.g., determining unambiguously how much energy flux contributes to the leading maximum at each stage of the propagation, or for how long self-accelerating transverse propagation survives. The analysis presented in this work thus provides an insight into the behavior of finite-energy Airy beams, and therefore is expected to contribute to the design and applications exploiting this singular type of beams., Comment: 15 pages, 5 figures; very minor changes with respect to previous version

Airy beams are known for displaying shape invariance and self-acceleration along the transverse direction while they propagate forwards. Although these properties could be associated with the beam coherence, it has been revealed that they also manifest in the case of partially coherent Airy-type. Here, these properties are further investigated by introducing and analyzing a class of partially coherent Airy beams under both infinite and finite energy conditions. The key element within the present approach is the so-called cross-spectral density, which enables a direct connection with the quantum density matrix, making the analysis exportable to the quantum realm to study the dynamics of Airy wave packets acted by both incoherence and decoherence. As it is shown, in the case of infinite energy beams both properties are preserved even under the circumstance of total incoherence provided the underlying structure of the beam remains equal to that of an Airy beam. In the case of finite energy beams, a situation closer to a realistic scenario, as experimental beams cannot have an infinite extension, it is shown that a propagation range along which both properties are preserved can be warranted. This is controlled by a critical distance, which depends on the spread range determined by the parameters ruling the extension of random field spatial fluctuations. Such a distance is determined by defining a position-dependent parameter that quantifies the degree of overlapping between the propagated beam and the input one displaced by an amount equivalent to the propagation distance., Comment: 13 pages, 6 figures (very minor corrections with respect to previous version)

In the analysis of the on-axis intensity for a highly focused optical field it is highly desirable to deal with effective relations aimed at characterizing the field behavior in a rather simple fashion. Here, a novel and adequate measure for the size of the region where the axial power content mainly concentrates is proposed on the basis of an uncertainty principle. Accordingly, a meaningful relationship is provided for both the spread of the incident beam at the entrance of the highly focused optical system and the size of the region where the on-axis power mainly concentrates., Comment: 15 pages, 6 figures