Difference of two Gaussian Schell-model cross-spectral densities

Random sources with rectangular coherence

A convenient method for modeling partially coherent sources with rectangular coherence is introduced by structuring the degree of coherence as two separable arbitrary functions with arbitrary dependence of variables. The included examples have demonstrated new opportunities of modeling random sources for beam shaping applications by coherence modulation. The first example discusses a class of rectangular sinc-correlated models generating radiating fields with self-focusing features. As a second example, we introduce a new type of partially coherent vortex beams, which has a unique feature of self-rotation around the optical axis upon propagation.

Linear Combinations of the Complex Degrees of Coherence

We propose a method for structuring the spatial coherence state of light via mixed linear combinations of N complex degrees of coherence (CDC) and specify the conditions under which such combinations represent a valid CDC. Several examples demonstrate that this method opens previously unknown avenues for modeling random sources, radiating to light fields with unique features.

Christoffel-Darboux sources

A new, to the best of our knowledge, class of partially coherent sources is introduced on the basis of the Christoffel-Darboux (CD) formula read as the expression of a possible cross-spectral density. It will be seen that such an interpretation is possible because the CD formula gives the reproducing kernel of a suitable Hilbert space. After discussing general properties of CD kernels, a specific example is worked out using Hermite polynomials. A connection with the density matrix will be highlighted.

Asymmetric coherence gratings

We introduce a class of partially coherent, Schell-type sources whose degree of coherence is represented by a finite series of complex-valued functions. The significance of implementing such a series is due to the fact that one can manipulate the weighting coefficients of its terms having a computationally trivial linear phase of the degree of coherence for obtaining the radiated beams of the same complexity as could only be previously achieved with analytically intractable nonlinear phases. Our examples illustrate new opportunities for modeling asymmetric coherence gratings and lattices.

Hyperbolic sine-correlated beams

An explicit expression is given for the cross-spectral density that characterizes a new family of partially coherence sources with hyperbolic sine correlated function. Beam conditions for such sources are established. The propagation properties of such partially coherent beams are studied by numerical simulations. It is demonstrated that, unlike the reciprocity theorems relating to radiation from classical Schell-model sources, such beams possess both the invariance of coherent distribution and of hollow intensity shape.

Random sources for rotating spectral densities

The propagating modes of a wide-sense stationary Schell-like source with arbitrary coherence state and a twist factor are determined. This suggests a convenient practical method for modeling novel classes of twisted partially coherent beam-like fields. The first example discusses the previously introduced twisted anisotropic Gaussian Schell-model source and verifies the feasibility of this method. As a second example, we introduce a new type of twisted partially coherent beam in which a radiated flat-top average intensity pattern remains invariant in shape (but not size) while it twists around the axis upon propagation.

Random sources for beams with azimuthal intensity variation

The structure of the degree of coherence of wide-sense statistically stationary planar sources radiating far fields with intensity patterns with azimuthal dependence is determined, to the best of our knowledge, for the first time. The Fourier series basis is used in decomposing arbitrary far-field intensity for obtaining the azimuthal modes, which are shown to correspond to the spatial frequency components of the source degree of coherence. The far fields with azimuthal structure can be produced experimentally with the help of spatial light modulators.

Convolution of degrees of coherence

The conditions under which convolution of two degrees of coherence represents a novel legitimate degree of coherence are established for wide-sense statistically stationary Schell-model beam-like optical fields. Several examples are given to illustrate how convolution can be used for generation of a far field being a modulated version of another one. Practically, the convolutions of the degrees of coherence can be achieved by programming the liquid crystal spatial light modulators.

Electromagnetic random source for circular optical frame and its statistical properties

A class of random source for circular optical frame is generalized to electromagnetic domain. Analytical formulas for the propagation of the electromagnetic source for circular frames combinations through atmospheric turbulence are derived. As two examples, the statistic characteristics of a single circular frame and two nested frames are comparatively studied in free space and in non-Kolmogorov's atmospheric turbulence. The evolutions of the degree of polarization and the degree of coherence of such circular frames exhibit unique features. The impacts, arising from the refractive-index structure constant, the fractal constant of the atmospheric spectrum and the upper index in the source degree of coherence, on the statistical characteristics are analyzed in detail.

Alternating series of cross-spectral densities

We establish a sufficient condition under which the alternation series of cross-spectral densities (CSD) constitutes a valid CSD. Using such a condition, we introduce a novel class of multi-sinc Schell-model sources with circular and Cartesian symmetries. It is demonstrated that far fields produced by the new sources carry interesting characteristics, being adjustable multi-rings and lattice patterns.

Splitting and combining properties of an elegant Hermite-Gaussian correlated Schell-model beam in Kolmogorov and non-Kolmogorov turbulence

Elegant Hermite-Gaussian correlated Schell-model (EHGCSM) beam was introduced in theory and generated in experiment just recently [Phys. Rev. A 91, 013823 (2015)]. In this paper, we study the propagation properties of an EHGCSM beam in turbulent atmosphere with the help of the extended Huygens-Fresnel integral. Analytical expressions for the cross-spectral density and the propagation factors of an EHGCSM beam propagating in turbulent atmosphere are derived. The statistical properties, such as the spectral intensity, the spectral degree of coherence and the propagation factors, of an EHGCSM beam in Kolmogorov and non-Kolmogorov turbulence are illustrated numerically. It is found that an EHGCSM beam exhibits splitting and combing properties in turbulent atmosphere, and an EHGCSM beam with large mode orders is less affected by turbulence than an EHGCSM beam with small mode orders or a Gaussian Schell-model beam or a Gaussian beam, which will be useful in free-space optical communications.

Shape-invariant difference between two Gaussian Schell-model beams

The difference between two Gaussian Schell-model cross-spectral densities can give a new genuine correlation function if suitable conditions are met. Generally speaking, the structure of such cross-spectral density changes in a complicated way upon propagation. We consider here the notable exception of shape-invariant beams, and we investigate their intensity and coherence properties. The modal analysis of this class of cross-spectral densities is exploited to devise a synthesis scheme for this type of beam.

Powers of the degree of coherence

We establish conditions under which a legitimate degree of coherence of a statistically stationary beam-like field raised to a power results in a novel legitimate degree of coherence. The general results and examples relate to scalar beams having uniform and non-uniform correlations.

Products of Schell-model cross-spectral densities

The condition under which a product of two cross-spectral densities (CSD) constitutes a valid correlation function is established. The results are obtained for the CSDs of two one-dimensional, scalar Schell-model sources, but can be readily generalized to other situations. It is shown via a number of numerical examples how new source classes of Schell-model type and the beam-like fields they radiate can be designed.

Position modulation with random pulses

A new class of sources generating ensemble of random pulses is introduced based on superposition of the mutual coherence functions of several Multi-Gaussian Schell-model sources that separately are capable of shaping the propagating pulse's average intensity into flat profiles with adjustable duration and edge sharpness. Under certain conditions that we discuss in detail such superposition allows for production of a pulse ensemble that after a sufficiently long propagation distance in a dispersive medium reshapes its average intensity from an arbitrary initial profile to a train whose parts have flat intensities of different levels and durations and can be either temporarily separated or adjacent.