Generalized partially coherent beams with nonseparable phases

Atmospheric turbulence effects on hollow Gaussian Schell-model array beams

Two types of hollow array beams with circular and rectangular distributions on propagating in atmospheric turbulence are investigated and analyzed comparatively with that in free space. Analytical formulas for the cross-spectral density function of two kinds of hollow array beam propagation in linear isotropic random media are derived and used to examine the behavior of the spectral densities. It is found that such beams possess stable hollow arrays with any dimension and lobes in free space, while such distributions only maintain small distances in atmospheric turbulence and ultimately tend to a Gaussian shape due to the turbulence destroying the hollow array profiles. The effects of the turbulence parameters on the behavior of the spectral density are analyzed in depth.

Virtual sources for structured partially coherent light fields

A virtual source (VS) is a hypothetical source instead of an actual physical entity, but provides a distinctive perspective to understand physical fields in a source-free area. In this work, we generalize the VS theory to structured partially coherent light fields (PCLFs) by establishing the partially coherent inhomogeneous Helmholtz equation, then demonstrate that PCLFs can be generated from the incoherent extended VS in imaginary space. Especially, we put forward an understanding of the Gaussian Schell-model beam, which consists of a group of partially coherent paraxial complex rays. The mutual coherence between these rays depends on the included angle between them. In previous studies, the analytical solution of the partially coherent Airy beam was obtained with difficulty by the Huygens-Fresnel integral; however, by applying the VS, we put forward, to our knowledge, an unprecedented analytical solution for a partially coherent Airy beam. We believe this example will qualify the VS as an important perspective to understand structured PCLFs.

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.

Asymmetrical inseparable coherent structures

A novel, to the best of our knowledge, class of coherent structures of inseparability, incorporating phases asymmetrically cross-coupled by two position vectors, is introduced in theory and experiment. These phases disappear in the environment of complete coherence, but the vanishment is avoidable in the coexistent state of extreme incoherence and full coherence. The radiated beams intrinsically possess a controllable rotation but undergo an intermediate process quite different from the twisted Gaussian Schell-model beams. Analysis shows a novel association between the magnitude and the phase of the coherent structure which displays both synergy and opposition. Our work further reveals the inner mechanism of the inseparable coherent structures and extends a new horizon for the optical twist.

Self-rotating beam in the free space propagation

We introduce a class of self-rotating beams whose intensity profile tends to self-rotate and self-bend in the free space propagation. The feature of the self-rotating beams is acceleration in the three-dimensional (3D) space. The acceleration dynamics of the self-rotating beams is controllable. Furthermore, multiple self-rotating beams can be generated by a combined diffractive optical element (DOE) simultaneously. Such a beam can be viewed as evolution of a vortex beam by changing the exponential constant of phase. We have generated this beam successfully in the experiment and observed the expected phenomenon, which is basically consistent with the result of the numerical simulation. Our results may provide new insight into the self-rotating beam and extend potential applications in optical imaging.

Twisted sinc-correlation Schell-model beams

We introduce a new class of twisted sinc-correlation Schell-model (TSCSM) beams and analyze the statistical characteristics of such novel sources during propagation. Several typical examples are given to specifically explore the distribution and twist effect of spectral density and degree of coherence (DOC). It is shown that the irradiance profile of light intensity always rotates to 90 degree. With appropriate light field adjustment, twist effect of DOC would be diverse. DOC can exhibit unidirectional or non-unidirectional rotation during propagation. Besides, the twist factor can make the spot show a tendency to split. And beam width and coherence length also have an impact on this splitting phenomenon of spectral density.

Generating non-uniformly correlated twisted sources

The inverse method of proving the twistability of cross-spectral density (CSD) inevitably falls into spontaneous difficulties. Based on a nonnegative self-consistent design guideline for generating genuine CSDs introduced by Gori and Santarsiero, we demonstrate a feasible way for twisting partially coherent sources by sticking a Schell-model function to CSDs, which also determines the upper bound of the twisting strength. Analysis shows that the degree of coherence of a new class of twisted pseudo-Gaussian Schell-model beam is neither shift invariant nor shift-circular symmetric. In the presence of a vortex phase, the two different types of chiral phases affect each other and together control the propagation behavior. We further carry out an experiment to generate this non-uniformly correlated twisted beam using weighted superposition of mutually uncorrelated pseudo modes. The result is beneficial for devising nontrivial twisted beams and offers new opportunities.

Special correlation model sources producing a self-focusing field

We evaluate the modes for non-Schell-model sources whose degrees of spectral coherence depend on the difference of the special function values of the position coordinated of two points. It is shown that such sources modulated by various function possess different spatial coherence properties, and cause them to produce the self-focusing fields with different characteristics. The results suggest a convenient method for modeling novel classes of partially coherent self-focusing optical fields.

Self-focusing vortex beams

A class of wide-stationary optical sources with a specially designed degree of coherence profile is introduced for radiating spectral densities with a vortex whose core's location and size can be controlled at a specified range. This is achieved by modeling of the source coherence state as a combination of a helicoidal separable phase and a Cartesian phase factor, depending on the separation between the $ n $th power of the radius-vectors of two points.

Progress on Studies of Beams Carrying Twist

Optical twist has always been a hot spot in optics since it was discovered in 1993. Twisted beams can be generated by introducing the twist phase into partially coherent beams, or by introducing the twisting phase into anisotropic beams, whose spectral density and degree of coherence will spontaneously rotate during propagation. Unlike conventional beams, twisted beams have unique properties and can be used in many applications, such as optical communications, laser material processing, and particle manipulation. In this paper, we present a review of recent developments on phase studies of beams carrying twist.

Generalized Schell-model sources

We evaluate the modes for generalized Schell-model planar source whose complex degree of coherence (CDC) is a function of the n-th power difference of two position coordinates instead of their direct distance between two source points. We discuss through two examples how new classes of CDCs can be devised and how they affect the radiation fields. It is demonstrated that the light beams generated by these families of sources carry interesting propagation characteristics, such as the lateral self-shifting and the self-focusing effect with controllable focal length determined by the non-trivial phase, power n and other source parameters.

Twisted anisotropic electromagnetic beams with Laguerre Gaussian-Schell model correlation

A class of twisted anisotropic electromagnetic beams with Laguerre Gaussian-Schell model correlation is introduced as an extension of the scalar beams into electromagnetic domain. The analytical formula for the cross-spectral density matrix of such a beam on propagation has been derived. Then the degree of coherence, the degree of polarization and the state of polarization are discussed in detail. Our results reveal that it is feasible and efficient to engineer the characteristics of beams via setting the anisotropy of the beam source, the topological charge, and specially the twisted factor. This provides us a method for synthesizing fields presenting peculiar coherence and polarization patterns.

Twisted space-frequency and space-time partially coherent beams

We present partially coherent sources that are statistically twisted in the space-frequency and space-time domains. Beginning with the superposition rule for genuine partially coherent sources, we derive source plane expressions for the cross-spectral density (CSD) and mutual coherence functions (MCFs) for twisted space-frequency and space-time Gaussian Schell-model (GSM) beams. Using the Fresnel approximation to the free-space Green's function, we then paraxially propagate the CSD and MCF to any plane [Formula: see text]. We discuss the beams' behavior as they propagate, with particular emphasis on how the beam shape rotates or tumbles versus z. To validate our analysis, we simulate the generation and subsequent propagation of twisted space-frequency and space-time GSM beams. We compare the simulated moments to the corresponding theoretical predictions and find them to be in excellent agreement. Lastly, we describe how to physically synthesize twisted space-frequency and space-time partially coherent sources.

Cross-spectral densities with helical-Cartesian phases

We introduce a class of planar, stationary sources whose cross-spectral densities carry a combination of helical and Cartesian phases. The helical phase is linear, separable in polar coordinates, resulting in a vortex-like average intensity with a dark area centered on the optical axis; the Cartesian counterpart is separable in the x- and y-coordinates and is responsible for asymmetric average intensity redistribution along the x- and y-axes. While endless possibilities exist for modeling of the Cartesian phase factor, in this paper we employ a superposition of linear phases with arbitrarily assigned weighing factors. Such construction is analytically and experimentally useful in general and, in particular, as we show, for generation of asymmetric vortex lattices.

Random source for generating Airy-like spectral density in the far field

A stationary beam forming an Airy-like spectral density in the far field is analyzed theoretically and experimentally. The Schell-model source that radiates such a beam is an extended version of a recently introduced source [O. Korotkova, et al., Opt. Lett.43, 4727 (2018)10.1364/OL.43.004727; X. Chen, et al., Opt. Lett.44, 2470 (2019)10.1364/OL.44.002470, in 1D and 2D, respectively]. We show, in particular, that the source degree of coherence, being the fourth-order root of a Lorentz-Gaussian function and having linear and cubic phase terms, may be either obtained from the Fourier transform of the far-field Airy-like pattern or at the source using the sliding function method. The spectral density of the beam is analyzed on propagation through paraxial ABCD optical systems, on the basis of the generalized Collins integral, by means of the derived closed-form expression. We show that the distribution of the side lobes in the Airy beam spectral density can be controlled by the parameters of the source degree of coherence. Further, an experiment involving a spatial light modulator (SLM) is carried out for generation of such a beam. We experimentally measure the complex degree of coherence of the source and observe the gradual formation of a high-quality Airy-like spectral density towards the far field. In addition, the trajectory of the intensity maxima of the beam after a thin lens is studied both theoretically and experimentally. The random counterpart of the classic, deterministic Airy beam may find applications in directed energy, imaging, beam shaping, and optical trapping.