Nonconventional Correlation Function Research Articles

Propagation Properties of a Twisted Hermite-Gaussian Correlated Schell-Model Beam in Free Space

We introduce a novel type of twisted partially coherent beams with a nonconventional correlation function, named the twisted Hermite-Gaussian correlated Schell-model (THGCSM) beam. The condition that a twist phase can be imposed on a partially coherent beam is addressed for Schell-model fields endowed with rectangular symmetry. Further, the analytical formula for the THGCSM beam propagating in free space has been derived with the help of the generalized Collins formula. The propagation properties, such as the spectral density and the spectral degree of coherence (SDOC) of the THGCSM beam, also have been studied in detail by some numerical examples. The numerical results show that the twist phase plays a role in resisting beam splitting, caused by the correlation structure, and induces the rotation of the distribution of the SDOC on propagation. Moreover, it is interesting to find that when the beam carries a twist phase, this will endow the beam the ability to maintain its distribution of the SDOC on propagation and enhance the self -reconstruction capability of the SDOC. Our results may provide new insights into nonconventional partially coherent beams with twisted phase and may be useful in some applications, such as optical communications and information recovery.

Review on partially coherent vortex beams

Ever since vortex beams were proposed, they are known for owning phase singularity and carrying orbital angular momentum (OAM). In the past decades, coherent optics developed rapidly. Vortex beams have been extended from fully coherent light to partially coherent light, from scalar light to vector light, from integral topological charge (TC) to fractional TC. Partially coherent vortex beams have attracted tremendous interest due to their hidden correlation singularity and unique propagation properties (e.g., beam shaping, beam rotation and self-reconstruction). Based on the sufficient condition for devising a genuine correlation function of partially coherent beam, partially coherent vortex beams with nonconventional correlation functions (i.e., non-Gaussian correlated Schell-model functions) were introduced recently. This timely review summarizes basic concepts, theoretical models, generation and propagation of partially coherent vortex beams.

Partially Coherent Flat-Topped Beam Generated by an Axicon

The intensity distribution of a partially coherent beam with a nonconventional correlation function, named the multi-Gaussian Schell-model (MGSM) beam, focused by an axicon was investigated in detail. Our numerical results showed that an optical needle with a flat-topped spatial profile and long focal depth was formed and that we can modulate the focal shift and focal depth of the optical needle by varying the width of the degree of coherence (DOC) and the parameters of the correlation function. The adjustable optical needle can be applied for electron acceleration, particle trapping, fiber coupling and percussion drilling.

Generation of an Adjustable Optical Cage through Focusing an Apertured Bessel-Gaussian Correlated Schell-Model Beam

An adjustable optical cage generated by focusing a partially coherent beam with nonconventional correlation function named the Bessel–Gaussian correlated Schell-model (BGCSM) beam is investigated in detail. With the help of the generalized Huygens–Fresnel integral and complex Gaussian function expansion, the analytical formula of the BGCSM beam passing through an apertured ABCD optical system was derived. Our numerical results show that the generated optical cage can be moderately adjusted by the aperture radius, the spatial coherence width, and the parameter β of the BGCSM beam. Furthermore, the effect of these parameters on the effective beam size and the spectral degree of coherence were also analyzed. The optical cage with adjustable size can be applied for particle trapping and material thermal processing.

Transmission of a polychromatic electromagnetic multi-Gaussian Schell-model beam in an inhomogeneous gradient-index fiber.

We derive analytical expressions for the cross-spectral density matrix of polychromatic electromagnetic multi-Gaussian Schell-model (EMGSM) beam transmission through a gradient-index fiber. The space-spectrum evolution properties for the spectral density, spectral shift, degree of polarization, and electromagnetic coherence state of a polychromatic EMGSM beam with Lorentzian line type spectrum and central wavelength λ0=1550 nm propagation in a silica-clad germania core inhomogeneous graded-index fiber are studied in detail. We show that these statistical properties exhibit periodicity in the fiber, caused by the focusing property of square-law media, which can be reminiscent of the self-imaging effect of optical fields. The effects of the nonconventional correlation functions of the polychromatic EMGSM beam on the transmission properties are also investigated.

Cascade self-splitting of a Hermite-cos-Gaussian correlated Schell-model beam

We propose a new kind of partially coherent beam with a nonconventional correlation function termed as Hermite-cos-Gaussian correlated Schell-model (HcGCSM) beam. The propagation properties of this novel model beam are investigated. It is found that the HcGCSM beam exhibits cascade self-splitting properties on propagation in free space, i.e., the initial single beam spot is proved to be successively split for two times during the whole propagation. Furthermore, we demonstrate that the cascade self-splitting phenomenon can be closely controlled through modulating the spectral degree of coherence of a HcGCSM beam in the source plane.

Propagation properties of nonuniform cosine-Gaussian correlated Bessel-Gaussian beam through paraxial ABCD system and generation of dark-hollow beam array

Partially coherent beams with nonconventional correlation functions have been extensively studied due to their wide and important applications in free-space optical communication, particle trapping, image transmission and optical encryption. Here, we study the propagation of nonuniform cosine-Gaussian correlated Bessel-Gaussian beam (cGBCB) in detail. Analytical expressions for the cross-spectral density function of cGBCBs through paraxial ABCD system are derived based on the extended Huygens-Fresnel integral. By use of the derived formulae, the intensity distribution properties of a nonuniform cGBCB on propagation in free space are analytically investigated. Some numerical calculation results are presented and discussed graphically. It is found that when the root-mean-square correlation width δ and the parameter controlling the degree of coherence profiles β are appropriately chosen, the intensity distribution of the nonuniform cGBCB displays self-splitting properties during propagation. We point out that rather than a simple duplication, the self-splitting behaviour consists of a complex process in which the dark hollow pattern for cGBCB is gradually filled in the centre at first, then starts to split with increasing the propagation distance, and most impressively, an evolution process from a single dark hollow beam in the source plane to quadruple dark hollow profiles in certain propagation ranges can be realized. The influence of correlation function on the evolution properties of the intensity distribution is investigated, demonstrating that the values of parameters δ and β of the correlation function play a critical role in inducing the self-splitting effect for nonuniform cGBCB on propagation in free space. Therefore, it is clearly shown that modulating the correlation function of a partially coherent beam can alter the coherence length and the degree of nonuniformity, and thus provides an effective way of manipulating its propagation properties. We also find the evolution speed of the intensity distribution can be greatly affected by the topological charge n of the beam function and the parameter R controlling the hollow size of cGBCB in source plane, e. g. the intensity distribution evolves into quadruple dark hollow profiles more slowly with larger n or smaller R. As is well known, the dark-hollow intensity configurations are useful in many applications and have been extensively studied both theoretically and experimentally. Therefore, the results drawn in the paper develop an alternative way to realize dark-hollow beam array, and further pave the way for dark hollow beam applications in long-distance free-space optical communications.

Effect of the correlation function on the focal shift of a partially coherent beam.

It is known that the focal shift is closely related with the Fresnel number of the lens system, the amplitude, and the spatial coherence of the light beam. In this paper, we investigate the focusing properties of a partially coherent beam with nonconventional correlation function, named a generalized multi-Gaussian correlated Schell-model beam, through deriving the analytical expression for its effective beam width. We find that the focal shift is also affected by the structure parameter of the correlation function. Our results provide a novel way for controlling the focal shift.

Theoretical and experimental studies of a rectangular Laguerre–Gaussian-correlated Schell-model beam

We introduce a new kind of partially coherent beam with nonconventional correlation function named rectangular Laguerre–Gaussian-correlated Schell-model (LGCSM) beam, whose degree of coherence is of rectangular symmetry, and analyze its propagation properties. We find that the rectangular LGCSM beam exhibits self-splitting properties on propagation in free space, i.e., the initial single beam spot evolves into (m + 1) × (n + 1) beam spots on propagation with m and n being the beam orders, which are totally different from that of a circular or elliptical LGCSM beam. The self-splitting properties of a rectangular LGCSM beam are also different from other self-splitting beam whose initial single beam spot only splits into two or four beam spots on propagation in free space. We also find that a focused rectangular LGCSM beam exhibits splitting and combining properties on propagation. Furthermore, we carry out experimental generation of a rectangular LGCSM beam and verify the splitting and combining properties of such beam focused by a thin lens. The rectangular LGCSM beam will be useful for manipulating multiple particles or attacking multiple targets simultaneously.

Nonuniform Laguerre-Gaussian correlated beam and its propagation properties

The conventional partially coherent beam has a Gaussian correlated Schell-model function. In 2007, Gori and Santarsiero[Gori F, Santarsiero M 2007 Opt. Lett. 32 3531] discussed the sufficient condition for devising a genuine correlation function of a partially coherent beam. Since then, a variety of partially coherent beams with nonconventional correlation functions, such as nonuniform correlated beam, Hermite-Gaussian correlated beam, Laguerre-Gaussian correlated beam and beam with optical coherence lattices, have been introduced, and such beams display many extraordinary propagation properties, such as self-focusing, self-shifting, self-splitting, self-shaping and periodicity reciprocity, and they have useful applications in many areas, such as free-space optical communication, particle trapping, image transmission and optical encryption.In most of previous studies, the correlation function of the partially coherent beam was assumed to be isotropic. In this paper, we introduce a new kind of partially coherent beam with anisotropic correlation function, which is named nonuniform Laguerre-Gaussian correlated（NLGC) beam. The NLGC beam has a nonuniform correlated function in the x-direction and Laguerre-Gaussian correlated Schell-model function in the y-direction. Furthermore, we explore the propagation properties of the NLGC beam in free space and in turbulent atmosphere comparatively with the help of the extended Huygens-Fresnel integral. In free space, it is found that the intensity distribution of the NLGC beam displays self-focusing and self-shifting behaviors in the x-direction and self-splitting properties in the y-direction during its propagation, which may be useful for particle trapping, and the distribution of the degree of coherence also varies during its propagation. In turbulent atmosphere, the NLGC beam displays similar propagation properties at short propagation distance because the influence of turbulence can be neglected, while with the further increase of the propagation distance, the influence of turbulence accumulates and both the intensity distribution and the degree of coherence distribution evolve into Gaussian profiles. We also find that the evolution properties of the intensity distribution and the degree of coherence are closely related to the mode order m of the correlation function, e.g. the intensity distribution and the degree of coherence distribution evolve into Gaussian profiles more slowly as the mode order m increases, which means that the NLGC beam with larger m is less affected by turbulence, which may be useful in free-space optical communication.Our results clearly show that modulating the correlation function of a partially coherent beam provides a novel way of manipulating its propagation properties, and will be useful in many applications, where light beam is required to possess a prescribed beam profile and controlled propagation properties. In this paper, only the NLGC beam is treated theoretically, and such a beam deserves further experimental investigation.

Dynamic reconstruction of heterogeneous materials and microstructure evolution.

Reconstructing heterogeneous materials from limited structural information has been a topic that attracts extensive research efforts and still poses many challenges. The Yeong-Torquato procedure is one of the most popular reconstruction techniques, in which the material reconstruction problem based on a set of spatial correlation functions is formulated as a constrained energy minimization (optimization) problem and solved using simulated annealing [Yeong and Torquato, Phys. Rev. E 57, 495 (1998)]. The standard two-point correlation function S2 has been widely used in reconstructions, but can also lead to large structural degeneracy for certain nearly percolating systems. To improve reconstruction accuracy and reduce structural degeneracy, one can successively incorporate additional morphological information (e.g., nonconventional or higher-order correlation functions), which amounts to reshaping the energy landscape to create a deep (local) energy minimum. In this paper, we present a dynamic reconstruction procedure that allows one to use a series of auxiliary S2 to achieve the same level of accuracy as those incorporating additional nonconventional correlation functions. In particular, instead of randomly sampling the microstructure space as in the simulated annealing scheme, our procedure utilizes a series of auxiliary microstructures that mimic a physical structural evolution process (e.g., grain growth). This amounts to constructing a series auxiliary energy landscapes that bias the convergence of the reconstruction to a favored (local) energy minimum. Moreover, our dynamic procedure can be naturally applied to reconstruct an actual microstructure evolution process. In contrast to commonly used evolution reconstruction approaches that separately generate individual static configurations, our procedure continuously evolves a single microstructure according to a time-dependent correlation function. The utility of our procedure is illustrated by successfully reconstructing nearly percolating hard-sphere packings and particle-reinforced composites as well as the coarsening process in a binary metallic alloy.

Self-splitting properties of a Hermite-Gaussian correlated Schell-model beam

We introduce one kind of partially coherent beam with a nonconventional correlation function named the Hermite-Gaussian correlated Schell-model (HGCSM) beam. It is found that a HGCSM beam exhibits self-splitting properties on propagation in free space, i.e., the initial single beam spot evolves into two or four beam spots in the far field depending on the initial beam orders, which are closely related to the beam width and coherence widths of the HGCSM beam, and a focused HGCSM beam exhibits splitting and combining properties near the focal plane. Furthermore, we report experimental generation of a HGCSM beam and demonstrate splitting and combining properties of a focused HGCSM beam in experiment. The phenomenon of correlation-induced self-splitting will be useful for attacking multiple targets, trapping multiple particles, and guiding atoms.

Generalized multi-Gaussian correlated Schell-model beam: from theory to experiment.

A new kind of partially coherent beam with non-conventional correlation function named generalized multi-Gaussian correlated Schell-model (GMGCSM) beam is proposed. The GMGCSM beam of the first or second kind is capable of producing dark hollow or flat-topped beam profile in the focal plane (or in the far field). Furthermore, we carry out experimental generation of a GMGCSM beam of the first or second kind, and measure its focused intensity. Our experimental results verify theoretical predictions. The GMGCSM beam will be useful for free-space optical communications, material thermal processing, particle or atom trapping.

Generation and propagation of partially coherent beams with nonconventional correlation functions: a review [invited

Partially coherent beams with nonconventional correlation functions have displayed many extraordinary properties, such as self-focusing and self-splitting, which are totally different from those of partially coherent beams with conventional Gaussian correlated Schell-model functions and are useful in many applications, such as optical trapping, free-space optical communications, and material thermal processing. In this paper, we present a review of recent developments on generation and propagation of partially coherent beams with nonconventional correlation functions.

Elliptical Laguerre-Gaussian correlated Schell-model beam.

A new kind of partially coherent beam with non-conventional correlation function named elliptical Laguerre-Gaussian correlated Schell-model (LGCSM) beam is introduced. Analytical propagation formula for an elliptical LGCSM beam passing through a stigmatic ABCD optical system is derived. The elliptical LGCSM beam exhibits unique features on propagation, e.g., its intensity in the far field (or in the focal plane) displays an elliptical ring-shaped beam profile, being qualitatively different from the circular ring-shaped beam profile of the circular LGCSM beam. Furthermore, we carry out experimental generation of an elliptical LGCSM beam with controllable ellipticity, and measure its focusing properties. Our experimental results are consistent with the theoretical predictions. The elliptical LGCSM beam will be useful in atomic optics.