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

Prime number factorization and degree of coherence of speckled light beams

We discover a connection between a Gauss sum of number theory and the degree of coherence (DOC) of the field in a transverse plane of structured speckled light beams. We theoretically demonstrate and experimentally validate that prime number factorization can be achieved by manipulating the source beam's DOC in Young's double-slit experiment. The determination of whether a number can be factored is based solely on the visibility of the resulting interference patterns. Our findings offer new insights into information encryption and decryption, data compression, etc.

Changes in the statistical properties of electromagnetic double multi-Gaussian Schell-model beams on propagation in free space

A class of electromagnetic random sources with a multi-Gaussian functional form in the spectral density and in the correlations part of the cross-spectral density matrix is introduced based on the genuine cross-spectral density-matrix theory. The analytic propagation formulas of the cross-spectral density matrix of such beams propagating in free space are derived by use of Collins' diffraction integral. With the help of analytic formulas, the evolution of statistical characteristics, i.e., the spectral density, the spectral degree of polarization, and the spectral degree of coherence for such beams in free space are analyzed numerically. Employing the multi-Gaussian functional form in the cross-spectral density matrix introduces one more freedom in the modeling of Gaussian Schell-model sources.

Propagation of radially polarized beams with a Hermite non-uniformly correlated array in free space and turbulent atmosphere

We introduce what we believe to be a novel class of radially polarized partially coherent beams in which the correlation function possesses a Hermite non-uniformly correlated array. The source parameter conditions required to generate a physical beam are derived. The statistical properties of such beam propagating in free space and turbulent atmosphere are thoroughly examined using the extended Huygens-Fresnel principle. It is shown that the intensity profile of such beams presents a controllable periodic grid distribution due to its multi-self-focusing propagation property and can keep the shape in free space while propagating in turbulent atmosphere, it exhibits self-combining properties over a long-ranges. Owing to the interaction between the non-uniform correlation structure and the non-uniform polarization, this beam can locally self-recover the polarization state after propagating a long distance in a turbulent atmosphere. Furthermore, the source parameters play essential roles in determining the distribution of spectral intensity, the state of polarization, and the degree of polarization of the RPHNUCA beam. Our results may benefit multi-particle manipulation and free-space optical communication applications.

Simulating random optical fields: tutorial

Numerous applications-including optical communications, directed energy, remote sensing, and optical tweezing-utilize the principles of statistical optics and optical coherence theory. Simulation of these phenomena is, therefore, critical in the design of new technologies for these and other such applications. For this reason, this tutorial describes how to generate random electromagnetic field instances or realizations consistent with a given or desired cross-spectral density matrix for use in wave optics simulations. This tutorial assumes that the reader has knowledge of the fundamental principles of statistical optics and optical coherence theory. An extensive reference list is provided where the necessary background information can be found. We begin this tutorial with a brief summary of the coherent-mode representation and the superposition rule of stochastic electromagnetic fields as these foundational ideas form the basis of all known synthesis techniques. We then present optical field expressions that apply these concepts before discussing proper sampling and discretization. We finally compare and contrast coherent-mode- and superposition-rule-based synthesis approaches, discussing the pros and cons of each. As an example, we simulate the synthesis and propagation of an electromagnetic partially coherent field from the literature. We compare simulated or sample statistics to theory to verify that we have successfully produced the desired field and are capturing its propagation behaviors. All computer programs, including detailed explanations of the source code, are provided with this tutorial. We conclude with a brief summary.

Coherent mode decomposition of multiple quantum well light emission

Developing a richer understanding of the various properties of light is central to the field of photonics. One often neglected degree of freedom (DOF) is the second-order correlation of the light field, known as coherence. To make proper use of this DOF, one needs to first obtain information about the field's coherence, which may be characterized through the cross spectral density (CSD) function. We present a measurement of the CSD of a ubiquitous, partially coherent source: a multiple quantum well device in its near-field region, where a photonic structure would commonly encounter the emitted field. We show a departure from the coherence area that is expected from an incoherent source and demonstrate the application of coherent mode decomposition as a way to further analyze the measured results.

Generation of a higher-order Poincaré sphere beam array with spatial coherence engineering

We propose a protocol to synthesize a class of vector beam array in the far field with periodic higher-order Poincaré sphere (HOPS) polarization states by engineering the second-order spatial coherence structure of a partially coherent light source. We show that the polarization state of a single HOPS beam at the source plane can be mapped into a beam array in the far field when the spatial coherence of the beam source is engineered to have a lattice-like distribution. We demonstrate that the degree of polarization of the generated HOPS beam array can be conveniently controlled by modulating the transverse spatial coherence width of the source. Our method provides an additional way to construct the structured beam array and may find applications, e.g., in multiparticle manipulations.

Complex and phase screen methods for studying arbitrary genuine Schell-model partially coherent pulses in nonlinear media

Partially coherent pulses, especially those with non-Gaussian correlated functions, have rarely been explored in nonlinear media because of the demanding procedure of the widely used coherent-mode representation method. This study develops temporal analogues of the complex screen and phase screen methods, which were recently introduced for the spatial counterpart of a partially coherent beam. These methods were employed to study the beam propagation properties of partially coherent pulses, and the obtained results show that they both are highly precise, convenient, and powerful. We believe that these protocols can effectively provide useful insight into the behavior of many coherence-related phenomena in nonlinear media.

Compact generation of robust Airy beam pattern with spatial coherence engineering

We present a class of partially coherent light sources having Airy-type amplitude and Airy-correlated spatial coherence. We show that the light beam generated by such sources can preserve the Airy beam pattern well during its propagation from source to far field. We demonstrate the robustness of the Airy beam pattern by introducing a hard aperture to largely block the beam source. We find that the coherence-induced Airy beam pattern can still be well reconstructed during propagation. We successfully synthesize such partially coherent source using the principle of complex random modes decomposition by using a single phase-only spatial light modulator. The proposed robust Airy beam pattern may find applications in information transmission through complex media.

Orientation-selective sub-Rayleigh imaging with spatial coherence lattices

The Rayleigh resolution criterion sets the minimum separation for two-point objects to be distinguishable in a classical optical imaging system. We demonstrate that the sub-Rayleigh resolution can be achieved in a telecentric imaging system with the help of a partially coherent illumination whose spatial coherence has lattice-like distribution. We show that the orientation-selective sub-Rayleigh imaging can be realized by controlling the spatial distribution of the coherence lattice into different symmetries. We carry out a proof-of-principle experiment to demonstrate the orientation-selective sub-Rayleigh imaging for a 1951 USAF resolution target. Our results indicate a flexible orientation-selective high-resolution imaging with spatial coherence engineering of the partially coherent light.

Modal Analysis of Pseudo-Schell Model Sources

All pseudo-Schell model sources have been shown to possess the same continuous set of circularly symmetric modes, all of them presenting a conical wavefront. For keeping energy at a finite level, the mode amplitude along the radial coordinate is modulated by a decreasing exponential function. A peculiar property of such modes is that they exist in the Laplace transform’s realm. After a brief discussion of the near-zone, we pass to the far-zone, where the field can be evaluated in closed form. The corresponding features of the intensity distribution are discussed.

Second-order statistical properties of conjugate mode "double-H" partially coherent beams in turbulence

We present an analysis of the propagation properties of a recently introduced class of conjugate mode partially coherent beams (called "double-H" beams) in a turbulent atmosphere using the extended Huygens-Fresnel integral. We investigate how the phase constant φ₀ between the modes plays a role in controlling the evolution of the intensity distribution and resisting the degradation effects of the atmosphere. Our results indicate that this new class of structured beams provides a new degree of freedom for controlling the shape of the beams and improves turbulence resistance, with potential application in free-space optical communications.

Three modal decompositions of Gaussian Schell-model sources: comparative analysis

Representation of the cross-spectral density (CSD) function of an optical source or beam as the incoherent superposition of mutually uncorrelated modes are widely used in imaging systems and in free space optical communication systems for simplification of the analysis and reduction of the time-consuming integral calculations. In this paper, we examine the equivalence and the differences among three modal representation methods: coherent-mode representation (CMR), pseudo-mode representation (PMR) and random mode representation (RMR) for the Gaussian Schell-model (GSM) source class. Our results reveal that for the accurate reconstruction of the CSD of a generic GSM source, the CMR method requires superposition of the least number of optical modes, followed by PMR and then by RMR. The three methods become equivalent if a sufficiently large number of optical modes are involved. However, such an equivalence is limited to the second-order statistics of the source, e.g., the spectral density (average intensity) and the degree of coherence, while the fourth-order statistics, e.g., intensity-intensity correlations, obtained by the three methods are quite different. Furthermore, the second- and the fourth- order statistics of the GSM beam propagating through a deterministic screen and dynamic random screens with fast and slow time cycling are investigated through numerical examples. It is found that the properties of the second-order statistics of the beams obtained by the three methods are the same, irrespectively of the characteristics of the screens, whereas those of the fourth-order statistics remain different.

Evolution of Spatiotemporal Intensity of Partially Coherent Pulsed Beams with Spatial Cosine-Gaussian and Temporal Laguerre–Gaussian Correlations in Still, Pure Water

A new family of partially coherent pulsed beams with spatial cosine-Gaussian and temporal Laguerre–Gaussian correlations, named spatial cosine-Gaussian and temporal Laguerre–Gaussian correlated Schell-model (SCTLGSM) pulsed beams, is introduced. An analytic propagation formula is derived for the SCTLGSM pulsed beam through the spatiotemporal ABCD optical system characterizing a continuous dispersive medium. As an example, the evolution of spatiotemporal intensity of the SCTLGSM pulsed beam in a still, pure water column is then investigated. It is found that the SCTLGSM pulsed beams simultaneously exhibit spatiotemporal self-splitting and self-focusing phenomena, which can be attributed to the special spatial/temporal coherence structures and the presence of pulse chirper in the source plane. The physical interpretation of the obtained phenomena is given. The results obtained in this paper will be of interest in underwater optical technologies, e.g., directed energy and communications.

Independently Controlling Stochastic Field Realization Magnitude and Phase Statistics for the Construction of Novel Partially Coherent Sources

In this paper, we present a method to independently control the field and irradiance statistics of a partially coherent beam. Prior techniques focus on generating optical field realizations whose ensemble-averaged autocorrelation matches a specified second-order field moment known as the cross-spectral density (CSD) function. Since optical field realizations are assumed to obey Gaussian statistics, these methods do not consider the irradiance moments, as they, by the Gaussian moment theorem, are completely determined by the field’s first and second moments. Our work, by including control over the irradiance statistics (in addition to the CSD function), expands existing synthesis approaches and allows for the design, modeling, and simulation of new partially coherent beams, whose underlying field realizations are not Gaussian distributed. We start with our model for a random optical field realization and then derive expressions relating the ensemble moments of our fields to those of the desired partially coherent beam. We describe in detail how to generate random optical field realizations with the proper statistics. We lastly generate two example partially coherent beams using our method and compare the simulated field and irradiance moments theory to validate our technique.

Modelling of Phase Contrast Imaging with X-ray Wavefront Sensor and Partial Coherence Beams

The Hartmann wavefront sensor is able to measure, separately and in absolute, the real δ and imaginary part β of the X-ray refractive index. While combined with tomographic setup, the Hartman sensor opens many interesting opportunities behind the direct measurement of the material density. In order to handle the different ways of using an X-ray wavefront sensor in imaging, we developed a 3D wave propagation model based on Fresnel propagator. The model can manage any degree of spatial coherence of the source, thus enabling us to model experiments accurately using tabletop, synchrotron or X-ray free-electron lasers. Beam divergence is described in a physical manner consistent with the spatial coherence. Since the Hartmann sensor can detect phase and absorption variation with high sensitivity, a precise simulation tool is thus needed to optimize the experimental parameters. Examples are displayed.

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.

Partially coherent Pearcey-Gauss beams

We investigate the dynamics of partially coherent Pearcey-Gauss beams propagating in free space, theoretically and experimentally. They are produced by introducing the degree of coherence (DOC) function with Gaussian Schell-model correlation into the light source in the frequency domain. Under a nearly incoherent state, the oscillation of the sidelobe turns smooth, and the intensity distribution concentrates on the mainlobe. Particularly, partially coherent Pearcey-Gauss beams would maintain the inherent properties of autofocusing performance and inversion effect without diminishing the autofocusing distance and form-invariable propagation. Moreover, the opening angle and the shift of peak intensity of the beams can be controlled by the binary parabola in the spectrum distribution of the Pearcey function. Our experimental results are in great agreement with the theoretical analysis.

Hollow rectangular multi-Gaussian Schell-model source

We introduced a new class of a partially coherent source that can generate a field with a hollow rectangular profile, named the hollow rectangular multi-Gaussian Schell-mode source. The dependence of distribution of far-zone spectral density on the properties of the proposed source was analyzed. The results show that one can control the distribution properties of the far-zone intensity profile, including the thickness of the hollow edge, the shape of the hollow, the size of the hollow, and the orientation of the hollow, by adjusting the corresponding structural parameters of the source.

Experimental demonstration of structural robustness of spatially partially coherent fields in turbulence

Structured fields that are spatially completely coherent have been extensively studied in the context of long-distance optical communication, as the structure in the intensity profile of such fields is used for encoding information. This method of doing optical communication works very well in the absence of turbulence. However, in the presence of turbulence, the intensity structures of such fields start to degrade because of the complete spatial coherence of the field, and this structural degradation increases with the increase in turbulence strength. On the other hand, several theoretical studies have now shown that the structured fields that are spatially only partially coherent are less affected by turbulence. However, to the best of our knowledge, no such experimental demonstration has been reported until now. In this Letter, we experimentally demonstrate the structural robustness of partially coherent fields in the presence of turbulence, and we show that for a given turbulence strength, the structural robustness of a partially coherent field increases as the spatial coherence length of the field is decreased.

Vector partially coherent beams with prescribed non-uniform correlation structure

We introduce a general strategy for the synthesis of vector partially coherent beams (PCBs) with a prescribed non-uniform correlation structure. With it, we characterize a specific family of such beams, termed radially polarized Hermite non-uniformly correlated (RPHNUC) beams. These beams possess unusual propagation properties compared to vector PCBs with uniform correlation structure; for example, they maintain their dark hollow core and evolve multi-ring structures. These beams may prove useful in free-space optical communications, optical trapping, and polarization-sensitive imaging.

Fast calculation of tightly focused random electromagnetic beams: controlling the focal field by spatial coherence

Focusing of a vectorial (electromagnetic) optical beam through a high numerical aperture can be investigated by means of the Richards-Wolf diffraction integral. However, such an integral extends from two-dimensional to four-dimensional, greatly increasing the computation time and therefore limiting the applicability, when light with decreased spatial coherence is considered. Here, we advance an effective protocol for the fast calculation of the statistical properties of a tightly focused field produced by a random electromagnetic beam with arbitrary state of spatial coherence and polarization. The novel method relies on a vectorial pseudo-mode representation and a fast algorithm of the wave-vector space Fourier transform. The procedure is demonstrated for several types of radially (fully) polarized but spatially partially coherent Schell-model beams. The simulations show that the computation time for obtaining the focal spectral density distribution with 512 × 512 spatial points for a low coherence beam is less than 100 seconds, while with the conventional quadruple Richards-Wolf integral more than 100 hours is required. The results further indicate that spatial coherence can be viewed as an effective degree of freedom to govern both the transverse and longitudinal components of a tightly focused field with potential applications in reverse shaping of focal fields and optical trapping control.

Propagation of temporal coherence gratings in dispersive medium with a chirper

In this paper, the propagation of Temporal Coherence Grating (TCG) pulse trains in a dispersive medium with a chirp is investigated for the first time. The two-time mutual coherence function of the TCG pulse trains propagating through extended dispersive medium specified by temporal ABCD matrix is derived and the evolution of their mean intensity and temporal degree of coherence (DOC) is explored. It is shown that the distribution of the mean intensity can be modulated freely by the number of grating lobes N, grating constant a, pulse duration T₀, power distributions vn, group-velocity dispersion coefficient β₂ and the medium chirper s. Upon dispersive-medium propagation, the single pulse splits into N+1 subpulses with the same or different peak intensities which depend on power distributions vn. What's more, during the propagation the pulse self-focusing occurs being the chirp-induced non-linear phenomenon. And the distribution of temporal DOC will degenerate into Gaussian form from initial periodic coherence distribution with increasing propagation distance z or adjusting incident pulse parameters and medium dispersion. The physical explanation and numerical illustrations relating to the pulse behavior are included.

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.

The evolution of spectral intensity and orbital angular momentum of twisted Hermite Gaussian Schell model beams in turbulence

We introduce a new class of twisted partially coherent beams with a non-uniform correlation structure. These beams, called twisted Hermite Gaussian Schell model (THGSM) beams, have a correlation structure related to Hermite functions and a twist factor in their degree of coherence. The spectral density and total average orbital angular momentum per photon of these beams strongly depend on the distortions applied to their degree of coherence. On propagation through free space, they exhibit both self-splitting and rotation of their spectral density profile, combining the interesting effects of twisted beams and non-uniformly correlated beams. We demonstrate that we can adjust both the beam order and the twist factor of THGSM beams to improve their resistance to turbulence.

Characterization of the electromagnetic Gaussian Schell-model beam using first-order interference

We propose a method for the characterization of electromagnetic Gaussian Schell-model (EMGSM) beams. This method utilizes the first-order interference consisting of polarization-state projections along with the two-point (generalized) Stokes parameters. The second-order field correlations employed in this method enable us to determine both the magnitude and the argument of the complex degree of electromagnetic coherence. We experimentally demonstrate this method by characterizing an EMGSM beam, which is synthesized using a laser beam passing through a rotating ground glass diffuser. This beam-characterization method is expected to be potentially useful for probing the partially coherent and partially polarized beams, and have tremendous applications in broad areas of optical communication and beam propagation.

Self-healing properties of Hermite-Gaussian correlated Schell-model beams

We study theoretically and experimentally the influence of the obstacle position separation from the source on the self-healing capacity of partially coherent beams using Hermite-Gaussian correlated Schell-model beams as a case in point. We establish that the shorter the distance between the obstacle and the source plane and the longer the distance between the obstacle and the observation (receiver) plane, the better the self-healing capacity of the beams. In addition, a similarity degree between the reconstructed and original beams is introduced to quantify the self-healing capacity of partially coherent beams. The derived interesting results may find applications in optical information processing, image transmission, and recovery.

Experimental generation of a kind of reversal rotating beams

A kind of reversal rotating beams with astigmatic phase is proposed, whose spectral density and degree of coherence both follow anisotropic Gaussian distribution. Unlike a general rotating beam, the spectral density and the degree of coherence of this beam can be reversal rotated during propagation, that is, the direction of rotation could change automatically. Such a beam can be viewed as having two elements with astigmatic phase and partial coherence of the beam, which can reshape the cross-spectral density, corresponding to two directions of rotation. We generated this beam successfully in experiment and observed the expected phenomenon, which is basically consistent with the result of the numerical simulation. The reversal rotating beam has certain requirements on the astigmatic phase, which is analyzed and verified. The effect of the main parameters in astigmatic phase on the reversal rotation is further studied from both simulation and experiment.

Stochastic complex transmittance screens for synthesizing general partially coherent sources

We develop a method to synthesize any partially coherent source (PCS) with a genuine cross-spectral density (CSD) function using complex transmittance screens. Prior work concerning PCS synthesis with complex transmittance screens has focused on generating Schell-model (uniformly correlated) sources. Here, using the necessary and sufficient condition for a genuine CSD function, we derive an expression, in the form of a superposition integral, that produces stochastic complex screen realizations. The sample autocorrelation of the screens is equal to the complex correlation function of the desired PCS. We validate our work by generating, in simulation, three PCSs from the literature-none has ever been synthesized using stochastic screens before. Examining planar slices through the four-dimensional CSD functions, we find the simulated results to be in excellent agreement with theory, implying successful realization of all three PCSs. The technique presented herein adds to the existing literature concerning the generation of PCSs and can be physically implemented using a simple optical setup consisting of a laser, spatial light modulator, and spatial filter.

Spatial-Temporal Self-Focusing of Partially Coherent Pulsed Beams in Dispersive Medium

Partially coherent pulsed beams have many applications in pulse shaping, fiber optics, ghost imaging, etc. In this paper, a novel class of partially coherent pulsed (PCP) sources with circular spatial coherence distribution and sinc temporal coherence distribution is introduced. The analytic formula for the spatial-temporal intensity of pulsed beams generated by this kind of source in dispersive media is derived. The evolution behavior of spatial-temporal intensity of the pulsed beams in water and air is investigated, respectively. It is found that the pulsed beams exhibit spatial-temporal self-focusing behavior upon propagation. Furthermore, a physical interpretation of the spatial-temporal self-focusing phenomenon is given. This is a phenomenon of optical nonlinearity, which may have potential application in laser micromachining and laser filamentation.

Experimental study of reducing beam wander by modulating the coherence structure of structured light beams

We study the beam wander of a class of structured light beams, Hermite-Gaussian correlated Schell-model (HGCSM) beams, in theory and in experiment. It is found that modulating the coherence structure of a structured light beam can reduce the turbulence-induced beam wander, i.e., a HGCSM beam with larger mode orders or lower coherence experiences smaller beam wander. Our experimental results are consistent with theoretical predictions, and the insights here suggest that HGCSM beams could be useful in free-space optical communications.

Besinc Pseudo-Schell Model Sources with Circular Coherence

Partially coherent sources with non-conventional coherence properties present unusual behaviors during propagation, which have potential application in fields like optical trapping and microscopy. Recently, partially coherent sources exhibiting circular coherence have been introduced and experimentally realized. Among them, the so-called pseudo Schell-model sources present coherence properties that depend only on the difference between the radial coordinates of two points. Here, the intensity and coherence properties of the fields radiated from pseudo Schell-model sources with a degree of coherence of the besinc type are analyzed in detail. A sharpening of the intensity profile is found for the propagated beam by appropriately selecting the coherence parameters. As a possible application, the trapping of different types of dielectric nanoparticles with this kind of beam is described.

Numerical Approach for Studying the Evolution of the Degrees of Coherence of Partially Coherent Beams Propagation through an ABCD Optical System

In this paper, we propose a numerical approach to simulate the degree of coherence (DOC) of a partially coherent beam (PCB) with a Schell-model correlator in any transverse plane during propagation. The approach is applicable for PCBs whose initial intensity distribution and DOC distribution are non-Gaussian functions, even for beams for which it is impossible to obtain an analytical expression for the cross-spectral density (CSD) function. Based on our approach, numerical examples for the distribution of the DOC of two types of PCBs are presented. One type is the partially coherent Hermite–Gaussian beam. The simulation results of the DOC agree well with those calculated from the analytical formula. The other type of PCB is the one for which it is impossible to obtain an analytical expression of CSD. The evolution of the DOC with the propagation distance and in the far field is studied in detail. Our numerical approach may find potential applications in optical encryption and information transfer.

Self-steering partially coherent vector beams

We introduce a class of self-steering partially coherent vector optical beams with the aid of a generalized complex Gaussian representation. We show that such partially coherent vector beams have mobile guiding centers of their intensity and polarization state distributions on the beam free space propagation that could be employed to generate far-field polarization arrays. Further, we introduce theoretically and realize experimentally a class of vector beams with inhomogeneous statistical and nontrivial far-field angular distributions, which we term cylindrically correlated partially coherent (CCPC) vector beams. We find that such novel beams possess, in general, cylindrically polarized, far-field patterns of an adjustable degree of polarization. The steering control of the intensity and polarization of the self-steering CCPC vector beam is also demonstrated in experiment. Our findings can find important applications, such as trapping of neutral microparticles and excitation of novel surface waves.

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.

Pseudo-Schell model sources

Partially coherent pseudo-Schell model sources are introduced and analyzed. They present radial symmetry and coherence characteristics depending on the difference between the radial distances of two points from the source center. As a consequence, all points belonging to circles centered on the symmetry center of the source are perfectly correlated. We show that such sources radiate fields with peculiar behaviors in paraxial propagation. In particular, when compared to beams produced by Gaussian Schell-model sources with comparable coherence parameters, their irradiance can present sharper profiles and higher peak valuesmono and a better beam quality parameter. Furthermore, when a pseudo-Schell model source presents a vortex, the propagated beam preserves a null of the intensity along its axis.

Generation and Propagation of a Hermite-Gaussian Correlated Schell-Model LG0l Beam

A partially coherent beam under the combined action of a Hermite-Gaussian correlated function and vortex phase, named the HGCSMLG0l beam has been explored both theoretically and experimentally. The statistical properties, such as the intensity and distribution of the degree of coherence (DOC) on propagation are analyzed in detail, based on the deduced equations. We find that the intensity is determined dominantly by the non-conventional correlated function when the coherence length is comparatively small and by vortex phase when the coherence length is large. The modulus of the DOC is not vulnerable to coherence width, rather, it is affected by both non-conventional correlated function and vortex phase. Our results are verified well by the experiment results.

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.

Evolution properties of the radially polarized multi-Gaussian Schell-model beam in uniaxial crystals

The evolution properties of the normalized intensity distribution, the spectral degree of coherence (SDOC), and the spectral degree of polarization (SDOP) of the radially polarized multi-Gaussian Schell-model (MGSM) beam in uniaxial crystals are illustrated. Numerical results show that the intensity distribution of the radially polarized MGSM beam gradually evolves from a doughnut shape into an elliptical symmetric flattop shape and retains its elliptical flattop shape on further propagation in anisotropic crystals. The evolution behavior of the SDOC and SDOP for the radially polarized MGSM beam is quite different from that of the linearly polarized one. In addition, the influences of the spatial coherence length δ₀, beam index M, and the ratio of the extraordinary refractive index to the ordinary refractive index n_e/n_o of the uniaxial crystals on the evolution properties of the normalized intensity distribution, the SDOC, and the SDOP of the radially polarized MGSM beam are discussed in detail.

Propagation of partially coherent beams with convex-shaped spatial coherence modulation in vertical turbulent links

The convex partially coherent beam (CPCB) is a special type of nonuniformly correlated beam with a convex-shaped complex degree of coherence (DoC) distributions. Previously our research has illustrated the potential of CPCBs with super-Gaussian DoCs in free-space optical communications (FSOC), mainly manifested as self-focusing which can be transferred into extra scintillation reduction and SNR gain. In this study, the effects of the DoC transition slopes are analyzed and more details about the turbulence propagation of CPCBs with super-Gaussian shaped DoC are revealed. By means of wave optics simulation, the longitudinal intensity evolution of the CPCB is explored, showing that the DoC slope has a profound influence on the self-focusing features such as the focusing plane and the peak intensity. Aperture scintillation and mean SNR at the receiver end of some short-range vertical turbulent links are numerically computed. The obtained results show that, with CPCBs, an ~2 dB SNR gain can be achieved as compared to conventional Gaussian Schell-modal (GSM) beams. However, CPCBs are preferred only in shorter links, which is found to be relevant to the power-in-the-bucket of the receiving aperture. Furthermore, the impacts of the ratio of the source coherence time to the detector integration time are investigated, implying that the CPCB is less susceptible than the GSM. We have also examined the off-axis scintillation of the CPCB. Due to its convex-shaped DoC, the CPCB has significantly reduced off-axis scintillation, which can be beneficial in the presence of pointing errors.

Partially coherent fractional vortex beam

We introduce a new kind of partially coherent vortex (PCV) beam with fractional topological charge named partially coherent fractional vortex (PCFV) beam and derive the propagation formula for such beam passing through a stigmatic ABCD optical system with the help of the convolution method. We calculate numerically the propagation properties of a PCFV beam focused by a thin lens, and we find that the PCFV beam exhibits unique propagation properties. The opening gap of the intensity pattern and the rotation of the beam spot disappear gradually and the cross-spectral density (CSD) distribution becomes more symmetric and more recognizable with the decrease of the spatial coherence width, being qualitatively different from those of the PCV beam with integral topological charge. Furthermore, we carry out experimental generation of a PCFV beam with controllable spatial coherence, and measure its focusing properties. Our experimental results are consistent with the theoretical predictions.

Twisted partially coherent array sources and their transmission in anisotropic turbulence

A new class of twisted Schell-model array correlated sources are introduced based on Mercer's expansion. It turns out that such sources can be expressed as superposition of fully coherent Laguerre-Gaussian modes, and the twistable condition is established. Furthermore, on the basis of a stretched coordinate system and a quadratic approximation, analytical expressions for the mutual coherence function of an anisotropic non-Kolmogorov turbulence and the cross-spectral density of a twisted Gaussian Schell-model array beam are rigorously derived. Due to the presence of the twist phase, the beam spot and the degree of coherence rotate as they propagate, but their rotation centers are different. It is shown that the anisotropy of turbulence causes an anisotropic beam spreading in the horizontal and vertical directions. However, impressing a twist phase on source beams can significantly inhibit this effect. For an anticipated atmospheric channel condition, a comprehensive selection of initial optical signal parameters, receiver aperture size and receiver capability, etc., is necessary. Our work is helpful for exploring new forms of twistable sources, and promotes guidance on optimization of partial coherent beam applications.

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.

Propagation Characteristics of a Twisted Cosine-Gaussian Correlated Radially Polarized Beam

Recently, partially coherent beams with twist phases have attracted growing interest due to their nontrivial dynamic characteristics. In this work, the propagation characteristics of a twisted cosine-Gaussian correlated radially polarized beam such as the spectral intensity, the spectral degree of coherence, the degree of polarization, the state of polarization, and the spectral change are investigated in detail. Due to the presence of the twisted phase, the beam spot, the degree of coherence, and the state of polarization experience rotation during transmission, but the degree of polarization is not twisted. Meanwhile, although their rotation speeds closely depend on the value of the twist factor, they all undergo a rotation of π / 2 when they reach the focal plane. Furthermore, the effect of the twist phase on the spectral change is similar to the coherence, which is achieved by modulating the spectral density distribution during transmission. The twist phase opens up a useful guideline for manipulation of novel vector structure beams and enriches potential applications in the field of beam shaping, optical tweezers, optical imaging, and free space optical communications.

Controlling the Spatial Coherence of an Optical Source Using a Spatial Filter

This paper presents the theory for controlling the spectral degree of coherence via spatial filtering. Starting with a quasi-homogeneous partially coherent source, the cross-spectral density function of the field at the output of the spatial filter is found by applying Fourier and statistical optics theory. The key relation obtained from this analysis is a closed-form expression for the filter function in terms of the desired output spectral degree of coherence. This theory is verified with Monte Carlo wave-optics simulations of spatial coherence control and beam shaping for potential use in free-space optical communications and directed energy applications. The simulated results are found to be in good agreement with the developed theory. The technique presented in this paper will be useful in applications where coherence control is advantageous, e.g., directed energy, free-space optical communications, remote sensing, medicine, and manufacturing.

Focus shaping of the radially polarized Laguerre-Gaussian-correlated Schell-model vortex beams

In this paper, we introduce a new kind of partially coherent vector beam with special correlation function and vortex phase named radially polarized Laguerre-Gaussian-correlated Schell-model (LGCSM) vortex beam as a natural extension of scalar LGCSM vortex beam. The realizability conditions for such beam are derived. The tight focusing properties of a radially polarized LGCSM vortex beam passing through a high numerical aperture (NA) objective lens are investigated numerically based on the vectorial diffraction theory. We find that not only the transverse component but also the longitudinal component of the focal field distributions can be shaped by regulating the structures of the correlation functions, which is quite different from that of the conventional radially polarized partially coherent beam. Moreover, a series of wildly used focal field with novel structure, e.g., focal spot, flat-topped or doughnut beam profiles, needle-like focal field and controllable three-dimensional (3D) optical cage, were obtained. These results indicate that the focus shaping can be achieved by combining the regulation of the structures of the correlation functions with the regulation of beam parameters effectively. Our results may be useful for potential applications in optical trapping, optical high-resolution microscopy and optical data storage.

Parabolic-Gaussian Schell-model sources and their propagations

In this work, a new class of partially coherent Schell-type sources is introduced by modifying its degree of coherence, which is a product of a parabolic function and a Gaussian function. Such sources are confirmed to be physically genuine and may be called parabolic-Gaussian Schell-model (PGSM) sources. The propagating expression of the cross-spectral density function of such PGSM sources is derived in a general linear optical system. In particular, the propagation properties of such PGSM beams in both free space and lens systems are discussed. The results show that such PGSM sources can produce dark-hollow intensity profiles in the regions of free space very near as well as far away from the source plane, or in the focal region of the lens system.

Far-field propagation of partially coherent laser light in random mediums

The irradiance of a partially coherent light propagated under the influence of multiple random effects is shown to be the convolution of the irradiance propagated in a vacuum with the system's point spread function representing the random effects. This is true regardless of whether the propagation is far-field or not. We also show that the far-field irradiance of any laser system, regardless of complexity, can be expressed in terms of three basic parameters; laser power, field area, and a pupil factor. A general analytical formula for the far-field irradiance distribution for partially coherent laser sources of any complexity is derived. The formula includes multiple random effects including strong turbulence, random beam jitter, partial coherence, in addition to laser system pupil effects. An efficient matrix based numerical solution is also developed to verify the accuracy of the formula. Applications to the propagation of clipped Gaussian or flat-top beams with an obscuration, both as a single beam or an array of beams, are shown to give accurate results over the whole range of weak to strong turbulence as compared to numerical modeling.

Partially coherent sources with radial coherence

Partially coherent sources with radial coherence are proposed. They present a circularly symmetric intensity profile and a degree of coherence whose absolute value only depends on the angular difference between the two considered points. In particular, the source is completely coherent at pairs of points belonging to the same radius. The modal structure of such sources is determined in the general case, and conditions are derived under which the field propagated in paraxial approximation remains radially coherent at any transverse plane. In such cases, the angular dependence of the correlation function is preserved upon propagation, although the intensity profile generally changes. An example of this kind of source has been experimentally synthesized by means of a simple setup, and its coherence characteristics have been tested by means of a Young interferometer.

Paraxial propagation of a class of Bessel-correlated fields

The propagation of a novel class of paraxial spatially partially coherent beams exhibiting Bessel-type correlations is studied in free space and in paraxial optical systems. We show that, under certain conditions, such beams can have functionally identical forms of the absolute value of the complex degree of spatial coherence not only at the source plane and in the far zone, but also at all finite propagation distances. Under these conditions the degree of spatial coherence properties of the field is a shape-invariant quantity, but the spatial intensity distribution is only approximately shape-invariant. The main properties of this class of model beams are demonstrated experimentally by passing a coherent Gaussian beam through a rotating wedge and measuring the coherence of the ensuing beams with a Young-type interferometer realized with a digital micromirror device.

Spatial coherence measurement and partially coherent diffractive imaging using self-referencing holography

The complete characterization of spatial coherence is extremely difficult because the mutual coherence function (MCF) is a complex-valued function of four independent Cartesian coordinates. This difficulty limits the ability to control and to optimize the spatial coherence in a broad range of key applications. Here we propose an efficient and robust scheme for measuring the complete MCF of an arbitrary partially coherent beam using self-referencing holography, which does not require any prior knowledge or making any assumptions about the MCF. We further apply our method to lensless diffractive imaging, and experimentally demonstrate the reconstruction of a phase object under spatially partially coherent illumination. This application is particularly useful for imaging at short wavelengths, where the illumination sources lack spatial coherence and no high-quality imaging optics are available.