Genuine cross-spectral densities and pseudo-modal expansions

Partially coherent laser beam shaping in a zoom homogenizer

This paper proposes a synthesis method including wave optics and geometric optics to design the zoom homogenizer for partially coherent laser beams and discusses the effects of the spatial coherence and system parameters on the beam performance. Based on the principles of pseudo-mode representation and matrix optics, a numerical model for fast simulation has been built and the parameter constraints for avoiding beamlet crosstalk have been presented. The relation of the size and the divergence angle of the highly uniform beams formed in the defocused plane with system parameters has been developed. The variations in the intensity profile and the uniformity of the variable-size beams during zooming have been explored.

Generating multi-focus beams with a spatial non-uniform coherence structure

We introduce a class of structured light beams, named multi-focus beams, which exhibit self-focusing at multiple propagation distances. We show that the proposed beams not only have the ability to produce multiple longitudinal focal spots, but also that the number, intensity, and position of the foci can be controlled by adjusting the initial beam parameters. Furthermore, we demonstrate that these beams still exhibit self-focusing in the shadow of an obstacle. We have experimentally generated such beams and the results are consistent with the theoretical predictions. Our studies may find application where fine control of the longitudinal spectral density is needed, such as longitudinal optical trapping and manipulation of multiple particles, and transparent material cutting.

Second-order statistics of a Hermite-Gaussian correlated Schell-model beam carrying twisted phase propagation in turbulent atmosphere

We investigate the second-order statistics of a twisted Hermite-Gaussian correlated Schell-model (THGCSM) beam propagation in turbulent atmosphere, including the spectral density, degree of coherence (DOC), root mean square (r.m.s.) beam wander and orbital angular momentum (OAM) flux density. Our results reveal that the atmospheric turbulence and the twist phase play a role in preventing the beam splitting during beam propagation. However, the two factors have opposite effects on the evolution of the DOC. The twist phase preserves the DOC profile invariant on propagation, whereas the turbulence degenerates the DOC. In addition, the influences of the beam parameters and the turbulence on the beam wander are also studied through numerical examples, which show that the beam wander can be reduced by modulating the initial parameters of the beam. Further, the behavior of the z-component OAM flux density in free space and in atmosphere is thoroughly examined. We show that the direction of the OAM flux density without the twist phase will be suddenly inversed at each point across the beam section in the turbulence. This inversion only depends on the initial beam width and the turbulence strength, and in turn, it offers an effective protocol to determine the turbulence strength by measuring the propagation distance where the direction of OAM flux density is inversed.

On z-coherence of Schell-model sources carrying a prescribed astigmatic phase

A simple expression for the correlations of beams radiated by Schell-model sources carrying a prescribed astigmatic phase (cross phase) in 3D space is derived. The z-coherence of such sources upon free-space propagation is investigated in detail. It is demonstrated that the z-coherence does not decrease to zero with an increasing separation of two axial points. Our results show that the initial cross phase, coherence, and correlation state of such sources affect the distribution of the z-coherence. Furthermore, the cross phase plays a role in maintaining z-coherence, which will be useful in applications where high z-coherence is required.

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.

Pulse-quality metric for nonstationary partially coherent fields

This paper generalizes a pulse-quality metric referred to as P², i.e., the time analogue of Siegman's beam quality factor M², to include pulsed (nonstationary) random fields of any state of coherence. The analysis begins with the derivation of a general P² relation, which we then specialize to the important cases of coherent and Schell-model pulsed beams. As examples, we derive the P² for two stochastic sources: (1) a cosine Gaussian-correlated Schell-model pulsed beam and (2) a nonuniformly correlated pulsed beam. For both of these sources, we generate (in simulation) random instances of each and compare the simulated (Monte Carlo) P², i.e., computed directly from its definition, to the theoretical quantity. The agreement is excellent, thereby validating our P² analysis.

Devising genuine cross-spectral densities by randomization of coherent fields

We propose a concept of devising the authentic cross spectral densities (CSDs) of partially coherent beam waves by averaging the second moment of a coherent field over a certain number of random parameters. Several CSD examples created by randomization are presented. We discuss application of randomization to generation of twisted Gaussian Schell model beams and show that compound randomization leads to a new, more general class of twisted beams. We revisit the necessary nonnegative definiteness condition of Opt. Lett.34, 1399 (2009)OPLEDP0146-959210.1364/OL.34.001399 and show that the two-dimensional space of parameters is necessary.

Super cosh-Gauss nonuniformly correlated radially polarized beam and its propagation characteristics

In this paper, a new kind of partially coherent vector beam termed as super cosh-Gauss nonuniformly correlated radially polarized (SCNRP) beam is introduced. Such beam source exhibits almost perfect coherence between two points that are within the beam center region or located on a ring concentric with the beam center. However, the coherence drops or even vanishes when the two points leave the central region and are located on the concentric rings with different radii. The second-order statistical properties, such as the spectral density, the state of polarization (SOP), and the degree of polarization (DOP) of such beam upon free-space propagation are studied through numerical examples. Our results reveal that the beam displays a self-focusing property during propagation. The focusing ability can be enhanced with increasing the beam index and decreasing the beam's spatial coherence width, whereas the DOP and SOP remain unchanged on propagation. Meanwhile, we establish an experimental system with the use of a radial polarization converter and a digital micro-mirror device to synthesize the SCNRP beam with controllable beam index and spatial coherence width. The spectral density and polarization properties of the synthesized beam during propagation are measured and analyzed in the experiment. The experimental results agree well with our theoretical predictions.

Polarization and coherence properties in self-healing propagation of a partially coherent radially polarized twisted beam

With the help of generalized Huygens-Fresnel integral, an analytical expression for the self-healing of a partially coherent radially polarized twisted (PCRPT) beam is derived. The coherence and polarization properties of the PCRPT beam in self-healing propagation are studied in detail. It shows that the existence of the twist phase is a double-edged sword for the self-healing properties of the beam. With the increase of the twist factor, the self-healing ability of beam intensity distribution decreases. However, the anti-disturbance performance of beam polarization improves at the same time. Besides, the polarization and coherence distribution of the beam are proved that own a slight self-healing ability when the obstacle is small. Our results will be helpful to the fields of optical tweezers, microscopy, optical communication, and so on.

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.

Fast calculation of orbital angular momentum flux density of partially coherent Schell-model beams on propagation

Optical coherence has recently become a degree of freedom to modulate the orbital angular momentum (OAM) flux density of a partially coherent beam during propagation. However, the calculation of the OAM flux density for the partially coherent beam involves partial differential and four-dimensional integral operations, which poses drawbacks for its fast numerical calculations. In this paper, we present an efficient numerical protocol for calculating the OAM flux density of any partially coherent Schell-model beam propagating through a paraxial ABCD optical system by only adopting two-dimensional (2D) Fourier transforms. The general formalism is established in detail for the fast numerical calculation of the OAM flux density. It is found that the operation number in the developed algorithm is independent on the spatial coherence states of the beam. To demonstrate the validity of our algorithm, we calculate the OAM flux density of the partially coherent Laguerre-Gaussian beams during propagation with both the analytical and numerical methods. The obtained results are consistent well with each other. Moreover, the OAM flux density properties of two other classes of Schell-model beams, having no analytical solutions, are investigated as the specific examples. Our method provides a convenient way for studying the correlation-induced OAM density changes for any Schell-model beam propagation through a paraxial optical system.

On z-coherence of beams radiated by Schell-model sources with Gaussian profile

The degree of coherence and the intensity distribution on the axis of the beam radiated by a planar partially coherent source of the Schell-model type are investigated. We present an expression for the on-axis cross-spectral density which is valid for a very general Schell-model source, with the only constraint that the intensity distribution across the source is Gaussian. Furthermore, we show that such an expression takes very simple analytical forms for several commonly used degrees of coherence of the source.

On z-coherence in self-focusing

Both the intensity distribution and the degree of coherence between pairs of points along the propagation axis (z-coherence) are derived in closed form for a phenomenon of self-focusing produced by circularly coherent light. The first confirms results previously obtained numerically, while the second exhibits new complex features. The physical interpretation is obtained by a suitable pseudo-modal expansion that suggests an analogy with a simple two-mode structure.

Pseudo-modal expansions for generating random electromagnetic beams

We derive two pseudo-modal expansions that provide insight into the structure of stationary electromagnetic sources and can be used for their physical realization and in computer simulations. Both expansions are derived from the vectorial version of Bochner's theorem of functional analysis. The first expansion employs the incoherent superposition of two completely polarized fields, while the second is based on the incoherent sum of three polarized fields. We generate, in simulation, two random electromagnetic beams from the literature using both expansions and compare the results to theory to validate our work. The primary utility of this research is twofold: in optical simulations involving partially coherent, partially polarized light beams and in the design/validation of new random electromagnetic sources.

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.

Radially polarized twisted partially coherent vortex beams

We introduce a new type of partially coherent vector beam, named the radially polarized twisted partially coherent vortex (RPTPCV) beam. Such a beam carries the twist phase and the vortex phase simultaneously, and the initial state of polarization (SOP) is radially polarized. On the basis of the pseudo-modal expansion and fast Fourier transform algorithm, the second-order statistics such as the spectral density, the degree of polarization (DOP) and the SOP, propagation through a paraxial ABCD optical system are investigated in detail through numerical examples. The results reveal that the propagation properties of the RPTPCV beam closely depends on the handedness of the twist phase and the vortex phase. When the handedness of the two phases is same, the beam profile is easier to remain a dark hollow shape and the beam spot rotates faster during propagation, compared to the partially coherent vortex beam or the RPTPCV beam with the opposite handedness of the two phases. In addition, the same handedness of two phases resists the coherence induced de-polarization of the beam upon propagation, and the SOP is also closely related to the handedness, topological charge of the vortex phase and the twist factor of the twist phase, providing an efficient way to modulate the beam's DOP and SOP in the output plane. Moreover, we establish an experiment setup to generate the RPTPCV beam. The average spectral density and the polarization properties are examined in the experiment. The experimental results agree reasonable well with the theoretical predictions. Our results will be useful for particle manipulating, free-space optical communications, and polarization lidar systems.

Analysis and experimental demonstration of propagation features of radially polarized specific non-uniformly correlated beams

With the development of the unified theory of coherence and polarization, the novel physical properties generated by different correlation structures of vector partially coherent beams (PCBs) have attracted much attention. Recently, a new class of structured beams have been proposed [Opt. Lett.45, 3824 (2020)10.1364/OL.397316], called vector specific non-uniformly correlated beams. These beams combine non-uniform polarization and non-uniform correlation, and they exhibit propagation features not seen in conventional vector PCBs. In this Letter, we continue the analysis of the previous work, taking radially polarized Hermite non-uniformly correlated (RPHNUC) beams as an example, and focus on the physical interpretation of the peculiar propagation features of such beams. We verify the predicted behavior of RPHNUC beams through experiment.

Introduction to the Special Issue on Structured Light Coherence

Statistical optics, and optical coherence in particular, developed into a stand-alone branch of physical optics in the second half of the 20th century and has found a number of ground-breaking applications in astronomical measurements, medical diagnostics, environmental remote sensing, and wireless communications [...]

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.

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.

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.

Efficient calculation of highly focused electromagnetic Schell-model beams

The calculation of the propagation of partially coherent and partially polarized optical beams involves using 4D Fourier Transforms. This poses a major drawback, taking into account memory and computational capabilities of nowadays computers. In this paper we propose an efficient calculation procedure for retrieving the irradiance of electromagnetic Schell-model highly focused beams. We take advantage of the separability of such beams to compute the cross-spectral density matrix by using only 2D Fourier Transforms. In particular, the number of operations depends only on the number of pixels of the input beam, independently on the coherence properties. To provide more insight, we analyze the behavior of a beam without a known analytical solution. Finally, the numerical complexity and computation time is analyzed and compared with some other algorithms.

Synthesis of vector nonuniformly correlated light beams by a single digital mirror device

We present a stable and flexible way to generate the vector nonuniformly correlated (NUC) beams with a compact optical system that involves only a single digital mirror device and a common-path interferometer. The system provides near real-time generation and accurate control of the phase difference between the orthogonal field components of the vector NUC beams. We discuss the methodology based on the vectorial pseudo-mode decomposition of the cross-spectral density matrix of the beam. The method is validated by experimentally generating a class of vector NUC beams, named electromagnetic cosh-Gauss NUC beams, which have not been previously synthesized. Such beams display self-focusing feature on propagation and can reduce to different types of scalar NUC beams by selecting out the linearly polarized components at different polarization angles.

Reproducing Kernel Hilbert spaces for wave optics: tutorial

An introduction to the Hilbert spaces that are endowed with a reproducing kernel is presented on using the mathematical tools of Fourier optics and coherence theory. After giving the basic definition of such spaces, some examples are worked out to show how the inner product can take different forms depending on the particular function space one works with. The basic rule to build a reproducing kernel Hilbert space (RKHS) is then presented together with the basic properties of those spaces. Eigenfunctions and eigenvalues of the reproducing kernel are then illustrated and lead to the important integral representation of the reproducing kernel. The latter is used to present pseudomodal expansions and generalized forms of sampling. The concluding section offers some thoughts on the applications of RKHSs in wave optics. An appendix presents an introduction to treatments using more advanced concepts of functional analysis.

A New Type of Shape-Invariant Beams with Structured Coherence: Laguerre-Christoffel-Darboux Beams

A new class of sources presenting structured coherence properties is introduced and analyzed. They are obtained as the incoherent superposition of coherent Laguerre-Gaussian modes with suitable coefficients. This ensures that the shape of the intensity profile and the spatial coherence features of the propagated beams are invariant during paraxial approximation. A simple analytical expression is obtained for the cross-spectral density of the sources of this class, regardless of the number of superposed modes. Properties of these sources are analyzed and described by several examples.

Optical coherence encryption with structured random light

Information encryption with optical technologies has become increasingly important due to remarkable multidimensional capabilities of light fields. However, the optical encryption protocols proposed to date have been primarily based on the first-order field characteristics, which are strongly affected by interference effects and make the systems become quite unstable during light-matter interaction. Here, we introduce an alternative optical encryption protocol whereby the information is encoded into the second-order spatial coherence distribution of a structured random light beam via a generalized van Cittert-Zernike theorem. We show that the proposed approach has two key advantages over its conventional counterparts. First, the complexity of measuring the spatial coherence distribution of light enhances the encryption protocol security. Second, the relative insensitivity of the second-order statistical characteristics of light to environmental noise makes the protocol robust against the environmental fluctuations, e.g, the atmospheric turbulence. We carry out experiments to demonstrate the feasibility of the coherence-based encryption method with the aid of a fractional Fourier transform. Our results open up a promising avenue for further research into optical encryption in complex environments.

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.

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.

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.

Genuine-field modeling of partially coherent X-ray imaging systems

A genuine representation of the cross-spectral density function as a superposition of mutually uncorrelated, spatially localized modes is applied to model the propagation of spatially partially coherent light beams in X-ray optical systems. Numerical illustrations based on mode propagation with VirtualLab software are presented for imaging systems with ideal and non-ideal grazing-incidence mirrors.

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.

Evolution properties of the radially polarized Laguerre-Gaussian-correlated Schell-model beams propagating in uniaxial crystals

In this paper, we discuss, both analytically and numerically, the paraxial propagation of the radially polarized Laguerre-Gaussian-correlated Schell-model (LGCSM) beams orthogonal to the optical axis in uniaxial crystals. The analytical expression for the cross-spectral density function and the second-order moments of the radially polarized LGCSM beams are derived, and the evolution properties of the normalized intensity distribution, the spectral degree of the coherence (SDOC), and the spectral degree of the polarization (SDOP) in uniaxial crystals are elucidated by numerical examples. It is found that the intensity distribution of the radially polarized LGCSM beams evolves from a doughnut shape into a solid shape and finally converts into an elliptical symmetric hollow-ring profile in uniaxial crystals due to the combined effect of special correlation functions and the anisotropy effect of the uniaxial crystals. The evolution of the SDOC and SDOP for the radially polarized LGCSM beams is quite different from that of the radially polarized Gaussian-Schell-model beams. In addition, the propagation properties of the radially polarized LGCSM beams are closely related to the spatial coherence length, the mode order, and the ratio of extraordinary and ordinary reflective indices. The results show that the uniaxial crystals could modulate the evolution properties of the radially polarized LGCSM beams.

Experimental synthesis of partially coherent sources

A flexible pseudo-mode sampling superposition method for synthesizing partially coherent sources has been introduced that can be thought of as an approximate discrete representation of Gori's nonnegative definiteness criterion for designing spatial correlation functions. Importantly, without performing formidable mode analysis, this method enables us to develop a convenient and efficient experimental technology to customize partially coherent sources without sacrificing theoretical accuracy. As an example, we experimentally generate a new, to the best of our knowledge, class of nontrivial pseudo-Schell model sources recently proposed by de Sande et al. Our approach opens up a useful avenue for manipulating nontrivial partially coherent beams and promotes applications for optical tweezers and photolithography.

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.

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.

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.

Generating dark and antidark beams using the genuine cross-spectral density function criterion

In this work, we demonstrate how to generate dark and antidark beams-diffraction-free partially coherent sources-using the genuine cross-spectral density function criterion. These beams have been realized in prior work using the source's coherent-mode representation and by transforming a J₀-Bessel correlated partially coherent source using a wavefront-folding interferometer. We generalize the traditional dark and antidark beams to produce higher-order sources, which have not been realized. We simulate the generation of these beams and compare the results to the corresponding theoretical predictions. The simulated results are found to be in excellent agreement with theory, thus validating our analysis. We discuss the pros and cons of our synthesis approach vis-à-vis the prior coherent modes work. Lastly, we conclude this paper with a brief summary, and a discussion of how to physically realize these beams and potential applications.

Higher-order twisted/astigmatic Gaussian Schell-model cross-spectral densities and their separability features

Adding a twist phase term to the cross-spectral density (CSD) function of a partially coherent source can be done if and only if the resulting function remains nonnegative definite. Constraints on the twist term that guarantee the validity of the resulting CSD have been derived only for Twisted Gaussian Schell-model (TGSM) sources. Here, an infinite family of higher-order TGSM sources is introduced, whose CSDs are expressed as products of the CSD of a standard TGSM source times Hermite polynomials of arbitrary orders and suitable arguments. All the members present the same twist term and for all of them the twist-coherence constraint keeps obeying the form valid for a standard TGSM source. They can be used as building blocks for constructing an endless number of valid twisted CSDs, with an assigned value of the twist parameter and intensity and/or coherence features that can be very different from those of a standard TGSM source. Through partial transposition, higher-order TGSM CSDs are converted into higher-order Astigmatic Gaussian Schell-model (AGSM) CSDs. The problem of the separability of higher-order TGSM and AGSM CSDs is addressed, and conditions ensuring their separability are derived.

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.

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.

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.

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.

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.

Temporal modes of stationary and pulsed quasistationary electromagnetic beams

We present a novel time-domain coherent-mode representation for random, stationary electromagnetic beams. We subsequently introduce random, quasistationary pulsed electromagnetic beams and develop an analogous (pseudo) mode decomposition for them as well. The former decomposition is valid provided the time window in which the field is considered is much longer than the coherence time, while the latter requires the field to vanish outside the window. For stationary beams, the theory is demonstrated by an example illustrating the role of polarization in the representation. In both cases, the data needed for the construction of the mode decomposition are straightforward to measure. The formalisms enable us to treat random vector-light beams in the time domain in terms of deterministic fields. We expect that the modal representations will find a wide range of applications in problems involving spatiotemporal propagation of temporally partially coherent light in optical systems.

Goos-Hänchen and Imbert-Fedorov shifts: relation with the irradiance moments of a beam

We present closed and simple expressions of the spatial and angular Goos-Hänchen and Imbert-Fedorov shifts in terms of the second-order irradiance moments of a beam. Our results are applicable to a general totally polarized partially coherent beam. One of the main advantages of this formalism is that it can be applied directly from the knowledge of the cross-spectral density function and the polarization state without using any modal beam expansion. The obtained expressions allow understanding of the relationship between the global spatial characteristics of the incident beam and the experimented shifts in the reflected beam. Cosine-Gaussian Schell-model beams with rectangular symmetry are used to exemplify results.

Complex Gaussian representations of partially coherent beams with nonconventional degrees of coherence

We adopt the recently introduced complex Gaussian function to expand the partially coherent beams with nonconventional degrees of coherence, and derive detailed representations of Hermite-Gaussian correlated Schell-model beam, elliptical Laguerre-Gaussian correlated Schell-model beam, and multi-Gaussian correlated Schell-model beam. Complex Gaussian representation of a partially coherent beam provides a convenient way for treating its propagation. As an application example, we explore the self-splitting properties of a Hermite-Gaussian correlated Schell-model beam propagating in a uniaxial crystal with the help of the complex Gaussian representation, and it is found that the uniaxial crystal can be used to control the splitting properties.

Quasi-monochromatic modes of quasi-stationary, pulsed scalar optical fields

We investigate the temporal coherence of random, pulsed, quasi-stationary scalar light fields and introduce a new type of expansion for the mutual coherence function in terms of fully coherent frequency-shifted quasi-monochromatic modes of identical shape. The mode representation is valid provided the pulse length is shorter and the coherence time is much shorter than the width of the time window in which the field is considered. The construction of the expansion is particularly straightforward since information is required only on the average spectrum and the average temporal intensity. The method enables us to assess the coherence properties of quasi-stationary light by analyzing the behavior of deterministic quasi-monochromatic fields. The frequency-domain counterpart of the representation is also given. The method is illustrated by application to a pulsed free-electron laser source.

Partially coherent sources with circular coherence

A new class of partially coherent light sources is introduced. At the source plane, they exhibit perfect coherence along any annulus that is concentric to the source center. Between two points at different distances from the center, coherence can be partial or even vanishing. Such sources can be synthesized by using a generalized form of van Cittert-Zernike theorem where axial sources are used. Beams radiated by this type of source are analyzed at the source plane and upon free propagation for some simple cases.