Random sources for rotating spectral densities

Twisted vortex Gaussian Schell-model beams, generalized ABCD systems, and multidimensional Hermite polynomials

We derive the cross-spectral density (CSD) function for a twisted vortex partially coherent beam at the output of a general ABCD system in terms of multidimensional Hermite polynomials (MDHPs). MDHPs offer notational and computational advantages over prior CSD function representations that use common (one-dimensional) Hermite polynomials. We explain how to compute MDHPs using the recurrence relation given in the literature and include MATLAB code to generate MDHPs of any order. Lastly, we validate our work experimentally by comparing the measured spectral density of a twisted vortex beam at the output of an asymmetric optical system to predictions from our theoretical CSD function.

Identifying the twist factor of twisted partially coherent optical beams

Twisted partially coherent light, characterized by its unique twist factor, offers novel control over the statistical properties of random light. However, the recognition of the twist factor remains a challenge due to the low coherence and the stochastic nature of the optical beam. This paper introduces a method for the recognition of twisted partially coherent beams by utilizing a circular aperture at the source plane. This aperture produces a characteristic hollow intensity structure due to the twist phase. A deep learning model is then trained to identify the twist factor of these beams based on this signature. The model, while simple in structure, effectively eliminates the need for complex optimization layers, streamlining the recognition process. This approach offers a promising solution for enhancing the detection of twisted light and paves the way for future research in this field.

Virtual sources for structured partially coherent light fields

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

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.

Generating a hollow twisted correlated beam using correlated perturbations

In this study, a twisted correlated optical beam with a dark hollow center in its average intensity is synthesized by correlated correlation perturbation and incoherent mode superposition. This new hollow beam has a topological charge (TC) mode with a zero value compared with a coherence vortex that has a TC mode with a nonzero value. We transform the twisted correlated beam from solid centered to dark hollow centered by constructing a correlation between the twist factor and the spot structure parameter. Theoretical and experimental results show that twist correlation makes the random optical beam an asymmetric orbital angular momentum spectral distribution and a tunable intensity center. Controlling the correlation parameters can make the focal spot of the twisted beam a dark core when the dominant mode of the TC is still zero. The new nontrivial beams and their proposed generation method provide important technical preparations for the optical particle manipulation with low coherence environment.

Tailoring the coherence-OAM matrices

We establish a general framework for introducing novel, to the best of our knowledge, classes of beams possessing precisely tailored coherence-orbital angular momentum (COAM) matrices, with the help of Bochner's theorem. The theory is illustrated by several examples relating to COAM matrices having a finite and infinite number of elements.

Evolution properties of twisted Hermite Gaussian Schell-model beams in non-Kolmogorov turbulence

A general form of twisted Hermite Gaussian Schell-model (THGSM) beams is introduced; analytical expressionsare obtained for cross-spectral density and M²-factor using the extended Huygens-Fresnel principle and Wigner function. The evolution of THGSM beams during propagation in non-Kolmogorov turbulence is shown numerically; the beams exhibit self-splitting and twist into two lobes. The intensity distribution evolves into a Gaussian shape and beam quality worsens with increasing distance; the intensity distribution and M²-factor are determined by the twist factor, beam orders, and other beam parameters. THGSM beams provide more degrees of freedom to regulate beam parameters, thereby enriching the types of partially coherent beams.

Twisted sinc-correlation Schell-model beams

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

Experimental synthesis of partially coherent beam with controllable twist phase and measuring its orbital angular momentum

Twist phase is a nontrivial second-order phase that only exists in a partially coherent beam. Such twist phase endows the partially coherent beam with orbital angular momentum (OAM) and has unique applications such as in super-resolution imaging. However, the manipulation and the detection of the twist phase are still far from easy tasks in experiment. In this work, we present a flexible approach to generate a famous class of twisted Gaussian Schell-model (TGSM) beam with controllable twist phase by the superposition of the complex field realizations using a single phase-only spatial light modulator. The precise control of the amplitude and phase of the field realizations allows one to manipulate the strength of the twist phase easily. In addition, we show that the twist factor, a key factor that determines the strength of twist phase and the amount of OAM, can be measured by extracting the real part of the complex degree of coherence of the TGSM beam. The experiment is carried out with the help of the generalized Hanbury Brown and Twiss experiment as the generated TGSM beam obeys Gaussian statistics. The flexible control and detection of the twist phase are expected to find applications in coherence and OAM-based ghost imaging.

Twisted electromagnetic elliptical multi-Gaussian Schell-model beams and their transmission in random media

We extend the scalar elliptical multi-Gaussian Schell-model (EMGSM) beams with twist phase to the electromagnetic domain and obtain the analytical expression for the propagation of the electromagnetic twisted EMGSM beams through random media. The twist phase-induced changes of the spectral density and degree of polarization of such beams on propagation are studied numerically. Results show that by adjusting the twist factor and the correlated parameters of the source, both the spectral density and degree of polarization not only rotate around the propagation axis but also exhibit diverse shapes. The flattopped ellipse-like and diamond-like shape maintain over a relatively long propagation distance and finally involve into Gaussian-like shape due to stronger atmospheric turbulence. The results will be useful in optical trapping and optical communication.

Twist phase and classical entanglement of partially coherent light

We demonstrate that the presence of a twist phase in a random light beam leads to classical entanglement between phase space degrees of freedom of the beam. We find analytically the bi-orthogonal decomposition of the Wigner function of a twisted Gaussian Schell-model (TGSM) source and quantify its entanglement by evaluating the Schmidt number of the decomposition. We show that (i) classical entanglement of a TGSM source vanishes concurrently with the twist in the fully coherent limit and (ii) entanglement dramatically increases as the source coherence level decreases. We also show that the discovered type of classical entanglement of a Gaussian Wigner function does not degrade on beam propagation in free space.

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.

Coherence theory for electromagnetic, planar, PT-symmetric sources

A class of planar, electromagnetic, stationary optical sources obeying PT-symmetry conditions is introduced. On deriving a set of Fourier reciprocity relations obeyed by such sources and far fields they radiate, we discover that the latter can only have linear polarization states.

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.

Progress on Studies of Beams Carrying Twist

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

Customizing twisted Schell-model beams

Since the celebrated twist phase was proposed by Simon and Mukunda, despite the tremendous progress made in theory over the past decades, developing a simple and flexible experimental method to customize this novel phase has long been a tricky challenge. In this Letter, we demonstrate a general experimental method for generating twisted Schell-model beams by implementing the continuous coherent beam integral function in a discrete form. Experimental results based on rigorous Laguerre-Gauss modes superposition are also demonstrated, indicating that our method is more convenient and of higher quality. The twist factor is also measured using the rotation characteristics during propagation, and the results agree well with the theoretical prediction. The method could serve as a general way for customizing bona fide twisted cross-spectral densities while facilitating certain applications.

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.

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.

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.

Statistical Characteristics of a Twisted Anisotropic Gaussian Schell-Model Beam in Turbulent Ocean

The analytical expression of the cross-spectral density function of a twisted anisotropic Gaussian Schell-model (TAGSM) beam transmitting in turbulent ocean is derived by applying a tensor method. The statistical properties, including spectral density, the strength of twist and beam width of the propagating beam are studied carefully through numerical examples. It is demonstrated that the turbulence of ocean has no effect on the rotation direction of the beam spot during propagation. However, the beam shape will degrade into a Gaussian profile under the action of oceanic turbulence with sufficiently long propagation distance, and a beam with a larger initial twist factor is more resistant to turbulence-induced degeneration. As oceanic turbulence becomes stronger, the beam spot spreads more quickly while the twist factor drops more rapidly upon propagation. The physical mechanisms of these phenomena are addressed in detail. The obtained results will be helpful in optical communication systems underwater.

Propagation of twisted EM Gaussian Schell-model array beams in anisotropic turbulence

The behavior of the twisted electomagnetic (EM) Gaussian Schell-model array beams in anisotropic random turbulence is investigated. An example illustrates that a twisted EM source can produce lattice-like patterns in degree of polarization with rotation or not, which depends on the setting of the initial twist phase. One also finds that the anisotropy of the medium leads to an anisotropic beam spreading, and we can effectively limit such turbulence-induced effects by optimizing the initial twist and source correlation widths. Moreover, after transmitting through the turbulence for sufficiently long distances, the intensity and coherence are mainly affected by turbulence statistics; however, for the case of polarization, the initial twist plays a dominant role in determining its distribution profile.

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.

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.

Ring-shaped twisted Gaussian Schell-model array beams

A new partially coherent source which can generate a beam field with a ring-shaped twisted array profile is presented, and the distribution characteristics of spectral density and degree of coherence of the field are discussed. It is shown that both the spectral density and degree of coherence will rotate along the propagating direction, but in opposite rotating directions. Furthermore, we find that the distribution properties of the ring-shaped array of the spectral density, including the number of the rings, the number of the lobes of each ring, and the distance of all adjacent lobes, can be properly controlled by adjusting structural parameters of the source.

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.

Twisted elliptical multi-Gaussian Schell-model beams and their propagation properties

We introduce a new kind of partially coherent source whose cross-spectral density (CSD) function is described as the incoherent superposition of elliptical twisted Gaussian Schell-model sources with different beam widths and transverse coherence widths, named twisted elliptical multi-Gaussian Schell-model (TEMGSM) beams. Analytical expression for the CSD function propagating through a paraxial ABCD optical system is derived with the help of the generalized Collins formula. Our results show that the TEMGSM beam is capable of generating a flat-topped elliptical beam profile in the far field, and the beam spot during propagation exhibits clockwise/anti-clockwise rotation with respect to its propagation axis. In addition, the analytical expressions for the orbital angular momentum (OAM) and the propagation factor are also derived by means of the Wigner distribution function. The influences of the twisted factor and the beam index on the OAM and the propagation factor are studied and discussed in detail.

Elliptical Laguerre Gaussian Schell-model beams with a twist in random media

A class of partially coherent elliptical sources with twisted Laguerre Gaussian Schell-model (TLGSM) correlation function is proposed, which are capable of producing beams whose intensity profiles may vary substantially. This kind of beam can be viewed as the generalization of the LGSM beams. Properties of the spectral density during propagation in free space and atmospheric turbulence are investigated with varying quantities related to the beam source and the medium. It is shown that the elliptical TLGSM beams evolve in a manner that is much more complex compared to the LGSM beams. In addition, the behaviour of the rotation angle is further analysed by quantitative examples.

Generalized partially coherent beams with nonseparable phases

A new, to the best of our knowledge, family of partially coherent beams incorporating a set of nonseparable phases is introduced. Due to the nonnegative definiteness of the cross-spectral density function, these phases cannot survive in the limit of full coherence, which distinguishes them from conventional phase terms. An example of a nonuniform model beam with a quartic nonseparable phase is presented. It is shown that the presence of such a phase in effect stretches the beam in a specific direction upon propagation. In particular, the interplay between the magnitude and phase of the source coherence state endows the beam with a wing-like structure. The fundamental concept developed here brings into evidence the countless nonseparable phases differing from twist, and can be exploited to generate customized partially coherent beams for applications in optics.

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.

Generating bona fide twisted Gaussian Schell-model beams

We experimentally realize bona fide twisted Gaussian Schell-model (TGSM) beams by converting an anisotropic GSM beam into a TGSM beam with a set of just three cylindrical lenses. In contrast to the previously reported experimental technique for generating TGSM beams, we are able to explicitly generate anisotropic GSM beams with simultaneously controlled beam widths and transverse coherence lengths along two mutually orthogonal directions, prior to converting them to genuine TGSM beams.

Phase structuring of 2D complex coherence states

It was shown in a recent publication [Opt. Lett.43, 4727 (2018)OPLEDP0146-959210.1364/OL.43.004727] that structuring the phase of 1D-sourced complex coherence states results in breaking the Cartesian symmetry for the far-field spectral density, leading to its one-sided radial acceleration, asymmetric splitting, and other effects. We extend this result from 1D to 2D (scalar) optical beams and consider examples of novel 2D sources with separable (in Cartesian coordinates) and nonseparable phase distributions of the complex coherence states, illustrating the possibilities in asymmetric manipulation of the 2D far-fields.

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.

Controllable rotating Gaussian Schell-model beams

Optical twists are the rotation of light structures along the beam axis, which can be caused by the quadratic twist phase of a partially coherent field. Here, we introduce a new class of partially coherent beams whose spectral density and degree of coherence tend to rotate during propagation. Unlike the previously reported twisted Gaussian Schell-model beams, this family of rotating beams is constructed without the framework of rotationally invariant cross-spectral density functions. Thus, these beams have different underlying physics and exhibit distinctive twist effects. It is shown that such beams can undergo a twist of more than 90 deg, providing larger degrees of freedom for flexibly tailoring the beam twist. Our results may pave the way toward synthesizing rotating beams for applications in optics and, in particular, inspire further studies in the field of twist phase proposed 25 years ago.

Twisted Laguerre-Gaussian Schell-model beam and its orbital angular moment

It is well known that a light beam with vortex phase or twist phase carries orbital angular momentum (OAM), while twist phase only exists in a partially coherent beam. In this paper, we introduce a new partially coherent beam, named twisted Laguerre-Gaussian Schell-model (TLGSM) beam. This TLGSM beam carries both vortex phase and twist phase. Further, the evolutional properties of the spectral density and spectral degree of coherence (SDOC)] of the TLGSM beam passing through a paraxial ABCD optical system are explored in detail. Our results reveal that the vortex and twist phases' handedness significantly affects the evolution properties. When the twist phase's handedness is the same as that of the vortex phase, the beam profile maintains a dark hollow shape during propagation and the side rings in SDOC are suppressed; however, when the handedness of two phases is opposite, then the beam shape evolves into a Gaussian shape and the side rings in SDOC are enhanced. Furthermore, we obtain the analytical expression for the OAM of the TLGSM beam. It is found that the vortex phase's and twist phase's contributions to the OAM are interrelated, which greatly increases the amount of OAM. In addition, the OAM variation of the TLGSM beam passing through an anisotropic optical system is also explored in detail. Our results will be useful for information transfer and optical manipulations.

Propagation properties of a twisted rectangular multi-Gaussian Schell-model beam in free space and oceanic turbulence

With the paraxial approximation, we derive the analytical formulas for the propagation of a twisted rectangular multi-Gaussian Schell-model (TRMGSM) beam in free space and oceanic turbulence. The spectral density and degree of coherence of the TRMGSM beam both in free space and oceanic turbulence are investigated. We found that the oceanic turbulence barely had any influence on the rotation effect both of the spectral density and degree of coherence. In addition, we found that the spot could degenerate into a Gaussian profile caused by a turbulence medium. Furthermore, for stronger turbulence, the beam spot spreads more rapidly, and the coherence region decreases faster. In this paper, we also study the effect of the twisted factor on the propagating properties.

Phase structuring of the complex degree of coherence

Simple conditions on the magnitude and phase of the complex degree of coherence of stationary, scalar, one-dimensional (1D) Schell-like sources are derived for the design of novel optical beam-like fields. Several examples of new sources obtained with the help of signum, odd monomials, and sine phase functions are given. A geometrical representation, termed coherence curve, for the complex coherence state of a 1D, Schell-like source with non-trivial magnitude and phase distributions by means of a parametric curve confined to the unit circle in a complex plane is also introduced.

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.

Twisted EM beams with structured correlations

The possibility of restructuring the spectral density of a random light beam up to three times along the propagation path in free space is demonstrated. This can be achieved by prescribing special correlations and different twist factors (by magnitude and/or direction) in the two mutually orthogonal electric field components in the source plane. We also show that the degree of polarization of such a light beam can rotate or not on propagation.

Twisted Gaussian Schell-model array beams

We introduce, to the best of our knowledge, a new class of twisted partially coherent sources for producing rotating Gaussian array profiles. The general analytical formula for the cross-spectral density function of a beam generated by such a novel source propagating in free space is derived, and its propagation characteristics are analyzed. It is shown that both the irradiance profile of each element of the array and the degree of coherence rotate during propagation, but in opposite directions. Further, the twist effects of the spectral density and the degree of coherence are quantified. It is shown that the direction of rotation is changeable upon propagation. Our results may provide new insight into the twist phase and may be applied in optical trapping.

Second-order moments of a twisted Gaussian Schell-model beam in anisotropic turbulence

Analytical formulas for the Wigner distribution functions and the second-order moments (SOMs) of a twisted Gaussian Schell-model (TGSM) beam propagating through anisotropic turbulence have been derived by means of a tensor method. It is found that the propagation law for the SOMs of a TGSM beam spreading in turbulence can be described as a first-order optical systems propagation law with an additional turbulence-induced effect. Based on the SOMs, second-order statistics in terms of the effective beam width, the M²-factor, the orbital angular momentum flux, and the effective radius curvature are analyzed in detail. One finds that the anisotropic turbulence leads to an anisotropic spreading of light beams, and a TGSM beam is less affected by turbulence than a Gaussian Schell-model beam without the twist phase. For short distance propagation, the light beam is more sensitive to the initial beam parameters than turbulence parameters, while for sufficiently long transmission distance, the beam characteristics are determined at most by turbulence statistics. The method can be extended to study the propagation characteristics of various coherent and partially coherent complex Gaussian beams, such as flat-topped beams, dark hollow beams, and array beams in turbulence, and promotes important supports in free-space optical communications and remote sensing.

Experimental investigation on a nonuniformly correlated partially coherent laser

We demonstrate, to the best of our knowledge, a new kind of laser, called a partially coherent digital laser, producing nonuniformly correlated partially coherent light beams by "playing a video" inside the cavity directly. In this laser, a spatial light modulator (SLM) with dynamic phase modulation acts as a cavity mirror. The coherence degree distribution of the output beams can be controlled simply by varying the waists of the computer-generated holograms on the SLM. The experimental results show that the coherence degree between two points on the observation plane is not only dependent on the distance between them, but also on the positions that are observed across the beam.

Sources for random arrays with structured complex degree of coherence

Simultaneous control of the far-field spectral density distribution, together with the modulus and the phase of the complex degree of coherence of a beam radiated by a stationary source, is hard to achieve in view of the intrinsic constraints on the source cross-spectral density. We tackle this problem by employing the Laguerre-Gaussian modes instead of the commonly used Gaussian mode for the source field realizations, in combination with structuring the degree of source coherence radiating to spectral density arrays. We show that the produced spectral density arrays with Cartesian and polar symmetries have special features in both the modulus and the phase of the complex degree of coherence.

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.

Devising genuine twisted cross-spectral densities

Sticking a twist to a partially coherent source cannot be done at will, since the result can violate the definiteness property of the corresponding cross-spectral density. As a matter of fact, the study of twisted sources has been mainly concentrated on the original case proposed by Simon and Mukunda [J. Opt. Soc. Am. A10, 95 (1993)JOAOD60740-323210.1364/JOSAA.10.000095] of circularly symmetric Gaussian Schell-model sources. Here, we discuss a modeling procedure that can be used to generate numberless genuine twisted sources without symmetry constraints. As geometrically simple examples, two cases of non-Gaussian twisted sources endowed with circular or rectangular symmetry are explicitly worked out.

Optical coherence grids and their propagation characteristics

A novel class of partially coherent light sources termed optical coherence grids (OCGs) are introduced that can yield stable optical grids in the far field. The optical grids, of which the light distributes in a network of straight lines crossing each other to form a series of hollow cages, can be seen as a better controlled optical lattice. Propagation properties of OCG beams in free space, including spectral density, transverse coherence, and M² factor, are investigated in detail. It is interesting that a periodic grid pattern is produced at a distance and remains stable on further propagation, and we stress that the structure of far-field optical grids can be flexibly tuned by modulating the correlation parameters of the source. In addition, by performing convolution of degree of coherence, we also propose perfect optical coherence grids (POCG). The far-field grid pattern of POCG is in a fully controllable fashion. This work is expected to find applications in cooling atoms, trapping microscopic particles, or assembling cells, etc.

Propagation of multi-cosine-Laguerre-Gaussian correlated Schell-model beams in free space and atmospheric turbulence

We introduce a class of random stationary, scalar source named as multi-cosine-Laguerre-Gaussian-correlated Schell-model (McLGCSM) source whose spectral degree of coherence (SDOC) is a combination of the Laguerre-Gaussian correlated Schell-model (LGCSM) and multi-cosine-Gaussian correlated Schell-model (McGCSM) sources. The analytical expressions for the spectral density function and the propagation factor of a McLGCSM beam propagating in turbulent atmosphere are derived. The statistical properties, such as the spectral intensity and the propagation factor, of a McLGCSM beam are illustrated numerically. It is shown that a McLGCSM beam exhibits a robust ring-shaped beam array with adjustable number and positions in the far field by directly modulating the spatial structure of its SDOC in the source plane. Moreover, we provide a detailed insight into the theoretical origin and characteristics of such a ring-shaped beam array. It is demonstrated that these peculiar shaping properties are the concentrated manifestation of the individual merits respectively associated with the Laguerre- and multi-cosine-related factors of the whole SDOC. Our results provide a novel scheme to generate robust and controllable ring-shaped beam arrays over large distances, and will widen the potentials for manipulation of multiple particles, free-space optical communications and imaging in the atmosphere.

Self-focusing of a partially coherent beam with circular coherence

In a recent publication [Opt. Lett.42, 1512 (2017)OPLEDP0146-959210.1364/OL.42.001512], a novel class of partially coherent sources with circular coherence was introduced. In this paper, we examine the propagation behavior of the spectral density and the spectral degree of spatial coherence of a beam generated by such a source in free space and in oceanic turbulent media. It is found that the beam exhibits self-focusing, which is dependent on the initial coherence and the parameters of oceanic turbulence. The self-focusing phenomenon disappears when the initial coherence is high enough or the oceanic turbulence is strong. The area of high coherence appears in the center and along two diagonal lines. With increasing turbulence, the coherence area reduces gradually along one diagonal line and is retained along the other one. A physical interpretation of the self-focusing phenomenon is presented, and potential applications in optical underwater communication and beam shaping are considered.