Radiation force of scalar and electromagnetic twisted Gaussian Schell-model beams

Measuring the orbital angular momentum of generalized higher-order twisted partially coherent beams

Recently a new family of partially coherent fields incorporating generalized inseparable cross-coupled phases named generalized higher-order twisted partially coherent beams (GHTPCBs) have been introduced. The twist factor u is a key parameter that not only quantifies the strength of the generalized cross-coupled phase for a given order, but also determines the amount of the concomitant orbital angular momentum (OAM). In this paper, we propose a simple and reliable method to measure the factor u using a two-pinhole mask. Without need of complicated optical system, it only requires to capture the far-field diffraction intensity distribution of the GHTPCB passing through the mask. By analyzing the Fourier spectrum of the intensity distribution, the value of twist factor can be derived nearly in real time. The influence of the separation distance between two pinholes and the pinholes' diameter and position on the measurement accuracy are thoroughly studied both in theory and experiment. The experimental results agree well with the theoretical results. Our methodology can also be extended to measure the sole factor of similar position dependent phases such as the topological charge of a vortex phase.

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.

Calculation of the optical forces exerted on a nano-dielectric sphere induced by a pulsed Laguerre-Gaussian beam

Theoretically, we investigated the optical force exerted on a nano-dielectric sphere illuminated by a pulsed Laguerre-Gaussian beam. In the frame of the dipole approximation, analytical expressions for the optical force were derived. Based on these analytical expressions, the effects of pulse duration τ and beam mode order (l,p) on the optical force were studied. It is recognized that the optical force values and the trapping regions are remarkably affected by changing pulse duration and mode parameters. Our results show good agreement with the results obtained by other authors for the use of a continuous Laguerre-Gaussian beam and pulsed Gaussian beam.

Classical entanglement of twisted random light propagating through atmospheric turbulence [Invited]

We examine the impact of the atmospheric turbulence on a recently discovered type of classical entanglement of partially coherent beams endowed with a twist phase. We derive a compact analytical expression for the Schmidt number of a bi-orthogonal decomposition of the Wigner function of a twisted Gaussian Schell-model (TGSM) beam propagating through the turbulent atmosphere. We elucidate conditions for a TGSM source to generate a strongly classically entangled paraxial field over a desired propagation distance in the turbulent atmosphere. Our results will find applications to free-space optical communications and motivate further research into classical entanglement with random light.

Changes in orbital angular momentum distribution of a twisted partially coherent array beam in anisotropic turbulence

Based on the generalized Huygens Fresnel integral, we derive the analytical formula of the cross-spectral density of a twisted partially coherent array beam propagating in non-Kolmogorov anisotropic turbulence, and investigate the changes in orbital angular momentum (OAM). The results show that the anisotropy of the turbulence causes different effects in horizontal and vertical directions. The spectral density distribution of twisted partially coherent array beam in turbulence presents self-splitting and rotation, which combines the interesting effects of the twist phase and coherent structure. Although OAM is conserved, the spatial distribution of OAM flux density can be changed by changing the propagation distance, power and anisotropy of turbulence, and the modulation of the twist phase affects not only the magnitude of OAM but also its distribution. Our work is helpful for exploring new forms of OAM sources, and promote the application of free-space optical communications and optical field modulation.

Twisted Gaussian Schell-model breathers and solitons in strongly nonlocal nonlinear media

Based on the Snyder-Mitchell linear model and the cross-spectral density (CSD) function, the analytical propagation formula of twisted Gaussian Schell-model (TGSM) beams in strongly nonlocal nonlinear medium (SNNM) is derived. Then the propagation characteristics of TGSM beam are studied. It is found that the soliton radius is jointly determined by the initial power, coherence length, and twist factor; the degree of spatial coherence is adjusted by changing the twist factor without affecting the soliton intensity. In the case of non-soliton properties, there is a threshold of coherence length which makes partially coherent beams have the same evolution law as completely coherent beams. Furthermore, increasing the twist factor, decreasing the coherence length and initial power can improve the beam quality of the beam propagating in SNNM.

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.

Degree of paraxiality of a twist electromagnetic Gaussian Schell-model beam

The definition of the degree of paraxiality (DOP) for a stochastic electromagnetic field is applied to a twist stochastic electromagnetic field. As an illustrative example, DOP for a wide class of model stochastic fields, i.e., twist electromagnetic Gaussian Schell-model (TEGSM) fields, is discussed. The dependence of the DOP of the light source on its properties is also studied in detail. The numerical results show that the DOP of a TEGSM beam is determined by the rms widths of auto-correlation functions and the twist factor as well as by the degree of polarization. To explain the behavior of DOP, the far-field divergence angle of this beam source is also discussed.

Scattering of Partially Coherent Vector Beams by a Deterministic Medium Having Parity-Time Symmetry

We study the scattering properties of the partially coherent vector beams with the deterministic media having the classic symmetric and parity-time (PT) symmetric scattering potential functions. The closed-form expressions for the intensity and polarization matrix of the far-zone scattered field are obtained, under first-order Born approximation, when the partially coherent vector beams are taken to be radially polarized and the deterministic media are assumed as the four-point scatterers. We demonstrate both analytically and numerically that the far-zone scattered field becomes noncentrosymmetric and the directionality appears in the scattering pattern when the scattering potential function is switched from classic symmetry to PT symmetry. We show the effect of spatial coherence of the incident partially coherent vector beam on the directionality in scattering. We find that by turning the symmetry property of the spatial coherence function of the incident beam, i.e., into PT symmetry, the directionality in the far-zone scattering can be suppressed or enhanced, depending on the joint effect from the symmetry of the scattering potential and the symmetry of the spatial coherence. Our findings may be useful in the application of dynamic control of the directionality in light scattering.

Rotation of degree of coherence and redistribution of transverse energy flux induced by non-circular degree of coherence of twisted partially coherent sources

It is known that a twisted Gaussian Schell-model (TGSM) beam with elliptical Gaussian amplitude will rotate its beam spot upon propagation because of the vortex structure of the transverse energy flux. In this paper, we study a special kind of twisted partially coherent beams named twisted Hermite-Gaussian correlated Schell model (HGCSM) beam whose degree of coherence (DOC) is non-circularly symmetric but the source amplitude is of the circular Gaussian profile. Our results reveal that the beam spot (average intensity distribution) does not rotate during propagation even if the circular symmetry of the beam spot is broken. However, the DOC pattern shows the rotation under propagation. From the investigation of the transverse energy flux and OAM density flux, we attribute the nontrivial rotation phenomenon to the redistribution of the transverse energy flux by non-circular DOC. Furthermore, based on Hyde's approach [J. Opt. Soc. Am. A37, 257 (2020)10.1364/JOSAA.381772], we introduce a method for the generation of this class of twisted partially coherent sources. The non-rotation of the beam spot and rotation of the DOC are demonstrated in experiment.

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.

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.

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.

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.

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.

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.

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 Characteristics of a Twisted Cosine-Gaussian Correlated Radially Polarized Beam

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

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.

Radiation forces of beams generated by Gaussian mirror resonator on a Rayleigh dielectric sphere

Optical trapping and manipulating of micron-sized particles have attracted enormous interests due to the potential applications in biotechnology and nanoscience. In this work, we investigate numerically and theoretically the radiation forces acting on a Rayleigh dielectric particle produced by beams generated by Gaussian mirror resonator (GMR) in the Rayleigh scattering regime. The results show that the focused beams generated by GMR can be used to trap and manipulate the particles with both high and low index of refractive near the focus point. The influences of optical parameters of the beams generated by GMR on the radiation forces are analyzed in detail. Furthermore, the conditions for trapping stability are also discussed in this paper.

Second-order statistics of a radially polarized partially coherent twisted beam in a uniaxial crystal

The Wigner function (WDF) has been used to study the second-order moments for a radially polarized partially coherent twisted (RPPCT) beam (i.e., a RPPC beam with a twist phase) propagating in a uniaxial crystal. With the help of the extended Huygens-Fresnel integral formula and definition of the WDF, the analytical formulas for propagation factor (M²-factor), effective radius of curvature (ERC), and Rayleigh range of a RPPCT beam propagating in a uniaxial crystal have been derived. Our numerical results show that the M²-factor of a RPPCT beam with larger absolute value of the twist factor or lower coherence is less affected by anisotropic diffraction in a uniaxial crystal. The dependence of the ERC and Rayleigh range of a RPPCT beam propagating in a uniaxial crystal on the parameter e along the x direction is much different from that along the y direction due to anisotropic diffraction. We can judge how much a RPPCT beam carries the twist factor by measuring the deviation percentage of the M²-factor of a RPPCT beam propagating in a uniaxial crystal and even can modulate the properties of a RPPCT beam by varying the beam parameters, which will be useful in some applications where a RPPCT beam is required.

Random sources for rotating spectral densities

The propagating modes of a wide-sense stationary Schell-like source with arbitrary coherence state and a twist factor are determined. This suggests a convenient practical method for modeling novel classes of twisted partially coherent beam-like fields. The first example discusses the previously introduced twisted anisotropic Gaussian Schell-model source and verifies the feasibility of this method. As a second example, we introduce a new type of twisted partially coherent beam in which a radiated flat-top average intensity pattern remains invariant in shape (but not size) while it twists around the axis upon propagation.

Propagation properties of a radially polarized partially coherent twisted beam in free space

We introduce a radially polarized partially coherent twisted beam based on the unified theory of coherence and polarization. We also derive expressions of this beam propagating in free space and examine the influence of twist factor and coherence of the source plane on its propagation properties. It is found that both twist factor and coherence of the source plane affect its average intensity, polarization, and coherence. Those results also show that the beam is rotating with beam propagation, and the beam's rotation is independent of coherence of the source plane.

Orbital angular moment of an electromagnetic Gaussian Schell-model beam with a twist phase

We derive the analytical formula for the orbital angular momentum (OAM) flux of a stochastic electromagnetic beam carrying twist phase [i.e., twisted electromagnetic Gaussian Schell-model (TEGSM) beam] in the source plane with the help of the Wigner distribution function. Furthermore, we derive the general expression of the OAM flux of a TEGSM beam on propagation with the help of a tensor method. As numerical examples, we explore the evolution properties of the OAM flux of a TEGSM beam propagating through a cylindrical thin lens or a uniaxial crystal. It is found that the OAM flux of a TEGSM beam closely depends on its twist factors and degree of polarization in the source plane, and one can modulate the OAM flux of a TEGSM beam by a cylindrical thin lens or a uniaxial crystal. Our results may be useful in some applications, such as particle manipulation and free-space optical communications, where light beam with OAM is preferred.

Generation and propagation of an electromagnetic Gaussian Schell-model vortex beam

We outline the propagation of an electromagnetic Gaussian Schell-model (EGSM) vortex beam through a paraxial ABCD optical system and analyze the vortex phase-induced changes of the statistical properties, such as average intensity, state of polarization, and degree of polarization (DOP), of a focused EGSM beam. It is found that one can shape the beam profile of an EGSM vortex beam in the focal plane through varying its initial topological charge, DOP, and coherence widths. Furthermore, we first report experimental generation of an EGSM vortex beam and measure its focusing properties in experiments. Our experimental results are consistent with the numerical results and may be useful in material thermal processing and particle trapping.

Twisted Schell-model beams with axial symmetry

The problem of when a twist can be impressed on a partially coherent beam is solved for Schell-model fields endowed with axial symmetry. A modal analysis can be performed for any such beam, thus permitting evaluation of whether it will withstand the twisting process. Beyond exemplifying some twistable beams, it is shown that, for certain correlation functions, the beam cannot be twisted, no matter how the numerical parameters are chosen.

Twist phase-induced changes of the statistical properties of a stochastic electromagnetic beam propagating in a uniaxial crystal

With the help of a tensor method, an analytical formula is derived for the cross-spectral density matrix of a twisted electromagnetic Gaussian Schell-model (EGSM) beam (i.e., EGSM beam with twist phase) propagating in a uniaxial crystal orthogonal to the optical axis. The twist phase-induced changes of the statistical properties, such as the spectral density, the degree of polarization and the degree of coherence, of an EGSM beam propagating in a uniaxial crystal are illustrated numerically. It is found that the distributions of the spectral density, the degree of polarization and the degree of coherence of a twisted EGSM beam in a uniaxial crystal all exhibit non-circular symmetries, which are quite different from those of a twisted EGSM beam in isotropic medium or in free space. One may use uniaxial crystal to determine whether an EGSM beam carries twist phase or not.

Experimental demonstration of ghost imaging with an electromagnetic Gaussian Schell-model beam

Recently, there has been a controversy about the dependence of the visibility of the ghost image on the degree of polarization (DOP) of a stochastic electromagnetic beam because of different definitions of the visibility. In this paper, we revisit ghost imaging with an electromagnetic Gaussian Schell-model (EGSM) beam. Through numerical examples based on the conventional definition of the visibility, we find that the visibility of the ghost image indeed increases or decreases with the increase of the DOP the beam source under certain conditions. We solve the controversy between literatures and the present paper through analyzing the r.m.s. widths of auto-correlation functions of the x component of the field and of the y component of the field. Furthermore, we carry out experimental demonstration of ghost imaging with an EGSM beam. Our experimental results verify the theoretical predictions.

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

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

Orbital angular moment of a partially coherent beam propagating through an astigmatic ABCD optical system with loss or gain

We derive the general expression for the orbital angular momentum (OAM) flux of an astigmatic partially coherent beam carrying twist phase [i.e., twisted anisotropic Gaussian-Schell model (TAGSM) beam] propagating through an astigmatic ABCD optical system with loss or gain. The evolution properties of the OAM flux of a TAGSM beam in a Gaussian cavity or propagating through a cylindrical thin lens are illustrated numerically with the help of the derived formula. It is found that we can modulate the OAM of a partially coherent beam by varying the parameters of the cavity or the orientation angle of the cylindrical thin lens, which will be useful in some applications, such as free-space optical communications and particle trapping.

Second-order moments of an electromagnetic Gaussian Schell-model beam in a uniaxial crystal

We derive the analytical expressions for the second-order moments of an electromagnetic Gaussian Schell-model (EGSM) beam propagating in a uniaxial crystal. With the help of the derived formulas, we study the evolution properties of the propagation factor, the effective radius of curvature and the Rayleigh range of an EGSM beam in a uniaxial crystal. It is found that the evolution properties of an EGSM beam in a uniaxial crystal are much different from its evolution properties in free space and are closely determined by the initial beam parameters and the parameters of the uniaxial crystal. The uniaxial crystal provides one way for modulating the properties of an EGSM beam.

Optical binding of cylinder photonic molecules in the near field of partially coherent fluctuating Gaussian Schell model sources: a coherent mode representation

We present a theory and computation method of radiation pressure from partially coherent light by establishing a coherent mode representation of the radiation forces. This is illustrated with the near field emitted from a Gaussian Schell model source, mechanically acting on a single cylinder with magnetodielectric behavior, or on a photonic molecule constituted by a pair of such cylinders. Thus after studying the force produced by a single particle, we address the effects of the spatial coherence on the bonding and antibonding states of two particles. The coherence length manifests the critical limitation of the contribution of evanescent modes to the scattered fields, and hence to the nature and strength of the electromagnetic forces, even when electric and/or magnetic partial wave resonances are excited.

State of polarization and propagation factor of a stochastic electromagnetic beam in a gradient-index fiber

With the help of a tensor method, we investigate the evolution properties of the state of polarization of an electromagnetic Gaussian Schell-model beam propagating through a gradient-index (GRIN) fiber. We find that the Stokes parameters and the polarization ellipse exhibit periodicity. The initial beam parameters affect the values of the Stokes parameters and the parameters of the polarization ellipse. Furthermore, based on the second-order moments of the Wigner distribution function, the explicit expression for the propagation factor (known as the M(2) factor) in the GRIN fiber is derived. It is shown that the M(2) factor remains invariant on propagation and is determined only by the initial beam parameters.

Partially coherent fluctuating sources that produce the same optical force as a laser beam

Inspired by a theory previously derived by Wolf and Collett [Opt. Commun. 25, 293 (1978)], we demonstrate that partially coherent Gaussian-Schell model fluctuating sources (GSMS) produce exactly the same optical forces as a fully coherent laser beam. We also show that this kind of source helps to control the magnitude of the light-matter mechanical interaction in biological samples, which are sensitive to thermal heating induced by higher intensities. The latter is a consequence of the fact that the same photonic force can be obtained with a low-intensity GSMS as with a high-intensity laser beam.

Nonparaxial propagation properties of a vector partially coherent dark hollow beam

Based on the generalized Raleigh-Sommerfeld diffraction integrals, analytical nonparaxial propagation formulas for the elements of the cross-spectral density matrix of a vector partially coherent dark hollow beam (DHB) in free space are derived. The effect of spatial coherence and beam waist sizes on the statistical properties of a nonparaxial vector DHB is studied numerically. It is found that one can modulate the statistical properties of a nonparaxial vector DHB by varying its initial spatial coherence, which will be useful in some applications where nonparaxial beams are commonly encountered.

Coherence and polarization properties of a radially polarized beam with variable spatial coherence

In a recent publication [Appl. Phys. Lett, 100, 051108 (2012)], a radially polarized (RP) beam with variable spatial coherence (i.e., partially coherent RP beam) was generated experimentally. In this paper, we derive the realizability conditions for a partially coherent RP beam, and we carry out theoretical and experimental study of the coherence and polarization properties of a partially coherent RP beam. It is found that after passing through a thin lens, both the degree of coherence and the degree of polarization of a partially coherent RP beam varies on propagation, while the state of polarization of the completely polarized part of such beam remains invariant. The variations of the degree of coherence and the degree of polarization depend closely on the initial spatial coherence. Our experimental results agree well with the theoretical predictions.

Statistical properties of a nonparaxial cylindrical vector partially coherent field in free space

Analytical nonparaxial propagation formula for the cross-spectral density matrix of a cylindrical vector partially coherent beam in free space is derived based on the generalized Raleigh-Sommerfeld diffraction integrals. Statistical properties, such as average intensity, degree of polarization and degree of coherence, of a nonparaxial cylindrical vector partially coherent field are illustrated numerically, and compared with that of a paraxial cylindrical vector partially coherent beam. It is found that the statistical properties of a nonparaxial cylindrical vector partially coherent field are much different from that of a paraxial cylindrical vector partially coherent beam, and are closely determined by the initial beam width and correlation coefficients. Our results will be useful for modulating the properties of a nonparaxial cylindrical vector partially coherent field.

Optical forces on small particles from partially coherent light

We put forward a theory on the optical force exerted upon a dipolar particle by a stationary and ergodic partially coherent light field. We show through a rigorous analysis that the ensemble averaged electromagnetic force is given in terms of a partial gradient of the space-variable diagonal elements of the coherence tensor. Further, by following this result we characterize the conservative and nonconservative components of this force. In addition, we establish the propagation law for the optical force in terms of the coherence function of light at a diffraction plane. This permits us to evaluate the effect of the degree of coherence on the force components by using the archetypical configuration of Young's two-aperture diffraction pattern, so often employed to characterize coherence of waves.

Modulation of spectral intensity, polarization and coherence of a stochastic electromagnetic beam

Analytical formula for the cross-spectral density matrix of a stochastic electromagnetic Gaussian Schell-model (EGSM) beam truncated by a circular phase aperture propagating in free space is derived with the help of a tensor method, which provides a reliable and fast way for studying the propagation and transformation of a truncated EGSM beam. Statistics properties, such as the spectral intensity, the degree of coherence, the degree of polarization and the polarization ellipse of a truncated EGSM beam in free space are studied numerically. The propagation factor of a truncated EGSM beam is also analyzed. Our numerical results show that we can modulate the spectral intensity, the polarization, the coherence and the propagation factor of an EGSM beam by a circular phase aperture. It is found that the phase aperture can be used to shape the beam profile of an EGSM beam and generate electromagnetic partially coherent dark hollow or flat-topped beam, which is useful in some applications, such as optical trapping, material processing, free-space optical communications.

Statistics properties of a cylindrical vector partially coherent beam

Cylindrical vector partially coherent beam is introduced as a natural extension of cylindrical vector coherent beam based on the unified theory of coherence and polarization. Analytical propagation formula for the cross-spectral density matrix of a cylindrical vector partially coherent beam passing through paraxial ABCD optical system is derived based on the generalized Collins integral formula. As an application example, the statistics properties, such as the average intensity, spreading and the degree of polarization, of a cylindrical vector partially coherent beam propagating in free space are studied in detail. It is found that the statistics properties of a cylindrical vector partially coherent beam are much different from a cylindrical vector coherent beam. Our results may find applications in connection with laser beam shaping and optical trapping.

M2-factor of a stochastic electromagnetic beam in a Gaussian cavity

With the help of a tensor method, an explicit expression for the M2-factor of a stochastic electromagnetic Gaussian Schell-model (EGSM) beam in a Gaussian cavity is derived. Evolution properties of the M2-factor of an EGSM beam in a Gaussian cavity are studied numerically in detail. It is found that the behavior of the M2-factor of an EGSM beam in a Gaussian cavity is determined by the statistical properties of the source beam and the parameters of the cavity. Thermal lens effect induced changes of the M2-factor of an EGSM beam in a Gaussian cavity is also investigated. Our results will be useful in many applications, such as free-space optical communications, laser radar system, optical trapping and optical imaging, where stochastic electromagnetic beams are required.

Second-order statistics of a twisted gaussian Schell-model beam in turbulent atmosphere

We present a detailed investigation of the second-order statistics of a twisted gaussian Schell-model (TGSM) beam propagating in turbulent atmosphere. Based on the extended Huygens-Fresnel integral, analytical expressions for the second-order moments of the Wigner distribution function of a TGSM beam in turbulent atmosphere are derived. Evolution properties of the second-order statistics, such as the propagation factor, the effective radius of curvature (ERC) and the Rayleigh range, of a TGSM beam in turbulent atmosphere are explored in detail. Our results show that a TGSM beam is less affected by the turbulence than a GSM beam without twist phase. In turbulent atmosphere the Rayleigh range doesn't equal to the distance where the ERC takes a minimum value, which is much different from the result in free space. The second-order statistics are closely determined by the parameters of the turbulent atmosphere and the initial beam parameters. Our results will be useful in long-distance free-space optical communications.

Propagation factor of a stochastic electromagnetic Gaussian Schell-model beam

Analytical formula is derived for the propagation factor (known asM(2)-factor) of a stochastic electromagnetic Gaussian Schell-model (EGSM) beam in free space and in turbulent atmosphere. In free space, the M(2)-factor of an EGSM beam is mainly determined by its initial degree of polarization, r.m.s. widths of the spectral densities and correlation coefficients, and its value remains invariant on propagation. In turbulent atmosphere, the M(2)-factor of an EGSM beam is also determined by the parameters of the turbulent atmosphere, and its value increases on propagation. The relative M(2)-factor of an EGSM beam with lower correlation factors, larger r.m.s. widths of the spectral densities and longer wavelength is less affected by the atmospheric turbulence. Under suitable conditions, an EGSM beam is less affected by the atmospheric turbulence than a scalar GSM beam (i.e. fully polarized GSM beam). Our results will be useful in long-distance free-space optical communications.