Review on partially coherent vortex beams

Abstract

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

Keywords

partially coherent vortex beam / phase singularity / correlation singularity / topological charge (TC) / coherence length / correlation function

1 Introduction

In 1974, Nye and Berry introduced a new kind fundamental property of light field, namely wave front dislocation (or phase defect) [1], which includes edge dislocation, screw dislocation and mixed edge-screw dislocation. The screw dislocation, the most common wave front dislocation will introduce a phase singularity at the center of the beam with zero amplitude and indefinite phase, and phase singularity of the wave function $\psi$ appears at the point where its modulus vanishes, i.e., when $Re(\psi )=Im(\psi )=\mathrm{0}$. Following this pioneering work, these dislocations, or phase singularities, become a separate area of investigation in modern optics, named singular optics [2]. The main object of singular optics is optical vortex (screw dislocation). As pointed out by Allen in 1992, a vortex beam with helical phase front, described by a phase term$\exp (il\theta )$, carries an orbital angular momentum (OAM) of $l\hslash$ per photon, where l is the topological charge (TC), $\theta$ is the azimuthal angle, and $\hslash$ is reduced Planck constant. Because this TC is quantized, it remains constant under small deformations. As pointed out by Gbur and Tyson, the TC of the vortex beams propagating through atmospheric turbulence is a robust quantity that could be used as an information carrier in optical communications [3]. In addition, because of conservation of TC of optical vortices, an unusual optical phenomenon (namely, existence of bound optical states in the radiation continuum) has been found in the radiation continuum [4]. Increasing attention has been paid to optical vortices (e.g., Gaussian-like beam [5], Bessel beam [6–8], anomalous vortex beam [9] and perfect vortex beam [10,11]) due to their important applications in free-space optical communications [12,13], optical manipulation [14–18], quantum information processing [19–21], optical measurements [22,23], super-resolution imaging [24,25] and so on. To introduce TC or OAM to light beam, several methods have been exploited [26–37]. Quantitative measurement of TC or OAM state of vortex beams is important in exploiting their applications. Many methods, such as Shack Hartmann wave front sensor [38], interference [39,40], diffraction [41–43], scattering [44], Fourier patterns of the intensity [45], mode transformation [46], OAM density [47], rotational Doppler effect [48], have been proposed for this purpose. Above mentioned literatures are limited to vortex beams with integral TC, and in fact, the value of TC can be non-integral (i.e., fractional TC). The vortex beam with fractional TC (i.e., fractional vortex beam), which possesses a radial opening in the annular intensity ring, was first observed by Basistiy et al. [49]. Later, many studies on fractional vortex beams have been reported [50–63].

In addition to intensity and phase, the beam itself has another important controllable parameter called coherence [64], which includes temporal coherence and spatial coherence. Temporal coherence is closely related to the monochromatism of a beam, while spatial coherence is closely related to the directivity of a beam. In this paper, we restrict our focus on spatial coherence. Laser beam is known for its high spatial coherence and is usually treated as a fully coherent beam. It is generally believed that with the decrease of spatial coherence, the divergence of the beam increases, which is not conducive to practical applications. Since 1970s, numerous efforts have been paid to the coherence of light beam [65–72]. Light beam with low spatial coherence, so-called partially coherent beam, exhibits some unique optical properties and superiority in many applications. For example, decreasing the spatial coherence of a light beam can increase the signal-noise ratio and reduce the bit-error rate in free-space optical communications [73,74]. Partially coherent beams can effectively overcome speckle in laser nuclear fusion [75], and such beam can be used to reduce noise in photograph [76] and to realize classic ghost interference [77]. In addition, partially coherent beams have advantages over coherent beams in particle trapping [78], atom cooling [79], second-harmonic generation [80,81], optical scattering [82,83] and laser scanning [84].

The research scope of singular optics has been extended from fully coherent vortex beam to partially coherent vortex beam since Gori et al. proposed the partially coherent sources with helicoidal modes [85]. When a light beam is partially coherent, its random fluctuations tend to move its singular points, leaving no zeros in the average intensity [86]. It was demonstrated that the zeros disappear as the coherence of the light beam decreases [87,88]. However, partially coherent vortex beams possess singularities in some hidden form, a good candidate for such hidden vortices is the singularities of two-point correlation functions described, so-called coherence vortices [89]. As pointed out by Gbur and Visser, coherence vortices is a generic feature of partially coherent light field [89]. Based on these basic studies, partially coherent vortex beams with conventional correlation functions (i.e., Gaussian correlated Schell-model functions) attract more and more attention [90–105].

Since Gori’s pioneering work about establishment of the sufficient condition for devising a genuine correlation function of a scalar partially coherent beam [106] and a genuine cross-spectral density matrix of an electromagnetic stochastic beam [107], various partially coherent beams with nonconventional correlation functions (i.e., non-Gaussian correlated Schell-model functions) were introduced [108–111]. The research scopes of partially coherent vortex beams have been extended to partially coherent vortex beams with nonconventional correlation functions [112–114] and vector cases [115–117], respectively. Modulating spatial coherence, topological charge and degree of polarization of a partially coherent vortex beam provides a novel way for shaping the focused intensity distribution (i.e., dark-hollow, flat-topped or Gaussian beam spots) [94,115,117], which is useful for optical trapping [118]. Partially coherent vortex beam (i.e., Gaussian–Schell model vortex beam) has appreciably smaller scintillation than a partially coherent beam without vortex phase (i.e., Gaussian–Schell model beam) [119], which will be useful in free-space optical communications. In addition, the complex degree of coherence of a partially coherent vortex beam exhibits self-reconstruction ability on propagation [102], which can be used for measuring the topological charge of partially coherent vortex beam and information encryption and decryption. Recently, a new kind of partially coherent vortex (PCV) beam with fractional topological charge [i.e., partially coherent fractional vortex (PCFV) beam] was introduced [120]. The opening gap of the intensity pattern and the rotation property of the PCFV beam spot disappear gradually on propagation, while the cross-spectral density (CSD) distribution becomes more symmetric and more recognizable with the decrease of spatial coherence [120]. Due to their interesting properties and potential applications, generation and measurement of partially coherent vortex beams have been explored extensively [113,115,121–127]. In this paper, we introduce recent developments on partially coherent vortex beams including basic concepts, theoretical models, generation and propagation.

2 Basic concept of singularity optics

The research subject of singular optics are optical vortex, wave front dislocation and wave front topological structure in the neighborhood of phase singularity. The research scope of singular optics has been extended from fully coherent light to partially coherent light, from scalar light wave field to vector light wave field, from theoretical research and computational simulation to experimental and applied researches. In this section, we give a brief introduction to the basic concept of singularity optics.

2.1 Phase singularities

According to the electromagnetic field theory, the complex electric field of any point $r$ can be expressed as

(1)$E(\mathit{r})=E_{\mathrm{R}}(\mathit{r})\mathrm{+i}{E}_{\mathrm{I}}(\mathit{r})=A(\mathit{r})\exp [i\phi (\mathit{r})],$

where $E_{\mathrm{R}}(\mathit{r})$ and $E_{\mathrm{I}}(\mathit{r})$ are real and imaginary parts of complex electric field, $A(\mathit{r})$ is the amplitude, $\phi (\mathit{r})\mathrm{=arctan}[E_{\mathrm{R}}(\mathit{r})/E_{\mathrm{I}}(\mathit{r})]$ denotes the phase which is between-p and p. Consider that E(r) in the space-time domain is a single value function, and the distribution of E(r) displays smooth transitions. Now, as it goes around a loop C, the magnitude of the change in phase $\phi$ must be integral multiple of 2p, namely 2pl (l is an integer). From a mathematical point of view, let’s assume that l (l≠0) is fixed, by compressing loop C indefinitely into a small enough area, one sees that the change rate of the phase $\phi$ is approaching infinity, that is, the loop surrounds around a singularity. Based on the smoothness of the field distribution function, the position of the phase singularity is the place where the electric field intensity is zero, i.e.,$E_{\mathrm{R}}(\mathit{r})=E_{\mathrm{I}}(\mathit{r})=\mathrm{0}$, and the phase $\phi (\mathit{r})\mathrm{=arctan}[E_{\mathrm{R}}(\mathit{r})/E_{\mathrm{I}}(\mathit{r})]$ is undefined.

2.2 Topological charge

According to the theory of crystal defects in materials science, we know that a single Burgers vector along a dislocation line is always conserved. Similar to the crystal structure, Nye and Berry have given the wave front dislocation (wave front defect) in the light wave field which is around the phase singularity along the dislocation line [1]

(2)${\displaystyle \frac{\lambda }{\mathrm{2}\mathrm{\pi }}}{\displaystyle \underset{C}{\propto }\nabla \phi ·\mathrm{d}r=l}\lambda ,$

where $\lambda$is wavelength, C is the closed curve around the phase singularity in the counterclockwise direction, $\phi$is the phase, l denotes winding number which is also named topological charge [2], the magnitude of l is equal to the number of starting points of the helix plane around the phase singularity. Positive l represents the right hand spiral direction, negative l represents the left hand spiral direction. In Figs. 1(a) and 1(b), the starting points of the helix alleles around the phase singularity are 1 and 3, respectively, and the directions of the helix are all right-handed, so we can figure out that the l which contains the magnitude and the sign are+ 1 and+ 3, respectively. According to above definition, one finds that l in Figs. 1(c) and 1(d) is equal to -1 and -3, respectively.

Fig.1 Wave front of spiral structure. (a) and (b) Wave front of right hand spiral structure with topological charge l = 1 and 3; (c) and (d) Wave front of left hand spiral structure with topological charge l = -1 and -3

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2.3 Wave front dislocation

Wave front dislocation (or phase defect) mainly includes three types [1], namely, screw dislocation, edge dislocation and mixed edge-screw dislocation. The light wave field of the mixed edge-screw dislocation includes both screw dislocation and edge dislocation.

Screw dislocation, is also known as optical vortex (or vortex beam) in the optical field. It is known that a fully coherent beam can be characterized by the electric field. We may write this electric field of a fully coherent optical vortex beam with an arbitrary initial phase$\beta$(normally set to zero) at the source plane in polar representation in terms of an amplitude $A(\mathit{r})$ and vortex phase structure $\exp (\mathrm{i}l\theta )$,

(3)$E_{\mathrm{V}}(\mathit{r},\theta )=A(\mathit{r})\exp (\mathrm{i}l\theta \mathrm{+i}\beta ),$

where l is the TC mentioned above, $\theta$ is the azimuthal angle. By modulating $A(\mathit{r})$, one can obtain different fully coherent vortex beam, such as Gaussian vortex (GV) beam [128], Laguerre-Gaussian (LG) beam [5,129], anomalous vortex (AV) beam [9], high order Bessel-Gaussian (BG) beam [8] and perfect vortex (PV) beam [10]. Compared with plane wave, the main characteristics of optical vortex are helical wave front and phase singularity. At the points of these singularities of optical vortex where the intensity of the wave is zero and the phase is undefined. Take the GV beam as an example, we calculate in Fig. 2 the intensity and phase distributions of the GV beam with different l at the source plane. It shows that the optical vortex is a point defect, and the center is dark, that is, the intensity is zero, which is caused by the phase singularity. In addition, the sign of l is positive if the phase increases along anti-clockwise direction, whereas, negative if the phase increases along clockwise direction. If the phase changes (increases) 2pl, the magnitude of TC is l [130], which can be an arbitrary value, both integral and fractional. All the phase lines intersect at the center of the optical vortex, namely, the center of the optical vortex and any other point are all on the same contour of constant phase. Therefore, the phase at the center of the vortex is undefined. Note that vortex beam with fractional topological charge possesses a radial opening in the annular intensity ring during propagation, which is different from that at the source plane.

Fig.2 Intensity and phase of Gaussian vortex beam with different l at the source plane. (a) and (d) l = 3; (b) and (e) l = -3; (c) and (f) l = 1.5

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Edge dislocation, namely, pure dislocation line, whose prominent feature is that there is a straight dislocation line (incision) in the transverse plane along the propagation direction, and it has a jump by p in the phase between the two sides of the incision. In Cartesian coordinates, the electric field of a beam with edge dislocation at the source plane can be expressed as

(4)$E_{\mathrm{E}}(\mathit{r})=(px-y+d)f(\mathit{r}),$

where p and d are the slope and the off-axis distance of edge dislocation, $f(\mathit{r})$ denotes the background light field distribution. We show the phase distribution of edge dislocation in Fig. 3.

Fig.3 Phase of edge dislocation

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2.4 Correlation singularities

It is well known that light fields which are partially coherent typically do not possess regions of zero intensity and hence do not possess any obvious phase singularities [87]. It is of interest to ask whether or not such fields possess singularities in some ‘hidden’ form. In 2003, Bogatyryova et al. [131] exhibited the vortex nature of partially coherent singular beam with a separable phase and studied the phase singularities of the spectral degree of coherence. Figure 4 shows the modulus and contours of constant phase of the spectral degree of coherence (correlation function) of a partially coherent beam with vortex phase superposed by LG₀₁ and LG₁₁ model beams. We can find from Fig. 4 when the modulus of the correlation function is 0, the phase will undergo a jump at that place, and is undefined as the correlation singularity. Later, Gbur et al. [86,89] further confirmed the existence of this hidden singularity (or hidden vortex), a good candidate for such hidden vortices is the singularities of two-point correlation functions described, so-called coherence vortices or correlation singularities [89]. Such vortices are pairs of points (${\mathit{r}}_{\mathrm{1}}\mathrm{and}{\mathit{r}}_{\mathrm{2}}$) at which the spectral degree of coherence of the field vanishes, i.e., where

(5)$u({\mathit{r}}_{\mathrm{1}},{\mathit{r}}_{\mathrm{2}})\equiv {\displaystyle \frac{W({\mathit{r}}_{\mathrm{1}},{\mathit{r}}_{\mathrm{2}})}{\sqrt{S({\mathit{r}}_{\mathrm{1}})S({\mathit{r}}_{\mathrm{2}})}}}=\mathrm{0},$

or, equivalently, where

(6)$\mathrm{Re}[W({\mathit{r}}_{\mathrm{1}},{\mathit{r}}_{\mathrm{2}})]=\mathrm{0},$

(7)$\mathrm{Im}[W({\mathit{r}}_{\mathrm{1}},{\mathit{r}}_{\mathrm{2}})]=\mathrm{0}.$

Here $W({\mathit{r}}_{\mathrm{1}},{\mathit{r}}_{\mathrm{2}})$ denotes the mutual coherence function (cross-spectral density [132]) which is used to characterize the statistic properties of the partially coherent beam, but at which the spectral density (often referred to as intensity)$S({\mathit{r}}_i)=W({\mathit{r}}_i,{\mathit{r}}_i)$ (i = 1,2) of the field is nonzero. Re and Im denote taking the real and imaginary parts, respectively.

In 2004, Palacios et al. verified the existence of a robust correlation singularity in experiment [133]. Figure 5 shows the intensity and cross correlation functions of a partially coherent vortex beam with l = 1 in the far-field plane with different coherence lengths. One sees that the dark vortex core disappears as the coherence length decreases. However, a ring dislocation appears in the cross-correlation function, and with the decrease of coherence length, the radial size of ring dislocation increases and becomes more recognizable. It is similar to Fig. 3, the place where the correlation function is 0 (i.e., dark ring dislocation), the phase will undergo a p phase jump across the boundary.

Therefore, a partially coherent vortex beams possess singularities with these two hidden forms, i.e., coherence vortices (or correlation vortices) and ring dislocations.

Fig.4 (a) Modulus and (b) contours of constant phase of the spectral degree of coherence (correlation function) of a partially coherent beam with vortex phase superposed by LG01 and LG11 model beams [131]

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Fig.5 Intensity (a−c) and cross correlation functions (d−f) of a partially coherent vortex beam with l = 1 in the far-field plane with different coherence lengths [133]

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3 Theoretical models of partially coherent vortex beams

Considering the polarization characteristics, we divide partially coherent vortex beams into scalar partially coherent vortex beam and vector partially coherent vortex beam (including vector partially coherent vortex beams with uniform state of polarization and nonuniform state of polarization). Considering the correlation function, we divide partially coherent vortex beams into partially coherent vortex beam with conventional correlation function (i.e., Gaussian-correlated Schell model correlation) and partially coherent vortex beam with nonconventional correlation function (i.e., non-Gaussian-correlated Schell model correlation). Considering the value of topological charge, we divide partially coherent vortex beams into partially coherent integral vortex (PCIV) beam and partially coherent fractional vortex (PCFV) beam.

3.1 Theoretical models of a scalar partially coherent vortex beam with conventional correlation function

Unlike the case of fully coherent optical vortex beam (optical vortex) described in the previous section (Eq. (3)), the initial phase $\beta$ of partially coherent vortex beam are random, it is generally accepted that a partially coherent beam can be characterized by its statistic properties (i.e., mutual coherence function (MCF) in the space-time domain or the CSD in the space-frequency domain) [132]. The CSD of a partially coherent vortex beam at the source plane is defined as a two-point correlation function, i.e.,

(8)$W({\mathit{r}}_{\mathrm{1}},r_{\mathrm{2}})=\langle E_{\mathrm{V}}({\mathit{r}}_{\mathrm{1}})E_{\mathrm{V}}^{*}({\mathit{r}}_{\mathrm{2}})\rangle ,$

where ${\mathit{r}}_{\mathrm{1}}\mathrm{and}{\mathit{r}}_{\mathrm{2}}$ are the position vectors at the source plane, the angular brackets denote an ensemble average and the asterisk denotes the complex conjugate.

On substituting from Eq. (3) into Eq. (8), we can obtain the theoretical model of a partially coherent vortex beam

(9)$W({\mathit{r}}_{\mathrm{1}},{\mathit{r}}_{\mathrm{2}})=A({\mathit{r}}_{\mathrm{1}})A({\mathit{r}}_{\mathrm{2}})g({\mathit{r}}_{\mathrm{1}}-{\mathit{r}}_{\mathrm{2}})\exp [il({\theta }_{\mathrm{1}}-{\theta }_{\mathrm{2}})],$

where $g({\mathit{r}}_{\mathrm{1}}-{\mathit{r}}_{\mathrm{2}})$ denotes the correlation function [132] between two points ${\mathit{r}}_{\mathrm{1}}\mathrm{and}{\mathit{r}}_{\mathrm{2}}$. It is generally accepted that for the Carter-Wolf source [134], the correlation function satisfying the Gaussian-correlated Schell-model function and can be expressed as follows

(10)$g({\mathit{r}}_{\mathrm{1}}-{\mathit{r}}_{\mathrm{2}})\mathrm{=exp}\left[-{\displaystyle \frac{{({\mathit{r}}_{\mathrm{1}}-{\mathit{r}}_{\mathrm{2}})}^{\mathrm{2}}}{\mathrm{2}{\sigma }_{\mathrm{g}}^{\mathrm{2}}}}\right],$

where ${\sigma }_{\mathrm{g}}^{}$ denotes the initial coherence length. If we set ${\sigma }_{\mathrm{g}}^{}\to \mathrm{0}$, the beam source reduces to an incoherent beam, If we set ${\sigma }_{\mathrm{g}}^{}\to \infty$, the beam source becomes to a fully coherent beam. By modulating$A(\mathit{r})$, one can obtain different partially coherent vortex beams with Gaussian correlated Schell-model function, such as Gaussian Schell-model vortex beam, Laguerre Gaussian Schell-model, Bessel Gaussian Schell-model beam.

If we set [128]

(11)$A(\mathit{r})=\exp \left(-{\displaystyle \frac{{\mathit{r}}^{\mathrm{2}}}{w_{\mathrm{0}}^{\mathrm{2}}}}\right),$

where $w_{\mathrm{0}}$ is the beam waist. By substituting Eqs. (10) and (11) into Eq. (9), the CSD function of the basic Gaussian Schell-model vortex (GSMV) beam takes the form

(12)$W_{GSMV}({\mathit{r}}_{\mathrm{1}},{\mathit{r}}_{\mathrm{2}})=\exp \left[-{\displaystyle \frac{{\mathit{r}}_{\mathrm{1}}^{\mathrm{2}}+{\mathit{r}}_{\mathrm{2}}^{\mathrm{2}}}{w_{\mathrm{0}}^{\mathrm{2}}}}-{\displaystyle \frac{{({\mathit{r}}_{\mathrm{1}}-{\mathit{r}}_{\mathrm{2}})}^{\mathrm{2}}}{\mathrm{2}{\delta }_g^{\mathrm{2}}}}\right]\exp [\mathrm{i}l({\theta }_{\mathrm{1}}-{\theta }_{\mathrm{2}})].$

If we set [5,129]

(13)$A(\mathit{r})={\left({\displaystyle \frac{\sqrt{\mathrm{2}}\mathit{r}}{w_{\mathrm{0}}}}\right)}^l{L}_p^l\left(-{\displaystyle \frac{\mathrm{2}{\mathit{r}}^{\mathrm{2}}}{w_{\mathrm{0}}^{\mathrm{2}}}}\right)\exp \left(-{\displaystyle \frac{{\mathit{r}}^{\mathrm{2}}}{w_{\mathrm{0}}^{\mathrm{2}}}}\right),$

where $L_p^l$ denotes the Laguerre polynomial with mode orders p and l. By substituting Eqs. (10) and (13) into Eq. (7), the CSD function of the Laguerre Gaussian Schell-model (LGSM) beam can be expressed as

$W_{\mathrm{L}\mathrm{G}\mathrm{S}\mathrm{M}}({\mathit{r}}_{\mathrm{1}},{\mathit{r}}_{\mathrm{2}})={\left({\displaystyle \frac{\mathrm{2}{\mathit{r}}_{\mathrm{1}}{\mathit{r}}_{\mathrm{2}}}{w_{\mathrm{0}}^{\mathrm{2}}}}\right)}^l{L}_p^l\left(-{\displaystyle \frac{\mathrm{2}{\mathit{r}}_{\mathrm{1}}^{\mathrm{2}}}{w_{\mathrm{0}}^{\mathrm{2}}}}\right)L_p^l\left(-{\displaystyle \frac{\mathrm{2}{\mathit{r}}_{\mathrm{2}}^{\mathrm{2}}}{w_{\mathrm{0}}^{\mathrm{2}}}}\right)$

(14)$\times \exp \left[-{\displaystyle \frac{{\mathit{r}}_{\mathrm{1}}^{\mathrm{2}}+{\mathit{r}}_{\mathrm{2}}^{\mathrm{2}}}{w_{\mathrm{0}}^{\mathrm{2}}}}-{\displaystyle \frac{{({\mathit{r}}_{\mathrm{1}}-{\mathit{r}}_{\mathrm{2}})}^{\mathrm{2}}}{\mathrm{2}{\sigma }_{\mathrm{g}}^{\mathrm{2}}}}\right]\exp [\mathrm{i}l({\theta }_{\mathrm{1}}-{\theta }_{\mathrm{2}})].$

If we set [9]

(15)$A(\mathit{r})={\left({\displaystyle \frac{\mathit{r}}{w_{\mathrm{0}}}}\right)}^{\mathrm{2}n+\left|l\right|}\exp \left(-{\displaystyle \frac{{\mathit{r}}^{\mathrm{2}}}{w_{\mathrm{0}}^{\mathrm{2}}}}\right),$

where n is the beam order of the anomalous vortex beam. By substituting Eqs. (10) and (15) into Eq. (9), the CSD function of the anomalous Gaussian Schell-model vortex(AGSMV) beam can be expressed as

(16)$W_{\mathrm{A}\mathrm{G}\mathrm{S}\mathrm{M}\mathrm{V}}({\mathit{r}}_{\mathrm{1}},{\mathit{r}}_{\mathrm{2}})={\left({\displaystyle \frac{{\mathit{r}}_{\mathrm{1}}{\mathit{r}}_{\mathrm{2}}}{w_{\mathrm{0}}^{\mathrm{2}}}}\right)}^{\mathrm{2}n+\left|l\right|}\exp \left[-{\displaystyle \frac{{\mathit{r}}_{\mathrm{1}}^{\mathrm{2}}+{\mathit{r}}_{\mathrm{2}}^{\mathrm{2}}}{w_{\mathrm{0}}^{\mathrm{2}}}}-{\displaystyle \frac{{({\mathit{r}}_{\mathrm{1}}-{\mathit{r}}_{\mathrm{2}})}^{\mathrm{2}}}{\mathrm{2}{\sigma }_{\mathrm{g}}^{\mathrm{2}}}}\right]\exp [\mathrm{i}l({\theta }_{\mathrm{1}}-{\theta }_{\mathrm{2}})].$

If we set [8]

(17)$A(\mathit{r})=J_l({\beta }_{\mathrm{0}}\mathit{r})\exp \left(-{\displaystyle \frac{{\mathit{r}}^{\mathrm{2}}}{w_{\mathrm{0}}^{\mathrm{2}}}}\right),$

where ${\beta }_{\mathrm{0}}$ is the radial frequency, $J_l$ is the Bessel function of first kind with order $l$. When $l\ge \mathrm{1}$, $J_l$ represents the high order Bessel function. By substituting Eqs. (10) and (17) into Eq. (9), one can obtain the CSD function of the Bessel Gaussian Schell-model (BGSM) beam as follows

(18)$W_{\mathrm{B}\mathrm{G}\mathrm{S}\mathrm{M}}({\mathit{r}}_{\mathrm{1}},{\mathit{r}}_{\mathrm{2}})=J_l({\beta }_{\mathrm{0}}{\mathit{r}}_{\mathrm{1}})J_l({\beta }_{\mathrm{0}}{\mathit{r}}_2)\exp \left[-{\displaystyle \frac{{\mathit{r}}_{\mathrm{1}}^{\mathrm{2}}+{\mathit{r}}_{\mathrm{2}}^{\mathrm{2}}}{w_{\mathrm{0}}^{\mathrm{2}}}}-{\displaystyle \frac{{({\mathit{r}}_{\mathrm{1}}-{\mathit{r}}_{\mathrm{2}})}^{\mathrm{2}}}{\mathrm{2}{\sigma }_{\mathrm{g}}^{\mathrm{2}}}}\right]\exp [\mathrm{i}l({\theta }_{\mathrm{1}}-{\theta }_{\mathrm{2}})].$

3.2 Theoretical models of a scalar partially coherent vortex beam with nonconventional correlation function

According to the Gori’s pioneering work [106], it is found that the correlation function of a partially coherent beam is not limited to the Gaussian correlated Schell-model function. In fact, partially coherent beams with conventional correlation function can be proposed and generated [135]. Chen et al. have introduced and generated one kind of partially coherent vortex beam with nonconventional correlation function named Laguerre-Gaussian correlated Schell-model vortex (LGCSMV) beam [113], whose CSD function is expressed as

(19)$W_{\mathrm{L}\mathrm{G}\mathrm{C}\mathrm{S}\mathrm{M}\mathrm{V}}(r_{\mathrm{1}},r_{\mathrm{2}})=\exp \left[-{\displaystyle \frac{r_{\mathrm{1}}^{\mathrm{2}}+r_{\mathrm{2}}^{\mathrm{2}}}{w_{\mathrm{0}}^{\mathrm{2}}}}-{\displaystyle \frac{{(r_{\mathrm{1}}-r_{\mathrm{2}})}^{\mathrm{2}}}{\mathrm{2}{\sigma }_g^{\mathrm{2}}}}\right]L_p^{\mathrm{0}}\left[{\displaystyle \frac{{(r_{\mathrm{1}}-r_{\mathrm{2}})}^{\mathrm{2}}}{\mathrm{2}{\sigma }_{\mathrm{g}}^{\mathrm{2}}}}\right]\exp [\mathrm{i}l({\theta }_{\mathrm{1}}-{\theta }_{\mathrm{2}})],$

where $L_p^{\mathrm{0}}$ represents the Laguerre polynomial with mode orders p and l = 0. We will use the experimental setup in Fig. 8 below to generate a LGCSMV beam represented by Eq. (19). If we set p = 0, Eq. (19) reduces to the GSMV beam. Due to the vortex phase, the LGCSMV beam exhibits unique propagation properties which are much different from those of the GSMV beam.

3.3 Theoretical models of a vector partially coherent vortex beam

The research scope of partially coherent vortex beam has been extended from scalar light field to vector light field. Unlike a scalar partially coherent beam, it is well known that a vector partially coherent beam can be characterized by the CSD matrix.

Electromagnetic Gaussian Schell-model vortex (EGSMV) beam is a typical kind of vector partially coherent vortex beam with uniform state of polarization, whose CSD matrix in the source plane is given as [115]

(20)${\overleftrightarrow{W}}_{\mathrm{EGSMV}}({\mathit{r}}_{\mathrm{1}},{\mathit{r}}_{\mathrm{2}})=\left(\begin{array}{ll}{W}_{xx}({\mathit{r}}_{\mathrm{1}},{\mathit{r}}_{\mathrm{2}})& W_{xy}({\mathit{r}}_{\mathrm{1}},{\mathit{r}}_{\mathrm{2}})\\ W_{yx}({\mathit{r}}_{\mathrm{1}},{\mathit{r}}_{\mathrm{2}})& W_{yy}({\mathit{r}}_{\mathrm{1}},{\mathit{r}}_{\mathrm{2}})\end{array}\right),$

where $W_{\alpha \beta }({\mathit{r}}_{\mathrm{1}},{\mathit{r}}_{\mathrm{2}})=\langle E_{\alpha }^{*}({\mathit{r}}_{\mathrm{1}})E_{\beta }({\mathit{r}}_{\mathrm{2}})\rangle ,(\alpha ,\beta =x,y)$. The elements of the CSD matrix are expressed as follows:

(21)$W_{\alpha \beta }({\mathit{r}}_{\mathrm{1}},{\mathit{r}}_{\mathrm{2}})=A_{\alpha }{A}_{\beta }{B}_{\alpha \beta }\exp \left[-{\displaystyle \frac{{\mathit{r}}_{\mathrm{1}}^{\mathrm{2}}}{\mathrm{4}{\delta }_{\alpha }^{\mathrm{2}}}}-{\displaystyle \frac{{\mathit{r}}_{\mathrm{2}}^{\mathrm{2}}}{\mathrm{4}{\delta }_{\beta }^{\mathrm{2}}}}-{\displaystyle \frac{{({\mathit{r}}_{\mathrm{1}}-{\mathit{r}}_{\mathrm{2}})}^{\mathrm{2}}}{\mathrm{2}{\sigma }_{\alpha \beta }^{\mathrm{2}}}}\right]\exp [\mathrm{i}l({\theta }_{\mathrm{1}}-{\theta }_{\mathrm{2}})],$

where $A_x$ and $A_y$ are the amplitudes of $x$ and $y$ components of the electric field, respectively. $B_{xx}=B_{yy}=\mathrm{1}$,$B_{xy}=\left|B_{xy}\right|\exp (i{\varphi }_{xy})$ is the complex correlation coefficient between the $x$ and $y$ components of the electric field with ${\varphi }_{xy}$ being the phase difference between the $x$ and $y$ components. ${\delta }_i^{}$ denotes the r.m.s width of the intensity distribution along the $i$ direction, ${\sigma }_{xx}$, ${\sigma }_{yy}$ and ${\sigma }_{xy}$ are the r.m.s widths of autocorrelation function of the x component of the electric field, of the y component of the electric field and of the mutual correlation function of $x$ and y components of the electric field, respectively.

Partially coherent radially polarized vortex beam is a typical vector partially coherent vortex beam with nonuniform state of polarization. The elements of the partially coherent radially polarized vortex beam are expressed as [117]

(22)$W_{xx}({\mathit{r}}_{\mathrm{1}},{\mathit{r}}_{\mathrm{2}})={\displaystyle \frac{r_{\mathrm{1}}{r}_{\mathrm{2}}\cos {\theta }_{\mathrm{1}}\cos {\theta }_{\mathrm{2}}}{w_{\mathrm{0}}^{\mathrm{2}}}}T,$

(23)$W_{xy}({\mathit{r}}_{\mathrm{1}},{\mathit{r}}_{\mathrm{2}})={\displaystyle \frac{r_{\mathrm{1}}{r}_{\mathrm{2}}\cos {\theta }_{\mathrm{1}}\sin {\theta }_{\mathrm{2}}}{w_{\mathrm{0}}^{\mathrm{2}}}}T,$

(24)$W_{yx}({\mathit{r}}_{\mathrm{1}},{\mathit{r}}_{\mathrm{2}})=W_{xy}^{*}({\mathit{r}}_{\mathrm{1}},{\mathit{r}}_{\mathrm{2}}),$

(25)$W_{yy}({\mathit{r}}_{\mathrm{1}},{\mathit{r}}_{\mathrm{2}})={\displaystyle \frac{r_{\mathrm{1}}{r}_{\mathrm{2}}\sin {\theta }_{\mathrm{1}}\sin {\theta }_{\mathrm{2}}}{w_{\mathrm{0}}^{\mathrm{2}}}}T,$

where

(26)$T=\exp \left(-{\displaystyle \frac{r_{\mathrm{1}}^{\mathrm{2}}+r_{\mathrm{2}}^{\mathrm{2}}}{w_{\mathrm{0}}^{\mathrm{2}}}}\right)exp\left[-{\displaystyle \frac{{({\mathit{r}}_{\mathrm{1}}-{\mathit{r}}_{\mathrm{2}})}^{\mathrm{2}}}{\mathrm{2}{\sigma }_{\mathrm{g}}^{\mathrm{2}}}}\right]\exp [\mathrm{i}l({\theta }_{\mathrm{1}}-{\theta }_{\mathrm{2}})].$

3.4 Theoretical models of a partially coherent fractional vortex beam

Above theoretical models are limited to partially coherent vortex beam with integral TC, and in fact, the value of TC can be non-integral (i.e., fractional TC), namely, l can be an arbitrary value, both integral and fractional. By choosing a fractional value of l, above equations represents the CSD functions of corresponding partially coherent fractional vortex beams. We have introduced partially coherent fractional vortex beam as a natural extension of coherent fractional vortex beam in Ref. [120], while it is hard to derive analytical propagation formula for a partially coherent fractional vortex beam due to fractional value of l, we have to resort to numerical integration.

4 Generation of partially coherent vortex beams

The conventional way of generating a partially coherent vortex beam is to produce a partially coherent source first, and then we add the vortex phase with the help of spiral phase plate (SPP) [28], spatial light modulator (SLM) [30] or digital micromirror device (DMD) [136] to a partially coherent source to generate a partially coherent vortex beam. It was shown in [137] that a partially coherent source can be produced starting with a spatial incoherent source and using a collimating lens and an amplitude filter. For creating the spatial incoherent source, people use, for example, the rotating ground glass plates (RGGD) [94], the SLM [138], and the DMD [136] to produce the incoherent beam. This paper mainly introduces the method for generating a partially coherent vortex beam by using a RGGD.

4.1 Generation of a scalar partially coherent vortex beam with conventional correlation function

Figure 6 shows the experimental setup for generating a Gaussian Schell-model vortex (GSMV) beam [94]. A light beam from a He-Ne laser first propagates through the neutral density filter (NDF). The transmitted beam is focused by a thin lens L₁ and then illuminates a rotating ground-glass disk (RGGD), producing a partially coherent beam with Gaussian statistics. The thin lens L₂ is used to collimate the transmitted light, and the Gaussian amplitude filter (GAF) is used to transform the intensity of the transmitted light into a Gaussian profile. The transmitted beam behind the GAF is a GSM beam. The transverse beam width of the GSM beam is determined by the transmission function of the GAF. The transverse coherence width of the GSM beam is determined by the roughness of the RGGD and the focused beam spot size on the RGGD together. In our experiment, the roughness of the RGGD is fixed. We mainly modulate the transverse coherence width of the GSM beam by varying the focused beam spot size on the RGGD. After passing through a spiral phase plate (SPP) with topological charge l = 1 located just behind the GAF, the GSM beam becomes a GSMV beam with l= 1.

Fig.6 Experimental setup for generating a Gaussian Schell-model vortex beam [94]. NDF, neutral density filter; RGGD, rotating ground-glass disk; GAF, Gaussian amplitude filter; SPP, spiral phase plate; L1, L2, lenses

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Figure 7(a) shows the experimental setup for generating a Laguerre Gaussian Schell-model (LGSM) beam [139]. In this way, the generation process is similar to that shown in Fig. 6, the only difference is that we used a spatial light modulator (SLM) to load the corresponding fork interference pattern with different topological charge l and radial mode p (i.e., hologram of interference between plane wave and LG beam). Similarly, if we replace above fork interference pattern with the corresponding holograms, we can also obtain the anomalous Gaussian Schell-model vortex (AGSMV) beam and Bessel Gaussian Schell-model (BGSM) beam.

Figure 7(b) shows our experimental setup for generating partially coherent fractional vortex beam [120]. The generation is the same as shown in Fig. 7(a), the biggest difference is that we used a fractional order fork interference pattern to replace an integral order fork interference pattern.

Fig.7 (a) Experimental setup for generating a LGpl beam [139]; (b) experimental setup for generating a partially coherent fractional vortex beam [120]. NDF, neutral density filter; BE, beam expander; RM, reflecting mirror; L1, L2, L3, thin lenses; PC1, personal computers; RGGD, rotating ground-glass disk; GAF, Gaussian amplitude filter; SLM, spatial light modulator; CA, circular aperture; BS, beam splitter; CGH, computer-generated holograms; CCD: charge-coupled device

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4.2 Generation of a scalar partially coherent vortex beam with nonconventional correlation function

In this section, we generate a LGCSMV beam [113] through conversion of a LGCSM beam with the help of a SPP, which means that we have to generate a LGCSM beam first, then to add a vortex phase on a LGCSM beam. The generation is the same as shown above, the biggest difference is that we need to transform the intensity distribution of the light source into a dark hollow beam before it reaches the lens L₁,the rationality of this experimental scheme has been confirmed by literature [140]. The pattern of the phase grating loaded on the SLM is obtained by computing the interference pattern between a plane wave (reference wave) and a dark hollow beam. The phase grating for generating a dark hollow beam with l = 1 is shown as inset in Fig. 8. The circular aperture (CA) put here is used to select out the first order of the beam from the SLM (i.e., a dark hollow beam with l = 1). After passing through a thin lens L₁, the generated dark hollow beam by a SLM illuminates a RGGD, producing an incoherent beam with dark hollow beam profile [140]. The GAF is used to transform generated incoherent dark hollow beam into a LGCSM beam. The SPP is used to load vortex phase to the generated LGCSM beam to produce LGCSMV beam. Similarly, if we change the above interference pattern, we can also obtain the partially coherent vortex beams with different nonconventional correlation functions [135,140].

Fig.8 Experimental setup for generating a Laguerre-Gaussian correlated Schell-model vortex beam [113]. BE, beam expander; SLM, spatial light modulator; CA, circular aperture; RGGD, rotating ground-glass disk; GAF, Gaussian amplitude filter; L1, L2, thin lenses; SPP, spiral phase plate;PC1, personal computers

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4.3 Generation of a vector partially coherent vortex beam

Vector partially coherent vortex beams can be divided into two types, i.e., a vector partially coherent vortex beam with uniform state of polarization and a vector partially coherent vortex beam with nonuniform state of polarization.

Electromagnetic Gaussian Schell model vortex beam is a typical vector partially coherent vortex beam with uniform state of polarization. If the antidiagonal elements of the CSD matrix are zero (i.e.,$B_{xy}=\mathrm{0}$), which means the x and y components of the beam source are uncorrelated, then we can use superposition of two independent of linear polarization beams to generate this kind of beam source as shown in Fig. 9(a). Here we use two different laser source and RGGD to make sure that the x and y components is uncorrelated. If the antidiagonal elements are nonzero (i.e.,$B_{xy}\ne 0$), we can use a Mach-Zehnder inter-ferrometer to generate this kind of beam source as shown in Fig. 9(b). The detailed information can be found in Ref. [115].

Partially coherent radially polarized vortex beam is a typical kind of vector partially coherent vortex beam with nonuniform state of polarization. Figure 10 shows the experiment setup for generating partially coherent radially polarized vortex beam, which is similar to the setup for generating GSMV beam. The only difference is that the addition of a radially polarized convertor (RPC) between GAF and SPP, which is used to convert the generated linearly polarized GSM beam to a radially polarized GSM beam. The detailed information can be found in Ref. [117].

Fig.9 Experimental setup for generating a vector partially coherent vortex beam with uniform state of polarization [115]. (a) Without antidiagonal elements; (b) with antidiagonal elements. NDF1, NDF2, neutral density filters; L1, L2, thin lenses; RGGD1, RGGD2, rotating ground-glass disks; PBS, polarization beam splitter; GAF, Gaussian amplitude filter; SPP, spiral phase plate; HP, half-wave plate; M1, M2, reflecting mirrors

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Fig.10 Experimental setup for generating a partially coherent radially polarized vortex beam [117]. M, reflecting mirror; BE, beam expander; L1, L2, thin lenses; RGGD, rotating ground-glass disk; GAF, Gaussian amplitude filter; RPC, radially polarized convertor; SPP, spiral phase plate

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5 Propagation properties and application of partially coherent vortex beams

Partially coherent vortex beams exhibit unique optical properties on propagation, such as beam shaping, beam rotation and self-reconstruction, which are much different from fully coherent vortex beam and partially coherent beam without vortex phase.

5.1 Beam shaping

Figures 11−13 show the intensity distribution and the corresponding cross line (y = 0) of focused partially coherent vortex beam in the focal plane with different values of the initial coherence length, the topological charge and the initial degree of polarization. It is found that we can shape the beam profile (i.e., dark hollow, flat-topped, and Gaussian beam spot) of the focused partially coherent vortex beam by varying its initial coherence length, topological charge and initial degree of polarization, which is useful for optical trapping [118].

Fig.11 Intensity distribution and the corresponding cross line (y = 0) of focused Gaussian Schell-model vortex beam in the focal plane for different values of initial coherence length ${\sigma }_g$ [94]

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Fig.12 Intensity distribution and the corresponding cross line (y = 0) of a focused partially coherent radially polarized vortex beam in the focal plane for different values of the topological charge m [117]

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Fig.13 Intensity distribution and the corresponding cross line (y = 0) of an electromagnetic Gaussian Schell-model beam without vortex phase and an electromagnetic Gaussian Schell-model vortex beam in the focal plane for different values of the initial degree of polarization [115]

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5.2 Beam rotation

Figure 14 shows the normalized intensity distributions of a partially coherent integral vortex (PCIV) beam and a partially coherent fractional vortex (PCFV) beam focused by a thin lens at several propagation distances z [120]. We infer from Fig. 14 that the intensity pattern of a PCFV beam is much different from that of a PCIV beam. The intensity pattern of a PCFV beam possesses a radial opening in the annular ring encompassing the dark core near the source plane (z = 0.1f), and the opening gap rotates clockwise as z increases (or rotates anticlockwise for negative l, which is not shown here), and up to 90° at the focal plane (z = f)

Fig.14 Normalized intensity distributions of a PCIV beam (l = 1) and a PCFV beam (l = 1.5) focused by a thin lens at several propagation distances [120]

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Figures 15−17 show the intensity distribution I and its components I_x and I_y of a focused partially coherent radially polarized vortex beam with different l (l= 0, 2 and -2) in the x-y plane at several propagation distances [117]. One sees that the beam spots of the I_x, I_y and I of a focused partially coherent radially polarized beam without vortex phase (l = 0) do not rotate on propagation (see Fig. 15). While the beam spots of the I_x, I_y and I of a focused partially coherent radially polarized vortex beam rotate in anti-clockwise with l = 2 (see Fig. 16) and in clockwise with l = -2 (see Fig. 17) on propagation, which means the beam spot of I also rotates on propagation although it is not demonstrated in Figs. 16(a1)−2(e1) and Fig. 17(a1)−3(e1) because the beam spot is of circular symmetry. The rotation of the beam spot is induced by the vortex phase, which imposes angular orbital angular momentum on the beam. Therefore, we can rotate the beam by modulating the topological charge, which is used for optical tweezers. In addition, in Fig. 18, it is found that we can realize beam rotation by using special optical system (i.e., cylindrical optical system) [141], which is used for determining topological charge.

Fig.15 Intensity distribution and its x- and y-components of a focused partially coherent radially polarized vortex beam with l = 0 in the x-y plane at several propagation distances. The green solid curve denotes the cross line (y = 0) [117]

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Fig.16 Intensity distribution and its x- and y-components of a focused partially coherent radially polarized vortex beam with l = 2 in the x-y plane at several propagation distances. The green solid curve denotes the cross line (y = 0) [117]

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Fig.17 Intensity distribution and its x- and y-components of a focused partially coherent radially polarized vortex beam with l = -2 in the x-y plane at several propagation distances. The green solid curve denotes the cross line (y = 0) [117]

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Fig.18 Normalized average intensity distribution of a partially coherent LG0l beam after passing through a couple of cylindrical lenses at different propagation distances [141]

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5.3 Self-reconstruction

In addition to above mentioned intensity properties (i.e., beam shaping and beam rotation) on propagation and the correlation singularity above, a partially coherent vortex beam has another propagation property, i.e., self-reconstruction, which includes self-reconstruction of intensity and self-reconstruction of correlation function.

Figures 19 and 20 show the normalized intensity distribution and modulus of the degree of coherence of a focused partially Laguerre Gaussian (LG_pl) beam with p = 1 and l = 1 obstructed by a sector shaped opaque obstacle with center angle a at several propagation distances. Figure 21 shows the normalized intensity distribution and modulus of the degree of coherence obstructed by a sector shaped opaque obstacle with center angle a = 90° in the focal plane for different values of initial coherence length. One finds from Figs. 19(a1)−19(d1) that a focused partially coherent LG_pl beam exhibits a dark hollow beam profile in the source plane as a = 0°, and with the increase of the propagation distance, the dark hollow beam profile gradually disappears, and finally becomes a Gaussian beam spot in the focal plane, while the intensity does not reveal any information. Fortunately, one sees that the distribution of the degree of coherence exhibits a Gaussian beam spot in the source plane, while it exhibits ring dislocation (i.e., correlation singularity) in the focal plane, and the number of ring dislocation is$N=\mathrm{2}p+\left|l\right|$, which reveals the information of topological charge. In Figs. 19(a2)−(d2), it is seen that although the intensity distribution is partly blocked in the source plane, the intensity distribution gradually self-reconstructs on propagation, and finally becomes a Gaussian beam spot. The distribution of the degree of coherence also exhibits self-reconstruction ability on propagation, although with some distortion, but it is not hard to recognize that the number of dislocation also is$\mathrm{2}p+\left|l\right|$. From Fig. 21, we can see that decreasing the initial coherence length will enhance the self-reconstruction ability and make the beam spot and ring dislocation more circular. Although self-reconstruction of intensity cannot reveal any information, but the self-reconstruction of correlation function can be used for measuring the topological charge of partially coherent vortex beam and information encryption and decryption.

Thus, in contrast with the non-diffracting beams (e.g., Airy or Bessel beams), the partially coherent beam has unique advantages in detecting information. Both of them have the property of self-reconstruction, however, the former only repair the intensity, and the intensity does not reveal all information of the beam source, such as topological charge. Fortunately, a partially coherent light cannot only repair the intensity, but also repair the correlation function which can reveal the information of topological charge.

Fig.19 Normalized intensity distribution of a focused partially coherent LGpl beam with p = 1 and l = 1 obstructed by a sector shaped opaque obstacle with center angle a at several propagation distances [102]

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Fig.20 Modulus of the degree of coherence of a focused partially coherent LGpl beam with p = 1 and l = 1 obstructed by a sector shaped opaque obstacle with center angle a at several propagation distances [102]

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Fig.21 Modulus of the degree of coherence of a focused partially coherent LGpl beam with p = 1 and l = 1 obstructed by a sector shaped opaque obstacle with center angle a = 90° in the focal plane for different values of initial coherence length [102]

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6 Conclusions

Recent developments on partially coherent vortex beam have been introduced both theoretically and experimentally. The basic concepts, theoretical models, generation and propagation of partially coherent vortex beams have been reviewed. Partially coherent vortex beams display some unique properties (i.e., beam shaping, beam rotation and self-reconstruction), which are useful in some applications, such as optical trapping, free-space optical communications, information encryption and decryption. From 2001 to 2017, several papers on vortex beams and partially coherent vortex beams have been published [2,48,130,142,143], and the progress of the research work in this field has been introduced systematically, we believe this field will grow further and expand rapidly, and more and more interesting results and potential applications will be revealed.

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Acknowledgements

Authors are thankful for the support of the National Natural Science Foundation of China (Grant Nos. 91750201, 11525418, 11774250 and 11804198), Project of the Priority Academic Program Development of Jiangsu Higher Education Institutions.

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