## Abstract

We have investigated the correlation singularities, coherence vortices of two-point correlation function in a partially coherent vector beam with initially radial polarization, i.e., partially coherent radially polarized (PCRP) beam. It is found that these singularities generally occur during free space propagation. Analytical formulae for characterizing the dynamics of the correlation singularities on propagation are derived. The influence of the spatial coherence length of the beam on the evolution properties of the correlation singularities and the conditions for creation and annihilation of the correlation singularities during propagation have been studied in detail based on the derived formulae. Some interesting results are illustrated. These correlation singularities have implication for interference experiments with a PCRP beam.

© 2015 Optical Society of America

## 1. Introduction

In recent years, there has been increasing interest in the subject of singular optics which deals with a wide class of effects occurring near the optical fields’ points where certain field parameters are undefined or “singular” [1–5]. In general, singular optics focuses on the study of the phase singularities occurring in the positions where the amplitude of the fields is zero and hence the phase is indeterminate [1, 2]. A typical example of a phase singularity presents in a Laguerre-Gaussian beam with azimuthal mode *l*≠0. Such field has a zero intensity on the central point in the cross-section of the beam, and the phase around that point exhibits the helical structure. Thus, such phase structure is generally referred to optical vortex. Owing to its unique phase structure, there are some nontrivial effects and has been found many applications in trapping and rotating micro particles [6], in phase contrast microscopy [7], and in free-space optical communications [8], etc. Besides the phase singularities, there are also some other types of singularities in optical fields. For example, polarization singularities arise at the positions where the state of the polarization is circular or linear polarization in the vector optical fields [9–12]. For circular polarization, the orientation of the major and minor axes of the polarization ellipse is undefined, and the handedness of the ellipse for the case of linear polarization becomes undefined. The Poynting vector singularities at which the direction of power flow is undefined have been observed in the focal plane of an aplanatic system and a sub-wavelength slit and the behavior near the singularities was analyzed in [13–15].

In 2003, the concept of the singularities was extended to partially coherent fields by Schouten *et al*. who study the singularities (named correlation singularities) in cross spectral density (CSD) function of partially coherent fields in Young’s interference experiment [16]. Since then, a significant amount of work has been devoted to explore the nontrivial behavior neighborhood the singularities in partially coherent fields both theoretically and experimentally [17–31]. Unlike complete coherent fields, the phase is not well-defined in partially coherent fields because one has to apply the “statistics” to quantify the phase. In the theory of partially coherence, the statistical properties of partially coherent fields are characterized by two-point coherence functions (CSD function in space-frequency domain or mutual coherence function (MCF) in space-time domain), which have a definite phase. In certain circumstance, the value of the coherence functions may equal zero, and hence the phase of the coherence functions is undefined, which exhibits the singular behavior. Such singularities are usually referred as correlation singularities or “coherence vortices” and have been found in multi-mode laser beams [17], in the fields generated by Mie scattering [18] and partially coherent vortex beams [20]. The evolution, representation and topological reaction of the correlation singularities in partially coherent vortex beams have been reported in [20–26]. Recently, C. S. D. Stahl and G. Gbur proposed a method to describe the geometric form of correlation singularities of partially coherent beams in the full six-variable correlation space, both in the transverse and the propagation directions [27]. The experimental study of the correlation singularities in partially coherent vortex beams was undertaken by Maleev *et al*. and Wang *et al.*, respectively [28, 29]. However, the study on the correlation singularities in partially coherent fields mentioned above was restricted on scalar fields, i.e., the optical fields are linearly polarized. Recently, Raghunathan and associates [32, 33] examined the correlation singularities in an electromagnetic Gaussian Schell-model (EGSM) field which is a typical class of partially coherent vector field. It was found that such singularities are existent in the partially coherent vector fields during propagation even though they are absent in the source plane. It is known the state of polarization (SOP) of EGSM beams across the source plane is uniform. Meanwhile, another class of random electromagnetic beam, named partially coherent vector beams with spatially variant SOP including partially coherent cylindrical vector (CV) beams and partially coherent beam with a spiral-like distribution of polarization, have been received considerable attention due to their special polarization structure [34–42]. A partially coherent radial polarized (PCRP) beam is a typical kind of partially coherent CV beams. In recent, the experimental generation of the PCRP beam and its focusing properties were investigated in [37]. The results were shown that one could shape the beam profile of the focused PCRP beam by varying its initially spatial coherence, which is very useful for particle trapping and material processing. The propagation characteristics and scintillation of the PCRP beam in atmospheric turbulence were extensively studied by several researchers [39–42]. It was found that the use of the PCRP beam may overcome the negative effect induced by the turbulence. Therefore, it has potential applications in free-space optical communication, remote sensing and laser radar. However, the possibility of the existence of the correlation singularities in the PCRP beam and the evolution properties of these singular structures on propagation have not been studied.

In this paper, we demonstrate the existence of correlation singularities in a PCRP beam during free space propagation. Analytical formulae for describing the propagation dynamics of the correlation singularities of the PCRP beam are derived. On the basis of the derived formulae, we study in detail the influences of the beam parameters such as spatial coherence length and the propagation distance on the properties of the correlation singularities. The conditions for creation and annihilation of the correlation singularities upon propagation are also derived. Our results may have an important implication in understanding the geometry and evolution properties of correlation singularities of the PCRP beam during propagation.

## 2. Theory of correlation singularities in partially coherent electromagnetic beams

Let us consider a quasi-monochromatic, stochastic electromagnetic beam which is statistical stationary, at least in wide sense, propagates close to *z* axis. The second-order statistical properties of the stochastic electromagnetic beam in the source plane (*z* = 0) may be characterized by the CSD matrix in space-frequency domain, given by [43]

**r**

_{1}≡(

*x*

_{1},

*y*

_{1}) and

**r**

_{2}≡(

*x*

_{2},

*y*

_{2}) denote two arbitrary position vectors in the source plane.

*E*and

_{x}*E*

_{y}represent two mutual orthogonal components of the random electric field along

*x*and

*y*directions respectively, perpendicular to

*z*axis. The angular brackets and the asterisk stand the ensemble average over the source fluctuation and the complex conjugate, respectively. In this paper, the dependence of the CSD matrix and other derived properties on

*ω*will be omitted for conciseness. Based on the unified theory of coherence and polarization of random electromagnetic beams [44], the spectral degree of coherence (SDOC) at the propagation distance

*z*is defined as

**ρ**

_{1}≡(

*ρ*

_{1}

*,*

_{x}*ρ*

_{1}

*) and*

_{y}**ρ**

_{2}≡(

*ρ*

_{2}

*,*

_{x}*ρ*

_{2}

*) are two arbitrary position vectors in the transverse plane*

_{y}*z*. Within the validity of the paraxial approximation, the relation between CSD matrix $\widehat{W}({\rho}_{1},{\rho}_{2},z)$ and ${\widehat{W}}^{(s)}\left({r}_{1},{r}_{2}\right)$ can be connected with the Huygens-Fresnel integral formula, and the elements of $\widehat{W}({\rho}_{1},{\rho}_{2},z)$ are given as follows [43]

In general, the function *η*(**ρ**_{1}, **ρ**_{2}, *z*) is complex. Correlation singularities are the phase singularities occurring at the pair of points at which the field is complete uncorrelated, i.e., *η*(**ρ**_{1}, **ρ**_{2}, *z*) = 0 [32,33]. The physical meaning of the correlation singularities is that when two points **ρ**_{1} and **ρ**_{2} where *η* (**ρ**_{1}, **ρ**_{2}, *z*) = 0 are combined in Young’s experiment, the visibility of the formed fringes on the observation screen will be zero. This is because the local modulations of ${\left|{E}_{x}\right|}^{2}$ and ${\left|{E}_{y}\right|}^{2}$ on the observation screen have equal magnitude and opposite sign, resulting in a zero visibility of the total spectral density. According to Eq. (3), correlation singularities occur in the transverse plane *z* when both

## 3. Correlation singularities in a PCRP beam upon propagation

Suppose that the stochastic electromagnetic beam generated from Gaussian Shell-mode source with initial radial polarization. Such beam has been demonstrated experimentally in our early work [37], and the CSD matrix of the PCRP beam in the source takes in the form

*σ*

_{0}is the beam width and

*δ*

_{0}is the spatial coherence length of the PCRP beam. when ${\delta}_{0}\to \infty $, the CSD matrix in Eq. (7) reduces to the polarization matrix of the complete coherent RP beam. Note that the spectral density of the PCRP beam at the on-axis point (

**r**= 0) vanishes. This singularity is commonly referred as

*polarization vortex*or vector vortex that has a space-variant polarization orientation evolving around that point [45,46]. However, the SDOC of the PCRP beam in the source plane is of a Gaussian function which is $\eta ({r}_{1},{r}_{2},0)=\mathrm{exp}\left[-{\left({r}_{1}-{r}_{2}\right)}^{2}/2{\delta}_{0}^{2}\right]$. Therefore, no correlation singularities will be found in the source plane. When the PCRP beam propagates to the half-plane (

*z*>0), the zero point of spectral density across the cross-section of the beam vanishes; whereas the correlation singularities will be created on propagation as we will analyze below.

On substituting from Eq. (7) into Eq. (4) and after tedious integrating, we obtain the elements of the CSD matrix of the PCRP beam at propagation distance *z*, i.e.,

*z*is

*z*vanishes during propagation (

*z*>0), which implies that the polarization singularity in the source plane what we mentioned above is annihilated when the PCRP beam starts to propagate (

*z*>0). By applying Eq. (8) into Eqs. (5)-(6), the real part and imaginary part of the trace of the CSD matrix become

**ρ**

_{2}in the transverse plane

*z*, the other point

**ρ**

_{1}which forms the pair of points with

**ρ**

_{2}for correlation singularities is always located on the circle whose radius is $R=\sqrt{{\rho}_{x2}^{2}+{\rho}_{y2}^{2}}$, independence of the propagation distance

*z*and beam parameters. By applying Eq. (11), Eq. (10) can be simplified by the following formula

**ρ**

_{2}, Eq. (12) is an equation of the straight line about

**ρ**

_{1}≡(

*ρ*

_{1}

*,*

_{x}*ρ*

_{1}

*) with its slope being*

_{y}*k*

_{0}= -

*ρ*

_{2}

_{x}/ ρ_{2}

*. Therefore, we come to the conclusion that*

_{y}*for the fixed point*

**ρ**

_{2}

*in the transverse plane z, the other point*

**ρ**

_{1}

*which the pair*(

**ρ**

_{1},

**ρ**

_{2})

*form the correlation singularity lies in the intersections of the straight line (*

*Eq. (12)*

*) and the circular line with*$R=\sqrt{{\rho}_{x2}^{2}+{\rho}_{y2}^{2}}$

*(*

*Eq. (11)*

*)*.

We plot in Fig. 1 the zeros of real part (solid line) and imaginary part (dashed line) of the SDOC of the PCRP beam at several propagation distances *z* when **ρ**_{2} = (1.0,1.0)mm is fixed. The other beam parameters in calculation are chosen to be *σ*_{0} = 1.0mm, *δ*_{0} = 0.5mm, *λ* = 632.8nm. The intersections with the solid straight line and the dashed circular line, labeled **ρ*** _{A}* and

**ρ**

*where the phase of*

_{B}*η*(

**ρ**

_{1},

**ρ**

_{2},

*z*) is undefined, indicate two correlation singularities. The corresponding contour graph of the spectral density of the PCRP beam at that propagation distance shows in the background of Fig. 1. One finds that the spectral densities at points

**ρ**

*and*

_{A}**ρ**

*are appreciable. No zero intensity point is found in the process of the moving of the singularities in the beam cross-section with the propagation distance. With the increase of the propagation distance, two singularities move along the dashed circular line. First, the moving direction of the point*

_{B}**ρ**

*is clockwise along the circle, while the other is anticlockwise. When the propagation distance is larger than 4.5m, the moving directions of two singularities turn opposite until they merge together at*

_{A}*z*= 19.93m, and annihilate for further propagation. The detailed propagation dynamics of two singularities upon propagation is shown in Media 1 in Fig. 1.

Figure 2 illustrates the corresponding contour plot of the phase of *η*(**ρ**_{1}, **ρ**_{2}, *z*) at the typical propagation distance in Fig. 1. It is evident that these points are indeed phase singularities, so-called coherence vortices. The phases of SDOC around the phase singularities **ρ*** _{A}* and

**ρ**

*display a helical structure evolving from 0 to 2*

_{B}*π*and from 2

*π*to 0 along the anti-clockwise direction, respectively, implying that

*η*(

**ρ**

_{A},

**ρ**

_{2},

*z*) and

*η*(

**ρ**

_{B},

**ρ**

_{2},

*z*) have the opposite topological charge, + 1 and −1, respectively. Note that the total topological charge is conserved during propagation.

The condition for existence of the singularities can be obtained from geometrical configuration of the straight line and the circular line shown in Fig. 1, which is that *the distance d from the original point* **ρ**_{0} = (0,0) *to the solid straight line is smaller than the radius of the dashed circular line R*. Following from Eq. (12), the distance *d* takes the form

*d*tends to zero if the propagation distance is $z\to 0$, which means that the distance of two created singularities in the transverse plane is 2

*R*independence of the beam parameters such as

*σ*

_{0},

*δ*

_{0}when the beam cross-section is close to the source plane. In addition, the dependence of

*d*on the reference point

**ρ**

_{2}is only through

*R*, implying that the distance for the annihilation of the correlation singularities (

*d*>

*R*) is independent on the specific values of

*ρ*

_{x}_{2}and

*ρ*

_{y}_{2}if the value of

*R*is kept same. Figure 3 gives the dependence of

*d*on the propagation distance with two different reference points

**ρ**

_{2}for different values of spatial coherence length. One finds that for the small value of

*δ*

_{0}(

*δ*

_{0}= 0.5mm), the evolution properties of

*d*with propagation distance for

*R*= 1.414mm and

*R*= 0.707mm are similar.

*d*first increases as the propagation distance increases, reaches a maximum value, and then decreases with the increase of propagation distance, reaches a minimum value zero, then starts to increase again. According to this variation, the evolution of the singularities in the transverse plane with the propagation distance is already shown in Fig. 1. However, for the case of

*δ*

_{0}= 1mm or 2mm (see in Fig. 3(b)), the distance

*d*increases monotonously with the increaseof the propagation distance

*z*for

**ρ**

_{2}= (0.5,0.5)mm, i.e.,

*R*= 0.707mm, indicating that the moving direction of the singularity

**ρ**

*or*

_{A}**ρ**

*keeps unchanged upon propagation. The corresponding propagation dynamics of the correlation singularities of the PCRP beam with*

_{B}*δ*

_{0}= 1.0mm is illustrated in Fig. 4. For other numerical calculations, it is found that two different propagation dynamics of correlation singularities is close related to the beam parameters and the position reference point

**ρ**

_{2}. We may obtain a criterion to distinguish two propagation dynamics from the term shown in the absolute symbol in Eq. (13). It is known that the parameter $\Delta $ increases monotonously with the propagation distance from its value of unity in the source plane. Therefore, the inequationis satisfied, the values in the absolute symbol will be first positive and then turn to negative when the propagation distance is taken to close to source plane and then moves away gradually. The change of the positive and minus sign in the absolute symbol leads to the change of moving direction of correlation singularity of

**ρ**

*or*

_{A}**ρ**

*on propagation. If Eq. (14) is not satisfied, the values in the absolute symbol keep negative during propagation. In this situation, the moving directions of two singularities remain unchanged on propagation. In our calculation, ${R}^{2}/2{\sigma}_{0}^{2}+{R}^{2}/{\delta}_{0}^{2}$ is always larger than one for the case of*

_{B}*R*= 1.414mm by varying

*δ*

_{0}, while the spatial coherent width should be smaller than 0.816mm to hold true for the inequation for the case of

*R*= 0.707mm. Figure 5 shows the 3D-plot of the evolution of the two singularities as a function of the propagation distance for different values of spatial coherence length. As expected, two singularities are created when the beam starts to propagate (

*z*>0), and at last merge together for the certain propagation distance which is dependent on the beam parameters and the reference point.

Now let us turn to investigate the propagation distance where two singularities annihilate. It follows from Eq. (13) that the condition for the annihilation of correlation singularities is *d*>*R*. By solving the equation *d* = *R*, we obtain the critical distance

*z*<

*z*, There are two singularities in the beam cross-section, while two singularities disappear for

_{m}*z*>

*z*. The dependence of this critical propagation distance

_{m}*z*on the spatial coherence length of the PCRP beam for several reference points

_{m}**ρ**

_{2}is illustrated in Fig. 6. It can be seen that the evolution of

*z*with the spatial coherence lengthfor different reference points is much different. For the case of

_{m}**ρ**

_{2}= (0.5,0.5)mm,

*z*first decreases as the spatial coherence length increases, reaches a minimum value when

_{m}*δ*

_{0}= 0.82mm, and then increases with the increase of the spatial coherence length. For the case of

**ρ**

_{2}= (1.0,1.0)mm and

**ρ**

_{2}= (1.0,2.0)mm,

*z*increases as the spatial coherence length increases monotonously. When the spatial coherence length is sufficient small, i.e., ${\delta}_{0}\to 0$, the critical propagation distance

_{m}*z*tends to $\sqrt{2}kR{\sigma}_{0}$ from Eq. (15), proportional to the initial beam width.

_{m}## 4. Summary

In summary, we have studied the properties of the SDOC of the PCRP beam during free space propagation, and demonstrated that electromagnetic correlation singularities occur in the PCRP beam. The analytical expressions for characterizing the correlation singularities on propagation are derived. Based on the derived formulae, the evolution properties of the correlation singularities with the propagation distance are investigated in detail. It is found that for the fixed reference point, the other points which form the singularities for the pair points lie in the intersection with a straight line and a circle. In general, there is a pair of correlation singularities with topological charge + 1 and −1, respectively. The moving directions of the singularities along the circle on propagation may change their directions, and close related by the beam parameters and the position of the reference point. The influences of the spatial coherence length and the reference point on the distance where the singularities annihilate are also discussed. The presence of these singularities may have a significant consequence for interference experiment performed with the PCRP beam.

## Acknowledgments

This work is supported by the National Natural Science Foundation of China under Grant Nos. 11474213&11274005&11474143, The Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions, and the Project Sponsored by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry.

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