## Abstract

Partially coherent radially polarized (PCRP) beam was introduced and generated in recent years. In this paper, we investigate the statistical properties of a PCRP beam embedded with a vortex phase (i.e., PCRP vortex beam). We derive the analytical formula for the cross-spectral density matrix of a PCRP vortex beam propagating through a paraxial ABCD optical system and analyze the statistical properties of a PCRP vortex beam focused by a thin lens. It is found that the statistical properties of a PCRP vortex beam on propagation are much different from those of a PCRP beam. The vortex phase induces not only the rotation of the beam spot, but also the changes of the beam shape, the degree of polarization and the state of polarization. We also find that the vortex phase plays a role of resisting the coherence-induced degradation of the intensity distribution and the coherence-induced depolarization. Furthermore, we report experimental generation of a PCRP vortex beam for the first time. Our results will be useful for trapping and rotating particles, free-space optical communications and detection of phase object.

© 2016 Optical Society of America

## 1. Introduction

As a typical kind of cylindrical vector beam with a spatially nonuniform state of polarization, radially polarized (RP) beam exhibits unique tight focusing properties [1–5], e.g., a strong longitudinal electric field appears and a much smaller focused beam spot can be formed. Since 2000, RP beam has been investigated in detail and found many applications, such as microscopy, lithography, electron acceleration, proton acceleration, material processing, optical data storage, high-resolution metrology, super-resolution imaging, plasmonic focusing and laser machining, free-space optical communications [1–10]. In the past several years, RP beam with controllable spatial coherence (i.e., PCRP beam) was introduced in theory and generated in experiment [11–15]. The propagation properties of a PCRP beam are quite different from those of a RP beam. The PCRP beam exhibits the properties of depolarization on propagation in free space, i.e., the degree of polarization decreases on propagation, while its state of polarization remains radial polarization on propagation. Through varying the spatial coherence width, one can shape the beam profile of the focused beam spot of a PCRP beam [11, 12], which is useful for particle trapping. Furthermore, it was found in [13] that a PCRP beam has an advantage over a linearly polarized partially coherent beam for reducing turbulence-induced scintillation, which is useful in free-space optical communications. Experimental observation of the spectral changes of a polychromatic PCRP beam was demonstrated in [14].Theoretical and experimental studies of the statistical properties in Young’s two-slit interference pattern formed with a PCRP beam were carried in [15]. PCRP beams with nonconventional correlation functions were also reported recently [16, 17].

On the other hand, light beam embedded a vortex phase (i.e., vortex beam) carries orbital angular momentum, which is useful in many applications, e.g., free-space information transfer and communications, optical trapping, optical manipulation, atom trapping, quantum information, astrophysics and detection of a spinning object [18–26]. Partially coherent vortex beam was introduced as a natural extension of coherent vortex beam [27–29], and such beam displays unique correlation singularities or coherence vortices [28–34], which can’t exist in a coherent vortex beam. The beam profile of the intensity distribution of a partially coherent vortex beam is modulated by its spatial coherence width [35], which is useful for particle trapping [36]. Compared to a partially coherent beam without vortex phase, partially coherent vortex beam has an advantage for reducing turbulence-induced scintillation [37]. Coherence properties of a partially coherent vortex beam were demonstrated in [38] with the help of a Shack-Hartmann wavefront sensor. Determination or measurement of the topological charge of a partially coherent vortex beam was reported in both theory and experiment recently [39–42]. Above mentioned studies are confined to scalar partially coherent vortex beams. In this paper, our aim is to investigate the statistical properties of a PCRP beam embedded with a vortex phase (i.e., PCRP vortex beam). Analytical propagation formula for a PCRP vortex beam passing through a stigmatic ABCD optical system is derived and the vortex phase-induced changes of the statistical properties on propagation are analyzed in detail. Experimental generation of a PCRP vortex beam is also reported. Some interesting and useful results are revealed.

## 2. Analytical propagation formula for a PCRP vortex beam

Based on the unified theory of coherence and polarization [43], the second-order statistical properties of a vector partially coherent beam can be characterized by the $2\times 2$ cross-spectral density (CSD) matrix $\overleftrightarrow{W}\left({\rho}_{1},{\theta}_{1},{\rho}_{2},{\theta}_{2}\right)$. For a PCRP beam generated by a Schell-model source, in the cylindrical coordinate system, the elements of its CSD matrix ${\overleftrightarrow{W}}_{0}\left({\rho}_{01},{\theta}_{01},{\rho}_{02},{\theta}_{02}\right)$in the source plane are defined as [11–13]

*ρ*

_{01},

*θ*

_{01}) and (

*ρ*

_{02},

*θ*

_{02}) are the radial and the azimuthal coordinates of two arbitrary points in the source plane,

*σ*

_{0}and

*δ*

_{0}represent the transverse beam width and the spatial coherence width, respectively, and the asterisk denotes the complex conjugate. Generation and propagation properties of a PCRP beam can be found in [11–13].

After passing through a spiral phase plate with transmission function$T\left(\theta \right)=\mathrm{exp}\left(im\theta \right)$, where *m* denotes the topological charge, the PCRP beam becomes a PCRP vortex beam and the elements of its CSD matrix can be expressed as follows

Propagation of the elements of the CSD matrix of a vector partially coherent beam through a paraxial ABCD optical system can be studied by the following generalized Collins formula [44, 45]

*k*= 2

*π*/

*λ*is the wavenumber with

*λ*being the optical wavelength,

*A*,

*B*,

*C*, and

*D*are the elements of the transfer matrix of the paraxial optical system.

Substituting Eq. (5) into Eq. (6), after tedious integration and operation, we obtain the following expression for the elements of the CSD matrix of the PCRP vortex beam in the output plane

_{1}

*F*

_{1}(﹒;﹒;) is a Kummer function.

In above derivations, we have used the following expansion and integral formulae

*J*(·) denotes the

_{l}*l*-th order Bessel function of the first kind and

*I*(·) denotes the

_{m}*m*-th order modified Bessel function of the first kind.

The spectral intensity and the degree of polarization of the PCRP vortex beam at point $\left(\rho ,\theta \right)$ in the output plane are obtained as

The CSD matrix of a partially coherent vector beam can be locally represented as a sum of a completely polarized portion and a completely unpolarized portion [46]

The spectral intensities of the completely unpolarized and completely polarized portions are expressed respectively as follows

The state of polarization of the completely polarized portion can be characterized by the polarization ellipse, whose major and minor semi-axes of the ellipse, *A*_{1} and *A*_{2}, as well as its degree of ellipticity, *ε*, and its orientation angle,$\vartheta $, can be related directly to the elements of the cross-spectral density matrix $\overleftrightarrow{W}\left(\rho ,\theta ,\rho ,\theta \right)$by the following relations [46]

*A*

_{1}(major semi axis) and

*A*

_{2}(minor semi-axis), respectively. Substituting Eqs. (7)-(10) into Eqs. (26)-(29), one can analyze the properties of the state of polarization of a PCRP vortex beam on propagation conveniently.

## 3. Vortex phase-induced changes of the statistical properties of a PCRP beam on propagation

In this section, with the formulae derived in section 2, we will study the vortex phase-induced changes of the statistical properties of a focused PCRP beam on propagation.

We assume that a PCRP vortex beam is focused by a thin lens with focal length *f* = 400mm, which is located at *z* = 0, and the output plane is located at *z*. The transfer matrix between the source plane and the output plane reads as

*λ*= 632.8nm and

*σ*

_{0}= 1mm.

We calculate in Figs. 1–3 the spectral intensity distribution *I* and its components *I _{x}* and

*I*of a focused PCRP vortex beam with

_{y}*δ*

_{0}= 0.5mm at several propagation distances for different values of the topological charge

*m*(

*m*= 0, 2, and −2), respectively. It is known that the dark hollow beam profile with zero intensity in the center point of a coherent RP beam always remains on propagation in free space. One finds that the evolution properties of spectral intensity distribution

*I*of a focused PCRP vortex beam (

*m*= 2 or −2) on propagation are quite different from those of a focused coherent RP beam, but are similar to those of a focused PCRP beam (

*m*= 0) in some way, e.g., the initial dark hollow beam profile evolves into flat-topped beam profile on propagation, and finally becomes a Gaussian beam profile in the focal plane [see Figs. 1(a1)-1(e1), Figs. 2(a1)-2(e1) and Figs. 3(a1)-3(e1)], which is induced by the decrease of the coherence width. While the evolution properties of the components

*I*and

_{x}*I*of the spectral intensity distribution of a focused PCRP vortex beam on propagation are quite different from those of a focused PCRP beam. The beam spots of the

_{y}*I*,

_{x}*I*and

_{y}*I*of a focused PCRP beam don’t rotate on propagation (see Fig. 1), but the beam spots of the

*I*and

_{x}*I*of a focused PCRP vortex rotates anti-clockwise for

_{y}*m*= 2 or clockwise for

*m*= −2 on propagation [see Figs. 2(a2)-2(e2), 2(a3)-2(e3) and Figs. 3(a2)-3(e2), 3(a3)-3(e3)], which means the beam spot of

*I*also rotates on propagation although it is not demonstrated in Fig. 2(a1)-2(e1) and Fig. 3(a1)-3(e1) because the beam spot is of circular symmetry. The rotation of the beam spot is induced the vortex phase, which imposes angular orbital angular momentum on the beam. The rotation angle reaches $-\pi /2$ or $-\pi /2$ in the focal plane for

*m*>0 or

*m*<0. This phenomenon also occurs in coherent radially polarized vortex beam, and can be seen more clearly in that case. In free space propagation, the beam spot of the PCPR vortex beam rotates on propagation, and the rotation angle reaches $-\pi /2$ or $-\pi /2$ in the far field for

*m*>0 or

*m*<0. The phenomenon of the coherence-induced beam shaping and the vortex phase-induced rotation of the beam spot will be useful for trapping and rotating particles.

Figure 4 shows the spectral intensity distribution *I* of a focused PCRP vortex beam in the focal plane for different values of the topological charge *m* with *δ*_{0} = 0.5mm. From Fig. 4, one sees that the beam profile in the focal plane is also shaped by the topological charge, e.g., the beam profile varies from a Gaussian beam profile to flat-topped beam and quasi-dark hollow beam profile as the value of *m* increases. Thus vortex phase also is useful for shaping the beam profile of a focused PCRP vortex beam, which is useful for tapping particle and material thermal processing. Furthermore, it is known from Refs [11,12]. and Fig. 1 that decreasing the coherence width of a RP beam induces degradation of its beam profile on propagation, e.g., the initial dark hollow beam profile evolves into flat-topped beam profile or Gaussian beam profile, and the results in Fig. 4 suggest that the vortex phase plays a role of resisting coherence-induced degradation of the intensity distribution *I*, which will be useful in some applications, such as free-space optical communication and laser radar, where the atmospheric turbulence will induce the decrease of the coherence width of laser beam and the degradation of its intensity distribution.

Now we analyze the polarization properties of a focused PCRP vortex beam on propagation. We calculate in Fig. 5 the degree of polarization of a focused PCRP vortex beam at point (0.05mm, 0.05mm) versus the propagation distance *z* and in Fig. 6 the degree of polarization (cross line *y* = 0) in the focal plane for different values of the coherence width *δ*_{0} and the topological charge *m*. One finds from Figs. 5(a) and 6(a) that degree of polarization of a coherent RP vortex beam (*δ*_{0} = Infinity) equals 1 and remains invariant on propagation, while the degree of polarization decreases on propagation, which means the coherence induces depolarization, and this phenomenon also exists in a focused PCRP beam [11]. From Figs. 5(b) and 6(b), it is clear that the vortex phase affects the degree of polarization on propagation and usually the degree of polarization increases as the value of topological charge *m* increases, which means the vortex phase plays a role of anti-depolarization on propagation.

To learn about the effect of anti-depolarization more clearly, now we study the evolution properties of the normalized powers of the completely polarized part and polarized part, which are defined as

*η*

^{(}

^{p}^{)}(

*z*) and

*η*

^{(}

^{u}^{)}(

*z*) represent the normalized powers of the completely unpolarized part and the completely polarized part, respectively. We calculate in Fig. 7 the variation of the normalized powers of the completely unpolarized part and the completely polarized part of a focused PCRP vortex beam versus the propagation distance

*z*for different values of the topological charge

*m*with

*δ*

_{0}= 0.5mm. From Fig. 7, we find that

*η*

^{(}

^{p}^{)}(

*z*) decreases on propagation, while the

*η*

^{(}

^{u}^{)}(

*z*) increases, which means that the power transits from the completely polarized part to the completely unpolarized part on propagation and the beam is depolarized as expected [11]. Furthermore, with the increase of

*m*,

*η*

^{(}

^{p}^{)}(

*z*) decreases more slower and

*η*

^{(}

^{u}^{)}(

*z*) increases more slower, which means the vortex phase indeed plays a role of anti-depolarization.

In [47] and [48], it was shown that the vortex phase plays a role of decreasing random wandering of the beam caused by atmospheric turbulence. The mechanisms of influence of the vortex phase on the resistance to the destructive action caused by the loss of coherence or the atmospheric turbulence have the similar character. In fact, the atmospheric turbulence not only caused random wandering of the beam, but also causes decrease of the coherence of the beam.

Furthermore, our numerical results have shown that the vortex phase with large topological charge is superior to the one with small topological charge for resisting the coherence-induced degradation of the intensity distribution and the coherence-induced depolarization. While it is known that the vortex with large topological charge is unstable when we discuss the correlation singularity. Thus, it would be interesting to investigate the correlation singularity and its stability of a PCRP vortex beam, and we leave this for future study.

We calculate in Fig. 8 the variation of the state of polarization of a focused PCRP vortex beam at several propagation distances in free space for different values of the topological charge *m* with *δ*_{0} = 0.5mm, and in Fig. 9 the variation of the state of polarization of a focused PCRP vortex beam in the focal plane for different values of the topological charge *m* with *δ*_{0} = 0.5mm. One finds from Fig. 8(a) that the state of polarization of a focused PCRP beam (*m* = 0) remains invariant on propagation and always displays radial polarization as expected [11]. When the vortex phase is embedded in the PCRP beam, the vortex phase induces dramatic changes of the state of polarization on propagation as shown in Figs. 8 and 9, e.g., radial polarization disappears and elliptical polarization appears on propagation. The state of polarization displays left-handed elliptical polarization around the beam center and right-handed elliptical polarization outside of the beam center for *m* = 2, and the handedness of the polarization ellipse is reversed for *m* = −2. Furthermore, the polarization ellipse rotates clockwise for *m* = 2 and anti-clockwise for *m* = −2 on propagation. From Fig. 9, one sees that the area where the state of polarization displays left-handed (or right-handed) elliptical polarization around the beam center increases when the absolute value of *m* increases. Thus modulating the vortex phase provides one way for modulating the degree of polarization and the state of polarization of a PCRP vortex beam. On the other hand, the phenomenon of vortex phase-induced changes of the state of polarization of a PCRP vortex beam may be used to detect a phase object, since the state of polarization of a PCRP beam remains invariant on propagation in free space.

## 4. Experimental generation of a PCRP vortex beam

In this section, we report experimental generation of a PCRP vortex beam. Figure 10 shows our experimental setup for generating a PCRP vortex beam and measuring its focused intensity. In our experiment, the PCRP vortex beam is generated through embedding a vortex phase in a PCRP beam. The PCRP beam is generated by the experimental setup, which is the same as that in [12]. A linearly polarized He-Ne laser beam (*λ* = 632.8nm) reflected by a reflecting mirror M transmits through a beam expander, then the transmitted beam is focused by the thin lens L_{1}, and next passes through the rotating ground-glass disk (RGGD), producing a linearly polarized partially coherent beam. After passing through a collimation thin lens L_{2} and a Gaussian amplitude filter, the transmitted beam becomes a linearly polarized Gaussian Schell-model (GSM) beam. Then the radial polarization converter (RPC) coverts the linearly polarized GSM beam into a PCRP beam. After passing through a spiral phase plate (SPP) with topological charge *m* = 2 which is located just behind the RPC, the generated PCRP beam becomes a PCRP vortex beam. The generated PCRP beam is focused by a thin lens with focal length *f*_{3} = 400mm, and a beam profile analyzer is used to measure the intensity distribution. In our experiment, the transverse beam width *σ*_{0} of the generated PCRP vortex beam just behind the SPP is measured to be 1mm. The coherence width *δ*_{0} of the generated PCRP vortex beam can be measured by the method described in [49], and is measured to be 0.8mm in our experiment.

Figure 11 shows our experimental results of the intensity distribution *I* and its components *I _{x}* and

*I*

_{y}of a focused PCRP vortex beam with

*m*= 2,

*σ*

_{0}= 1mm and

*δ*

_{0}= 0.8mm at several propagation distances. To measure

*I*or

_{x}*I*

_{y}, we just put a linear polarizer whose transmission axis forms an angle $\phi =0$or $\phi =\pi /2$ with

*x*-axis. From Fig. 11, one finds that the intensity distribution

*I*gradually transforms from a dark hollow beam profile into a flat-topped beam profile and finally becomes a Gaussian beam profile on propagation, which is consistent with the results shown in Fig. 2, and the beam spots of

*I*and

_{x}*I*

_{y}rotate anti-clockwise as expected. The degree of polarization of a PCRP vortex beam can measured by the method proposed in [50], which has been used to measure the DOP of a PCPR beam successfully [11]. Figure 12 shows our experimental result (dotted curve) of the degree of polarization of a focused PCRP vortex beam with

*m*= 2,

*σ*

_{0}= 1mm and

*δ*

_{0}= 0.8mm versus the coordinate

*x*in the focal plane. For the convenience of comparison, the corresponding theoretical result (solid curve) is also shown in Fig. 12. One sees that the vortex phase indeed induces changes of the degree of polarization, and our experiment result agrees well with the theoretical result.

## 5. Summary

We have obtained the analytical formula for the CSD matrix of a PCRP vortex beam propagating through a paraxial optical system, and have studied the statistical properties of a focused PCRP vortex beam on propagation in detail. We have found that embedding a vortex phase in a PCRP beam will play a role of resisting coherence-induced degradation of the intensity distribution and the coherence-induced depolarization, which may be useful in free-space optical communication and laser radar. We also have found that the vortex phase will induce rotation of the beam spot and modulating the topological charge can be used to shape the beam profile of a focused PCRP vortex beam in the focal plane, which will be useful for trapping and rotating particles. Furthermore, it was revealed that the vortex phase will induce changes of the state of polarization, and this phenomenon may be useful for detection of phase object. We have carried out experimental generation of a PCRP vortex beam and verified our theoretical results.

## Acknowledgments

This work is supported by the National Natural Science Fund for Distinguished Young Scholar under Grant No. 11525418, the National Natural Science Foundation of China under Grant Nos.11274005, 11404067 and 11404234, the China Postdoctoral Science Foundation under Grant No. 2015M570473, the Project of the Priority Academic Program Development (PAPD) of Jiangsu Higher Education Institutions.

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