Partially coherent vortex beam with periodical coherence properties

Accepted Manuscript

Partially coherent vortex beam with periodical coherence properties Xianlong Liu , Leixin Liu , Xiaofeng Peng , Lin Liu , Fei Wang , Yaru Gao , Yangjian Cai PII: DOI: Reference:

S0022-4073(18)30591-0 https://doi.org/10.1016/j.jqsrt.2018.10.024 JQSRT 6261

To appear in:

Journal of Quantitative Spectroscopy & Radiative Transfer

Received date: Revised date: Accepted date:

13 August 2018 16 October 2018 16 October 2018

Please cite this article as: Xianlong Liu , Leixin Liu , Xiaofeng Peng , Lin Liu , Fei Wang , Yaru Gao , Yangjian Cai , Partially coherent vortex beam with periodical coherence properties, Journal of Quantitative Spectroscopy & Radiative Transfer (2018), doi: https://doi.org/10.1016/j.jqsrt.2018.10.024

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Highlights  Partially coherent vortex beam with periodical coherence properties is introduced theoretically and generated experimentally  Intensity lattices with controllable beamlet can be formed through varying initial coherence width  Partially coherent vortex beam with periodical coherence properties is useful for particle trapping and information transfer.

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Partially coherent vortex beam with periodical coherence properties Xianlong Liua, Leixin Liub, Xiaofeng Pengb, Lin Liub, Fei Wangb, Yaru Gaoa*, Yangjian Caia,b,** a

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Center of Light Manipulations and Applications & Shandong Provincial Key Laboratory of Optics and Photonic Device, School of Physics and Electronics, Shandong Normal University, Jinan 250014, China b School of Physical Science and Technology, Soochow University, Suzhou 215006, China *Corresponding author: [email protected] **Corresponding author: [email protected] Abstract: We introduce a new kind of partially coherent vortex beam with periodical

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coherence properties, named optical coherence vortex lattices (OCVLs). With the help of the generalized Collins formula, we explore the propagation properties of the intensity and the complex degree of coherence of OCVLs focused by a thin lens. Compared to conventional partially coherent vortex beam, OCVLs display extraordinary propagation properties, i.e., a Gaussian beam spot evolves into multiple

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beam spots (i.e., intensity lattices) in the focal plane (i.e., in the far field). The intensity lattices with solid or hollow beamlets can be formed through manipulating

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the coherence width in the source plane. We also find that the topological charge of the OCVLs can be determined from the distribution of its complex degree of

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coherence in the focal plane. Furthermore, we report experimental generation of OCVLs. The OCVLs will be useful for particle trapping and information transfer.

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Keywords: Partially coherent vortex beam; Optical coherence lattices; Beam

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characterization; Propagation.

1. Introduction Phase is an important property of a light beam characterized by the wavefront. Light beam with helical wavefront is called vortex beam, such as higher-order Laguerre-Gaussian beam and Bessel beam. Since Allen et al. found that each photon of a vortex beam carries orbital angular momentum (OAM) of l

with l being the

topological charge, numerous efforts have been paid to various vortex beams due to their important applications in optical communications [2, 3], particle trapping and manipulations [4-8], object detection [9], image edge enhancement [10], and so on.

ACCEPTED MANUSCRIPT Coherence is another important property of a light beam [11-13]. Laser beam with low coherence, named partially coherent beam, displays some interesting properties and notably a super strong self-reconstruction and resistance to turbulence [13, 14]. The latter feature is preferred in many applications, such as free-space optical communications [15, 16], laser radar systems [17, 18], optical imaging [19-21], optical trapping [22, 23]. The degree of coherence (i.e., coherence structure) of the conventional partially coherent beam satisfies Gaussian distribution [11-13]. Since

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Gori et al. discussed the sufficient condition for devising a genuine correlation function of a partially coherent beam [24], various partially coherent beams with non-conventional

coherence

structures

were

proposed

[25-35],

and

some

extraordinary properties caused by non-conventional coherence structures were found, such as self-shaping, self-focusing and self-splitting [25-35]. As a typical kind of

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partially coherent beam with non-conventional coherence structure, partially coherent beam with periodical coherence properties [i.e., optical coherence lattices (OCLs)] were introduced in [32, 33] and generated in [34]. OCLs were found to be useful for information encryption and free-space optical communications [32-35].

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Since Gori et al. proposed partially coherent beam with helical wavefront named partially coherent vortex beam [36], such beam has drawn a lot of attention due to its

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interesting propagation properties and unique coherence singularities [37-53]. Different methods were developed to generate various partially coherent vortex beams [41-45]. The conventional methods for determining the topological charge of a vortex

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beam is invalid for a partially coherent vortex beam, while one can infer the information of the topological charge of a partially coherent vortex beam from its

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coherence singularities [46-49]. The distribution of the degree of coherence of a partially coherent vortex beam obstructed by an opaque obstacle displays the property

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of self-reconstruction in the far field [50]. Partially coherent vortex beam has advantage over coherent vortex beam and partially coherent Gaussian Schell-model beam for reducing turbulence-induced scintillation [51]. Partially coherent vortex beam focused by a thin lens is also useful for trapping a particle whose refractive index is larger or smaller than the ambient [52]. A review of the theoretical models, propagation properties, experimental generation, topological charge determination and application of partially coherent vortex beam is presented in [53]. The degrees of coherence of the partially coherent vortex beams in the source plane are of Gaussian distributions in above literatures. In this paper, we carry out theoretical and

ACCEPTED MANUSCRIPT experimental study of a new kind of partially coherent vortex beam with periodical coherence properties, called optical coherence vortex lattices (OCVLs). The OCVLs display unique propagation properties, and are useful for particle trapping and information transfer.

2. Theoretical model for a partially coherent vortex beam with periodical

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coherence properties and its focusing properties Partially coherent beam is characterized by the correlation function of the electric field, i.e., the cross-spectral density (CSD) function in the space-frequency domain. To be a genuine correlation function, the CSD function can be expressed as follows [24]

(1)

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Wo  r1 , r2    I  v H  r1 , v  H  r2 , v  d 2 v,

where I  v  denotes an arbitrary nonnegative function. H  r, v  is an arbitrary kernel. Various partially coherent beams can be obtained by choosing suitable expressions of I  v  and H  r, v  [25-35]. For the convenience of experimental

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generation, one can rewrite Eq. (1) in the following alternative form [28] Wo  r1 , r2    Wi  v1 , v 2 H  r1 , v1  H  r2 , v 2  d 2 v1d 2 v 2

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with

Wi  v1 , v2   I  v1  I  v 2   v1  v 2  ,

(2)

(3)

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where Wi  v1 , v 2  and Wo  r1 , r2  denote the CSD functions in the input plane and output plane, respectively. I  v  and   v1  v 2  are the intensity and delta function,

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respectively. H  r, v  can be regarded as the response function of the optical system between the input and output planes. Thus one can generate a partially coherent beam

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with prescribed beam properties from an incoherent light source through propagation. Assume that the intensity I  v  of the incoherent source displays the

distribution of a uniform Gaussian beam array, i.e., M 1 2   v  v 0 m 2  2   I  v   exp   2 / 2  , M m M 1 2   0

(4)

where M is a non-negative integer denoting the number of Gaussian beam spots,

0 is the beam waist size of each Gaussian beam spot, v 0 m   v0 mx , v0 my    md , md 

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with v0mx and v0my being Gaussian beam spot displacements along x- and ydirections, respectively. d is the separation distance between adjacent Gaussian beam spots. In the following text, we set 0  0.1mm and d  1mm . H  r, v  is the response function of a Fourier transform optical system, given by  i  i H  r, v   T  r  exp  v 2  2r  v   .  f  f 

(5)

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On substituting from Eqs.(4) and (5) into Eqs.(2) and (3), after some integrations, one can derive the CSD function of a partially coherent beam with periodical coherence properties (i.e., OCLs) as follows [32-34]

 r2  r2   r  r  WOCLs  r1 , r2   exp   1 2 2    1 2  ,  4 0    f 

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incoherent source, i.e.,

1 2   r1 , r2   2 2   f M

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where   r1  r2  /  f  denotes the complex degree of coherence (CDOC), which comes from the Fourier transform of the intensity distribution IGBA  v  of the

  v  v 0 m 2   i 2 v   r2  r1   2 exp    exp    d v 2 0 / 2   f m  M 1 2     M 1

2

(7)

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  r2  r1 2   M 1 2 2  exp     exp  iV0 m   r2  r1  , M 2 02  m M 1 2 

with V0 m  kv 0 m / f  V0 mx ,V0 my  ,  0 and  0 denote the transverse beam waist

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size and the coherence width of the OCLs source. By adding a vortex phase

exp  il  to the OCLs source through a spiral phase plate, where l denotes the

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topological charge and   arctan  y / x  is the azimuthal angle, we can obtain the CSD function of a partially coherent vortex beam with periodical coherence

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properties (i.e., OCVLs) as follows

  r1  r2 2   r12  r22  2 WOCVLs  r1 , r2   exp   exp    2  M 2 02    4 0 

(8)

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 exp iV0 m   r1  r2   exp  il 1  2   . m 1

According to above theoretical analysis, we can propose the schematic for the generating OCVLs as shown in Fig. 1. The incoherent beam with prescribed intensity distribution given by Eq. (4) propagates a distance of f in free space, a thin lens with

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focal length f and a Gaussian amplitude filter, then becomes OCVs. After passing a

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spiral phase plate, the OCVs become OCVLs.

Fig. 1. Schematic for generating OCVLs. L, thin lens; GAF, Gaussian Amplitude filter; SPP, spiral phase plate.

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We can treat the paraxial propagation of the OCVLs through a stigmatic ABCD optical system by the following generalized Collins formula [40, 54]

WOCVLs  ρ1 , ρ2  

 ikD 2 2  exp   ρ1  ρ2   2B B 1

2

   

(9)

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 ikA 2 2   ik       WOCVLs  r1 , r2  exp   r1  r2  exp   r1  ρ1  r2  ρ2  d 2r1d 2r2 ,   2B  B     

where WOCVLs  r1 , r2  and WOCVLs  ρ1 , ρ2  are the CSD functions of the OCVLs in the

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source plane and output plane, respectively, ρ1 and ρ 2 are two arbitrary transverse position vectors in the output plane, A, B, C and D are the elements of the transfer

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matrix of the stigmatic optical system. The intensity and CDOC of the OCVLs in the

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output plane are obtained as

I  ρ   WOCVLs  ρ1 , ρ2  ,

  ρ1 , ρ2  

WOCVLs  ρ1 , ρ 2  I OCVLs  1  I OCVLs  ρ 2 

(10)

.

(11)

Now we study the focusing properties of the OCVLs. We assume the OCVLs is

focused by a thin lens with focal length f1 located in the source plane, and the output plane is at z as shown in Fig. 2. The transfer matrix of the optical system between the source plane and the output plane reads as

0  1  z / f1 z   A B  1 z  1 ,     1/ f 1    1/ f 1  C D   0 1  1   1

(12)

applying Eqs. (8)-(12), one can calculate numerically the propagation properties of

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OCVLs focused by the thin lens.

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Fig. 2. Schematic for focusing OCVLs. L1, thin lens.

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Fig. 3. Density plots of (a-1)-(d-1) the normalized intensity distribution I  v  / I  v max

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of the incoherent source for generating OCVLs, (a-2)-(d-2) the square of the modulus of the CDOC of the OCVLs in the source plane, and (a-3)-(d-3) the normalized intensity distribution I  ρ  / I  ρ max of the OCVLs in the focal plane with l  2 ,

0  2mm and z  f1  400mm for different M. Figure 3 shows the density plots of the normalized intensity distribution

I  v  / I  v max of the incoherent source for generating OCVLs, the square of the modulus of the CDOC of the OCVLs in the source plane, and the normalized intensity distribution I  ρ  / I  ρ max of the OCVLs in the focal plane with l  2 , 0  2mm

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and z  f1  400mm for different M. One finds from Fig. 3 that through varying the intensity distribution I  v  of the incoherent source by varying the value of M, we can obtain various OCVLs with different distributions of the CDOC in the source plane. Due to different distributions of the CDOC, the OCVLs display different intensity lattices in the focal plane (i.e., in the far field). Different from the intensity distribution of the OCLs in the focal plane, which also exhibits intensity lattices while

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each beamlet is of Gaussian beam profile, each beamlet of the intensity lattices of the OCVLs in the focal plane in our case displays hollow beam profile due to the vortex phase. The hollow beam spot array generated by the OCVLs is useful for simultaneously trapping multiple particles whose refractive indices are smaller than

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that of the ambient.

Fig. 4. Density plot of the normalized intensity distribution I  ρ  / I  ρ max of the focused OCVLs at several propagation distances with 0  2mm and M=4 for different values of the topological charge l .

To learn more about the influence of the vortex phase on the propagation properties of the OCVLs focused by a thin lens, we calculate in Figs. 4 and 5 the

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density plots of the normalized intensity distribution I  ρ  / I  ρ max and the square of the modulus of the CDOC of the focused OCVLs with 0  2mm and M=4 at several propagation distances for different values of the topological charge l . One sees from Fig. 4 that the Gaussian beam spot of the OCVLs gradually evolves into intensity lattices on propagation, and finally becomes Gaussian beam spot array for the case of l =0 and hollow beam spot array for the case of l  0 . The hollow size

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of each beamlet of the intensity lattices in the focal plane depends on l , which increases with the increase of l . On the other hand, one can infer from Fig. 5 that the lattices structure of the CDOC also disappears on propagation, and finally the CDOC of the OCVLs with l  0 displays cross like brighten spot arrays along x- and ydirections. It is interesting to find that the number of the bright spots along x- or y-

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direction is related with l by the relation N  2l 1 when l  0 [Fig. 3(e-2)-(e-4)]. Thus one can determine the topological charge of the OCVLs from the CDOC in the focal plane (or in the far field). In other words, the CDOC of the OCVLs can be used

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for information transfer if information is encoded into the CDOC.

Fig. 5. Density plot of the square of the modulus of the CDOC of the focused OCVLs with 0  2mm and M=4 at several propagation distances for different values of the

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topological charge l .

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Fig. 6. Density plots of (a-1)-(d-1) the normalized intensity distribution I  ρ  / I  ρ max and (a-2)-(d-2) the square of the modulus of the CDOC of the focused OCVLs with l  3 and M=4 in the focal plane for different values of the coherence width  0 .

Fig. 7. Density plots of (a-1), (b-1) the normalized intensity distribution I  ρ  / I  ρ max and (a-2), (b-2) the square of the modulus of the CDOC of the focused OCVLs with l  3 and 0  3mm in the focal plane for two different values of M.

Figures 6 and 7 show the density plots of the normalized distribution and the square of the modulus of the CDOC of the focused OCVLs in the focal plane for

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different values of the coherence width  0 and non-negative number M, respectively. One finds from Fig. 6 (a-1)-(d-1) that the intensity lattices of the OCVLs in the focal plane can be flexiblely controlled by the initial value of  0 , e.g., each beamlet gradually evolves from a hollow beam profile to a flat-topped profile and finally to a Gaussian beam profile as the value of  0 decreases gradually. The focused beam spot of OCVLs will be useful for trapping particles with different refractive indices,

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e.g., the intensity lattices with Gaussian beamlets or hollow beamlets in the focal plane can be used to simultaneously trap multiple particles whose refractive indices are larger or smaller than that of the ambient. From Fig. 7(a-1) and (b-1), we find that the number of the beamlets in the intensity lattices in the focal plane is controlled by the non-negative number M and increases as M increases. It is also interesting to find

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from Fig. 6(a-2)-(d-2), Fig. 7(a-2) and (b-2) that the square of the modulus of the CDOC of the OCVLs in the focal plane varies slightly as  0 or M varies, and the relationship between the bright spot number N along x- or y-direction and the

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topological charge l also exists.

3. Experimental generation of a partially coherent vortex beam with periodical

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coherence properties

In this section, we carry out experimental generation of OCVLs. Figure 8 shows the experimental setup for generating OCVLs and measuring the focused intensity. A

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beam emitted from a diode-pumped solid-state laser with wavelength   532nm

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passes through a beam expander and illuminates an amplitude mask after being reflected by a mirror, then goes towards the rotating ground-glass disk, producing an

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incoherent light with a prescribed intensity distribution. The generated incoherent light goes through a thin lens L and a Gaussian amplitude filter, producing OCVs [34]. After passing through a spiral phase plate with l=2, the OCVs becomes OCVLs. The generated OCVLs are focused by a thin lens L1 with focal length f1, and the focused beam intensity is recorded by the CCD camera. The neutral density filter is used to adjust the amplitude of the laser beam in our experiments. The method and detailed process for measuring the spatial coherence width and CDOC of partially coherent beam can be found in [13, 25, 48-50, 55].

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Fig. 8. Experimental setup for generating OCVLs and measuring the focused intensity. NDF, neutral density filter; BE, beam expander; M, mirror; AM, amplitude mask; RGGD, rotating ground-glass disk; L, L1, thin lens; GAF, Gaussian amplitude filter; SPP, spiral phase plate; CCD, camera; PC, computer.

Figure 9 shows our experimental results of the square of the modulus of the CDOC of the generated OCVLs in the source plane, and the normalized intensity distribution in the focal plane with 0  2.0mm and l  2 for different M. One

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finds from Fig. 9 that through varying the amplitude mask (i.e., varying M), as expected we indeed generate various OCVLs with different periodical coherent

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properties, which produce different intensity lattices in the focal plane and are useful for particle trapping. Our experimental results agree well with the simulation results

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shown in Fig. 3.

Fig. 9. Experimental results of (a-1)-(d-1) the square of the modulus of the CDOC of the generated OCVLs in the source plane, and (a-2)-(d-2) the normalized intensity

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distribution in the focal plane with 0  2.0mm and l  2 for different M.

Fig. 10. Experimental result of the normalized intensity distribution of the generated OCVLs in the focal plane for different transverse coherence widths  0 with M=4 and l  2.

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Figures 10 shows our experimental results of the normalized intensity distribution of the generated OCVLs in the focal plane for different transverse coherence widths  0 with M=4 and l  2 . One finds from Fig. 10 that through varying the value of the initial coherence width  0 , various intensity lattices, such as

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Gaussian spot array, flat-topped spot array, semi-hollow spot array, hollow spot array

4. Conclusion

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in the far field indeed can be generated as expected by Fig. 6.

We have introduced a new kind of partially coherent vortex beam with periodical

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coherence properties named OCVLs, and study the focusing properties both numerically and experimentally. We have found that the OCVLs display unique

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focusing properties, where are much different from those of OCLs and conventional partially coherent vortex beam. The OCVLs produce intensity lattices in the focal

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plane and the profile of each beamlet is controlled by the initial coherence width. One can obtain Gaussian array, flat-topped spot array, semi-hollow spot array and hollow spot array through varying initial coherence width, which will be useful for simultaneously trapping multiple particles whose refractive indices are larger or smaller than that of the ambient. Furthermore, we have found that the CDOC of the OCVLs contains the information of the topological charge l , i.e., the topological charge l is related with the number of the bright spots of the CDOC along x- or ydirection through the relation N  2l 1 for l  0 . Our results will be useful for

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particle trapping and information transfer. 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. 91750201, 11774251, 11474213 & 11804198, the Qing

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Lan Project of Jiangsu Province. References

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