Vector optical coherence lattices generating controllable far-field beam profiles

We introduce partially coherent vector sources with periodic spatial coherence properties, which we term vector optical coherence lattices (VOCLs), as an extension of recently introduced scalar OCLs. We derive the realizability conditions and propagation formulae for radially polarized VOCLs (i.e., a typical kind of VOCLs). We show that radially polarized VOCLs display nontrivial propagation properties and generate controllable intensity lattices in the far zone of the source (or in the focal plane of a lens). By adjusting source coherence, one can obtain intensity lattices with bright or dark nodes. The latter can be employed to simultaneously trap multiple particles or atoms as well as in free-space optical communications. 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Introduction
Optical lattices with periodic intensity, polarization and phase patterns have been examined theoretically and generated experimentally [1][2][3][4][5][6].They have also found numerous applications to atom cooling and trapping [7], ultracold quantum gas trapping [8], atomic clocks [9], microfluidic sorting [10], lattice light-sheet microscopy [11] and photonic crystal engineering [12].Besides the methods mentioned in [1][2][3][4][5][6], one can also generate intensity lattices through Tablot imaging of optical gratings because the optical gratings display transverse periodic properties [13].The above mentioned optical lattices are generated by coherent light fields.Recently, a new kind of optical lattices named optical coherence lattices (OCLs) was introduced in [14,15], where the OCLs are generated by partially coherent light sources with periodic coherence properties.It was shown in [14,15] that the OCLs exhibit periodicity reciprocity: the initial periodic degree of coherence transfers its periodicity to the periodic intensity distribution on propagation in free-space.In other words, the OCLs yield intensity lattices with bright nodes in the far field (or in the focal plane of an imaging lens).This property, unique to partially coherent sources, is useful for simultaneously trapping multiple particles whose refractive indices are larger than that of the ambient.In the turbulent atmosphere, it was shown [16] that the OCLs also display the periodicity reciprocity over long propagation distances and they have scintillation indices lower than those of Gaussian beams, even though the lattices are eventually affected by the turbulence.The discovered periodicity reciprocity makes OCLs attractive for robust free-space optical communications [14][15][16].More recently, experimental generation of OCLs was reported and it was shown that the OCLs may find applications to image transmission and optical encryption [17].However, all research on OCLs has so far focused on scalar light fields.
At the same time, polarization is a fundamental property of light fields.Depending on its polarization state, electromagnetic beams can be classified as uniformly polarized vector beams and non-uniformly polarized vector beams.Cylindrical vector beams, such as radially polarized and azimuthally polarized beams, are typical kinds of non-uniformly polarized vector beams [18].Due to their tight focusing properties [19,20] and self-healing potential [21], cylindrical vector beams have found applications to optical trapping, microscopy, optical data storage, lithography, proton acceleration, electron acceleration, material processing, plasmonic focusing, dark imaging, high-resolution metrology, free-space optical communication, super-resolution imaging and laser machining [18], among other venues.Partially coherent vector beams can be described in terms of their beam coherencepolarization (BCP) matrices in the space-time representation [22] or their cross-spectral density matrices in the space-frequency representation [23].A unified theory of coherence and polarization for partially coherent vector beams was developed by Wolf in 2003 [24], and the conditions for devising bona fide cross-spectral density matrices were introduced by Gori et al. in 2009 [25].Different definitions of the degree of coherence of partially coherent vector beams were developed [24,26].Partially coherent vector beams with either uniform or non-uniform states of polarization (SOP) were explored theoretically and generated experimentally [27][28][29][30][31][32][33][34][35].One of the key properties of a partially coherent vector beam SOP is its ability to change on propagation in free space [36].Further, the degree of polarization was shown to regain its magnitude at the source over sufficiently long distance in the turbulent atmosphere [37,38], making such beams attractive for free-space optical communications, laser radar system and remote sensing [39][40][41].By adjusting the coherence width of a nonuniformly polarized partially coherent vector beam, we can generate a Gaussian-like beam spot or a flat-topped beam spot or a dark hollow beam spot in the far field [32][33][34], which can come in handy for material thermal processing and for trapping a Rayleigh particle whose refractive index is larger or smaller than that of the ambient [34,42].In [35,43], a class of partially coherent vector beams, partially correlated azimuthal vortices, were employed in illuminating an imaging system.It was discovered that the image contrast can be thereby improved, which carries potential for metrology, microscopy, and lithography.
The coherence properties of previously studied vector beams display no periodicity at the source.In this work, we propose partially coherent vector beams with periodic spatial coherence properties, vector optical coherence lattices (VOCLs) as a natural extension of recently introduced scalar optical coherence lattices.The most interesting property of VOCLs is that the initial single radially polarized beam spot evolves into multiple radially polarized beam spots with bright or dark beam profiles in the far zone of the source (or in the focal plane of a lens), in other words, the VOCLs generate intensity lattices with bright or dark nodes depending on the magnitude of a controlled source coherence parameter.The scalar OCLs in our previous papers can only generate intensity lattices with bright nodes [14][15][16][17].The introduced VOCLs are expected to be useful for simultaneous trapping of multiple particles or atoms and for free-space optical communications.

Theoretical models for vector optical coherence lattices and realizability conditions
In this section, we will introduce a theoretical model for VOCLs and discuss their realizability conditions.The degree of coherence of scalar OCLs can be expressed as [14,17] ( ) ( ) ( ) Here δ and 0m V are coherence and phase parameters, respectively.One can synthetize scalar OCLs through, for instance, a superposition of multiple Schell-model beams with prescribed degrees of coherence [17].
Consider a BCP matrix ( ) ˆ, Γ r r of the source, defined as [22] ( ) ( ) ( ) ( ) Here the angle brackets denote ensemble averaging and ( ) , x y ≡ r are transverse position vectors in the source plane; ( ) To be a bona fide BCP matrix, the elements of the BCP matrix can be expressed as [25] ( ) ( ) ( ) ( ) where ( ) , H α is an arbitrary kernel, ( ) To obtain VOCLs, we define H α and P αβ as follows ( ) Here H α can be regarded as a transfer function of an optical path consisting of free space of distance f, a thin lens with the focal length f and a spatial filter with a transmission function T α .Further, λ is the wavelength of the light field.

( )
P αβ v can be regarded as a superposition of multiple circ functions with a radius a αβ and off-axis displacements ( ) ; d denotes the separation between adjacent circ functions, M and N stand for the numbers of the  5) and ( 6) into Eq.( 3), we obtain the elements of the BCP matrix of VOCLs as where 0 C is a constant, , αβ γ r r represent the correlation functions given by ( ( ) Here 0 2 f ka are referred to as coherence parameters.Comparing Eqs. ( 1) and ( 8), we infer that the expressions of the correlation functions of VOCLs are similar to the expression of the degree of coherence of scalar OCLs.
The VOCL SOP is closely related to T α .If we set and s w being a constant and the beam width, respectively, VOCLs exhibit uniform SOP (i.e., the state of polarization of any point in the source plane is the same) and are called uniformly and are termed radially or azimuthally polarized VOCLs.
Hereafter, we mainly focus on radially polarized VOCLs, which can be readily generated in the laboratory.First, we discuss the realizability conditions.To be a physically realizable partially coherent vector beam, it is known that the BCP matrix of VOCLs should be quasi-Hermitian [25], namely , , r r .To meet this requirement, the following conditions should be satisfied To check the non-negativity conditions, substituting from Eq. ( 6) into Eq.( 4), we obtain the following inequality Applying the equation , it is not difficult to derive the following inequality from Eq. ( 10) Equation ( 11) is equivalent to the following inequality ( ) ( ) ( ) ( ) To satisfy the additional conditions (a) and (b), substituting the elements of the BCP matrix of radially polarized VOCLs into Eqs.( 13)-( 15), we obtain the equalities Equations ( 9), ( 12) and ( 16) are the realizability conditions for radially polarized VOCLs.
In this work, we set 1 x y A A = = .By applying Eq. ( 16), we find the correlation functions of the radially polarized VOCLs satisfy the following equalities , , , , .
The electromagnetic degree of coherence of radially polarized VOCLs is defined as [26] ( ) ( ) ( ) ( ) Substituting from Eqs. ( 7) and ( 17) into Eq.( 18), we obtain It follows from Eq. ( 19) that the degree of coherence of the radially polarized VOCLs and the correlation functions are the same.To save space, we only display theoretical and experimental results for the degree of coherence , μ r r hereafter.Figure 1 shows the density plot of the square of the degree of coherence ( ) , ,1mm,1mm x y μ of radially polarized VOCLs for different values of M and N. We see from Fig. 1 that the degree of coherence of radially polarized VOCLs does display a latticeslike behavior when M>1 and N>1, and the lattice structure becomes more complex as the magnitudes of M and N increase.

Propagation properties of radially polarized vector optical coherence lattices
In this section, we will derive the analytical propagation formulae for radially polarized VOCLs and explore their propagation properties in free space numerically.The BCP matrix of a partially coherent vector beam passing through a stigmatic ABCD optical system can be expressed using the so-called generalized Collins formula as [44] ( where ( ) Substituting the elements of BCP matrix of radially polarized VOCLs in the source plane into Eq.( 20), we obtain (after integration) the following expressions for the elements of the BCP matrix in the receiver plane where Here t H denotes a Hermite polynomial of order t.The average intensity of radially polarized VOCLs is obtained as [22,23] ( xx yy x y Applying Eqs. ( 13)-( 15) and ( 21)-( 26), we can study the propagation properties (e.g., average intensity and SOP) of radially polarized VOCLs in free space numerically by setting the elements of the transfer matrix of free space as 1, , 0, We can study the focusing properties of radially polarized VOCLs focused by a thin lens with focal length f by setting 1 / , , 1/ , 0, = Here we assumed the distances from the source plane to the thin lens and from the thin lens to the receiver plane to be f and z, respectively.We note that the beam parameters in the far field and the focal plane are the same.
We now proceed to numerically study propagation properties of radially polarized VOCLs in free space.To this end, we set 0.4mm, =1mm a δ with M = N = 3.We infer from Figs. 2-4 that the beams display instructive propagation features in free space.In particular, dark hollow beam profiles at the source gradually evolve into lattice distributions on propagation, implying that the degree of coherence transfers its periodicity to beam intensity distributions.The intensity lattices in the far field are controlled by the initial coherence parameter 0 δ .Thus, we can obtain intensity lattices with dark nodes (i.e., dark hollow beam arrays) when the coherence parameter 0 δ is large (see Fig. 2), and we can obtain intensity lattices with bright nodes (i.e., flat-topped or Gaussian-like beam arrays) as we decrease the coherence parameter 0 δ (see Figs. 3 and 4).In addition, we deduce from Figs. 2-4 that the radially polarized VOCLs SOP varies on propagation, and a single radial polarized beam spot at the source evolves into multiple radially polar polarization.δ , and we find that the number of dark cores equals to M N × .Numerical results (not shown here) also show that the initial parameter d only affects the separation of the dark or solid cores in the far-field lattices and doesn't affect the SOP of each core.The intensity periodicity in the focal plane of a lens (or in the far zone of the source) in this work is caused by the spatial coherence periodicity in the transverse plane of the source, and this periodicity reciprocity is specific to partially coherent sources.Thus, one can modulate the distribution of the intensity lattices in the far field (or in the focal plane) conveniently through varying its initial coherence properties, which will be useful for particle trapping, e.g., the obtained intensity lattices with solid cores or dark cores in the focal plane can be used to simultaneously trap multiple particles whose refractive indices are larger or smaller than that of the ambient.The obtained intensity lattices with dark cores in the focal plane can also be utilized for simultaneous trapping of multiple atoms if the light field is blue-detuned.Furthermore, it is known that both scalar OCLs and radially polarized beams have advantage over Gaussian beams for mitigating the effect of atmospheric turbulence [16,41].One may expect to further mitigate the effect of atmospheric turbulence using radially polarized VOCLs, which will be useful in free-space optical communications.In our previous papers [14][15][16][17], the proposed scalar OCLs do not carry phase vortices.Thus, they generate intensity lattices with only bright nodes in the far field.In principle, one may expect to generate intensity lattices with dark nodes from scalar OLCs with phase vortices.In any case, our work clearly shows that manipulating coherence properties of a radially polarized beam provides a novel way to produce far-zone intensity lattices with bright or dark nodes.

Experimental generation of radially polarized vector optical coherence lattices
In this section, we report the experimental generation and characterization of radially polarized VOCLs.We employ a radial polarization converter (RPC) to convert linearly polarized VOCLs into radially polarized VOCLs.Figure 6 shows our experimental setup for generating radially polarized VOCLs and measuring its degree of coherence and focused intensity.In our experiment, a laser beam emitted by an Nd:YAG laser ( 532nm λ = ) first passes through a linear polarizer and a beam expander, followed by an amplitude mask (AM), which is used to modulate the intensity distribution of the incident beam.Here the AM is a M N × circular aperture array; a is the radius of each aperture and d the separation between adjacent apertures.The modulated beam from the AM illuminates a rotating ground-glass disk (RGGD), producing an incoherent beam with a prescribed intensity distribution.Here the speed of the RGGD is controlled by a motion controller.Having passed through a thin lens L 2 with the focal length 1 25cm f = and the Gaussian amplitude filter (GAF), the incoherent beam from the RGGD transforms into linearly polarized VOCLs [17].We then convert it into a radially polarized VOCLs by the RPC.The coherence parameter of the generated beam is controlled by varying the radius of the aperture through the relation 0 / 2 f ka δ = . The experimental setup for generating linearly polarized VOCLs is the same as that reported in [17], and the key point for generating radially polarized VOCLs in our experiment is the RPC use.Fig. 6.Experimental setup for generating radially polarized VOCLs, measuring the degree of coherence and the focused intensity.Laser, Nd: YAG laser; LP, linear polarizer; BE, beam expander; AM, amplitude mask; L 1 , L 2 and L 3 , thin lenses; RGGD, rotating ground-glass disk; MC, motion controller; GAF, Gaussian amplitude filter; RPC, radial polarization converter; BS, beam splitter; CCD, charge-coupled device; BPA, beam profile analyzer; PC, personal computer.
To measure the degree of coherence and the focused intensity of the generated radially polarized VOCLs, we use a beam splitter (BS) to split the generated beam into two beams.The transmitted beam from the BS passes through the thin lens L 2 with the focal length 2 15cm f = arriving at a charge-coupled device (CCD).Both distances from RPC to L 2 and from L 2 to CCD are 2 2 f (i.e., 2f imaging system) such that the VOCLs degree of coherence in the CCD plane is the same as that in the source plane (just behind the RPC).Here the CCD is used to measure the modulus of the VOCLs degree of coherence.A detailed measurement protocol can be found in [33].The beam reflected from the BS is then transmitted through a thin lens L 3 with the focal length 3 15cm f = .The resulting beam then arrives at the beam profile analyzer (BPA), which is used to measure the focused intensity distribution.The distances from the BPA to L 3 and from L 3 to BPA are 3 f and z, respectively.The elements of the transfer matrix of the optical system between the RPC and BPA are    evolves into an intensity lattice in the focal plane (or the far field), and one can obtain lattices with bright or dark nodes by varying the source coherence parameter.Furthermore, each bright or dark node of the lattice features radial polarization as expected.Our experimental results are consistent with our theoretical predictions.

Summary
We have introduced vector optical coherence lattices as a natural extension of recently introduced scalar OCLs, and we have explored the propagation properties of radially polarized VOCLs as a numerical example.In contrast with the scalar OCLs, which generate intensity lattices with bright nodes in the far zone of the source, the radially polarized VOCLs can generate intensity lattices with bright or dark nodes depending on the magnitude of the source coherence parameter.In addition, we have reported the experimental generation of radially polarized VOCLs and characterized their focusing properties.Our experimental results verify our theoretical predictions.Engineering spatial coherence properties of vector beams paves a way for manipulating their propagation properties and for beam shaping.These tools can be useful for trapping multiple particles whose refractive indices are larger or smaller than that of their surroundings, for simultaneous multiple atom trapping and for freespace optical communications.

Funding
α r is a fluctuating electric field component along the α axis at point r .
circ functions along x and y directions, respectively.coefficients between the field components E α and E β .Substituting from Eqs. (

1 A r and ( ) 2 A
To realize radially polarized VOCLs, two additional conditions should be imposed: (a) any point in the source plane is linearly polarized, (b) the orientation angle of the polarization at any point in the source plane should satisfy ( ) ( ) known that the BCP matrix of a partially coherent vector beam can be represented as a sum of the BCP matrix of a completely polarized beam and the BCP matrix of a completely unpolarized beam[23,36].The SOP of the completely polarized beam can be characterized by the polarization ellipse, with the major and minor semi-axes of the polarization ellipse ( ) r , the degree of ellipticity ( ) ε r and the orientation angle ( ) θ r related with the elements of the BCP matrix as[23,36]

Fig. 1 .
Fig. 1.Density plot of the square of the the degree of coherence()2 1 1, ,1mm,1mm x y μ ; A, B, C, and D denote optical system transfer matrix elements.

Fig. 4 ..Fig. 5 .
Fig. corres the SO N = 3 components of the focused intensity x I and y I can be measured by adding a linear polarizer between L 3 and BPA.

Figure 7
Figure 7 shows our experimental results for the squared modulus of the radially polarized VOCLs degree of coherence ( ) 2 Figure 8 shows our experimental results for the intensity distribution and corresponding components x several propagation distances.Figure 9 shows our experimental results for the VOCLs intensity distribution with 3 M N = = and 1mm d = in the focal plane for different values of the coherence parameter 0 δ .One infers from Figs. 8 and 9 that a single source beam

National
Natural Science Fund for Distinguished Young Scholar (11525418); National Natural Science Foundation of China (11474213, 11374222, 11274005); Project of the Priority Academic Program Development (PAPD) of Jiangsu Higher Education Institutions; National Science and Engineering Research Council of Canada (NSERC).