Radially polarized twisted partially coherent vortex beams

Leixin Liu; Haiyun Wang; Lin Liu; Yan Ye; Fei Wang; Fei Wang; Yangjian Cai; Yangjian Cai; Yangjian Cai; Xiaofeng Peng; Xiaofeng Peng

doi:10.1364/OE.452147

1. Introduction

Polarization is one of the most salient features of a laser beam, which manifests the oscillation state of electric field at a spatial point. Beams with spatially varying state of polarization (SOP) across the beam’s cross section are referred to vector beams [1]. As a typical kind of vector beams, radially polarized beam whose SOP is linearly polarized but its oscillation direction at any point is along the radial direction has attracted wide attention due to its unique tight focusing properties, and has important applications in plasmonic focusing, material processing, second-harmonic generation, particle trapping and detection [1–7]. In addition to polarization, spatial coherence is another intrinsic characteristic in a light field, describing the correlations of randomly fluctuating electromagnetic fields at two or more spatial points [8]. Light beams with low coherence (i.e., partially coherent beam) have attracted intensive attention for their widely applications in beam shaping, optical coherence encryption and robust far-field imaging [9–11]. Moreover, as a natural extension of coherent vector beams, partially coherent vector beams have been widely investigated both theoretically and experimentally [8,12–16]. Perhaps, radially polarized partially coherent (RPPC) beam is one of the most known kind of partially coherent vector beams. Such the beam has radial polarization and the degree of coherence (DOC) is of a Gaussian function in the source plane. The propagation characteristics of the RPPC beam have been extensively studied recently [17–23]. It was found that the RPPC beam presents interesting features on propagation [17–19], e.g., depolarization effects and spatial coherence dependence intensity evolution. Compared with a scalar partially coherent beam, RPPC beam can reduce the intensity scintillation caused by atmospheric turbulence [19], making it a promise source for free space optical communications. Recently, RPPC beams with nonconventional correlation functions were introduced [21–23], and the self-reconstruction and tight focusing properties have been explored. It was shown that these beams exhibit strong self-reconstruction of its intensity profile and the SOP scattered by an opaque obstacle [22]. Both transverse and longitudinal field distributions can be shaped by engineering the structure of the correlation functions of the beams [23].

In the aforementioned studies for the RPPC beams, the phase distribution was not considered i.e., the phase distribution in the initial plane is uniform. In fact, phase is the most important characteristic in a light field, which plays a significant role in beam shaping, optical communications, optical trapping and so on. For instance, light beams with vortex phase (known as vortex beams) carry an amount of orbital angular momentum (OAM) that is quantified by the topological charge l [24,25], and have found applications in free-space information transfer, detection of spinning object and edge detection in image processing [26–28]. The study of the vortex beams also extended from complete spatial coherence to partial spatial coherence, i.e., partially coherent vortex (PCV) beams [29]. The statistical properties of PCV beams are quite different with their coherent counterparts. The phase singularities in the vortex core will disappear while correlation singularities (i.e., ring dislocations) appear in its DOC function on propagation [30,31]. Further, PCV beams have more advantages in reducing turbulent-induced scintillation, beam wander and depolarization, and exhibit stronger self-reconstruction [32–35]. PCV beams with fractional vortex phase also were proposed recently [36,37]. In addition to the vortex phase, there is another nontrivial phase (named twist phase) that induces the beam carrying OAM [38,39]. Twist phase was first introduced by Simon, which only exists in partially coherent beams [40]. Twist phase not only induces the rotation of beam spot on propagation, but also have advantages in resisting coherence (or turbulence)-induced degeneration, depolarization, overcoming the classical Rayleigh limit, information transfer through the turbulent atmosphere and quantum information processing [41–46]. In addition, the vector twisted Gaussian Schell model (TGSM) beams also has been studied. It was shown that the twist phase can be used to control the polarization state of the soliton [47]. The restriction has been solved whether the twist phase could add on partially coherent beams with nonconventional correlation functions or not [48,49]. Friberg et al. first realized the experimental generation of a partially coherent beam carrying twist phase [50], Wang et al. have simplified the experimental scheme for generating a bona fide twisted Gaussian Schell-model beam, which provide a new way for generating other types of partially coherent beams carrying twist phase [51].

Recently light beams carrying both vortex phase and twist phase has been introduced [52,53], and it was found that one can control the distribution of the intensity, the DOC, the OAM and the capability of self-reconstruction by modulating the chirality of these two phases. In this paper, we combine the spatial varying polarization, two nontrivial phase and partial spatial coherence in a light beam simultaneously. A new kind of partially coherent vector beam named radially polarized twisted partially coherent vortex (RPTPCV) beam is introduced. The effects of twist phase, vortex phase, polarization and coherence on the second-order statistical properties such as average spectral density (intensity distribution), the DOP and the SOP, of the RPTPCV beam are explored in detail. Furthermore, we have generated the RPTPCV beam in experiment, and measure the intensity distribution and polarization properties experimentally. The experimental results agree well with the theoretical results. Our strategy provides a new way to modulate the statistical properties of the partially coherent vector beams and promote the application of the partially coherent vector beams in the particles trapping and the optical communication.

2. Theoretical mode and statistical properties of a RPTPCV beam through a paraxial ABCD optical system

Let us consider a quasi-chromatic, statistically stationary stochastic beam-like field, propagation along z-axis. According to the unified theory of coherence and polarization [8], the second-order statistical characteristics of such vector source (z = 0) can be characterized by a 2 × 2 cross-spectral density (CSD) matrix $\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over W} ({{{\textbf r}_1},{{\textbf r}_2};\omega } )$ in the space-frequency domain, defined by the formula

(1)$$\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over W} ({{{\textbf r}_1},{{\textbf r}_2};\omega } )= \left( {\begin{array}{{cc}} {{W_{xx}}({{\textbf r}_1},{{\textbf r}_2};\omega )}&{{W_{xy}}({{\textbf r}_1},{{\textbf r}_2};\omega )}\\ {{W_{yx}}({{\textbf r}_1},{{\textbf r}_2};\omega )}&{{W_{yy}}({{\textbf r}_1},{{\textbf r}_2};\omega )} \end{array}} \right),$$

where ${{\textbf r}_1} \equiv ({x_1},{y_1})$ and ${{\textbf r}_2} \equiv ({x_2},{y_2})$ are two arbitrary position vectors in the source plane. The elements is expressed as ${W_{\alpha \beta }}({{\textbf r}_1},{{\textbf r}_2};\omega ) = \left\langle {E_\alpha^\ast ({{\textbf r}_1};\omega ){E_\beta }({{\textbf r}_2};\omega )} \right\rangle$, (α, β = x, y), where E_x and E_y represent randomly fluctuating electric field with respect to two mutually orthogonal x and y direction, perpendicular to z-axis. Here, the asterisk and angle brackets denote the complex conjugate and ensemble averaging over the source field fluctuations. ω is the angular frequency of the light beam. For brevity, the dependence of the derived quantities on the angular frequency is suppressed in the following analysis. For the RPTPCV source, the elements of the CSD matrix takes the form

(2)$$\begin{aligned} {W_{\alpha \beta }}({{{\textbf r}_1},{{\textbf r}_2}} )&= \frac{{{\alpha _1}{\beta _2}}}{{4\sigma _0^2}}\textrm{exp} \left( { - \frac{{{\textbf r}_1^2 + {\textbf r}_2^2}}{{4\sigma_0^2}}} \right)\textrm{exp} \left( { - \frac{{{{|{{{\textbf r}_1} - {{\textbf r}_2}} |}^2}}}{{2\delta_0^2}}} \right)\textrm{exp} [{ik{\mu_0}({{x_2}{y_1} - {x_1}{y_2}} )} ]\\ &\times \textrm{exp} [{ - il({{\varphi_1} - {\varphi_2}} )} ]\textrm{, }({\alpha ,\beta = x,y} ), \end{aligned}$$

where ${\sigma _0}$ and ${\delta _0}$ represent the initial beam width and transverse coherence width, respectively. The term $\textrm{exp} [{ik{\mu_0}({{x_2}{y_1} - {x_1}{y_2}} )} ]$ denotes the twist phase depending on the two non-separable spatial points. k = 2π / λ is the wavenumber with λ being the wavelength. μ₀ is the real-valued twist factor, a measurement of the strength of the twist phase. Note that the twist factor is bounded by the inequality $\mu _0^2 \le {({{k^2}\delta_0^4} )^{ - 1}}$. Hence, the twist phase vanishes when the beam is completely coherent (${\delta _0} \to \infty$). The last term $\textrm{exp} [{ - il({{\varphi_1} - {\varphi_2}} )} ]$ stands for the vortex phase with ${\varphi _i} = \arctan ({y_i}/{x_i}),(i = 1,2)$ being the azimuthal angle and l being the topological charge. One can see that the RPTPCV beam in the source plane contains the two nontrivial phases, i.e., twist phase and vortex phase, and the polarization distribution is radially polarized. In the following analysis, we focus on the influences of the initial phases, polarization and coherence state on the second-order statistical properties upon propagation.

It has been shown in [54,55] that the CSD of the TGSM beam can be written as the incoherent superposition of Laguerre-Gaussian (LG) modes, given by

(3)$$\begin{aligned} {W_T}({{{\bf r}_1},{{\bf r}_2}} )&= \textrm{exp} \left( { - \frac{{{\bf r}_1^2 + {\bf r}_2^2}}{{4w_0^2}}} \right)\textrm{exp} \left( { - \frac{{{{|{{{\bf r}_1} - {{\bf r}_2}} |}^2}}}{{2\delta_0^2}}} \right)\textrm{exp} ({ - iu({{x_1}{y_2} - {x_2}{y_1}} )} )\\ &= \sum\limits_{m ={-} \infty }^\infty {\sum\limits_{n = 0}^\infty {{\kappa _{nm}}[} } \Theta _{nm}^{}({r_1},{\varphi _1}){]^\ast }\Theta _{nm}^{}({r_2},{\varphi _2}) \end{aligned}$$

with

(4)$$\Theta _{nm}^{}({r,\varphi } )= \frac{1}{\omega }\sqrt {\frac{{2n!}}{{\pi ({n + |m |} )!}}} {\left( {\frac{{\sqrt 2 r}}{\omega }} \right)^{|m |}}L_n^{|m |}\left( {\frac{{2{r^2}}}{{{\omega^2}}}} \right)\textrm{exp} \left( { - \frac{{{r^2}}}{{{\omega^2}}}} \right)\textrm{exp}(im\varphi ).$$

Comparing the expression of the CSD function of the TGSM beam and the elements of the CSD matrix of the RPTPCV beam, the elements can be written as

(5)$$\begin{aligned} {W_{\alpha \beta }}({{{\textbf r}_1},{{\textbf r}_2}} )&= \frac{{{\alpha _1}{\beta _2}}}{{4\sigma _0^2}}\textrm{exp} [{ - il({{\varphi_1} - {\varphi_2}} )} ]{W_T}({{{\textbf r}_1},{{\textbf r}_2}} )\\ &= \sum\limits_{m ={-} \infty }^\infty {\sum\limits_{n = 0}^\infty {{\kappa _{nm}}\frac{{{\alpha _1}}}{{2{\sigma _0}}}\textrm{exp} ({ - il{\varphi_1}} )[} } \Theta _{nm}^{}({r_1},{\varphi _1}){]^\ast }\frac{{{\beta _2}}}{{2{\sigma _0}}}\textrm{exp} ({il{\varphi_2}} )\Theta _{nm}^{}({r_2},{\varphi _2}) \end{aligned}$$

After some tedious operation, we have

(6)$${W_{\alpha \beta }}({{{\textbf r}_1},{{\textbf r}_2}} )= \sum\limits_{m ={-} \infty }^\infty {\sum\limits_{n = 0}^\infty {{\lambda _{nm}}} } {[{\Phi _{n{\alpha_1}}^m({{{\bf r}_1}} )} ]^\ast }\Phi _{n{\beta _2}}^m({{{\bf r}_2}} ),(\alpha ,\beta = x,y),$$

where the modes $\Phi _{n\alpha }^m({\bf r} )$ take the form

(7)$$\Phi _{n\alpha }^m({r,\varphi } )= {T_{mn}}\frac{\alpha }{\omega }{\left( {\frac{{\sqrt 2 r}}{\omega }} \right)^{|m |}}L_n^{|m |}\left( {\frac{{2{r^2}}}{\omega }} \right)\textrm{exp} \left( { - \frac{{{r^2}}}{{{\omega^2}}}} \right)\textrm{exp} [{i({l + m} )\varphi } ]\textrm{,}$$

with

(8)$${T_{mn}} = \frac{2}{\omega }{\left[ {\sum\limits_{q = 0}^n {\frac{{{{({\varepsilon^{\prime}} )}_q}{{({\beta^{\prime}} )}_q}}}{{q!{{({\gamma^{\prime}} )}_q}}}{{[{{f^{(n )}}(h )} ]}_{h = 0}}} } \right]^{ - 1/2}}\sqrt {\frac{{2n!n!|m |!}}{{\pi ({n + |m |} )!({|m |+ 1} )!}}} ,$$

(9)$${[{{f^{(n )}}(h )} ]_{h = 0}} = {\left\{ {\frac{{{d^n}}}{{d{h^n}}}\left[ {\frac{{{{({4h} )}^q}}}{{{{({1 - h} )}^{ - 1}}{{({1 + h} )}^{2 + |m |+ 2q}}}}} \right]} \right\}_{h = 0}},{\omega ^2} = \frac{1}{{\sqrt {{a^2} + 2ab + {u^2}/4} }},$$

(10)$${(\tau )_q} = \frac{{\Gamma ({\tau + q} )}}{{\Gamma (\tau )}},({\tau = \varepsilon^{\prime},\beta^{\prime},\gamma^{\prime}} ),\varepsilon ^{\prime} = ({|m |+ 2} )/2,\beta ^{\prime} = 1 + ({|m |+ 1} )/2,\gamma ^{\prime} = 1 + |m |,$$

where r and φ are radial and azimuthal angle in the polar coordinate system. $a = 1/4\sigma _0^2$, $b = 1/2\delta _0^2$, $u = k{\mu _0}$ is a parameter related to the twist factor. $L_n^{|m |}$ denotes the associated Laguerre polynomial with mode indices m and n. $\Gamma (. )$ and${f^{(n )}}(. )$ represent the Gamma function and nth-order derivatives, respectively. The mode weights ${\lambda _{nm}}$ is easily derived and defined as

(11)$${\lambda _{nm}} = \frac{{a\pi }}{2}{\omega ^2}({1 - \xi } ){t^m}{\xi ^{|m |/2 + n}}\frac{{{\omega ^2}}}{4}\frac{{({|m |+ 1} )!}}{{n!|m |!}}\left[ {\sum\limits_{q = 0}^n {\frac{{{{({\varepsilon^{\prime}} )}_q}{{({\beta^{\prime}} )}_q}}}{{q!{{({\gamma^{\prime}} )}_q}}}{{[{{f^{(n )}}(h )} ]}_{h = 0}}} } \right],$$

with $t = \sqrt {({b + u/2} )/({b - u/2} )} $, $\xi = ({a + b - 1/{\omega^2}} )/({a + b + 1/{\omega^2}} )$. It is worth mentioning that the modes $\Phi _{n\alpha }^m({\bf r} )$ are not the standard form of Laguerre-Gaussian (LG) modes, and therefore, they are not the exact solution of the paraxial wave equation in the polar system. In addition, the modes are not mutually orthogonal with each other, which is different from the modal expansion in [48]. Such expansion is called pseudo-model expansion [56], pertinent to reproducing kernel Hilbert spaces (RKHSs) in mathematics [57]. Further, it is shown in Eq. (6) that to exactly represent the elements of the CSD matrix, one must involve the infinite number of the modes $\Phi _{n\alpha }^m({\bf r} )$. Nevertheless, the weights ${\lambda _{nm}}$ in general decrease with the increase of the mode order m and n, which means that one could use the limited number of the modes to approximately represent the RPTPCV source.

To assess how many modes should be involved in practical situation, Fig. 1(a) illustrates the variation of the normalized mode weights defined as ${D_{nL}} = {\lambda _{nm}}/\sum\limits_{m ={-} \infty }^\infty {\sum\limits_{n = 0}^\infty {{\lambda _{nm}}} } $ against the mode order L = l + m and n. The beam parameters used in the calculations are λ= 632.8nm, δ₀ = 0.5mm, σ₀ = 1mm, and l = 4. As expected, the normalized mode weights decrease as the mode order n or L increases, implying that those modes with larger mode orders have less contribution on the elements of the CSD matrix. The mode weight distributions are also affected by the twist factor, the topological charge l and the coherence width (not shown here). In general, the mode weight distributions become more dispersive as the coherence width decreases, resulting in more modes should be considered in the calculation. In Fig. 1(b), the spectral density ${S_0}({\textbf r}) = Tr[\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over W} ({\textbf r},{\textbf r})]$ at the cross-line y = 0, where Tr denotes the trace of the matrix, with different truncation parameters λ_c are presented. The truncation parameter means that only the modes which the normalized mode weights is larger than λ_c, i.e., D_nL> λ_c, are involved in the calculation. One finds that when λ_c <0.001, the spectral density calculated from the pseudo-mode expansion are exactly the same with that calculated from the theoretical mode directly.

Fig. 1. (a) Distribution of normalized mode weights D_nL of the RPTPCV beam with l = 4 as a function of n and L = l + m for three different values of twist factor μ₀. (b) The spectral density at the cross-line y = 0, with different truncation parameter λ_c.

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On the basis of the modal expansion, the elements of the CSD matrix of the RPTPCV beam propagating through a paraxial ABCD optical system also can be expressed as the incoherent superposition of spatially coherent modes

(12)$${W_{\alpha \beta }}({{{\mathbf \rho }_1},{{\mathbf \rho }_2},z} )= \sum\limits_{m ={-} \infty }^\infty {\sum\limits_{n = 0}^\infty {{\lambda _{nm}}} } {[{\Phi _{n{\alpha_1}}^m({{{\mathbf \rho }_1},z} )} ]^\ast }\Phi _{n{\beta _2}}^m({{{\mathbf \rho }_2},z} ),$$

where ρ₁ and ρ₂ are two position vectors in the output plane. $\Phi _{n\alpha }^m({{\mathbf \rho },z} )$, (α = x, y) denote the optical modes generated by the corresponding modes $\Phi _{n\alpha }^m({\bf r} )$ propagation from the source plane to the output plane. Within the accuracy of the paraxial approximation, the two modes is connected with the Collins integral [58]

(13)$$\begin{aligned} \Phi _{n\alpha }^m({{\mathbf \rho },z} )&= \frac{{ik}}{{2\pi B}}\textrm{exp} \left( {\frac{{ikD{{\mathbf \rho }^2}}}{{2B}}} \right)\int {\int {\Phi _{n\alpha }^m({\bf r} )} } \\ &\times \textrm{exp} \left( {\frac{{ikA}}{{2B}}{{\bf r}^2}} \right)\textrm{exp} \left( { - \frac{{ik}}{B}{\bf r}\cdot {\mathbf \rho }} \right){d^2}{\bf r}, \end{aligned}$$

where A, B, D are the elements of transfer matrix of the optical system. To evaluate Eq. (13), we define a new function, i.e.,

(14)$$f_{n\alpha }^m({\bf r} )= \Phi _{n\alpha }^m({\bf r} )\textrm{exp} \left( {\frac{{ikA}}{{2B}}{{\bf r}^2}} \right),$$

and therefore, Eq. (13) can be written as the form of Fourier transform

(15)$$\Phi _{n\alpha }^m({{\mathbf \rho },z} )= \frac{{ik}}{{2\pi B}}\textrm{exp} \left( {\frac{{ikD{{\mathbf \rho }^2}}}{{2B}}} \right)F[{f_{n\alpha }^m({\bf r})} ]\left\{ {\frac{{\mathbf \rho }}{{\lambda B}}} \right\},$$

where $F[. ]$ represents the Fourier transform of the function $f_{n\alpha }^m({\bf r})$. Equation (15) can be solved numerically with the help of fast Fourier transform (FFT) algorithm.

The spectral density and the DOP of the RPTPCV beam in the output plane are obtained from the CSD matrix, given by

(16)$$\begin{aligned} S({{\mathbf \rho },z} ) &= \textrm{Tr[}\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over W} ({{\mathbf \rho },{\mathbf \rho },z} )\textrm{]}\\ &= \sum\limits_{m ={-} \infty }^\infty {\sum\limits_{n = 0}^\infty {{\lambda _{nm}}} } {|{\Phi _{nx}^m({{\mathbf \rho },z} )} |^2} + \sum\limits_{m ={-} \infty }^\infty {\sum\limits_{n = 0}^\infty {{\lambda _{nm}}{{|{\Phi _{ny}^m({{\mathbf \rho },z} )} |}^2}} } , \end{aligned}$$

(17)$$P({\mathbf \rho } )\textrm{ = }\sqrt {1 - \frac{{4Det\left[ {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over W} ({{\mathbf \rho },{\mathbf \rho },z} )} \right]}}{{{{\left\{ {Tr\left[ {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over W} ({{\mathbf \rho },{\mathbf \rho },z} )} \right]} \right\}}^2}}}} ,$$

where Det denote the determinant of the matrix.

For a partially coherent vector beam with any state of coherence, the CSD matrix can be represented as a sum of a completely unpolarized portion and a completely polarized portion [8], i.e.,

(18)$$\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over W} ({{\mathbf \rho },{\mathbf \rho },z} )= {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over W} ^{(u )}}({{\mathbf \rho },{\mathbf \rho },z} )+ {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over W} ^{(p )}}({{\mathbf \rho },{\mathbf \rho },z} ),$$

where

(19)$${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over W} ^{(u )}}({{\mathbf \rho },{\mathbf \rho },z} )\textrm{ = }\left( {\begin{array}{{cc}} {A({{\mathbf \rho },{\mathbf \rho },z} )}&0\\ 0&{A({{\mathbf \rho },{\mathbf \rho },z} )} \end{array}} \right),{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over W} ^{(p )}}({{\mathbf \rho },{\mathbf \rho },z} )= \left( {\begin{array}{{cc}} {B({{\mathbf \rho },{\mathbf \rho },z} )}&{D({{\mathbf \rho },{\mathbf \rho },z} )}\\ {{D^ \ast }({{\mathbf \rho },{\mathbf \rho },z} )}&{C({{\mathbf \rho },{\mathbf \rho },z} )} \end{array}} \right),$$

with

(20)$$A({{\mathbf \rho },{\mathbf \rho },z} )\textrm{ = }\frac{1}{2}\left[ {{W_{xx}}({{\mathbf \rho },{\mathbf \rho },z} )+ {W_{yy}}({{\mathbf \rho },{\mathbf \rho },z} )- \sqrt {{{[{{W_{xx}}({{\mathbf \rho },{\mathbf \rho },z} )- {W_{yy}}({{\mathbf \rho },{\mathbf \rho },z} )} ]}^2} + 4{{|{{W_{xy}}({{\mathbf \rho },{\mathbf \rho },z} )} |}^2}} } \right],$$

(21)$$B({{\mathbf \rho },{\mathbf \rho },z} )\textrm{ = }\frac{1}{2}\left[ {{W_{xx}}({{\mathbf \rho },{\mathbf \rho },z} )- {W_{yy}}({{\mathbf \rho },{\mathbf \rho },z} )+ \sqrt {{{[{{W_{xx}}({{\mathbf \rho },{\mathbf \rho },z} )- {W_{yy}}({{\mathbf \rho },{\mathbf \rho },z} )} ]}^2} + 4{{|{{W_{xy}}({{\mathbf \rho },{\mathbf \rho },z} )} |}^2}} } \right],$$

(22)$$C({{\mathbf \rho },{\mathbf \rho },z} )\textrm{ = }\frac{1}{2}\left[ {{W_{yy}}({{\mathbf \rho },{\mathbf \rho },z} )- {W_{xx}}({{\mathbf \rho },{\mathbf \rho },z} )+ \sqrt {{{[{{W_{xx}}({{\mathbf \rho },{\mathbf \rho },z} )- {W_{yy}}({{\mathbf \rho },{\mathbf \rho },z} )} ]}^2} + 4{{|{{W_{xy}}({{\mathbf \rho },{\mathbf \rho },z} )} |}^2}} } \right],$$

(23)$$D({{\mathbf \rho },{\mathbf \rho },z} )= {W_{xy}}({{\mathbf \rho },{\mathbf \rho },z} ).$$

The average intensities of the completely polarized portion and the completely unpolarized portion are then given by

(24)$${I^{(u )}}({{\mathbf \rho },{\mathbf \rho },z} )= Tr{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over W} ^{(u )}}({{\mathbf \rho },{\mathbf \rho },z} ),{I^{(p )}}({{\mathbf \rho },{\mathbf \rho },z} )= Tr{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over W} ^{(p )}}({{\mathbf \rho },{\mathbf \rho },z} ).$$

In addition, the SOP of a partially coherent vector beam (denoting from completely polarized portion) can be characterized by the polarization ellipse [8]

(25)$$\begin{aligned} {A_{1,2}}({{\mathbf \rho },{\mathbf \rho },z} )&= \frac{1}{{\sqrt 2 }}\left\{ {\sqrt {{{[{{W_{xx}}({{\mathbf \rho },{\mathbf \rho },z} )- {W_{yy}}({{\mathbf \rho },{\mathbf \rho },z} )} ]}^2} + 4{{|{{W_{xy}}({{\mathbf \rho },{\mathbf \rho },z} )} |}^2}} } \right.\\ &{\left. { \pm \frac{1}{2}\sqrt {{{[{{W_{xx}}({{\mathbf \rho },{\mathbf \rho },z} )- {W_{yy}}({{\mathbf \rho },{\mathbf \rho },z} )} ]}^2} + 4{{|{Re [{{W_{xy}}({{\mathbf \rho },{\mathbf \rho },z} )} ]} |}^2}} } \right\}^{1/2}}, \end{aligned}$$

(26)$$\varepsilon ({{\mathbf \rho },{\mathbf \rho },z} )= {A_2}({{\mathbf \rho },{\mathbf \rho },z} )/{A_1}({{\mathbf \rho },{\mathbf \rho },z} ),$$

(27)$$\theta ({{\mathbf \rho },{\mathbf \rho },z} )= \frac{1}{2}\arctan \left\{ {\frac{{2Re [{{W_{xy}}({{\mathbf \rho },{\mathbf \rho },z} )} ]}}{{{W_{xx}}({{\mathbf \rho },{\mathbf \rho },z} )- {W_{yy}}({{\mathbf \rho },{\mathbf \rho },z} )}}} \right\},$$

where the signs “+”and “-” in Eq. (25) correspond to A₁ (major semi-axis) and A₂ (minor semi- axis) of the polarization ellipse respectively. Further, $\varepsilon$ and $\theta $ represent the degree of ellipticity and the orientation angle of the polarization ellipse.

3. Numerical results

In this section, as numerical examples, we will study the second-order statistical characteristics of the RPTPCV beam passing through a paraxial ABCD optical system. The beam parameters are chosen to be $\lambda = 632.8\textrm{nm,}$ ${\sigma _0} = 1\textrm{mm}$ and ${\delta _0} = 0.5\textrm{mm}$ in the following analysis unless other values are specified. The truncated normalized mode weight λ_c is chosen to be 0.001 in the following numerical examples.

Let us consider a thin lens focusing system in which a thin lens with focal length f = 400 mm is placed in the source plane of the RPTPCV beam, and the output plane is located at the distance z from the lens. In this situation, the elements of the transfer matrix are $A = 1 - z/f$, $B = z$, and$D = 1$. We first pay attention to the influence of the twist phase on the spectral density of the RPTPCV beam during propagation. Figure 2 illustrates the evolution of the normalized spectral density $S({{\mathbf \rho },z} )/{[S({{\mathbf \rho },z} )]_{\max }}$ of the RPTPCV beam with propagation distance z. The topological charge l is 4 in the calculation. It can be seen that the spectral density upon propagation are closely related the sign (handedness) of the twist factor. When the twist phase is negative (the same handedness with the vortex phase), i.e., μ₀=−0.08m⁻¹, the beam is more likely to maintain the dark hollow shape upon propagation (see the first row). In the focal plane, the beam intensity on the axis is still not the intensity maxima. For the case of ${\mu _0} = 0$, i.e., the RPTPCV beam reduces to the radially polarized PCV beam, the beam profile degenerates from the dark hollow beam shape in the source plane to a Gaussian beam profile on propagation owing to the effects of partial spatial coherence (see the second row). This result is consistent with that reported in [34]. However, the beam profile quickly evolves into Gaussian shape when the twist factor is positive (see the third row). If the topological charge is negative, i.e., l = −4, the situation of the spectral density evolution is inversed, implying that the beam profile with negative twist factor quickly turns to be Gaussian shape upon propagation, while it more likely to maintain dark hollow shape for positive twist factor. One may come to a conclusion that when the handedness of the twist phase and vortex phase are opposite (see the third row), the evolution process of the beam profile from dark hollow to Gaussian is accelerated. On the contrary, if the handedness of the two phases is same (see the first row), the beam is more likely to maintain a dark hollow beam shape. This phenomenon can be explained from the point view of modal expansion that the beam modes with vortex phase have ring shape beam profile during propagation, and the larger the topological is, the larger radius of the ring is and the stronger the ability of remaining dark hollow shape is. When the handedness of the two phases is opposite (or the same), the proportion of the mode with large topological charge (L) is decreased (or increased), As a result, RPTPCV beam with the same chiral phases are more capable of maintaining hollow beam profile during propagation.

Fig. 2. Density plot of the normalized spectral density of a focused RPTPCV beam with l = 4 for different values of the twist factor μ₀ at several propagation distances. The green solid curve denotes the cross line of spectral density at y = 0.

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To examine the behavior of the spectral densities of x and y components during propagation, we plot in Fig. 3(a)-(c) the density plots of the normalized spectral densities S_x / S_xmax and S_y /S_ymax of the focused RPTPCV beam at distance z = 350mm and 375mm with different twist factors. The topological charge in the calculation is l = 4. One can see that the beam spot of x and y component rotates during propagation, which is consistent with the results reported in [34]. In addition, the angle formed by the long axis of the x and y component spectral density is close to 90, independent of the propagation distance. The rotation angle of the beam spot is also closely related to the twist fact. Figure 3(d) illustrates the variation of the rotation angle of x component spectral density with the propagation distance z for three different values of twist factor. One can find that when the handedness of two phases is the same, the beam spot rotates faster than the beam with opposite handedness of two phases, and the rotation angle reaches to π / 2 in the focal plane. In Fig. 3(e), the dependence of the rotation angle of x component spectral density on the twist factor at z = 375mm is presented. It is shown that one can control the direction and magnitude of the rotation angle through adjusting the handedness and numerical values of the twist factor.

Fig. 3. (a)-(c) Density plot of the normalized spectral densities S_x / S_xmax and S_y / S_ymax of a focused RPTPCV beam with l = 4 for different values of the twist factor μ₀ at different propagation distances. (d) Variation of the rotation angle of the RPTPCV beam with (d) the propagation distance z after the lens for different twist factors and (e) the value of twist factor at z = 375 mm.

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Let us now turn to investigate the polarization properties of the RPTPCV beam passing through a thin lens focusing system. According to the definition of the DOP in Eq. (17), the DOP of the RPTPCV beam in source plane (z = 0) is uniform and equals to 1 (complete polarization) except for the on-axis point at which the DOP is undefined. Figure 4(a) and 4(b) show the variation of the DOP at the point r = (0.6mm,0.6mm) as a function of propagation distance with different topological charges and twist factors, respectively. As the propagation distance increases, the DOP of the reference point is gradually depolarized owing to the effect of partial coherence, independence of the twist factor and topological charge. Nevertheless, the twist factor and the topological charge affect the speed of the depolarization of the reference point upon propagation. In the far field (focal plane), the polarization of the reference point for l = 0 [solid curve in Fig. 4(a)] and μ₀= 0.1mm⁻¹ [dashed curve in Fig. 4(b)] is nearly unpolarized. Figure 4(c)-(d) shows the DOP distribution of the RPTPCV beam at the cross-line y = 0 in the focal plane. It can be seen that the twist factor and the topological charge have a significant influence on the DOP distribution. As the topological charge increases or the twist factor decreases, the DOP of the beam near the on-axis region increases, which indicate that one could modulate the DOP of the beam through controlling the twist factor and topological charge.

(28)$${\eta ^{(i )}}(z )= \frac{{\int {{I^{(i )}}({{\mathbf \rho },{\mathbf \rho },z} ){d^2}{\mathbf \rho }} }}{{\int {I({{\mathbf \rho },{\mathbf \rho },z} ){d^2}{\mathbf \rho }} }}, ({i = p,u} ),$$

where η with the superscript i = p and i = u denote the ratio of polarized and unpolarized portion of energy, respectively. Based on the definition, the larger the value of ${\eta ^{(p )}}$ (${\eta ^{(u )}}$) is, the more polarized (unpolarized) the beam is. Figure 5 shows the variation of the normalized power of the polarized/unpolarized portions of the RPTPCV beam versus the propagation distance z with different values of the topological charge l and twist factor μ₀. The energy of the polarized portion gradually transfers into unpolarized portion under propagation owing the effect of partial coherence, as expected. Nevertheless, the beam with large topological charge or large value of twist factor resists the transform of the polarized portion into the unpolarized portion. As a result, the beam with larger l and μ₀ in the far filed become more polarization.

Fig. 4. (a)-(b) Degree of polarization of a focused RPTPCV beam at point (0.06 mm, 0.06 mm) versus the propagation distance for different values of the topological charge l and twist factor μ₀. (c)-(d) Dependence of the degree of polarization of a focused RPTPCV beam on the topological charge l and twist factor μ₀ in the focal plane.

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Fig. 5. Variation of the normalized powers of the completely polarized portion and the completely unpolarized portion of a RPTPCV beam versus the propagation distance z for different values of the topological charge l and twist factor μ₀.

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Finally, we examine the evolution of the SOP of the RPTPCV beam upon propagation. Figure 6(a) presents the evolution of the polarization ellipse associated with spectral density with propagation distance for different twist factors when the topological charge is l = 2. The green and red ellipses stand for the right-handed and left-handed elliptical polarizations. The SOP of the RPTPCV beam remains spatially non-uniform and varies during propagation, irrespectively of the sign of the twist factor. In the case for μ₀= 0, the SOP in the central region of the beam is right-handed polarization, while in the outside it becomes left-handed polarization. The SOP with μ₀=−0.1mm⁻¹ is similar with that with μ₀= 0, whereas the situation is versed for μ₀= 0.1mm⁻¹. The SOP in the central region is right-handed polarization. We should emphasize that the evolution of the SOP of the RPTPCV beam is quite different for that of the radially polarized beam (l = 0, μ₀= 0). It is known that the SOP of the radially polarized beam for any position is linearly polarized with its direction along the radial direction [see in Fig. 6(b)], and keeps unchanged upon propagation. Figure 6(c) shows the SOP of the focused RPTPCV beam in the focal plane with three different values of the twist factor. It is interesting to find that for an appropriate value of twist factor μ₀= 0.05 mm⁻¹, the beam becomes nearly right-handed circular polarization, which provide a way to control the SOP through controlling the twist phase under a certain condition.

Fig. 6. (a) Variation of the state of polarization of a focused RPTPCV beam at several propagation distances with different values of the topological charge l and the twist factor μ₀. (b)-(c) Dependence of the state of polarization of a focused RPTPCV beam on the topological charge l and twist factor μ₀ in the focal plane. The white line denotes linear polarization and the green (or red) ellipse denotes right-handed (or left-handed) elliptical polarization.

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4. Experiment generation of a RPTPCV beam

In this section, we carry out an experiment for the generation of a RPTPCV beam. The schematic for the experimental setup is shown in Fig. 7(a). A linearly polarized Nd:YAG laser beam (λ = 532nm) is expanded by a beam expander (BE) and then passes through a cylindrical lens (CL₁) with its generatrix along x-axis, illuminating on a rotating ground glass disk (RGGD). Through carefully adjusting the distance from the CL₁ to the RGGD, the ratio of the beam size ω_x along the x axis and ω_y along the y axis can be controlled. After the RGGD, the scattering light is collimated by a lens L₁ with focal length f₁ placed the distance f₁ after he RGGD and impinges on a spatial light modulator (SLM). In the experiment, the grid size of the RGGD is far smaller than the spot size of the light beam on the RGGD. In this case, the scattering light after the RGGD can be considered as the incoherent light. According to the Van Cittert-Zernike theorem, the light after the L₁ is an elliptical Gaussian Schell-model (GSM) source whose DOC function is the elliptical Gaussian function. The coherence width along the x and y directions are δ_x= 2π / ω_x and δ_y= 2π / ω_y, respectively. The SLM acts as an amplitude-only screen which transforms the beam profile from uniform distribution to elliptical profile. The transmittance function loaded on the SLM is an elliptical function with beam width σ_x and σ_y along x and y direction, respectively. After the SLM, the elliptical GSM beam enters a 4f-optical system composed of lens L₂ and L₃ with their focal lengths f₂= f₃. The distance from the SLM to L₂, from L₂ to the circular aperture (CA), from the CA to L₃ and from L₃ to CL₂ are all f₂. Hence, the elliptical GSM source with elliptical beam profile on the SLM plane is imaged on the CL₂ plane with unit magnification. The CA is used to block the unwanted diffraction orders and background noise. The generated elliptical GSM source in the CL₂ plane then passes through the three CLs system which is used to transform the elliptical GSM source into the desired twisted GSM source (the beam carries twist phase) in the CL₄ plane [44]. The beam parameters of the generated twisted GSM source are σ₀= 0.233mm, δ₀= 0.09 and μ₀= 0.0027m⁻¹. Finally, the RPTPCV source is produced by passing the twisted GSM beam through a spiral phase plate (SPP) with topological charge l and a radial polarization converter (RPC). A beam profile analysis (BPA) is placed in the focal plane of L₄ to measure the spectral density of the focused beam.

Fig. 7. (a) Experimental setup for generating a RPTPCV beam and measuring its focused spectral intensity. L₁, L₂, L₃, L₄ thin lenses; CL₁, CL₂, CL₃, CL₄ thin cylindrical lenses; M, reflecting mirror; CA, circular aperture; BE, beam expander; RGGD, rotating ground-glass disk; RPC, radial polarization converter; SLM, spatial light modulator; SPP, spiral phase plate; BPA, beam profile analyzer. (b)-(c) Experimental and theoretical results of the normalized spectral density S and its y component S_y of a focused RPTPCV beam with different values of the topological charge l at z = 0.6f, where z is the distance from L₄ to the BPA.

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Figure 7(b) shows the experimental results (the first row) of the normalized spectral densities with three different topological charge l. The white dotted curves denote the spectral densities at the cross line y = 0. For convenience of comparison, the corresponding theoretical results are plotted in the second row of Fig. 7(b). One can see that the handedness of the vortex phase has a significant influence on the beam profile in the focal plane. When the handedness of the twist phase and vortex phase (positive value of twist factor and negative value of the topological charge) are the same, the beam profile is still a dip in the center region, whereas it is a quasi-Gaussian shape for the opposite handedness between two phases. To investigate the polarization properties of the beam in the focal plane, we insert a linear polarizer after L₄ and observe the spectral density in the focal plane. The experimental and corresponding theoretical results are shown in Fig. 7(c) when the angle between the transmission axis of the polarizer and the x-axis is π / 2. Our results (not shown here) also demonstrate that the beam shape remain nearly unchanged and only the long axis rotates with the transmission axis of the polarizer if we rotates the polarizer with respect to the beam propagation axis, implies that the polarization distribution is rotational symmetry. The experimental results agree reasonably with the theoretical predictions.

5. Summary

In conclusion, we have introduced a new kind of radially polarized partially coherent beam, carrying both vortex phase and twist phase, named radially polarized twisted partially coherent vortex beam. With the help of the pseudo-modal expansion and the fast Fourier transform algorithm, the second-order statistics such as the average spectral density, DOP and SOP of the RPTPCV beam passing through a paraxial focusing system are studied through numerical examples. Our results reveal that the twist phase and the vortex phase have important effects on the propagation properties of the beam. When the handedness of the two phases is same, not only the ability of the beam profile to maintain a dark hollow shape is enhanced, but also the rotation speed of the beam spot during propagation is accelerated. However, these situations are reversed, if the handedness of the two phases are opposite. In addition, the twist phase and vortex phase also have important effects on the DOP and SOP upon propagation. The identical handedness of two phases plays a key role in resisting the depolarization caused by the partial spatial coherence of the beam. The twist phase and vortex phase provide an effect way to modulate the average spectral density, DOP and SOP of the beam. Further, an experiment is carried out to generated the RPTPCV source, and the spectral density and its polarization properties of the focused beam in the focal plane are measured. The experimental results are agree well with the theoretical results. Our finds will found potential applications in particle trapping, optical communication, and laser radar.

Funding

National Key Research and Development Program of China (2019YFA0705000); National Natural Science Foundation of China (11774251, 11874046, 11974218, 12104263, 12174279, 12192254, 62075149); Local Science and Technology Development Project of the Central Government (YDZX20203700001766); Natural Science Foundation of Shandong Province (ZR2021QA093); Innovation Group of Jinan (2018GXRC010); Natural Science Foundation of Jiangsu (BK20201406).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

References

1. Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photonics 1(1), 1–57 (2009). [CrossRef]

2. C. Sheppard, H. Wang, L. Shi, C. T. Chong, and B. Lukyanchuk, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2(8), 501–505 (2008). [CrossRef]

3. W. Chen, D. C. Abeysinghe, R. L. Nelson, and Q. Zhan, “Plasmonic Lens Made of Multiple Concentric Metallic Rings under Radially Polarized Illumination,” Nano Lett. 9(12), 4320–4325 (2009). [CrossRef]

4. M. Meier, V. Romano, and T. Feurer, “Material processing with pulsed radially and azimuthally polarized laser radiation,” Appl. Phys. A 86(3), 329–334 (2007). [CrossRef]

5. E. Y. S. Yew and C. J. R. Sheppard, “Second harmonic generation polarization microscopy with tightly focused linearly and radially polarized beams,” Opt. Commun. 275(2), 453–457 (2007). [CrossRef]

6. Q. Zhan, “Trapping metallic Rayleigh particles with radial polarization,” Opt. Express 12(15), 3377–3382 (2004). [CrossRef]

7. S. Roy, K. Ushakova, Q. van den Berg, S. F. Pereira, and H. P. Urbach, “Radially polarized light for detection and nanolocalization of dielectric particles on a planar substrate,” Phys. Rev. Lett. 114(10), 103903 (2015). [CrossRef]

8. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007).

9. Y. Chen, F. Wang, and Y. Cai, “Partially coherent light beam shaping via complex spatial coherence structure engineering,” Adv. phys-x 7(1), 2009742 (2022). [CrossRef]

10. D. Peng, Z. Huang, Y. Liu, Y. Chen, F. Wang, S. A. Ponomarenko, and Y. Cai, “Optical coherence encryption with structured random light,” PhotoniX 2(1), 6 (2021). [CrossRef]

11. L. Yonglei, Y. Chen, F. Wang, Y. Cai, C. Liang, and O. Korotkova, “Robust far-field imaging by spatial coherence engineering,” OEA 4(12), 210027 (2021). [CrossRef]

12. M. W. Hyde IV, S. Bose-Pillai, D. G. Voelz, and X. Xiao, “Generation of Vector Partially Coherent Optical Sources Using Phase-Only Spatial Light Modulators,” Phys. Rev. Appl. 6(6), 064030 (2016). [CrossRef]

13. D. F. V. James, “Change of polarization of light beams on propagation in free space,” J. Opt. Soc. Am. A 11(5), 1641–1643 (1994). [CrossRef]

14. F. Gori, “Matrix treatment for partially polarized, partially coherent beams,” Opt. Lett. 23(4), 241–243 (1998). [CrossRef]

15. E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312(5-6), 263–267 (2003). [CrossRef]

16. O. Korotkova, “Scintillation index of a stochastic electromagnetic beam propagating in random media,” Opt. Commun. 281(9), 2342–2348 (2008). [CrossRef]

17. F. Wang, Y. Cai, Y. Dong, and O. Korotkova, “Experimental generation of a radially polarized beam with controllable spatial coherence,” Appl. Phys. Lett. 100(5), 051108 (2012). [CrossRef]

18. G. Wu, F. Wang, and Y. Cai, “Coherence and polarization properties of a radially polarized beam with variable spatial coherence,” Opt. Express 20(27), 28301–28318 (2012). [CrossRef]

19. F. Wang, X. Liu, L. Liu, Y. Yuan, and Y. Cai, “Experimental study of the scintillation index of a radially polarized beam with controllable spatial coherence,” Appl. Phys. Lett. 103(9), 091102 (2013). [CrossRef]

20. W. Fu and H. Zhang, “Propagation properties of partially coherent radially polarized doughnut beam in turbulent ocean,” Opt. Commun. 304(1), 11–18 (2013). [CrossRef]

21. Y. Chen, F. Wang, L. Liu, C. Zhao, Y. Cai, and O. Korotkova, “Generation and propagation of a partially coherent vector beam with special correlation functions,” Phys. Rev. A 89(1), 013801 (2014). [CrossRef]

22. F. Wang, Y. Chen, X. Liu, Y. Cai, and S. A. Ponomarenko, “Self-reconstruction of partially coherent light beams scattered by opaque obstacles,” Opt. Express 24(21), 23735–23746 (2016). [CrossRef]

23. C. Ping, C. Liang, F. Wang, and Y. Cai, “Radially polarized multi-Gaussian Schell-model beam and its tight focusing properties,” Opt. Express 25(26), 32475–32490 (2017). [CrossRef]

24. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992). [CrossRef]

25. A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photonics 3(2), 161–204 (2011). [CrossRef]

26. G. Gibson, J. Courtial, M. Padgett, M. Vasnetsov, V. Pas’Ko, S. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express 12(22), 5448–5456 (2004). [CrossRef]

27. M. P. J. Lavery, F. C. Speirits, S. M. Barnett, and M. J. Padgett, “Detection of a Spinning Object Using Light's Orbital Angular Momentum,” Science 341(6145), 537–540 (2013). [CrossRef]

28. X. Qiu, F. Li, W. Zhang, Z. Zhu, and L. Chen, “Spiral phase contrast imaging in nonlinear optics: seeing phase objects using invisible illumination,” Optica 5(2), 208–212 (2018). [CrossRef]

29. F. Gori, M. Santarsiero, R. Borghi, and S. Vicalvi, “Partially coherent sources with helicoidal modes,” J. Mod. Optic. 45(3), 539–554 (1998). [CrossRef]

30. D. M. Palacios, I. D. Maleev, A. S. Marathay, and G. A. Swartzlander, “Spatial Correlation Singularity of a Vortex Field,” Phys. Rev. Lett. 92(14), 143905 (2004). [CrossRef]

31. Y. Yang, M. Mazilu, and K. Dholakia, “Measuring the orbital angular momentum of partially coherent optical vortices through singularities in their cross-spectral density functions,” Opt. Lett. 37(23), 4949–4951 (2012). [CrossRef]

32. X. Liu, Y. Shen, L. Liu, F. Wang, and Y. Cai, “Experimental demonstration of vortex phase-induced reduction in scintillation of a partially coherent beam,” Opt. Lett. 38(24), 5323–5326 (2013). [CrossRef]

33. G. Wu, W. Dai, H. Tang, and H. Guo, “Beam wander of random electromagnetic Gaussian-shell model vortex beams propagating through a Kolmogorov turbulence,” Opt. Commun. 336, 55–58 (2015). [CrossRef]

34. L. Guo, Y. Chen, X. Liu, L. Liu, and Y. Cai, “Vortex phase-induced changes of the statistical properties of a partially coherent radially polarized beam,” Opt. Express 24(13), 13714–13728 (2016). [CrossRef]

35. X. Liu, X. Peng, L. Liu, G. Wu, C. Zhao, F. Wang, and Y. Cai, “Self-reconstruction of the degree of coherence of a partially coherent vortex beam obstructed by an opaque obstacle,” Appl. Phys. Lett. 110(18), 181104 (2017). [CrossRef]

36. J. Zeng, X. Liu, F. Wang, C. Zhao, and Y. Cai, “Partially coherent fractional vortex beam,” Opt. Express 26(21), 26830–26844 (2018). [CrossRef]

37. J. Zeng, C. Liang, H. Wang, F. Wang, C. Zhao, G. Gbur, and Y. Cai, “Partially coherent radially polarized fractional vortex beam,” Opt. Express 28(8), 11493–11513 (2020). [CrossRef]

38. J. Serna and J. M. Movilla, “Orbital angular momentum of partially coherent beams,” Opt. Lett. 26(7), 405–407 (2001). [CrossRef]

39. Y. Cai and S. Zhu, “Orbital angular moment of a partially coherent beam propagating through an astigmatic ABCD optical system with loss or gain,” Opt. Lett. 39(7), 1968–1971 (2014). [CrossRef]

40. R. Simon and N. Mukunda, “Twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 10(1), 95–109 (1993). [CrossRef]

41. G. Wu, “Propagation properties of a radially polarized partially coherent twisted beam in free space,” J. Opt. Soc. Am. A 33(3), 345–350 (2016). [CrossRef]

42. X. Peng, L. Liu, J. Yu, X. Liu, Y. Cai, Y. Baykal, and W. Li, “Propagation of a radially polarized twisted Gaussian Schell-model beam in turbulent atmosphere,” J. Opt. 18(12), 125601 (2016). [CrossRef]

43. Z. Tong and O. Korotkova, “Beyond the classical Rayleigh limit with twisted light,” Opt. Lett. 37(13), 2595–2597 (2012). [CrossRef]

44. Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89(4), 041117 (2006). [CrossRef]

45. F. Wang, Y. Cai, H. T. Eyyuboglu, and Y. Baykal, “Twist phase-induced reduction in scintillation of a partially coherent beam in turbulent atmosphere,” Opt. Lett. 37(2), 184–186 (2012). [CrossRef]

46. S. A. Ponomarenko, “Twist phase and classical entanglement of partially coherent light,” Opt. Lett. 46(23), 5958–5961 (2021). [CrossRef]

47. S. A. Ponomarenko and G. P. Agrawal, “Asymmetric incoherent vector solitons,” Phys. Rev. E 69(3), 036604 (2004). [CrossRef]

48. R. Borghi, F. Gori, G. Guattari, and M. Santarsiero, “Twisted Schell-model beams with axial symmetry,” Opt. Lett. 40(19), 4504–4507 (2015). [CrossRef]

49. R. Borghi, “Twisting partially coherent light,” Opt. Lett. 43(8), 1627–1630 (2018). [CrossRef]

50. A. T. Friberg, E. Tervonen, and J. Turunen, “Interpretation and experimental demonstration of twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 11(6), 1818–1826 (1994). [CrossRef]

51. H. Wang, X. Peng, L. Liu, F. Wang, Y. Cai, and S. A. Ponomarenko, “Generating bona fide twisted Gaussian Schell-model beams,” Opt. Lett. 44(15), 3709–3712 (2019). [CrossRef]

52. X. Peng, L. Liu, F. Wang, S. Popov, and Y. Cai, “Twisted Laguerre-Gaussian Schell-model beam and its orbital angular moment,” Opt. Express 26(26), 33956–33969 (2018). [CrossRef]

53. X. Peng, H. Wang, L. Liu, F. Wang, S. Popov, and Y. Cai, “Self-reconstruction of twisted Laguerre-Gaussian Schell-model beams partially blocked by an opaque obstacle,” Opt. Express 28(21), 31510–31523 (2020). [CrossRef]

54. S. A. Ponomarenko, “Twisted Gaussian Schell-model solitons,” Phys. Rev. E 64(3), 036618 (2001). [CrossRef]

55. F. Gori and M. Santarsiero, “Twisted Gaussian Schell-model beams as series of partially coherent modified Bessel-Gauss beams,” Opt. Lett. 40(7), 1587–1590 (2015). [CrossRef]

56. R. Martínez-Herrero, P. M. Mejías, and F. Gori, “Genuine cross spectral densities and pseudo-modal expansions,” Opt. Lett. 34(9), 1399–1401 (2009). [CrossRef]

57. F. Gori and R. Martínez-Herrero, “Reproducing Kernel Hilbert spaces for wave optics: tutorial,” J. Opt. Soc. Am. A 38(5), 737–748 (2021). [CrossRef]

58. S. A. Collins, “Lens-System Diffraction Integral Written in Terms of Matrix Optics,” J. Opt. Soc. Am. A 60(9), 1168–1177 (1970). [CrossRef]

Radially polarized twisted partially coherent vortex beams

Abstract

1. Introduction

2. Theoretical mode and statistical properties of a RPTPCV beam through a paraxial ABCD optical system

3. Numerical results

4. Experiment generation of a RPTPCV beam

5. Summary

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (7)

Equations (28)

Optics Express