Statistical properties of a partially coherent vector beam with controllable spatial coherence, vortex phase, and polarization

Hao Zhang; Haiyun Wang; Xingyuan Lu; Xingyuan Lu; Xingyuan Lu; Xuechun Zhao; Bernhard J. Hoenders; Chengliang Zhao; Yangjian Cai; Yangjian Cai; Yangjian Cai; Yangjian Cai

doi:10.1364/OE.465274

1. Introduction

In recent years, the amplitude, polarization, and phase of light beams have been extensively studied [1]. Polarization, a crucial vector property of light, has been widely used in optical science. In general, beams can be divided into scalar and vector beams according to their polarization distributions, which correspond to spatially homogeneous and spatially inhomogeneous state of polarization (SOP) distributions, respectively [2]. A radially polarized beam is a type of classical vector beam with a polarization singularity (the point where the polarization orientation is indeterminate) in the beam center [3]. A radially polarized beam can be focused into a smaller spot than a scalar beam under tight focusing conditions with a high numerical aperture objective [4]. Because of this unique focusing feature, radially polarized beams have been studied in diverse applications, such as the optical trapping of nanoparticles [5,6], high-capacity optical communication [7], and super-resolution imaging [8]. In addition to polarization, the phase modulation of light beams, such as vortex beams possessing orbital angular momentum (OAM), is also of great interest [9,10]. A conventional vortex beam is characterized by a helical wavefront and hollow intensity structure [11,12]. Such beams have various applications, including optical measurements [13,14], optical tweezers [15–17], and OAM-based optical communication [18,19].

Vector vortex beams, which possess the characteristics of vector beams and vortex beams, such as the radial polarization state and helical wavefront, have gradually become a research hotspot in the optics field [20–22]. Vector vortex beams can be generated inside a laser cavity with high purity at the source via geometric phase control [23]. In general, vector vortex beams are generated by coherent superposition of two vortex beams with orthogonal circular polarization and different topological charges (TCs) [24]. Based on this method, vector vortex beams can be produced by interferometers [25,26], space-variant subwavelength gratings [27], and liquid crystal devices [28]. Furthermore, vector vortex beams show advantages in some optical applications, such as redistributing the energy flow for optical trapping [29], spin-orbit Hall effect based on a radially polarized vortex beam [30], and conformal frequency conversion in nonlinear optics [31].

Furthermore, coherence is another crucial property of a beam, which can be used to reduce the turbulence induced light degradation in optical communication [32], more secure encryption of information [33] and robust far-field imaging [34]. However, the coherence of the beam has been neglected in the aforementioned studies. In fact, polarization and coherence are interrelated and should be studied together, based on the unified theory of coherence and polarization first proposed by Wolf in 2003 [35]. Since then, researchers have begun to consider the polarization and coherence properties of light beams simultaneously [36–43]. For a fully coherent radially polarized beam, the degree of polarization (DOP) and SOP are invariant during propagation in free space. In contrast to a fully coherent beam, the DOP of a partially coherent radially polarized beam decreases and the SOP remains radially polarized during propagation [44]. Significantly, a partially coherent radially polarized beam can be reshaped from a hollow intensity distribution to a Gaussian light spot by modulating the coherence width and correlation function [38], which can be used to trap particles [45]. Further, the tight focusing properties of a partially coherent radially polarized beam with multi-Gaussian correlation function are investigated to explore the application of optical tweezers [46]. Coherence modulation can also reduce the effect of turbulence and increase the channel capacity of a radially polarized beam [47]. Furthermore, the self-healing properties of the partially coherent radially polarized beam scattered by opaque obstacles are studied and have verified that the beam will be helpful to optical communication [48,49].

In 2016, for the first time researchers introduced the vortex phase with an integer TC for a partially coherent radially polarized beam and proposed a partially coherent radially polarized integer vortex beam [50]. Interestingly, the introduction of the vortex phase can counteract the depolarization effect caused by reduced coherence. In addition, coherence-induced polarization changes of a vector vortex beam have been studied both theoretically and experimentally [51]. Compared to the fully coherent radially polarized vortex beam, the partially coherent radially polarized vortex beam has the advantage that it can be used to decrease the on-axis scintillation index in a turbulent atmosphere [52]. Furthermore, a partially coherent radially polarized vortex beam with a fractional TC has been proposed, which has an asymmetric intensity distribution [53]. The partially coherent radially polarized fractional vortex beam has proven useful for trapping irregular particles [54]. Moreover, a partially coherent radially polarized beam with multiple off-axis vortices has been proposed, which provides an additional degree of freedom for focal intensity shaping [55]. Recently, the propagation properties of a partially coherent radially polarized vortex beam influenced by the twist phase is studied [43]. And the tight focusing properties of a partially coherent radially polarized vortex beam with conventional power-exponent-phase exhibits potential application in optical trapping [56]. Although, the total phase is asymmetric like the fractional vortex phase, but each phase section from 0 to 2π is of uniformly variation. A partially coherent radially polarized vortex beam exhibits these properties in combination, plays an important role in many optical applications. However, such beams possess only a uniformly varying vortex phase; the modulation freedom of the vortex has therefore not been fully developed, which limits further applications.

To address this limitation, we introduce a unique vortex phase [57] with exponential phase variation (referred to as the power-exponent-phase) to the partially coherent radially polarized beam. By combining the modulation of coherence, polarization, and phase, we report on a partially coherent radially polarized power-exponent-phase vortex (PC-RP-PEPV) beam. The average intensity of the beam was analyzed by numerical simulation and experiment. In contrast to conventional partially coherent radially polarized vortex beams, the PC-RP-PEPV beam can be reshaped as a polygonal intensity distribution. Furthermore, the SOP and DOP properties of the beam during propagation were investigated. Compared with the previous partially coherent radially polarized vortex beams, the PC-RP-PEPV beam possess various mode distribution and complex polarization distribution. We also discuss the specific properties of the PC-RP-PPEV beam and its potential applications such as optical tweezers and optical measurements.

2. Theory of the PC-RP-PEPV beam

We begin with a discussion of the statistical properties of the PC-RP-PEPV beam using the unified theory of coherence and polarization [35,58]. In general, the cross-spectral density (CSD) matrix $\overleftrightarrow W({{\boldsymbol r}_1},{{\boldsymbol r}_2})$ is used to characterize the statistical properties of a partially coherent vector beam that propagates along the z-axis. Here, r₁ and r₂ represent two arbitrary coordinate vectors in a plane at propagation distance z. The CSD matrix in the source plane (z = 0) can be expressed as

(1)$${W_{\alpha \beta }}({{\boldsymbol r}_1},{{\boldsymbol r}_2}) = \left\langle {E_\alpha^{\ast }({{\boldsymbol r}_1}){E_\beta }({{\boldsymbol r}_2})} \right\rangle , ({\alpha ,\beta = x,y} ),$$

where E_x(r) and E_y(r) respectively denote two electric fields with orthogonal polarization directions along the x and y axis. Here, the angular brackets represent the ensemble average and the asterisk denotes the complex conjugate. Here, we consider a specific electric field with a power-exponent-phase [57]:

(2)$${E_\alpha }(r,\theta ) = \frac{\alpha }{{{w_0}}}\exp \left( { - \frac{{{r^2}}}{{w_{^0}^2}}} \right)\exp \left( {\textrm{i}2\mathrm{\pi }{{\left[ {\frac{{\textrm{rem}(m\theta ,2\mathrm{\pi })}}{{2\mathrm{\pi }}}} \right]}^n}} \right),(\alpha = x,y),$$

where (r, θ) denotes polar coordinates, r is the radial coordinate vector, θ is the azimuthal angle, w₀ is the beam width, rem() is the remainder function, m denotes the value of TC, and n represents the power order of the power-exponent-phase. The third term in Eq. (2) indicates the power-exponent-phase, which is different from the conventional vortex phase term exp(imθ) and the conventional power-exponent-phase exp(i2mθ(φ/2π)ⁿ) [56]. Note that the power-exponent-phase in Eq. (2) is produced by modulating the conventional power-exponent-phase by the rem function, which will induce a rotational symmetry of the phase, just like the conventional vortex phase, but the conventional power-exponent-phase will not [57,59]. It should be noted that the power order n can be an integer or fraction, and Eq. (2) reduces to a conventional vortex beam when the power order n = 1. When the power order n > 1, the phase is fundamentally different from the vortex phase exp(imθ) in Ref. [43,53]. Based on Eq. (2), the elements of the CSD matrix of the PC-RP-PEPV beam with a Gaussian Schell-model correlation can be defined as follows:

(3)$$\begin{aligned} {W_{\alpha \beta }}({{\boldsymbol r}_1},{{\boldsymbol r}_2}) &= \frac{{\alpha \beta }}{{{w_0}}}\exp \left( { - \frac{{{\boldsymbol r}_1^2 + {\boldsymbol r}_2^2}}{{w_{^0}^2}}} \right){\mu _{\alpha \beta }}({{{\boldsymbol r}_1} - {{\boldsymbol r}_2}} )\\ &\exp \left( {\textrm{i}2\mathrm{\pi }{{\left[ {\frac{{\textrm{rem}(m{\theta_1},2\mathrm{\pi })}}{{2\mathrm{\pi }}}} \right]}^n} - \textrm{i}2\mathrm{\pi }{{\left[ {\frac{{\textrm{rem}(m{\theta_2},2\mathrm{\pi })}}{{2\mathrm{\pi }}}} \right]}^n}} \right),(\alpha ,\beta = x,y),\end{aligned}$$

with correlation function:

(4)$${\mu _{\alpha \beta }}({{{\boldsymbol r}_1} - {{\boldsymbol r}_2}} )= {B_{\alpha \beta }}\exp \left[ { - \frac{{{{({{{\boldsymbol r}_1} - {{\boldsymbol r}_2}} )}^2}}}{{2\sigma_{\alpha \beta }^2}}} \right],(\alpha ,\beta = x,y),$$

where w₀ represents the beam width in the source plane, and ${B_{\alpha \beta }} = |{{B_{\alpha \beta }}} |\exp ({\textrm{i}{\phi_{\alpha \beta }}} )$ and ϕ_αβ respectively are the complex correlation coefficient and phase difference between the α and β components. σ_αβ denotes the coherence width of the Gaussian correlation function for α-β components. When the power order n is greater than one, Eq. (3) represents the CSD matrix of the proposed PC-RP-PEPV beam in the source plane. It should be noted that Eq. (3) reduces to a partially coherent radially polarized vortex beam with an integer or fractional TC [53] when the power order n = 1. More generally, the beam source reduces to a partially coherent radially polarized beam when the TC m = 0 [50]. In this study, we set the complex correlation coefficient B_αβ = 1 and coherence width ${\sigma _{\alpha \beta }} = \sigma$[53]. Hence, the CSD matrix of the PC-RP-PEPV beam in the source plane can be expressed as

(5)$$\begin{aligned} {W_{\alpha \beta }}({{\boldsymbol r}_1},{{\boldsymbol r}_2}) &= \frac{{\alpha \beta }}{{{w_0}}}\exp \left( { - \frac{{{\boldsymbol r}_1^2 + {\boldsymbol r}_2^2}}{{w_{^0}^2}}} \right)\exp \left[ { - \frac{{{{({{{\boldsymbol r}_1} - {{\boldsymbol r}_2}} )}^2}}}{{2{\sigma^2}}}} \right]\\ &{\exp \left( {\textrm{i}2\mathrm{\pi }{{\left[ {\frac{{\textrm{rem}(m{\theta_1},2\mathrm{\pi })}}{{2\mathrm{\pi }}}} \right]}^n} - \textrm{i}2\mathrm{\pi }{{\left[ {\frac{{\textrm{rem}(m{\theta_2},2\mathrm{\pi })}}{{2\mathrm{\pi }}}} \right]}^n}} \right),(\alpha ,\beta = x,y)}, \end{aligned}$$

where σ is the initial coherence width of the PC-RP-PEPV beam. Subsequently, based on an ABCD optical system, we analyzed the statistical properties of the PC-RP-PEPV beam propagation through a thin lens. In general, the propagation properties of each CSD matrix element through a paraxial ABCD optical system (such as a thin lens) can be studied using the generalized Collins formula [53,60]. The CSD matrix elements in the observation plane can be expressed as [58]

(6)$$\begin{aligned} {W_{\alpha \beta }}({{\boldsymbol \rho }_1},&{{\boldsymbol \rho }_2})\textrm{ = }\frac{{{k^2}}}{{4{\pi ^2}{B^2}}}\int {\int {{\textrm{d}^2}{{\boldsymbol r}_1}{\textrm{d}^2}{{\boldsymbol r}_2}{W_{\alpha \beta }}({{\boldsymbol r}_1},{{\boldsymbol r}_2})} } \\ &\times\exp \left\{ { - \frac{{\textrm{i}k}}{{2B}}[{A{\boldsymbol r}_1^2 - 2{{\boldsymbol r}_1}{\rho_1} + D{\boldsymbol \rho }_1^2} ]} \right\}\exp \left\{ {\frac{{\textrm{i}k}}{{2B}}[{A{\boldsymbol r}_2^2 - 2{{\boldsymbol r}_2}{\rho_2} + D{\boldsymbol \rho }_2^2} ]} \right\}, \end{aligned}$$

where ρ₁ and ρ₂ are the vector coordinates in the observation plane, and A, B, C, and D are the transfer matrix elements of the optical system. As an example, we analyzed the propagation properties of the PC-RP-PEPV beam when it passes through a thin lens. The transfer matrix of the ABCD optical system can then be written as

(7)$$\left( {\begin{array}{cc} A&B\\ C&D \end{array}} \right) = \left( {\begin{array}{cc} 1&z\\ 0&1 \end{array}} \right)\left( {\begin{array}{cc} 1&0\\ { - 1/f}&1 \end{array}} \right) = \left( {\begin{array}{cc} {1 - z/f}&z\\ { - 1/f}&1 \end{array}} \right),$$

where f and z are the focal length of the thin lens and propagation distance from the original plane to the observation plane, respectively. Note that the CSD matrix of a PC-RP-PEPV beam in the observation plane can be calculated numerically via the convolution method to avoid onerous integral calculations [53]. The CSD matrix can be written as [58]

(8)$${W_{\alpha \beta }}({{\boldsymbol \rho }_1},{{\boldsymbol \rho }_2})\textrm{ = }\left( {\begin{array}{cc} {{W_{xx}}({{\boldsymbol \rho }_1},{{\boldsymbol \rho }_2})}&{{W_{xy}}({{\boldsymbol \rho }_1},{{\boldsymbol \rho }_2})}\\ {{W_{yx}}({{\boldsymbol \rho }_1},{{\boldsymbol \rho }_2})}&{{W_{yy}}({{\boldsymbol \rho }_1},{{\boldsymbol \rho }_2})} \end{array}} \right).$$

From the CSD matrix of a PC- RP-PEPV beam in the observation plane, we can obtain the corresponding average intensity and DOP distributions to study the statistical properties of the PC-RP-PPEV beam upon propagation. By setting ρ₁ = ρ₂ = ρ, the average intensity in any plane can be obtained [58]

(9)$$I({\boldsymbol \rho } )= {W_{xx}}({\boldsymbol \rho },{\boldsymbol \rho }) + {W_{yy}}({\boldsymbol \rho },{\boldsymbol \rho }).$$

Note that the W_xx(ρ, ρ) = I_xx and W_yy(ρ, ρ) = I_yy. The DOP plays a crucial role in analyzing an electromagnetic beam, and can be expressed as:

(10)$$P({\boldsymbol \rho } )= \sqrt {1 - \frac{{4\textrm{Det}[{\overleftrightarrow W({{\boldsymbol \rho },{\boldsymbol \rho }} )} ]}}{{{{\{{\textrm{Tr}[{\overleftrightarrow W({{\boldsymbol \rho },{\boldsymbol \rho }} )} ]} \}}^2}}}} ,$$

where Det and Tr are the determinant and trace, respectively, of the CSD matrix. Furthermore, the CSD matrix can be regarded as the sum of a polarized and an unpolarized part [61]:

(11)$$\overleftrightarrow W({{\boldsymbol \rho },{\boldsymbol \rho }} )= {\overleftrightarrow W^p}({{\boldsymbol \rho },{\boldsymbol \rho }} )+ {\overleftrightarrow W^u}({{\boldsymbol \rho },{\boldsymbol \rho }} ),$$

where ${\overleftrightarrow W^p}({{\boldsymbol \rho },{\boldsymbol \rho }} )$ and ${\overleftrightarrow W^u}({{\boldsymbol \rho },{\boldsymbol \rho }} )$ represent the polarized and unpolarized parts of the CSD matrix, respectively, and can be expressed as:

(12)$${\overleftrightarrow W^p}({{\boldsymbol \rho },{\boldsymbol \rho }} )= \left( {\begin{array}{cc} {B({{\boldsymbol \rho },{\boldsymbol \rho }} )}&{D({{\boldsymbol \rho },{\boldsymbol \rho }} )}\\ {{D^\ast }({{\boldsymbol \rho },{\boldsymbol \rho }} )}&{C({{\boldsymbol \rho },{\boldsymbol \rho }} )} \end{array}} \right),$$

(13)$${\overleftrightarrow W^u}({{\boldsymbol \rho },{\boldsymbol \rho }} )= \left( {\begin{array}{cc} {A({{\boldsymbol \rho },{\boldsymbol \rho }} )}&0\\ 0&{A({{\boldsymbol \rho },{\boldsymbol \rho }} )} \end{array}} \right),$$

with

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

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

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

(17)$$D({{\boldsymbol \rho },{\boldsymbol \rho }} )= {W_{xy}}({{\boldsymbol \rho },{\boldsymbol \rho }} ).$$

From the CSD matrix of the polarized and unpolarized parts, we can obtain the spectral intensities of the polarized part I^p(ρ) and unpolarized part I^u(ρ) as follows:

(18)$${I^p}({\boldsymbol \rho }) = W_{xx}^p({\boldsymbol \rho },{\boldsymbol \rho }) + W_{yy}^p({\boldsymbol \rho },{\boldsymbol \rho })$$

(19)$${I^u}({\boldsymbol \rho }) = W_{xx}^u({\boldsymbol \rho },{\boldsymbol \rho }) + W_{yy}^u({\boldsymbol \rho },{\boldsymbol \rho }).$$

Finally, we studied the SOP distribution of the proposed PC-RP-PPEV beam using the polarization ellipse characterized by the Stokes parameters S₀, S₁, S₂, and S₃. Based on the CSD matrix, the Stokes parameters can be written as [58]

(20)$${S_0}({\boldsymbol \rho }) = {W_{xx}}({{\boldsymbol \rho },{\boldsymbol \rho }} )+ {W_{yy}}({{\boldsymbol \rho },{\boldsymbol \rho }} ),$$

(21)$${S_1}({\boldsymbol \rho }) = {W_{xx}}({{\boldsymbol \rho },{\boldsymbol \rho }} )- {W_{yy}}({{\boldsymbol \rho },{\boldsymbol \rho }} ),$$

(22)$${S_2}({\boldsymbol \rho }) = {W_{xy}}({{\boldsymbol \rho },{\boldsymbol \rho }} )+ {W_{yx}}({{\boldsymbol \rho },{\boldsymbol \rho }} ),$$

(23)$${S_3}({\boldsymbol \rho }) = \textrm{i}[{{W_{yx}}({{\boldsymbol \rho },{\boldsymbol \rho }} )- {W_{xy}}({{\boldsymbol \rho },{\boldsymbol \rho }} )} ].$$

3. Propagation properties of a PC-RP-PEPV beam

In this section, we investigate the propagation properties of a PC-RP-PEPV beam propagation through a thin Fourier lens at five propagation distances z with different coherence widths σ. Figures 1 and 2 illustrate the orthogonal intensity components I_x and I_y and the total average intensity distribution I of PC-RP-PEPV beams with high and low coherence widths, respectively. The common parameters used in the numerical calculations of Figs. 1 and 2 are as follows: λ = 532 nm, w₀ = 1 mm, f = 300 mm, power order n = 2, and TC m = 4. The values for the coherence width parameters are σ = 3 mm for Fig. 1 and σ = 0.5 mm for Fig. 2. As shown in Figs. 1 and 2, the PC-RP-PEPV beam possesses a circular dark core and four radial intensity lines at an approximate initial plane (z = 0.1f). Four radial intensity lines arise owing to the nonlinear variation in the power-exponent phase in the source plane. From Figs. 1(c1)–(c5), we can see the rich evolution of the intensity patterns. This is because the influence of the power-exponent-phase on the intensity distribution appears only after propagation. At the focal plane (z = f), a cruciform dark core appears in the center of the quadrilateral PC-RP-PEPV beam. The structure of the dark core is determined by modulation caused by the combination of the power-exponent-phase and polarization. Furthermore, the total average intensity rotated during propagation from the source plane to the focal plane owing to the influence of the vortex phase.

Fig. 1. Numerical-simulation orthogonal intensity components I_x, I_y and the total average intensity I of a PC-RP-PEPV beam at several propagation distances z with high coherence width σ = 3 mm and TC m = 4. (a1)–(a5) Intensity patterns of the component I_x. (b1)–(b5) Intensity patterns of the component I_y. (c1)–(c5) Total average intensity patterns I.

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Fig. 2. Numerical-simulation orthogonal intensity components I_x, I_y and the total average intensity I of a PC-RP-PEPV beam at several propagation distances z with low coherence width σ = 0.5 mm and TC m = 4. (a1)–(a5) Intensity patterns of the component I_x. (b1)–(b5) Intensity patterns of the component I_y. (c1)–(c5) Total average intensity patterns I.

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Subsequently, we studied the characteristics of a PC-RP-PEPV beam with low coherence width (σ = 0.5 mm), as shown in Fig. 2. When the propagation distance z is less than or equal to 0.6f, the intensity distribution of the PC-RP-PEPV beam is similar to that in the high coherence width case. This indicates that the modulation of coherence occurs after propagation. Furthermore, the dark core of the cruciform structure at the focal plane disappears gradually and finally becomes a quadrangular light spot, as shown in Fig. 2(c5). The intensity distributions of the orthogonal components I_x and I_y show a complex structure in the case of high coherence width, whereas they appear as slanting spots in the case of low coherence width. The PC-RP-PEPV beam has been found to differ from a partially coherent radially polarized beam and a partially coherent radially polarized vortex beam with integer and fractional TC [50,53]. This phenomenon reveals that we can realize various beam modes by combining the modulation of coherence, radial polarization, and power-exponent-phase, which may have potential applications in complex optical manipulation.

We then studied the evolution of the SOP distribution of the PC-RP-PEPV beam as the propagation distance increases. The SOP of the PC-RP-PPEV beam is calculated using Eqs. (20)–(23) and is shown in the bottom rows of Figs. 1 and 2. The yellow and red ellipses with different ellipticities indicate left-handed and right-handed elliptical polarizations, respectively. As shown in Fig. 1(c1), the main SOP distribution remains in the initial radially linear polarization. Subsequently, the radially linear polarization gradually transforms into elliptical polarization with different angles and ellipticity, which means that the power-exponent-phase begins to modulate the beam during propagation. As the propagation distance increases, the radial polarization gradually changes into azimuthal elliptical polarization. As shown in Fig. 1(c5), the SOP distribution varies along the angular direction. Then, with a decrease in the coherence width, the SOP distributions become more ordered, as shown in Figs. 2(c1)–(c5). We observed right-handed elliptical polarization around the beam center and left-handed elliptical polarization outside it. Moreover, a gradual change from radial polarization to azimuthal elliptical polarization is observed in the case of a positive TC. Note that the SOP will have opposite handedness when the TC is negative (not discussed in this work).

4. Modulation properties of the PC-RP-PEPV beam

The value of the TC is a crucial parameter that can be used to modulate the intensity distribution of a PC-RP-PPEV beam. Figure 3 shows the numerical simulation intensity patterns of the PC-RP-PEPV beam at the focal plane with varying TC, power order and coherence width. In the area with a blue background, the beam width w₀ = 1 mm, power order n = 2, and TC m = 0, 1, 2, and 3. As depicted in Fig. 3(a1), in the case of high coherence width (σ = 3 mm), a PC-RP-PEPV beam with TC m = 0 is a classical radially polarized beam with an annular intensity distribution and radial polarization states, and a polarization singularity is formed in the beam center. Subsequently, the hollow structure of the beam transforms into a solid light spot owing to the combined action of the vortex phase with TC m = 1 and radial polarization [53], as shown in Fig. 3(a2). When the TC m >1, as shown in Figs. 3(a3) and (a4), the intensity distribution of the PC-RP-PEPV beam is reshaped into elliptic and triangular structures. The beam properties are different from those of scalar beams [56]. The coherence widths in the middle and bottom rows are σ = 1 mm and σ = 0.5 mm. When the coherence width decreases, a PC-RP-PPEV beam with a hollow structure gradually transforms into a solid structure. When the TC value m is 0 or 1, the PC-RP-PEPV beams both have a circular intensity distribution, whereas for m = 2 and 3, the PC-RP-PEPV beams will form a polygonal solid intensity distribution. Additional polygonal beam structures can be realized by modulating the TC value (not shown here). Furthermore, the beam will degenerate into a classical radially polarized vortex beam (possessing only a circular intensity distribution) when the power order n = 1, as shown in the area with a yellow background in Fig. 3.

Fig. 3. Numerical-simulation intensity patterns of a PC-RP-PEPV beam with varying TC (m = 0, 1, 2, 3), and power order (n = 1, 2). Here, power order n = 2 for the area with blue background and n = 1 for the area with yellow background. (a1)–(a4) and (d1), (d2) Intensity patterns with high coherence width (σ = 3 mm). (b1)–(b4) and (e1), (e2) Intensity patterns with middle coherence width (σ = 1 mm). (c1)–(c4) and (f1), (f2) Intensity patterns with low coherence width (σ = 0.5 mm). All results are in the focal plane.

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The SOP distributions are also shown in Fig. 3. The beams with m = 0 appear to be radially polarized and maintain their SOP as the coherence width decreases. When m = 1, the beams with n = 1 and 2 have similar intensity distributions, but their SOP distributions differ. When the power order n = 2 and the TC m = 2 or 3, the radial polarization transforms into an azimuthal elliptical polarization with an elliptical and triangular distribution owing to the polygonal intensity structure. With a decrease in coherence, the ellipticity of the polarization ellipse in the center of the beam decreases. A different behavior is clearly shown for the SOP distribution if the TC m = 2 and the power order n = 1; under these conditions, the SOP distribution is symmetric for three coherence widths σ = 3 mm, 1 mm, 0.5 mm.

We then studied the properties of the PC-RP-PPEV beam with a power order n > 2. Figure 4 depicts the intensity distributions of the beam with TC m = 4 when the power order n increases from 4 to 10 at intervals of 2. As shown in Figs. 4(a1)–(a4), the sidelobes around the quadrangular hollow intensity pattern gradually disappear as the power order increases, and the quadrangular intensity distribution gradually changes into an approximate circle. When the coherence width decreases, the intensity changes into a solid structure if the power order n = 4, and reshapes from a quadrangular pattern to a circular light spot, as shown in Figs. 4(b1)–(b4). From the SOP distribution, it can be observed that the azimuthal elliptical polarization states transform into radially polarization states with an increase in the power order n. This transformation occurs because the power-exponent vortex phase gradually assumes a constant value [56]. In addition, we plotted the intensity patterns of the orthogonal components I_x and I_y with power order n = 10 and high and low coherence widths, as shown in Figs. 4(c1), (c2) and (d1), (d2), respectively. The directions of the orthogonal components I_x and I_y are along the x and y axis, respectively, indicating that the beam is radially polarized.

Fig. 4. Numerical simulation intensity patterns of a PC-RP-PEPV beam with TC m = 4, and power order n = 4, 6, 8, 10. (a1)–(a4) Intensity patterns with high coherence width (σ = 3 mm). (b1)–(b4) Intensity patterns with low coherence width (σ = 0.5 mm). (c1), (c2) Orthogonal components I_x and I_y with n = 10 and σ = 3 mm. (d1), (d2) Orthogonal components I_x and I_y with n = 10 and σ = 0.5 mm. All results are in the focal plane.

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5. Degree of polarization for the PC-RP-PEPV beam

As shown in Fig. 5, the DOP (crossline y = 0) of a PC-RP-PEPV beam with varying TCs and power orders was calculated to study the polarization properties. We found that the distribution of the DOP of the beam with power order n = 2 differs from that of a partially coherent radially polarized vortex beam (n = 1). Here, the coherence width σ is equal to 0.5 mm and the TC m = 1 for Fig. 5(a) and (m) = 4 for Fig. 5(b). Note that the value of DOP will be a constant (equal to 1) when the beam has a high coherence width [50]. However, this case is not discussed in this study. In Fig. 5(a), the DOP distribution is similar for the PC-RP-PEPV beam and the partially coherent radially polarized beam. The DOP exhibits a peak at point (0, 0) when n = 1. However, the peak is located to the right of point (0, 0) when n = 2. Comparing the cases m = 1 and m = 4, we observe that the peak value of the DOP increases as the value of the TC increases when the power order n = 1, as shown in Fig. 5(b). In contrast to the partially coherent radially polarized vortex beam (n = 1), the PC-RP-PEPV beam with n = 2 has a flat trough near the point (0, 0).

Fig. 5. Degree of polarization of a PC-RP-PEPV beam (crossline) with TC m = 1, 4, power order n = 1, 2 and fixed coherence width σ = 0.5 mm.

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Figure 6 depicts the calculated crosslines of the normalized intensity distributions I(x, 0)/max[I(x, 0)], I^u(x, 0)/max[I(x, 0)], and I^p(x, 0)/max[I(x, 0)] of a PC-RP-PEPV beam at several propagation distances with power orders n = 1, 2. Here, TC m = 4, f = 300 mm, λ = 532 nm, and σ = 0.5 mm. I, I^u and I^p are the normalized intensity distributions of the total intensity, the unpolarized part, and the polarized intensity part, respectively. Note that the total intensity consists of the unpolarized and polarized components. When the propagation distance is short (i.e., z = 150 mm), the polarized intensity is the main component. The unpolarized component increases with increasing propagation distance. For a partially coherent radially polarized vortex beam (n = 1), as shown in Figs. 6(a)–(c), the components of the polarized and unpolarized parts gradually approached each other. The polarized part is larger than the unpolarized part for a long propagation distance (i.e., z = 300 mm). However, for the PC-RP-PEPV beam (n = 2), the unpolarized component gradually overtakes the polarized part, as shown in Figs. 6(d)–(f). This phenomenon demonstrates that the power-exponent-phase influences depolarization and can be used to modulate the components of the polarized and unpolarized parts and then reshape the structure of the intensity.

Fig. 6. Crosslines of the normalized intensity distributions I(x, 0)/max[I(x, 0)] (short dashed red line), I^u(x, 0)/max[I(x, 0)] (long dashed black line), I^p(x, 0)/max[I(x, 0)] (solid green line) of a PC-RP-PEPV beam at several propagation distances z with TC m = 4 and power order n = 1, 2.

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For a better understanding of the variation law for the intensities of the polarized and unpolarized parts during propagation, we studied the evolution properties of the normalized polarized and unpolarized intensity parts, which can be defined as [38]

(24)$${\eta ^l}\textrm{ = }\frac{{\int {{I^l}({\boldsymbol \rho }){\textrm{d}^2}{\boldsymbol \rho }} }}{{\int {I({\boldsymbol \rho }){\textrm{d}^2}{\boldsymbol \rho }} }},(l = p,u),$$

where η^p and η^u are the integrated normalized intensities of the polarized and unpolarized parts, respectively.

Figure 7 shows the variations in η^p and η^u for a PC-RP-PPEV beam with different values of TC (m = 1, 2, 4), propagation distance (z increases from 150 to 300 mm), power order (n = 1, 2, 4), and coherence width σ = 1, 0.5, 0.25 mm. It can be seen from this figure that the polarized part gradually decreases and the unpolarized part gradually increases. For a partially coherent radially polarized vortex beam with coherence width σ = 0.5 mm, the unpolarized part will be larger than the polarized part with TC m = 1 and a long propagation distance z = 300 mm, as shown in Fig. 7(a). The effect of depolarization can be mitigated during propagation by increasing the value of the TC, and we can observe that the polarized part is larger than the unpolarized part with m = 4. As shown in Fig. 7(b), the PC-RP-PEPV beam (n = 2) exhibits similar trends to the beam with n = 1 shown in Fig. 7(a). However, the unpolarized part of the PC-RP-PPEV beam increases faster. The effect caused by the power order n can be clearly seen with further study of Fig. 7(c). With an increase in the power order n from 1 to 4, the unpolarized part increases, and the polarized part becomes smaller than the unpolarized part for a constant TC m = 4. Finally, the polarized part gradually decreases with a decrease in the coherence width from 1 mm to 0.25 mm, as demonstrated in Fig. 7(d). This also implies that the PC-RP-PEPV beam becomes depolarized during propagation. These results are also supported by the results shown in Fig. 6, and the behavior of the PC-RP-PPEV beam differs from those of coherent and partially coherent radially polarized vortex beams.

Fig. 7. Variation of the integrated normalized intensities of the polarized part (η^p) and unpolarized part (η^u) of a PC-RP-PEPV beam with different propagation distance z from 150–300 mm for (a) varying TC m = 1, 2, 4 and power order n = 1, (b) varying TC m = 1, 2, 4 and power order n = 2, (c) varying power order n = 1, 2, 4 and TC m = 4, (d) varying coherence width σ = 1, 0.5, 0.25 mm and TC m= 4, power order n = 2. (a)–(c) Coherence width σ = 0.5 mm.

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6. Experimental results

Figure 8 shows a schematic of the experimental setup which was used to generate a PC-RP-PEPV beam. A coherent laser beam with a wavelength of λ = 532 nm is emitted from a solid-state laser and then expanded using a beam expander (BE). The expanded beam is focused on the rotating ground glass disk (RGGD) by lens L₁ (f₁ = 100 mm) and then collimated by lens L₂ (f₂ = 100 mm). A Gaussian amplitude filter (GAF) is used to obtain a partially coherent Gaussian Schell model beam. The coherence width can be controlled by modulating the focused beam spot size on the RGGD [62]. The first half-wave plate HWP₁ is used to convert the beam to a horizontal polarization state, so that it can be modulated by a spatial light modulator (SLM) which responds only to horizontally polarized light. As shown in Fig. 8(a), a specific fork grating loaded on the SLM was designed using computer-generated holograms [63]. Subsequently, the power-exponent phase was added to the partially coherent Gaussian shell model beam with a horizontal polarization state. A radial polarization converter (RPC) is used to convert the linear polarization state to radial polarization. A second half-wave plate HWP₂ is used to convert the horizontal polarization state to a vertical polarization state. Finally, the PC-RP-PEPV beam is generated by lens L₃ (f₃ = 300 mm), and the intensity is recorded by a charge-coupled device (CCD), as shown in Fig. 8(b).

Fig. 8. Schematic diagram for generating a PC-RP-PEPV beam. (a) Phase mask written into the SLM and (b) intensity pattern of the PC-RP-PEPV beam captured by CCD. LP, linear polarization plate; BE, beam expander; L₁-L₃, lenses; HWP, half wave plate; RGGD, rotating ground glass disk; SLM, spatial light modulators; BS, beam splitter; RPC, radial polarization converter; CCD, charge-coupled device.

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Figure 9 shows the experimental results for the orthogonal intensity components I_x and I_y and the total average intensity distribution I of the PC-RP-PEPV beam with high coherence width (σ = 3 mm, yellow background area) and low coherence width (σ = 0.5 mm, blue background area), respectively. The three propagation distances were set as z = 0.8f, 0.95f, f. The other parameters were the same as those shown in Figs. 1 and 2. To measure the orthogonal intensity components I_x and I_y, a linear polarizer was temporarily placed in front of the CCD with the 0^° and 90^° angles between the transmission axis and the x-axis. The components I_x and I_y rotated anticlockwise by 90° at the focal plane, and the directions of the intensities are orthogonal with the x and y axis, respectively. Furthermore, the average intensity of the PC-RP-PEPV beam transforms into a quadrangular light spot. Components I_x and I_y have the same direction as the high-coherence width results. Our experimental results agree with the numerical simulation results. The directions of components I_x and I_y verify that the SOP has converted from radial polarization to azimuthal elliptical polarization because of the introduction of the vortex phase.

Fig. 9. Experimental results for the orthogonal intensity components I_x, I_y and total average intensity I of a PC-RP-PEPV beam with TC m = 4 in the x-y plane at several propagation distances z with high (σ = 3 mm, yellow background area) and low (σ = 0.5 mm, blue background area) coherence width. (a1)–(a3) and (d1)–(d3) Intensity patterns of component I_x. (b1)–(b3) and (e1)–(e3) Intensity patterns of component I_y. (c1)–(c3) and (f1)–(f3) Total average intensity patterns I.

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Figure 10 shows experimental results for a PC-RP-PEPV beam with high (σ = 3 mm, top row), middle (σ = 1 mm, middle row), and low coherence width (σ = 0.5 mm, bottom row). The other parameters are the same as those in Fig. 3, and all results are recorded by the CCD at the focal plane. A PC-RP-PEPV beam with a high coherence width forms an annular intensity spot and a solid light spot when the TC m = 0 and 1, respectively. Then, with decreasing coherence width, both transform into Gaussian light spots. When the TC m > 1, the intensity of the beam forms a polygonal hollow structure with high coherence and a polygonal solid structure with a decreasing coherence width. The experimental results are in accordance with the numerical simulation results shown in Fig. 3.

Fig. 10. Experimental intensity patterns of a PC-RP-PEPV beam with different TC values m = 0, 1, 2, 3, and fixed power order n = 2. (a1)–(a4) Intensity patterns with high coherence width (σ = 3 mm). (b1)–(b4) Intensity patterns with middle coherence width (σ = 1 mm). (c1)–(c4) Intensity patterns with low coherence width (σ = 0.5 mm). All results are in the focal plane.

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In addition, the intensity distributions of a PC-RP-PEPV beam with TC m = 4 and different power orders n = 4, 6, 8, and 10 are shown in Fig. 11. With an increase in the power order and high coherence, the sidelobe of the quadrangular hollow intensity distribution gradually disappears, and its quadrangular shape gradually changes into an approximately circular shape, as shown in Figs. 11(a1)–(a4). The hollow structure then changes into a solid structure for a low coherence width, as shown in Figs. 11(b1)–(b4). To verify that the vortex phase is fading away, we measured the intensity components I_x and I_y with high and low coherence widths, as shown in Figs. 11(c1), (c2) and (d1), (d2), respectively. The orthogonal components are the same as those of the partially coherent radially polarized beam. The experimental intensities are in accordance with the numerical simulation results, with only a slight difference between them. This may be caused by imperfections in optical components.

Fig. 11. Experimental intensity patterns of a PC-RP-PEPV beam with TC m = 4, and different power order n = 4, 6, 8, 10. (a1)–(a4) Intensity patterns with high coherence width (σ = 3 mm). (b1)–(b4) Intensity patterns with low coherence width (σ = 0.5 mm). (c1), (c2) Orthogonal components I_x and I_y with n = 10 and σ = 3 mm. (d1), (d2) Orthogonal components I_x and I_y with n = 10 and σ = 0.5 mm. The arrows in the white circle represent the direction of the linear polarizer. All results are in the focal plane.

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7. Conclusion

In conclusion, we combined both polarization and coherence modulations owing to the statistical fluctuations of the electromagnetic field for a power-exponent-phase vortex beam. This combination led to a PC-RP-PEPV beam. The focusing properties of the PC-RP-PEPV beam were studied numerically and experimentally. By adjusting the value of the TC, the intensity distribution of the PC-RP-PEPV beam can be manipulated from circular to polygonal structures, such as ellipses, triangles, and quadrangles. As the coherence width decreases, the dark core in the center of the beam gradually disappears. This PC-RP-PEPV beam also affects depolarization during propagation, which is similar to partially coherent radially polarized vortex beams, and the depolarization effect gradually increases as the power order increases. Our experimental results are in good agreement with the numerical simulation results. The power-exponent-phase induces changes in the SOP distributions, which may be useful for the detection of a phase object. Furthermore, the power-exponent-phase can be used to modulate the intensity structure of the beam, which can be used to trap particles with high and low refractive indices using coherence modulation. We believe that our results will be useful for optical manipulation, optical measurement, and optical information processing.

Funding

National Key Research and Development Program of China (2019YFA0705000); National Natural Science Foundation of China (11974218, 12174280, 12192254); Innovation Group of Jinan (2018GXRC010); Priority Academic Program Development of Jiangsu Higher Education Institutions; Tang Scholar; Local Science, Technology Development Project of the Central Government (YDZX20203700001766); Key Lab of Modern Optical Technologies of Jiangsu Province (KJS2138); Postgraduate Research & Practice Innovation Program of Jiangsu Province (KYCX22_3183).

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.

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Statistical properties of a partially coherent vector beam with controllable spatial coherence, vortex phase, and polarization

Abstract

1. Introduction

2. Theory of the PC-RP-PEPV beam

3. Propagation properties of a PC-RP-PEPV beam

4. Modulation properties of the PC-RP-PEPV beam

5. Degree of polarization for the PC-RP-PEPV beam

6. Experimental results

7. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (11)

Equations (24)

Optics Express