Propagation properties of phase-locked radially-polarized vector fields array in turbulent atmosphere

Lu Lu; Zhiqiang Wang; Zhiqiang Wang; Zhiqiang Wang; Yangjian Cai; Yangjian Cai; Yangjian Cai

doi:10.1364/OE.427003

1. Introduction

Free space optical communication (FSOC) has been widely researched owing to its advantages of larger bandwidth, better privacy, smaller antenna and component sizes, and lower component costs [1]. A significant challenge in FSOC is the beam distortion induced by atmospheric turbulence [1]. With the increasing demand for information transmission, methods to increase capacity and reliability must be developed. Currently, several methods exist for increasing capacity, such as increasing the transmitted power [2], optimizing receiver aperture [3], using multiple receiver apertures [4], and using the orbital angular momentum multiplexing [5].

A vector optical field was proposed in 1961 [6]. In 2000, this type of optical field re-emerged in research owing to its tight-focusing property [7]. Vector optical fields with spatially inhomogeneous states of polarization (SoPs) at the field cross-section have garnered significant attention, which are applicable to nonlinear optics [8], optical tweezers [9], optical micro-manipulation [10], quantum information [11], optical coherence encryption [12], and other related fields. In 2013, Lou et al. [13,14] proposed a vector field array and applied it to the laser micro-nano processing field. Hitherto, many works have used the focusing behavior of vector optical field array, such as multiple holes fabrication [13], multiple focal spots manipulation [14,15], and generation of arbitrary radially polarized array [16].

In 2008, the propagation properties of a vector beam in a turbulent atmosphere were investigated [17]. In recent years, many studies and applications pertaining to vector optical fields in atmospheric turbulence have been reported [18–23]. The effect of turbulence on vector optical fields is less prominent than the effect of the Gaussian or Laguerre Gaussian beam [18]. Vector modes as transmission bases in FSOC are more robust in turbulence compared with scalar modes [19]. The FSOC of high-dimensional structured optical fields is important for developing channel models in urban environment [20]. Experiments regarding the propagation of vector optical beams through random phase screens mimicking turbulence have been conducted [21]. Multi-wave mixing using a single-vector optical field has been demonstrated [22]. Recently, turbulence aberration correction for vector vortex beams has been researched on experimental data using deep neural networks [23].

One of the most significant advantages of the laser array is to increase the power, which is beneficial to high-power systems, inertial confinement, and high-energy weapons [24,25]. Hitherto, various scalar laser arrays propagating in turbulent environments have been investigated [26–32]. In FSOC links, increasing power is an option for increasing capacity [2], and some studies have shown that the transmission performance of vector beams is better than that of scalar beams [18–20]. It is essential to further investigate the propagation properties of vector optical field arrays in atmospheric turbulence. However, only a few reports are available regarding the propagation properties of vector field arrays in atmospheric turbulence.

In this study, based on a typical kind of cylindrical vector beam array, i.e., the radially-polarized vector field (RPVF) array, the propagation characteristics of atmospheric turbulence are investigated. On the basis of the extended Huygens–Fresnel principle, the analytical formulae for the average intensity, degree of polarization (DOP), and local SoPs of a phase-locked RPVF array are derived. Changes in the average intensity and DOP with several parameters, i.e., propagation distance, strength of turbulence, beam radius, and beamlet number, are obtained via numerical analysis. In particular, local SoPs with respect to the propagation distance are investigated to reduce the effect of turbulence and increase the channel capacity. This study may provide a theoretical basis for improving the information capacity and reliability of FSOC, as well as benefit potential applications in laser lidar, remote sensing, and related areas.

2. Theory

Under a paraxial approximation, the RPVF is expressed as the superposition of a TEM₀₁ with polarization along the x-axis and a TEM₁₀ with polarization parallel to the y-axis, as follows [33,34]:

(1)$$E(x,y) = {E_x}{{\mathbf e}_x} + {E_y}{{\mathbf e}_y} = {E_0}\left[ {\frac{x}{{{w_0}}}\exp \left( { - \frac{{{r^2}}}{{w_0^2}}} \right){{\mathbf e}_x} + \frac{y}{{{w_0}}}\exp \left( { - \frac{{{r^2}}}{{w_0^2}}} \right){{\mathbf e}_y}} \right],$$

where ${E_0}$ is a constant, ${w_0}$ is the beam waist associated with Gaussian beam, ${r^2} = {x^2} + y{}^2$, ${{\mathbf e}_x}$ and ${{\mathbf e}_y}$ are two unit vectors.

Assuming that a radial array comprises $N$ equal RPVF and is located symmetrically on a ring with an effective beam radius ${r_0}$, the separation angle between two adjacent beamlets is ${\alpha _n} = 2\pi /N$, and the spacing of adjacent beamlets is ${d_0}$, as shown in Fig. 1.

Fig. 1. Schematic diagram of RPVF array at the source plane (i.e., z = 0).

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The beam coherence-polarization (BCP) matrix provides the information of polarization and spatial correlation, and the BCP matrix is [35]

(2)$$\hat{{\mathrm{\boldsymbol {\Gamma}}}}({{{\mathbf r}_1},{{\mathbf r}_2},z} )= \left( {\begin{array}{cc} {{\Gamma _{xx}}({{{\mathbf r}_1},{{\mathbf r}_2},z} )}&{{\Gamma _{xy}}({{{\mathbf r}_1},{{\mathbf r}_2},z} )}\\ {{\Gamma _{yx}}({{{\mathbf r}_1},{{\mathbf r}_2},z} )}&{{\Gamma _{yy}}({{{\mathbf r}_1},{{\mathbf r}_2},z} )} \end{array}} \right),$$

where ${\Gamma _{ij}}({{{\mathbf r}_1},{{\mathbf r}_2},z} )= \left\langle {{E_i}({{{\mathbf r}_1},{{\mathbf r}_2},z} )E_j^ \ast ({{{\mathbf r}_1},{{\mathbf r}_2},z} )} \right\rangle ,({i,j = x,y} )$, $\left\langle \bullet \right\rangle$ denotes an ensemble average over the medium statistics, $({\ast} )$ is the conjugation operator, ${{\mathbf r}_1}$ and ${{\mathbf r}_2}$ are observation points on the receiver plane.

Substituting the Eq. (1) into Eq. (2), the BCP matrix for the RPVF array on the source plane can be expressed as

(3)$$\hat{{\mathrm{\boldsymbol {\Gamma}}}}({{{\mathbf r}_1},{{\mathbf r}_2},0} )= \frac{{E_0^2}}{{w_0^2}}\exp \left[ { - \frac{{{{({{{\mathbf r}_1} - {{\mathbf r}_0}} )}^2} + {{({{{\mathbf r}_2} - {{{\mathbf r^{\prime}}}_0}} )}^2}}}{{w_0^2}}} \right]\left( {\begin{array}{cc} {({{x_1} - {a_n}} )({{x_2} - {a_m}} )}&{({{x_1} - {a_n}} )({{y_2} - {b_m}} )}\\ {({{y_1} - {b_n}} )({{x_2} - {a_m}} )}&{({{y_1} - {b_n}} )({{y_2} - {b_m}} )} \end{array}} \right).$$

where ${{\mathbf r}_0} = {a_n}{{\mathbf e}_x} + {b_n}{{\mathbf e}_y}$, ${{\mathbf r^{\prime}}_0} = {a_m}{{\mathbf e}_x} + {b_m}{{\mathbf e}_y}$, ${a_n} = {r_0}\cos {\alpha _n}$, ${b_n} = {r_0}\sin {\alpha _n}$, ${a_m} = {r_0}\cos {\alpha _m}$, ${b_m} = {r_0}\sin {\alpha _m}$, and n ($m$) represents the n-th (or m-th) beamlet.

Based on the extended Huygens-Fresnel principle, supposing ${{\mathbf r}_1} = {{\mathbf r}_2} = {\mathbf r}$, the element of BCP matrix propagating through atmospheric turbulence on the receiver plane is expressed as [36]

(4)$$\begin{aligned} \left\langle {{{\Gamma }_{ij}}({\mathbf r},{\mathbf r},z)} \right\rangle & = {\left( {\frac{k}{{2\pi z}}} \right)^2}\int_{ - \infty }^\infty {\int_{ - \infty }^\infty {\int_{ - \infty }^\infty {\int_{ - \infty }^\infty {{{\Gamma }_{ij}}\left( {{{{\mathbf r'}}_1},{{{\mathbf r'}}_2},0} \right)} } } } \exp \left[ { - \frac{{\textrm{i}k}}{{2z}}{{\left( {{{{\mathbf r'}}_1} - {\mathbf r}} \right)}^2} + \frac{{\textrm{i}k}}{{2z}}{{\left( {{{{\mathbf r'}}_2} - {\mathbf r}} \right)}^2}} \right]\\ & \times \exp \left[ { - \frac{1}{{{\rho _0}}}{{\left( {{{{\mathbf r'}_1}} - {{{\mathbf r'}_2}}} \right)}^2}} \right] \textrm{d}{{{\mathbf r'}}_1}\textrm{d}{{{\mathbf r'}}_2}, \end{aligned}$$

where ${\rho _0} = {({0.545C_n^2{k^2}z} )^{ - 3/5}}$ is the coherence length of a spherical wave propagation in turbulent medium, and $C_n^2$ is the refractive index structure constant [36].

To perform integration on Eq. (4), the following integral is applied [37]

(5)$$\int {{x^l}} \exp ( - \beta {x^2} + 2\gamma x)\textrm{d}x = l!\exp \left( {\frac{{{\gamma^2}}}{\beta }} \right){\left( {\frac{\gamma }{\beta }} \right)^l}\sqrt {\frac{\pi }{\beta }} \sum\limits_{{l_i} = 0}^{[l/2]} {\frac{{{{({\beta /4{\gamma^2}} )}^l}}}{{(l - 2{l_i})!({l_i})!}}} , $$

After a series of calculations, the BCP matrix elements of the RPVF array propagating through atmospheric turbulence are derived as follows:

(6)$${\Gamma _{xx}} = CS^{\prime}\left\{ {\frac{{({{A^2} - {B^2}} )S}}{4} - ASJ + \frac{{BVL}}{2}} \right. + \frac{{VW}}{{8T}} + S\left[ {\frac{{{A^2} + w_0^2}}{4}} \right. + O\left. {\left. { - \frac{{{k^2}w_0^4}}{{16{z^2}}}\left( {\frac{{{P^2}}}{{2{R^2}}} + \frac{1}{R}} \right)} \right]} \right\},$$

(7)$${\Gamma _{yy}} = CS\left\{ {\frac{{({{{A^{\prime}}^2} - {{B^{\prime}}^2}} )S^{\prime}}}{4} - A^{\prime}S^{\prime}J^{\prime} + \frac{{B^{\prime}V^{\prime}L^{\prime}}}{2} + \frac{{V^{\prime}W^{\prime}}}{{8T}} + S^{\prime}\left[ {\frac{{{{A^{\prime}}^2} + w_0^2}}{4} + O^{\prime}\left. { - \frac{{{k^2}w_0^4}}{{16{z^2}}}\left( {\frac{{{{P^{\prime}}^2}}}{{2{R^2}}} + \frac{1}{R}} \right)} \right]} \right.} \right\},$$

(8)$${\Gamma _{xy}} = C\left\{ {SS^{\prime}JJ^{\prime} + \frac{{SVJL^{\prime}}}{2} - \frac{{S^{\prime}V^{\prime}J^{\prime}L}}{2} - \frac{{VV^{\prime}LL^{\prime}}}{4} - {b_m}SS^{\prime}J + \frac{{{b_m}VV^{\prime}L}}{2} - {a_n}SS^{\prime}J^{\prime} - \frac{{{a_n}VV^{\prime}L^{\prime}}}{2} - {a_n}{b_m}S} \right\},$$

(9)$${\Gamma _{yx}} = C\left\{ {SS^{\prime}JJ^{\prime} - \frac{{SVJL^{\prime}}}{2} + \frac{{S^{\prime}V^{\prime}J^{\prime}L}}{2} - \frac{{VV^{\prime}LL^{\prime}}}{4} - {b_n}SS^{\prime}J + \frac{{{b_n}VV^{\prime}L}}{2} - {a_m}SS^{\prime}L^{\prime} - \frac{{{a_m}VV^{\prime}L^{\prime}}}{2} + {a_m}{b_n}S} \right\},$$

where

A = {a_m} + {a_n},\;B = {a_m} - {a_n},\;A^{\prime} = {b_m} + {b_n},\;B^{\prime} = {b_m} - {b_n},

C = {({k/2\pi z} )^2}\exp [{ - ({a_m^2 + b_m^2 + a_n^2 + b_n^2} )/w_0^2} ],

T = 1/2w_0^2 + 1/\rho _0^2,\;H = 2/w_0^2 + {k^2}/4{z^2}T,\;R = T + {k^2}w_0^2/8{z^2},

G = B/w_0^2 - \textrm{i}kx/z,\;G^{\prime} = B^{\prime}/w_0^2 - \textrm{i}ky/z,

O\textrm{ = i}kw_0^2P/8zR,\;O^{\prime}\textrm{ = i}kw_0^2P^{\prime}/8zR,

J = A/2 + O,\;J^{\prime} = A^{\prime}/2 + O^{\prime},

K = 2A/w_0^2 + \textrm{i}kG/2Tz,\;K^{\prime} = 2A^{\prime}/w_0^2 + \textrm{i}kG^{\prime}/2Tz,

L = G - \textrm{i}kK/4HTz,\;L^{\prime} = G^{\prime} - \textrm{i}kK^{\prime}/4HTz,

P = B/w_0^2 + \textrm{i}kA/z - \textrm{i}kx/z,\;P^{\prime} = B^{\prime}/w_0^2 + \textrm{i}kA^{\prime}/z - \textrm{i}ky/z,

S = \sqrt {\pi /2} \sqrt {\pi /R} {w_0}\exp ({{A^2}/2w_0^2 + {P^2}/4R} ),\;S^{\prime} = \sqrt {\pi /2} \sqrt {\pi /R} {w_0}\exp ({{{A^{\prime}}^2}/2w_0^2 + {{P^{\prime}}^2}/4R} ),

V = \sqrt {\pi /{T^3}} \sqrt {\pi /H} \exp ({{G^2}/4T + {K^2}/4H} ),\;V^{\prime} = \sqrt {\pi /{T^3}} \sqrt {\pi /H} \exp ({{{G^{\prime}}^2}/4T + {{K^{\prime}}^2}/4H} ),

W = {k^2}{K^2}/4{H^2}{z^2} + {k^2}/2H{z^2} - T + \textrm{i}kGK/Hz - {G^2}/4, \;W^{\prime} = {k^2}{K^2}/4{H^2}{z^2} + {k^2}/2H{z^2} - T + \textrm{i}kG^{\prime}K^{\prime}/Hz - {G^{\prime2}}/4.

The average intensity of the phase-locked RPVF array propagating through atmospheric turbulence can be expressed as [35]

(10)$$I({\mathbf r},z) = {\Gamma _{xx}}({{\mathbf r},{\mathbf r},z} )+ {\Gamma _{yy}}({{\mathbf r},{\mathbf r},z} ).$$

Degree of polarization is used to describe a portion of a polarized laser beam. A perfectly polarized beam has a DOP of 1, whereas an unpolarized beam has a DOP of 0. The DOP of the RPVF array propagating through atmospheric turbulence is expressed as [35]

(11)$$P({\mathbf r},z) = \sqrt {1 - \frac{{4\det \hat{\Gamma }({\mathbf r},{\mathbf r},z)}}{{{{[{\textrm{Tr}\hat{\Gamma }({\mathbf r},{\mathbf r},z)} ]}^2}}}} = \sqrt {1 - \frac{{4[{{\Gamma _{xx}}({\mathbf r},{\mathbf r},z){\Gamma _{yy}}({\mathbf r},{\mathbf r},z) - {\Gamma _{xy}}({\mathbf r},{\mathbf r},z){\Gamma _{yx}}({\mathbf r},{\mathbf r},z)} ]}}{{{{[{{\Gamma _{xx}}({\mathbf r},{\mathbf r},z) + {\Gamma _{yy}}({\mathbf r},{\mathbf r},z)} ]}^2}}}} ,$$

The ellipticity can be described by the Stokes parameters ${S_3}$ [38], which can be expressed as

(12)$${S_3} = \textrm{i}({{E_x}E_y^ \ast{-} {E_y}E_x^ \ast } )= \textrm{i}[{{\Gamma _{xy}}({\mathbf r},{\mathbf r},z) - {\Gamma _{yx}}({\mathbf r},{\mathbf r},z)} ].$$

This parameter characterizes the local SoPs [39]; if ${S_3} = 0$, then the elliptic polarization state degenerates into linear, whereas if ${S_3} = 1$ (${S_3} ={-} 1$), then the SoP becomes the right-handed (left-handed) circularly polarized state [40].

3. Numerical calculation and analysis

3.1 Normalized average intensity

In this section, the normalized average intensity is investigated comprehensively based on the propagation distance z, strength of turbulence $C_n^2$, size of beam waist ${w_0}$, and number of beamlets $N$. The universal numerical calculation parameters used were as follows: wavelength $\lambda = 1064$ nm, grid number $Q = 512$, spacing of the beamlet ${d_0} = {{5{w_0}} / {\sqrt 2 }}$, and other parameters listed in Table 1. The variation in the average intensity of the RPVF array with the propagation distance is investigated. When $z = 4$km (Fig. 2(a)), unlike the hollow shape on the initial source plane, the beamlet became a solid beam, which is consistent with the propagation property of the hollow beam in atmospheric turbulence [41]. As the propagation distance increased, the energy of the beamlet concentrates gradually at the center of the beam array, e.g., $z = 6$km (Fig. 2(b)). At the $z = 8$ km plane, the beam spot evolves into a Gaussian type, as shown in Fig. 2(c). In Figs. 2(d)–(f), a crossline (i.e., $y = 0$ in Figs. 2(a)–(c)) of the average intensity distribution with various propagation distances is demonstrated. Furthermore, the energy of the beamlet transfers gradually to the center of the RPVF array.

Fig. 2. Normalized average intensity of RPVF array varying with propagation distance: (a) $z = 4$ km, (b) $z = 6$km, and (c) $z = 8$km, (d)-(f) show intensity distribution for $y = 0$.

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Table 1. Calculation parameters related to the Figs. 2–5

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Changes in the normalized average intensity of the RPVF array under various turbulence strengths are investigated. The normalized intensity distributions are shown in Figs. 3(a)–(c). In Figs. 3(d)–(f), a crossline ($y = 0$ in Figs. 3(a)–(c)) of the average intensity distribution with various turbulence strengths is shown. It is discovered that a higher turbulence strength accelerates the spreading of beamlets, which is consistent with the results in [28,29,31,42,43], and decreases the hollow zone because the energy of the beamlets gradually flows towards the effective beam center. Thus, the beam coherent combination with a stronger turbulence strength is faster than that with a relatively weaker one.

Fig. 3. Normalized average intensity of RPVF array varying with strength of turbulence: (a) $C_n^2 = 0.2 \times {10^{ - 15}}{\textrm{m}^{\textrm{ - 2/3}}}$, (b) $C_n^2 = {10^{ - 15}}{\textrm{m}^{\textrm{ - 2/3}}}$, and (c) $C_n^2 = 5 \times {10^{ - 15}}{\textrm{m}^{\textrm{ - 2/3}}}$, where (d)-(f) show intensity distribution for $y = 0$.

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The relationship between the average intensity of the RPVF array and beam-waist-related Gaussian beam is investigated. The normalized intensity distributions are shown in Figs. 4(a)–(c), and the corresponding intensity distributions for $y = 0$ are shown in Figs. 4(d)–(f). When ${w_0} = 3$ mm, the beamlets are close to each other, and the energy of the beamlets concentrates gradually toward the center of the beam array. With the beam waist increased, the beamlets propagate separately, such as ${w_0} = 4$mm, and ${w_0} = 5$mm. For the small beam waist, the beam combination has appeared at $z = 4$km; for the large beam waist, the beamlet has not been combined at $z = 4$km. It is clear that the beam combination of small beam waist is earlier than that for large beam waist, which indicates the onset of beam combination can be manipulated by the initial beam waist.

Fig. 4. Normalized average intensity of RPVF array with various beam waist: (a) ${w_0} = 3$mm, (b) ${w_0} = 4$mm, and ${w_0} = 5$mm, where (d)-(f) show intensity distribution for $y = 0$.

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The relationship between the average intensity and beamlet number is investigated. In this study, the effective beam radii are ${r_0} = 2.5{w_0}$ for $N = 4$, ${r_0} = {{5{w_0}} / {\sqrt 2 }}$ for $N = 6$, and ${r_0} = {{5{w_0}} / {\sqrt {4 - 2\sqrt 2 } }}$ for $N = 8$ (ensuring the same spacing of the beamlet ${d_0} = {{5{w_0}} / {\sqrt 2 }}$). Figure 5 shows the normalized average intensity distribution for various beamlet numbers: $N = 4$ (Figs. 5(a)–(c)), $N = 6$ (Figs. 5(d)–(f)), and $N = 8$(Figs. 5(g)–(i)). When $z = 4$km (i.e., the first column), the beamlet evolves into a solid beam, and the outline of every beamlet can be distinguished for three cases. However, the beamlet shape could not be distinguished when $z = 6$km (i.e., the second column). On the $z = 8$ km plane (i.e., the third column), the beam spots gradually transform into a Gaussian shape for the three cases. Owing to the increase in the beamlet number, the size of the transmission aperture increases, resulting in an increase in the size of the receiver aperture. As shown in Fig. 5, the larger the beamlet number is, the larger the beam spot on the receiver plane is.

Fig. 5. Normalized average intensity of RPVF array with various beamlet number: (a)-(c) $N = 4$, (d)-(f) $N = 6$, and (g)-(i) $N = 8$, where the 1^st column $z = 4$ km, 2^nd column $z = 6$km, and 3^rd column $z = 8$ km.

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3.2 Degree of polarization

The numerical parameters used for the investigation in this subsection are the same as those described in Section 3.1. By definition, it is clear that the DOP is equal to 1 on the source plane. To investigate the polarization properties of the RPVF array with several significant parameters in a turbulent atmosphere, we analyze the relationship between the DOP and the propagation distance, strength of turbulence, beam waist, and beamlet number. Figure 6(a) shows the DOP (i.e., $P(x,0,z)$) varying with the propagation distance. It is clear that the DOP on the axis is zero, whereas that on the off-axis increases gradually toward $\textrm{DOP} = 1$. In addition, the width of the valley increases during propagation. The results show that the DOP of the RPVF array is depolarized when it propagates through atmospheric turbulence, which is consistent with the result presented in [17]. Figure 6(b) shows the variation in the DOP with the refractive index structure constant. As shown, a stronger turbulence strength depolarizes the DOP more significantly. The effect of the beam waist size on the DOP is simulated, and the results are as shown in Fig. 6(c). It is found that a smaller beam waist causes the DOP to be destroyed more severely. Figure 6(d) shows the relation between the DOP and the number of beamlets. In the near-on-axis region, the variation in the DOP becomes less significant as the beam number increased, whereas the effect of the larger beamlet number on the DOP is more significant in the near-edge region.

Fig. 6. Degree of polarization $P(x,0,z)$ of RPVF array for different parameters: (a) propagation distance, (b) refractive index structure constant, (c) beam waist, and (d) beamlet number.

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3.3 Local states of polarization

To understand the mechanism of the RPVF array propagating through a turbulent atmosphere more effectively, the local SoPs (or local ellipticity) are investigated. The refractive index structure constant is $C_n^2 = {10^{ - 15}}{\textrm{m}^{\textrm{ - 2/3}}}$. To avoid duplication of the average intensity images, we set the propagation distance as $z = 0$, $z = 3$, $z = 5$, and $z = 7$km. It is noteworthy that other propagation distances are applicable. When the beamlet number $N = 4$ and the effective beam radius ${r_0} = 2.5{w_0}$, the average intensity distributions for the aforementioned propagation distance are shown in Figs. 7(a)–(d). For the crossline (i.e., $y ={-} {r_0}$) in Figs. 7(a)–(d), the corresponding local SoPs are depicted in Fig. 7(e). The ellipticity is -1 (or 1) at $x ={-} {r_0}$ (or $x = {r_0}$), whereas it is zero at $x = 0$. Furthermore, it is clear that ellipticity is not a monotonic function; in fact, it exhibits a center symmetry at the coordinate $(0,0)$, where the positive and negative signs of ellipticity represent the right-handed (RH) or left-handed (LH) rotations, respectively [37]. As the propagation distance increased, the SoPs do not change at the special locations, i.e., $x ={\pm} {r_0}$ and $x = 0$, whereas the absolute of SoPs in other locations increases for both LH and RH rotations. The inset illustrates the sign of ellipticity in Fig. 7(e); the local SoP is a LH rotation in the third quadrant and a RH rotation in the fourth quadrant. In addition, although the energy flows continuously to the center of the beam array as the propagation distance increases, the phase-locked RPVF array imposes a slight effect on the evolution of the local SoPs during propagation, and the sign of ellipticity at different receiver planes is consistent with that of the source field. Therefore, the SoPs of specific locations remain the same, and the range of local SoPs evolving into LH or RH polarization will increase with propagation distance. The local SoPs with different N are investigated, i.e., six beamlets (as shown in Fig. 8) and eight beamlets (as shown in Fig. 9), where the effective beam radii are ${r_0} = {{5{w_0}} / {\sqrt 2 }}$ for $N = 6$ and ${r_0} = {{5{w_0}} / {\sqrt {4 - 2\sqrt 2 } }}$ for $N = 8$. Similarly, the local SoPs demonstrate the similar performance as the propagation distance increases.

Fig. 7. Normalized average intensity varying with propagation distance for beam number $N = 4$: (a) $z = 0$, (b) $z = 3$km, (c) $z = 5$km, (d) $z = 7$km, and (e) the local ellipticity (or local SoPs) for crossline $y ={-} {r_0}$ (dashed white line).

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Fig. 8. Normalized average intensity varying with propagation distance for beam number $N = 6$: (a) $z = 0$, (b) $z = 3$km, (c) $z = 5$km, (d) $z = 7$km, and (e) the local ellipticity (or local SoPs) for crossline $y ={-} {r_0}$ (dashed white line).

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Fig. 9. Normalized average intensity varying with propagation distance for beam number $N = 8$: (a) $z = 0$, (b) $z = 3$km, (c) $z = 5$km, (d) $z = 7$km, and (e) the local ellipticity (or local SoPs) for crossline $y ={-} {r_0}$ (dashed white line).

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Based on the principle of coherent combination, every beamlet interacts with each other, and the energy of beamlets gradually concentrates to the center of effective beam spot. Due to the symmetric configuration of beamlets, the intensity distribution still exhibits axial symmetry. With the beam combination, spatial SoPs affect each other simultaneously. For the location on the symmetry axis, there is no polarization component, so the local polarization state is zero; If a line is parallel to the axis of symmetry, taking two symmetrical positions on it, the local polarization state of two locations is not equal to zero because of the existence of polarization component, and the sign is opposite due to the LH rotation and a RH rotation distribution. Thus, it can be seen that the SoPs do not change at the aforementioned special locations, and the sign of ellipticity at different receiver planes is agreement with that of the source field location.

Based on numerical analysis, the SoPs of several special locations remain constant for the phase-locked RPVF array propagation in atmospheric turbulence. This is beneficial to applications in FSOC and related areas owing to the following reasons: (i) the turbulence imposes minimal effect on the SoPs at specific locations, and the transmission performance might be improved; (ii) different SoPs (i.e., linear, LH, and RH rotation polarizations) appear in the corresponding specific positions, and the polarization (the degree of freedom of a light beam) is used as a carrier basis of signals for FSOC links [44,45]—this may increase the channel capacity. The use of the RPVF array may improve the FSOC performance as well as realize local SoP manipulation at specific locations in an atmospheric turbulent environment.

4. Conclusion

Based on the extended Huygens–Fresnel principle, analytical formulae for the average intensity, DOP, and local SoPs of a phase-locked RPVF array propagating through atmospheric turbulence were derived. The average intensity and DOP varying with propagation distance, turbulent strength, beam waist, and beamlet number were investigated. These results indicated that the RPVF array can accelerate the rate of beam coherent combination, whereas the DOP depolarizes more significantly when the parameters are selected for longer propagation distance, higher turbulent strength, smaller beam waist, and larger beamlet number. In particular, one of the more significant findings to emerge from this study is that the sign of local SoPs at different receiver planes is consistent with that of the source field, and that the SoPs do not change at specific locations, which indicates that the local SoPs are slightly affected by turbulence; Meanwhile, at the corresponding locations, three SoPs are observed, thereby enabling the channel capacity to be increased using the polarization (the degree of freedom of a light beam) as a carrier basis of signals. This study may provide a theoretical basis for improving the information capacity and reliability of FSOC links, thereby enabling potential applications in laser lidar, remote sensing, and related areas.

Funding

National Key Research and Development Program of China (2019YFA0705000); National Natural Science Foundation of China (11804234, 11903062, 11974218, 91750201); Innovation Group of Jinan (2018GXRC010); Local Science and Technology Development Project of the Central Government (YDZX20203700001766).

Acknowledgments

We are very grateful to the aforementioned funding. We appreciate the reviewers for their valuable comments and suggestions.

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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Propagation properties of phase-locked radially-polarized vector fields array in turbulent atmosphere

Abstract

1. Introduction

2. Theory

3. Numerical calculation and analysis

3.1 Normalized average intensity

3.2 Degree of polarization

3.3 Local states of polarization

4. Conclusion

Funding

Acknowledgments

Disclosures

Data availability

References

Data availability

Cited By

Figures (9)

Tables (1)

Equations (24)

Optics Express