Changes in orbital angular momentum distribution of a twisted partially coherent array beam in anisotropic turbulence

Jing Wang; Jing Wang; Yakai Zhang; Yinlong Guo; Xianmei Qian; Wenyue Zhu; Jinhong Li; Jinhong Li

doi:10.1364/OE.463642

1. Introduction

In recent years, orbital angular momentum (OAM) has attracted extensive attention because of its great application prospects in the optical communication field [1–5]. OAM has an infinite number of mutually orthogonal modes, and modulating each signal on a different mode will greatly enhance the channel capacity and spectrum utilization of optical communication systems. Gibson et al. encoded the information with OAM to achieve 15m free-space optical communication, and the information encoded in this way has strong confidentiality [1]. Tamburini et al. made OAM multiplexing in the experiment for the first time and realized 442m wireless optical communication [2]. Bozinovic et al. achieved 1.6 Tbit/s information transmission by using a specially designed ring fiber to transmit the OAM beam [3]. Xie et al. designed an ultra-wideband OAM transmitter for 1.2Tbit/s OAM communication [4]. Pang et al. effectively reduce power loss and modal crosstalk in free-space optical communication by simultaneous transmission of four signals through mode orthogonalization [5]. In addition to the communication field, OAM also has broad application prospects in many other fields, such as optical wrenches [6], designing optical micromachines to manipulate particles [7], quantum memory [8], detecting rotating objects [9], etc.

A simple way to make the beam carry OAM is to modulate the phase of the beam. As an unconventional phase structure, the vortex phase has attracted the interest of researchers because of its spiral wavefront property during transmission [10]. The number of OAM that can be carried by each photon of a fully coherent vortex beam is lћ, l is the topological charge number, ћ is the approximate Planck constant. Unlike the vortex phase, in the partially coherent light field, there is also a special phase that can carry OAM called twist phase [11–18]. The twist phase is a position-dependent quadratic function that cannot be simply decomposed into the product of two one-dimensional covariates, therefore, can only exist in partially coherent beams. It is interesting to note, partially coherent beams carrying twist phase can be generated by superposition of vortex beams [12]. In 1993, Simon et al. introduced the concept of twist phase and introduced it into the Gaussian Schell-model beam to obtain the twisted Gaussian Schell-model(TGSM) beam [13], which rotate around the axis during transmission. Then the TGSM beam was generated by Friberg using an elliptical Gaussian Schell-model beam and two sets of cylindrical lenses [14]. When a twisted beam is used as an illumination source, the resulting image resolution can exceed the Rayleigh limit [15]. In addition, twisted beams also have potential applications in ghost imaging [16] and particle capture [17]. It is now well appreciated that twisted partially coherent beams can suppress the negative effects induced by turbulence [18]. However, due to the difficulty of the construction of the new correlation structure, the above studies are confined to the TGSM beam. Many thanks to Gori et al. for proposing a method for constructing special correlated partially coherent beams [19] and extended it to vector beams [20], variety of interesting new special correlated beams had been proposed [21–24]. Multi-Gaussian Schell-model beams can produce a flat-top distribution in the far field [21]. Hermite-Gaussian correlated beam splits itself during propagation [22]. Self-focusing of non-uniformly correlated beams [23]. Arbitrarily manipulable partially coherent array beams [24]. When adding consideration of twist phase, Borghi et al. provided the proof condition of whether the axisymmetric partially coherent beam can carry the twist phase based on the modal analysis method [25]. Gori et al. proposed a method to generate a twisted beam without symmetric constraints [26]. Many special correlated twisted partially coherent beams have also been generated [27–32], such as radially polarized twisted beams [27], twisted Laguerre-Gaussian Schell-model beams [28], twisted vortex Gaussian Schell-model beams [29], twisted array beams [30,31], and twisted elliptical multi-Gaussian Schell-model beams [32].

In the past, the calculation of partially coherent beam OAM mainly used the method of finding second-order moment parameters, which required complex matrix operations [11]. In 2012, Kim et al. proposed another method of calculating OAM, which can be realized only by simple differentiation [33], compared with the second-moment method, the calculation process was greatly simplified. Later, this method was used to calculate the OAM of twisted Laguerre-Gaussian Schell-model beams [34] and twisted Hermite Gaussion Schell-model beams [35] in free space and atmospheric turbulence, and the results showed that the total average OAM per photon was conserved when transmitting in both free space and atmospheric turbulence. Zhang et al. found that different OAM flux density distributions can be controlled by changing the initial coherence of the beam [36]. Due to the important applications of lasers for free-space-optical communication [37–39], their interaction with turbulence has become an important issue that researchers must consider. It is shown that partially coherent beams can well suppress the negative effects of beam propagation in atmospheric turbulence [40–48]. For a long time, the isotropic Kolmogorov power spectrum model has been used to study atmospheric turbulence. However, results have shown that the turbulence above the atmospheric boundary layer (1 to 2km from the earth's surface) does not always follow the Kolmogorov power spectrum model, and the general non-Kolmogorov turbulence statistics should be considered [49–51]. Moreover, turbulence in the free atmosphere has strong inhomogeneity and anisotropy at large scales [52,53]. Therefore, the non-Kolmogorov anisotropic turbulence power spectrum model is a better choice for simulating beam propagating in the atmosphere turbulence [54].

In our previous work, we proposed a class of twisted Schell-model array correlated sources [31], which gradually split the spot into arrays during propagating and can suppress the effects of atmospheric turbulence. In this paper, the OAM variation of twisted array beams have attracted our interest, the OAM evolution characteristics of twisted radially and rectangularly symmetric array beams in anisotropic turbulence are further investigated, and the results have potential applications in the field of laser atmospheric communication.

2. Theoretical model

First, we recall the cross-spectral density (CSD) function of a twisted radial array beam with an elliptical amplitude profile at the source plane can be expressed as [31]

(1)$$\begin{array}{l} {W_{cir}}({{\mathbf r}_1},{{\mathbf r}_2}) = \frac{1}{{{C_{ra}}}}\sum\limits_{n = 0}^{{N_{ra}}} {\sum\limits_{m = 1}^{{M_{ra}}} {\exp } } \left( { - \frac{{x_1^2 + x_2^2}}{{4\sigma_x^2}} - \frac{{y_1^2 + y_2^2}}{{4\sigma_y^2}}} \right)\exp \left[ { - \frac{{{{({{{\mathbf r}_1} - {{\mathbf r}_2}} )}^2}}}{{2\delta_0^2}}} \right]\\ \times \exp [{i2\pi m{r_0}\cos {\theta_n}({{x_1} - {x_2}} )} ]\exp [{i2\pi m{r_0}\sin {\theta_n}({{y_1} - {y_2}} )} ]\exp ({ - ik\mu {{\mathbf r}_1} \times {{\mathbf r}_2}} ), \end{array}$$

where ${{\mathbf r}_i} \equiv ({{x_i},{y_i}} )({i = 1,2} )$, ${\sigma _x}$, ${\sigma _y}$ denotes the width of the waist in the x, y direction, respectively. ${\delta _0}$ is the coherence length of the beam, $\mu$ denotes twist factor and must be bounded by the inequality $\mu \le 1/k\delta _0^2$ [25]. Array structure parameter ${r_0}$ is the spacing parameter of the central ring, ${\theta _n} = 2\pi n/({N_{ra}} + 1)$ is the angular spacing, ${C_{ra}} = {M_{ra}}({N_{ra}} + 1)$ is the normalization factor. ${M_{ra}}$ is the number of rings, ${N_{ra}} + 1$ indicates the number of spots per ring, when ${M_{ra}} = {N_{ra}} = 0$ the beam degenerates to a TGSM beam [13].

According to the generalized Huygens-Fresnel principle, the CSD function of the beam propagation in atmospheric turbulence can be expressed as [18,38]

(2)$$\begin{array}{l} W({{\mathbf \rho }_1},{{\mathbf \rho }_2}) = {\left( {\frac{k}{{2\pi z}}} \right)^2}\int {\int {W({{\mathbf r}_1},{{\mathbf r}_2})\left\langle {\exp [{\psi ({{\mathbf r}_1},{{\mathbf \rho }_1}) + {\psi^\ast }({{\mathbf r}_2},{{\mathbf \rho }_2})} ]} \right\rangle } } \\ \times \exp \left[ { - \frac{{ik}}{{2z}}[{({\mathbf r}_1^2 - {\mathbf r}_2^2) - 2({{\mathbf r}_1} \cdot {{\mathbf \rho }_1} - {{\mathbf r}_2} \cdot {{\mathbf \rho }_2}) + ({\mathbf \rho }_1^2 - {\mathbf \rho }_2^2)} ]} \right]{d^2}{{\mathbf r}_1}{d^2}{{\mathbf r}_2}, \end{array}$$

where ${{\mathbf \rho }_i} \equiv ({{u_i},{v_i}} )({i = 1,2} )$ are the position vectors in the receiver plane, $\lambda$ is the wavelength, $k = 2\pi /\lambda$ is wavenumber, $\psi ({\mathbf r},{\mathbf \rho })$ is the phase perturbation of a spherical wave passing through turbulence, ensemble average $\left\langle \cdot \right\rangle$ represent the two-point mutual coherence function. For non-Kolmogorov anisotropic turbulence, it can be expressed as [31]

(3)$$\begin{array}{l} \left\langle {\exp [{\psi ({{\mathbf r}_1},{{\mathbf \rho }_1}) + {\psi^ \ast }({{\mathbf r}_2},{{\mathbf \rho }_2})} ]} \right\rangle \\ \cong \exp \left[ { - \frac{1}{{\rho_0^2(z)}}\left( {\frac{{u_d^2 + {u_d}{x_d} + x_d^2}}{{\varepsilon_x^2}} + \frac{{v_d^2 + {v_d}{y_d} + y_d^2}}{{\varepsilon_y^2}}} \right)} \right], \end{array}$$

(4)$${\rho _0}(z) = {\left[ {\frac{{\xi \Gamma (\xi - 1)\Gamma ( - \xi /2)\cos (\mathrm{\pi }\xi \textrm{/2)}}}{{{2^\xi }(\xi - 1)\Gamma (\xi /2)}}\tilde{C}_n^2{k^2}z} \right]^{1/(2 - \xi )}},$$

where ${x_d} = {x_1} - {x_2},\;{y_d} = {y_1} - {y_2},\;{u_d} = {u_1} - {u_2},\;{v_d} = {v_1} - {v_2}$, ${\rho _0}(z)$ denotes the spatial coherence radius of spherical waves in turbulence. $\tilde{C}_n^2$ represents the refractive index structure constant of atmospheric turbulence in the unit ${m^{3 - \xi }}$. $\xi$ denotes the power exponent of turbulence, ${\varepsilon _x}$ and ${\varepsilon _y}$ denotes the anisotropy coefficient of turbulence. $\Gamma ({\cdot} )$ is the Gamma function. When ${\varepsilon _x}\textrm{ = }{\varepsilon _y}\textrm{ = }1$, turbulence is isotropic, other cases ${\varepsilon _x} \ne {\varepsilon _y}$ the turbulence is anisotropic.

In this paper, we study a twisted partially coherent beam which carrying OAM. For scalar partially coherent beams, the OAM flux density represents the spatial distribution of OAM in the beam cross-section, and the OAM flux density along the z-axis can be expressed as [33]

(5)$$O({\mathbf \rho },z) ={-} \frac{{{\varepsilon _0}}}{k}{\mathop{\textrm {Im}}\nolimits} {[{v_1}{\partial _{{u_2}}}W({{\mathbf \rho }_1},{{\mathbf \rho }_2},z) - {u_1}{\partial _{{v_2}}}W({{\mathbf \rho }_1},{{\mathbf \rho }_2},z)]_{{{\mathbf \rho }_1} = {{\mathbf \rho }_2} = {\mathbf \rho }}},$$

where ${\varepsilon _0}$ represents the free space dielectric constant, ${\partial _{{u_2}}}$ and ${\partial _{{v_2}}}$ represents the partial derivative with respect to ${u_2}$ and ${v_2}$. The value of this quantity is closely related to the spectral intensity. In order to better understand the distribution of OAM over the beam cross section, the normalized OAM flux density is defined as

(6)$${O_n}({\mathbf \rho },z) = \frac{{\hbar \omega O({\mathbf \rho },z)}}{{{S_p}({\mathbf \rho },z)}},$$

where $\omega \textrm{ = }2\pi c/\lambda$ is the angular frequency of the beam, ${S_p}({\mathbf \rho },z)$ is the z component of the Poynting vector, which is of the form

(7)$${S_p}({\mathbf \rho },z) = \frac{k}{{{\mu _0}\omega }}I({\mathbf \rho },z),$$

where µ₀ is the vacuum permeability. The spectral intensity is given by $I({{\mathbf \rho },z} )= W({{\mathbf \rho },{\mathbf \rho },z} )$, and the degree of coherence (DOC) is defined as [18–21,31]

(8)$$\eta ({{{\mathbf \rho }_1},{{\mathbf \rho }_2},z} )= \frac{{W({{{\mathbf \rho }_1},{{\mathbf \rho }_2},z} )}}{{\sqrt {I({{{\mathbf \rho }_1},z} )I({{{\mathbf \rho }_2},z} )} }}.$$

Substituting Eq. (1) and Eq. (3) into Eq. (2), the analytical formula for the CSD function of a twisted radial array beam propagating in non-Kolmogorov anisotropic turbulence is obtained

(9)$$\begin{array}{l} {W_{cir}}({{\mathbf \rho }_1},{{\mathbf \rho }_2}) = \frac{{{k^2}}}{{4{z^2}{C_{ra}}\sqrt {{X_1}{Y_1}{X_2}{Y_2}} }}\sum\limits_{n = 0}^{{N_{ra}}} {\sum\limits_{m = 1}^{{M_{ra}}} {\exp [ - \frac{{ik}}{{2z}}({\mathbf \rho }_1^2 - {\mathbf \rho }_2^2)]} } \\ \;\;\;\; \times \exp [ - \frac{{u_d^2}}{{\varepsilon _x^2\rho _0^2(z)}} - \frac{{v_d^2}}{{\varepsilon _y^2\rho _0^2(z)}} + \frac{{\gamma _{x1}^2}}{{4{X_1}}} + \frac{{\gamma _{y1}^2}}{{4{Y_1}}} + \frac{{\Omega _x^2}}{{4{X_2}}} + \frac{{\Omega _y^2}}{{4{Y_2}}}], \end{array}$$

where

(10)$$\begin{array}{l} {X_1} = \frac{1}{{4\sigma _x^2}} + \frac{1}{{2{\delta ^2}}} + \frac{1}{{\varepsilon _x^2\rho _0^2(z)}} + \frac{{ik}}{{2z}},{Y_1} = \frac{1}{{4\sigma _y^2}} + \frac{1}{{2{\delta ^2}}} + \frac{1}{{\varepsilon _y^2\rho _0^2(z)}} + \frac{{ik}}{{2z}},\\ {\gamma _{x1}} = \frac{{ik{u_1}}}{z} + i2\pi {r_m}\cos {\theta _n} - \frac{{{u_d}}}{{\varepsilon _x^2\rho _0^2(z)}},{\gamma _{x2}} = \frac{{ik{u_2}}}{z} + i2\pi {r_m}\cos {\theta _n} - \frac{{{u_d}}}{{\varepsilon _x^2\rho _0^2(z)}},\\ {\gamma _{y1}} = \frac{{ik{v_1}}}{z} + i2\pi {r_m}\sin {\theta _n} - \frac{{{v_d}}}{{\varepsilon _y^2\rho _0^2(z)}},{\gamma _{y2}} = \frac{{ik{v_2}}}{z} + i2\pi {r_m}\sin {\theta _n} - \frac{{{v_d}}}{{\varepsilon _y^2\rho _0^2(z)}},\\ {\varDelta _x} = \frac{1}{{{\delta ^2}}} + \frac{2}{{\varepsilon _x^2\rho _0^2(z)}},{\varDelta _y} = \frac{1}{{{\delta ^2}}} + \frac{2}{{\varepsilon _y^2\rho _0^2(z)}},\\ {X_2} = \frac{1}{{4\sigma _x^2}} + \frac{1}{{2{\delta ^2}}} + \frac{1}{{\varepsilon _x^2\rho _0^2(z)}} - \frac{{ik}}{{2z}} - \frac{{\Delta _x^2}}{{4{X_1}}} + \frac{{{k^2}{\mu ^2}}}{{4{Y_1}}},\\ {Y_2} = \frac{1}{{4\sigma _y^2}} + \frac{1}{{2{\delta ^2}}} + \frac{1}{{\varepsilon _y^2\rho _0^2(z)}} - \frac{{ik}}{{2z}} - \frac{{\Delta _y^2}}{{4{Y_1}}} + \frac{{{k^2}{\mu ^2}}}{{4{X_1}}} + \frac{{{k^2}{\mu ^2}}}{{4{X_2}}}{\left( {\frac{{{\Delta _y}}}{{2{Y_1}}} - \frac{{{\Delta _x}}}{{2{X_1}}}} \right)^2},\\ {\Omega _x} = \frac{{{\gamma _{x1}}{\varDelta _x}}}{{2{X_1}}} - {\gamma _{x2}} + \frac{{ik\mu {\gamma _{y1}}}}{{2{Y_1}}},\;\;{\Omega _y} = \frac{{{\gamma _{y1}}{\varDelta _y}}}{{2{Y_1}}} - {\gamma _{y2}} - \frac{{ik\mu {\gamma _{x1}}}}{{2{X_1}}} + \frac{{ik\mu {\Omega _x}}}{{4{X_2}}}\left( {\frac{{{\Delta _y}}}{{{Y_1}}} - \frac{{{\Delta _x}}}{{{X_1}}}} \right). \end{array}$$

The expression of the CSD function of the twisted rectangular array beam is obtained by referring to [31]

(11)$$\begin{array}{l} {W_{rec}}({{\mathbf \rho }_1},{{\mathbf \rho }_2}) = \frac{{{k^2}}}{{4{z^2}{C_0}\sqrt {{X_1}{Y_1}{X_2}{Y_2}} }}\sum\limits_{n ={-} N}^N {\sum\limits_{m ={-} M}^M {\exp [ - \frac{{ik}}{{2z}}({\mathbf \rho }_1^2 - {\mathbf \rho }_2^2)} } ]\\ \;\;\;\;\;\;\; \times \exp [ - \frac{{u_d^2}}{{\varepsilon _x^2\rho _0^2(z)}} - \frac{{v_d^2}}{{\varepsilon _y^2\rho _0^2(z)}} + \frac{{\gamma _{x1}^2}}{{4{X_1}}} + \frac{{\gamma _{y1}^2}}{{4{Y_1}}} + \frac{{\Omega _x^2}}{{4{X_2}}} + \frac{{\Omega _y^2}}{{4{Y_2}}}], \end{array}$$

(12)$$\begin{array}{l} {\gamma _{x1}} = \frac{{ik{u_1}}}{z} + i\frac{{\pi n}}{a} - \frac{{{u_d}}}{{\varepsilon _x^2\rho _0^2(z)}},\;\;\;{\gamma _{x2}} = \frac{{ik{u_2}}}{z} + i\frac{{\pi n}}{a} - \frac{{{u_d}}}{{\varepsilon _x^2\rho _0^2(z)}},\\ {\gamma _{y1}} = \frac{{ik{v_1}}}{z} + i\frac{{\pi m}}{a} - \frac{{{v_d}}}{{\varepsilon _y^2\rho _0^2(z)}},\;\;{\gamma _{y2}} = \frac{{ik{v_2}}}{z} + i\frac{{\pi m}}{a} - \frac{{{v_d}}}{{\varepsilon _y^2\rho _0^2(z)}}, \end{array}$$

Except for Eq. (12), other parameter substitutions are similar to the radial array beam, ${C_0} = (2M + 1)(2N + 1)$ is the normalization factor, a denotes row spacing of array. By substituting the W_cir or W_rec into Eqs. (5)-(8), the spatial distribution of OAM flux density, spectral intensity and DOC of twisted array beams in atmospheric turbulence can be calculated.

3. Numerical simulation and analysis

In this section, we simulate the propagation characteristics of twisted array beams in anisotropic turbulence. Unless otherwise specified, the parameters are set to σ_x= 4 cm, σ_y= 2 cm, δ₀= 1 cm, λ=632.8 nm, ε_x= 1, ε_y= 3, ξ=11/3, μ=-0.5km^-1, $C_n^2 = 5 \times {10^{ - 15}}{m^{3 - \xi }}$, ${M_{ra}} = 0$, ${N_{ra}} = 3$, M = N = 1, r₀= 0.15mm^-1, a = 5 mm. Figure 1 depict the spatial distribution of the spectral intensity, the modulus of the DOC $\eta ({x,y,0,0} )$ at ${{\mathbf r}_1}\textrm{ = }({x,y} )$ and ${{\mathbf r}_2}\textrm{ = }({0,0} )$, OAM flux density, and normalized OAM flux density at the source plane for the twisted radial array and rectangular array beam, respectively. By observing Figs. 1(a2), (c2) and (d2), we can see that the DOC $\eta ({x,y,0,0} )$ of twisted radial array beam have the same distribution for different σ_y and µ, and in Fig. 1(b2) the rectangular array beam has different lattice coherence distribution at the source plane, this is attributed to the difference in the structure of the correlation [31]. It is of interest to find from Figs. 1(a3)–1(b3), for circularly symmetrical amplitude profiles, the distribution of OAM flux density at the source plane is circular with a dark hollow in the center, which is very similar to the OAM flux density of a partially coherent vortex beam [33]. This phenomenon further verifying the conclusion [12] that a twisted partially coherent beam can be generated by the superposition of vortex beams. As can be seen from Fig. 1(c), when the beam waist width in x and y directions are different, the distribution of spectral intensity and OAM flux density at the source plane becomes elliptical, while the DOC are not affected. With the addition of the twist factor in Fig. 1(d), we find that both the OAM flux density and the normalized OAM flux density will increase by a fixed multiple, indicating that modulation of the twist phase can control the magnitude of OAM.

Fig. 1. The spatial distribution of spectral intensity, DOC $\eta ({x,y,0,0} )$, OAM flux density and normalized OAM flux density of twisted array beams at the source plane. (a) Twisted radial array beam σ_y= 4 cm; (b) Twisted rectangular array beam σ_y= 4 cm; (c) Twisted radial array beam σ_y= 2 cm; (d) Twisted radial array beam σ_y= 2 cm, µ=-0.75km^-1

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Figure 2 illustrates the normalized spectral intensity distribution of the twisted radial array beam propagating in atmospheric turbulence. The spectral intensity gradually splits from a single spot to four spots distributed in a radial array, which is the same as [24]. Compare Figs. 2(a) and 2(b), we find that the intensity distribution in Fig. 2(b) gradually becomes elliptical. The reason for this phenomenon is that when the beam propagating in anisotropic turbulence, the spectral intensity distribution in the near field is mainly influenced by the initial beam parameters. After a certain propagation distance, the effect of turbulence will dominate due to the cumulative effect of turbulence, which will cause anisotropic beam spreading due to the different statistical properties in the horizontal and vertical directions. In Figs. 2(c)–2(d), the beam waist width is set to σ_x= 4 cm, σ_y= 2 cm in order to better observe the rotation of the spot. We can clearly observe that the beam is self-splitting while the individual sub spots are accompanied by rotation, which is consistent with the general twisted GSM beam [13].

Fig. 2. Normalized spectral intensity distribution of the twisted radial array beam propagating in turbulence. (a) σ_y= 4 cm, ε_y= 1; (b)σ_y= 4 cm, ε_y= 3; (c)σ_y= 2 cm, ε_y= 1; (d)σ_y= 2 cm, ε_y= 3

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As we all know, the twisted beam can rotate around the propagation axis during propagating and carrying OAM. Figure 3 shows the DOC $\eta ({u,v,0,0} )$ of the twisted radial array beam propagating in turbulence, the parameter settings correspond to Fig. 2. It is clearly seen that as the propagation distance increases, the initial lattice-like DOC is disrupted. The DOC distribution rotates due to twisted phase, but different source parameters result in different rotation angles. Compared with Figs. 2(a)–2(b) and Figs. 3(a)–3(b), one finds that the effect of anisotropic turbulence on the DOC distribution is more intuitive compared to the intensity, the rotation center of the spectral density splits as the beam splits during propagating, while the rotation center of the DOC remains invariant.

Fig. 3. The modulus of the DOC $\eta ({u,v,0,0} )$ of the twisted radial array beam propagating in turbulence. Same parameters as Fig. 2: (a) σ_y= 4 cm, ε_y= 1; (b)σ_y= 4 cm, ε_y= 3; (c)σ_y= 2 cm, ε_y= 1; (d)σ_y= 2 cm, ε_y= 3

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The OAM flux density distribution of the twisted radial array beam at different distances in turbulence is shown in Fig. 4, the evolution of the OAM flux density is similar to that of the spectral intensity, which splits into a multi-petal distribution, rotates and diffuses with the increase of propagation distance. Unlike the uniform distribution of the source field in Figs. 1(a3)-1(d3), the OAM flux density has negative values during transmission, where the OAM rotation direction is opposite to the positive region. The OAM negative region gradually increases with the increase of the distance. This related to the array distribution, which redistributes the OAM over a larger radial distance [34]. Integrating the OAM flux density in the plane yields the total OAM of the beam, which is conserved according to [33]. The spatial distribution of different OAM flux densities can be produced by tuning the anisotropy parameter of turbulence, but it does not affect the magnitude of the total OAM.

Fig. 4. OAM flux density distribution of the twisted radial array beam propagating in turbulence. Same parameters as Fig. 2: (a) σ_y= 4 cm, ε_y= 1; (b)σ_y= 4 cm, ε_y= 3; (c)σ_y= 2 cm, ε_y= 1; (d)σ_y= 2 cm, ε_y= 3

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Furthermore, Fig. 5 shows the characteristics in the far field, parameters correspond to Fig. 2(b), 3(b) and 4(b). It is worth noting that the twisted beam rotation reaches π/2 as $z \to \infty$ in free space [13,14]. As shown in Figs. 5(a) and 5(b), the rotation of intensity and DOC distribution is no longer apparent in the far-field. Observation of Figs. 5(a1)–5(a4) reveals that when the beam is transmitted to the far field, the cumulative effect of turbulence becomes stronger as the propagation distance increases, and the beam spreading caused by the anisotropy of the turbulence causes the beam to lose its array distribution [31]. Due to the decoherence effect of turbulence, the contour of DOC distribution gradually decreases, and shows elliptical distribution due to the anisotropy of turbulence in Figs. 5(b3)–5(b4). The beam spreading causes the OAM flux density to redistribute and obtains an array OAM flux density distribution in the far field as Fig. 5(c4).

Fig. 5. Normalized spectral intensity, DOC and OAM flux density distribution in the far field

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Figure 6 shows the normalized OAM flux density distribution of the twisted radial array beam at different distances in atmospheric turbulence. The normalized OAM flux density represents the average OAM per photon at a specific location in space. It is clear from Eq. (6) that the calculation of the normalized OAM flux density OAM depends on the ratio of OAM flux density to intensity, this results in regions with very low or 0 intensity have a large normalized OAM. In the numerical simulation, we filtered these invalid ranges, these erroneous high values of normalized OAM flux density near the edges/corners are not displayed. In Fig. 6, the distribution of normalized OAM flux density is generally low in the central region and high in the edge region. Which is similar to the distribution of the multi-Gaussian Schell mode vortex beam in [36]. In addition, due to the effects of anisotropic turbulence, negative values of the normalized OAM flux density appear in the central region.

Fig. 6. Normalized OAM flux density distribution of twisted radial array beam propagating in turbulence. Same parameters as Fig. 2 (a) σ_y= 4 cm, ε_y= 1; (b)σ_y= 4 cm, ε_y= 3; (c)σ_y= 2 cm, ε_y= 1; (d)σ_y= 2 cm, ε_y= 3

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Figure 7 demonstrates the normalized spectral intensity, OAM flux density, and normalized OAM flux density distributions of the twisted radial array beam propagating in anisotropic turbulence at z = 5 km, with different turbulence power exponent $\xi$. For non-Kolmogorov anisotropic turbulence as described by Eq. (3), we restrict it to $3 < \xi < 4$ since beyond this interval the wave structure function is not defined [50] and some of the integrals involved in propagation laws may diverge. Comparing Figs. 7(a1)–7(c1) and Figs. 7(a4)–7(c4), it is found that in the case near the extreme, both the normalized spectral intensity, OAM flux density and normalized OAM flux density distributions have significantly difference. The physical interpretation is on the extreme of $\xi \to 3$, the turbulence tends to vanish. On the other extreme of $\xi \to 4$, the phase effects dominate in the form of random tilts, which cause beam wander [50,54].

Fig. 7. Normalized intensity, OAM flux density and normalized OAM flux density distributions of twisted radial array beam in anisotropic turbulence with different turbulence power exponent, z = 5 km.

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Figure 8 demonstrates the normalized intensity, OAM flux density, and normalized OAM flux density distribution for an elliptical twisted radial array beam with different twist factors in anisotropic turbulence at z = 5 km. By observing Figs. 8(a1)–8(a4), it is obtained that as the twist factor increases bringing about the spread of the beam [31], which requires a reasonable choice of the initial optical parameters in practical applications. As can be seen from Figs. 8(b1) and 8(c1), when the distortion factor is 0, the region of OAM flux density and normalized OAM flux density is equal in size and opposite in direction, indicating that the beam does not carry OAM at this time. As the distortion factor increases, the positive region of OAM flux density is larger than the negative region, and the total OAM of the beam is positive and shows a counterclockwise rotation, which can explain the rotation of the light spot phenomenon. As an important conclusion, we found that the modulation of the twist phase affects not only the magnitude of OAM but also its distribution.

Fig. 8. Normalized spectral intensity, OAM flux density, and normalized OAM flux density distribution for elliptical twisted radial array beams with different twist factors in anisotropic turbulence, z = 5 km.

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In order to visualize the effect of array structure distributions on the evolution of the OAM in anisotropic turbulence. Based on Eq. (11) and (12), Fig. 9 demonstrates the normalized spectral intensity, OAM flux density, and normalized OAM flux density distribution of twisted rectangular array beam in anisotropic turbulence at z = 5 km. Figures 9(a1)–9(c1) N = M = 0, i.e. there is only one spot and no array, Figs. 9(a4)–9(c4) N = M = 1, for 3 × 3 array. As in Figs. 9(a1)–9(a4), the number of intensity array spots gradually increases by modulating the initial coherent structure. Moreover, arbitrary array intensity distributions can be generated. From Figs. 9(b1)–9(b4), due to the effect of anisotropic turbulence, the distributions of the OAM flux density is not a simple summation of the array beam. As we observe Figs. 9(b)–9(c), it is found that the change of the array structure not only affects the distribution structure of the OAM flux density and normalized OAM flux density, but also affects their numerical magnitudes.

Fig. 9. Normalized spectral intensity, OAM flux density, normalized OAM flux density distribution for twisted rectangular array beam in anisotropic turbulence, z = 5 km. (a1)-(c1) N = M = 0; (a2)-(c2) N = 1 & n≠0, M = 0; (a3)-(c3) N = 1, M = 0; (a4)-(c4) N = M = 1.

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4. Conclusion

In this paper, we simulate the propagating of twisted partially coherent array beams in non-Kolmogorov anisotropic turbulence, and analyze the evolution of the spectral intensity and OAM properties in turbulence. The results show when all parameters are ideally isotropic (circularly symmetric), the OAM flux density of the twisted array beam exhibits a circular distribution with a dark hollow in the center as show at the source plane. which is the same as that of the partially coherent vortex beam, further verifying the conclusion that the twisted beam can be generated by the superposition of vortex beams. However, the anisotropy of the turbulence causes anisotropic intensity distribution in horizontal and vertical directions. and the spectral density distribution rotates with increasing distance in turbulence. The twisted array beams have both the rotational characteristics of the TGSM beam and the self-splitting characteristics of the array beam. Although OAM is conserved, the spatial distribution of OAM flux density can be changed by changing the propagation distance, power and anisotropy of turbulence, and the modulation of the twist phase affects not only the magnitude of OAM but also its distribution. Our work is helpful for exploring new forms of OAM sources. The results of this paper have potential applications in free-space optical communication and optical field modulation.

Funding

National Natural Science Foundation of China (11904253); Open Research Fund of Key Laboratory of Atmospheric Optics, Chinese Academy of Sciences (HTAD-JJ-19-02); Taiyuan University of Science and Technology Scientific Research Initial Funding (20202013); Shanxi Provincial Central Government Guides Local Science and Technology Development Fund Project (YDZX20201400001386).

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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Changes in orbital angular momentum distribution of a twisted partially coherent array beam in anisotropic turbulence

Abstract

1. Introduction

2. Theoretical model

3. Numerical simulation and analysis

4. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (9)

Equations (12)

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