Self-focusing effect analysis of a perfect optical vortex beam in atmospheric turbulence

ShuaiLing Wang; MingJian Cheng; XiHua Yang; JingPing Xu; JingPing Xu; YaPing Yang; YaPing Yang

doi:10.1364/OE.492275

1. Introduction

Optical vortex beams are structured light fields carrying orbital angular momentum (OAM) [1,2]. OAM can theoretically form an infinite N qubit basis; different OAM modes are orthogonal to each other [3,4], allowing the realization of high-capacity wireless optical communication. When optical vortex beams are employed as information carriers to transmit information in a turbulent environment, the turbulence disturbance severely affects the beam's transmission quality and reduces the communication system's performance [5,6]. At present, an anti-diffraction vortex beam is used as a system beam source for wireless optical communication to mitigate the turbulence effect [7,8]. Bessel-Gaussian beam (BGB) are typical anti-diffraction vortex beam, and they can improve communication quality better than Laguerre-Gaussian beam [9] due to their anti-diffraction and self-healing behavior. The OAM carried by a BGB can realize high-capacity data transmission based on OAM coding [10].

However, the anti-diffraction characteristics of BGB are affected by the transmission distance and beam shape parameters. Cheng et al. [11] pointed out that the anti-diffraction characteristics of BGB have a certain effective distance: beyond a certain range, they gradually lose efficacy, and the transmission quality of BGB decreases significantly. Therefore, BGB are more suitable for short-range application scenarios. As a unique feature of self-focusing Airy beams, the self-focusing effect has been extensively studied [12–14] since it can compensate for the beam loss caused by the diffraction effect to a certain extent. Thus, utilizing a self-focusing Airy beam as a background beam field for loading optical vortices helps reduce the impact of turbulence disturbance on the propagation of optical vortices, achieving a long-distance propagation of self-focusing Airy beam [15]. Zhang et al. [16] designed and experimentally demonstrated an “optical pin beam” that exhibited self-focusing dynamics during propagation. Experimental results also indicated that the self-focusing effect could help propagate the optical pin beam steadily over long distances in artificial atmospheric turbulence. As a kind of optical vortex beam, the profile size of a self-focusing beam strongly depends on topological charge, causing severe losses when multimodal beams are multiplexed [17].

Ostrovsky et al. [18] experimentally produced a perfect optical vortex beam (POVB), solving the problem of the dependence of the initial profile size on topological charge. Indeed, an ideal POVB is an annular ring with an arbitrary helical phase, maintaining a fixed initial radial profile [19]. Vaity et al. [20] formed a POVB through the Fourier transform of a BGB of different orders and derived a mathematical description of the POVB. These marks of the POVB were verified experimentally and mathematically. Researchers in wireless optical communication were quickly drawn to this novel structured light fields, finding many relevant applications [21–24]. Long-distance transmission in turbulent environments can be achieved using the self-focusing effect of POVB while keeping the initial profile size unaffected by the changes in topological charge. However, to our knowledge, the self-focusing effect of a POVB propagating in atmospheric turbulence has not yet been thoroughly investigated.

In this article, we investigated the self-focusing effect of a POVB exerting the atmospheric power spectrum. Firstly, the correlation function for the POVB in atmospheric turbulence was established by employing the Rytov approximation. On that basis, an analytical expression for signal OAM mode probability density for the POVB was obtained. Further, we investigated the influence of the transmission distance, topological charge, and the ratio of the ring radius to the Gaussian beam waist on the self-focusing effect by the intensity distribution of the POVB in the x-y plane. Subsequently, the comparison of the received probability between BGB and POVB was analyzed and discussed using numerical simulation. Finally, we derived conclusions based on the obtained research data.

2. Complex amplitude of POVB and BGB in the turbulence-free channel

At the original plane, the complex amplitude of the well-studied BGB can be expressed as [25,26]

(1)$${E_{BG}}({\rho ,\varphi } )= {J_{{m_0}}}\left( {\frac{{\eta \rho }}{{{w_b}}}} \right)\exp \left( { - \frac{{{\rho^2}}}{{w_b^2}}} \right)\exp ({i{m_0}\varphi } ),$$

where ${J_{{m_0}}}({\cdot} )$ represents the Bessel function of first kind, $({\rho ,\varphi } )$ is the two dimensional position vector at the original plane, ${m_0}$ denotes the topological charge associated to the optical vortex beam, $\eta = k\sin ({{\theta_0}} ){w_b}$ is a constant that determines the profile of the BGB, $k = {{2\pi } / \lambda }$ means the wave number, $\lambda$ denotes the wavelength, ${w_b}$ is the beam width, ${\theta _0}$ is defined as Bessel cone angle.

Utilizing a simple lens that acts as an optical Fourier transformer, BGB can be transformation into the POVB through Fourier transform by the optical Fourier transformer. This transformation for the field ${E_{BG}}({\rho ,\varphi } )$ into ${E_P}({\tilde{r},\tilde{\theta }} )$ can be written mathematically in the original plane as [20]

(2)$${E_P}({\tilde{r},\tilde{\theta }} )= \frac{{{i^{ - 1}}k}}{{2\pi f}}\int_0^\infty {\int_0^{2\pi } {{E_{BG}}({\rho ,\varphi } )} } \rho d\rho d\varphi \exp \left[ { - \frac{{ik}}{f}\rho \tilde{r}\cos ({\tilde{\theta } - \varphi } )} \right],$$

where f is the focal length of the optical Fourier transformer.

Based on the Bessel function identity [27] and standard integral, by substituting Eq. (1) into Eq. (2), the complex amplitude of POVB having constant amplitude ${A_0}$ in the original plane can be expressed as [20]

(3)$${E_P}({\tilde{r},\tilde{\theta }} )= {A_0}\exp \left( { - \frac{{{{\tilde{r}}^2}}}{{w_p^2}}} \right){I_{{m_0}}}\left( {\frac{{2\tilde{r}{r_0}}}{{w_p^2}}} \right)\exp ({i{m_0}\tilde{\theta }} ),$$

where ${w_p} = {{2f} / {k{w_0}}}$ is the Gaussian beam waist at the focus, ${r_0}$ is defined as the ring radius of the POVB.

When POVB propagates through turbulence-free channel to a receiver located at transmission distance z from the transmitter, using the extended Huygens-Fresnel principle [28], the complex amplitude of POVB can be expressed as

(4)$${E_P}({r,\theta ,z} )={-} \frac{{ik}}{{2\pi z}}\exp ({ikz} )\int_0^\infty {\int_0^{2\pi } {{E_P}({\tilde{r},\tilde{\theta }} )} } \exp \left\{ {\frac{{ik}}{{2z}}[{{r^2} + {{\tilde{r}}^2} - 2r\tilde{r}\cos ({\theta - \tilde{\theta }} )} ]} \right\}\tilde{r}d\tilde{r}d\tilde{\theta }.$$

Similarly, the complex amplitude of BGB in turbulence-free channel can be obtained,

(5)$${E_{BG}}({r,\theta ,z} )={-} \frac{{ik}}{{2\pi z}}\exp ({ikz} )\int_0^\infty {\int_0^{2\pi } {{E_{BG}}({\rho ,\varphi } )} } \exp \left\{ {\frac{{ik}}{{2z}}[{{r^2} + {\rho^2} - 2r\rho \cos ({\theta - \varphi } )} ]} \right\}\rho d\rho d\varphi .$$

Further, with the help of the listed below mathematical integral expressions [27]

(6)$$\int_0^{2\pi } {\exp [{iv\tilde{\theta } + ix\cos ({\tilde{\theta } - \theta } )} ]} d\tilde{\theta } = {i^v}2\pi \exp ({i\nu \theta } ){J_v}(x ),$$

(7)$$\int_0^\infty {\exp ({ - n{x^2}} )} {I_v}({\alpha x} ){J_v}({\beta x} )xdx = \frac{1}{{2n}}\exp \left( {\frac{{{\alpha^2} - {\beta^2}}}{{4n}}} \right){J_v}\left( {\frac{{\alpha \beta }}{{2n}}} \right),$$

(8)$$\; \int_0^\infty {\exp ({ - n{x^2}} )} {J_v}({\alpha x} ){J_v}({\beta x} )xdx = \frac{1}{{2n}}\exp \left( { - \frac{{{\alpha^2} + {\beta^2}}}{{4n}}} \right){I_v}\left( {\frac{{\alpha \beta }}{{2n}}} \right),$$

the complex amplitude of POVB and BGB in the turbulence-free channel can be calculated, separately

(9)$${E_P}({r,\theta ,z} )={-} \frac{{{i^{{m_0} + 1}}k{A_0}}}{{2{n_p}z}}\exp \left[ {i\left( {kz + {m_0}\theta + \frac{{k{r^2}}}{{2z}}} \right) + \frac{{\alpha_p^2 - \beta_p^2}}{{4{n_p}}}} \right]{J_{{m_0}}}\left( {\frac{{{\alpha_p}{\beta_p}}}{{2{n_p}}}} \right),$$

(10)$${E_{BG}}({r,\theta ,z} )={-} \frac{{{i^{{m_0} + 1}}k}}{{2{n_b}z}}\exp \left[ {i\left( {kz + {m_0}\theta + \frac{{k{r^2}}}{{2z}}} \right) - \frac{{\alpha_b^2 + \beta_b^2}}{{4{n_b}}}} \right]{I_{{m_0}}}\left( {\frac{{{\alpha_b}{\beta_b}}}{{2{n_b}}}} \right),$$

with

(11)$${n_p} = \frac{1}{{w_p^2}} - \frac{{ik}}{{2z}},\textrm{ }{\alpha _p} = \frac{{2{r_0}}}{{w_p^2}},\textrm{ }{\beta _p} ={-} \frac{{kr}}{z},\textrm{ }{n_b} = \frac{1}{{w_b^2}} - \frac{{ik}}{{2z}},\textrm{ }{\alpha _b} = \frac{\eta }{{{w_b}}},\textrm{ }{\beta _b} ={-} \frac{{kr}}{z}.$$

Figure 1 and Fig. 2 show the normalized intensity distribution of the POVB and the BGB in the x-y plane with the transmission distance, respectively. The maximum beam profiles of the POVB and the BGB were set to be similar by adjusting the beam profile parameters. The BGB exhibits better anti-diffraction than the POVB for short-range transmission distances within 500 m. Compared with the POVB, the BGB intensity can be better maintained. However, Fig. 1 indicates that the POVB shows a significant self-focusing effect. When the POVB is transmitted to about 500 m, the beam profile size greatly reduces due to the self-focusing effect. At the same time, the beam intensity increases instead of decreasing, having the maximum value at about 900 m. For the BGB, the beam intensity begins to decrease as the transmission distance increases, and the beam profile expands at about 900 m. When $z = 900\textrm{ m}$, the normalized intensity of the POVB recovers to 0.7751 due to its self-focusing effect, while the normalized intensity of the BGB decays to 0.1832. Compared with the BGB, the POVB exhibits better application results in long-range transmission due to its unique self-focusing effect.

3. Received probability of POVB propagate in atmospheric turbulence

Consider the scenario that atmospheric turbulence exists in a communication link without an optical element between the source and detector, and modified Rytov approximation can be used to analyze such channels under turbulence fluctuation. In the atmospheric turbulence environment of transmission, the correlation function of POVB can be defined by ensemble average [29]

(12)$$\begin{aligned} R({r,\theta ,\theta^{\prime},z} )&= \left\langle {v({r,\theta ,z} ){v^\mathrm{\ast }}({r,\theta^{\prime},z} )} \right\rangle \\ &= {E_p}({r,\theta ,z} )E_p^\mathrm{\ast }({r,\theta^{\prime},z} )\left\langle {\exp [{iS({r,\theta ,z} )- i{S^\ast }({r,\theta^{\prime},z} )} ]} \right\rangle , \end{aligned}$$

where $\left\langle {\exp [{iS({r,\theta ,z} )- i{S^\ast }({r,\theta^{\prime},z} )} ]} \right\rangle$ is expressed as the coherence function of the phase fluctuations in anisotropic turbulence of atmosphere, given by [30]

(13)$$\left\langle {\exp [{iS({r,\theta ,z} )- i{S^\ast }({r,\theta^{\prime},z} )} ]} \right\rangle = \exp \left\{ {\frac{{ - 2{r^2}[{1 - \cos ({\theta - \theta^{\prime}} )} ]}}{{\rho_{o\zeta }^2}}} \right\}.$$

Fig. 1. Normalized intensity distribution of POVB in the x-y plane $x \in [{ - 0.12,0.12} ]\textrm{ m,}\,y \in [{ - 0.12,0.12} ]\textrm{ m}$ for (a) $z = 0\textrm{ m}$, (b) $z = 500\textrm{ m}$, (c) $z = 1000\textrm{ m}$, (d) $z = 1500\textrm{ m}$, (e) $z = 2000\textrm{ m}$ and (f) the normalized intensity distribution of POVB varies with the transmission distance $z$.

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Fig. 2. Normalized intensity distribution of BGB in the x-y plane $x \in [{ - 0.12,0.12} ]\textrm{ m,}\,y \in [{ - 0.12,0.12} ]\textrm{ m}$ for (a) $z = 0\textrm{ m}$, (b) $z = 500\textrm{ m}$, (c) $z = 1000\textrm{ m}$, (d) $z = 1500\textrm{ m}$, (e) $z = 2000\textrm{ m}$ and (f) the normalized intensity distribution of BGB varies with the transmission distance $z$.

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Here, ${\rho _{o\xi }}$ is the spatial coherence radius of the spherical wave in the anisotropic atmospheric turbulence and given by [31,32]

(14)$${\rho _{o\zeta }} = {\left[ {\frac{{{\pi^2}{k^2}z}}{3}\int_0^\infty {{\kappa^3}} {\Phi _n}(\kappa )d\kappa } \right]^{ - {1 / 2}}}.$$

Considering the anisotropic atmospheric turbulence vortex in vertical and horizontal direction there is a certain difference, the spatial power spectrum model with refraction-index fluctuations is given by [11]

(15)$${\Phi _n}(\kappa )= A(\alpha )C_n^2{\mu _x}{\mu _y}\exp \left( { - \frac{{\mu_x^2\kappa_x^2 + \mu_y^2\kappa_y^2 + \kappa_z^2}}{{\kappa_l^2}}} \right){({\mu_x^2\kappa_x^2 + \mu_y^2\kappa_y^2 + \kappa_z^2 + \kappa_0^2} )^{ - \frac{\alpha }{2}}},$$

where ${\mu _x}$ and ${\mu _y}$ are anisotropic factors of atmospheric turbulence in the x and y direction, respectively, spatial frequency can be expressed as $\kappa = \sqrt {\kappa _x^2 + \kappa _y^2 + \kappa _z^2}$ and we can set ${\kappa _z} = 0$ inside the power spectrum by Markov approximation [33], ${\kappa _l} = {{c(\alpha )} / {{l_0}}}$, ${\kappa _0} = {{2\pi } / {{L_0}}}$ denotes the outer scale parameter, characterizing the outer scale ${L_0}$ of turbulent cells, ${l_0}$ is defined as the inner scale, $A(\alpha )$ and $c(\alpha )$ are given by

(16)$$A(\alpha )= \Gamma ({\alpha - 1} )\frac{{\cos ({{{\pi \alpha } / 2}} )}}{{4{\pi ^2}}},$$

(17)$$c(\alpha )= {\left[ {A(\alpha )\Gamma \left( {\frac{{5 - \alpha }}{2}} \right)\frac{{2\pi }}{3}} \right]^{\frac{1}{{\alpha - 5}}}},$$

where $\Gamma ({\cdot} )$ represents the Gamma function, $\alpha$ is defined as the power spectrum index, $C_n^2$ is a generalized structure parameter, typically from ${10^{ - 17}}\textrm{ }{\textrm{m}^{ - {2 / 3}}}$ to ${10^{ - 12}}\textrm{ }{\textrm{m}^{ - {2 / 3}}}$ [31].

Substituting Eq. (15) into Eq. (14) and utilizing the Markov approximation, yields

(18)$$\begin{aligned} {\rho _{o\zeta }} &= \left\{ {\frac{{\mu_x^2 + \mu_y^2}}{{\mu_x^2\mu_y^2}}\frac{{{k^2}{\pi^2}zA(\alpha )C_n^2}}{{6({\alpha - 2} )}}} \right.\\ &\textrm{ } \times {\left. {\left[ {\kappa_l^{2 - \alpha }({2\kappa_0^2 + \alpha \kappa_l^2 - 2\kappa_l^2} )\exp \left( {\frac{{\kappa_0^2}}{{\kappa_l^2}}} \right)\Gamma \left( {2 - \frac{\alpha }{2},\frac{{\kappa_0^2}}{{\kappa_l^2}}} \right) - 2\kappa_0^{4 - \alpha }} \right]} \right\}^{ - {1 / 2}}}. \end{aligned}$$

Substitution of Eq. (9), Eq. (13) and Eq. (18) in Eq. (12) yields the correlation function of POVB in the atmospheric turbulence considering asymmetric anisotropy:

(19)$$\begin{aligned} R({r,\theta ,\theta^{\prime},z} )&= \frac{{{k^2}A_0^2}}{{4{n_p}n_p^\ast {z^2}}}\exp \left[ {i{m_0}({\theta - \theta^{\prime}} )+ \frac{{\alpha_p^2 - \beta_p^2}}{{4{n_p}}} + \frac{{\alpha_p^2 - \beta_p^2}}{{4n_p^\ast }}} \right]\\ &\textrm{ } \times {J_{{m_0}}}\left( {\frac{{{\alpha_p}{\beta_p}}}{{2{n_p}}}} \right){J_{{m_0}}}\left( {\frac{{{\alpha_p}{\beta_p}}}{{2n_p^\ast }}} \right)\exp \left\{ {\frac{{ - 2{r^2}[{1 - \cos ({\theta - \theta^{\prime}} )} ]}}{{\rho_{o\zeta }^2}}} \right\}. \end{aligned}$$

In the turbulent channels of an atmosphere, the complex amplitude of the POVB can be expanded in terms of its spiral harmonic $\exp ({im\theta } )$ to expound OAM content [34]

(20)$${E_p}({r,\theta ,z} )= \frac{1}{{\sqrt {2\pi } }}\sum\limits_{m ={-} \infty }^\infty {{a_{p,m}}} ({r,z} )\exp ({im\theta } ),$$

where ${a_{p,m}}({r,z} )$ means the superposition coefficients and can be written as

(21)$${a_{p,m}}({r,z} )= \frac{1}{{\sqrt {2\pi } }}\int_0^{2\pi } {{E_p}({r,\theta ,z} )} \exp ({ - im\theta } )d\theta .$$

Therefore, we can obtain the signal OAM mode probability density of POVB propagation in the atmospheric turbulence

(22)$$\begin{aligned} &\left\langle {{{|{{a_{p,m}}({r,z} )} |}^2}} \right\rangle \\ &= \frac{1}{{2\pi }}\int_0^{2\pi } {\int_0^{2\pi } {R({r,\theta ,\theta^{\prime},z} )} } \exp [{ - im({\theta - \theta^{\prime}} )} ]d\theta d\theta ^{\prime}\\ &= \frac{1}{{2\pi }}\int_0^{2\pi } {\int_0^{2\pi } {\frac{{{k^2}A_0^2}}{{4{n_p}n_p^\ast {z^2}}}} } \exp [{i({{m_0} - m} )({\theta - \theta^{\prime}} )} ]\exp \left( {\frac{{\alpha_p^2 - \beta_p^2}}{{4{n_p}}} + \frac{{\alpha_p^2 - \beta_p^2}}{{4n_p^\ast }}} \right)\\ &\quad \times {J_{{m_0}}}\left( {\frac{{{\alpha_p}{\beta_p}}}{{2{n_p}}}} \right){J_{{m_0}}}\left( {\frac{{{\alpha_p}{\beta_p}}}{{2n_p^\ast }}} \right)\exp \left\{ {\frac{{ - 2{r^2}[{1 - \cos ({\theta - \theta^{\prime}} )} ]}}{{\rho_{o\zeta }^2}}} \right\}d\theta d\theta ^{\prime}. \end{aligned}$$

By integrating the following formula [27]

(23)$$\int_0^{2\pi } {\exp [{ - iy\theta + x\cos ({\theta - \theta^{\prime}} )} ]} d\theta = 2\pi \exp ({ - iy\theta^{\prime}} ){I_y}(x ),$$

we can transform the integral expression of signal OAM mode probability density into a more convenient analytical expression

(24)$$\begin{aligned} &\left\langle {{{|{{a_{p,m}}({r,z} )} |}^2}} \right\rangle \\ &= \frac{{{k^2}A_0^2\pi }}{{2{n_p}n_p^\ast {z^2}}}\exp \left( {\frac{{\alpha_p^2 - \beta_p^2}}{{4{n_p}}} + \frac{{\alpha_p^2 - \beta_p^2}}{{4n_p^\ast }} - \frac{{2{r^2}}}{{\rho_{o\zeta }^2}}} \right){J_{{m_0}}}\left( {\frac{{{\alpha_p}{\beta_p}}}{{2{n_p}}}} \right){J_{{m_0}}}\left( {\frac{{{\alpha_p}{\beta_p}}}{{2n_p^\ast }}} \right){I_{m - {m_0}}}\left( {\frac{{2{r^2}}}{{\rho_{o\zeta }^2}}} \right). \end{aligned}$$

At the receiver plane, the OAM detection probability of POVB is determined by the integral of r as follows:

(25)$$P({m|{m_0}} )= \frac{{\int_0^{{D / 2}} {\left\langle {{{|{{a_{p,m}}({r,z} )} |}^2}} \right\rangle rdr} }}{{\sum\nolimits_{m ={-} \infty }^\infty {\int_0^{{D / 2}} {\left\langle {{{|{{a_{p,m}}({r,z} )} |}^2}} \right\rangle rdr} } }},$$

where D is the receiving aperture.

4. Numerical simulation

In this section, the POVB self-focusing effect was discussed by numerical simulation of the POVB propagation model in atmospheric turbulence. Unless otherwise specified, we used the following typical parameter values in the following numerical simulation [35–37]: $C_n^2 = {10^{ - 14}}\textrm{ }{\textrm{m}^{ - {2 / 3}}}$, ${\mu _x} = 2$, ${\mu _y} = 2$, ${r_0} = 0.04\textrm{ m}$, ${w_p} = 0.008\textrm{ m}$, ${m_0} = 1$, $m = 1$, $\alpha = 3.2$, $\eta = 7$, ${w_b} = 0.04\textrm{ m}$, ${L_0} = 1\textrm{ m}$, $\lambda = 1.55\mathrm{\ \mu m}$, ${l_0} = 0.001\textrm{ m}$, $D = 0.08\textrm{ m}$.

The influence of topological charge on the side-view propagation of the POVB in a turbulence-free channel is depicted in Fig. 3. The POVB propagation can be divided into two stages: in the first stage, the profile size of POVB is well-preserved because the POVB is derived from the Fourier transform of the BGB so that the POVB still retains the anti-diffraction properties of the BGB. We refer to this stage as the anti-diffraction stage; in the second stage, the POVB profile shrinks and focuses, the beam intensity decreases first and then increases with the increase of z, and then continues to decrease when it reaches the peak intensity of the self-focusing effect. We refer to this stage as the self-focusing stage. Simultaneously, the POVB changes from the single-ring intensity mode of the anti-diffraction stage to the multi-ring intensity mode. As shown in Fig. 3, the profile and intensity of POVB remain practically unchanged with the change of ${m_0}$ in the anti-diffraction stage. In the self-focusing stage, the POVB behaves like any other ordinary vortex beam, i.e., the profile and intensity of POVB expand and decrease with the increase of ${m_0}$. Furthermore, it can be observed that in the far-field region, POVBs degrade into a class of Bessel-like beams, which is similar to the conclusion drawn from the optical vortices formed by using a coherent laser beam array [38].

Fig. 3. Normalized intensity distribution of POVB in turbulence-free channel for different topological charges (a) ${m_0} = 1$, (b) ${m_0} = 2$, (c) ${m_0} = 3$ and (d) ${m_0} = 4$.

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In order to study the POVB variation in different stages in more detail, in Fig. 4, we plot the intensity distribution of different transmission distance locations as a function of ${m_0}$ in the x-y plane. At $z = 50\textrm{ m}$, the POVB's profile and intensity hardly change in the x-y plane with the increase of ${m_0}$, indicating that in the anti-diffraction stage, the POVB at different ${m_0}$ values can have practically identical transmission effects. Conversely, the beam profile and intensity are unstable when the transmission distance z is 750 m. From Fig. 4(d) to Fig. 4(e), and then to Fig. 4(f), it is observed that the central dark spot of the POVB increases as the ${m_0}$ increases, and the beam intensity is attenuated. This phenomenon leads to the attenuation of the beam transmission quality with the increase of ${m_0}$ in the self-focusing stage.

Fig. 4. In turbulence-free channel, the normalized intensity and phase distribution of POVB at $z = 50\textrm{ m}$ for different topological charges (a) ${m_0} = 1$, (b) ${m_0} = 2$, (c) ${m_0} = 3$ and when $z = 750\textrm{ m}$, the normalized intensity and phase distribution for different topological charges (d) ${m_0} = 1$, (e) ${m_0} = 2$, (f) ${m_0} = 3$.

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Figure 5 shows the variation in the intensity distribution of the POVB carrying OAM in the y-z plane with ${r_0}$. Compared with Fig. 5(b), the region required for the beam shrinking and focusing is larger when the POVB transitions from the anti-diffraction stage to the self-focusing stage in Fig. 5(a). We refer to this region as the self-focusing region. As the ${r_0}$ decreases, the self-focusing region shrinks. Simultaneously, we find that this phenomenon is related to the ratio of ${r_0}$ to ${w_p}$. In fact, here, we set the ratio ${{{r_0}} / {{w_p}}}$ to 10, 5, 4, 3, 2, and 1 at fixed ${w_p} = 0.008\textrm{ m}$. As shown in Fig. 5(e), at ${{{r_0}} / {{w_p}}} = 2$, the self-focusing effect is no longer obvious. When the ${{{r_0}} / {{w_p}}}$ tends to 1, the self-focusing region disappears, and the POVB degenerates into a BGB-like. In addition, it is observed that the profile size of POVB rises with the increase of ${r_0}$, and the number of rings in the self-focusing stage accordingly increases.

Fig. 5. Normalized intensity distribution of POVB in turbulence-free channel under different z and y when (a) ${r_0} = 0.08\textrm{ m}$, (b) ${r_0} = 0.04\textrm{ m}$, (c) ${r_0} = 0.032\textrm{ m}$, (d) ${r_0} = 0.024\textrm{ m}$, (e) ${r_0} = 0.016\textrm{ m}$ and (f) ${r_0} = 0.008\textrm{ m}$.

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The influence of atmospheric turbulence on the side-view propagation of the POVB in the communication link is depicted in Fig. 6. We chose nine observing planes at different propagation distances to investigate the changes in the signal OAM mode probability density of POVB along the propagation path. As shown in Fig. 6, although signal OAM mode probability density of POVB decreases rapidly due to turbulence perturbation of the POVB helical phase structure. However, accounting for the self-focusing effect, signal OAM mode probability density of POVB is restored to a certain extent in the self-focusing stage. As shown in the observation plane, the signal OAM mode probability density at $z = 750\textrm{ m}$ m and $z = 1000\textrm{ m}$ m is stronger than that at $z = 250\textrm{ m}$ and $z = 500\textrm{ m}$. Choosing a suitable propagation distance can help improve the POVB transmission quality in atmospheric turbulence.

Fig. 6. Signal OAM mode probability density of POVB propagation in the atmospheric turbulence under different z and $y$.

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In Fig. 7(a), the received probability of POVB and BGB are plotted according to Eq. (25). When ${m_0} = m$, the detection probability $P({m|{m_0}} )$ is expressed as the received probability of POVB and BGB at the signal receiver after the beams have propagated through atmospheric turbulence. According to Fig. 1 and Fig. 2, the BGB anti-diffraction effect is better than that of the POVB in the anti-diffraction stage, which is reflected in the received probability. This shows that the BGB can have better communication quality than the POVB in short-range transmission application scenarios. The property that the initial profile size of the POVB is not affected by topological charge does not improve the POVB received probability in wireless optical communication. However, due to the POVB unique self-focusing effect, when the beam further propagates in atmospheric turbulence, the POVB can have a higher received probability than the BGB in the self-focusing stage. Figure 7(b) shows the effect of ${m_0}$ on received probability; different ${m_0}$ values of the POVB exhibit different performances in different transmission stages. In the anti-diffraction stage, the influence of the change of ${m_0}$ on the POVB received probability is minimal because the size and intensity of the POVB are negligibly affected by the change of ${m_0}$ in the anti-diffraction stage, as shown in Fig. 4. However, when entering the self-focusing stage, the POVB received probability begins to be affected by ${m_0}$, similar as for ordinary vortex beams. For ${m_0} = 4$, the effect of the POVB self-focusing on the recovery of the POVB received probability is no longer obvious, which is consistent with the conclusion in Fig. 3. The influence of the self-focusing effect on the beam profile still exists, but the recovery effect of the beam intensity is almost negligible. In addition, the beam profile size increases with ${m_0}$ in the self-focusing stage, making the higher-order POVB more susceptible to the disturbance of its helical phase structure caused by atmospheric turbulence. Thus, this phenomenon leads to a decrease in the POVB received probability as the ${m_0}$ increases in the self-focusing stage, as shown in Fig. 7(b).

Fig. 7. (a) Received probability of OAM signal modes carried by POVB and BGB versus transmission distance in atmospheric turbulence and (b) received probability of POVB versus transmission distance for different ${m_0}$.

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5. Conclusions

In this work, we investigated the POVB self-focusing effect based on the POVB propagation model. According to the numerical simulation results, we divided the POVB transmission process into two stages: (1) anti-diffraction stage, where the POVB profile size can be well preserved, and the transmission distance and topological charge have a less effect on the beam profile in a turbulence-free channel; (2) Self-focusing stage, in a non-turbulent channel, the beam profile size expands with the transmission distance and topological charge after the POVB is focused in the self-focusing region. It is worth mentioning that the influence of topological charge on the beam intensity is very limited in the anti-diffraction stage, but the beam intensity decreases significantly with the increase of topological charge in the self-focusing stage. We also found that the POVB self-focusing effect is related to the ratio of the ring radius to the Gaussian beam waist. When the ratio tends to 1, the POVB self-focusing effect disappears, and the POVB degenerates into a BGB-like. Simultaneously, the received probability of BGB and POVB in atmospheric turbulence show that the POVB self-focusing effect endows the POVB with a better propagation effect than the BGB in the long-range transmission scenario. However, for short-range transmission application scenarios, the BGB exhibits a higher received probability in atmospheric turbulence than the POVB whose initial profile size is not affected by topological charge. We expect to find a new beam type with good anti-diffraction characteristics in short-range transmission and improve the transmission quality through the self-focusing effect in long-range transmission.

Funding

National Natural Science Foundation of China (12174243, 12174288, 12274326, 61905186); National Key Research and Development Program of China (2021YFA1400602); Shanghai Aerospace Science and Technology Innovation Foundation (SAST-2022-069); Fundamental Research Funds for the Central Universities (ZYTS23078).

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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Self-focusing effect analysis of a perfect optical vortex beam in atmospheric turbulence

Abstract

1. Introduction

2. Complex amplitude of POVB and BGB in the turbulence-free channel

3. Received probability of POVB propagate in atmospheric turbulence

4. Numerical simulation

5. Conclusions

Funding

Disclosures

Data Availability

References

Data Availability

Cited By

Figures (7)

Equations (25)

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