Expand this Topic clickable element to expand a topic
Skip to content
Optica Publishing Group
About All Issues Feature Issues

Spiral spectrum of the phase singularity beam in the source plane and atmospheric turbulence

Penghui Gao and Lu Bai

Open Access Open Access

Abstract

The phase singularity beam has a unique wavefront phase structure—a screw dislocation (optical vortex) or edge dislocation—which is closely related to the orbital angular momentum (OAM). Screw dislocation refers to the phase change of integer multiples of 2π around a point, while edge dislocation refers to the existing π phase shift along a curve in the wavefront phase distribution. In this study, we derived the OAM detection probability expression and mode probability density expression of the phase singularity beam propagating through atmospheric turbulence. These expressions were used to examine the spiral spectrum and the OAM mode probability density of the phase singularity beam. It was determined that the phase singularity beam carrying a noncanonical optical vortex or an edge dislocation has more than one OAM state, whereas the phase singularity beam carrying a canonical optical vortex has only one OAM state. In atmospheric turbulence transmission, atmospheric disturbances will cause crosstalk between OAM states. The larger is the detection probability of the OAM state at the source plane, the closer is the maximum value of the OAM mode probability density throughout the transmission process to the beam center. The conclusions obtained have important applications in optical communications.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Wavefront dislocations—edge and screw dislocations—are related to orbital angular momentum (OAM) [1,2] and have important applications in optical communications [3,4]. Edge dislocation refers to the existing π phase shift along the curve in the wavefront phase distribution of a beam, and screw dislocation is the optical vortex [5]. When the phase distribution of an optical vortex changes linearly, the optical vortex is a canonical optical vortex; otherwise, it is a noncanonical optical vortex [6]. In recent studies on the spiral spectrum, researchers have mainly focused on the spiral spectrum characteristics of various beam-carrying OAMs in turbulence. Jiang et al. and Ou et al. respectively reported the spiral spectrum characteristics of the Laguerre−Gaussian beam and Bessel−Gaussian beam [7,8]. Cheng et al. compared the spiral spectrum characteristics of the Laguerre−Gaussian beam with that of the Bessel−Gaussian beam in atmospheric turbulence [9]. Fu and Gao studied the multiplexed Laguerre−Gaussian beam and Bessel−Gaussian beam [10]. Meanwhile, Chen et al., Zhang et al., and Zhu et al. respectively studied the OAM detection probability of the vortex Gaussian beam, Whittaker−Gaussian beam, and hypergeometric−Gaussian beam propagating through atmospheric turbulence [11–13]. Zhu et al. reported the spiral spectrum characteristics of the Airy vortex beam in maritime atmosphere turbulence [14]. A partially coherent beam has also been reported. For example, Chen et al. and Li et al. respectively studied the spiral spectrum characteristics of the partially coherent modified Bessel correlated vortex beam and elegant Laguerre−Gaussian beam in ocean turbulence [15,16].

Furthermore, the evolution characteristics of wavefront dislocations have also been studied. It was shown that the higher-order optical vortex can be split, the fractional-order optical vortex can evolve into an integer-order optical vortex, and the edge dislocation can also evolve into an optical vortex [17–20]. To the best of our knowledge, OAM detection probability has not been recently investigated in terms of the phase distribution of wavefront dislocations. The objective of the present study was thus to explore the effect of wavefront distortion on the beam-carried OAM in turbulence.

The remainder of this paper is organized as follows. In Section 2, the OAM probability expression and mode probability density expression for the phase singularity beam are derived based on the Huygens−Fresnel diffraction integral formula and the Rytov approximation theory. In Sections 3 to 5, the spiral spectrum and the OAM mode probability density of the phase singularity beam are discussed. In Section 6, the main conclusions of the paper are provided.

2. Theoretical model

In the cylindrical coordinate system, the light field of the phase singularity beam in the source plane can be written as follows [21]

$${E_0}({r,\theta ,z = 0} )\textrm{ = }r({a\cos \theta + \textrm{i}\sin \theta } )\exp \left( { - \frac{{{r^2}}}{{2w_0^2}}} \right),$$
where r and θ are the radial and azimuthal coordinates in the source plane. w0 is the beam waist width of the phase singularity beam, z is the propagation distance and a denotes a dimensionless parameter. When a = ±1, the phase singularity beam carries a canonical vortex; when a ≠ ±1 and a ≠ 0, the phase singularity beam carries a noncanonical optical vortex; and when a = 0, the phase singularity beam carries an edge dislocation.

Using the Euler formula [22]

$${e^{\textrm{i}\theta }}\textrm{ = }\cos \theta + \textrm{i}\sin \theta ,$$
Equation (1) can also be written as follows
$${E_0}({r,\theta ,z = 0} )\textrm{ = }\frac{{({a\textrm{ + }1} )r}}{2}\exp \left( { - \frac{{{r^2}}}{{2w_0^2}}\textrm{ + i}\theta } \right) + \frac{{({a - 1} )r}}{2}\exp \left( { - \frac{{{r^2}}}{{2w_0^2}} - \textrm{i}\theta } \right).$$
It can be observed from Eqs. (1) and (3) that the light field of the phase singularity beam carrying a noncanonical optical vortex with a topological charge of ±1 or an edge dislocation is obtained by the light field of the canonical vortex beam with a topological charge of +1 plus that of the canonical vortex beam with a topological charge of −1.

Based on the Huygens−Fresnel diffraction integral formula [23], the light field of the phase singularity beam in free space transmission is

$$\begin{array}{ll} {E_{free}}({\rho ,\varphi ,z} )&= - \frac{\textrm{i}}{{\lambda z}}\exp ({\textrm{i}kz} )\int {\int {{E_0}({r,\theta ,z = 0} )} } \\ &\times \exp \left\{ {\frac{{\textrm{i}k}}{{2z}}[{{r^2} + {\rho^2} - 2r\rho \cos ({\varphi - \theta } )} ]} \right\}r\textrm{d}r\textrm{d}\theta , \end{array}$$
where ρ and φ are the radial and azimuthal coordinates at propagation distance z. λ is the wavelength and k = 2π / λ is the wave number.

Using the following formula [22]

$$\textrm{exp}({\textrm{i}z\cos \varphi } )\textrm{ = }\sum\limits_{j ={-} \infty }^\infty {{\textrm{i}^j}} {J_j}(z )\exp ({\textrm{i}j\varphi } ),$$
$$\int\limits_0^{2\mathrm{\pi }} {\textrm{exp}({\textrm{i}m\theta } )} \textrm{ = }\left\{ {\begin{array}{cc} {2\mathrm{\pi }}&{m = 0}\\ 0&{m \ne 0} \end{array}} \right.,$$
$${J_{ - j}}(z )\textrm{ = }{({ - \textrm{1}} )^j}{J_j}(z ),$$
$$\int\limits_0^\infty {{x^{j + 1}}} \textrm{exp}({ - \alpha {x^2}} ){J_j}({\beta x} )\textrm{d}x\textrm{ = }\frac{{{\beta ^j}}}{{{{({2\alpha } )}^{j + 1}}}}\exp \left( { - \frac{{{\beta^2}}}{{4\alpha }}} \right),$$
where Jj (·) denotes the j order Bessel function of the first kind. Substituting Eq. (3) into Eq. (4), through integration operations, the light field of the phase singularity beam in free space transmission is
$$\begin{array}{ll} {E_{free}}({\rho ,\varphi ,z} )&= \left( {\frac{{\textrm{i}\mathrm{\pi }k\rho }}{{4{A^2}\lambda {z^2}}}} \right)\exp \left( {\textrm{i}kz\textrm{ + }\frac{{\textrm{i}k{\rho^2}}}{{2z}} - \frac{{{k^2}{\rho^2}}}{{4A{z^2}}}} \right)\\ &\times [{ - ({a + 1} ){{(\textrm{i} )}^{ - 1}}\exp ({\textrm{i}\varphi } )+ \textrm{i}({a - 1} )\exp ({ - \textrm{i}\varphi } )} ], \end{array}$$
where
$$A\textrm{ = }\frac{1}{{2w_0^2}} - \frac{{\textrm{i}k}}{{2z}}.$$
Using Rytov approximation [23], the light field of the phase singularity beam in atmospheric turbulence can be expressed as
$$E({\rho ,\varphi ,z} )\textrm{ = }{E_{free}}({\rho ,\varphi ,z} )\exp [{\psi ({\rho ,\varphi ,z} )} ],$$
where ψ (ρ, φ, z) represents the phase disturbance caused by atmospheric turbulence.

To better elucidate the OAM spectrum, also known as the spiral spectrum, the light field is expanded by spiral harmonics exp(imφ), which can be written as [24]

$$E({\rho ,\varphi ,z} )\textrm{ = }\frac{1}{{\sqrt {2\mathrm{\pi }} }}\sum\limits_{m ={-} \infty }^\infty {{a_m}({\rho ,z} )} \exp ({\textrm{i}m\varphi } ),$$
where
$${a_m}({\rho ,z} )\textrm{ = }\frac{1}{{\sqrt {2\mathrm{\pi }} }}\int\limits_0^{2\mathrm{\pi }} {E({\rho ,\varphi ,z} )} \exp ({ - \textrm{i}m\varphi } )\textrm{d}\varphi .$$
Hence, the mode probability density of OAM state m can be obtained as follows
$$\begin{array}{ll} \left\langle {{{|{{a_m}({\rho ,z} )} |}^2}} \right\rangle &= \frac{1}{{2\mathrm{\pi }}}\int {\int {{E_{free}}({\rho ,{\varphi_1},z} )} } E_{free}^\ast ({\rho ,{\varphi_2},z} )\\ &\times \left\langle {\exp [{\psi ({\rho ,{\varphi_1},z} )\textrm{ + }{\psi^{\ast }}({\rho ,{\varphi_2},z} )} ]} \right\rangle \exp [{ - \textrm{i}m({{\varphi_1} - {\varphi_2}} )} ]\textrm{d}{\varphi _1}\textrm{d}{\varphi _2}, \end{array}$$
where * denotes the complex conjugate. Moreover, <exp [ψ (ρ, φ, z) + ψ * (ρ, φ, z)] > can be expressed as follows [25]
$$\left\langle {\exp [{\psi ({\rho ,{\varphi_1},z} )\textrm{ + }{\psi^{\ast }}({\rho ,{\varphi_2},z} )} ]} \right\rangle { = \exp}[{ - 2{\rho^2}T({\alpha ,z} )+ 2{\rho^2}T({\alpha ,z} )\cos ({{\varphi_1} - {\varphi_2}} )} ],$$
where
$$T({\alpha , {\kern 1pt} {\kern 1pt} {\kern 1pt} z} )= \frac{{{\mathrm{\pi }^2}{k^2}z}}{3}\int_0^\infty {{\kappa ^3}} {\varPhi _n}({\kappa ,{\kern 1pt} {\kern 1pt} \alpha } )d\kappa .$$
T(α, z) is the physical quantity describing the turbulence intensity, Φn(κ, α) is the spatial power spectrum of the refractive index in the turbulent medium andα is the generalized exponential parameter. In this paper, non-Kolmogorov spectrum is used to simulate atmospheric turbulence and T(α, z) is expressed as follows [8]
$$\begin{array}{ll} T({\alpha , {\kern 1pt} {\kern 1pt} {\kern 1pt} z} )&= \frac{{{\mathrm{\pi }^2}{k^2}z}}{{6({\alpha - 2} )}}\chi (\alpha )\tilde{C}_n^2\\ &\times \left\{ {\kappa_m^{2 - \alpha }[{({\alpha - 2} )\kappa_m^2 + 2\kappa_0^2} ]\exp \left( {\frac{{\kappa_0^2}}{{\kappa_m^2}}} \right)\Gamma \left( {2 - \frac{\alpha }{2},{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \frac{{\kappa_0^2}}{{\kappa_m^2}}} \right) - 2\kappa_0^{4 - \alpha }} \right\} \end{array}({{\kern 1pt} {\kern 1pt} {\kern 1pt} 3 < \alpha < 4} ),$$
where
$$\chi (\alpha )\textrm{ = }\frac{{\Gamma ({\alpha - 1} )}}{{4{\mathrm{\pi }^2}}}\cos \left( {\frac{{\alpha \mathrm{\pi }}}{2}} \right),$$
$${\kappa _0}\textrm{ = }\frac{{2\mathrm{\pi }}}{{{L_0}}},$$
$${\kappa _m}\textrm{ = }\frac{1}{{{l_0}}}{\left[ {\Gamma \left( {\frac{{5 - \alpha }}{2}} \right)\frac{{2\mathrm{\pi }\chi (\alpha )}}{3}} \right]^{1/({\alpha - 5} )}}.$$
Here, l0 and L0 are the inner and outer scales of atmospheric turbulence, respectively. Moreover, $\tilde{C}_n^2$ is the generalized structural parameter of atmospheric turbulence, and Γ(·) denotes the gamma function.

We use the following integral formula [22]

$$\int\limits_0^{2\mathrm{\pi }} {\exp [{ - \textrm{i}j{\varphi_1} + \eta \cos ({{\varphi_1} - {\varphi_2}} )} ]} \textrm{d}{\varphi _1}\textrm{ = }2\mathrm{\pi }{{I}_j}(\eta )\exp ({ - \textrm{i}j{\varphi_2}} ),$$
where Ij(·) is the modified Bessel function of the first kind, and the analytical expression of the OAM mode probability density is derived as
$$\left\langle {{{|{{a_m}({\rho ,z} )} |}^2}} \right\rangle \textrm{ = }2\mathrm{\pi }\exp ({ - 2{\rho^2}T} )[{B{B^\ast }{I_{m - 1}}({2{\rho^2}T} )+ C{C^\ast }{I_{m + 1}}({2{\rho^2}T} )} ],$$
where
$$B\textrm{ = }[{ - ({a + 1} ){{(\textrm{i} )}^{ - 1}}\exp ({\textrm{i}\varphi } )} ]\left( {\frac{{\textrm{i}\mathrm{\pi }k\rho }}{{4{A^2}\lambda {z^2}}}} \right)\exp \left( {\textrm{i}kz\textrm{ + }\frac{{\textrm{i}k{\rho^2}}}{{2z}} - \frac{{{k^2}{\rho^2}}}{{4A{z^2}}}} \right),$$
$$C\textrm{ = }[{\textrm{i}({a - 1} )\exp ({ - \textrm{i}\varphi } )} ]\left( {\frac{{\textrm{i}\mathrm{\pi }k\rho }}{{4{A^2}\lambda {z^2}}}} \right)\exp \left( {\textrm{i}kz\textrm{ + }\frac{{\textrm{i}k{\rho^2}}}{{2z}} - \frac{{{k^2}{\rho^2}}}{{4A{z^2}}}} \right).$$

When the phase singularity beam propagates in atmospheric turbulence, the detection probability of OAM state m at the receiving plane is [26]

$${P_m}\textrm{ = }\frac{{\int\limits_0^R {\left\langle {{{|{{a_m}({\rho ,z} )} |}^2}} \right\rangle \rho \textrm{d}\rho } }}{{\sum\limits_{n ={-} \infty }^\infty {\int\limits_0^R {\left\langle {{{|{{a_n}({\rho ,z} )} |}^2}} \right\rangle \rho \textrm{d}\rho } } }},$$
where R is the radius of the receiving aperture. When a = ±1, Eq. (25) degenerates into the OAM detection probability expression of the phase singularity beam carrying a canonical optical vortex. When a ≠ ±1 and a ≠ 0, Eq. (25) is the OAM detection probability expression of the phase singularity beam carrying a noncanonical optical vortex. When a = 0, Eq. (25) degenerates into the OAM detection probability expression of the phase singularity beam carrying an edge dislocation.

3. Spiral spectrum of the phase singularity beam in the source plane

Figure 1 depicts the spiral spectrum (a, b) and phase distribution (c, d) of the phase singularity beam carrying a noncanonical optical vortex at the source plane. The calculated parameters are w0 = 2 cm, R = 5 cm, a = 5 (a, c), and a = −5 (b, d). In Figs. 1(a) and 1(c), it is observed that the phase singularity beam has a noncanonical optical vortex with a topological charge of +1, and the detection probabilities are 0.308 and 0.692 when the OAM states are −1 and 1, respectively. Figures 1(b) and 1(d) indicate that the phase singularity beam has a noncanonical optical vortex with a topological charge of −1 in the source plane, and it also has two OAM states, −1 and +1, whose detection probabilities are 0.692 and 0.308, respectively. Thus, it is evident from Fig. 1 that the noncanonical optical vortex with a topological charge of ±1 is obtained by the superposition of the two OAM states, −1 and 1.

 figure: Fig. 1.

Fig. 1. Spiral spectrum (a, b) and phase distribution (c, d) of the phase singularity beam carrying a noncanonical optical vortex at the source plane; sx = rcosθ, sy = rsinθ; a = 5 (a, c), a = −5 (b, d).

Download Full Size | PDF

Figure 2 shows the spiral spectrum (a) and phase distribution (b) of the phase singularity beam carrying an edge dislocation at the source plane. The calculated parameter is a = 0; the other parameters are shown in Fig. 1. Figure 2 indicates that a linear edge dislocation with a zero slope exists in the source plane of the phase singularity beam. Moreover, the detection probabilities of OAM states +1 and −1 are both 0.5. Hence, it can be determined from Fig. 2 that the linear edge dislocation with a zero slope is obtained by superposition of OAM states +1 and −1, and the detection probabilities of these two OAM states are equal.

 figure: Fig. 2.

Fig. 2. Spiral spectrum (a) and phase distribution (b) of the phase singularity beam carrying an edge dislocation at the source plane; sx = rcosθ, sy = rsinθ; a = 0.

Download Full Size | PDF

Figure 3 shows the spiral spectrum (a, b) and phase distribution (c, d) of the phase singularity beam carrying a canonical optical vortex in the source plane. The calculated parameters are a = 1 (a, c) and a = −1 (b, d). The other parameters are shown in Fig. 1. From Figs. 3(a) and 3(c), we can observe that, when the phase singularity beam has a canonical optical vortex with a topological charge of +1, the detection probability of the +1 OAM state is 1. However, Figs. 3(c) and 3(d) show that the phase singularity beam has a canonical optical vortex with a topological charge of −1, and the detection probability of the −1 OAM state is 1. Therefore, we can observe that the canonical optical vortex corresponds to only one OAM state.

 figure: Fig. 3.

Fig. 3. Spiral spectrum (a, b) and phase distribution (c, d) of the phase singularity beam carrying a canonical optical vortex at the source plane; sx = rcosθ, sy = rsinθ; a = 1 (a, c), a = −1 (b, d).

Download Full Size | PDF

4. Spiral spectrum of the phase singularity beam in atmospheric turbulence transmission

Figure 4 shows the spiral spectrum of the phase singularity beam carrying a noncanonical optical vortex in atmospheric turbulence. The calculated parameters are λ = 1.06 µm, w0 = 2 cm, R = 5 cm, $\tilde{C}_n^2$ = 10−15 m3-α, l0 = 10 mm, L0 = 10 m, α = 3.1, a = 5 (a, c), and a = −5 (b,d). Figure 4(a) indicates that the phase singularity beam carrying a noncanonical optical vortex with a topological charge of +1 exhibits crosstalk between the OAM states in atmospheric turbulent transmission, and the spiral spectrum significantly spreads. Figure 4(c) shows that the detection probabilities of OAM states −1 and +1 gradually decrease with the increase of the transmission distance. Furthermore, during the whole process of transmission, the detection probability of OAM state −1 is always smaller than that of OAM state +1.

 figure: Fig. 4.

Fig. 4. Spiral spectrum of the phase singularity beam carrying a noncanonical optical vortex in atmospheric turbulence; a = 5 (a, c), a = −5 (b, d).

Download Full Size | PDF

Figures 4(b) and 4(d) indicate that the propagation property is similar to that shown in Figs. 4(a) and 4(c) when the topological charge of the noncanonical optical vortex is −1. The difference is that the detection probability of OAM state −1 is always greater than that of OAM state +1. Hence, Fig. 4 shows that, when the phase singularity beam carrying a noncanonical optical vortex with a topological charge of ±1 propagates through atmospheric turbulence, the disturbance of atmospheric turbulence will lead to crosstalk between the OAM states, and the detection probabilities of OAM states ±1 are always greater than those of OAM states ∓1.

Figure 5 depicts the spiral spectrum of the phase singularity beam carrying an edge dislocation in atmospheric turbulence. The calculated parameter is a = 0; the other calculated parameters are shown in Fig. 4. From a comparison of Figs. 5(a) and 2(a), it can be observed that the spiral spectrum spreads when the phase singularity beam carrying an edge dislocation propagates. Figure 5(b) shows that the detection probabilities of OAM states +1 and −1 decrease with the increase of the transmission distance. In addition, at the same transmission distance, the detection probabilities of both are always equal. Therefore, Fig. 5 shows that the detection probabilities of OAM states +1 and −1 are always equal at the same transmission distance for the phase singularity beam carrying a linear edge dislocation with a zero slope in atmospheric turbulence.

 figure: Fig. 5.

Fig. 5. Spiral spectrum of the phase singularity beam carrying an edge dislocation in atmospheric turbulence; a = 0.

Download Full Size | PDF

Figure 6 shows the spiral spectrum of the phase singularity beam carrying a canonical optical vortex in atmospheric turbulence propagation. The calculated parameters are a = 1 (a, c) and a = −1 (b, d). The other parameters are shown in Fig. 4. From Figs. 6(a) and 6(b), it is evident that the phase singularity beam carrying a canonical optical vortex exhibits crosstalk between the OAM states. Figures 6(c) and 6(d) show that the detection probabilities of OAM states +1 and −1 decrease gradually with the increase of the transmission distance. Hence, Fig. 6 indicates that the disturbance of atmospheric turbulence causes crosstalk between the OAM states of the phase singularity beam carrying a canonical optical vortex.

 figure: Fig. 6.

Fig. 6. Spiral spectrum of the phase singularity beam carrying a canonical optical vortex in atmospheric turbulence; a = 1 (a, c), a = −1 (b, d).

Download Full Size | PDF

5. OAM mode probability density of the phase singularity beam

Figure 7 shows the OAM mode probability density when the phase singularity beam carrying an edge dislocation, a canonical optical vortex, or a noncanonical optical vortex propagates to 1 km. The calculated parameters are a = 5 (a, b), a = 0 (c, d), a = 1 (e), m = 1 (a, c, e), and m = −1 (b, d). The other parameters are shown in Fig. 6.

 figure: Fig. 7.

Fig. 7. Mode probability density of OAM state m; z = 1 km, a = 5 (a, b), a = 0 (c, d), a = 1 (e), m = 1 (a, c, e), m = −1 (b, d).

Download Full Size | PDF

As shown in Figs. 7(a)–7(e), along the beam radius, the OAM mode probability density increases gradually at first, reaches the maximum value, and then gradually decreases. Figures 7(a) and 7(b) show that, when a = 5, the maximum value of the mode probability density of OAM state m = +1 is at ρ = 1.959 cm, and the maximum value is at ρ = 1.980 cm for m = −1. We can observe from Figs. 7(c) and 7(d) that the maximum mode probability density of OAM states m = 1 or −1 is at ρ = 1.966 cm when a = 0. Figure 7(e) shows that the maximum value of the mode probability density of OAM state m = 1 is at ρ = 1.954 cm for a = 1. We already know from Figs. 1–3 that, at the source plane, when a = 5, the detection probabilities of OAM states m = +1 and −1 are 0.692 and 0.308, respectively; when a = 0, the detection probabilities of OAM states m = +1 and −1 are both 0.5; and when a = 1, the detection probability of OAM state m = 1 is 1.

Therefore, in Figs. 1−3 and 7, in comparing the OAM detection probability with the OAM mode probability density, it is found that the greater the OAM detection probability at the source plane is, the closer is the maximum value of the OAM mode probability density to the beam center during transmission.

6. Conclusions

Based on the Huygens−Fresnel diffraction integral formula, the light field of the phase singularity beam in free space transmission was derived in this paper. The atmospheric turbulence disturbance was then introduced to derive the analytical expression of the OAM detection probability of the phase singularity beam in atmospheric turbulence transmission. The phase distribution of the phase singularity beam in the source plane was discussed when dimensionless parameter a of the light field was different. According to the different values of a, the phase singularity beam can carry different wavefront dislocations. At the source plane, the study of the spiral spectrum of the phase singularity beam revealed that the phase singularity beam carrying a noncanonical optical vortex has multiple OAM states, the phase singularity beam with an edge dislocation also has multiple OAM states, and the phase singularity beam carrying a canonical optical vortex has only one OAM state. Therefore, the noncanonical optical vortex and the edge dislocation are representations of the superposition of OAM states.

In atmospheric turbulence propagation, the spiral spectrum of the phase singularity beam spreads. In terms of the OAM carried by the beam itself, the size relationship of the OAM detection probability in the source plane is the same as that during transmission. In addition, we compared the correlation between the OAM detection probability and the OAM mode probability density. In addition, the conclusions obtained are still valid when the wavelength and the intensity of atmospheric turbulence change.

The conclusions obtained have important applications in wireless optical communications and enable a better understanding of the effects of turbulence on wavefront dislocations and OAM.

Funding

National Natural Science Foundation of China (61875156, 61475123); Higher Education Discipline Innovation Project (B17035).

Disclosures

The authors declare no conflicts of interest.

References

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

2. M. Krenn, R. Fickler, M. Fink, J. Handsteiner, M. Malik, T. Scheidl, R. Ursin, and A. Zeilinger, “Communication with spatially modulated light through turbulent air across Vienna,” New J. Phys. 16(11), 113028 (2014). [CrossRef]  

3. C. Liu, K. Pang, Z. Zhao, P. Liao, R. Zhang, H. Song, Y. Cao, J. Du, L. Li, H. Song, Y. Ren, G. Xie, Y. Zhao, J. Zhao, S. M. H. Rafsanjani, A. N. Willner, J. H. Shapiro, R. W. Boyd, M. Tur, and A. E. Willner, “Single-end adaptive optics compensation for emulated turbulence in a bi-directional 10-Mbit/s per channel free-space quantum communication link using orbital-angular-momentum encoding,” Research (Washington, DC, U. S.) 2019, 1–10 (2019). [CrossRef]  

4. N. Mphuthi, L. Gailele, I. Litvin, A. Dudley, R. Botha, and A. Forbes, “Free-space optical communication link with shape-invariant orbital angular momentum Bessel beams,” Appl. Opt. 58(16), 4258–4264 (2019). [CrossRef]  

5. M. Soskin and M. Vasnetsov, “Singular optics,” Prog. Opt. 42(4), 219–276 (2001). [CrossRef]  

6. F. S. Roux, “Coupling of noncanonical optical vortices,” J. Opt. Soc. Am. B 21(3), 664–670 (2004). [CrossRef]  

7. Y. Jiang, S. Wang, J. Zhang, J. Ou, and H. Tang, “Spiral spectrum of Laguerre–Gaussian beam propagation in non-Kolmogorov turbulence,” Opt. Commun. 303, 38–41 (2013). [CrossRef]  

8. J. Ou, Y. Jiang, J. Zhang, H. Tang, Y. He, S. Wang, and J. Liao, “Spreading of spiral spectrum of Bessel–Gaussian beam in non-Kolmogorov turbulence,” Opt. Commun. 318, 95–99 (2014). [CrossRef]  

9. M. Cheng, L. Guo, J. Li, and Q. Huang, “Propagation properties of an optical vortex carried by a Bessel–Gaussian beam in anisotropic turbulence,” J. Opt. Soc. Am. A 33(8), 1442–1450 (2016). [CrossRef]  

10. S. Fu and C. Gao, “Influences of atmospheric turbulence effects on the orbital angular momentum spectra of vortex beams,” Photonics Res. 4(5), B1–B4 (2016). [CrossRef]  

11. Y. Zhang, M. Cheng, Y. Zhu, J. Gao, W. Dan, Z. Hu, and F. Zhao, “Influence of atmospheric turbulence on the transmission of orbital angular momentum for Whittaker-Gaussian laser beams,” Opt. Express 22(18), 22101–22110 (2014). [CrossRef]  

12. Y. Zhu, L. Zhang, Z. Hu, and Y. Zhang, “Effects of non-Kolmogorov turbulence on the spiral spectrum of Hypergeometric-Gaussian laser beams,” Opt. Express 23(7), 9137–9146 (2015). [CrossRef]  

13. C. Chen, H. Yang, S. Tong, and Y. Lou, “Changes in orbital-angular-momentum modes of a propagated vortex Gaussian beam through weak-to-strong atmospheric turbulence,” Opt. Express 24(7), 6959–6975 (2016). [CrossRef]  

14. Y. Zhu, Y. Zhang, and Z. Hu, “Spiral spectrum of Airy beams propagation through moderate-to-strong turbulence of maritime atmosphere,” Opt. Express 24(10), 10847–10857 (2016). [CrossRef]  

15. M. Chen and Y. Zhang, “Effects of anisotropic oceanic turbulence on the propagation of the OAM mode of a partially coherent modified Bessel correlated vortex beam,” Wave. Random Complex 29(4), 694–705 (2019). [CrossRef]  

16. Y. Li, Y. Han, Z. Cui, and Y. Hui, “Performance analysis of the OAM based optical wireless communication systems with partially coherent elegant Laguerre–Gaussian beams in oceanic turbulence,” J. Opt. 21(3), 035702 (2019). [CrossRef]  

17. M. P. Lavery, “Vortex instability in turbulent free-space propagation,” New J. Phys. 20(4), 043023 (2018). [CrossRef]  

18. S. Maji and M. M. Brundavanam, “Controlled noncanonical vortices from higher-order fractional screw dislocations,” Opt. Lett. 42(12), 2322–2325 (2017). [CrossRef]  

19. J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys. 6, 71 (2004). [CrossRef]  

20. P. Gao, L. Bai, Z. Wu, and L. Guo, “Evolution of linear edge dislocation in atmospheric turbulence and free space,” J. Mod. Opt. 66(1), 17–25 (2019). [CrossRef]  

21. V. V. Kotlyar, A. A. Kovalev, and A. P. Porfirev, “Elliptic Gaussian optical vortices,” Phys. Rev. A 95(5), 053805 (2017). [CrossRef]  

22. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic Press, 2007).

23. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Rrandom Media (The International Society for Optical Engineering Press, 2005).

24. G. Molina-Terriza, J. P. Torres, and L. Torner, “Management of the angular momentum of light: preparation of photons in multidimensional vector states of angular momentum,” Phys. Rev. Lett. 88(1), 013601 (2001). [CrossRef]  

25. Y. Zhu, M. Chen, Y. Zhang, and Y. Li, “Propagation of the OAM mode carried by partially coherent modified Bessel–Gaussian beams in an anisotropic non-Kolmogorov marine atmosphere,” J. Opt. Soc. Am. A 33(12), 2277–2283 (2016). [CrossRef]  

26. Y. D. Liu, C. Gao, M. Gao, and F. Li, “Coherent-mode representation and orbital angular momentum spectrum of partially coherent beam,” Opt. Commun. 281(8), 1968–1975 (2008). [CrossRef]  

Cited By

Optica participates in Crossref's Cited-By Linking service. Citing articles from Optica Publishing Group journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (7)
Fig. 1.
Fig. 1. Spiral spectrum (a, b) and phase distribution (c, d) of the phase singularity beam carrying a noncanonical optical vortex at the source plane; sx = rcosθ, sy = rsinθ; a = 5 (a, c), a = −5 (b, d).

View in Article | Download Full Size | PDF

Fig. 2.
Fig. 2. Spiral spectrum (a) and phase distribution (b) of the phase singularity beam carrying an edge dislocation at the source plane; sx = rcosθ, sy = rsinθ; a = 0.

View in Article | Download Full Size | PDF

Fig. 3.
Fig. 3. Spiral spectrum (a, b) and phase distribution (c, d) of the phase singularity beam carrying a canonical optical vortex at the source plane; sx = rcosθ, sy = rsinθ; a = 1 (a, c), a = −1 (b, d).

View in Article | Download Full Size | PDF

Fig. 4.
Fig. 4. Spiral spectrum of the phase singularity beam carrying a noncanonical optical vortex in atmospheric turbulence; a = 5 (a, c), a = −5 (b, d).

View in Article | Download Full Size | PDF

Fig. 5.
Fig. 5. Spiral spectrum of the phase singularity beam carrying an edge dislocation in atmospheric turbulence; a = 0.

View in Article | Download Full Size | PDF

Fig. 6.
Fig. 6. Spiral spectrum of the phase singularity beam carrying a canonical optical vortex in atmospheric turbulence; a = 1 (a, c), a = −1 (b, d).

View in Article | Download Full Size | PDF

Fig. 7.
Fig. 7. Mode probability density of OAM state m; z = 1 km, a = 5 (a, b), a = 0 (c, d), a = 1 (e), m = 1 (a, c, e), m = −1 (b, d).

View in Article | Download Full Size | PDF

Equations (25)

Equations on this page are rendered with MathJax. Learn more.

E 0 ( r , θ , z = 0 )  =  r ( a cos ⁡ θ + i sin ⁡ θ ) exp ⁡ ( − r 2 2 w 0 2 ) ,
e i θ  =  cos ⁡ θ + i sin ⁡ θ ,
E 0 ( r , θ , z = 0 )  =  ( a  +  1 ) r 2 exp ⁡ ( − r 2 2 w 0 2  + i θ ) + ( a − 1 ) r 2 exp ⁡ ( − r 2 2 w 0 2 − i θ ) .
E f r e e ( ρ , φ , z ) = − i λ z exp ⁡ ( i k z ) ∫ ∫ E 0 ( r , θ , z = 0 ) × exp ⁡ { i k 2 z [ r 2 + ρ 2 − 2 r ρ cos ⁡ ( φ − θ ) ] } r d r d θ ,
exp ( i z cos ⁡ φ )  =  ∑ j = − ∞ ∞ i j J j ( z ) exp ⁡ ( i j φ ) ,
∫ 0 2 π exp ( i m θ )  =  { 2 π m = 0 0 m ≠ 0 ,
J − j ( z )  =  ( − 1 ) j J j ( z ) ,
∫ 0 ∞ x j + 1 exp ( − α x 2 ) J j ( β x ) d x  =  β j ( 2 α ) j + 1 exp ⁡ ( − β 2 4 α ) ,
E f r e e ( ρ , φ , z ) = ( i π k ρ 4 A 2 λ z 2 ) exp ⁡ ( i k z  +  i k ρ 2 2 z − k 2 ρ 2 4 A z 2 ) × [ − ( a + 1 ) ( i ) − 1 exp ⁡ ( i φ ) + i ( a − 1 ) exp ⁡ ( − i φ ) ] ,
A  =  1 2 w 0 2 − i k 2 z .
E ( ρ , φ , z )  =  E f r e e ( ρ , φ , z ) exp ⁡ [ ψ ( ρ , φ , z ) ] ,
E ( ρ , φ , z )  =  1 2 π ∑ m = − ∞ ∞ a m ( ρ , z ) exp ⁡ ( i m φ ) ,
a m ( ρ , z )  =  1 2 π ∫ 0 2 π E ( ρ , φ , z ) exp ⁡ ( − i m φ ) d φ .
⟨ | a m ( ρ , z ) | 2 ⟩ = 1 2 π ∫ ∫ E f r e e ( ρ , φ 1 , z ) E f r e e ∗ ( ρ , φ 2 , z ) × ⟨ exp ⁡ [ ψ ( ρ , φ 1 , z )  +  ψ ∗ ( ρ , φ 2 , z ) ] ⟩ exp ⁡ [ − i m ( φ 1 − φ 2 ) ] d φ 1 d φ 2 ,
⟨ exp ⁡ [ ψ ( ρ , φ 1 , z )  +  ψ ∗ ( ρ , φ 2 , z ) ] ⟩ = exp [ − 2 ρ 2 T ( α , z ) + 2 ρ 2 T ( α , z ) cos ⁡ ( φ 1 − φ 2 ) ] ,
T ( α , z ) = π 2 k 2 z 3 ∫ 0 ∞ κ 3 Φ n ( κ , α ) d κ .
T ( α , z ) = π 2 k 2 z 6 ( α − 2 ) χ ( α ) C ~ n 2 × { κ m 2 − α [ ( α − 2 ) κ m 2 + 2 κ 0 2 ] exp ⁡ ( κ 0 2 κ m 2 ) Γ ( 2 − α 2 , κ 0 2 κ m 2 ) − 2 κ 0 4 − α } ( 3 < α < 4 ) ,
χ ( α )  =  Γ ( α − 1 ) 4 π 2 cos ⁡ ( α π 2 ) ,
κ 0  =  2 π L 0 ,
κ m  =  1 l 0 [ Γ ( 5 − α 2 ) 2 π χ ( α ) 3 ] 1 / ( α − 5 ) .
∫ 0 2 π exp ⁡ [ − i j φ 1 + η cos ⁡ ( φ 1 − φ 2 ) ] d φ 1  =  2 π I j ( η ) exp ⁡ ( − i j φ 2 ) ,
⟨ | a m ( ρ , z ) | 2 ⟩  =  2 π exp ⁡ ( − 2 ρ 2 T ) [ B B ∗ I m − 1 ( 2 ρ 2 T ) + C C ∗ I m + 1 ( 2 ρ 2 T ) ] ,
B  =  [ − ( a + 1 ) ( i ) − 1 exp ⁡ ( i φ ) ] ( i π k ρ 4 A 2 λ z 2 ) exp ⁡ ( i k z  +  i k ρ 2 2 z − k 2 ρ 2 4 A z 2 ) ,
C  =  [ i ( a − 1 ) exp ⁡ ( − i φ ) ] ( i π k ρ 4 A 2 λ z 2 ) exp ⁡ ( i k z  +  i k ρ 2 2 z − k 2 ρ 2 4 A z 2 ) .
P m  =  ∫ 0 R ⟨ | a m ( ρ , z ) | 2 ⟩ ρ d ρ ∑ n = − ∞ ∞ ∫ 0 R ⟨ | a n ( ρ , z ) | 2 ⟩ ρ d ρ ,
Select as filters
Select Topics Cancel
Top
       
© Copyright 2024 | Optica Publishing Group. All rights reserved, including rights for text and data mining and training of artificial technologies or similar technologies.
Privacy | Terms of Use