Self-recovery effect of orbital angular momentum mode of circular beam in weak non-Kolmogorov turbulence

Tao Zhang; Yi-Dong Liu; Jiandong Wang; Pusheng Liu; Yuanjie Yang

doi:10.1364/OE.24.020507

1. Introduction

In general, the OAM-carrying beam has the helical phase front with regard to a phase term of exp(ilφ), where l is an arbitrary integer and φ refers to azimuthal coordinate [1]. Different coaxial OAM eigen-modes of vortex beam are orthogonal, which has important application in free space communication [2,3]. While, it is well known that the effect of atmosphere turbulence is one of the key factors that limit the development of OAM-based communication, because the turbulence effects cause the spreading of the spiral spectrum [4–6]. Recently, the effect of Kolmogorov turbulence on single photon and entangled photon pairs has been studied in detail [7,8]. Moreover, the effect of non-Kolmogorov turbulence on some vortex beams, such as hypergeometric-Gaussian Beam [9], Hankel-Bessel beam [10] and Whittaker-Gaussian beam [11], has been discussed as well.

Very recently, some attention is paid to circular beam (CiB) [12,13], which is a very general solution of the paraxial wave equation. The equivalent form, special cases, normalization, Laguerre-Gaussian expansion, free-space divergence and parameter constraints of CiB have been discussed in previous works [12,13]. By assigning particular values to three complex beam parameters, the CiB is reduced to some well-known vortex beams such as standard Laguerre-Gaussian beam, elegant Laguerre-Gaussian beam, generalized Laguerre-Gaussian beam, hypergeometric beams, fractional-order elegant Laguerre-Gaussian beams, hypergeometric-Gaussian Beam and Whittaker-Gaussian beam [12].

The focusing property of beam can be employed in many fields, such as trapping and guiding microparticles [14] and realizing optical micromanipulation [15]. To our knowledge, currently, there is no further research to the influence of focusing property on the propagation of vortex beams in atmosphere turbulence. In this work, we discuss how the beam parameters affect the self-focusing property of CiB. Furthermore, we characterize the influence of non-Kolmogorov turbulence on the OAM modes of self-focusing CiB and reveal its self-recovery effect.

2. Power weight of OAM state for CiB in weak non-Kolmogorov turbulence

The normalized CiB propagating along z axis in cylindrical coordinates reads [13]

\begin{array}{l} {CiB}_{p, l_{0}}^{(q_{0}, q_{1})} (r, φ, z) & = {(i \sqrt{2} \frac{z_{0}}{W_{0}})}^{| l_{0} | + 1} {[π | l_{0} |! Ψ_{p, l_{0}}^{(ξ)}]}^{- \frac{1}{2}} \frac{1}{q (z)} exp [- \frac{i k r^{2}}{2 q (z)}] \\ \times {[(1 + ξ) \frac{\tilde{q} (z)}{q (z)}]}^{\frac{p}{2}} {[\frac{r}{q (z)}]}^{| l_{0} |} {{}_{1}F}_{1} (- \frac{p}{2}, | l_{0} | + 1; \frac{r^{2}}{χ^{2} (z)}) . \\ \times exp (i l_{0} φ) \end{array}

In Eq. (1), there are three complex parameters q₀, q₁ and p, where q₀ is given by q₀ = iz₀ − d₀. W₀ is a real number similar to the waist radii of Gaussian beam and hence $z_{0} = k W_{0}^{2} / 2$ is similar to the Rayleigh range of wave number k. d₀ is the position of waist of Gaussian envelope. l₀ is an integer corresponding to the initial OAM quantum number. Besides, the other four complex parameters in Eq. (1) are defined as q(z) = q₀ + z, q̃(z) = q₁ + z, $ξ = (q_{1} - q_{0}) / (q_{0}^{*} - q_{1})$ and 1/χ²(z) = ik [1/q(z) − 1/q̃(z)]/2. q(z) and q̃(z) are similar to the q-parameter of Gaussian beam and 1/χ²(z) is the scale factor. $Ψ_{p, l_{0}}^{(ξ)} = {{}_{1}F}_{2} (- p / 2, - p^{*} / 2, | l_{0} | + 1; {| ξ |}^{2})$ is the normalization factor. ₁F₁ and ₂F₁ denote the confluent hypergeometric function and hypergeometric function, respectively.

In weak atmospheric turbulence, the intensity fluctuation is very small that can be neglected, so we only consider the phase aberration on the complex amplitude of CiB. Supposing propagation distance z > 0, the complex amplitude of distorted CiB here can be expressed as

Ψ_{p, q_{0}}^{(q_{0}, q_{1})} (r, φ, z) = {CiB}_{p, l_{0}}^{(q_{0}, q_{1})} (r, φ, z) \cdot exp [i ψ (r, φ, z)],

where ψ(r, φ, z) is the term corresponding to complex phase perturbation caused by the turbulence. To elucidate the weight of all OAM modes, the complex amplitude of distorted CiB in weak turbulence can be expressed as the superposition of spiral harmonics [16]

Ψ_{p, l_{0}}^{(q_{0}, q_{1})} (r, φ, z) = \frac{1}{\sqrt{2 π}} \sum_{l = - \infty}^{\infty} β_{p, l_{0}, l}^{(q_{0}, q_{1})} (r, z) exp (i l φ)

with the expansion coefficient

β_{p, l_{0}, l}^{(q_{0}, q_{1})} (r, z) = \frac{1}{\sqrt{2 π}} \int_{0}^{2 π} {CiB}_{p, l_{0}}^{(q_{0}, q_{1})} (r, φ, z) exp [i ψ (r, φ, z)] exp (- i l φ) d φ .

Instead of the random variable $β_{p, l_{0}, l}^{(q_{0}, q_{1})} (r, z)$ , we are usually interested in the ensemble average over the turbulence statistics, i.e.

\begin{array}{l} 〈 {| β_{p, l_{0}, l}^{(q_{0}, q_{1})} (r, z) |}^{2} 〉 = & \frac{1}{2 π} \int_{0}^{2 π} \int_{0}^{2 π} {CiB}_{p, l_{0}}^{(q_{0}, q_{1})} (r, φ_{1}, z) {CiB}_{p, l_{0}}^{(q_{0}, q_{1}) *} (r, φ_{2}, z) \\ \times exp [- i l (φ_{1} - φ_{2})] \\ \times 〈 exp {i [ψ (r, φ_{1}, z) - ψ (r, φ_{2}, z)]} 〉 d φ_{1} d φ_{2} \end{array}

The ensemble average in right hand of Eq. (5) reads [17]

〈 exp {i [ψ (r, φ_{1}, z) - ψ (r, φ_{2}, z)]} 〉 = exp [- \frac{2 r^{2} - 2 r^{2} cos (φ_{1} - φ_{2})}{ρ_{0}^{2}}],

where ρ₀ denotes the spatial coherence radius of a spherical wave propagating in non-Kolmogorov turbulence [18]

ρ_{0} = {[\frac{8}{α - 2} Γ (\frac{2}{α - 2})]}^{\frac{1}{2}} \cdot {[\frac{2 (α - 1) Γ (\frac{3 - α}{2})}{\sqrt{π} Γ (\frac{2 - α}{2}) k^{2} C_{n}^{2} z}]}^{\frac{1}{α - 2}} (3 < α < 4),

where

C_{n}^{2}

denotes refractive-index structure constant and α denotes non-Kolmogorov turbulence parameter. Equation (7) implies that ρ₀ → ∞ if α → 3, and ρ₀ → 0 if α → 4. Following Eqs. (5)–(7) and using Eq. 8.411.1 in [19], we can obtain

\begin{array}{l} 〈 {| β_{p, l_{0}, l}^{(q_{0}, q_{1})} (r, z) |}^{2} 〉 = & \frac{1}{2^{| l_{0} |} | l_{0} |! Ψ_{p, l_{0}}^{(ξ)}} {(\frac{k W_{0}}{| q (z) |})}^{2 | l_{0} | + 2} | {[(1 + ξ) \frac{\tilde{q} (z)}{q (z)}]}^{p} | \\ \times exp (- \frac{k^{2} r^{2} W_{0}^{2}}{2 {| q (z) |}^{2}}) {| {{}_{1}F}_{1} (- \frac{p}{2}, | l_{0} | + 1; \frac{r^{2}}{χ^{2} (z)}) |}^{2}, \\ \times r^{2 | l_{0} |} exp (- \frac{2 r^{2}}{ρ_{0}^{2}}) I_{l - l_{0}} (\frac{2 r^{2}}{ρ_{0}^{2}}) \end{array}

where I_l−l₀ is the modified Bessel function of the first kind with order l − l₀. By normalizing the detected power

P_{l} = \int_{0}^{+ \infty} 〈 {| β_{p, l_{0}, l}^{(q_{0}, q_{1})} (r, z) |}^{2} 〉 r d r

in spiral harmonics with azimuthal number l, the corresponding power weight reads [16]

C_{l} = \frac{P_{l}}{\sum_{m = - \infty}^{\infty} P_{m}} .

Here C_l means the power weight of OAM mode with azimuthal number l and in OAM-based optical communication it’s also the detection probability of OAM state in receiver. Especially, C_l₀ is the power weight corresponding to the original OAM mode with azimuthal number l₀. Without turbulence and with infinite aperture, C_l₀ = 1 but C_l≠l₀ = 0. In the presence of turbulence, the transmitted power leak into near modes and therefore C_l≠l₀ is called as crosstalk power weight in context.

Because the crosstalk power weight is symmetric about l₀ due to I_n(x) = I₋_n(x), it’s convenient to discuss the case of l ≥ l₀ only. Under the condition of l ≥ l₀, the infinite series form of C_l can be obtained using Eq. 6.622 in [19] as

\begin{array}{l} C_{l} = & \frac{i {(- 1)}^{| l_{0} | + 1} k W_{0} ρ_{0}}{\sqrt{2 π} | q (z) | \cdot | l_{0} |! Ψ_{p, l_{0}}^{(ξ)}} {(\frac{η - 1}{η + 1})}^{\frac{| l_{0} |}{2} + \frac{1}{4}} | {[(1 + ξ) \frac{\tilde{q} (z)}{q (z)}]}^{p} | \\ \times \sum_{m = 0}^{\infty} \sum_{n = 0}^{\infty} \frac{{(- \frac{p^{*}}{2})}_{m} {(- \frac{p}{2})}_{n}}{{(| l_{0} | + 1)}_{m} {(| l_{0} | + 1)}_{n} n! m!} \\ \times {[- \frac{ρ_{0}^{2}}{2 {(χ^{2})}^{*} {(η^{2} - 1)}^{\frac{1}{2}}}]}^{m} {[- \frac{ρ_{0}^{2}}{2 χ^{2} {(η^{2} - 1)}^{\frac{1}{2}}}]}^{n} \\ \times Q_{l - l_{0} - \frac{1}{2}}^{| l_{0} | + n + m + \frac{1}{2}} (η) \end{array},

where

η = \frac{k^{2} W_{0}^{2} ρ_{0}^{2}}{4 {| q (z) |}^{2}} + 1

, (*)_n denotes the Pochhammer symbol and

Q_{ν}^{μ} (*)

denotes the associated Legendre functions of the second kind.

3. Numerical results and discussion

In this section, we discuss the relation of self-focusing effect to the parameters of CiB at first and then investigate the influence of non-Kolmogorov turbulence on the original OAM mode. W₀ relates to the scale of beam and is taken as 0.04m here. The imaginary part of q₁, which is similar to that of q₀, is taken as 1200m here. Besides, wavelength is taken as λ = 1550nm in the paper. For quantitative analysis of the focusing property of CiB, we use the second order momentum to describe the beam width as

W^{2} (z) = 4 〈 r^{2} 〉 = 4 \iint r^{2} {| {CiB}_{p, l_{0}}^{(q_{0}, q_{1})} (\vec{r}, z) |}^{2} d^{2} \vec{r} (z \geq 0),

where r⃗ denotes the point vector on transverse plane and * denotes the ensemble average. The waist, i.e. the minimal beam width, is found at z = z_e. z_e is also the position where the energy of beam is most concentrated. If z_e > 0m, the CiB converges at z_e and we call the beam is self-focusing in context. If z_e=0m, the CiB is not self-focusing.

In Fig. 1, We investigate the effect of some parameters of the CiB on z_e. This curves In Figs. 1(a)–1(d) are non-differentiable at the joint point of z_e=0m and z_e > 0m due to the mandatory condition z ≥ 0m in Eq. (11) resulting in z_e ≥ 0m. In Figs. 1(a) and 1(c), we plot the waist position z_e against beam parameters Re(q₀) and Re(q₁). z_e decreases as Re(q₁) increases or Re(q₀) decreases. We observe an approximate linear relation between Re(q₁) and z_e, which shows how the waist of Gaussian envelope affects the waist of the CiB. If the value of Re(q₁) is greater than about 2000m, z_e reaches 0m that means the CiB is not self-focusing. From Figs. 1(a) and 1(c), We suppose that the value of z_e can be changed greatly if Re(q₁) is varied adequately. In Figs. 1(b) and 1(d), we plot the waist position z_e against beam parameter p. As Im(p) increases from 0 to 20, z_e increases at first and then decreases. z_e increases as Re(p) increases. It is clear that when the value of Im(p) is smaller than about 0, z_e is 0m, that is seemingly independent of Re(p). It seems that the value of z_e can but be changed into a limited range if only Im(p) is varied. The information contained in Fig. 1 indicate that one can manipulate z_e by varying p, Re(q₀) and Re(q₁).

Fig. 1 The waist position z_e against (a) Re(q₀) and Re(q₁) with p=2+20i; (b) p with Re(q₀)=Re(q₁)=0m; (c) Re(q₀) and Re(q₁) with p=2+10i; (d) p with Re(q₀)=Re(q₁)=1000m. Other parameters: l₀ = 1, $C_{n}^{2} = 10^{- 15} m^{3 - α}$ and α = 3.67, respectively.

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Figure 2(a) shows the location of waist of one kind of self-focusing CiB in weak non-Kolmogorov turbulence. The location of waist of this CiB is at z_e=2264m. The difference between the beam width at z_e and the one at z=0m is about 100mm. The phase patterns of distorted CiB in weak non-Kolmogorov turbulence are shown in Figs. 2(b)–2(d). In these phase patterns, The gray scales from black to white correspond to the phase varying from 0 to 2π. Figure 2(b) shows the phase pattern at z=0m. This undistorted azimuthal phase increases counterclockwise while the phase contour lines twine clockwise. Figure 2(c) shows the distorted phase pattern at the waist position z = z_e=2264m. A serious distorted azimuthal spiral phase is difficult to recognize at farther position such as z=5000m shown in Fig. 2(d).

Fig. 2 (a) beam width W (z) of the CiB against propagation distance z in non-Kolmogorov turbulence. Phase patterns of the CiB on x − y plane of (b) z=0m, (c) z=2264m and (d) z=5000m in non-Kolmogorov turbulence. Other parameters: $C_{n}^{2} = 10^{- 15} m^{3 - α}$ , α=3.67, q₁=1200i m, l₀=1 and p=2+20i, respectively.

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In Fig. 3, we plot the received power weight C_l₀ against propagation distance z in different cases. Figure 3(a) is plotted for different values of parameter p. All the lines show the similar character that as z increases, C_l₀ decreases except for an arch near z_e. Since the previous works show that detection probability of initial OAM state decreases as the relative beam width W (z)/r₀ increases [7], where r₀ is Fried parameter, here the presence of arch can be explained as the case that the ratio W (z)/ρ₀ decreases as z increases. Around z_e, C_l₀ varies slowly. This mechanism proposes a novel method to alleviate the effect of turbulence on free space communication. In other words, one can manipulate z_e according to the position of the receiver to obtain maximal C_l₀ or alleviate the influence of atmospheric turbulence.

Fig. 3 The received power weight C_l₀ for the CiB against propagation distance z (a) with different Re(p) where $C_{n}^{2} = 10^{- 15} m^{3 - α}$ , α=3.97, Im(p) = 20 and l₀=1; (b) with different α where $C_{n}^{2} = 10^{- 15} m^{3 - α}$ , l₀=1, p=2+20i; (c) with different $C_{n}^{2}$ where α=3.67, l₀=1, p=2+20i; (d) with different l₀ where $C_{n}^{2} = 10^{- 15} m^{3 - α}$ , α=3.67, p=2+20i. Other parameter: q₁=1200i m.

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Figure 3(a) also shows that as Re(p) increases, z_e increases, but meanwhile the value of C_l₀ at z_e decreases. For example, Re(p) increases from 2 to 8, z_e expands from about 2.25km to about 3.0km while C_l₀ at z_e decreases from about 0.93 to 0.89. Parameter Re(p) of CiB has similar meaning to the radial index of LGB, and bigger value of Re(p) results in larger beam width that makes the OAM of CiB be more vulnerable to spatial aberrations in atmospheric turbulence.

Figure 3(b) shows that C_l₀ decreases as α increases. With different α, the curves also show an arch near z_e, But with smaller α the curve is flatter and the arch is not clear. Figure 3(c) shows similar character that C_l₀ decreases as $C_{n}^{2}$ increases. While turbulence is weak, e.g. $C_{n}^{2} = 7.5 \times 10^{- 16} m^{3 - α}$ , C_l₀ is almost 1 from 0km to 5km. In Figs. 3(b) and 3(c), it is obvious that the self-recovery effect of self-focusing CiB is more clear in the strong perturbation case i.e. large $C_{n}^{2}$ or α than that in the weak perturbation case. Figures 3(b) and 3(c) also show that z_e is almost the same and not relevant to the weak atmospheric turbulence. This conclusion is approximative because the weak turbulence is presupposed to affect the phase of the beam only in Eq. (2). But this presupposition is unreasonable in strong turbulence.

Figure 3(d) is plotted for different l₀. C_l₀ decreases as l₀ increases, because bigger l₀ means larger beam width W(z) that makes the OAM of beam be more vulnerable to turbulence, which is similar to LGB propagating in atmospheric turbulence in [20]. From Fig. 3(d) we also find that as l₀=1 and 2, the arch emerges and the recovery effect of CiB arises. As l₀=3,4 and 5, the local arches are not so clear and C_l₀ descends very slowly around z_e. Therefore, the resistance ability of OAM mode is weaker with larger l₀. To obtain better signal quality, it is better to employ lower OAM modes as channels in OAM-based free space communication. For example, the OAM modes with l₀=−4,−3...3,4 can be employed for coaxial transmission in condition of Fig. 3(d).

In Fig. 4, we consider the crosstalk power weight C_l of CiB mode. C_l decreases as Δl = |l − l₀| increases. As z increases, C_l increases except a slight declining near z_e=2.264km. The reason for this declining is that received power weight C_l₀ reaches its maximum near z_e and then less power from initial mode falls into other modes. As Δl = 1, this declining is most obvious. It is clear that the crosstalk power weights for Δl=2,3 or 4 are very small within short propagation distance and they can be omitted comparing with the crosstalk power weights for Δl=1. If some communication systems utilize CiB modes for short distance message transmission, the crosstalk is very small when nonadjacent modes are employed.

Fig. 4 The crosstalk power weight C_l for the CiB against propagation distance z with Δl=|l − l₀|=1,2,3,4. Other parameters: l₀=1, p=2+20i, q₁=1200i m, α=3.67 and $C_{n}^{2} = 10^{- 15} m^{3 - α}$ .

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4. Conclusions

In this paper, we have revealed the self-focusing property and the self-recovery effect of the CiB in weak non-Kolmogorov turbulence. This work is carried out theoretically and numerically. Our results show that the appearance of self-focusing property of the CiB strongly depends on Re(q₁) and Im(p), and the location of waist of the CiB can be changed by varying Re(q₀),Re(q₁), Re(p) and Im(p). The key conclusion shows that the self-recovery effect may arise while the CiB has self-focusing property, which implies that self-focusing CiB may have more excellent resistance ability to turbulence effect on the position of waist than vortex beam with no self-focusing property. It is noticeable that the self-recovery effect is only available for the CiB with small azimuthal number l₀. Anyway, it proposes an opportunity to alleviate the turbulent effects on OAM-based optical communication.

Funding

We acknowledge the support of the National Natural Science Foundation of China (NSFC) under Grant nos. 60908034, 61205122 and 11474048, and Fundamental Research Funds for the Central Universities under grant nos. ZYGX2013J052 and ZYGX2015J042.

References and links

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

2. A. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. Lavery, M. Tur, S. Ramachandran, A. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photon. 7, 66–106 (2015).

3. G. Gibson, J. Courtial, M. Padgett, M. Vasnetsov, V. Pas’ko, S. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express 12(22), 5448–5456 (2004). [CrossRef] [PubMed]

4. B. Rodenburg, M. P. J. Lavery, M. Malik, M. N. O’Sullivan, M. Mirhosseini, D. J. Robertson, M. Padgett, and R. W. Boyd, “Influence of atmospheric turbulence on states of light carrying orbital angular momentum,” Opt. Lett. 37(17), 3735–3737 (2012). [CrossRef] [PubMed]

5. G. A. Tyler and R. W. Boyd, “Influence of atmospheric turbulence on the propagation of quantum states of light carrying orbital angular momentum,” Opt. Lett. 34(2), 142–144 (2009). [CrossRef] [PubMed]

6. J. A. Anguita, M. Neifeld, and B. Vasic, “Turbulence-induced channel crosstalk in an orbital angular momentummultiplexed free-space optical link,” Appl. Opt. 47(13), 2414–2429 (2008). [CrossRef] [PubMed]

7. C. Paterson, “Atmospheric turbulence and orbital angular momentum of single photons for optical communication,” Phys. Rev. Lett. 94(15), 153901 (2005). [CrossRef] [PubMed]

8. C. Gopaul and R. Andrews, “The effect of atmospheric turbulence on entangled orbital angular momentum states,” New J. Phys. 9, 94 (2007). [CrossRef]

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

10. Y. Zhu, X. J. Liu, J. Gao, Y. X. Zhang, and F. S. Zhao, “Probability density of the orbital angular momentum mode of Hankel-Bessel beams in an atmospheric turbulence,” Opt. Express 22(7), 7765–7772 (2014). [CrossRef] [PubMed]

11. Y. X. Zhang, M. J. Cheng, Y. Zhu, J. Gao, W. Y. Dan, Z. D. Hu, and F. S. 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] [PubMed]

12. M. A. Bandres and J. C. Gutierrez-Vega, “Circular beams,” Opt. Lett. 33(2), 177–179 (2008). [CrossRef] [PubMed]

13. G. Vallone, “On the properties of circular beams: normalization, Laguerre-Gauss expansion, and free-space divergence,” Opt. Lett. 40(8), 1717–1720 (2015). [CrossRef] [PubMed]

14. P. Zhang, J. Prakash, Z. Zhang, M. S. Mills, N. K. Efremidis, D. N. Christodoulides, and Z. Chen, “Trapping and guiding microparticles with morphing autofocusing Airy beams,” Opt. Lett. 36(15), 2883–2885 (2011). [CrossRef] [PubMed]

15. N. K. Efremidis and D. N. Christodoulides, “Abruptly autofocusing waves,” Opt. Lett. 35(23), 4045–4047 (2010). [CrossRef] [PubMed]

16. L. Torner, J. P. Torres, and S. Carrasco, “Digital spiral imaging,” Opt. Express 13(3), 873–881 (2005). [CrossRef] [PubMed]

17. L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media (SPIE, 2005). [CrossRef]

18. C. H. Rao, W. H. Jiang, and N. Ling, “Spatial and temporal characterization of phase fluctuations in non-Kolmogorov atmospheric turbulence,” J. Mod. Opt. 47(6), 1111–1126 (2000). [CrossRef]

19. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 7th ed. (Academic, 2007).

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

Self-recovery effect of orbital angular momentum mode of circular beam in weak non-Kolmogorov turbulence

Abstract

1. Introduction

2. Power weight of OAM state for CiB in weak non-Kolmogorov turbulence

3. Numerical results and discussion

4. Conclusions

Funding

References and links

Cited By

Figures (4)

Equations (11)

Optics Express