It is generally true that the orbital angular momentum (OAM) mode persistently degenerate when a vortex beam propagates in the atmospheric turbulence. Here, however, we unveil an interesting self-recovery effect of OAM mode of the circular beam (CiB) in weak non-Kolmogorov turbulence. We show that the CiB displays the self-focusing effect and has clear focus in the weak non-Kolmogorov turbulence if we choose proper complex parameters, and the detection probability of the original OAM mode reaches the maximum at the focus. Our study proposes a method to alleviate the turbulent effects on OAM-based communication.
© 2016 Optical Society of America
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 . 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 , Hankel-Bessel beam  and Whittaker-Gaussian beam , 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 .
The focusing property of beam can be employed in many fields, such as trapping and guiding microparticles  and realizing optical micromanipulation . 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 
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 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, 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. 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 as16]
Instead of the random variable , we are usually interested in the ensemble average over the turbulence statistics, i.e.18] 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 , we can obtain 16]
Here Cl 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, Cl0 is the power weight corresponding to the original OAM mode with azimuthal number l0. Without turbulence and with infinite aperture, Cl0 = 1 but Cl≠l0 = 0. In the presence of turbulence, the transmitted power leak into near modes and therefore Cl≠l0 is called as crosstalk power weight in context.
Because the crosstalk power weight is symmetric about l0 due to In(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 Cl can be obtained using Eq. 6.622 in  as
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
In Fig. 1, We investigate the effect of some parameters of the CiB on ze. This curves In Figs. 1(a)–1(d) are non-differentiable at the joint point of ze=0m and ze > 0m due to the mandatory condition z ≥ 0m in Eq. (11) resulting in ze ≥ 0m. In Figs. 1(a) and 1(c), we plot the waist position ze against beam parameters Re(q0) and Re(q1). ze decreases as Re(q1) increases or Re(q0) decreases. We observe an approximate linear relation between Re(q1) and ze, 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, ze reaches 0m that means the CiB is not self-focusing. From Figs. 1(a) and 1(c), We suppose that the value of ze can be changed greatly if Re(q1) is varied adequately. In Figs. 1(b) and 1(d), we plot the waist position ze against beam parameter p. As Im(p) increases from 0 to 20, ze increases at first and then decreases. ze increases as Re(p) increases. It is clear that when the value of Im(p) is smaller than about 0, ze is 0m, that is seemingly independent of Re(p). It seems that the value of ze 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 ze by varying p, Re(q0) and Re(q1).
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 ze=2264m. The difference between the beam width at ze 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 = ze=2264m. A serious distorted azimuthal spiral phase is difficult to recognize at farther position such as z=5000m shown in Fig. 2(d).
In Fig. 3, we plot the received power weight Cl0 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, Cl0 decreases except for an arch near ze. Since the previous works show that detection probability of initial OAM state decreases as the relative beam width W (z)/r0 increases , 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 ze, Cl0 varies slowly. This mechanism proposes a novel method to alleviate the effect of turbulence on free space communication. In other words, one can manipulate ze according to the position of the receiver to obtain maximal Cl0 or alleviate the influence of atmospheric turbulence.
Figure 3(a) also shows that as Re(p) increases, ze increases, but meanwhile the value of Cl0 at ze decreases. For example, Re(p) increases from 2 to 8, ze expands from about 2.25km to about 3.0km while Cl0 at ze 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 Cl0 decreases as α increases. With different α, the curves also show an arch near ze, But with smaller α the curve is flatter and the arch is not clear. Figure 3(c) shows similar character that Cl0 decreases as increases. While turbulence is weak, e.g. , Cl0 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 or α than that in the weak perturbation case. Figures 3(b) and 3(c) also show that ze 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. Cl0 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 . 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 Cl0 descends very slowly around ze. 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 Cl of CiB mode. Cl decreases as Δl = |l − l0| increases. As z increases, Cl increases except a slight declining near ze=2.264km. The reason for this declining is that received power weight Cl0 reaches its maximum near ze 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.
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.
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.
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