## Abstract

The radial average-power distribution and normalized average power of orbital-angular-momentum (OAM) modes in a vortex Gaussian beam after passing through weak-to-strong atmospheric turbulence are theoretically formulated. Based on numerical calculations, the role of the intrinsic mode index, initial beam radius and turbulence strength in OAM-mode variations of a propagated vortex Gaussian beam is explored, and the validity of the pure-phase-perturbation approximation employed in existing theoretical studies is examined. Comparison between turbulence-induced OAM-mode scrambling of vortex Gaussian beams and that of either Laguerre-Gaussian (LG) beams or pure vortex beams has been made. Analysis shows that the normalized average power of OAM modes changes with increasing receiver-aperture size until it approaches a nearly stable value. For a receiver-aperture size of practical interest, OAM-mode scrambling is severer with a larger mode index or smaller initial beam radius besides stronger turbulence. Under moderate-to-strong turbulence condition, for two symmetrically-neighboring extrinsic OAM modes, the normalized average power of the one with an index closer to zero may be greater than that of the other one. The validity of the pure-phase-perturbation approximation is determined by the intrinsic mode index, initial beam radius and turbulence strength. It makes sense to jointly control the amplitude and phase of a fundamental Gaussian beam for producing an OAM-carrying beam.

© 2016 Optical Society of America

## 1. Introduction

To date, it has been widely known that the orbital angular momentum (OAM) of a light wave is associated with optical vortices, which are related to phase singularities in an optical field. From a quantum perspective, the number of OAM eigenstates that are available in optical fields is in principle unbounded. As a result, much interest in the use of OAM for increasing the capacity of free-space optical (FSO) communication systems has recently appeared. Generally speaking, there are two different ways in which OAM can be utilized: One is that information is encoded by the OAM states of photons carried by a vortex beam [1,2]; the other is that the OAM is treated as a degree of freedom of a vortex-beam wave for multiplexing [3–7 ]. So far, many experimental demonstrations related to OAM-based FSO communications under both laboratory and outdoor conditions have been reported [5–8 ]. Although, the compositions of OAM modes of a vortex beam remain unchanged as it propagates in a vacuum, this behavior is no longer maintained when there is atmospheric turbulence along the propagation path. The turbulence results in OAM-mode scattering and hence spreads the power contained in an OAM mode onto other neighboring OAM modes. The reason for this phenomenon is that turbulence-induced phase fluctuations distort the specific helical phase structure associated with a given OAM mode, thus leading to OAM-mode scrambling.

Turbulence-induced OAM-mode scrambling incurs variations in the compositions of OAM modes of a propagated vortex beam. This may result in deleterious effects on OAM-based FSO communications. Among different types of OAM-carrying beams, Laguerre-Gaussian (LG) beams have been under intensive study in the literature [2,9–12 ] partially because other kinds of OAM-carrying beams can theoretically be represented as a linear superposition of LG beams with various radial mode indices and the same OAM mode index [13]. On the other hand, many experimental demonstrations generated OAM-carrying beams by employing an optical device, e.g., a spatial light modulator, to impress a helical phase structure onto a fundamental Gaussian beam [5–8 ]. It is noted that the amplitude of an OAM-carrying LG beam at the transmitter plane indeed depends on the OAM mode index of the beam [9]. Accordingly, for producing a real LG beam in experiments, the phase and amplitude of the fundamental Gaussian beam should be controlled jointly [13]. With this in mind, the beam generated by only impressing a helical phase structure onto a fundamental Gaussian beam is in essence not an LG beam, which is here referred to as a vortex Gaussian beam in contrast with the LG beam. Now that vortex Gaussian beams can be readily produced in experiments, it is worthwhile examining the changes in the OAM modes of a propagated vortex Gaussian beam through atmospheric turbulence.

Numerous published works concerning the changes in OAM modes of a propagated beam carrying an optical vortex have treated the turbulence-induced distortion of the beam as a pure phase perturbation by making an assumption of weak fluctuations and ignoring both the scintillations and additional beam spreading due to the turbulence [2,13–16
]. However, the turbulence strength in many practical situations may go beyond the weak-turbulence regime. As a result, it makes sense to develop theoretical formulations that are applicable to both the weak- and strong-turbulence conditions. Moreover, like LG beams with a radial mode index of zero and an OAM mode index of *m*, later denoted by LG_{0}
* _{m}* beams, vortex Gaussian beams after propagating a distance in free space also feature an annular “ring” intensity profile. A natural question is which of these two types of beams is more resistant to the turbulence-induced OAM-mode scrambling. This is another issue that will be treated in this paper.

## 2. Theoretical formulations

At the transmitter plane, the field of a vortex Gaussian beam is defined by

**s**represents a two-dimensional vector in the transmitter plane at angle

*φ*,

*s*=|

**s**|,

*a*

_{0}is the on-axis amplitude of the field,

*w*

_{0}denotes the initial beam radius, and

*m*is the OAM mode index. We emphasize once again that the amplitude of a vortex Gaussian beam at the transmitter plane is independent of the OAM mode index, whereas that of an LG beam at the transmitter plane does depend on the OAM mode index. Gu and Gbur [17] studied the feasibility of measuring the refractive-index structure constant by propagating a vortex Gaussian beam with

*m*= 1 through atmospheric turbulence. When a vortex Gaussian beam, defined by Eq. (1) at the transmitter plane, propagates through atmospheric turbulence, new OAM modes with indices different from

*m*may arise in the beam field due to the turbulence-induced perturbations; in this sense, for convenience of description, in what follows, we will refer to the OAM mode with index

*m*as the intrinsic OAM mode and those OAM modes with other indices as the extrinsic OAM modes. In the absence of atmospheric turbulence, by employing the Huygens-Fresnel principle [18,19], the field of the vortex Gaussian beam at a receiver plane separated a distance

*L*from the transmitter plane can be formulated as follows:

**r**is a two-dimensional vector in the receiver plane at angle

*θ*,

*r*=|

**r**|, ${\alpha}_{0}=2/\left(k{w}_{0}^{2}\right)$,

*k*= 2

*π/λ*is the optical wavenumber with

*λ*being the wavelength,

*I*(·) is a modified Bessel function of the first kind and order

_{p}*p*. The superscript

*m*enclosed between parentheses on the quantities above shows the dependence of these quantities on the OAM mode index

*m*; similar notation will also be used below; for succinctness, we will not explain it anymore. To formulate the turbulence-induced changes in the OAM modes of a propagated vortex Gaussian beam in atmospheric turbulence, below we follow the approach used by Tyler and Boyd [16]. In the presence of atmospheric turbulence, the field of a propagated vortex Gaussian beam can be formally expressed by

*ψ*

^{(}

^{m}^{)}(

**r**) denotes the turbulence-induced complex phase perturbation. The term exp[

*ψ*

^{(}

^{m}^{)}(

**r**)] in Eq. (5) can be expanded into a complex-valued Fourier series

*ψ*

^{(}

^{m}^{)}(

*·*) is expressed in circular cylindrical coordinates. The field

*U*

^{(}

^{m}^{)}(

**r**,

*L*) can also be expanded into a complex-valued Fourier series

*U*

^{(}

^{m}^{)}(

*·*) is expressed in circular cylindrical coordinates. Substitution of Eq. (5) with exp[

*ψ*

^{(}

^{m}^{)}(

**r**)] given by Eq. (6) into Eq. (9) leads to

Equation (8) manifests that the field of a propagated vortex Gaussian beam at the receiver plane is a linear superposition of various OAM modes with different indices; the spatial irradiance distribution of OAM mode *l* is described by
${I}_{l}^{(m)}(r)={u}_{l}^{(m)}(r){u}_{l}^{(m)*}(r)$ with the asterisk denoting complex conjugate, which is a random quantity due to the stochastic nature of atmospheric turbulence. The normalized average power of OAM mode *l* contained in a propagated vortex Gaussian beam received by an aperture can be defined as follows:

*R*stands for the radius of the receiver aperture, and ${\overline{P}}^{(m)}(R)$ represents the total average power collected by the receiver which can be determined by integrating the average irradiance of the beam over the aperture. The normalized average power of OAM modes defined by Eq. (11) can be used to quantify the turbulence-induced changes in the compositions of OAM modes of a received vortex Gaussian beam; for a vortex Gaussian beam with intrinsic mode index

*m*, in an average sense, a larger ${\widehat{P}}_{m}^{(m)}(R)$ means more optical power remained in the intrinsic OAM mode, i.e., slighter OAM-mode scrambling; it is apparent that ${\widehat{P}}_{m}^{(m)}(R)=1$ if there is no atmospheric turbulence along the propagation path. Similar to [2], the integrand $r\u3008{I}_{l}^{(m)}(r)\u3009$ appearing in Eq. (12) indeed characterizes the radial average-power distribution of OAM mode

*l*; notice that, the superscript

*m*denotes the intrinsic OAM mode index of the beam. More specifically, the quantity $r\u3008{I}_{l}^{(m)}(r)\u3009$ can be interpreted as the average optical power of OAM mode

*l*in an annulus of radius

*r*; based on this quantity, one can examine the turbulence-induced variations in the compositions of OAM modes of a propagated vortex Gaussian beam in an annulus of radius

*r*at the receiver plane. Based on Eqs. (7) and (10), after some mathematical manipulation and a change of the region of integration, one finds

We note that Eq. (14) is the mutual coherence function (MCF) associated with two observation points (*r,θ*
_{1},*L*) and (*r,θ*
_{1} − *θ _{d},L*) for a vortex Gaussian beam with intrinsic OAM mode

*m*propagating in isotropic atmospheric turbulence, which can be referred to as the rotational MCF because it is actually determined by the angle difference

*θ*, irrespective of

_{d}*θ*

_{1}. The rotational MCF for a vortex Gaussian beam with intrinsic OAM mode

*m*after propagating through atmospheric turbulence can be developed to give

*κ*= |

**K**|,

*θ*is the azimuthal angle of the two-dimensional vector $\mathbf{K},{\rho}_{0}={\left(0.55{C}_{n}^{2}{k}^{2}L\right)}^{-3/5}$ is the spherical-wave coherence length due to Kolmogorov atmospheric turbulence, and ${C}_{n}^{2}$ is the refractive-index structure constant. The use of either the extended Huygens-Fresnel principle or the parabolic equation method together with the Markov approximation [19] can lead to the above results, which are in essence applicable to both weak- and strong-turbulence conditions. Furthermore, in arriving at Eq. (15), the quadratic approximation for the two-point spherical wave structure function has been employed. Under the far-field condition $2L/\left(k{w}_{0}^{2}\right)\gg 1$, i.e.,

_{κ}*W*

_{0}

*≈ w*

_{0}, Eq. (15) with

*m*= 1 is consistent with Eq. (11) of [17]. It is difficult to find an analytical solution in closed form for the double integral in Eq. (15); we numerically evaluate it in our subsequent treatment. Examination of Eqs. (15) and (19) reveals that

*μ*(

*κ*) acts like a truncating factor, and the integrand in Eq. (15) has an appreciable value only when

*κ*= |

**K**| ≲ 2

^{1}

^{/}^{2}/(

*πρ*

_{0}). In the special case of

*kr/*(2

*πL*) ≫ 2

^{1}

^{/}^{2}/(

*πρ*

_{0}), viz.,

*ρ*

_{0}≫

*L/*(

*kr*) with the constant multiplicative factors ignored, one can use the two approximations [

*k/*(2

*πL*)]

**r**

_{0}−

**K**

*≈*[

*k/*(2

*πL*)]

**r**

_{0}and [

*k/*(2

*πL*)](

**T**·

**r**

_{0}) −

**K**

*≈*[

*k/*(2

*πL*)](

**T**·

**r**

_{0}) to simplify Eq. (15) and find that

It should be pointed out that
${\widehat{I}}^{(m)}(r,L)$ is actually the irradiance of a propagated vortex Gaussian beam with intrinsic OAM mode *m* at point (**r**,*L*) in the absence of atmospheric turbulence. Furthermore, notice that there exists a formal resemblance between Eq. (7) of [2] and Eq. (20). By comparing these two expressions, we recognize the term *H* (·) in Eq. (20) as the rotational coherence function of the phase perturbations under the quadratic approximation in the situation that the turbulence is weak enough that the pure-phase-perturbation approximation can be properly employed. Hence, the substitution of Eq. (20) into Eq. (13) yields the average irradiance of OAM mode *l* under the condition that atmospheric turbulence is sufficiently weak. Equations (11) – (13) and (15) – (22) show the main theoretical formulations derived by us, which can be used to determine the variations in the OAM content of a propagated vortex Gaussian beam in atmospheric turbulence.

The OAM-mode density of a received vortex beam after propagating through atmospheric turbulence characterizes the ratio of the power contained in an OAM mode to that carried by the whole received beam field. In the literature [2,16], the average OAM-mode density has been treated in special cases where all the power carried by a vortex beam can be collected by a receiver aperture. In general cases, the average OAM-mode density can be defined as follows:

*R*= ∞; nevertheless, in the cases where this condition is not satisfied, it is too complicated to make a rigorous analysis of the deviation of ${\widehat{P}}_{l}^{(m)}(R)$ from ${\overline{p}}_{l}^{(m)}(R)$ because ${\overline{p}}_{l}^{(m)}(R)$ is defined as an ensemble average of the ratio of the random received power of an OAM mode to the random total received power of all OAM modes.

## 3. Numerical calculations and analysis

Here, we explore the turbulence-induced changes in the OAM modes of a propagated vortex Gaussian beam by numerical examples based on the theoretical models developed in the preceding section. To begin, we define two nondimensional parameters *q _{c}* =

*ρ*

_{0}

*/q*and

_{F}*q*=

_{w}*w*

_{0}

*/q*with

_{F}*q*= (

_{F}*L/k*)

^{1/2}denoting the Fresnel zone. These two nondimensional parameters will be used in the display and analysis of our calculation results. Doing so offers a little more generality to us than directly using the parameters

*ρ*

_{0},

*w*

_{0},

*k*and

*L*, which will become apparent later. In the literature, the Rytov variance is often used as a measure of turbulence strength, which can be expressed in terms of

*q*as follows: ${\sigma}_{R}^{2}={(1.23/0.55)}_{{q}_{c}}{}^{-5/3}$. Notice that, the Rytov variance is uniquely determined by

_{c}*q*. It is customary to describe the weak turbulence by the condition ${\sigma}_{R}^{2}<1$ ([18], p. 210), i.e., ${(1.23/0.55)}_{{q}_{c}}{}^{-5/3}<1$. As a result, we consider that

_{c}*q*> 1.62 corresponds to weak atmospheric turbulence and

_{c}*q*≤ 1.62 corresponds to moderate-to-strong atmospheric turbulence. The basic parameters used in the calculations below, associated with the Fresnel zone

_{c}*q*, are

_{F}*L*= 5 km and

*λ*= 800 nm.

Figure 1 shows the scaled radial power distribution of the intrinsic OAM mode of a propagated vortex Gaussian beam in the absence of atmospheric turbulence with various *q _{w}* and

*m*. It is seen from Fig. 1 that with the same intrinsic mode index

*m*, a larger

*q*corresponds to the case with the optical power concentrated in an annulus of smaller radius, meaning that the aperture size required for the receiver to collect most of the optical power can be smaller. The reason for this phenomenon is that diffraction-induced beam spreading is greater when

_{w}*q*becomes smaller. In contrast with Fig. 1, Figs. 2 and 3 illustrate the scaled radial average-power distribution of both the intrinsic and extrinsic OAM modes of a propagated vortex Gaussian beam through atmospheric turbulence with different

_{w}*q*,

_{c}*q*and

_{w}*m*. The turbulence strength corresponding to Fig. 2 is very weak, whereas that corresponding to Fig. 3 is moderately strong. As could have been expected, it is found by comparing the results shown in Figs. 1 – 3 that atmospheric turbulence spreads the optical power contained in the intrinsic OAM mode onto its neighboring extrinsic OAM modes, leading to a reduction and increase in the average optical power contained in the intrinsic and extrinsic OAM modes, respectively; moreover, the optical power leaking out from the intrinsic OAM mode grows with increasing turbulence strength.

Examination of Figs. 2 and 3 also reveals some aspects that cannot be found by intuition. By analyzing the variations in the curves with increasing intrinsic mode index *m*, one can find that, in both weak and moderately strong turbulence, the optical power leaking out from the intrinsic OAM mode becomes more significant as *m* turns larger; in other words, the strength of scattering of intrinsic OAM mode is dependent on *m*. Moreover, under the weak-turbulence condition (*q _{c}* = 6.5), for a propagated vortex Gaussian beam with intrinsic OAM mode

*m*, in the case of

*q*= 0.5, the curves corresponding to each pair of extrinsic OAM modes with indices

_{w}*l*=

*m ± δ*basically merge together (

_{l}*δ*= 1 or 2); that is, a pair of extrinsic OAM modes located symmetrically around the intrinsic OAM mode almost feature the same radial average-power distribution. However, when

_{l}*q*= 1 or 1.5, the said phenomenon is no longer perfectly maintained for a small intrinsic mode index

_{w}*m*(e.g.,

*m*= 1 or 2). In what follows, for description convenience, we will refer to two extrinsic OAM modes located symmetrically around the intrinsic OAM mode as a pair of symmetrically-neighboring extrinsic OAM modes. One can see from Figs. 2(d), 2(e), 2(g) and 2(h) that there is an observable difference in the shape of the radial average-power distribution between two extrinsic OAM modes

*l*=

*m ± δ*(

_{l}*δ*= 1 or 2); notice that the aforementioned difference in the case of

_{l}*l*=

*m ±*2 can be observed clearly if we display only these two curves in a figure. This finding indicates that

*q*has an impact on whether a pair of symmetrically-neighboring extrinsic OAM modes can feature the same radial average-power distribution. On the other hand, when the intrinsic mode index

_{w}*m*increases to 10, even for cases of

*q*= 1 and 1.5, two curves corresponding to a pair of symmetrically-neighboring extrinsic OAM modes do merge together. Consequently, a smaller

_{w}*q*or a greater

_{w}*m*leads to a more negligible difference in the radial average-power distribution between a pair of symmetrically-neighboring extrinsic OAM modes. Comparison of Fig. 2 with Fig. 3 shows that the difference in the radial average-power distribution between a pair of symmetrically-neighboring extrinsic OAM modes becomes more considerable when atmospheric turbulence becomes stronger in the case of a relatively small intrinsic mode index

*m*. It is seen clearly from Figs. 3(a), 3(d), 3(e), 3(g) and 3(h) that, for a pair of symmetrically-neighboring extrinsic OAM modes, at a relatively small

*r*, the average power of the extrinsic OAM mode with an index closer to zero is greater than that of the other one; however the situation is reversed at a relatively large

*r*. This fact manifests that it is both the intrinsic and extrinsic mode indices rather than the absolute value

*δ*of their difference that play a role in determining the radial behavior of the average-power spread from an intrinsic OAM mode onto its neighboring extrinsic OAM modes.

_{l}In practice, an aperture is generally used to collect a vortex-beam wave propagated through atmospheric turbulence to the receiver plane. To determine the normalized average power of both the intrinsic and extrinsic OAM modes of the wave field collected by a receiver aperture with radius *R*, we need to integrate the radial average-power distribution of corresponding OAM modes over the range 0 *≤ r ≤ R*. This operation is manifested by Eqs. (11) and (12). At this point, we examine how the receiver aperture size affects the normalized average power of received OAM modes. Figures 4 and 5 illustrate the normalized average power of OAM modes in a propagated vortex Gaussian beam through atmospheric turbulence in terms of the scaled receiver-aperture radius with various combinations of *q _{w}* and

*m*. It is seen from Figs. 4 and 5 that ${\widehat{P}}_{l}^{(m)}(R)$ initially changes rapidly with increasing

*R/q*until it arrives at a nearly stable value. The reason for this variation behavior can be found by recalling the definition of ${\widehat{P}}_{l}^{(m)}(R)$ given by Eq. (11) and analyzing the results shown by Figs. 2 and 3. Within a vicinity of

_{F}*r*= 0, the rate of increase in $r\u3008{I}_{l}^{(m)}(r)\u3009$ with enlarging

*r*depends on the index

*l*, resulting in that the same increment in

*R*does not lead to the same growth in ${P}_{l}^{(m)}(R)$ for OAM modes with different indices; on the other hand, the radial average-power distribution of OAM modes with various indices tends to approach zero when

*r*becomes sufficiently large, implying that ${P}_{l}^{(m)}(R)$ changes more and more slightly when

*R*increases beyond a large enough value for all OAM modes.

It should be pointed out that a large enough aperture is generally required for the receiver in practical applications to collect sufficient optical power. Consequently, below we mainly pay attention to the normalized average power of OAM modes within the range of *R* where it does not change rapidly. As expected, it is found by comparing Figs. 4 and 5 that stronger turbulence makes the OAM-mode scrambling more significant and hence causes a lower normalized average power of the intrinsic OAM mode. Furthermore, it is observed from Figs. 4 and 5 that a larger *q _{w}* leads to a greater normalized average power of the intrinsic OAM mode, implying that the intrinsic OAM mode is more resistant to turbulence-induced perturbations; nevertheless, increasing the intrinsic mode index

*m*curtails the normalized average power of the intrinsic OAM mode, meaning that the intrinsic OAM mode becomes more susceptible to turbulence-induced perturbations. Furthermore, some subtle differences between the OAM-mode scrambling under weak-turbulence condition and that under relatively-strong-turbulence condition can be found by making a thoroughgoing comparison of Fig. 4 with Fig. 5. When atmospheric turbulence is very weak, two symmetrically-neighboring extrinsic OAM modes

*l*=

*m ±*1 almost have the same normalized average power; however, in relatively strong atmospheric turbulence, this behavior no longer holds, and the normalized average power of the extrinsic OAM mode

*l*=

*m*− 1 even reaches the level of that of the intrinsic one. Hence, the average spread of optical power from an intrinsic OAM mode onto its two symmetrically-neighboring extrinsic OAM modes may be different from each other under strong-turbulence conditions.

To gain a deeper insight into the average-power spread of intrinsic OAM mode caused by atmospheric turbulence, in Figs. 6 and 7, we plot the normalized average power of successive OAM modes in a propagated vortex Gaussian beam through atmospheric turbulence in terms of the mode index *l* with various *q _{c}* and

*m*. For an OAM-based multiplexing FSO communication system, the results shown by Figs. 6 and 7 can be used to characterize the average crosstalk observed on sub-channels associated with OAM modes

*l*≠

*m*resulting from the transmitted OAM mode

*m*. Notice that, for the cases of

*q*= 6.5, 3.5 and 2.5, the turbulence can generally be considered weak; on the other hand, when

_{c}*q*= 1.5 and 1, the turbulence should be thought of as relatively strong. It is seen clearly from Figs. 6 and 7 that more neighboring extrinsic OAM modes have a value of normalized average power which cannot be neglected as compared with the one of the intrinsic OAM mode when the turbulence gets stronger or the intrinsic mode index grows larger. Under weak-turbulence conditions, ${\widehat{P}}_{l}^{(m)}(R)$ remains peaked at the intrinsic OAM mode

_{c}*m*. However, the normalized average power of the extrinsic OAM mode

*l*=

*m−*1 may increase to almost the same or even higher level of that of the intrinsic one in the case of relatively strong atmospheric turbulence. Furthermore, for a relatively small intrinsic mode index

*m*, the normalized average power of OAM modes gradually gets asymmetric with respect to the intrinsic OAM mode as the turbulence strength increases; in fact, even for weak turbulence with

*q*= 3.5 and 2.5, this asymmetry can be observed clearly. Based on the above observations, a statement can be made that the average-power spread from an intrinsic OAM mode onto its neighboring extrinsic OAM modes is dependent on both the intrinsic and extrinsic mode indices instead of only on the OAM mode separation

_{c}*δ*= |

_{l}*m*−

*l*|. This is consistent with the analysis of Figs. 2 and 3.

Many existing results with respect to the turbulence-induced changes in OAM modes of a propagated vortex beam have been obtained by making use of the pure-phase-perturbation approximation [2,13–16
]; it has been claimed that this approximation is valid under the weak-turbulence condition. As mentioned previously, if Eq. (20) is introduced into Eq. (13), one can yield the version of the theoretical models for a propagated vortex Gaussian beam under the pure-phase-perturbation approximation. Now, we make comparisons of the radial average-power distribution of OAM modes calculated according to Eqs. (13) – (19) with those obtained based on the pure-phase-perturbation approximation. Doing so permits us to briefly examine the variations in the accuracy of the theoretical models based on the pure-phase-perturbation approximation with varying *m*, *q _{w}* and

*q*. Figure 8 illustrates the radial average-power distribution of OAM modes of a propagated vortex Gaussian beam with

_{c}*q*= 6.5 fixed and various combinations of

_{c}*q*,

_{w}*m*and

*l*. For comparison purposes, the results obtained based on the pure-phase-perturbation approximation are also presented in Fig. 8. It is found from Fig. 8 that the use of the pure-phase-perturbation approximation does not incur discernable loss of accuracy when

*q*= 0.5, whereas the loss of accuracy becomes apparent when

_{w}*q*gets relatively large (e.g.,

_{w}*q*= 1 and 1.5). On the other hand, Fig. 9 shows the radial average-power distribution of OAM modes of a propagated vortex Gaussian beam with

_{w}*q*= 1 fixed and various combinations of

_{w}*q*,

_{c}*m*and

*l*. Similar to Fig. 8, the results calculated based on the pure-phase-perturbation approximation are also given in Fig. 9. As expected, the degree of accuracy of the theoretical models based on the pure-phase-perturbation approximation decreases with increasing turbulence strength. It is also seen from Fig. 9 that with the same turbulence strength, the degree of accuracy of the theoretical models based on the pure-phase-perturbation approximation becomes better when the intrinsic mode index

*m*grows larger. Based on the above observations, it can be found that the validity of the pure-phase-perturbation approximation depends on both

*q*and

_{w}*m*besides

*q*.

_{c}In what follows, we elucidate why the intrinsic mode index *m* and scaled initial beam radius *q _{w}* play an important role in determining the normalized average power of both the intrinsic and extrinsic OAM modes of a propagated vortex Gaussian beam through atmospheric turbulence. To better describe the underlying physics, the theoretical models under the pure-phase-perturbation approximation will be employed. The case of strong turbulence can be considered by using the multiple-phase-screen concept [20]. Introduction of Eq. (12) into Eq. (11), with
$\u3008{I}_{l}^{(m)}(r)\u3009$ obtained by substituting Eq. (20) into Eq. (13), gives us

*ω*

^{(}

^{m}^{)}(

*r,L*) can be regarded as the normalized radial power distribution of a propagated vortex beam at the receiver plane in the absence of atmospheric turbulence,

*C*(

_{t}*R*) denotes the total optical power collected by the receiver aperture. In fact, ${\widehat{P}}_{l}^{(m)}(R)$ given by Eq. (24) can be viewed as a weighted integration of

*ω*

^{(}

^{m}^{)}(

*r,L*) with a

*r*-dependent weighting coefficient denoted by Θ(

*r,m − l*); it is apparent that

*ω*

^{(}

^{m}^{)}(

*r,L*) is weighted more heavily at a radial distance

*r*where Θ(

*r,m − l*) achieves a greater value. Notice that, the quantity

*ω*

^{(}

^{m}^{)}(

*r,L*) is equivalent to the term “ ${\left|\tilde{R}(r,z)\right|}^{2}r$” in Eq. (9) of [2], where we use the notation “ $\tilde{R}(r,z)$” for the normalized radial LG beam profile to avoid confusion with the receiver-aperture radius denoted by

*R*in this paper. Figure 10 shows Θ(

*r,m − l*) as a function of

*r/ρ*

_{0}, where both the quadratic-approximation-based results with

*H*(

*r,θ*) given by Eq. (22) and the accurate results with

_{d}*H*(

*r,θ*) = exp{−6.88

_{d}*×*2

^{2}

^{/}^{3}[

*r/*(2.1

*ρ*

_{0})]

^{5}

^{/}^{3}|sin(

*θ*/2)|

_{d}^{5}

^{/}^{3}} [2] are presented. It can be seen from Fig. 10 that there is only very small difference between the quadratic-approximation-based results and the accurate ones.

Note that
${\widehat{P}}_{l}^{(m)}(\infty )={\overline{p}}_{l}^{(m)}(\infty )$. It has been claimed in [2] that
${\overline{p}}_{l}^{(m)}(\infty )$ does not depend on the intrinsic mode index *m* with “
$\tilde{R}(r,z)$” fixed. This seems to be in contradistinction to our previous findings. Here, *R* = ∞ corresponds to the situation in which almost all the energy carried by the beam is collected by a large enough aperture. Now, we deal with this contradiction. Examination of Fig. 10 reveals that Θ(*r,m − l*) achieves its peak value at *r* = 0 when *l* = *m*, but its peak moves further away from *r* = 0 when *δ _{l}* = |

*m – l*| becomes greater. With this in mind, it is straightforward to find that the average unscattered portion of the power contained in the initial intrinsic OAM mode becomes smaller if the peak of

*ω*

^{(}

^{m}^{)}(

*r,L*) moves further away from

*r*= 0, which in turn means that the average scattered portion of the power contained in the initial intrinsic OAM mode by atmospheric turbulence gets larger. Figure 11 shows

*ω*

^{(}

^{m}^{)}(

*r,L*) in terms of

*r/ρ*

_{0}with various combinations of

*q*and

_{w}*m*for the cases of vortex Gaussian beams, pure vortex beams and LG

_{0}

*beams. The pure vortex beam, i.e., truncated plane-wave vortex beam, is defined by Eq. (1) of [16]; different from the treatment in [16], here we take free-space diffraction into account. By carefully comparing Figs. 11(a) and 11(b), one can find that unlike a vortex Gaussian beam,*

_{m}*ω*

^{(}

^{m}^{)}(

*r,L*) of a pure vortex beam has an oscillating tail due to free-space diffraction; with the same parameter values,

*ω*

^{(}

^{m}^{)}(

*r,L*) of a pure vortex beam is peaked at a slightly greater radial distance and has a sharper peak than that associated with a vortex Gaussian beam; this behavior becomes more apparent with a smaller

*q*or larger

_{w}*m*. By inspecting the results shown in Fig. 11, one can infer that the average power scattered form intrinsic OAM mode

*m*into extrinsic OAM modes

*l*=

*m ± δ*will become more significant if

_{l}*m*becomes larger, because a larger

*m*means

*ω*

^{(}

^{m}^{)}(

*r,L*) is peaked at a greater radial distance. As a result, ${\overline{p}}_{l}^{(m)}(\infty )$ in the cases of both

*l*=

*m*and

*l*≠

*m*is dependent on the intrinsic mode index

*m*. The fundamental reason for the defective deduction obtained in [2] is that the author neglected the fact that the field distribution of propagated LG beams in the absence of atmospheric turbulence depends on the intrinsic mode index

*m*, resulting in that

*ω*

^{(}

^{m}^{)}(

*r,L*), i.e., “| ${\left|\tilde{R}(r,z)\right|}^{2}r$” in [2], has

*m*-dependence; this can be verified by inspecting Fig. 11(c). By performing numerical propagation of LG

_{0}

*beams in atmospheric turbulence, Anguita*

_{m}*et al.*[4] also questioned the defective deduction in [2]; nevertheless, they did not give physical elucidation similar to that presented above. Moreover, it is observed from Fig. 11 that a smaller

*q*makes the peak of

_{w}*ω*

^{(}

^{m}^{)}(

*r,L*) move further away from

*r*= 0. For the same reason as mentioned above, the scaled initial beam radius

*q*plays a role in determining ${\overline{p}}_{l}^{(m)}(\infty )$. By making comparison of Figs. 11(a) – 11(c), we find that, with the same

_{w}*q*and

_{w}*m*,

*ω*

^{(}

^{m}^{)}(

*r,L*) associated with either the vortex Gaussian beams or pure vortex beams is peaked at a larger radial distance and has a tail decaying slower than that associated with the LG

_{0}

*beams, implying that the OAM carried by either the vortex Gaussian beams or pure vortex beams is more susceptible to atmospheric turbulence than that carried by the LG*

_{m}_{0}

*beams. Severer OAM-mode scrambling generally means greater crosstalk in an OAM-based multiplexing FSO communication system. Hence, it is easy to find that the turbulence-induced average crosstalk in an OAM-based multiplexing FSO communication system employing LG*

_{m}_{0}

*beams is smaller than the one in that using either vortex Gaussian beams or pure vortex beams. It should be pointed out that*

_{m}*C*(∞) equals the total transmitted power; thus, Fig. 11 actually shows the radial power distribution of vortex beams scaled by the total transmitted power. By recognizing this fact, as far as the power loss is concerned, it is obvious that when the receiver aperture cannot be large enough, it is better to use LG

_{t}_{0}

*beams than either vortex Gaussian beams or pure vortex beams in practical systems, because the power of LG*

_{m}_{0}

*beams at the receiver plane is concentrated within a smaller circle than that of either vortex Gaussian beams or pure vortex beams. Consequently, in practical OAM-based applications, it does make sense to generate an LG*

_{m}_{0}

*beam by jointly controlling the amplitude and phase of a fundamental Gaussian beam.*

_{m}It is noted that Θ(*r,m − l*) is independent of the intrinsic mode index *m* and depends only on *δ _{l}* = |

*m*−

*l*|. Consequently, based on Eq. (24), one finds that the normalized average power of the extrinsic mode

*l*=

*m − δ*is equal to that of the extrinsic mode

_{l}*l*=

*m*+

*δ*for a given

_{l}*m*. The results shown by Figs. 6(a), 6(f), 7(a) and 7(f) associated with

*q*= 6.5 almost agree with this prediction; however, it is not supported by our calculation results with a greater turbulence strength (see, e.g., Figs. 6(b) – 6(e) and 6(g) – 6(j)). In fact, when atmospheric turbulence becomes relatively strong, one can conceptually employ multiple phase screens distributed over the propagation path to model the turbulence, with each phase screen behaving in the same manner as that used in the pure-phase-perturbation approximation; in this sense, for any phase screen located somewhere on the propagation path, all the OAM modes in the incident beam may be viewed as intrinsic modes even though the beam initially contains only one OAM mode. According to the preceding analysis, it is apparent that the fraction of the power remaining in an intrinsic OAM mode passing through a phase screen is larger if its index is closer to zero. With this in mind, it is straightforward to find that, after the beam propagates through a series of phase screens, the fraction of the average power remaining in OAM mode

_{c}*l*=

*m − δ*will be greater than that in OAM mode

_{l}*l*=

*m*+

*δ*. This gives a plausible physical explanation of why two symmetrically-neighboring extrinsic OAM modes in a propagated vortex beam may have different values of normalized average power when atmospheric turbulence gets strong. The experimental results given in [6] also show that the average power in OAM mode

_{l}*l*= 3

*− δ*is different from that in OAM mode

_{l}*l*= 3 +

*δ*for

_{l}*δ*= 1,2,⋯,5 (see the lower right subplot of Fig. 5 in [6]). We note that the mentioned results in [6] are not completely consistent with our above statements; this may be caused by the underlying difference between the phase-screen-plate-based turbulence emulator and general extended atmospheric turbulence; strictly speaking, when the pure-phase-perturbation approximation is not valid, extended atmospheric turbulence should be viewed as a series of phase screens distributed separately over the propagation path instead of one phase screen located at a fixed position.

_{l}Theoretically, adaptive optics (AO) can be used to compensate for turbulence-induced OAM-mode scrambling. The main challenges facing the AO-based turbulence compensation of OAM beams are the existence of phase singularities in the beam fields. Recently, several reports about turbulence compensation of OAM beams have been published [21,22]; the said challenges are overcome by employing different approaches. We note that the instantaneous spatial irradiance distribution of OAM modes depends on
${u}_{l}^{(m)}(r)$ given by Eq. (10), which is indeed a function of the complex phase perturbation *ψ*
^{(}
^{m}^{)} (**r**). Because *ψ*
^{(}
^{m}^{)} (**r**) involves both amplitude and phase perturbations, current AO-based phase correction carried out at the receiver end is somewhat difficult to completely eliminate turbulence-induced deleterious effects on OAM beam propagation. As a final comment, although fixed values of *L* and *λ* have been used in our numerical calculations, the figures presented above can also be applied to cases where *L* and *λ* are specified as other values for the reason that nondimensional parameters have been employed in the display of the results.

## 4. Conclusions

In this paper, theoretical formulations for both the radial average-power distribution and normalized average power of OAM modes in a propagated vortex Gaussian beam through weak-to-strong atmospheric turbulence have been developed. With the help of the formulations obtained, we have explored the effects of the intrinsic mode index, initial beam radius and turbulence strength on the changes in OAM modes of a propagated vortex Gaussian beam, and have also given the corresponding physical explanations. For a vortex Gaussian beam propagating in atmospheric turbulence, what has been found includes the following. The receiver-aperture size has an important impact on the normalized average power of OAM modes; as the aperture radius varies from zero to a large enough value, the normalized average power of OAM modes initially changes rapidly until it reaches a nearly stable value. Within the range where the normalized average power of OAM modes does not varies rapidly, stronger turbulence leads to severer OAM-mode scrambling; a larger initial beam radius makes the intrinsic OAM mode more resistant to atmospheric turbulence; intrinsic OAM modes with a larger index are more susceptible to turbulence-induced perturbations; when turbulence is relatively strong, a pair of symmetrically-neighboring extrinsic OAM modes may have different values of normalized average power; more specifically, the one with an index closer to zero may have a greater normalized average power than the other one.

Based on numerical examples, the accuracy of the pure-phase-perturbation approximation used in earlier investigations has also been examined. It has been found that, besides the turbulence strength, both the initial beam radius and intrinsic OAM mode index play a role in determining the validity of this approximation. Moreover, comparisons of turbulence-induced OAM-mode scrambling of vortex Gaussian beams with that of pure vortex beams and LG_{0}
* _{m}* beams have been made. It has been revealed that OAM-carrying beams generated by only impressing a helical phase structure onto a fundamental Gaussian beam or truncated plane-wave beam are not the optimal ones for practical OAM-based FSO communication systems impaired by atmospheric turbulence, and the control of the initial beam-field amplitude is useful for weakening the OAM-mode scrambling.

## Acknowledgments

The authors are very grateful to the reviewers for valuable comments. This work was supported by the National Natural Science Foundation of China (NSFC) (61007046, 61275080 and 61475025), the Natural Science Foundation of Jilin Province of China (20150101016JC), and the Specialized Research Fund for the Doctoral Program of Higher Education of China (20132216110002).

## References and links

**1. **Y. Zhang, I. B. Djordjevic, and X. Gao, “On the quantum-channel capacity for orbital angular momentum-based free-space optical communications,” Opt. Lett. **37**(15), 3267–3269 (2012). [CrossRef] [PubMed]

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

**3. **J. H. Shapiro, S. Guha, and B. I. Erkmen, “Ultimate channel capacity of free-space optical communications,” J. Opt. Netw. **4**(8), 501–516 (2005). [CrossRef]

**4. **J. A. Anguita, M. A. Neifeld, and B. V. Vasic, “Turbulence-induced channel crosstalk in an orbital angular momentum-multiplexed free-space optical link,” Appl. Opt. **47**(13), 2414–2428 (2008). [CrossRef] [PubMed]

**5. **J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photon. **6**(7), 488–496 (2012). [CrossRef]

**6. **Y. Ren, H. Huang, G. Xie, N. Ahmed, Y. Yan, B. I. Erkmen, N. Chandrasekaran, M. P. J. Lavery, N. K. Steinhoff, M. Tur, S. Dolinar, M. Neifeld, M. J. Padgett, R. W. Boyd, J. H. Shapiro, and A. E. Willner, “Atmospheric turbulence effects on the performance of a free space optical link employing orbital angular momentum multiplexing,” Opt. Lett. **38**(20), 4062–4065 (2013). [CrossRef] [PubMed]

**7. **G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, V. Pas’ko, S. M. 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]

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

**9. **A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photon. **3**(2), 161–204 (2011). [CrossRef]

**10. **F. Wang, Y. Cai, and O. Korotkova, “Partially coherent standard and elegant Laguerre-Gaussian beams of all orders,” Opt. Express **17**(25), 22366–22379 (2009). [CrossRef]

**11. **V. P. Aksenov, F. Y. Kanev, and C. E. Pogutsa, “Spatial coherence, mean wave tilt, and mean local wave-propagation vector of a Laguerre-Gaussian beam passing through a random phase screen,” Atm. Ocean. Opt. **23**(5), 344–352 (2010). [CrossRef]

**12. **G. Gbur and R. K. Tyson, “Vortex beam propagation through atmospheric turbulence and topological charge conservation,” J. Opt. Soc. Am. A **25**(1), 225–230 (2008). [CrossRef]

**13. **A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photon. **7**(1), 66–106 (2015).

**14. **Y. Zhu, X. Liu, J. Gao, Y. Zhang, and F. 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]

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

**16. **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]

**17. **Y. Gu and G. Gbur, “Measurement of atmospheric turbulence strength by vortex beam,” Opt. Commun. **283**(7), 1209–1212 (2010). [CrossRef]

**18. **L. C. Andrews and R. L. Phillips, *Laser Beam Propagation through Random Media*, 2nd ed. (SPIE, 2005). [CrossRef]

**19. **M. Charnotskii, “Extended Huygens-Fresnel principle and optical waves propagation in turbulence: discussion,” J. Opt. Soc. Am. A **32**(7), 1357–1365 (2015). [CrossRef]

**20. **F. S. Roux, “Entanglement evolution of twisted photons in strong atmospheric turbulence,” Phys. Rev. A **92**(1), 012326 (2015). [CrossRef]

**21. **Y. Ren, G. Xie, H. Huang, N. Ahmed, Y. Yan, L. Li, C. Bao, M. P. J. Lavery, M. Tur, M. A. Neifeld, R. W. Boyd, J. H. Shapiro, and A. E. Willner, “Adaptive-optics-based simultaneous pre- and post-turbulence compensation of multiple orbital-angular-momentum beams in a bidirectional free-space optical link,” Optica **1**(6), 376–382 (2014). [CrossRef]

**22. **G. Xie, Y. Ren, H. Huang, M. P. J. Lavery, N. Ahmed, Y. Yan, C. Bao, L. Li, Z. Zhao, Y. Cao, M. Willner, M. Tur, S. J. Dolinar, R. W. Boyd, J. H. Shapiro, and A. E. Willner, “Phase correction for a distorted orbital angular momentum beam using a Zernike polynomials-based stochastic-parallel-gradient-descent algorithm,” Opt. Lett. **40**(7), 1197–1200 (2015). [CrossRef] [PubMed]