Changes in orbital-angular-momentum modes of a propagated vortex Gaussian beam through weak-to-strong atmospheric turbulence

Chunyi Chen; Huamin Yang; Shoufeng Tong; Yan Lou

doi:10.1364/OE.24.006959

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the compositions of OAM modes of a vortex beam remain unchanged as it propagates in a vacuum, this behavior is no longer maintained when there is atmospheric turbulence along the propagation path.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 LG₀ _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

U_{0}^{(m)} (s, 0) = a_{0} \exp (- \frac{s^{2}}{w_{0}^{2}}) \exp (i m φ),

where 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:

U_{0}^{(m)} (r, L) = a_{0} C_{0}^{(m)} A^{(m)} (r) \exp (i m θ)

with

C_{0}^{(m)} = \frac{\sqrt{π} k^{2}}{8 L^{2}} {(- i)}^{m + 1} \exp (i k L) {[- \frac{i k}{2 L} (1 + i α_{0} L)]}^{- 3 / 2},

\begin{array}{l} A^{(m)} (r) = r \exp [\frac{i k r^{2}}{2 L} - \frac{i k r^{2}}{4 L (1 + i α_{0} L)}] \\ \times [I_{m / 2 - 1 / 2} (\frac{i k r^{2}}{4 L (1 + i α_{0} L)}) - I_{m / 2 + 1 / 2} (\frac{i k r^{2}}{4 L (1 + i α_{0} L)})], \end{array}

where r is a two-dimensional vector in the receiver plane at angle θ, r =|r|,

α_{0} = 2 / (k w_{0}^{2})

, k = 2π/λ is the optical wavenumber with λ being the wavelength, I_p (·) is a modified Bessel function of the first kind and order 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

U^{(m)} (r, L) = a_{0} C_{0}^{(m)} A^{(m)} (r) \exp (i m θ) \exp [ψ^{(m)} (r)],

where 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

\exp [ψ^{(m)} (r, θ)] = \sum_{n = - \infty}^{\infty} c_{n}^{(m)} (r) \exp (i n θ)

with

c_{n}^{(m)} (r) = \frac{1}{2 π} \int_{0}^{2 π} d θ \exp [ψ^{(m)} (r, θ)] \exp (- i n θ),

where 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)} (r, θ, L) = \sum_{l = - \infty}^{\infty} u_{l}^{(m)} (r) \exp (i l θ)

with

u_{l}^{(m)} (r) = \frac{1}{2 π} \int_{0}^{2 π} d θ U^{(m)} (r, θ, L) \exp (- i l θ)

where 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

u_{l}^{(m)} (r) = a_{0} C_{0}^{(m)} A^{(m)} (r) c_{l - m}^{(m)} (r) .

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:

{\hat{P}}_{l}^{(m)} (R) = \frac{〈 P_{l}^{(m)} (R) 〉}{{\bar{P}}^{(m)} (R)}

with

〈 P_{l}^{(m)} (R) 〉 = 2 π \int_{0}^{R} d r r 〈 I_{l}^{(m)} (r) 〉,

where the angle brackets denote an ensemble average, R stands for the radius of the receiver aperture, and

{\bar{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

{\hat{P}}_{m}^{(m)} (R)

means more optical power remained in the intrinsic OAM mode, i.e., slighter OAM-mode scrambling; it is apparent that

{\hat{P}}_{m}^{(m)} (R) = 1

if there is no atmospheric turbulence along the propagation path. Similar to [2], the integrand

r 〈 I_{l}^{(m)} (r) 〉

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 〈 I_{l}^{(m)} (r) 〉

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

〈 I_{l}^{(m)} (r) 〉 = \frac{1}{2 π} \int_{0}^{2 π} d θ_{d} Γ_{2}^{(m)} (r, θ_{d}, L) \exp (- i l θ_{d}),

where

Γ_{2}^{(m)} (r, θ_{d}, L) = 〈 U^{(m)} (r, θ_{1}, L) U^{(m) *} (r, θ_{1} - θ_{d}, L) 〉 .

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_d, irrespective of 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

\begin{array}{l} Γ_{2}^{(m)} (r, θ_{d}, L) = a_{0}^{2} {(\frac{k}{2 π L})}^{2} \exp [- \frac{3 r^{2}}{2 ρ_{0}^{2}} (1 - \cos θ_{d})] π ρ_{0}^{2} \\ \times \int \int_{- \infty}^{\infty} V_{1}^{(m)} (T \cdot r_{0}, K) V_{1}^{(m) *} (r_{0}, K) μ (κ) d^{2} K \end{array}

with

T = [\begin{matrix} \cos θ_{d} & - \sin θ_{d} \\ \sin θ_{d} & \cos θ_{d} \end{matrix}], r_{0} = [\begin{matrix} r \\ 0 \end{matrix}],

V_{1}^{(m)} (r, K) = {\tilde{V}}_{0}^{(m)} (\frac{k r}{2 π L} - K) \exp (i π K \cdot r)

\begin{matrix} {\tilde{V}}_{0}^{(m)} (K) = \frac{{(- i)}^{m} π^{5 / 2}}{2} κ W_{0}^{3} \exp (- \frac{π^{2} κ^{2} W_{0}^{2}}{2}) \exp (i m θ_{κ}) \\ \times [I_{m / 2 - 1 / 2} (\frac{π^{2} κ^{2} W_{0}^{2}}{2}) - I_{m / 2 + 1 / 2} (\frac{π^{2} κ^{2} W_{0}^{2}}{2})], \end{matrix}

μ (κ) = \exp (- π^{2} ρ_{0}^{2} κ^{2}),

where

W_{0} = {[1 / w_{0}^{2} - i k / (2 L)]}^{- 1 / 2}

, κ = |K|, θ_κ is the azimuthal angle of the two-dimensional vector

K, ρ_{0} = {(0.55 C_{n}^{2} k^{2} L)}^{- 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

2 L / (k w_{0}^{2}) ≫ 1

, i.e., 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| ≲ 2¹ ^/ ²/(πρ ₀). In the special case of kr/(2πL) ≫ 2¹ ^/ ²/(πρ ₀), viz., ρ ₀ ≫ L/(kr) with the constant multiplicative factors ignored, one can use the two approximations [k/(2πL)]r ₀ − K ≈ [k/(2πL)]r ₀ and [k/(2πL)](T · r ₀) − K ≈ [k/(2πL)](T · r ₀) to simplify Eq. (15) and find that

Γ_{2}^{(m)} (r, θ_{d}, L) = {\hat{I}}^{(m)} (r, L) \exp (i m θ_{d}) H (r, θ_{d})

with

{\hat{I}}^{(m)} (r, L) = a_{0}^{2} C_{0}^{(m)} C_{0}^{(m) *} A^{(m)} (r) A^{(m) *} (r),

H (r, θ_{d}) = \exp [- \frac{4 r^{2} \sin^{2} (θ_{d} / 2)}{ρ_{0}^{2}}] .

It should be pointed out that ${\hat{I}}^{(m)} (r, L)$ 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:

{\bar{p}}_{l}^{(m)} (R) = 〈 \frac{P_{l}^{(m)} (R)}{\sum_{l^{'} = - \infty}^{\infty} P_{l^{'}}^{(m)} (R)} 〉 .

Because, in general, both the numerator and denominator in Eq. (23) are random quantities and they may be partially correlated, it is very difficult to develop an analytical expression for Eq. (23). Conceptually, there is a degree of resemblance between

{\hat{P}}_{l}^{(m)} (R)

and

{\bar{p}}_{l}^{(m)} (R)

. However, these two quantities are basically not identical. On the other hand, it is noted that under the condition that the total power collected by a receiver aperture remains nearly constant, the relation

{\hat{P}}_{l}^{(m)} (R) \approx {\bar{p}}_{l}^{(m)} (R)

should hold true. In fact, it is evident that the said condition is satisfied if the aforementioned pure-phase-perturbation approximation is applicable or R = ∞; nevertheless, in the cases where this condition is not satisfied, it is too complicated to make a rigorous analysis of the deviation of

{\hat{P}}_{l}^{(m)} (R)

from

{\bar{p}}_{l}^{(m)} (R)

because

{\bar{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_F and q_w = w ₀ /q_F with 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_c as follows: $σ_{R}^{2} = {(1.23 / 0.55)}_{q_{c}}^{- 5 / 3}$ . Notice that, the Rytov variance is uniquely determined by q_c. It is customary to describe the weak turbulence by the condition $σ_{R}^{2} < 1$ ([18], p. 210), i.e., ${(1.23 / 0.55)}_{q_{c}}^{- 5 / 3} < 1$ . As a result, we consider that q_c > 1.62 corresponds to weak atmospheric turbulence and q_c ≤ 1.62 corresponds to moderate-to-strong atmospheric turbulence. The basic parameters used in the calculations below, associated with the Fresnel zone q_F, are 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_w 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 q_w 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 q_c, q_w and 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.

Fig. 1 Scaled radial power distribution of the intrinsic OAM mode of a propagated vortex Gaussian beam in the absence of atmospheric turbulence. $I_{m}^{(m)} (r)$ is not a random quantity.

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Fig. 2 Scaled radial average-power distribution of OAM modes of a propagated vortex Gaussian beam through atmospheric turbulence with q_c = 6.5 fixed.

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Fig. 3 Scaled radial average-power distribution of OAM modes of a propagated vortex Gaussian beam through atmospheric turbulence with q_c = 1.5 fixed.

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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_w = 0.5, the curves corresponding to each pair of extrinsic OAM modes with indices l = m ± δ_l basically merge together (δ_l = 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 q_w = 1 or 1.5, the said phenomenon is no longer perfectly maintained for a small intrinsic mode index 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_l = 1 or 2); notice that the aforementioned difference in the case of l = m ± 2 can be observed clearly if we display only these two curves in a figure. This finding indicates that q_w 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 m increases to 10, even for cases of q_w = 1 and 1.5, two curves corresponding to a pair of symmetrically-neighboring extrinsic OAM modes do merge together. Consequently, a smaller q_w or a greater 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_l 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.

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 ${\hat{P}}_{l}^{(m)} (R)$ initially changes rapidly with increasing R/q_F until it arrives at a nearly stable value. The reason for this variation behavior can be found by recalling the definition of ${\hat{P}}_{l}^{(m)} (R)$ given by Eq. (11) and analyzing the results shown by Figs. 2 and 3. Within a vicinity of r = 0, the rate of increase in $r 〈 I_{l}^{(m)} (r) 〉$ 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.

Fig. 4 Normalized average power of OAM modes in a propagated vortex Gaussian beam through atmospheric turbulence in terms of the scaled receiver-aperture radius with q_c = 6.5 fixed. The curves associated with l = m correspond to intrinsic OAM modes and the other ones to extrinsic OAM modes.

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Fig. 5 Normalized average power of OAM modes in a propagated vortex Gaussian beam through atmospheric turbulence in terms of the scaled receiver-aperture radius with q_c = 1.5 fixed. The curves associated with l = m correspond to intrinsic OAM modes and the other ones to extrinsic OAM modes.

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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_c = 6.5, 3.5 and 2.5, the turbulence can generally be considered weak; on the other hand, when q_c = 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, ${\hat{P}}_{l}^{(m)} (R)$ remains peaked at the intrinsic OAM mode 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_c = 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 δ_l = |m − l|. This is consistent with the analysis of Figs. 2 and 3.

Fig. 6 Normalized average power of both intrinsic and extrinsic OAM modes in a propagated vortex Gaussian beam through atmospheric turbulence with q_w = 1 fixed. The shaded circles with red color represent intrinsic modes and those with blue color denote extrinsic modes. The receiver-aperture radius R is specified as the root-mean-square radius of a propagated vortex Gaussian beam with intrinsic mode index m at the receiver plane in the absence of atmospheric turbulence.

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Fig. 7 Average density of both intrinsic and extrinsic OAM modes in a propagated vortex Gaussian beam through atmospheric turbulence with q_w = 1 fixed. The shaded circles with red color represent intrinsic modes and those with blue color denote extrinsic modes. The receiver-aperture radius R is specified as the root-mean-square radius of a propagated vortex Gaussian beam with intrinsic mode index m at the receiver plane in the absence of atmospheric turbulence.

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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_c. Figure 8 illustrates the radial average-power distribution of OAM modes of a propagated vortex Gaussian beam with q_c = 6.5 fixed and various combinations of 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_w = 0.5, whereas the loss of accuracy becomes apparent when q_w gets relatively large (e.g., q_w = 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 q_w = 1 fixed and various combinations of 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_w and m besides q_c.

Fig. 8 Scaled radial average-power distribution of OAM modes of a propagated vortex Gaussian beam through atmospheric turbulence with q_c = 6.5 fixed. The curves annotated with “Numerical” in the legend represent the results calculated according to Eqs. (13) – (19), and those with “PPP approx.” in the legend denote the results obtained based on the pure-phase-perturbation approximation.

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Fig. 9 Scaled radial average-power distribution of OAM modes of a propagated vortex Gaussian beam through atmospheric turbulence with q_w = 1 fixed. The curves annotated with “Numerical” in the legend represent the results calculated according to Eqs. (13) – (19), and those with “PPP approx.” in the legend denote the results obtained based on the pure-phase-perturbation approximation.

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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 $〈 I_{l}^{(m)} (r) 〉$ obtained by substituting Eq. (20) into Eq. (13), gives us

{\hat{P}}_{l}^{(m)} (R) = \int_{0}^{R} ω^{(m)} (r, L) Θ (r, m - l) d r

with

ω^{(m)} (r, L) = 2 π {\hat{I}}^{(m)} (r, L) r / C_{t} (R),

Θ (r, m - l) = \frac{1}{2 π} \int_{0}^{2 π} d θ_{d} H (r, θ_{d}) \exp [- i (l - m) θ_{d}],

C_{t} (R) = 2 π \int_{0}^{R} d r r {\hat{I}}^{(m)} (r, L),

where 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,

{\hat{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 “

{| \tilde{R} (r, z) |}^{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_d) given by Eq. (22) and the accurate results with H (r,θ_d) = exp{−6.88 × 2² ^/ ³[r/(2.1ρ ₀)]⁵ ^/ ³ |sin(θ_d/2)|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.

Fig. 10 Changes in Θ(r,m − l) with varying scaled radial distance. The curves annotated with “Quad.” in the legend represent the results calculated with H (r,θ_d) given by Eq. (22), and those with “Accu.” denote the results obtained with H (r,θ_d) = exp{−6.88 × 2² ^/ ³[r/(2.1ρ ₀)]⁵ ^/ ³|sin(θ_d/2)|⁵ ^/ ³}. ρ ₀ = 6.5q_F.

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Note that ${\hat{P}}_{l}^{(m)} (\infty) = {\bar{p}}_{l}^{(m)} (\infty)$ . It has been claimed in [2] that ${\bar{p}}_{l}^{(m)} (\infty)$ 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_w and m for the cases of vortex Gaussian beams, pure vortex beams and LG₀ _m 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, 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_w or larger 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_l will become more significant if m becomes larger, because a larger m means ω ⁽ ^m ⁾ (r,L) is peaked at a greater radial distance. As a result, ${\bar{p}}_{l}^{(m)} (\infty)$ 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., “| ${| \tilde{R} (r, z) |}^{2} r$ ” in [2], has m-dependence; this can be verified by inspecting Fig. 11(c). By performing numerical propagation of LG₀ _m beams in atmospheric turbulence, Anguita 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_w makes the peak of ω ⁽ ^m ⁾ (r,L) move further away from r = 0. For the same reason as mentioned above, the scaled initial beam radius q_w plays a role in determining ${\bar{p}}_{l}^{(m)} (\infty)$ . By making comparison of Figs. 11(a) – 11(c), we find that, with the same q_w and 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 LG₀ _m 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 LG₀ _m 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 LG₀ _m beams is smaller than the one in that using either vortex Gaussian beams or pure vortex beams. It should be pointed out that C_t (∞) 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 LG₀ _m beams than either vortex Gaussian beams or pure vortex beams in practical systems, because the power of LG₀ _m 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 LG₀ _m beam by jointly controlling the amplitude and phase of a fundamental Gaussian beam.

Fig. 11 Changes in ω ⁽ ^m ⁾ (r,L) with varying scaled radial distance when R = ∞. (a) vortex Gaussian beams; (b) pure vortex beams; (c) LG₀ _m beams. The parameter w ₀ associated with LG₀ _m beams is the radius of the Gaussian term included in the expression used to describe LG₀ _m beams at the transmitter plane [18]. ρ ₀ = 6.5q_F.

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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_l is equal to that of the extrinsic mode l = m + δ_l for a given m. The results shown by Figs. 6(a), 6(f), 7(a) and 7(f) associated with q_c = 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 l = m − δ_l will be greater than that in OAM mode l = m + δ_l. 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 = 3 − δ_l is different from that in OAM mode l = 3 + δ_l for δ_l = 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.

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 LG₀ _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).

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Changes in orbital-angular-momentum modes of a propagated vortex Gaussian beam through weak-to-strong atmospheric turbulence

Abstract

1. Introduction

2. Theoretical formulations

3. Numerical calculations and analysis

4. Conclusions

Acknowledgments

References and links

Cited By

Figures (11)

Equations (27)

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