Characterizing the radial content of orbital-angular-momentum photonic states impaired by weak-to-strong atmospheric turbulence

Chunyi Chen; Huamin Yang

doi:10.1364/OE.24.019713

1. Introduction

Over the recent years, orbital angular momentum (OAM) of light waves has attracted much attention because of its potential for implementing spatial-mode multiplexing in classical free-space optical (FSO) communication systems or photonic qudits in quantum FSO communication links [1–5]. Atmospheric turbulence causes OAM-mode scrambling; that is, the power of a transmitted OAM mode is spread onto adjacent OAM modes. The OAM mode density, defined as the ratio of the power contained in an OAM mode to that carried by the whole wave field, is a fundamental quantity for understanding the deleterious effects of atmospheric turbulence on both classical and quantum OAM-based FSO communications. In fact, from a quantum point of view, the OAM mode density actually describes the probability of finding one photon in a given OAM state. Up to now, several studies have been devoted to characterization of this quantity [6–12].

In the existing literature [6–8,13–16], the Laguerre-Gaussian (LG) modes, which form a complete orthogonal basis set of spatial modes [17,18], have been widely used to describe optical wave fields carrying OAM. An LG mode is characterized by both the radial and azimuthal indices, which are also referred to as the radial and azimuthal quantum numbers of a photonic state [15]. However, it is only the azimuthal index that plays a role in determining the amount of OAM associated with an LG photonic state, irrespective of the radial one. The transmitted OAM modes considered in many studies (see, e.g., [8,15]) are indeed LG modes with a radial index of zero, hereafter denoted by ${LG}_{0}^{l}$ modes with l being the azimuthal index. Intuitively, one can predict that when an OAM-carrying beam associated with an ${LG}_{0}^{l}$ mode propagates in atmospheric turbulence, the power contained in the initial ${LG}_{0}^{l}$ mode will be scattered into adjacent LG modes with different radial or azimuthal indices; however, strictly speaking, the power scattering from the ${LG}_{0}^{l}$ mode into those LG modes with an azimuthal index of l and any nonzero radial index should not be regarded as a contribution to OAM-mode scrambling, because they belong to the same OAM mode (or, in other words, only the azimuthal quantum number needs to be tracked in the detection of OAM photonic states, keeping the radial one unobserved). In many experimental demonstrations [19–22], the radial profile of OAM modes that can be measured is actually fixed in a way by the optical components used to collect the wave field of the modes; for instance, in the case that the wave field is coupled into a single-mode fiber followed by a photodetector [19–22], only the radial content of OAM modes that is able to enter the fiber can be measured; hence, it is of practical interest to consider the radial content of photonic states in investigations into the effects that atmospheric turbulence has on OAM-based FSO communications. Although the issue concerning OAM-mode scrambling has been addressed in the literature, up to now, very few efforts have been made specifically to characterize the radial content of OAM photonic states impaired by atmospheric turbulence. This paper is intended to address this problem.

The pure-phase-perturbation approximation has been extensively employed in theoretical treatments of the evolution of OAM photonic states in atmospheric turbulence (see, e.g., [6,9,14,15]). Nevertheless, this approximation is only applicable under the condition that the atmospheric turbulence is weak enough that scintillations in the wave field can be appropriately ignored. When this condition is not satisfied, other approaches, such as the parabolic equation method and extended Huygens-Fresnel principle [23], should be utilized to develop theoretical formulations of average mode density. However, generally speaking, the use of these approaches does not lead to simple closed-form expressions for the average mode density, which will become apparent shortly. Accordingly, numerical simulations based on multiple random phase screens will be used in producing graphical results for quantitative analysis. On the other hand, to render the graphical results as general as possible, it is useful to find some nondimensional parameters that can completely determine the evolution of OAM photonic states in atmospheric turbulence; viz., the nondimensional parameters incorporate all the adjustable dimensional ones. To this end, in Sec. 2, we begin by developing analytical expressions for both the LG and OAM mode densities in terms of some nondimensional parameters, which are just the ones we want to obtain. Details about numerical simulations and corresponding analysis are depicted in Sec. 3. Finally, conclusions are given in Sec. 4.

2. Analytical formulation for turbulence-induced effects on OAM photonic states

As mentioned previously, the transmitted OAM photonic states considered in many studies correspond to ${LG}_{0}^{l}$ modes. However, for the moment, we do not restrict ourselves to the LG modes with a radial index of zero and will treat the general case instead. The average mode density of the photonic field will be used to characterize the turbulence-induced change in the state of a propagated photon. Let us consider that a single-photon field associated with an ${LG}_{q}^{l}$ mode propagates along the positive z-axis in atmospheric turbulence from the source plane at z = 0 to an observation plane at z = L, where ${LG}_{q}^{l}$ denotes the LG mode with a radial index q and an azimuthal index l. By making use of the extended Huygens-Fresnel principle [23], the field at the observation plane can be expressed by

U_{q, l} (r, z = L) = - \frac{i k}{2 π L} exp (i k L) \iint_{- \infty}^{\infty} d^{2} s U_{q, l}^{(0)} (s) exp [\frac{i k {| s - r |}^{2}}{2 L} + ψ (r, s)],

where r is a point in the observation plane, s is a point in the source plane, k = 2π/λ is the optical wavenumber with λ being the wavelength, ψ (·) represents the complex phase perturbation of a spherical wave propagating from the point (s, z = 0) to the point (r, z = L),

U_{q, l}^{(0)} (s)

stands for the field of a normalized

{LG}_{q}^{l}

mode in the source plane which is given by [7,17]

U_{q, l}^{(0)} (s, θ) = \frac{1}{\sqrt{2 π}} \cdot \frac{N_{q, l}}{w_{0}} \cdot {(\frac{s \sqrt{2}}{w_{0}})}^{| l |} \cdot exp (i l θ) \cdot exp (- \frac{s^{2}}{w_{0}^{2}}) \cdot L_{q}^{| l |} (\frac{2 s^{2}}{w_{0}^{2}})

with (s, θ) being the polar coordinates corresponding to s, N_q,l = 2{q!/[(q + |l|)!]}^1/2, w₀ representing the beam waist radius, and

L_{q}^{| l |} (\cdot)

denoting an associated Laguerre polynomial. The mutual coherence function for the wave field described by Eq. (1) can be formulated by

\begin{array}{l} Γ_{q, l} (r, r^{'}, z = L) & = {(\frac{k}{2 π L})}^{2} \int_{- \infty}^{\infty} d^{2} s \int_{- \infty}^{\infty} d^{2} s^{'} Γ_{q, l}^{(0)} (s, s^{'}) \\ \times exp [\frac{i k {| s - r |}^{2}}{2 L} - \frac{i k {| s^{'} - r^{'} |}^{2}}{2 L}] \\ \times exp (- \frac{r_{d}^{2} + r_{d} \cdot s_{d} + s_{d}^{2}}{ρ_{0}^{2}}), \end{array}

where r_d = |r_d| with r_d = r − r′, s_d = |s_d| with s_d = s − s′,

ρ_{0} = {(0.55 C_{n}^{2} k^{2} L)}^{- 3 / 5}

is the spherical-wave coherence radius with

C_{n}^{2}

being the refractive-index structure constant, and

Γ_{q, l}^{(0)} (s, s^{'}) = U_{q, l}^{(0)} (s, θ) U_{q, l}^{(0) *} (s^{'}, θ^{'})

with the asterisk denoting the complex conjugate. Here, we subscript the quantities given by Eqs. (1) – (4) with “q, l” to emphasize their dependence on both radial and azimuthal indices of the transmitted

{LG}_{q}^{l}

mode; similar notation will also be employed later. In arriving at Eq. (3), the quadratic approximation for the two-source spherical wave structure function [23] has been employed. By making the change of variables r̂ = r/w₀, r̂′ = r′/w₀, ŝ = s/w₀ and ŝ′= s′/w₀, Eq. (3) becomes

Γ_{q, l} (r, r^{'}, z = L) = \frac{N_{q, l}^{2}}{π^{3} Λ_{0}} \cdot \frac{k}{4 L} \cdot {\tilde{Γ}}_{q, l} (\hat{r}, {\hat{r}}^{'}, z = L)

with

\begin{array}{l} {\tilde{Γ}}_{q, l} (\hat{r}, {\hat{r}}^{'}, z = L) & = exp (- \frac{{2.1}^{2} {| \hat{r} - {\hat{r}}^{'} |}^{2}}{Ω^{2}}) \iint_{- \infty}^{\infty} d^{2} \hat{s} \iint_{- \infty}^{\infty} d^{2} {\hat{s}}^{'} \\ \times F (\hat{s}, {\hat{s}}^{'}, \hat{r}, {\hat{r}}^{'}, Λ_{0}, Ω, q, l), \end{array}

\begin{array}{l} F (\hat{s}, {\hat{s}}^{'}, \hat{r}, {\hat{r}}^{'}, Λ_{0}, Ω, q, l) & = {(\hat{s} \sqrt{2})}^{| l |} exp (i l θ) exp (- {\hat{s}}^{2}) L_{q}^{| l |} (2 {\hat{s}}^{2}) \\ \times {({\hat{s}}^{'} \sqrt{2})}^{| l |} exp (i l θ^{'}) exp (- {\hat{s}}^{' 2}) L_{q}^{| l |} (2 {\hat{s}}^{' 2}) \\ \times exp [\frac{i {| \hat{s} - \hat{r} |}^{2}}{Λ_{0}} - \frac{i {| {\hat{s}}^{'} - {\hat{r}}^{'} |}^{2}}{Λ_{0}} - \frac{{2.1}^{2} {| \hat{s} - {\hat{s}}^{'} |}^{2}}{Ω^{2}}] \\ \times exp [- \frac{{2.1}^{2} (\hat{r} - {\hat{r}}^{'}) \cdot (\hat{s} - {\hat{s}}^{'})}{Ω^{2}}], \end{array}

where the Fresnel ratio

Λ_{0} = 2 L / (k w_{0}^{2})

and coherence-width-to-beam-radius (CWBR) ratio Ω = r₀/w₀ are two nondimensional parameters with r₀ = 2.1ρ₀, (ŝ, θ) and (ŝ′, θ′) denote the polar coordinates corresponding to ŝ and ŝ′, respectively. Note that r₀ is the Fried’s spherical-wave atmospheric coherence width, which is widely used as a measure of turbulence strength in the literature [6,9,14,15,19,20].

The field given by Eq. (1) can be expanded in the form

U_{q, l} (r, ϕ, z = L) = \sum_{q^{'} = 0}^{\infty} \sum_{l^{'} = - \infty}^{\infty} a_{q, l} (q^{'}, l^{'}) U_{q^{'}, l^{'}}^{(0)} (r, ϕ, z = L),

where

a_{q, l} (q^{'}, l^{'}) = \int_{0}^{2 π} \int_{0}^{\infty} U_{q, l} (r, ϕ, z = L) U_{q^{'}, l^{'}}^{(0) *} (r, ϕ, z = L) r d r d ϕ,

U_{q^{'}, l^{'}}^{(0)} (r, ϕ, z = L) = \frac{1}{\sqrt{2 π}} ρ_{q^{'}, l^{'}} (r, L) exp (i l^{'} ϕ)

with (r, ϕ) being the polar coordinates corresponding to r,

ρ_{q^{'}, l^{'}} (r, L) = (N_{q^{'}, l^{'}} / w_{L}) \times {(r \sqrt{2} / w_{L})}^{| l^{'} |} \times exp {i k r^{2} L / [2 (L^{2} + z_{R}^{2})]} \times exp [- i (2 q^{'} + | l^{'} | + 1) {tan}^{- 1} (L / z_{R})] \times exp (- r^{2} / w_{L}^{2}) \times L_{q^{'}}^{| l^{'} |} (2 r^{2} / w_{L}^{2})

,

w_{L} = w_{0} {[(L^{2} + z_{R}^{2}) / z_{R}^{2}]}^{1 / 2}

and

z_{R} = 0.5 k w_{0}^{2}

[17]. Note that,

U_{q^{'}, l^{'}}^{(0)}

(r, ϕ, z = L) is actually the field of a normalized

{LG}_{q^{'}}^{l^{'}}

mode at the observation plane in the absence of atmospheric turbulence. The average LG mode density, i.e., the probability of finding one photon in an LG mode, for the field U_q,l (r, z = L) can be defined by

p_{q, l} (q^{'}, l^{'}) = 〈 {| a_{q, l} (q^{'}, l^{'}) |}^{2} 〉,

where the angle brackets represent an ensemble average. By introducing Eq. (9) into Eq. (11), one can find

\begin{array}{l} p_{q, l} (q^{'}, l^{'}) & = \frac{N_{q, l}^{2} N_{q^{'}, l^{'}}^{2}}{4 π^{6} Λ_{0}^{4}} \iint_{- \infty}^{\infty} d^{2} \hat{r} \iint_{- \infty}^{\infty} d^{2} {\hat{r}}^{'} \\ \times {\tilde{Γ}}_{q, l} (\hat{r}, {\hat{r}}^{'}, z - L) {\tilde{Γ}}_{q^{'}, l^{'}}^{(0)} (\hat{r}, {\hat{r}}^{'}, z = L), \end{array}

where

{\tilde{Γ}}_{q^{'}, l^{'}}^{(0)}

(r̂, r̂′, z = L) is equal to

{\tilde{Γ}}_{q^{'}, l^{'}}

(r̂, r̂′, z = L) with Ω = ∞. The average OAM mode density, i.e., the probability of finding one photon in an OAM mode, for the field U_q,l (r, z = L) is

P_{q, l} (l^{'}) = \sum_{q^{'} = 0}^{\infty} p_{q, l} (q^{'}, l^{'}) .

Equation (13) indicates the fact that the average OAM mode density is a sum of various average LG mode densities with the same azimuthal index and different radial indices. Put differently, an OAM photonic state l′ of the field U_q,l (r, z = L) can be expressed as a superposition of LG photonic states with the same azimuthal quantum number l′ and different radial quantum numbers. As a result, p_q,l (q′, l′) with fixed l′ and varying q′ can be used to characterize the radial content of the OAM state l′ of a photonic field arriving at the observation plane due to a single photon with an ${LG}_{q}^{l}$ mode transmitted at the source plane. It can be found from Eqs. (6), (7) and (12) that if the parameter pairs (q, l) and (q′, l′) are fixed, the average LG mode density p_q,l (q′, l′) depends only on the two nondimensional parameters Λ₀ and Ω, by which one can reduce the parameter space from (k, L, w₀, r₀) to (Λ₀, Ω). According to Eq. (13), it is apparent that the average OAM mode density can also be regarded as a function of the nondimensional parameter pair (Λ₀, Ω). This result is different from the one found by using the pure-phase-perturbation approximation [6,7,9], which reveals that the average OAM mode density depends only on Ω.

3. Numerical simulations and analysis

Equation (12) is formidable to evaluate analytically; indeed, it is also not easy to determine it numerically because of the existence of an eightfold integral. For high-dimensional integrals, in principle, the quasi-Monte Carlo technique can be used to estimate their values. On the other hand, the use of numerical simulations based on multiple random phase screens can generate random realizations of propagated wave fields in atmospheric turbulence; in fact, the quantity a_q,l (q′, l′) can be readily computed for each wave-field realization. Furthermore, generating the wave-field realizations can provide the foundation for us to investigate, in the future, other optical-wave statistics of LG beams propagating through atmospheric turbulence. For this reason, below the numerical simulation method is first employed to generate a set of realizations of wave fields due to a single photon with an LG mode propagating through atmospheric turbulence. Then, Eq. (9) is utilized to extract the coefficient a_q,l (q′, l′) of different modes from each of the realizations; i.e., for each realization of U_q,l (·), a_q,l (q′, l′) is obtained by numerically evaluating the double integral in Eq. (9). Finally, according to Eq. (11), p_q,l (q′, l′) is computed by averaging all |a_q,l (q′, l′)|². Notice that, $U_{q^{'}, l^{'}}^{(0)} (\cdot)$ appearing in Eq. (9) is determined by Eq. (10) when extracting a_q,l (q′, l′). Since, in an orthogonal LG basis, for a fixed l, the ${LG}_{q}^{l}$ mode with q = 0 has a minimum space-bandwidth product [24], practical communication systems would generally transmit photons with ${LG}_{0}^{l}$ modes. Consequently, here we only consider the case that a photon with a normalized ${LG}_{0}^{l}$ mode is transmitted at the source plane.

Numerical simulations of propagation of an ${LG}_{0}^{l}$ beam along a horizontal path in atmospheric turbulence performed here follow the procedures presented in [25]. Turbulence is represented by a series of random phase screens located separately with uniform spacing along the propagation path. Each random phase screen, whose statistics obey the Kolmogorov turbulence theory, is created by using the fast-Fourier-transform (FFT) method augmented with subharmonics. The split-step-based angular-spectrum propagation method is used to simulate the random fields at the observation plane. The numerical grid consists of 1024 × 1024 elements with grid spacing appropriately specified according to both the source- and observation-plane spot sizes of a propagated beam. The Fried’s spherical-wave atmospheric coherence width r₀ is employed to characterize the absolute turbulence strength. Martin and Flatté [26] suggested that the scintillation index occurring over an inter-screen distance should be, on the one hand, smaller than 0.1 and, on the other hand, less than 10% of the total scintillation index. This can be employed as a guideline to determine the number of phase screens. For the simulations presented below, the number of phase screens is chosen within the range from 16 to 50 according to the turbulence strength; stronger turbulence leads to more phase screens. For all the simulations, L = 2 km and λ = 800 nm are fixed; different values of Λ₀ and Ω are obtained by only altering the two parameters w₀ and r₀.

To confirm that numerical simulations are performed properly, we compare the average irradiance distribution of propagated ${LG}_{0}^{l}$ beams at the observation plane obtained by numerical simulations and that computed according to analytical formulae. The reason for choosing the average irradiance distribution is that, unlike other statistics, analytically evaluating it is a straightforward task. In accordance with [27], the average irradiance of an ${LG}_{0}^{l}$ beam at the observation plane can be written by

〈 I_{0, l} (r, z = L) 〉 = ℱ^{- 1} {ℱ {I_{0, l}^{(0)} (r, z = L)} \cdot ℱ {η (r, z = L)}},

where

I_{0, l}^{(0)} (r, z = L) = U_{0, l}^{(0)} (r, z = L) U_{0, l}^{(0) *} (r, z = L)

with

U_{0, l}^{(0)} (\cdot)

being the field of an

{LG}_{0}^{l}

beam at the observation plane in the absence of atmospheric turbulence, ℱ represents the Fourier transform, ℱ⁻¹ denotes the inverse Fourier transform, η (·) is the turbulent point spread function. The Fourier transform of η (·) is [27]

ℱ {η (r, z = L)} = exp [- \frac{1}{2} D_{s p} (κ \frac{L}{k})],

where D_sp (·) is the spherical wave structure function, and κ denotes the spatial frequency vector. The analytical expression given by Eq. (14) can be easily determined by using FFT algorithms.

All of the numerical simulations presented thereinafter are verified by the aforementioned approach. Figure 1 exemplifies a comparison between the average irradiance distributions of ${LG}_{0}^{l}$ beams propagating through atmospheric turbulence obtained by numerical simulations and those computed according to the analytical solution shown by Eq. (14). It is found from Figs. 1(d) – 1(f) that the simulation results closely match the analytical ones except for a small difference at the two peaks when Ω = 1.78. This difference is caused by the fact that the average irradiance distribution is computed by averaging a finite number of simulated realizations. Indeed, for the case of Λ₀ = 1 and Ω = 1.78, atmospheric turbulence is relatively strong with the Rytov variance $σ_{l}^{2} = 1.23 C_{n}^{2} k^{7 / 6} L^{11 / 6} \approx 1.65$ , where $C_{n}^{2} = {[2 L Ω^{2} / ({2.1}^{2} k Λ_{0})]}^{- 5 / 6} \times {(0.55 k^{2} L)}^{- 1}$ . The annular “ring” of the average irradiance pattern shown by Fig. 1(c) will become smoother if more simulated realizations are averaged, hence making the simulation results match the analytical ones more closely. Note that, the said guideline proposed by Martin and Flatté [26] has been used to make sure the chosen number of phase screens is large enough for a given turbulence condition. For example, the number of phase screens is set as 20 when we perform the simulations corresponding to Figs. 1(c) and 1(f), where the Rytov variance corresponding to the whole propagation path is 1.65 and the Rytov variance corresponding to an inter-screen distance is 0.0068.

Fig. 1 Average irradiance distribution of ${LG}_{0}^{l}$ beams propagating through atmospheric turbulence with l = 2 and Λ₀ = 1, where Ω = 5.62 for both (a) and (d), Ω = 3.16 for both (b) and (e), Ω = 1.78 for both (c) and (f). The subplots (a) – (c) are average irradiance patterns obtained by numerical simulations; the subplots (d) – (f) are cross-sectional profiles of the average irradiance patterns scaled by the maximum average irradiance obtained according to the analytical solution given by Eq. (14). 600 simulated field realizations are used to compute the average irradiance distribution.

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Figure 2 demonstrates the azimuthally-adjacent-mode spread in terms of Ω⁻¹ for an ${LG}_{0}^{l}$ -mode field propagating through atmospheric turbulence, where l ≡ 4. It is observed from Fig. 2 that, on the one hand, p_0,l (0, l′ = l) decreases with increasing Ω⁻¹; on the other hand, as Ω⁻¹ becomes large, p_0,l (0, l′ ≠ l) initially increases until it approaches the maximum value and then drops off gradually; all the curves tend to merge together when Ω⁻¹ increases beyond a large enough value. The reported experimental results (see Figs. 2 and 3 in [10] and Fig. 4 in [11]) concerning the OAM-mode spread show a similar general behavior, even though there is a difference in the rate at which all the curves tend to merge together between the results in Fig. 2 and those in [10,11]. Note that, as pointed out in [12], in most experimental researches, the transmitted OAM beams are generally not real LG beams, which may deserve a consideration when performing comparisons between experimental and theoretical results. In fact, by making a comparison of Fig. 2(a) with Fig. 2(b), it is found that different Λ₀ leads to different rate at which all the curves tend to merge together, meaning that the initial beam parameter affects the azimuthally-adjacent-mode spread. In addition, our results illustrate that the two curves corresponding to l′ = l ± Δ_l do not completely coincide (Δ_l = 1, 2, and 3); this shows an important distinction between our results and those in [10,11]. It may arise from the fact that the experimental measurements have generally used a single phase screen, realized by a spatial light modulator or phase plate, to emulate the turbulence, whereas we essentially employ multiple phase screens to simulate the turbulence. Indeed, the multiple-phase-screen based numerical simulation results presented in Fig. 6 of [8] have shown that the scattered optical power contained in OAM mode l − Δ_l may be different from that contained in OAM mode l + Δ_l when OAM mode l is transmitted.

Fig. 2 Azimuthally-adjacent-mode spread in terms of Ω⁻¹ for an ${LG}_{0}^{l}$ -mode field propagating through atmospheric turbulence with l ≡ 4. The curves are smoothing-spline-based fitted results obtained according to the simulation ones denoted by the circles and asterisks. (a) Λ₀ = 0.1; (b) Λ₀ = 1.

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Fig. 3 Probability of finding a photon in various radial modes of an intended OAM photonic state in terms of the relative turbulence strength $\sqrt{| l | + 1} Ω^{- 1}$ . The photonic field is due to a single photon with an ${LG}_{0}^{l}$ mode propagating from the source to observation planes in atmospheric turbulence. The curves are smoothing-spline-based fitted results obtained according to the simulation ones denoted by the circles, asterisks and triangles. Indeed, the results for l = l′ = 0 or 4 are also obtained, which show similar features and are omitted to make the displayed curves distinguishable. (a) Λ₀ = 0.1; (b) Λ₀ = 1; (c) Λ₀ = 10.

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Fig. 4 Probability of finding a photon in various radial modes of extrinsic OAM states in terms of the relative turbulence strength $\sqrt{| l | + 1} Ω^{- 1}$ with Λ₀ = 0.1 being fixed. The photonic field is due to a single photon with an ${LG}_{0}^{l}$ mode propagating from the source to observation planes in atmospheric turbulence. The curves are smoothing-spline-based fitted results obtained according to the simulation ones denoted by the circles and asterisks. (a) l = 1; (b) l = 2; (c) l = 3; (d) l = 4. The results with Λ₀ = 1 or 10 actually show similar features and are omitted for saving space.

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The probability of finding a photon in various radial modes of an intended OAM photonic state impaired by atmospheric turbulence is shown by Fig. 3 in terms of the relative turbulence strength $\sqrt{| l | + 1} Ω^{- 1}$ with different Λ₀. It is noted that p_0,l (0, l) can be regarded as the detection probability of an ${LG}_{0}^{l}$ photonic state undergoing turbulence-induced distortions. One finds from Fig. 3 that, with an increasing value of $\sqrt{| l | + 1} Ω^{- 1}$ , i.e., a growth in the relative turbulence strength, the curves corresponding to p_0,l (0, l) fall monotonically, whereas the other curves first rise to a certain value and after that begin to fall gradually; all the curves in Fig. 3 obviously tend to approach each other as $\sqrt{| l | + 1} Ω^{- 1}$ turns large enough. This fact means that when atmospheric turbulence gets stronger, radial modes with nonzero indices contained in the intended OAM state become more noticeable; that is, turbulence-caused mixing of radial modes in the intended OAM state gets more remarkable. Moreover, for different transmitted ${LG}_{0}^{l}$ modes with the same Λ₀, the curves corresponding to p_0,l (0, l) with various l are hardly distinguishable (viz., they almost merge together) when $\sqrt{| l | + 1} Ω^{- 1}$ decreases below a certain value; similar phenomenon has been found for the detection probability of OAM states of photons propagating in extremely weak atmospheric turbulence [6]. However, this behavior is not retained for a group of curves corresponding to p_0,l (q′, l) with an identical nonzero q′ and different l; they basically follow the same trend but the curve with a smaller l lies above that with a larger l. Note that, $\sqrt{| l | + 1} Ω^{- 1} = w_{0, l} / r_{0}$ with $w_{0, l} = \sqrt{| l | + 1} w_{0}$ being the initial spot radius of an ${LG}_{0}^{l}$ beam. As a result, a statement can be made that the ratio of the initial spot radius of an ${LG}_{0}^{l}$ beam to the Fried’s spherical-wave atmospheric coherence width plays a key role in determining the detection probability of the initial ${LG}_{0}^{l}$ photonic state in atmospheric turbulence. This reveals that the degree of deleterious effects that atmospheric turbulence has on the propagation of an ${LG}_{0}^{l}$ photon depends on the relative sizes of the initial beam-spot radius w_0,l and Fried’s spherical-wave atmospheric coherence width r₀ rather than the absolute value of turbulence strength described by r₀.

Comparison between the three subplots of Fig. 3 reveals that all the curves are dependent on both $\sqrt{| l | + 1} Ω^{- 1}$ and Λ₀. As the relative turbulence strength $\sqrt{| l | + 1} Ω^{- 1}$ increases, a larger Λ₀ makes the curves corresponding to p_0,l (0, l) drop off more rapidly and those corresponding to p_0,l (q′ ≠ 0, l) rise to their peaks faster, respectively. Moreover, it is seen from Fig. 3 that, when Λ₀ gets greater, all the curves begin to gradually merge together at a smaller value of $\sqrt{| l | + 1} Ω^{- 1}$ , implying that the probabilities of finding a photon in various radial modes of an intended OAM state tend to become identical to each other at a lower relative turbulence strength level. The physical explanation of this result is as follows. Indeed, only the spatial phase fluctuations in the propagating beam’s wave front impressed by those turbulent eddies whose scale size is smaller than the transverse diameter of the beam at the positions where the eddies are located can damage the specific spatial structure related to an ${LG}_{0}^{l}$ mode. Beam spreading enlarges the transverse beam diameter and consequently increases the number of turbulent eddies that can impair the said specific spatial structure. For fixed w₀ and k, a larger Λ₀ actually implies a longer propagation path and, in turn, a greater amount of beam spreading, thus resulting in a faster decrease in p_0,l (0, l) with the increasing relative turbulence strength $\sqrt{| l | + 1} Ω^{- 1}$ .

It is known that atmospheric turbulence causes phase distortions in the propagating wave field of an OAM photon; the resultant photonic state is a superposition of the intended and extrinsic OAM states. The probability of finding a photon in various radial modes of the extrinsic OAM states is shown by Fig. 4 in terms of $\sqrt{| l | + 1} Ω^{- 1}$ with Λ₀ = 0.1. It is observed that, in general, the curves in Fig. 4 follow the trend of those associated with q′ ≠ 0 in Fig. 3(a). If we deal with the evolution of the photonic state in terms of an orthonormal LG basis, for a transmitted photon with an ${LG}_{0}^{l}$ mode, all the ${LG}_{q^{'}}^{l^{'}}$ modes appearing in the propagated photonic field with either q′ ≠ 0 or l′ ≠ l are actually the outcome of turbulence-induced distortions. Figure 5 depicts the probability of finding a photon in various ${LG}_{q^{'}}^{l^{'}}$ modes with different combinations of Λ₀, Ω and l. One can see from Figs. 4(a) – 4(d) that two curves corresponding to the same q′ and different l′ = l ± 1 do not completely coincide, implying that it is both l and l′ instead of the azimuthal-index separation |l − l′| that determines the transition probability from the original ${LG}_{0}^{l}$ photonic state to the extrinsic ${LG}_{q^{'}}^{l^{'}}$ photonic state. This behavior has been noted in Fig. 2 and can be observed very clearly from Fig. 5, in which the 3D bars exhibit an asymmetry about l′ = l when l > 0, especially in the case of Ω = 1. Of course, one can also see from Fig. 4 that all the curves tend to approach each other when $\sqrt{| l | + 1} Ω^{- 1}$ grows large enough. In addition, it can be inferred from Fig. 5 that the power scattering from a transmitted ${LG}_{0}^{l}$ mode into its radially adjacent modes is generally weaker than into its azimuthally adjacent modes. The radially-adjacent-mode spread is relatively slight for Figs. 5(a) – 5(c) with Ω = 5.62; however, it does become remarkable when Ω = 1 for the cases of Figs. 5(d) – 5(f).

Fig. 5 Probability of finding a photon in various ${LG}_{q^{'}}^{l^{'}}$ modes belonging to the same orthonormal LG basis. The photonic field is due to a single photon with an ${LG}_{0}^{l}$ mode propagating from the source to observation planes in atmospheric turbulence. (a) Λ₀ = 1, Ω = 5.62, l = 0; (b) Λ₀ = 1, Ω = 5.62, l = 2; (c) Λ₀ = 1, Ω = 5.62, l = 4; (d) Λ₀ = 1, Ω = 1, l = 0; (e) Λ₀ = 1, Ω = 1, l = 2; (f) Λ₀ = 1, Ω = 1, l = 4.

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Fig. 6 Probability of finding a photon in different radial modes of various intended OAM photonic states impaired by atmospheric turbulence. All the transmitted photonic states in each subplot are ${LG}_{0}^{l}$ modes that belong to the same orthonormal LG basis. The ${LG}_{0}^{l}$ modes with different Λ₀ belong to different orthonormal LG bases. The absolute turbulence strengths for (a) – (c) are identical with r₀ = 7.1 cm, and those for (d) – (f) are identical with r₀ = 22.6 cm. (a) Λ₀ = 0.1, Ω = 1; (b) Λ₀ = 1, Ω = 3.2; (c) Λ₀ = 10, Ω = 10; (d) Λ₀ = 0.1, Ω = 3.2; (e) Λ₀ = 1, Ω = 10; (f) Λ₀ = 10, Ω = 31.6.

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For a given OAM-based FSO communication link, if the parameter w₀ appearing in Eq. (2) is specified as an identical value for all the modes (i.e., Λ₀ is the same for all the modes), it is apparent that r₀, in turn, Ω should be the same for propagation of all the modes. In this situation, we emphasize that the relative turbulence strength $\sqrt{| l | + 1} Ω^{- 1}$ indeed takes different values for transmitted modes with different l. Figure 6 exemplifies the turbulence-induced variations in the radial content of different intended OAM states in an OAM-based FSO communication link transmitting various ${LG}_{0}^{l}$ modes. It is seen from Fig. 6 that, at the receiver plane, the probability that a photon with a nonzero radial quantum number is found in the intended OAM states can be nonzero because of the turbulence-caused distortions; with the same Λ₀, stronger turbulence makes the radial modes with q′ ≠ 0 in the intended OAM states more significant. It can be clearly observed from Fig. 6 that, with fixed Λ₀ and Ω, the detection probabilities of various transmitted ${LG}_{0}^{l}$ photonic states that belong to the same orthonormal LG basis depend on the azimuthal index l; more specifically, the detection probability of a transmitted ${LG}_{0}^{l}$ photonic state with a smaller l is higher than that with a larger l. In accordance with Eq. (13), the detection probability of an intended OAM state l is a sum of p_0,l (q′, l) with various q′. It can be deduced from Fig. 6 that the detection probability of an intended OAM state with a smaller l is higher than that with a larger one. Based on these observations, one can infer that the azimuthal index l plays a role in determining the detection probability of the intended OAM states, meaning that the bit error probability of an OAM-based FSO communication link is dependent on what digit is sent. This fact deserves attention in the studies of OAM-based FSO communications. Moreover, one can also reason from Fig. 6 that, with the same r₀, the probability of finding a photon in the intended OAM state l with Λ₀ = 1 is obviously higher than that with Λ₀ = 0.1 or 10. We note that this finding does not contradict the one described in the analysis of Fig. 3, i.e., a larger Λ₀ makes the curves therein descend more rapidly, because the relative turbulence strengths associated with the subplots in the same row of Fig. 6 are not identical. The above analysis reveals that, for an OAM-based FSO communication system, the value of the parameter w₀ can be appropriately chosen to reduce the turbulence-induced deleterious effects.

In some experimental demonstrations of OAM-mode propagation in atmospheric turbulence [19–22], the helical-phase-removed wave field has been coupled into a single-mode fiber for detection; there, the single-mode fiber indeed acts as a spatial mode filter, which can isolate the helical-phase-removed wave from other waves still carrying helical phase fronts [5]. At this point, we briefly quantify the effects of the use of a single-mode fiber on detection of various OAM photonic states containing many different radial modes. Following Winzer and Leeb [28], the efficiency of coupling a collimated ${LG}_{q}^{l}$ -mode wave with the helical phase removed into a single-mode fiber can be defined by

ξ (q, l) = \frac{{| \iint_{𝒜} d^{2} r U_{q, l}^{(0)} (r, z = L) Θ (r) U_{m}^{*} (r) |}^{2}}{\iint_{𝒜} d^{2} r {| U_{q, l}^{(0)} (r, z = L) |}^{2}},

where

U_{q, l}^{(0)} (\cdot)

given by Eq. (10) is here expressed in Cartesian coordinates,

Θ (r) = exp {- i k r^{2} L / [2 (L^{2} + z_{R}^{2})]} exp (- i l ϕ)

is used to remove the helical phase and collimate the resultant wave, U_m (r) is the backpropagated fiber-mode profile at the plane of the receiving aperture 𝒜 which is given by U_m (r) = (2/π)^1/2 · a · exp(−a²r²) with a = πW_m/(λf), W_m is the radius of the fiber mode field at the fiber end face, and f is the focal length of the receiving lens.

An effect of the single-mode fiber on OAM measurements is to attenuate the power that can arrive at the detector, leading to a loss of information in the OAM photonic field. According to Fig. 6, the radial mode with an index of zero in intended OAM modes is most important. Hence, it is assumed here that, for detecting a given intended OAM state l, the receiving optics is adjusted to maximize the efficiency of coupling the collimated ${LG}_{0}^{l}$ -mode wave with the helical phase removed into a single-mode fiber. Under this condition, below, we explore the fiber-coupling efficiency ξ (q, l) in terms of q and l. It is noted that, with fixed $U_{q, l}^{(0)} (\cdot)$ , ξ (q, l) is only dependent on the parameter a and the diameter of the aperture 𝒜. To simplify the treatment, here we further assume the receiving aperture is large enough that it does not impose an important limit on both $U_{q, l}^{(0)}$ (r, z = L) and U_m (r), implying that the region of integration in Eq. (16) can be changed from the aperture 𝒜 to the whole plane where the aperture is located without significant error. For any given intended OAM state l, the value of the parameter a that can maximize ξ (0, l) is used to evaluate ξ (q ≠ 0, l). Figure 7 illustrates the obtained results for ξ (q, l). Figure 8 shows the probability of detecting a photon in different radial modes of various intended OAM states with the use of a single-mode fiber. As expected, one can find from Fig. 7 that it is the radial modes with a zero index that can be most efficiently coupled into the single-mode fiber. It is also seen from Fig. 7 that ξ (0, l) decreases with increasing l; i.e., with a larger l, less power contained in a collimated and helical-phase-removed ${LG}_{0}^{l}$ -mode wave can be coupled into the single-mode fiber; indeed, ξ (0, 8) is only just 56.7% of ξ (0, 0). On the other hand, some radial modes with nonzero indices in an OAM state can have a relatively higher fiber-coupling efficiency when l becomes larger. By comparison with Fig. 6, we find from Fig. 8 that the radial modes that can be effectively detected are the ones with a zero index. In this sense, roughly speaking, the orthogonal basis, in terms of which measurements of OAM states are made with the use of a single-mode fiber, can be approximately thought of as an LG basis with a fixed radial index of zero. It is apparent that the probability of detecting a photon in an OAM state l′ with fixed radial dependence is lower than that in the full OAM state l′. This corresponds to photon loss in an OAM-based FSO communication system, which may affect the transition probability of OAM photonic states and, in turn, may play a role in determining the capacity of OAM-based quantum channels.

Fig. 7 Fiber-coupling efficiency ξ (q, l) in terms of q and l. The value of the parameter a that can maximize ξ (0, l) for a given intended OAM state l is used to calculate ξ (q ≠ 0, l).

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Fig. 8 Probability of detecting a photon in different radial modes of various intended OAM states impaired by atmospheric turbulence with the use of a single-mode fiber. Each subplot is obtained under the same conditions as its counterpart in Fig. 6, except for the use of a single-mode fiber. p̂_0,l (q′, l′)=p_0,l (q′, l′) ξ(q′, l′). (a) Λ₀ = 0.1, Ω = 1; (b) Λ₀ = 1, Ω = 3.2; (c) Λ₀ = 10, Ω = 10; (d) Λ₀ = 0.1, Ω = 3.2; (e) Λ₀ = 1, Ω = 10; (f) Λ₀ = 10, Ω = 31.6.

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Here, only the results with respect to propagation of ${LG}_{0}^{l}$ -mode photons in atmospheric turbulence have been presented due to the reason stated at the beginning of this section. Nevertheless, we have performed numerical simulations of propagation of LG beams with a nonzero radial index through atmospheric turbulence. In fact, it is found that atmospheric turbulence also causes radial-mode scrambling if an LG mode with a nonzero radial index is transmitted; for relatively weak atmospheric turbulence, the average LG mode density is obviously peaked at the radial index of the transmitted LG mode; radially-adjacent-mode mixing becomes severer when the turbulence strength grows greater.

4. Conclusions

In this paper, we have explored the variations in the radial content of OAM photonic states propagating through weak-to-strong atmospheric turbulence by employing numerical simulations. Most reported analytical treatments of OAM-mode scrambling made the pure-phase-perturbation approximation. Unlike the existing results, we have found that two nondimensional parameters Λ₀ and Ω are needed to completely determine both the average LG and OAM mode densities of a field due to an LG-mode photon propagating through atmospheric turbulence. The propagation of OAM photons described by ${LG}_{0}^{l}$ modes is considered in our numerical simulations, and the ratio of the beam-spot radius to Fried’s spherical-wave coherence width is utilized to characterize the relative turbulence strength.

It has been shown that, on the one hand, atmospheric turbulence causes the propagated photonic state to become a superposition of both intended and extrinsic OAM states, on the other hand, atmospheric turbulence also raises the appearance of radial modes with nonzero indices in all OAM states. Stronger turbulence results in more noticeable radial modes with nonzero indices and correspondingly a less remarkable radial mode with the zero index. In terms of an LG basis, for a transmitted OAM photon through atmospheric turbulence, the radially-adjacent-mode mixing is relatively slighter than the azimuthally-adjacent-mode mixing. With the same Λ₀, the probabilities of finding a photon in the zero-index radial mode of the intended OAM states as a function of the relative turbulence strength can be displayed by curves that partially merge together; further, a larger Λ₀ makes the curves drop off more rapidly as the relative turbulence strength increases. The probabilities of finding a photon in various radial modes of an OAM state tend to become identical when the relative turbulence strength turns extremely large.

If OAM-based FSO communication systems transmit photonic modes belonging to the same orthogonal LG basis, with a given absolute turbulence strength, the probability of finding a photon in the transmitted photonic mode is dependent on its azimuthal quantum number. Indeed, a larger azimuthal quantum number makes the photon have more mixed azimuthal and radial modes, meaning a lower detection probability of the initial photonic state. The OAM measurement with the use of a single-mode fiber can be roughly regarded as that made in terms of an LG basis with a fixed radial index of zero, leaving the radial modes with nonzero indices approximately unobserved. This may lead to photon loss in OAM-based FSO communication links through atmospheric turbulence and hence alter the probability of observable transitions between various OAM states.

Funding

National Natural Science Foundation of China (61007046, 61275080 and 61475025); Natural Science Foundation of Jilin Province of China (20150101016JC); Specialized Research Fund for the Doctoral Program of Higher Education of China (20132216110002).

Acknowledgments

The authors are very grateful to the reviewers for valuable comments.

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Characterizing the radial content of orbital-angular-momentum photonic states impaired by weak-to-strong atmospheric turbulence

Abstract

1. Introduction

2. Analytical formulation for turbulence-induced effects on OAM photonic states

3. Numerical simulations and analysis

4. Conclusions

Funding

Acknowledgments

References and links

Cited By

Figures (8)

Equations (16)

Optics Express