Influence of atmospheric turbulence on OAM-based FSO system with use of realistic link model

Optics Communications 364 (2016) 50–54

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Influence of atmospheric turbulence on OAM-based FSO system with use of realistic link model Ming Li a,n, Zhongyuan Yu b, Milorad Cvijetic c a

College of Electronic and Communication Engineering, Tianjin Normal University, Tianjin 300387, China State Key Laboratory of Information Photonics and Optical Communications, Beijing University of Posts and Telecommunications, Beijing 100876, China c College of Optical Sciences, University of Arizona, Tucson 85721, USA b

art ic l e i nf o

a b s t r a c t

Article history: Received 20 October 2015 Received in revised form 6 November 2015 Accepted 15 November 2015

We study the influence of atmospheric turbulence on OAM-based free-space optical (FSO) communication by using the Pump turbulence spectrum model which accurately characterizes the realistic FSO link. A comprehensive comparison is made between the Pump and Kolmogorov spectrum models with respect to the turbulence impact. The calculated results show that obtained turbulence-induced crosstalk is lower, which means that a higher channel capacity is projected when the realistic Pump spectrum is used instead of the Kolmogorov spectrum. We believe that our results prove that performance of practical OAM-based FSO is better than one predicted by using the original Kolmogorov turbulence model. & 2015 Elsevier B.V. All rights reserved.

Keywords: Free-space optical communication Orbital angular momentum Atmospheric turbulence Channel capacity

1. Introduction It has been well known that a photon that propagates along þz axis can carry both spin angular momentum (SAM) and orbital angular momentum (OAM) [1]. The SAM is associated with circular polarization since a circularly polarized light beam carries SAM of 7ℏ per photon (þ sign corresponds to left circular polarization, and–sign refers to right circular polarization), where ℏ is the Planck constant divided by 2π. On the other side, the OAM is related to the helical phase front since the light beam carrying OAM contains a helical phase factor of exp(iℓθ), where θ is the azimuthal angle, and ℓ denotes the number of winding helices (ℓ is an integer that can take positive-, negative- or zero-value corresponding to left-handed, right-handed, and no phase helices, respectively). Accordingly, each eigenmode carries the corresponding OAM of ℓℏ per photon. These modes form an infinite-dimensional Hilbert space, thus can be used to effectively increase information capacity of the classical optical communications by the OAM-encoding modulation [2] or OAM-multiplexing technique [3,4]. OAM has significant applications as well as in the quantum optical communications region. As compared to the conventional two-dimensional polarization scheme, OAM-based quantum key distribution can transfer more information per photon and offer n

Corresponding author. E-mail addresses: [email protected] (M. Li), [email protected] (Z. Yu), [email protected] (M. Cvijetic). http://dx.doi.org/10.1016/j.optcom.2015.11.041 0030-4018/& 2015 Elsevier B.V. All rights reserved.

more tolerance to eavesdropping [5]. Since the first experiment using OAM modes to transfer information over an atmospheric link was demonstrated [2], the free-space optical (FSO) communication based on OAM has been under considerable consideration [3,6–13]. However, due to the nature of helical phase profile, the OAM modes can be easily scrambled by the atmospheric turbulence effects, while crosstalk and signal fading would be induced. Numerous literatures have investigated the impact of atmospheric turbulence on the OAM-based FSO data channel. As an example, Paterson studied the turbulence-induced crosstalk among OAM modes by employment of semi-classical theory [10]. The results in [10] showed that the transmitted OAM mode would eventually transfer its energy into the neighboring modes due to the deleterious turbulence effects, thus leading to the crosstalk among them. The pure OAM mode propagating through the turbulent atmosphere was analyzed by the Fourier series method [14], which was experimentally validated in [15]. A similar conclusion to one presented in [10] was obtained in [14,15]. In parallel, the crosstalk was calculated by using the split-step beam propagation with angular-spectrum method [7]. The results in [7] showed that the crosstalk was proportional to the strength of turbulence as well as the index number of transmitted OAM modes. To the best of our knowledge, however, the vast majority of research results mentioned above, for the sake of mathematical tractability, are based on turbulence spectrum model that follows the basic Kolmogorov statistics, which is more appropriate only

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within the initial subrange of spectral frequencies. Nevertheless, the realistic atmospheric turbulence, in addition to the initial subrange, includes both the low and high spatial frequency effects. In this paper, we employ the Pump atmospheric turbulence spectrum model to study the impact of turbulence on the propagating OAM modes. The Pump spectrum model has been accurately validated by experiments [16–18] and is expected to lead to more realistic results with respect to induced crosstalk and channel capacity.

2. FSO channel model based on the Pump spectrum In the near-ground area, air density fluctuates randomly due to wind and temperature gradient, thus leading to the atmospheric turbulence. The atmospheric turbulence, in turn, gives rise to small fluctuations of the atmospheric refractive-index. As a result, the phase of the light beam propagating through turbulent atmosphere will be distorted. Although the phase distortion is a random process, it obeys a specific statistic rule. Several spectrum models have been proposed so far to represent that statistics ever since the framework of Kolmogorov spectrum model was introduced. The Kolmogorov model, although fundamental, is the basic and simplest one, since it is assumed that outer scale of

Fig. 1. Phase structure functions based on the Kolmogorov and Pump spectrum models. Without loss of generality, the parameters used here are set to: L ¼1000 m; λ¼ 1550 nm; Cn2 ¼10  14 m  2/3; linner ¼0.003 m; Louter ¼20 m, respectively.

model is applied. By doing so, the analytical expression of phase structure function becomes [22]

⎧ ⎛ ⎡ ⎡ 2 2 ⎞ ⎤⎫ 2 2 ⎞⎤ ⎛ ⎛ ⎪ Γ ⎜ −5 ⎞⎟ ⎢ 1 − F ⎜ −5 ; 1; −κ l r ⎟ ⎥ + 1.802Γ ⎛⎜ −1 ⎞⎟ ⎢ 1 − F ⎜ −1 ; 1; −κ l r ⎟ ⎥⎪ 1 1 1 1 ⎪ ⎝ 3 ⎠ ⎢⎣ 4 ⎠ ⎥⎦ 4 ⎠ ⎥⎦⎪ ⎝ 6 ⎝ 3 ⎪ ⎝ 6 ⎠ ⎢⎣ ⎪ ⎬, DPump (r ) = 3.08r0−5/3 κ l−5/3 ⎨ ϕ 2 2 ⎞⎤ ⎛ −1 ⎪ ⎪ ⎛ −1 ⎞ ⎡ −κ l r 9 2 5/3 1/3 ⎟ ⎢ 1 − 1 F1⎜ ⎟ ⎥ − κl κ0 r ; 1; ⎪ − 0.254Γ ⎜ ⎪ ⎝ ⎠ ⎪ ⎪ ⎢ ⎥ 4 4 4 5 ⎝ ⎠⎦ ⎣ ⎩ ⎭

turbulence is infinite, while the inner scale of turbulence is 0. However, in practice, the large turbulence cells are not stable and can be split into smaller ones. On the other side, the energies of small turbulence cells will be reduced, eventually reaching zero value due to the dissipation effect. Accordingly, based on the facts above, the Kolmogorov spectrum model is appropriate only within the initial subrange (i.e. spatial range between the outer scale and inner scale). Yet, the realistic turbulence spectrum model should also consider the effects of the outer and inner scales so that it would be applicable to both low and high spatial frequencies in addition to the initial subrange. We believe that the Pump spectrum model, whose accuracy has been validated by number of experiments [16–18], describes realistic turbulence spectrum [19,20]. The phase structure function, based on the Kolmogorov spectrum model, can be expressed as [21] 5 ⎞3

⎛ r DKol ⎟ , ϕ (r ) = 6.88 ⎜ ⎝ r0 ⎠

(1)

where r denotes the spatial distance separated by two points over phase front; r0 is the atmospheric coherent diameter (also referred to as Freid's parameter) calculated as r0 ¼(0.423k2 Cn2L)  3/5, where k, L and Cn2 are wavenumber, propagation distance, and atmospheric refractive-index structure parameter, respectively. Parameter Cn2 is commonly used to measure the strength of atmospheric turbulence. For a horizontal FSO link, in particular, Cn2 can be treated as a constant [19]. Now, the Eq. (1) can be extended to include the entire spatial range if the Pump turbulence spectrum

(2)

where Γ(∙) and 1F1(∙) represent the gamma function and the confluent hypergeometric function of the first kind, respectively; k0 ¼2π/Louter, kl ¼3.3/linner, with Louter and linner representing the outer and inner scale of turbulence, respectively. As an example, in Fig. 1 we plotted the difference between the Pump spectrum and Kolmogorov spectrum phase structure functions. As we can clearly see, the Pump spectrum phase structure function shows the deflection from the Kolmogorov function when the spatial distance is larger than the outer scale factor or lower than the inner scale factor. Accordingly, the Pump spectrum appears to be more appropriate to the entire spatial frequency range when characterizing the realistic atmospheric link.

3. Turbulence-induced crosstalk and channel capacity of FSO link Since the Laguerre–Gaussian (LG) beam carrying OAM was readily generated in the laboratory [1], it has been widely used in numerous applications [23,24]. In this paper, we consider the LG beams with radial mode number equal 0 as the carriers of optical signals. They have the optical field distribution given as [24]

Uℓ (r , θ , z ) = Aℓ (r , z ) exp (iℓθ ),

(3)

where (r, θ, z) represents cylindrical coordinates; ℓ is an angular index of the OAM mode; Aℓ(r, z) has the expression

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Aℓ (r , z ) =

ℓ ⎡ r2 ⎤ 2 1 ⎡ 2r ⎤ ⎢ ⎥ exp ⎢ − 2 ⎥ ⎣ π ℓ ! ω (z ) ⎣ ω (z ) ⎦ ω (z ) ⎦

⎡ ikr 2z × exp ⎢ ⎢⎣ 2 z2 + zR2

(

)

the integral variable can be expressed as

⎤ ⎡ ⎤ ⎥ exp ⎢ −i ( ℓ + 1) tan−1 z ⎥, ⎥⎦ ⎣ zR ⎦

P ( ℓ ℓ0 ) = (4)

where ω(z) ¼ ω0[1þ (z/ZR) ] is the diffraction limited spot size of the fundamental Gaussian beam (i.e. TEM00 mode) at propagation distance z (ω0 is the beam waist, and ZR is the Rayleigh range). Note that Eq. (3) is reduced to the TEM00 mode under the condition of ℓ ¼0. After the LG beam propagates through the turbulent atmosphere, its helical wavefront phase profile will be distorted, thus leading to the degradation of OAM mode purity. Specifically, in the OAM-based FSO, the transmitted OAM mode will transfer its energy into the other OAM modes. The optical field U(r, θ, z) at the receiving plane z originating from a single OAM mode with index ℓ 0 can be now regarded as a superposition of all OAM modes. As a result, the crosstalk among OAM modes will be induced, which can be characterized by the conditional probability representing a measurement of obtaining OAM mode with index ℓ [Ref. [10], the equation above Eq. (3)] 2 1/2

P ( ℓ ℓ0 ) =

∫ ∫ ∫ U * (r , θ, z ) U (r , θ′, z ) exp [iℓ(θ − θ′)] r dr dθ′dθ I0

,

(5)

where operator ∗ denotes the conjugate; I0 is the intensity of transmitted LG beam with OAM mode index ℓ0, which is,

I0 =

∫

2

Uℓ0 (r , θ ) r dr dθ .

(6)

Atmospheric turbulence gives rise to the random wavefront phase distortions, thus the optical field U(r, θ, z) at the receiving plane z can be written as

U (r , θ , z ) = Uℓ0 (r , θ , z ) exp [iξ (r , θ )],

(7)

where ξ(r,θ) represents distorted wavefront phase resulting from the turbulence. Since the distortions are random, P(ℓ|ℓ0) is measured by taking its ensemble average practically. Now substituting Eq. (7) into Eq. (5) yields P ( ℓ ℓ0) =

∫ ∫ ∫ A ℓ 0 (r, z )

2

exp ⎡⎣ i ( ℓ − ℓ0) (θ − θ ′) ⎤⎦ exp {i [ξ (r, θ ′) − ξ (r, θ )]} r dr dθ ′dθ I0

,

(8)

where operator 〈∙〉 represents the ensemble average. Considering the statistics of the turbulence-induced fluctuations to be isotropic and the fact that the OAM mode profile is rotationally symmetric, we let Δθ = θ − θ′. By doing so, Eq. (8) reduces to a double-integral form,

1 I0

∫0

×

∫0

∞

2π

Δθ to φ, the conditional probability P(ℓ|ℓ0) 2

r A ℓ0 (r , z ) dr

⎡ 1 ⎛ φ ⎞⎤ exp ⎢ − Dφ ⎜ 2r sin ⎟ ⎥ exp ⎡⎣ i ( ℓ − ℓ0 ) φ⎤⎦ dφ, ⎣ 2 ⎝ 2 ⎠⎦

(12)

Now, by using Eqs. (1), (2), (3) and (12), we can calculate the crosstalk under the Kolmogorov and Pump spectrum turbulence models, respectively. Without any loss of generality, we assume that Louter ¼5 m, linner ¼6 mm for the case when Cn2 ¼ 10  15 m  2/3 for optical wavelength λ ¼1550 nm and the propagation distance L¼5000 m; the launching OAM mode index ℓ falls in the range [  10, 10]. Given L, λ and ℓ, there would be an optimum ω0 to assure that the LG beam has the minimum spot size at the receiver side [13], which is highly desired for a practical OAM-based FSO design. For that reason, we adopt that the beam waist equals ω0 ¼(λz/π)1/2 ¼0.0497 m. As an example, the Fig. 2 shows the calculated crosstalk when the OAM mode with index ℓ ¼  1 is transmitted, which clearly indicates the better energy preservation of transmitted OAM mode when the Pump spectrum model is applied. That means the crosstalk among OAM modes could be overestimated if the Kolmogorov spectrum model is employed. Specifically, the calculated probability of receiving the OAM mode with index ℓ ¼  1 (i.e. the original transmitted OAM mode) is 0.4012 under the Pump spectrum turbulence model while it is only 0.3618 if Kolmogorov spectrum turbulence model is applied, which is ∼20% difference. We will now evaluate the channel capacity of FSO link based on crosstalk models described above. Assuming that FSO link is a discrete memoryless channel, the channel capacity can be calculated as

Cchannel = max[H (Y ) − H (Y X )] { pi } ⎛ ∑ pi Pji ⎞ ⎟, = max⎜⎜ −∑ ∑ pi Pji log2 i Pji ⎟⎠ { Pi } ⎝ i j

(13)

where H(X) and H(Y|X) denote the source entropy and conditional entropy, respectively; X and Y are transmitted and received symbols, respectively; pi is probability of the transmitting OAM mode with index i; Pji is the channel transfer probability (corresponding to an element of the channel transform matrix), where j is the received OAM mode index. In light of Eq. (7), the channel

2

P ( ℓ ℓ0 ) =

∫ ∫ A ℓ 0 (r , z ) exp ⎡⎣ i ( ℓ − ℓ0 ) Δθ ⎤⎦ exp {i [ξ (r , Δθ ) − ξ (r , 0) ] } r dr dΔθ I0

.

(9)

By the large number theorem, we can assume the turbulenceinduced distortions are normal random variables. As a result, the ensemble average term in Eq. (9) can be given as

exp {i [ξ (r , Δθ ) − ξ (r , 0)]} ⎧ 1 ⎫ = exp ⎨ − [ξ (r , Δθ ) − ξ (r , 0)]2 ⎬ , ⎩ 2 ⎭

(10)

0)]2

where the term [ξ (r , Δθ ) − ξ (r , is well known as the phase structure function Dϕ(x). Here, the distance x separated by two points over phase front can be deduced, x ¼2rsin(Δθ∕2). Thus,

⎛ Δθ ⎞⎟ . [ξ (r , Δθ ) − ξ (r , 0)]2 = Dφ ⎜ 2r sin ⎝ 2 ⎠

(11)

Eventually, by combining Eqs. (9), (10) and (11) and changing

Fig. 2. Crosstalk for the transmitted OAM mode with index ℓ ¼  1 under the realistic (Pump) and Kolmogorov spectrum turbulence models.

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4. Conclusions In conclusion, we have studied the impact of atmospheric turbulence on OAM-based FSO channel by using the Pump turbulence spectrum model which accurately characterizes the realistic FSO link. A comprehensive comparison is made between the Pump and Kolmogorov spectrum models with respect to the turbulence impact. The calculated results showed that use of Kolmogorov turbulence model leads to underestimation of the total channel capacity by 20–80% depending of the number of OAM modes that are deployed. This is significant difference and we believe that the Pump spectrum model is much more appropriate in any analysis of OAM-based FSO systems.

Acknowledgments

Fig. 3. The OAM based FSO channel capacities for Kolmogorov and Pump models applied.

We acknowledge the support from National Natural Science Foundation of China (NSFC) (61372037). We would like to thank Professor Qilian Liang for his helpful comments.

References

Fig. 4. The percentages of OAM-based FSO channel capacity increase when the Pump spectrum turbulence model is employed relative to the case of Kolmogorov model.

capacities can be evaluated by the Blahut–Arimoto algorithm [25]. The calculated results are shown in Fig. 3. As we can see, the channel capacity values are higher for all of the number of transmitted OAM modes if the realistic turbulent link model is applied as compared with case corresponding to the Kolmogorov turbulent link model. This reveals that the use of Kolmogorov turbulence model underestimates the channel capacity of OAM-based FSO system. Specifically, the quantities of underestimation are calculated. Then the percentages of channel capacity increase when the realistic Pump spectrum turbulence model is employed, relative to the case of Kolmogorov model, is shown in Fig. 4. As one can see, even if we launch all of the 20 OAM modes (the OAM mode with index 0, generally, is left for alignment), an channel capacity increase of 18.45% is shown when the realistic Pump turbulence model is used relative to the results obtained by using the Kolmogorov model.

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