## Abstract

We have evaluated the channel capacity of OAM-based FSO link under a strong atmospheric turbulence regime when adaptive optics (AO) are employed to correct the wavefront phase distortions of OAM modes. The turbulence is emulated by the Monte-Carlo phase screen method, which is validated by comparison with the theoretical phase structure function. Based on that, a closed-loop AO system with the capability of real-time correction is designed and validated. The simulation results show that the phase distortions of OAM modes induced by turbulence can be significantly compensated by the real-time correction of the properly designed AO. Furthermore, the crosstalk across channels is reduced drastically, while a substantial enhancement of channel capacity can be obtained when AO is deployed.

© 2014 Optical Society of America

## 1. Introduction

An optical beam traveling along z axis can carry a well-defined z-component orbital angular momentum (OAM) of $\ell \hslash $ that is associated with the azimuthal phase of the optical field. It has an eigenmode associated with a helical phase factor exp ($i\ell \theta $), where $\ell $, which is referred to as an OAM angular mode index, can take any integer, with either positive or negative value (corresponding to left-handed or right-handed phase helices, respectively). Due to the structure of helical phase, the optical beam carrying OAM has an intensity null in its core and a phase singularity around which the phase circulates. These OAM modes form an infinite-dimensional Hilbert space, which in practice can be used to increase the channel information capacity through higher data transmission rates over both the quantum and classical channels [1–8]. Moreover, OAM-based quantum key distribution (QKD) application can provide higher tolerance to eavesdropping as compared to the conventional QKD schemes [8–10]. The free-space optical communication (FSO) employing OAM modes encoding, including both the quantum and classical regimes, has attracted considerable attention [3, 5, 7, 8, 11–16]. Although it has been proven that the vortices of OAM modes are robust against the atmospheric turbulence [11], the transverse spatial profiles of OAM modes are susceptible to turbulence effects [12, 13, 17–20], which eventually causes performance degradation of the OAM-based FSO.

In our previous work [18], we have evaluated
the channel capacities of OAM-based FSO links under more realistic atmospheric
turbulence model that follows modified Von Karman spectrum. A conclusion has been
drawn that the capacity is reduced significantly in the presence of atmospheric
turbulence. However, due to the nature of signal distortion, we envisioned that,
based on the fact that adaptive optics (AO) have been used to successfully correct
the distorted wavefronts, an essential enhancement in channel capacity could be
achieved when an AO is deployed. To date, rather limited amount of works has been
reported for correction of the distorted phase of OAM modes in the FSO link
operating over turbulent atmosphere. Ren *et al*. [21] investigated improvement of purity of the
OAM modes and reduction of crosstalk by the Gerchberg-Saxton (GS) algorithm. In
addition, a comparison between the GS algorithm and the Shack-Hartmann sensor method
with respect to wavefront correction has been done in [22]. On the other hand, Rodenburg *et al*.
[14] experimentally demonstrated in a lab
environment that the impact of atmospheric turbulence on OAM modes could be
mitigated by an AO system. In parallel, Ren *et al*. [23] experimentally studied the performance
enhancement of the OAM-multiplexed FSO by using AO to simultaneously compensate
multiple OAM beams. However, to the best of our knowledge, the specific processes of
real-time correcting turbulence-induced phase distortions of the OAM modes by an AO
system has not been analyzed.

In this paper, we perform the simulations of the real-time correction of wavefront distortions of OAM modes by employing a closed-loop AO system model. The model is applied to the FSO link operating over a strong air turbulence channel, and it is realized by employing the Monte-Carlo phase screen method. Moreover, the temporal properties of the correction process are shown reflecting a dynamically evolving turbulent atmosphere. We evaluated the residual distorted wavefront phase after the AO real-time compensation is performed and calculated the ensemble average of the crosstalk between OAM modes. Finally, the FSO channel capacity has been evaluated based the crosstalk estimation.

## 2. Modelling of atmospheric turbulence

Atmospheric turbulence causes perturbation of the atmospheric refractive-index, thus giving rise to the wavefront phase distortion of the light that propagates through it. Since light carrying OAM has the helicoidal phase profile, OAM modes are eventually scrambled by the turbulence.

The phase variations induced by atmospheric turbulence are a random process that obeys a specific statistical rule. Based on the turbulence theory developed by Kolmogorov, several models for the phase power spectrum density (PSD) have been proposed so far. In our previous work [18, 25], we applied the modified Von Karman PSD given as [24]

*κ*denotes spatial frequency; ${\kappa}_{0}$ = 2π/

*L*

_{outer}, ${\kappa}_{m}$ = 5.92/

*l*

_{inner}, while

*L*

_{outer}and

*l*

_{inner}are the outer and inner scale of turbulence;

*r*

_{0}= (0.423

*k*

^{2}C

_{n}

^{2}

*L*)

^{-3/5}is the atmospheric coherent diameter (also refered to as Freid parameter);

*k*,

*L*and

*C*

_{n}

^{2}are wavenumber, propagation distance, and atmospheric refractive-index structure parameter, respectively. This representation of PSD offers the possibility of including the effects of both low and high spatial frequency factors as compared with the Kolmogorov PSD version and can be used to build more realistic atmospheric turbulence model (and that is the reason that the Von Karman PSD model will be also used in our analysis presented below). The

*C*

_{n}

^{2}parameter measures the strength of turbulence, and is regarded as a constant for any horizontal signal path. Without loss of generality, we can assume that

*L*

_{outer}= 20m,

*l*

_{inner}= 3mm for the case when

*C*

_{n}

^{2}= 10

^{−14}m

^{-2/3}(which typically corresponds to a strong turbulence). Next, we utilize the Monte-Carlo phase screen method [26] based on the Fourier transform to effectively express the atmospheric turbulence. With this method, turbulence-induced phase is represented by a computer-generated array of the random sample points that have the statistics in accordance with Eq. (1). Moreover, we additionally applied the subharmonic method to generate the phase screen more accurately to reflect the low spatial frequency regime as well.

It is well known that the phase structure function D* _{ϕ}*(

*r*) can be deduced from the phase PSD, which leads to expression

*r*is the spatial distance separating by two points on the phase front. Now combining Eq. (1) and Eq. (2), the analytical expression of the phase structure function becomes [27],

_{1}

*F*

_{1}(∙) is the confluent hypergeometric function of the first kind. Eventually, we have to validate the random draw of turbulence-induced phase screen by comparing it with values obtained from Eq. (3). For that purpose, we define the error ratio

*(*

_{ϕ}*r*)

_{theory}are values calculated by Eq. (3), and D

*(*

_{ϕ}*r*)

_{fitting}are values of fitting curve based on the random data of generated phase screen. We adopt that the phase screen will be regarded as an acceptable model if Δ is less than 10%.

## 3. Real-time correction of wavefront distortions

The scheme of the OAM-based FSO system with AO compensation considered in our case is presented in Fig. 1. At the transmitter side, the OAM modes are generated by computer-generated holograms (CGH). These modes, which carry the information data, are propagating through turbulent atmosphere before coming to receiving side. At the receiver side, the OAM mode sorter and CCD camera(s) are employed to identify the transmitted OAM modes. The wavefronts of OAM modes are distorted while traveling through the turbulent atmosphere, which leads to system performance degradation. The distortions can be substantially suppressed when the AO system is employed, thus leading to improvements in system performance. The wavelength of the beacon beam from Fig. 1, which is a plane wave used for sampling the atmospheric turbulence, is 1500nm, while the optical signal has a wavelength of 1550nm. Thus, the beacon beam with sampling information can be easily separated by a beam splitter. Accordingly, the AO system will use the obtained information from the beacon beam wavefront to perform the correction of wavefront distortions of OAM modes.

As shown in Fig. 1, the AO system is composed of a deformable mirror, the Shack-Hartmann wavefront sensor, and a control system. In our model we assume that: (i) the continuous-surface deformable mirror has 177 actuators over the aperture; (ii) the Shack-Hartmann sensor consists of a lenslet of 277 lenses and a high-quality CCD array with multiple pixels; (iii) the control system has a gain of 0.3; and (iv) a correction rate of 1 kHz is set. The wavefront phase is reconstructed by polynomials, which can be expressed by [28]:

where*a*is the coefficient of polynomial basis function

_{m}*Z*(∙) and

_{m}*N*is the number of polynomials used. In terms of our Shack-Hartmann sensor model, the wavefront is divided into 277 subparts over the aperture, resulting in 277 spot positions on the CCD. In accordance with that, the wavefront slopes are measured given that the coordinates (

*x*,

_{i}*y*) are centroid of the

_{i}*i-*th spot (

*i*= 1, 2, … 277).

The relationship between the wavefront slopes and the polynomial gradients can be described by the following system of linear equations:

Since in our model the circular pupil is considered, the Zernike polynomials are the
most suitable for employment since they are mutually orthogonal over a circle, so
that the derivatives of polynomials in Eq.
(6) have analytical forms. In order to reconstruct the wavefront more
accurately, we will use 56 Zernike polynomials (i.e. *N* = 56). As a
result, by using the singular value decomposition method to solve Eq. (6), the coefficient vector
[*a*_{1}, $\cdots $, *a*_{56}]^{T} is obtained. Now,
based on coefficients we calculated, the wavefront is reconstructed in accordance
with Eq. (5). Next, the control system
translates the information of reconstructed wavefront into the control signals that
drive the actuators, thus altering the shape of the deformable mirror in order to
compensate for distortions.

According to Fig. 1, a plane wave beacon beam is used to sample the atmospheric turbulence. By using the sampling information, the designed closed-loop AO system will perform the real-time correction. We assume that the diameter of the telescope is 0.24m, and the propagation distance is 1000m. The phase screen of size of 242 × 242 with spacing 0.001m is generated by the Monte Carlo phase screen method, and validated after that. As a result, Fig. 2(a) represents the generated phase screen, while Fig. 2(b) shows the validation results producing the ratio Δ = 3.44%. In Fig. 2(b), the circles represent the data of generated phase screen by the Monte-Carlo phase screen method; the solid line is the curve given by Eq. (3); and the dash line is the fitting curve based on the phase screen data. Therefore, we consider that the generated phase screen is an accurate realization of the turbulent process.

To further evaluate the quality of the turbulence compensation by the designed AO
system, we will use the Strehl ratio *S* as a key merit parameter. It
is expressed as [28]:

*ρ*, $\phi $) are polar coordinates over the unit pupil. Parameter

*S*takes values from 0 to 1; the smaller the value of

*S*is, the more severe the wavefront distortions are. As one can see from the Fig. 3, in our case the AO system responds to make a correction after a time period of [0, 0.01] sec. As a result, the Strehl ratio takes the value

*S*= 0.07 at the outset since the turbulence-induced distortions are severe. However, once the AO system starts to perform the correction, the Strehl ratio increases substantially, and eventually reaches the value

*S*= 0.88. With this, we convincingly confirm that the quality of the FSO system will be significantly improved.

The wavefront phase distortions for uncompensated and compensated cases are shown in Fig. 4. As we can see from Fig. 4(a), if there is no AO correction the distortions are ranging from −3.84 to 5.67 radians with rms of 2.29 radians. However, after AO correction, the wavefront distortions are reduced to fall within the range [-0.78, 0.99] rad with rms of 0.24 rad, as seen in Fig. 4(b). This result confirms that the proper design of the AO system can significantly compensate for the phase distortions even under the impact of strong atmospheric turbulence.

## 4. Evaluation of channel capacity

In this paper, we consider that the family of Laguerre-Gaussian (LG) beams carrying
OAM is used to carry information data, as it has been considered in numerous
applications after the lab generation was performed in [29]. As we mentioned, we consider the computer generated
holograms (CGH) to excite the LG beams. The holograms are created by computing the
interference patterns formed between a reference beam and the desired LG beams.
Accordingly, after the incident reference beam/mode (such as TEM_{00} one)
goes through selected hologram, a corresponding LG beam is being generated. We
assume that the LG beam with a radial node number of 0 is employed, with the field
distribution given as [30],

*r*,

*θ, z*) are cylindrical coordinates;

*ω*(z) =

*ω*

_{0}[1 + (

*z*/

*Z*

_{R})

^{2}]

^{1/2}is the diffraction limited spot size of the fundamental Gaussian beam;

*ω*

_{0}is the beam waist; and

*Z*

_{R}is Rayleigh range. For $\ell $ = 0, Eq. (8) is reduced to the zero-order Gaussian beam (i.e. TEM

_{00}). The diffraction limited radius of the LG beam at the propagation distance

*z*is given as [31]:

For a given propagation distance and OAM mode index, radius ${r}_{\ell}$ has a minimum for *ω*_{0} =
(*λz*/π)^{1/2}. As an example, for *λ* =
1550nm and z = 1000m, *ω*_{0} would be 0.022m. On the other
hand, the LG beam will be broadened as the OAM mode index increases in accordance
with Eq. (9). In such a case, for the
telescope aperture with a diameter of 0.24m that we applied in our model, we could
launch the OAM modes with indexes in the range [-10, 10] so that all of OAM modes
will fall within the aperture.

In the presence of atmospheric turbulence, there will be a transfer of energy between transmitted OAM modes, thus causing the channel crosstalk [17]. The received optical field originating from a single OAM mode with index ${\ell}_{0}$ can be now regarded as a superposition of all OAM modes, so we have that,

where ${C}_{\ell}$ is the weighted coefficient of the OAM mode with index $\ell $. Assuming the original orthogonality of OAM modes expressed as*I*

_{0}is the intensity of transmitted OAM mode with the index ${\ell}_{0}$ calculated as

*D*is the diameter of the telescope. Our focus in this paper is the evaluation of temporal properties and performance enhancement of the OAM-based FSO link under a strong turbulence impact after the AO compensation is applied. Accordingly, we will assume that distortions of the OAM modes are induced by turbulence only (i. e. transmitter and receiver are perfect and well aligned). In such a case, received optical field

*u*(

*r*,

*θ*) can be written as

*ξ*(

*r*,

*θ*) represents distorted wavefront phase resulting from the turbulence. Note that the atmospheric turbulence is sampled by the beacon beam at the wavelength of 1500nm, thus

*ξ*(

*r,θ*) must be scaled down to the carrier wavelength of the OAM mode,where

*ξ*

_{0}(

*r,θ*) is the distorted phase of the beacon beam. Accordingly, if there is no AO correction,

*ξ*

_{0}(

*r,θ*) would present the generated phase screen, while with the AO correction applied

*ξ*

_{0}(

*r,θ*) presents a residual distorted phase.

In order to obtain ensemble averages of the crosstalk, we performed 1000 processes of
AO correction over the *M _{a}* = 1000 different phase
screens, which represents

*M*realizations of the same atmospheric turbulence process. For each process, the crosstalk with and without AO correction can be obtained. The ensemble averages of crosstalk are calculated by averaging over the 1000 crosstalk cases. As we can clearly see from Fig. 5, the strong turbulence induces severe crosstalk, which is significantly reduced by the applied AO compensation.

_{a}Based on the calculated ensemble averages of crosstalk, we can obtain the channel transfer matrices and calculate the channel capacity of OAM-based FSO link. Considering a discrete memoryless model, the channel capacity can be expressed as

*H*(∙) denotes the entropy [32];

*X*and

*Y*are transmitted and received symbols, respectively;

*p*is the probability of transmitting OAM modes with index

_{i}*i*; and

*P*is the channel transfer matrix, where

_{ji}*j*is the index of received OAM mode. We can evaluate the channel capacities by the Blahut-Arimoto algorithm [32]. The results are shown in Fig. 6. As we can see, the channel capacity values with no AO correction in place are much lower than those associated with an ideal case (i.e. in the absence of atmospheric turbulence). However, after deploying the AO system, the channel capacity even under strong turbulence is substantially enhanced (representing curves are approaching to those belonging to the ideal case). As an example, when the number of transmitted OAM modes is large (i.e.

*N*

_{OAM}= 20), the obtained capacity is 3.99 bits/symbol, as compared to an ideal case with a capacity of 4.32 bits/symbol, while it is only 1.34 bits/symbol if there is no AO correction.

As far as dynamically evolving atmosphere is concerned, we can also evaluate the
temporal properties of the OAM-based FSO with AO in place. In such a case, the phase
screen has to be moved in the transverse dimension as the wind evolves with the
time. We should note that the size of the phase screen is larger than that of the
static case, so that the turbulence can be sampled by the beacon beam during the
process of AO correction. We generated a phase screen of size 1024 × 1024 with a
spacing of 0.001m, with the same atmospheric turbulence parameters as in the above
case, which is validated by using Eq.
(4), while obtaining the error ratio Δ = 1.60%. Given that the turbulence
phase screen will be moved by a wind speed of (2, −2) m/s along *x*
and *y* axis, respectively, we performed the AO correction from 0 to
0.31 sec.

The calculated Strehl ratio fluctuates with the time, as presented in Fig. 7(a). For convenience purposes (but without loss of any generality), we assumed that the number of transmitted OAM modes is 20. The variations of channel capacity in time are shown in Fig. 7(b). As one can see, the AO limits the fluctuations in both cases. After 0.05 secs, the Strehl ratio with AO applied has a mean value of 0.79 and an rms of 0.03, as compared to a mean value of 0.10 and an rms of 0.08 for the without AO compensation, which indicates the effectiveness of the wavefront distortions by the AO employment. As for the change of channel capacity with the time, the significant enhancement can be observed as well. The maximum improvement of channel capacity is 2.72 bits/symbol, while an average improvement of 1.90 bits/symbol has been obtained.

## 5. Conclusion

We have studied the enhancement of the channel capacity of the OAM-based FSO link under the impact of strong atmospheric turbulence by considering the use of adaptive optics system to correct the phase distortions of OAM modes. By performing numeric simulations in out modelling, we also demonstrated specifics of the real-time correction processes. The results show that optimally designed AO system can effectively correct the wavefront phase distortions caused by the atmospheric turbulence, thus drastically reducing the crosstalk across the OAM-based channels. Consequently, the channel capacity of the FSO link is substantially improved. Furthermore, we have also demonstrated the modelling of a real-time compensation process under dynamically evolving atmosphere conditions and confirmed highly reduced fluctuation of both the Strehl ratio and the channel capacity during the AO real-time correction.

## Acknowledgments

The authors are thankful for the help of Dr. Jinhun Kim on AO modeling. M. Li acknowledges the support from BUPT Excellent Ph.D. Students Foundation (Grant No.CX201333) and the program of China Scholarship Council (Grant No.201306470039). This work was supported in part by the NSF under Grant CCF-0952711, NSF CIAN ERC under grant EEC-0812072.

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