Evaluation of channel capacities of OAM-based FSO link with real-time wavefront correction by adaptive optics Ming Li,1,2 Milorad Cvijetic,1,* Yuzuru Takashima,1 and Zhongyuan Yu2 2
1 College of Optical Sciences, University of Arizona, Tucson, 85721, USA State Key Laboratory of Information Photonics and Optical Communications, Beijing University of Posts and Telecommunications, Beijing 100876, China * [email protected]
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 OCIS codes: (060.2605) Free-space optical communication; (010.1330) Atmospheric turbulence; (010.1080) Active or adaptive optics.
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#224770 - $15.00 USD Received 10 Oct 2014; revised 25 Nov 2014; accepted 26 Nov 2014; published 11 Dec 2014 © 2015 OSA 15 Dec 2014 | Vol. 22, No. 25 | DOI:10.1364/OE.22.031337 | OPTICS EXPRESS 31337
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1. Introduction An optical beam traveling along z axis can carry a well-defined z-component orbital angular momentum (OAM) of that is associated with the azimuthal phase of the optical field. It has an eigenmode associated with a helical phase factor exp ( iθ ), where , 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
#224770 - $15.00 USD Received 10 Oct 2014; revised 25 Nov 2014; accepted 26 Nov 2014; published 11 Dec 2014 © 2015 OSA 15 Dec 2014 | Vol. 22, No. 25 | DOI:10.1364/OE.22.031337 | OPTICS EXPRESS 31338
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] Φφ (κ ) = 0.49r0−5/3
exp(− κ 2 κ m2 )
(κ
2
+ κ 02 )
11/ 6
, for 0 ≤ κ < ∞,
(1)
where κ denotes spatial frequency; k 0 = 2π/Louter, k m = 5.92/linner, while Louter and linner are the outer and inner scale of turbulence; r0 = (0.423k2Cn2L)-3/5 is the atmospheric coherent diameter (also refered to as Freid parameter); k, L and Cn2 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 Cn2 parameter measures the strength of turbulence, and is regarded as a constant for any horizontal signal
#224770 - $15.00 USD Received 10 Oct 2014; revised 25 Nov 2014; accepted 26 Nov 2014; published 11 Dec 2014 © 2015 OSA 15 Dec 2014 | Vol. 22, No. 25 | DOI:10.1364/OE.22.031337 | OPTICS EXPRESS 31339
path. Without loss of generality, we can assume that Louter = 20m, linner = 3mm for the case when Cn2 = 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 Dφ(r) can be deduced from the phase PSD, which leads to expression ∞ sin κ r Dφ (r ) = 8π κ 2 Φφ (κ ) 1 − dκ , 0 κ r
(2)
where 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], −5 −5 −κ 2 r 2 Dφ (r ) = 3.08r0−5/3 Γ κ m−5/3 1 − 1 F1 ;1; m 4 6 6
9 1/3 2 − κ 0 r , 5
(3)
where 1F1(·) 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 Dφ (r ) theory − Dφ (r )fitting Δ = max max {Dφ (r ) theory }
,
(4)
where Dφ(r)theory are values calculated by Eq. (3), and Dφ(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 computergenerated 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 ShackHartmann 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 ShackHartmann 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]:
#224770 - $15.00 USD Received 10 Oct 2014; revised 25 Nov 2014; accepted 26 Nov 2014; published 11 Dec 2014 © 2015 OSA 15 Dec 2014 | Vol. 22, No. 25 | DOI:10.1364/OE.22.031337 | OPTICS EXPRESS 31340
N
φ p = am Z m ( x, y ),
(5)
m =1
where am is the coefficient of polynomial basis function Zm(·) and 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 (xi, yi) are centroid of the i-th spot (i = 1, 2, … 277).
Fig. 1. Schematic of OAM-based FSO system with AO compensation.
The relationship between the wavefront slopes and the polynomial gradients can be described by the following system of linear equations: ∂Z1 ( x, y ) ∂Z N ( x, y ) ∂φ p ∂x ∂x ∂x ( x1 , y1 ) ( x1 , y1 ) ( x1 , y1 ) ∂Z ( x, y ) φ ∂ , ∂ Z x y ) N ( 1 p a ∂x ∂x 1 ∂x ( x277 , y277 ) ( x277 , y277 ) ( x277 , y277 ) = . (6) ∂Z1 ( x, y ) ∂φ p ∂Z N ( x, y ) aN ∂y ∂y ∂y ( x1 , y1 ) ( x1 , y1 ) ( x1 , y1 ) ∂Z x, y ∂φ p ∂Z N ( x, y ) ( ) 1 y ∂ ∂y ∂ y , x y ( 277 277 ) ( x277 , y277 ) ( x277 , y277 ) 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 [a1, , a56]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
#224770 - $15.00 USD Received 10 Oct 2014; revised 25 Nov 2014; accepted 26 Nov 2014; published 11 Dec 2014 © 2015 OSA 15 Dec 2014 | Vol. 22, No. 25 | DOI:10.1364/OE.22.031337 | OPTICS EXPRESS 31341
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.
Fig. 2. (a) Phase screen generated by Monte Carlo phase screen method; (b) validation results.
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]:
S=
1
π
2
1
2π
0
0
2
exp ( ikφ p ) ρ dρ dϕ ,
(7)
where (ρ, ϕ ) 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.
Fig. 3. Strehl ratio of OAM-based FSO with AO compensation.
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 #224770 - $15.00 USD Received 10 Oct 2014; revised 25 Nov 2014; accepted 26 Nov 2014; published 11 Dec 2014 © 2015 OSA 15 Dec 2014 | Vol. 22, No. 25 | DOI:10.1364/OE.22.031337 | OPTICS EXPRESS 31342
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.
Fig. 4. Wavefront phase distortions resulting from atmospheric turbulence (a) without AO correction; (b) with AO correction.
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 TEM00 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], u ( r , θ , z ) =
ikr 2 z 2 1 2r r2 exp exp − 2 π ! ω ( z ) ω ( z ) 2 ( z 2 + zR2 ) ω ( z )
(8)
z ×exp −i ( + 1) tan −1 exp ( iθ ) , zR
where (r, θ, z) are cylindrical coordinates; ω(z) = ω0[1 + (z/ZR)2]1/2 is the diffraction limited spot size of the fundamental Gaussian beam; ω0 is the beam waist; and ZR is Rayleigh range. For = 0, Eq. (8) is reduced to the zero-order Gaussian beam (i.e. TEM00). The diffraction limited radius of the LG beam at the propagation distance z is given as [31]: r ( z ) =
ω0 + λ z 2
2
2
π 2ω02
+ 1.
(9)
For a given propagation distance and OAM mode index, radius r has a minimum for ω0 = (λz/π)1/2. As an example, for λ = 1550nm and z = 1000m, ω0 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 0 can be now regarded as a superposition of all OAM modes, so we have that,
#224770 - $15.00 USD Received 10 Oct 2014; revised 25 Nov 2014; accepted 26 Nov 2014; published 11 Dec 2014 © 2015 OSA 15 Dec 2014 | Vol. 22, No. 25 | DOI:10.1364/OE.22.031337 | OPTICS EXPRESS 31343
u ( r , θ ) = C u ( r , θ ) ,
(10)
where C is the weighted coefficient of the OAM mode with index . Assuming the original orthogonality of OAM modes expressed as u ( r , θ ) 2 rdrdθ , if = 0 u 0 ( r , θ ) u ( r , θ ) = 0 , 0,if ≠ 0
(11)
the OAM mode crosstalk can be described by the conditional probability given by: u ( r , θ ) u ( r , θ )
P ( 0 ) =
I0
2 2
= C ,
(12)
where I0 is the intensity of transmitted OAM mode with the index 0 calculated as
I0 =
D/2
0
2π
0
2
u 0 ( r , θ ) rdrdθ .
(13)
where 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
u ( r , θ ) = u 0 ( r , θ ) exp iξ ( r ,θ ) ,
(14)
where 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,
ξ ( r ,θ ) =
1500 ξ0 ( r ,θ ) , 1550
(15)
where ξ0(r,θ) is the distorted phase of the beacon beam. Accordingly, if there is no AO correction, ξ0(r,θ) would present the generated phase screen, while with the AO correction applied ξ0(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 Ma = 1000 different phase screens, which represents Ma 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.
#224770 - $15.00 USD Received 10 Oct 2014; revised 25 Nov 2014; accepted 26 Nov 2014; published 11 Dec 2014 © 2015 OSA 15 Dec 2014 | Vol. 22, No. 25 | DOI:10.1364/OE.22.031337 | OPTICS EXPRESS 31344
Fig. 5. Crosstalk (a) without AO correction; (b) with AO correction.
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 Cchannel
= max H (Y ) − H (Y | X ) = max − pi Pji log 2 { pi } {Pi } i j
pP i
i
Pji
ji
,
(16)
where H(·) denotes the entropy [32]; X and Y are transmitted and received symbols, respectively; pi is the probability of transmitting OAM modes with index i; and Pji is the channel transfer matrix, where 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. NOAM = 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.
Fig. 6. Channel capacities of OAM-based FSO.
#224770 - $15.00 USD Received 10 Oct 2014; revised 25 Nov 2014; accepted 26 Nov 2014; published 11 Dec 2014 © 2015 OSA 15 Dec 2014 | Vol. 22, No. 25 | DOI:10.1364/OE.22.031337 | OPTICS EXPRESS 31345
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.
Fig. 7. Temporal properties of OAM-based FSO under dynamically evolving atmosphere: (a) Strehl ratio; (b) channel capacity.
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.
#224770 - $15.00 USD Received 10 Oct 2014; revised 25 Nov 2014; accepted 26 Nov 2014; published 11 Dec 2014 © 2015 OSA 15 Dec 2014 | Vol. 22, No. 25 | DOI:10.1364/OE.22.031337 | OPTICS EXPRESS 31346
1 College of Optical Sciences, University of Arizona, Tucson, 85721, USA State Key Laboratory of Information Photonics and Optical Communications, Beijing University of Posts and Telecommunications, Beijing 100876, China * [email protected]
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 OCIS codes: (060.2605) Free-space optical communication; (010.1330) Atmospheric turbulence; (010.1080) Active or adaptive optics.
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#224770 - $15.00 USD Received 10 Oct 2014; revised 25 Nov 2014; accepted 26 Nov 2014; published 11 Dec 2014 © 2015 OSA 15 Dec 2014 | Vol. 22, No. 25 | DOI:10.1364/OE.22.031337 | OPTICS EXPRESS 31337
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1. Introduction An optical beam traveling along z axis can carry a well-defined z-component orbital angular momentum (OAM) of that is associated with the azimuthal phase of the optical field. It has an eigenmode associated with a helical phase factor exp ( iθ ), where , 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
#224770 - $15.00 USD Received 10 Oct 2014; revised 25 Nov 2014; accepted 26 Nov 2014; published 11 Dec 2014 © 2015 OSA 15 Dec 2014 | Vol. 22, No. 25 | DOI:10.1364/OE.22.031337 | OPTICS EXPRESS 31338
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] Φφ (κ ) = 0.49r0−5/3
exp(− κ 2 κ m2 )
(κ
2
+ κ 02 )
11/ 6
, for 0 ≤ κ < ∞,
(1)
where κ denotes spatial frequency; k 0 = 2π/Louter, k m = 5.92/linner, while Louter and linner are the outer and inner scale of turbulence; r0 = (0.423k2Cn2L)-3/5 is the atmospheric coherent diameter (also refered to as Freid parameter); k, L and Cn2 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 Cn2 parameter measures the strength of turbulence, and is regarded as a constant for any horizontal signal
#224770 - $15.00 USD Received 10 Oct 2014; revised 25 Nov 2014; accepted 26 Nov 2014; published 11 Dec 2014 © 2015 OSA 15 Dec 2014 | Vol. 22, No. 25 | DOI:10.1364/OE.22.031337 | OPTICS EXPRESS 31339
path. Without loss of generality, we can assume that Louter = 20m, linner = 3mm for the case when Cn2 = 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 Dφ(r) can be deduced from the phase PSD, which leads to expression ∞ sin κ r Dφ (r ) = 8π κ 2 Φφ (κ ) 1 − dκ , 0 κ r
(2)
where 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], −5 −5 −κ 2 r 2 Dφ (r ) = 3.08r0−5/3 Γ κ m−5/3 1 − 1 F1 ;1; m 4 6 6
9 1/3 2 − κ 0 r , 5
(3)
where 1F1(·) 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 Dφ (r ) theory − Dφ (r )fitting Δ = max max {Dφ (r ) theory }
,
(4)
where Dφ(r)theory are values calculated by Eq. (3), and Dφ(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 computergenerated 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 ShackHartmann 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 ShackHartmann 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]:
#224770 - $15.00 USD Received 10 Oct 2014; revised 25 Nov 2014; accepted 26 Nov 2014; published 11 Dec 2014 © 2015 OSA 15 Dec 2014 | Vol. 22, No. 25 | DOI:10.1364/OE.22.031337 | OPTICS EXPRESS 31340
N
φ p = am Z m ( x, y ),
(5)
m =1
where am is the coefficient of polynomial basis function Zm(·) and 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 (xi, yi) are centroid of the i-th spot (i = 1, 2, … 277).
Fig. 1. Schematic of OAM-based FSO system with AO compensation.
The relationship between the wavefront slopes and the polynomial gradients can be described by the following system of linear equations: ∂Z1 ( x, y ) ∂Z N ( x, y ) ∂φ p ∂x ∂x ∂x ( x1 , y1 ) ( x1 , y1 ) ( x1 , y1 ) ∂Z ( x, y ) φ ∂ , ∂ Z x y ) N ( 1 p a ∂x ∂x 1 ∂x ( x277 , y277 ) ( x277 , y277 ) ( x277 , y277 ) = . (6) ∂Z1 ( x, y ) ∂φ p ∂Z N ( x, y ) aN ∂y ∂y ∂y ( x1 , y1 ) ( x1 , y1 ) ( x1 , y1 ) ∂Z x, y ∂φ p ∂Z N ( x, y ) ( ) 1 y ∂ ∂y ∂ y , x y ( 277 277 ) ( x277 , y277 ) ( x277 , y277 ) 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 [a1, , a56]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
#224770 - $15.00 USD Received 10 Oct 2014; revised 25 Nov 2014; accepted 26 Nov 2014; published 11 Dec 2014 © 2015 OSA 15 Dec 2014 | Vol. 22, No. 25 | DOI:10.1364/OE.22.031337 | OPTICS EXPRESS 31341
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.
Fig. 2. (a) Phase screen generated by Monte Carlo phase screen method; (b) validation results.
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]:
S=
1
π
2
1
2π
0
0
2
exp ( ikφ p ) ρ dρ dϕ ,
(7)
where (ρ, ϕ ) 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.
Fig. 3. Strehl ratio of OAM-based FSO with AO compensation.
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 #224770 - $15.00 USD Received 10 Oct 2014; revised 25 Nov 2014; accepted 26 Nov 2014; published 11 Dec 2014 © 2015 OSA 15 Dec 2014 | Vol. 22, No. 25 | DOI:10.1364/OE.22.031337 | OPTICS EXPRESS 31342
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.
Fig. 4. Wavefront phase distortions resulting from atmospheric turbulence (a) without AO correction; (b) with AO correction.
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 TEM00 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], u ( r , θ , z ) =
ikr 2 z 2 1 2r r2 exp exp − 2 π ! ω ( z ) ω ( z ) 2 ( z 2 + zR2 ) ω ( z )
(8)
z ×exp −i ( + 1) tan −1 exp ( iθ ) , zR
where (r, θ, z) are cylindrical coordinates; ω(z) = ω0[1 + (z/ZR)2]1/2 is the diffraction limited spot size of the fundamental Gaussian beam; ω0 is the beam waist; and ZR is Rayleigh range. For = 0, Eq. (8) is reduced to the zero-order Gaussian beam (i.e. TEM00). The diffraction limited radius of the LG beam at the propagation distance z is given as [31]: r ( z ) =
ω0 + λ z 2
2
2
π 2ω02
+ 1.
(9)
For a given propagation distance and OAM mode index, radius r has a minimum for ω0 = (λz/π)1/2. As an example, for λ = 1550nm and z = 1000m, ω0 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 0 can be now regarded as a superposition of all OAM modes, so we have that,
#224770 - $15.00 USD Received 10 Oct 2014; revised 25 Nov 2014; accepted 26 Nov 2014; published 11 Dec 2014 © 2015 OSA 15 Dec 2014 | Vol. 22, No. 25 | DOI:10.1364/OE.22.031337 | OPTICS EXPRESS 31343
u ( r , θ ) = C u ( r , θ ) ,
(10)
where C is the weighted coefficient of the OAM mode with index . Assuming the original orthogonality of OAM modes expressed as u ( r , θ ) 2 rdrdθ , if = 0 u 0 ( r , θ ) u ( r , θ ) = 0 , 0,if ≠ 0
(11)
the OAM mode crosstalk can be described by the conditional probability given by: u ( r , θ ) u ( r , θ )
P ( 0 ) =
I0
2 2
= C ,
(12)
where I0 is the intensity of transmitted OAM mode with the index 0 calculated as
I0 =
D/2
0
2π
0
2
u 0 ( r , θ ) rdrdθ .
(13)
where 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
u ( r , θ ) = u 0 ( r , θ ) exp iξ ( r ,θ ) ,
(14)
where 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,
ξ ( r ,θ ) =
1500 ξ0 ( r ,θ ) , 1550
(15)
where ξ0(r,θ) is the distorted phase of the beacon beam. Accordingly, if there is no AO correction, ξ0(r,θ) would present the generated phase screen, while with the AO correction applied ξ0(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 Ma = 1000 different phase screens, which represents Ma 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.
#224770 - $15.00 USD Received 10 Oct 2014; revised 25 Nov 2014; accepted 26 Nov 2014; published 11 Dec 2014 © 2015 OSA 15 Dec 2014 | Vol. 22, No. 25 | DOI:10.1364/OE.22.031337 | OPTICS EXPRESS 31344
Fig. 5. Crosstalk (a) without AO correction; (b) with AO correction.
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 Cchannel
= max H (Y ) − H (Y | X ) = max − pi Pji log 2 { pi } {Pi } i j
pP i
i
Pji
ji
,
(16)
where H(·) denotes the entropy [32]; X and Y are transmitted and received symbols, respectively; pi is the probability of transmitting OAM modes with index i; and Pji is the channel transfer matrix, where 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. NOAM = 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.
Fig. 6. Channel capacities of OAM-based FSO.
#224770 - $15.00 USD Received 10 Oct 2014; revised 25 Nov 2014; accepted 26 Nov 2014; published 11 Dec 2014 © 2015 OSA 15 Dec 2014 | Vol. 22, No. 25 | DOI:10.1364/OE.22.031337 | OPTICS EXPRESS 31345
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.
Fig. 7. Temporal properties of OAM-based FSO under dynamically evolving atmosphere: (a) Strehl ratio; (b) channel capacity.
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.
#224770 - $15.00 USD Received 10 Oct 2014; revised 25 Nov 2014; accepted 26 Nov 2014; published 11 Dec 2014 © 2015 OSA 15 Dec 2014 | Vol. 22, No. 25 | DOI:10.1364/OE.22.031337 | OPTICS EXPRESS 31346
