Hybrid Atmospheric Compensation in FreeSpace Optical Communication
 Author: Wang Tingting, Zhao Xiaohui
 Publish: Journal of the Optical Society of Korea Volume 20, Issue1, p13~21, 25 Feb 2016

ABSTRACT
Since the directgradient (DG) method uses the ShackHartmann wave front sensor (SHWFS), based on the phaseconjugation principle, for atmospheric compensation in freespace optical (FSO) communication, it cannot effectively correct highorder aberrations. While the stochastic parallel gradient descent (SPGD) can compensate the distorted wave front, it requires more calculations, which is sometimes undesirable for an FSO system. A hybrid compensation (HC) method is proposed by properly using the DG method and SPGD algorithm to improve the performance of FSO communication. Simulations show that this method can well compensate wavefront aberrations and upgrade the coupling efficiency with few computations, preferable correction results, and rapid convergence rate.

KEYWORD
Freespace optical communication , Hybrid compensation , Coupling efficiency , Convergence rate

I. INTRODUCTION
Freespace optical (FSO) communication is an advanced technology to implement lineofsight transmission of light signals. It transmits laser carrying signals through a freespace channel with high capacity, high bandwidth, and a flexible network [1]. FSO is one of the most promising alternative schemes for addressing the ‘last mile’ communication bottleneck in emerging broadbandaccess markets [2]. However, atmospheric disturbance can easily affect the laser beam. The refractive index of the atmosphere changes randomly because of variations in temperature, humidity, and wind speed in the atmosphere. This will lead to beam wandering, scattering, scintillation, and power fluctuations [3]. The phase and intensity of the laser are distorted, and the coupling efficiency decreases [4, 5]. Consequently the bit error rate (BER) of the communication system is degraded. This arouses the interest of researchers worldwide in studying phase correction, and effective methods are in high demand for FSO communication systems.
Adaptive optical (AO) systems have been successfully applied to FSO to compensate the distorted wave front [39]. Based on the phaseconjugation principle, traditionally there are two main branches. One in [6] is based on a wavefront sensor, such as the SHWFS, to detect the local slope of the wave front and reconstruct the wave front in the Zernike or some other model. With a model, a deformable mirror (DM) can be controlled to construct a conjugated wave front, to offset the aberrations and obtain an approximately planar wave. In this method, the computational cost is very high for large numbers of subapertures in the sensor. Therefore the typically used sensors will fall short when high speed of detection and correction are needed, to implement fast adaptive optics in the FSO system. The other branch is the wavefront sensorless optimization method [3, 9], which aims to optimize the performance metrics of the received laser, such as Strehl ratio (SR), root mean square (RMS) and image sharpness functions, etc. The AO system searches for the suitable voltage to control the DM to optimize the performance metric. Many algorithms, such as simulated annealing (SA), hill climbing, and stochastic parallel gradient descent (SPGD) have been developed [5]. Among them, the SPGD algorithm is widely considered for its simple mechanism and rapid convergence, although hundreds of iterations are needed.
Recently, some new methods have been proposed. Reference [7] proposes a method to process ShackHartmann data by a focalplane approach. This method is more favorable in noise propagation, compared to classical ShackHartmann, and senses more phase modes with fewer subapertures under a comparable computation burden. A phaseretrieval method is used instead of the wavefrontslope method to reconstruct the wave front in reference [8]. The method provides more accurate estimation of aberrations in nearly flat wave fronts. Interestingly, reference [10] proposes a combined approach involving SPGD and DMmodelbased algorithms, to achieve similar correction results to those of SPGD with many fewer iterations. But in this method a SHWFS is not effective, and both SPGD and DMmodelbased algorithms need parameter settings that will increase the complexity of the method. In addition, a trustregion method has been proposed that is superior to both SA and SPGD algorithms, with respect to convergence rate for slowly changing wavefront aberrations, in reference [11].
Considering the rapidly changing atmospheric environment in an FSO system, we propose an HC scheme for wavefront corrections. This method combines the DG method [12] and SPGD algorithm to improve both the results of wavefront compensation and convergence rate.
This paper is organized as follows: In section II, the principles of the DG method and SPGD algorithm are briefly introduced and their deficiencies discussed, then we propose an HC scheme based on a suitable combination of the methods discussed above. In section III, computer simulations using MATLAB are carried out and the results are analyzed, to investigate the compensation capability of the HC scheme. Finally, we give our conclusion in section IV.
II. PROPOSED HYBRID COMPENSATION METHOD
The fundamental scheme of FSO communication is shown in Fig. 1. We adapt optical intensity modulation, and the data signals directly modulate the light source to generate optical signals. When the optical signals are transmitted to an atmospheric channel, they may suffer from atmospheric turbulence, and the receiver may receive optical signals with distorted wave fronts. Therefore, at the receiver we use an AO system to compensate for the aberrations of the distorted wave front. Then the optical signals with corrected wave fronts are coupled into the fiber and detected by the photodetector, to recover the original data signals.
As we know, a wave front distorted by atmospheric turbulence is commonly described by the Zernike polynomials [13], which are a set of polynomials defined on a unit circle. It is convenient to use polar coordinates, so that the polynomials are a product of angular functions and radial polynomials. The wave front is described as [14]
where (
r ,θ ) are the polar coordinates of the pupil,Z_{i} (r ,θ ) is thei th order Zernike polynomial,a i is thei th coefficient, andq is the highest order Zernike polynomial considered. The expressionϕ (r ,θ ) can be transformed into rectangular coordinatesϕ (x ,y ) for calculation.To compensate a distorted wave front, the DG method is frequently used to detect and calculate a wavefront gradient matrix
using SHWFS and the optimal controlG of DM, which is the leastsquares solution ofV where
= [G G _{1x},G _{2x},⋯ ,G_{Mx} ,G _{1y},G _{2y},⋯ G_{My} ,]^{T} contains the local slopes in the horizontal directionx and vertical directiony , detected in every subaperture, and the subscriptM is the number of subapertures of the SHWFS. The matrix K = ( _{1},G _{2},G ⋯ , ) is defined as the gradient response matrix of the DM, andG _{N}N is the number of control elements of the DM.For later use, we give a brief interpretation of Eq. (2). We suppose that the voltages applied to the actuators of the DM are linear, and let
Ψ (x ,y ) be the wave front generated by theN actuators, i.e.where (
x ,y ) are the coordinates of the wave front generated by the DM andI_{n} (x ,y ), n = 1, 2,⋯ ,N is the influence function [5] of then th actuator, andwhere (
x_{n} ,y_{n} ) are the central coordinates of then th actuator,ω is the coupling coefficient,b is the normalized interval between the adjacent actuators, andα is the Gaussian index.On the other hand, the mean local gradient of the distorted wave front in the
m th (m =1, 2,⋯ ,M ) subaperture is regarded as the mean of the partial derivative of the wave frontϕ (x ,y ) in the areaS_{m} , which is the local area of them th subaperture [6]. Based on Eq. (3) and (4), we obtainwhere
Since this method directly calculates the optimal control voltage by the gradient of the wave front, without wavefront reconstruction, it can reduce the computational cost of the AO system. From Eq. (2) we can see that the influence function of DM is a key factor in compensation. To investigate the compensation capacity of the DM, we introduce the definition of compensation error as
where RMS_{o} is the RMS value of the original distorted wave front, and RMS_{c} is the RMS value of the corrected wave front. To test the general correction ability of the DM, the first 35 Zernike polynomials are taken as the targets for the correction, and all of their amplitudes are normalized. Here we consider a DM with 61 elements for the analysis. The compensation error of the 61element DM for each Zernike order from 3 to 35 is shown in Fig. 2. From this figure, we can see that the compensation error increases with increasing Zernike order, which means the DG method is relatively incapable of correcting highorder aberrations.
For this reason, we introduce the SPGD algorithm for the sensorless wavefront compensation given in [3] to correct highorder aberrations. In this algorithm the object of optimization is a performance metric
J for the received laser. This algorithm searches for optimal control = (u u _{1},u _{2},⋯ ,u_{N} ) and obtains the optimalJ iteratively. In addition, the SPGD controller generates a set of statistically independent control perturbations {δu_{j} ^{(k)}},j = 1, 2,⋯ ,N for each iterationk to update the control . The search procedure is carried out as follows:u where