Hybrid Atmospheric Compensation in Free-Space Optical Communication

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  • ABSTRACT

    Since the direct-gradient (DG) method uses the Shack-Hartmann wave front sensor (SH-WFS), based on the phase-conjugation principle, for atmospheric compensation in free-space optical (FSO) communication, it cannot effectively correct high-order 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 wave-front aberrations and upgrade the coupling efficiency with few computations, preferable correction results, and rapid convergence rate.


  • KEYWORD

    Free-space optical communication , Hybrid compensation , Coupling efficiency , Convergence rate

  • I. INTRODUCTION

    Free-space optical (FSO) communication is an advanced technology to implement line-of-sight transmission of light signals. It transmits laser carrying signals through a free-space 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 broadband-access 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 [3-9]. Based on the phase-conjugation principle, traditionally there are two main branches. One in [6] is based on a wave-front sensor, such as the SH-WFS, 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 wave-front 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 Shack-Hartmann data by a focal-plane approach. This method is more favorable in noise propagation, compared to classical Shack-Hartmann, and senses more phase modes with fewer subapertures under a comparable computation burden. A phase-retrieval method is used instead of the wave-front-slope 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 DM-model-based algorithms, to achieve similar correction results to those of SPGD with many fewer iterations. But in this method a SH-WFS is not effective, and both SPGD and DM-model-based algorithms need parameter settings that will increase the complexity of the method. In addition, a trust-region method has been proposed that is superior to both SA and SPGD algorithms, with respect to convergence rate for slowly changing wave-front aberrations, in reference [11].

    Considering the rapidly changing atmospheric environment in an FSO system, we propose an HC scheme for wave-front corrections. This method combines the DG method [12] and SPGD algorithm to improve both the results of wave-front 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]

    image

    where (r, θ) are the polar coordinates of the pupil, Zi(r, θ) is the ith order Zernike polynomial, ai is the ith coefficient, and q is the highest order Zernike polynomial considered. The expression ϕ(r, θ) can be transformed into rectangular coordinates ϕ(x, y) for calculation.

    To compensate a distorted wave front, the DG method is frequently used to detect and calculate a wave-front gradient matrix G using SH-WFS and the optimal control V of DM, which is the least-squares solution of

    image

    where G = [G1x, G2x, , GMx, G1y, G2y, GMy,]T contains the local slopes in the horizontal direction x and vertical direction y, detected in every subaperture, and the subscript M is the number of subapertures of the SH-WFS. The matrix K = (G1, G2, , GN) is defined as the gradient response matrix of the DM, and 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 the N actuators, i.e.

    image

    where (x, y) are the coordinates of the wave front generated by the DM and In(x, y), n = 1, 2, , N is the influence function [5] of the n th actuator, and

    image

    where (xn, yn) are the central coordinates of the nth actuator, ω is the coupling coefficient, b is the normalized interval between the adjacent actuators, and α is the Gaussian index.

    On the other hand, the mean local gradient of the distorted wave front in the mth (m=1, 2,, M) subaperture is regarded as the mean of the partial derivative of the wave front ϕ(x, y) in the area Sm, which is the local area of the m th subaperture [6]. Based on Eq. (3) and (4), we obtain

    image

    where

    image

    Since this method directly calculates the optimal control voltage by the gradient of the wave front, without wave-front 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

    image

    where RMSo is the RMS value of the original distorted wave front, and RMSc 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 61-element 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 high-order aberrations.

    For this reason, we introduce the SPGD algorithm for the sensorless wave-front compensation given in [3] to correct high-order aberrations. In this algorithm the object of optimization is a performance metric J for the received laser. This algorithm searches for optimal control u = (u1, u2, , uN) and obtains the optimal J iteratively. In addition, the SPGD controller generates a set of statistically independent control perturbations {δuj(k)}, j = 1, 2, , N for each iteration k to update the control u. The search procedure is carried out as follows:

    image

    where

    image
    image