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In aero optics, the linking equation proposed by Sutton is an important equation which can link the fluid-mechanic statistical parameters to the statistical optical degradation parameters. However, in the application of simplified linking equation (SLE) to subsonic flowfields, the weighting function is often ignored. The supersonic mixing layer flowfield is generated in the supersonic wind tunnel. The nanoparticle-based planar laser scattering technology is used to obtain the density field of flowfield. The optics errors between supersonic mixing layer wave-front variances calculated from the SLE and the generalized linking equation are analyzed. The results indicate the validity of using the SLE to estimate the wave-front variance of supersonic mixing layer flowfield. Moreover, the SLE with weighting function has better fitting accuracy than the SLE without weighting function. The weighting function for the application of SLE to the high correlated regions in the supersonic mixing layer is necessary.

Bessel beam is an important member of the family of non-diffracting beams and has some unique properties which can be used in many areas, such as micro particle manipulating, material processing and optical communication. However, the source of Bessel beam generated by the existing methods can be used only in a short distance due to its low power. In this paper, according to the coherent combining technology, we propose a method to generate a second-order Bessel-Gaussian (BG) beam by loading discrete vortex phase on specific spatially distributed Gaussian beam array. The coherent combining technology can enhance the output power by increasing the number of beams and use the phase-locking technique to maintain the beam quality. The experimental scheme is described as follows. The expanded Gaussian beam is first split by an amplitude-based spatial light modulator, then the Gaussian beam array is incident on a phase-only spatial light modulator to load the discrete vortex phase, and finally the Gaussian beam array loaded with phase can synthesize BG beam in free space. Due to the diffraction effect of the sub-beams, the optical field distribution between the adjacent sub-beams which are loaded with phase differences, are superimposed. As a result, the optical field distribution of the approximate beam can be obtained by coherent synthesis in free space. After that, the degree of similarity between simulated results and theoretical data is analyzed by correlation coefficient, including the comparison of light intensity between experiment and simulation, and the power-in-the-bucket is used to evaluate beam quality. In addition, the topological charge of the synthesized BG beams is verified by the interference method. By studying the number of beams, the waist radius and the radius of the ring, we find some interesting results which are summarized as follows. Firstly, the closed arrangement of Gaussian beam arrays can improve the quality of the synthesized BG beam. Secondly, the smaller the phase difference between the sub-beams, the more easily the discontinuous piston phase approaches to the vortex phase. Therefore, increasing the number of sub-beams can significantly improve the beam quality of the synthesized BG beam and obtain a higher order synthetic BG beam. Finally, we define the parameter k to represent the tightness of a circular array of Gaussian beams. The present study shows that when the parameter k is close to 1, the best experimental results can be obtained. Therefore, the proposed method has important guidance in generating various vortex beams or enhancing the vortex beam power.