We have experimentally generated a radially polarized (RP) beam with controllable state of spatial coherence by using a rotating ground-glass plate and a radial polarization converter. Furthermore, experimental and theoretical analysis of the focusing properties of a partially coherent RP beam is carried out. Our results show that we can shape the beam profile of the focused RP beam by varying its initial spatial coherence, which is useful for material thermal processing and particle trapping. The experimental results are in complete agreement with the theoretical predictions.

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In order to improve the effects of denoising, this paper introduces the basic principles of wavelet threshold denoising and traditional structures threshold functions. Meanwhile, it proposes wavelet threshold function and fixed threshold formula which are both improved here. First, this paper studies the problems existing in the traditional wavelet threshold functions and introduces the adjustment factors to construct the new threshold function basis on soft threshold function. Then, it studies the fixed threshold and introduces the logarithmic function of layer number of wavelet decomposition to design the new fixed threshold formula. Finally, this paper uses hard threshold, soft threshold, Garrote threshold, and improved threshold function to denoise different signals. And the paper also calculates signal-to-noise (SNR) and mean square errors (MSE) of the hard threshold functions, soft thresholding functions, Garrote threshold functions, and the improved threshold function after denoising. Theoretical analysis and experimental results showed that the proposed approach could improve soft threshold functions with constant deviation and hard threshold with discontinuous function problems. The proposed approach could improve the different decomposition scales that adopt the same threshold value to deal with the noise problems, also effectively filter the noise in the signals, and improve the SNR and reduce the MSE of output signals.

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This article investigates the consensus tracking control problem of the leader-follower spacecraft formation system in the presence of model uncertainties, external disturbances, and actuator saturation, where the relative motion of the leader and the follower need to track a desired time-varying trajectory given in advance. First, the six-degree-of-freedom relative-coupled translational and rotational dynamics models are derived using the exponential coordinates on the Lie group SE(3). Then, a fast terminal sliding mode control law is proposed to guarantee the tracking control objective come true in finite time robust against all the aforementioned drawbacks. As a stepping stone, an extended state observer is designed to estimate and compensate the total composed disturbances of the system, and it is proved that the estimate errors can converge to a really small neighborhood of the origin in finite time. Based on the observer information, a less-conservative modified controller is furthermore developed to eliminate the chattering caused by the signum function. The stability of the closed-loop system is shown by theoretical analysis. Finally, the validity of the proposed schemes is illustrated through numerical simulations.