The effects of the jet engine turbulence on a Gaussian Schell-Model (GSM) beam array propagating in a jet engine plume are studied theoretically. The analytical expressions of the cross-spectral density (CSD) function, the spectral degree of coherence (DOC) and the mean-squared beam width for a GSM beam array propagating through a jet engine plume region are derived based on the extended Huyens-Fresnel integral. It is found that the profile of spectral density and the DOC of a GSM beam array gradually transfer to an elliptical shape as the propagation distance increases, and the elliptical degree becomes greater with increasing source coherence length. Besides, as the source coherence length decreases, the effect of the jet engine turbulence on the beam broadening is smaller and the profile of the DOC appears sidelobes at short distance. It is also found that the values of effective coherence width nearly remain the same for different values of source coherence length after propagating a certain distance. Furthermore, the effects of the number of beamlets on the beam width and effective coherence width are investigated in detail.

The exhaust from engines close to the fuselage can cause extreme turbulence and have a significant impact on the performance of airborne laser systems. In this paper, the combined effect of the jet engine and atmospheric turbulence on propagation of a Gaussian Schell-model (GSM) array beam over long distances is studied in detail. Based on the two-step propagation method, the analytical expressions of the cross-spectral density (CSD) function, mean-squared beam width and spectral degree of spatial coherence (DOC) of GSM array beams are derived. It is shown that the jet engine turbulence has a significant effect on the beam broadening even at a very short distance. As the propagation distance in the jet engine turbulence increases, the beam width is less effected by the spatial coherence. Meanwhile, the resistance of lower spatially coherent beam to the turbulence increases as the distance of the jet engine turbulence or the strength of the atmospheric turbulence increases. It is demonstrated that the effect of jet engine turbulence on the beam spreading can be equivalent to moderate or even severe atmospheric turbulence. Moreover, the DOC is a function with oscillatory phenomenon when the propagation distance is short or the strength of turbulence is low.

We investigate the focusing properties of cylindrical vector beams (CVBs) generated from the combination of an array of beams, each with sub-apertures and controllable polarization. The analytical expression of the tight focusing field of the combined CVBs has been derived based on the Richard-Wolf vector diffraction integral. To obtain a desired focal spot size which includes efficient sidelobe suppression, the required parameters, such as the exit sub-aperture, numerical aperture and truncation parameter, have been studied in detail. The result shows that the combined CVB distribution has a good match with the theoretical ideal CVB distribution. However, compared with the ideal CVBs, the focal spot width produced by the combined radially polarized beams is smaller. With the increase of initial polarization rotation of sub-aperture, the focal spot width increases, and the focal shape shifts from Gaussian-like to a flat-topped distribution and then to an annular distribution. Furthermore, flexible focal field tailoring can also be realized by adjusting the initial polarization rotation of each sub-aperture. These results might provide a valuable reference for material processing, microlithography and multi-particle manipulation.

We correct a few typographical errors in the original manuscript. In our paper [1], some inadvertent typing errors have been found and corrected as below. The word "Huygens-Fresnel" is misspelled in the Abstract section and in the last paragraph of Introduction section. "1 − 10 × 10 10 m −2/3 " should be revised to "1 − 10 × 10 −10 m −2/3 " in the first paragraph of Introduction section. The title of section 3.2 should be revised to "3.2 The Spectral Degree of Spatial Coherence".

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We theoretically propose and experimentally demonstrate a flexible approach to generate an arbitrary shape-invariant structured array beam during propagation with the prescribed intensity distribution. An algorithm for generating flexible desired phase is presented according the reverse engineering and Huygens-Fresnel theory. In this paper, we outline the theory for creating the shape-invariant array beams. The numerical simulation results show that the shape of generated array beams is invariant and its size increases gradually from the near field to the far field. The generation of shape-invariant structured array beams can be realized by the phase-only and amplitude-only elements encoded on the spatial light modulators. The generated arbitrary shape-invariant structured array beams are verified and are in excellent agreement with the theory.