[1] Lü Baida, Ma Hong. Coherent and incoherent combinations of off-axis Gaussian beams with rectangular symmetry[J]. Optics Communications, 171, 185-194(1999).

[2] Ji Xiaoling, Li Xiaoqing. The far-field divergence angle and the far-field radiant intensity distribution of Gaussian Schell-model array beams[J]. Acta Physica Sinica, 58, 4624-4629(2009).

[3] Lü Baida, Ma Hong. Coherent and incoherent off-axis Hermite–Gaussian beam combinations[J]. Applied Optics, 39, 1279-1289(2000).

[4] Ren Zhijun, Wu Qiong, Shi Yile, et al. Production of accelerating quad Airy beams and their optical characteristics[J]. Optics Express, 22, 15154-15164(2014).

[5] [5] Rao Ruizhong. Modern atmospheric optics[M]. Beijing: Science Press, 2012

[6] [6] rews L C, Phillips R L. Laser beam propagation through rom media[M]. 2nd ed. Bellingham: SPIE Press, 2005.

[7] [7] Wang Yingjian, Fan Chengyu, Wei Heli. Laser beam propagation applications through the atmosphere sea water[M]. Beijing: National Defense Industry Press, 2015

[8] Sprangle P, Hafizi B, Ting A, et al. High-power lasers for directed-energy applications[J]. Applied Optics, 54, F201-F209(2015).

[9] Wu Shuyun, Luo Xi, Li Xinyang. Adaptive optics for reduction of thermal blooming effects by the phase compensation[J]. Journal of Russian Laser Research, 41, 413-423(2020).

[10] Weyrauch T, Vorontsov M A, Carhart G W, et al. Experimental demonstration of coherent beam combining over a 7 km propagation path[J]. Optics Letters, 36, 4455-4457(2011).

[11] Baykal Y. Field correlations of laser arrays in atmospheric turbulence[J]. Applied Optics, 53, 1284-1289(2014).

[12] Zhou Pu, Ma Yanxing, Wang Xiaolin, et al. Average intensity of a partially coherent rectangular flat-topped laser array propagating in a turbulent atmosphere[J]. Applied Optics, 48, 5251-5258(2009).

[13] Yuan Yangsheng, Cai Yangjian. Scintillation index of a flat-topped beam array in a weakly turbulent atmosphere[J]. Journal of Optics, 13, 125701(2011).

[14] Wang Kuilong, Zhao Chengliang. Propagation properties of a radial phased-locked partially coherent anomalous hollow beam array in turbulent atmosphere[J]. Optics & Laser Technology, 57, 44-51(2014).

[15] Li Xiaoqing, Ji Xiaoling. Angular spread and directionality of the Hermite-Gaussian array beam propagating through atmospheric turbulence[J]. Applied Optics, 48, 4338-4347(2009).

[16] Li Xiaolei, Ji Xiaoling, Eyyuboğlu H T, et al. Turbulence distance of radial Gaussian Schell-model array beams[J]. Applied Physics B, 98, 557-565(2010).

[17] [17] Gebhardt F G. Twentyfive years of thermal blooming: an overview[C]Proceedings of SPIE 1221, Propagation of HighEnergy Laser Beams Through the Earth’s Atmosphere. 1990: 225.

[18] Smith D C. High-power laser propagation: Thermal blooming[J]. Proceedings of the IEEE, 65, 1679-1714(1977).

[19] Fleck J A, Morris J R, Feit M D. Time-dependent propagation of high energy laser beams through the atmosphere[J]. Applied Physics, 10, 129-160(1976).

[20] Qiang Xiwen, Wang Tieliang, Wu Naiqing. Multi-laser-beams atmospheric propagation[J]. Optoelectronic Technology, 19, 167-172(1999).

[21] Banakh V A, Falits A V. Numerical simulation of propagation of laser beams formed by multielement apertures in a turbulent atmosphere under thermal blooming[J]. Atmospheric and Oceanic Optics, 26, 455-465(2013).

[22] [22] Spencer M F, Hyde IV M W. Phased beam projection from tiled apertures in the presence of turbulence thermal blooming[C]Proceedings of SPIE 8877, Unconventional Imaging Wavefront Sensing 2013. 2013: 887703.

[23] Zhang Yuqiu, Hou Tianyue, Chang Hongxiang, et al. Thermal blooming effect and the scaling laws of partial spatially coherent beam array propagating through the atmosphere[J]. Results in Physics, 26, 104444(2021).

[24] Qiu Die, Tian Boyu, Ting He, et al. Mitigation of thermal blooming by rotating laser beams in the atmosphere[J]. Applied Optics, 60, 8458-8465(2021).

[25] Li Xiaoqing, Cao Jianyong, Ding Zhoulin, et al. Influence of fill factors on the thermal blooming of array laser beams in the air[J]. Optik, 182, 314-323(2019).

[26] Ding Zhoulin, Li Xiaoqing, Cao Jianyong, et al. Influence of thermal blooming on the beam quality of an array of Hermite–Gaussian beams propagating in the atmosphere[J]. Applied Optics, 59, 10944-10952(2020).

[27] Li Xiaoqing, Cao Jianyong, Ding Zhoulin, . Thermal blooming effect of flat-topped laser beam array propagating through atmosphere[J]. Acta Optica Sinica, 39, 0126020(2019).

[28] [28] Liu Zejin, Zhou Pu, Xu Xiaojun, et al. Coherent beam combining of high average power fiber lasers[M]. Beijing: National Defense Industry Press, 2016

[29] Shirai T, Dogariu A, Wolf E. Mode analysis of spreading of partially coherent beams propagating through atmospheric turbulence[J]. Journal of the Optical Society of America A, 20, 1094-1102(2003).

[30] Ji Xiaoling, Zhang Entao, Lü Baida. Changes in the spectrum of Gaussian Schell-model beams propagating through turbulent atmosphere[J]. Optics Communications, 259, 1-6(2006).

[31] Ji Xiaoling, Li Xiaoqing. Influence of turbulence on the coherent and incoherent combinations of off-axis Gaussian beams[J]. Acta Physica Sinica, 57, 7674-7679(2008).

[32] Ji Xiaoling, Zhang E T, Lü B D. Propagation of multi-Gaussian beams in incoherent combination through turbulent atmosphere and their beam quality[J]. Journal of Modern Optics, 53, 2111-2127(2006).

[33] Dragoman D. Higher-order moments of the Wigner distribution function in first-order optical systems[J]. Journal of the Optical Society of America. A, 11, 2643-2646(1994).

[34] Chu Xiuxiang. Evolution of an Airy beam in turbulence[J]. Optics Letters, 36, 2701-2703(2011).

[35] Li Xiaoqing, Ji Xiaoling. Propagation of higher-order intensity moments through an optical system in atmospheric turbulence[J]. Optics Communications, 298-299, 1-7(2013).

[36] Dan Youquan, Zhang Bin. Second moments of partially coherent beams in atmospheric turbulence[J]. Optics Letters, 34, 563-565(2009).

[37] Wang S C H, Plonus M A, Ouyang C F. Irradiance scintillations of a partially coherent source in extremely strong turbulence[J]. Applied Optics, 18, 1133-1135(1979).

[38] Leader J C. Atmospheric propagation of partially coherent radiation[J]. Journal of the Optical Society of America, 68, 175-185(1978).

[39] Wigner E. On the quantum correction for thermodynamic equilibrium[J]. Physical Review, 40, 749-759(1932).

[40] Walther A. Radiometry and coherence[J]. Journal of the Optical Society of America, 58, 1256-1259(1968).

[41] [41] Ozaktas H M, Kutay M A, Zalevsky Z. The fractional Fourier transfm with applications in optics signal processing[M]. New Yk: John Wiley & Sons, 2000.

[42] Martínez-Herrero R, Mejías P M, Weber H. On the different definitions of laser beam moments[J]. Optical and Quantum Electronics, 25, 423-428(1993).

[43] Weber H. Propagation of higher-order intensity moments in quadratic-index media[J]. Optical and Quantum Electronics, 24, S1027-S1049(1992).

[44] Li Xiaoqing, Ji Xiaoling, Wang Tao, et al. Matrix formulation of higher-order moments of partially coherent beams propagating through atmospheric turbulence along a slanted path[J]. Journal of Optics, 15, 125720(2013).

[45] Yang Ting, Ji X Liaoling, Li Xiaoqing, et al. Changes of skewness and sharpness of partially coherent decentered annular beams on propagation[J]. Optics Communications, 359, 146-156(2016).

[46] Jiang Shaoen, Sun Jingwen. Simulation and analysis of nonlinear effect of laser atmospheric propagation[J]. Chinese Journal of Lasers, 23, 144-150(1996).

[47] [47] Herman B J, Strugala L A. Method f inclusion of lowfrequency contributions in numerical representation of atmospheric turbulence[C]Proceedings of SPIE. 1990: 183192.

[48] Dai Guangming, Mahajan V N. Zernike annular polynomials and atmospheric turbulence[J]. Journal of the Optical Society of America A, 24, 139-155(2007).

[49] Gebhardt F G, Smith D C. Self-induced thermal distortion in the near field for a laser beam in a moving medium[J]. IEEE Journal of Quantum Electronics, 7, 63-73(1971).

[50] Chen Dongquan, Li Youkuan, Xu Xishen, . Numerical simulation of thermal blooming in atmospheric laser propagation[J]. High Power Laser and Particle Beams, 5, 243-252(1993).

[51] Young C Y, Gilchrest Y V, Macon B R. Turbulence induced beam spreading of higher order mode optical waves[J]. Optical Engineering, 41, 1097-1103(2002).

[52] Cai Yangjian, He Sailing. Average intensity and spreading of an elliptical Gaussian beam propagating in a turbulent atmosphere[J]. Optics Letters, 31, 568-570(2006).

[53] Eyyuboğlu H T, Baykal Y, Sermutlu E. Convergence of general beams into Gaussian intensity profiles after propagation in turbulent atmosphere[J]. Optics Communications, 265, 399-405(2006).

[54] Li Xiaoqing, Chen Xiaowen, Ji Xiaoling. Influence of atmospheric turbulence on the propagation of superimposed partially coherent Hermite-Gaussian beams[J]. Optics Communications, 282, 7-13(2009).

[55] Mahdieh M H. Numerical approach to laser beam propagation through turbulent atmosphere and evaluation of beam quality factor[J]. Optics Communications, 281, 3395-3402(2008).

[56] Dan Youquan, Zhang Bin. Beam propagation factor of partially coherent flat-topped beams in a turbulent atmosphere[J]. Optics Express, 16, 15563-15575(2008).

[57] Shirai T, Dogariu A, Wolf E. Directionality of Gaussian Schell-model beams propagating in atmospheric turbulence[J]. Optics Letters, 28, 610-612(2003).

[58] Ji Xiaoling, Pu Zhengcai. Angular spread of Gaussian Schell-model array beams propagating through atmospheric turbulence[J]. Applied Physics B, 93, 915-923(2008).

[59] Ji Xiaoling, Eyyuboğlu H T, Baykal Y. Influence of turbulence on the effective radius of curvature of radial Gaussian array beams[J]. Optics Express, 18, 6922-6928(2010).

[60] [60] Lü Baida. Laser optics[M]. 3rd ed. Beijing: Higher Education Press, 2003

[61] Wolf E, Collett E. Partially coherent sources which produce the same far-field intensity distribution as a laser[J]. Optics Communications, 25, 293-296(1978).

[62] Ji Xiaoling. Influence of the atmospheric turbulence on propagation properties of laser beams[J]. Journal of Sichuan Normal University (Natural Science), 35, 127-136(2012).

[63] Porras M A, Alda J, Bernabeu E. Complex beam parameter and ABCD law for non-Gaussian and nonspherical light beams[J]. Applied Optics, 31, 6389-6402(1992).

[64] Ricklin J C, Davidson F M. Atmospheric turbulence effects on a partially coherent Gaussian beam: implications for free-space laser communication[J]. Journal of the Optical Society of America A, 19, 1794-1802(2002).

[65] Gbur G, Wolf E. Spreading of partially coherent beams in random media[J]. Journal of the Optical Society of America A, 19, 1592-1598(2002).

[66] Lu Lu, Ji Xiaoling, Deng Jinping, . Influence of non-Kolmogorov turbulence on the spreading of Gaussian array beams[J]. Acta Physica Sinica, 63, 014207(2014).

[67] Van Zandt N R, Cusumano S J, Bartell R J, et al. Comparison of coherent and incoherent laser beam combination for tactical engagements[J]. Optical Engineering, 51, 104301(2012).

[68] Zhou Pu, Wang Xiaolin, Ma Yanxing, et al. Optimal truncation of element beam in a coherent fiber laser array[J]. Chinese Physics Letters, 26, 044206(2009).

[69] [69] Siegman A E. How to (Maybe) measure laser beam quality[C]OSA Trends in Optics Photonic. 1998: 184199.

[70] Wan Min, Su Yi. Computation and analysis on focus shift of laser caused by atmospheric thermal blooming[J]. Chinese Journal of Computational Physics, 19, 449-452(2002).