[1] GBUR G, VISSER T D. Coherence vortices in partially coherent beams[J]. Optics Communications, 222, 117-125(2003).

[2] GBUR G, VISSER T D, WOLF E. ‘Hidden’ singularities in partially coherent wavefields[J]. Journal of Optics A： Pure and Applied Optics, 6, S239-S242(2004).

[3] SOSKIN M S, VASNETSOV M V. Singular optics[J]. Progress in Optics, 42, 219-276(2001).

[4] WANG J, YANG J Y, FAZAL I M et al. Terabit free-space data transmission employing orbital angular momentum multiplexing[J]. Nature Photonics, 6, 488-496(2012).

[5] WANG Jian. Advances in communications using optical vortices[J]. Photonics Research, 4, B14-B28(2016).

[6] HANSEN A, SCHULTZ J T, BIGELOW N P. Singular atom optics with spinor Bose-Einstein condensates[J]. Optica, 3, 355-361(2016).

[7] PATERSON L, MACDONALD M P, ARLT J et al. Controlled rotation of optically trapped microscopic particles[J]. Science, 292, 912-914(2001).

[8] LEONHARD J D, SHATOKHIN V N, BUCHLEITNER A. Universal entanglement decay of photonic oribital angular momentum qubit states in atmospheric turbulence[J]. Physics Review A, 91(2015).

[9] LIU Dajun, YIN Hongming, WANG Guiqiu et al. Propagation of partially coherent Lorentz-Gauss vortex beam through oceanic turbulence[J]. Applied Optics, 56, 8785-8792(2017).

[10] WANG Xinguang, YANG Zhen, ZHAO Shengmei. Influence of oceanic turbulence on propagation of Airy vortex beam carrying orbital angular momentum[J]. Optik, 176, 49-55(2019).

[11] LUO Meilan, CHEN Qi, HUA Limin et al. Propagation of stochastic electromagnetic vortex beams through the turbulent biological tissues[J]. Physics Letter A, 378, 308-314(2014).

[12] DING Chaoliang, Baida LÜ, PAN Liuzhan. Evolution of phase singularities of higher-order Bessel-Gaussian pulsed beams[J]. High Power Laser and Particle Beams, 21, 177-182(2009).

[13] GE Xiaolu, WANG Benyi, GUO Liping等. Behavior of phase singularities for laser beam propagating through uplink and downlink atmospheric turbulence paths[J]. High Power Laser and Particle Beams, 30, 121001(2018).

[14] GE Xiaolu, YUE Xifu, WANG Benyi等. Beam spreading and phase singularities’ behavior of non-diffracting vortex beams through turbulent atmosphere[J]. Acta Optica Sinica, 39(2019).

[15] GAO Penghui, LI Jinhong, GUO Miaojun等. Dynamic evolution and propagation trajectories of high order coherent vortices and optical vortices[J]. Journal of Optoelectronics Laser, 27, 773-781(2016).

[16] ZENG Jun, LI Jinhong. Dynamic evolution and classification of coherent vortices in atmospheric turbulence[J]. Optica Applicata, 45, 299-308(2015).

[17] LI Jinhong, GAO Penghui, CHENG Ke et al. Dynamic evolution of circular edge dislocations in free space and atmospheric turbulence[J]. Optics Express, 25, 2895-2908(2017).

[18] YAN Jiawei, YONG Kangle, TANG Shanfa等. Comparison of propagation characteristics between super Gaussian and Gaussian vortex beams in Air[J]. Acta Optica Sinica, 40(2020).

[19] ZAUDERER E. Complex argument Hermite-Gaussian and Laguerre Gaussian beams[J]. Journal of the Optical Society of America A, 3, 465-469(1986).

[20] KIMEL I, ELIAS L R. Relations between Hermite and Laguerre Gaussian modes[J]. IEEE Journal of Quantum Electronics, 29, 2562-2567(1993).

[21] ISHIMARU A. Phase fluctuations in a turbulent medium[J]. Applied Optics, 16, 3190-3192(1977).

[22] ANDREWS L C, PHILLIPS R L[M]. Laser beam propagation through random media(2005).

[23] SCHMITT J M, KUMAR G. Turbulent nature of refractive-index variations in biological tissue[J]. Optics Letters, 21, 1310-1312(1996).

[24] WANG S C H, PLONUS M A. Optical beam propagation of a partially coherent source in the turbulent atmosphere[J]. Journal of the Optical Society of America, 69, 1297-1304(1979).

[25] GRADSHTEYN I S, RYZHIK I M[M]. Table of integrals， series and products(2014).

[26] MANDEL L, WOLF E[M]. Optical Coherence and Quantum Optics(1995).

[27] FREUND I, SHVARTSMAN N. Wave-field phase singularities： The sign principle[J]. Physics Review A, 50, 5164-5172(1994).