基于广义惠更斯-菲涅尔原理、交叉谱密度函数，利用修正von Karmon 谱模型以及
ITU-R 颁布的大气折射率结构模型，首先推导出部分相干高斯-谢尔光束在大气湍流中斜程和
水平传输时的光强分布以及光束扩展解析式，然后分别从归一化光强分布和相对束宽的角度
讨论并对比了斜程和水平传输时天顶角、湍流强度、光束相干长度等参数对光束光强分布和
扩展的影响，并给予相应的物理解释. 研究表明：1) 部分相干高斯-谢尔光束在自由空间中水
平传输时的归一化光强与斜程传输时的光强分布相同；2) 天顶角越小，光束的归一化强度分
布越集中，峰值光强越强，同时所受湍流的影响也越小. 3) 在斜程传输情况下，天顶角、湍流
强度、光束相干长度等参数对光束扩展的影响与水平传输几乎一致.
Based on the generalized Huygens-Fresnel principle, we derive the cross-spectral
density function using the modified von Karmon spectrum and atmospheric refractive-index
structure constant issued by ITU-R. We also derive analytic expressions of intensity distribution
and beam width of partially coherent Gaussian-Schell beam propagating in the
atmosphere along slant and horizontal paths. The impact on the beam propagating in
the two ways due to turbulence intensity, zenith, coherence length of the beam and other
parameters are discussed. They are compared respectively from the perspective of normalized
intensity distribution and relative expansion. The results show that the normalized
intensity distribution of the beam in a free space along a horizontal path is the same as
that along a slant path. The normalized intensity distribution is more concentrated, with
the peak much stronger when zenith is smaller, while the effects of the turbulence become
weaker. Impact on the zenith expansion, turbulence, coherence of the beam is almost consistent
with the slant path.
[1]WU J. Propagation of Gaussian-Schell beam through turbulent media [J]. Journal of Modern Optics, 1990, 37(4): 671-684
[2]GREG G, EMIL W. Spreading of partially coherent beams in random media [J]. Journal of the Optical Society of America A-Optics Image Science and Vision, 2002, 19(8): 1592-1598.
[3] 张逸新，王高刚. 斜程大气传输激光束的平均光强与短期光束扩展[J]. 红外与激光工程，2007, 36(2): 167.
ZHANG Y X, WANG G G. Average intensity and short-term beam spread of a laser beam propagating in a slant-path atmosphere [J]. Infrared and Laser Engineering, 2007, 36(2): 167. (in Chinese)
[4] ZHANG Y T, LIU L, WANG F, CAI Y J, DING C L , PAN L Z. Average intensity and spectral shifts of a partially coherent standard or elegant Laguerre-Gaussian beam beyond paraxial approximation [J]. Optical Quant Electron, 2014, 46: 365-379
[5]李晓峰. 星地激光通信链路原理与技术[M]. 北京：国防工业出版社, 2007: 1-46.
[6]Deva K. Borah, David G. Voelz. Spatially partially coherent beam parameter optimization for free space optical communications[J].Optics Express,2010,20(18):20746
[7] 张晓欣，但有权，张彬.湍流大气中斜程传输部分相干光的光束扩展[J].光学学报，2012, 32(12).
Zhang X X, Dan Y Q, Zhang b. Spreading of Partially Coherent Flat-Topped Beams Propagating along a Slant Path in Turbulent Atmosphere [J]. Acta Optica Sinica, 2012, 32(12). (in Chinese)
[8]Duan M L，Li J H ，Wei J L. Influence of different propagation paths on the propagation of laser in atmospheric turbulence[J]. Optoelectronics Letters, 2013, 9(6):1.
[9] 李成强，张合勇，王挺峰，刘立生，郭劲.高斯-谢尔模光束在大气湍流中传输的相干特性的研究[J].物理学报，2013，62（22），224203
Li C Q, Zhang H Y, Wang T F, Liu L S, Guo J. Investigation on coherence characteristics of
Gauss-Schell model beam propagating in atmospheric Turbulence [J]. Acta Physica Sinica , 2013, 62 (22),224203. (in Chinese)
[10] Dan Y Q, Zhang B. Beam propagation factor of partially coherent flat-topped beams in a turbulent atmosphere [J]. Optics Express, 2008, 16 (20):15563.
[11]Andrews L C, Phillips R L. Laser beam propagation through random media [M] . Bellingham: SPIE optical Engineering Press, 2005:195.
[12] 李晋红，吕百达.部分相干涡旋光束通过大气湍流上行和下行传输的比较研究[J].物理学报，2011, 60 (7):4205.
Li J H, Lü B D. Comparative study of partially coherent vortex beam propagations through atmospheric turbulence along a uplink path and a downlink path [J]. Acta Physica Sinica, 2011, 60 (7):4205. (in Chinese)
[13]Wei L, Liu L R, Sun J F，Yang Q G, Zhu Y J. Change in degree of coherence of partially coherent electromagnetic beams propagating through atmosphere turbulence [J].Optics Communications, 2007, 27(1):1-8.
[14] Tomohiro S, Aristide D, Emil W. Mode analysis of spreading of partially coherent beams propagating through atmospheric turbulence [J]. Journal of the Optical Society of America A-Optics Image Science and Vision, 2003, 20(6): 1094-1102.
[15]Korotkova O, Andrews L C. Phillips R L, Speckle propagation through atmospheric turbulence: effects of a random phase screen at the source[C]//2002: 3-11.
