Propagation factor of a truncated partially coherent flat-topped beam in turbulent atmosphere

Analytical expression for the propagation factor of a truncated partially coherent flat-topped (FT) beam in turbulent atmosphere is derived based on the extended Huygens-Fresnel integral and the second-order moments of the Wigner distribution function. Numerical results show that the radius of the aperture (i.e., truncation parameter) has strong influence on the evolution properties of the propagation factor of a truncated partially coherent FT beam, and the advantage of a truncated partially coherent FT beam over a truncated Gaussian Schell-model (GSM) beam or a truncated coherent FT beam for overcoming the turbulence-induced degradation disappears gradually with the decrease of the radius of the aperture.     

Partially coherent beam and its applications

The recent progresses on the propagation, generation and application of partially coherent beam (PCB) are reviewed in this paper. A new tensor ABCD law for treating the propagation of partially coherent Gaussian-Schell model (GSM) beams through a paraxial optical system is derived. The focusing, spectral shift and fractional Fourier transform of the GSM beam are investigated by using the tensor method. The generation of PCB with special optical resonator is studied both theoretically and experimentally. Furthermore, the ghost imaging and superluminal propagation of the PCB is discussed. The results show clearly that the coherence of light have strong influences on the ghost imaging and superluminal propagation.      

Off-axis Gaussian Schell-model beam and partially coherent laser array beam in a turbulent atmosphere

The propagation of an off-axis Gaussian Schell-model (GSM) beam in a turbulent atmosphere is investigated based on the extended Huygens-Fresnel integral formula. Analytical formulae for the cross-spectral density and corresponding partially coherent complex curvature tensor of an off-axis GSM beam propagating in a turbulent atmosphere are derived. Based on these formulae, the propagation properties of such kind of beam in a turbulent atmosphere are investigated in detail. Furthermore, the methods are extended to investigate the propagation properties of a partially coherent laser array beam in a turbulent atmosphere. The properties of an off-axis GSM beam and a partially coherent laser array beam in a turbulent atmosphere are closely related with the beam parameters and the structure constant of the turbulent atmosphere.     

Circular partially coherent flattened Gaussian beam

An alternative theoretical model called circular partially coherent flattened Gaussian beam (FGB) is developed to describe a circular partially coherent beam with a flat-topped spatial profile. Explicit expression for the propagation factor of a circular partially coherent FGB is derived. We drive the analytical formulae for the cross-spectral density and mean-squared beam width of a circular partially coherent FGB propagating through a paraxial ABCD optical system based on the generalized Collins formula. The intensity, spreading and directionality properties of a circular partially coherent FGB propagating in free space are studied as numerical examples. The propagation properties of a circular partially coherent FGB agree well with those of a partially coherent flat-topped beam reported in the literature. Thus, our model provides an alternative but reliable model for describing a circular partially coherent beam with flat-topped profile.     

Intensity distribution and focal depth of axicon illuminated by Gaussian Schell-model beam

In view of second-order coherence theory, the generalized diffractive axicon illuminated by partially coherent Gaussian Schell-model (GSM) beams is analyzed. An analytical equation for focal depth of axicon without aperture illuminated by spatial partially coherent GSM beam is derived. We showed that the off-axis images can be evaluated by using the method of stationary phase at any correlation state, hence we demonstrated that the intensity distribution of diffracted beam is a superposition of Bessel Beams of various orders. We also showed that the diffracted beam is nondiffractive only on the boundaries of the focal depth.     

Twist phase-induced reduction in scintillation of a partially coherent beam in turbulent atmosphere

The scintillation index of a Gaussian Schell-model beam with twist phase (i.e., twisted GSM beam) in weak turbulent atmosphere is formulated with the help of a tensor method. Variations of the scintillation index of a twisted GSM beam on propagation in turbulent atmosphere are studied in detail. It is interesting to find that the scintillation index of a twisted GSM beam can be smaller than that without twist phase in weak turbulent atmosphere. Thus, modulation of the twist phase of a partially coherent beam provides a new way to reduce turbulence-induced scintillation.     

部分相干光对周期性局域空心光束的影响

用旋转毛玻璃和光阑把激光变成部分相干光, 再经过双轴棱锥系统把一束平行光变成两束同频率但不同径向波矢分量的无衍射贝塞尔光, 相干叠加产生了部分相干的周期性局域空心光束. 通过干涉理论与实验结果相互佐证, 得出局域空心光束的周期为2.5 mm. 进一步探究入射光场相干度对产生局域空心光束的影响, 发现随着相干度的降低局域空心光束中心暗斑与周围光强的衬比度会降低, 但不影响局域空心光束的周期以及中心暗斑尺寸.     

Effects of different correlations of partially coherent electromagnetic beam on three-dimensional spectral intensity distribution in the focal region

The effects of position-dependent correlations between the x- and y-components of the partially coherent incident electromagnetic beam on the spectral intensity distribution in the focal plane and along the z-axis of an optical imaging system with a high-numerical aperture are studied. The electromagnetic Gaussian Schell-model (GSM) beam is employed as an example to illustrate the analysis.     

伪部分相干高斯-谢尔模型光束在湍流大气中传播的闪烁孔径平滑效应

采用光传播的数值模拟方法, 对伪部分相干高斯-谢尔模型光束大气传播过程进行数值模拟, 统计分析不同接收口径内的光强起伏特性,计算闪烁的孔径平滑因子, 讨论了表征伪部分相干光的调制相位屏的相对变化频率对闪烁指数的影响, 并将其与充分发展的部分相干高斯-谢尔模型光束及完全相干光的闪烁指数对比. 结果表明: 降低光源相干性可大幅度降低闪烁指数, 但同时其闪烁指数的孔径平滑效应减弱, 在相同接收孔径下, 伪部分相干光闪烁指数的孔径平滑效果要比完全相干光差; 增大调制相位屏的相对变化频率可在一定程度上降低闪烁指数, 随着其相对频率的增大, 伪部分相干光的闪烁指数与部分相干光的结果趋于一致. 关键词： 伪部分相干高斯-谢尔模型光束 湍流大气 孔径平滑闪烁指数 数值模拟     

Paraxial propagation of a partially coherent flattened Gaussian beam through apertured ABCD optical systems

By expanding the hard aperture function into a finite sum of complex Gaussian functions, some approximate analytical formulae for the cross-spectral density of a partially coherent flattened Gaussian beam (FGB) propagating through apertured aligned and misaligned ABCD optical systems are derived based on the generalized Collins formula. The results obtained by using the approximate analytical formula are in good agreement with those obtained by using the numerical integral calculation. As a numerical example, the focusing properties (including average irradiance distribution and focal shift) of a partially coherent FGB focused by an apertured thin lens are studied, and it is found that the focusing properties of a partially coherent FGB are closely related to its initial coherence and the aperture width. Our results provide an effective and fast way for studying the paraxial propagation of a partially coherent FGB through apertured ABCD optical systems.     

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

基于广义惠更斯-菲涅耳原理,以高斯-谢尔模型(GSM)涡旋光束作为典型的部分相干涡旋光束,推导出GSM涡旋光束通过大气湍流斜程传输的平均光强、均方根束宽和交叉谱密度函数的解析表达式,并用以研究了大气湍流中上行和下行对GSM涡旋光束传输和对相干涡旋的影响.结果表明,在相同条件下,GSM涡旋光束下行传输受大气湍流的影响要小于上行传输,下行传输时相干涡旋拓扑电荷守恒距离要长于上行传输.对所得结果做了物理解释. 关键词： 部分相干涡旋光束 相干涡旋 大气湍流 上行和下行传输     

M 2-factors of a non-circular partially coherent flat-topped beam

Based on the extended Huygens-Fresnel integral and the second-order moments of the Wigner distribution function, analytical formulae for the propagation factors (known as M ²-factors) of a non-circular (i.e., rectangular or elliptical) partially coherent flat-topped beam in turbulent atmosphere are derived. The properties of the M ²-factors of a non-circular partially coherent flat-topped beam in turbulent atmosphere and in free space are studied numerically and comparatively. It is found that the evolution properties of the M ²-factors are mainly determined by the parameters of the beam and the turbulent atmosphere. The relative M ²-factors of a non-circular partially coherent flat-topped beam can be smaller than a circular partially coherent flat-topped beam and a Gaussian Schell-model beam, particularly at long propagation ranges in turbulent atmosphere. Our results will be useful in long-distance free-space optical communications.     

Characteristics of a partially coherent Gaussian Schell-model beam propagating in slanted atmospheric turbulence

On the basis of the extended Huygens-Fresnel principle and the model of the refractive-index structure constant in the atmospheric turbulence proposed by the International Telecommunication Union-Radio Communication Sector,the characteristics of the partially coherent Gaussian Schell-model(GSM) beams propagating in slanted atmospheric turbulence are studied.Using the cross-spectral density function(CSDF),we derive the expressions for the effective beam radius,the spreading angle,and the average intensity.The variance of the angle-of-arrival fluctuation and the wander effect of the GSM beam in the turbulence are calculated numerically.The influences of the coherence degree,the propagation distance,the propagation height,and the waist radius on the propagation characteristics of the partially coherent beams are discussed and compared with those of the fully coherent Gaussian beams.     

激光器的部分相干性对光纤激光器阵列相干合成远场图样的影响

考虑光纤激光器阵列相干合成的远场分布时，一般都认为单个激光器的输出光束是理想的高斯光束，实际上受各种因素的影响，一般激光器的输出光束都是一部分相干光.从部分相干光的高斯-谢尔模型出发，首次推导出了当单个激光器的输出光束是部分相干光时，激光器阵列相干合成的远场光强分布模型，并给出了数值模拟计算结果. 关键词： 光纤激光器阵列 相干合成 部分相干光     

Generalized beam propagation factors of truncated partially coherent cosine-Gaussian and cosh-Gaussian beams

Based on the method of truncated second-order moments, the analytical expressions of the generalized beam propagation factor for truncated partially coherent cosine-Gaussian and cosh-Gaussian beams are derived. The beam propagation factors of the truncated partially coherent cosine-Gaussian and cosh-Gaussian beams depend on the acentric parameter, the truncation parameter, and the coherent parameter. According to the derived expressions, the beam propagation factors are illustrated and analyzed with numerical examples, and the influences of the different parameters on the beam propagation factors are also discussed in detail.     

截断部分相干厄米-高斯光束的广义M2G因子

基于广义截断二阶矩法,推导出了截断部分相干厄米-高斯(H-G)光束的广义M²_G因子的解析表达式. 截断完全相干H-G光束、截断高斯谢尔模型(GSM)光束以及截断高斯光束可以做为本文结果的特例给出. 研究表明:截断部分相干H-G光束的广义M²_G因子与截断参数δ,模阶数m以及相干参数α有关. 当δ非常小时,M^{2关键词： 截断部分相干厄米高斯光束 M2_G因子')" href="#">广义M2_G因子 广义截断二阶矩法}     

Effect of spatial coherence on the scintillation properties of a dark hollow beam in turbulent atmosphere

With the help of a tensor method, we derive an explicit expression for the on-axis scintillation index of a circular partially coherent dark hollow (DH) beam in weakly turbulent atmosphere. The derived formula can be applied to study the scintillation properties of a partially coherent Gaussian beam and a partially coherent flat-topped (FT) beam. The effect of spatial coherence on the scintillation properties of DH beam, FT beam and Gaussian beam is studied numerically and comparatively. Our results show that the advantage of a DH beam over a FT beam and a Gaussian beam for reducing turbulence-induced scintillation increases particularly at long propagation distances with the decrease of spatial coherence or the increase of the atmospheric turbulence, which will be useful for long-distance free-space optical communications.     

The propagation of radially polarized partially coherent beam through an optical system in turbulent atmosphere

The propagation of a radially polarized partially coherent beam through an optical system in turbulent atmosphere is investigated. Analytical propagation formulae for the radially polarized partially coherent beam through an optical system in turbulent atmosphere are derived, and some analyses are illustrated by numerical examples related to propagation properties of the radially polarized partially coherent beam through a double-lens system in turbulent atmosphere. It is shown that the optical system, structure constant of turbulent atmosphere and the initial coherent length have significant influence on intensity, degree of polarization and degree of coherence for radially polarized partially coherent beam.     

Propagation properties of the beam generated by Gaussian mirror resonator passing through a paraxial ABCD optical system

By expanding a hard aperture function into a finite sum of complex Gaussian functions, approximate propagation formula is derived in the situation that the beam generated by Gaussian mirror resonator passes through a paraxial ABCD optical system with an annular aperture. The corresponding forms for a circular aperture and a circular black screen are also given. Some numerical simulations are shown to illustrate propagation properties and focusing properties of the beam passing through a paraxial ABCD optical system with the three different kinds of aperture.     

Experimental observation of lensless coincidence fractional Fourier transform with a Gaussian Schell-model beam

A modified lensless optical system for implementing coincidence fractional Fourier transform (FRT) is proposed, and the conditions for the optical system to implement the coincidence FRT with incoherent or partially coherent light are discussed. Furthermore, we report the experimental observation of lensless coincidence FRT of an object (double slits) with a typical partially coherent beam - Gaussian Schell-model (GSM) beam. The experimental results are analyzed and agree reasonably well with the theoretical results.