Angular spread of Gaussian Schell-model array beams propagating through atmospheric turbulence

The closed-form expression for the angular spread of Gaussian Schell-model (GSM) array beams propagating through atmospheric turbulence is derived. It is shown that the angular spread θ _sp of GSM array beams for the superposition of the cross-spectral density function is smaller than of those for the superposition of the intensity. However, the θ _sp of GSM array beams for the superposition of the intensity is less sensitive to turbulence than that for the superposition of the cross-spectral density function. For the superposition of the cross-spectral density function, θ _sp of GSM array beams with smaller coherence length σ ₀, smaller waist width w ₀, smaller beam number N, and larger separation distance x _d are less affected by turbulence than of those with larger σ ₀,w ₀,N, and smaller x _d ; while, for the superposition of the intensity, the effect of turbulence on θ _sp is independent of N and x _d . In addition, the angular spread is nearly the same for the two types of superposition when σ ₀ or w ₀ is small enough, or x _d is large enough. On the other hand, it is found that there exist equivalent GSM array beams for the two types of superposition which may have the same directionality as the corresponding fully coherent Gaussian beam in free space and also in turbulence.     

Directionality of Gaussian Schell-model beams propagating in atmospheric turbulence

It is known that some partially coherent Gaussian Shell-model beams may generate, in free space, the same angular distribution of radiant intensity as a fully coherent laser beam. We show that this result also holds even if the beams propagate in atmospheric turbulence, irrespective of the particular model of turbulence used. The result is illustrated by an example.     

大气湍流对斜程传输准单色高斯-谢尔光束空间相干性的影响

由湍流大气中斜程传输时准单色高斯-谢尔(GSM)光束互相干函数的解析式导出了该光束的复相干度.然后，利用表征光束空间相干性的横向相干长度，研究了斜程传输时大气湍流对准单色GSM光束空间相干性的影响.研究结果表明：1)当传输路径偏离水平方向较大(即θ≤88°)时，准单色GSM光束横向相干长度随传输距离均为先迅速增加，后缓慢增加，最后基本保持不变.2)当传输路径接近水平方向(即θ≥89°)时，准单色GSM光束横向相干长度随传输距离均为先增大，达到一个最大值后开始下降并持续减小.3)     

大气湍流对斜程传输准单色高斯-谢尔光束空间相干性的影响

由湍流大气中斜程传输时准单色高斯-谢尔(GSM)光束互相干函数的解析式导出了该光束的复相干度.然后,利用表征光束空间相干性的横向相干长度,研究了斜程传输时大气湍流对准单色GSM光束空间相干性的影响.研究结果表明：1)当传输路径偏离水平方向较大(即θ≤88°)时,准单色GSM光束横向相干长度随传输距离均为先迅速增加,后缓慢增加,最后基本保持不变.2)当传输路径接近水平方向(即θ≥89°)时,准单色GSM光束横向相干长度随传输距离均为先增大,达到一个最大值后开始下降并持续减小.3) 关键词： 大气光学 空间相干性 高斯-谢尔光束 斜程传输     

Cross-polarization properties of two Gaussian Schell-model beams through non-Kolmogorov turbulence

General expressions are derived for the spectral degree of cross-polarization (SDCP) of a beam generated by the superposition of two Gaussian Schell-model (GSM) beams, which illuminated with the same Gaussian Schell-mode source propagating in non-Kolmogorov turbulent atmosphere by adopting beam cross-spectral density matrix and Young's interference theory. In particular, through numerical examples based on our analytical formal the SDCP of two GSM beams is analyzed. Detailed analysis demonstrate that the SDCP is closely to the spacing of two beams on source plane as well as the strength of the atmospheric turbulent, but the fractal constant α has no affect on the SDCP.     

倾斜离轴高斯-谢尔模型光束在大气湍流中通过猫眼光学镜头反射光的光强特性

本文研究了大气湍流和探测光空间相干性对倾斜离轴激光束通过猫眼光学 镜头反射光光强特性的影响. 研究表明: 由于光阑效应以及光束倾斜或离轴, 反射光光强出现振荡和非对称分布现象. 大气湍流不会改变其非对称性, 但湍流中其光强不再振荡. 当猫眼光学镜头存在微弱正离焦δ_max情况下, 轴上光强可达到最大值. 猫眼光学镜头焦距越大, 所需δ_max越大. 但是, 大气湍流和探测光相干性好坏都不会改变所需δ_max值. 在大气湍流中传输距离足够远时, 反射光强会成为离轴类高斯分布. 随着传输距离的增大, 相干性越差的探测光的反射光束扩展可以更小, 这与部分相干光自由空间传输规律完全不同. 探测光相干性越好, 其反射光强受湍流的影响越大, 但差异不大. 本文所得结果对激光主动探测的应用具有意义. 关键词： 猫眼光学镜头 大气湍流 部分相干光 光强     

Effective Rayleigh range of Gaussian array beams propagating through atmospheric turbulence

The analytical expressions for the effective Rayleigh range z_R of Gaussian array beams in turbulence for both coherent and incoherent combinations are derived. It is shown that z_R of Gaussian array beams propagating through atmospheric turbulence depends on the strength of turbulence, the array beam parameters and the type of beam combination. For the coherent combination z_R decreases due to turbulence. However, for the incoherent combination there exists a maximum of z_R as the strength of turbulence varies. The z_R of coherently combined Gaussian array beams is larger than that of incoherently combined Gaussian array beams, but for the coherent combination case z_R is more sensitive to turbulence than that for the incoherent combination case. The larger the beam number is, the longer z_R is, and the more z_R is affected by turbulence. For the coherent combination z_R is not monotonic versus the relative beam separation distance, and the effect of turbulence on z_R is appreciable within a certain range of the relative beam separation distance.     

Changes in the coherence of quasi-monochromatic electromagnetic Gaussian Schell-model beams propagating through turbulent atmosphere

Based on the extended Huygens-Fresnel principle, the mutual coherence function of quasi-monochromatic electromagnetic Gaussian Schell-model (EGSM) beams propagating through turbulent atmosphere is derived analytically. By employing the lateral and the longitudinal coherence length of EGSM beams to characterize the spatial and the temporal coherence of the beams, the behavior of changes in the spatial and the temporal coherence of those beams is studied. The results show that with a fixed set of beam parameters and under particular atmospheric turbulence model, the lateral coherence of an EGSM beam reaches its maximum value as the beam propagates a certain distance in the turbulent atmosphere, then it begins degrading and keeps decreasing along with the further distance. However, the longitudinal coherence length of an EGSM beam keeps unchanging in this propagation. Lastly, a qualitative explanation is given to these results.     

Beam quality changes of Gaussian Schell-model beams propagating through axicons

Beam quality changes of Gaussian Schell-model (GSM) beams propagating through axicons are investigated in detail. The expressions showing beam quality changes of GSM beams are derived in terms of Wigner distribution function. It is shown that, when a GSM beam passes through a divergent or convergent axicon, the beam quality factor (M2 factor) becomes larger. The increment of the M2 factor is dependent on the incident beam size, wave number, refractive index of the axicon material and the angle of the axicon; but it is independent of the coherence of the beam. The explanation for the increment of the M2 factor is also given.     

Long-distance propagation of pseudo-partially coherent Gaussian Schell-model beams in atmospheric turbulence

Propagation properties of spatially pseudo-partially coherent Gaussian Schell-model beams through the atmo- spheric turbulence over a long-distance uplink path are studied by numerical simulation. A linear coordination trans-formation is introduced to overcome the window effect and the loss-of-resolution problem. The beam spreading, beam wandering, and intensity scintillation as functions of turbulence strength, source correlation length, and change frequency of random phase that models the partial coherence of the source are analyzed. It is found that the beam spreading and the intensity scintillation of the partially coherent beam are less affected by the turbulence than those of the coherent one, but it suffers from a more severe diffractive effect, and the change frequency of random phase has no evident influence on it. The beam wandering is insensitive to the variation of source correlation length, and decreases firstly then goes to a fixed value as the change frequency increases.     

Partially coherent vortex beams propagating through slant atmospheric turbulence and coherence vortex evolution

Using the extended Huygens-Fresnel principle and the quadratic approximation of the phase structure function, and taking the Gaussian Schell-model (GSM) vortex beam as a typical example of partially coherent vortex beams, the explicit expressions for the cross-spectral density function and average intensity of GSM vortex beams with topological charge m=+1 propagating through slant atmospheric turbulence are derived, and used to study the propagation properties of GSM vortex beams in atmospheric turbulence along a slant path and evolution behavior of coherence vortices. It is shown that the spreading of GSM vortex beams along a horizontal path is larger than that along a slant path in the long atmospheric propagation. The propagation through horizontal atmospheric turbulence can be treated as a special case of the altitude-independent structure constant. The position of coherence vortices in slant atmospheric turbulence does not coincide with that in horizontal atmospheric turbulence, and the dependence of position of coherence vortices on the zenith angle, wavelength and reference point is illustrated by numerical examples. A comparison with the previous work is also made.     

大气湍流对多色高斯-谢尔模型光束扩展的影响

基于广义惠更斯-菲涅耳原理,采用Rytov相位结构函数二次近似和硬边窗口函数的复高斯展开法,推导了受光阑限制的多色高斯-谢尔模型(GSM)光束在大气湍流中的二阶矩束宽公式。研究表明:二阶矩束宽随着大气湍流折射率结构常数、源光谱带宽和光束传输距离的增加而增大,随着光束截断参数和光束相干参数的增加而减小;并且,当源光谱带宽越大、光束截断参数和光束相干参数越小,则湍流对受光阑限制的多色GSM光束扩展的影响就越小。关键词:      

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.     

Change in degree of coherence of partially coherent electromagnetic beams propagating through atmospheric turbulence

We study the change in the degree of coherence of partially coherent electromagnetic beam (so called electromagnetic Gaussian Schell-model beam). It is shown analytically that with a fixed set of source parameters and under a particular atmospheric turbulence model, an electromagnetic Gaussian Schell-model beam propagating through atmospheric turbulence reaches its maximum value of coherence after the beam propagates a particular distance, and the effective width of the spectral degree of coherence also has its maximum value. This phenomenon is independent of the used turbulence model. The results are illustrated by numerical curves.     

大气湍流对多色高斯-谢尔模型光束扩展的影响

基于广义惠更斯-菲涅耳原理,采用Rytov相位结构函数二次近似和硬边窗口函数的复高斯展开法,推导了受光阑限制的多色高斯-谢尔模型(GSM)光束在大气湍流中的二阶矩束宽公式。研究表明：二阶矩束宽随着大气湍流折射率结构常数、源光谱带宽和光束传输距离的增加而增大,随着光束截断参数和光束相干参数的增加而减小;并且,当源光谱带宽越大、光束截断参数和光束相干参数越小,则湍流对受光阑限制的多色GSM光束扩展的影响就越小。关键词：      

Spatial correlation properties of apertured partially coherent beams propagating through atmospheric turbulence

The spatial correlation properties of apertured partially coherent beams propagating through atmospheric turbulence are studied in detail. By using the quadratic approximation of the Rytov’s phase structure function and the finite complex Gaussian expansion of the aperture function, the closed-form expression for the spectral degree of coherence of apertured partially coherent beams in turbulence is derived. It is shown that the larger the aperture width h is, and the smaller the spatial correlation length σ₀ of the source is, the less the spatial correlation is affected by turbulence. Specially, the influence of turbulence on the spatial correlation for unapertured partially coherent beams is less than that for apertured ones. The spectral degree of coherence is shown to possess phase singularities for apertured partially coherent beams, but the phase singularities behavior disappears when the turbulence is strong enough, which is very different from the behavior of unapertured partially coherent beams. In addition, a comparison between the width of the spectral degree of coherence and that of the spectral intensity of apertured partially coherent beams in turbulence is also given, and some interesting results are obtained. The results are interpreted physically. PACS  42.68.Bz; 42.79.Ag; 42.25.Fx     

Changes in the spectrum of Gaussian Schell-model beams propagating through turbulent atmosphere

Based on the extended Huygens-Fresnel principle, the spectrum of Gaussian Schell-model (GSM) beams propagating through turbulent atmosphere is derived analytically. It is shown that, if the scaling law is valid, the normalized spectrum S(ω) of GSM beams propagating through turbulent atmosphere is the same as the normalized source spectrum S⁽⁰⁾(ω), whether GSM source is quasi-homogenous or not. On the other hand, if the scaling law fails, S(ω) of GSM is different from S⁽⁰⁾(ω). The structure constant of the refractive index, transverse coordinate of observation point and spatial correlation length of the source affect the spectrum, which is illustrated numerically.     

Changes in the spectrum of twist anisotropic Gaussian Schell-model beams propagating through turbulent atmosphere

Based on the extended Huygens-Fresnel principle, the spectrum of twist anisotropic Gaussian Schell-model (TAGSM) beams propagating through turbulent atmosphere is derived analytically by using the partially coherent complex curvature tensor. The relative spectral shift of TAGSM beams propagating through turbulent atmosphere is closely related with the strength of atmospheric turbulence, the beam’s parameter and the radial coordinate. The on-axis spectral shift of TAGSM beams propagating through turbulent atmosphere changes from blue shift to red shift with the increasing of the propagation distance z, and at a certain propagation distance z, there exists a rapid transition of the spectrum at the critical position r_c.     

Directionality of partially coherent annular flat-topped beams propagating through atmospheric turbulence

The closed-form expression for the angular spread of partially coherent annular flat-topped beams propagating through atmospheric turbulence is derived by using the integral transform technique. In comparison with previous work, the results obtained in this paper are more general. It is shown that, except for equivalent Gaussian Schell-model (GSM) beams as pointed out in Ref. [T. Shirai, A. Dogariu, E. Wolf, Opt. Lett. 28 (2003) 610], there also exist equivalent annular and non-annular beams which may generate the same directionality as the corresponding fully coherent Gaussian beam both in free space and in turbulence. The theoretical results are illustrated by numerical examples and interpreted physically.     

Intensity distribution and coherence property for the broadband Gaussian Schell-model array beams in free space

The intensity distribution and the equal-time complex degree of coherence for the broadband Gaussian Schell-model array beams with different bandwidth propagating in free space are analyzed by the numerical calculation. Results show that the intensity distribution for the broadband Gaussian Schell-model array beams by uncorrelated superposition is rarely affected by the bandwidth; the intensity distribution in the far field for the broadband Gaussian Schell-model array beams by correlated superposition is affected by the bandwidth more obviously than that in the near field. With the increasing of the propagation distance, differences in the equal-time complex degree of coherence for the broadband Gaussian Schell-model array beams with different bandwidth become prominent.