Scintillation behavior of cos, cosh and annular Gaussian beams in non-Kolmogorov turbulence

For a non-Kolmogorov spectrum, scintillation aspects of cos, cosh and annular Gaussian beams are investigated. The appropriate mathematical formulation is developed, the derived scintillation index is evaluated and its variation is plotted in graphs. We find that, when the values of the power coefficient of the spectrum are just above 3, low scintillation is encountered, then as the power coefficient is increased, rises will occur with a peak being reached around 3.21. From there onwards, scintillation will drop, as the power coefficient approaches a value of 5. For extreme off-axis positions, there will be slight increases in scintillation at high power coefficient values. At points near on-axis and when the beams have small width sizes, cosh Gaussian beam having a bigger displacement parameter will offer the lowest scintillation. At large width sizes, this advantage will switch to the side of the cos Gaussian beam. In this study, the variation of scintillation with other sources and propagation parameters is examined as well.     

Scintillations of multiwavelength Gaussian, cos, cosh and annular Gaussian beams

We provide the scintillation formulation for a multiwavelength source. Within this context, the scintillation aspects of Gaussian, cos, cosh and annular Gaussian beams are investigated. For all situations examined, it is found that for a source comprising many wavelengths, there will be less scintillations as compared to a single wavelength source of the lowest wavelength and but the reverse will be true if the comparison is with respect to the single wavelength source of the highest wavelength. The same is observed at all propagation distances, source sizes, on-axis and off-axis positions considered. Additionally, it is seen that the scintillation characteristics of multiwavelength sources will follow similar trends of single wavelength sources. The analysis is based on the Rytov approximation, therefore our results are valid for conditions of weak atmospheric turbulence.     

Scintillation behavior of Laguerre Gaussian beams in strong turbulence

In strong atmospheric turbulence, the asymptotic on-axis scintillation behaviors of Laguerre Gaussian (LG) beams are examined. To arrive at the strong-turbulence solution, we utilize the existing filtering approach for weak-turbulence solutions and our recently reported weak-turbulence scintillation index formula for LG beams. In the limiting case, our solution correctly predicts the asymptotic strong-turbulence behavior of Gaussian beam wave scintillation. Investigation of the scintillations versus the propagation distance, source size, wavelength and refractive index structure parameter lead to the conclusion that the LG beams with higher order radial modes can provide less scintillation. The results are applicable to long-haul atmospheric optical communication links.     

Scintillation index of the Gaussian beams propagated through the paths with strong turbulence

Theoretical and experimental studies of the local statistical characteristics of Gaussian beams propagated through the paths with strong turbulence were carried out and their results are presented. It is ascertained that the applicability of the scintillation index meanings are substantially limited for the assessment of the performance quality of the optical data transmitting channels. On the basis of the experimental data got under different propagation conditions the peculiarities of the localized intensity distributions and those of the scintillation index are revealed.     

Field correlations of flat-topped Gaussian and annular beams in turbulence

Starting from the second order moment formulation for multi-beam incidence, field correlations at the receiver plane of flat-topped Gaussian and annular beams are found in turbulence. Reflecting the information on both the randomness due to turbulence and the field profile of the incident field, field correlations of the flat-topped Gaussian beams are found to become larger at larger source size, smaller flatness parameters and smaller turbulence strengths. For the annular beam structures, field correlations are larger for thicker beams. Field correlations of larger primary beam sized annular incidences are smaller at smaller diagonal distances and larger at larger diagonal distances. As expected, annular beam field correlations are found to be larger at smaller structure constants and at smaller wavelengths. However, at large link lengths, field correlations could be larger than at smaller link lengths due to fact that for annular beams, the field at the centre of the receiver attains very small value at smaller link lengths, however, on propagation in turbulence, receiver field distribution changes to a Gaussian profile.     

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.     

Beam spreading of truncated Gaussian beams in Kolmogorov turbulence

Based on the extended Huygens-Fresnel principle and the hard-edge aperture function expanded as the sum of finite-term complex Gaussian function, analytical expression for average intensity of truncated Gaussian beam in Kolmogorov turbulence is derived, and some limiting cases are discussed. The influences of many factors, such as Fresnel number, wavelength, truncation parameter and structure constant on beam spreading are studied with the help of the average intensity formula. We found that the peak value of average intensity decreases and the beam spot spreads with the decrease of Fresnel number. The change of peak intensity against Fresnel number is slower with large aperture than that with small aperture. When Fresnel number is not small enough the influence of turbulence on intensity profiles is so small that can be neglected if structure constant is small.     

Field correlations of annular beams in extremely strong turbulence

The field correlations of annular beams are formulated when the atmosphere assumes extremely strong turbulence. Thicker and larger ring sized annular beams are found to exhibit larger absolute field correlations. For the same transverse distance at the receiver plane, annular beams attain larger field correlations if the transverse distance starts from the receiver origin. Comparisons of the annular beam absolute field correlations in extremely strong turbulence with the no turbulence results show that the absolute field correlation variations follow similar trends, except that the magnitudes of the absolute field correlations are much smaller in extremely strong turbulence and the annular fields become decorrelated at very short transverse distances. When the inner scale of turbulence becomes smaller, the absolute field correlations of the annular beams in extremely strong turbulence become smaller.     

Propagation of higher order Bessel–Gaussian beams in turbulence

The propagation characteristics of higher order Bessel–Gaussian beams travelling in turbulent atmosphere are investigated. Using extended Huygens–Fresnel principle, I formulated receiver plane intensity and solved it down to a double integral stage. Source beam plots are made illustrating the variation of intensity against order and width parameter. From the examination of receiver intensity graphs, it is seen that Bessel–Gaussian beam are converted into modified Bessel–Gaussian beams at intermediate propagation ranges eventually ending up as Gaussian profiles. The impacts of order and turbulence levels on beam profile are analysed. Focusing effects and beam size change along the propagation axis are studied. PACS 41.85.Ew; 42.68.Bz     

Propagation characteristics of higher-order annular Gaussian beams in atmospheric turbulence

The propagation characteristics of higher-order annular Gaussian (HOAG) beams in turbulence are investigated. From a HOAG source plane excitation, the average intensity of the receiver plane is developed analytically. This formulation is verified against the previously derived HOAG beam solution in free space. The graphical outputs indicate that, upon traveling in turbulent atmosphere, the HOAG beam will undergo different stages of evolution. At intermediate propagation distances, it will attempt to concentrate the energy near the origin. In this process, the appearance of the single higher-order primary beam will be encountered. Eventually HOAG originated beam will become a pure Gaussian beam after propagating an excessive distance in the turbulent medium.     

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.     

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.     

Effect of spherical aberration on scintillations of Gaussian beams in atmospheric turbulence

The effect of spherical aberration on scintillations of Gaussian beams in weak, moderate and strong turbulence is studied using numerical simulation method. It is found that the effect of the negative spherical aberration on the on-axis scintillation index is quite different from that of the positive spherical aberration. In weak turbulence, the positive spherical aberration results in a decrease of the on-axis scintillation index on propagation, but the negative spherical aberration results in an increase of the on-axis scintillation index when the propagation distance is not large. In particular, in weak turbulence the negative spherical aberration may cause peaks of the on-axis scintillation index, and the peaks disappear in moderate and strong turbulence, which is explained in physics. The strong turbulence leads to less discrepancy among scintillations of Gaussian beams with and without spherical aberration.     

非Kolmogorov大气湍流对高斯列阵光束扩展的影响

本文推导出了高斯列阵光束在非Kolmogorov大气湍流中传输的瑞利区间zR、湍流距离zT和远场发散角θ的解析表达式,研究了非Kolmogorov湍流的广义指数α和列阵光束的合成方式对高斯列阵光束扩展的影响.研究表明:不论相干还是非相干合成高斯列阵光束,zR,zT和θ均随着α的增加而呈非单调变化.当α=3.108时,zR和zT取极小值,而θ取极大值,即当α=3.108时高斯列阵光束扩展最厉害,光束扩展受湍流影响也最厉害.非相干合成高斯列阵光束扩展比相干合成的要大,但受非Kolmogorov湍流影响却要小.特别值得指出的是:当自由空间光束衍射较小时,有zTzR,即在瑞利区间范围内大气湍流就对光束扩展有影响;而当自由空间光束衍射较大时,有zTzR,即在瑞利区间范围内大气湍流对光束扩展几乎没有影响.     

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.     

Propagation properties of partially coherent Hermiteben Gaussian beams through non-Kolmogorov turbulence

The propagation properties of partially coherent Hermite—Gaussian beams through non-Kolmogorov atmospheric turbulence are studied. The effects of non-Kolmogorov turbulence and beam nonparaxiality on the average intensity evolution and the beam-width spreading are stressed. It is found that the evolution of the average intensity distribution and the beam-width spreading depends on the generalized exponent factor, namely, on the non-Kolmogorov turbulence strength for the paraxial case. For the non-paraxial case the effect of the turbulence is negligible, while the beam-width spreading becomes very large. The analytical results are illustrated numerically and interpreted physically.     

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.     

Influence of oceanic turbulence on propagation characteristics of Gaussian array beams

Based on the power spectrum of oceanic turbulence proposed by Nikishov, the influence of oceanic optical turbulence on propagation of Gaussian array beams is studied in detail by using five propagation characteristic parameters (i.e., mean-squared beam width, Rayleigh range, turbulence distance, power in the bucket and Strehl ratio). It is shown that the influence of oceanic optical turbulence on propagation of laser beams becomes stronger as τ (ratio of temperature to salinity contribution to the refractive index spectrum) and χ_T (rate of dissipation of mean-squared temperature) increase and ɛ (rate of dissipation of kinetic energy per unit mass of fluid) decreases, which is in agreement with the further analysis of oceanic turbulence shown in this paper. Furthermore, it is concluded that propagation of laser beams is more affected by oceanic optical turbulence in abyssal region than that in active region or oceanic surface.     

Beam wander characteristics of cos and cosh-Gaussian beams

Within the context of a general beam formulation, beam wander characteristics of cos and cosh-Gaussian beams are derived and numerically evaluated. In our graphs, the fundamental Gaussian beam is used as a benchmark for comparisons. The associated plots reveal that at small source sizes, a cos-Gaussian beam has the lowest beam wander, while this property is enhanced with increasing values of the displacement parameter. At large source sizes however, this advantage is taken over by cosh-Gaussian beam. Joint examination against the changing source sizes and propagation lengths shows that the range of source sizes, where the beam wander of cos-Gaussian beam remains lower, is enlarged as we go toward higher propagation lengths. Asymmetric beams tend to exhibit higher beam wanders both at small and large source sizes, but for the intermediate source size ranges, the beam wanders of asymmetric beams will fall below those of the symmetric beams. Explanations concerning these behaviors are offered. A historical account of beam wander formulation is also included.     

湍流对离轴列阵高斯光束相干与非相干合成的影响

研究了湍流对离轴列阵高斯光束相干与非相干合成的影响.推导出了相干合成光束的传输方程.采用二阶矩束宽、桶中功率和参数β作为光束质量评价参数比较了离轴列阵高斯光束通过湍流大气的相干与非相干合成,并对主要结果给予了合理的物理解释.研究表明：一方面,不论是相干合成还是非相干合成,湍流都使得合成光束扩展、峰值光强下降,并且子光束数越多,合成光束受湍流影响就越小.另一方面,非相干合成光束较相干合成光束受到湍流的影响要小. 关键词： 相干与非相干合成 湍流大气 离轴列阵高斯光束