Propagation properties of partially coherent dark hollow beam in inhomogeneous atmospheric turbulence

Abstract

Propagation properties of a partially coherent dark hollow beam (PC-DHB) in inhomogeneous atmospheric turbulence are studied in detail. Analytical formulae for the root-mean-square (rms) spatial width, rms angular width, M²-factor of PC-DHB in inhomogeneous turbulence are derived based on the extended Huygens–Fresnel integral. It is found that PC-DHB spreads in inhomogeneous turbulence more rapidly than the free space, and the saturation propagation distances (SPDs) of relative spatial and angular spreadings for uplink slant paths with zenith angles of 45° or less are about 5 and 0.6 km, respectively. M²-factor of PC-DHB in turbulence depends on beam order, waist width, inner scale of the turbulence and the SPD of the normalized M²-factor for the propagation with zenith angles of 45° or less is about 30 km. Our results are useful for the free space optical communications and the beam propagation in the slant path.     

Coupling characteristics and kurtosis parameter of partially coherent beams in turbulent atmosphere

Coupling properties and kurtosis parameter (K parameter) of arbitrary beams propagating through atmospheric turbulence are investigated. A correlation factor (C⁴-factor) is introduced to describe the influence of turbulence on coupling characteristics. The general analytical expression for C⁴-factor of arbitrary beams in atmospheric turbulence is derived. It is shown that C⁴-factor of arbitrary beams in the turbulent atmosphere depends on the initial second-order moments and fourth-order moments and turbulence quantities. Taking the partially coherent anomalous elliptical hollow Gaussian (PCAEHG) beam as an example, we can obtain that C⁴-factor decreases as structure constant of the refractive index fluctuations and inner scale increase, and waist width and transverse coherence length decrease when z?>?5?km. Moreover, K parameter of PCAEHG beam in turbulent atmosphere converges to 2 when propagation distance is large enough. It indicates that the profile of PCAEHG beams in turbulent atmosphere finally evolves into fundamental Gaussian distribution.     

Beam wander and M2-factor of partially coherent electromagnetic hollow Gaussian beam propagating through non-Kolmogorov turbulence

Propagation formulae for M²-factor and beam wander of partially coherent electromagnetic hollow Gaussian (PCEHG) beam in non-Kolmogorov turbulence are derived based on the extended Huygens–Fresnel principle and the second-order moments of the Wigner distribution function. Our results indicate that the normalized M²-factors of PCEHG beam with larger beam order, waist width, inner scale of turbulence, the generalized exponent parameter, and smaller transverse coherent widths, outer scale of turbulence, the generalized structure parameter are less affected by the turbulence. The root mean square beam wander and relative beam wander are more obvious for PCEHG beam with smaller beam order, larger inner and outer scales of turbulence, exponent parameter, transverse coherent widths, and the generalized structure parameter. What is more, the beam wander properties of PCEHG beam in non-Kolmogorov turbulence are very different from M²-factor and spreading properties of beam in turbulence.     

A comparative study of standard and elegant Hermite–Gaussian beams propagating through turbulent atmosphere

Based on the extended Huygens–Fresnel principle and the definition of the second-order moments of the Wigner distribution function (WDF), the analytical expressions for the root-mean-square (rms) beam width and far-field divergence angle, curvature radius and M ²-factor of standard Hermite–Gaussian (SHG) and elegant Hermite–Gaussian (EHG) beams passing through turbulent atmosphere are derived and compared. It is shown that in turbulent atmosphere the far-field divergence angle of SHG and EHG beams is equal under the same conditions, but the rms beam width, curvature radius and the M ²-factor of SHG and EHG beams are different except for beam orders m?=?0 and m?=?1. The relative rms beam width, relative curvature radius and relative M ²-factor of SHG beams are less than those of EHG beams. Therefore, the conclusion that SHG beams are less influenced by turbulence than EHG beams can be drawn if we examine one of the above three relative beam parameters.     

Propagation of Gaussian-Schell beam in turbulent atmosphere of three-layer altitude model

We propose a method that is used to derive the moment radius of intensity distribution in a turbulent atmosphere. From this study, we have found that the second moment radius is affected only by the first-order expansion coefficient of the wave structure function. If our attention is directed to a higher moment radius, a higher order approximation of the expansion needs to be used. As an example, the propagation of a Gaussian-Schell beam in a slant path has been studied based on the turbulent atmosphere of a three-layer model. The variation of some beam properties, such as the relative waist width, angular spread, and kurtosis parameter with the initial waist width, wavelength, and zenith angle, has been analyzed and discussed in detail. The study shows that there is little difference between the three-layer model and the Kolmogorov model in studying uplink propagation, and the difference is large for downlink propagation. The intensity profile of the Gaussian beam in turbulence does not keep a Gaussian shape unless the beam spreading due to turbulence is very large or very small.     

The M 2 factor matrix of a paraxial Laguerre–Gaussian beam

The M ² factor matrix of a paraxial Laguerre–Gaussian beam has been derived based on the combination of the M ² factor definition and introduced quantities such as the coupled beam’s half width square, the coupled M ² factor and 2×2 rotational matrix, when the beam is rotated around the propagation z-axis by an arbitrary azimuth angle α. The beam’s half widths at different distances from the beam waist along the z-axis, the M ² factors of the beam with different orders, and the track of M ² factor matrix elements versus α have been obtained by numerical simulation and analytical derivation. Results illustrate that by using the invariability of the M ² factor matrix, the beam quality of the mode with an arbitrary rotational angle can be easily evaluated. Like the M ² factor matrix of the Hermite–Gaussian mode, this M ² factor matrix is meaningful in the evaluation of the beam quality of a two-dimensional astigmatic or asymmetric laser beam.     

Propagation based on second-order moments for partially coherent Laguerre–Gaussian beams through atmospheric turbulence

Abstract

The Wigner distribution function (WDF) has been used to study the propagation properties of partially coherent Laguerre Gaussian (PCLG) beams through atmospheric turbulence. Based on the extended Huygens–Fresnel principle, an analytical formula of the propagation matrixes in terms of the second-order moments of the WDF for PCLG Beams in the receiving plane is derived. And then the analytical formulae for the curvature radii of PCLG Beams propagating in turbulence are given by the second-order moments of the WDF. The numerical results indicate that the curvature radius of PCLG Beams changes more rapidly in turbulence than that in the free space. The influence of the transverse coherence width and the beam waist width on the curvature radius of PCLG Beams is obvious, while the laser wavelength and the inner scale of turbulence have a slight effect. The study results may be useful for remote sensing and free space optical communications.     

The beamwidth of bessel-modulated gaussian beams

Abstract

Based on the second-order irradiance moment definition, a closed-form expression for the beam width of Bessel-modulated Gaussian beams (QBG beams) is derived, from which the expressions for the waist width, its position, and beam propagation factor (M ² factor) of QBG beams with complex parameter μ are obtained. The dependence of the beam width and waist position on the parameter μ is analysed and illustrated with numerical examples.     

Beam combination of a linear array: Hermite-Gaussian superposition model

Abstract

Starting from the generalized Huygens-Fresnel diffraction integral and using a Hermite-Gaussian superposition model, an analytical propagation equation of the beam emitted from a linear array is derived and illustrated numerically. Furthermore, the beam quality of the resulting beam in terms of the M ²-factor and power in the bucket (PIB) is analysed. Interestingly, the M ²-factor in the × direction is dependent upon the separation distance of beamlets, but the PIB is not, because the latter is determined by the irradiance distribution in the far field and the bucket size chosen.     

A further comparative study of flattened gaussian beams and super-Gaussian beams

Abstract

A further comparison between flattened Gaussian beams (FGBs) and super-Gaussian beams (SGBs) is performed. It is shown that the two FGB and SGB having the same beam propagation factor (M²?factor) but different waist widths demonstrate a similar irradiance profile at the position of the equal generalized Fresnel number, while they propagate through a paraxial ABCD system. Furthermore, the focusing by a spherically aberrated lens is studied, and nearly the same propagation characteristics are achieved for the two FGB and SGB with the same M²?factor and waist width.     

Scattering of a partially coherent Gaussian-Schell beam from a diffuse target in slant atmospheric turbulence

On the basis of the extended Huygens-Fresnel principle, the scattering of partially coherent Gaussian-Schell-model (GSM) beams from a diffuse target in slant double-passage atmospheric turbulence is studied and compared with that of fully coherent Gaussian beams. Using the cross-spectral density function of the GSM beams, we derive the expressions of the mutual coherence function, angle-of-arrival fluctuation, and covariance and variance of the intensity of the scattered field, taking into account the fluctuations of both the log-amplitude and phase. The numerical results are presented, and the influences of the wavelength, propagation distance, and waist radius on scattering properties are discussed. The perturbation region of the normalized intensity variance of the partially coherent GSM beam is smaller than that of the fully coherent Gaussian beam at the middle turbulence level. The normalized intensity variance of long-distance beam propagation is smaller than that of beam propagation along a short distance.     

Propagation of a general multi-Gaussian beam in turbulent atmosphere in a slant path

The propagation of a multi-Gaussian beam in turbulent atmosphere in a slant path is studied. The analytical expression for the average intensity of a general multi-Gaussian beam is derived. As special cases the average intensities of a two- and a four-Gaussian beam are investigated and numerically calculated. The investigation reveals that at lower altitude and with large sigma the intensity distribution at the receiver plane can have a shape (multiple peaks) similar to that at the source plane. But with increase in altitude or decrease in sigma, the multiple peaks gradually disappear and evolve into the profile of a fundamental Gaussian beam. From the comparisons between the different propagations we can see that the beam spreading due to wavelength and initial waist width in a slant path is much slower than that in a horizontal path.     

Second-order moments representation of nonparaxial TM flattened Gaussian beams

By using the angular spectrum representation of the Hertz vector and the theory of nonparaxial vectorial moments of light beam propagation, expressions for the width matrix, curvature and divergence matrices and M ² factor of nonparaxial TM flattened Gausssian beams (FGBs) are derived and their matrix elements are expressed in terms of a series sum. Generally, nonparaxial TM FGBs possess different beam widths, divergence angles and M ² factors in the x and y directions, which are dependent on the beam order N and f parameter. For 1/f → 0 the divergence angles θ _x and θ _y in the x and y directions approach their maximum values θ _x = 54.736° and θ _y = 63.435°, respectively. The M ² factor can be less than 1 and take arbitrarily small values if 1/f is small enough.     

Propagation of Gaussian Schell-model Beams in Dispersive and Absorbing Media

Abstract

The spectral properties and the coherence properties of Gaussian Schell-model beams, propagating in dispersive and absorbing media, are discussed. Unlike in free space, the ratio of the transverse spectral correlation length to the beam width is found to increase on propagation. Consequently upon propagation the beam becomes spatially more coherent at each frequency. Numerical results for the spectrum and for the degrees of spectral and of temporal coherence of the field are presented for some selected values of the beam parameters and for several values of the propagation distance. Propagation in a gain medium is also briefly discussed.     

Beam quality characterization of a radial laser array

Abstract

With the Hermite-Gaussian mode used to describe radial laser arrays and the M²-factor and power in the bucket (PIB) chosen as criteria for characterizing beam quality, a detailed study of beam quality of coherent and incoherent radial laser arrays is made. The closed-form expression for the M²-factor of incoherent radial arrays, which depends upon the Hermite-Gaussian order m, inverse radial fill factor r' and aspect ratio w _0x /w _0y , is derived. Numerical examples are given to illustrate and compare beam quality of coherent and incoherent radial arrays.     

Evolution of decentred laser beams propagating through atmospheric turbulence

The changes of the average intensity, the centre of beam gravity and the position of intensity maximum of decentred laser beams propagating through atmospheric turbulence are examined in detail. It is shown that the decentred intensity distribution is amended gradually with increasing the propagation distance and the strength of turbulence, and it becomes an off-axis Gaussian-like beam when the propagation distance and the strength of turbulence become large enough. The centre of beam gravity is independent of both the propagation distance and the strength of turbulence. On the other hand, there are two intensity maxima, and their positions are symmetrical around the propagation z-axis when the propagation distance z is small. With increasing z, there is only one intensity maximum. As z further increases, position of the intensity maximum is further shifted towards the z-axis. When z is large enough, the position of the intensity maximum is unchanged. The unchanged position of the intensity maximum moves further away from the z-axis with an increase in the refraction index structure constant, the decentred parameter and the waist width.     

复宗量双曲正弦高斯光束的传输特性

对复宗量双曲正弦高斯(EshG)光束的一些基本特性进行了研究。由Collins公式出发推导出了光束通过近轴ABCD光学系统传输的场分布函数,根据二阶矩的定义和光束在近轴ABCD光学系统中的传输公式推导出了光束通过近轴ABCD光学系统光斑尺寸的解析表达式。得到了复宗量双曲正弦高斯光束在自由空间传输时束腰宽度和位置的解析表达式以及M2因子的表达式。对光强分布、光斑尺寸、束腰宽度及其位置和M2因子进行了数值计算,并对计算结果作了分析。本文所得结果具有广泛的意义,因为正弦高斯光束、双曲正弦高斯光束和复宗量正弦高斯光束均可视为其特例。     

Statistical distribution of the optical intensity obtained using a Gaussian Schell model for space-to-ground link laser communications

Abstract

Based on the characteristics of the laser device and the inevitable error of the processing technique, a laser beam emitted from a communication terminal can be represented by the Gaussian Schell model (GSM). In space-to-ground link laser communications, the optical intensity is affected by the source coherence parameter and the zenith angle. With full consideration of these two parameters, the statistical distribution model of the optical intensity with a GSM laser in both downlink and uplink is derived. The simulation results indicate that increasing the source coherence parameter has an effect on the statistical distribution of the optical intensity; this effect is highly similar to the effect of a larger zenith angle. The optical intensity invariably degrades with increasing source coherence parameter or zenith angle. The results of this work can promote the improvement of the redundancy design of a laser communication receiver system.