Propagation of multi-Gaussian beams in incoherent combination through turbulent atmosphere and their beam quality

Based on the extended Huygens–Fresnel principle, the propagation of M?×?N multi-Gaussian beams in incoherent combination through turbulent atmosphere, and their beam quality are studied. The power in the bucket (PIB), β parameter and strehl ratio (S_R ) are taken as the characteristic parameters of beam quality, and analytical expressions for PIB and S_R are derived. It is shown that multi-Gaussian beams in turbulent atmosphere undergo three stages of evolution with increasing propagation distance z, and turbulence accelerates the evolution of the three stages which multi-Gaussian beams undergo. The turbulence results in a degradation of the beam quality, and multi-Gaussian beams with higher numbers M and N are less sensitive to the effects of turbulence than those with lower M and N.     

Propagation property of a nonuniformly polarized beam array in turbulent atmosphere

A nonuniformly polarized beam array (NUBPA) is modeled by coherent superposition of a pair of orthogonally polarized spatial modes. The propagation of a NUPBA in turbulent atmosphere is investigated based on the extended Huygens-Fresnel method. An analytical expression for the intensity profile of a NUBPA in turbulent atmosphere is presented. The influence of polarization degree, intensity of turbulence, array number, and distance between adjacent elements on the intensity profile in the receiving plane is evaluated numerically and analyzed in detail.     

Propagation properties of partially coherent radially polarized beam in a turbulent atmosphere

Based on the Huygens–Fresnel principle and the unified theory of coherence and polarization of partially coherent beams, we investigate the propagation characteristics of a partially coherent radially polarized doughnut (PCRPD) beam in a turbulent atmosphere. It is found that, after propagating through a turbulent atmosphere, the doughnut beam spot is changed into a circular Gaussian beam. Moreover, the degree of coherence, the degree of polarization and the degree of cross-polarization of the beam will change on propagation, and this change is dependent upon the degree of coherence of the source and atmospheric turbulence.     

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.     

Propagation of a random electromagnetic beam through a misaligned optical system in turbulent atmosphere

On the basis of the generalized diffraction integral formula for misaligned optical systems in the spatial domain, an analytical propagation expression for the elements of the cross-spectral density matrix of a random electromagnetic beam passing through a misaligned optical system in turbulent atmosphere is derived. Some analyses are illustrated by numerical examples relating to changes in the state of polarization of an electromagnetic Gaussian Schell-model beam propagating through such an optical system. It is shown that the misalignment has a significant influence on the intensity profile and the state of polarization of the beam, but the influence becomes smaller for the beam propagating in strong turbulent atmosphere. The method in this paper can be applied for sources that are either isotropic or anisotropic. It is shown that the isotropic sources and the anisotropic sources have different polarization properties on beam propagation.     

Propagation of a partially coherent beam from an unstable resonator through turbulent atmosphere

The influence of atmospheric turbulence on the propagation of a partially coherent beam from an unstable resonator was studied numerically. The resonant mode of the unstable resonator is obtained by iterative calculation using the Huygens–Fresnel formula. Also, using the extended Huygens–Fresnel integral, the intensity distribution of a propagating laser beam is calculated for different conditions. The influence of turbulence on the profile of partially coherent beams of an unstable resonator is studied. The effects of geometrical parameters of the resonator on the far-field beam profile are investigated. The results show that an unstable resonator with higher magnification has a superior far-field beam profile under partial coherency and turbulence conditions.     

Propagation properties of flat-topped beams in a turbulent atmosphere

Based on the extended Huygens–Fresnel principle and the quadratic approximation for phase structure function, the analytical formulas for the average irradiance of the phase-locked and the non-phase-locked radial modified flat-topped array beams in turbulence have been derived. The propagation properties of radial modified flat-topped array beams have been investigated. It is found that the irradiance distributions of the radial flat-topped array beams propagating in turbulence are different from those in free space. The propagation properties depend on the refractive index structure constant, the beam order and the array radius. The power in the bucket (PIB) for the array beams increases with the decrease of the array radius for a given propagation distance. For a certain bucket radius, there exhibits a PIB maximum upon propagation. The optimal propagation distance becomes shorter along with the increasing refractive index structure constant, and the PIB maximum increases with bucket radius.     

Scintillation index of a multiwavelength beam in turbulent atmosphere

We characterize the scintillation index of a multiwavelength plane-wave optical beam that is subjected to a turbulent optical channel. It is assumed that the level of turbulence in the atmosphere ensures a weak-turbulence scenario and that the turbulence is due to the fluctuations in the index of refraction of the medium. It is assumed that the propagation path is nearly horizontal and that the heights of the transmitter and receiver justify a near-ground propagation assumption.     

Propagation characteristics of partially coherent anomalous elliptical hollow Gaussian beam propagating through atmospheric turbulence along a slant path

Based on the extended Huygens–Fresnel principal and the Wigner distribution function, the root mean square (rms) angular width and propagation factor (M²-factor) of partially coherent anomalous elliptical hollow Gaussian (PCAEHG) beam propagating through atmospheric turbulence along a slant path are studied in detail. Analytical formulae of the rms angular width and M²-factor of PCAEHG beam are derived. Our results show that the rms angular width increases with increasing of wavelength and zenith angle and with decreasing of transverse coherence length, beam waist sizes and inner scale. The M²-factor increases with increasing of zenith angle and with decreasing of wavelength, transverse coherence length, beam waist sizes and inner scale. The saturation propagation distances (SPDs) increase as zenith angle increases. The numerical calculations also indicate that the SPDs of rms angular width and M²-factor for uplink slant paths with zenith angle of π/12 are about 0.2 and 20 km, respectively.     

Propagation and transformation of flat-topped multi-Gaussian beams in a general nonsymmetrical apertured double-lens system

On the basis of expanding a hard-edged aperture function as a finite sum of complex Gaussian functions, an approximate analytical expression for the propagation of an input complex amplitude distribution passing through a general nonsymmetrical apertured double-lens system is derived. Then, the propagation result for two-dimensional flat-topped multi-Gaussian beams is given. It is shown that the apertured Lohmann's symmetrical double-lens system for fractional Fourier transform is a special case of the general apertured double-lens system. The numerical calculation, graphical illustration, and some discussions for the transformation of the two-dimensional flat-topped multi-Gaussian beam in apertured Lohmann's symmetrical double-lens systems are also presented.     

Propagation of partially coherent flat-topped beams through a turbulent atmosphere

Based on the modified beam model for flat-topped beams and the Schell model for partially coherent light, an expression for partially coherent flat-topped (PCFT) beams has been proposed. The propagation characteristics of PCFT beams with circular symmetry through a turbulent atmosphere have been studied. By using the generalized Huygens-Fresnel integral and Fourier transform method, the expressions for the cross-spectral density function and the average intensity have been given and the analytical expression for the root-mean-square width has been derived. The effects of the beam order, the spatial coherence, and the turbulent parameter on the intensity distributions and beam spreading have been discussed in detail. Our results show that the on-axis intensity of the beams decreases with increasing turbulence and decreasing coherence of the source, whereas the on-axis intensity of the beams in the far field decreases slightly with increasing beam order. The relative spreading of PCFT beams is smaller for beams with a higher order, a lower degree of global coherence of the source, a larger inner scale, and a smaller outer scale of the turbulence.     

Propagation factors of laser array beams in turbulent atmosphere

The propagation factors of phased locked laser array beams of radial and rectangular symmetries in a turbulent atmosphere are investigated based on the extended Huygens–Fresnel integral and the Wigner distribution function. Analytical propagation formulae for the propagation factors are derived and numerical examples are illustrated. We find that unlike their propagation invariant properties in free space, the propagation factors of laser array beams increase when propagating in turbulent atmosphere, and are closely related to the parameters of initial beams and the atmosphere.     

Arbitrary moments of elliptical Gaussian-Schell beam in turbulent atmosphere

Arbitrary moments of elliptical Gaussian-Schell beams in turbulent atmosphere is studied. An analytical formula for arbitrary moments of elliptical Gaussian-Schell beams in turbulent atmosphere is derived. As an example, the kurtosis parameter for elliptical Gaussian-Schell beams in turbulent atmosphere is investigated in detail. The comparison between the variations of the kurtosis parameter for elliptical Gaussian-Schell beams with different parameters is made. Meanwhile, the difference between the kurtosis parameter in vacuum and in turbulent atmosphere is analyzed and discussed.     

Propagation of flat-topped multi-Gaussian laser beams

The multi-Gaussian beam shape is proposed as a model for aperture functions and laser beam profiles that have a nearly flat top but whose sides decrease continuously. Beams and apertures of this type represent a simple, elegant, and intuitive alternative to super-Gaussian beams, which are important in a number of applications such as laser resonator design. Analytical formulas are developed for the propagation of these beams through free space and optical systems representable by ABCD matrices.     

Hermite-cosine-Gaussian laser beam and its propagation characteristics in turbulent atmosphere

Hermite-cosine-Gaussian (HcosG) laser beams are studied. The source plane intensity of the HcosG beam is introduced and its dependence on the source parameters is examined. By application of the Fresnel diffraction integral, the average receiver intensity of HcosG beam is formulated for the case of propagation in turbulent atmosphere. The average receiver intensity is seen to reduce appropriately to various special cases. When traveling in turbulence, the HcosG beam initially experiences the merging of neighboring beam lobes, and then a TEM-type cosh-Gaussian beam is formed, temporarily leading to a plain cosh-Gaussian beam. Eventually a pure Gaussian beam results. The numerical evaluation of the normalized beam size along the propagation axis at selected mode indices indicates that relative spreading of higher-order HcosG beam modes is less than that of the lower-order counterparts. Consequently, it is possible at some propagation distances to capture more power by using higher-mode-indexed HcosG beams.     

Wind profiling based on the optical beam intensity statistics in a turbulent atmosphere

Reconstruction of the wind profile from the statistics of intensity fluctuations of an optical beam propagating in a turbulent atmosphere is considered. The equations for the spatiotemporal correlation function and the spectrum of weak intensity fluctuations of a Gaussian beam are obtained. The algorithms of wind profile retrieval from the spatiotemporal intensity spectrum are described and the results of end-to-end computer experiments on wind profiling based on the developed algorithms are presented. It is shown that the developed algorithms allow retrieval of the wind profile from the turbulent optical beam intensity fluctuations with acceptable accuracy in many practically feasible laser measurements set up in the atmosphere.     

Effects of finite inner and outer scales on the scintillation index of turbulent slant path

In this paper, the effects of nonzero inner scales and finite outer scales are investigated, in the context of Gaussian beam propagation along a slant path under general turbulence conditions. Theoretical expressions for the cut-off spatial frequencies are derived with an approach method, and thereby a modified scintillation model is developed to incorporate inner scale and outer scale parameters in the analysis. Then, inner and outer scale effects on the downlink are analysed with respect to the zenith angle, the altitude of the transmitter, the initial beam radius, as well as the turbulence strength. Numerical results indicate that the effects of a finite outer scale mainly influence transmission that occurs at large zenith angles or high altitudes, while the inner scale effects are more prevalent. This study may be helpful to improve the accuracy of calculation of slant-path scintillation index, and thus benefit the characterization and optimization of space/air-ground laser communication systems.     

Evolution properties of the intensity distribution of an anisotropic Gaussian Schell-model beam in a turbulent atmosphere

An analytical expression for the cross-spectral density of an anisotropic Gaussian Schell-model (AGSM) beam propagating in a turbulent atmosphere is derived, which is featured by its clear physical meaning. The evolution properties of the intensity distribution of an AGSM beam in a turbulent atmosphere are studied thoroughly. It is found that owing to the anisotropy of the source coherence, the intensity distribution can have many different evolution processes; but under the influence of the turbulent atmosphere, it will finally take on a circular shape. The effects of lowering the source coherence on the circularization speed of the intensity distribution are also investigated. It is found that, when the anisotropy of the source coherence is taken into account, lowering the source coherence can accelerate or decelerate the circularization of the intensity distribution. We propose and demonstrate five kinds of conditions, and, under each condition, lowering the source coherence has a unique and definite effect on the circularization speed of the intensity distribution. In particular, we give an analysis about the equivalence between the problem studied in one of our five conditions and that in the work of Cai and He [Appl. Phys. Lett. 89, 041117 (2006)] and show that their relevant conclusion is incorrect.     

Simulation of laser propagation in a turbulent atmosphere

The split-step Fourier-transform algorithm for numerical simulation of wave propagation in a turbulent atmosphere is refined to correctly include the effects of large-scale phase fluctuations that are important for imaging problems and many beam-wave problems such as focused laser beams and beam spreading. The results of the improved algorithm are similar to the results of the traditional algorithm for the performance of coherent Doppler lidar and for plane-wave intensity statistics because the effects of large-scale turbulence are less important. The series solution for coherent Doppler lidar performance converges slowly to the results from simulation.     

Measurement of geometric distortion in a turbulent atmosphere

A technique for determining the vector displacement that maximizes the spatially local cross-correlation between an image and a reference image as a continuous function of the image space is presented. This is applied to solar observations of granulation made during a condition of rapidly changing atmospheric distortion, a particular turbulence condition wherein features are remapped without loss of spatial resolution.