A three-dimensional refractive index model for simulation of optical wave propagation in atmospheric turbulence

Numerical modeling of optical wave propagation in atmospheric turbulence is traditionally performed with using the so-called “split”-operator method, when the influence of the propagation medium’s refractive index inhomogeneities is accounted for only within a system of infinitely narrow layers (phase screens) where phase is distorted. Commonly, under certain assumptions, such phase screens are considered as mutually statistically uncorrelated. However, in several important applications including laser target tracking, remote sensing, and atmospheric imaging, accurate optical field propagation modeling assumes upper limitations on interscreen spacing. The latter situation can be observed, for instance, in the presence of large-scale turbulent inhomogeneities or in deep turbulence conditions, where interscreen distances become comparable with turbulence outer scale and, hence, corresponding phase screens cannot be statistically uncorrelated. In this paper, we discuss correlated phase screens. The statistical characteristics of screens are calculated based on a representation of turbulent fluctuations of three-dimensional (3D) refractive index random field as a set of sequentially correlated 3D layers displaced in the wave propagation direction. The statistical characteristics of refractive index fluctuations are described in terms of the von Karman power spectrum density. In the representation of these 3D layers by corresponding phase screens, the geometrical optics approximation is used.     

Statistical characteristics of a wave propagating through a layer with random irregularities

Statistical characteristics of a wave propagating through a layer with random irregularities are investigated by a simulation procedure. The investigation is carried out within the geometrical optics approximation in its validity range. It is shown that when the irregular layer is a long distance from the source and observer, a significant role in the formation of eikonal (phase path) fluctuations is then played by trajectory fluctuations in regions of the propagation medium, free from irregularities before and after the irregular layer. With these variations taken into account, which are neglected in conventional perturbation theory, we obtained approximate expressions for the dispersion and the correlation function of the eikonal. We investigate the behaviour of the eikonal dispersions, the angles and correlation functions of the eikonal and field for different disturbances of the medium, and for different distances of the receiver and transmitter from the layer boundaries.     

Generation of infinitely long phase screens for modeling of optical wave propagation in atmospheric turbulence

Numerical modeling of optical wave propagation in atmospheric turbulence is traditionally performed by using the so-called 'split'-operator method, where the influence of the propagation medium's refractive index inhomogeneities is accounted for only within a set of infinitely narrow phase distorting layers (phase screens). These phase screens are generated on a numerical grid of finite size, which corresponds to a rather narrow slice (spatial area) of atmospheric turbulence. In several important applications including laser target tracking, remote sensing, adaptive optics, and atmospheric imaging, optical system performance depends on atmospheric turbulence within an extended area that significantly exceeds the area associated with the numerical grid.

In this paper we discuss methods that allow the generation of a family of long (including infinitely long) phase screens representing an extended (in one direction) area of atmospheric turbulence-induced phase distortions. This technique also allows the generation of long phase screens with spatially inhomogeneous statistical characteristics.     

Spherical wave propagation through inhomogeneous, anisotropic turbulence: log-amplitude and phase correlations

Inhomogeneity and anisotropy are intrinsic characteristics of daytime and nighttime turbulence in the atmospheric boundary layer. In the present paper, line-of-sight sound propagation through inhomogeneous, anisotropic turbulence with temperature and velocity fluctuations is considered. Starting from a parabolic equation and using the Markov approximation, formulas are derived for the correlation functions and variances of log-amplitude and phase fluctuations of a spherical sound wave. These statistical moments of a sound field are important for many practical applications in atmospheric acoustics. The derived formulas for the correlation functions and variances generalize those already known in the literature for two limiting cases: (a) homogeneous, isotropic turbulence, and (b) inhomogeneous, anisotropic turbulence with temperature fluctuations only. Furthermore, the formulas differ from those for the case of plane wave propagation. Using the derived formulas and Mann's spectral tensor of velocity fluctuations for shear-driven turbulence, the correlation functions and variances of log-amplitude and phase fluctuations are studied numerically. The results obtained clearly show that turbulence inhomogeneity and anisotropy significantly affect sound propagation in the atmosphere.     

在大气湍流斜程传输中拉盖高斯光束的轨道角动量的研究

大气湍流引起大气折射率随机变化,导致空间不均匀性.拉盖高斯光束在大气湍流中传输时,空间不均匀性会使光子波函数改变,引起轨道角动量的变化.本文讨论了拉盖尔高斯光束在大气斜程传输时,湍流介质改变光子轨道角动量而形成不同的光子态.计算了螺旋谐波各分量所占光束总能量的权重,分析了拉盖高斯光束的轨道角动量的变化规律.     

Study of the sound-vortex interaction: direct numerical simulations and experimental results

We present a study of sound propagation through a single vortex by direct numerical simulations (DNS) compared to experimental measurements. We analyse the amplitude and the phase shift of the sound wave when it interacts with the vortical flow and we display the focusing effects produced by the vortex. We show that the turbulent fluctuations have a little effect on the sound phase shift whereas they induce a strong defocusing effect on the sound amplitude. Received 9 October 2002 / Received in final form 20 January 2003 Published online 1st April 2003 RID="a" ID="a"e-mail: rberthet@lps.ens.fr RID="b" ID="b"UMR CNRS 8550     

Correlation characteristics of waves in lossy media with smooth fluctuations of the refractive index with oblique incidence on boundary

The features of coherence function (and angular spectrum) and also the fluctuation of amplitude and phase of wavefield in a random strongly absorptive medium are investigated in the case of arbitrary angle of incidence at the surface.

In this paper it is elucidated that with oblique incidence a dissipation in the random medium can accelerate the accumulation of wave fluctuations and its incoherence. This effect strongly depends on the rate of decrease of the 'wings' of the scattering indicatrix. An analytical theory (Rytov's approximation and modified method of parabolic equation) has been modelled by Monte Carlo simulation of wave propagation, and also by numerical solution of the model transfer equation. It is revealed that the width of an angle spectrum can nonmonotonically change with the immersion to the absorptive medium.     

Fluctuations of the propagating beam field in a lens waveguide

It is well known that light beams propagated through a turbulent medium such as the atmosphere or the air between optical lenses suffer fluctuations of the field in respect of its amplitude and phase. This paper investigates the fluctuation distributions of a laser beam propagating through a lens waveguide which consists of a sequence of optical lenses placed in a randomly inhomogeneous medium. By using a Rytov approximation approach, the mean-square values of fluctuations are obtained in terms of log-amplitude and phase of the beam field. The results show that as the light beam propagates through the lens waveguide, the mean-square fluctuations increase with periodic variations corresponding to a period of lens spacing and that a singular phenomenon of fluctuation reduction at each focus of the beam is evident at larger beam spot sizes.     

Sound-wave coherence in atmospheric turbulence with intrinsic and global intermittency

The coherence function of sound waves propagating through an intermittently turbulent atmosphere is calculated theoretically. Intermittency mechanisms due to both the turbulent energy cascade (intrinsic intermittency) and spatially uneven production (global intermittency) are modeled using ensembles of quasiwavelets (QWs), which are analogous to turbulent eddies. The intrinsic intermittency is associated with decreasing spatial density (packing fraction) of the QWs with decreasing size. Global intermittency is introduced by allowing the local strength of the turbulence, as manifested by the amplitudes of the QWs, to vary in space according to superimposed Markov processes. The resulting turbulence spectrum is then used to evaluate the coherence function of a plane sound wave undergoing line-of-sight propagation. Predictions are made by a general simulation method and by an analytical derivation valid in the limit of Gaussian fluctuations in signal phase. It is shown that the average coherence function increases as a result of both intrinsic and global intermittency. When global intermittency is very strong, signal phase fluctuations become highly non-Gaussian and the average coherence is dominated by episodes with weak turbulence.     

Fluctuations of spherical waves in a turbulent atmosphere: effect of the axisymmetric approximation in computational methods

The validity of the axisymmetric parabolic-equation (PE) method for line-of-sight sound propagation in a turbulent atmosphere is investigated. The axisymmetric PE method is a finite-difference method for solving a 2D parabolic wave equation, which follows from the 3D wave equation by the assumption of axial symmetry around the vertical axis through the source. It is found that this axisymmetric approximation has a considerable spurious effect on the fluctuations of the sound field. This is concluded from analytical expressions for the log-amplitude and phase variances, derived both for isotropic turbulence and for axisymmetric turbulence. The expressions for axisymmetric turbulence are compared with the results of numerical computations with the PE method.     

Focusing of sound pulses using the time reversal technique on 100-km paths in a deep sea

Numerical and analytical studies are performed on how unstable fluctuations of the parameters of the medium in a deep sea affect the focusing of sound pulses using the time reversal method. The simplest situation, when point sources and receivers are used for emission and reception, is considered. Pulse propagation in the direct and backward directions is numerically simulated by the parabolic equation method. Calculations are performed for sound signals with frequencies of several tens of hertz. It is shown that, in the presence of sound velocity fluctuations caused by random internal waves, noticeable attenuation of the field amplitude at the center of the focal spot can be observed beginning from distances of 200 to 400 km. As the central frequency of the pulsed signal increases, the effect of nonstationarity of the perturbation on the focusing is amplified. This phenomenon is explained qualitatively and quantitatively in the geometrical optics approximation.     

基于轴棱锥产生近似无衍射Mathieu光束的新方法

提出了一种基于轴棱锥产生零阶近似无衍射Mathieu光束的新方法,利用轴棱锥聚焦具有椭圆高斯振幅调制的平面波,得到近似零阶无衍射Mathieu光束.根据椭圆高斯平面波经轴棱锥衍射的衍射积分公式,对光强分布进行了数值模拟,依据几何光学模型计算了近似无衍射Mathieu光束的最大无衍射距离,并设计了实验对理论模拟的结果进行了验证.实验采用柱透镜和准直扩束系统变换圆高斯光束产生具有椭圆高斯振幅调制的平面波,用轴棱锥聚焦该平面波后得到近似无衍射Mathieu光束,实验结果与理论模拟和计算相符.     

Slant path average intensity of finite optical beam propagating in turbulent atmosphere

The average intensity of finite laser beam propagating through turbulent atmosphere is calculated from the extended Huygens Fresnel principle. Formulas are presented for the slant path average intensity from an arbitrarily truncated Gaussian beam. The new expressions are derived from the modified von Karman spectrum for refractive-index fluctuations, quadratic approximation of the structure function,and Gaussian approximation for the product of Gaussian function and Bessel function. It is shown that the form of average intensity is not a Gaussian function but a polynomial of the power of the binomial function, Gaussian function, and the incomplete gamma function. The results also show that the mean irradiance of a finite optical beam propagating in slant path turbulent atmosphere not only depends on the effective beam radius at the transmitting aperture plane, propagation distance, and long-term lateral coherence length of spherical wave, but also on the radius of emit aperture.     

Propagation of Helmholtz-Gauss beams in weak turbulent atmosphere

Based on the Rytov approximation of light propagation in weak turbulent atmosphere,the closed-form expressions of field and average irradiance of each one of the four fundamental families of Helmholtz-Gauss (HzG)beams:cosine-Gauss beams,stationary Mathieu-Gauss beams,stationary parabolic-Gauss beams,and Bessel-Gauss beams,which are propagating in weak turbulent atmosphere,are obtained.The results show that the field and average irradiance can be written as the product of four factors:complex amplitude depending on the z-coordinate only,a Ganssian beam.a factor of complex phase perturbation induced by atmospheric turbulence,and a complex scaled version of the transverse shape of the non-diffracting beam.The effect of weak atmospheric turbulence on irradiance distribution of the HzG beam can be ignored.     

Scintillation of a laser beam propagation through non-Kolmogorov strong turbulence

Atmospheric turbulence causes strong irradiance fluctuations of propagating optical wave under the severe weather conditions in long-distance free space optical communication. In this paper, the scintillation index for a Gaussian beam wave propagation through non-Kolmogorov turbulent atmosphere is derived in strong fluctuation regime, using non-Kolmogorov spectrum with a generalized power law exponent and the extended Rytov theory with a modified spatial filter function. The analytic expressions are obtained and then used to analyze the effect of power law, refractive-index structure parameter, propagation distance, phase radius of curvature, beam width and wavelength on scintillation index of Gaussian beam under the strong atmospheric turbulence. It shows that, with the increasing of structure parameter or propagation distance, scintillation index increases sharply up to the peak point and then decreases gradually toward unity at rates depending on power law. And there exist optimal value of radius of curvature and beam width for minimizing the value of scintillation index and long wavelength for mitigating the effect of non-Kolmogorov strong turbulence on link performance.     

Theoretical prediction and experimental validation of the angle-of-arrival probability density of a laser beam in a strong plane-flame turbulence

The propagation of a laser beam through a plane turbulent flame is studied using geometrical optics approximation. The random fluctuations of the refractive index caused by a strong thermal turbulence in the flame create random perturbations of the laser beam direction. From the Markovian process model applied along the whole random path of the beam, the theoretical probability density of the laser beam angle-of-arrival is derived from the analytical solution of the Einstein-Fokker-Kolmogorov equation, which we have determined in terms of a series expansion of spherical harmonics. An experimental setup and a method for measuring this probability density are described. The experimental results obtained are shown to agree with the theoretical predictions.     

Transverse-longitudinal coherence function of a sound field for line-of-sight propagation in a turbulent atmosphere

Using the narrow-angle and Markov approximations, a formula for the transverse-longitudinal coherence function of a sound field propagating in a turbulent atmosphere with temperature and wind velocity fluctuations is derived. This function, which applies to observation points that are arbitrarily located in space, generalizes the transverse coherence function (coherence when the observation points are in a plane perpendicular to the sound propagation path), which has been studied extensively. The new result is expressed in terms of the transverse coherence function and the extinction coefficient of the mean sound field. The transverse-longitudinal coherence function of a plane sound wave is then calculated and studied in detail for the Gaussian and von Kármán spectra of temperature and wind velocity fluctuations. It is shown, for relatively small propagation distances, that the magnitude of the coherence function decreases in the longitudinal direction but remains almost constant in the transverse direction. On the other hand, for moderate and large propagation distances, the magnitude of the coherence decreases faster in the transverse direction than in the longitudinal. For some parameters of the problem, the coherence function has relatively large local maxima and minima as the transverse and longitudinal coordinates are varied. With small modifications, many results obtained in the paper can be applied to studies of electromagnetic wave propagation in a turbulent atmosphere.     

Ray theory for a locally layered random medium

We consider acoustic pulse propagation in inhomogeneous media over relatively long propagation distances. Our main objective is to characterize the spreading of the travelling pulse due to microscale variations in the medium parameters. The pulse is generated by a point source and the medium is modelled by a smooth three-dimensional background that is modulated by stratified random fluctuations. We refer to such media as locally layered .

We show that, when the pulse is observed relative to its random arrival time, it stabilizes to a shape determined by the slowly varying background convolved with a Gaussian. The width of the Gaussian and the random travel time are determined by the medium parameters along the ray connecting the source and the point of observation. The ray is determined by high-frequency asymptotics (geometrical optics). If we observe the pulse in a deterministic frame moving with the effective slowness , it does not stabilize and its mean is broader because of the random component of the travel time. The analysis of this phenomenon involves the asymptotic solution of partial differential equations with randomly varying coefficients and is based on a new representation of the field in terms of generalized plane waves that travel in opposite directions relative to the layering.     

Numerical modeling of a point-source image under relative motion of radiation receiver and atmosphere

A procedure is proposed for numerical modeling of instantaneous and averaged (over various time intervals) distant-point-source images perturbed by a turbulent atmosphere that moves relative to the radiation receiver. Examples of image calculations under conditions of the significant effect of atmospheric turbulence in an approximation of geometrical optics are presented and analyzed.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 37, No. 2, pp. 200–209, February, 1994.