Intensity fluctuations of flat-topped beam in non-Kolmogorov weak turbulence

Results obtained on the intensity fluctuations of flat-topped Gaussian beams in weakly turbulent non-Kolmogorov horizontal atmospheric optics links are represented. Effects on the scintillation index of the power law α that describes the non-Kolmogorov spectrum are examined. Our results correctly reduce to the existing intensity fluctuations of flat-topped beams in Kolmogorov turbulence. Variation of the scintillation index against non-Kolmogorov power law α exhibits a peak at the worst power law α(w), which happens to be smaller than the Kolmogorov power law of 11/3. If the power law is smaller (larger) than α(w), increase in α will increase (decrease) the intensity fluctuations. Evaluation of the scintillation index at the worst power law results in smaller fluctuations for a Gaussian beam at short propagation distances; however, at long propagation distances flatter beams happen to possess smaller fluctuations. The scintillation change versus the source size follows a similar trend regardless whether the flat-topped beam propagates in a Kolmogorov or non-Kolmogorov medium.     

Generalized modified atmospheric spectral model for optical wave propagating through non-Kolmogorov turbulence

A new generalized modified atmospheric spectral model is derived theoretically for wave propagating through non-Kolmogorov turbulence, which has been reported recently by increasing experimental evidence and theoretical investigation. The generalized, modified atmospheric spectrum considers finite turbulence inner and outer scales and has a spectral power law value in the range of 3 to 5 instead of the standard power law value of 11/3. When the inner scale and outer scale are set to zero and infinity, respectively, this spectral model is reduced to the classical non-Kolmogorov spectrum.     

Measurements of laser phase fluctuations induced by atmospheric turbulence over 2 km and 17.5 km distances

A laser heterodyne system was used to measure the phase fluctuations imposed on a 1.5?μm wavelength laser beam when double-passed over long atmospheric paths. Two distances were used: 2 and 17.5?km. Results are given for intensity scintillation, phase fluctuation time series and spectra, and phase structure function. The results are found to agree well with theory: the spectrum of phase fluctuations follows the 8/3 power law predicted for Kolmogorov turbulence over 3 orders of magnitude in frequency. The methods reported here could be used to investigate large-scale temperature variations in the atmosphere.     

Channel correlation of free space optical communication systems with receiver diversity in non-Kolmogorov atmospheric turbulence

Spatial diversity as an effective technique to mitigate the turbulence fading has been widely utilized in free space optical (FSO) communication systems. The received signals, however, will suffer from channel correlation due to insufficient spacing between component antennas. In this paper, the new expressions of the channel correlation coefficient and specifically its components (the large- and small-scale channel correlation coefficients) for a plane wave with aperture effects are derived for horizontal link in moderate-to-strong turbulence, using a non-Kolmogorov spectrum that has a generalized power law in the range of 3–4 instead of the fixed classical Kolmogorov power law of 11/3. And then the influence of power law variations on the channel correlation coefficient and its components are analysed. The numerical results indicated that various value of the power law lead to varying effects on the channel correlation coefficient and its components. This work will help with the further investigation on the fading correlation in spatial diversity systems.     

Dual-channel heterodyne measurements of atmospheric phase fluctuations

A dual-channel fiber-coupled laser heterodyne system operating at a 1.55-microm wavelength is used to investigate phase fluctuations induced on a laser beam by propagation through turbulent air. Two receivers are used to characterize spatial and temporal variations produced by a turbulent layer of air in the laboratory. The system is also used for measurements through extended turbulence along an 80-m outdoor atmospheric path. Phase structure functions, power spectral densities, and cross correlations are presented.     

Spatial coherence function of partially coherent Gaussian beams in atmospheric turbulence

We introduce a new method of estimating the coherence function of a Gaussian-Schell model beam in the inertial subrange of atmospheric turbulence. It is compared with the previously published methods based on either the quadratic approximation of the parabolic equation or an assumed independence between the source's randomness and the atmosphere using effective beam parameters. This new method, which combines the results of the previous two methods to account for any random source/atmospheric coupling, was shown to more accurately estimate both the coherence radius and coherence functional shape across much of the relevant parameter space. The regions of the parameter space where one method or another is the most accurate in estimating the coherence radius are identified along with the maximum absolute estimation error in each region. By selecting the appropriate estimation method for a given set of conditions, the absolute estimation error can generally be kept to less than 5%, with a maximum error of 7%. We also show that the true coherence function is more Gaussian than expected, with the exponential power tending toward 9/5 rather than the theoretical value of 5/3 in very strong turbulence regardless of the nature of the source coherence.     

大气湍流随机相位屏的数值模拟和验证

本文使用傅立叶变换法对符合Kolmogorov谱的大气随机相位屏进行了数值模拟,采用三层随机相位屏叠加模拟大气湍流.通过比较模拟相位屏的相位结构函数和理论值的符合度对相位屏的统计特性进行了验证,利用长曝光传递函数和相位结构函数计算大气相干长度r0.结果表明,用傅立叶变换法模拟的随机相位屏是正确的,但相位结构函数存在明显的低频空间频率成分不足,采用三层随机相位屏叠加对相位结构函数和理论值的符合度有所改善,实际r0值比设计值偏大.     

大气湍流随机相位屏的数值模拟和验证大气湍流随机相位屏的数值模拟和验证

本文使用傅立叶变换法对符合Kolmogorov谱的大气随机相位屏进行了数值模拟，采用三层随机相位屏叠加模拟大气湍流。通过比较模拟相位屏的相位结构函数和理论值的符合度对相位屏的统计特性进行了验证，利用长曝光传递函数和相位结构函数计算大气相干长度r0。结果表明，用傅立叶变换法模拟的随机相位屏是正确的，但相位结构函数存在明显的低频空间频率成分不足，采用三层随机相位屏叠加对相位结构函数和理论值的符合度有所改善，实际r0值比设计值偏大。     

大气湍流畸变相位屏的数值模拟方法研究

利用功率谱反演法和Zernike多项式展开法对符合Kolmogonov统计规律的大气湍流畸变波前相位屏进行了数值模拟研究。通过对比模拟相位屏的相位结构函数与理论结果的差异分析模拟相位屏的准确性。结果表明,功率谱反演法产生的相位屏在高空间频率部分与理论相符,在低空间频率部分明显偏离理论值,通过次谐波补偿有效改善低频不足,次谐波级数达到4级足够;Zernike多项式展开法产生的相位屏的低空间频率成分与理论相符,而高空间频率成分不足随着所用Zernike阶数的增加而有所改善,但同时也带来计算量增大的问题。     

Average spreading of a radial gaussian beam array in non-Kolmogorov turbulence

The analytical expression for the rms beam width of the radial gaussian beam array propagating in non-Kolmogorov turbulence is derived, where the coherent combination is considered. The influences of the beam number, the generalized exponent, and the ring radius on the rms beam width are investigated. The results indicate that the rms beam width depends greatly on the generalized exponent and the beam number. Further, an optimum ring radius, which leads to a minimum beam width, is proved to exist within a certain traveling distance and the optimum ring radius increases when the beam number increases.     

Measurement of telescope aberrations through atmospheric turbulence by use of phase diversity

We conduct computer simulations of the reconstruction of a wave front at a telescope pupil with the phase-diversity method. An instantaneous wave front is reconstructed from focused and defocused specklegrams of a point star. In the wave-front reconstruction we do not fit the wave front to Zernike polynomials but retrieve the phase with a phase-unwrapping procedure. Averaging over many atmospherically perturbed wave fronts leads to the residual phase error, namely, the aberration of the telescope. The scintillation effect, nonuniformity of amplitude on a telescope pupil, is also discussed.     

Kurtosis parameter K of arbitrary electromagnetic beams propagating through non-Kolmogorov turbulence

General analytical formulae for the kurtosis parameters K (K parameters) of the arbitrary electromagnetic (AE) beams propagating through non-Kolmogorov turbulence are derived, and according to the unified theory of polarization and coherence, the effect of degree of polarization (DOP) of an electromagnetic beam on the K parameter is studied. The analytical formulae can be given by the second-order moments and fourth-order moments of the Wigner distribution function for AE beams at source plane, the two turbulence quantities relating to the spatial power spectrum, and the propagation distance. Our results can also be extended to the arbitrary beams and the arbitrary spatial power spectra of Kolmogorov turbulence or non-Kolmogorov turbulence. Taking the stochastic electromagnetic Gaussian Schell-model (SEGSM) beam as an example, the numerical examples indicate that the K parameters of a SEGSM beam in non-Kolmogorov turbulence depend on propagation distance, the beam parameters and turbulence parameters. The K parameter of a SEGM beam is more sensitive to effect of turbulence with smaller inner scale and generalized exponent parameter. A non-polarized light has the strongest ability of resisting turbulence (ART), however, a fully polarized SEGSM beam has the poorest ART.     

Method for simulating atmospheric turbulence phase effects for multiple time slices and anisoplanatic conditions

Simulating the effects of atmospheric turbulence on optical imaging systems is an important aspect of understanding the performance of these systems. Simulations are particularly important for understanding the statistics of some adaptive-optics system performance measures, such as the mean and variance of the compensated optical transfer function, and for understanding the statistics of estimators used to reconstruct intensity distributions from turbulence-corrupted image measurements. Current methods of simulating the performance of these systems typically make use of random phase screens placed in the system pupil. Methods exist for making random draws of phase screens that have the correct spatial statistics. However, simulating temporal effects and anisoplanatism requires one or more phase screens at different distances from the aperture, possibly moving with different velocities. We describe and demonstrate a method for creating random draws of phase screens with the correct space-time statistics for a bitrary turbulence and wind-velocity profiles, which can be placed in the telescope pupil in simulations. Results are provided for both the von Kármán and the Kolmogorov turbulence spectra. We also show how to simulate anisoplanatic effects with this technique.     

Laser beam propagation in atmospheric turbulence

The optical effects of atmospheric turbulence on the propagation of low power laser beams are reviewed in this paper. The optical effects are produced by the temperature fluctuations which result in fluctuations of the refractive index of air. The commonly-used models of index-of-refraction fluctuations are presented. Laser beams experience fluctuations of beam size, beam position, and intensity distribution within the beam due to refractive turbulence. Some of the observed effects are qualitatively explained by treating the turbulent atmosphere as a collection of moving gaseous lenses of various sizes. Analytical results and experimental verifications of the variance, covariance and probability distribution of intensity fluctuations in weak turbulence are presented. For stronger turbulence, a saturation of the optical scintillations is observed. The saturation of scintillations involves a progressive break-up of the beam into multiple patches; the beam loses some of its lateral coherence. Heterodyne systems operating in a turbulent atmosphere experience a loss of heterodyne signal due to the destruction of coherence.     

Propagation of coherently combined truncated laser beam arrays with beam distortions in non-Kolmogorov turbulence

The propagation properties of coherently combined truncated laser beam arrays with beam distortions through non-Kolmogorov turbulence are studied in detail both analytically and numerically. The analytical expressions for the average intensity and the beam width of coherently combined truncated laser beam arrays with beam distortions propagating through turbulence are derived based on the combination of statistical optics methods and the extended Huygens-Fresnel principle. The effect of beam distortions, such as amplitude modulation and phase fluctuation, is studied by numerical examples. The numerical results reveal that phase fluctuations have significant influence on the spreading of coherently combined truncated laser beam arrays in non-Kolmogorov turbulence, and the effects of the phase fluctuations can be negligible as long as the phase fluctuations are controlled under a certain level, i.e., a>0.05 for the situation considered in the paper. Furthermore, large phase fluctuations can convert the beam distribution rapidly to a Gaussian form, vary the spreading, weaken the optimum truncation effects, and suppress the dependence of spreading on the parameters of the non-Kolmogorov turbulence.     

Zernike annular polynomials and atmospheric turbulence

Imaging through atmospheric turbulence by systems with annular pupils is discussed using the Zernike annular polynomials. Fourier transforms of these polynomials are derived analytically to facilitate the calculation of variance and covariance of the aberration coefficients. Zernike annular shape functions are derived and used to calculate the Strehl ratio and the residual phase structure and mutual coherence functions when a certain number of modes are corrected using, say, a deformable mirror. Special cases of long- and short-exposure images are also considered. The results for systems with a circular pupil are obtained as a special case of the annular pupil.     

Statistics of the fractal structure and phase singularity of a plane light wave propagation in atmospheric turbulence

Numerical experiments are carried out for a plane wave propagating in the atmospheric turbulence for a weak to strong fluctuation condition, i.e., the Rytov index being in a large range of 2x10(-3) to 20. Mainly two categories of propagation events are explored for the same range of Rytov index. In one category the propagation distance and also the Fresnel length are kept fixed with the turbulence strength changing. In the other the turbulence strength is kept fixed with the distance changing. The statistical characteristics of the scintillation index, the maximum and minimum of the intensity, the fractal dimension of the intensity image, and the number density of the phase singularity are analyzed. The behaviors of the fractal dimension and the density of the phase singularity present obvious differences for the two categories of propagation. The fractal dimension depends both on the Rytov index and the Fresnel length. In both weak and strong fluctuation conditions the dimension generally increases with the Rytov index, but is at minimum at the onset region. The phase singularity density is coincident with the theoretical results under a weak fluctuation condition, and has a slowly increasing manner with the Rytov index in the strong fluctuation condition. The dependence on the Fresnel size is confident and there is no saturation for the phase singularity.     

Spatial versus temporal phase shifting in electronic speckle-pattern interferometry: noise comparison in phase maps

Temporal and spatial phase shifting in electronic speckle-pattern interferometry are compared quantitatively with respect to the quality of the resultant deformation phase maps. On the basis of an analysis of the noise in sawtooth fringes a figure of merit is defined and measured for various in-plane and out-of-plane sensitive electronic speckle-pattern interferometry configurations. Varying quantities like the object-illuminating intensity, the beam ratio, the speckle size and shape, and the fringe density allows characteristic behaviors of both phase-shifting methods to be explored.     

Effect of anisotropy on intensity fluctuations in oceanic turbulence

For an optical spherical wave propagating in an oceanic turbulent medium, the effect of anisotropy on the received intensity fluctuations is investigated. For different anisotropy factors, the variations of the scintillation index vs. the ratio that determines the relative strength of temperature and salinity in the index fluctuations, the rate of dissipation of the mean squared temperature, the rate of dissipation of the turbulent kinetic energy, viscosity, link length and the wavelength are plotted. It is found that, for all the oceanic turbulence and the link parameters of interest, as the medium becomes more anisotropic, the intensity of the optical spherical wave fluctuates less. It is concluded that the performance of an optical wireless communication systems (OWCS) operating in anisotropic oceanic turbulence is better than the performance of OWCS operating in isotropic oceanic turbulence.     

Threshold detection in the presence of atmospheric turbulence

Recently there has been increased interest in threats to spacecraft from ground-based lasers. It has been suggested that some spacecraft should use laser-threat-warning receivers. We consider the effects of atmospheric turbulence on threshold detection of optical signals by an exoatmospheric receiver. The results are applicable to both cw and pulsed optical illumination that results from ground-based lasers. In particular we obtain accurate analytical expressions, over a wide range of conditions of practical interest, that yield the required signal-to-noise ratio for a given (single-event) probability of detection, false-alarm rate, and turbulence-induced log-intensity variance. The degrading effects of atmospheric turbulence on threshold detection are most important for large zenith angles in the blue-green region of the visible. As an illustrative example, a false-alarm rate of 1 in 3 years is assumed, and specific numerical results are presented for the required signal-to-noise ratio necessary to obtain a detection probability of at least 95% over a range of optical wavelengths and propagation conditions of interest.