Modified moment-matching method for estimating pointing parameters in the presence of atmospheric turbulence

An accurate pointing system is required in free-space optical (FSO) communication links. Low energy-transmission efficiency caused by pointing errors would decline the communication system's performance. The statistics of the detected signal or return signal values could be used to estimate the pointing parameters, whereas atmospheric turbulence brings in serious challenges. A modified moment-matching estimation method is presented in this paper. The irradiance fluctuation caused by the atmospheric turbulence is considered, and the probability density function (PDF) in a weak turbulence condition is assumed to be lognormal. This modified approach is evaluated with wave-propagation simulation data and shows significant improvement over the conventional approach. The estimation accuracy and the properties of this new approach are also discussed. Although our method is based on lognormal irradiance PDF under a weak turbulence condition, the irradiance PDF would tend to be lognormal with aperture averaging effect under moderate to strong turbulence, and the ideas can be extended with appropriate PDF models to satisfy different conditions.     

Average BER performance of FSO SIM-QAM systems in the presence of atmospheric turbulence and pointing errors

This paper presents the exact average bit error rate (BER) analysis of the free-space optical system employing subcarrier intensity modulation (SIM) with Gray-coded quadrature amplitude modulation (QAM). The intensity fluctuations of the received optical signal are caused by the path loss, atmospheric turbulence and pointing errors. The exact closed-form analytical expressions for the average BER are derived assuming the SIM-QAM with arbitrary constellation size in the presence of the Gamma–Gamma scintillation. The simple approximate average BER expressions are also provided, considering only the dominant term in the finite summations of obtained expressions. Derived expressions are reduced to the special case when optical signal transmission is affected only by the atmospheric turbulence. Numerical results are presented in order to illustrate usefulness of the derived expressions and also to give insights into the effects of different modulation, channel and receiver parameters on the average BER performance. The results show that the misalignment between the transmitter laser and receiver detector has the strong effect on the average BER value, especially in the range of the high values of the average electrical signal-to-noise ratio.     

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.     

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.     

Mode coupling approach to beam propagation in atmospheric turbulence

A mode coupling approach based on the modal theory of coherence is suggested for the study of partially coherent beams in atmospheric turbulence. An approximate expression is derived for the mode power coupling coefficients, and some specific cases are studied using numerical methods. Several general results derived from the properties of the coupling coefficients are also presented.     

大气湍流对厄米-高斯光束光束质量的影响

采用二阶矩束宽、桶中功率和光束质量参数作为特征参数,研究了大气湍流对厄米-高斯(H-G)光束远场光束质量造成的影响,并对其作了详细的数值计算和分析。研究结果表明,在大气湍流中,TEM10模光束的相对展宽小于TEM00模光束的相对展宽。在湍流不太强的情况下,例如Rytov变量小于9时,TEM10模的PIB小于TEM00模光束的桶中功率,TEM10模的光束质量参数大于TEM00模光束的光束质量参数。但是,随着湍流的进一步增强,TEM10模的PIB和光束质量参数值逐渐趋近于TEM00模的PIB和光束质量参数值。     

Double passage enhancement of thin beam wandering through atmospheric turbulence

Abstract

Beam wandering (random lateral displacement) enhancement of thin beams by a double passage through turbulence is theoretically and experimentally investigated. A relationship is found between the random lateral displacements of double passage and single passage, in the case of homogeneous turbulence. Results of laboratory measurements are presented and suggestions for practical use given.     

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.     

Vortex beam propagation through atmospheric turbulence and topological charge conservation

The propagation of vortex beams through weak-to-strong atmospheric turbulence is simulated and analyzed. It is demonstrated that the topological charge of such a beam is a robust quantity that could be used as an information carrier in optical communications. The advantages and limitations of such an approach are discussed.     

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.     

Propagation properties of a partially coherent radially polarized beam in atmospheric turbulence

Based on the extended Huygens–Fresnel integral, second-order moments of the Wigner distribution function of a partially coherent radially polarized beam propagating through atmospheric turbulence are derived. Besides, propagation properties such as the mean-squared beam width, angular width, effective radius of curvature, beam propagation factor and Rayleigh range can also be obtained and calculated numerically. It is shown that the propagation properties are dependent on the spatial correlation length, refraction index structure constant and propagation distance.     

Optimum truncation of a Gaussian beam for propagation through atmospheric turbulence

The mean on-axis far-field (or focal-plane) irradiance of a Gaussian beam that is truncated by a circular aperture in the presence of atmospheric turbulence is considered. In the absence of turbulence, an accurate analytic approximation for the irradiance distribution that is valid within the main central lobe of the beam is presented. Based on this approximation, the mean on-axis far-field irradiance and the corresponding turbulence Strehl ratio for the truncated Gaussian beam are then obtained. By maximization of the on-axis irradiance, the optimum ratio of the beam diameter to the aperture diameter in the presence of turbulence is obtained, and the results for the corresponding maximum on-axis irradiance as a function of the strength of turbulence are presented. In particular, for D/r(0) > 1, where D is the aperture diameter and r(0) is Fried's coherence length, optimum truncation of a Gaussian beam and uniform illumination of a circular aperture (where the same total power isuniformly distributed over the aperture) result in the same on-axis irradiance in the presence of uncompensated turbulence.     

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.     

Average intensity and spreading of an astigmatic sinh-Gaussian beam with small beam width propagating in atmospheric turbulence

Propagation properties of astigmatic sinh-Gaussian beams (ShGBs) with small beam width in turbulent atmosphere are investigated. Based on the extended Huygens–Fresnel integral, analytical formulae for the average intensity and the effective beam size of an astigmatic ShGB are derived in turbulent atmosphere. The average intensity distribution and the spreading properties of an astigmatic ShGB propagating in turbulent atmosphere are numerically demonstrated. The influences of the beam parameters and the structure constant of atmospheric turbulence on the propagation properties of astigmatic ShGBs are also discussed in detail. In particular, for sufficiently small beam width and sinh-part parameter as well as suitable astigmatism, we show that the average intensity pattern converts into a perfect dark-hollow profile from initial two-petal pattern when ShGBs with astigmatic aberration propagate through atmospheric turbulence.     

Estimation of laser system pointing performance by use of statistics of return photons

Strategic laser systems are subject to residual pointing errors arising from vibrations and atmospheric turbulence, estimates of which may allow improved system performance. Field data from the Air Force Research Laboratory Floodbeam Experiments suggested a linear relationship between the mean and standard deviation of the shot-by-shot signals and the jitter. An ideal analytic solution and Monte Carlo simulations confirmed this result for a relatively large number of returns. A refined approach using statistical chi(2) techniques, which simultaneously estimates jitter and boresight, was developed to address results from satellite passes with relatively few returns and provides excellent jitter and boresight predictions.     

Study on the propagation parameters of Bessel-Gaussian beams carrying optical vortices through atmospheric turbulence

Based on the integral representation of Bessel function and the extended Huygens-Fresnel principle, an integral expression of the Wigner distribution function (WDF) for partially coherent Bessel-Gaussian beams (PBGBs) propagating through turbulent atmosphere has been obtained. Also, the analytical formulas of the M2-factor for PBGB propagation in such a medium have been derived, which can be applied to cases of different spatial power spectra of the refractive index fluctuations. The performed numerical results reveal that the M2-factor of a PBGB in turbulent atmosphere depends on the beam parameters of the initial input beam, the structure constants of the turbulent atmosphere, and the propagation distance. These results may be useful in long-distance optical communications in free space or in turbulent atmosphere.     

LADAR detection statistics in the presence of pointing errors

The probability of detection of optically rough targets with pulsed LADAR systems that use direct detection is considered. It is assumed that the LADAR operates under conditions of both unintentional pointing offset bias (i.e., bore-sight error) andjitter. Under these conditions the probabilities of detection of targets in both the near field and the far field of the collecting aperture (i.e., for resolved, partially resolved, and unresolved targets) and for both large and small photoelectron counts are derived, and in many cases of practical interest accurate, elementary analytic approximations that are useful for parametric system studies are obtained. A number of technical references are appended, in which some of the key results are derived. In particular, an interesting new mathematical result involving the complementary incomplete gamma function and an analytic expression for the probability distribution function of a signal photoelectron count obeying BoseEinstein statistics (such as that arising from unresolved targets) immersed in Poisson noise is derived.     

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.     

Branch-point reconstruction in laser beam projection through turbulence with finite-degree-of-freedom phase-only wave-front correction

Wave-front sensing and deformable mirror control algorithms in adaptive optics systems are designed on the premise that a continuous phase function exists in the telescope pupil that can be conjugated with a deformable mirror for the purpose of projecting a laser beam. However, recent studies of coherent wave propagation through turbulence have shown that under conditions where scintillation is not negligible, a truly continuous phase function does not in general exist as a result of the presence of branch points in the complex optical field. Because of branch points and the associated branch cuts, least-squares wave-front reconstruction paradigms can have large errors. We study the improvement that can be obtained by implementing wave-front reconstructors that can sense the presence of branch points and reconstruct a discontinuous phase function in the context of a laser beam projection system. This study was conducted by fitting a finite-degree-of-freedom deformable mirror to branch-point and least-squares reconstructions of the phase of the beacon field, propagating the corrected field to the beacon plane, and evaluating performance in the beacon plane. We find that the value of implementing branch-point reconstructors with a finite-degree-of-freedom deformable mirror is significant for optical paths that cause saturated log-amplitude fluctuations.