Wave optics simulation of atmospheric turbulence and reflective speckle effects in CO2 lidar

Laser speckle can influence lidar measurements from a diffuse hard target. Atmospheric optical turbulence will also affect the lidar return signal. We present a numerical simulation that models the propagation of a lidar beam and accounts for both reflective speckle and atmospheric turbulence effects. Our simulation is based on implementing a Huygens-Fresnel approximation to laser propagation. A series of phase screens, with the appropriate atmospheric statistical characteristics, are used to simulate the effect of atmospheric turbulence. A single random phase screen is used to simulate scattering of the entire beam from a rough surface. We compare the output of our numerical model with separate CO(2) lidar measurements of atmospheric turbulence and reflective speckle. We also compare the output of our model with separate analytical predictions for atmospheric turbulence and reflective speckle. Good agreement was found between the model and the experimental data. Good agreement was also found with analytical predictions. Finally, we present results of a simulation of the combined effects on a finite-aperture lidar system that are qualitatively consistent with previous experimental observations of increasing rms noise with increasing turbulence level.     

Target-in-the-loop beam control: basic considerations for analysis and wave-front sensing

Target-in-the-loop (TIL) wave propagation geometry represents perhaps the most challenging case for adaptive optics applications that are related to maximization of irradiance power density on extended remotely located surfaces in the presence of dynamically changing refractive-index inhomogeneities in the propagation medium. We introduce a TIL propagation model that uses a combination of the parabolic equation describing coherent outgoing-wave propagation, and the equation describing evolution of the mutual correlation function (MCF) for the backscattered wave (return wave). The resulting evolution equation for the MCF is further simplified by use of the smooth-refractive-index approximation. This approximation permits derivation of the transport equation for the return-wave brightness function, analyzed here by the method of characteristics (brightness function trajectories). The equations for the brightness function trajectories (ray equations) can be efficiently integrated numerically. We also consider wave-front sensors that perform sensing of speckle-averaged characteristics of the wave-front phase (TIL sensors). Analysis of the wave-front phase reconstructed from Shack-Hartmann TIL sensor measurements shows that an extended target introduces a phase modulation (target-induced phase) that cannot be easily separated from the atmospheric-turbulence-related phase aberrations. We also show that wave-front sensing results depend on the extended target shape, surface roughness, and outgoing-beam intensity distribution on the target surface. For targets with smooth surfaces and nonflat shapes, the target-induced phase can contain aberrations. The presence of target-induced aberrations in the conjugated phase may result in a deterioration of adaptive system performance.     

Propagation properties of rectangular Laguerre–Gaussian-correlated Schell-model beams in atmospheric turbulence

Based on the extended Huygens–Fresnel principle and the second-order moments of the Wigner distribution function (WDF), the analytical expressions for the cross-spectral density (CSD) and the propagation factor of a rectangular Laguerre–Gaussian-correlated Schell-model (LGCSM) beam propagating in atmospheric turbulence are derived. The statistical properties, such as the average intensity, the spectral degree of coherence (SDOC) and the propagation factor, of a rectangular LGCSM beam in free space and atmospheric turbulence are comparatively analysed. It is illustrated that a rectangular LGCSM beam exhibits self-splitting and combing properties on propagation in atmospheric turbulence, and the self-splitting properties of such beam are closely related to its beam orders m and n, which is quite different from other self-splitting beams. In addition, the rectangular LGCSM beam has an advantage for reducing the turbulence-induced degradation compared with the conventional partially coherent beams.     

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.     

Effects of atmosphere on spectrum of modified partially coherent flat-topped beams

The analytical formulas for the spectrum of modified partially coherent flat-topped (MPCFT) beams propagating in a turbulent atmosphere have been derived under strong fluctuation condition of turbulence. The spectral properties of MPCFT beams propagating in a strong turbulence have been investigated, and the changes of on-axis and off-axis spectral distributions have been analyzed quantitatively. The results show that the on-axis relative spectral shift of the modified partially coherent flat-topped beam exhibits spectral switch for the beam order M > 1, and with the increasing propagation distance the blue-shift gradually decreases in atmospheric turbulence. A rapid transition for the off-axis spectrum appears at one critical position in turbulence, and the position decreases with the increasing beam order for the lower beam order and the nearer propagation distance. The spectral properties and the spectral shifts of MPCFT beams also depend on the inner scale, the spatial coherence parameter.     

Feasibility study for the simulation of beam propagation: consideration of coherent lidar performance

To analyze the effects of atmospheric refractive turbulence on coherent lidar performance in a realistic way it is necessary to consider the use of simulations of beam propagation in three-dimensional random media. The capability of the split-step solution to simulate the propagation phenomena is shown, and the limitations and numerical requirements for a simulation of given accuracy are established. Several analytical theories that describe laser beam spreading, beam wander, coherence diameters, and variance and autocorrelation of the beam intensity are compared with results from simulations. Although the analysis stems from a study of coherent lidar performance, the conclusions of the method are applicable to other areas related to beam propagation in the atmosphere.     

Wavefront propagation in turbulence: an unified approach to the derivation of angular correlation functions

A general expression of the spatial correlation functions of quantities related to the phase fluctuations of a wave that have propagated through the atmospheric turbulence are derived. A generalization of the method to integrand containing the product of an arbitrary number of hypergeometric functions is presented. The formalism is able to give the coefficients of phase-expansion functions orthogonal over an arbitrary circularly symmetric weighting function for an isotropic turbulence spectrum, as well as to describe the effect of the finite outer and inner scales of the turbulence and to describe the spherical propagation or to derive the effects of the analytical operators acting on the phase such as the derivatives of any order. The derivation of the generalized integrals with multiparameters is based on the Mellin transforms integration method.     

Numerical simulation of an adaptive optics system with laser propagation in the atmosphere

A comprehensive model of laser propagation in the atmosphere with a complete adaptive optics (AO) system for phase compensation is presented, and a corresponding computer program is compiled. A direct wave-front gradient control method is used to reconstruct the wave-front phase. With the long-exposure Strehl ratio as the evaluation parameter, a numerical simulation of an AO system in a stationary state with the atmospheric propagation of a laser beam was conducted. It was found that for certain conditions the phase screen that describes turbulence in the atmosphere might not be isotropic. Numerical experiments show that the computational results in imaging of lenses by means of the fast Fourier transform (FFT) method agree well with those computed by means of an integration method. However, the computer time required for the FFT method is 1 order of magnitude less than that of the integration method. Phase tailoring of the calculated phase is presented as a means to solve the problem that variance of the calculated residual phase does not correspond to the correction effectiveness of an AO system. It is found for the first time to our knowledge that for a constant delay time of an AO system, when the lateral wind speed exceeds a threshold, the compensation effectiveness of an AO system is better than that of complete phase conjugation. This finding indicates that the better compensation capability of an AO system does not mean better correction effectiveness.     

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.     

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.     

Heterodyne lidar returns in the turbulent atmosphere: performance evaluation of simulated systems

Simulations of beam propagation in three-dimensional random media were used to study the effects of atmospheric refractive turbulence on coherent lidar performance. By use of the two-beam model, the lidar return is expressed in terms of the overlap integral of the transmitter and the virtual (backpropagated) local oscillator beams at the target, reducing the problem to one of computing irradiance along the two propagation paths. This approach provides the tools for analyzing laser radar with general refractive turbulence conditions, beam truncation at the antenna aperture, beam-angle misalignment, and arbitrary transmitter and receiver configurations. Simplifying assumptions used in analytical studies, were tested and treated as benchmarks for determining the accuracy of the simulations. The simulation permitted characterization of the effect on lidar performance of the analytically intractable return variance that results from turbulent fluctuations as well as of the heterodyne optical power and system-antenna efficiency.     

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.     

Surface roughness evaluation via ultrasonic scanning

Despite extensive applications of ultrasonic waves to various nondestructive testing and evaluation of materials, scattering of focused ultrasonic waves due to surface roughness has not been fully investigated. This paper presents an analytical and experimental evaluation of surface roughness measurement using focused ultrasonic beams. The characteristics of focused ultrasonic waves are analyzed by using the impulse response method with a sine-modulated Gaussian pulse as source. First, the beam profile in the focal plane of the focused ultrasonic transducer is analyzed both numerically and experimentally. Second, peak amplitude distribution and reflected waveforms from a flat surface with various incident angles are analytically generated and compared with experimental results. Then, the peak amplitudes of the ultrasonic waves reflected from cusped surfaces which are easily found among machined surfaces are analyzed and compared with experimental data for the first time. The analysis shows good agreement between analytical and experimental results. The excellent correlation between the measurements using a profilometer and the proposed ultrasonic system demonstrates a good potential for surface roughness measurement by ultrasonic sensing.     

Fiber-coupling efficiency for free-space optical communication through atmospheric turbulence

High-speed free-space optical communication systems have recently used fiber-optic components. The received laser beam in such a system must be coupled into a single-mode fiber at the input of the receiver module. However, propagation through atmospheric turbulence degrades the spatial coherence of a laser beam and limits the fiber-coupling efficiency. We numerically evaluate the fiber-coupling efficiency for laser light distorted by atmospheric turbulence. We also investigate the use of a coherent fiber array as a receiver structure and find that a coherent fiber array that consists of seven subapertures would significantly increase the fiber-coupling efficiency.     

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.     

Analysing the behaviour of partially coherent divergent Gaussian beams propagating through oceanic turbulence

Theoretical study of propagation behaviour of partially coherent divergent Gaussian beams through oceanic turbulence has been performed. Based on the previously developed knowledge of propagation of a partially coherent beam in atmosphere, the spatial power spectrum of the refractive index of ocean water, extended Huygens–Fresnel principle and the unified theory of coherence and polarization, analytical formulas for cross-spectral density matrix elements are derived. The analytical formulas for intensity distribution, beam width and spectral degree of coherence are determined by using cross-spectral density matrix elements. Then, the effects of some source factors and turbulent ocean parameters on statistical properties of divergent Gaussian beam propagating through turbulent water are analysed. It is found that beam’s statistical propagation behaviour is affected by both environmental and source parameters variations.     

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.     

Effects of different beacon wavelengths on atmospheric compensation in strong scintillation

During strong scintillation, the number and location of branch points in a distorted optical field induced by atmospheric turbulence are closely related to the characteristic parameters of the turbulence effect, propagation distance, and wavelength. It is necessary to consider the effect of the beacon's wavelength on the adaptive optics system that is used to compensate for atmospheric turbulence. Our analytical results show that the performance of adaptive optics can be improved by nearly a factor of 2 when the beacon's wavelength is chosen slightly longer than the wavelength of the main laser in the branch points considered.     

Effective radius of curvature of partially coherent Hermite–Gaussian beams propagating through non-Kolmogorov turbulence

Based on the extended Huygens–Fresnel principle and non-Kolmogorov spectrum, the analytical expression for the effective radius of curvature of partially coherent Hermite–Gaussian (PCHG) beams propagating through non-Kolmogorov turbulence is derived, and the relative effective radius of curvature is used to describe the effect of turbulence on the effective radius of curvature. It is shown that the effective radius of curvature of PCHG beams depends on the beam and non-Kolmogorov turbulence parameters and on the propagation distance. The variation of relative effective radius of curvature with increasing generalized exponent parameter α of non-Kolmogorov turbulence is non-monotonic. The longer the propagation distance is, the larger the effect of turbulence on the effective radius of curvature of PCHG beams is. The effective radius of curvature of PCHG beams with shorter wavelength, smaller beam order, larger beam waist width or better spatial coherence is more affected by the non-Kolmogorov turbulence. The results are interpreted physically.     

Reliability analysis of the flat-topped array beam FSO communication link

Free-space optical communication systems are affected by turbulent atmosphere. The atmospheric transmission is affected by absorption, scattering and turbulence. In this paper, the effects of absorption and scattering are taken into account using Beer’s law and the effects of turbulence are considered in calculating the average intensity distribution. An analytical expression for the average intensity distribution of a partially coherent flat-topped array (PCFTA) beam in turbulent atmosphere is derived based on the extended Huygens–Fresnel principle. The average intensity, power in bucket, signal to noise ratio, and bit error rate of this kind of beam are investigated in details. It is shown by numerical results and analytical methods that the average intensity and link parameters of PCFTA beams change during propagation and these changes are dependent upon both source parameters and weather conditions.