Probability density function of the intensity for a laser-generated speckle field after propagation through the turbulent atmosphere

System-level noise performance of coherent imaging systems

We provide an in-depth analysis of noise considerations in coherent imaging, accounting for speckle and scintillation in addition to "conventional" image noise. Specifically, we formulate closed-form expressions for total effective noise in the presence of speckle only, scintillation only, and speckle combined with scintillation. We find analytically that photon shot noise is uncorrelated with both speckle and weak-to-moderate scintillation, despite their shared dependence on the mean signal. Furthermore, unmitigated speckle and scintillation noise tends to dominate coherent-imaging performance due to a squared mean-signal dependence. Strong coupling occurs between speckle and scintillation when both are present, and we characterize this behavior by fitting a scale factor capable of generating variances in closed form. We verify each of these claims through a series of wave-optics simulations, and we see strong agreement in general between numerical results and theoretical predictions. Our findings allow us to confidently gauge signal-to-noise ratio (SNR) expectations when active illumination produces coherent noise.

Spatiotemporal characteristics of dynamic speckle from a 3D target in atmospheric turbulence

Spatiotemporal correlation function is the basic characteristic of dynamic laser speckle and the basis of various applications. The correlation function of the speckle intensity from a 3D target in turbulent atmosphere is derived based on the model of random phase screen and Fresnel Kirchhoff Diffraction Formula, and a fast algorithm based on FFT is developed. The particularity of dynamic speckle in turbulence is numerically analyzed and discussed. The results show that the speckle intensity fluctuates at two independent scales both in space and time domain, which are affected by target size and atmospheric turbulence respectively. In particular, the time scale caused by turbulence is also affected by the target translating velocity and the wind direction. The theory and algorithm developed in this paper can play important roles in applications of laser speckle such as remote detection in atmospheric environment.

Dependence of the probability density function of laser radiation power on the scintillation index and the size of a receiver aperture

Power scintillations of a Gaussian laser beam propagated through a 7 km long horizontal atmospheric path for a wide range of turbulence strengths and different sizes of the receiver aperture were studied experimentally. The probability density function (PDF) and its properties were analyzed for a wide range of scintillation conditions. It was shown that the PDF can be described by the fractional exponential distribution in the strong scintillation regime (scintillation index of power measured on aperture σPIB2>1) for apertures with diameter d < a, where a is the size of the isoplanatic region, and by the gamma distribution in the weak scintillation regime (σPIB2<1), as well as by the lognormal distribution for σPIB2≪1. More than one distribution can be considered as a good approximation for experimental data for some ranges of d/a and σPIB2, but the transition from one distribution to another as the best approximation occurs at certain characteristic values of these parameters. In the strong scintillation regime, the aperture averaging effect resulted in the transition from the fractional exponential distribution to the fractional gamma (FG) distribution when the aperture diameter is about the size of the isoplanatic region (d/a∼1). The FG distribution better approximates the experimental PDF because it accounts for the fact that the probability of zero power values becomes zero due to the averaging effect of the aperture. The fractional gamma distribution is the best approximation of the PDF for a finite aperture when σPIB2>σPIB,crit2=[0.7-0.8], while the gamma distribution is the best approximation for σPIB2<σPIB,crit2.

Image reconstructions with active illumination in strong-turbulence scenarios with single-frame blind deconvolution approaches

Image formation over long horizontal or slant paths is of interest in surveillance and remote sensing. Image reconstructions of isolated objects are presented using active illumination in long-path, strong-turbulence conditions using a wave-optics simulation to produce the images. Fast-running reconstruction algorithms are used, including a novel single-frame blind iterative deconvolution algorithm and a generalized expectation maximization algorithm. Significant improvements in image quality and image recognizability can be found for spherical-wave Rytov variances up to 0.4 and for up to 10 atmospheric coherence lengths across the aperture in uniform-turbulence scenarios over a 30 km range. For these conditions, the isoplanatic patch angle is comparable to the diffraction angle, and there are 20 or more isoplanatic patches across the objects considered. The results are compared with idealized atmospheric phase correction with an incoherent beacon. Several image quality metrics are considered. Results for strongest turbulence are explained in terms of a local average of the point spread function and the central limit theorem for cases in which there are many isoplanatic patches across the object.

Probability distribution of intensity fluctuations of arbitrary-type laser beams in the turbulent atmosphere

Statistical characteristics of radiation intensity in the cross section of laser beams propagating in the turbulent atmosphere are analyzed by using numerical simulations. It is shown that for arbitrary values of the propagation distance, Rytov parameter, beam type, and position of the analyzed point in the beam cross section, the probability density function (PDF) of radiation intensity is fully determined by the scintillation index and the average intensity. In the case of moderate and weak intensity fluctuations characterized by the scintillation index smaller than unity, the probability density function is determined by the gamma distribution. For the case of strong fluctuations (with the scintillation index larger than unity), a new analytical expression for PDF is proposed, which well approximates PDFs obtained in numerical experiments under different conditions of propagation of different-type beams.

Circular Bessel statistics: derivation and application to wave propagation in random media

We present a family of circular Bessel probability density functions that are capable of describing the intensity, amplitude, and field statistics of waves in any random medium, with only the assumption of circularity. The well-known zero-mean circular Gaussian statistics break down in the Anderson localization and the weakly scattering regimes, where the field can no longer be regarded as the sum of a multitude of independent random phasors. We find that in such scenarios circular Bessel statistics apply because the field can be modeled as a random phasor sum with a random number of contributing phasors. The validity of our density functions is verified through numerical simulations of electromagnetic waves propagating in 2D random media. Having a set of density functions that work in all scattering regimes provides a framework for modeling wave propagation in random media, facilitating random media characterization, imaging in and through scatter, and random laser design.

Zero-mean circular Bessel statistics and Anderson localization

We demonstrate that a circular Bessel density function describes the electromagnetic field statistics in the Anderson localization regime using example numerical terahertz field data in strongly scattering media. This density function for localized fields provides a measure that allows identification and description in a manner akin to the Gaussian density function for weakly interacting scatterers, the mathematical framework to date for statistical optics. Our theory provides a framework for improved understanding of wave propagation in random media, random scattering media characterization, and imaging in and through randomly scattering media.

Probability density function estimation of laser light scintillation via Bayesian mixtures

A method for probability density function (PDF) estimation using Bayesian mixtures of weighted gamma distributions, called the Dirichlet process gamma mixture model (DP-GaMM), is presented and applied to the analysis of a laser beam in turbulence. The problem is cast in a Bayesian setting, with the mixture model itself treated as random process. A stick-breaking interpretation of the Dirichlet process is employed as the prior distribution over the random mixture model. The number and underlying parameters of the gamma distribution mixture components as well as the associated mixture weights are learned directly from the data during model inference. A hybrid Metropolis-Hastings and Gibbs sampling parameter inference algorithm is developed and presented in its entirety. Results on several sets of controlled data are shown, and comparisons of PDF estimation fidelity are conducted with favorable results.

Speckle statistics and transverse coherence of an x-ray laser with fluctuations in its active medium

It is shown that the statistics of the intensity distribution in the output beam of a collisional X-ray laser, analysed in terms of the degree of freedom or equivalently the number of the coherence modes in the beam cross-section, has non-Gaussian character. The non-Gaussian character seems to be caused by the small-scale plasma/medium fluctuations. It was assumed that these overlap the modal structure imposed by the geometry of the medium and considered as equivalent to a large-scale inhomogeneity. Thus, the fluctuations decide about the character of the output beam transverse coherence. It is also shown that the relevant to this model compound statistics of the intensity fluctuations in the output beam is well described by the m-m-distribution, a specific form of the K-distribution. The deviation from the Gaussian statistics was confirmed by the field correlation function at the laser exit plane, retrieved from the experimental data.

Radiometric calibration of an airborne CO2 pulsed Doppler lidar with a natural earth surface

Radiometric calibration of an airborne CO2 pulsed Doppler lidar has been accomplished with surface retroreflection signals from the White Sands National Monument, New Mexico. Two circular passes were made at altitudes of 6.3 and 9.3 km. The computed calibration factors for both altitudes are in excellent agreement with the value derived from standard ground-based measurements involving a fixed sandpaper target of known reflectance. This finding corroborates a previous study that successfully calibrated an airborne cw Doppler lidar with a variety of natural Earth surfaces. The present results indicate that relatively uniform Earth surface targets can be used for in-flight calibration of CO2 pulsed airborne and, in principal, other infrared lidars.

Description and simulation of an active imaging technique utilizing two speckle fields: root reconstructors

Quasi-monochromatic light will form laser speckle upon reflection from a rough object. This laser speckle provides information about the shape of the illuminated object. Further information can be obtained if two colors of coherent light are used, provided that the colors are sufficiently close in wavelength that the interference is also measurable. It is shown that no more than two intensities of two speckle patterns and their interference are required to produce an unambiguous band-limited image of an object, to within an overall spatial translation of the image, in the absence of measurement errors and in the case where all roots of both fields and their complex conjugates are distinct. This result is proven with a root-matching technique, which treats the electric fields as polynomials in the pupil plane, the coefficients of which form the desired complex object. Several root-matching algorithms are developed and tested. These algorithms are generally slow and sensitive to noise. So motivated, several other techniques are applied to the problem, including phase retrieval, expectation maximization, and probability maximization in a sequel paper [J. Opt. Soc. Am. A 19, 458 (2002)]. The phase-retrieval and expectation-maximization techniques proved to be most effective for reconstructions of complex objects larger than 10 pixels across.

Numerical simulation of the effect of refractive turbulence on coherent lidar return statistics in the atmosphere

We propose an algorithm and the results of a numerical study of random realizations and statistics of a pulsed coherent lidar return that allow for refractive turbulence. We show that, under conditions of refractive turbulence, the relative variance of the lidar return power can exceed unity by a factor of as much as 1.5. Clear manifestations of the turbulent effect of backscattering amplification have been revealed from simulations of space-based lidar sensing of the atmosphere with coherent lidar. Under conditions of strong optical turbulence in the atmospheric boundary layer, as a result of the backscattering amplification effect, the mean lidar return power can exceed the return power in the absence of turbulence by a factor of 3.

Four-Element Receiver for Pulsed 10-mum Heterodyne Doppler Lidar

A four-element photomixer receiver has been tested in a 10-mum heterodyne Doppler lidar. It addresses a reduction of the variance of the power scattered off distributed aerosols targets at ranges as long as 8 km. An improvement in performance is expected when the four independent signals recorded on every single shot are combined. Two summation techniques of the four signals have been implemented: a coherent summation of signal amplitude and an incoherent summation of intensities. A phasing technique for the four signals is proposed. It is based on a more suitable correlation time with discernible self-consistent packets (SCP's). The SCP technique has been successfully tested, and the results obtained with a coherent summation of the four signals, i.e., variance reduction, carrier-to-noise ratio improvement, and velocity accuracy improvement, are in agreement with theory.

Measurement of integrated speckle statistics for CO2 lidar returns from a moving, nonuniform, hard target

A pulsed, dual CO(2) laser lidar was used to measure return signal statistics as a function of the number of speckles integrated by the lidar receiver per laser pulse. A rotating target generated statistically independent speckle patterns on each laser pulse. Data were collected for a wide range of receiver aperture sizes. A statistical model is developed that predicts the probability density of the return lidar pulse energy, which includes speckle, depolarization by the target, and albedo sampling. The predictions of this model are compared with the measured probability density function of the return pulse energies. Very good agreement is found between the geometrically calculated number of integrated speckles and the number predicted by the model.

Coherent optical array receivers for the mitigation of atmospheric turbulence and speckle effects

A description is given of the design, operation, and test over a 2-km path (roundtrip) of a continuous wave, coherent laser array receiver that uses two independent aperture-receivers whose intermediate frequencies are electro-optically co-phased in real time and then added as a proposed way to overcome effective aperture limitations imposed by atmospheric turbulence and to mitigate signal fading associated with atmospheric turbulence and speckle effects. The experiment resulted in a mean carrier-to-noise ratio increase of 1.8, which is within 1% of the theoretical predictions, when the two signals were phase locked, versus no increase without phase locking. Further, the carrier fading strength, or normalized carrier-to-noise ratio variance, was reduced by a factor of 0.53, which is within 2% of the theoretical prediction. The bandwidth of the electro-optic phase-locked loop was measured to be of the order of 600 Hz, which is adequate to compensate for atmospheric refractive turbulence fluctuations.

Unwrapped-phase distribution model for speckle and turbulence

A model for the unwrapped phase of a speckle field that propagates through clear air turbulence is proposed, and a formulation for the probability density function of the unwrapped phase is developed. A method is given to obtain the parameters of the unwrapped density function from parameters of the diffuse target, the laser source, and the atmosphere. Unwrapped-phase measurements at an atmospheric test site with a CO(2) laser show agreement with the model.

Probability distributions for the integrated intensity and photocount associated with a K-distribution for laser intensity

A numerical approach is used for evaluating the probability density functions of the integrated intensity and photocount when the received laser intensity obeys a K-distribution. Different scattering strengths are considered for comparisons. Also, those probability distribution functions based on a generalized K-distribution and a lognormally modulated exponential distribution for the laser intensity are computed and compared with those based on the K-distribution. Some general trends are drawn. The accuracy of our numerical computations is also discussed.

Measured signal amplitude distributions for a coherent FM-cw CO2 laser radar

Measurements of signal amplitude distributions with a FM-cw CO2 laser radar have been made against various targets in both imaging and staring modes. Data show good agreement with theoretical distributions. From the measurements conclusions are drawn about the atmospheric- as well as target-induced effects. Beam wandering effects are shown to be of importance in the staring mode.