Effect of the log-amplitude covariance function on the statistics of speckle 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.

Effects of source spatial partial coherence on temporal fade statistics of irradiance flux in free-space optical links through atmospheric turbulence

The temporal covariance function of irradiance-flux fluctua-tions for Gaussian Schell-model (GSM) beams propagating in atmospheric turbulence is theoretically formulated by making use of the method of effective beam parameters. Based on this formulation, new expressions for the root-mean-square (RMS) bandwidth of the irradiance-flux temporal spectrum due to GSM beams passing through atmospheric turbulence are derived. With the help of these expressions, the temporal fade statistics of the irradiance flux in free-space optical (FSO) communication systems, using spatially partially coherent sources, impaired by atmospheric turbulence are further calculated. Results show that with a given receiver aperture size, the use of a spatially partially coherent source can reduce both the fractional fade time and average fade duration of the received light signal; however, when atmospheric turbulence grows strong, the reduction in the fractional fade time becomes insignificant for both large and small receiver apertures and in the average fade duration turns inconsiderable for small receiver apertures. It is also illustrated that if the receiver aperture size is fixed, changing the transverse correlation length of the source from a larger value to a smaller one can reduce the average fade frequency of the received light signal only when a threshold parameter in decibels greater than the critical threshold level is specified.

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.

Speckle-field propagation in 'frozen' turbulence: brightness function approach

Speckle-field long- and short-exposure spatial correlation characteristics for target-in-the-loop (TIL) laser beam propagation and scattering in atmospheric turbulence are analyzed through the use of two different approaches: the conventional Monte Carlo (MC) technique and the recently developed brightness function (BF) method. Both the MC and the BF methods are applied to analysis of speckle-field characteristics averaged over target surface roughness realizations under conditions of 'frozen' turbulence. This corresponds to TIL applications where speckle-field fluctuations associated with target surface roughness realization updates occur within a time scale that can be significantly shorter than the characteristic atmospheric turbulence time. Computational efficiency and accuracy of both methods are compared on the basis of a known analytical solution for the long-exposure mutual correlation function. It is shown that in the TIL propagation scenarios considered the BF method provides improved accuracy and requires significantly less computational time than the conventional MC technique. For TIL geometry with a Gaussian outgoing beam and Lambertian target surface, both analytical and numerical estimations for the speckle-field long-exposure correlation length are obtained. Short-exposure speckle-field correlation characteristics corresponding to propagation in 'frozen' turbulence are estimated using the BF method. It is shown that atmospheric turbulence-induced static refractive index inhomogeneities do not significantly affect the characteristic correlation length of the speckle field, whereas long-exposure spatial correlation characteristics are strongly dependent on turbulence strength.

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.

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 1-microm lidar measurements of atmospheric-turbulence-induced spatial decorrelation using a multielement heterodyne detector array

We have employed a coherent 1-microm Nd:YAG lidar system to measure directly, for the first time to our knowledge, the reduced spatial coherence length, p(0), of the lidar returns caused by atmospheric turbulence. Our experiments were conducted by using a 2 x 2 heterodyne detector array, which permitted real-time spatial correlation measurements of the lidar returns at two different detector spacings. The spatial correlation coefficients and spatial coherence length of the lidar returns from a hard target weremeasured during a day-to-night time period when the atmospheric turbulence parameter, C(n)(2), was measured to vary from 2 x 10(-13) to 2 x 10(-14) m(-1/3). These directly measured values of p(0) as a function of C(n)(2) were found to be in good agreement with theoretical predictions.

Covariance of the received intensity of a partially coherent laser speckle pattern in the turbulent atmosphere

A theoretical expression for the covariance of the received intensity of a partially coherent laser speckle pattern after propagation through the turbulent atmosphere is developed. It is shown that the atmospheric perturbation on a partially coherent speckle pattern can be decomposed into a coherent term and an incoherent term. The dependence of contributions of these components on the level of turbulence, vacuum speckle-contrast ratio, and detector spacing is studied in detail and the results are compared with the available experimental data.

Enhanced detection of atmospheric-turbulence-distorted 1-microm coherent lidar returns using a two-dimensional heterodyne detector array

We have employed a two-dimensional multielement heterodyne detector array and demonstrated, for the first time to our knowledge, the enhanced detection efficiency of atmospheric-turbulence-distorted 1-microm coherent lidar returns. The heterodyne lidar signal intensity and statistical signal fluctuation were measured for both a 2 x 2 detector array and a single detector as a function of the atmospheric turbulence parameter C(n)(2). The detector array improved the lidar detection efficiency by a factor of approximately 4, and the statistical signal distribution changed from Rayleigh to Gaussian. This improvement is shown to be consistent with a firstorder analysis.

Time-delayed statistics for a bistatic coherent lidar operating in atmospheric turbulence

Analytical expressions are derived for the time-delayed statistics and the heterodyne signal power as a function of the transmitter-receiver spacing for the general case of a bistatic coherent lidar. The effect of the turbulence-induced correlation between the outgoing and return paths was included and the 5/3 law form of the wave structure function was used.

Optimal heterodyne detector array size for 1-microm coherent lidar propagation through atmospheric turbulence

The heterodyne detection efficiency for a 1-microm coherent atmospheric backscatter lidar was numerically calculated using a Monte Carlo technique which included a simple model for the effects of atmospheric turbulence. The results show that the heterodyne detection efficiency of a single-element detector is severely reduced by the effects of atmospheric turbulence, but that the use of an appropriate sized, multiple-element heterodyne detector array can overcome these effects. In addition, the statistical fluctuation (signal-to-noise ratio) of the lidar signal was also calculated and showed that the use of a heterodyne detector array can increase the accuracy to that for direct detection.

Heterodyne Doppler 1-microm lidar measurement of reduced effective telescope aperture due to atmospheric turbulence

We performed an experimental study on the effect of atmospheric turbulence on heterodyne and direct detection lidar at 1 microm, employing a pulsed Nd:YAG bistatic focused beam lidar that permitted simultaneous heterodyne and direct detection of the same lidar returns. The average carrier-to-noise ratio and statistical fluctuation level in the lidar return signals were measured in various experimental and atmospheric conditions. The results showed that atmospheric turbulence could reduce the effective receiver telescope diameter of the l-microm heterodyne lidar to <5cm at a relatively short range of approximately 450 m near the ground. The observed effective telescope aperture and heterodyne detection efficiency varied during the day as the atmospheric turbulence level changed. At this time, we are not able to compare our experimental lidar data to a rigorous atmospheric turbulence and lidar detection theory which includes independently variable transmitter, receiver, and detector geometry. It is interesting to note, however, that the observed limitation of the effective receiver aperture was similar in functional form with those predictions based on the heterodyne wavefront detection theory by D. L. Fried [Proc. IEEE 55, 57-67 (1967)] and the heterodyne lidar detection theory for a fixed monostatic system by S. F. Clifford and S. Wandzura [Appl. Opt. 20, 514-516 (1981)]. We have also applied such an effective receiver aperture limitation to predict the system performance for a heterodyne Ho lidar operating at 2 microm.

Time delayed covariance of the received intensity of a monochromatic speckle pattern in the turbulent atmosphere

An approximate numerical evaluation of the previously formulated time delayed covariance of the received intensity of a laser speckle pattern is described and the results are compared with experimental data.

Remote sensing of atmospheric winds using speckleturbulence interaction, a CO(2) laser, and optical heterodyne detection

Speckle-turbulence interaction can be utilized to measure the vector wind in a plane perpendicular to the line of sight from a laser transmitter to a target. A continuous wave source of around 1 W and operating at 10.6 microm, in conjunction with an optical heterodyne receiver, has been used to measure atmospheric winds along horizontal paths. A theoretical basis, the experimental apparatus, processing techniques, and experimental results are presented. The technique has been demonstrated for remote sensing of atmospheric winds along horizontal paths but also has potential for global remote sensing of atmospheric winds and for onboard wind shear detection systems for aircraft. The results show that rms accuracies of the order of 0.5 m/s are possible with averaging times as short as 2 s.

Experimental comparison of heterodyne and direct detection for pulsed differential absorption CO2 lidar

A pulsed dual-wavelength dual-CO2-laser differential-absorption lidar (DIAL) system has been developed which permits simultaneous heterodyne and direct detection of the same lidar returns. This system has been used to make an experimental comparison of the SNRs and statistical and temporal characteristics of the DIAL returns from several topographic targets. These results were found to be in general agreement with theory and were used to quantify the relative merits of the two detection techniques. The measured parameter values were applied to an analytical treatment to predict system trade-offs for the remote sensing of atmospheric species, with application to both path-averaged and range-resolved measurements.

Heterodyne laser radar SNR from a diffuse target containing multiple glints

A coherent laser radar performance model is formulated using the Huygens-Fresnel principle for a near-field or far-field diffuse target which may contain a number of glints. The received signal is calculated for a truncated Gaussian-transmitted beam mixed with a matched Gaussian or a uniform local oscillator (LO) beam. In general, the uniform LO beam provides a good or better mixing than the matched Gaussian beam. The effects of beam truncation and receiver aperture size on the received signal level are presented. In the case of a far-field diffuse target, an increase in SNR is obtained if the receiver aperture increases to about three times the transmitter aperture, or if the transmitter aperture increases to about three times the receiver aperture. The target signature statistics and the weather statistics are considered as random parameters in the evaluation of receiver performance characteristics.