Stellar scintillation statistics and the impact of aperture averaging on space-to-ground optical communications.

Statistical probability distributions characterizing received optical power fluctuations, or scintillation, enable performance predictions of space-to-ground optical communication systems. In this paper, we present measurements of stellar scintillation over a wide range of elevation angles and turbulence conditions collected simultaneously with a 5 cm and 40 cm telescope aperture, which allows a comparison between minimal and significant aperture averaging conditions. The measured data is compared to a reasonable set of candidate probability distribution functions (PDFs), including lognormal, which is most often cited in the literature for weak to moderate scintillation. For scintillation indices (SIs) less than about 0.2, the Nakagami-m distribution provides the best representation of the collected data for both apertures and imposes a greater lasercom link penalty than a lognormal distribution, which has been inaccurately implemented as the default probability distribution in the literature. For larger values of the SI, the scintillation is best characterized by a Gamma-Gamma distribution. Additionally, the measured temporal covariance for weak to moderate scintillation conditions is found to be in reasonably good agreement with theoretical predictions.

Optical downlink propagation from space-to-earth: aperture-averaged power fluctuations, temporal covariance and power spectrum.

In this work, we consider optical downlink from space-based laser sources and develop a consistent quantitative analysis of the collected power fluctuations by finite receiving apertures, and both the corresponding temporal covariance and power spectral density (PSD). Here we assume weak to moderate scintillation conditions where lognormal statistics are valid. We derive both exact solutions and highly accurate engineering easy to implement approximations for the downlink aperture-averaging factor, and both the corresponding aperture-averaged signal temporal covariance and PSD. Additionally, highly accurate elementary analytic scaling relations are derived for the corresponding aperture-averaged characteristic correlation time and scintillation bandwidth, which are in good agreement with available experimental observations. Finally closed form expressions for the so-called quasi-frequency that is central to the determination of level crossing rates and duration of fades and surges in a propagation channel are derived. Wherever possible, we endeavor to derive "user friendly" accurate engineering approximations for the various statistical quantities of interest.

Temporal averaging of atmospheric turbulence-induced optical scintillation.

Based on the Rytov approximation we have developed for weak scintillation conditions a general expression for the temporal averaged variance of irradiance. The present analysis provides, for what we believe is the first time, a firm theoretical basis for the often-observed reduction of irradiance fluctuations of an optical beam due to atmospheric turbulence. Accurate elementary analytic approximations are presented here for plane, spherical and beam waves for predicting the averaging times required to obtain an arbitrary value of the ratio of the standard deviation to the mean of an optical beam propagating through an arbitrary path in the atmosphere. In particular, a novel application of differential absorption measurement for the purpose of measuring column-integrated concentrations of various so-called greenhouse gas (GHG) atmospheric components is considered where the results of our analysis indicates that relatively short averaging times, on the order of a few seconds, are required to reduce the irradiance fluctuations to a value precise enough for GHG measurements of value to climate related studies.

Exponentiated Weibull distribution family under aperture averaging Gaussian beam waves: comment.

Recently, an exponentiated Weibull distribution model was presented for describing the effects of aperture averaging on scintillation of Gaussian beams propagating through atmospheric turbulence. The model uses three parameters that are derived from physical quantities so that in principle the model could be used to predict optical link performance. After reviewing this model, however, we find several inconsistencies that render it unusable for this purpose under any scintillation conditions.

Spatial filtering velocimetry of objective speckles for measuring out-of-plane motion.

This paper analyzes the dynamics of objective laser speckles as the distance between the object and the observation plane continuously changes. With the purpose of applying optical spatial filtering velocimetry to the speckle dynamics, in order to measure out-of-plane motion in real time, a rotational symmetric spatial filter is designed. The spatial filter converts the speckle dynamics into a photocurrent with a quasi-sinusoidal response to the out-of-plane motion. The spatial filter is here emulated with a CCD camera, and is tested on speckles arising from a real application. The analysis discusses the selectivity of the spatial filter, the nonlinear response between speckle motion and observation distance, and the influence of the distance-dependent speckle size. Experiments with the emulated filters illustrate performance and potential applications of the technology.

Comment: absolute measurement of roughness and lateral-correlation length of random surfaces by use of the simplified model of image-speckle contrast.

This comment demonstrates several errors in the derivation of speckle contrast and corresponding measurements reported by Cheng et at. (Appl. Opt. 41, 4148 (2002)] for a 4f optical system. In particular, the theoretical derivation is wrong: It is used outside of its domain of validity, and the experimental results do not support the theoretical analysis.

Analysis of optical coherence tomography systems based on the extended Huygens-Fresnel principle.

We have developed a new theoretical description of the optical coherence tomography (OCT) technique for imaging in highly scattering tissue. The description is based on the extended Huygens-Fresnel principle, valid in both the single- and multiple-scattering regimes. The so-called shower curtain effect, which manifests itself in a standard OCT system, is an inherent property of the present theory. We demonstrate that the shower curtain effect leads to a strong increase in the heterodyne signal in a standard OCT system. This is in contrast to previous OCT models, where the shower curtain effect was not taken into account. The theoretical analysis is verified by measurements on samples consisting of aqueous suspensions of microspheres. Finally, we discuss the use of our new theoretical model for optimization of the OCT system.

Closed-form solution for the Wigner phase-space distribution function for diffuse reflection and small-angle scattering in a random medium.

Within the paraxial approximation, a closed-form solution for the Wigner phase-space distribution function is derived for diffuse reflection and small-angle scattering in a random medium. This solution is based on the extended Huygens-Fresnel principle for the optical field, which is widely used in studies of wave propagation through random media. The results are general in that they apply to both an arbitrary small-angle volume scattering function, and arbitrary (real) ABCD optical systems. Furthermore, they are valid in both the single- and multiple-scattering regimes. Some general features of the Wigner phase-space distribution function are discussed, and analytic results are obtained for various types of scattering functions in the asymptotic limit s >> 1, where s is the optical depth. In particular, explicit results are presented for optical coherence tomography (OCT) systems. On this basis, a novel way of creating OCT images based on measurements of the momentum width of the Wigner phase-space distribution is suggested, and the advantage over conventional OCT images is discussed. Because all previous published studies regarding the Wigner function are carried out in the transmission geometry, it is important to note that the extended Huygens-Fresnel principle and the ABCD matrix formalism may be used successfully to describe this geometry (within the paraxial approximation). Therefore for completeness we present in an appendix the general closed-form solution for the Wigner phase-space distribution function in ABCD paraxial optical systems for direct propagation through random media, and in a second appendix absorption effects are included.

Laser-speckle angular-displacement sensor: theoretical and experimental study.

A novel, to our knowledge, method for the measurement of angular displacement for arbitrarily shaped objects is presented in which the angular displacement is perpendicular to the optical axis. The method is based on Fourier-transforming the scattered field from a single laser beam that illuminates the target. The angular distribution of the light field at the target is linearly mapped on a linear image sensor placed in the Fourier plane. Measuring this displacement facilitates the determination of the angular displacement of the target. It is demonstrated both theoretically and experimentally that the angular-displacement sensor is insensitive to object shape and target distance if the linear image sensor is placed in the Fourier plane. A straightforward procedure for positioning the image sensor in the Fourier plane is presented. Any transverse or longitudinal movement of the target will give rise to partial speckle decorrelation, but it will not affect the angular measurement. Furthermore, any change in the illuminating wavelength will not affect the angular measurements. Theoretically and experimentally it is shown that the method has a resolution of 0.3 mdeg ( approximately 5 murad) for small angular displacements, and methods for further improvement in resolution is discussed. No special surface treatment is required for surfaces giving rise to fully developed speckle. The effect of partially developed speckle is considered both theoretically and experimentally.

Effects of target structure on the performance of laser time-of-flight velocimeter systems.

The consequences of partially developed speckle and the effects giving rise to bias errors in velocity determination are discussed with respect to robustness of a classical laser time-of-flight velocimetry (LTV) system. It is demonstrated that surface regimes exist that define the degree of partially developed speckle. These regimes are explored both theoretically and experimentally; surface models are developed to predict the resulting cross covariance from which velocity estimations can be obtained. The surface models describe the behavior of the cross covariance caused by reflection structures and with disparate lateral-roughness scales. In particular, it is shown that it is possible to obtain a twin-Gaussian cross covariance as a result of the presence of partially developed speckle. All models described are compared with experimental observations of the cross covariance for differing surface regimes. The objects are solid targets having lateral spatial correlations in reflection amplitude, height, or both, generally giving rise to partially developed speckle. In almost all cases good agreement with the corresponding theoretical predictions are found. Decorrelation caused by velocity misalignment is shown to shift the peak of the cross covariance significantly, giving a velocity bias. A corresponding theoretical model is developed and verified experimentally. Cross-talk measurements have been performed and compared with a theory developed herein. Both measurements and theory indicate that only spot sizes comparable with or larger than their corresponding separation will lead to a measurable peak shift of the time lag for the maximum of the cross covariance. We conclude that LTV systems will provide accurate velocity estimates under a wide variety of practical conditions.

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.

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.

Gaussian beam transfer through hard-aperture optics.

We consider Gaussian beam diffraction by hard circular and rectangular-slit apertures. Both numerical results and accurate elementary analytic approximations are derived for the fraction of transmitted power (or energy) contained within the main central lobe of the far-field (or focal-plane) irradiation distribution as a function of the truncation ratio.

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.

Optical system defect propagation in ABCD systems.

We describe how optical system defects (tilt/jitter, decenter, and despace) propagate through an arbitrary paraxial optical system that can be described by an ABCD ray transfer matrix. A pedagogical example is given that demonstrates the effect of alignment errors on a typical optical system.

Zernike-polynomial expansion of turbulence-induced centroid anisoplanatism.

An expansion of image centroid position after propagation through atmospheric turbulence in terms of Zernike-polynomial coefficients is shown to consist of a tilt term plus coma terms. Centroid anisoplanatism therefore arises from turbulence-induced coma distortions. By correcting for coma effects to a sufficiently high order, the deleterious effects of centroid anisoplanatism can be reduced to any desired level. The Strehl ratio is given by the asymptotic formula [1 + A(D/r(0))(5/3)](-1), where A = 0.018(I + 1)(-7/3) and I is the number of coma terms corrected.

Laser vector velocimetry: a 3-D measurement technique.

A technique is described which will provide both magnitude and direction of the transverse component of velocity of a remote target. A laser beam is transmitted toward a remote diffuse target, and the backscattered light is collected and mixed with a suitable local oscillator reference field. The method is based on the general spatiotemporal correlation function of the signal currents from two heterodyne detectors. The time derivative of this function, evaluated at zero time delay, is directly proportional to the component of target velocity parallel to the separation of the detector elements. A two-element by two-element detector array can therefore be used to measure the two orthogonal transverse velocity components. Each component is found from the correlation properties of the signals from diagonally opposed pairs of detector elements. Since the radial velocity component can be found from conventional laser Doppler techniques, all three components of the velocity vector can be measured simultaneously. Practical considerations and experimental results are discussed.