Larry C. Andrews

Time-of-Arrival Fluctuations of a Space-Time Gaussian Pulse in Weak Optical Turbulence: an Analytic Solution.

An analytic expression is derived for the long-term temporal broadening (fluctuations of arrival time) of a collimated space-time Gaussian pulse propagating along a horizontal path through weak optical turbulence. General results are presented for nominal parameter values characterizing laser communication through the atmosphere. Specific examples are calculated for both upper-atmosphere and ground-level cross links. It is shown that, for upper-atmosphere cross links, pulses shorter than 100 fs have considerable broadening, whereas at ground level, broadening is predicted in pulses as long as 1 ps.

Near ground surface turbulence measurements and validation: a comparison between different systems

Recently, the number of optical systems that operate along near horizontal paths within a few meters of the ground has increased rapidly. Examples are LIDAR or optical sensors imbedded in a vehicle, long range surveillance or optical communication systems, a LIFI network, new weather monitoring stations, as well as directed energy systems for defense purposes. Near ground turbulence distortion for optical waves used in those systems cannot be well described by conventional turbulence and beam propagation theory. Phenomena such as anisotropy, micro mirage effects, a temporal negative relation between diurnal dips and altitude, and condensation induced measurement errors are frequently involved. As a result, there is a high risk of defective designs or even failures in those optical systems if the near ground turbulence effects are not well considered. To illustrate such risk, we make Cn2 measurements by different approaches and cross compare them with associated working principles. By demonstrating the reasons for mismatched Cn2 results, we point out a few guidelines regarding how to use the general anisotropy theorem and the risk of ignoring it. Our conclusions can be further supported by an advanced plenoptic sensor that provides continuous wavefront data.

Near ground measurements of beam shaping and anisotropic turbulence over concrete runway and grass range

Researchers from the University of Central Florida recently carried out a series of measurements over a concrete runway and a grass range using a 632.8 nm Gaussian beam propagated for 100 or 125 m at a height of 2 m. Mean intensity and scintillation index contours varied significantly throughout these measurements in ways that corresponded to more than simple isotropy or anisotropy of optical turbulence. A simple theory is developed to show the effect of a nonlinear index of refraction gradient in addition to the possibility of anisotropic turbulence. Theoretical contours are compared to experimental results which seem to indicate the presence of a beam shaping phenomena near the ground in addition to anisotropy.

Analysis of optical turbulence evolution over the Space Shuttle Landing Facility

Ground to air temperature gradients drive the creation and evolution of optical turbulence in the atmospheric boundary layer. Ground composition is an important factor when observing and measuring the generated optical turbulence. Surface roughness and thermal characteristics influence the formation of optical turbulence eddies. The Space Shuttle Landing Facility (SLF) at The Kennedy Space Center offers a unique opportunity to measure the generation and evolution of these turbulent eddies, while also providing a temperature gradient “Step Function” after which turbulence evolution can be analyzed. We present the analysis of data collected on the SLF during May of 2018. Mobile towers instrumented with sonic anemometers are used to examine the statistics of turbulent eddies leaving the increased heat gradient of the runway. This data is compared to an optical scintillometer and other local weather station data. Point and path average Cn2 data are calculated and attention is given to turbulence spectrum as a function of height above ground.

Propagation Through Random Phase Screens

Overview: The notion of a thin turbulent layer along a propagation path has been used for many years to model radio wave propagation through the ionosphere, scattering from a rough sea surface, or propagation of an optical wave between a satellite and the Earths surface, among other settings. Such a turbulent layer is widely known as a phase screen, although this term generally refers to only a âx80x9cvery thinâx80x9d turbulent layer. In this chapter we develop a general model for a layer of optical turbulence between a transmitter and receiver along a horizontal propagation path. If the layer is fairly thick,x80x9d it is treated much like an extended medium. However, when the ratio of the turbulent layer thickness to the propagation distance from the turbulent layer to a receiver is sufficiently small, we classify the turbulent layer as a thin phase screen. Basically, this means that only the phase of the optical wave is disrupted as it passes through the turbulent layer - x80x94not its amplitude. Consequently, it is not necessary to integrate over the thickness of the layer, thus simplifying some of the expressions for various statistical quantities concerning a laser beam propagating over a path in which only a thin phase screen exists.nnIn our analysis we neglect the presence of extended optical turbulence and concentrate on the effects generated by the phase screen itself, taking into account the placement of the screen with respect to the transmitter and receiver. It is a straightforward extension of our model to embed the phase screen directly in an extended turbulence medium, although we dont do so here. Statistical quantities, like the mutual coherence function and scintillation index developed in Chaps. 6 and 8 for optical turbulence everywhere along the propagation path, are calculated here for the case of a single phase screen. In particular, we show how proper placement of the phase screen between the input and output planes can lead to essentially the same numerical results as that obtained from an extended turbulence model. In addition, we briefly treat the case of multiple thin phase screens that can be arbitrarily located along the propagation path. All results in this chapter, however, are limited to weak irradiance fluctuations for which the Rytov approximation is valid.

Classical Theory for Propagation Through Random Media

Overview: In this chapter we introduce the stochastic Helmholtz equation as the governing partial differential equation for the scalar field of an optical wave propagating through a random medium. However, we provide only the foundational material here for the classical methods of solving the Helmholtz equation. It is interesting that all such methods are based on the same set of simplifying assumptions - x80x94backscattering and depolarization effects are negligible, the refractive index is delta correlated in the direction of propagation (Markov approximation), and the paraxial approximation can be invoked.nnThe Born and Rytov perturbation methods for solving the stochastic Helmholtz equation are introduced first. Whereas the Born approximation has limited utility in optical wave propagation, the Rytov approximation has successfully been used to predict all relevant statistical parameters associated with laser propagation throughout regimes featuring weak irradiance fluctuations. We also illustrate that the Rytov approximation can be generalized to include wave propagation through a train of optical elements that are all characterized by ABCD matrix representations. Methods applicable also under strong irradiance fluctuations are briefly discussed here but formulated in greater detail in Chap. 7. These methods are the parabolic equation method, which is based on the development of parabolic equations for each of the statistical moments of the field, and the extended Huygens-Fresnel principle.nnEarly probability density function (PDF) models developed for the irradiance of the optical wave include the modified Rician distribution, which follows from the Born approximation, and the lognormal model, which follows directly from the first Rytov approximation. Of these two PDFs, only the lognormal PDF model compares well with the lower-order irradiance moments calculated from experimental data under weak fluctuation conditions. Hence, in this regime it has been the most often-used model for calculating fade statistics associated with a fading communications channel. Nonetheless, more recent investigations of the lognormal PDF suggest that it may be optimistic in predicting fade probabilities, even in weak fluctuation regimes.nnWe end the chapter with a modification of the Rytov method called the extended Rytov theory that utilizes the two-scale behavior of the propagating wave encountered in regimes of strong irradiance fluctuations. The formalism of the method presented here permits the development of new models for beam wander and scintillation in subsequent chapters that are applicable under strong fluctuations.

Fourth-Order Statistics: Strong Fluctuation Theory

Overview: In this chapter we extend our examination begun in Chap. 8 of various fourth-order statistical quantities, like the scintillation index and the irradiance covariance function, to the strong fluctuation regime. We develop separate scintillation models for plane waves, spherical waves, and Gaussian-beam waves. These models evolve from the extended Rytov theory (Chap. 5) by taking into account the role of decreasing spatial coherence of the optical wave as it propagates further and further through the random medium. The net result is a modification of the atmospheric spectrum to an âx80x9ceffective spectrumâx80x9d arising in the form of a multiplicative spatial filter function that eliminates the effects of moderate-sized refractive-index scales (or turbulent âx80x9ceddiesâx80x9d) under strong fluctuation conditions. This is similar to the use of spatial filters in adaptive optics applications to eliminate piston and tilt effects (among others) in the received wave front.nnUnder the general irradiance fluctuation theory developed here, the covariance function acquires a two-scale behavior in the strong fluctuation regime, consistent with earlier theories. From the frozen-turbulence hypothesis, we can infer the temporal covariance function from which we calculate the temporal spectrum of irradiance fluctuations. As shown in Chap. 8, the spectral width is determined by the transverse wind velocity scaled by the first Fresnel zone under weak irradiance fluctuations, but the power becomes concentrated at higher and higher frequencies as the strength of turbulence increases. Nonetheless, under strong irradiance fluctuations the two-scale behavior in the covariance function is also evident in the power spectrum.nnIn the last two sections, we review probability distribution models proposed for the irradiance fluctuations, including the gamma-gamma distribution that is theoretically valid under all fluctuation conditions. A favorable characteristic of the gamma-gamma distribution is that it has two parameters that are completely determined by atmospheric conditions.

Imaging Systems Analysis

Overview: In this chapter we discuss a fundamental area of application involving imaging systems. Of course, this application area is far more encompassing than our cursory treatment here. Imaging systems are typically classified as passive or active. A passive system is one that is based on receiving emitted radiation by the target (e.g., blackbody radiation), or reflected radiation by the target from natural sources (e.g., reflected sunlight or moonlight). An active system refers to one in which the target is intentionally illuminated by a source such as a laser. Our treatment here is primarily for active systems.nnA coherent imaging system is one in which the illumination wave and reflected wave are both coherent radiation. Such imaging systems are linear with respect to the optical electromagnetic field and can therefore be analyzed by conventional linear shift-invariant (LSI) principles, which involve the notions of impulse response and transfer function. In the open atmosphere a coherent illumination wave will suffer irradiance and phase distortions, leading to a partially coherent beam. Even in the partially coherent case, the imaging system is called an incoherent imaging system. Incoherent systems are linear with respect to irradiance instead of optical field. The useful parameters in this case are the point spread function (PSF) and optical transfer function (OTF), which are related through two-dimensional Fourier transforms. The modulus of the OTF, known as the modulation transfer function (MTF), is used to describe image quality whereas the phase transfer function (PTF) determines image position and orientation.nnAdaptive optics (AO) methods are commonly used to improve images. Imaging performance measures of an AO system, such as resolution and Strehl ratio, are defined in terms of the PSF or MTF. These particular metrics involve Frieds atmospheric coherence width, also known as the x80x9cseeing parameter.x80x9d In the use of a beacon or guide star, the isoplanatic angle arises as another important parameter that describes the useable field of view.

Fourier Series, Eigenvalue Problems, and Green's Function

In this chapter we introduce the method of Fourier series for the analysis of periodic waveforms (e.g., power signals). This approach reduces the signal being studied to a spectral representation in which the distribution of power is found to be concentrated at specific frequencies that are harmonically related to a fundamental frequency. In addition, we discuss the related notion of eigenvalue problem for homogeneous boundary value problems, the eigenfunctions of which are used to develop generalized Fourier series. Last, the method of Greens function is introduced for solving nonhomogeneous problems (including eigenvalue problems). By representing the Greens function in a x80x9cbilinearx80x9d representation, we amalgamate the theory of Fourier series and eigenvalue problems with that of the Greens function method.

Optical Wave Propagation in Random Media: Background Review

When an optical wave propagates through the atmosphere of the earth, it experiences distortions caused by small temperature variations related to the suns heating of the earth and the turbulent motion of the air due to winds and convection. The most well-known manifestation of this phenomenon is the twinkling of stars, observed long before the invention of the laser.nnLaser beam propagation through the atmosphere, which is of great interest to a variety of scientists and engineers, is a subset of the more general study of optical wave propagation through random media. By random medium, we mean one whose basic properties are random functions of space and time. Astronomers (including Sir Isaac Newton) were among the first scientists to show interest in certain atmospheric effects, like the quivering of the image of an astronomical object at the focus of a telescope and temporal fluctuations in received irradiance (intensity), the latter commonly called scintillation. During the 1950s, Russian scientists Obukhov [1] and Tatarskii [2] began theoretical studies of scintillation. These early theoretical studies were soon followed by a series of measurements of optical scintillation, the results of which were published mostly in astronomy journals. With the invention of the laser in 1960, theoretical investigations of optical wave propagation went beyond the interest of astronomers by focusing on characteristics of laser beams propagating through atmospheric turbulence. A brief history of scintillation studies by Russian scientists throughout the decades of the 1950s, 1960s, and 1970s can be found in an article by Gurvich [3].