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_{n}

^{2}data are calculated and attention is given to turbulence spectrum as a function of height above ground.

_{n}

^{2}measurements by different approaches and cross compare them with associated working principles. By demonstrating the reasons for mismatched C

_{n}

^{2}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.

^{2}and wind speed.

*z*direction with circular symmetry maintained in the orthogonal

*xy*-plane throughout the path. In the present analysis, however, the anisotropic spectrum model is no longer based on a single anisotropy parameter—instead, two such parameters are introduced in the orthogonal xyplane so that circular symmetry in this plane is no longer required. In addition, deviations from the 11/3 power-law behavior in the spectrum model are allowed by assuming power-law index variations 3 <

*α*< 4 . In the current study we develop theoretical models for beam spot size, spatial coherence, and scintillation index that are valid in weak irradiance fluctuation regimes as well as in deep turbulence, or strong irradiance fluctuation regimes. These new results are compared with those derived from the more specialized anisotropic spectrum used in previous analyses.

^{2}has proven to be a reliable, well-used means of quantifying and characterizing the atmospheric turbulence in a given environment. This model relies on various meteorological parameters such as temperature, wind speed, relative humidity, and time of day in order to procure the resulting Cn

^{2}quantity. Using experimentally confirmed results from a desert environment, the utility of this model may be extended to other climates by adapting temporal hour weights used within the model. The adaptation of these weighted parameters are shown to have a relationship with the unique weather conditions of a given region which are demonstrated by data points collected from two testing ranges located in Florida in addition to archived weather data. The resulting extended model is then compared to commercial scintillometer data for validation.

^{2}) data was also taken at various ranges using two commercial scintillometers. This parameter is used to characterize the strength of atmospheric turbulence, which induces scintillation effects on the laser beam, and is a vital input parameter to the mathematical model. Data was taken and analyzed using a 4-detector board array. The material presented in this paper outlines the verification and validation of the theoretical scintillation model, and steps to improve the scintillation fluctuation effects on the laser beam through additional detectors and a longer transmitting wavelength. Experimental data was post processed and analyzed for scintillation fluctuations of the two transmitting wavelengths. The results demonstrate the benefit of additional detectors and validate a mathematical model that can be scaled for use in a variety of communications or defense applications. Scintillation is a problem faced by every free space laser communication system and the verification of an accurate mathematical model to simulate these effects has strong application across the industry.

_{n}

^{2}and the inner-scale of atmospheric turbulence, l

_{0}. The atmospheric parameters inferred from the collected laser data and the commercial instruments were compared. Mean and variance of the fade times were found to agree well with theory for smaller apertures where effects of aperture averaging are not present and in cases where scintillation is weak to moderate. It is suggested that a more appropriate PDF, with a heavier focus on aperture averaging, may be applied in future studies of free space optical communication system fade statistics.

Creating a Cn2 profile as a function of altitude using scintillation measurements along a slant path

_{n}

^{2}profile model as a function of altitude up to (and possibly beyond) the maximum altitude of a laser beam along the propagation slant path. This technique was demonstrated recently in June 2011 on a beacon beam transmitted between Hollister Airport in California and Fremont Peak at a slant range of 17 km. Although the primary experiment was to test a hybrid optical RF communication system (FOENEX), the beacon signal at the transmitter was intercepted by the TASS from which weighted path-average values of C

_{n}

^{2}, inner scale l

_{0}, and outer scale L

_{0}were determined. Path-average values were then entered into an algorithm that determines the parameters of the HAP C

_{n}

^{2}profile model (a variation of the HV profile model). In this paper we report on these recent measurements and how this method of constructing the HAP model can be used over other propagation paths.

_{n}

^{2}. Histograms were formed with the data and compared to the Log-Normal and Gamma-Gamma PDF models. As expected, neither PDF model was applicable under all conditions of aperture averaging. Hypotheses are made as to why the models were unable to completely capture the effects of aperture averaging on received irradiance data.

Impact of ground profile on scintillation: 50km and 200km slant paths from airplane to Antelope Peak

_{n}^{2}profile model as a function of altitude.

_{n}

^{2}[m

^{-2/3}] ranging from 10

^{-15}to 10

^{-13}.

^{2}

_{n}and also from ground to space, the latter based on the Hufnagle-Valley C

^{2}

_{n}. We consider cases of tracked beams and untracked beams, both of which involve a certain amount of beam wander. Theoretical models of scintillation and corresponding probability density function (pdf) models are compared with simulation data over a broad range of beam diameters. We include both collimated and focused beams. For the uplink from ground to space path, we also examine some fade statistics.

*C*

_{n}

^{2},

*l*

_{0}, and

*L*

_{0}. Parallel to the three-aperture data collection was a commercial scintillometer unit which reported

*C*

_{n}

^{2}and crosswind speed. There was also a weather station positioned at the receiver side which provided point measurements for temperature and wind speed. The

*C*

_{n}

^{2}measurement obtained from the commercial scintillometer was used to infer

*l*

_{0},

*L*

_{0}, and the scintillation index. Those values were then compared to the inferred atmospheric parameters from the experimental data. Finally, the optimal aperture sizes for data collection with the three-aperture receiver were determined.

_{0}to Fried's parameter r

_{0}(

*W*

_{0}/r

_{0}<<1,

*W*

_{0}/r

_{0}~ 1,

*W*

_{0}/r

_{0}>> 1), but not generally in the other regimes. This is true for tracked beams as well as untracked beams. Two new pdf models, developed here as a modulation of either the gamma-gamma pdf or the gamma pdf, are shown to provide excellent fits to the simulation data over all three regimes defined above.

_{00}Gaussian beam. We verify our results by comparison with a computer simulated model for the flattened beam.

*w*is on the order of the atmospheric coherence width

_{0}*r*. It has been shown that such departures from Rytov theory are primarily a consequence of beam wander and other low-order aberrations. In this paper we discuss modeling of the probability density function (PDF) for uplink beams. In particular, we show how the PDF transitions from lognormal statistics when

_{0}*w*/

_{0}*r*<< 1 to the negative exponential distribution when

_{0}*w*>> 1. The most interesting regime is the transition region near

_{0}/r_{0}*w*= 1, where the statistical behavior of tracked and untracked beams differs significantly.

_{0}/r_{0}*C*,

_{n}^{2}*l*and

_{o}*L*, were inferred from these optical measurements. Simultaneously, a commercial scintillometer, which recorded values for

_{o}*C*, was set up parallel to the optical path. In this paper, a numerical scheme is used to infer the three atmospheric parameters and comparisons are made with the

_{n}^{2}*C*readings from the scintillometer.

_{n}^{2}_{n}

^{2}is constant and use our recently developed gamma-gamma model and the well known lognormal model to consider the fading statistic associated with a spherical wave model for simplicity. The results are similar to a Gaussian-beam wave with perfect pointing. Our analysis show that compared to the gamma-gamma model, the lognormal model predicts optimistic values of probability of fade, underestimate the number of fades per second and consequently does not measure the mean fade time correctly.

^{2}

_{n}is constant and use our recently developed gamma-gamma model and the well known lognormal model to consider the fading statistics associated with a spherical wave model for simplicity. The results are similar to a Gaussian-beam wave with perfect pointing. Our analysis show that compared to the gamma-gamma model, the lognormal model predicts optimistic values of probability of fade, underestimate the number of fades per second and consequently does not measure the mean fade time correctly.

_{1}

^{2}equals (sigma)

_{x}

^{2}+ (sigma)

_{y}

^{2}+ (sigma)

_{x}

^{2}(sigma)

_{y}

^{2}where (sigma)

_{x}

^{2}denotes large-scale scintillation and (sigma)

_{y}

^{2}denotes small-scale scintillation. By applying a modification of the Rytov method that incorporates an amplitude spatial frequency filter function under strong fluctuation conditions, tractable expression are developed for the scintillation index of a Gaussian beam wave that are valid under moderate- to-strong irradiance fluctuations. The expected scintillation of Gaussian beams predicted by these analytic models is compared to the experimental data previously published.

*L/k*)

^{1/2}are used to construct simple interpolation formulas, which are generally within 5% of exact numerical-integration values in the case of a plane wave and within 12% of exact values for a spherical wave. On comparison with published experimental data on spherical waves, we found similar agreement, with maximum error 8%.

^{1/2}to inner scale l

_{0}. Simple interpolation formulas are developed here for all such ratios, and comparisons of results with numerical integration values and experimental data show excellent agreement.

_{o}

^{2}and (Omega)

_{o}equals 1 - L/R

_{o}associated with the beam when transmitted, and the second pair given by (Lambda) equals (lambda) L/(pi) W

^{2}and (Theta) equals 1 + L/R associated with the received beam. Here, (lambda) is wavelength, L is path length, R

_{o}and R are radii of curvature of the phase front at the transmitter and receiver, respectively, and W

_{o}and W are the beam radii at the transmitter and receiver. With the addition of (lambda) and L, either of these beam parameter pairs completely characterizes the diffractive propagation environment for a lowest- order paraxial Gaussian beam, and is fundamental in the analytic expression of the irradiance variance. Special attention is paid to differences between the perfectly focused beam and the nearly focused beam. We also show that every beam has a convergent counterpart with identical diffractive irradiance behavior at the receiver, but decreased irradiance variance.

_{0}for the critical diameter of a telescope, are also examined.

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This course describes beam wave propagation through optical turbulence. Satellite communication systems, laser radar, remote sensing, and adaptive optics are some of the applications affected by optical turbulence. Tractable analytic equations are provided for calculating Gaussian-beam wave statistical quantities affecting system performance. The mutual coherence function (MCF), mean intensity, degree of coherence, and intensity fluctuations (scintillation) are presented. Videos of actual experiments show how to gather data. Examples are presented using MATHEMATICA software programs. Copies of these programs are available in the text.

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