Theory And Experiments On The Probability Density Of The Irradiance In Atmospheric Turbulence

AbstractA universal math model has been developed for the irradiance fluctuations of an optical beam propagating through atmospheric turbulence. This new model was developed under the assumption that the field irradiance consists of two principal components, each of which has an amplitude that is m distributed. By comparing this model with our own experimental data, as well as comparable data of other researchers, we found both qualitative and quan­ titative agreement over all ranges tested and strengths of turbulence. The statistics used in making these comparisons were the first five normalized moments of the bearr intensity. Our own data were obtained over a very homogeneous environment with ranges from 180 m to 3 km. The data so obtained correspond to many conditions of turbulence, from weak where lognormal statistics are ordinarily valid, to superstrong turbulence where negative expo­ nential statistics exist in the limit. In the weak turbulence regime we found that the data support the lognormal and universal model, which both predict essentially the same re­ sults. For conditions of moderate to strong turbulence, both the K distribution and the universal model match the data. However, for very strong turbulence prior to the limiting form of the negative exponential distribution, only our universal model matched the data. One of the interesting observations that we made in our data was that the normalized mo­ ments follow a looped path in the vicinity of their maximum values when they are plotted as functions of the normalized second moment. None of the standard theoretical models (i.e., lognormal, K, and negative exponential distributions) predict this looping effect since they are only two-parameter models whereas our universal model is a three-parameter model.IntroductionIt is widely known that the propagation of an optical beam through clear-air turbulence will result in fluctuations of the beam intensity, commonly called irradiance fluctuations. These fluctuations are due almost exclusively to atmospheric temperature variations caused by heating of the earth's surface. Over the years there have been numerous experimental and theoretical investigations into the phenomenon of irradiance fluctuations,1 '18 irost of which have been directed at the statistics associated with the phenomenon. There is an abundance of experimental evidence that validates the Tatarskii theory, or lognormal model, for both the amplitude and intensity of the beam under conditions of weak turbulence.1 > 2 » u » 7 > 12 For conditions of moderate turbulence the K distribution matches much of the experi­ mental data,7 * 8 ' 12 and in the regime of super saturation the theoretically predicted nega­ tive exponential distribution is now widely accepted.12"15 Unfortunately, none of these standard distributions universally fit all the available data on intensity fluctuations.During March and August, 1980, experimental data on the statistics of optical beam in­ tensity fluctuations were gathered over various path lengths up to 3 km. The propagation experiments were carried out at the Kennedy Space Center Space Shuttle Landing Facility, a concrete runway 4.9 km long and 90 m wide. The runway is extremely level and the edges totally clear of all structures and vegetation (except grass) for distances up to several hundred meters. These conditions made for a near-perfect test site ensuring homogeneous conditions the full length of the runway. The refractive-index-structure parameter C£ varied by several orders of magnitude, from 10"14 in"2/3 to 2 x 10"10 nT2/3. These values correspond to many different conditions of turbulence -- from very weak to superstrong into the saturation regime. Such a wide range of data over various conditions of turbulence and path lengths gave us new insight into the phenomenon of irradiance fluctuations.In this paper we discuss the experiments conducted at the Kennedy Space Center as well as a universal math model that we developed that is seemingly applicable to all conditions of turbulence for which data is available. This universal model has characteristics of the lognormal distribution in the regime of weak turbulence, characteristics of the K distribu­ tion in the regime of moderate turbulence, and asymptotically approaches the limiting nega­ tive exponential distribution in the deep saturation regime.