Radio Science, Volume 10, Number 1, pages 129-137, January 1975 The effects of atmosphericturbulence on the propagation of pulsed laser beams C. S. Gardner Department of Electrical Engineering, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801 M. A. Plonus Department of Electrical Engineering, Northwestern University, Evanston, Illinois 60201 (Received October 3, 1974.) The effects of turbulence on the propagation of pulsed laser beams are examined. Using the Rytov theory, general expressions for the pulse fluctuations are derived in terms of arbitrary beam geometries and pulse shapes. Physical interpretations of the pulse distortion and the effects of the beam geometry on the pulse statisticsare discussed. 1. INTRODUCTION The study of propagation of transient signals throughthe turbulent atmosphere has receivedin- creasing attentionin recent years. The problem has important applications in optical communications, laser ranging, and laser radar since many of the developing and existingsystems in these areas em- ploy sho.rt opticalpulses. The error performance of optical communication systems and the accuracy of laser radars and ranging systems are highly de- pendent on the duration and statisticsof these pulses. The propagation of monochromatic plane, spher- ical, and gaussian beam waves in media described by random spatial and temporal fluctuations of the refractive index has been studied by many authors. Variousmethods have been proposed to analyze the statistics of the propagating wave.Strohbehn [1968], Lawrence and Strohbehn [1970], and Barabanenkov et al. [1971] have given excellent reviews of these methods and have summarized the major results. Probablythe most widely used technique for study- ing optical waves in random media is perturbation theory, particularly the method of Rytov, whichwas first introduced for this purpose in the monographs o,fChernov [1960] and Tatarskil [1961]. O'ne of the earliest treatments of pulse propagation waspresented by Mintzer [1953], who used the Born approximation to calculate the time-averaged statis- Copyright ¸ 1975 by the American Geophysical Union. ticsof acoustic pulses in ocean turbulence. Mintzer's approach has been used by a number of investigators to studyopticalpulses propagating in atmo.spheric turbulence [Su and Plonus, 1971; Gardner and Plonus, 1974]. Shirokova [1963] and Knollman [1965] investigated the effectsof pulse distortion using results derived from the Rytov approximation, while just recentlyLiu et al. [1974] usedthe par- abolicequation methodto analyze the propagation of pulse trainsin randomly ionized media. In thispaper we examine the effects of turbulence on the propagation of pulsed beamwaves. General expressions for the pulse fluctuations are derived in terms of arbitrary beam geometries and pulse shapes. Physical interpretations of the pulse distortion and the effects of the beam geometry are discussed. 2. BEAM WAVE RESPONSE IN WEAK TURBULENCE If polarization fluctuations areneglected, theprop- agation of theelectric field in atmospheric turbulence is described by the scalar waveequation XZ2E(r, t) -- (1/c2)(O2/Ot •) ß [her, t)E(r, t)] = --4•r/(r, t) (1) where n is the random refractive index and f is the source distribution. By applyingthe method of smooth perturbations, thesolution of (1) is given by E(r, t) = Eo exp (E1/YEo) = Eo exp ((/•1) (2) Eo is the unperturbed field and Ex is the single 129