September 2011 Phys. Chem. News 61 (2011) 24-33 PCN 24 EFFECTS OF ATMOSPHERIC TURBULENCE ON THE PROPAGATION OF Li’s FLAT-TOPPED OPTICAL BEAMS A. Kinani 1 , L. Ez-zariy 1 , A. Chafiq 2 , H. Nebdi 3 , A. Belafhal 1, * 1 Laboratoire de Physique Nucléaire, Atomique et Moléculaire Département de Physique, Faculté des Sciences, Université Chouaïb Doukkali, B. P 20, 24000 El Jadida, Morocco 2 CPR Mohammed V, 46000 Safi, Morocco 3 Physics Department, College of Applied Sciences, Umm Al-Qura University, P.O. Box : 715 , Makkah 21955, Saudi Arabia. E-mail: hbnebdi@uqu.edu.sa * Corresponding author. E-mail: belafhal@gmail.com Received: 15 August 2011; revised version accepted:29 September 2011 Abstract In this paper, we analyze the propagation of the Flat-topped beams (FTB) introduced by Li in turbulent atmosphere within the regime of weak fluctuations. We use the paraxial approximation and the Rytov theory to determine some parameters, and factors characterizing fluctuations of amplitude and phase of beams during its propagation in turbulent atmosphere. Thus, the statistical moments as the scintillation index, the probability of fade, the excepted number of fades per unit time, and the mean duration of fade are found. The obtained theoretical results are illustrated by some numerical calculations and discussions. Keywords: Flat-topped beams; Rytov theory; Scintillation index; Probability of fade; Mean duration of fade. 1. Introduction Laser propagation through the atmosphere is currently a very active area of research, with its various applications such as optical communications, imaging, and remote sensing [1]. During its propagation in the atmosphere, the amplitude and phase of the electromagnetic field representing the optical/IR wave experience random fluctuations caused by random changes of the refractive index. In the pioneer works, the propagation of the laser in a turbulent atmosphere are based on the standard Gaussian beam [1], while recently other classes of lasers have been studied in terms of propagation in a turbulent atmosphere, and has proved more performance than the standard Gaussian beam. From these beams, we found the higher-order and elliptical Gaussian beams [2-5], cosh-Gaussian beams [6-7], Hermite-cosh-Gaussian laser beams [8-9], partially coherent twisted anisotropic Gaussian Schell-model beams [10], higher order Bessel- Gaussian beams [11], dark hollow beams [12], Helmholtz-Gauss beams [13] and so on. On other hand, Flat-topped beams (FTB), considered here, are described by different formulations [14-21], from which we found the Li’s model [16] which is described by a summation of Gaussian terms. In our study, we use this formulation to derive the parameters characterizing the propagation of beams through turbulent atmosphere. In literature, we found a similar work where D. Cowan [22] has used a Flat-topped beam modeled on the Flattened- Gaussian profiles of Gori [14]. Both of these models are therefore preferable to the Super- Gaussian model [16-17], Multi-Gaussian beams [18], and Baykal’s model [19-21], since it has the advantage of analytical solutions in propagations scenarios. In this paper, we investigate the propagation of the FTBs represented by Li’s model in turbulent atmosphere within the regime of weak fluctuations, where we use the paraxial approximation and the Rytov theory. And we give the mathematical expressions of basic statistical quantities of the wave field propagating through turbulent atmosphere. This paper is organized as follows: in Section 2, we give the theory of the propagating FTBs in turbulent atmosphere, and in Section 3, we perform the theoretical expressions of some statistical moments in Rytov approximation. In Section 4, we list the expressions of these statistical moments in the case of FTBs. In order to illustrate the obtained results, we present in Section 5 some numerical simulations and discussions. Finally, we outlined this paper by a conclusion. 2. The propagation of FTB in the presence of atmospheric turbulence 2.1 Perturbation theories for wave propagation in turbulent atmosphere In the presence of atmospheric turbulence, an electromagnetic wave experiences fluctuations caused by random changes in the index of refraction. This spatio-temporal variation of refraction index makes the wave equation more complicated. Thus, consider a scalar