Turbulence induced changes in spectrum and time shape of fully coherent Gaussian pulses propagating in atmosphere D. Razzaghi a, * , F. Hajiesmaeilbaigi a , M. Alavinejad b a Laser and Optics Research School, P.O. Box 14155-1339, Tehran, Iran b Photonics Laboratory, Physics Department, Iran University of Science and Technology, Tehran, Iran article info Article history: Received 4 October 2009 Accepted 22 January 2010 abstract Propagation of fully coherent pulses with Gaussian spatial distribution through turbulent atmosphere is studied. Turbulence induced changes in spectrum of propagated pulse and its effect on temporal behavior of the signal is discussed and has been showed that signal is widened in time domain as propagating in turbulent atmosphere. It is also proved that various points in observation plane experience different effects. Ó 2010 Elsevier B.V. All rights reserved. 1. Introduction Propagation induced changes in the spectrum of partially coher- ent light through atmospheric turbulence have attracted consider- able attention recently [1–5]. It has also been realized that the broad spectrum of partially coherent light can lead to propagation through atmospheric turbulence induced changes in the shape of an optical beam [6,7]. It was shown in 1990 that a partially coher- ent, continuous-wave, Gaussian beam whose spot size at the beam waist is frequency independent does not remain Gaussian on prop- agation in atmospheric turbulence [3]. One may ask if that would also happen for a coherent Gaussian beam consisting of ultra short optical pulses such that the field spectrum is quite broad. Surpris- ingly, a definite answer to this question is not provided in the lit- eratures. Of course, propagation of ultra short optical pulses in a linear optical medium has been studied extensively in recent years [8–13]. The propagation properties of ultra short pulsed beams with constant diffraction length in free space and dispersive media have been studied both analytically and numerically [14–18]. However, because of the mathematical difficulty in treating ul- tra short pulsed beams propagated through atmospheric turbu- lence, only numerical results or an approximate propagation expression in the far field are found. So the purpose of the present work is to study the propagation properties of the ultra short pulsed beams through atmospheric turbulence. Specially, a pulse with Gaussian spatial distribution is considered and some features of turbulence induced changes in temporal behavior of the pulse is predicted. 2. Basic theory Consider a pulse which has a spatial Gaussian form as below: Eðr 0 ; z ¼ 0; tÞ¼ A exp  r 2 0 w 2 0   ; ð1Þ where w 0 is the beam waste, r 0 is radial distance, and A is constant which is supposed to be one for simplicity. Moreover temporal part of the pulse is assumed to be Gaussian form at source plane so that spatiotemporal form of the pulse can be written as following form: Eðr 0 ; z ¼ 0; tÞ¼ f ðtÞ exp  r 2 0 w 2 0   ; ð2Þ where f(t), the temporal part, is: f ðtÞ¼ exp  a g t T c   2 " # cosðx c tÞ; ð3Þ where a g ¼ð2 ln 2Þ 1 2 , x c is the carrier frequency, T c is the pulse duration. We now consider pulse propagation through atmospheric tur- bulence using the paraxial form of the extended Huygens–Fresnel principle [2] so that each Fourier component of the pulse is prop- agated as below: Eðr; z; xÞ¼ k 2pz   2 ZZ d 2 r 0 Eðr 0 ; z ¼ 0; xÞ exp ik ðr  r 0 Þ 2 2z " #  exp½wðr; r 0 ; z; xÞ; ð4Þ where r 0 and r are position vector in source plane and observation plane, respectively, E(r 0 , z = 0,x) and E(r, z, x) represent Fourier components of input and propagated pulse, and finally w is a ran- dom phase factor representing the turbulence effect. 0030-4018/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2010.01.048 * Corresponding author. E-mail address: davrazzaghi@yahoo.com (D. Razzaghi). Optics Communications 283 (2010) 2318–2323 Contents lists available at ScienceDirect Optics Communications journal homepage: www.elsevier.com/locate/optcom