July 2014 Phys. Chem. News 73 (2014) 14-20 PCN 14 GENERALIZED BEAM PROPAGATION FACTOR OF HOLLOW SINH-GAUSSIAN BEAMS PASSING THROUGH A HARD-EDGED APERTURE F. Khannous, A.A.A. Ebrahim, H. Nebdi, A. Belafhal* Laboratoire de Physique Nucléaire, Atomique et Moléculaire Département de Physique, Faculté des Sciences, Université Chouaïb Doukkali, B. P.: 20, 24000 El Jadida, Morocco * Corresponding author. E-mail: belafhal@gmail.com Received: 11 February 2014; revised version accepted: 19 May 2014 Abstract The generalized M²-factor of hollow sinh-Gaussian beams through a hard-edged aperture is derived based on the truncated second-order moments definition. The effects of the beam truncation parameter δ and the beam order n on the beam propagation factor have been investigated numerically. Three special cases have been obtained from the closed-form of the M²-factor of the truncated hollow sinh-Gaussian beams; the non-truncated hollow sinh-Gaussian beams, the truncated and non-truncated Gaussian beams. The power fraction of this beams family is also illustrated analytically and numerically. Keywords: Generalized M²-factor; Hard-edged aperture; Hollow-sinh-Gaussian beams; Power fraction. 1. Introduction In recent years, many theoretical models have been introduced to describe dark-hollow beams, such as the TEM* 01 beam which is the most popular model [1], the superposition of off-axis Gaussian beams which have been introduced by Zhu et al. [2], the higher order Bessel-Gaussian beams [3], the hollow Gaussian beams [4], and the controllable dark beams [5]. More recently, hollow sinh-Gaussian (HsG) beams, which has been proposed by Sun et al. [6], is considered one of mathematical methods to describe dark-hollow beams. These beams have attracted some interests because of their increasing applications in many fields, such as atomic optics, binary optics, optical trapping of particles and medical sciences [7]. Meanwhile, the corresponding characteristics of those ray beams also have took the same attention of optical researchers. On the other hand, several parameters and factors were introduced in laser optical domain, among these factors, the so-called beam propagation factor (M²-factor) [8, 9], which is an important parameter for characterizing the laser beams. The realizable beams are more or less limited by aperture effects, therefore, it is suitable to study M²-factor of the truncated beams through a hard-edged aperture. As it is well known, there are a variety of methods for studying the parametric characterization of the truncated laser beam through a hard-edged aperture [10-14]. There are many publications which have been focused on the investigation of the generalized M²-factor of laser beams through a hard-edged aperture [15-20]. For example, Lü et al. [16] have been investigated the beam propagation factor of cosh-Gaussian beams, Mei et al. [17,18] have been determined the generalized M 2 factor of dark- hollow beams and Bessel–Gaussian beams, and Deng [19] has been introduced the 2 G M -factor of hollow Gaussian beams. In our work, based on the truncated second- order moments method, the generalized M²-factor of HsG beams truncated by a hard-edged aperture is derived mathematically and illustrated numerically. The method used for analysing this factor is characterized by a competitive advantage which enables us to describe the expression as the untrancated case, if the truncation parameter approaches infinity. In addition, the power fraction of truncated HsG beams in the waist plane is investigated both analytically and numerically. 2. M²-factor of truncated HsG beams In the polar system, the electric field distribution ( ) 0 , r E n of a HsG beam at the initial plane is given by [6] ( ) ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = 2 0 2 0 exp sinh 0 , w r w r r E n n , (1) where 0 w is the waist width of a Gaussian beam, n (=0,1,2,…) is the beam order and r is the radial coordinate variable. Eq. (1) can be rewritten in the following form [6] ( ) ( ) ∑ = ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ + − = n m m m m n w c r b a r E 0 2 0 2 exp 0 , , (2.a) where the coefficients a m and b m are given by