Propagation of doughnut-shaped super-Gaussian beams, convolution theorem and Hankel transform S. BOLLANTIy, P. DI LAZZAROy, D. MURRAy, A. TORREy and C. E. ZHENGz yCentro Ricerche Ente Nazimale Energia Altoniva Frascati, PO Box 65, 00044 Frascati, Italy; email: dilazzaro@frascati.enea.it zElectronic Engineering SpA, via Baldanzese 17, 50041 Calenzano, Italy (Received 9 February 2004; revisison received 3 May 2004) Abstract. The most direct and simple method to calculate analytically the propagation in the far field of a coherent beam with a rectangular symmetry and a super-Gaussian-like irradiance profile involves the use of the convolution theorem. In this paper we extend this method to a circularly symmetric beam, and hence from the Fourier transform to the zero-order Hankel transform. Then, we examine the more complex case of super-Gaussian-like beam irradiance shapes with an axial shadow, as those emitted by unstable resonators with diffraction output coupling or with a variable-reflectivity mirror. The analytical results obtained by approximating the super-Gaussian-like beam with a convolution function and applying the Hankel transform are in excellent agreement with those obtained by numerical simulations. 1. Introduction The saturation of the active medium gain in most high-gain lasers causes a flattening effect on the Gaussian-shaped intracavity irradiance, thus producing a super-Gaussian (SG-) like shaped output beam. Basically, SG-like profiles consist of a nearly flat central region with a smooth continuous transition to zero. Usually, the SG-like beam yields a better energy extraction from the laser active volume than that of the equivalent Gaussian beam. Moreover, there are many applications requiring a nearly uniform beam irradiance distribution within a certain area on a given plane, and vanishing outside, like a SG-like beam. As a consequence, much research work has been devoted to study the SG-like beam’s generation and propagation [1–10]. The propagation of coherent SG beams has been numerically simulated [8, 9], while analytical studies were mainly devoted to a family of SG-like beams, called ‘flattened Gaussian beams’, which are dealt with a linear superposition of Laguerre–Gauss functions under the paraxial approximation [3–5]. More recently, a new family of SG-like beams was introduced as a fundamental Gaussian modes superposition [10]. A simpler analytical approach has been described in [7], which uses the square of the convolution product of a Gaussian function with a rectangle function to approximate the SG-shaped beam irradiance. Then the convolution theorem, according to which the convolution of two functions is Fourier journal of modern optics, 10 march 2005 vol. 52, no. 4, 551–561 Journal of Modern Optics ISSN 0950–0340 print/ISSN 1362–3044 online # 2005 Taylor & Francis Ltd http://www.tandf.co.uk/journals DOI: 10.1080/09500340410001725973