Theoretical Introduction and Generation Method of a Novel Nondiffracting Waves: Olver Beams

Abdelmajid Belafhal, Lahcen Ez-Zariy, Salima Hennani, Hamid Nebdi
2015 Optics and Photonics Journal  
In this paper, we introduce a new class of scalar nondiffracting Helmholtz-equation solution. We demonstrate that this novel wave-equation solution has some specific orders; among these ordinary Airy beams which are regarded as the zeroth order. Moreover, a general expression of these novel beams, which are named Olver Beams and referred to OBs, is developed. The zeroth and the first high orders of the incident OBs are presented theoretically and numerically in this paper. Yet, based on a
more » ... er generated holograms method, the generation's masks of the Finite OBs in first orders are given in this work. Also, the incident transverse intensity distribution in 1-D and 2-D of the first orders of OBs is performed. A. Belafhal et al. 235 theoretically and experimentally by Siviloglu et al. [2] [3] in 2007. Recently, for physical realization, this beams family has been modulated by an exponential term to obtain the finite Airy beam which is expressed as a product of the classical Airy mode and the linear exponential function. For many reasons, researchers have tried to properly study these laser beams and to be interested in their most remarkable properties. These beams are already considered as a single type of nondiffracting beams that are currently described in literature as a one-dimensional system [2]-[10]. The concept of Airy beams has been deeply explored on optical transforms for light beams; Airy transform can be applied in optics laser in many contexts by using Fourier Transform, Hankel Transform and Hilbert Transform. These transforms are all useful in beam shaping, beam analysis, image processing and others [11]- [22] . Since 2013, our research laboratory group has started its researches over the Airy class beams through different optical systems. At first, we found a theoretical formalism for the study of the propagation of these nondiffracting beams family through an ABCD optical system with a rectangular annular aperture [23] . Another study has demonstrated the propagation characteristics of Airy-Gaussian beams passing through a misaligned ABCD optical system with aperture [24] . Thereafter, our studies have gone further than the previous work to do the same study of the propagation through an optical ABCD system, but this time with a new form of Airy-Hermite-Gaussian beam [25] . This series of studies continue to give an interesting work that explores another new form of Finite Airy-Gaussian-Hermite-Gaussian Beams propagating through a paraxial ABCD optical system [26] , which is a special case of the Olver beams. The present work will be an extension of the study of nondiffracting Airy beams. We will give a general analytical expression of a novel beams family, called Olver beams and is a solution of Schrödinger equation. This solution can be expressed in terms of Olver functions obtained directly by solving Olver's differential equation given by [27] :
doi:10.4236/opj.2015.57023 fatcat:i2o237n2bzgutdt7x4tmkjdroe