This special issue of the Physical and Chemical News Journal is dedicated to some recent research subjects, in the field of modern laser physics, realized in our Laboratory by some of our graduate students in the PhD and Masters programs of Faculty of Sciences at the Chouab Doukkali University, or by our partners in other institutions.


These recent developments covered areas from the nano-scale to the atmospheric one. In the present papers, we report for the first time our analytical calculations and numerical simulations related to the following studies:


1- Beam shape coefficients and scattering of zeroth order Mathieu beam by homogeneous sphere: The scattering of the linearly polarized zeroth Mathieu beam by a dielectric homogenous sphere is investigated, and an exact analytical expression of the beam shape coefficients (BSCs) of this beam are obtained for the first time by using the integral localized approximation. These BSCs are used to derive the scattered intensity in far-field domain. Through numerical simulations, we have demonstrated that the normalized intensity depends on the ratio between the diameter of the sphere and the wavelength and the conical angle characterizing the zeroth order Mathieu beams.


2- Propagation of a novel beams through a paraxial ABCD optical system: Finite Airy-Gaussian Hermite-Gaussian beams (FAGHGB). First, we give the analytical form of the electric field at the origin of the optical axis for this new family of the paraxial optical waves. Their evolutions in free space and through an ABCD optical system are studied.


3- The generalized beam propagation M-factor of hollow sinh-Gaussian beams through a hard-edged aperture: This is derived based on the truncated second-order moments definition. The effects of the beam truncation parameter d and the beam order n on the beam propagation factor have been investigated. Three special cases have been obtained from the closed-form of the M-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.


4- Diffraction of Finite Airy-Hermite-Gaussian beams by an apertured misaligned optical system: The propagation properties of such beams through an apertured aligned and misaligned circular thin lens are developed. Some particular cases are deduced from the main theoretical formalism. The results of this finding show that the output beam shape remains unchanged and similar to that of the incident beam. However, it is slightly decentred in x or/and y directions and its deflection is related to the misalignment parameters.


5- Transformation of hollow-Gaussian beam (HGB) by an ABCD optical system coupled with optical systems of annular aperture basis: The corresponding closed-forms of the transformation of such beam by an unapertured ABCD optical system, and by this last coupled with a circular aperture or circular black screen are treated as particular cases of the principal finding. After propagating in optical systems cited above, it is observed that the HGB will be transformed into a Laguerre-Gaussian beam. From these results, we found the previous study concerning the propagation of pure Gaussian laser through the above optical systems, and it be pointed out that the receiver electric field is a doughnut beam in the near of optical system, and had a top intensity on the far-field of the source plane position of annular aperture or black screen. 


6- The propagation of a two-dimensional finite Airy beam through a rectangular aperture followed by a paraxial ABCD optical system: Based on the fact that a rectangular hard-edged aperture function can be expanded into a finite sum of complex Gaussian functions, and by using the generalized Collins diffraction integral, an approximate analytical expression of propagation properties is derived. From the main finding, the propagation characteristics formulas of the finite Airy beam through an unapertured ABCD optical system, or this last with a rectangular aperture, or a rectangular black screen are deduced. According to the closed-form expressions developed, the propagation characteristics of finite Airy beam through a free space with a square annular aperture or with a square aperture are illustrated.


7- Generalized M-factor of hard-edged diffracted Hypergeometric-Gaussian type-II beams: In the cylindrical coordinate system, and through a hard-edged circular aperture, the effects of the beam truncation parameter and the beam order on the propagating beam factor are examined. We showed that this factor can be simplified to four special cases, among of them, the truncated and untruncated Gaussian beams. The power fraction of such beams is also considered.


8- Action of misalignment of an annular aperture and an optical system on diffraction of an hyperbolic-cosine-gaussian beam: Based principally on the generalized Collins integral diffraction, the propagation properties of such beam passing through a misaligned annular aperture coupled with a misaligned optical system is devoted as the main result. Numerical examinations of the impact of misalignment elements, especially those of the annular and circular apertures, on shape and profile of the beam exiting the optical system captured at the output plane are performed.


9- Lis Flattened Gaussian beams propagating in a maritime medium: The power spectrum of refractive index introduced by Grayshan is used to study the effect of atmospheric weak turbulence on these beams. The propagation parameters, such as the long term beam spread, the beam wander and the Strehl ratio are calculated.


10- The propagation analysis of the superposition of Kummer beams in a turbulent atmosphere: By using the extended Huygens-Fresnel integral formula in the paraxial approximation, and based on the expression of the hard aperture, we explore the axial intensity distribution changes with the variation of the order N, the topological charge, the wavelength, the beam waist width, and for different values of the turbulence strength.


11- On the Humbert confluent hypergeometric function used in laser field: Important properties of the Humbert confluent hypergeometric function  as integral representations, special differentiation formulas, recurrence relationships, and combinations of powers are developed from those of Gauss hypergeometric function. These results can be used in many physical areas, particularly in the description of the donut laser beams.


     We hope that our results will be useful and helpful for researchers and graduate students interested to the above subjects. The reader with a mathematical eye can find also many theoretical developments based on the use of some special functions and their applications in the various studies contained herein.


Pr. H. Nebdi and Pr. A. Belafhal