Papers by EZ-ZARIY LAHCEN
An analytical expression to study the propagation properties of an Airy-Hermite-Gaussian beam
pas... more An analytical expression to study the propagation properties of an Airy-Hermite-Gaussian beam
passing through an apertured misaligned optical system is developed, in this paper, by using the
generalized Huygens-Fresnel diffraction integral and by means of expansion of the hard-edged aperture
into a finite sum of complex-Gaussian functions. Some particular cases are deduced from the main
theoretical formalism of our work. The propagation characteristics of the considered beam through an
apertured aligned and misaligned circular thin lens are investigated numerically. 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.
Keywords: Generalized Huygens-Fresnel diffraction integral; Airy-Hermite-Gaussian beam; Aligned optical system; Misalignment;
Circular aperture; Circular thin lens.
An analytical expression to study the propagation properties of an Airy-Hermite-Gaussian beam pas... more An analytical expression to study the propagation properties of an Airy-Hermite-Gaussian beam passing through an apertured misaligned optical system is developed, in this paper, by using the generalized Huygens-Fresnel diffraction integral and by means of expansion of the hard-edged aperture into a finite sum of complex-Gaussian functions. Some particular cases are deduced from the main theoretical formalism of our work. The propagation characteristics of the considered beam through an apertured aligned and misaligned circular thin lens are investigated numerically. 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.
Based on the generalized Collins integral diffraction and by means of the expansion method of a h... more Based on the generalized Collins integral diffraction and by means of the expansion method of a hardedged aperture into finite sum of complex Gaussian terms, an approximate analytical expression of the propagation properties of Hyperbolic-Cosine-Gaussian (ChG) beam passing through a misaligned annular aperture coupled with a misaligned optical system is devoted as the main result of this work. The propagation properties of the fundamental Gaussian beam through a misaligned annular aperture followed by a misaligned optical system and those of ChG and Gaussian beams passing through an unapertured misaligned or aligned optical system or truncated by an aligned or misaligned: annular aperture, circular aperture and black screen are derived in the article as particular cases of the main finding. Some numerical simulations are performed in the study to examine 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. Some numerical calculations are performed in the work.
This paper deals with the propagation of a two-dimensional finite Airy beam through a rectangular... more This paper deals with 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 numerically in the work.
Based on the generalized Collins-Huygens diffraction integral, and by means of the expansion of a... more Based on the generalized Collins-Huygens diffraction integral, and by means of the expansion of a hard aperture function into a finite sum of complex Gaussian functions a main approximate analytical expression of hollow-Gaussian light, propagating through a paraxial ABCD optical system coupled with an annular aperture, is devoted in this paper. The corresponding closed-forms of the transformation of a hollow-Gaussian beam (HGB) by an unapertured ABCD optical system, and by this last coupled with a circular aperture or circular black screen are also treated in this work 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 light through the above optical systems. Using the analytical expressions, the transverse intensity distribution of propagation characteristics in the annular aperture, and in black screen are discussed and investigated numerically in detail in this work.
The aim of this paper is to develop the propagation characteristics of the four-petal Gaussian (F... more The aim of this paper is to develop the propagation characteristics of the four-petal Gaussian (FPG) beam through a misaligned circular annular aperture followed by a misaligned optical system. Based on the diffraction theory, analytical expressions of the FPG beam transformation through this considered system were obtained. Several particular cases corresponding to the transformation of FPG beam by an unapertured misaligned optical system or the latter with misaligned circular aperture or misaligned black screen and other special cases concerning the transformation of a FPG beam by aligned optical system with aligned/misaligned: annular aperture, circular aperture or black screen could be deduced in this paper. Some numerical simulations describing the effect of misalignment of annular aperture, circular aperture and black screen on propagation properties are also investigated. The present work generalizes about 31 investigations of FPG and fundamental Gaussian beams through misaligned and aligned various kinds of apertures followed by aligned or misaligned optical system.
Based on the generalized Huygens-Fresnel diffraction integral and by expanding a hard-edged apert... more Based on the generalized Huygens-Fresnel diffraction integral and by expanding a hard-edged aperture function into a finite sum of complex Gaussian functions, an approximate analytical expression for description of transformation properties of two-dimensional Hermite-Gaussian beam by a misaligned annular aperture followed by a misaligned paraxial ABCD optical system is developed. The propagation of the Hermite-Gaussian beam through an unapertured misaligned optical system or this latter with a misaligned circular aperture or a misaligned black screen cases are deduced here as particular cases of the misaligned annular apertured misaligned optical system which can be regarded as the general case. So, other particular cases concerning the Hermite-Gaussian beam and the pure Gaussian beam traveling an unapertured aligned optical system or coupled with aligned annular aperture, circular aperture or black screen can be derived. Some numerical simulations are also performed in this work.
Integer or non-integer topological charge Apertured spiral phase plate system a b s t r a c t Thi... more Integer or non-integer topological charge Apertured spiral phase plate system a b s t r a c t This paper comments on a recently published work dealing with: "Generation of dark hollow beams by using a radial Hilbert transform system" by Q. Xie and D. Zhao, Opt. Commun. 275 (2007) 394. The authors have applied an integral, valid only for the integer orders, in cases of fractional ones. In addition, in their main result established by Eq. (11), the authors have neglected the dependence of some terms on the parameter summation. So, in the actual comment, we will give the main exact closed-form of the considered theory.
In this paper, an approximate analytical expression for description of the transformation propert... more In this paper, an approximate analytical expression for description of the transformation properties of the 2D-Li's Flattened-Gaussian beam by a misaligned annular aperture followed by a misaligned paraxial ABCD optical system is developed. The study of the fundamental Gaussian beam through the considered system is derived as a particular case. Yet, the propagation of the Li's Flattened-Gaussian and the pure Gaussian beams through an unapertured misaligned optical system or this last with a misaligned circular aperture or a misaligned black screen cases are deduced here, independently, also as special cases of the misaligned annular apertured misaligned optical system case which considered the generalized one. So, other particular cases concerning the Li's Flattened-Gaussian beam and pure Gaussian beam travelling an unapertured misaligned (aligned) optical system or coupled with misaligned (aligned) annular aperture, circular aperture or black screen can be derived in the investigation. Some numerical simulations are also performed in this work.
Based on Huygens-Fresnel diffraction integral, by means of the expansion of a hard-edged aperture... more Based on Huygens-Fresnel diffraction integral, by means of the expansion of a hard-edged aperture function into a finite sum of complex Gaussian function and with help of the Rytov approximation, an approximate analytical expression of modified Bessel modulated Gaussian beam with quadrature radial dependence (MQBG beam) passing through an apertured turbulent atmosphere is derived in this work. The influence of some factors, including incident beam parameters, size aperture and turbulence parameters on propagation characteristics are numerically analyzed in this study.
By using the Collins integral formula and the expanding of a hard aperture function into a finite... more By using the Collins integral formula and the expanding of a hard aperture function into a finite sum of complex Gaussian terms, a closed-form expression for transformation of Modified Bessel beam of quadratic radial dependence (MQBG beam) by a paraxial ABCD optical system with annular aperture is etablished in this paper. An approximate analytical expression of a diffracted MQBG beam through a circular apertured ABCD optical system is deduced as particular case of the main result. Some numerical simulations are given for studying the response of the laser beam influenced by the propagation distance and the annular aperture sizes.
In this work, using a Spiral Phase Plate (SPP) the superposition of Kummer beams as a novel famil... more In this work, using a Spiral Phase Plate (SPP) the superposition of Kummer beams as a novel family of doughnut beams and solution of Helmholtz equation is generated from Gori's flattened-Gaussian (GFTG) beams. Based on Collins-Huygens integral formula an analytical expression of the produced superposition of Kummer beams through a paraxial optical ABCD is developed. The propagations of GFTG beams through an SPP followed by a free space and by a lens system are numerically investigated respectively. The propagation of GTFG beams through an optical system without SPP and the propagation of Gaussian beams travelling an SPP followed by an optical system are derived as particular cases of our main finding. The propagation of GTFG beams in an SPP, followed by a free space and through a telescope, is also given, in the work, as examples of numerical simulations.
Using Collins-Huygens integral formula, a method for generation of Superposition of Kummer (SK) b... more Using Collins-Huygens integral formula, a method for generation of Superposition of Kummer (SK) beams as new kind of doughnut beams is demonstrated. The method is based on the conversion of phase of the non-doughnut Li's flattened Gaussian (LFTG) beams, which are the incident beams, by a Spiral Phase Plate (SPP) system. It is shown that doughnut beams can be produced from non-doughnut beams. Mathematical approaches of propagation properties of generated laser light are established for integer and fractional topological charges of SPP. The above results concerning the formation of Kummer beams by an SPP, also about propagation of LFTG and fundamental Gaussian beams through an ABCD optical system without SPP and other results are deduced from ours as particular cases when the topological charges are integer. Some numerical simulations of propagation of novel beams in free space (in the cases of an integer (χ=l=1) and a fractional topological charge (χ≠1)) and through a fractional Fourier transform (FRFT) system (for l=1) are also performed in this work.
Using the generalized diffraction integral formula for paraxial misaligned optical systems, an an... more Using the generalized diffraction integral formula for paraxial misaligned optical systems, an analytical propagation expression is derived for the flattened Mathieu Gauss beam (FTMG) passing through any misaligned optical system. The propagation of: Mathieu Gauss (MG), Mathieu, Flattened Bessel-Gauss, Bessel-Gauss, pure Bessel, flat-toped Gauss and Gaussian beams through any misaligned or aligned system are deduced from our main result as particular cases. Based on derived principle formula, the propagation properties of FTMG beam traveling through a misaligned circular thin lens are illustrated analytically and numerically. The results show that the shape of the out-put beam from the misaligned system remains unchanged and similar to that of the incident one. However, the center of the beam is shifted and its deviation is related to the misalignment parameters.
The formalism of generalized diffraction integral for paraxial misaligned optical systems is used... more The formalism of generalized diffraction integral for paraxial misaligned optical systems is used to investigate the propagation of the Modified Bessel-Gaussian (MBG) beam through a misaligned thin lens. The properties of the propagation of MBG beam traveling through this misaligned ABCD optical system are discussed. A special case of misaligned circular thin lens is illustrated analytically and numerically. The shape of the MBG beam at the exit of the misaligned optical system is unchanged; however the center of the beam is shifted from the propagation axis in correlated manner with the design parameters of the optical system.
Characteristics of the propagation of Li's flattened Mathieu-Gaussian (FTMG) beams traveling a fr... more Characteristics of the propagation of Li's flattened Mathieu-Gaussian (FTMG) beams traveling a free paraxial ABCD optical system with: annular aperture, circular aperture and black screen are studied theoretically in this work, respectively. By means of expanding of a hard aperture function into a finite sum of complex Gaussian functions, and based on the Collins integral formula, a main approximate analytical expression of propagation of FTMG beams through an annular apertured ABCD optical system is developed. From this general result, we investigate the propagation of FTMG beams through an ABCD optical system with a circular aperture system, a circular black screen and unapertured optical system as particular cases. The actual work generalizes also the above studies concerning the Mathieu, Mathieu-Gauss, flattened Bessel, Bessel-Gauss, pure Bessel, flattened Gaussian, and pure Gaussian beams passing through the considered optical systems. Some numerical calculations of the cases of propagation of FTMG through an annular apertured ABCD optical system and with a circular black are given and discussed in the present study.
Based on the generalized Collins integral formula and using the expansion of the hard aperture fu... more Based on the generalized Collins integral formula and using the expansion of the hard aperture function into a finite sum of complex Gaussian functions, an approximate analytical expression of Mathieu-Gauss (MG) beams passing through a paraxial ABCD optical system with an annular aperture is derived. From this general analytical expression, the traveling of MG beams through free space, circular aperture and circular black screen are deduced as particular cases. Using the theoretical expressions developed in this work, some numerical simulations of normalized intensity distribution of MG beams are investigated.
We investigate the propagation characteristics of truncated Bessel-modulated Gaussian beams in tu... more We investigate the propagation characteristics of truncated Bessel-modulated Gaussian beams in turbulence. Based on the extended Huygens-Fresnel integral formula in the paraxial approximation, an approximate analytical axial intensity of the considered beams family is formulated. By using our theoretical results, numerical examples are given to illustrate the axial intensity distribution in turbulence which is compiled numerically for different beam parameters and for different values of the turbulence strength. Our result is very accurate in the far field, because the use of the paraxial approximation and by using the expression of the hard-edge aperture function expanded as a sum of finite-term complex Gaussian function.
An exact analytical solution for the scattering of a non-diffracting beams by a rigid sphere plac... more An exact analytical solution for the scattering of a non-diffracting beams by a rigid sphere placed on their axis is derived. From our generalized formula, the form functions for scattering by X waves and helicoidal beams are deduced as particular cases. Some examples are given for X waves to show the effect of varying the scattering angle θ . The scattering field is investigated also versus the wavenumberradius product (ka) as well as the polar angle ϕ and half-cone angle β .
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Papers by EZ-ZARIY LAHCEN
passing through an apertured misaligned optical system is developed, in this paper, by using the
generalized Huygens-Fresnel diffraction integral and by means of expansion of the hard-edged aperture
into a finite sum of complex-Gaussian functions. Some particular cases are deduced from the main
theoretical formalism of our work. The propagation characteristics of the considered beam through an
apertured aligned and misaligned circular thin lens are investigated numerically. 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.
Keywords: Generalized Huygens-Fresnel diffraction integral; Airy-Hermite-Gaussian beam; Aligned optical system; Misalignment;
Circular aperture; Circular thin lens.
passing through an apertured misaligned optical system is developed, in this paper, by using the
generalized Huygens-Fresnel diffraction integral and by means of expansion of the hard-edged aperture
into a finite sum of complex-Gaussian functions. Some particular cases are deduced from the main
theoretical formalism of our work. The propagation characteristics of the considered beam through an
apertured aligned and misaligned circular thin lens are investigated numerically. 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.
Keywords: Generalized Huygens-Fresnel diffraction integral; Airy-Hermite-Gaussian beam; Aligned optical system; Misalignment;
Circular aperture; Circular thin lens.