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In this paper, an exact analytical propagation formula of Finite Olver-Gaussian Beams (FOGBs) passing through a paraxial ABCD optical system is developed and some numerical examples are performed. The propagation properties of the FOGBs through general optical systems characterized by given ABCD matrix are studied on the basis of the generalized Huygens-Fresnel diffraction integral, which permits to show the behavior of this laser beams family and its properties de-pending of the laser parameters. This research is of interest to prove some investigations done in the past by other researchers.

More freshly, Belafhal et al. [

On the other hand, the studies of the diffraction of apertured and unapertured laser beams by optical systems including aligned and misaligned ones and through turbulent media, are very vital to physical optics and propagation properties of the studied beams whatever Gaussian, nondiffracting or quasi-nondiffracting beams. For this purpose, several literature researches are elaborated within the context [

The diffraction features of the Finite Olver beams (FOBs) and the Finite Olver-Gaussian beams (FOGBs) by optical systems have not been studied elsewhere. The current paper is the first one of a series of works that are interested in the treatment of the Olver beams family in aligned or misaligned optical systems, turbulence, crystals and others which are within the interest of our research group. In the actual work, we start the series by a theoretical and numerical examination of the propagation properties of FOGBs through a simple paraxial ABCD optical system. An analytical formula is developed in the coming section using the Collins diffraction integral formula. Some special cases correspond to the Finite Airy-Gaussian beams, FOBs, Olver-Gaussian beams and Airy-Gaussian beams passing through an ABCD optical system are derived from our main finding in Section 3. Several numerical calculations are performed in Section 4 to analyze the action of some parameters on the transverse intensity distribution and shape of the beams class exiting the ABCD optical system. A simple conclusion is Plainfield in Section 5 of the paper.

The propagation of the light beam passing through a paraxial optical system, described by an ABCD matrix, obeys to the generalized Huygens-Fresnel integral, which connects the output electric field

where

In the Cartesian coordinates system, the incident electrical field distribution of Finite Olver-Gaussian beam is given by [

where ^{th}-order Olver function of real x with n = 0; 1; 2; …

with

Introducing Equation (2) and Equation (3) into Equation (1) and recalling the integral formulas [

and after some calculations, the final analytical expression of the output electrical field expression of FOGBs is given by

This is the main result of this paper; it permits us to study the propagation properties of FOGBs through any ABCD optical system. It is to note that the output beam is of the same family as the input beam. This finding is regarded as a generalization of several studies of literature which are derived as particular cases of our investigation.

1) Ordinary Finite Airy-Gaussian Beams through an ABCD Optical System

This case is obtained when

By the use of Equation (7) and under the above condition, the output electrical field of the Finite Airy- Gaussian beam travelling any ABCD optical system reads to

This result is similar to that found by Bandres et al. [

2) Olver-Gaussian Beams through an ABCD Optical System

This case can be obtained when

This beam passes through an ABCD optical system to give another beam of the same family where its form is obtained from Equation (7) by replacing a_{0} by 0 and is expressed as

Yet, if

This result is a particular case, when

3) Finite Olver Beam through an ABCD Optical System

When

Under this condition, the output electrical field of the Finite Olver beams through an ABCD optical system is given by

If

which corresponds to a Finite Airy beam. The output field of this beam through an ABCD is deduced from our principle result established in Equation (7) taking into account the parameters chosen above and it is expressed as

This expression is the same of the principle result of Ref. [

Equation (7) will be simulated numerically in three particular cases: Fractional Fourier Transform (FFT) system, Free space system and Thin Lens system.

1) FOGBs through a FFT System

The transfer matrix corresponding to a FFT system is given by:

with

The intensity distribution of the Finite zeroth-order Olver-Gauss beams is illustrated numerically in

In _{0} (b_{0} = 0 and 1) and Fractional order P on the propagation characteristics of Olver beams through a FFT optical system. The calculations parameters, used in the compilation of these figures, are:

From the plots of

2) FOGBs through a Free Space

The transfer matrix which represent the Free Space of the axial distance z is found to be given as

From Equation (7), one obtains the output field as:

In _{0} = 0.1 for the beam order n = 2 and for two values of ω_{0}: 4 and 12 mm. From this figure one can deduced that the amplitude profile of the intensity distribution expands completely with an increase of the beam spot size with seeing the effect of the Gaussian parameter b_{0}. It can be seen that much as ω_{0} increases as the beam spot becomes wider either with or without Gaussian term. Always, the secondary lobes vanish if the Finite Olver beams are modulated by a Gaussian envelope.

3) FOGBs through a Thin Lens

We consider an optical system formed with a thin lens followed by a free space. The matrix corresponding to the considered optical system is characterised by

In this case, the output field can be written as

In

through thin lenses of focal lengths f = 150 mm, 200 mm and 500 mm, respectively. According to the graphs of these figures, the choice of f acts mostly on the propagation distance of the beam. Also, one remarks that the Gaussian term has a major effect on the elimination of secondary lobes of the beam leaving the thin lens outgoing. The side lobes totally vanish for b_{0} = 1. From the illustrations of these figures, it is provided that as far as the thin lens focal length f increases as much as the spot width of the beam leaving the thin lens deceases. Usually, the oscillations of the beam exiting in the thin lens disappear with a modulation of the Finite Olver beams by a Gaussian transmittance.

The numerical simulations performed above show that the secondary lobes present in the intensity distribution of a FOGBS, at the receiver plane optical system, vanish with the presence of the Gaussian modulation whatever the FOGBs order and whatever the optical system.

In this study, by means of the Collins diffraction integral formula, a general exact analytical expression of the properties of the propagation of a Finite Olver-Gaussian beam through any paraxial complex ABCD optical sys- tem is developed. This formula considered the main finding of the work is applied for the Fractional Fourier transform, the free space and the thin lens as examples of optical systems and three analytical expressions are developed, respectively. The properties of Finite Airy-Gaussian, Finite Olver, Finite Airy, Olver-Gaussian, Airy-Gaussian beams traveling an ABCD optical system are derived as particular cases of our principle investi- gation. In order to examine the impact of some parameters, such as the propagation distance, the fractional dis- tance and the focal length on the propagation characteristics of the FOGBs through the above optical systems, several numerical calculations are performed in the study. The obtained formulae developed in the present work form the basis for the propagation of an important beams family through any paraxial complex ABCD optical system and can be extended in a future study to any medium as misaligned optical systems or turbulent atmos- phere.

SalimaHennani,LahcenEz-Zariy,AbdelmajidBelafhal, (2015) Propagation Properties of Finite Olver-Gaussian Beams Passing through a Paraxial ABCD Optical System. Optics and Photonics Journal,05,273-294. doi: 10.4236/opj.2015.59026