ABSTRACT Space Geodetic techniques based on dual frequency signals have already proven their usef... more ABSTRACT Space Geodetic techniques based on dual frequency signals have already proven their usefulness for the study of the ionosphere and are nowadays widely used by the ionosphere community. The so-called “geometric free” combination of the measurements made on these dual frequency signals allows to retrieve the Total Electron Content which is the integral of the electron concentration along the receiver-to-satellite path from the bottom of the ionosphere (about 50 km) up to the altitude of the satellite considered. This geodetic technique does not give a direct access to the electron concentration which is the most interesting data for ionospherists. Nevertheless, the combination of several techniques at the same location allows to retrieve more detailed information. The paper illustrates the interest of such collocations for ionospheric studies. First, the collocation of GPS (altitude 20 000 km) and DORIS (altitude from 800 to 1336 km) allows to give information about the integrated electron content of the protonosphere (ionised atmosphere above 1 000 km) by simply making the difference between GPS and DORIS electron contents. In addition, the collocation of GPS, DORIS and an ionosonde allows to retrieve the electron concentration profile which could be used to validate ionospheric models. The installation of a DORIS beacon at the Geophysical Centre of Dourbes (Belgium) would allow to take profit of such a collocation.
This work focuses on the interdisciplinary mathematics-physics practices in classroom at the last... more This work focuses on the interdisciplinary mathematics-physics practices in classroom at the last year of secondary school regarding mathematic notions used in the electricity's program. The present study was elaborated following the analysis of the practices of ten future physics teachers during their practical training in schools. Most pupils had difficulties in apply algebraic formulas used in courses of the electrical resistance, power and energy and with graphic representation for defining the characteristics of an electric dipole. The three components of the didactic triangle were considered. Indeed, classification of semiotic registers in the most used textbooks was completed; pupil's difficulties were identified and discussed with physics and mathematics teachers then a focus groups and individual interviews semi-structured with the teachers of the two disciplines were held. The study reveals serious deficiencies in knowledge interdisciplinary hindering the construct...
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.
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 t...
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. K...
Propagation characteristics of finite Airy-Gaussian beams through an apertured misaligned firstor... more Propagation characteristics of finite Airy-Gaussian beams through an apertured misaligned firstorder ABCD optical system are studied. In this work, the generalized Huygens-Fresnel diffraction integral and the expansion of the hard aperture function into a finite sum of complex Gaussian functions are used. The propagation of Airy-Gaussian beam passing through: an unapertured misaligned optical system, an apertured aligned ABCD optical system and an unapertured aligned ABCD optical system are derived here as particular cases of the main finding. Some numerical simulations are performed in the paper. L. Ez-Zariy et al. 326 definitely [4]. The original Airy beam which contains infinite energy is not realizable in practice. However, in 2007, Siviloglou et al. [3] and Siviloglou and Christodoulides [5] have started the first observation of Airy optical beam that presents a finite energy and demonstrates experimentally the unusual features of the new finite Airy beam. In the literature, several methods were used to produce the finite Airy beam, including cubic phase, 3/2 phase only pattern [6]-[9], and three-wave mixing processes in an asymmetric nonlinear photonic crystals [10] [11]. In the past few years, the propagation characteristics of Airy family have been examined widely in free space [12] [13], in fractional Fourier transform and quadratic index medium [13]-[16], in turbulence [17] [18], in a unixial crystals [19] and in other media [20]-[22]. Among of these, in [13], Bandres and Gutierrez-Vega have introduced for the first time, the so-called generalized Airy-Gaussian beam and treated its propagation properties through different complex paraxial optical systems characterized by ABCD matrices. This generalized Airy-Gaussian beam carries a finite energy and can be realized experimentally. The Airy beam devoted by Berry and Balazs [1] and the finite Airy invented and produced by Siviloglou et al. [3] [4] are regarded as special cases of the study of Bandres and Gutiérrez-Vega [13].
ABSTRACT This paper comments on a recently published work dealing with: “Generation of dark hollo... more ABSTRACT 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. Keywords: Fractional radial Hilbert transform system; Integer or non-integer topological charge; Apertured spiral phase plate system
Based on the truncated second-order moments definition, the generalized M²-factor of the hypergeo... more Based on the truncated second-order moments definition, the generalized M²-factor of the hypergeometric-Gaussian type-II beams in the cylindrical coordinate system through a hard-edged circular aperture is derived. The effects of the beam truncation parameter δ and the beam orders on the beam propagation factor have been investigated. The theoretical results show that the generalized M²factor can be simplified to four special cases, among of them, the truncated and untruncated Gaussian beams. The power fraction of the hypergeometric-Gaussian type-II beams is calculated analytically and illustrated numerically.
ABSTRACT Space Geodetic techniques based on dual frequency signals have already proven their usef... more ABSTRACT Space Geodetic techniques based on dual frequency signals have already proven their usefulness for the study of the ionosphere and are nowadays widely used by the ionosphere community. The so-called “geometric free” combination of the measurements made on these dual frequency signals allows to retrieve the Total Electron Content which is the integral of the electron concentration along the receiver-to-satellite path from the bottom of the ionosphere (about 50 km) up to the altitude of the satellite considered. This geodetic technique does not give a direct access to the electron concentration which is the most interesting data for ionospherists. Nevertheless, the combination of several techniques at the same location allows to retrieve more detailed information. The paper illustrates the interest of such collocations for ionospheric studies. First, the collocation of GPS (altitude 20 000 km) and DORIS (altitude from 800 to 1336 km) allows to give information about the integrated electron content of the protonosphere (ionised atmosphere above 1 000 km) by simply making the difference between GPS and DORIS electron contents. In addition, the collocation of GPS, DORIS and an ionosonde allows to retrieve the electron concentration profile which could be used to validate ionospheric models. The installation of a DORIS beacon at the Geophysical Centre of Dourbes (Belgium) would allow to take profit of such a collocation.
This work focuses on the interdisciplinary mathematics-physics practices in classroom at the last... more This work focuses on the interdisciplinary mathematics-physics practices in classroom at the last year of secondary school regarding mathematic notions used in the electricity's program. The present study was elaborated following the analysis of the practices of ten future physics teachers during their practical training in schools. Most pupils had difficulties in apply algebraic formulas used in courses of the electrical resistance, power and energy and with graphic representation for defining the characteristics of an electric dipole. The three components of the didactic triangle were considered. Indeed, classification of semiotic registers in the most used textbooks was completed; pupil's difficulties were identified and discussed with physics and mathematics teachers then a focus groups and individual interviews semi-structured with the teachers of the two disciplines were held. The study reveals serious deficiencies in knowledge interdisciplinary hindering the construct...
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.
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 t...
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. K...
Propagation characteristics of finite Airy-Gaussian beams through an apertured misaligned firstor... more Propagation characteristics of finite Airy-Gaussian beams through an apertured misaligned firstorder ABCD optical system are studied. In this work, the generalized Huygens-Fresnel diffraction integral and the expansion of the hard aperture function into a finite sum of complex Gaussian functions are used. The propagation of Airy-Gaussian beam passing through: an unapertured misaligned optical system, an apertured aligned ABCD optical system and an unapertured aligned ABCD optical system are derived here as particular cases of the main finding. Some numerical simulations are performed in the paper. L. Ez-Zariy et al. 326 definitely [4]. The original Airy beam which contains infinite energy is not realizable in practice. However, in 2007, Siviloglou et al. [3] and Siviloglou and Christodoulides [5] have started the first observation of Airy optical beam that presents a finite energy and demonstrates experimentally the unusual features of the new finite Airy beam. In the literature, several methods were used to produce the finite Airy beam, including cubic phase, 3/2 phase only pattern [6]-[9], and three-wave mixing processes in an asymmetric nonlinear photonic crystals [10] [11]. In the past few years, the propagation characteristics of Airy family have been examined widely in free space [12] [13], in fractional Fourier transform and quadratic index medium [13]-[16], in turbulence [17] [18], in a unixial crystals [19] and in other media [20]-[22]. Among of these, in [13], Bandres and Gutierrez-Vega have introduced for the first time, the so-called generalized Airy-Gaussian beam and treated its propagation properties through different complex paraxial optical systems characterized by ABCD matrices. This generalized Airy-Gaussian beam carries a finite energy and can be realized experimentally. The Airy beam devoted by Berry and Balazs [1] and the finite Airy invented and produced by Siviloglou et al. [3] [4] are regarded as special cases of the study of Bandres and Gutiérrez-Vega [13].
ABSTRACT This paper comments on a recently published work dealing with: “Generation of dark hollo... more ABSTRACT 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. Keywords: Fractional radial Hilbert transform system; Integer or non-integer topological charge; Apertured spiral phase plate system
Based on the truncated second-order moments definition, the generalized M²-factor of the hypergeo... more Based on the truncated second-order moments definition, the generalized M²-factor of the hypergeometric-Gaussian type-II beams in the cylindrical coordinate system through a hard-edged circular aperture is derived. The effects of the beam truncation parameter δ and the beam orders on the beam propagation factor have been investigated. The theoretical results show that the generalized M²factor can be simplified to four special cases, among of them, the truncated and untruncated Gaussian beams. The power fraction of the hypergeometric-Gaussian type-II beams is calculated analytically and illustrated numerically.
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