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Research paper thumbnail of Electromagnetic tensors

The electromagnetic potential A i is a quadrivector in Minkowski space-time x k and its gradient ... more The electromagnetic potential A i is a quadrivector in Minkowski space-time x k and its gradient a i k is a tensor of rank two whose elements are the sixteen partial derivatives ∂A i /∂x k. We study in this article the properties of a family of tensors resulting from [a i k ]. We first introduce the covariant tensor [a ki ]. Four initial tensors are obtained by separating [a i k ] on the one hand, and [a ki ] on the other hand into their symmetric and antisymmetric parts. These are (s i k , [f i k ], [S ki ], [F ki ]). As the lowering-raising index operations and symmetrization-antisymmetrization operations do not commute, these four tensors are different. We associate a Lagrangian density L to the determinant of [a i k ] which is invariant in an operation of symmetry of the Poincaré group. In the first part of the article, we show that there is a particular coordinate system where the scalar potential obeys the Hemholtz equation. The solutions allow to describe the "electromagnetic particles", characterized by three quantum numbers n, and m. We give the tensors corresponding to the first five solutions. They describe energy and electric charge distributions. The condition of existence of these particles is related to a property of the electron described in the Wheeler-Feynman's absorber theory. In a second part, we first check that [F ki ] is the usual electromagnetic tensor whose components are the electric and magnetic fields. We prove that Maxwell's equations are obtained by applying the principle of least action to the 4-potential endowed with L. The source terms (ρ and − → j) are expressed in terms of the components of [S ki ]. The results obtained are covariant. The formulation of these tensors being independant of scale, they unify the human and the electron scales, giving a new way to understand elementary particles.

Research paper thumbnail of Tenseurs Electromagnétiques

Research paper thumbnail of Particules Electromagnetiques

The covariant derivative of the 4-components electromagnetic potential in a flat Minkowski space-... more The covariant derivative of the 4-components electromagnetic potential in a flat Minkowski space-time is split into its antisymmetric and symmetric parts. While the former is well known to describe the electromagnetic field, we show that the latter describes the associated particles. When symmetry principles are applied to the invariants in operations of the Poincaré group, one finds equations which describe the structure of the particles. Both parts of the tensor unify the concept of matter-wave duality. Charge and mass are shown to be associated to the potential.La dérivée covariante du potentiel électromagnétique à 4 composantes dans l'espace plat de Minkowski est séparée en ses deux parties antisymmétrique et symmétrique. Alors que la première représente le champ électromagnétique, nous montrons que la seconde décrit les particules associées. Quand les principes de symétrie sont appliqués aux invariants dans une opération du groupe de Poincaré, on trouve les équations permet...

Research paper thumbnail of Theory of the µ and the τ structures

Research paper thumbnail of Potential theory, Maxwell Equations and the Lorentz Force

We show that Maxwell equations and the Lorentz force can be expressed in terms of the spatial and... more We show that Maxwell equations and the Lorentz force can be expressed in terms of the spatial and temporal derivatives of the electromagnetic potential. 1 Introduction. Maxwell equations are the basis of electromagnetism. They express the relationships between electromagnetic fields and their sources. There are several ways to write these equations [1] but all formulations distinguish fields from sources. The fields can be expressed as functions of the spatial and temporal derivatives of the electromagnetic potential. Sources are electrical charges and currents. The first aim of this study is to show that sources can also be expressed in terms of these derivatives. In other words, we show that Maxwell equations can be expressed with the potential derivatives only. The second objective is to show that the Lorentz force is obtained as a direct application of these new expressions for sources. The theory starts with a 4-potential A i at each event M in Minkowski's spacetime. M is defined by its coordinates x k. The 16 partial derivatives ∂A i /∂x k are the components of the gradient tensor D(A i). The antisymmetric part of D(A i) is the usual electromagnetic tensor. The Lagrangian density L at M is proportional to the determinant of D(A i). It allows the calculation of the induction tensor. The antisymmetric part of this tensor contains the usual electromagnetic induction. The symmetric part contains the sources. An application of Euler-Lagrange equations gives the second pair of Maxwell equations. In the last section we use the new expressions for the charge and current densities to deduce the Lorentz force. The theory is very simple and does not need any specific form of the Lagrangian which confers a wide generality to it. Associated to this subject is the question of the preeminence of potentials and fields which arises in standard textbooks on classical electromagnetism : In ref.[2] , electric and magnetic fields are deduced from a 4-potential. In refs.[3, 4] , the 4-potential is deduced from the fields. In quantum theory, it is the potential which is more fundamental as illustrated by the Aharonov-Bohm effect[5, 6]. Recently, two articles[7, 8] have been published where the authors develop Richard Feynman's idea of introducing potentials before fields[9]. The theory which is presented here brings the proof that the potential is also more fundamental than the fields in classical electromagnetism. 2 Electromagnetic tensors. 2.1 Electromagnetic potential and its derivatives. The aim of this section is to describe the notations and to give the relations between the components of the different tensors and the pseudovectors which appear in Maxwell equations. From the beginning, it is important to stress the fact that we are dealing with quantities, scalars, vectors and tensors having a dimensioned physical reality. These quantities can be represented in direct (or real space), or in inverse (or reciprocal) space. A vector can be represented by its covariant or contravariant components and both quantities are related by the metric tensor. A 2x2 tensor has 4 representations following those of its constitutive vectors.

Research paper thumbnail of Non degenerate relaxation frequencies in VCSELs: proposal of a new theoretical model

Technical Digest. Summaries of papers presented at the Conference on Lasers and Electro-Optics. Postconference Technical Digest (IEEE Cat. No.01CH37170), 2001

ABSTRACT

Research paper thumbnail of Polarization switching of an er :yb :cr: phosphate glass microchip laser

Annales Des Télécommunications, 1997

Abstract We show that the light of a commercially available Er: Yb: Cr.-phosphate glass microchip... more Abstract We show that the light of a commercially available Er: Yb: Cr.-phosphate glass microchip laser end-pumped at 977 nm by an InGaAs laser diode can be switched between two linear polarization eigenstates when it is submitted to a polarized optical feedback. ...

Research paper thumbnail of Spectral properties in laser patterns

Quantum and Semiclassical Optics: Journal of the European Optical Society Part B, 1998

A description of multimode lasers is presented in the frequency domain where the laser Airy funct... more A description of multimode lasers is presented in the frequency domain where the laser Airy function plays a central role. The laser linewidths are calculated in the case of relaxed modes.

Research paper thumbnail of Bipolarized Nd-doped fibre laser: effects of localized and distributed dichroisms

Quantum and Semiclassical Optics: Journal of the European Optical Society Part B, 1998

We present a static analysis of a general model for a bipolarized Nd-doped fibre laser including ... more We present a static analysis of a general model for a bipolarized Nd-doped fibre laser including the slow longitudinal variations of the dynamical variables. Approximate analytical results are derived for the thresholds and the output intensities. A good agreement with the numerical simulations is achieved over a large range of parameters. The combined effects of pump-induced, distributed and localized dichroisms

Research paper thumbnail of Stability of polarized modes in gas lasers

Quantum and Semiclassical Optics: Journal of the European Optical Society Part B, 1998

This tutorial paper is based on lectures delivered by G M St?phan at PELS'97. The idea was to... more This tutorial paper is based on lectures delivered by G M St?phan at PELS'97. The idea was to tell the students about the methodology used by the authors to understand and to quantitatively compute the flips observed between two polarization states in a single-frequency He-Ne laser. The study is limited to this phenomenon. Contents are: introduction, quasi-isotropic lasers and polarization flips; the passive Fabry-Perot interferometer, an illustrative example; a vectorial description of a two energy-level system with magnetic degeneracy; two useful tools: adiabatic elimination of fast variables and use of Lyapunov exponents to study the stability of a solution; medium polarization in the adiabatic approximation; the laser equations for the field. These contents are organized for people coming into the field: rather than being too general, we have preferred to focus on limited and well defined examples from which other cases can be studied.

Research paper thumbnail of Bipolarized Nd-doped fibre laser: isomorphism between the two-site ion distribution and the truncated multi-site approaches

Quantum and Semiclassical Optics: Journal of the European Optical Society Part B, 1998

This paper presents a synthetic formulation of the dynamical equations of a fibre laser in which ... more This paper presents a synthetic formulation of the dynamical equations of a fibre laser in which vectorial properties of the field as well as longitudinal aspects are preserved. The model is shown to reduce to previous and more restricted models which thus become linked together through a simple linear transformation we call isomorphism.

Research paper thumbnail of Line shapes of a « sensitive » laser

Journal de Physique Lettres, 1984

2014 Les inhomogénéités de saturation et de population sont à l'origine d'effets de lentille dans... more 2014 Les inhomogénéités de saturation et de population sont à l'origine d'effets de lentille dans les lasers à gaz. Ceci a pour conséquence de donner un Lamb dip dissymétrique. En général, l'une des inhomogénéités est prédominante et c'est elle qui fixe le sens de cette dissymétrie. Cependant, les paramètres géométriques du laser modifient les inhomogénéités et il peut arriver qu'elles soient du même ordre de grandeur et de signe opposé ; elles entrent alors en compétition et la forme de raie peut varier beaucoup pour un faible changement d'un paramètre. Nous appelons un tel laser un laser « sensible ». Dans cet article, nous donnons l'exemple d'un laser quasi hémisphérique à He-Ne dont la longueur (57 cm) est presque égale au rayon de courbure du miroir concave (60 cm). Nous montrons expérimentalement que le sens de la dissymétrie peut s'inverser avec le courant de décharge et que la forme de raie dépend aussi du diamètre du diaphragme. Ceci est révélateur des deux effets en compétition. Nous calculons théoriquement les formes de raies en utilisant des méthodes précédemment mises au point et nous constatons un excellent accord qualitatif avec l'expérience. Abstract. 2014 Inhomogeneities of population and saturation are responsible for lens effects in gas lasers. They give rise to an asymmetric Lamb dip. Generally, one of these inhomogeneities is predominant and thus determines the sense of the asymmetry. However, since the two kinds of inhomo

Research paper thumbnail of Vectorial bistability in a quasi-isotropic laser

Journal de Physique, 1986

2014 Nous montrons théoriquement et expérimentalement que le domaine de bistabilité vectorielle o... more 2014 Nous montrons théoriquement et expérimentalement que le domaine de bistabilité vectorielle observable lors du basculement de polarisation dans un laser quasiisotrope diminue quand l'intensité de la lumière augmente. Abstract.2014 The domain of vectorial bistability observable in the polarization flip of a quasi-isotropic laser is theoretically and experimentally shown to decrease when light intensity increases.

Research paper thumbnail of �tudes de l'effet laser dans un micro r�sonateur � modes de Galerie en verre ZBLALiP dop� Erbium

Research paper thumbnail of TUTORIAL: Stability of polarized modes in gas lasers

J Opt B Quantum Semicl Opt, 1998

Research paper thumbnail of Injected semiconductor laser

SPIE Proceedings, 2002

ABSTRACT

Research paper thumbnail of Anisotropy of optical transition in direct bandgap AIII-BV bulk semiconductor

Conference Digest. 2000 International Quantum Electronics Conference (Cat. No.00TH8504)

ABSTRACT

Research paper thumbnail of Experimental bifurcation diagrams of a dual-wavelength erbium-doped fiber laser

Conference Digest. 2000 International Quantum Electronics Conference (Cat. No.00TH8504)

w d c (IDIL), ENSSAT, lrrboratoirr d'optmnique, 6 m u p Kcrampant, 22305 LANNION, France; Emnil :... more w d c (IDIL), ENSSAT, lrrboratoirr d'optmnique, 6 m u p Kcrampant, 22305 LANNION, France; Emnil : p"l.bemard~enasat.fr An erbium-doped fiber laser exhibits self-pulsing behavior [I] when the ion-pair coneentratinn 5 is-large enough (z > 5%). Far high concentration (z z lo%), thQ behavior

Research paper thumbnail of Surface-Emitting Lasers Coupled to Extended Cavities with a Phase Plate: Dependence on the Orientation of the Plate Axes

CLEO/Europe Conference on Lasers and Electro-Optics

Research paper thumbnail of Proprietes optiques et spectre electronique du MgF2 et du CaF2 de 10 a 48 eV

Solid State Communications, 1968

Research paper thumbnail of Electromagnetic tensors

The electromagnetic potential A i is a quadrivector in Minkowski space-time x k and its gradient ... more The electromagnetic potential A i is a quadrivector in Minkowski space-time x k and its gradient a i k is a tensor of rank two whose elements are the sixteen partial derivatives ∂A i /∂x k. We study in this article the properties of a family of tensors resulting from [a i k ]. We first introduce the covariant tensor [a ki ]. Four initial tensors are obtained by separating [a i k ] on the one hand, and [a ki ] on the other hand into their symmetric and antisymmetric parts. These are (s i k , [f i k ], [S ki ], [F ki ]). As the lowering-raising index operations and symmetrization-antisymmetrization operations do not commute, these four tensors are different. We associate a Lagrangian density L to the determinant of [a i k ] which is invariant in an operation of symmetry of the Poincaré group. In the first part of the article, we show that there is a particular coordinate system where the scalar potential obeys the Hemholtz equation. The solutions allow to describe the "electromagnetic particles", characterized by three quantum numbers n, and m. We give the tensors corresponding to the first five solutions. They describe energy and electric charge distributions. The condition of existence of these particles is related to a property of the electron described in the Wheeler-Feynman's absorber theory. In a second part, we first check that [F ki ] is the usual electromagnetic tensor whose components are the electric and magnetic fields. We prove that Maxwell's equations are obtained by applying the principle of least action to the 4-potential endowed with L. The source terms (ρ and − → j) are expressed in terms of the components of [S ki ]. The results obtained are covariant. The formulation of these tensors being independant of scale, they unify the human and the electron scales, giving a new way to understand elementary particles.

Research paper thumbnail of Tenseurs Electromagnétiques

Research paper thumbnail of Particules Electromagnetiques

The covariant derivative of the 4-components electromagnetic potential in a flat Minkowski space-... more The covariant derivative of the 4-components electromagnetic potential in a flat Minkowski space-time is split into its antisymmetric and symmetric parts. While the former is well known to describe the electromagnetic field, we show that the latter describes the associated particles. When symmetry principles are applied to the invariants in operations of the Poincaré group, one finds equations which describe the structure of the particles. Both parts of the tensor unify the concept of matter-wave duality. Charge and mass are shown to be associated to the potential.La dérivée covariante du potentiel électromagnétique à 4 composantes dans l'espace plat de Minkowski est séparée en ses deux parties antisymmétrique et symmétrique. Alors que la première représente le champ électromagnétique, nous montrons que la seconde décrit les particules associées. Quand les principes de symétrie sont appliqués aux invariants dans une opération du groupe de Poincaré, on trouve les équations permet...

Research paper thumbnail of Theory of the µ and the τ structures

Research paper thumbnail of Potential theory, Maxwell Equations and the Lorentz Force

We show that Maxwell equations and the Lorentz force can be expressed in terms of the spatial and... more We show that Maxwell equations and the Lorentz force can be expressed in terms of the spatial and temporal derivatives of the electromagnetic potential. 1 Introduction. Maxwell equations are the basis of electromagnetism. They express the relationships between electromagnetic fields and their sources. There are several ways to write these equations [1] but all formulations distinguish fields from sources. The fields can be expressed as functions of the spatial and temporal derivatives of the electromagnetic potential. Sources are electrical charges and currents. The first aim of this study is to show that sources can also be expressed in terms of these derivatives. In other words, we show that Maxwell equations can be expressed with the potential derivatives only. The second objective is to show that the Lorentz force is obtained as a direct application of these new expressions for sources. The theory starts with a 4-potential A i at each event M in Minkowski's spacetime. M is defined by its coordinates x k. The 16 partial derivatives ∂A i /∂x k are the components of the gradient tensor D(A i). The antisymmetric part of D(A i) is the usual electromagnetic tensor. The Lagrangian density L at M is proportional to the determinant of D(A i). It allows the calculation of the induction tensor. The antisymmetric part of this tensor contains the usual electromagnetic induction. The symmetric part contains the sources. An application of Euler-Lagrange equations gives the second pair of Maxwell equations. In the last section we use the new expressions for the charge and current densities to deduce the Lorentz force. The theory is very simple and does not need any specific form of the Lagrangian which confers a wide generality to it. Associated to this subject is the question of the preeminence of potentials and fields which arises in standard textbooks on classical electromagnetism : In ref.[2] , electric and magnetic fields are deduced from a 4-potential. In refs.[3, 4] , the 4-potential is deduced from the fields. In quantum theory, it is the potential which is more fundamental as illustrated by the Aharonov-Bohm effect[5, 6]. Recently, two articles[7, 8] have been published where the authors develop Richard Feynman's idea of introducing potentials before fields[9]. The theory which is presented here brings the proof that the potential is also more fundamental than the fields in classical electromagnetism. 2 Electromagnetic tensors. 2.1 Electromagnetic potential and its derivatives. The aim of this section is to describe the notations and to give the relations between the components of the different tensors and the pseudovectors which appear in Maxwell equations. From the beginning, it is important to stress the fact that we are dealing with quantities, scalars, vectors and tensors having a dimensioned physical reality. These quantities can be represented in direct (or real space), or in inverse (or reciprocal) space. A vector can be represented by its covariant or contravariant components and both quantities are related by the metric tensor. A 2x2 tensor has 4 representations following those of its constitutive vectors.

Research paper thumbnail of Non degenerate relaxation frequencies in VCSELs: proposal of a new theoretical model

Technical Digest. Summaries of papers presented at the Conference on Lasers and Electro-Optics. Postconference Technical Digest (IEEE Cat. No.01CH37170), 2001

ABSTRACT

Research paper thumbnail of Polarization switching of an er :yb :cr: phosphate glass microchip laser

Annales Des Télécommunications, 1997

Abstract We show that the light of a commercially available Er: Yb: Cr.-phosphate glass microchip... more Abstract We show that the light of a commercially available Er: Yb: Cr.-phosphate glass microchip laser end-pumped at 977 nm by an InGaAs laser diode can be switched between two linear polarization eigenstates when it is submitted to a polarized optical feedback. ...

Research paper thumbnail of Spectral properties in laser patterns

Quantum and Semiclassical Optics: Journal of the European Optical Society Part B, 1998

A description of multimode lasers is presented in the frequency domain where the laser Airy funct... more A description of multimode lasers is presented in the frequency domain where the laser Airy function plays a central role. The laser linewidths are calculated in the case of relaxed modes.

Research paper thumbnail of Bipolarized Nd-doped fibre laser: effects of localized and distributed dichroisms

Quantum and Semiclassical Optics: Journal of the European Optical Society Part B, 1998

We present a static analysis of a general model for a bipolarized Nd-doped fibre laser including ... more We present a static analysis of a general model for a bipolarized Nd-doped fibre laser including the slow longitudinal variations of the dynamical variables. Approximate analytical results are derived for the thresholds and the output intensities. A good agreement with the numerical simulations is achieved over a large range of parameters. The combined effects of pump-induced, distributed and localized dichroisms

Research paper thumbnail of Stability of polarized modes in gas lasers

Quantum and Semiclassical Optics: Journal of the European Optical Society Part B, 1998

This tutorial paper is based on lectures delivered by G M St?phan at PELS'97. The idea was to... more This tutorial paper is based on lectures delivered by G M St?phan at PELS'97. The idea was to tell the students about the methodology used by the authors to understand and to quantitatively compute the flips observed between two polarization states in a single-frequency He-Ne laser. The study is limited to this phenomenon. Contents are: introduction, quasi-isotropic lasers and polarization flips; the passive Fabry-Perot interferometer, an illustrative example; a vectorial description of a two energy-level system with magnetic degeneracy; two useful tools: adiabatic elimination of fast variables and use of Lyapunov exponents to study the stability of a solution; medium polarization in the adiabatic approximation; the laser equations for the field. These contents are organized for people coming into the field: rather than being too general, we have preferred to focus on limited and well defined examples from which other cases can be studied.

Research paper thumbnail of Bipolarized Nd-doped fibre laser: isomorphism between the two-site ion distribution and the truncated multi-site approaches

Quantum and Semiclassical Optics: Journal of the European Optical Society Part B, 1998

This paper presents a synthetic formulation of the dynamical equations of a fibre laser in which ... more This paper presents a synthetic formulation of the dynamical equations of a fibre laser in which vectorial properties of the field as well as longitudinal aspects are preserved. The model is shown to reduce to previous and more restricted models which thus become linked together through a simple linear transformation we call isomorphism.

Research paper thumbnail of Line shapes of a « sensitive » laser

Journal de Physique Lettres, 1984

2014 Les inhomogénéités de saturation et de population sont à l'origine d'effets de lentille dans... more 2014 Les inhomogénéités de saturation et de population sont à l'origine d'effets de lentille dans les lasers à gaz. Ceci a pour conséquence de donner un Lamb dip dissymétrique. En général, l'une des inhomogénéités est prédominante et c'est elle qui fixe le sens de cette dissymétrie. Cependant, les paramètres géométriques du laser modifient les inhomogénéités et il peut arriver qu'elles soient du même ordre de grandeur et de signe opposé ; elles entrent alors en compétition et la forme de raie peut varier beaucoup pour un faible changement d'un paramètre. Nous appelons un tel laser un laser « sensible ». Dans cet article, nous donnons l'exemple d'un laser quasi hémisphérique à He-Ne dont la longueur (57 cm) est presque égale au rayon de courbure du miroir concave (60 cm). Nous montrons expérimentalement que le sens de la dissymétrie peut s'inverser avec le courant de décharge et que la forme de raie dépend aussi du diamètre du diaphragme. Ceci est révélateur des deux effets en compétition. Nous calculons théoriquement les formes de raies en utilisant des méthodes précédemment mises au point et nous constatons un excellent accord qualitatif avec l'expérience. Abstract. 2014 Inhomogeneities of population and saturation are responsible for lens effects in gas lasers. They give rise to an asymmetric Lamb dip. Generally, one of these inhomogeneities is predominant and thus determines the sense of the asymmetry. However, since the two kinds of inhomo

Research paper thumbnail of Vectorial bistability in a quasi-isotropic laser

Journal de Physique, 1986

2014 Nous montrons théoriquement et expérimentalement que le domaine de bistabilité vectorielle o... more 2014 Nous montrons théoriquement et expérimentalement que le domaine de bistabilité vectorielle observable lors du basculement de polarisation dans un laser quasiisotrope diminue quand l'intensité de la lumière augmente. Abstract.2014 The domain of vectorial bistability observable in the polarization flip of a quasi-isotropic laser is theoretically and experimentally shown to decrease when light intensity increases.

Research paper thumbnail of �tudes de l'effet laser dans un micro r�sonateur � modes de Galerie en verre ZBLALiP dop� Erbium

Research paper thumbnail of TUTORIAL: Stability of polarized modes in gas lasers

J Opt B Quantum Semicl Opt, 1998

Research paper thumbnail of Injected semiconductor laser

SPIE Proceedings, 2002

ABSTRACT

Research paper thumbnail of Anisotropy of optical transition in direct bandgap AIII-BV bulk semiconductor

Conference Digest. 2000 International Quantum Electronics Conference (Cat. No.00TH8504)

ABSTRACT

Research paper thumbnail of Experimental bifurcation diagrams of a dual-wavelength erbium-doped fiber laser

Conference Digest. 2000 International Quantum Electronics Conference (Cat. No.00TH8504)

w d c (IDIL), ENSSAT, lrrboratoirr d'optmnique, 6 m u p Kcrampant, 22305 LANNION, France; Emnil :... more w d c (IDIL), ENSSAT, lrrboratoirr d'optmnique, 6 m u p Kcrampant, 22305 LANNION, France; Emnil : p"l.bemard~enasat.fr An erbium-doped fiber laser exhibits self-pulsing behavior [I] when the ion-pair coneentratinn 5 is-large enough (z > 5%). Far high concentration (z z lo%), thQ behavior

Research paper thumbnail of Surface-Emitting Lasers Coupled to Extended Cavities with a Phase Plate: Dependence on the Orientation of the Plate Axes

CLEO/Europe Conference on Lasers and Electro-Optics

Research paper thumbnail of Proprietes optiques et spectre electronique du MgF2 et du CaF2 de 10 a 48 eV

Solid State Communications, 1968