Pierre PELLAT-FINET - Academia.edu (original) (raw)

Papers by Pierre PELLAT-FINET

$Research paper thumbnail of Effect of Diffraction on Wigner Distributions of Optical Fields and how to Use It in Optical Resonator Theory. III -Ray Tracing in Resonators$

HAL (Le Centre pour la Communication Scientifique Directe), Apr 2, 2022

The third part of the paper is devoted to ray tracing in optical resonators. The employed method ... more The third part of the paper is devoted to ray tracing in optical resonators. The employed method for dealing with the issue uses the elliptical or hyperbolic rotations that Wigner distributions associated with optical fields undergo during propagation from one spherical mirror of a resonator to the other. It is illustrated by various examples concerning meridional or skew rays, reentrant in stable resonators or propagating in unstable resonators. A classification of optical resonators is eventually deduced.

arXiv (Cornell University), Apr 7, 2023

A light ray in space is characterized by two vectors: (i) a transverse spatial-vector associated ... more A light ray in space is characterized by two vectors: (i) a transverse spatial-vector associated with the point where the ray intersects a given spherical cap; (ii) an angular-frequency vector which defines the ray direction of propagation. Given a light ray propagating from a spherical emitter to a spherical receiver, a linear equation is established that links its representative vectors on the emitter and on the receiver. The link is expressed by means of a matrix which is not homogeneous, since it involves both spatial and angular variables (having distinct physical dimensions). Indeed, the matrix becomes a homogeneous rotation-matrix after scaling the previous variables with appropriate dimensional coefficients. When applied to diffraction, in the framework of a scalar theory, the scaling operation results directly in introducing fractional-order Fourier transformations as mathematical expressions of Fresnel diffraction phenomena. Linking angular-frequency vectors and spatial frequencies results in an interpretation of the notion of a spherical angular-spectrum. Accordance of both inhomogeneous and homogeneous ray-matrices with Huygens-Fresnel principle is examined. The proposed ray-matrix representation of diffraction is also applied to coherent imaging through a lens.

Fractal and Fractional

A light ray in space is characterized by two vectors: (i) a transverse spatial vector associated ... more A light ray in space is characterized by two vectors: (i) a transverse spatial vector associated with the point where the ray intersects a given spherical cap; (ii) an angular-frequency vector which defines the ray direction of propagation. Given a light ray propagating from a spherical emitter to a spherical receiver, a linear equation is established that links its representative vectors on the emitter and on the receiver. The link is expressed by means of a matrix which is not homogeneous, since it involves both spatial and angular variables (having distinct physical dimensions). Indeed, the matrix becomes a homogeneous rotation matrix after scaling the previous variables with appropriate dimensional coefficients. When applied to diffraction, in the framework of a scalar theory, the scaling operation results directly in introducing fractional-order Fourier transformations as mathematical expressions of Fresnel diffraction phenomena. Linking angular-frequency vectors and spatial fr...

Notion d'espace des formes de lumiere. La mise en evidence de la structure quaternionienne de... more Notion d'espace des formes de lumiere. La mise en evidence de la structure quaternionienne de la lumiere polarisee debouche sur la representation des formes de lumiere et des operateurs de polarisation par des quaternions

Mise en evidence de la structure geometrique de la lumiere polarisee. La forme quadratique de Lor... more Mise en evidence de la structure geometrique de la lumiere polarisee. La forme quadratique de Lorentz est invariante lorsque la lumiere traverse un milieu anisotrope pour la polarisation. Un tel milieu se traduit alors par une transformation de Lorentz

$Research paper thumbnail of Effect of Fresnel Diffraction on Wigner Distributions of Optical Fields and how to Use It in Optical Resonator Theory$

According to a scalar theory, diffraction affects the electric ﬁeld amplitude as well as its angu... more According to a scalar theory, diffraction affects the electric ﬁeld amplitude as well as its angular spectrum, providing a natural space–frequency phase space on which the Wigner distribution can be deﬁned. The phase space is equipped with an Euclidean structure, so that the effect of diffraction is a rotation of the Wigner distribution associated with the optical ﬁeld. The rotation is shown to split into two speciﬁc rotations. Wigner distributions associated with transverse modes of a resonator are invariant in such rotations, and a complete theory of optical resonators is developed on the basis of this property, including waist existence and related formulae, and naturally introducing the Gouy phase.

$Research paper thumbnail of The Fractional Order Fourier Transform and Fresnel Diffraction$

Fresnel diffraction phenomenae between spherical emitters and receivers are mathematically repres... more Fresnel diffraction phenomenae between spherical emitters and receivers are mathematically represented by fractional Fourier transforms whose orders are related to the emitter-receiver geometries. Analyzing and solving some problems in the scalar theory of diffraction and light propagation are carried out by only manipulating fractional orders without explicitely writing integral forms of the involved fractional transforms. Applications to image formation, optical resonator and Gaussian beam theories illustrate the method.

Optik, 1990

An introduction to a vectorial calculus for polarization optics. The calculus of quaternions is u... more An introduction to a vectorial calculus for polarization optics. The calculus of quaternions is used to both represent partially polarized light waves and polarization operators and the way it is introduced explains how it connects the Jones and Stokes-Mueller formalisms. This calculus, which is equivalent to the Stokes-Muller formalism, improves some classical methods based on the Poincare shpere. The given examples should illustrate how to actually use this vectorial calculus in polarization optics. A differential expression of a polarization operator is provided with applications to Neumann's equations and rotating axis birefringent

$Research paper thumbnail of Un panorama de la óptica de Fourier fraccionaria: resultados recientes y aplicaciones$

Bistua Revista De La Facultad De Ciencias Basicas, 2008

Optik, 1991

Geometrical approach to polarizatioo optics: II. Quaternionic representation of polarized light. ... more Geometrical approach to polarizatioo optics: II. Quaternionic representation of polarized light. Since the four-dimensional vector space of forms of light has a Minkowski space structure, and since pure dephasors form a group isomorphic to the Lorentz proper rotation group, it is possible to represent forms of light and polarization operators by complex quaternions, once chosen a reference basis on the vector space of polarization states. Changing this basis differently affects the representative quatemions of forms of light and polarization operators, but has a simple effect on diagonable operators

$Research paper thumbnail of Generator Of Optical Fibre Modes Using A Photorefractive Crystal$

Progress in Holography, 1987

A photorefractive BSO crystal used as a light amplifier within an optical fibre ring cavity forms... more A photorefractive BSO crystal used as a light amplifier within an optical fibre ring cavity forms an optical oscillator whose transverse modes are the modes of the optical fibre. Several intracavity filtering techniques can thence be considered in order to select any particular transverse mode among the others, thus achieving an active generator of pure and isolated optical fibre modes.

Optica Acta: International Journal of Optics, 1984

On presente un formalisme de la polarisation de la lumiere dans lequel les etats et les operateur... more On presente un formalisme de la polarisation de la lumiere dans lequel les etats et les operateurs de polarisation sont representes par des quaternions construits sur le corps des nombres complexes. On exprime ainsi simplement les polariseurs, les birefringents et les dichroiques a partir d'un meme operateur, mettant en evidence l'aspect unitaire de ces operateurs. On applique cette methode de calcul a la birefringence elliptique.

$Research paper thumbnail of Numerical holography computing by using the fractional Fourier transform: a ping-pong-pang algorithm$

SPIE Proceedings, 2004

An iterative algorithm for computing holograms is presented: it is based on the fractional order ... more An iterative algorithm for computing holograms is presented: it is based on the fractional order Fourier transform. It allows us to generate diffraction patterns which are not in the Fourier plane. It also makes possible the design of computer generated holograms providing two beforehand given illumination patterns on two preselected planes. Holograms have been calculated according to the proposed method and have been engraved by photolithography. Experimental results are given. A speckle reduction technique is also explained.

$Research paper thumbnail of Sampling theorem in fractional Fourier domains$

SPIE Proceedings, 2004

ABSTRACT The standard Fourier transform operator F is a particular case of the fractional order F... more ABSTRACT The standard Fourier transform operator F is a particular case of the fractional order Fourier transform operator Falpha (it is obtained for alpha = 1). In the same way that has been done for F, a sampling theorem for Falpha is proposed for signals whose fractional Fourier transforms of order alpha have finite supports. An interpolation formula for fractional bandlimited functions is also deduced.

L'optique de Fourier doit son nom à l'emploi délibéré de la transformation de Fourier d... more L'optique de Fourier doit son nom à l'emploi délibéré de la transformation de Fourier dans la représentation de phénomènes fondés sur la diffraction de la lumière1. Inscrite dans les limites d'une théorie scalaire, elle est, traditionnellement, liée à l'optique cohérente et les sujets développés dans ce livre se rattachent à ce thème. Le domaine et les applications ussuelles de l'optique de Fourier concernent la formation des images, la résolution des instruments d'optique, le traitement du signal optique, l'holographie, le transfert de la cohérence. Nous verrons comment y inclure la théorie des résonateurs optiques et celle des faisseaux gaussiens; celle de la dispersion dans les fibres optiques. L'optique de Fourier fournit ainsi un cadre général à la modélisation d'un grand nombre de phénomènes optiques2.

$Research paper thumbnail of An optical diffractive method for characterizing ssflc field induced director patterns in the smectic C* phase$

Ferroelectrics, 1996

Chevron and quasi-bookshelf textures in Surface Stabilized Ferroelectric Liquid Crystals, as well... more Chevron and quasi-bookshelf textures in Surface Stabilized Ferroelectric Liquid Crystals, as well as intermediate transition textures, correspond to distinct birefringence modulations and result in distinct diffraction properties which are theoretically analysed. The experimental study of the polarization and illumination properties of diffraction patterns allows us to distinguish the actual texture of a liquid crystal among the above possible textures.

Optique de Fourier, 2009

Traitement du signal optique Le traitement du signal optique s'applique it des signaux d'origines... more Traitement du signal optique Le traitement du signal optique s'applique it des signaux d'origines variees-signaux optiques bien sur, mais aussi acoustiques (sonar), electriques, signaux radar-pour peu qu'on sache transcrire ces derniers sur un support « lisible » par la lumiere, que ce soit sous forme monodimensionnelle (traitements acousto-optiques par exemple) ou bi-dimensionnelle (emploi de modulateurs de lumiere spatiaux), en eclairage coherent ou en eclairage incoherent. II s'agit d'un domaine particulierement vaste dont Ie developpement est lie, historiquement et techniquement, it l'optique de Fourier. II n'etait pas concevable d'ignorer dans cet ouvrage un tel domaine d'applications, meme si l'optique metaxiale, et encore moins l'optique de Fourier fractionnaire, n'ont penetre Ie domaine. Nous nous bornerons it citer quelques exemples qui nous paraissent bien illustrer it la fois les capacites de l'optique et Ie role de la transformation de Fourier. D'autres applications sont decrites dans de nombreux ouvrages [38,59,85,97,233,235,250]. 16.1 G€meralites sur Ie traitement du signal optique Au CCBur du traitement des signaux par voie optique figure la capacite qu'offre la lumiere de realiser une transformation de Fourier (analogique, it deux dimensions en general) par propagation. Cette capacite a suscite de grands espoirs dans les annees 60, it une epoque OU les traitements numeriques n'etaient pas ce qu'ils sont aujourd'hui, et a conduit it la conception et la realisation de nombreux processeurs optiques. L'emploi de l'optique se fonde sur les considerations suivantes.

$Research paper thumbnail of Topics in fractional Fourier optics and applications$

Optoelectronic Information Processing: Optics for Information Systems: A Critical Review, 2001

Fourier analysis has been revealing itself as powerful for dealing with coherent optics and the f... more Fourier analysis has been revealing itself as powerful for dealing with coherent optics and the fractional order Fourier transform is an extension of this tool. There exists a method of the fractional Fourier transform in coherent optics which legitimizes the name of fractional Fourier optics. This fact is illustrated by examples in light propagation, coherent imaging, image formation, and optical resonator theory. A fractional correlation operation is deduced and the designs of optical fractional correlators are proposed.

Latin America Optics and Photonics Conference, 2016

We propose a method for calculating a bandlimited diffuser with smooth spectrum in the Fresnel do... more We propose a method for calculating a bandlimited diffuser with smooth spectrum in the Fresnel domain without any 2π phase ambiguities using the fractional Fourier transform. Such diffusers are necessary to avoid problems of speckles.