UMR 5224


Brigitte Bidégaray-Fesquet — Optics

Optics is one of my major themes. I am currently following two directions : Quantum Optics and Sub-wavelength optics.

Quantum Optics

Previous work

I have studied at length the Maxwell–Bloch model in different aplicative contexts.
See also the
Maxwell–Bloch page and Maxwell–Bloch gallery.

  • Centro-symmetric materials (gases, glasses).
    We have selected mathematical and numerical midels which correctly (conservation of certain physical variables) account for some relaxation phenomena in Bloch equations [BBR03]. These equations describe matter at the quantum level. The same issues are of interest in the coupling with an electromagnetic wave, yielding the Maxwell–Bloch equations [Bid03], allowing to treat many experimental phenomena [Bid02]. We also have derived asymptotic models from the Bloch equations resulting in rate equations [BCD04] All these works have been exhautively presented in a book at the Master level [Bid06].
  • Crystals.
    The case of crystal was the subject of Olivier Saut's PhD to which I contributed concerning the quantum description [BBBDS04]. This new case introduces technical problems dues to (a) the complexity of the derivation of the model for each symmetru class (still to be done, only KDP has been traeted) (b) the eed to have full 3D models to describe anisometries.
  • Quantum dots.
    In the case of quantum dots, new theoretical problems arise, due to the fact that we have two species of electron and we have to derive the models in the Heisenberg picture of quatum mechanics (the Schrödinger picture was used in the previous studies). A first preliminary work is devoted to degeneracies and the Pauli exclusion principle with a very reduced model [Bid10]. A second in collaboration with Kole Keita deals with the introduction of Coulomb interaction, leading to nonlinear Bloch equations [BK14].

Ongoing work

I am currently working on

  • the introduction of electron-phonon interactions (and associated nonlinear effects);
  • the coupling with the wetting layer (quantum well model);
  • the introduction of other phenomenological phenomena;
  • the numerical implementation (multi-scale computing).

References

[BBBDS06] Christophe Besse, Brigitte Bidégaray-Fesquet, Antoine Bourgeade, Pierre Degond, and Olivier Saut. A Maxwell–Bloch model with discrete symmetries for wave propagation in nonlinear crystals: an application to KDP. Mathematical Modelling and Numerical Analysis, 38(2), 321–344, 2004.
[Bid02] Brigitte Bidégaray. Modèles d'interaction radiation–matière en milieu résonnant. in Compte-rendus de la 5ème Rencontre du non linéaire 2002, 14–15 mars 2002, Paris, 1–6, Non linéaire Publications, Orsay, 2002.
[Bid03] Brigitte Bidégaray. Time discretizations for Maxwell--Bloch equations. Numerical Methods for Partial Differential Equations, 19(3), 284–300, 2003.
[Bid06] Brigitte Bidégaray-Fesquet. Hiérarchie de modèles en optique quantique. De Maxwell–Bloch à Schrödinger non linéaire. XIV+175 pages, Collection Mathématiques et Applications 49, Springer, 2006.
[Bid10] Brigitte Bidégaray-Fesquet. Positiveness and Pauli exception principle in raw Bloch equations for quantum boxes. Annals of Physics, 325(10), 2090–2102, 2010.
[BBR03] Brigitte Bidégaray, Antoine Bourgeade, and Didier Reignier. Introducing physical relaxation terms in Bloch equations. Journal of Computational Physics, 170(2), 603–613, 2001.
[BCD04] Brigitte Bidégaray-Fesquet, François Castella, and Pierre Degond. From the Bloch model to the rate equations. Discrete and Continuous Dynamical Systems A, 11(1), 1–26, 2004.
[BCDG04] Brigitte Bidégaray-Fesquet, François Castella, Éric Dumas, and Marguerite Gisclon. From Bloch model to the rate equations II: the case of almost degenerate energy levels. Mathematical Models and Methods in Applied Sciences, 14(12), 1785–1817, 2004.
[BK14] Brigitte Bidégaray-Fesquet and Kole Keita. A nonlinear Bloch model for Coulomb interaction in quantum dots. Journal of Mathematical Physics, 55, 021501, 2014.

Sub-wavelenth Optics

Partners

LJK Brigitte Bidégaray-Fesquet, Éric Bonnetier, and Faouzi Triki
and formerly Jean-François Babadjian
Inst. Fourier Éric Dumas
Inst. Néel Aude Barabara and Pascal Quémerais

Previous work

I am responsible of the MADISON (Modèles Asymptotiques pour la DIffraction Sub-longueur d'ONde de surfaces rugueuses) which un funded by the M-STIC pole of the Joseph Fourier University. This funding was aimed at the organization of the pluri-disciplinary workshop Polaritons 2009, but we also work on the study of diffractive phenomena on sub-wavelength surfaces. We more specifically address the asymptotic and numerical analysis of the diffraction by metallic rectangular gratings which are experimentally studied at the Institut Néel.

My colleagues at the LJK have performed the asymptotic study of a system of cavities which allow to replace them, as their width tends to zero, by dipoles [BT10]. This work is using perfect metal boundary conditions to treat the surfaces.

In parallel, I studied with Éric Dumas the impact of surface conditions on the computation of refected modes and cavity modes. These computations suppose a modal expansion of the electromagnetic fields and have been communicated at the Waves 2009 Conference (9th International Conference on Mathematical and Numerical Aspects of Waves) in June 2009 and are submitted to the special issue dediacted to this event [BD09]. This work show that real metal boundary conditions (surface impedance conditions) are more adequate and we explain how asymptotic expansions with respect to the dielectric permittivity of the metal can make computations effectient. We also diplay that for some frequencies (or incident angles) the incident wave is absorbed by the cavities, thus yielding high energy concentrations usueful for applications.

Ongoing work

We are currently working or intend to work on

  • the comparison of modal computations with other modal techniques or full Helmholtz finite difference computations;
  • the combinaison of our cavity modes with the asymptotic study of equivalent dipoles;
  • the derivation of diffractive geometry models when the cavities are fille with nonlinear (Kerr) materials.

References

[BD09] Brigitte Bidégaray-Fesquet and Éric Dumas. Impact of metallic interface description on sub-wavelength cavity mode computations. submitted to publication, 2009.
[BT10] Éric Bonnetier and Faouzi Triki. Asymptotic of the Green function for the diffraction by a perfectly conducting plane perturbed by a sub-wavelength rectangular cavity. Mathematical Methods in the Applied Sciences, 33 (6), 772–798, 2010.
(Last updated January 2015)