Modélisation et simulation photonique:RSM
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Calculation method RSM :
- The Radiation Spectrum Method (RSM) [1], solves same problem than the well known Beam Propagation Method (BPM). It permits the simulation of the propagation of light in an integrated optical component of arbitrary shape. The use of the modal approach in replacement of the finite differences or of the finite elements is the basic idea at the origin of the method. In this problem, it must be possible to define arbitrarily both the component's geometry (distribution of the refractive index in space) and the excitation field (transverse components of the electromagnetic fields at the input of the component). Contrary to FDTD where any temporal form of excitation is possible,in the RSM, the hypothesis of a permanent harmonic field is made. This method makes use of the analytical results on the normalization of radiation modes of multilayer slab dielectric waveguides [2], [3]. The fact that there is no need to close laterally the waveguide with metal walls is the main originality of the RSM compared to others competing modal BPM. This gives a true advantage in calculation speed and makes the RSM tool compatible with a direct acceleration using fast Fourier transforms.
Principle :
- The two dimensional geometry of the integrated optical component is first sampled in a succession of straight waveguides segments.Each straight waveguide which refractive index profile can be a continuous function is approximated into an equivalent slab multilayer waveguide. The excitation electromagnetic field defined in the input plane of the component is projected on the guided and radiation modes of the first waveguide. In order to keep the vectorial aspect of the method, this projection must be done on the incident and reflected modes. The field at the end of this straight segment is then easily calculated in propagating the field coming of all the so excited forward and backward modes. This field is then used as the new excitation field for the following straight waveguide segment for which the same operation is repeated. This so as to reach the output of the component. For components presenting reflexions, we make use of an iterative approach process with go and back from the input to the output. The limit conditions are then recalled at these input or ouput faces of the element.
Advantages :
- modal approach : permits a physical interpretation of the operating principle of component thanks to the analysis of the power distribution and exchange on the different modes (spectrum of the modes)
- rigorous vectorial simulation : this means wide angle propagation but also that polarization effects and reflexion are taken into account
- highly guiding conditions accepted
- high calculation speed
Limitations :
- Harmonic established field
- 2D modelling
- material with real refractive indices
Exemples d’utilisation :
- Interféromètre de Mach-Zehnder dans l'état off
Géométrie
Superposition des courbes de propagation du champ (couleur) et profil d'indice de réfraction (gris)
- Interféromètre de Mach-Zehnder dans l'état on
Géométrie
Superposition des courbes de propagation du champ (couleur) et profil d'indice de réfraction (gris)
- Cristal photonique
Géométrie
Superposition des courbes de propagation du champ (couleur) et profil d'indice de réfraction (gris)
- Micro lentille
Géométrie
Superposition des courbes de propagation du champ (couleur) et profil d'indice de réfraction (gris)
Logiciel disponible :
- Nous avons ces dernières années réalisé un travail de développement logiciel autour de l’outil RSM. La motivation principale a été de disposer d’un logiciel très simple d’utilisation apte à démontrer toutes les possibilités liées au choix de cette approche modale. Le logiciel est pour ce faire muni d’une interface utilisateur. Nous disposons à ce jour d’une première version non libre car basée pour la visualisation sur le logiciel commercial IDL. Une version de démonstration tournant sous MacOS X est d’ores et déja disponible et accessible à la demande. Nous réfléchissons actuellement au développement d’une version libre qui serait basée sur le logiciel de visualisation Visit. Si la version actuelle ne tourne que sous MacOS X, l’architecture logicielle a cependant été pensée pour simplifier le portage ultérieur vers Windows ou Linux.
Caractéristiques principales :
- -définition de la géométrie du guide de 3 façons :
- fichier DAO (dxf)
- script
- fichier direct
- - conditions d’éclairement :
- champs E et H du mode fondamental d’un guide
- champs E et H gaussiens
- champs E et H échantillonnés dans un fichier
- - sorties graphiques :
- carte de profil 3D de l’indice de réfraction
- carte de profil 3D du champ propagé
- spectre propagé des modes
- - sauvegarde de tous les paramètres du calcul courant
- - sorties fichiers pour la visualisation à l’aide d’outils de tierces parties
Other developments around the RSM software :
- -Accelerated version by use of FFT (fast Fourier transform) : in order to underline the good performances of the software concerning the calculation speed and because such a possibility was not permitted by the other competing modal BPM, we have developed a version of the method accelarated by use of fast Fourier transforms [5]. As the main result we noticed the transition from a O(N2) algorithm to a O(N) algorithm. Of that way, for a total number of modes equal to 128, corresponding to a standard RSM calculation, the measured speedup was of a factor 10. One have to notice that this acceleration adds itself to the one estimated to a factor 30 linked to the use of the opened waveguide approach in compairison with other modal BPM tools.
- -Coupling FDTD-RSM : the RSM tool has been used in a project aiming at designing diffractive optical element (DOE) with the use of the effective medium technique [6]. The rigorous vectorial propagation of the electromagnetic field in the DOE is done with FDTD. The RSM in its fast Fourier transform accelerated form is used as a rigorous vectorial plane waves propagator. This permits starting from the field at the output of the DOE to calculate the diffracted field in the reconstruction plane of the element.
- -Reflections :the rigorous electromagnetic character, that is to say taking into account all the continuity equations for the electric and magnetic fields at the interfaces (interfaces between dielectric slab dielectric layers for the modes solver, interfaces between staight waveguides sections issued of the geometry discretization) permits a direct treatment of the reflected light. To do so, we have investigated two approaches. The first one, also the most direct solution, consists of a full matrix formulation and leads to the resolution of a linear system of equations. In that case we obtain a O(N3) algorithm. This is also the approach chosen in the Camfr software. The second one is an iterative method that proceeds with go and back between the input and the output of the component. In this case, the limit conditions are reimposed at each iteration. We have observed the conbergency of this method with test cases [7].
- -Extension to lossy or active waveguides : in this work we started establishing the properties of the radiations modes of a multilayer slab waveguide presenting complex refractive indices. In the following, we detail the main properties of such modes with the assumption that the waveguide provides external layers with pure real refractive index. The propagation constant of these radiation modes is real as it was the case for lossless waveguides. However, they are no more orthogonal to each others and the transverse field pattern associated to a radiation mode presents points that are not oscillating in phase. Including all these result, a variation of the RSM algotithm has been established. It makes use of the Gram Schmidt orthogonalization procedure. The here expressed properties of the radiation modes and the results obtained by the modified RSM on the tilted waveguide test problem have been published in ref [8].
Références :
- [1] P. Gérard, P. Benech, D. Khalil, R. Rimet, S. Tedjini, “Towards a full vectorial and modal technique for the analysis of integrated optics structures : the Radiation Spectrum Method (RSM)”, Optics Communications , Vol 140, july 1997, pp 128-145.
- [2] H. Ding, P. Gérard and P. Benech "Radiation modes of lossless multilayer dielectric waveguides". IEEE Journal of Quantum Electronics, Vol. 31, n°2, February 1995, pp 411-416.
- [3] P. Gérard, P. Benech, H. Ding and R. Rimet. "A simple method for the determination of orthogonal radiation modes in planar multilayer structures". Optics Communication 108, June 1994 , pp 235-238.
- [5] P. Gérard, “Vers une méthode du faisceau propagé modale et rapide : RSM-FFT”, actes de la conférence Journées Nationales de l’Optique Guidée, 12-14 novembre 2003, Valence, p231-233.
- [6] V. Raulot, P. Gérard, B. Serio, M. Flury, B. Kress, P. Meyrueis, “Modeling of the angular tolerancing of an effective medium diffractive lens using combined finite difference time domain and radiation spectrum method algorithms”, Optics Express, vol. 18, n°. 17, August 2010, p 17974-17982.
- [7] K. P. Fakhri, P. Benech “A new technique for the analysis of planar optical discontinuities : an iterative modal method”, Optics Communications, Vol. 177, 15 april 2000, pp 233-243.
- [8] P. Gérard, “Radiation modes of lossy or active slab waveguides”, Optics Communications , Vol 151, may 1998, pp 110-116.