Raman laser : mathematical and numerical analysis of a model
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 38 (2004) no. 3, p. 457-475

In this paper we study a discrete Raman laser amplification model given as a Lotka-Volterra system. We show that in an ideal situation, the equations can be written as a Poisson system with boundary conditions using a global change of coordinates. We address the questions of existence and uniqueness of a solution. We deduce numerical schemes for the approximation of the solution that have good stability.

DOI : https://doi.org/10.1051/m2an:2004022
Classification:  65L10,  65L20
Keywords: optical device, Raman gain, Poisson system, integro-differential equations
@article{M2AN_2004__38_3_457_0,
     author = {Castella, Fran\c cois and Chartier, Philippe and Faou, Erwan and Bayart, Dominique and Leplingard, Florence and Martinelli, Catherine},
     title = {Raman laser : mathematical and numerical analysis of a model},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {38},
     number = {3},
     year = {2004},
     pages = {457-475},
     doi = {10.1051/m2an:2004022},
     zbl = {1078.78012},
     mrnumber = {2075755},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2004__38_3_457_0}
}
Castella, François; Chartier, Philippe; Faou, Erwan; Bayart, Dominique; Leplingard, Florence; Martinelli, Catherine. Raman laser : mathematical and numerical analysis of a model. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 38 (2004) no. 3, pp. 457-475. doi : 10.1051/m2an:2004022. http://www.numdam.org/item/M2AN_2004__38_3_457_0/

[1] M. Achtenhagen, T. Chang and B. Nyman, Analysis of a multiple-pump Raman amplifiers. Appl. Phys. Lett. 78 (2000) 1322-1324.

[2] U. Ascher, R. Mattheij and R. Russel, Numerical Solution of Boundary Value for Ordinary Differential Equations. Prentice Hall, Englewood Cliffs (1988). | MR 1000177 | Zbl 0671.65063

[3] F. Castella, P. Chartier and E. Faou, Analysis of a Poisson system with boundary conditions. C. R. Acad. Sci. Paris, Sér. I 336 (2003) 703-708. | Zbl 1030.45006

[4] G. Golub and C. Van Loan, Matrix Computations. The Johns Hopkins University Press (1989). | MR 1002570 | Zbl 0733.65016

[5] E. Hairer, C. Lubich, and G. Wanner, Geometric numerical integration Springer-Verlag 31 Springer Ser. Comput. Math. Berlin (2002). Structure-preserving algorithms for ordinary differential equations. | MR 1904823 | Zbl 0994.65135

[6] H. Kidorf, K. Rottwitt, M. Nissov, M. Ma and E. Rabarijaona, Pump Interactions in a 100-nm Bandwidth Raman Amplifier. IEEE Photonics Technology Letters 11 (1999) 530-532.

[7] J. Pocholle, M. Papuchon, J. Raffy and E. Desurvire, Non linearities and optical amplification in single mode fibers. Revue Technique Thomson-CSF 22 (1990) 187-268.

[8] M. Rini, I. Christiani and V. Degiorgio, Numerical modeling and optimization of cascaded Raman fiber lasers. IEEE Journal of Quantum Electronics 36 (2000) 1117-1122.