Tamtam Proceedings - lamsin

510 Labidi et al.pk+k+ppkpFigure 2. After performing (m − k) implicitly shifted QR steps on H m, the middle set ofpictures illustrates V mQH + m + f me T mQ. The last (m − k) columns of f me T mQ are non zerobecause of the QR iteration.k+kkFigure 3. An implicitly restarted Arnoldi factorisation of length k results after discardingthe last (m − k) columns of Figure 2.3. Application of IRAM for computing guided modes in anoptical fibreWe are interested in computing the propagation constants and their associated electromagneticfields of guided modes that propagate, under the conditions of weak guidance, inan optical fibre with an arbitrary shape and a general refractive-index profile. An efficientmethod coupling a local boundary condition of Robin type to a finite element techniqueis proposed in [2]. This consists on solving the following non-linear eigenvalue problem.⎧⎪⎨(P Σ )⎪⎩F ind λ Σ ∈]0, V 2 [ and u Σ ∈ H 1 (Ω Σ ); u Σ ≠ 0 such that− △ u Σ + q(x)u Σ = −λ Σ u Σ dans Ω Σ∂u Σ∂ν + (√ λ Σ + K 2 )u Σ = 0 sur ΣΩ Σ is a bounded domain of R 2 , with Σ = ∂Ω Σ . We place Σ in such way that Ω Σ containsthe core of considered fibre, and hence the support of the function q ∈ L ∞ (R 2 ).Σ canhave an arbitrary shaped, but must be convex (cf. [2]). K denotes the curvature of theconsidered artificial boundary Σ. −→ ν denotes the outward directed unitary normal vector.(λ Σ , u Σ ) approximates a guided mode that propagates in the considered optical fibre.TAMTAM –Tunis– 2005