Numerical Modeling of Optical Fibers Using the Finite Element
Method and an Exact Non-reflecting Boundary Condition

Rafail Z. Dautov; Evgenii M. Karchevskii

doi:10.1515/cmam-2017-0049

Published by De Gruyter November 17, 2017

Numerical Modeling of Optical Fibers Using the Finite Element Method and an Exact Non-reflecting Boundary Condition

Rafail Z. Dautov and Evgenii M. Karchevskii

From the journal Computational Methods in Applied Mathematics

https://doi.org/10.1515/cmam-2017-0049

Showing a limited preview of this publication:

Abstract

The original problem for eigenwaves of weakly guiding optical fibers formulated on the plane is reduced to a convenient for numerical solution linear parametric eigenvalue problem posed in a disk. The study of the solvability of this problem is based on the spectral theory of compact self-adjoint operators. Properties of dispersion curves are investigated for the new formulation of the problem. An efficient numerical method based on FEM approximations is developed. Error estimates for approximate solutions are derived. The rate of convergence for the presented algorithm is investigated numerically.

Keywords: Finite Element Method; Exact Non-Reflecting Boundary Condition; Spectral Problem; Optical Fiber

MSC 2010: 65N30; 65N25; 65Z05

Funding source: Russian Foundation for Basic Research

Award Identifier / Grant number: 16-01-00408

Funding statement: This work was partly supported by the Russian Foundation for Basic Research, project no. 16-01-00408 (R. Z. Dautov) and was performed according to the Russian Government Program of Competitive Growth of Kazan Federal University (E. M. Karchevskii).

A The Proof of Theorem 5.4

Thanks to Lemma 5.3 we can use Theorem 5.1 with NV=∞. Therefore all the assertions instead of (c) and the statement on the monotonicity of the functions p→βi2⁢(p)-p2, i≥1, are true.

Let p>0. Using the Courant–Fischer minimum-maximum principle, we write

βi2⁢(p)=minVi⊂V~⁡maxv∈Vi\{0}⁡R⁢(p,v),R⁢(p,v):=a⁢(p,v,v)b⁢(v,v).

Therefore,

(A.1)βi2⁢(p)-p2=minVi⊂V~⁡maxv∈Vi\{0}⁡R~⁢(p,v),R~⁢(p,v):=R⁢(p,v)-p2.

It is easy to see that

R~⁢(p,v)=1b⁢(v,v)⁢(∫Ω(|∇⁡v|2+p2⁢v2)⁢𝑑x+s∞⁢(p,v,v)).

Now it follows from (A.1) that the functions p→βi2⁢(p)-p2, i≥1, are non-negative and increasing.

The first statement in (c) follows from (βi⁢(p),p)∈K (see Remark 1). Now we shall prove the second assertion. By Br we denote the circle with a small radius r and the center at the point x+∈Ω¯i such that

ε+=ε⁢(x+)=maxx∈Ω¯i⁡ε⁢(x),δr:=maxx∈B¯r⁡ε+-ε⁢(x)ε∞.

Let Vr be the set of all functions in the space V that are equal to zero outside of Br. We note that s∞⁢(p,u,u)=0 on Vr; σ⁢(x):=ε⁢(x)ε∞≤ε+ε∞ on Br,

σ(x)-1=ε+ε∞-1-ε+-ε⁢(x)ε∞≥ε+-ε∞-ε∞⁢δrε∞=:drε∞.

It is easy to see that for any v∈Vr the next estimate is true:

R⁢(p,v)≤ε∞dr⁢(∫Br|∇⁡v|2⁢𝑑x)⁢(∫Brv2⁢𝑑x)-1+ε+⁢p2dr.

By (λri,ui) we denote the eigenpairs of the Laplace operator on the circle Br with the Dirichlet boundary conditions (we extend the function ui outside of Br by zero). Since λri=r-2⁢λ1i, we see that

βi2⁢(p)=minVi⊂V~⁡maxv∈Vi\{0}⁡R⁢(p,v)≤ε∞⁢λ1ir2⁢dr+ε+⁢p2dr.

Therefore,

k02≤βi2⁢(p)p2≤ε∞⁢λ1ip2⁢r2⁢dr+ε+dr.

Taking the limit as r→0 and p→∞ such that p⁢r→∞, we get the second assertion in (c), since δr→0, ε+dr→k02.

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Received: 2017-04-21

Revised: 2017-10-10

Accepted: 2017-10-23

Published Online: 2017-11-17

Published in Print: 2018-10-01

Numerical Modeling of Optical Fibers Using the Finite Element Method and an Exact Non-reflecting Boundary Condition

Abstract

A The Proof of Theorem 5.4

References

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