Efficient numerical methods for initialvalue ... - Wiley Online Library

PAMM · Proc. Appl. Math. Mech. 12, 655 – 656 (2012) / DOI 10.1002/pamm.201210316

Efficient numerical methods for initial-value solid-state laser problems Fan Feng1,∗ and Christoph Pflaum1,∗∗ 1

Department of Computer Science and Erlangen Graduate School in Advanced Optical Technologies (SAOT), Friedrich-Alexander University Erlangen-Nuernberg, Cauerstr. 11, D-91058 Erlangen, Germany

The difficulties of solving initial-value solid-state laser problems numerically arise from both stiffness of the problems and near-to-zero nonnegative exact solutions. Stability and non-negativity must be maintained simultaneously in the numerical solutions. Backward differentiation formulas (BDFs) is capable of dealing with stiff problems, but is of small oscillation when time-step is large. Therefore unfortunately BDFs suffers from severe time-step restriction. In this paper, we present an optimized numerical approach, with which 3-dimensional laser problems can be solved faster and much more efficiently. These techniques can not only be used for solid-state laser systems, but can also be applied to solve other stiff problems which have near-to-zero nonnegative exact solutions. c 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim

1

Introduction

Essentially, our laser system [1] is of the following form: (Laser system [2] and [3] are of the similar “λ − d, H” form.) dN (t) dt dNQ (t) dt dΦ(t) dt dΦ(2ω) (t) dt

=

λ1 (Φ(t))N (t) + d1 ,

(1)

=

λ2 (Φ(t))NQ (t) + d2 ,

(2)

=

H(N (t), NQ (t), Φ(t))Φ(t),

(3)

=

λ3 · Φ(2ω) (t) + d3 (Φ(t)),

(4)

where λ1 , λ2 , λ3 < 0 ; d1 , d2 , d3 > 0 ; Sometimes H >= 0 , sometimes H < 0 . We can prove that if N (0), NQ (0), Φ(0), Φ(2ω) (0) > 0 , then ∀t > 0, N (t), NQ (t), Φ(t), Φ(2ω) (t) >= 0

2

Efficient numerical methods

With certain parameter values, this initial value problem is stiff. So we consider BDFs a hopeful high order method for it. Unfortunately, Φ(t) and Φ(2ω) (t) are sometimes very close to 0 and BDFs is of small oscillation when step-size is large. This oscillation may sometimes oscillate Φ(t) or Φ(2ω) (t) to negative values. Therefore BDFs suffers a time-step restriction. The problem in application is that for a set of given parameter values, we firstly want to know the type of the solution (e.g. it approaches zero steady state; it approaches positive steady state; it is a periodic or chaotic solution;etc.) instead of the precise values of the solution. In our laser model, Φ(t) = 0 is indeed a steady state, which with certain parameter values is stable. However, we generally do not know whether the dynamic solution will approach this zero steady state before we get the numerical result. So if we adopt BDFs directly, the possible case is that very long time is consumed (because very small step-size has to be used to preserve the non-negativity of the exact solutions) to get the information that the dynamic solution will approach this zero steady state. In this paper,we present an optimized numerical approach, with which 3-dimensional laser problems can be solved faster and much more efficiently. These techniques can not only be used for solid-state laser system [3], passively Q-switched laser system [2], passively Q-switched intracavity frequency-doubling laser system [1], but can also be applied to solve other stiff problems which have near-to-zero nonnegative exact solutions. Our idea is to design two algorithms, which we call “Step 1” and “Step 2”. “Step 1” is a low order algorithm, which can accept large step-size and with which we get rough information. “Step 2” is a high order algorithm, which cannot accept large step-size but with which we get more precise values. For each set of parameter values, we firstly perform “Step 1”. After knowing the rough information, if more precise solution is necessary, then we spend longer time on “Step 2” to get more precise values. ∗ ∗∗

E-mail [email protected], phone +49 9131 85 67286, fax +49 9131 85 28928 E-mail [email protected]

c 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim

656

Section 18: Numerical methods of differential equations

Euler method is capable of preserving the non-negativity of exact solutions and BDFs is of high order and stable. Our algorithms are based on these two methods. In “Step 1”, we firstly use very small step-size and Euler method to get several initial values which is required by BDFs. Then order 6 BDF is applied to N and NQ ; Euler method is applied to Φ and Φ(2ω) . (Here Upwind Euler means when H >= 0, we use explicit Euler, else we use implicit Euler): Do Predictor twice dN (t) ∗ = λ1 (Φn )N (t) + d1 , order 6 BDF −→ Nn+1 dt dNQ (t) ∗ = λ2 (Φn )NQ (t) + d2 , order 6 BDF −→ NQn+1 dt dΦ(t) ∗ ∗ , NQn+1 , Φn )Φ(t), Upwind Euler −→ Φ∗n+1 = H(Nn+1 dt Do Corrector once dN (t) = λ1 (Φ∗n+1 )N (t) + d1 , order 6 BDF −→ Nn+1 dt dNQ (t) = λ2 (Φ∗n+1 )NQ (t) + d2 , order 6 BDF −→ NQn+1 dt dΦ(t) ¯ n+1 = H(Nn+1 , NQn+1 , Φ∗n+1 )Φ(t), Upwind Euler −→ Φn+1 , order 6 BDF → Φ dt dΦ(2ω) (t) (2ω) = λ3 · Φ(2ω) (t) + d3 (Φn+1 ), Implicit Euler −→ Φn+1 dt ¯ because the distance between Φ and Φ ¯ can be used to offer We can see from above that order 6 BDF is also applied to Φ, adaptive step-size: ¯ n+1 | |Φn+1 − Φ Distance= |Φn+1 | ∆tn · AllowableErr , where 0 < ǫ ≪ 1 is used to avoid zero denominator ∆tn+1 = Distance + ǫ In “Step 2”, we firstly also use very small step-size and Euler method to get several initial values which is required by ¯ In far-from-zero-region, BDFs. Then order 6 BDF is applied to N , NQ , Φ and Φ(2ω) . Upwind Euler is applied to Φ. we adopt larger step-size bound value; in near-zero-region, we adopt smaller step-size bound value. Generally, in order to preserve the non-negativity of the exact solutions, the step-size applied in “Step 2” has to be smaller than in “Step 1”.

3

Numerical result

A simulation of solid-state laser system [3] is carried out in this section (See Fig. 1.). Firstly, use the lower order algorithm for rough information (“Step 1”). Secondly, use the high order algorithm for more precise result (“Step 2”).

Step1 N Step1 Output power Step2 N Fig. 1. Use the algorithms of this paper to simulate solid-state laser system [3].

Step2 Output power

Step2 Distance

Step2 Step-size

Remark : “N” and “Output power” in both Step 1 and Step 2 approaches the same steady state. So if people only want to know the values of this steady state, Step 2 is not necessary. Acknowledgements The first author is grateful to her mentor, Professor C. Pflaum, for the instruction, encouragement and very helpful discussion. The research is funded by Erlangen Graduate School in Advanced Optical Technologies (SAOT), Friedrich-Alexander University Erlangen-Nuernberg and China Scholarship Council (CSC).

References [1] F. Feng and C. Pflaum, Dynamic multimode analysis (DMA) of passively Q-switched intracavity frequency-doubling solid state laser, Proc. SPIE 8434, 8434-3 (2012). [2] C. Pflaum, Z. Rahimi and F. Feng, Dynamic multi-mode analysis of passive Q-switched lasers, Proc. SPIE 8236, 823612 (2012). [3] M. Wohlmuth, C. Pflaum, K. Altmann, M. Paster and C. Hahn, Dynamic multimode analysis of Q-switched solid state laser cavities, Optics Express Vol.17 (2009). c 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim

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