Stiffness analysis in the numerical solution of Raman amplifier propagation equations

Song Hu; Hanyi Zhang; Yili Guo

doi:10.1364/OPEX.12.001656

Optics Express
Vol. 12,
Issue 8,
pp. 1656-1664
(2004)
•https://doi.org/10.1364/OPEX.12.001656

Stiffness analysis in the numerical solution of Raman amplifier propagation equations

Song Hu, Hanyi Zhang, and Yili Guo

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Song Hu, Hanyi Zhang, and Yili Guo, "Stiffness analysis in the numerical solution of Raman amplifier propagation equations," Opt. Express 12, 1656-1664 (2004)

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Abstract

For the first time, the stiffness of Raman amplifier propagation equations is analyzed. And based on this analysis, a novel method for propagation equations is proposed to enhance the stability of numerical simulation. To verify the reliability of this method, simulation experiments are employed by using our method and the existent predictor-corrector method with comparison. The results show that our backward differentiation formulae method behaves much better in stability with a comparative accuracy.

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Fig. 1. The stiffness ratio of PERA along the fiber distance.

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Fig. 2. The maximum error of ‖P⃗‖ varied with the iterating step based on model (2) and (3).

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Fig. 3. Pump power evolution along the fiber distance obtained from three methods.

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Fig. 4. Net gain of FRA obtained from three methods.

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Fig. 5. Signal power evolution along the fiber calculated by different methods, in which (a) for BDF method based on model (2), (b) for PC method, (c) for BDF method based on model (3) and (d) for VPI.

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Tables (2)

Table 1. Comparison of the largest difference of the values calculated by three methods

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Table 2. Coefficients of zero-stable BDF methods for p=0,1,…,5.

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Equations (13)

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\frac{d P_{ν}^{\pm}}{d z} = F_{ν}^{\pm} (z, \vec{P})

F_{ν}^{\pm} (z, \vec{P}) = \mp α_{ν} P_{ν}^{\pm} \pm ε_{ν} P_{ν}^{\pm} \pm P_{ν}^{\pm} \sum_{ς > ν} \frac{g_{ς ν}}{Γ_{ς υ}} \cdot (P_{ς}^{+} + P_{ς}^{-}) \mp P_{ν}^{\pm} \sum_{ς < ν} \frac{ν}{ς} \cdot \frac{g_{ν ς}}{Γ_{ν ς}} \cdot (P_{ς}^{+} + P_{ς}^{-})

\pm 2 h ν \sum_{ς > ν} g_{ς ν} \cdot (P_{ς}^{+} + P_{ς}^{-}) \cdot [1 + \frac{1}{\exp [\frac{h (ς - ν)}{k T}] - 1}] \cdot Δ ν

\mp 4 h ν P_{ν}^{\pm} \sum_{ς < ν} \frac{ν}{ς} \cdot g_{ν ς} \cdot [1 + \frac{1}{\exp [\frac{h (ν - ς)}{k T}] - 1}] \cdot Δ ς

{\begin{matrix} d \vec{P} ⁄ d z = \vec{F} (z, \vec{P}) \\ \vec{P} = {(P_{1} (z), P_{2} (z), \dots, P_{n + m} (z))}^{T} \\ \vec{F} = {(F_{1} (z, \vec{P}), F_{2} (z, \vec{P}), \dots, F_{n + m} (z, \vec{P}))}^{T} \end{matrix}

{\begin{matrix} d \ln (\vec{P}) ⁄ d z = {\vec{F}}^{*} (z, \vec{P}) \\ \vec{P} = {(P_{1} (z), P_{2} (z), \dots, P_{n + m} (z))}^{T} \\ {\vec{F}}^{*} = {(F_{1} (z, \vec{P}) ⁄ P_{1}, F_{2} (z, \vec{P}) ⁄ P_{2}, \dots, F_{n + m} (z, \vec{P}) ⁄ P_{n + m})}^{T} \end{matrix}

P_{p i}^{\pm} (z_{k \pm 1}) = \exp {\frac{48}{25} \ln [P_{p i}^{\pm} (z_{k})] - \frac{36}{25} \ln [P_{p i}^{\pm} (z_{k \mp 1})] + \frac{16}{25} \cdot \ln [P_{p i}^{\pm} (z_{k \mp 2})] - \frac{3}{25} \ln [P_{p i}^{\pm} (z_{k \mp 3})] + \frac{12}{25} F_{p i}^{*} (z_{k \pm 1}, \vec{P})}

P_{s j}^{+} (z_{k + 1}) = \exp {\frac{48}{25} \ln [P_{s j}^{+} (z_{k})] - \frac{36}{25} \ln [P_{s j}^{+} (z_{k - 1})] + \frac{16}{25} \cdot \ln [P_{s j}^{+} (z_{k - 2})] - \frac{3}{25} \ln [P_{s j}^{+} (z_{k - 3})] + \frac{12}{25} F_{s j}^{*} (z_{k + 1}, \vec{P})}

\frac{d \vec{y}}{dt} = A \vec{y} (t) + ϕ (t), t \in [a, b]

Re (λ_{j}) < 0, j = 1, 2, \dots, n

s = max_{1 \leq j \leq n} ∣ Re (λ_{j}) ∣ ⁄ min_{1 \leq j \leq n} ∣ Re (λ_{j}) ∣ □ 1

\frac{d \vec{y}}{dt} = f (t, \vec{y} (t)), t \in [a, b]

\vec{y} (z_{k + 1}) = \sum_{i = 0}^{p} q_{i} \vec{y} (z_{k - i}) + h b_{- 1} {\vec{f}}_{k + 1}

Compared items	PC vs. BDF	PC vs. VPI	BDF vs. VPI
Compared methods	PC vs. BDF	PC vs. VPI	BDF vs. VPI
Largest difference of pump power (W)	2.46×10^-3	4.96×10^-4	2.23×10^-3
Largest difference of net gain (dB)	0.099	0.263	0.282
Computational time	1 : 1.3	—	—

p	a ₀	a ₁	a ₂	a ₃	a ₄	a ₅	b _-1
0	1	0	0	0	0	0	1
1	4/3	-1/3	0	0	0	0	2/3
2	18/11	-9/11	2/11	0	0	0	6/11
3	48/25	-36/25	16/25	-3/25	0	0	12/25
4	300/137	-300/137	200/137	-75/137	12/137	0	60/137
5	360/147	-450/147	400/147	-225/147	72/147	-10/147	60/147

Compared items	PC vs. BDF	PC vs. VPI	BDF vs. VPI
Compared methods	PC vs. BDF	PC vs. VPI	BDF vs. VPI
Largest difference of pump power (W)	2.46×10^-3	4.96×10^-4	2.23×10^-3
Largest difference of net gain (dB)	0.099	0.263	0.282
Computational time	1 : 1.3	—	—

p	a ₀	a ₁	a ₂	a ₃	a ₄	a ₅	b _-1
0	1	0	0	0	0	0	1
1	4/3	-1/3	0	0	0	0	2/3
2	18/11	-9/11	2/11	0	0	0	6/11
3	48/25	-36/25	16/25	-3/25	0	0	12/25
4	300/137	-300/137	200/137	-75/137	12/137	0	60/137
5	360/147	-450/147	400/147	-225/147	72/147	-10/147	60/147

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Tables (2)

Equations (13)

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