Effective shooting algorithm and its application to fiber amplifiers

Xueming Liu; Byoungho Lee

doi:10.1364/OE.11.001452

Optics Express
Vol. 11,
Issue 12,
pp. 1452-1461
(2003)
•https://doi.org/10.1364/OE.11.001452

Effective shooting algorithm and its application to fiber amplifiers

Xueming Liu and Byoungho Lee

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Xueming Liu and Byoungho Lee, "Effective shooting algorithm and its application to fiber amplifiers," Opt. Express 11, 1452-1461 (2003)

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Abstract

A series of new methods based on the Runge-Kutta (RK) formula are proposed, which not only retain the merit of RK methods, in that the adaptive stepsize is easily implemented, but also dramatically decrease the error under the same conditions. Based on the new methods, an effective shooting algorithm is also proposed. A two-point boundary value problem for the Raman amplifier propagation equations is solved using the proposed algorithm. Our algorithm markedly increases the simulating speed for Raman amplifier propagation equations, as well as improves the accuracy, compared to the traditional algorithm.

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Fig. 1. Relationship of relative error with t, for (a) the fourth-order RK method, and (b) the fourth-order ORK method.

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Fig. 2. Calculation procedure for the shooting algorithm based on the fourth-order ORK method, where (a) for y ₁(t), (b) for y ₂(t), and (c) for the relative error of y ₁(t) and y ₂(t) in the last iteration.

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Fig. 3. Raman gain spectrum g_R (Δv) of the fiber, which was used in the simulation.

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Fig. 4. Iteration procedure and numerical results for a distributed pump Raman amplifier, for (a) iteration procedures of a pump, (b) iteration procedures of a signal, and (c) results of all pumps and signals at the last iteration. The calculating procedure is from z=L to z=0.

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Equations (25)

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\frac{d y}{d t} = f (t, y),

y (t = t_{0}) = y_{0} .

y_{n + 1} = y_{n} + \sum_{i = 1}^{m} w_{i} k_{i},

k_{i} = h f (t_{n} + c_{i}, y_{n} + \sum_{j = 1}^{m} d_{ij} k_{j}), i = 1, 2, \dots m .

\frac{d z}{d t} = g (t, \exp (z)) .

z_{n + 1} = z_{n} + \sum_{i = 1}^{m} w_{i} q_{i},

q_{i} = h g (t_{n} + c_{i}, \exp (z_{n}) (1 + \sum_{j = 1}^{m} d_{ij} q_{j})), i = 1, 2, \dots m .

y_{n + 1} = y_{n} \exp (\sum_{i = 1}^{m} w_{i} q_{i}),

q_{i} = h g (t_{n} + c_{i}, y_{n} + y_{n} \sum_{j = 1}^{m} d_{ij} q_{j}), i = 1, 2, \dots m .

y_{n + 1} = y_{n} \exp [\frac{(q_{1} + 2 q_{2} + 2 q_{3} + q_{4}) h}{6}],

q_{1} = g (t_{n}, y_{n}),

q_{2} = g (t_{n} + \frac{h}{2}, y_{n} + y_{n} q_{1} \frac{h}{2}),

q_{3} = g (t_{n} + \frac{h}{2}, y_{n} + y_{n} q_{2} \frac{h}{2}),

q_{4} = g (t_{n} + h, y_{n} + y_{n} q_{3} h),

g (t_{n}) = f \frac{(t_{n})}{y_{n}} .

ε = C h^{5} {[In (y)]}^{(5)} (ξ) .

ε = C' h^{5} y^{(5)} (ξ) .

\frac{dy}{dt} = y (\frac{t^{3}}{200} + \frac{t^{2}}{50} + t), y (0) = 1, and t \in [0, 3.6] .

y = \exp (\frac{t^{4}}{800} + \frac{t^{3}}{150} + \frac{t^{2}}{2}) .

\frac{d y_{1} (t)}{d t} = f_{1} (t, y_{1}, y_{2}),

\frac{d y_{2} (t)}{d t} = f_{2} (t, y_{1}, y_{2}),

y_{1} (a) = α,

y_{2} (b) = β .

A_{j + 1} = {\begin{matrix} A_{j - 1} + \frac{(β - β_{j - 1}) (A_{j} - A_{j - 1})}{(β_{j} - β_{j - 1})}, (if j > 1) \\ \frac{A_{j} β}{β_{j}}, (if j = 1) \end{matrix}

\pm \frac{d P_{i}}{d z} = [- α_{i} + \sum_{j = 1}^{i - 1} \frac{g_{R} (v_{j} - v_{i})}{Γ A_{eff}} P_{j} - \sum_{j = i + 1}^{m} \frac{v_{i}}{v_{j}} \frac{g_{R} (v_{i} - v_{j})}{Γ A_{eff}} P_{j}] P_{i}, (i = 1, 2, \dots, m),

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Equations (25)

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