A fast and stable method for Raman amplifier propagation equations

Xueming Liu; Byoungho Lee

doi:10.1364/OE.11.002163

Optics Express
Vol. 11,
Issue 18,
pp. 2163-2176
(2003)
•https://doi.org/10.1364/OE.11.002163

A fast and stable method for Raman amplifier propagation equations

Xueming Liu and Byoungho Lee

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Xueming Liu and Byoungho Lee, "A fast and stable method for Raman amplifier propagation equations," Opt. Express 11, 2163-2176 (2003)

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Abstract

A novel predictor-corrector method for the coupled equations for ultrabroad-band Raman amplifiers with multiple pumps is proposed and derived, for the first time, based on the Adams formula. The proposed algorithm is effective in solving Raman amplifier equations that include pumps, signals, noises, and their backscattering waves. The detail procedure is given, and proves the excellence of our algorithm. Simulation results show that, in designing the Raman amplifier, our multistep method can effectively improve the accuracy and stability compared with the one-step method and explicit multistep method. The numerical results show that the power of backscattering pumps and signals is lower by ~30 dB and 20 dB than their original power, respectively, and the power of forward and backward noises is less than that of input signals by ~30 dB under our simulation conditions.

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Fig. 1. Illustration of frequency intervals for the numerical calculation. The entire spectral range of interest is represented by frequency v ₀ to v_n (Note that the horizontal axis in the ascending order of wavelength. Hence, the frequency is descending with horizontal axis, i.e., v ₀ > v_n .) (a) for input power P(0, v_i ) at the position of z=0, and (b) for input power P(z, v_i ) at the position of z=L.

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Fig. 2. Absolute error for one-step method, four-step method in [39] and PCM in this paper. The step size Δz=0.1.

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Fig. 3. The stability for one-step method, four-step method in [39] and PCM in this paper. The exact solution almost coincides with the solution of PCM.

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Fig. 4. Spectrum at the fiber input, output and backscattering, where (a) for input pump (backward) and signal (forward); (b) for output pump, backscattering signal and backward noise at the input-end of the fiber (i.e., z=0); (c) for output signal, forward noise and backscattering pump at the output-end of the fiber (i.e., z=L).

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Fig. 5. The propagation for signals, pumps, noises and their backscattering waves along the transmission fiber, where for (a) pumps, (b) backscattering pumps, (c) signals, (d) backscattering signals, (e) forward noises, and (f) backward noises. The fiber parts are 29. The dashes in Fig.5(a) are the projection of eight pumps.

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Fig. 6. Convergence for the reported four-step method [39] and our proposed PCM in designing DRA, where (a) for PCM and (b) for four-step method. All parameters are same with those in Fig.5 except that the single-pump power is 1.3 W at the wavelength of 1470 nm and there are only fifteen signals, from 192.15 THz to 194.95 THz.

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Fig. 7. Evolution of all signals along the fiber in each iteration under the starting boundary conditions of z=L (L=80 km). Figures (a), (b), (c), and (d) are the first-, second-, third-, and fourth-iteration for our algorithm, and their corresponding maximum of relative error of all signals are 0.3388, 0.0591, 0.0019, 9.844×10^-4, respectively. The initially assumed value for each signal at z=80 km is 0.15 mW (i.e., -8.239 dBm), which are shown in figure (a). After only 4 iterations, the required relative error of <10^-3 is contented [see figure (d)]. In the simulations, all parameters are same with Fig.5 and the noises are neglected (i.e., the first step of two-step procedure for the simulation algorithm in Section 3.2).

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Equations (24)

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\frac{d p^{\pm} (z, v_{i})}{d z} = \mp α (v_{i}) P^{\pm} (z, v_{i}) \pm η (v_{i}) P^{\mp} (z, v_{i})

\pm P^{\pm} (z, v_{i}) \sum_{m = 1}^{i - 1} \frac{g_{R} (v_{m} - v_{i})}{Γ A_{eff}} [P^{\pm} (z, v_{i}) + P^{\mp} (z, v_{i})]

\pm h v_{i} \sum_{m = 1}^{i - 1} \frac{g_{R} (v_{m} - v_{i})}{Γ A_{eff}} [P^{\pm} (z, v_{i}) + P^{\mp} (z, v_{i})] [1 + {(e^{\frac{h (v_{m} - v_{i})}{k T}} - 1)}^{- 1}] Δ v

\mp P^{\pm} (z, v_{i}) \sum_{m = i + 1}^{n} \frac{v_{i}}{v_{m}} \frac{g_{R} (v_{i} - v_{m})}{Γ A_{eff}} [P^{\pm} (z, v_{i}) + P^{\mp} (z, v_{i})]

\mp 2 h v_{i} P^{\pm} (z, v_{i}) \sum_{m = i + 1}^{n} \frac{v_{i}}{v_{m}} \frac{g_{R} (v_{i} - v_{m})}{Γ A_{eff}} [1 + {(e^{\frac{h (v_{i} - v_{m})}{k T}} - 1)}^{- 1}] Δ μ

\frac{d P^{\pm} (z, v_{i})}{d z} = P^{\pm} (z, v_{i}) F (z, v_{i}),

F (z, v_{i}) = \mp α (v_{i}) \pm \frac{η (v_{i}) P^{\mp} (z, v_{i})}{P^{\pm} (z, v_{i})}

\mp \sum_{m = i + 1}^{n} \frac{v_{i}}{v_{m}} \frac{g_{R} (v_{i} - v_{m})}{Γ A_{eff}} [P^{\pm} (z, v_{i}) + P^{\mp} (z, v_{i}) + 2 h v_{i} (1 + {(e^{\frac{h (v_{i} - v_{m})}{k T}} - 1)}^{- 1}) Δ μ]

\pm \sum_{m = 1}^{i - 1} \frac{g_{R} (v_{m} - v_{i})}{Γ A_{eff}} [P^{\pm} (z, v_{i}) + P^{\mp} (z, v_{i})] [1 + \frac{h v_{i}}{P^{\pm} (z, v_{i})} (1 + {(e^{\frac{h (v_{m} - v_{i})}{k T}} - 1)}^{- 1}) Δ v]

P^{\pm} (z_{j + 1}, v) = P^{\pm} (z_{j}, v) \exp (F (z_{j}, v) Δ z),

\bar{P^{\pm} (z_{j + 1}, v)} = P^{\pm} (z_{j}, v) \exp [(55 \cdot F (z_{j}, v) - 59 \cdot F (z_{j - 1}, v) + 37 \cdot F (z_{j - 2}, v) - 9 \cdot F (z_{j - 3}, v)) \frac{Δ z}{24}] .

P^{\pm} (z_{j + 1}, v) = P^{\pm} (z_{j}, v) \exp [(9 \cdot \bar{F (z_{j + 1}, v)} + 19 \cdot F (z_{j}, v) - 5 \cdot F (z_{j - 1}, v) + F (z_{j - 2}, v)) \frac{Δ z}{24}],

P^{\pm} (z_{1}, v) = P^{\pm} (z_{0}, v) \exp (F (z_{0}, v) Δ z),

\bar{P^{\pm} (z_{2}, v)} = P^{\pm} (z_{1}, v) \exp [\frac{(3 \cdot F (z_{1}, v) - F (z_{0}, v)) Δ z}{2}],

P^{\pm} (z_{2}, v) = P^{\pm} (z_{1}, v) \exp ⌊ \frac{(\bar{F (z_{2}, v)} + F (z_{1}, v)) Δ z}{2} ⌋,

\bar{P^{\pm} (z_{3}, v)} = P^{\pm} (z_{2}, v) \exp [\frac{(23 \cdot F (z_{2}, v) - 16 \cdot F (z_{1}, v) + 5 F (z_{0}, v)) Δ z}{12}],

P^{\pm} (z_{3}, v) = P^{\pm} (z_{2}, v) \exp [\frac{(5 \cdot \bar{F (z_{3}, v)} + 8 \cdot F (z_{2}, v) - F (z_{1}, v)) Δ z)}{12}],

\frac{d y}{d z} = - y + z + 1, and y (0) = 1 .

\frac{d y}{d z} = y \cdot \ln (y) \cdot (z^{2} + z + 1), and y (0) = 2 .

y (z) = \exp {\exp [\frac{1}{3} z^{3} + \frac{1}{2} z^{2} + z + \ln (\ln (2))]} .

\bar{P^{\pm} (z_{j + 1}, v)} = P^{\pm} (z_{j}, v) \exp [(1901 \cdot F (z_{j}, v) - 2774 \cdot F (z_{j - 1}, v)

+ 2616 \cdot F (z_{j - 2}, v) - 1274 \cdot F (z_{j - 3}, v) + 251 \cdot F (z_{j - 4}, v)) \frac{Δ z}{720}],

P^{\pm} (z_{j + 1}, v) = P^{\pm} (z_{j}, v) \exp [(251 \cdot \bar{F (z_{j + 1}, v)} + 646 \cdot F (z_{j}, v)

- 264 \cdot F (z_{j - 1}, v) + 106 \cdot F (z_{j - 2}, v) - 19 \cdot F (z_{j - 3}, v)) \frac{Δ z}{720}] .

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Equations (24)

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