Uncertainty relation for the optimization of optical-fiber transmission systems simulations

A. A. Rieznik; T. Tolisano; F. A. Callegari; D. F. Grosz; H.L. Fragnito

doi:10.1364/OPEX.13.003822

Optics Express
Vol. 13,
Issue 10,
pp. 3822-3834
(2005)
•https://doi.org/10.1364/OPEX.13.003822

Uncertainty relation for the optimization of optical-fiber transmission systems simulations

A. A. Rieznik, T. Tolisano, F. A. Callegari, D. F. Grosz, and H.L. Fragnito

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A. A. Rieznik, T. Tolisano, F. A. Callegari, D. F. Grosz, and H.L. Fragnito, "Uncertainty relation for the optimization of optical-fiber transmission systems simulations," Opt. Express 13, 3822-3834 (2005)

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Abstract

The mathematical inequality which in quantum mechanics gives rise to the uncertainty principle between two non commuting operators is used to develop a spatial step-size selection algorithm for the Split-Step Fourier Method (SSFM) for solving Generalized Non-Linear Schrödinger Equations (G-NLSEs). Numerical experiments are performed to analyze the efficiency of the method in modeling optical-fiber communications systems, showing its advantages relative to other algorithms.

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Fig. 1. Number of FFTs versus global relative error for a second order soliton propagating through ~81.2 km of fiber. Results form the UPM, LEM, and NPRM are shown. The lines joining the points are just aids to the eyes in identifying the general behavior of the methods.

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Fig. 2. Number of FFTs versus global relative error for a two first-order soliton collision propagating through 400 km of fiber. Results from the UPM, LEM, and NPRM are shown.

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Fig. 3. Number of FFTs versus global relative error for a WDM eight-channel system simulated using the UPM, LEM and WOM for propagating distances of (a) 10 km and (b) 50 km.

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Fig. 4. Global relative error as a function of the method parameter (ε or δ), for the UPM and the LEM, and linear fit of each curve. We show the results from the systems simulated in Section 3. The slopes given by the linear fits of the curves are shown in the labels.

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Tables (1)

Table 1. Main parameters used in the simulation of an eight-channel WDM system

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Equations (21)

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i \frac{\partial A}{\partial z} = - \frac{1}{2} β_{2} \frac{\partial^{2} A}{\partial t^{2}} - \frac{i}{6} β_{3} \frac{\partial^{3} A}{\partial t^{3}} + γ {∣ A ∣}^{2} A

i \frac{\partial A (z, t)}{\partial z} = (D + N) A (z, t),

D = \frac{1}{2} β_{2} \frac{\partial^{2}}{\partial t^{2}} + \frac{i}{6} β_{3} \frac{\partial^{3}}{\partial t^{3}}

N = γ {∣ A (z, t) ∣}^{2}

A (z + h, t) ≅ e^{- i h D} e^{- i h N} = F^{- 1} e^{- i h \tilde{D}} F e^{- i h N} A (z, t),

A (z + h, t) = e^{- i h (D + N)} A (z, t) .

e^{D} e^{N} = e^{D + N} + \frac{1}{2} [D, N] + \frac{1}{3!} {(N + 2 D) [D, N] + [D, N] (D + 2 N)} + \dots,

δ \equiv \frac{∥ A_{n} - A_{a} ∥}{∥ A_{a} ∥},

δ = (1 ⁄ 2) h^{2} {(\frac{{\int ∣ [D, N] A (t) ∣}^{2} d t}{{\int ∣ A (t) ∣}^{2} d t})}^{\frac{1}{2}} .

ε \equiv (1 ⁄ 2) h^{2} ∣ 〈 [D, N] 〉 ∣ \equiv (1 ⁄ 2) h^{2} \frac{∣ \int A^{*} (z, t) [D, N] A (z, t) d t ∣}{\int {∣ A (t) ∣}^{2} d t},

∣ 〈 [D, N] 〉 ∣ \leq 2 Δ D Δ N,

Δ B = \sqrt{〈 {(B - 〈 B 〉)}^{2} 〉} with 〈 B 〉 = \frac{\int A^{*} (z, t) B A (z, t) d t}{\int {∣ A (t) ∣}^{2} d t}

h = \sqrt{\frac{ε}{Δ D Δ N}}

〈 B 〉 = 〈 \tilde{B} 〉 = \frac{\int \tilde{A} * B \tilde{A} d ω}{{\int ∣ \tilde{A} ∣}^{2} d ω},

Δ D^{2} = Δ {\tilde{D}}^{2} = 〈 {[\frac{1}{2} β_{2} (ω^{2} - 〈 ω^{2} 〉) + \frac{1}{3!} β_{3} (ω^{3} - 〈 ω^{3} 〉)]}^{2} 〉 .

i \frac{\partial A}{\partial z} = (L + N) A,

\tilde{L} (z, ω) = \tilde{D} (z, ω) - \frac{1}{2} i α (z, ω),

ε = (1 ⁄ 2) h^{2} ∣ 〈 [L, N] 〉 ∣ = (1 ⁄ 2) h^{2} \sqrt{{∣ 〈 [D, N] 〉 ∣}^{2} + {∣ 〈 [(1 ⁄ 2) α, N] 〉 ∣}^{2}} .

ε \leq (1 ⁄ 2) h^{2} Δ N \sqrt{Δ D^{2} + (1 ⁄ 4) Δ α^{2}},

h = \sqrt{\frac{ε}{Δ D Δ N \sqrt{1 + {(Δ α ⁄ 2 Δ D)}^{2}}}}

A (z + h, t) ≅ e^{- i h D ⁄ 2} e^{- i h N} e^{- i h D ⁄ 2} A (z, t) .

Fiber parameters
γ	2 W^-1/km
β₂	-0.006 ps²/km
β₃	0.1 ps³/km
α	0 dB/km
Input field parameters
Channel frequencies	(1875+n*200) GHz, n=0,1…,7
Average power per channel	0 dBm
Noise power	-50 dBm @ 10 GHz
Numerical parameters
Sample points	2¹⁶=65536
Total time windows	3200 ps

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Equations (21)

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