Parametric amplification in presence of dispersion fluctuations

Mitra Farahmand; Martijn de Sterke

doi:10.1364/OPEX.12.000136

Optics Express
Vol. 12,
Issue 1,
pp. 136-142
(2004)
•https://doi.org/10.1364/OPEX.12.000136

Parametric amplification in presence of dispersion fluctuations

Mitra Farahmand and Martijn de Sterke

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Mitra Farahmand and Martijn de Sterke, "Parametric amplification in presence of dispersion fluctuations," Opt. Express 12, 136-142 (2004)

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Abstract

Parametric amplification in fibers with dispersion fluctuations is analyzed. The fluctuations are modelled as a stochastic process, with their size at a given position modelled as a Gaussian, and the autocorrelation decreasing exponentially. Two models are studied: in one the dispersion is piecewise constant, while in the other it is continuous. We find that the amplification does not depend on the models’ details and that only fluctuations with long correlation lengths affect the amplification significantly.

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Fig. 1. (a) Example of dispersion fluctuations according to Model I. (b) Same as (a), but for Model II.

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Fig. 2. Numerical results for the expectation value of the gain versus γP ₀ L_c , for γP ₀ L=4. The horizontal solid line refers to result (5) whereas the dotted line indicates result (10). The red coloring refers to Model I, whereas green refers to Model II. The error bars indicate the variance of the distribution. In (a) and (b) Δ/(γP ₀)=0.25; in (c) and (d) Δ/(γP ₀)=2.00; in (e) and (f) Δ/(γP ₀)=1.00. Further, in (a), (c) and (e) σ/(γP ₀)=0.50, while in (b), (d) and (f) σ/(γP ₀)=1.00.

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Equations (10)

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Δ β = 2 β_{p} - β_{s} - β_{i},

A_{p} = \sqrt{P_{0}} \exp (i γ P_{0} z),

\frac{d A}{dz} = (\begin{matrix} i Δ k & i γ P_{0} \\ - i γ P_{0} & - i Δ k \end{matrix}) A .

A (z) = MA (0) \equiv [\begin{matrix} \cosh α z + \frac{i Δ k}{α} \sinh α z & \frac{i γ P_{0}}{α} \sinh α z \\ - \frac{i γ P_{0}}{α} \sinh α z & \cosh α z - \frac{i Δ k}{α} \sinh α z \end{matrix}] A (0),

G = \cosh^{2} α L + \frac{{(Δ k)}^{2}}{α^{2}} \sinh^{2} α L,

f_{G} (δ k) = \frac{1}{σ \sqrt{2 π}} \exp [- \frac{1}{2} {(\frac{δ k}{σ^{2}})}^{2}],

C (z) = σ^{2} \exp (- ∣ z ∣ ⁄ L_{c}),

P (L_{s}) \propto e^{- L_{s} ⁄ L_{c}} .

f (δ k (z_{1}) ∣ δ k (z_{0})) = \frac{1}{σ \sqrt{2 π (1 - r^{2})}} e^{- \frac{1}{2 σ^{2} (1 - r^{2})} {(δ k (z_{1}) - r δ k (z_{0}))}^{2}},

G (L_{c} ≫ L) = \frac{1}{\sqrt{2 π} σ} \int_{- \infty}^{+ \infty} d (Δ k) e^{- {(Δ k - Δ)}^{2} ⁄ 2 σ^{2}} \log [\cosh^{2} α L + \frac{{(Δ k)}^{2}}{α^{2}} \sinh^{2} α L] .

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Equations (10)

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