Analytical characterization of optical power and noise figure of forward pumped Raman amplifiers

Malin Premaratne

doi:10.1364/OPEX.12.004235

Optics Express
Vol. 12,
Issue 18,
pp. 4235-4245
(2004)
•https://doi.org/10.1364/OPEX.12.004235

Analytical characterization of optical power and noise figure of forward pumped Raman amplifiers

Malin Premaratne

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Malin Premaratne, "Analytical characterization of optical power and noise figure of forward pumped Raman amplifiers," Opt. Express 12, 4235-4245 (2004)

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Abstract

We show that it is possible to find analytic expressions for characterizing the evolution of signal and noise photon numbers along the active fiber of a forward-pumped Raman amplifier with unequal signal and pump loss coefficients. We confirm the validity of the result by comparing the analytical solutions with numerical calculations and by analytically deriving the well-known 3 dB noise figure limit for high Raman gain. Apart from aiding the analysis and design of forward pumped Raman amplifiers, these results also enable one to find approximate analytical solutions for bidirectional Raman amplifiers and backward pumped Raman amplifiers with Rayleigh backscattering and Brillouin scattering.

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Fig. 1. A schematic diagram of forward pumped Raman amplifier.

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Fig. 2. Photon number, n(z), against fiber length for two different pump powers P_P and pump loss coefficients α_P

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Fig. 3. I(z) against fiber length for two different pump loss coefficients,α_P = 0.4 dB/km and α_P = 0.3 dB/km.

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Fig. 4. Noise Figure against fiber length for two different pump loss coefficients: (a) α_P = 0.4 dB/km (b) α_P = 0.3 dB/km

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Equations (40)

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\frac{dn (z)}{dz} = γ n_{p} (z) (n (z) + 1) - α_{S} n (z)

\frac{d n_{P} (z)}{dz} = - γ n_{p} (z) (n (z) + 1) - α_{P} n_{P} (z)

n_{P} (z) = n_{P} (0) \exp (- α_{p} z)

\frac{dn (z)}{dz} + (α_{S} - γ n_{P} (0) \exp (- α_{P} z)) n (z) = γ n_{p} (0) \exp (- α_{P} z)

n (z) = \sum_{i = 0}^{\infty} n_{i} (z) α^{i}

\frac{d n_{0} (z)}{dz} + (α_{P} - γ n_{P} (0) \exp (- α_{P} z)) n_{0} (z) = γ n_{p} (0) \exp (- α_{P} z)

\frac{d n_{i} (z)}{dz} + (α_{P} - γ n_{P} (0) \exp (- α_{P} z)) n_{i} (z) = α_{P} n_{i - 1} (z), i \neq 0

n_{0} (z) = H_{P} (z) \exp (u_{0}) n (0) + H_{P} (z) (Ei (u_{0}) - Ei (u (z))) u_{0}

Ei (x) = P . V . \int_{- \infty}^{x} u^{- 1} \exp (u) du, x > 0

n_{1} (z) = α_{P} H_{P} (z) (\exp (u_{0}) n (0) + u_{0} Ei) u_{0})) z - α_{P} H_{P} (z) u_{0} \int_{0}^{z} Ei (u (z)) dz

\int_{0}^{z} Ei (u (z)) dz \approx (C + \ln (u_{0})) z - \frac{α_{P}}{2} z^{2} + \frac{u_{0}^{\ln (u_{0})}}{α_{P}} (1 - \frac{u^{2} (z)}{u_{0}^{2}})

n_{1} (z) = α α_{P} \times (\binom{\exp (u_{0}) n (0) z + \frac{α_{P}}{2} u_{0} z^{2} + (Ei (u_{0})}{- C - \ln (u_{0}))) u_{0} z - \frac{u_{0}^{1 + \ln (u_{0})}}{α_{P}} (1 - \frac{u^{2} (z)}{u_{0}^{2}})}) H_{P} (z)

\frac{n (z)}{H_{P} (z)} \approx (\exp (u_{0}) n (0) + u_{0} (Ei (u_{0}) - Ei (u (z)))

+ α α_{P} \times (\binom{\exp (u_{0}) n (0) z + \frac{α_{P}}{2} u_{0} z^{2} + (Ei (u_{0})}{- C - \ln (u_{0}))) u_{0} z - \frac{u_{0}^{1 + \ln (u_{0})}}{α_{P}} (1 - \frac{u^{2} (z)}{u_{0}^{2}})})

+ \sum_{i = 2}^{\infty} \frac{{(α_{P} z)}^{i}}{i!} (\exp (u_{0}) n (0) + u_{0} Ei (u_{0})) α^{i}

n (z) = G_{S} (z) n (0) + (α_{P} - α_{S}) \sqrt{H_{P} (z) H_{S} (z)} ℘ (z)

+ H_{S} (z) u_{0} Ei (u_{0}) - H_{P} (z) u_{0} Ei (u (z))

℘ (z) = \frac{α_{P}}{2} u_{0} z^{2} - u_{0} (C + \ln (u_{0})) z - \frac{u_{0}^{1 + \ln (u_{0})}}{α_{P}} (1 - \frac{u^{2} (z)}{u_{0}^{2}})

n (z) ∣_{α = 0} = G_{S} (z) n (0) + u_{0} H_{S} (z) \times (Ei (u_{0}) - Ei (u (z)))

n (z) = G_{S} (z) \times n (0)

n (z) = G_{S} (z) \times \hat{n} (z)

\frac{d \hat{n} (z)}{dz} = α_{P} u_{0} \exp (- α_{P} αz - u_{0} (1 - \exp (- α_{P} z)))

\hat{n} (0) = n (0) + u_{0} \exp (- u_{0}) \times I (z)

I (z) = \frac{1}{u_{0}^{α}} \int_{u (z)}^{u_{0}} v^{α - 1} \exp (v) dv

I (z) = \sum_{k = 0}^{\infty} \frac{{(α)}_{k} u_{0}^{k}}{{(α + 1)}_{k} k!} - {(\frac{u (z)}{u_{0}})}^{α} \sum_{k = 0}^{\infty} \frac{{(α)}_{k} u^{k} (z)}{{(α + 1)}_{k} k!}

Φ (a; b; z) = \sum_{k = 0}^{\infty} \frac{{(a)}_{k} z^{k}}{{(b)}_{k} k!}

n (z) = G_{S} (z) n (0) + \frac{u_{0} H_{S} (z)}{α} (Φ (α; α + 1; u_{0}) - Φ (α; α + 1; u (z)) {(\frac{u (z)}{u_{0}})}^{α})

n_{noise} (0) = \frac{u_{0} \exp (- u_{0})}{α} (Φ (α; α + 1; u_{0}) {- Φ (α; α + 1; u (z)) (\frac{u (z)}{u_{0}})}^{α})

n (z) ∣_{α = 0} = G_{S} (z) n (0) +

u_{0} H_{S} (z) \times lim_{α \to 0} \frac{d}{dα} (Φ (α; α + 1; u_{0}) - Φ (α; α; + 1; u (z)) {(\frac{u (z)}{u_{0}})}^{α})

n (z) ∣_{α = 0} = G_{S} (z) n (0) + u_{0} H_{S} (z) \times (\sum_{k = 1}^{\infty} \frac{u_{0}^{k}}{k \cdot k!} + \ln (u_{0}) - \sum_{k = 1}^{\infty} \frac{u^{k} (z)}{k \cdot k!} - \ln (u (z)))

Ei (x) = C + \ln (x) + \sum_{k = 1}^{\infty} \frac{x^{k}}{k \cdot k!}

NF (z) = \frac{1 + 2 (n (z) - G_{S} (z) n (0))}{G_{S} (z)}

{NF}_{a} (z) = \frac{1 + 2 (α_{P} - α_{S}) \sqrt{H_{P} (z) H_{S} (z)} ℘ (z) + 2 (H_{S} (z) u_{0} Ei (u_{0}) - H_{P} (z) u (z) Ei (u (z)))}{\exp (u_{0}) H_{S} (z)}

{NF}_{e} (z) = \frac{1}{G (z)} + 2 \frac{u_{0} \exp (- u_{0})}{α} (Φ (α; α + 1, u_{0}) - Φ (α; α + 1, u (z)))

{NF}_{a} (z) \approx \frac{2 (u_{0} Ei (u_{0}) - u (z) Ei (u (z)))}{\exp (u_{0})}

Ei (x) = \frac{\exp (x)}{x} (1 + O (\frac{1}{x}))

\frac{dn (z)}{dz} = (\sum_{k = 1}^{M} γ_{k} n_{Pk} (z)) (n (z) + 1) - α_{S} n (z)

\frac{dn P_{i} (z)}{dz} = γ_{i} n P_{i} (z) ({\sum_{k = 1}}_{k \neq i}^{M} nPk (z) + n (z) + 1) - α_{Pi} n_{Pi} (z), i = 1,2, \dots, M

n (z) = \sum_{i = 0}^{\infty} n_{i} (z) {(1 - \frac{αM}{\sum_{k = 1}^{M} α_{k}})}^{i}

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