The temperature dependent performance analysis of EDFAs pumped at 1480 nm: A more accurate propagation equation

Cüneyt Berkdemir; Sedat Özsoy

doi:10.1364/OPEX.13.005179

Optics Express
Vol. 13,
Issue 13,
pp. 5179-5185
(2005)
•https://doi.org/10.1364/OPEX.13.005179

The temperature dependent performance analysis of EDFAs pumped at 1480 nm: A more accurate propagation equation

Cüneyt Berkdemir and Sedat Özsoy

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Cüneyt Berkdemir and Sedat Özsoy, "The temperature dependent performance analysis of EDFAs pumped at 1480 nm: A more accurate propagation equation," Opt. Express 13, 5179-5185 (2005)

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About this Article
History
- Original Manuscript: May 3, 2005
- Revised Manuscript: June 21, 2005
- Manuscript Accepted: June 22, 2005
- Published: June 27, 2005

Abstract

An analytically expression for the temperature dependence of the signal gain of an erbium-doped fiber amplifier (EDFA) pumped at 1480 nm are theoretically obtained by solving the propagation equations with the amplified spontaneous emission (ASE). It is seen that the temperature dependence of the gain strongly depends on the distribution of population of Er³⁺-ions in the second level. In addition, the output pump power and the intrinsic saturation power of the signal beam are obtained as a function of the temperature. Numerical calculations are carried out for the temperature range from -20 to +60 °C and the various fiber lengths. But the other gain parameters, such as the pump and signal wavelengths and their powers, are taken as constants. It is shown that the gain decreases with increasing temperature within the range of L≤27 m.

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Fig. 1. Two level amplification system and main transitions of erbium ion.

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Fig. 2. Simulation setup for measurement of the co-propagating ASE power in an Er³⁺-doped optical fiber amplifier (from OptiAmplifier 4.0).

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Fig. 3. Gain as a function of fiber length. P_p (0)=30 mW and P_s (0)=10 µW.

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Tables (2)

Table 1. Typical fiber parameters for an Al/P-silica erbium-doped fiber (from Ref.[12]).

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Table 2. The relevant fiber parameters as a function of temperature.

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Equations (20)

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β = \frac{N_{2 +}}{N_{2 -}} = \frac{C_{nr}^{+}}{C_{nr}^{-}} = \exp (- \frac{Δ E_{2}}{k_{B} T})

\frac{d N_{2 +}}{dt} = R_{p}^{a} N_{1} - R_{p}^{e} N_{2 +} + C_{nr}^{-} N_{2 -} - C_{nr}^{+} N_{2 +},

\frac{d N_{2 -}}{dt} = S_{12} N_{1} - S_{21} N_{2 -} - N_{2 -} γ - C_{nr}^{-} N_{2 -} + C_{nr}^{+} N_{2 +},

\frac{d N_{1}}{dt} = R_{p}^{e} N_{2 +} - R_{p}^{a} N_{1} + S_{21} N_{2 -} - S_{12} N_{1} + N_{2 -} γ .

N_{2 -} = τ [(σ_{p}^{a} N_{1} - β σ_{p}^{e} N_{2−}) \frac{I_{p}}{h v_{p}} + (σ_{s}^{a} N_{1} - σ_{s}^{e} N_{2 -}) \frac{(I_{s} + I_{ASE}^{\pm})}{h v_{s}}],

\frac{N_{2 -}}{N} = \frac{\frac{I_{p}}{b_{p}^{a}} + \frac{(I_{s} + I_{ASE}^{\pm})}{b_{s}^{a}}}{(1 + β) \frac{I_{p}}{b_{p}^{a}} + β \frac{I_{p}}{b_{p}^{e}} + (1 + β + η) \frac{(I_{s} + I_{ASE}^{\pm})}{b_{s}^{a}} + 1}

\frac{d P_{s}}{d z} = 2 π \int_{0}^{\infty} I_{s} [σ_{s}^{e} N_{2 -} (r) - σ_{s}^{a} N_{1} (r)] rdr,

\frac{d P_{p}}{dz} = \pm 2 π \int_{0}^{\infty} I_{p} [β σ_{p}^{e} N_{2 -} (r) - σ_{p}^{a} N_{1} (r)] rdr,

\frac{d P_{ASE}^{\pm}}{dz} = \pm 2 h v_{s} \int_{0}^{\infty} 2 π σ_{s}^{e} N_{2 -} f_{ASE}^{\pm} (r) rdr \pm 2 π \int_{0}^{\infty} [σ_{s}^{e} N_{2−} (r) - σ_{s}^{a} N_{1} (r)] P_{ASE}^{\pm} f_{ASE}^{\pm} rdr,

P_{ASE}^{\pm} = P_{ASE}^{+} + P_{ASE}^{-} .

\frac{d P_{s}}{dz} = 2 π σ_{s}^{a} P_{s} (1 + β + η) \int_{0}^{\infty} N_{2 -} f_{2} (r) rdr - P_{s} α_{s},

\int_{0}^{\infty} N_{2 -} rdr = \int_{0}^{\infty} \frac{τ I_{p}}{h v_{p}} (σ_{p}^{a} N_{1} - β σ_{p}^{e} N_{2 -}) rdr + \int_{0}^{\infty} \frac{τ I_{s}}{h v_{s}} (σ_{s}^{a} N_{1} - σ_{s}^{e} N_{2}) rdr

+ \int_{0}^{\infty} \frac{τ I_{ASE}^{+}}{h v_{s}} (σ_{s}^{a} N_{1} - σ_{s}^{e} N_{2 -}) rdr,

\int_{0}^{\infty} N_{2 -} rdr = - \frac{τ}{2 π h v_{p}} \frac{d P_{p}}{dz} - \frac{τ}{2 π h v_{s}} \frac{d P_{s}}{dz} - \frac{τ}{2 π h v_{s}} \frac{d P_{ASE}^{+}}{dz} + 2 τ σ_{s}^{e} \int_{0}^{\infty} N_{2 -} f (r) rdr,

\int_{0}^{\infty} N_{2 -} f (r) rdr = - \frac{τ}{2 π (A ⁄ Γ - 2 τ σ_{s}^{e})} [\frac{1}{h v_{p}} \frac{d P_{p}}{dz} + \frac{1}{h v_{s}} (\frac{d P_{s}}{dz} + \frac{d P_{ASE}^{+}}{dz})],

\frac{d P_{s}}{dz} = - P_{s} (α_{s} + \frac{h v_{s}}{P_{s}^{int}} [\frac{1}{h v_{p}} \frac{d P_{p}}{dz} + \frac{1}{h v_{s}} (\frac{d P_{s}}{dz} + \frac{d P_{ASE}^{+}}{dz})]),

P_{s}^{int} = h v_{s} (A - 2 τ σ_{s}^{e} Γ) ⁄ τ σ_{s}^{a} Γ (1 + β + η) .

\frac{P_{s} (L)}{P_{s} (0)} = \exp (- α_{s} L) \exp (\frac{h v_{s}}{P_{s}^{int}} [\frac{P_{p} (0) - P_{p} (L)}{h v_{p}} + \frac{(P_{s} (0) + P_{ASE}^{+} (0)) - (P_{s} (L) + P_{ASE}^{+} (L))}{h v_{s}}]) .

G = \exp (- α_{s} L) \exp (\frac{h v_{s}}{P_{s}^{int}} [\frac{P_{p} (0) - P_{p} (L)}{h v_{p}} - \frac{P_{s} (0)}{h v_{s}} (1 - G) - \frac{P_{ASE}^{+} (L)}{h v_{s}}]),

P_{p} (L) = \frac{1}{R (η ⁄ b_{p}^{a} - β ⁄ b_{p}^{e})},

Symbols	Definitions	Values
$σ_{s}^{e}$	signal emission cross-section	5.7x10^-25 m ²
$σ_{s}^{a}$	signal absorption cross-section	6.6x10^-25 m ²
$σ_{p}^{e}$	pump emission cross-section	0.87x10^-25 m ²
$σ_{p}^{a}$	pump absorption cross-section	2.44x10^-25 m ²
τ	life time	10.8 ms
N	erbium concentration	3.86x10²⁴ m ^-3
λ_s	signal wavelength	1530 nm
λ_p	pump wavelength	1480 nm
$P_{ASE}^{+}$ (L)	copropagating ASE power	0.15 mW

Temperature (°C)	β	η	P_p (L)	$P_{s}^{int}$
-20	0.306	0.845	2.308 mW	0.493 mW
+20	0.357	0.862	2.311 mW	0.474 mW
+60	0.406	0.879	2.314 mW	0.451 mW

Symbols	Definitions	Values
$σ_{s}^{e}$	signal emission cross-section	5.7x10^-25 m ²
$σ_{s}^{a}$	signal absorption cross-section	6.6x10^-25 m ²
$σ_{p}^{e}$	pump emission cross-section	0.87x10^-25 m ²
$σ_{p}^{a}$	pump absorption cross-section	2.44x10^-25 m ²
τ	life time	10.8 ms
N	erbium concentration	3.86x10²⁴ m ^-3
λ_s	signal wavelength	1530 nm
λ_p	pump wavelength	1480 nm
$P_{ASE}^{+}$ (L)	copropagating ASE power	0.15 mW

Temperature (°C)	β	η	P_p (L)	$P_{s}^{int}$
-20	0.306	0.845	2.308 mW	0.493 mW
+20	0.357	0.862	2.311 mW	0.474 mW
+60	0.406	0.879	2.314 mW	0.451 mW

Abstract

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Equations (20)

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