1-

Introduction

The operational properties of erbium-doped fiber amplifiers (EDFAs), which have an important role in optical fiber communication systems operating at the 1.55 μm window, directly depend on the pump and signal characteristics. It is well-known that an erbium-doped fiber amplifier demonstrates the property of two-level amplification system when it is pumped at 1480 nm wavelength [1]. Particularly, the distribution of Er³+-ions within the metastable level (⁴I_13/2) changes continuously with temperature according to Boltzmann’s distribution law, owing to the closeness of the pump and signal rates within this level. Thus, the population of metastable level shows a highly temperature sensitive signal gain for 1480 nm pumping regime [2, 3]. In this study, we analyzed theoretically the signal gain performance of EDFAs pumped at 1480 nm pump wavelength by using the temperature-sensitive rate equation model in the practical temperature range of - 20 to 60 °C.

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Modified Rate Equation Model

The solution of the two-level amplification system can be carried out by using the modified rate equations in terms of the pump absorption and emission rates (R_1-22 and R_22-1, respectively), the signal absorption and emission rates (S_1-21 and S_21-1, respectively) and the amplified

spontaneous emission rate γ_sp. Then, an expression for the relative populations in the

metastable level, N₂ / N, at steady state conditions is found as

where v_p and v_s are the pump and signal frequencies, respectively; h is Planck’s constant; τ is the lifetime of metastable level; Γ_p and Γ_s are the pump and signal overlap factors, respectively; A_eff is the effective core area. The relevant parameters used in Eq.(1) are as bellows: σ_1-22 and σ_22-1 are the stimulated absorption and emission cross sections for the pump wavelength, while σ_1-21 and σ_21-1 are for the signal wavelength; η = σ_21-1/σ_1-21 (from McCumber’s theory). P_p and P_s are the pump and signal powers, respectively. The parameter β in Eq.(1) is characterized by the Boltzmann’s distribution of Er³+-ions within the ⁴I_{13 / 2} energy state and given by β = N₂₂/N₂₁ = exp(-ΔE₂/k_BT). k_B is the Boltzmann’s constant, ΔE₂ = E₂₂ - E₂₁ is the energy difference between the sublevels N₂₂ and N₂₁ and its value is nearly 200 cm^-1 at temperature range of-20 to 60 ^oC. The gain parameter G(λ_s) is defined by G(λ_s) = e^γ.L, where γ = σ_1-21(ν₂ -N₁)Γ, γ is small-signal gain coefficient and L is the doped fiber length.

Results and Discussions

Using EDFA parameters given in Table 1, we have plotted the dependence of gain on the launched pump powers at temperature values of-20°C, 20°C and 60°C in Fig.1.

Fig. 1.

The variation of signal gains versus launched pomp powers for the signal power regime 50 μW. (a), (b) and (c) curves correspond to - 20°C, 20°C and 60 °C, respectively. τ is taken as 10 ms and L = 15 m.

Table 1.

EDFA parameters used for the calculations.

σ1-22 (10-25 m2)	σ22-1 (10-25 m2)	σ1-21 (10-25 m2)	σ21-1 (10-25 m2)	Γp; Γs	Aeff (10-12 m2)	N (1023 m-
1,86	0,42	2,85	5,03	0,43; 0,37	12,6	9,28

The value ofσ_21-1/σ_1-21 at room temperature (20°C) for 1550 nm signal wavelength is obtained as 1.76, because there is a relationship between signal emission and absorption cross sections in the form of η = exp [hc/k_BT(1/λ₀-1/λ_s)] according to McCumber’s theory. The wavelength λ₀ depends on the electronic structure of the ground and the excited state of the Er³+-ion and its value is calculated as 1522, 7 nm. Thus, the values of η at – 20 and 60°C are 1.93 and 1.65 and also the values of β at – 20, 20 and 60°C are calculated as 0.33, 0.38, 0.43, respectively.

Fig. 1 shows in 50 μW signal power regime that, the gain values increase an amount of 3-5 dB when the temperature is decreased from 60 °C to -20 °C. Thus, it is seen from the gain analysis that the performance of EDFAs pumped at 1480 nm pump wavelength is affected by the temperature. In addition, it is seen from the modified rate equation (Eq.1) that the smaller signal powers are more efficient than the higher signal powers. Moreover, this model which involves β and η is more accurate to describe the signal gain for a wide range of temperatures for EDFAs pumped at 1480 nm pump wavelength.