Optimization of Multiple Active Ion Doped Fiber Amplifiers for Three Communication Windows

Abstract

We present for the first time a theoretical model of Er³⁺-Tm³⁺-Pr³⁺ codoped fiber pumped with both 800 nm and 980 nm lasers to explore possibility of this co-doped system as all-wave fiber amplifier. The rate and power propagation equations of the model are solved numerically and the dependence of the gains at 1310, 1470, 1530, 1600, 1650 nm windows on fiber length is calculated. The results show that with pump power of 200 mW/200 mW, when the concentrations of Pr³⁺, Tm³⁺, Er³⁺ are around 1.7×10²⁴, 3.9×10²⁴, 1.2×10²⁴ (ions/m³), respectively, the signals at 1310, 1470, 1530, 1600, 1650 nm may be nearly equally amplified with gain of 13–16.0 dB in the active fiber with length of 23.5 m; the co-doping concentrations and fiber length and pump powers may be further optimized to reduce the ripple.

1. Introduction

All-wave fiber in which OH group was suppressed is attracting increasing interest in optical transmission system and network because it has low loss windows of 400 nm covering the range 1250–1650 nm. Wavelength Division Multiplexing (WDM) has been the most important technology of large capacity optical transmission system, and optical amplifiers are the key devices of WDM system. Although Fiber Raman Amplifier (FRA) is a promising candidate for long haul and large capacity transmission system, it requires high pump power due to its lower pump efficiency; thus, the solution scheme for all-wave fiber transmission system is not available yet. Compared to FRA, rare-earth doped fiber amplifier has high gain and high pump efficiency, and in the past decade, the research on the rare-earth doped fiber amplifier has been focusing on the single-rare earth-doped fiber amplifiers, and all the amplifiers have their own bandwidths. Er³⁺-doped fiber amplifier (EDFA) with new split-band configuration [1, 2] was reported providing gain bandwidth of more than 100 nm covering the range 1500–1600 nm; Tm³⁺-doped fiber amplifiers (TDFAs) [3–5] and Pr³⁺-doped fiber amplifiers (PDFA) [6] separately provided amplification in range of 1450–1520 nm and 1280–1340 nm, respectively.

Recent research on emission properties of Er³⁺-Tm³⁺co-doped and Pr³⁺-Er³⁺ co-doped fibers showed that the combination of the emission at 1530 nm window due to Er³⁺:⁴I_13/2→⁴I_15/2 transition with the emission at 1470 nm window due to Tm³⁺:³H₄-³F₄ transition may generate a larger seamless emission spectrum up to 200 nm in the co-doped system [7–12]. Meanwhile, the research on emission properties of Pr³⁺-Er³⁺ co-doped fiber showed that the combination of the emission at 1530 nm window due to Er³⁺:⁴I_13/2→⁴I_15/2 transition with the emission at 1310 nm window due to Pr³⁺:³F₄-³H₅ transition may generate an emission spectrum having two peaks centered at 1310 nm and 1530 nm windows [7–13]. In this article, we present a theoretical model of Er³⁺-Tm³⁺-Pr³⁺ co-doped fiber amplifier for the first time to explore the possibility of this multiple rare-earth doped system for all-wave fiber transmission system application. After the rate and power propagation equations of the doped system are solved numerically and analyzed, the parameters of doped fiber are optimized to achieve the equalized gains for 1310, 1470, 1530, 1600, 1650 nm bands.

2. Theoretical Model

Figure 1 shows the schematic of the energy levels and electron transitions and energy transfer process of Er³⁺-Tm³⁺- Pr³⁺-co-doped system pumped by both 800 nm and 980 nm lasers. Following the diagram, the rate equations can be written as an equation group:𝜕𝑁₁𝜕𝑡=−(𝑊¹³+𝑊₁₄)𝑁₁+𝐴₂₁𝑁₂+(𝑊₃₁+𝐴₃₁)𝑁₃𝜕𝑁₂+(𝑊₄₁+𝐴₄₁)𝑁₄−𝑊_ET63𝑁₁𝑁₆−𝑊_ET74𝑁₁𝑁₇,𝜕𝑡=−(𝑊²⁴𝜕𝑁₃+𝐴₂₁)𝑁₂+(𝑊₄₂+𝐴₄₂)𝑁₄,𝜕𝑡=𝑊¹³𝜕𝑁₄𝑁₁−(𝑊₃₁+𝐴₃₁)𝑁₃+𝑊_ET63𝑁₁𝑁₆,𝜕𝑡=𝑊²⁴𝜕𝑁₅𝑁₂−(𝑊₄₁+𝑊₄₂+𝐴₄₁)𝑁₄+𝑊_ET74𝑁₁𝑁₇,𝜕𝑡=−(𝑊⁵⁷+𝑊₅₈)𝑁₅+𝑊₆₈𝑁²⁶+𝑊_ET63𝑁₁𝑁₆+𝑊_ET74𝑁₁𝑁₇+(𝑊₆₅+𝐴₆₅)𝑁₆+𝑊_ET6−10𝑁₆𝑁₉𝜕𝑁₆+𝑊_ET7−11𝑁₇𝑁₉−𝑊_ET12−8𝑁₅𝑁₁₂,𝜕𝑡=−𝑊^ET63𝑁₁𝑁₆−(𝑊₆₅+𝐴₆₅)𝑁₆𝜕𝑁₇−𝑊_ET6−10𝑁₆𝑁₉+𝐴₇₆𝑁₇−2𝑊₆₈𝑁²⁶,𝜕𝑡=𝑊⁵⁷𝑁₅+𝐴₈₇𝑁₈−(𝑊_ET74𝑁₁+𝐴₇₆)𝑁₇𝜕𝑁₈−𝑊_ET7−11𝑁₇𝑁₉,𝜕𝑡=𝑊⁵⁸𝜕𝑁₉𝑁₅−𝑊_ET12−8𝑁₁₂𝑁₅−𝐴₈₇𝑁₈+𝑊₆₈𝑁²⁶,𝜕𝑡=−(𝑊⁹⁻¹²+𝑊_ET6−10𝑁₆+𝑊_ET7−11𝑁₇)𝑁₉𝜕𝑁₁₀+𝑊_ET12−8𝑁₅𝑁₁₂+(𝑊₁₀₋₉+𝐴₁₀₋₉)𝑁₁₀−𝑊₉₋₁₀𝑁₉,𝜕𝑡=−(𝑊¹⁰⁻⁹+𝐴₁₀₋₉)𝑁₁₀+𝑊₉₋₁₀𝑁₉+𝑊_ET6−10𝑁₆𝑁₉+𝐴₁₁₋₁₀𝑁₁₁+(𝑊₁₂₋₁₀+𝐴₁₂₋₁₀)𝑁₁₂𝜕𝑁₁₁−𝑊₁₀₋₁₂𝑁₁₀−𝑊₁₀₋₁₃𝑁₁₀,𝜕𝑡=𝑊^ET7−11𝜕𝑁₁₂𝑁₇𝑁₉−𝐴₁₁₋₁₀𝑁₁₁+𝐴₁₂₋₁₁𝑁₁₂,𝜕𝑡=𝑊⁹⁻¹²𝑁₉+𝐴₁₃₋₁₂𝑁₁₃𝜕𝑁₁₃−(𝑊_ET12−8𝑁₅+𝐴₁₂₋₁₁+𝑊₁₂₋₁₀)𝑁₁₂+𝑊₁₀₋₁₂𝑁₁₀,𝜕𝑡=𝑊¹⁰⁻¹³𝑁₁₀−𝐴₁₃₋₁₂𝑁₁₃,(1) where 𝑁₁-𝑁₄ are the population densities of Pr³⁺ ion at energy levels ³H₄, ³H₅, ³F₄, ¹G₄, 𝑁₅-𝑁₈ are the population densities of Er³⁺ ion at energy levels ⁴I_15/2, ⁴I_13/2, ⁴I_11/2, and ⁴I_9/2, 𝑁₉-𝑁₁₃ are the population densities of Tm³⁺ ions at energy levels ³H₆, ³F₄, ³H₅, ⁴H₄, ¹G₄. 𝑊₁₃, 𝑊₃₁, 𝐴₃₁ are the stimulated absorption and emission rates, spontaneous emission rate between ³H₄ and ³F₄ levels of Pr³⁺, respectively. 𝑊₂₄, 𝑊₄₂, 𝐴₄₂ are the stimulated absorption and emission rates, spontaneous emission rate between the ³H₅ and ¹G₄ levels of Pr³⁺, respectively. 𝐴₂₁ is the spontaneous emission rate between the ³H₄ and ³H₅ levels of Pr³⁺. 𝑊₅₆, 𝑊₆₅, 𝐴₆₅ are the stimulated absorption and emission rates, spontaneous emission rate between the ⁴I_15/2 and ⁴I_13/2 levels of Er³⁺, respectively. 𝑊₅₈, 𝐴₈₇, 𝐴₇₆ are the 800 nm- pump absorption rate, spontaneous emission rate from ⁴I_9/2 to ⁴I_11/2 levels, spontaneous emission rate from ⁴I_11/2 to ⁴I_13/2 levels of Er³⁺, respectively. 𝑊_10-12, 𝑊_12-10, 𝐴_12-10 are the stimulated absorption and emission rates, spontaneous emission rate between the ³H₄ and ³F₄ levels of Tm³⁺, respectively. 𝑊_9-12, 𝑊_9-10, 𝑊_10-9, 𝐴_10-9, 𝐴_12-11, 𝐴_11-10 are the 800 nm pump absorption rate, stimulated absorption rate, stimulated emission rate, spontaneous emission rate between ³H₆ and ³F₄ levels of Tm³⁺, nonradiation transition rate from ³H₄ to ³H₅, nonradiation transition rate from ³H₅ to ³F₄, respectively. 𝑊_1-4, 𝑊_5-8, 𝑊_10-13 are 980 nm pump absorption rates between the ³H₄ and ¹G₄ levels of Pr³⁺, between the ⁴I_15/2 and ⁴I_19/2 levels of Er³⁺, and between the ³H₆ and ³H₄ levels of Tm³⁺, respectively. 𝑊_ET6-3, 𝑊_ET7-4, 𝑊_ET6-10, 𝑊_ET7-11, 𝑊_ET12-8 stand for the transfer rates from Er³⁺:⁴I_13/2, Pr³⁺:³H₄ to Er³⁺:⁴I_15/2, Pr³⁺:³F₄, from Er³⁺:⁴I_11/2, Pr³⁺:³F₄ to Er³⁺:⁴I_15/2, Pr³⁺:¹G₄, from Er³⁺:⁴I_11/2, Tm³⁺:³H₆ to Er³⁺:⁴I_15/2, Tm³⁺:³H₅, from Er³⁺:⁴I_13/2, Tm³⁺:³H₆ to Er³⁺:⁴I_15/2, Tm³⁺:³F₄, from Tm³⁺:³H₄, Er³⁺:⁴I_15/2 to Tm³⁺:³H₆, and from Er³⁺:⁴I_9/2, Tm³⁺:³H₆ to Er³⁺:⁴I_15/2, Tm³⁺:³F₄, respectively. The transition rates: 𝑊_𝑖𝑗=𝜎^𝑖𝑗𝑃_𝑘ℎ𝜈_𝑘𝐴_eff(𝑖,𝑗=1∼12,𝑘=𝑠,𝑝),(2) where 𝜎_𝑖𝑗 is cross-section of the transition between 𝑖 and 𝑗 level, and 𝐴_eff is the effective cross-section area. Propagation of the pump and signal and ASE power along the fiber is described by the differential equation group:𝑑𝑃_𝑆1𝑑𝑧=Γ¹³¹⁰𝑑𝑃_𝑆2(𝑁₃𝜎_prse𝐵−𝑁₂𝜎_prsa𝐵)𝑃_𝑆1−𝛼₁₃₁₀𝑃_𝑆1,𝑑𝑧=Γ¹⁴⁷⁰𝑑𝑃_𝑆3(𝑁₁₂𝜎_tmse𝐴−𝑁₁₁𝜎_tmsa𝐴)𝑃_𝑆2−𝛼₁₄₇₀𝑃_𝑆2,𝑑𝑧=Γ¹⁵⁵⁰𝑑𝑃_𝑆4(𝑁₆𝜎_erse−𝑁₅𝜎_ersa𝐴−𝑁₈𝜎_ersa𝐵)𝑃_𝑆3−𝛼₁₅₅₀𝑃_𝑆3,𝑑𝑧=Γ¹⁶⁰⁰𝑑𝑃_𝑆5(𝑁₃𝜎_prse𝐴−𝑁₁𝜎_prsa𝐴)𝑃_𝑆4−𝛼₁₆₀₀𝑃_𝑆4,𝑑𝑧=Γ¹⁶⁵⁰𝑑𝑃𝑝₁(𝑁₁₀𝜎_tmse𝐵−𝑁₉𝜎_tmsa𝐵)𝑃_𝑆5−𝛼₁₆₅₀𝑃_𝑆5,𝑑𝑧=−Γ⁸⁰⁰[(+𝑁₅(𝑁𝜎_erpa9−𝑁₈𝜎_erpe)𝑑𝑃𝑝₂𝜎_tmpa−𝑁₁₂𝜎_tmpe)]𝑃𝑝₁−𝛼₈₀₀𝑃𝑝₁,𝑑𝑧=−Γ⁹⁸⁰[(+𝑁₁(𝑁𝜎_prpa10−𝑁₄𝜎_prpe)+(𝑁₅𝜎_erpa−𝑁₇𝜎_erpe)𝑑𝑃_ASE1𝜎_tmpa−𝑁₁₃𝜎_tmpe)]𝑃𝑝₂−𝛼₉₈₀𝑃𝑝₂,𝑑𝑧=Γ¹³¹⁰(𝑁₃𝜎_prse𝐵−𝑁₂𝜎_prsa𝐵)𝑃_ASE1𝑑𝑃_ASE2+2ℎ𝜐Δ𝜐𝑁₃𝜎_prse𝐵−𝛼₁₃₁₀𝑃_ASE1,𝑑𝑧=Γ¹⁴⁷⁰(𝑁₁₂𝜎_tmse𝐴−𝑁₁₁𝜎_tmsa𝐴)𝑃_ASE2𝑑𝑃_ASE3+2ℎ𝜐Δ𝜐𝑁₁₂𝜎_tmse𝐴−𝛼₁₄₇₀𝑃_ASE2,𝑑𝑧=Γ¹⁵⁵⁰(𝑁₆𝜎_erse−𝑁₅𝜎_ersa𝐴−𝑁₈𝜎_ersa𝐵)𝑃_ASE3𝑑𝑃_ASE4+2ℎ𝜐Δ𝜐𝑁₆𝜎_erse−𝛼₁₅₅₀𝑃_ASE3,𝑑𝑧=Γ¹⁶⁰⁰(𝑁₃𝜎_prse𝐴−𝑁₁𝜎_prsa𝐴)𝑃_ASE4𝑑𝑃_ASE5+2ℎ𝜐Δ𝜐𝑁₃𝜎_prse𝐴−𝛼₁₆₀₀𝑃_ASE4,𝑑𝑧=Γ¹⁶⁵⁰(𝑁₁₀𝜎_tmse𝐵−𝑁₉𝜎_tmsa𝐵)𝑃_ASE5+2ℎ𝜐Δ𝜐𝑁₁₀𝜎_tmse𝐵−𝛼₁₆₅₀𝑃_ASE5,(3) where 𝑃_𝑝1, 𝑃_𝑝2 are the pump powers at 800 nm, 980 nm, respectively. 𝑃_𝑆1, 𝑃_𝑆2, 𝑃_𝑆3, 𝑃_𝑆4, 𝑃_𝑆5 are the powers of the signals at 1310, 1470, 1530, 1600, and 1650 nm bands, respectively. 𝑃_ASE1, 𝑃_ASE2, 𝑃_ASE3, 𝑃_ASE4, 𝑃_ASE5 are the powers of the ASE at 1310, 1470, 1530, 1600, and 1650 nm bands, respectively. Γ₁₃₁₀, Γ₁₄₇₀, Γ₁₅₃₀, Γ₁₆₀₀, Γ₁₆₅₀ are overlapping factors at 1310, 1470, 1530, 1600, 1650 nm bands, respectively, and calculated from [14], and 𝜐 is signal frequency. 𝛼(𝑣) is the frequency dependent background loss of the active fiber. 𝜎_tmse𝐴, 𝜎_tmse𝐵, 𝜎_erse are the emission cross-sections of ³H₄-³F₄ (1470 nm) and ³F₄-³H₆ (1650 nm) in Tm³⁺ ions and ⁴I_13/2-⁴I_15/2 (1530 nm) in Er³⁺ ions, respectively. 𝜎_tmsa𝐴, 𝜎_tmsa𝐵, 𝜎_ersa are the absorption cross-sections of ³H₄-³F₄ (1470 nm) and ³F₄-³H₆ (1650 nm) in Tm³⁺ ions and ⁴I_13/2-⁴I_15/2 (1530 nm) in Er³⁺ ions, respectively. 𝜎_prse𝐴, 𝜎_prsa𝐴 and 𝜎_prse𝐵, 𝜎_prsa𝐵 are the emission cross-sections of ¹G₄-³H₅ (1310 nm) and ³F₄-³H₄ (1600 nm) in Pr³⁺ ions.

Figure 1 

Schematic of energy levels and transition configurations of Er3+-Tm3+-Pr3+ co-doped telluride fiber amplifier pumped with both 800 nm and 980 nm laser diodes.

The above differential equation group is solved by numerical integration along the active fiber using Newton iterative method and Runge-Kutta method. It was assumed that the energy transfer rates (𝑊_ET63, 𝑊_ET74, 𝑊_ET6-10,𝑊_ET7-11, 𝑊_ET12-8) were linearly increasing functions of 𝑁^𝑡_Er, 𝑁^𝑡_Tm, 𝑁^𝑡𝑃𝑟, respectively, [15, 16] and are expressed with equations:𝑊_ET63=×𝑊10_ET74−49=1.0×10⁻²²+4.0𝑊_ET6-10(𝑁_𝑃𝑟𝑁_Er)^1/2−1.0×10²⁵,=×𝑊10_{ET7_11−49}=1.0×10⁻²²+4.0𝑊_ET12-8(𝑁_Tm𝑁_Er)^1/2−1.0×10²⁵,=×1.010×1₋₄₉0⁻²²+4.0(𝑁_Tm𝑁_Er)^1/2−1.0×10²⁵.(4)

3. Result and Discussion

Figure 2 shows the variation of the gains at the five bands (1310, 1470, 1530, 1600, 1650 nm) with the fiber length and with optimized dopant concentrations: Pr³⁺ concentration at 1.7×10²⁴, Tm³⁺ concentration at 3.9×10²⁴, Er³⁺ concentration at 1.2×10²⁴ ions/m³ and fixed pump powers at 200 mW/200 mW for 800 nm/980 nm.

Figure 2 

Variation of the gain at 1310, 1470, 1530, 1600, 1650 nm with fiber length. Pump power, input signal power are 200 mW/200 mW, −30 dB m, respectively. The optimal concentration Pr3+=1.9×1024, Tm3+=3.9×1024, Er3+=1.2×1024 ions/m3, and energy transfer coefficient Wet6-3=63×10−20, Wet6-10=200×10−22.

When fiber length increases from 0.0 to 30.0 m, the gain at 1310 nm increases monotonically from 0.0 to 17.6 dB, and the gains at 1470, 1530, and 1600 nm increase from 0.0 to 17.7 dB, 27.0 dB, 16.0 dB at the fiber lengths 18.0, 14.0, 23.0 m, respectively; after these lengths, they drop. The gain at 1650 nm decreases from 0.0 to −4.8 dB; after the length 14.0 m, it rises. We think that the variation of the gains at 1310–1600 nm with fiber length is reasonable. With fixed pump power and fiber length increasing from 0 to certain level, pump powers are so high that the number of population inversions between the ¹G₄ and ³H₅ level of Pr³⁺ ions, the ³H₄ and ³F₄ level of Tm³⁺ ions, the ⁴I_13/2 and ⁴I_15/2 levels of Er³⁺ ions increases; thus, the gains increase. When fiber length is over the level, pump powers are comsumpted so much that the inversion number drops; thereby the gain decreases. For the channel at 1650 nm originating from the transition from the ³F₄ to ³H₆ levels of Tm³⁺, it shares same level (³F₄) with the channel at 1470 nm arising from the transition from ³H₄ and ³F₄ level of the ions, but the shared level acts as the upper level for 1650 nm channel and the terminated level for 1470 nm; therefore, the gains at 1470 nm and 1650 nm have opposite variation trend with increased fiber length.

4. Conclusions

In conclusion, we have presented a theoretical model of Er³⁺-Tm³⁺-Pr³⁺ co-doped fiber amplifier pumped with 800 nm and 980 nm lasers. The rate and power propagation equations of the model have been solved numerically and the dependence of the gains at 1310, 1470, 1530, 1600, 1650 nm windows on the fiber length has been calculated. The results showed that with pump power of 200 mW/200 mW, when concentrations of Pr³⁺, Tm³⁺, Er³⁺ are around 1.7 × 10²⁴, 3.9 × 10²⁴, 1.2 × 10²⁴ (ions/m³), respectively, the signals at 1310, 1470, 1530,1600, 1650 nm may be nearly equally amplified with gain of 13–16.0 dB in the active fiber with fiber length of 23.5 m. The co-doping concentrations and fiber length and pump powers of the co-doped system may be further optimized to reduce the ripple.

Table 1 

Spectral parameters of Er³⁺-doped and Tm³⁺ and Pr³⁺ doped telluride fiber for numerical calculation.

Acknowledgments

This work is supported by National Natural Science Foundation of China (Grants no. 60377023 and no. 60672017) and Program for New Century Excellent Talents in University and Shanghai Optical Science and Technology (no. 05DZ22009) and sponsored by Shanghai Pujiang Program.