IEEE PHOTONICS TECHNOLOGY LE’ITERS, VOL. 5, NO. 12, DECEMBER 1993 1427 zyx Performance Reduction and Design Modification of Erbium-Doped Fiber Amplifiers Resulting from Pair-Induced Quenching J. Nilsson, B. Jaskorzynska, and P. Blixt Abstract-We model the deleterious effect of pair-induced quenching (PIQ) in Erbium-doped zyxwvuts fiber amplifiers (EDFA’S) and describe it in terms of pump power, amplification, and noise factor penalties. Moreover, with a simple model for how PIQ increases with concentration, we show that to achieve a desired amplification, shorter fiber lengths will require higher pump powers, and that there is a lower length limit with realistic pump powers. Rate equations for Erbium-ion pairs are formu- lated with the assumption of one ion per pair completely quenched. Although our results were obtained for EDFA’s, the methods and conclusions are also relevant for high concentra- tion erbium doped integrated waveguides. INTRODUCTION VIDENCE of Erbium-ion pair formation in silica- E glass EDFAs has recently been presented [1]-[3]. In a pair, the lifetime of the state where both ions are excited (doubly excited state) is exceedingly short, zyxwvuts so that one ion in the pair is effectively quenched for pump powers below the pair saturation power in the order of a watt [l]. Thus, it is not possible to invert the population of paired Erbium ions with moderate pump powers. The effect is called Pair-Induced Quenching (PIQ). It leads to a nonsaturable absorption of, e.g., a 980-nm pump [l] as well as to EDFA performance degradation [2], [3]. In this letter, we present a more detailed analysis of the effect of PIQ on the performance of EDFAs in terms of the implied penalties of pump power, gain, and noise, depending on the ratio of Erbium ions in pairs to the total number of Erbium ions (the pair ratio). We also indicate the existence of a lower limit set on fiber length, when a given gain is required, by using a simple model for how the pair ratio increases with concentration. Similar to [l], we use a two-level model to describe both the pairs and the ions themselves, assuming one ion per pair completely quenched. We find (1) that the steady-state solution to the rate equations used is consistent with that we obtain from Manuscript received July 8, 1993; revised October 1, 1993. This work was supported by the Swedish National Board for Industrial and Techni- cal Development and Ericsson Telecom AB. J. Nilsson is with the Department of Physics 11, Royal Institute of Technology, S-100 44, Stockholm, Sweden. B. Jaskorzynska and P. Blixt are with the Institute of Optical Re- search, S-100 44 Stockholm, Sweden. IEEE Log Number 9214044. the more complete model of [2], and (2) how to generalize the model to describe k-ion clusters. RATE EQUATIONS Following [l], we consider two states of Erbium-ion pairs to be populated: Either none (per state 1) or one (per state 2) of the ions in the pair can be in its excited state 4Z13/2. When both ions become excited, an excitation is rapidly transferred from one ion to the other, resulting in one ion in the ground state zyxw 4Z15/2 and the other upconverted to level 4Z11/2. The upconverted ion quickly relaxes to the metastable state 4Z13/2, so that pair state 2 is established. For moderate pump powers, this cross-relaxa- tion process is much faster (5 to 50 ps, according to [l], and 1 ps in [3]) than the pumping rate. Thus, the doubly excited state can be disregarded, and the rate equations for the pair states can be written as: dP, dP2 dt dt zyxwv -= -- = ApP2 - R,P, + WpP, (1) where PI and P2 are the concentrations of pairs in states 1 and 2, P, + P2 = Po is the total pair concentration, R, and Wp are the pump and the stimulated emission rates for pairs, respectively, and A, is the pair spontaneous emission rate: (3) A, = zyxw 1/r =A, (4) In (2)-(4), the subscript s signifies rates for single ions. zy Z(u) is the intensity spectrum, including pump, signal, and ASE intensities. Furthermore, U& zyxw v) denotes the ground- state absorption cross section, and q2 denotes the stimu- lated emission cross section of 4Z13/2. We assume the same cross sections for paired and isolated ions. For modeling EDFA’S, it is convenient to express (1) in terms of single-ion states because once the populations of states 4Z15/2 and 4Z13/2 are known, the coexisting paired and isolated ions can be treated together. If &.P, i = 1,2 1041-1135/93$03.00 0 1993 IEEE