Low-frequency (∼ kHz) amplitude modulation of a digital lightwave communication signal, referred to here as overmodulation, has been investigated as a means of propagating line-monitoring and control information in erbium-doped fiber amplifier (EDFA) systems (Freeman and Conradi, 1993; Murakami <etal>,1996; Shimizu <etal>, 1993; Gieske and Bystrowski, 2001). In particular, Freeman and Conradi (1993) describe a numerical model for overmodulation behavior that considers the modulation as a perturbation applied to steady-state EDFA operation. Their results present the essential character of the EDFA overmodulation response, but details of their analysis are not provided and extending their results to other operating conditions is uncertain. We develop the perturbation concept of Freeman and Conradi, using methods described by Bononi and Rusch (1998) for modeling transient effects in EDFAs, which are in turn based on the time-varying gain equation of Sun <etal> (1996). Two models have been developed. An exact numerical solution is presented with supporting experimental data in a companion paper by Novak and Gieske (2002). The present paper considers a small-modulation-index approximation and derives explicit analytic solutions for the overmodulated pump-to-signal and signal-to-signal EDFA transfer functions,which are first order in the modulation index. These are compared with results from the exact numerical solution, with excellent agreement within the limits of the approximation. To our knowledge, this is the first application of the Bononi-Rusch-Sun approach to overmodulation studies. The analytic expressions we obtain are somewhat more complex than the zero and pole equivalents suggested by Freeman and Conradi. The resulting model is practical and accurately represents the overmodulation dynamics in an EDFA under varying conditions of gain saturation. Our results do not at this time take into account amplifier spontaneous emission, which has some effect on predictions in the high gain/low saturation regime. However, our expressions are otherwise valid even into the deeply saturated gain region of the EDFA. They correctly predict a limit and reversal in the overmodulation transfer function magnitudes as the saturation level is increased and are further seen to capture nuances in experimental data reported by Freeman-Conradi, but missed by their own model. The more exact understanding of overmodulation dynamics should prove useful, e.g.,when considering long-haul DWDM systems, in which the EDFA saturation level is high and predictions are complicated by the cascade of amplifiers.
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