Modeling of gain dynamics in erbium-doped fiber amplifiers

Abstract

The dynamics of EDFA's and networks of EDFA's may have an important impact on the performance of multi channel optical communication systems and networks with wavelength routing'. A model of EDFA dynamics is needed to understand the transient behavior in large systems or networks. Most previous EDFA models are represented by sets of coupled partial differential equations, which can be solved only through iterative, computationally intensive numerical calculations, especially for multi-channel WDM systems with counter-propagating pump or signals. An exception is a steady state model2 which reduces the coupled partial differential equations to a single transcendental equation. In this letter we show that the coupled time dependent partial differential equations can be dramatically simplified, first to a set of coupled ordinary differential equations and then further to a single ordinary differential equation. The simulation results agree with experimental data. atoms in the first excited state Nz(z, t) and the photon equation for the kth channel are2, respectively, Under conditions that are commonly satisfied in WDM systems, the rate equation for the fraction of where p is the number density of the active erbium atoms in the fiber core of cross-section S, and T is the spontaneous lifetime of the upper level. There are N optical channels each of wavelength and optical power p k ( Z , t) at location z and time t, with confinement factor r k and emission and absorption cross sections ut and o: , respectively. The optical powers are expressed in units of number of photons per unit time. Beams traveling along the z direction are indicated by unit vector Uk = f l ; beams moving the opposite direction by U k = -1. The upper state population can be found from the photon equation, which is then substituted into the rate equation to obtain, -a k aPj(z, t ) az N 8 1 a P k ( z , t ) T u k -at (-p k az ) = -U k where f & = p r k a g is the absorption constant and P~S = s/rk ( o~ + 0 ; ) r is the intrinsic saturation power of the kth channel. After integration over z from 0 to L, we have, N pis [ T y f G k ( t ) f Ak] = -Pj'"(t) [exp (Gj(t)) -11, j=1 where G k ( t ) = In [ F t ( ( t ) / P p ( t ) ]is the overall gain parameter, Ak = a k h is the total absorption constant where L is the length of the erbium-doped fiber, and PF(t) and p t ( t ) are the input and output powers of the kth channel at time t. With given input power @ ( t ) and initial conditions G k ( 0 ) = for all the channels, the above set of coupled ordinary differential equations c m be numerically solved.These coupled ordinary differential equations can be further reduced to a single equation. Since the right side of the above equation is the same for all the channels, we can express the gain of channel j in terms of 365

Doped-fiber amplifier dynamics: a system perspective

“…Habbab et al [6] arrived at a linear approximation for the small signal gain in single-channel amplifiers. More recently, Sun et al [1] at Bell Laboratories have succeeded in reducing the system of coupled differential equations into a single ordinary differential equation (ODE).…”

Large power swings in doped-fiber amplifiers with highly variable data

“…This model has been verified experimetally and provides a good description of transients on time scales of fractions of microseconds [1]- [4].…”

“…The EDFA transient time response to input power change was calculated numerically solving a system of differential equations by using a fast Runge-Kutta method [1], calculated in 10 points per individual cell. The calculated output power on slots carrying a cell was arranged in a histogram comprising 500 bins of width 0.04 dBm, uniformly tiling the range between 1 and 21 dBm.…”

“…1041-1135/99$10.00 © 1999 IEEE Excited state-absorption and ion-ion cooperative upconversion are not taken into account in the model [1] used here. This model has been verified experimetally and provides a good description of transients on time scales of fractions of microseconds [1]- [4].…”

Large core highly doped fiber amplifiers.

“…For the PG amplifier where the amplifier fiber length is about 10 cm, the propagation time is about 0.5 ns, which may be safely ignored for the input pulse duration of several nanometers. It is also noteworthy that the effect of propagation is in many cases ignorable so that many published modeling/simulation do not even include the time derivative in the power propagation from the beginning [10], [28], [29], [30], [31]. We also developed a full partial differential equation solver using the upwind and downwind combo method where the forward signal and pump is solved on upwind scheme whereas the backward signal and pump is solved on downwind scheme.…”