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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Theoretical model for nonlinear homogeneous upconversion and clustering in Er 3+ -doped waveguide amplifiers

Sergey V. Sergeyev, B. Jaskorzyńska

Published Mar 9, 2001• 0 citations
Publisher

Abstract

We model upconversion and migration in an ensemble of inhomogeneously distributed erbium ions. Treating Er- clusters as fractal objects we find the dependence of the upconversion rate on the population inversion for different fractal dimensions.
DOI
10.1117/12.418822

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We present an extension to our earlier proposed statistical model [Phys. Rev. B 62, 15628 (2002)] for studying migration-assisted homogeneous upconversion in erbium-doped fibers. The extension takes into account minimum proximity distance between erbium ions randomly distributed in the host material and the nonuniformity of the excitation distribution among them. We derive a transcendental equation for the population inversion and find the dependence of the upconversion rate on the population inversion and the pump power for the entire range of feasible Er concentrations. We verify the validity and accuracy of the model by means of time-resolved Monte Carlo simulations.
Based on an approach for averaging solution of rate equations over possible ensemble configurations, we derive integral equation for population inversion taking into account nonlinear homogeneous upconversion and migration in Er3+{\mathrm{Er}}^{3+}Er3+-doped silica waveguides for the case of 1480-nm CW excitation switch-off (CWES) regime. Because of pump for CWES is off, only migration tends to refill the holes in excitation distribution caused by upconversion. Thus the upconversion is weaker than that for the CW operation, except for t=0t=0t=0 and $\stackrel{\ensuremath{\rightarrow}}{t}\ensuremath{\infty},$ upconversion.
From careful studies of the 1530-nm fluorescence decay, we obtain the rate of energy transfer upconversion as a function of the inverted population for a series of nine highly Er-doped silica fibers (with concentrations from 0.3 to 8.6/spl middot/10/sup 25/ Er/sup 3+/-ions per m/sup 3/). The results demonstrate that the slow component (microsecond to millisecond scale) of the upconversion, usually referred to as the homogeneous upconversion, is clearly nonquadratic in its dependence on the inverted population, contrary to previous assumptions in the literature. In a second part, we present a new detailed model for energy transfer upconversion, permitting-to our knowledge for the first time-calculation of the rate of migration accelerated upconversion for any given spatial distribution of Er/sup 3+/-ions. We demonstrate that the results from the decay measurements may be explained with this model. Next, we review the results from a CW green fluorescence detection experiment for the determination of the degree of clustering, which was previously performed on five of the nine fibers. We find accordance between these results and our model, with parameters consistent with those needed to fit the results of the decay experiment, and we arrive at a new conclusion about the nature of clustering.
Measurements of time evolution of amplified spontaneous emission for square-wave modulated pump were made on a highly erbium-doped silica photonic crystal fiber and fitted to theoretical results calculated using different formalisms to describe theEr3+{\rm Er}^{3+}Er3+-ion migration-assisted co-operative upconversion mechanism in the population densities rate equations. The determined characterization parameters are the constant homogeneous upconversion coefficient, 3.0×10−24 m3/s\times 10^{-24}~{\rm m}^{3}/{\rm s}×10−24 m3/s, the critical radii for upconversion and migration, 10.5 and 14Å, respectively, and the rate of clustered ions, 14%. These parameters are analysed and compared to those obtained with continuous-wave pump characterization techniques. Although similar values are obtainable for many parameters using steady-state and dynamic characterization techniques, if a careful enough modelling is carried out, dynamic techniques are preferable to assess the presence of pair-induced quenching.
The effect of homogeneous upconversion on the performance of an erbium-doped fiber amplifier (EDFA) is analyzed numerically, by means of a set of rate equations model. In particular, the influence of pump power, signal power, and Er/sup 3+/ concentration on the degradation, due to upconversion, is examined. Forward pumping is considered and amplified spontaneous emission, as well as scattering loss are taken into account. In addition, approximate analytical solutions, for the rate equations, which include the effect of the cooperative upconversion, are presented. It is shown that under the considered conditions, these analytical solutions provide a good approximation to the exact numerical solution and thus may be used for the design of such amplifiers.
We investigate homogeneous upconversion (HUC) assisted decay of the population inversion in Er-doped silica after delta-pulse excitations by means of Monte Carlo simulations. We derive an analytic expression for this decay when energy migration among Er is negligible and show that it is essentially an extension of the Fo/spl uml/rster-type donor-acceptor quenching decay for the case when donor and acceptor ions are the same species. We compare HUC during the decay to HUC while pumping CW and show that they considerably differ at the low and moderate Er concentrations. Based on Monte Carlo (MC) simulations, we construct a heuristic approximation to the HUC assisted decay, which makes our MC results readily available for the future experimental and theoretical studies and provides simple means for determination of HUC parameters by fitting to experimental data.