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Journal of Lightwave Technology

Comments on “Study of the Complex Atomic Susceptibility of Erbium-Doped Fiber Amplifiers”

Jean-Fran�ois Bisson, Dmitrii Kouznetsov

Published Jan 1, 2008• 1 citation
Publisher

Abstract

This comment aims to report and correct a mistake in the paper by Desurvire (J. Lightw. Technol., vol. 8, no. 10, pp. 1517-1527, Oct. 1990) with regard to the expression of the complex susceptibility of quasi-three-level laser systems. The derived susceptibility formula contradicts the assumption of thermal equilibrium between sublevels within each manifold and leads to the wrong conclusion that the real and imaginary parts of the susceptibility of saturated materials do not verify the Kramers-Kronig relations (KKR). The corrected formula for the susceptibility is hereby proposed. The poles of the susceptibility obtained with the new formula are all located in the upper half plane of the complex plane, in accordance with the principle of causality. Hence, the KKR should be valid in the saturated regime.

DOI
10.1109/jlt.2007.909904

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This comment aims to report and correct a mistake in the paper by Desurvire (J. Lightw. Technol., vol. 8, no. 10, pp. 1517-1527, Oct. 1990) with regard to the expression of the complex susceptibility of quasi-three-level laser systems. The derived susceptibility formula contradicts the assumption of thermal equilibrium between sublevels within each manifold and leads to the wrong conclusion that the real and imaginary parts of the susceptibility of saturated materials do not verify the Kramers-Kronig relations (KKR). The corrected formula for the susceptibility is hereby proposed. The poles of the susceptibility obtained with the new formula are all located in the upper half plane of the complex plane, in accordance with the principle of causality. Hence, the KKR should be valid in the saturated regime.
A study of the complex atomic susceptibility of erbium-doped fiber amplifiers operating near the 1.53- mu m transition (/sup 4/I/sub 13/2/-/sup 4/I/sub 15/2/) developed from a semiclassical theoretical description is presented. Expressions for the emission cross section sigma /sub e/( lambda ) and absorption cross section sigma /sub a/( lambda ) of erbium:glass as a quasi-three-level laser system with Stark-split sublevel manifolds are derived. The results are used to calculate the cross sections of aluminosilicate erbium-doped fibers at low temperatures (T=4.2, 77 K) and at room temperature. It is demonstrated that an expression for the gain coefficient of the fiber amplifier can be derived from the expression of the susceptibility. Using Kramers-Kronig relations, an expression for the refractive index change in the fiber is derived. The theoretical spectral gain profile of the fiber amplifier and concurrent refractive index changes are analyzed for different pumping regimes.>
A basic understanding of the susceptibility of fiber amplifiers (SOFA) is required to undertake device optimization, e.g., for uses in optical communications systems. In recent work Desurvire1 has developed a semiclassical treatment of the SOFA for erbium-doped fiber amplifiers (EDFAs) taking account of the Stark splitting of the energy levels in the Er-doped glass. That work, which utilized the standard Kramers-Kronig relationships, was applicable to the unsaturated regime of operation. In this contribution, generalized Kramers-Kronig relations2 are utilized to obtain analytical expressions for the SOFA valid even in the saturation regime. It is then shown how, given experimentally measurable absorption cross-sections, the SOFA may be calculated in any operating conditions. In this way gain and refractive-index changes occurring in EDFAs may be deduced, again having universal validity.
We investigate nonanalyticities (e.g., zeros and poles) of refractive index $n(\ensuremath{\omega})$ and group index ${n}_{g}(\ensuremath{\omega})$ in different optical media. We first demonstrate that while a Lorentzian dielectric has no nonanalyticity in the upper half of the complex frequency plane (CFP), its group index---which governs the pulse-center propagation---possesses zeros in the same region. Thus we classify the nonanalyticities as in the (a) first order (refractive index or reflection) and (b) second order (group index or group delay). The latter contains the derivative of the former. Then, we study a possible connection between the negative superluminal velocities and the presence of nonanalyticities in the upper half of the CFP. We show that the presence of nonanalyticities in the upper half of the CFP for (a) the first-order and (b) the second-order responses are accompanied by the appearance of negative (a) phase and (b) group velocities, respectively. We also distinguish between two kinds of superluminosity, v>cv>cv>c and v<0v<0v<0, where we show that the second one (v<0v<0v<0) appears with the violation of KKRs.
We present an analytical study of susceptibility in pump–probe laser spectroscopy for an atomic medium composed of two-level atoms in the Doppler-broadened limit. We derive an accurate analytical formula for susceptibility up to the first (arbitrary) order at the Rabi frequency of the probe (pump) beam. This is expressed in a succinct single term rather than as a complicated summation, as presented in previous papers. We also derive analytical solutions for the linewidth and peak value of the absorption spectrum of the probe beam.