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

Rate equations for an erbium laser system: a numerical approach

Florin C. Maciuc, Costin I. Stere, Andreea-Rodica P. Sterian

Published Jun 29, 2001• 7 citations
Publisher

Abstract

In our work, we develop a numerical model for studding the variation of phenomenological and optical parameters in an Er+3 system of 6 excited levels that is subject to a lasing condition. The model is using a four order Runge- Kutta method to solve a system of eight time-dependent variables. The simulations, based on the change of pumping wavelength, prove that the highest efficiency is reached for a 970 nm pump wavelength, concordant with the existing literature results. Different processes, including: three ion-ion up-conversion processes, two pumping processes and the inherent spontaneous transition, are considered.

DOI
10.1117/12.432832

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In our work, we develop a numerical model for studding the variation of phenomenological and optical parameters in an Er+3 system of 6 excited levels that is subject to a lasing condition. The model is using a four order Runge- Kutta method to solve a system of eight time-dependent variables. The simulations, based on the change of pumping wavelength, prove that the highest efficiency is reached for a 970 nm pump wavelength, concordant with the existing literature results. Different processes, including: three ion-ion up-conversion processes, two pumping processes and the inherent spontaneous transition, are considered.
"Doping a part of the optical fiber core by (Er3+ ) ions in presence of external pumping power will lead to form the erbium-doped fiber amplifier (EDFA).The performance of this optical amplifier depends on (the power and the wavelength of the pumping laser, the power and wavelength of the input signal, amplifier length, ion concentration). These parameters will affect the characteristics of EDFA such as amplifier gain, gain saturation, noise figure and output power. However, these characteristics can be determined by solving the EDFA propagation and rate equations. The solution of these equations of two-level laser medium can be done numerically. In this paper, we are proposed a novel method to solve these equations. The reconstructed results are perfectly coincided the well known numerical results.
Doping a part of the optical fiber core by (Er3+ ) ions in presence of external pumping power will lead to form the erbium-doped fiber amplifier (EDFA).The performance of this optical amplifier depends on (the power and the wavelength of the pumping laser, the power and wavelength of the input signal, amplifier length, ion concentration). These parameters will affect the characteristics of EDFA such as amplifier gain, gain saturation, noise figure and output power. However, these characteristics can be determined by solving the EDFA propagation and rate equations. The solution of these equations of two-level laser medium can be done numerically. In this paper, we are proposed a novel method to solve these equations. The reconstructed results are perfectly coincided the well known numerical results.
A detailed analysis of the non-exponential behaviour of the kinetics of the initial laser level for three-micron generation 4I11/2, in ErAG (Erbium Aluminium Garnet) crystals (laser transition 4I11/2 →4I13/2), excited in the 4H11/2 level with short laser pulses of variable intensity, is performed. The observed non-exponentiality of the 4I11/2 decay and its dependence on the energy of the exciting pulse cannot be satisfactorily described with a rate equation modelling which includes only two-ion energy transfer processes among Er3+ ions as up-conversion from the terminal laser level (4I13/2), up-conversion from the initial one. and cross-relaxation from 4I9/2 and 4S3/2 (thermalised, at room temperature, with 2H11/2) Our analysis has shown that the inclusion of three-ion up-conversion processes from both initial and terminal three-micron laser levels leads to an excellent agreement between experimental and theoretical non-exponentiality of the 4I11/2 decay for various pumping intensities and allows the estimation of the three-ion up-conversion rates The best agreement with the experimental non-exponentialities was obtained ω111 = ω222 = 2 x 10-34 cm6s-1 and q2 = 0.9, where ω111 and ω222 represent the rates of three-ion upconversion from 4I13/2 and 4I11/2, respectively, and q2 is the ratio o two-ion up-conversio rates.
Based on the energy level of erbium ions in the glass matrix,we analyzed the work mechanism of erbium-doped fiber super-fluorescent light source. The physical model of light source was set up,which was described by propagation equations of pump light and signal light along the erbium-fiber. By dividing the fluorescence spectra into 10 wavelengths,the issue was solving the boundary value problem of 21-dimensional ordinary differential equations. We translated the boundary value problems into an initial value problem using shooting method. The equations were solved using Runge-Kutta method. We wrote a program to implement the algorithms using Visual C++. The numerical simulation and software can provide directions for researching on erbium-doped fiber amplifiers and lasers.
In Chapter 1, we introduced the standard rate equations (SRE) for a laser containing two-level atoms between which stimulated emission is possible. But real lasers exhibit much more complicated energy level schemes. Throughout this chapter, we consider several models in which the light–matter interaction is described by population equations for all the levels involved in the laser operation. Although they are mathematically more complicated, we shall investigate these equations in the same way as in Chapter 1, i.e. by formulating dimensionless equations and by analyzing the stability properties of the steady states. The basic ingredients of the SRE model used up to here, namely pumping and relaxation processes, play key roles in the efficiency of lasers. But already during the pioneering days of the laser, it appeared important to investigate three- and four-level models in order to obtain more reliable information on quantities such as the power conversion efficiency or the response time. The common extensions of the two-level SRE typically consider an open two-level system, or three- or four-level systems, depending on the nature of the active medium. Restriction to as few as four levels is again a crude simplification of the complex population dynamics occurring in most lasers. But it is rather surprising to see how good these simple kinetic models are. As we demonstrate by studying specific examples, the solution of the three- or four-level rate equations differs only slightly from the solution of the SRE.