JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 20,NO. 2,FEBRUARY 2002 243 Spectral Functional Forms for Gain and Noise Characterization of Erbium-Doped Fiber Amplifiers Evgeny V. Vanin, Ulf Persson, Member, IEEE, and Gunnar Jacobsen Abstract—We have, for the first time, experimentally verified the black box model based upon the knowledge in the form of charac- teristic tilt functions and demonstrated accurate characterization of erbium-doped fiber amplifiers (EDFAs) in wavelength division multiplexing (WDM) system operation. Index Terms—Erbium, optical communication, optical fiber am- plifiers, optical noise. I. INTRODUCTION E RBIUM-DOPED fiber optical amplifiers (EDFAs) are very important elements of modern optical transmission systems that use wavelength division multiplexing (WDM) techniques. This type of fiber amplifier is capable of com- pensating for optical fiber losses in broad wavelength ranges and allowing a long-distance transmission of more than a hundred wavelength channels. The performance of the entire transmission system is strongly influenced by the spectral gain and spectral noise figure of installed EDFAs. Therefore, detailed knowledge about these spectral characteristics of the amplifier is the key for advanced design of WDM systems. The spectral gain and the spectral noise figure of EDFAs can be completely described by the propagation and rate equations modeling the interaction of the optical field with erbium ions [1], [2]. A numerical solution of these equations can determine the spectral gain and noise figure for specified amplifier param- eters such as the emission and absorption cross sections of er- bium, its concentration and distribution in the doped fiber, the fiber length, the fiber loss parameter, the pump power, and the signal input spectral density, as well as the parameters of passive optical components like optical isolators, couplers, gain-flat- tening filters, and the like [2]. This approach is rather efficient for design of optical amplifiers, but requires accurate character- istic data for all amplifier components. When EDFAs are used in WDM systems, the detailed infor- mation about parameters of the erbium-doped fiber and internal design of the amplifier is not always available. Moreover, even if it is available, the model outlined in [2] is very complicated and requires many input parameters that cause the accurate charac- terization of the amplifier to become a difficult problem. Another approach to characterize EDFA is to apply the so- called black-box model [3]–[6]. The main idea of applying black-box modeling is to simplify the characterization of the EDFA. The black-box model is based upon input–output exper- Manuscript received December 27, 2000; revised November 7, 2001. The authors are with the Optical Networks Research Laboratory, Ericsson Telecom, Stockholm SE-126 25, Sweden (e-mail: evgeny.vanin@etx.eric- sson.se). Publisher Item Identifier S 0733-8724(02)00495-4. imental data obtained in a simple test measurement of a certain amplifier unit and does not require access to internal details of the amplifier construction. The physical background of the black-box model is the same as for most rate-equation-based models [1], [2]. Both approaches rely on the quasi-two-level description of erbium ions and the homogeneous broadening assumption. However, rather than generating the spectral gain and amplified spontaneous emission (ASE) noise curves from the propagation and rate equations, the black-box model uses a superposition of two measured spectral gain or ASE curves measured for two different saturation conditions to estimate the gain and noise figure of the amplifier in a WDM system operation [3]–[5]. The first publication on the black-box model [3] demon- strated the efficiency of this approach for the spectral gain characterization of a simple EDFA operated at a fixed pumping power. The method of evaluating of the spectral gain from two measured reference curves, which was formulated in [3], has been derived from the rate equations [4]. In the [4], it was also demonstrated that the black-box model is also applicable for an EDFA with internal optical filters and other passive optical components with wavelength-dependent losses. The spectral gain tilt function that is the merit of the amplifier homogeneous broadening (independent of the amplifier saturation conditions) has been introduced. The extension of the model for the amplifier operated under arbitrary pumping power levels and for the noise figure characterization has been proposed and experimentally verified in [5]. In this paper, we show as an addition to the results reported in [3]–[5]—using the two-level approximation—that the ampli- fier gain and noise can be estimated by simple analytical equa- tions using three spectral tilt functions, which are independent of the amplifier saturation condition. We use this new formula- tion to specify the accuracy of estimated gain and noise figure for the amplifier in WDM operation, and to see to what extent the measurements are influenced by inhomogeneous effects, such as spectral hole burning [7]. We also present the verification of the specific WDM black-box model formulated in [5] by comparing the estimated gain and noise figure with experimental data mea- sured for the amplifier saturated by a 32-channel WDM signal. II. THEORY The black-box model is based on the assumption of homoge- neous saturation of the amplifier. In the frame of this assump- tion, the spectral gain of the amplifier is entirely determined by the averaged distribution of the population inversion, which is maintained in the active fiber due to balancing between reso- nance optical pumping and deactivation of erbium ions. It means 0733–8724/02$17.00 © 2002 IEEE