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## Brazilian Journal of Physics

##
*versão impressa* ISSN 0103-9733*versão On-line* ISSN 1678-4448

### Braz. J. Phys. v.33 n.1 São Paulo mar. 2003

#### https://doi.org/10.1590/S0103-97332003000100008

**Simplified slab waveguide three-level model for short length EDFA pumped at 1.48 mm**

**Angela Maria Guzmán ^{I}; Hypolito José Kalinowski^{II}**

^{I}Universidad Nacional de Colombia Departamento de Física, Bogotá, Colombia

^{II}Centro Federal de Educação Tecnológica do Paraná, Av. Sete de Setembro, 3165, 80230-901 Curitiba, PR, Brazil

**ABSTRACT**

We adopt a three–level scheme for the pump – amplification process in 1.48 mm pumped Erbium-Doped Fiber Amplifiers, which reproduces the main experimental features of amplifiers with short and long fiber lengths. Continuous wave amplification in a simplified slab waveguide structure is simulated by means of the scalar Beam Propagation Method, taking into account signal and pump propagation through the waveguide. Results from the simulation are compared with measurements done by the COST 217 Project Group. The method may be well suited for the project of Integrated Optics Optical Amplifiers based on rare earth doped waveguides.

** I Introduction**

Erbium Doped Fiber Amplifiers (EDFA) play an important role in lightwave communication systems operating in the 1.55 m*m* window. The production of Erbium Doped Fiber (EDF) is based on well established CVD fiber technologies, so that their reliability and most of the operational properties directly depend on the pump and signal characteristics. Of the several parameters involved, the wavelength and power of pump and signal fields, as well as the *Er* ^{3+} profile, are some of the more important to take into account in the amplifier design.

Pumping of EDFA can be done at several wavelengths were commercial lasers are available (0.514 m*m*, 0.8 m*m*, 0.98 m*m*, 1.48 m*m* ...) However, some of these bands have reduced efficiency in signal amplification due to Excited State Absorption (ESA). The 1.48 m*m* band shows good characteristics for amplification in the 1.53 - 1.55 m*m* window, as there is no strong competitive ESA. The same occurs for EDFA pumped in the 0.98 m*m* band, so that commercial devices are based with pump lasers at those two wavelengths.

The study of EDFA by numerical simulation could lead to better optimization of the amplifier design, reducing costs of production and development time of new products and systems. Modeling of EDFA can be done by several approaches, most of them discussed in a paper by Giles and Desurvire [1]. EDFA pumped with 1.48 m*m* light are usually considered as a two level system [2], while 0.98 m*m* pumped devices are modeled after a three level system [3]. However, it is also shown that 0.98 m*m* pumped amplifiers can be modeled as a two level system when the population of the excited state manifold is negligible [1].

A comparison of measurements in EDFA was carried out by a group of European laboratories in the framework of COST 217 Project [4]. The small signal gain was measured as function of pump intensity for different fibers and several signal and pump wavelengths. The signal wavelength varied from 1.530 m*m* to 1.550 m*m* and the pumping wavelengths from 1.478 m*m* to 1.486 m*m*. Gain curves for short and long fibers were reported.

We propose a theoretical model based on a three level system for the optical pumping cycle, and use the scalar Beam Propagation Method (BPM) [5] to simulate the continuous wave gain for an 1.48 m*m* pumped *Al/Er* doped core fiber made by the Technical Research Center of Finland – VTT(SF), whose parameters were specified in [4]. We don't take in account the presence of the amplified spontaneous emission (ASE) in the amplifier.

** II Theoretical Model**

We model the mentioned EDFA by a planar waveguide structure whose thicknesses are the same as the core – cladding diameters of the original cylindrical fiber. The two-dimensional scalar wave equation for an * Er* doped planar waveguide reads

where * P* includes both the host and dopant polarizations. We introduce slowly varying complex amplitudes by means of

where * j* = 1,2 stands for signal and pump field respectively, and * k _{j} * =

*k*

_{0j}

*n*

_{0j}, with

*k*

_{0j}=

*w*/

_{j}*c*. Neglecting field second order derivatives with

*z*we obtain the scalar wave equation in the parabolic approximation,

In order to evaluate the *Er*-polarizations , a model for the pump – amplification cycle is needed. The measured absorption spectrum of the ^{4}*I*_{13/2 }« ^{4}*I*_{15/2} transition has maxima at 1.49m*m* and 1.53 m*m*, and is well fitted by Lorentzians. Therefore we consider the ^{4}*I*_{13/2} state as compossed by two sublevels (1 and 3) and adopted a three-level energy diagram as sketched in Fig. 1. The 1.48m*m* pump promotes a transition to the sublevel 1 from the fundamental state (level 2) which is followed by a fast non-radiative transition to the metastable sublevel 3. From this last state the system may decay by stimulated emission of an 1.53 – 1.55 m*m* photon. Oscillator strengths, linewidths, and absorption coefficients can be inferred from the measured absorption spectrum and fluorescence lifetime of the metastable state [4]. We assume homogeneous line broadening and use population rate equations since absorption profiles are Lorentzians and linewidths are much larger than the pump rate:

where * N _{l}, l* = 1,2,3 are level populations. g

_{l},

*l*= 1,3 and G

_{c}are longitudinal decay rates. The signal and pump transition rates

*W*= 1,2 are related to Rabi frequencies W

_{j}, j_{j}by

with

where Ã_{j} and G_{j} are respectively the electric dipole moments and lorentzian linewidths associated with the transitions 2 « 3 (*j* = 1) and 1 « 2 (*j* = 2). D_{1} = w_{32}– w_{1} and D_{2} = w_{21 }+ w_{2} are signal and pump detunings with w_{lj} being the proper frequency of the transition * l *« *j*. We assume that G_{c} is much larger than any * W _{j} * or g

_{j}, and obtain from Eq. 4 the steady state population inversion

The corresponding polarizations for c.w. gain are given by

where ^{Er}(*x*) is the * Er* density profile and

_{j} = (G_{j} + iD_{j})^{–1}

We do not take into account the in-phase polarizations, which would give negligible refractive index variations at the considered field powers. By replacing Eq. 8 in Eq. 3 we obtain nonlinear wave equations, which include signal and pump saturation. In terms of Rabi frequencies they read

where

In Eq. 10, a_{0j} is the absorption coefficient and ^{Er}(*x*) a dopant profile factor. Pump and signal fields are coupled through * N _{Tj} * appearing in the last term of the right hand side of Eqs. 9 which describe signal amplification as well as pump absorption.

** III Numerical Simulation Results**

We solved Eqs. 9 by means of the Finite Difference Beam Propagation Method (FD-BPM) [6,7], where partial differential equations are replaced by their finite difference approximation^{1}:

where

The signal and pump electromagnetic fields inside the optical waveguide were propagated from single-mode transversal profiles at the launching facet. The absorption coefficients a_{0j} were inferred from a lorentzian fit of the measured absorption spectrum [4]. A step profile with the core width was assumed for ^{Er}(*x*). The step value was adjusted in order to obtain the measured signal absorption in a short length fiber for zero pump field. To normalize the input power in the slab waveguide to the corresponding power launched into the fiber amplifier, we used the electromagnetic field intensity that reproduced, in our model, the measured small signal gain threshold in the EDFA.

Figure 2 shows the short length (*L * =* 2.2 m* ) amplifier gain as a function of the pump power. The obtained curve fits nicely to the experimental data reported by the COST 217 group [4]. The threshold pump powers * I _{th} * is essentially determined by intrinsic properties of the dopant, as well as by the pump wavelength and the overlap of pump and Er density profile. For long amplifier lengths (

*L*=

*14.5 m*), there is only qualitative agreement with the experimental data. The theoretical pump power threshold is lower than the experimental value, but the saturation gain is nearly the same (Fig.3). Small differences either in the Er density profile, linewidths or absorption coefficients having almost no effect in the short length gain could be strongly enhanced as the amplifier length increases. The presence of ASE, which was discarded in our approximation, contributes to deplete the metastable level and this may also decrease the gain of the amplifier (for the same pump power). The effect will be more noticeable as the fiber length increases, consistent with the results displayed in Fig. 2 and 3.

Figure 4 shows signal and pump powers as function of the amplifier length for different input pump powers, all of them above * I _{th}*. Initially the signal field gets amplified meanwhile the pump field is absorbed. Eventually (curves P1 and P2) the pump field drops below the threshold value

*I*» 2

_{th}*mW*and both fields get absorbed from there on. Therefore the output gain curve as well as threshold pump power depend strongly on the amplifier length.

3-D Beam Propagation curves corresponding to curves P2 and S2 in Fig. 4 are shown in Fig. 5 left and right, respectively. The monomode profiles are preserved for both fields as they propagate through the waveguide. It suggests that there is no need for solving the complete wave equation (Eq. 9). Instead we can obtain equations for the field intensities * I _{j}*, assuming monomode beam structure for both fields and integrating Eq. 9 over the transverse coordinate. Disregarding the transverse structure of N

_{Tj}, we obtain:

where the effective absorption coefficient depends on the overlap of the *Er* density profile and *j*-field profile.

We believe that a good procedure for device design using arbitrary Er density profiles would be to solve the complete set of Eq. 9 for short distances, and to obtain from the gain curve the effective absorption coefficient to be used in long distance propagation. Following this procedure we calculated gain curves for different *Er*-density profiles and signal wavelengths. The results for short and long amplifier lengths are shown in Figs 6 and 7 respectively. They give information relevant to amplifier design. Let us first consider amplification at 1.53 m*m*. By extending the dopant profile further into the cladding, a shift of the threshold pump power to higher values is produced without increasing significantly the amplifier gain. Instead, if the dopant concentration in the core is increased, the short length gain is increased as expected, since the effective length * ^{Er }* increases. The same saturation gain is obtained for longer amplifiers with the disadvantage of shifting the threshold pump power to higher values. Therefore for higher dopant concetrations, short amplifiers can be more efficient.

The simulation for the short amplifier with signal at 1.55 m*m* gives less gain due to the signal detuning from resonance with the metastable state. This fact was also observed experimentally, as mentioned by the COST 217 project [4].

** IV Conclusion**

We use a three-level model for the optical pumping cycle of Erbium-Doped Fiber Amplifiers pumped at 1.48 m*m*, to obtain the non linear wave propagation equations for signal and pump fields, disconsidering the ASE. Those equations are solved by Finite Difference Beam Propagation Method to obtain the gain, pump and signal powers in a slab-waveguide approximation for the Erbium-Doped Fiber. Results from the simulation agree very well with experimental data of CW-gain for short length amplifiers, while providing usefull insigth for amplifiers with longer lengths of fiber. A time saving procedure for the simulation of EDFA under such approximation is given.

As the method is particularly fitted to simulate short length EDFA and due to the slab waveguide structure used in the simulation, we expect that this method will be usefull to simulate Integrated Optics Optical Amplifiers based on rare earth doped waveguides.

**Acknowledgments**

The authors acknowledge financial suport from the following agencies: CNPq *(Brazil)*, FPIT *(Colombia)* and ICTP *(Italy)*. Part of this work was done when Dr. Guzmán was a visiting researcher at CEFET under the sponsorship of project RHAE.

**References**

[1] C. R. Giles and E. Desurvire, *"Modeling Erbium-Doped Fiber Amplifiers"*, J. Lightwave Technol. **9(2)**, 271-283 (1991). [ Links ]

[2] E. Desurvire, *''Analysis of Erbium-Doped Fiber Amplifiers Pumped in the * ^{4}*I*_{15/2} * - * ^{4}*I*_{13/2} * Band''*, IEEE Photon. Technol. Lett. **1(10)**, 293-296 (1989). [ Links ]

[3] B. Pedersen, A. Bjarklev, J. Hedegaard, K. Dybdal and C. C. Larsen, *''The Design of Erbium-Doped Fiber Amplifiers''*, J. Lightwave Technol. **9(9)**, 1105-1112 (1991). [ Links ]

[4] P. Kiiveri and COST 217 group, *''COST 217 Intercomparison of Measurements on Er-doped Fibres''*. Symposium on Optical Fiber Measurements, Boulder (USA), 183-186 (September, 1992). [ Links ]

[5] J. van Roey, J. van der Dook and P. E. Lagasse, *Beam Propagation Method: analysis and assessment"*, J. Opt. Soc. Am. **71** , 803-810 (1981). [ Links ]

[6] Y. Chung and N. Dagli, *''An Assessment of Finite Difference Beam Propagation Method"*, IEEE J. Quantum Electr. **26(8)**, 1335-1339 (1990). [ Links ]

[7] R. Scarmozzina and R. M. Osgood Jr., *''Comparison of finite difference and Fourier transform solutions of the parabolic wave equation with emphasis on integrated optics applications"*, J. Opt. Soc. Am. **8(5)**, 724-731 (1991). [ Links ]

Received on 27 April, 2002

^{1} In order to obtain a system of linear equations for the unknown variables at the lefthand side of Eq., we assumed N_{Tj}(z+Dz) » N_{Tj}(z)