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

Brazilian Journal of Physics, vol. 33, no. 1, March, 2003 104 Simplified Slab Waveguide Three-Level Model for Short Length EDFA Pumped at 1.48 m Angela Maria Guzmán Universidad Nacional de Colombia Departamento de Fı́sica, Bogotá, Colombia and Hypolito José Kalinowski Centro Federal de Educação Tecnológica do Paraná Av. Sete de Setembro, 3165, 80230-901 Curitiba, PR, Brazil Received on 27 April, 2002 We adopt a three-level scheme for the pump – amplification process in 1.48 m 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 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, 0.8 m, 0.98 m, 1.48 m . . . ) However, some of these bands have reduced efficiency in signal amplification due to Excited State Absorption (ESA). The 1.48 m band shows good characteristics for amplification in the 1.53 - 1.55 m window, as there is no strong competitive ESA. The same occurs for EDFA pumped in the 0.98 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 light are usually considered as a two level system [2], while 0.98 m pumped devices are modeled after a three level system [3]. However, it is also shown that 0.98 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 to 1.550 m and the pumping wavelengths from 1.478 m to 1.486 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 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 twodimensional scalar wave equation for an Er doped planar waveguide reads @2E @2E + @x2 @z 2 1 @2E @2P = 0 2 2 2 c @t @t (1) Angela Maria Guzmán and Hypolito José Kalinowski 105 where P includes both the host and dopant polarizations. We introduce slowly varying complex amplitudes by means of X2 u (x; z) exp [{(k z ! t)] + c:c: =1 X2 [P (x; z) + P (x; z)]  P (x; z; t) = E (x; z; t) = j j pump rate: N_ 3 = N_ 2 = 3 N3 + N1 + W1 (N2 N3 ) 3 N3 + 1 N1 W1 (N2 N3 ) + + W2 (N1 N2 ) _ N1 = N1 W2 (N1 N2 ) 1 N1 j j j =1 host j Er j  exp [{(k z ! t)] + c:c: j (2) j where j = 1; 2 stands for signal and pump field respectively, and kj = k0j n0j , with k0j = !j =c. Neglecting field second order derivatives with z we obtain the scalar wave equation in the parabolic approximation, c c where Nl , l = 1; 2; 3 are level populations. l , l = 1; 3 and c are longitudinal decay rates. The signal and pump transition rates Wj , j = 1; 2 are related to Rabi frequencies j by W = j with j j2 L1 j 2 j @u @z j j = @2u + [k02 n2 (x; z ) k2 ]u + @x2 + 0 !2 P (3) j j j 1> j j Er j Γc L1 = 2 +
2 j j j (6) j where }j and j are respectively the electric dipole moments and lorentzian linewidths associated with the transitions 2 $ 3 (j = 1) and 1 $ 2 (j = 2).
1 = !32 !1 and
2 = !21 + !2 are signal and pump detunings with !lj being the proper frequency of the transition l $ j . We assume that c is much larger than any W j or j , and obtain from Eq. 4 the steady state population inversion N 1 3> T E2 γ3 (5) j 2 j j = j} ~u j ; j 2{k (4) N 2 T E1  N2 N3 = 2W 3+ WW2+ 1 2 3  N2 N1  2W 3++WW1+ 1 2 (7) 3 The corresponding polarizations for c.w. gain are given by 2> Figure 1. Three level model for Er3+ optical pumping cycle. P (x; z ) = { ( 1) N (x)} D N where N (x) is the Er density profile and D = ( + {
) 1 Er j j j j j (8) Tj Er j In order to evaluate the Er-polarizations P jEr , a model for the pump – amplification cycle is needed. The measured absorption spectrum of the 4 I13=2 $ 4 I15=2 transition has maxima at 1:49m and 1:53m, and is well fitted by Lorentzians. Therefore we consider the 4 I13=2 state as compossed by two sublevels (1 and 3) and adopted a threelevel energy diagram as sketched in Fig. 1. The 1:48m pump promotes a transition to the sublevel 1 from the fundamental state (level 2) which is followed by a fast nonradiative transition to the metastable sublevel 3. From this last state the system may decay by stimulated emission of an 1:53 1:55m 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 Er j j 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 @ { @2 = f + [k02 n2 k2] @z 2k @x2 j j j j (...truncated)