Modeling High-Concentration L-Band EDFA at High Optical Powers Based on Inversion Function

Transcription

1 IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 8, NO. 3, MAY/JUNE Modeling High-Concentration L-Band EDFA at High Optical Powers Based on Inversion Function Vladimir Chernyak and Li Qian, Member, IEEE Abstract High-concentration erbium-doped fibers used in -band erbium-doped fiber amplifier (EDFA) are known to experience additional losses due to effects of energy transfer between Er 3+ ions and possibly quenchers (e.g., homogeneous and clustering-induced up-conversion, quenching, etc.). These interactions between Er 3+ ions make modeling of inversion dynamics in -band EDFA more complicated. In this paper, we present a modeling approach for high-concetration doped fiber based on lumping of all complex phenomena of energy transfer in EDFA into a function of two variables hereafter referred to as the inversion function. The inversion function is evaluated for high optical powers (which constitutes a typical situation for fiber amplifiers) using a cluster expansion and assuming that upconversion is the leading loss mechanism. The modeling results show a close agreement with the measurement data for various input power conditions and inversion levels. Index Terms Concentration quenching, EDFA, -band, upconversion. I. INTRODUCTION ERBIUM-DOPED fiber amplifiers (EDFAs) are widely used in telecommunication industry [1], [2]. The most common tool for modeling the EDFA performance is based on the approach developed in early 90s [3], [4] (see also [1], [2] for details) that rests on the following assumptions: 1) The state of the Er ions can be described on the level of populations (inversions) whereas the coherences (i.e., off diagonal elements of the density matrixes) can be integrated out, which is always the case in EDFA; 2) Any direct interactions between the ions may be neglected; and 3) ions are spectrally homogeneous, i.e., have identical absorption and emission spectra. Studies of effects of spectral inhomogenity (spectral hole burning) go beyond the scope of the present manuscript which implies that condition 3) is assumed to be satisfied. The validity of condition 2) depends crucially on concentrations since it is satisfied provided the - interaction induced energy transfer rates are small compared to the rates induced by interactions with the optical field, hereafter referred to as the optical rates. The latter are proportional to the optical power propagating in the amplifier whereas the energy transfer rates are determined by the distanses between the ions and are strongly dependent on concentrations. Degradation of the EDFA performance with the increase of concentration has been studied and attributed to additional losses due to energy transfer upconversion between ions [5] [13]. Accounting for upconversion-induced Manuscript received January 31, 2002; revised April 8, The authors are with Corning Inc., Corning, NY USA. Publisher Item Identifier S X(02) quenching in modeling of the gain and noise figure performance becomes especially important for design and optimization of -band EDFA, since they use higher erbium concentrations compared to -band EDFA. The models that have been used to interpret the experimental data [5] [13] are based on adding quadratic or higher order terms to the rate equation for erbium populations (homogeneous upconversion) or/and assuming a fraction of to be in clusters, the latter being treated explicitly (pair-induced upconversion). Similar approaches have been used to treat energy transfer for modeling of the -sensitized erbium-doped amplifiers [14], [15]. However, the aforementioned models do not provide sufficient accuracy in predicting the gain shape and noise figure of -band EDFA across various input conditions to use them as a practical tool for amplifier optimization and design. This is not too surprising since adding higher order terms to the equations for single-ion populations constitutes an example of an approximation known as mean-field or local-field applroximation (see, e.g., [16]) that does not account for a collective nature of the energy transfer. The collective nature of homogeneous upconversion erbium-doped fibers has been observed [17] and studied theoretically using a Monte Carlo approach [18]. In this manuscript, we present an approach that is capable of accounting for the collective nature of complex energy transfer dynamics in high-concenration -band EDFA by lumping it into a function of two variables (namely the absorption and emission rates), hereafter referred to as the inversion function. This means that in terms of modeling the complex part that involves solving a many-body problem is to calculate the inversion function. In Section II, we introduce the inversion function and formulate a closed system of equation that allow to model EDFA performance. The inversion function can be either obtained from additional exhaustive measurements, or computed using numerically expensive Monte Carlo simulations. To avoid that, in Section III, we develop an approach to evaluating the asymtotic behavior of the inversion function when the optical rates are high compared to energy transfer rates for typical distances between the ions, which is a typical situation for EDFA. We also derive analytical expressions for in this limit assuming random spatial distribution of. In Section IV, we compare the modeling results with the measurement data an -band fiber amplifier across various input conditions. II. MODEL AND BASIC EQUATIONS Our model of single-mode -doped gain fiber includes optical field in the fiber modes accounted for explicitly that interacts with the system of ions treated as two-level systems. For the sake of simplicity, we do not consider the X/02$ IEEE

2 570 IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 8, NO. 3, MAY/JUNE 2002 double-degeneracy of the fiber mode (this can be incorporated in a standard way). All ions are assumed to be spectrally homogeneous (spectral hole-burning is neglected). The basic equations that we present here generalize the standard equations of the homogeneous model (HM) for EDFA [1], [2] in a fairly straightforward although very useful way. In principle, they can be derived starting from a fully microscopic picture by utilizing the path-integral representation of the Liouville space dynamics (see e.g., [19]) followed by integrating over fast trajectories (see [20] for the technique). These equations, however, have a transparent meaning and can be rationalized using simple physical arguements. As in the case of the HM the optical field can be described on the power level in terms of discrete powers, where stand for forward and backward propagating powers and ASE power spectral densities. The optical powers satisfy the following equations: with for. In (1) and (2), and are the Giles parameters corresponding to absorption, emission, excited state absorption, and passive loss at frequency. The inversion can be naturally defined as the fraction of excited ions averaged across the fiber. These equations have the same form as the equations for optical powers in the HM. The only difference is that may not be found by solving simple rate equations as in the case of the HM, since the erbium inversion dynamics becomes collective in nature because of energy transfer. However, being still the population dynamics (although collective in nature), it is affected by the optical field only through the total absorption and emission rates and, respectively. Measured in the units of the radiative lifetime of the excited state, they can be expressed in terms of the optical powers in the same way as for the HM (see, e.g., [1], [2]): with being the saturation parameter. This implies that in the steady state of EDFA the inversion is some functional of the absorption and emission rates: Although the inversion is generally speaking a functional of the absorption and emission rates, the functional is actually local, since those rates change substantially on a macroscopic length scale. This means that (1) (2) (3) (4) the inversion at point is determined by the values of the rates at the same only. This implies Stated differently, this means that the functional is determined by a function, the one in the right-hand side of (5). We refer to this function as the inversion function. In principle, a well-defined perturbation theory can be developed in a straightforward way that expands the inversion functional in the -derivatives of and, the expansion starting with the local form [ (5)]. However, we do not pursue this approach since the corrections are negligible for all practical purposes. Finally, (1), (2) together with (3), (4), and (5) constitute a closed system of equations, hereafter referred to as the generalized homogeneous model (GHM) that allow to model high-concentration EDFA. The inversion function is considered to be an input that contains all relevant information on the energy transfer dynamics. III. INVERSION FUNCTION AT HIGH VALUES OF OPTICAL RATES In this section, we develop a procedure of calculating the asymtotic behavior of the inversion function at high optical rates that corresponds to the high-power regime of EDFA. We first note that model that combines homogeneous upconversion [5] with the pair-induced quenching [7] (see, also, [2]) is a particular case of the GHM with where is the homogeneous upconversion constant and is the fraction of ions in clusters. The inversion function in (6) considered as a function with is analytical in with the expansion coefficients depending on for large values of, i.e.,, which corresponds to high optical powers. In particular, it describes saturation of the concentration-induced losses at high optical powers. We now consider a general model where upconversion is the only energy transfer mechanism that is described by a set of upconversion rates, with denoting the ions, and calculate the desired asymtotic behavior by applying a cluster expansion when accounting for energy transfer in a situation when typically upconversion is slow compared to optical transitions:. This implies that typically upconversion can be accounted pertubatively. However, since do not form a crystal lattice, but are rather distributed more or less randomly, there is always a probability to find some ions at short enough distances so that the upconversion rate is comparable to the optical rates and upconversion may not be treated pertubatively. On the other hand, if the optical rates are high enough, the probability of clustering is low, and clustering may be considered as an expansion in the number of ions in a cluster starting with two-particle clusters only. (5) (6)

3 CHERNYAK AND QIAN: MODELING HIGH-CONCENTRATION -BAND EDFA 571 To put this simple idea on a more firm theoretical basis, we introduce some value of the upconversion rate so that typically and the probability of finding a pair of ions with is low. We then connect any two ions if and define a cluster as a set of connected ions in a sense that you can travel between any two particles in a cluster using the connections introduced above. The zeroth-order of the cluster expansion corresponds to considering only single ions. We will restrict ourselves to the first order which implies that we consider single ions and clusters of two ions. Denoting by the transverse two-dimensional (2-D) coordinate a closed system of rate equation can be written for the following variables: inversion of single ions, and the populations and of the states in twoparticle clusters with the upconversion constant when both ions are in the ground and excited states, respectively. The relevant information on Er positions is described by the distribution functions and for single ions and clusters, respectively, that are normalized to one where is the power profile of the fiber mode normalized to the unit power: and we have neglected the wavelength dependence of the fiber mode in the spectral region of the pumps, signals, and ASE which is somewhat between 1480 and 1620 nm ( -band EDFA are pumped at 1480 nm). The coefficient 2 in the normalization condition reflects the fact that there are two ions in a cluster. The value of the inversion function can be expressed in terms of the relevant variables introduced above in the following way: (7) (8) where the coefficients and are: (13) (14) and with being the effective erbium concentration defined as the overlap of the function that describes the concentration dependence across the fiber with. The first term in the rhs of (10) is exactly the same as for the standard HM. The second term represents losses due to radiative decay and homogeneous upconversion, i.e., upconversion that involves pairs of ions at typical distances, it constitutes the first term in the expansion of the inverson function in powers of and does not depend on the power profile of the fiber mode. The last term describes the losses due to clustering-induced upconversion that includes upconversion processes between pairs of ions that are close enough so that the upconversion is competitive with optically induced processes and in general depends on in a nonanalytical way. Equation (10) describes the leading terms of the asymptotic expansion of the inversion function at high optical powers where the terms up to are retained and effects of collective upconversion energy transfer are taken into account explicitly [i.e., the retained terms in the asymptotic expansion are evaluated exactly and only the higher order terms of the expansion are neglected in (10)]. More generally, as stated above, at high values of optical rates collective effects of energy transfer that occur between ions at typical distances can be treated pertubatively and a regular asymptotic expansion of the inversion function in powers of can be developed. We now turn to a more particular case of locally random distribution of ions (that corresponds to the distribution of the distances between the ions in clusters) and a power-like scaling of the upconversion constant the distance between the ions, which yields with (15) (9) Treating the clusters explicitly [the second term in (9)] and accounting for upconversion at typical distances [the first term in (9)] pertubatively yields the inversion function in a form (10) Substituting (15) into (12) yields, after some straightforward transformations with (16) with (11) and (17) (18) (12) It is possible to show that is -independent (the dependence of the individual terms on is cancelled out when they are

4 572 IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 8, NO. 3, MAY/JUNE 2002 Fig. 1. Gain differences between experimental results and modeling results by LBCM and DRCM for the same Er fiber inversion level under low and high optical powers. The input signal power to the coil is 01 dbm, and the forward and backward pump powers are 175 and 69 mw, respectively. The input signal power to the coil is 5 dbm, and the forward and backward pump powers are 175 and 224 mw, respectively. added up). The asymptotic expression by (16) has four parameters:, and, only being dependent on the power distribution in the fiber mode. Consider a particular case of the GHM with the inversion function that satisfies the following requirements: Its asymptotic behavior at large values of is given by (16), for low concentration it reduces to the standard HM, it has one concentration-related parameter [a counterpart of in (16)] that describes effects of clustering that are the most important at high optical powers. A model with the aforementioned properties will be referred to as distributed random cluster model (DRCM). IV. EXPERIMENTAL To experimentally validate our model, we performed gain and noise figure measurement on an -band EDFA, consisting of a single high concentration coil (3700 ppm), bidirectionally pumped at 1480 nm. A comb of wavelengths ( nm) at 100-GHz spacing was used as the input signal to the amplifier. The gain was obtained by measuring the input and output powers of each wavelength channel, and the noise figure was obtained from the gain and the ASE power density at each wavelength channel, measured on an optical spectrum analyzer. The measurement inaccuracies are estimated to be 0.3 db for gain and 0.5 db for noise figure. Fig. 2. Noise figure comparison between experimental results and modeling results by LBCM and DRCM for the same Er fiber inversion level under low and high optical powers. See Fig. 1 caption for signal and pump powers for and. Since the purpose of this work is to allow accurate modeling of EDFA performance, we obtained the experimental results under various input powers from 1 to 10 dbm, which is the typical input power range of an EDFA. We also performed measurements on the erbium fiber under various inversion levels typical for -band amplifiers. For a multistage EDFA, the first fiber coil is usually operating at high inversion to keep noise figure low, whereas the last coil is usually operating at high power and a relatively low inversion. Simulation results obtained from two methods are compared to the experimental results here, which will be referred to as the -band cluster model (LBCM) and the DRCM. Both methods use (1) and (2) to calculate the optical powers, where the Giles parameters, and were measured on the fiber under test. These parameters are kept the same for all model calculations and therefore do not affect the comparison of the modeling results calculated by the two methods. The two methods differ in the way they calculate the inversion function: The LBCM uses (6) with set to 0, which corresponds to the cases at high optical powers where homogeneous upconversion is negligible compared to clustering. The coefficient is the only fitting parameter used in this model, determined from a single gain measurement, and kept constant for the various modeling cases presented below. The DRCM calculates the inversion using the method described in the previous section. The

5 CHERNYAK AND QIAN: MODELING HIGH-CONCENTRATION -BAND EDFA 573 Fig. 3. Gain difference between experimental results and modeling results by LBCM and DRCM for different Er inversion levels. High input power (10 dbm) and low inversion level (0.375). The forward and backward pump powers are 119 and 103 mw, respectively. Low input power (0 dbm) and high inversion level (0.48). The forward and backward pump powers are 175 and 204 mw, respectively. only fitting parameter for the DRCM is, which is also determined by a single gain measurement and kept constant for the various modeling cases presented below. First, we test the models for varying power levels while keeping the inversion constant. Fig. 1 illustrates the gain discrepancies calculated by the two models under both low and high optical powers, with an average gain of 18.2 db in the -band. Experimentally, a constant inversion level, which approximately corresponds to the average inversion of an -band amplifier, is used purposely for both cases. The gain discrepancies shown in Fig. 1 include measurement error in gain, model error in the inversion level, as well as the error in the Giles parameters. We use the low power case to determine the fitting parameters and for the LBCM and DRCM, respectively. Both models predict gain in good agreement (within 0.5 db) with the experiment for both lowand high-power cases. Furthermore, the spectral shapes of the gain discrepancy in both cases are very similar, indicating that both models predict consistent inversion level for the range of optical powers used, and the discrepancy seen is largely due to the error in the Giles parameters and/or systematic error in the measurement. Noise figure spectra calculated by both models agree well with experimental results (Fig. 2). We observe a nearly constant Fig. 4. Noise figure comparison between experimental results and modeling results by LBCM and DRCM for different Er inversion levels. See Fig. 3 caption for signal and pump powers for and. offset of 0.3 db across the spectral range between experiment and the DRCM model results, the offset being within the error margin of our measurement. The good agreement in the spectral shape of the noise figure calculated by the DRCM gives us good confidence in the model, since it indicates that the model calculate ASE power density correctly, a parameter depends not only on the average inversion over the fiber length, but also on the variation of inversion along the Er fiber. Next, we test the model validity for varying inversion levels. The gain and noise figure comparisons between experimental and modeling results for low and high inversions are illustrated in Fig. 3 and 4, respectively. The low inversion case is chosen to represent the typical inversion level of the last coil in a multistage -band amplifier. It has an estimated inversion level of and an average gain of 10.4 db in the -band. The high inversion case has an estimated inversion level of 0.48, typical for the first coil of an -band amplifier. The average gain in this case is 21.7 db in the -band. Both LBCM and DRCM produce results in good agreement with experiments for the low inversion level [Fig. 3 and 4]. For the high inversion case, the LBCM gives slightly better gain prediction [Fig. 3 and 4]. It should be noted that, for the high inversion case, the measured gain and noise figure are much higher than in the low inversion case, and therefore the relative discrepancy in the high inversion case is comparable to or better than that observed for the low inversion case.

6 574 IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 8, NO. 3, MAY/JUNE 2002 V. CONCLUSION Neglecting effects of spectral hole burnuing all relevant information on collective energy transfer processes that cause gain degradation in high concentration -band EDFA is contained in the inversion function. At large values of, the inversion function can be calculated analytically by combining a perturbative approach with a clustering expansion. Comparison of the modeling results with measurements data demonstrates that both, the DRCM based on the assumption of locally random spatial distribution (statistical clustering) and LBCM that assumes clusters as pairs of ions at very short distances (physical clustering) allow accurate modeling of the high-concentration -band EDFA performance across a broad range of relevant power and inversion conditions. This demonstrates that performance of high-concenration -band EDFA is determined mostly by the concentration, rather than by the details of spatial distribution. ACKNOWLEDGMENT The authors would like to thank A. Brown, S. Guy, D.E. Goforth, J.D. Minelly, and M. Mlejnek for stimulating discussions and useful comments. REFERENCES [1] E. Desurvire, Erbium-Doped Fiber Amplifiers. New York: Wiley, [2] P. C. Becker, N. A. Olsson, and J. R. Simpson, Erbium-Doped Fiber Amplifiers: Fundamentals and Technology. New York: Academic, [3] B. Pedersen, A. Bjarklev, J. H. Povlsen, K. Dybdal, and C. C. Larsen, The design of erbium-doped fiber amplifiers, J. Lightwave Technol., vol. 9, pp , Sept [4] C. R. Giles and E. Desurvire, Modeling erbium-doped fiber amplifiers, J. Lightwave Technol., vol. 9, pp , Feb [5] P. Blixt, J. Nilsson, T. Carlnas, and B. Jaskorzynska, Concentrationdependent upconversion in Er -doped fiber amplifiers: Experiments and modeling, IEEE Photon. Technol. 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London, U.K.: Oxford Univ. Press, [17] J. L. Philipsen, J. Broeng, A. Bjarklev, S. Helmfrid, D. Bremberg, B. Jaskorzynska, and B. Palsdottir, Observation of strongly nonquadratic homogeneous upconversion in Er -doped silica fibers and reevaluation of the degree of clustering, IEEE J. Quantum Electron., vol. 35, pp , Nov [18] J. L. Philipsen and A. Bjarklev, Monte Carlo simulations of homogeneous upconversion in erbium-doped silica glasses, IEEE J. Quantum Electron., vol. 33, pp , May [19] V. Chernyak and S. Mukamel, Path-integral formulation of retardation effects in nonlinear optics, J. Chem. Phys., vol. 100, pp , [20] V. N. Popov, Path Integrals in Quantum Field Theory and Statistical Mechanics: Reidel-Kluwer, Vladimir Chernyak was born in Moscow U.S.S.R., received his M.S. and Ph.D. degrees from the Moscow Institute for Physics and Technology and U.S.S.R. Academy of Sciences Institute of Spectroscopy, in 1979 and 1983, respectively. He worked at several research institutions of the U.S.S.R. State Bureau of Standards and U.S.S.R. Academy of Sciences. In , he was with the Department of Chemistry of the University of Rochester and joined Corning Inc. in 2000 where his work is mainly related to modeling of fiber amplifiers. He is an author of more than 100 publications in the areas of nonlinear optics and spectroscopy, mathematical physics, condensed matter physics, disordered systems, femtosecond electronic and vibrational spectroscopy, and quantum chemistry. Li Qian (S 99 M 00) was born in Shanghai. She received the B.A.Sc., M.A.Sc., and Ph.D. degrees from the Department of Electrical and Computer Engineering at the University of Toronto, ON, Canada, in 1993, 1996, and 2000, respectively. Her graduate work focused in the fields of ultrafast photonics and nonlinear optics. She designed and built a picosecond, high-power, optical parametric oscillator, and demonstrated subpicosecond to picosecond all-optical switching capable of ultrahigh repetition rate based on large band gap nonlinearity in bulk semiconductors with intentionally induced defects. In 2000, she joined Corning Inc., Corning, NY, and her work mainly involves the development of the first commercial extended L-band erbium-doped fiber amplifier.