An Improved Algorithm to Solve Erbium Doped Fiber Source Propagation equations.

Transcription

1 An Improved Algorithm to Solve Erbium Doped Fiber Source Propagation equations. Abu Thomas, Anil Prabhakar, Hari Ramachandran, Department of Electrical Engineering, Indian Institute of Technology-Madras, Chennai, India ABSTRACT An improved scheme has been developed to solve the standard propagation equations (SPE) 1 for Erbium doped fiber sources (EDFS). 2 The scheme uses Rosenbrock algorithm based on implicit differentiation methods. 3 The profiles of Forward Amplified Spontaneous Emission (FASE), Backward Amplified Spontaneous Emission (BASE) and Pump power along the fiber are presented and compared with the results of conventional fixed step and embedded Runge- Kutta methods, specifically RK The proposed multi-step method is stable even when designing EDFS for fiber lengths at which the above methods fail to converge. Keywords: EDFA,RK 45, Rosenbrock, Amplified Spontaneous Emission 1. INTRODUCTION EDFS are required as low temporal coherence sources in fiber sensor applications like fiberoptic gyroscopes (FOG) and interferometers. 2 The broad spectrum of such sources helps in the elimination of coherent errors due to Rayleigh back-scattering, 4 polarization cross coupling 5 and Kerr effect. 6 The use of fibers as sources help in the efficient coupling of power to systems using optical fibers for sensor applications. They also facilitate the use of the fiber directional couplers to perform beam-splitting and hence avoid diffraction losses associated with bulk optics and micro-optics. The standard propagation equations used for the design of EDFS are coupled nonlinear equations. All solution approaches use the relaxation method 7 to solve these equations. However the conventional fixed step method does not give an estimate of the error accumulated during each step and the accumulated error limits the accuracy of the solution for long fibers. This is dramatically highlighted in the studies carried out in this paper. The RK 45 provides for an adaptive step size based on a local error estimate and hence control over the global error. But for long fibers, under steady state conditions, certain portions of the FASE spectrum display exponential growth even while other portions (and the pump) exponentially decay. This results in coupled equations with very different scales of variation, as a result of which the problem becomes stiff. The solution of such a stiff problem proves intractable for the fixed step RK4 method, and becomes very slow for the adaptive step RK 45 method. The Rosenbrock method, Experimental Optics Lab, Telephone: +91 (44)

2 however, overcomes this limitation by means of implicit backward differencing resulting in faster integration and better accuracy for longer length EDFS simulations. Indeed, while relatively slow for shorter fiber lengths, the algorithm scales very well for longer lengths of fiber. 2. MODEL EQUATIONS AND ALGORITHMS This section describes the Standard Propagation (SPE) Equations and the issues that arise in their analysis. The equations are similar to those used by Wysocki. 1 dp ± s (z, λ i ) dz = ± γ s (z, λ i )P s ± (z, λ i ) + γ es (z, λ i )2hλ i ( δν h n ) (1) dp p dz = γ p(z, λ p )P p (z, λ p ) (2) where z is the position along the Erbium doped fiber, λ i and λ p are the wavelengths of the i th source and the pump channel respectively and P s and P p are the radially integrated intensities of the source and pump respectively. The gain and loss coefficients are given as follows: γ s (z, λ i ) = A 0 A shp [σ e (λ i )N u (z) σ a (λ i )N l (z)] (3) γ p (z, λ p ) = γ es = A 0 A shp [σ e (λ i )N u (z)] (4) A 0 A p [σ pa (λ p )N l (z)], A 0 A p [σ pa N l (z) σ pe N u (z)], λ P = 980nm λ P = 1480nm The P s + and Ps represents the FASE and BASE respectively. A 0, A p and A shp are the core, pump and signal mode areas respectively. The occupation levels of the upper lasing level is N u = N d N l where N d represents the Er3+ concentration in Silica and N l is given by n P s (z, λ i ) N l (z) = 1 + N d n i=1 i=1 I SEsat (λ i)a s P s I Ssat (λ + P p(z, λ p ) i)a s I P Esat (λ + P p(z, λ p ) p)a p I P Asat (λ + 1 p)a p 1 + N d n i=1 n i=1 P s (z, λ i ) I SEsat (λ + P p(z, λ p ) i)a s I P Esat (λ p)a p P s I Ssat (λ i)a s + P p(z, λ p ) I P Esat (λ p)a p + P p(z, λ p ) I P Asat (λ p)a p + 1 2, λ P = 980nm, λ P = 1480nm (5) (6)

3 The SPE system of equations assumes a three level atomic system. Pump power at 980 nm on absorption raises electrons to the 4 I 11/2 level, which has a very short life time of the order of nanoseconds. The excited electrons non-radiatively relax to the band of energy levels between the upper and lower lasing levels. Hence, the model assumes that the Erbium ions will be distributed in states lying between the upper lasing level and the lower lasing level. The lasing levels are Stark split because of the perturbation of the Erbium energy levels by the silica host. The propagation equations for ASE power at different wavelengths are only coupled through the population of the two lasing levels. The boundary conditions of the Eq. (1) are P s + (z = 0, λ i ) = 0 and Ps (z = L, λ i ) = 0, where L is the length of the fiber. The boundary condition of the pump equation. Eq. (2) is P p (λ p, z = 0) = P P 0. The problem to be solved is therefore a set of 2n + 1 equations for 2n+1 uknowns, of which n+1 of the unknowns have their boundary value specified at z = 0 and the remaining n unkowns have their boundary value specified at z = L, where n is the number of wavelengths. Each equation in this set of equations is a first-order differential equation specified in terms of an unknown function, namely, N l (z). The difficulty in solving this set of equations lies in the way errors in P s (z, λ i ) can feedback to P s (z, λ j ) through N l (z). As is typical in such hyperbolic systems of equations, numerical integration of the equations is stable as long as the quantities are growing or static. Errors tend to grow exponentially when the quantities are decaying exponentially along the direction of integration. Since the FASE grows in z while the BASE grows with L z, the standard prescription is to integrate the FASE from z = 0 to z = L, while the BASE is integrated from z = L to z = 0. During the FASE integration, the BASE is held constant, and during the BASE integration, the FASE is held constant. The above procedure runs into problems when the pump depletes. When this happens, the FASE also begins to deplete and errors begin to contaminate the solution. Further, since the problem becomes progressively stiff, even small errors push the operating point far from the expected solution. The approach suggested in this paper solves both these problems.the stiffness of the problem is tackled by using an implicit scheme. As is well known, implicit schemes are stable even when integrating an exponentially decaying solution, due to the negative feedback introduced by the implicit step. While the use of an implicit algorithm increases the cost of the computation, the algorithm becomes unconditionally stable to the errors commonly encountered when solving Eq. (1)-(6). During the forward step, the BASE equations are kept frozen, and the pump and the FASE signal powers are integrated. During the return step, the FASE signal powers and the pump are kept frozen, and the BASE signal powers are integrated. In the conventional approach, the step size is pre-decided so that the information of the frozen signals are available during each step. However, this results in poor or no error control. Hence, in the current implementation, an adaptive step size algorithm is used to determine the step size that is consistent with the desired end-to-end error. While this is optimal from the point of view of minimizing the number of function calls, it requires the frozen signals at points other than where they are available. To handle this problem, interpolation is carried out. 3

4 Interpolation introduces its own error into the scheme, and system accuracy is now a combination of interpolation error and the error due to integrating the differential equation. It is worth noting that Eq. (6) actually involves integrals over λ which have been approximated as sums. There is therefore an error due to the discretization and truncation in λ in addition to the two errors mentioned here. The discretization error is minimized by choosing λ = 1 nm, which is below the scale on which the spectrum varies. The truncation error is minimized by choosing a spectral range large enough that the edge signal powers are 40 db below the powers in the peak channels. Currently, the interpolation scheme used is cubic spline interpolation. Higher order spline interpolation has been tried and does not yield better results. It is planned to implement higher order interpolation using a tableau which will also yield an error estimate for the interpolation. Currently, the step size h is maintained small enough so that the error (which is of order h 4 per step) is sub-dominant compared to the errors in the remaining portions of the algorithm. The largest error that is present is the iteration error. After each cycle of calculating FASE and BASE, the values of FASE powers at z = L and the BASE powers at z = 0 are compared to their previous values. Convergence is assumed to have been reached when the differences are small enough. This convergence criterion is known to be unreliable and often results in convergence to limit cycles rather than to the actual solution. However, for such nonlinear systems, it is difficult to devise a better scheme. As described above, any adaptive step size algorithm could be used. In this paper we discuss the performance of two adaptive step size algorithms, namely the Runge Rutta-45 algorithm which is an explicit adaptive step size algorithm, and a Rosenbrock alorithm which is also based on runge kutta but is an implicit adaptive step size algorithm. 3. RESULTS Both algorithms were observed to converge to the same solution for short lengths of fiber. This solution was also that which was obtained using fixed step size algorithms. Table 1 compares the performance the different algorithms for different lengths. For short lengths, the fixed step algorithm outperforms the adaptive algorithms. The main reason for this is the interpolation step required for the adaptive step algorithms. However, for lengths greater than 5 meters, the fixed step RK4 algorithm failed to converge. The per-iteration error plateaued and failed to reduce even after 10 4 iterations. For the longer lengths, the adaptive step size Runge Kutta algorithm converges extremely slowly and only the Rosenbrock algorithm is able to cope. Algorithm 5 meters 10 meters 15 meters 20 meters 25 meters Fixed Step RK4 51 NA NA NA NA Adaptive Step RK NA NA Rosenbrock

5 Table 1: Table showing the execution times in seconds of three different algorithms. The runs were for a particular type of Erbium doped fiber with perfect isolation at both ends. The required error was 10 5 relative to the peak power. A 1480 nm pump of 15 mw was used. 100 channels spaced 1 nm apart were used to track the ASE. The RK4 algorithm failed to converge for distances greater than 5 meters Evolution of Amplified Spontaneous Emission and its Spectrum The integration of the SPE gives as output, the evolution of the FASE, BASE and their spectra along the fiber. Simulation results for a typical Erbium Doped Fiber are shown in Figure 1 and Figure 2. The simulation was done using Rosenbrock algorithm with a user specified tolerance of 10 4 of peak power at each step. The iterations were continued till a user specified tolerance of 10 4 was attained. Figure 2 shows that the spectrum of the FASE has a peak at 1558 nm instead of the usual one at 1532 nm. This is expected for an Erbium doped fiber long enough to deplete the pump. The pump depletion causes reabsorption of shorter wavelengths (in 1530 nm range) and the power is coupled to higher wavelegths. As Figure1 shows evolution of FASE along the fiber has a peak at around 7m. However the onset of reabsorption will be at different points for different wavelengths. In contrast, the growth of BASE is more steady and reaches a peak at the pumping end of the fiber. Fig. 1: Evolution of the pump, the FASE and the BASE along an Erbium Doped Fiber. The simulation is done for an Erbium doped fiber of length 10 m pumped at 15 mw. At a length of 10m most of the pump is depleted and the power in the spontaneous emission is mostly contained in the BASE. Fig. 2: The spectra of the FASE at the output and the BASE at the pumping end. The power in FASE is contained at longer wavelegths compared to BASE which is typical of any EDF operated in the depleted pump region. The simulation is done for an Erbium doped fiber of length 10 m pumped at 15 mw. 5

6 3.2. Scaling of error with step size for RK 45 and Rosenbrock The performance of the two adaptive step size algorithms was further investigated by studying the dependence of per-step error on step size. A fiber of length 15 meters was iterated till convergence was achieved. A point 12 meters from the beginning was then chosen for investigation. This location was one where the pump power had depleted and backward emission (BASE) was growing at the expense of the already fully developed forward emission (FASE). A single step of differing size was taken and the estimated error tabulated. Figure 3 presents the results of this study. For large step sizes, both algorithms scale as h 2.8. This observed scaling is at variance with the expected h 5 from the two algorithms. However, as expected, the implicit nature of the Rosenbrock algorithm allows the error to be two orders of magnitude lower for such stiff conditions. Fig. 3: Error-scaling with step size for RK 45 and Rosenbrock methods. The simulation is done for an Erbium doped fiber of length 15 m pumped at 15 mw. Fig. 4: Number of function calls for RK 45 and Rosenbrock methods for different lengths. The user specified tolerance was 10 4.The simulation is done for an Erbium doped fiber pumped at 15 mw. For smaller step sizes, there is a separation in the behaviours. The Rosenbrock algorithm continues to scale as h 2.8 as the step size is reduced, while the RK 45 algorithm on the other hand, stagnates in error with a scaling of h 1 for error. These two behaviours can be understood if we note the following. Firstly, the equations are stiff at this point in the fiber. Figure 6 presents the condition number of the Jacobian used in the Rosenbrock algorithm as a function of position. As can be seen, the system of equations is stiff, with R varying between 10 3 and 10 5, upto 12 meters, which is where all the FASE spectral lines begin to decay. Once that happens, the condition number stagnates at around 20. The RK 45 algorithm succeeds in integrating the equations. However, as the accuracy requirement is increased, smaller and smaller powers need to be taken into account. These are infact the powers that are decaying at a different rate from the primary channels. As a result, the scaling for RK 45 stagnates. 6

7 Fig. 5: The execution times of Rosenbrock and RK 45 algorithms for various lengths of the Erbium Doped Fiber.The user specified tolerance was 10 4.The simulation is done for an Erbium doped fiber pumped at 15 mw. Fig. 6: Evolution of condition number (the ratio of the largest to the smallest singular value) of the Jacobian along the fiber. The simulation is done for an Erbium doped fiber of length 15 m pumped at 15 mw. There are two other possible reasons for the observed behaviours. The BASE grows (along L z) with a 1/e scalelength of about L e 80 cm. Thus step sizes in excess of 80 cm are meaningless, as the error in these algorithms scales as (h/l e ) 5 only when h L e. For longer step sizes, the error is determined by other considerations. Also, in the process of solution, the BASE was computed at roughly every 10 cm near z = 12 meters. Thus, when the step size exceeds this length, higher order discontinuities in interpolations could also limit the accuracy of the Rosenbrock algorithm. To analyse these possibilities, the BASE data was sub-sampled to eliminate discontinuities in the region where this study was conducted. The scaling was completely unaffected. Further, both higher and lower order spline interpolation was tried. Once again, the scaling behaviour proved to be independent of the spline order. It appears that the scaling of h 2.8 is a feature of the system of equations and not a consequence of interpolation Computational Effort The previous section clearly established that fewer steps were required for the Rosenbrock algorithm than for the RK 45 algorithm, in regions where the equation is stiff. However, the equations have different stiffness at different portions of the fibre, as seen in Figure 6. Thus, this advantage does not necessarily translate into a speed differential. Figure 4 presents the number of function calls required to solve for the profiles of fibers of different lengths. It is clear that the Rosenbrock algorithm outperforms the RK 45 algorithm for all lengths. The Rosenbrock algorithm involves inverting the Jacobian of the system of equations and using that inverse to determine the new values. Thus, each function call in the Rosenbrock algorithm is significantly costlier than in the RK 45 or the fixed step RK4 algorithms. Figure 5 presents the actual time taken by the different algorithms for different lengths. This is a plot 7

8 of essentially the same information that is in Table 1. As the figure shows, the Rosenbrock algorithm is significantly superior in performance for longer lengths of fiber. 4. CONCLUSION This study has explored the usefulness of implicit methods to solve the SPE equations for Erbium doped fibre source problems. The Rosenbrock class of algorithms is seen to perform better than the conventionally used RK4 and RK 45 algorithms. It is the only algorithm that works for longer lengths of fiber, which are of interest when developing L Band fiber sources. The relatively slow convergence of all the algorithms is a cause for concern. The authors believe that the principal obstacle to improving speed lies in the 1/e scaling distance for the emission. Any polynomial-based solver (and both RK 45 and Rosenbrock are based on Taylor approximation of systems of differential equations) will struggle with larger step sizes. The obvious alternative is to look at the equations in log space, where the exponential variation will appear as straight lines. Such a suggestion has already been made in the literature. 8 This work is currently under progress and the authors hope to report signficant success shortly. REFERENCES 1. P. F. Wysocki, M. J. Digonnet, B. Y. Kim, H.J.Shaw, and E. Teller, Charecteristics of Erbium doped fiber sources in interferometric sensor applications, IEEE J. Lightwave. Tech. 12, pp , M. J. F. Digonnet, Theory of super-fluorescent fiber lasers, IEEE J. of Lightwave. Tech. 4, pp , W. H. Press, S. A. Teukolsky, W. H. Vetterling, and B. P. Flannery, Numerical Recipes in C, Cambridge University Press, Cambridge, C. C. Cutler, S. A. Newton, and H. J. Shaw, Limitation of rotation sensing by scattering, Optical Letters 5, pp , W. K. Burns and R. P. Moeller, Polariser requirements for Fiber Gyroscopes with high bi-refrigence fiber and broad band sources, IEEE J. of Lightwave Tech 2, p. 430, R. A. Bergh, B. Culshaw, C. C. Cutler, H. C. Lefevre, and H. J. Shaw, Source statistics and the Kerr effect in Fibre-optic Gyroscopes, Optical Letters 7, p. 563, E. Desurvire, Erbium Doped Fiber Amplifiers, Principles and Applications, Wiley-Interscience, New York, X. Liu and B. Lee, Effective shooting algorithm and its applicaton to fiber amplifiers, Optics Express 11, pp ,

9 List of Figures Fig. 1 Evolution of the pump, the FASE and the BASE along an Erbium Doped Fiber.The simulation is done for a fiber of length 10 m pumped at 15 mw using Rosenbrock method. Fig. 2 The spectra of the FASE at the output and the BASE at the pumping end. The simulation conditions are same as in Figure1. Fig. 3 Error-scaling with step size for RK 45 and Rosenbrock methods. The simulation is done for an Erbium doped fiber of length 15 m pumped at 15 mw. Fig. 4 Number of calls to slope routine for RK 45 and Rosenbrock methods for different lengths. The user specified tolerance was The simulation is done for an Erbium doped fiber pumped at 15 mw. Fig. 5 Actual time taken by different algorithms to solve for fibers of different lengths. The user specified tolerance was 10 4 of the peak power, and a pump of 15 mw was used. Fig. 6 Variation of the condition number of the Jacobian of the system of equation with position along the fiber. The data was collected for the same conditions as those of Fig. 3. List of Tables Table 1 Table showing the execution times of three different algorithms. The runs were for a particular type of Erbium doped fiber with perfect isolation at both ends. The required error was 10 5 relative to the peak power. A 1480 nm pump of 15 mw was used. 100 channels spaced 1 nm apart were used to track the ASE. 9