Formulation of adaptive Adam-Bashforth method for solving ordinary differential equations: modeling of highly doped waveguide amplifiers

Accurate modeling of dynamic optical interactions such as saturation and re-absorption in highly doped waveguide amplifiers requires solving a stiff system of ordinary differential equations (ODEs). Traditional ODE solvers including Range-Kutta methods are computationally ill-suited for such applications. In this paper, we derive and apply predictor - corrector adaptive Adam-Bashforth scheme for modeling the population dynamics in Erbium - Doped Fiber Amplifiers (EDFA). Predictor and corrector equations for adaptive Adam-Bashforth have been derived by using Lagrange polynomial as basis rather than the Newton polynomials used in constant stepsize Adam-Bashforth scheme. Convergence and stability analysis conducted on the scheme shows that the method has similar characteristics as that of constant step-size conventional Adam-Bashforth methods for small changes in step sizes. Solutions have been validated by re-generating the absorption and emission coefficients for doped fibers with two different doping concentrations, which is found to match with the manufacturer datasheet. This method is compared with other method like Euler and the optimum order of predictor and corrector is estimated. The result show that this modified form of the scheme results in 75% reduction in step-size to maintain an relative accuracy level of 10 -3 as compared to adaptive Euler method. Finally, different orders were compared by using ratio of step-size and number of operations per step as a metric for Figure of Merit (FOM). FOM analysis shows that use of higher order methods are not efficient in reducing the number of steps required to obtain accurate results. It is found that the scheme with both second order predictor and corrector is the most efficient computationally. However, in terms of accuracy second order predictor and third order corrector is more suitable with only a marginal degradation of FOM.