Analysis and Optimization of the Numerical Calculation in the Slowly Decaying Imaginary Distance Beam Propagation Method

The detailed analyses of different numerical schemes are presented for solving the governing equation of the slowly decaying imaginary distance beam propagation method (SD-ID-BPM), including their convergence speed and stability. It will be demonstrated that the fully implicit scheme has great advantages over the Crank-Nicholson scheme to obtain the eigenmodes using SD-ID-BPM. The converged solution of the eigenmodes can be obtained tremendously faster using the fully implicit scheme than using the Crank-Nicholson scheme with similarly good accuracy. In addition the fully implicit scheme will be shown to be easier to implement, more reliable and robust than the Crank-Nicholson scheme. The use of the fully implicit scheme in the SD-ID-BPM results in great improvements of the modeling capability of this method. It will also be demonstrated that both the fully implicit and Crank-Nicholson schemes are numerically stable.