A Finite-difference Time Domain Method for Analysis in Optical Waveguides: The Time Stability Determination

This paper proposes a method how to determine the critical time step for use in Finite-Difference Time Domain (FDTD) analysis. Typically, the method is requires an appropriate time step in the calculation. The method is based on the criterion of stability increasing time must less than the time for the increasing propagation distance used in the calculation. For the large time step, the calculation would be diverged whereas the small time step would consumed huge execution time for getting a result. Therefore, we propose a method to determine the value of the critical time step of stability in the FDTD calculation for the optical slab waveguide. We investigate the maximum time step to provide the minimum time expensed in the calculation with maintaining the stability in the beam propagation. A ratio of the transversal step to the longitudinal step or vice versa, as greater than unity, F, and the fraction, A, of the stability time step are established for this method. Both parameters would simplified the derivatives terms of the electric and magnetic fields in the Maxwell's Equations. The execution time would reasonably decreased. Alternatively, the relationship between the A and the fraction number F was calculated and represented by A = 0.003F⁴ - 0.135F³ + 2.24F² - 17.73F + 66.78. It is found that the maximum time step is limited at 50% of the propagation time through the guide. The critical time step, which is used in the FDTD calculation, can be determined by the fractional critical time step A. With the guide length/width ratio > 10, the fractional critical time step should be not more than 10%. To implement the graph, we chose F of 4, corresponding to A of 19%. So, the simulation does diverge when A of 20%. The simulated results at the critical A and the small A of 9.5% have the same beam patterns.