Effectiveness of a second order CPML absorbing boundary condition within the periodic FDTD method

The analysis of periodic structures is of great interest for engineers and scientists in a variety of applications. EMC design, antenna design, photonics, and LO design are examples of such areas. In previous work [1], [2], [3], [4], [5], [6] it has been shown by these authors and others that such analyses may be accurately accomplished in the frequency or time domain. While previous works have been successful for a variety of problem scenarios, time domain implementations suffer a loss of accuracy at very low frequencies due to a breakdown of PML performance. To understand this deficiency, consider the expressions below which embody the split-field periodic FDTD where convolutional PML (CMPL) has been employed.