Absorbing boundary conditions for adjoint problems in the design sensitivity analysis with the FDTD method

In this paper, absorbing boundary conditions (ABCs) for adjoint problems with a backward time variable are derived from first principles. It is shown that all single-layer ABCs for the adjoint backward time problem, which are based on the one-way wave equation, have the same form as for the original forward time problem. In the case of the adjoint perfectly matched layer (PML) ABC, the signs before the spatial derivatives are opposite to those in the PML ABC of the original forward time problem. To verify the theoretical findings, the numerical reflections from the adjoint ABCs are investigated in a microstrip-line example. The reflections from the ABCs of the forward- and backward time schemes are shown to be identical for the same type of ABCs.