Beam-propagation method for analysis of z -dependent structures that uses a local oblique coordinate system

In the most current beam-propagation method (BPM), the coupling between the transverse and the longitudinal fields in z -dependent structures is ignored under the staircase approximation, which results in violation of power conservation. We propose a novel BPM that is derived in a local oblique coordinate system to analyze z -dependent structures accurately and efficiently without taking the staircase approximation. The coupling between the transverse and the longitudinal fields is automatically included in the local oblique coordinate system, and power is thus conserved if the reflection is neglected.