  We investigate bound states in the continuum (BICs) in a planar dielectric
waveguide structure consisting of a gold grating on a dielectric layer with a
back layer of metal. In this structure, Friedrich-Wintgen (FW) BICs caused by
the destructive interference between the radiations from two waveguide modes
appear near the anti-crossing point of the dispersion curves. In this study, it
is revealed that the branch at which the BIC appears changes according to the
polarization of incident radiation. Based on a temporal coupled mode theory, it
is shown that the BIC branch is determined by the sign of the product of the
coupling coefficients between the two waveguide modes and external radiation,
which is consistent with FW theory. The signs of the coupling coefficients are
estimated by the waveguide-mode decomposition of the numerically obtained
electric fields and are confirmed to vary depending on the polarization.
