Localized-mode evolution in a curved planar waveguide with combined Dirichlet and Neumann boundary conditions. 
We present a theoretical study of a planar waveguide with a uniformly curved section.  Opposite sides of the channel satisfy different boundary conditions.  It is shown that if the Dirichlet condition is applied to the inner side of the strip and the Neumann one to the outer wall, then properties of such a system in many respects resemble those with the Dirichlet requirements on both surfaces.  Namely, in both cases a propagation threshold for the curved section is smaller than its counterpart for the straight channel.  As a consequence, a localized mode exists with its energy below the propagation threshold of the straight waveguide.  Analysis of such states is presented as a function of the bend parameters.  For the transport in the fundamental mode an interaction of a quasibound level split off from the higher-lying threshold, with its degenerate continuum counterpart, causes a dip in the transmission.  Such a resonance is characterized by a location of its zero minimum E(min) and the half width Gamma.  Changing the bend angle and radius, one varies E(min) and Gamma.  In particular, for some critical parameters of the bend it is possible to turn the half width to zero, i.e., to eliminate the dip in the transmission.  This corresponds to the absence of the interaction between the split-off level and the continuum, and, consequently, to the formation of the true bound state in the continuum.  Vortex structure of the currents flowing in the waveguide near the resonance is also shown to strongly resemble the analogous results for the Dirichlet case.  It is pointed out that the properties of the waveguide with the Neumann inner condition and the Dirichlet outer one mimic the duct with the Neumann requirements on the two sides, since for both these cases the propagation threshold in the curved section is greater than in the straight channel.