Length optimization of an S-shaped transition between offset optical waveguides

We derive expressions for the radiation loss of an S-shaped waveguide transition used to connect two straight integrated optics waveguides that are offset with respect to each other. It is assumed that the diffused integrated optics waveguides are produced with the help of an electron beam machine that allows beam positioning in the y direction only in discrete steps. We thus must consider staircase approximations to the desired smooth S-shaped curves. A waveguide whose axis consists of a staircase suffers radiation losses due to the quasi-periodic deformation of its axis. A second loss contribution comes from the S-shape of the waveguide axis. The sum of these loss contributions assumes a minimum that defines the optimum length of the transition waveguide.