Modelling of circular wave guides

In many circumstances the design of interconnects in a photonic integrated circuit can be simplified by using low loss curved wave guides in the shapes of circular arcs. Radiative losses associated with the curvature have been computed as a function of the radius of curvature. The technique takes advantage of the effective index method to reduce the problem from two dimensions to one dimension (1D) and uses a change of coordinate that transforms an arc of circle into a straight line. This transformation results in a monotonous increase of the refractive index as function of r (the distance from the centre of the circle) for original constant index regions. The new system is solved by discretizing this varying effective index onto many small layers of constant index over a window large enough to contain the region where the field is not negligible. A multilayer algorithm in 1D is then used to find complex propagation constants in which the imaginary part is related to the fundamental energy loss owing to the curvature. The solution also gives the shape of the field necessary to match the mode profiles at the junction between the straight and curved part of the wave guide. The basic change of variable has been extended to the finite difference solution of the scalar wave equation and to the beam propagation method.