Exact equivalent-profile formulation for bent optical waveguides

A widespread, intuitive and computationally inexpensive method to analyze light guidance through waveguide bends is by introducing an equivalent straight waveguide with refractive index profile modified to account for actual waveguide bend. Here we revise the commonly used equivalent-index formula, ending up with its simple extension that enables rigorous treatment of one- and two-dimensionally confined, uniformly bent waveguides, including tightly coiled microstructure fibers, curved ridge waveguides and ring microresonators. We also show that such technique is applicable only to waveguides composed of isotropic or uniaxially anisotropic materials, with anisotropy axis directed perpendicular to the curvature plane. To understand and to predict light wave behavior at waveguide bends was of major importance and interest to integrated- and fibre-optics community from around 1970s and onwards. Today this interest is stimulated largely by two developments: (i) increasing the packaging density of integrated-optic circuits, backed by minimizing integrated waveguide bend radii while keeping bend losses at a tolerable level; and (ii) the advent of photonic crystal fibers possessing quite complicated, high index-contrast dielectric profiles as compared to step- or graded-index fibers for which early theoretical methods to treat bend losses were developed. Numerical methods to simulate light propagation through waveguide bends, such as beam propagation method (1), the method of lines (2, 3), as well as more general-purpose finite-element and finite-difference techniques, can be very demanding computationally: in one recent example (4), a 64-processors cluster was used in full-vector finite-element modelling. No wonder one leans to analytic techniques to reduce computation burden whenever possible; one such technique, perhaps the simplest and most widely used in modelling of microstructure fibre bends today (5, 6, 7, 8), is equivalent-profile method (9, 10). It reduces dimensionality of the problem by replacing actual waveguide bend by a straight piece with refractive index profile