Mode coupling at avoided crossings in slab waveguides with comparison to optical fibers: tutorial

Abstract

Avoided crossings are important in many waveguides and resonators. That is particularly the case in modern-day solid-core and air-core optical fibers that often have a complex geometry. The study of mode coupling at avoided crossings often leads to a complicated analysis. In this tutorial, we aim to explain the basic features of avoided crossings in a simple slab waveguide structure so that the modes can be found analytically with simple sinusoidal and exponential forms. We first review coupled-mode theory for the guided mode in a slab waveguide, which has a higher index in the core. We study the effective index of the guided true mode for a five-layer slab waveguide including two core layers with higher indices compared to the indices in the three cladding layers. Then, we study the same structure by using the overlap between approximate modes confined in the two individual core slabs. When the two individual core slabs are not near each other, the avoided crossing using the true modes within the two-slab waveguide agrees well with the results using the overlap between the two approximate modes. We also study coupled-mode theory and avoided crossings for leaky modes in an antiresonant slab waveguide. We obtain good agreement between the results using the true leaky mode and the results using the overlap between approximate modes. We then discuss examples of avoided crossings in solid-core and air-core optical fibers. We describe the similarities and differences between the optical fibers and simple slab waveguides that we have analyzed in detail.