  At a fixed frequency, a general wave field in an open optical waveguide is a
sum of finite number of guided modes and an integral of radiation modes. Most
studies on open lossless dielectric waveguides are concerned with regular
guided modes which are confined around the waveguide core, have a real
propagation constant, and propagate along the waveguide axis without
attenuation. However, there may be guided modes with a complex propagation
constant even though the waveguide consists of lossless isotropic dielectric
materials, and they are the so-called complex modes. The complex modes are
proper modes confined around the waveguide core, and are different from the
leaky modes which have divergent wave fields in the transverse plane. For
waveguides with discontinuities and defects, the complex modes can be excited
and must be included in eigenmode expansions. However, existing studies on
complex modes in open lossless dielectric waveguides are very limited. In this
Letter, we consider circular fibers and silicon waveguides, study the formation
mechanism of complex modes, and calculate the dispersion relations for several
complex modes in each waveguide. For circular fibers, we also determine the
minimum refractive-index ratio for the existence of complex modes. Our study
fills a gap in optical waveguide theory and provides a basis for realizing
potential applications of complex modes.
