  We theoretically investigate the dispersion relation of small-amplitude
optical waves superimposing upon a beam of polarized monochromatic light
propagating along a single-mode channel waveguide characterized by an
instantaneous and spatially local Kerr nonlinearity. These small luminous
fluctuations propagate along the waveguide as Bogoliubov elementary excitations
on top of a one-dimensional dilute Bose quantum fluid evolve in time. They
consequently display a strongly renormalized dispersion law, of Bogoliubov
type. Analytical and numerical results are found in both the absence and the
presence of one- and two-photon losses. Silicon and silicon-nitride waveguides
are used as examples. We finally propose an experiment to measure this
Bogoliubov dispersion relation, based on a stimulated four-wave mixing and
interference spectroscopy techniques.
