Propagation of Gaussian beams in piano-curved optical waveguides with a parabolic refractive index profile

An analysis is made of the propagation of paraxial Gaussian beams in optical waveguides arbitrarily curved in one plane and having a parabolic refractive index profile. The propagation function is derived for this case and by analyzing this function, it is shown that the center of the beam moves along a specific trajectory. Moreover, the Gaussian intensity distribution with a periodically varying parameter characterizing the beam width is conserved along the trajectory in the transverse cross section of the beam.