Electromagnetic inverse scattering theory is applied to the synthesis of optical waveguides and devices from specified reflection and transmission characteristics. The permittivity profiles in the inhomogeneous core regions of the devices are reconstructed from their transverse reflection coefficients. Two applications of this theory are demonstrated with examples using specified reflection coefficients: design of a single-mode inhomogeneous optical waveguide and design of an optical logic gate. The previous inverse scattering theory which used reflection coefficients that are rational functions of the transverse wavenumber has been supplemented with an inverse scattering theory that uses nonrational reflection coefficients. This inverse scattering theory uses an iterative discretized space-time finite-difference method with the “leapfrog” algorithm. The initial values for the iterative method are obtained from the inverse theory that uses rational function reflection coefficients. The inverse scattering calculations have been verified by finite-difference frequency domain direct scattering techniques.