Heterostructure GaAs-Al<sub>x</sub>Ga<sub>1-x</sub>As-GaAs waveguides for 10.6 µm wavelength grown by liquid-phase epitaxy have a linear variation of dielectric constant in their substrates. On the basis of their mode equation, a computer program was devised for obtaining optimum values of the six parameters that define the index profiles, given a sufficient number and distribution of mode-angle measurements. The optimized parameters agreed well with the measured values for a variety of waveguides, illustrating the generality of the parameter-optimization procedure. Two types of leaky modes were observed, whose effective index β/<i>k</i> is less than the seed index, (1) Lummer—Gehrcke modes that are incompletely reflected at the substrate-seed interface and (2) low-loss totally reflected modes whose evanescent fields radiate into high-index seeds.
W. W. Rigrod, J. H. McFee, M. A. Pollack, and R. A. Logan, "Index-profile determination of heterostructure GaAs planar waveguides from mode-angle measurements at 10.6 µm wavelength," J. Opt. Soc. Am. 65, 46-55 (1975)