Synthesis of the Thickness Profile of the Waveguide Layer of the Thin Film Generalized Waveguide Luneburg Lens

A local variation in the thickness of the waveguide layer of integrated optics waveguide causes a local decrease of phase velocity, and hence bending of rays and of the wave front. The relationship of the waveguide layer thickness profile h (y, z) with the distribution of the effective refractive index of the waveguide β (y, z) is described in terms of a particular model of waveguide solutions of the Maxwell equations. In the model of comparison waveguides the support of the thickness irregularity of the waveguide layer Δh coincides with the support of inhomogeneity of the effective refractive index Δβ. A more adequate but more cumbersome model of the adiabatic waveguide modes allows them to mismatch supp Δh ⊃ supp Δβ. In this paper, we solve the problem of the Δh reconstruction on the base of given Δβ of the thin film generalized waveguide Luneburg lens in a model of adiabatic waveguide modes. The solution is found in the form of a linear combination of Gaussian exponential functions and in the form of a cubic spline for the cylindrically symmetric Δh (r) and in the form of a cubic spline for Δβ (r).