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Optical metasurfaces (subwavelength-patterned surfaces typically described byvariable effective surface impedances) are typically modeled by anapproximation akin to ray optics: the reflection or transmission of an incidentwave at each point of the surface is computed as if the surface were "locallyuniform", and then the total field is obtained by summing all of these localscattered fields via a Huygens principle. (Similar approximations are found inscalar diffraction theory and in ray optics for curved surfaces.) In thispaper, we develop a precise theory of such approximations forvariable-impedance surfaces. Not only do we obtain a type of adiabatic theoremshowing that the "zeroth-order" locally uniform approximation converges in thelimit as the surface varies more and more slowly, including a way to quantifythe rate of convergence, but we also obtain an infinite series of higher-ordercorrections. These corrections, which can be computed to any desired order byperforming integral operations on the surface fields, allow rapidly varyingsurfaces to be modeled with arbitrary accuracy, and also allow one to validatedesigns based on the zeroth-order approximation (which is often surprisinglyaccurate) without resorting to expensive brute-force Maxwell solvers. We showthat our formulation works arbitrarily close to the surface, and can evencompute coupling to guided modes, whereas in the far-field limit ourzeroth-order result simplifies to an expression similar to what has been usedby other authors.

Sideways adiabaticity: Beyond ray optics for slowly varying metasurfaces