# General Refraction Problems with Phase Discontinuity

## TL;DR

This paper develops a mathematical framework for analyzing metasurfaces with phase discontinuities on curved surfaces, establishing conditions linking surface curvature and refraction directions, applicable to near and far fields.

## Contribution

It introduces analytical conditions connecting surface curvature and refraction directions, extending phase discontinuity analysis to non-flat geometries with a vector Snell law formulation.

## Key findings

- Derived conditions for phase discontinuities on curved metasurfaces
- Unified approach for near and far field cases
- Formulated a vector Snell law with discontinuities

## Abstract

This paper provides a mathematical approach to study metasurfaces in non flat geometries. Analytical conditions between the curvature of the surface and the set of refracted directions are introduced to guarantee the existence of phase discontinuities. The approach contains both the near and far field cases. A starting point is the formulation of a vector Snell law in presence of abrupt discontinuities on the interfaces.