Focal-plane wavefront sensing with photonic lanterns I: theoretical framework

The photonic lantern (PL) is a tapered waveguide that can efficiently couple light into multiple single-mode optical fibers. Such devices are currently being considered for a number of tasks, including the coupling of telescopes and high-resolution, fiber-fed spectrometers, coherent detection, nulling interferometry, and vortex-fiber nulling (VFN). In conjunction with these use cases, PLs can simultaneously perform low-order focal-plane wavefront sensing. In this work, we provide a mathematical framework for the analysis of the photonic lantern wavefront sensor (PLWFS), deriving linear and higher-order reconstruction models as well as metrics through which sensing performance --- both in the linear and non-linear regimes --- can be quantified. This framework can be extended to account for additional optics such as beam-shaping optics and vortex masks, and is generalizable to other wavefront sensing architectures. Lastly, we provide initial numerical verification of our mathematical models, by simulating a 6-port PLWFS. In a companion paper, we provide a more comprehensive numerical characterization of few-port PLWFSs, and consider how the sensing properties of these devices can be controlled and optimized.