Focal-plane wavefront sensing with photonic lanterns: 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. 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 a PL wavefront sensor (PLWFS), deriving linear and higher-order reconstruction models as well as metrics through which sensing performance-in both the linear and nonlinear regimes-can be quantified. This framework can be extended to account for additional optics such as beam-shaping optics and vortex masks, and can be generalized for other wavefront sensing architectures. Finally, we provide initial numerical verification of our mathematical models by simulating a six-port PLWFS. In a forthcoming companion paper (Lin and Fitzgerald), 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.